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abstract: 'Arcminute-resolution radio continuum images at 408 and 1420 MHz from the Canadian Galactic Plane Survey (CGPS) have been used to reexamine radio sources listed in the @kal80 catalogue. This catalogue is of particular interest to Galactic studies as it lists both extended and compact radio sources found in the second Galactic quadrant. We have determined the nature (extended vs. compact, Galactic vs. extragalactic) of all of these bright radio sources. A number of large regions with no optical counterparts are highlighted along with a sample of large radio galaxies. Many sources previously thought to be extended Galactic objects are shown to be point sources. A sample of point sources with flat or rising spectra between 408 and 1420 MHz has been compiled, and within this sample likely Gigahertz Peaked Spectrum sources have been identified.'
author:
- |
C. R. Kerton,$^1$[^1]\
$^1$Iowa State University, Department of Physics and Astronomy, Ames, IA, 50011, USA
title: 'A sharper view of the outer Galaxy at 1420 and 408 MHz from the Canadian Galactic Plane Survey I: Revisiting the KR catalogue and new Gigahertz Peaked Spectrum sources'
---
surveys – catalogues – Galaxy: disc – radio continuum: general.
Introduction {#sec:intro}
============
Radio continuum observations of the second quadrant of our Galaxy ($90\degr < l < 180\degr$) provide an unmatched opportunity for studying the structure and content of a spiral arm in detail. The Perseus Arm dominates Galactic structure in this quadrant and is viewed almost perpendicular to its long axis over the entire longitude range. The more distant Outer Arm is also well placed for study in this quadrant and in both cases confusion from Local Arm sources is minimal (cf. the view of the Galaxy around $l\sim
75\degr$ looking along the Local Arm). The previously best view of this region in the radio continuum (at 1420 MHz) was a series of surveys done by the Effelsberg 100-m telescope at 9-arcmin resolution. The surveys were summarized in the @kal80 and @rrf97 catalogues (KR and RRF respectively). RRF provides a listing of small diameter sources ($<16$ arcmin in extent) with an 80 mJy flux density limit (for point sources). The KR catalogue has a higher flux density limit (0.3 Jy) but is of particular interest to Galactic studies as it lists both compact and extended objects.
The new Canadian Galactic Plane Survey (CGPS; @tay03) data provide an unprecedented view of the continuum radiation at both 1420 and 408 MHz from the outer Galaxy. The data have arcminute-scale resolution and have full spatial frequency sensitivity crucial for the detection of extended structures.
In this paper we first revisit the sources found in the KR catalogue. @fic86 obtained high resolution VLA images of the sources originally classified as point sources in KR. For these sources we are primarily interested in observing the few of them that had poor VLA observations and to look for inverted spectrum sources. @tru90 obtained one-dimensional scans at 7.6 and 31.3 cm of most of the extended KR sources using the RATAN-600 telescope and found that many of the apparently extended KR objects were compact sources ($\leq$ 1-arcmin scale). @tru90 also suggested that a number of the KR objects were previously unknown compact Galactic supernova remnants (SNRs). We have reexamined all of these sources using the higher resolution and regular beamshape of the CGPS data and have been able to better determine the nature of all of the extended KR objects.
In the course of this study a new sample of extragalactic Gigahertz Peaked Spectrum (GPS) sources has been compiled. CGPS data have also revealed numerous new extended emission features in the second quadrant including both low-surface brightness extended emission and narrow filamentary features – both of which tend to be missed in the lower resolution surveys. The second paper in this series will present a complete catalogue of all extended emission features seen in the CGPS radio continuum data thus providing an updated version of the comprehensive catalogue compiled by @fic86.
In the next section we review the properties of the CGPS 1420 and 408 MHz data. In Sections \[sec:kr\] and \[sec:kr-p\] the CGPS view of the KR sources is presented. Flat and inverted spectrum sources are discussed in Section \[sec:fiss\] and conclusions are presented in Section \[sec:conc\].
Observations {#sec:observe}
============
The goal of the CGPS is to enhance the study of our Galaxy by obtaining arcminute-resolution images of all of the major components of the interstellar medium (ISM) in our Galaxy. Radio continuum observations made as part of this project were obtained using the seven-element interferometer at the Dominion Radio Astrophysical Observatory (DRAO) in Penticton, Canada [@lan00]. Details of the CGPS radio continuum observations, data reduction and data distribution are discussed at length in @tay03. CGPS observations currently cover $65\degr < l < 175\degr$ between $-3\fdg5
< b < +5\fdg5$ encompassing almost the entire second quadrant. The 1420 MHz observations have a nominal 1-arcmin resolution and both the 1420 and 408 MHz survey images were constructed with full spatial frequency coverage by combining the interferometer data with data from surveys using the Effelsberg single-dish and the Stockert single-dish telescopes. This provides sensitivity to extended structure which is very important for Galactic studies.
The simultaneous 408 MHz images, with nominal 3-arcmin resolution, provide invaluable data on the shape of the radio continuum spectrum as parameterized by the spectral index ($\alpha_{408}^{1420}$) between 408 and 1420 MHz (where flux density F$_\nu \propto \nu^\alpha$). In this paper we refer to inverted-spectrum sources as those with $\alpha_{408}^{1420} \geq +0.25$ and flat-spectrum sources as those with $|\alpha_{408}^{1420}| < 0.25$.
We also make use of the Mid-infrared Galaxy Atlas (MIGA; @ker00) and Infrared Galaxy Atlas (IGA; @cao97) arcminute resolution infrared images which make up part of the larger CGPS data collection. These infrared images are very useful in the identification of Galactic regions in cases where there is no associated optical emission or available radio recombination line observations.
Flux density measurements were made using software contained in the DRAO Export Software Package. Point source flux densities were obtained using the “fluxfit” program which fits Gaussians to the image and makes use of the beam shape information available in the CGPS data. Extended sources were measured using the “imview” program which allows the user to interactively derive background levels to use in determining the flux densities.
Extended sources in the KR catalogue {#sec:kr}
====================================
The KR catalogue is based on 1420 MHz radio continuum observations made at 9-arcmin resolution with the Effelsburg 100-m telescope. @kal80 identified 236 radio sources with flux density $F_\nu > 0.3$ Jy including point sources and extended objects up to 30-arcmin in diameter. The catalogue covered $l=93\degr$ to $l=162\degr$ and $|b| < 4\degr$. Extended sources were subdivided into three categories depending upon their apparent size: EP (partially extended), E (extended) and VE (very extended). EP sources had a greatest extent of $<$ 9-arcmin, E sources had greatest extents between 11-arcmin and 20-arcmin, while VE sources had greatest extents between 20-arcmin and 30-arcmin.
![1420 MHz images of KR 1, an enormous region in the Perseus Arm. The top panel shows the full extent of the region including extensive filamentary structure seen between $l=92\degr$ and $l=92\fdg5$. KR 4 is located in the lower left corner of this panel around $l=93\fdg75$. The lower panel shows the central region and reveals an intricate combination of filaments and bubble-like structures.[]{data-label="fig:kr1"}](fig1.eps){width="84mm"}
{width="140mm"}
Very-extended (VE) sources {#sec:kr-ve}
--------------------------
Data on the twelve very-extended (VE) objects identified by @kal80 are listed in Table \[tab:ve\]. The first column gives the KR catalogue number. Letters following the KR number are used in cases where the object is actually a multiple source at arcminute resolution and are not part of the original classification (e.g., KR206A). Columns 2 through 5 give the flux density measurements and 1$\sigma$ error estimates at 1420 and 408 MHz from the CGPS data. The spectral index between 408 and 1420 MHz ($\alpha_{408}^{1420}$) is given in column 6 followed by the angular scale of the source as seen in the 1420 MHz images in column 7. The final column provides extra information about the source, such as an association with well-known optically visible region or SNR. For extended (at 1-arcmin resolution) sources the RRF catalogue number is given if applicable, and for all of the arcminute-scale point sources the NRAO VLA Sky Survey (NVSS; @con98) catalogue designation is provided.
------ ------------------ ----------------- ------------------ ----------------- ----------------------- ---------- ---------------------
KR F$_\nu$ (1420) $\sigma$ (1420) F$_\nu$ (408) $\sigma$ (408) $\alpha_{408}^{1420}$ Diameter Notes
(mJy) (mJy) (mJy) (mJy)
1 $3.26\times10^4$ $9.9\times10^2$ $3.35\times10^4$ $7.9\times10^2$ $-0.02$ 120 RRF 861; Region
3 $4.48\times10^3$ $1.0\times10^2$ $4.29\times10^3$ $2.5\times10^2$ $+0.03$ 18 RRF 863; Region
6 $7.92\times10^2$ $5.0\times10^1$ $5.01\times10^2$ $2.9\times10^1$ $+0.4$ 12 Region
20 $1.01\times10^3$ $5.7\times10^1$ $9.68\times10^2$ $1.1\times10^2$ $+0.03$ 15 Region
47 $2.99\times10^3$ $1.3\times10^1$ $2.08\times10^3$ $7.5\times10^1$ $+0.3$ 20 Sh 2-135
65 $1.10\times10^3$ $5.4\times10^1$ $9.68\times10^2$ $2.6\times10^2$ $+0.1$ 12 Sh 2-151
122 $6.43\times10^2$ $3.9\times10^1$ $4.36\times10^2$ $1.8\times10^1$ $+0.3$ 24 Region
166A $7.35\times10^3$ $2.2\times10^2$ $1.52\times10^4$ $4.6\times10^2$ $-0.6$ 1 NVSS J032719+552029
166B $1.23\times10^3$ $3.9\times10^1$ $2.77\times10^3$ $8.4\times10^1$ $-0.7$ 1 NVSS J032744+552226
175A $2.31\times10^3$ $7.0\times10^1$ $4.86\times10^3$ $1.5\times10^2$ $-0.6$ 1 NVSS J032952+533236
175B $7.45\times10^1$ $5.3\times10^0$ $1.51\times10^2$ $4.5\times10^0$ $-0.6$ 1 NVSS J033003+532944
180 $4.5\times10^ 2$ $1.4\times10^1$ $1.03\times10^3$ $3.1\times10^1$ $-0.7$ 1 NVSS J035927+571706
206A $3.37\times10^2$ $1.0\times10^1$ $4.96\times10^2$ $1.5\times10^1$ $-0.3$ 1 NVSS J043523+511422
206B $2.28\times10^2$ $6.8\times10^0$ $1.08\times10^2$ $3.2\times10^0$ $+0.6$ 1 NVSS J043621+511253
210A $1.84\times10^2$ $5.6\times10^0$ $5.22\times10^2$ $1.6\times10^1$ $-0.8$ 1 NVSS J043342+502428
210B $7.89\times10^1$ $2.7\times10^0$ $1.52\times10^2$ $6.3\times10^0$ $-0.5$ 1 NVSS J043357+502420
------ ------------------ ----------------- ------------------ ----------------- ----------------------- ---------- ---------------------
Seven of these sources are Galactic regions. These sources all have flat or inverted spectral indices and have extensive infrared emission visible in the *IRAS* images. Five of the regions have no optical counterparts. KR 1 is an enormous region stretching up to 2in size (see Figure \[fig:kr1\]). Radio recombination line emission has been detected from the region at V$_\mathrm{LSR} \sim -60$ km s$^{-1}$ [@fic86] yielding a kinematic distance (accounting for known streaming motions) of $\sim 4.5$ kpc, which implies that the region is also physically large ($\sim 200$ pc). Note that the RRF 861 source associated with the region refers only to a compact source making up only a small portion of this extensive region.
KR 3, often incorrectly classified as a SNR, is a Galactic region with a blister morphology which was extensively studied by @fos01. In addition to the flat radio spectrum and extensive associated infrared emission, radio recombination line emission from the region has also been detected [@fos01] solidifying its classification as an region. RRF 863 is centered on the bright radio emission associated with the region/molecular cloud interface while the entire region extends up to 03 in size.
KR 6, KR 20 and KR 122 are all classified as extended Galactic regions on the basis of their radio spectrum and associated infrared emission. None of these regions have known optical counterparts. Finally there are two radio sources associated with optically visible regions. KR 47 is radio emission, about 20-arcmin in extent, associated with the Sh 2-135 region, while KR 65 is diffuse radio emission, about 12-arcmin in extent, that is apparently associated with Sh 2-151.
The remaining five VE sources turn out to be point sources at arcmin-scale resolution. KR 180 appears to have been misclassified because of nearby diffuse radio emission associated with Sh 2-214. This object was also listed by @tru90 as being extended and being a possible SNR but the CGPS data show this is not the case. The other sources tend to be pairs of point sources with separations $<$9-arcmin. All but one of the point sources have a non-thermal spectral index and no detectable infrared emission, consistent with them being distant extragalactic objects. The exception is the compact massive star-forming region KR 206B (NVSS J043621+511254) which has an inverted spectrum ($\alpha =
+0.6$) and is associated with the bright infrared source IRAS 04324+5106 (RAFGL 5124).
Extended (E) sources {#sec:kr-e}
--------------------
@kal80 listed 48 of these sources. Table \[tab:e\] summarizes the CGPS view of this sample using the same notation as in Table \[tab:ve\]. Note that KR 86 was not observed in the CGPS and KR 35 is apparently a spurious source; no bright point source or region of diffuse emission was detected near its catalogued position.
----- ------------------ ----------------- ------------------ ----------------- ----------------------- ---------- ---------------------
KR F$_\nu$ (1420) $\sigma$ (1420) F$_\nu$ (408) $\sigma$ (408) $\alpha_{408}^{1420}$ Diameter Notes
(mJy) (mJy) (mJy) (mJy)
4 $1.06\times10^3$ $3.7\times10^1$ $9.01\times10^2$ $1.7\times10^1$ $+0.1$ 12 RRF 865; Region
7 $2.69\times10^3$ $8.1\times10^1$ $2.45\times10^3$ $7.3\times10^2$ $+0.07$ 12 RRF 874; Region
19A $1.70\times10^2$ $7.3\times10^0$ $7.85\times10^2$ $3.6\times10^0$ $+0.1$ 5 RRF 903; Region
19B $1.47\times10^2$ $3.4\times10^0$ $1.29\times10^2$ $3.9\times10^0$ $+0.1$ 4 RRF 903; Region
21A $3.78\times10^2$ $1.1\times10^1$ $9.09\times10^2$ $2.7\times10^1$ $-0.7$ 1 NVSS J214343+523958
21B $3.64\times10^2$ $1.1\times10^1$ $1.05\times10^2$ $3.1\times10^1$ $-0.8$ 1 NVSS J214418+524501
----- ------------------ ----------------- ------------------ ----------------- ----------------------- ---------- ---------------------
Table \[tab:e\] is presented in its entirety in the electronic edition of the journal.
One source, KR 196, is a very large ($\sim$25-arcmin diameter) region of bright radio emission associated with the optical region Sh 2-206. Seven other sources match the original classification (diameters between 11-arcmin and 20-arcmin). Three of these (KR 55, 91 and 98) are associated with radio emission from known optical regions, while three others (KR 4, 7, and 80) are regions with no optical counterparts. All of these objects have flat or inverted radio spectra and have associated infrared emission. Finally KR 101 is the well-studied SNR 3C 10 (Tycho’s SNR).
Five other regions (KR 19, 46, 48, 171 and 198) are smaller extended regions. KR 19 consists of two compact regions with the western (19A) region being associated with IRAS 21336+5333 and the eastern one (19B) being associated with two infrared sources IRAS 21340+5339 and IRAS 21340+5337 (see Figure \[fig:small\_ex\]). KR 46 is a compact region that shows hints of a blister morphology at 1-arcmin resolution. The radio spectrum is thermal and there is bright infrared emission associated with the region. @tru90 suggested that KR 48 and KR 171 were possible Galactic supernova remnants. However the CGPS data show the regions have inverted (KR 48) and flat (KR 171) radio spectra and are associated with bright diffuse infrared emission and IRAS point sources. Thus it is more likely that they are both Galactic regions. Finally KR 198 is associated with the optical region Sh 2-207.
KR 168 consists of two slightly elongated sources separated by $\sim
4.5$ arcmin. It is likely that these sources are extragalactic jets that are just barely resolved at 1-arcmin resolution. It is not clear that the two sources are physically associated. KR 188 also consists of two elongated sources with a similar point source plus faint jet structure with the point sources being separated by $\sim$4 arcmin. In this case the two objects do share common diffuse emission and the jet structures both point back to a common point suggesting that they are physically related. In Table \[tab:e\] the NVSS designations for the point-like portions of these objects are given.
The remaining “extended” KR sources are all actually point sources at 1-arcmin resolution. The majority of these sources are extragalactic as they have strongly non-thermal spectral indices, are unresolved at 1-arcmin resolution, and have no associated infrared emission. Three of the sources have flat spectra (KR 63, 189 and 192A) and two have inverted spectra (KR 53 and 60A). None of the flat spectrum sources have associated infrared emission and, given that they all have $\alpha_{408}^{1420} = -0.2$, they are also most likely extragalactic objects. KR 53 is associated with the optical region Sh 2-138. Finally, KR 60A is apparently a flat-spectrum radio galaxy. There is no associated infrared emission and, combining the CGPS flux density measurements with data obtained using SPECFIND [@vol05], we find a very flat spectral index of $+0.09\pm0.05$ over the range from 325 to 4800 MHz as illustrated in Figure \[fig:kr60a\].
![KR 60A, a flat-spectrum radio galaxy. CGPS data are at 408 and 1420 MHz. Other data points were obtained from @vol05.[]{data-label="fig:kr60a"}](fig3.eps){width="84mm"}
Partially-extended (EP) sources {#sec:kr-ep}
-------------------------------
The KR catalogue lists 41 of these sources. Table \[tab:ep\] summarizes the CGPS view of this sample using the same notation as in the previous tables. One source (KR 145) appears to have been a spurious object as there are no strong point sources or regions of extended emission near the catalogued coordinates.
Three of the sources have diameters greater than 11-arcmin. KR 200 is a large ($\sim 30$ arcmin) region of radio emission a portion of which is directly associated with the optical region Sh 2-209. KR 140 is a 12-arcmin scale region and KR 130 is the well-studied SNR 3C 58.
There are 13 sources which are not point sources but have diameters $<9$ arcmin. Nine of these objects are radio sources associated with known small-diameter optical regions and one is associated with the nearby galaxy Maffei 2.
----- ------------------ ----------------- ------------------ ----------------- ----------------------- ---------- ---------------------
KR F$_\nu$ (1420) $\sigma$ (1420) F$_\nu$ (408) $\sigma$ (408) $\alpha_{408}^{1420}$ Diameter Notes
(mJy) (mJy) (mJy) (mJy)
13 $1.13\times10^3$ $2.6\times10^2$ $7.32\times10^2$ $9.4\times10^1$ $+0.3$ 6 RRF 888; BFS 6
15 $3.10\times10^2$ $9.3\times10^0$ $5.55\times10^2$ $1.7\times10^1$ $-0.5$ 1 NVSS J212305+550027
17 $6.45\times10^2$ $1.9\times10^1$ $5.46\times10^2$ $1.6\times10^1$ $+0.1$ 2 RRF 899; Sh 2-187
18 $6.52\times10^2$ $8.7\times10^0$ $4.30\times10^2$ $1.5\times10^1$ $+0.3$ 6 RRF 929; BFS 8
28A $2.56\times10^2$ $7.9\times10^0$ $7.27\times10^2$ $2.3\times10^1$ $-0.8$ 1 NVSS J213932+554030
28B $1.64\times10^2$ $5.3\times10^0$ $5.03\times10^2$ $1.7\times10^1$ $-0.9$ 1 NVSS J213934+554445
28C $5.32\times10^1$ $2.5\times10^0$ $\cdots$ $\cdots$ $\cdots$ 1 NVSS J213943+554340
----- ------------------ ----------------- ------------------ ----------------- ----------------------- ---------- ---------------------
Table \[tab:ep\] is presented in its entirety in the electronic edition of the journal.
KR 45 (RRF 981) is a combination of extended and point source emission (see Figure \[fig:kr45\]). The extended radio emission is associated with the distant region IRAS 22181+5716. Molecular line (CO) observations towards this source detect emission at V$_\mathrm{LSR} = -63$ km s$^{-1}$ placing the region at a heliocentric distance of $\sim 7$ kpc. There is also a close double point source (denoted 45A and 45B) which is unresolved in the lower resolution 408 MHz images. These non-thermal point sources have no infrared counterparts and are apparently just background extragalactic sources. The remaining two extended objects (KR 144 and 172) both appear to be radio galaxies with a distinct core/lobe morphology (see Figure \[fig:rgals\]). The objects shown in Figure \[fig:rgals\] appear to be similar to the giant radio source WN 1626+5153 discovered in the Westerbork Northern Sky Survey [@rot96].
![KR 45 at 1420 MHz. The original single source is actually a Galactic region and a pair of bright extragalactic sources. Contours are at 7, 8, 9, 10, 20, and 30 K. The cross indicates the position of the infrared source IRAS 22181+5716.[]{data-label="fig:kr45"}](fig4.eps){width="84mm"}
Finally the remaining EP sources are all point sources at 1-arcmin resolution. All but one (KR 58) are likely extragalactic sources having a non-thermal spectral index and no detectable infrared emission. KR 58 has an inverted spectrum and is the planetary nebula NGC 7354 (IRAS 22384+6101).
{width="140mm"}
The nature of the point sources in the KR Catalogue {#sec:kr-p}
===================================================
All of the KR point sources (135 in total) except one (KR 195) were observed by the CGPS. Table \[tab:p\] summarizes the CGPS view of this sample using the same notation as in the previous tables.
The vast majority of these sources are point sources at 1-arcmin resolution. As first demonstrated by @fic86 most of these are extragalactic sources as indicated in this study by their strongly negative spectral index between 408 and 1420 MHz and lack of associated infrared emission.
There are a few small extended sources in this subsample. KR 77, 212 and 228 are all regions of extended thermal emission associated with the optical regions Sh 2-159, Sh 2-212 and Sh 2-217 respectively. Perhaps more interesting are the extended extragalactic sources KR 2 and KR 226. Both of these objects are clearly radio galaxies (see Figure \[fig:rgals\]) and were noted by @fic86 as being overresolved in his VLA images. KR 2 extends for about 10-arcmin in its longest direction. Optical spectroscopy of this source places it at a redshift of z=0.02 [@mas04]. KR 226 extends for about 5-arcmin and no studies of this object beyond cataloging have been made.
---- ------------------ ----------------- ------------------ ----------------- ----------------------- ---------- ---------------------
KR F$_\nu$ (1420) $\sigma$ (1420) F$_\nu$ (408) $\sigma$ (408) $\alpha_{408}^{1420}$ Diameter Notes
(mJy) (mJy) (mJy) (mJy)
2 $2.87\times10^3$ $8.6\times10^1$ $6.16\times10^3$ $1.8\times10^2$ $-0.6$ 6 RRF 862
5 $4.37\times10^2$ $1.3\times10^1$ $1.39\times10^3$ $4.2\times10^1$ $-0.9$ 1 NVSS J213646+495318
8 $1.77\times10^3$ $5.3\times10^1$ $1.07\times10^3$ $3.3\times10^1$ $+0.4$ 1 NVSS J213701+510136
9 $3.22\times10^2$ $9.9\times10^0$ $7.56\times10^2$ $2.4\times10^1$ $-0.7$ 1 NVSS J213158+521415
10 $6.69\times10^2$ $2.0\times10^1$ $1.24\times10^3$ $3.7\times10^1$ $-0.5$ 1 NVSS J213340+521951
11 $7.72\times10^2$ $2.3\times10^1$ $1.49\times10^3$ $4.5\times10^1$ $-0.5$ 1 NVSS J213833+513550
---- ------------------ ----------------- ------------------ ----------------- ----------------------- ---------- ---------------------
Table \[tab:p\] is presented in its entirety in the electronic edition of the journal.
There are 14 flat spectrum sources of which three (KR 23, 208, and 212) are associated with optical regions (Sh 2-148, Sh 2-211 and Sh 2-212 respectively). The remaining 11 sources have no associated infrared emission and thus inferred to be extragalactic sources. We examined the four flat spectrum sources with positive spectral indices in more detail. CGPS data were combined with data from @vol05 and @fic86 to obtain the spectra shown in Figure \[fig:ps-flat-pos\].
The radio spectrum of KR 24 is very flat over a wide frequency range, and certainly flatter than expected just from 408 and 1420 MHz data. A least absolute deviation fit to the data gives an overall spectral index of $\alpha = -0.06$. KR 178 is another very flat spectrum source with least absolute deviation spectral index of $\alpha = +0.04$ over the entire range of observations. KR 30 shows a slightly rising spectrum with $\alpha = +0.2$. The highest frequency point suggests that the spectrum may be flattening above 10 GHz. Finally the KR 234 radio spectrum has a shallow negative slope spectrum of $\alpha = -0.2$. The low frequency data points for KR 234 are in good agreement but there is increased scatter at the higher frequencies. The large scatter observed in the spectra of KR 24, 178 and 234 at particular wavelengths suggests that these sources are variable. This is the likely reason that the overall spectral index for these three sources is shallower than the spectral index determined by the simultaneous CGPS observations.
{width="140mm"}
There are also eight inverted spectrum point sources. Three of the sources (KR 61, 67 and 72) are associated with optical regions (Sh 2-146, Sh 2-152 and Sh 2-156 respectively) and KR 138 is the compact region IRAS 02044+6031. Molecular line emission at V$_\mathrm{LSR} \sim -55$ km s$^{-1}$ has been detected towards this *IRAS* source placing it at a kinematic distance of $\sim 5.5$ kpc. Unfortunately the velocity field model of @bra93 is quite uncertain around this longitude ($l\sim
130\degr$) for this velocity making corrections for streaming motions problematic. Given its small angular size it it quite possible that KR 138 lies beyond the Perseus Arm. The remaining four sources have no infrared counterpart and are most likely extragalactic.
Such extragalactic radio sources with inverted spectra are interesting because of the possibility that they are Gigahertz Peaked Spectrum (GPS) sources. Astronomically these objects are of interest because they may represent an early stage in the evolution of radio galaxies [@ort06; @ode98]. Observationally these objects are defined as having a convex radio spectrum that peaks between 500 MHz and 10 GHz. The shape of the spectrum is most likely due to synchrotron self-absorption [@ort06]. Below the peak frequency the average spectral index is $0.51\pm0.03$ and above the peak it is $-0.73\pm0.06$ [@dev97].
For each of the extragalactic inverted spectrum sources we combined flux density measurements at other wavelengths from @vol05 and @fic86 with the CGPS measurements. The spectra are shown in Figure \[fig:kr-invert\]. Following @mar99 we fit a second order polynomial of the form $ \log F_\nu = a + b \log \nu - c(\log \nu
)^2$. This curve is not physically motivated, rather it simply allows us to easily identify sources with sufficiently high spectral curvature. Sources with $c > 1.0$ have sufficient spectral curvature to be considered GPS sources.
{width="140mm"}
{width="140mm"}
KR 8 does appear to have a convex spectra but the data above the peak has a large amount of scatter and the curvature is not as high as one would expect for a true GPS source ($c = 0.6$). KR 125 has a very low curvature spectrum ($c= 0.25$) with the curvature arising almost entirely from the highest frequency data point. Except for this point the spectrum is consistent with a rising spectrum with $\alpha = +0.3$ from 300 to 4800 MHz. KR 135 has a very steep low frequency spectral index and the cuvature of the spectrum is quite high ($c=0.96$). Unfortunately the data above the apparent peak in the spectrum are quite scattered and its status as a GPS source is very uncertain. Finally, KR 182 shows a rising spectrum with $\alpha = +0.3$ with no signs of any spectral curvature. There is a large amount of scatter in the spectrum at both low and high frequency.
{width="140mm"}
Flat and Inverted-spectrum point sources {#sec:fiss}
========================================
The presence of extragalactic sources with both flat and inverted spectra within the KR sample led us to examine all of the CGPS second quadrant data for similar sources. To rapidly search for other point sources with flat or inverted spectra the 1420 MHz images were first convolved to the 408 MHz resolution. The brightness of the convolved 1420 MHz images were then scaled to the expected brightness at 408 MHz assuming an optically thin thermal spectrum between 408 and 1420 MHz. The true 408 images were then subtracted from the scaled images resulting in a series of difference images. Point sources with steep negative spectral indices show up as distinct negative-valued sources on the difference images thus allowing the rapid identification of flat and inverted-spectrum sources. After candidate sources were identified in this manner, flux densities were measured at 1420 and 408 MHz. Sources in the final sample had both measurable 408 flux densities (complete to $\sim 50$ mJy at 408 MHz) and no visible infrared emission in the ancillary CGPS infrared images.
Table \[tab:if\] shows the resulting sample of flat-spectrum and inverted-spectrum sources. Column 1 gives the NVSS catalogue designation, columns 2-5 give the flux density and error estimates at 1420 and 408 MHz, and column 6 gives the spectral index.
---------------- ------------------ ----------------- ------------------ ----------------- -----------------------
NVSS F$_\nu$ (1420) $\sigma$ (1420) F$_\nu$ (408) $\sigma$ (408) $\alpha_{408}^{1420}$
(mJy) (mJy) (mJy) (mJy)
J054044+391612 $1.53\times10^2$ $4.7\times10^0$ $4.74\times10^1$ $4.2\times10^0$ $+0.9$
J054052+372847 $1.74\times10^2$ $5.3\times10^0$ $1.30\times10^2$ $8.2\times10^0$ $+0.2$
J050905+352817 $3.85\times10^2$ $1.2\times10^1$ $1.49\times10^2$ $3.8\times10^1$ $+0.8$
J050920+385046 $9.40\times10^1$ $2.9\times10^0$ $8.22\times10^1$ $7.5\times10^0$ $+0.1$
J051346+400618 $3.55\times10^2$ $1.1\times10^1$ $3.35\times10^2$ $1.1\times10^1$ $+0.0$
J050948+395154 $7.83\times10^1$ $2.4\times10^0$ $3.52\times10^1$ $1.5\times10^0$ $+0.6$
---------------- ------------------ ----------------- ------------------ ----------------- -----------------------
Table \[tab:if\] is presented in its entirety in the electronic edition of the journal.
In order to identify potential GPS sources we examined in more detail 43 of the sources which had $\alpha_{408}^{1420} \geq +0.4$. As before, radio data from the compilation of @vol05 were used to construct spectra over as wide a range of frequencies as possible. Of these objects eight of them were found to have a curvature of $c >
+1$. The radio spectra of these objects are shown in Figure \[fig:curve\].
We also found four other objects in the sample that had rising spectra ($\alpha \geq +0.3$ over the entire spectral range) combined with little scatter (see Figure \[fig:rise\]). These sources may be examples of, relatively rare, GPS sources with a peak above 5 GHz similar to the point source 71P 52 (NVSS 213551+471022) examined by @hig01.
Conclusions {#sec:conc}
===========
The KR catalogue is very useful for Galactic studies as it contains information on both compact and extended radio sources in the outer Galaxy. Unfortunately the relatively low resolution of the survey means that it overestimates the number of extended sources in the outer Galaxy. This paper updates this catalogue based primarily on new higher resolution images of the outer Galaxy at 1420 MHz obtained as part of the CGPS. We have clearly identified sources that were misclassified as extended objects and have determined which sources remain unresolved at 1-arcmin scale resolution. The simultaneous 408 MHz CGPS observations, combined with ancillary infrared data, also have allowed the nature of all of the observed KR sources to be determined with some confidence. Attention has been drawn particularly to a large number of unstudied Perseus Arm regions (including the extremely large KR 1 complex), objects previously considered to be SNR candidates (e.g., KR 171), and a sample of large radio galaxies (e.g., KR 144).
In addition, through the examination of the 408 and 1420 MHz CGPS images, this study has identified a sample of flat-spectrum and inverted-spectrum extragalactic radio sources based upon their 408 and 1420 MHz flux densities. A subset of these objects was examined in more detail and a new sample of GPS sources has been compiled.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank ISU undergraduate students Jason Murphy and Jon Patterson for their assistance on this project. The Dominion Radio Astrophysical Observatory is operated by the National Research Council of Canada. The Canadian Galactic Plane Survey is supported by a grant from Natural Science and Engineering Research Council of Canada.
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[^1]: E-mail: [email protected]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'For the size of the largest component in a supercritical random geometric graph, this paper estimates its expectation which tends to a polynomial on a rate of exponential decay, and sharpens its asymptotic result with a central limit theory. Similar results can be obtained for the size of biggest open cluster, and for the number of open clusters of percolation on a box, and so on.'
title: The Asymptotic Size of The Largest Component in Random Geometric Graphs with some applications
---
Introduction
============
The size of the largest component is a basic property for random geometric graphs (RGGs) and has attracted much interest during the past years, including both theoretical studies [@penrose1995][@penrose1996][@penrose2001][@p10] and various applications [@Glauche2003][@p17][@p16][@Pishro2009]. This paper firstly investigates the asymptotic size of the largest component of RGG in the supercritical case.
Given a set $\mathcal {X}\subset\mathbb{R}^d$, let $G(\mathcal
{X};r)$ denote the undirected graph with vertex set $\mathcal {X}$ and with undirected edges which connect all those pairs $\{X,Y\}$ with $\parallel Y-X\parallel\leq r$, where $\|\cdot\|$ denotes the Euclidean norm ($l_2-norm$). The basic model of RGGs can be formulated as $G(\mathcal {X}_n;r_n)$, where $\mathcal{X}_n$ denotes $n$ points which are independently and uniformly distributed in a $d$-dimensional unit cube. To overcome the lack of spatial independence for the binomial point process $\mathcal{X}_n$, the model of continuum percolation must be introduced. Following Section 1.7 in [@p10], let $\mathcal{H}_{\lambda}$ be a homogeneous Poisson process of intensity $\lambda$ on $\mathbb{R}^{d}$. For $s>0$, define $B(s):=[0,s]^{d}$ and $\mathcal{H}_{\lambda,s}:=\mathcal{H}_{\lambda}\cap B(s).$ Following [@p10], we write the Poisson Boolean model as $G(\mathcal
{H}_{\lambda,s};1)$.
There exist some notations related to percolation must be introduced. Following Section 9.6 in [@p10], let $\mathcal{H}_{\lambda,0}$ denote the point process $\mathcal{H}_{\lambda}\cup\{\textbf{0}\}$, where $\textbf{0}$ is the origin in $\mathbb{R}^d$, and for $k\in\mathds{N}$, let $p_{k}(\lambda)$ denote the probability that the order of the component in $G(\mathcal{H}_{\lambda,0};1)$ containing the origin is equal to $k$. The $percolation$ $probability$ $p_{\infty}(\lambda)$ is defined to be the probability that $\textbf{0}$ lies in an infinite component of the graph $G(\mathcal{H}_{\lambda,0};1)$. Therefore, we have $p_{\infty}(\lambda)=1-\sum\limits_{k=1}^{\infty}p_{k}(\lambda)$. Let $$\begin{aligned}
\label{lc}
\lambda_c=inf\{\lambda>0:p_{\infty}(\lambda)>0\}\end{aligned}$$ denote the critical intensity of continuum percolation. It is well known that $0<\lambda_c<\infty$ for $d\geq 2$ [@p12][@p13][@p14].
Following Section 9.6 in [@p10], let $L_j(G)$ denote the order of its $j$th-largest component for any graph $G$. Then $L_1(G(\mathcal {H}_{\lambda,s};1))$ denotes the order of the largest component of $G(\mathcal {H}_{\lambda,s};1)$. The asymptotic properties of $L_1(G(\mathcal{H}_{\lambda,s};1))$ have been well studied by Penrose. The basic asymptotic result about $L_1
(G(\mathcal{H}_{\lambda,s};1))$ is provided by Penrose (Theorem 10.9 in [@p10]), that if $\lambda\neq\lambda_c$ then $$\begin{aligned}
\label{o1}
s^{-d}L_1(G(\mathcal{H}_{\lambda,s};1))\xrightarrow{P}\lambda
p_{\infty}(\lambda)\quad as \quad s\rightarrow \infty.\end{aligned}$$ Also, Penrose has given a central limit theorem for $L_1(G(\mathcal{H}_{\lambda,s};1))$ in the supercritical case $\lambda>\lambda_c$ (Theorem 10.22 in [@p10]), that $$\begin{aligned}
\label{clt_1}
s^{-d/2}(L_1(G(\mathcal{H}_{\lambda,s};1))-E[L_1(G(\mathcal{H}_{\lambda,s};1))])
\xrightarrow{D} \mathcal {N}(0,\sigma^2).\end{aligned}$$ However, the question as how large $E[L_1(G(\mathcal{H}_{\lambda,s};1))]$ should be still remains unsolved. By (\[o1\]) it can be deduced that $E[L_1(G(\mathcal{H}_{\lambda,s};1))]=\lambda p_{\infty}(\lambda)
s^d +o(s^d)$, where $f(s)=o(g(s))$ indicates that $\lim_{s\rightarrow\infty}\frac{f(s)}{g(s)}=0$. This result is not precise enough for some theoretic analysis and practical applications.
The corresponding asymptotic results and central limit theorem for $G(\mathcal {X}_n;r_n)$ have also been established by Peorose (Theorems 11.9 and 11.16 in [@p10]), but we may ask similar questions. This paper will study the problem and give a more precise description for the asymptotic sizes of $L_1(G(\mathcal{H}_{\lambda,s};1))$ and $L_1(G(\mathcal {X}_n;r_n))$. Our method can be adapted to study some other models and problems.
Main Results
============
Our main results can be formulated as the following two theorems.
\[t1\] Suppose $d\geq 2$ and $\lambda>\lambda_c$. Then there exist constants $c=c(d,\lambda)>0$ and $\tau_i=\tau_i(d,\lambda)$, $1\leq i \leq d$, with $\tau_1>0$, such that for all $s$ large enough, $$\begin{aligned}
\label{order_t1_00}
E[L_1(G(\mathcal{H}_{\lambda,s};1))]=\lambda
p_{\infty}(\lambda) s^d-\sum_{i=1}^d \tau_i s^{d-i}+o\left(e^{-c s}\right).\end{aligned}$$ Also, there exists a constant $\sigma=\sigma(d,\lambda) > 0$, such that $$\begin{aligned}
\label{order_t1_01}
L_1(G(\mathcal{H}_{\lambda,s};1))s^{-d/2}- \lambda
p_{\infty}(\lambda) s^{d/2}+ \sum_{i=1}^{\lfloor\frac{d}{2} \rfloor}\tau_i s^{d/2-i}
\xrightarrow{D} \mathcal {N}(0,\sigma^2)\end{aligned}$$ as $s\rightarrow\infty$.
\[t2\] Suppose $d\geq 2$ and $\lambda>\lambda_c$. Let $\sigma$ and $\tau_i$ be the same constants appearing in Theorem \[t1\]. There exists a constant $\delta=\delta (d,\lambda)$, with $0<\delta\leq\sigma$, such that $$\begin{aligned}
L_1\left(G\left(\mathcal{X}_n;(n/\lambda)^{-1/d}\right)\right)\left(n/\lambda\right)^{-1/2}-
p_{\infty}(\lambda) \left(\lambda n\right)^{1/2}+\sum_{i=1}^{\lfloor\frac{d}{2} \rfloor} \tau_i
\left(n/\lambda\right)^{\frac{1}{2}-\frac{i}{d}}
\xrightarrow{D} \mathcal {N}(0,\delta^2)\end{aligned}$$ as $n\rightarrow\infty$.
To prove the two theorems, we estimate the value of $E[L_1(G(\mathcal{H}_{\lambda,s};1))]$ firstly, and then using the central limit theorems for $L_1(G(\mathcal{H}_{\lambda,s};1))$ and $L_1(G(\mathcal{X}_n;(n/\lambda)^{-1/d}))$, we can prove (\[order\_t1\_01\]) and Theorem \[t2\].
Some notations must be stated before the proof of our results. For any $x\in \mathbb{R}^d$, we write its $l_{\infty}$ norm with $\|x\|_{\infty}$ given by the maximum absolute value of its coordinates. For any finite set $A\subset \mathbb{R} ^d$, we set the diameter of $A$ by diam$(A)=\sup_{x,y\in A}\|x-y\|_{\infty}.$ Also, let $|A|$ denote the cardinality of $A$.
Let $\oplus$ denote the Minkowski addition of sets. Let $Leb(\cdot)$ denote the Lebesgue measure. For $s \geq 0$, let $\lfloor s
\rfloor$ denote the smallest integer not smaller than $s$.
To simplify the expression, we will omit the dependence of all constants on $d$ and $\lambda$, for example, the constant $c$ stands for $c(d,\lambda)$.
Given $\lambda>\lambda_c$, by the uniqueness of the infinite component in continuum percolation (Theorem 9.19 in [@p10]), the infinite graph $G(\mathcal{H}_{\lambda};1)$ has precisely one infinite component $\mathcal {C}_{\infty}$ with probability $1$. Let $C_1,C_2,...,C_M$ denote the components of $G(\mathcal
{C}_{\infty}\cap B(s);1)$, taken in a decreasing order. We give a result on the rate of sub-exponential decay of the difference between $E[L_1(G(\mathcal{H}_{\lambda,s};1))]$ and $E[|C_1|]$.
\[temp1\] Suppose $d\geq 2$ and $\lambda>\lambda_c$. The exists a constant $c>0$, such that for large enough $s$, $$\begin{aligned}
\label{temp1_0}
0\leq E[L_{1}(G(\mathcal{H}_{\lambda,s};1))]-E[|C_1|] \leq e^{-cs}.\end{aligned}$$
By the definition of $L_{1}(G(\mathcal{H}_{\lambda,s};1))$ and $C_1$, obviously $E[L_{1}(G(\mathcal{H}_{\lambda,s};1))]\geq
E[|C_1|]$. Thus it just remains to prove the second inequality of (\[temp1\_0\]).
Given any $x\in\mathbb{R}^d$, let $C_{\infty}(x)$ denote the infinite connected component of $G(\mathcal{H}_{\lambda}\cup\{x\};1)$. By Palm theorem for Poisson processes (Theorem 1.6 in [@p10]), we have $$\begin{aligned}
\label{order_t1_6}
E[L_{1}(G(\mathcal{H}_{\lambda,s};1))]=\lambda\int_{B(s)}P[x\in
V_{1}(x)]dx,\end{aligned}$$ where $V_{1}(x)$ denotes the largest component of $G(\mathcal{H}_{\lambda,s}\cup\{x\};1)$, and $$\begin{aligned}
\label{order_t1_7}
E[|C_1|]=\lambda\int_{B(s)}P[x\in C_{1}(x)]dx,\end{aligned}$$ where $C_{1}(x)$ denotes the largest component of $C_{\infty}(x)
\cap B(s)$. Therefore, $$\begin{aligned}
\label{order_t1_8}
\begin{aligned}
E[L_{1}(G(\mathcal{H}_{\lambda,s};1))]-E[|C_1|]&=\lambda\int_{B(s)}(P[x\in
V_{1}(x)]-P[x\in C_{1}(x)])dx\\
&\leq \lambda\int_{B(s)} P[\{x\in V_{1}(x)\} \cap \{ x\notin
C_1(x)\}]dx\\
&= \lambda\int_{B(s)} P[\{x\in V_{1}(x)\} \cap \{ x\notin
C_{\infty}(x)\}]dx.
\end{aligned}\end{aligned}$$ Suppose $0<\varepsilon<\frac{1}{2}$. By Theorem 10.19 in [@p10], there exist constants $c_{1}>0$ and $s_{1}>0$, such that if $s>s_1$ then $$\begin{aligned}
\label{order_t1_9}
\begin{aligned}
P\left [|V_{1}(x)|<(1-\varepsilon)\lambda s^{d}p_{\infty}(\lambda)
\right ] &\leq P\left
[L_{1}(G(\mathcal{H}_{\lambda,s};1))<(1-\varepsilon)\lambda
s^{d}p_{\infty}(\lambda)\right ]\\
&\leq \exp \left (-c_{1}s^{d-1} \right ).
\end{aligned}\end{aligned}$$ Also, by Theorem 10.15 in [@p10], there exists a constant $c_2>0$ such that for $s$ large enough, $$\begin{aligned}
\label{order_t1_10}
\sum\limits_{k\geq \lceil(1-\varepsilon)\lambda
s^{d}p_{\infty}(\lambda)\rceil}p_{k}(\lambda)<\exp \left
(-c_2[(1-\varepsilon)\lambda s^{d}p_{\infty}(\lambda)]^{(d-1)/d}
\right ).\end{aligned}$$ Therefore, from (\[order\_t1\_9\]) and (\[order\_t1\_10\]) we can obtain $$\begin{aligned}
\label{order_t1_11}
&&P[\{x\in V_{1}(x)\}\cap\{x\not\in
C_{\infty}(x)\}]\nonumber\\
&&~~~~\leq P[|V_{1}(x)|<(1-\varepsilon)\lambda
s^{d}p_{\infty}(\lambda)]\nonumber\\
&&~~~~~~~~~+P[\{x\in V_{1}(x)\}\cap\{x\not\in
C_{\infty}(x)\}\cap\{|V_{1}(x)|\geq(1-\varepsilon)\lambda
s^{d}p_{\infty}(\lambda)\}]\nonumber\\
&&~~~~\leq \exp \left (-c_{1}s^{d-1} \right )+\sum\limits_{k\geq
\lceil(1-\varepsilon)\lambda
s^{d}p_{\infty}(\lambda)\rceil}p_{k}(\lambda)\nonumber\\
&&~~~~< \exp \left (-c_{1}s^{d-1} \right )+ \exp \left
(-c_2[(1-\varepsilon)\lambda p_{\infty}(\lambda)]^{(d-1)/d} s^{d-1}
\right )~~as~s\rightarrow\infty.\end{aligned}$$ Combined with (\[order\_t1\_8\]) this yields our result.
To estimate the value of $E[L_1(G(\mathcal{H}_{\lambda,s};1))]$, by Lemma \[temp1\] we just need to get the value of $E[|C_1|]$ instead. Actually, by Palm theory for infinite Poisson process (Theorem 9.22 in [@p10]), $$\begin{aligned}
\label{comm_6}
E\left [\sum_{i=1}^M |C_i| \right ]=E[\left|\mathcal
{C}_{\infty}\cap B(s)\right|]=\lambda p_{\infty}(\lambda)s^d,\end{aligned}$$ so we just need to estimate the value of $E[\sum_{i=2}^M |C_i|]$. Let $L(s):=B(s)\backslash[1,s-1]^d$. For any $2\leq i \leq M$, since $C_i\subset \mathcal {C}_{\infty}$, therefore there exists at least one point in $L(s)\cap C_i$ which connects to $\mathcal
{C}_{\infty} \setminus B(s)$ directly; we choose the nearest one to the boundary of $B(s)$ as the $out-connect$ $point$. We can see that each component of $C_2,...,C_M$ contains exactly one out-connect point.
For any region $R\subseteq B(s)$ and $2\leq i \leq M$, define $$\label{chidef}
\chi_{i}(R):=\left\{
\begin{array}{ll}
1, & \mbox{if the out-connect point of $C_i$ is contained by } R, \\
0, & \mbox{otherwise},
\end{array}
\right.$$ and define $$\label{xidef}
\xi(R)=\xi(R,s):= \sum\limits_{i=2}^{M}\chi_{i}(R)|C_i|.$$ By the definition of $\xi(\cdot)$, it is easy to see that for any $R,\widetilde{R}\subset B(s)$, if $Leb(R\cap \widetilde{R})=0$, then $E[\xi(R\cap \widetilde{R})]=0$ and $E[\xi(R\cup
\widetilde{R})]=E[\xi(R)]+E[\xi(\widetilde{R})].$
For $0 \leq i \leq d-1$, define $$R_i=R_i(s):=[0,1]\times\underbrace{[0,s/2]\times\cdots\times[0,s/2]}\limits_{d-1-i} \times \underbrace{[1,s/2]\times\cdots\times[1,s/2]}\limits_{i}.$$ Noted that $[1,s/2]^d \cap L(s)=\emptyset$, then by symmetry, $$\begin{aligned}
\label{total_1}
\begin{aligned}
&E\left [\sum\limits_{i=2}^M |C_i| \right
]=E\left[\xi(B(s))\right]=2^dE\left[\xi\left(\left[0,\frac{s}{2}\right]^d\right)\right]\\
&=2^d\left\{E\left[\xi(R_0)\right]+E\left[\xi\left(\left[1,\frac{s}{2}\right]\times\left[0,\frac{s}{2}\right]^{d-1}\right)\right]\right\} = 2^d\sum\limits_{i=0}^{d-1} E\left[\xi(R_i)\right].
\end{aligned}\end{aligned}$$ Thus, we just need to estimate the value of $E\left[
\xi\left(R_i \right) \right]$. The following Lemmas \[exponent2\]-\[limit\] are introduced to get the desired estimation.
\[exponent2\] Suppose $d\geq 2$ and $\lambda>\lambda_c$. Let $V_x=V_x(s)$ denote the connected component containing $x$ of $G(\mathcal{H}_{\lambda,s}\cup \{x\};1)$. There exist constants $c>0$ and $n_0>0$, such that if $n>n_0$ and $s>2n$ then for any point $x \in B(s)$, $$\begin{aligned}
\label{exp00}
P \left [ n \leq
\mbox{diam}(V_x) \leq s/2 \right ] < e^{ -c n },\end{aligned}$$ and $$\begin{aligned}
\label{exp01}
P \left [ \left \{ |V_x| \geq n \right \} \cap \left \{
\mbox{diam}(V_x) \leq s/2 \right \} \right ] < \exp \left ( -c
n^{(d-1)/d} \right ).\end{aligned}$$
The proof uses ideas from the latter part of the proof of Theorem 10.18 in [@p10]. Given $x\in \mathbb{R}^d$, let $\widetilde{z}$ denote the point in $B_{\mathbb{Z} }'(n(s))$ satisfying $x\in
B_{\widetilde{z}}$, where the definition of $B_{\mathbb{Z} }'(n(s))$ and $
B_{\widetilde{z}}$ is given in pp.216 and pp.217 of [@p10] respectively. Also, $C_x$, $D_{ext}C_x$, $M_0$, $n(s)$ and $M(s)$ are defined as same as those appearing in pp.218-219 of [@p10]. Penrose has proved that $D_{ext}C_x$ is $*-$connected and if $|C_x|<n(s)^d/2$ then $$\begin{aligned}
\label{exp22_1}
\begin{aligned}
|D_{ext}C_x| \geq (2d)^{-1} (1- ({\textstyle \frac{2}{3}})^{1/d})|C_x|^{(d-1)/d},
\end{aligned}\end{aligned}$$ see pp.219 of [@p10].
Let $\mathcal{A}_{m,s}$ denote the collection of $*-$connected subsets of cardinality $m$ which disconnects the point $\widetilde{z}$ from the giant component of $B_{\mathbb{Z}
}'(n(s))$. Then $\mathcal{A}_{m,s}$ is restricted by the box of $B_{\mathbb{Z} }'(n(s))\cap ([-m,m]^d \oplus \widetilde{z})$ and $D_{ext}C_x \in \mathcal{A}_{|D_{ext}C_x|,s}$. By a Peierls argument (Corollary 9.4 in [@p10]), the cardinality $|\mathcal{A}_{m,s}|$ is bounded by $(2m+1)^d \gamma^m$, with $\gamma:=2^{3^d}$. Therefore, there exists a constant $k_0$ such that for any integer $k>k_0$, $$\begin{aligned}
\label{exp23}
\begin{aligned}
P \left [|D_{ext}C_x|\geq k \right ] &\leq P \left [
\bigcup\limits_{m \geq k} \bigcup\limits_{\sigma \in
\mathcal{A}_{m,s} } \{X_z=0, \forall z\in\sigma\} \right
]\\
&\leq \sum\limits_{m \geq k } (2m+1)^d \gamma^m (1-p_1)^m <
(\frac{2}{3})^{k}.
\end{aligned}\end{aligned}$$
By the definition of $C_x$ and $D_{ext}C_x$, if $n\leq$diam$(V_x)\leq
s/2$ then $$\frac{n}{M(s)}-1\leq\mbox{diam}(C_x)\leq \frac{n(s)}{2}+2,$$ and therefore we can get $|C_x| < n(s)^d/2$ and $|D_{ext}C_x|\geq \frac{n}{M(s)}-1$ for large $s$. Therefore, by (\[exp23\]), there exists a constant $n_0>0$, such that if $n>n_0$ then, $$\begin{aligned}
P \left [ n\leq
\mbox{diam}(V_x) \leq \frac{s}{2}\right ] \leq P \left
[|D_{ext}C_x|\geq \frac{n}{M(s)}-1 \right ]
<\left (\frac{2}{3} \right )^{\frac{n}{2M_0}-1}.\end{aligned}$$ This yields (\[exp00\]).
It remains to consider the case of $|V_x|>n$. Since $C_x$ is a $*-$connected component containing $\widetilde{z}$ in $B_{\mathbb{Z}
}'(n(s))$, by a Peierls argument (Lemma 9.3 in [@p10]), for all $k$, the number of $*-$ connected subsets of $B_{\mathbb{Z}
}'(n(s))$ of cardinality $k$ containing $\widetilde{z}$ is at most $\gamma^k$. Let $c_2\geq e^2 (2M_0)^d \lambda$. If $|C_x|<k$ and $|V_x|\geq c_2 k+1$, then for at least one of these subsets of $B_{\mathbb{Z} }'(n(s))$ the union of the associated boxes $B_z$ contains at least $c_2 k$ points of $\mathcal{H}_{\lambda}$. Therefore, by Lemma 1.2 in [@p10], we have $$\begin{aligned}
\label{exp24}
\begin{aligned}
P[\{|C_x|<k\}\cap\{|V_x|\geq c_2 k+1\}] &< \gamma^k
P \left [Po \left (k(2M_0)^d\lambda \right )\geq c_2 k \right ]\\
&\leq \gamma^k \exp \left \{-\left (\frac{c_2 k}{2} \right ) \log
\left ( \frac{c_2}{(2M_0)^d \lambda } \right ) \right \}.
\end{aligned}\end{aligned}$$ So if $c_2$ is chosen large enough, this probability decays exponentially in $k$.
Set $\beta:=(2d)^{-1} (1- (\frac{2}{3})^{1/d})$. By (\[exp22\_1\]) and (\[exp23\]), we have $$\begin{aligned}
P[\{\mbox{diam}(V_x) \leq s/2 \} \cap \{ |C_x| \geq k \} ] \leq
P \left [|D_{ext}C_x| \geq \beta k^{(d-1)/d} \right ]<
(\frac{2}{3})^{\beta k^{(d-1)/d}}.\end{aligned}$$ Combined with (\[exp24\]), this gives (\[exp01\]).
\[expectation\] Suppose $d\geq 2$ and $\lambda>\lambda_c$. Then $$\begin{aligned}
0<\sum\limits_{n=1}^{\infty}nP(|V_0|=n)<\infty.\end{aligned}$$
By Lemma \[exponent\], there exist two constants $c>0$ and $n_0>0$, such that for all $n>n_0$, $$\begin{aligned}
n^{-(d-1)/d}\log P( n\leq |V_0|<\infty)<-c.\end{aligned}$$ From this it can be deduced that $$\begin{aligned}
\sum\limits_{n=n_0+1}^{\infty}P(n\leq |V_0|<\infty)<
\sum\limits_{n=n_0+1}^{\infty}e^{-cn^{(d-1)/d}}<\infty.\end{aligned}$$ Therefore, we have $$\begin{aligned}
\begin{aligned}
\sum\limits_{n=1}^{\infty}nP(|V_0|=n)&=\sum\limits_{n=1}^{\infty}P(n\leq
|V_0|<\infty)\\
&\leq n_0+\sum\limits_{n=n_0+1}^{\infty}P(n\leq
|V_0|<\infty)\\
&< \infty.
\end{aligned}\end{aligned}$$ In the following we prove that $P(1\leq |V_0|<\infty)>0$.
Define $\widetilde{\tau}=\widetilde{\tau}
(\lambda):=\sum\limits_{n=1}^{\infty}nP(|V_0|=n)$.
For $x\in B(s)$ and $0<a\leq 1$, define the box $$\begin{aligned}
B_i(x,a):=x\oplus
\big(\underbrace{[0,1]\times\cdots\times[0,1]}\limits_i \times
\underbrace{[0,a]\times\cdots\times[0,a]}\limits_{d-i}\big).\end{aligned}$$ Also, for any region $R\subseteq B(s)$, define $$\begin{aligned}
\label{prop1_1}
D(R)=D(R,s):=\max_{2\leq j \leq M,\chi_{j}(R)=1} \mbox{diam} (C_j).\end{aligned}$$
\[prop1\] Suppose $d\geq 2$ and $\lambda>\lambda_c$. There exist constants $c>0$ and $n_0>0$, such that if $x\in B(s)$, $a\in(0,1]$ and $n>n_0$ then $$\begin{aligned}
\label{prop1_000}
P[D(B_i(x,a))\geq n]<e^{-c n},\end{aligned}$$ and $$\begin{aligned}
\label{prop1_00}
P[\xi(B_i(x,a))\geq n]<\exp \left (-c n^{(d-1)/d} \right )+e^{-c
s}.\end{aligned}$$
Let $W_1$ denote the number of the connected components which intersect with $B_i(x,a)$, and have metric diameter not greater than $s/2$ but not smaller than $n$. By Markov’s inequality, $$\begin{aligned}
\label{prop1_02}
P \left [\left \{D(B_i(x,a)) \geq n \right \} \cap \left \{
D(B_i(x,a)) \leq s/2 \right \} \right ]\leq P[W_1>0] \leq
E[W_1].\end{aligned}$$ By Palm theory for Poisson process and Lemma \[exponent2\], if $n>n_0$ then $$\begin{aligned}
\label{prop1_03}
\begin{aligned}
E[W_1]&=\lambda\int_{B_i(x,a)} P \left [ \left \{
\mbox{diam}(V_x(s)) \geq n \right \} \cap \left \{
\mbox{diam}(V_x(s)) \leq s/2 \right \} \right ]dx\\
&<\lambda a^{d-i} e^{-c n}.
\end{aligned}\end{aligned}$$ Also, $C_i$ ($2\leq i \leq M$) is not the largest component of $G(\mathcal{H}_{\lambda,s};1)$, then by Proposition 10.13 in [@p10], there exist constants $c_1>0$ and $s_1>0$, such that if $s>s_1$ then $$\begin{aligned}
\label{prop1_04}
&& P \left [ D(B_i(x,a)) > s/2 \right ] <e^{-c_1 s} .\end{aligned}$$ Together with (\[prop1\_02\]), (\[prop1\_03\]) and (\[prop1\_04\]), we obtain $$\begin{aligned}
P[D(B_i(x,a))\geq n]<e^{-c n}+e^{-c_1 s}.\end{aligned}$$ Since $P[D(B_i(x,a))>s]=0$, thus (\[prop1\_000\]) follows.
Note that $B_i(x,a)$ contains at most $2^d$ connected components. Thus, if $\xi(B_i(x,a)) \geq n$, by the definition of $\xi(\cdot)$, there exists at least one component intersecting with $B_i(x,a)$ such that it contains no less than $2^{-d}n$ points. Let $W_2$ be the number of the connected components which intersect with $B_i(x,a)$, and have more than $2^{-d}n$ elements and not larger than $s/2$ metric diameter. With the similar argument as (\[prop1\_02\]) and (\[prop1\_03\]), we get if $n>n_0$ then $$\begin{aligned}
&&P \left [\left \{ \xi(B_i(x,a)) \geq n \right \} \cap \left \{
D(B_i(x,a)) \leq s/2 \right \} \right ] \leq E[W_2]\\
&&~~=\lambda\int_{B_i(x,a)} P \left [ \left \{ |V_x(s)| \geq
2^{-d}n \right \} \cap \left \{ \mbox{diam}(V_x(s)) \leq
s/2 \right \} \right ]dx\\
&&~~<\lambda a^{d-i} \exp\left(-c2^{-d} n\right),\end{aligned}$$ together with (\[prop1\_04\]) this gives (\[prop1\_00\]).
Let real numbers $s_1>2$ and $s_2>2$ be given. Let points $x=(x_1,x_2,\ldots,x_d)\in [0,s_1/2]^d$ and $\widetilde{x}=(\widetilde{x}_1,\widetilde{x}_2,\ldots,\widetilde{x}_d)\in
[0,s_2/2]^d$ be given. For all $1\leq j \leq d$, define $$\begin{aligned}
N_{x,\widetilde{x}}^j(s_1,s_2):=\left\{
\begin{array}{ll}
\min(s_1,s_2)-x_j-1, & \mbox{if } x_j=\widetilde{x}_j, \\
\min(x_j,\widetilde{x}_j,s_1-x_j-1,s_2-\widetilde{x}_j-1), &
\mbox{otherwise},
\end{array}
\right.\end{aligned}$$ and let $$\begin{aligned}
\label{prop1_1}
N_{x,\widetilde{x}}(s_1,s_2):=\min_{1\leq j\leq d}\lfloor
N_{x,\widetilde{x}}^j(s_1,s_2)\rfloor.\end{aligned}$$
\[argument\] Let us assume $d \geq 2$, $\lambda>\lambda_c$, $1\leq i \leq d$ and $0<a\leq 1$. There exist constants $c>0$ and $n_0>0$, such that if $x\in [0,s_1/2]^d$, $\widetilde{x}\in [0,s_2/2]^d$ and $N_{x,\widetilde{x}}(s_1,s_2)>n_0$ then $$\begin{aligned}
&&\left| E\left[\xi(B_i(x,a),s_1)\right]-
E\left[\xi(B_i(\widetilde{x},a),s_2)\right]\right|<\exp\left(-c
N_{x,\widetilde{x}}(s_1,s_2) \right).\end{aligned}$$
Let $B'(s_2):=B(s_2)\oplus\{x-\widetilde{x}\},$ and let $\widetilde{C}_1,\widetilde{C}_2,\ldots,\widetilde{C}_{\widetilde{M}}$ denote the components of $G(\mathcal{C}_{\infty}\cap B'(s_2);1),$ taking in order of decreasing order. For any region $R\subseteq
B'(s_2)$ and $2\leq i \leq \widetilde{M}$, define $$\widetilde{\chi_{i}}(R):=\left\{
\begin{array}{ll}
1, & \mbox{if the out-connect point of $\widetilde{C}_i$ is contained by } R, \\
0, & \mbox{otherwise}.
\end{array}
\right.$$ Let $\widetilde{\xi}(R,s_2):=
\sum_{i=2}^{\widetilde{M}}\widetilde{\chi}_{i}(R)|\widetilde{C}_i|$ and define $$\begin{aligned}
\widetilde{D}(R,s_2):=\max_{2\leq j \leq
\widetilde{M},\widetilde{\chi}_{j}(R)=1} \mbox{diam}
(\widetilde{C}_j).\end{aligned}$$ According to the ergodicity of Poisson point processes, we can get $$\begin{aligned}
\label{argu_1}
P\left[\widetilde{\xi}\left(B_i(x,a),s_2\right)=k\right]=P\left[\xi\left(B_i(\widetilde{x},a),s_2\right)=k\right],~~~~\forall~k\geq
1.\end{aligned}$$
![If $C_k$ connects with $\mathcal{H}_{\lambda}\cap \Delta$, the event of $\xi(B_i(x,a),s_1)\neq
\widetilde{\xi}(B_i(x,a),s_2)$ may happen.[]{data-label="Lemmagraph"}](13295fig1.eps){width="3in"}
Let $\Delta:=B(s_1)\cup B'(s_2)-B(s_1)\cap B'(s_2).$ If $\xi(B_i(x,a),s_1)\neq \widetilde{\xi}(B_i(x,a),s_2)$, then there exists at least one component among $C_2,\ldots,C_M,\widetilde{C}_2,\ldots,\widetilde{C}_{\widetilde{M}}$ which connects directly with $\mathcal{H}_{\lambda}\cap \Delta$, see Figure \[Lemmagraph\]. For simplicity of exposition, we take $N=N_{x,\widetilde{x}}(s_1,s_2)$, $\xi_1=\xi(B_i(x,a),s_1)$ and $\xi_2=\widetilde{\xi}(B_i(x,a),s_2)$. Therefore, by (\[prop1\_000\]), if $N>n_0+1$ then $$\begin{aligned}
\label{argu_2}
\begin{aligned}
P\left[\xi_1 \neq \xi_2\right]&\leq P\left[ \left\{D(B_i(x,a),s_1)\geq N-1 \right\} \cup
\left\{\widetilde{D}(B_i(x,a),s_2)\geq N-1 \right\} \right]\\
&< 2e^{-c(N-1)}.
\end{aligned}\end{aligned}$$ Also, $$\begin{aligned}
\label{argu_2_1}
\begin{aligned}
&P\left[\left\{\xi_1=k\right\} \cap \left\{\xi_2\neq k \right\} \right]+P\left[\left\{\xi_1\neq k\right\} \cap \left\{\xi_2=k \right\} \right]\\
&~~=P\left[\left\{\xi_1=k\right\}\right]+P\left[\left\{\xi_2=k\right\}\right]-2P\left[\left\{\xi_1=k\right\} \cap \left\{\xi_2=k \right\} \right]\\
&~~\geq |P\left[\left\{\xi_1=k\right\}\right]-P\left[\left\{\xi_2=k\right\}\right]|,
\end{aligned}\end{aligned}$$ so by (\[argu\_2\]) and (\[argu\_2\_1\]) we have $$\begin{aligned}
\label{argu_3}
\begin{aligned}
&\sum_{k=1}^{\infty} \left|P\left[\left\{\xi_1=k\right\}\right]-P\left[\left\{\xi_2=k\right\}\right]\right|\\
&~~\leq \sum_{k=1}^{\infty} \left( P\left[\left\{\xi_1=k\right\} \cap \left\{\xi_1\neq \xi_2 \right\} \right]+P\left[\left\{\xi_2=k \right\} \cap \left\{\xi_1 \neq \xi_2 \right\} \right] \right)\\
&~~=P\left[\left\{\xi_1\geq 1 \right\} \cap \left\{\xi_1\neq \xi_2 \right\} \right]+P\left[\left\{\xi_2\geq 1 \right\} \cap \left\{\xi_1 \neq \xi_2 \right\} \right]<4e^{-c(N-1)}.
\end{aligned}\end{aligned}$$ Thus, by (\[argu\_1\]) and (\[argu\_3\]) we can get $$\begin{aligned}
\label{argu_4}
\begin{aligned}
&\left| E\left[\xi_1\right]-E\left[
\xi(B_i(\widetilde{x},a),s_2)\right]\right|=\left| \sum_{n=1}^{\infty}\sum_{k=n}^{\infty}
\left(P\left[\xi_1=k\right]-P\left[
\xi_2=k \right]\right) \right|\\
&< 4N^{d/(d-1)}e^{-c(N-1)}+ \sum_{n=N^{d/(d-1)}}^{\infty}\left(
P\left[\xi_1\geq n\right]+P\left[
\xi_2\geq n) \right]\right).
\end{aligned}\end{aligned}$$
In the following we estimate the upper bound of $\sum_{n=N^{d/(d-1)}}^{\infty} P[\xi_1 \geq n]$. Firstly, by (\[prop1\_00\]), for $N$ large enough, we can obtain $$\begin{aligned}
\label{argu_5}
\begin{aligned}
\sum\limits_{n=N^{d/(d-1)}}^{e^2 \lambda s_1^d} P[\xi_1
\geq n]<\sum\limits_{n=N^{d/(d-1)}}^{e^2 \lambda s_1^d} \exp
\left (-c n^{(d-1)/d} \right )+ e^2 \lambda s_1^d e^{-c
s_1 }.
\end{aligned}\end{aligned}$$ Set $\alpha:=\exp (-c N)$, then $$\begin{aligned}
\label{argu_6}
\begin{aligned}
&\sum\limits_{n=N^{d/(d-1)}}^{e^2 \lambda s_1^d} \exp \left (-c
n^{(d-1)/d} \right )=\sum\limits_{n=N^{d/(d-1)}}^{e^2 \lambda
s_1^d} \alpha^{(nN^{-d/(d-1)})^{(d-1)/d}}\\
&~~<N^{d/(d-1)}\sum\limits_{k=1}^{\infty} \alpha ^{k^{(d-1)/d}}=
N^{d/(d-1)} \alpha \sum\limits_{k=1}^{\infty}
\alpha^{k^{(d-1)/d}-1}<M N^{d/(d-1)} \alpha, \end{aligned}\end{aligned}$$ where $M=\sum_{k=1}^{\infty} \exp(-c (k^{(d-1)/d}-1))<\infty$ is a constant.
Secondly, by Lemma 1.2 in [@p10], $$\begin{aligned}
\label{argu_7}
\begin{aligned}
&\sum\limits_{n=e^2 \lambda s_1^d+1}^{\infty} P[\xi_1
\geq n] < \sum\limits_{n=e^2 \lambda s_1^d+1}^{\infty} P[Po(\lambda
s_1^d)\geq
n]\\
&~~ \leq \sum\limits_{n=e^2 \lambda s_1^d+1}^{\infty} \exp \left (
-\left (\frac{n}{2} \right ) \log \left ( \frac{n}{\lambda s_1^d}
\right ) \right ) < \frac{e^{-(e^2 \lambda s_1^d +1)}}{1-e^{-1}}.
\end{aligned}\end{aligned}$$ Thus, by (\[argu\_5\]), (\[argu\_6\]) and (\[argu\_7\]), there exists a constant $c_1>0,$ such that for large $N$, $$\begin{aligned}
\label{argu_8}
\sum\limits_{n=N^{d/(d-1)}}^{\infty} P[\xi_1 \geq n]<
e^{-c_1 N}.\end{aligned}$$ Using the ergodicity of Poisson point processes, similarly, we can get $$\begin{aligned}
\label{argu_9}
\sum\limits_{n=N^{d/(d-1)}}^{\infty} P[\xi_2
\geq n]< e^{-c_1 N}.\end{aligned}$$ Combining (\[argu\_4\]), (\[argu\_8\]) and (\[argu\_9\]) gives us the result.
\[limit\] Suppose $d \geq 2$ and $\lambda>\lambda_c$. Let integer $i\in [1,d]$, and constants $a\in(0,1]$ and $x_j\in [0,\infty)$, $1\leq j \leq i$. Define the point $$\widetilde{x}_{s,a}=\widetilde{x}_{s,a}(x_1,\ldots,x_i):=\left(x_1,\ldots,x_i,\frac{s}{2}-a,\ldots,\frac{s}{2}-a\right)\in
\mathbb{R}^d,$$ then the limit of $E[\xi(B_i(\widetilde{x}_{s,a},a))]$ exists and $$\begin{aligned}
\label{lim_new_00}
\lim_{s\rightarrow\infty}E[\xi(B_i(\widetilde{x}_{s,a},a))]=a^{d-i}\lim_{s\rightarrow\infty}E[\xi(B_i(\widetilde{x}_{s,1},1))].\end{aligned}$$ Also, if $\min_{1\leq j \leq i}\{x_j\}=0$, then $\lim_{s\rightarrow\infty}E[\xi(B_i(\widetilde{x}_{s,a},a))]>0$.
For $s_1$ and $s_2$ large enough, suppose $s_2>s_1$. By (\[prop1\_1\]), it is easy to get $N_{\widetilde{x}_{s_1,a},\widetilde{x}_{s_2,a}}(s_1,s_2)>s_1/2-2.$ Therefore by Lemma \[argument\] and Cauchy’s criterion for convergence, the limit of $E[\xi(B_i(\widetilde{x}_{s,a},a))]$ exists as $s\rightarrow\infty$.
For any constant $b\in[0,1]$, let $$\begin{aligned}
y_{s,b}=y_{s,b}(x_1,\ldots,x_i):=\left(x_1,\ldots,x_i,\frac{s}{2}-1,\ldots,\frac{s}{2}-1,\frac{s}{2}-b\right)\in\mathbb{R}^d.\end{aligned}$$ Similarly, by Lemma \[argument\] and the Cauchy’s criterion we have the limit of $E[\xi(B_{d-1}(y_{s,b},b))]$ exists. Define $$f_{x_1,\ldots,x_i}(b):=\lim_{s\rightarrow\infty}E[\xi(B_{d-1}(y_{s,b},b))].$$ Since $Leb(B_{d-1}(y_{s,b},b)\cap B_{d-1}(y_{s,1},1-b))=0$, then by the definition of $\xi$ we have $$\begin{aligned}
\label{lim_new_01}
E\left[\xi(B_{d-1}(y_{s,1},1))\right]=E[\xi(B_{d-1}(y_{s,1},1-b))]+E[\xi(B_{d-1}(y_{s,b},b))].\end{aligned}$$ By (\[prop1\_1\]), $N_{y_{s,1},y_{s,1-b}}(s,s)>s/2-2$. Using Lemma \[argument\] and Cauchy’s criterion we have $$\begin{aligned}
\lim_{s\rightarrow\infty}E[\xi(B_{d-1}(y_{s,1},1-b))]=\lim_{s\rightarrow\infty}E[\xi(B_{d-1}(y_{s,1-b},1-b))]=f_{x_1,\ldots,x_i}(1-b).\end{aligned}$$ Therefore, taking the limits of the both sides on (\[lim\_new\_01\]), we can get $$\begin{aligned}
f_{x_1,\ldots,x_i}(1)=f_{x_1,\ldots,x_i}(1-b)+f_{x_1,\ldots,x_i}(b),\end{aligned}$$ which indicates that $f_{x_1,\ldots,x_i}(b)=bf_{x_1,\ldots,x_i}(1)$. With the similar method, we can get $$\begin{aligned}
\lim_{s\rightarrow\infty}E[\xi(B_i(\widetilde{x}_{s,a},a))]=a^{d-i}f_{x_1,\ldots,x_i}(1),\end{aligned}$$ which gives (\[lim\_new\_00\]).
It remains to prove that $\lim_{s\rightarrow\infty}E[\xi(B_i(\widetilde{x}_{s,a},a))]>0$ if $\min_{1\leq j
\leq i}\{x_j\}=0$. For simplicity of exposition, we restrict ourselves to the case of $d=2$, and the proof of this result has no essential difficulty when $d\geq 3$.
Let $\partial B(s)$ denote the boundary of $B(s)$. If $\min_{1\leq j
\leq i}\{x_j\}=0$, then $\widetilde{x}_{s,a}\in\partial B(s)$. For $x\in B_i(\widetilde{x}_{s,a},a)$, let $d_x$ to be the Euclid distance from $x$ to $\partial B(s)$, then $0\leq d_x\leq 1$. Let $V_x$ denote the connected component containing $x$ of $G(\mathcal{H}_{\lambda,s}\cup \{x\};1)$. Firstly, we will show that there exists a constant $c>0$, such that $$\begin{aligned}
\label{bound_2}
\begin{aligned}
&P\left[\{|V_x|=1\} \cap \{x\in
\mathcal{C}_{\infty}\}\right]\\
&~~\geq c\left[1-\exp\left(\lambda\left(d_x\sqrt{1-d_x^2}-\arccos
d_x\right)\right)\right]p_{\infty}(\lambda).
\end{aligned}\end{aligned}$$ Define
![The placements of $B_x^-,B_x^+,R_1$ and $R_2$ are shown.[]{data-label="box2"}](13295fig2.eps){width="2in"}
$B_x^-$ to be the rectangle of $(1+d_x)\times 2$ centred at $x$ and $B_x^+$ to be the rectangle of $(\frac{7}{3}+d_x)\times
\frac{10}{3}$ centred at $x$. Divide the region of $B_x^+\backslash
B_x^-$ into 64 small rectangles with two diffrent sizes: one size recorded $R_1$ is $\frac{1}{3}\times \frac{1}{3}$, and the other size recorded $R_2$ is $\frac{1+d_x}{6}\times \frac{1}{3}$, see Figure \[box2\]. The number of small rectangles with size $R_1$ is $40$, and the number of small rectangles with size $R_2$ is $24$. Define $A_1$ to be the event that each of these 64 small rectangles includes at least one point of $\mathcal{H}_{\lambda}$. By the properties of Poisson point processes, we have $$\begin{aligned}
\label{bound_3}
\begin{aligned}
P(A_1)&=\left(1-e^{-\lambda/9}\right
)^{40}\cdot\left(1-e^{-\lambda(1+d_x)/18}\right )^{24}\\
&\geq \left(1-e^{-\lambda/9}\right
)^{40}\cdot\left(1-e^{-\lambda/18}\right )^{24}.
\end{aligned}\end{aligned}$$ If $A_1$ happens, there exists a connected component in $B_x^+\backslash B_x^-$ which contains all the points in these small rectangles. Also, for any point in $\mathbb{R}^d \backslash B_x^-$ which can connect directly with a point in $B_x^-$, it must connect directly with this connected component. Let $A_2$ denote the event that there exists at least one point in $B_x^+\backslash B_x^-$ contained by $\mathcal{C}_{\infty}$. So according to above discussion, the event $A_1\cap A_2$ is independent with the distribution of the points of $\mathcal{H}_{\lambda}$ in $B_x^-$. Therefore, $$\begin{aligned}
\label{bound_4}
P(A_1\cap A_2)=P(A_1)P(A_2|A_1) \geq P(A_1) p_{\infty}(\lambda).\end{aligned}$$ Denote $A_3$ to be the event that there exists at least one point of $\mathcal{H}_{\lambda}$ in $B(x;1) \cap B(s)^c$, where $B(x;1)$ denotes the $d-dimensional$ unit ball centred at point $x$. By the properties of Poisson point processes it can be computed that $$\begin{aligned}
\label{bound_5}
P(A_3)=1-\exp\left(\lambda\left(d_x\sqrt{1-d_x^2}-\arccos
d_x\right)\right).\end{aligned}$$ Because $A_3$ and $A_1\cap A_2$ are both increasing events in $G(\mathcal{H}_{\lambda};1)$, by FKG inequality (Theorem 2.2 in [@p12]) we have $$\begin{aligned}
\label{bound_6}
P(A_3\cap A_1 \cap A_2) \geq P(A_3)P(A_1 \cap A_2).\end{aligned}$$ If the event $A_3\cap A_1 \cap A_2$ happens, it must be true that $x\in \mathcal{C}_{\infty}$. Also, the event $A_3$ is independent with the distribution of the points of $\mathcal{H}_{\lambda}$ in $B_x^-$, so we have $$\begin{aligned}
\label{bound_6_temp}
\begin{aligned}
P\left[\{|V_x|=1\} \cap \{x\in \mathcal{C}_{\infty}\}\right] &\geq
P[A_3\cap A_1 \cap A_2 \cap \{\mathcal{H}_{\lambda} \cap B_x^-
= \emptyset\}]\\
&=e^{-2(1+x)\lambda}P(A_3\cap A_1 \cap A_2)\\
&\geq e^{-4\lambda}P(A_3\cap A_1 \cap A_2).
\end{aligned}\end{aligned}$$ Set $c:=e^{-4\lambda}\cdot \left(1-e^{-\lambda/9}\right
)^{40}\cdot\left(1-e^{-\lambda/18}\right )^{24}$, together with (\[bound\_3\]), (\[bound\_4\]), (\[bound\_5\]), (\[bound\_6\]) and (\[bound\_6\_temp\]) we can get (\[bound\_2\]).
Let $W$ denote the number of the points of $\mathcal{H}_{\lambda}
\cap B_i(\widetilde{x}_{s,a},a)$ which belong to $\mathcal{C}_{\infty}$ but are isolated in $B(s)$. By the definition of $\xi(B_i(\widetilde{x}_{s,a},a))$ and Palm theory for Poisson processes, we have $$\begin{aligned}
E[\xi(B_i(\widetilde{x}_{s,a},a))]\geq E[W]=\lambda
\int_{B_i(\widetilde{x}_{s,a},a))} P\left[\{|V_x|=1\} \cap \{x\in
\mathcal{C}_{\infty}\}\right] dx.\end{aligned}$$ Combining this with (\[bound\_2\]), we can get $E[\xi(B_i(\widetilde{x}_{s,a},a))]>\frac{1}{2}c\left(1-e^{(1-\pi)\lambda/4}\right)\lambda
p_{\infty}(\lambda).$ Our result follows.
For simplicity of exposition, we shall prove (\[order\_t1\_00\]) only in the case of $d=3$, and this proof has no essential difficulty in the case of $d=2$ or $d\geq 4$.
Let $\eta_{ij}(s):=E\left[\xi\left([0,1]\times [i,i+1]\times [j,j+1],s \right)\right]$ and take $n=\lfloor \frac{s}{2} \rfloor$. By symmetry we have $\eta_{ij}(s)=\eta_{ji}(s)$, and therefore $$\begin{aligned}
\label{t1_1}
\begin{aligned}
E\left[\xi\left([0,1]\times[0,n]^2\right)\right]&=\sum_{i=0}^{n-1}\sum_{j=0}^{n-1}\eta_{ij}(s)\\
&=\eta_{00}(s)+\sum_{k=1}^{n-1}\left(2\sum_{i=0}^{k-1}\eta_{ik}(s)+\eta_{kk}(s) \right).
\end{aligned}\end{aligned}$$ Set $$a_1(s):=\eta_{00}(s)+\sum_{k=1}^{n-1}\left(2\sum_{i=0}^{k-1}\left(\eta_{ik}(s)-\eta_{i,n-1}(s)\right)+\eta_{kk}(s)-\eta_{k,n-1}(s) \right),$$ then for large $s$ and $s_2$ satisfying $s_2>s$, by Lemma \[argument\] we have $$\begin{aligned}
\label{t1_2}
\begin{aligned}
&|a_1(s)-a_1(s_2)|<2n^2 e^{-cs/2}\\
&~~~~~~~~+\sum_{k=n}^{n_2-1}\left(2\sum_{i=0}^{k-1}|\eta_{ik}(s_2)-\eta_{i,n_2-1}(s_2)|+|\eta_{kk}(s_2)-\eta_{k,n_2-1}(s_2)| \right)\\
&<2n^2 e^{-cs/2}+ \sum_{k=n}^{n_2-1}\left(2\sum_{i=0}^{k-1}e^{-ck}+e^{-ck}\right)=o\left(e^{-cs/3}\right),
\end{aligned}\end{aligned}$$ where $n_2=\lfloor\frac{s_2}{2} \rfloor$ and $c$ is the same constant appearing in Lemma \[argument\]. Then by Cauchy’s criterion the limit of $a_1(s)$ exists.
Define the point $y_i=(0,i,n)\in \mathbb{R}^3$. For any $i\in [0,n-1]$ and large $s$, using Lemmas \[argument\] and \[limit\] we can get $$\begin{aligned}
\label{t1_3}
\begin{aligned}
&\big|E\left[\xi\left(B_{2}\left(y_{i},{\textstyle \frac{s}{2}}-n\right)\right)\right]-({\textstyle \frac{s}{2}}-n)\eta_{i,n-1}(s)\big|\\
&\leq \big|E\left[\xi\left(B_{2}\left(y_{i},{\textstyle \frac{s}{2}}-n\right)\right)\right]-({\textstyle \frac{s}{2}}-n)E\left[\xi\left([0,1]\times[i,i+1]\times[{\textstyle \frac{s}{2}}-1,{\textstyle \frac{s}{2}}]\right)\right]\big|\\
&~~~~~~~~+({\textstyle \frac{s}{2}}-n)\big|E\left[\xi\left([0,1]\times[i,i+1]\times[{\textstyle \frac{s}{2}}-1,{\textstyle \frac{s}{2}}]\right)\right]-\eta_{i,n-1}(s)\big|\\
&=o\left(e^{-cs/3}\right).
\end{aligned}\end{aligned}$$ Similarly, we can get $$\begin{aligned}
\label{t1_4}
E\left[\xi\left([0,1]\times\left[n,\frac{s}{2}\right]^2\right)\right]=\left(\frac{s}{2}-n\right)^2\eta_{n-1,n-1}(s)+o\left(e^{-cs/3}\right).\end{aligned}$$ We recall that $R_0=[0,1]\times [0,s/2]^2$, then together with (\[t1\_1\]), (\[t1\_2\]), (\[t1\_3\]) and (\[t1\_4\]), $$\begin{aligned}
\label{t1_5}
\begin{aligned}
E\left[\xi\left(R_0\right)\right]&=E\left[\xi\left([0,1]\times[0,n]^2\right)\right]+2\sum_{i=0}^{n-1}E\left[\xi\left(B_{2}\left(y_{i},\frac{s}{2}-n\right)\right)\right]\\
&~~~~~~~~~~~~+E\left[\xi\left([0,1]\times\left[n,\frac{s}{2}\right]^2\right)\right]\\
&=\sum_{k=1}^{n-1}\left(2\sum_{i=0}^{k-1}\eta_{i,n-1}(s)+\eta_{k,n-1}(s) \right)+\left(s-2n\right)\sum_{i=0}^{n-1}\eta_{i,n-1}(s)\\
&~~~~~~~~~~~~+\left(\frac{s}{2}-n\right)^2\eta_{n-1,n-1}(s)+a_1+o\left(e^{-cs/3}\right),
\end{aligned}\end{aligned}$$ where $a_1:=\lim_{s\rightarrow\infty}a_1(s)$. Let $b_i(s):=\eta_{i,n-1}(s)-\eta_{n-1,n-1}(s)$, then by (\[t1\_5\]) we have $$\begin{aligned}
\label{t1_6}
\begin{aligned}
E\left[\xi\left(R_0\right)\right]
&=\left(\frac{s^2}{4}-1\right)\eta_{n-1,n-1}(s)+\sum_{k=1}^{n-1}\left(2\sum_{i=0}^{k-1}b_i(s)+b_k(s) \right)\\
&~~~~~~~~~~~~+\left(s-2n\right)\sum_{i=0}^{n-1}b_i(s)+a_1+o\left(e^{-cs/3}\right)\\
&=\left(\frac{s^2}{4}-1\right)\eta_{n-1,n-1}(s)+s\sum_{i=0}^{n-2}b_i(s)-2b_0(s)-\sum_{i=1}^{n-2}(2i+1)b_i(s)\\
&~~~~~~~~~~~~+a_1+o\left(e^{-cs/3}\right).
\end{aligned}\end{aligned}$$ Set $$\begin{aligned}
a_2(s):=\sum_{i=0}^{n-2}b_i(s)~~~\mbox{and}~~~a_3(s):=2b_0(s)+\sum_{i=1}^{n-2}(2i+1)b_i(s).\end{aligned}$$ With the similar argument as (\[t1\_3\]), we can get that the exist constants $a_2$ and $a_3$ such that $$\begin{aligned}
|a_2(s)-a_2|<3ne^{-cs/2}~~~\mbox{and}~~~|a_3(s)-a_3|=o\left(e^{-cs/3}\right).\end{aligned}$$ Also, by Lemmas \[argument\] and the Cauchy’s criterion, there exists a constant $a_0>0$ such that $$|\eta_{n-1,n-1}(s)-a_0|<e^{-c(n-1)}.$$ Taking $a_0,a_2$ and $a_3$ into (\[t1\_6\]) we have $$\begin{aligned}
E\left[\xi\left(R_0\right)\right]=\left(\frac{s^2}{4}-1\right)a_0+s a_2-a_3+a_1+o\left(e^{-cs/3}\right).\end{aligned}$$ with the similar argument as above, there exist constants $a_4, a_5, a_6$ and $a_7$, such that $$\begin{aligned}
E\left[\xi\left(R_1\right)\right]=\frac{s^2}{4}a_0+s a_4+a_5+o\left(e^{-cs/3}\right),\end{aligned}$$ and $$\begin{aligned}
E\left[\xi\left(R_2\right)\right]=\frac{s^2}{4}a_0+s a_6+a_7+o\left(e^{-cs/3}\right).\end{aligned}$$ Combined these with (\[comm\_6\]), (\[total\_1\]) and Lemma \[temp1\], (\[order\_t1\_00\]) has been deduced, where $\tau_1=6a_0>0$.
With the results of Theorem 10.22 and Theorem 11.16 (which shows that $\delta>0$) in [@p10], (\[order\_t1\_00\]) is followed by (\[order\_t1\_01\]).
Given the discussion in the proof of Theorem 11.16 in [@p10], (2.45) in [@p10] is followed by $$\begin{aligned}
\left(n/\lambda\right)^{-1/2}
\left(L_1\left(G\left(\mathcal{X}_n;(n/\lambda)^{-1/d}\right)\right)-E[L_1(G(\mathcal{H}_{\lambda,s};1))]\right)
\xrightarrow{D} \mathcal {N}(0,\delta^2),\end{aligned}$$ where $s=(n/\lambda)^{1/d}$. Combining this and (\[order\_t1\_00\]) our result follows.
Some Applications
=================
Our method used in the proof of Theorem \[t1\] can be applied to estimate the expectation of many other random variables restricted to a box $B$ as $B$ becomes large, for example, the size of the biggest *open cluster* for percolation, the coverage area of the largest component for Poisson Boolean model, the number of open clusters or connected components for percolation and Poisson Boolean model, the number of open clusters or connected components with order $k$ for percolation and Poisson Boolean model, the final size of a spatial epidemic mentioned in [@p10] and so on. We will give the similar results as Theorem \[t1\] for the size of the biggest open cluster and the number of open clusters for site percolation but the method can be adapted to bond percolation.
Following Chapter 1 of [@p13], let $\mathbb{L}^d=(\mathbb{Z}^d,\mathbb{E}^d)$ denote the integer lattice with vertex set $\mathbb{Z}^d$ and edges $\mathbb{E}^d$ between all vertex pairs at an $l_1$-distance of 1. For $d\geq 2$ we take $X =(X_x,x\in \mathbb{Z}^d)$ to be a family of i.i.d. Bernoulli random variables with parameter $p\in (0,1)$. Sites $x\in\mathbb{Z}^d$ with $X_x = 1(0)$ are denoted *open* (*closed*). The corresponding probability measure of on $\{0,1\}^{\mathbb{Z}^d}$ is denoted by $P_p$. The open clusters are denoted by the connected components of the subgraph of $\mathbb{L}^d$ induced by the set of open vertices. Let $C_{\textbf{0}}$ denote the open cluster containing the origin. The percolation probability is $\theta(p)=P_p(|C_{\textbf{0}}|=\infty)$ and the critical probability is $p_c=p_c(d):=\sup\{p:\theta(p)=0\}.$ It is well known [@p13] that $p_c\in(0,1)$. If $p>p_c$, by Theorem 8.1 in [@p13], with probability $1$ there exists exactly one infinite open cluster $\mathcal{C}_{\infty}$.
Given integer $n>0$, we denote by open clusters in $B(n)$ the connected components of the subgraph of the integer lattice $\mathbb{L}^d$ induced by the set of open vertices lying in $B(n)$. Similar results as Theorem \[t1\] concerned with the order of the biggest open cluster in $B(n)$ can be given as follows.
\[t3\] Suppose $d\geq 2$ and $p\in (p_c,1)$. Let $H(X;B(n))$ be the order of the biggest open cluster in $B(n)$. Then there exist constants $c=c(d,p)>0$ and $\tau_i=\tau_i(d,p)$, $1\leq i \leq d$, with $\tau_1>0$, such that for all large enough $n$, $$\begin{aligned}
\label{t3_00}
E_p[H(X;B(n-1))]=\theta(p)n^d-\sum_{i=1}^d\tau_i n^{d-i}+o\left(e^{-c n}\right).\end{aligned}$$ Also, there exists a constant $\sigma=\sigma(d,p) > 0$, such that $$\begin{aligned}
\label{t3_01}
H(X;B(n-1))n^{-d/2}- \theta(p) n^{d/2}+\sum_{i=1}^{\lfloor\frac{d}{2}\rfloor}\tau_i n^{d/2-i}
\xrightarrow{D} \mathcal {N}(0,\sigma^2)\end{aligned}$$ as $n\rightarrow\infty$.
Similar to the above, $E_p[|\mathcal {C}_{\infty}\cap
B(n-1)|]=\theta(p)n^d$. Let $C_1,C_2,...,C_M$ denote the components of $\mathcal {C}_{\infty}\cap B(n-1)$, taken in a decreasing order. Let $L(n-1)=B(n-1)\backslash[1,n-2]^d$. For any $2\leq i \leq M$, since $C_i\subset \mathcal {C}_{\infty}$, therefore there exists at least one point in $L(n-1)\cap C_i$ which connects to $\mathcal
{C}_{\infty}$ directly; we choose the smallest one according to the lexicographic ordering on $\mathbb{Z}^d$ as the $out-connect$ $point$. For any $x\in \mathbb{Z}^d \cap L(n-1)$, define $$\xi(x):=\left\{
\begin{array}{ll}
|C_i|, &\mbox{if there exists } i\in[2,M] \mbox{ such that $x$ is the out-connect point of $C_i$}, \\
0, & \mbox{otherwise},
\end{array}
\right.$$ Also, for integer $j\in[0,d-1]$, let $$\begin{aligned}
R_j:=\left([0,1]\times [0,n-1]^{d-1-j} \times [1,n-2]^j \right) \cap \mathbb{Z}^d,\end{aligned}$$ then $$E\left[\sum_{i=2}^M |C_i|\right]=\sum_{x\in \mathbb{Z}^d \cap L(n-1)
}E[\xi(x)]=2\sum_{j=0}^{d-1}\sum_{x\in R_j
}E[\xi(x)].$$ With the similar process as the proof of Theorem \[t1\], (\[t3\_00\]) can be deduced, where $$\tau_1=2d\lim_{n\rightarrow\infty}E\left[\xi\left(\left(0,\lfloor\frac{n}{2}\rfloor,\ldots,\lfloor\frac{n}{2}\rfloor\right)\right)\right]>0.$$ Using Theorem 3.2 in [@penrose2001], (\[t3\_01\]) follows.
Following Chapter 1.5 of [@p13], we define the *number of open clusters per vertex* by $$\kappa(p)=E_p(|C_{\textbf{0}}|^{-1})=\sum_{n=1}^{\infty}\frac{1}{n}P_p(|C_{\textbf{0}}|=n),$$ with the convention that $1/\infty=0$. Similar results as Theorem \[t1\] concerning with the number of the open clusters in $B(n)$ can also be given as follows.
\[t4\] Suppose $d\geq 2$ and $p\in (0,p_c)\cup (p_c,1)$. Let $H(X;B(n))$ be the number of the open clusters in $B(n)$. Then there exist constants $c=c(d,p)>0$ and $\tau_i=\tau_i(d,p)>0$, $1\leq i \leq d$, with $\tau_1>0$, such that for all large enough $n$, $$\begin{aligned}
\label{t4_00}
E_p[H(X;B(n-1))]=\kappa(p)n^d+\sum_{i=1}^d \tau_i n^{d-i}+o\left(e^{-cn}\right).\end{aligned}$$ Also, there exists a constant $\sigma=\sigma(d,p) > 0$, such that $$\begin{aligned}
\label{t4_01}
H(X;B(n-1))n^{-d/2}- \kappa(p) n^{d/2}- \sum_{i=1}^{\lfloor \frac{d}{2} \rfloor}\tau_i n^{d/2-i}
\xrightarrow{D} \mathcal {N}(0,\sigma^2)\end{aligned}$$ as $n\rightarrow\infty$. Moreover, for any constant $\varepsilon\in(0,d/2)$, $$\begin{aligned}
\label{t4_03}
\begin{aligned}
&P_p\left(\frac{H(X;B(n-1))-\kappa(p)n^d-\sum_{i=1}^d \tau_i n^{d-i}}{\mbox{Var}(H(X;B(n-1)))}\leq x \right)\\
&=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}} e^{-y^2/2}dy+o\left(n^{-\frac{d}{2}+\varepsilon} \right),
\end{aligned}\end{aligned}$$ where Var$(\cdot)$ denotes the variance.
Let $L(n-1)=B(n-1)\backslash[1,n-2]^d$. For any $x\in B(n-1)\cap
\mathbb{Z}^d$, let $C_x$ denote the open cluster including $x$, and let $C_x(B(n-1))$ denote the open cluster including $x$ in $B(n-1)$. Then $C_x(B(n-1))\subseteq C_x$. For all open clusters $C$ in $B(n-1)$, if $C\cap L(n-1)\neq \emptyset,$ according to the lexicographic ordering on $\mathbb{Z}^d$ we choose the smallest element of $C\cap L(n-1)$ as the *indicated vertex* of $C$. For any $x\in \mathbb{Z}^d \cap L(n-1)$, define $$\xi(x,B(n-1)):=\left\{
\begin{array}{ll}
1-\frac{|C_x(B(n-1))|}{|C_x|}, &\mbox{if $x$ is the idicated vertex of $C_x(B(n-1))$}, \\
0, & \mbox{otherwise}.
\end{array}
\right.$$ Noted that for any $y\in \mathbb{Z}^d \cap B(n-1)$, $$\sum_{x\in
C_y(B(n-1))} \left(|C_y(B(n-1))|^{-1}-|C_y|^{-1}\right)=1-
\frac{|C_y(B(n-1))|}{|C_y|},$$ then by (4.7) in [@p13], we have $$\begin{aligned}
\label{t4_1}
\begin{aligned}
H(X;B(n-1))&=\sum_{x\in \mathbb{Z}^d \cap B(n-1)}
|C_x(B(n-1))|^{-1}\\
&= \sum_{x\in \mathbb{Z}^d \cap B(n-1)} |C_x|^{-1}+ \sum_{x\in
\mathbb{Z}^d \cap L(n-1)} \xi(x,B(n-1)).
\end{aligned}\end{aligned}$$ Therefore, take the expectation for the both sides of (\[t4\_1\]), we can get $$E_p[H(X;B(n-1))]=\kappa(p)n^d + \sum_{x\in \mathbb{Z}^d
\cap L(n-1)} E_p[\xi(x,B(n-1))].$$
Suppose $1\leq i \leq d$ and $x_j\in [0,K/2-1]\cap \mathbb{Z}$ for $1\leq j \leq i$. For large integers $n_1,n_2$, let $x=(x_1,\ldots,x_i,\lfloor \frac{n_1}{2} \rfloor, \ldots,\lfloor
\frac{n_1}{2} \rfloor) \in \mathbb{Z}^d$ and $\widetilde{x}=(x_1,\ldots,x_i,\lfloor \frac{n_2}{2} \rfloor,
\ldots,\lfloor \frac{n_2}{2} \rfloor ) \in \mathbb{Z}^d$. Set $\widetilde{B}(n_2):=B(n_2)\oplus\{x-\widetilde{x}\}.$ Since $\xi$ is stationary under translations of the lattice $\mathbb{L}^d$, then $\xi(\widetilde{x},B(n_2))$ and $\xi(x,\widetilde{B}(n_2))$ have the same distribution function. However, let $n_0=\min\{\lfloor\frac{n_1}{2}\rfloor,\lfloor\frac{n_2}{2}\rfloor\}$, by the definition of $\xi$ we have $$\begin{aligned}
&&P_p\left[\xi(x,B(n_1))\neq \xi(x,\widetilde{B}(n_2)) \right]=P_p\left[\xi(x,B(n_1))\neq \xi(x,\widetilde{B}(n_2)),C_x\neq C_\infty \right]\nonumber\\
&&~~\leq P_p\left[\mbox{diam}(C_x)\geq n_0,C_x\neq C_\infty\right]<
e^{-cn_0},\end{aligned}$$ where the last inequality follows from Theorem 6.1 of [@p13] for $p<p_c$ and Theorem 8.18 of [@p13] for $p>p_c$ respectively. Thus, $$\begin{aligned}
&&\left|E_p\left[\xi(x,B(n_1))\right]-E_p\left[\xi(\widetilde{x},B(n_2))\right]\right|\\
&&~~\leq \sum_{t} t \left|
P_p\left[\xi(x,B(n_1))=t\right]-P_p\left[\xi(x,\widetilde{B}(n_2))=t\right]
\right|\\
&&~~\leq \sum_{t} \left( P_p\left[\xi(x,B(n_1))=t,\xi(x,B(n_1))\neq
\xi(x,\widetilde{B}(n_2))
\right] \right.\\
&&~~~~~~~~~~\left.+P_p\left[\xi(x,\widetilde{B}(n_2))=t,\xi(x,B(n_1))\neq
\xi(x,\widetilde{B}(n_2))\right]
\right)\\
&&~~=2P_p\left[\xi(x,B(n_1))\neq \xi(x,\widetilde{B}(n_2))
\right]<2e^{-c n_0}.\end{aligned}$$ Therefore, $\lim_{n\rightarrow\infty}E_p[\xi(x,B(n)]$ exists. In fact, a similar result as Theorem \[argument\] can be deduced. Let $$\begin{aligned}
\widetilde{\tau}_i(K)={{d}\choose{i}}\sum_{x_j\in [0,K-1]\cup
[n-K,n-1],1\leq j\leq i }
\lim_{n\rightarrow\infty}E_p\left[\xi\left(\left(x_1,\ldots,x_i,\left\lfloor
\frac{n}{2} \right\rfloor,\ldots,\left\lfloor \frac{n}{2}
\right\rfloor\right)\right)\right],\end{aligned}$$ and let $\tau_i(K)=\sum_{j=1}^i \widetilde{\tau}_j(K)
{{d-j}\choose{i-j}} (-2K)^{i-j}.$ In a similar way, (\[t4\_00\]) is obtained.
Combining (\[t4\_00\]) with Theorem 3.1 in [@penrose2001], (\[t4\_01\]) follows immediately.
By Theorem 2.1 in [@JJP2010], Theorem 3.1 in [@penrose2001] and (\[t4\_00\]), (\[t4\_03\]) can be deduced.
It is worth noting that our results do have significance for some practical applications. In fact, the initial motivation of this paper is to provide theoretical foundation and guidance for the design of *wireless multihop networks*. The wireless multihop networks, e.g., vehicular ad hoc networks, mobile ad hoc networks, and wireless sensor networks, typically consists of a group of decentralized and self-organized nodes that communicate with each other in a peer-to-peer manner over wireless channels, and are increasingly being used in military and civilian applications [@p16]. The large scale wireless multihop networks are usually formulated by the random geometric graphs, and the size of the largest component is a fundamental variable for a network, which plays a key role for the topology control in wireless multihop networks. However, this variable can not be described very precisely by both former theoretic results and even computer simulations as the scale of the network grows to very large. Theorem \[t1\] and Theorem \[t2\] provides a precise estimation for this variable respectively. Using simulations the approximative values of the parameters $p_{\infty}(\lambda)$, $\tau_i$, $\sigma$ and $\delta$ can be obtained, and thus the expression of the asymptotic size of the largest component can be well established, which has guiding significance to the topology control in wireless multihop networks.
This research was Supported by the National Natural Science Foundation of China under Grants No. 61203141 and 71271204, and the Innovation Program of the Chinese Academy of Sciences under Grant No. kjcx-yw-s7.
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| {
"pile_set_name": "ArXiv"
} |
**Total variance and invariant information**
**in complementary measurements**
[Bin Chen,$^{\star}$]{} [^1] [Shao-Ming Fei$^{\natural,\sharp}$]{}
*$^\star$ College of Mathematical Science, Tianjin Normal University, Tianjin 300387, China*
*$^\natural$ School of Mathematical Sciences, Capital Normal University, Beijing 100048, China*
*$^\sharp$ Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany*
Abstract
We investigate the total variance of a quantum state with respect to a complete set of mutually complementary measurements and its relation to the Brukner-Zeilinger invariant information. By summing the variances over any complete set of mutually unbiased measurements and general symmetric informationally complete measurements respectively, we show that the Brukner-Zeilinger invariant information associated with such types of quantum measurements is equal to the difference between the maximal variance and the total variance obtained. These results provide an operational link between the previous interpretations of the Brukner-Zeilinger invariant information.
Introduction
============
In a classical measurement, the observation removes our ignorance about the state by revealing the properties of the system which are considered to be pre-existing and independent of the observation. The Shannon information is a desirable measure to quantify the amount of information carried by a classical system. It is also a natural measure of our ignorance regarding the properties of a classical system. However, the situation is quite different for the case of quantum measurements, for which one cannot say that quantum measurements reveal a pre-existing property of a quantum system. Therefore, the Shannon entropy could be thought of as ¡°conceptually¡± inadequate in quantum physics. Brukner and Zeilinger introduced a quantity as a new measure of total information obtained by summing individual measures over a complete set of mutually complementary measurements [@BZ1; @BZ2]. This measure of quantum information takes into account that the only features of quantum systems known before a measurement are the probabilities for various events to occur. This quantity can be expressed as $$I(\rho):=\mathrm{Tr}(\rho^{2})-\frac{1}{d},$$ where $d$ is the dimension of the quantum system.
Moreover, the “total" uncertainty related a quantum measurement is also defined. For $d$ measurement outcomes with probabilities $p_1,p_2,...,p_d$, the lack of information about the $j$-th outcome with respect to a single experimental trial is given by $p_j(1-p_j)$. The total lack of information regarding all $d$ possible experimental outcomes is then given by $\sum_{j=1}^{d} p_j(1-p_j)=1-\sum_{j=1}^{d} p_j^2$, which is nothing but $1-\mathrm{Tr}(\rho^{2})$, where $\rho$ is the state after a quantum (projective) measurement, the linear entropy of the measured state. The “total" uncertainty related to a quantum measurement is then given by $$U(\rho):=1-\mathrm{Tr}(\rho^{2}),$$ which is the sum of all individual measurement uncertainties over a complete set of mutually complementary measurements.
It can be seen that $I(\rho)$ and $U(\rho)$ are two mutually complementary quantities, and both of them are invariant under unitary transformations. These two quantities have many useful applications in various issues of quantum mechanics and quantum information theory [@1; @2; @3]. In the rest of this paper we call $I(\rho)$ the BZ invariant information and $U(\rho)$ the BZ invariant uncertainty.
In Ref. [@Luo], Luo presented an alternative interpretation of the BZ invariant information. By summing the variances of a quantum state over any complete orthogonal set of observables, he obtained a new quantity $$V(\rho):=d-\mathrm{Tr}(\rho^{2}).$$ This quantity, called total variance, is related to the BZ invariant information and the BZ invariant uncertainty. One can see that $V(\rho)$ attains its minimum value $V_{\mathrm{min}}(\rho):=d-1$ iff $\rho$ is a pure state, and attains its maximum value $V_{\mathrm{max}}(\rho):=d-1/d$ iff $\rho=I/d$ is the maximally mixed state. Hence the BZ invariant information is equal to the difference between the maximal variance and the total variance, while the BZ invariant uncertainty is exactly the difference between the total variance and the minimal variance, i.e., $$I(\rho)=V_{\mathrm{max}}(\rho)-V(\rho),~~~U(\rho)=V(\rho)-V_{\mathrm{min}}(\rho).$$ To this point, the BZ invariant information (uncertainty) can be viewed as a renormalized version of the total variance of a quantum state [@Luo].
After that, Rastegin reinterpreted the BZ invariant information from another perspective [@Ras]. By introducing the index of coincidence [@co] $$C(\mathcal{P}|\rho):=\sum_{j}p_{j}(\mathcal{P}|\rho)^{2},$$ the BZ information measure can be rewritten as $$I(\mathcal{P}|\rho)=C(\mathcal{P}|\rho)-C(\mathcal{P}|\rho_{\ast}),$$ where $\mathcal{P}$ denotes a set of complementary measurements, $\{p_{j}(\mathcal{P}|\rho)\}_{j}$ is the probability distribution given by $\mathcal{P}$ on the quantum state $\rho$, and $\rho_{\ast}$ is the maximally mixed state. When a complete set of $d+1$ mutually unbiased bases (MUBs) [@WF; @Durt] in $d$-dimensional quantum system is taken into account, one gets [@La; @Iv] $$C(\mathcal{P}|\rho)=1+\mathrm{Tr}(\rho^{2}),$$ and in this case $$I(\mathcal{P}|\rho)=\mathrm{Tr}(\rho^{2})-\mathrm{Tr}(\rho_{\ast}^{2})=\mathrm{Tr}(\rho^{2})-\frac{1}{d},$$ which is in consistent with the original definition of the BZ invariant information.
Besides MUBs, there are several special types of quantum measurements, the mutually unbiased measurements (MUMs, as a generalization of MUBs) [@KG1], the symmetric informationally complete positive operator-valued measures (SIC-POVMs) [@Re], and the general symmetric informationally complete measurements (general SIC measurements, as a generalization of SIC-POVMs) [@Ap; @GK2]. Both of the MUMs and the general SIC measurements form a complete set of mutually complementary measurements respectively. It has been shown that for a complete set of $d+1$ MUMs $\mathcal{P}$ with the parameter $\kappa$, the BZ information measure gives rise to [@Ras] $$I(\mathcal{P}|\rho)=\frac{\kappa d-1}{d-1}[\mathrm{Tr}(\rho^{2})-\mathrm{Tr}(\rho_{\ast}^{2})],$$ while for a complete set of $d^{2}$ general SIC measurements $\mathcal{M}$ with the parameter $a$, one has [@Ras], $$I(\mathcal{M}|\rho)=\frac{ad^{3}-1}{d(d^{2}-1)}[\mathrm{Tr}(\rho^{2})-\mathrm{Tr}(\rho_{\ast}^{2})].$$
Then a natural question is what the relationship between these two interpretations of the BZ invariant information is. Inspired by Luo’s work, in this paper, we define the notion of total variance in a set of complementary measurements. By summing over all variances of a quantum state for a complete set of MUMs and general SIC measurements, respectively, we show that the resulted total variance is related to BZ invariant information for such types of quantum measurements given in [@Ras]. In particular, such BZ invariant information is precisely the difference between the maximal variance (total variance of the maximally mixed state) and the total variance. Our results provide an operational link between the two interpretations of BZ invariant information.
BZ invariant information in MUMs
================================
Let us first review some basic concepts about MUBs and MUMs. Two orthonormal bases in $d$-dimensional Hilbert space $\mathcal{H}_{d}$ are said to be mutually unbiased if the absolute values of the inner products of any vector from one basis and any vector from another basis are $1/\sqrt{d}$. A set of orthonormal bases in $\mathcal{H}_{d}$ is called a set of MUBs if each two bases from the set are mutually unbiased. It has been shown that the maximum number $N(d)$ of MUBs is no more than $d+1$, and $N(d)=d+1$ when $d$ is a prime power [@WF]. But $N(d)$ is still unknown when $d$ is not a prime power [@Durt]. In Ref. [@KG1], Kalev and Gour generalize the concept of MUBs to MUMs. They show that there always exists a complete set of $d+1$ MUMs for arbitrary $d$, which can be explicitly constructed. Two POVM measurements $\mathcal{P}^{(b)}=\{P_{n}^{(b)}\}_{n=1}^{d}$, $b=1,2$, are said to be MUMs if $$\begin{split}
\mathrm{Tr}(P_{n}^{(b)})&=1,\\
\mathrm{Tr}(P_{n}^{(b)}P_{n'}^{(b')})&=\frac{1}{d},~~~b\neq b',\\
\mathrm{Tr}(P_{n}^{(b)}P_{n'}^{(b)})&=\delta_{n,n'}\,\kappa+(1-\delta_{n,n'})\frac{1-\kappa}{d-1},
\end{split}$$ where $\frac{1}{d}<\kappa\leq1$, and $\kappa=1$ if and only if all $P_{n}^{(b)}$’s are rank one projectors, which gives rise to two MUBs. The construction of a complete set of $d+1$ MUMs is as follows. Let $\{F_{n,b}:n=1,2,\ldots,d-1,b=1,2,\ldots,d+1\}$ be a set of $d^{2}-1$ Hermitian, traceless operators acting on $\mathcal{H}_{d}$, satisfying $\mathrm{Tr}(F_{n,b}F_{n',b'})=\delta_{n,n'}\delta_{b,b'}$. Define $d(d+1)$ operators $$\label{2}
F_{n}^{(b)}=
\begin{cases}
F^{(b)}-(d+\sqrt{d})F_{n,b},&n=1,2,\ldots,d-1;\\[2mm]
(1+\sqrt{d})F^{(b)},&n=d,
\end{cases}$$ where $F^{(b)}=\sum_{n=1}^{d-1}F_{n,b}$, $b=1,2,\ldots,d+1$. Then the operators $$\label{3}
P_{n}^{(b)}=\frac{1}{d}I+tF_{n}^{(b)},$$ with $b=1,2,\cdots,d+1,n=1,2,\cdots,d$, give rise to $d+1$ MUMs, where $t$ should be chosen such that $P_{n}^{(b)}\geq0$. Thus the parameter $\kappa$ is given by $$\label{5}
\kappa=\frac{1}{d}+t^{2}(1+\sqrt{d})^{2}(d-1)$$ from the construction. On the other hand, any $d+1$ MUMs can be expressed in such form [@KG1].
A complete set of $d+1$ MUMs is exactly a complete set of mutually complementary measurements, and can be used to characterize the BZ invariant information in measurements. To this end, we define the quantity $V(\mathcal{P}_{\mathrm{MUM}}|\rho)$ as a measure of total variance of a quantum state $\rho$ in a complete set of MUMs $\mathcal{P}_{\mathrm{MUM}}=\{\mathcal{P}^{(b)}\}_{b=1}^{d+1}$ with the parameter $\kappa$, $$V(\mathcal{P}_{\mathrm{MUM}}|\rho):=\sum_{b=1}^{d+1}V(\mathcal{P}^{(b)}|\rho)=\sum_{b=1}^{d+1}\sum_{n=1}^{d}V(P_{n}^{(b)}|\rho),$$ where $V(X|\rho)=\langle X^{2}\rangle_{\rho}-\langle X\rangle_{\rho}^{2}$ is the variance of the observable $X$. In the next, we calculate the quantity $V(\mathcal{P}_{\mathrm{MUM}}|\rho)$.
We first notice that $$\begin{split}
\sum_{b=1}^{d+1}\sum_{n=1}^{d}V(P_{n}^{(b)}|\rho) & = \sum_{b=1}^{d+1}\sum_{n=1}^{d}\langle(P_{n}^{b})^{2}\rangle_{\rho}-\sum_{b=1}^{d+1}\sum_{n=1}^{d}\langle P_{n}^{b}\rangle_{\rho}^{2}\\
& := \sum_{b=1}^{d+1}\sum_{n=1}^{d}\langle(P_{n}^{b})^{2}\rangle_{\rho}-C(\kappa,\rho),
\end{split}$$ where $C(\kappa,\rho)$ denotes the index of coincidence for probability distribution generated by $\{P_{n}^{(b)}\}$. It has been shown that [@Chen], $$\label{c}
C(\kappa,\rho)=\frac{(\kappa d-1)[d\mathrm{Tr}(\rho^{2})-1]+d^{2}-1}{d(d-1)}.$$ Thus we only need to compute $\sum_{b,n}\langle(P_{n}^{b})^{2}\rangle_{\rho}$. Taking into account that $\sum_{n=1}^{d}F_{n}^{(b)}=0,~\forall b$, we have $$\label{p}
\begin{split}
\sum_{b=1}^{d+1}\sum_{n=1}^{d}\langle(P_{n}^{b})^{2}\rangle_{\rho} & = \sum_{b=1}^{d+1}\sum_{n=1}^{d}\left\langle
\left(\frac{1}{d}I+tF_{n}^{(b)}\right)^{2}\right\rangle_{\rho}\\[2mm]
& = \frac{d+1}{d}+\frac{2t}{d}\sum_{b=1}^{d+1}\sum_{n=1}^{d}\langle F_{n}^{(b)}\rangle_{\rho}+t^{2}\sum_{b=1}^{d+1}\sum_{n=1}^{d}\langle (F_{n}^{(b)})^{2}\rangle_{\rho}\\
& = \frac{d+1}{d}+t^{2}\sum_{b=1}^{d+1}\sum_{n=1}^{d}\langle (F_{n}^{(b)})^{2}\rangle_{\rho}.
\end{split}$$
On the other hand, $$\label{f}
\begin{split}
\sum_{b=1}^{d+1}\sum_{n=1}^{d}\langle (F_{n}^{(b)})^{2}\rangle_{\rho} & =
\sum_{b=1}^{d+1}\sum_{n=1}^{d-1}\left\langle\left[F^{(b)}-(d+\sqrt{d})F_{n,b}\right]^{2}\right\rangle_{\rho}+(1+\sqrt{d})^{2}\sum_{b=1}^{d+1}\langle (F^{(b)})^{2}\rangle_{\rho}\\
& = \sum_{b=1}^{d+1}\sum_{n=1}^{d-1}[\langle (F^{(b)})^{2}\rangle_{\rho}-(d+\sqrt{d})\langle F^{(b)}F_{n,b}+F_{n,b}F^{(b)}\rangle_{\rho}\\
&\quad +(d+\sqrt{d})^{2}\langle F_{n,b}^{2}\rangle_{\rho}]+(1+\sqrt{d})^{2}\sum_{b=1}^{d+1}\langle (F^{(b)})^{2}\rangle_{\rho}\\
& = (d-1)\sum_{b=1}^{d+1}\langle (F^{(b)})^{2}\rangle_{\rho}-2(d+\sqrt{d})\sum_{b=1}^{d+1}\langle (F^{(b)})^{2}\rangle_{\rho}\\
&\quad +(d+\sqrt{d})^{2}\left\langle\sum_{b=1}^{d+1}\sum_{n=1}^{d-1}F_{n,b}^{2}\right\rangle_{\rho}+(1+\sqrt{d})^{2}\sum_{b=1}^{d+1}\langle (F^{(b)})^{2}\rangle_{\rho}\\
& = (d+\sqrt{d})^{2}\left\langle\sum_{b=1}^{d+1}\sum_{n=1}^{d-1}F_{n,b}^{2}\right\rangle_{\rho}\\
& = (d+\sqrt{d})^{2}(d-\frac{1}{d}).
\end{split}$$ In the last equality, we have used the fact that $\sum_{b=1}^{d+1}\sum_{n=1}^{d-1}F_{n,b}^{2}=(d-\frac{1}{d})I$ [@Luo]. Combining Eqs. (\[c\]), (\[p\]) and (\[f\]), we get $$V(\mathcal{P}_{\mathrm{MUM}}|\rho)=\frac{\kappa d-1}{d-1}(d-\mathrm{Tr}(\rho^{2})).$$
It can be seen that $V(\mathcal{P}_{\mathrm{MUM}}|\rho)$ reaches its minimum value $V_{\mathrm{min}}(\mathcal{P}_{\mathrm{MUM}}|\rho)=\kappa d-1$ when $\rho$ is a pure state, and reaches its maximum value $V_{\mathrm{max}}(\mathcal{P}_{\mathrm{MUM}}|\rho)=(\kappa d-1)(d+1)/d$ when $\rho=I/d$ is the maximally mixed state. Following Luo’s perspective in Ref. [@Luo], the BZ invariant information in MUMs can be characterized by $$I(\mathcal{P}_{\mathrm{MUM}}|\rho):=V_{\mathrm{max}}(\mathcal{P}_{\mathrm{MUM}}|\rho)-V(\mathcal{P}_{\mathrm{MUM}}|\rho)=
\frac{\kappa d-1}{d-1}[\mathrm{Tr}(\rho^{2})-\frac{1}{d}].$$ which is coincide with the illustration of the BZ invariant information in MUMs given in [@Ras]. Furthermore, the BZ invariant uncertainty in MUMs is formulated as $$U(\mathcal{P}_{\mathrm{MUM}}|\rho):=V(\mathcal{P}_{\mathrm{MUM}}|\rho)-V_{\mathrm{min}}(\mathcal{P}_{\mathrm{MUM}}|\rho)=
\frac{\kappa d-1}{d-1}[1-\mathrm{Tr}(\rho^{2})].$$
BZ invariant information in general SIC measurements
====================================================
A set of $d^{2}$ rank one operators acting on $\mathcal{H}_{d}$ is called a SIC-POVM, if every operator is of the form $$P_{j}=\frac{1}{d}|\phi_{j}\rangle\langle\phi_{j}|,~~~j=1,2,\ldots,d^{2},$$ where the vectors $|\phi_{j}\rangle$ satisfy the following relation $$\mid\langle\phi_{j}|\phi_{k}\rangle\mid^{2}=\frac{1}{d+1},~~~j\neq k.$$ Similar to MUBs, the existence of SIC-POVMs in arbitrary dimension $d$ is still unknown. It has been only proved that there exist sets of SIC-POVMs for all dimensions up to 67 [@Sco]. In Ref. [@GK2], Gour and Kalev generalize the concept of SIC-POVMs to general SIC measurements. A set of $d^{2}$ positive-semidefinite operators $\{P_{\alpha}\}_{\alpha=1}^{d^{2}}$ on $\mathcal{H}_{d}$ is said to be a general SIC measurements, if\
$$\begin{split}
\sum_{\alpha=1}^{d^{2}}P_{\alpha}&=I,\\[1mm]
\mathrm{Tr}(P_{\alpha}^{2})&=a,~\forall\alpha\in\{1,2,\ldots,d^{2}\},\\
\mathrm{Tr}(P_{\alpha}P_{\beta})&=\frac{1-ad}{d(d^{2}-1)},~\forall\alpha,\beta\in\{1,2,\ldots,d^{2}\},~\alpha\neq\beta,
\end{split}$$ where the parameter $a$ satisfies $\frac{1}{d^{3}}<a\leq\frac{1}{d^{2}}$, and $a={1}/{d^{2}}$ if and only if all $P_{\alpha}$ are rank one operators, which gives rise to a SIC-POVM.
Like the MUMs, there always exists a general SIC measurements for arbitrary $d$, and can be explicitly constructed [@GK2]. Let $\{F_{\alpha}\}_{\alpha=1}^{d^{2}-1}$ be a set of $d^{2}-1$ Hermitian, traceless operators acting on $\mathcal{H}_{d}$, satisfying $\mathrm{Tr}(F_{\alpha}F_{\beta})=\delta_{\alpha,\beta}$. Define $F=\sum_{\alpha=1}^{d^{2}-1}F_{\alpha}$, then the $d^{2}$ operators $$\begin{split}
P_{\alpha}&=\frac{1}{d^{2}}I+t[F-d(d+1)F_{\alpha}],~~~\alpha=1,2,\ldots,d^{2}-1,\\
P_{d^{2}}&=\frac{1}{d^{2}}I+t(d+1)F,
\end{split}$$ form a general SIC measurements, where $t$ should be chosen such that all $P_{\alpha}\geq0$, and the parameter $a$ is given by $$a=\frac{1}{d^{3}}+t^{2}(d-1)(d+1)^{3}.$$ In the next, we characterize the BZ invariant information in general SIC measurements.
We define the quantity $V(\mathcal{P}_{\mathrm{GSM}}|\rho)$ as a measure of total variance in a general SIC measurements $\mathcal{P}_{\mathrm{GSM}}=\{P_{\alpha}\}_{\alpha=1}^{d^{2}}$ with the parameter $a$ as follows $$V(\mathcal{P}_{\mathrm{GSM}}|\rho):=\sum_{\alpha=1}^{d^{2}}V(P_{\alpha}|\rho).$$ By using the fact that [@Ras1] $$\sum_{\alpha=1}^{d^{2}}\langle P_{\alpha}\rangle_{\rho}^{2}=\frac{(ad^{3}-1)\mathrm{Tr}(\rho^{2})+d(1-ad)}{d(d^{2}-1)},$$ we obtain $$V(\mathcal{P}_{\mathrm{GSM}}|\rho)=\sum_{\alpha=1}^{d^{2}}\langle P_{\alpha}^{2}\rangle_{\rho}-\frac{(ad^{3}-1)\mathrm{Tr}(\rho^{2})+d(1-ad)}{d(d^{2}-1)}.$$ On the other hand, $$\begin{aligned}
\sum_{\alpha=1}^{d^{2}}\langle P_{\alpha}^{2}\rangle_{\rho} & = &
\sum_{\alpha=1}^{d^{2}-1}\left\langle\left[\frac{1}{d^{2}}I+t(F-d(d+1)F_{\alpha})\right]^{2}\right\rangle_{\rho}
+\left\langle\left[\frac{1}{d^{2}}I+t(d+1)F\right]^{2}\right\rangle_{\rho}\\
& = & \frac{1}{d^{2}}+t^{2}\sum_{\alpha=1}^{d^{2}-1}\langle(F-d(d+1)F_{\alpha})^{2}\rangle_{\rho}+t^{2}(d+1)^{2}\langle F^{2}\rangle_{\rho}\\
& = & \frac{1}{d^{2}}+t^{2}d^{2}(d+1)^{2}\sum_{\alpha=1}^{d^{2}-1}\langle F_{\alpha}^{2}\rangle_{\rho}\\
& = & \frac{1}{d^{2}}+t^{2}d^{2}(d+1)^{2}(d-\frac{1}{d})\\
& = & ad.\end{aligned}$$ Again, in the last equality, we have used the fact that $\sum_{\alpha=1}^{d^{2}-1}F_{\alpha}^{2}=(d-\frac{1}{d})I$ [@Luo]. Hence we have $$V(\mathcal{P}_{\mathrm{GSM}}|\rho)=\frac{ad^{3}-1}{d(d^{2}-1)}(d-\mathrm{Tr}(\rho^{2})).$$
The quantity $V(\mathcal{P}_{\mathrm{GSM}}|\rho)$ also enables us to characterize the BZ invariant information $I(\mathcal{P}_{\mathrm{GSM}}|\rho)$ and the BZ invariant uncertainty $U(\mathcal{P}_{\mathrm{GSM}}|\rho)$. Taking into account that $$V_{\mathrm{max}}(\mathcal{P}_{\mathrm{GSM}}|\rho)=\frac{ad^{3}-1}{d(d^{2}-1)}(d-\frac{1}{d}),~~~
V_{\mathrm{min}}(\mathcal{P}_{\mathrm{GSM}}|\rho)=\frac{ad^{3}-1}{d(d^{2}-1)}(d-1),$$ we have $$I(\mathcal{P}_{\mathrm{GSM}}|\rho)=V_{\mathrm{max}}(\mathcal{P}_{\mathrm{GSM}}|\rho)-V(\mathcal{P}_{\mathrm{GSM}}|\rho)=
\frac{ad^{3}-1}{d(d^{2}-1)}[\mathrm{Tr}(\rho^{2})-\frac{1}{d}],$$ which is also coincide with the illustration of the BZ invariant information in general SIC measurements given in [@Ras]. Moreover, we have $$U(\mathcal{P}_{\mathrm{GSM}}|\rho)=V(\mathcal{P}_{\mathrm{GSM}}|\rho)-V_{\mathrm{min}}(\mathcal{P}_{\mathrm{GSM}}|\rho)=
\frac{ad^{3}-1}{d(d^{2}-1)}[1-\mathrm{Tr}(\rho^{2})].$$
Conclusion
==========
We have studied the Brukner-Zeilinger invariant information (uncertainty) with respect to MUMs and general SIC measurements. By summing the variances over a complete set of mutually unbiased measurements and general symmetric informationally complete measurements respectively, we have defined a new measure of information (uncertainty) in quantum measurements, and reinterpreted the invariant information in an alternative perspective. These results can also be used to provide an operational link between the previous interpretations of the Brukner-Zeilinger invariant information. Taking into consideration that our new quantity of information content in measurements is defined in a simple and intuitive way, and is invariant under unitary transformations of the quantum states, one may expect that it brings significant applications in quantum information theory.
[**Acknowledgments**]{} This work is supported by the National Natural Science Foundation of China under Grant Nos. 11805143 and 11675113, Beijing Municipal Commission of Education under Grant No. KZ201810028042.
[18]{}
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[^1]: Corresponding author: [email protected]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Droplet migration in a Hele–Shaw cell is a fundamental multiphase flow problem which is crucial for many microfluidics applications. We focus on the regime at low capillary number and three-dimensional direct numerical simulations are performed to investigate the problem. In order to reduce the computational cost, an adaptive mesh is employed and high mesh resolution is only used near the interface. Parametric studies are performed on the droplet horizontal radius and the capillary number. For droplets with an horizontal radius larger than half the channel height the droplet overfills the channel and exhibits a pancake shape. A lubrication film is formed between the droplet and the wall and particular attention is paid to the effect of the lubrication film on the droplet velocity. The computed velocity of the pancake droplet is shown to be lower than the average inflow velocity, which is in agreement with experimental measurements. The numerical results show that both the strong shear induced by the lubrication film and the three-dimensional flow structure contribute to the low mobility of the droplet. In this low-migration-velocity scenario the interfacial flow in the droplet reference frame moves toward the rear on the top and reverses direction moving to the front from the two side edges. The velocity of the pancake droplet and the thickness of the lubrication film are observed to decrease with capillary number. The droplet velocity and its dependence on capillary number cannot be captured by the classic Hele–Shaw equations, since the depth-averaged approximation neglects the effect of the lubrication film.'
author:
- Yue Ling
- 'Jose-Maria Fullana'
- Stéphane Popinet
- Christophe Josserand
title: 'Droplet migration in a Hele–Shaw cell: Effect of the lubrication film on the droplet dynamics'
---
INTRODUCTION {#sec:intro}
============
Droplet-based microfluidics is a promising tool for performing biochemical or chemical assays or to observe liquid-liquid extraction [@Stone_2004a; @Fair_2007a; @Teh_2008a]. Droplets are unit systems of controlled volume and content, within which mixing, reacting and/or transferring can be achieved. Therefore, a comprehensive understanding of droplet migration in microchannels is essential to many microfluidics applications. Rectangular channels with a width significantly larger than the height are commonly used in microfluidics devices. In such cases, the confinement of the droplet is only in the vertical direction and the droplet is exposed to a two-dimensional Poiseuille flow (as in a Hele–Shaw cell). The migration of the droplet is driven by the ambient fluid with an average inflow velocity $U_f$ and a constant terminal migration velocity $U_d$ will be reached.
The ratio between the horizontal radius of the droplet ($R$) and half of the channel height ($H$) has been shown to have a strong effect on the droplet migration velocity $U_d$ in a microchannel (hereafter we simply refer to $U_d$ as the droplet velocity since the transient process is not of interest in the present study). The overall variation of $U_d$ with $R/H$ can be seen in the experiments by Roberts [*et al.*]{}[@Roberts_2014a], (the experimental results will be shown later in section \[sec:3D\]) and can be divided into three different regimes according to the dominant mechanism:
- Poiseuille-dominated regime (Regime P), $R/H\ll 1$. In this regime the droplet velocity is dominated by the Poiseuille flow in the Hele–Shaw cell. In the limit of $R/H\to 0$ (as the droplet is a point mass) the droplet at the centerline of the channel will move with the maximum fluid velocity ($U_{f,max}=1.5 U_f$). As $R$ increases the droplet is exposed to the parabolic fluid velocity profile with a decreasing velocity toward the top and bottom walls and the droplet slows down. Theoretical studies on the migration of droplets much smaller than the channel height have been conducted in many previous works, see for example Nadim and Stone [@Nadim_1991a] and Hudson [@Hudson_2010a].
- Film-dominated regime (Regime F), $R/H\gg 1$: In this regime, the droplet overfills the channel and a thin lubrication film is formed between the droplet and the wall. The droplet loses its roughly spherical geometry and becomes “pancake-like”. Furthermore, the droplet velocity $U_d$ becomes independent of $R/H$ and is mainly dictated by the lubrication film. The film thickness $h$ is governed by the balance between surface tension and viscous stresses. As the former tends to minimize the interface curvature and pushes the droplet interface closer to the wall, the latter tends to open up the gap between the droplet and the wall. Therefore the capillary number ($\mathrm{Ca}=\mu_f U_d/\sigma$, where $\mu_f$ and $\sigma$ is the viscosity of the ambient fluid and the surface tension, respectively), which is the ratio between the viscous and surface tension forces, is critical in determining the film thickness and the droplet mobility in this regime [@Bretherton_1961a]. Beyond the capillary number, the viscosity ratio between the droplet and the ambient fluid has also been shown to have a significant influence on the lubrication film thickness and the flow pattern [@Hodges_2004a].
- Transition regime (Regime T), $R/H \approx 1$: This regime connects the previous two regimes. As the droplet radius is comparable to half of the channel height, the effect of confinement on the droplet becomes significant. As a result, the droplet velocity decreases dramatically with $R/H$ in this regime.
The three droplets shown in Fig. \[fig:problem\_description\] schematically represent these three regimes. The axes corresponding to the channel length, height, and width are defined as $x$, $y$, and $z$, respectively. The experimental and numerical results of Roberts [*et al.*]{}[@Roberts_2014a] clearly show the variation of droplet velocity in the Poiseuille-dominated regime, and close agreement with theoretical predictions has been achieved. The experimental data in the film-dominated regime is less documented however. Furthermore, simulation results for the transition and film-dominated regimes are not provided, due to the numerical challenge in resolving the thin film [@Roberts_2014a]. The present study aims at improving the current understanding of droplet motion in microchannels, in particular the droplet dynamics in the transition and film-dominated regimes.
![Schematics of droplet migration in a Hele–Shaw cell for different droplet radii. (a) Side-view and (b) top-view. []{data-label="fig:problem_description"}](problem_description){width="0.6\columnwidth"}
In contrast with scenarios in which the droplet is much smaller than the channel height, the dynamics of droplets in the film-dominated regime ([*i.e.*]{}, $R/H\gg1$) are more complicated and less understood. Early efforts in predicting the velocity of a flat bubble in a Hele–Shaw cell go back to the pioneering theoretical work by Taylor and Saffman [@Taylor_1959a]. The Hele–Shaw equations are obtained by averaging the Stokes equations along the channel height (depth), and it can be shown that the bubble velocity $U_d$ is always larger than the average inflow velocity $U_f$. For low capillary numbers, the horizontal cross-section of the bubble is close to a circle (exhibiting a pancake shape similar to the present problem) [@Kopf-Sill_1988a], then $U_d/U_f=2$ according to the Taylor–Saffman theory [@Taylor_1959a; @Tanveer_1986a; @Tanveer_1987a]. In the later experiments by Kopf-Sill and Homsy [@Kopf-Sill_1988a], the bubble velocity was observed to be about an order of magnitude lower than the Taylor–Saffman prediction. Possible reasons for the discrepancy have been explored, such as an excessive drag due to a moving contact line within the film and Marangoni effects due to surfactant transport [@Saffman_1989a; @Park_1994a; @Maruvada_1996a], yet the underlying mechanisms responsible for the low mobility of the bubble is still not fully understood. In the limit when the 2D droplet is perfectly circular, the Hele–Shaw equations have also been solved analytically by Afkhami and Renardy [@Afkhami_2013a] and Gallaire [*et al.*]{}[@Gallaire_2014a] to obtain the droplet velocity for arbitrary viscosity ratios as $$\frac{U_d}{U_f}= \frac{2}{1+\mu_d/\mu_f}\, , \label{eq:HS_drop_vel}$$ where $\mu_d$ is the viscosity of the droplet fluid. Equation [(\[eq:HS\_drop\_vel\])]{} recovers the Taylor–Saffman solution $U_d/U_f=2$ for bubbles ($\mu_d/\mu_f=0$). Nevertheless, since the Hele–Shaw equations ignore the lubrication film, Eq. [(\[eq:HS\_drop\_vel\])]{} does not capture the effect of the film dynamics on the droplet velocity.
Due to the small scale of microchannels, experimental measurements of the detailed flow structure are very difficult. Recently Huerre [*et al.*]{}[@Huerre_2015a] presented quantitative optical measurements of lubrication film thicknesses as small as 20 nm for a channel height $2H=25 \mu$m (For such a thin film, the intermolecular forces become important for the film dynamics and droplet velocity. In the present study, we focus on the cases where the minimum film thickness is always larger than one micrometer. As a result, the effects of intermolecular forces can be ignored.) However it is very challenging to probe local values of velocity and pressure, and consequently the flow structure inside and outside of the droplet is hard to obtain. Numerical simulations which provide a complete description of the flow field can shed light on the present problem. It has been shown that numerical approximations of the Navier-Stokes equations can represent flow physics in many microfluidics applications, such as droplets/bubbles migration and the breakup of droplets at T-junctions [@Afkhami_2011a; @Bedram_2013a; @Hoang_2013a; @Hoang_2013b].
Nevertheless 3D direct numerical simulations of droplet migration in a microchannel are challenging as well. This is in particular true for droplets with large ratio $R/H$ and small Ca. Since capillary effects are dominant, the time step used in the temporal integration is controlled by the surface tension term in the momentum equation. For small capillary numbers the time step becomes very small and thus the number of steps for the terminal velocity of droplet to be reached is huge. In addition, at low capillary numbers the thickness of the lubrication film becomes very small (several orders of magnitude smaller than the channel height) [@Huerre_2015a]. Since the mesh resolution and the time step restriction are dictated by the smallest length scale in the problem, the total number of cells and integration steps in 3D simulations are both very large. Three-dimensional simulation results of droplet/bubble in microchannels are rarely seen until the very recent works by Hoang [*et al.*]{}[@Hoang_2013b], Roberts [*et al.*]{}[@Roberts_2014a] and Zhu and Gallaire [@Zhu_2016a]. However, as far as the authors are aware, a 3D direct numerical solution to the Navier-Stokes equations for droplets with a large $R/H$ and with a lubrication film formed has never been reported in the literature.
The goal of the present study is to investigate the droplet migration in a Hele–Shaw cell through 3D direct numerical simulations. The capillary number of the droplet is considered to be small enough so that the horizontal shape of the droplet is near-circular all the time. Particular attention will be paid to the effect of the lubrication film on the droplet dynamics. A two-phase flow solver, Gerris[@Popinet_2003a; @Popinet_2009a], is employed to solve the variable-density Navier-Stokes equation with surface tension. In the Gerris solver, the Volume-of-Fluid (VOF) method is used to resolve the interface between the two phases and an adaptive mesh is used to refine the mesh only in the lubrication film and near the interface, so that the computational cost can be reduced. The governing equations and the numerical methods are presented in section \[sec:equations\]. The simulation results will be shown in sections \[sec:2D\] and \[sec:3D\]. We will first perform simulations of 2D droplets migrating in a microchannel with parameters similar to the 3D case for the purpose of validation, see \[sec:2D\]. The simulation results and discussions for the 3D droplet are then presented in section \[sec:3D\]. Finally, we draw conclusions on the present study.
Governing equations and numerical methods {#sec:equations}
=========================================
Governing equations {#sec:gov_eqns}
-------------------
The one-fluid approach is employed to resolve the two-phase flow, where the phases corresponding to the droplet and the ambient fluid are treated as one fluid with material properties that change abruptly across the interface. The incompressible, variable-density, Navier-Stokes equations with surface tension can be written as $$\begin{aligned}
\rho (\partial_t{\boldsymbol{u}} + {\boldsymbol{u}} \cdot \nabla {\boldsymbol{u}}) = -\nabla p
+ \nabla \cdot (2\mu {\boldsymbol{D}}) + \sigma \kappa \delta_s {\boldsymbol{n}}\, ,
\label{eq:mom} \\
\nabla \cdot {\boldsymbol{u}} = 0 \, ,
\label{eq:divfree}\end{aligned}$$ where $\rho$, $\mu$, ${\boldsymbol{u}}$, and $p$ represent density and viscosity, velocity and pressure, respectively. The deformation tensor is denoted by ${\boldsymbol{D}}$ with components $D_{ij}=(\partial_i u_j + \partial_j u_i)/2$. The third term on the right hand side of Eq. [(\[eq:mom\])]{} is a singular term, with a Dirac distribution function $\delta_s$ localized on the interface, and it represents the surface tension force. The surface tension coefficient is $\sigma$, and $\kappa$ and ${\boldsymbol{n}}$ are the local curvature and unit normal of the interface.
The volume fraction $C$ is introduced to distinguish the different phases, in particular $C=1$ in the computational cells with only the ambient fluid (respectively $C=0$ in the droplet), and its time evolution satisfies the advection equation $$\begin{aligned}
\partial_t C + {\boldsymbol{u}} \cdot \nabla C =0 \, .
\label{eq:color_func}\end{aligned}$$ The fluid density and viscosity are then defined by $$\begin{aligned}
\rho & = C \rho_f + (1-C) \rho_d \, ,
\label{eq:density} \\
\mu & = C \mu_f + (1-C) \mu_d \, .
\label{eq:viscosity}\end{aligned}$$ where the subscripts $f$ and $d$ represent the ambient fluid and the droplet respectively.
Numerical methods {#sec:numerics}
-----------------
The Navier-Stokes equations (Eqs. [(\[eq:mom\])]{} and [(\[eq:divfree\])]{}) are solved by the open-source package Gerris[@Popinet_2003a; @Popinet_2009a]. In Gerris, a finite volume approach based on a projection method is employed. A staggered-in-time discretisation of the volume-fraction/density and pressure leads to a formally second-order accurate time discretisation. The interface between the different fluids are tracked and followed using a Volume-of-Fluid (VOF) method [@Scardovelli_1999a; @Tryggvason_2011a]. A quad/octree spatial discretisation is used, which gives a very important flexibility allowing dynamic grid refinement into user-defined regions [@Popinet_2003a]. Finally the height-function (HF) method is utilized to calculate the local interface curvature, and a balanced-force surface tension discretization is used [@Francois_2006a; @Popinet_2009a].
Time step restriction {#sec:numerics}
---------------------
Both temporal and spatial schemes of integration are explicit. Therefore, numerical stability requires the time step for integration to satisfy the stability conditions corresponding to the advection, viscous, and surface tension terms, respectively. Due to the small capillary number in the present problem, the time step restriction is dominated by the contribution from the surface tension force. The conventional capillary time step restriction is given by Brackbill [*et al.*]{}[@Brackbill_1992a] as, $$\begin{aligned}
\Delta t \le \Delta t_{BKZ} = \sqrt{ { \rho \Delta^3 \over \pi \sigma } }\, \label{eq:dt_surf_tension}\, ,\end{aligned}$$ where $\Delta$ represents the cell size. For droplets with large $R/H$, the thickness of the lubrication film is the smallest scale in the problem and thus dictates the smallest cell size and the time step to be used. Taking the conventional scaling of film thickness [@Landau_1942a; @Bretherton_1961a], the grid size can be estimated as $\Delta \sim h\sim H Ca^{2/3}$. (The scaling of the film thickness on capillary number will be discussed in detail in the sections \[sec:2D\] and \[sec:3D\].) Then Eq. [(\[eq:dt\_surf\_tension\])]{} becomes $$\begin{aligned}
\Delta t_{BKZ} = \sqrt{ { \rho H^3 \mathrm{Ca}^2 \over \pi \sigma } }\, \label{eq:dt_surf_tension_BKZ}\, . \end{aligned}$$ If we assume that the terminal migration velocity is reached after a time of order $H/U_d$, then the total number of time steps can be estimated as $$\begin{aligned}
\mathcal{N}_t \ge \sqrt{ {\pi \sigma \over \rho H \mathrm{Ca}^2 U_d^2} } \sim \mathrm{Ca}^{-3/2} \mathrm{Re}^{-1/2}\, \label{eq:Nt_surf_tension_BKZ}\, , \end{aligned}$$ where $\mathrm{Re}=\rho_f U_d H/ \mu_f$. As the total number of grid points (on a regular Cartesian grid) will scale like $\mathcal{N}_{\Delta} \sim (H/\Delta)^D \sim \mathrm{Ca}^{-2D/3}$ with $D$ the dimension of the problem. Then the total work (number of grid points times number of time steps) for a 3D problem will thus scale as $$\begin{aligned}
\mathcal{N}_{\Delta} \mathcal{N}_t \sim \mathrm{Ca}^{-7/2} Re^{-1/2}\, \label{eq:totalwork_surf_tension_BKZ}\, . \end{aligned}$$
Given that capillary numbers for microfluidics applications can be as low as $10^{-5}$, we see that direct numerical simulations will be extremely expensive. The adaptive mesh refinement in Gerris allows to refine the mesh only in the lubrification film, which greatly decreases this cost, however this is still not sufficient to reach such low capillary numbers.
A different capillary time step restriction has been proposed by Galusinski and Vigneaux [@Galusinski_2008a]. They suggested that when both $\mathrm{Re}$ and $\mathrm{Ca}$ are very small, the capillary time step should satisfy $$\begin{aligned}
\Delta t \le \max(\Delta t_{BKZ}, \Delta t_{STK})\, \label{eq:dt_GV}\, ,\end{aligned}$$ where $\Delta t_{STK} = \mu_f \Delta / \sigma$ is the capillary time step related to the Stokes equation (when inertial effects are negligible). It can then easily be shown that the ratio $\Delta t_{BKZ}/\Delta t_{STK}\sim \mathrm{Re}^{-1/2}\mathrm{Ca}^{-1/6}$. Therefore, if $\mathrm{Re}< \mathrm{Ca}^{1/3}$ the larger time step $\Delta t_{STK}$ can be used. In the present study, however, we are interested in the cases where $\mathrm{Re}\sim O(1)$ and $\mathrm{Ca}$ varies from $10^{-3}$ to $10^{-1}$ (see the detailed values in section \[sec:3D\]). As a consequence, the criterion $\mathrm{Re}< \mathrm{Ca}^{1/3}$ is never satisfied and Eqs. [(\[eq:dt\_surf\_tension\])]{}-[(\[eq:totalwork\_surf\_tension\_BKZ\])]{} remain valid to estimate the time step and computational cost.
2D simulations of droplet migration in a microchannel {#sec:2D}
=====================================================
Before we present the 3D results, 2D simulations for a droplet moving in a microchannel are performed for validation purpose. The simulation setup is schematically shown in Fig. \[fig:problem\_description\_2d\]. The flow is symmetric in the $y$ direction with regard to the axis $y=0$, thus only the upper half of the channel is considered. The inflow from the left boundary is a parabolic velocity profile with mean value $U_f$ and maximum value $U_{f,max}=1.5 U_f$. An outflow boundary condition is applied at the right boundary of the domain; while the top boundary is taken as a no-slip wall condition. The length of the bubble is denoted by $2R$, and $R/H=2$. As the droplet is constrained by the channel, a lubrication film is formed between the droplet and the wall. The channel length is $L_x=16H$, which is long enough for the droplet to reach its terminal velocity $U_d$ before being affected by the outflow boundary.
The physical properties of the ambient fluid and the droplet are listed in Table \[tab:phys\_parameters\], which are similar to the recent experiments by Roberts [*et al.*]{}[@Roberts_2014a] The Reynolds number based on the channel height and average inflow velocity, $\mathrm{Re}_f=\rho_f U_f H/\mu_f$, is taken to be 1.35. The surface tension is varied from 0.0002 to 0.0159 N/m.
![Two-dimensional simulation setup for a droplet in a planar microchannel.[]{data-label="fig:problem_description_2d"}](problem_description_2d){width="0.8\columnwidth"}
$\hat{\rho}_f$ (kg/m$^3$) $\hat{\rho}_d$ (kg/m$^3$) $\hat{\mu}_f$ (Pa s) $\hat{\mu}_d$ (Pa s) $\hat{H}$ ($\mu$m) $\hat{U}_f$ (m/s)
--------------------------- --------------------------- ---------------------- ---------------------- -------------------- -------------------
750 1000 $7.4\times 10^{-4}$ $ 10^{-3}$ $50$ 0.0267
: \[tab:phys\_parameters\] Physical properties of the ambient fluid and the droplet in the present study. [^1]
Bretherton [@Bretherton_1961a] studied theoretically and experimentally the dynamics of a long bubble in a cylindrical channel. In the limit of small capillary numbers, [*i.e.*]{}, $Ca=\mu_f U_d/\sigma \ll 1$, it is shown that the ratio between the film thickness $h$ and the half-height of the channel $H$ scales as $Ca^{2/3}$, as does the ratio between the excess of droplet velocity compared to the mean inflow velocity of the ambient fluid: $$\begin{aligned}
{ h \over H } = { (U_d-U_f) \over U_f} \sim \mathrm{Ca}^{2/3}\end{aligned}$$ the proportionality constant depends on the geometrical configuration (planar or cylindrical channels) and the viscosity ratio, although temperature and surfactants may also modify the surface tension. It can be shown that $h/H$ is identical to $(U_d-U_f)/U_f$ due to mass fluxes balance. In the original work of Bretherton [@Bretherton_1961a], the viscosity of the fluid within the bubble is considered to be zero. An extension to droplets of arbitrary viscosity was given later by Hodges [*et al.*]{}[@Hodges_2004a] The theory for a slender droplet in a cylindrical channel is readily modified for a 2D droplet confined by two parallel plates. The film thickness and the excess of droplet velocity for a 2D droplet in the limits of $\mu_d\to 0$ (bubbles) and $\mu_d\to \infty$ (very viscous droplets) are given by $$\frac{h}{H} = \frac{U_d - U_f }{U_f} \sim 0.643 \ (3 \mathrm{Ca})^{2/3}\, .
\label{eq:scaling_2d_inviscid}$$ and $$\frac{h}{H} = \frac{U_d - U_f }{U_f} \sim 0.51 \ (3 \mathrm{Ca})^{2/3}\, ,
\label{eq:scaling_2d_visc}$$ respectively [@Afkhami_2011a]. Similar scaling relations have also been derived for moving foams as well as bubble trains in micro-devices [@Baroud_2010a].
The multi-scale nature of the problem can be seen through Eqs. [(\[eq:scaling\_2d\_inviscid\])]{} and [(\[eq:scaling\_2d\_visc\])]{}. For a capillary number of $10^{-4}$, $h/H=0.00288$ and therefore the length scales involved in the system span over three orders of magnitudes. This fact highlights the importance of the adaptive mesh refinement technique for the present study, which reduces the overall computational costs by refining the mesh only in the lubrication film and near the interface. The shapes of the droplets and the corresponding streamlines in the droplet reference frame are shown in Fig. \[fig:2d\_results\](a) for different capillary numbers. The color of the streamlines denotes the velocity magnitude in the droplet reference frame. The droplet shape and flow pattern are in full agreement with the theory [@Bretherton_1961a; @Hodges_2004a]. Under the effect of the viscous stress, the radius of curvature at the front meniscus is smaller than that at the rear. A small bump is visible on the top-rear of the droplet. The four stagnation points on the droplet interface (A: top-rear; B: top-front; C: rear; D: front) and the other two inside the droplet (E: near the rear; F: near the front) are clearly seen. For small Ca, the major recirculation (A-E-F-B-A) is located at the center of the droplet, with two minor ones sitting on both sides (A-E-C-A and B-F-D-B). When Ca increases the major recirculation shifts toward the rear of the droplet and the two minor recirculation zones become smaller. When the capillary number reaches $\mathrm{Ca}=0.116$ the stagnation points C and E get very close and the minor recirculation near the rear disappears.
![(a) Droplet shape and the streamlines in the droplet reference frame. The color of the streamlines denotes the velocity magnitude in the droplet reference frame. (b) Relative droplet velocity and film thickness as functions of the capillary number. []{data-label="fig:2d_results"}](2d_results){width="0.99\columnwidth"}
Figure \[fig:2d\_results\](b) shows a quantitative comparison between the numerical and the theoretical results for the droplet velocity and the film thickness. The excess of droplet velocity $(U_d - U_f)/U_f$ (which is the same as $h/H$) are plotted as functions of the droplet capillary number $\mathrm{Ca}$ in log-log scale, for $\mathrm{Ca}$ varying from $1.24 \times 10^{-3}$ to $9.87 \times 10^{-2}$. The droplet velocity in the simulation is evaluated by a weighted-integral over all the cells occupied by the droplet fluid, [*i.e.*]{}, $ U_d = \int_S u\ (1-C) \ dS / \int_S (1-C)\ dS$. It is seen that the simulation results match the theoretical results quite well. For large $\mathrm{Ca}$ the simulation results are slightly lower than the theoretical prediction. This observation is consistent with the previous studies by Afkhami [*et al.*]{}[@Afkhami_2011a]
3D simulations of droplet migration in a microchannel (Hele–Shaw cell) {#sec:3D}
======================================================================
In this section we present the 3D simulation results of droplet migration in a Hele–Shaw cell. The physical properties of the ambient fluid and the droplet are identical to those of the previous 2D simulations, but the droplet is 3D and the cross-section in the $x$-$z$ plane is close to a circle.
Simulation setup
----------------
The computational domain for a 3D droplet migrating in a Hele–Shaw cell is shown in Fig. \[fig:3d\_setup\]. It is assumed that the droplet motion and the corresponding flow is symmetrical in the $y$ and $z$ directions, with respect to the planes $y=0$ and $z=0$, respectively. Therefore, only a quarter of the overall domain is actually computed. Taking the half-height of the channel $H$ as the typical length scale, the length, height, and width of the computational domain are taken as $L_x=16H$, $L_y=H$, and $L_z=8H$, respectively.
![Three-dimensional simulation setup for a droplet in a Hele–Shaw cell.[]{data-label="fig:3d_setup"}](3d_setup){width="0.9\columnwidth"}
An inflow boundary condition is imposed on the left of the domain with a parabolic velocity profile. Correspondingly an outflow boundary condition is used on the right. The top of the domain is taken to be a no-slip wall moving with a velocity $U_w$ in the opposite direction with respect to the flow. Therefore the simulation is indeed conducted in a moving reference frame with a mean inflow velocity of $U_f-U_w$. The purpose of using a moving reference frame is to keep the droplet near the center of the domain and to avoid the influence of the boundaries. Ideally the terminal droplet velocity in the lab reference frame $U_d$ should be used for $U_w$, but since $U_d$ is unknown *a priori*, an initial guess of $U_d$ is used for $U_w$. Nevertheless as long as the droplet does not get too close to the inflow and outflow boundaries, the specific value of $U_w$ is immaterial to the simulation results. Finally the other boundaries are set to symmetric boundary conditions.
The droplet is taken to be initially stationary in the simulation frame. Driven by the ambient fluid flow the droplet approaches the terminal droplet velocity as time evolves. The initial shape of droplets with $R/H < 1$ is taken to be a sphere with radius $R$. For $R/H \gtrsim 1$ the cross-section at $y=0$ is a circle with radius $R_0$. The top of the droplet is initially flat with a constant distance of $0.1H$ to the top wall. The “edge" of the droplet is formed by a quarter of a circle of radius of $0.9H$ rotated with respect to the $y$ axis, see Fig. \[fig:3d\_setup\]. Similar to the droplet velocity, the droplet will evolve to the final shape for which the surface tension and viscous forces at the interface are in balance. The final value of $R$, based on the horizontal cross-section area $S_{y=0}$ (at the plane $y=0$) as $R=\sqrt{2S_{y=0}/\pi}$, is slightly different from its initial value $R_0$. Due to the initial configuration we impose, the difference between $R$ and $R_0$ is generally small. (Therefore from here on we do not distinguish $R_0$ and $R$ unless otherwise indicated.) Similar to the velocity of the top wall, the specific details of the initial shape of the droplet are irrelevant to the final results as long as the terminal droplet velocity and shape are reached.
An adaptive mesh is used to discretize the spatial domain. The finest cell size used is $\Delta_{\min}=H/128$. The mesh is refined at the interface based on the gradient of the volume fraction in each cell. Mesh refinement is also conducted using an *a posteriori* error estimate of all three velocity components as a cost function for adaptation. A mesh refinement study was performed in the 2D simulations, showing that $\Delta_{\min}$ is sufficient for the range of capillary numbers considered in the present study.
$r=\rho_f$/$\rho_d$ $m=\mu_f/\mu_d$ $\mathrm{Re}_f=\rho_f U_f H/\mu_f$ $R/H$ $\mathrm{Ca}_f=\mu_f U_f/\sigma$
--------------------- ----------------- ------------------------------------ ---------- --------------------------------------------
0.75 0.74 1.35 0.5 to 2 $6.17\times10^{-3}$ to $9.87\times10^{-2}$
: \[tab:dmls\_parameters\] Key dimensionless parameters in 3D simulations.
The physical parameters are identical to the 2D simulation presented in section \[sec:2D\] (see Table \[tab:phys\_parameters\]). Parametric studies are performed by varying the surface tension coefficient $\sigma$ and the droplet radius $R$. Taking the ambient fluid density $\rho_f$, the half of the channel height $H$, and the mean inflow velocity $U_f$ as characteristic scales, the unknown variables and physical parameters can be non-dimensionalized and the governing equations (Eqs. [(\[eq:mom\])]{}–[(\[eq:color\_func\])]{}) are solved in dimensionless form. The key dimensionless parameters of the present problem are given in Table \[tab:dmls\_parameters\]. As shown in the table, the ratio $R/H$ and the capillary number based on the average inflow velocity $\mathrm{Ca}_f$ are chosen as the tuning parameters. The capillary numbers in the 3D simulation are significantly larger than that in experiments by Roberts [*et al.*]{}[@Roberts_2014a], in which $\mathrm{Ca}_{f,\mathrm{Exp}}=9.87\times 10^{-4}$. The smallest $\mathrm{Ca}_f$ considered here is about an order of magnitude larger than this experimental value. As illustrated in the previous 2D results, the lubrication film formed is very small for small capillary numbers. It is prohibitively difficult to simulate droplets with a larger $R/H\gtrsim 1$ and a small $\mathrm{Ca}_f$ comparable to the experimental value with current computational resources. Nevertheless it is expected that as long as the thickness of the lubrication film does not reach the regime where van der Waals forces are important, the effect of the lubrication film on the droplet dynamics can still be captured with simulation results at larger $\mathrm{Ca}_f$. At the end of this section an extensive discussion on the effect of surface tension will be given.
Effect of droplet horizontal radius on droplet dynamics
-------------------------------------------------------
![(a) Temporal evolution of mean droplet velocity. (b) Terminal mean droplet velocity $U_d$, (c) thickness (height) of the gap (film) between the droplet interface and the top channel wall, denoted by $h$, and (d) relative $u$-velocities at the top and the side of the droplet ($U_y$ and $U_z$) as a function of the ratio between the droplet radius $R$ and half of the channel height $H$. The experimental and theoretical results are shown for comparison [@Roberts_2014a; @Nadim_1991a; @Hudson_2010a] . []{data-label="fig:udvsR"}](udvsR){width="0.96\columnwidth"}
The temporal evolution of the droplet velocity is shown in Fig. \[fig:udvsR\](a) for $\mathrm{Ca}_f=9.87\times 10^{-2}$ and $R/H$ varying from 1 to 2. When $R/H$ increases, $U_d$ decreases and in contrast, the time it takes for the droplet to reach its terminal velocity increases (while identical initial conditions are imposed). From $R/H=1$ to 1.25, $U_d$ decreases significantly (about 9%). The decrease of $U_d$ with $R/H$ becomes more gradual for larger $R/H$. It is seen that the difference between $U_d$ for $R/H=1.75$ and $R/H=2$ becomes invisible as $t>2$.
The simulation results of the droplet velocity as a function of $R/H$ are shown in Fig. \[fig:udvsR\](b). The experimental and simulation results of Roberts [*et al.*]{}[@Roberts_2014a] and the theoretical results of Nadim and Stone [@Nadim_1991a] for small $R/H$ are also shown for comparison. The present predictions of $U_d$ are in good agreement with both the theoretical and experimental results for $R/H<1$.
For $R/H>1$ the simulation results of $U_d$ are larger than those measured in the experiment. The discrepancy is expected since smaller surface tension coefficients (or larger capillary numbers) are considered in the simulation. Nevertheless the simulation results capture well the trend of variation of $U_d$ with $R/H$, and the three different regimes (Poiseuille-dominated, transition, and film-dominated regimes) can be clearly seen. In particular, the variation of $U_d$ in the transition and the film-dominated regimes is much better illustrated. In the transition regime ($0.75\lesssim R/H \lesssim 1.25$), the droplet just overfills the channel forming a lubrication film. Correspondingly, the droplet velocity decreases rapidly. In the film-dominated regime, for $R/H>1.5$, $U_d$ is observed to reach a plateau gradually. The plateau value of $U_d$ decreases with $\mathrm{Ca}_f$, approaching the experimental value which corresponds to a much smaller $\mathrm{Ca}_f$.
The three different regimes can also be identified from the variation of film thickness with $R/H$ as shown in Fig. \[fig:udvsR\](c). In the Poiseuille-dominated regime, as the gap between the droplet and the top wall is relatively large, its thickness is not affected by the wall. The droplet remains close to a sphere, so $h/H$ closely follows $1-R/H$. As $R/H$ approaches 1, (in the transition regime,) the film thickness starts to deviate from $1-R/H$ and becomes governed by the viscous stress and the surface tension. When $R/H\gg 1$, (in the film-dominated regime,) the film thickness becomes completely independent from the horizontal radius and becomes dependent on $\mathrm{Ca}_f$.
At last the droplet circulation velocities in the three different regimes are shown in Fig. \[fig:udvsR\](d). The $u$-velocities at the top (the point with maximum $y$ on the droplet) and the side of the droplet (the point with minimum $z$ on the droplet) are denoted as $U_y$ and $U_z$, respectively. The simulation results of Roberts [*et al.*]{}[@Roberts_2014a] and the theoretical results of Hudson [@Hudson_2010a] are shown for comparison. For $R/H \ll 1$, namely in the Poiseuille-dominated regime, $U_y$ and $U_z$ vary as parabolic functions with respect to $R/H$. The present predictions of $U_y$ are in good agreement with both the previous numerical results and the theoretical prediction. In the film-dominated regime ($R/H \gg 1$), it is observed that both $U_y$ and $U_z$ reach constant values, similar to $U_d$. It is also interesting to notice that the relative velocities $U_y-U_d$ and $U_z-U_d$ for different $\mathrm{Ca}_f$ seem to overlap.
A simple model for droplet dynamics
-----------------------------------
An interesting observation from the simulation results is that the velocity of a 3D droplet can vary in a much wider range than the velocity of a 2D droplet. In particular, the droplet velocity in the film-dominated regime is observed to be generally smaller than the average inflow velocity. This low droplet velocity has also been observed in experiments [@Roberts_2014a; @Huerre_2015a].
From Fig. \[fig:udvsR\](b) it can be seen that when $R/H$ increases from 0.5 to 2, the droplet velocity $U_d$ for $\mathrm{Ca}_f=2.47\times 10^{-2}$ decreases from 1.4 to 0.8. In this section, we propose a simple model of mass fluxes balance to qualitatively illustrate the droplet mobility in the film-dominated regime and the resulting 3D flow patterns.
\[sec:model\]
![Schematics of mass fluxes budget for a droplet in (a) 2D and (b) 3D channels. []{data-label="fig:mass_cons"}](mass_cons){width="0.85\columnwidth"}
In the 2D planar (or axisymmetric) geometry the ambient fluid which enters the channel can only move along the flow direction, and thus must push the droplet to move faster than the average inflow velocity to ensure mass conservation. If $h$ is given the corresponding value of $U_d$ can be obtained by the balance of mass fluxes [@Bretherton_1961a]. Taking a control volume (of zero thickness) near the rear of the droplet as shown in Fig. \[fig:mass\_cons\](a), the mass fluxes are identical to zero on both the $y$ and $z$ boundaries and thus the fluxes from the left and right must cancel. Considering a bubble with large $R/H$ and small $h/H$, with a stress-free interface (and thus a constant velocity within the film), the relation between $h$ and $U_d$ can be expressed as $$\frac{U_d-U_f}{U_d}\frac{H}{h} = 1\, .
\label{eq:2D_mass_cons}$$ It is seen that $U_d > U_f$ for non-zero $h$. For small $h/H$ the excess of droplet velocity is small and the droplet velocity $U_d$ is close to the average inflow velocity $U_f$. In the 3D analogue, since the ambient fluid can flow around the droplet from the sides (or edge) of the droplet, the mass fluxes balance and the flow pattern becomes more complex. Similarly to the 2D case we look at a thin layer of small thickness $\epsilon \ll h$ along the symmetric plane ($z\in (0,\epsilon)$) near the rear of the droplet. As there is zero flux on the symmetric plane $z=0$, the transverse mass flux is only non-zero at the plane $z=\epsilon$, normalized by the flux from the right, [*i.e.*]{}, $U_d h \epsilon$, which can be expressed as $$F_{z,rear} = 1 - \frac{U_d-U_f}{U_d}\frac{H}{h} \, ,
\label{eq:3D_mass_cons}$$ assuming the variation of $U_f$ in $z\in (0,\epsilon)$ is negligible. The transverse flux $F_{z,rear}$ is not uniformly distributed on the plane $z=\epsilon$, instead it is concentrated near the stagnation point at the rear of the droplet. A similar analysis can be performed for the front of the droplet and it can be shown that $F_{z,rear}$ and $F_{z,front}$ have the same magnitude but different signs, which creates a circulation around the edge of the droplet.
For a given $h$, $F_{z,rear}$ is related to $U_d$ through mass conservation, however the fluxes balance is not sufficient to determine $U_d$. This extra spatial degree of freedom of mass flux thus allows a much wider variation of droplet velocity. In contrast to the 2D case, where $U_d$ is always slightly larger than $U_f$, in the 3D case $U_d/U_f$ has been observed to vary from as low as $0.1$ to about 2 in experiments [@Park_1994a; @Huerre_2015a].
Furthermore the two scenarios of $U_d>U_f$ and $U_d<U_f$ correspond to two qualitatively different three-dimensional flow patterns over the droplet. Assuming $h\ll H$, it can be shown that $F_{z,rear}<0$ when $U_d>U_f$. The “pure" bubble experiment by Park [*et al.*]{}[@Park_1994a] falls in this scenario of high-migration-velocity (HMV). In the droplet reference frame the sweeping motion of the ambient fluid on the edge of the droplet is from the front to the rear. Combined with the fact that the flow on the top of the droplet is always from the front to the rear, this means that the overall flow in the droplet reference frame is in the rear-to-front direction.
On the other hand the scenario of low-migration-velocity (LMV), [*i.e.*]{}, $U_d<U_f$ and thus $F_{z,rear} > 0$, is very different. The droplets in the present simulation and in the experiments of Roberts [*et al.*]{}[@Roberts_2014a] and Huerre [*et al.*]{}[@Huerre_2015a] fall in this category. The positive transverse flux near the droplet rear ($F_{z,rear}>0$) indicates that the fluid actually goes around the droplet from the rear to the front in the droplet reference frame. As a consequence, the relative flows on the top and the side of the droplets are in opposite directions, (which are also illustrated in Fig. \[fig:udvsR\](d)). A complex interfacial flow sweeping over the droplet is then formed, in which the flow (in the droplet reference frame) first goes from the front to the rear on the top of the droplet then returns from the rear back to the front on the edge of the droplet. (A visualization of this interfacial flow will be presented in Fig. \[fig:interfacial\_vel-vector\_dropframe\].)
Effect of three-dimensional flow structure on droplet dynamics
--------------------------------------------------------------
### Streamlines and flow circulation
![Streamlines inside and outside of the droplet in the droplet reference frame for $\mathrm{Ca}_f=9.87\times 10^{-2}$ and $R/H=0.5$, 1 and 2. The color of the streamlines denotes the corresponding relative velocity magnitude. The pink line indicates the droplet interface.[]{data-label="fig:streamlines"}](streamlines){width="1\columnwidth"}
![Streamlines for $\mathrm{Ca}_f=9.87\times 10^{-2}$ and $R/H=2$ in the droplet reference frame: (a) Internal circulation and external circulation near the front of the droplet. (b) Flow within the film between the droplet and the wall and the internal circulation inside the droplet. The color of the streamlines denotes the magnitude of the relative velocity. The angular coordinate $\theta$ is defined in the droplet reference frame with the origin located at the center of the horizontal cross-section.[]{data-label="fig:streamlines_3D"}](streamlines_3D){width="1.0\columnwidth"}
The three-dimensional flow structure inside and outside of the droplet is shown in Fig. \[fig:streamlines\] for $\mathrm{Ca}_f=9.87\times 10^{-2}$ and $R/H=0.5$, 1 and 2. The streamlines on the two symmetric planes $z=0$ and $y=0$ are plotted in the droplet reference frame and colored by the relative velocity magnitude. As the droplet moves faster for smaller droplets the relative velocity magnitude is larger for smaller $R/H$. The flow pattern for $R/H<1$ here is also consistent with the simulation results of Roberts [*et al.*]{}[@Roberts_2014a].
It is interesting to compare the 3D flow pattern for the droplet with $R/H=2$ to its 2D analogue shown in Fig. \[fig:2d\_results\]. In the vertical plane $z=0$ the ambient fluid is moving from right to left at the top ($y=H$) and in a reverse direction at the center plane ($y=0$). As a result two external circulations develop outside of the droplet, similar to the 2D analogue shown in Fig. \[fig:2d\_results\](a). The external flow will also drive internal circulation inside the droplet. Similarly to the 2D case, six stagnation points (A, B, C, D on the droplet interface and E, F inside the droplet) can be seen, however the stagnation flow around these points becomes three dimensional. For example in the 2D case, the flow reaches the rear stagnation point C and can only turn toward point A and then turn again to the left and leave the domain, forming a counter-clockwise external circulation, see Fig. \[fig:2d\_results\](a). In 3D the fluid has an extra degree of freedom and can go in the $z$ direction around the droplet toward the stagnation point D, see the streamlines on the $x$-$z$ plane in Fig. \[fig:streamlines\](c). This transverse fluid motion near the stagnation point C is characterized by $ F_{z,rear} $ in the model presented in section \[sec:model\]. Due to the 3D flow structure the minor internal circulations (A-E-C-A and B-D-F-B) in the 2D case are invisible here.
The transverse fluid motion significantly affects the external and internal circulating flow. As shown in Fig. \[fig:streamlines\_3D\](a), the 3D streamlines inside and outside of the droplet become very complicated and are very dependent on the angular coordinate $\theta$. (Here $\theta$ is defined as the angle between the normal to the interface in the $x$-$z$ plane and the $x$ axis.) For $\theta=0$ the circulation is perfectly aligned with the symmetric $x$-$y$ plane, and the flow pattern is similar to the 2D case. As $\theta$ increases the external circulation is twisted in a complex manner. The incoming flows for nonzero $\theta$ near the top of the channel are observed to turn toward the center plane $z=0$, (indicated by black arrows in Fig. \[fig:streamlines\_3D\](a)). As the flow approach the droplet interface, it turns to the side to adjust to the droplet shape, (see the magenta arrow). At about $\theta \approx 60$ degrees the streamlines outside of the droplet are almost tangential to the droplet interface. Therefore, in contrast with the typical quasi-2D flows in a Hele–Shaw cell, droplet migration in the microchannel generates a fully 3D flow.
### Velocity field on the droplet interface
![(a) Top- (b) front- and (c) rear-views of relative velocity in the droplet reference frame $\{u-U_d,v,w\}$ on the droplet interface for $\mathrm{Ca}_f=9.87\times 10^{-2}$ and $R/H=0.5$, 1 and 2. The color denotes the magnitude of the relative velocity. []{data-label="fig:interfacial_vel-vector_dropframe"}](interfacial_vel-vector_dropframe){width="0.99\columnwidth"}
The vectors of the interfacial velocity in the droplet reference frame are shown in Fig. \[fig:interfacial\_vel-vector\_dropframe\]. The color of the interface indicates the relative velocity magnitude. Regions with white color indicate zero velocity, [*i.e.*]{}, stagnation regions on the droplet interface. The four stagnation points (A, B, C, D) on the droplet interface and 3D stagnation flows in the vicinity can be clearly seen.
Driven by the flow within the film, the interfacial flow on the top of the droplet moves to the left; while the flow on the edge near the center plane $y=0$ moves in the opposite direction. An interesting and complex interfacial flow is then formed on the droplet interface, which is highlighted with yellow dashed lines in Fig. \[fig:interfacial\_vel-vector\_dropframe\]. At first the flow starts from the vicinity of stagnation point B (indicated by the blue dashed line $\mathrm{B}_1$$\mathrm{B}_2$ ), and goes toward the rear. It is seen that the velocity vectors on the top interface are not parallel to the $x$ axis, instead they turn toward the side of the droplet. The angle between the top interfacial flow and the $-x$ axis is denoted by $\alpha$ here, and it can be seen that $\alpha$ varies significantly with the angle $\theta$ of the relative flow near the wall (in the droplet reference frame). For $\theta=0$, $\alpha=0$, but then $\alpha$ increases significantly with $\theta$ up to about 45 degrees. Then top interfacial flow is observed to curve back toward the center plane ($z=0$) before reaching stagnation point A. After reaching point A the flow reverses its direction and moves toward the front stagnation point D along the edge of the droplet. For the top interfacial flow leaving from $\mathrm{B}_1$$\mathrm{B}_2$ with a large $\theta$, the flow may not get a chance to turn back to stagnation point A before returning to the front. As a result a ring of zero interfacial velocity (the region in white color) is formed on the edge of the top where the interfacial flow reverses direction. Near the rear of the droplet the expected up-going interfacial flow from stagnation points C to A is not seen due to the strong interfacial flow along the edge. The stagnation flows near C and A seem to merge, forming a stagnation area at the droplet rear.
Furthermore, the streamlines in Fig. \[fig:streamlines\_3D\](b) show that the flow within the lubrication film is influenced by the complex interfacial flow on the droplet interface. Similar to the interfacial flow on the top of droplet (see Fig. \[fig:interfacial\_vel-vector\_dropframe\]), when the flow passes through the lubrication film, the streamlines off the center plane $z=0$ also first turn toward the side and then return back to the streamwise direction.
Configuration and dynamics of the lubrication film
--------------------------------------------------
![(a) Thickness of the lubrication film between the interface and the wall $h/H$ and (b) shear $\partial u/\partial y$ on the interface for $\mathrm{Ca}_f=9.87\times 10^{-2}$ and $R/H=0.5$, 1 and 2. The film for $R/H=2$ is highlighted by pink dotted lines. []{data-label="fig:interface_height_shear"}](interface_height_shear){width="0.85\columnwidth"}
As the film between the droplet and the wall serves to “lubricate" the droplet, when the film thickness decreases (for example due to increasing surface tension) it is expected the droplet will slow down correspondingly. The increase of droplet velocity with the film thickness has been confirmed in the experiments by Huerre [*et al.*]{}[@Huerre_2015a]. To better understand the effect of the lubrication film on the droplet dynamics, we plot the thickness of the lubrication film and the shear on the droplet top interface in Fig. \[fig:interface\_height\_shear\] for $\mathrm{Ca}_f=9.87\times 10^{-2}$ and $R/H=0.5$, 1 and 2.
As $R/H$ is small the gap between the droplet and the wall is large and thus strictly speaking there exists no film, the term is retained only for convenience. When $R/H$ is larger or comparable to unity a thin lubrication film with small thickness can be clearly seen. In particular for $R/H=2$ the film area (highlighted by the pink dotted line) becomes quite large compared to the cross-section of the droplet at the $y=0$ plane. Instead of being centered, the lubrication film is shifted toward the rear (see Fig. \[fig:interfacial\_vel-vector\_dropframe\]). The overall film exhibits a “peach" shape.
Values of the shear $\partial u/\partial y$ on the droplet interface are shown in Fig. \[fig:interface\_height\_shear\](b). When $R/H$ is small the droplet is away from the top wall and the shear on the top of the droplet is indeed quite small. The shear increases dramatically as $R/H$ becomes comparable or larger than unity. For $R/H=2$ it is observed that the shear on the droplet is not uniformly distributed and the large negative shear is located near the rear. The region of high shear has a croissant shape, and minimum shear rate reaches about $(\partial u/\partial y)_{\min}=-8$ (in dimensionless unit), the magnitude of which is more than twice larger than the minimum shear in the ambient 2D Poiseuille flow, [*i.e.*]{}$(\partial u/\partial y)_{\min, \mathrm{Poiseuille}}=-3$. It is obvious that the strong shear stress induced by the film is closely related to the low mobility of droplets with larger $R/H$.
Effect of capillary number on film and droplet dynamics
-------------------------------------------------------
As mentioned at the beginning of this section, the capillary numbers considered in the present 3D simulations are larger than that in experiments. To justify that the simulation results presented above at least qualitatively reflect the correct flow physics of droplet migration in a Hele–Shaw cell, the capillary number is varied to investigate its effect on the results. From the droplet velocity previously shown in Fig. \[fig:udvsR\](b), the change in $U_d$ is invisibly small for $R/H=0.5$ as $\mathrm{Ca}_f$ varies from $9.87\times 10^{-2}$ to $6.17\times 10^{-3}$. Though smaller capillary numbers are used, the surface tension is still sufficient to keep the droplet spherical and the droplet velocity is then mainly determined by the viscous stress. The droplet velocity obtained in simulations for $\mathrm{Ca}_f=9.87\times 10^{-2}$, which is two orders of magnitude larger than that in experiments, still agrees quite well with the experimental and theoretical results. On the other hand, as $R/H\gtrsim 1$, the effect of the capillary number on the droplet velocity becomes much stronger. As shown in Fig. \[fig:udvsR\](b) and (c), the droplet velocity and film thickness decrease by about $12.2\%$ and $53.3\%$ for $R/H=2$, when $\mathrm{Ca}$ decreases from $9.87\times 10^{-2}$ to $2.47\times 10^{-2}$.
The film thickness and shear on the droplet interface for $\mathrm{Ca}_f=2.47\times 10^{-2}$ and $R/H=0.5$, 1 and 2 are shown in Fig. \[fig:interface\_height\_shear\_SmallST\]. The surface tension here is four times that in Fig. \[fig:interface\_height\_shear\] while the other parameters remain the same. (Correspondingly $\mathrm{Ca}_f$ in Fig. \[fig:interface\_height\_shear\_SmallST\] is a quarter of that in Fig. \[fig:interface\_height\_shear\].) When $R/H=0.5$ there is no difference in the gap thickness. For $R/H=2$ the film becomes significantly thinner as surface tension increases. (Therefore different legends for film thickness are used in in Figs. \[fig:interface\_height\_shear\](a) and \[fig:interface\_height\_shear\_SmallST\](a) for better visualization.) The minimum film thickness for $\mathrm{Ca}_f=2.47\times 10^{-2}$ is about $h_{\min}=0.015H$ while for $\mathrm{Ca}_f=9.87\times 10^{-2}$, $h_{\min}=0.048H$. It is also observed that when $\mathrm{Ca}_f$ decreases, the lubrication film still shifts to the rear, however the shift becomes smaller and the droplet shape becomes more symmetric with regard to the $y$-$z$ plane. Furthermore, the overall “peach" shape of the film remains, but the smallest film thickness for $R/H=2$ seems to move to the two sides of the film.
When $\mathrm{Ca}_f$ decreases the general distribution of the shear rate within the film is similar. The high-shear region still holds a croissant shape. The minimum shear near the rear decreases to about $-10$ which is about 25% larger than that for $\mathrm{Ca}_f=9.87\times 10^{-2}$ in magnitude. This indicates that the decreases of $\mathrm{Ca}_f$ results in a decrease of $h$ which in turn introduces a stronger shear drag on the droplet and reduces the droplet migration velocity.
![(a) Thickness of the lubrication film between the interface and the wall $h/H$ and (b) shear $\partial u/\partial y$ on the interface for $\mathrm{Ca}_f=2.47\times 10^{-2}$ and $R/H=0.5$, 1 and 2. []{data-label="fig:interface_height_shear_SmallST"}](interface_height_shear_SmallST){width="0.85\columnwidth"}
The minimal film thickness on the symmetric $x$-$y$ plane is plotted as a function of $\mathrm{Ca}$ in Fig. \[fig:ud-h\_vs\_Ca\](a), where $\mathrm{Ca}=\mu_f U_d/\sigma$ is the capillary number based on the droplet velocity $U_d$. The 2D theory, [*i.e.*]{}, Eq. [(\[eq:scaling\_2d\_inviscid\])]{}, is also plotted for comparison. It is seen that $h_{z=0,\min}/H$ in the 3D case is much smaller than in the 2D case. For $\mathrm{Ca}=9.87\times 10^{-2}$, $h_{z=0,\min}/H$ in 3D is an order of magnitude smaller than in 2D, as already visualized in Figs. \[fig:2d\_results\] and \[fig:streamlines\]. The difference between 3D and 2D film thickness seems to decrease with $\mathrm{Ca}$. The film thickness variation with $\mathrm{Ca}$ seems to follow a power law but the power coefficient is about 0.4 instead of 2/3 as in 2D.
The droplet velocity $U_d$ decreases with $\mathrm{Ca}_f$ as shown in Fig. \[fig:ud-h\_vs\_Ca\](b). Here the capillary number based on the average inflow velocity is used following previous works [@Kopf-Sill_1988a; @Park_1994a]. The droplet velocity obtained by the Hele–Shaw equations (Eq. [(\[eq:HS\_drop\_vel\])]{}), which is about 0.85, is also plotted for comparison. The depth-averaged approximation misses the effect of the lubrication film on the droplet dynamics, and thus Eq. [(\[eq:HS\_drop\_vel\])]{} fails to capture the variation of the droplet velocity with capillary number. It is seen that the Hele–Shaw solution under-predicts $U_d/U_f$ for larger $\mathrm{Ca}_f$ and over-predicts $U_d/U_f$ for small $\mathrm{Ca}_f$. The droplet velocity only matches the Hele–Shaw solution at $\mathrm{Ca}_f\approx 0.05$. Similar observations have also been made by Zhu and Gallaire [@Zhu_2016a]. The range of $\mathrm{Ca}_f$ considered in the present simulation is an order of magnitude larger than the experimental value. It is hard to tell if the simulation results would match the experimental measurements [@Roberts_2014a] if the same $\mathrm{Ca}_f$ is used. Nevertheless it seems like the computed $U_d$ would approach the experimental value as $\mathrm{Ca}_f$ decreases.
As discussed previously the mass fluxes balance in 3D is substantially different from 2D. In the 2D case it is known that $(U_d-U_f)/U_f=h/H$ (for $h/H\ll1$). In 3D, $(U_d-U_f)/U_f$ becomes negative. It is shown that $U_d$ also decreases with $h/H$, but the decrease is non-linear and follows a power law about $(h/H)^{1/4}$ as indicated in Fig. \[fig:ud-h\_vs\_Ca\](c).
The simple model of transverse flux near the rear of the droplet (Eq. [(\[eq:3D\_mass\_cons\])]{}) can serve as a measure of the three-dimensional “strength” of the flow. The variation of $F_{z,rear}$ is plotted as a function of the film thickness in Fig. \[fig:ud-h\_vs\_Ca\](d). When $h$ decreases it is observed that $F_{z,rear}$ increases rapidly. As the film becomes thinner, the drag induced by the lubrication film becomes stronger and it is more difficult for the ambient fluid to push the droplet forward. As a result the ambient fluid entering the channel tends to go around the droplet from the transverse edge instead of pushing the droplet. Therefore it is quite clear that both the 3D flow effect and the lubrication film dynamics control the mobility of a droplet in a Hele–Shaw cell.
![(a) The minimum film thickness at the $z=0$ plane and (b) droplet velocity as functions of the capillary number. The droplet velocity obtained by the Hele–Shaw equations (Eq. [(\[eq:HS\_drop\_vel\])]{}) is also plotted for comparison. (c) Excess of droplet velocity compared to the average inflow velocity and (d) Normalized transverse flux near the rear of the droplet as a function of the film thickness. []{data-label="fig:ud-h_vs_Ca"}](ud-h_vs_Ca){width="0.99\columnwidth"}
Discussions of computational costs
----------------------------------
The present simulations are performed on 8 to 24 processors for 3 to 25 days depending on cases. The adaptive meshes used for $\mathrm{Ca}_f=9.87\times 10^{-2}$ and $2.47\times 10^{-2}$ and $R/H=2$ at the plane $z=0$ are shown in Fig. \[fig:2D-interface\_with-mesh\]. The main constraint on the computational efficiency comes from the small time step which is in turn restricted by the surface tension (or capillary number), see Eq. [(\[eq:dt\_surf\_tension\])]{}. When surface tension increases four times, the time step is divided by two and namely the computational time doubles assuming the mesh is unchanged. However, as indicated in Fig. \[fig:2D-interface\_with-mesh\](b) that if $\mathrm{Ca}_f$ is further decreased, the minimum mesh size in the adaptive mesh, [*i.e.*]{}, $\Delta_{min}=H/128$, will not be sufficient and thus must be refined. Therefore, even with an adaptive mesh, it is too expensive to simulate droplets at capillary number comparable to experiments for large $R/H$. To perform 3D direct simulation of droplets at low capillary number, novel numerical schemes to compute the surface tension are required. If the surface tension term in Eq. [(\[eq:mom\])]{} can be treated in an implicit manner similar to what is done for the viscous term in Stokes flow (at low Reynolds number), then the time-step restriction due to the explicit integration of the surface tension term can be lifted (the authors are not aware of this type of scheme in the literature). Nevertheless the development of an implicit treatment of surface tension is out of the scope of the present paper and will be relegated to future work.
![The adaptive mesh on the plane $z=0$ for $R/H=2$ and (a) $\mathrm{Ca}_f=9.87\times 10^{-2}$ and (b) $\mathrm{Ca}_f=2.47\times 10^{-2}$. []{data-label="fig:2D-interface_with-mesh"}](2D-interface_with-mesh){width="1.0\columnwidth"}
CONCLUSIONS
===========
The migration of a droplet in a Hele–Shaw cell has been investigated by 3D direct numerical simulations. We focused on the regime at low capillary number where the droplet remains circular in the horizontal plane. Parametric studies were performed by varying the droplet horizontal radius and the capillary number. For droplets with an horizontal radius $R$ larger than the half-height of the channel $H$, the droplet overfills the channel and a thin lubrication film is formed between the droplet and the wall. An adaptive two-phase flow solver is utilized for the simulations and mesh is locally refined to accurately resolve the thin film. The ratio $R/H$ has a significant impact on the droplet velocity and controls three regimes: the Poiseuille-dominated regime, the film-dominated regime, and the transition regime. The droplet velocity decreases dramatically with $R/H$ in the transition regime. In the film-dominated regime, the droplet horizontal radius is much larger than the channel height and the droplet holds a pancake-like shape. Then the migration velocity is found to become independent of the radius and is mainly dictated by the capillary number. The simulation also shows that the droplet velocity in the film-dominated regime decreases with the capillary number and is generally lower than the average inflow velocity. The lubrication film dynamics and the three-dimensional flow structure seem to both contribute to the low mobility of the droplet. In this low-migration-velocity scenario the interfacial flow on the top of the droplet moves from front toward the rear in the droplet reference frame, but reverses its direction moving toward the front from the two sides of the the droplet. As the depth-averaged Hele–Shaw equations ignore the effect of the lubrication film on the droplet dynamics, their solution fails to capture the dependence of droplet velocity on capillary number. As the shear induced by the lubrication film has a strong influence on the droplet dynamics, the viscosity ratio can be an important parameter for the present problem, and we intend to investigate its influence in future work.
Acknowledgements {#acknowledgements .unnumbered}
================
This project has been supported by the the ANR TRAM project (ANR-13-BS09-0011). The simulations of this paper are conducted on our laboratory cluster and on the CINES Occigen machine for which we gratefully acknowledge grant x20152b7325 from GENCI.
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[^1]: Symbol with $\hat{}$ denotes dimensional parameters.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Observed chemical (anti)correlations in proton-capture elements among globular cluster stars are presently recognized as the signature of self-enrichment from now extinct, previous generations of stars. This defines the multiple population scenario. Since fluorine is also affected by proton captures, determining its abundance in globular clusters provides new and complementary clues regarding the nature of these previous generations, and supplies strong observational constraints to the chemical enrichment timescales. In this paper we present our results on near-infrared CRIRES spectroscopic observations of six cool giant stars in NGC 6656 (M22): the main objective is to derive the F content and its internal variation in this peculiar cluster, which exhibits significant changes in both light- and heavy- element abundances. We detected F variations across our sample beyond the measurement uncertainties and found that the F abundances are positively correlated with O and anticorrelated with Na, as expected according to the multiple population framework. Furthermore, our observations reveal an increase in the F content between the two different sub-groups, $s$-process rich and $s$-process poor, hosted within M22. The comparison with theoretical models suggests that asymptotic giant stars with masses between 4 and 5[M$_{\odot}$]{} are responsible for the observed chemical pattern, confirming evidence from previous works: the difference in age between the two sub-components in M22 must be not larger than a few hundreds Myr.'
author:
- 'Valentina D’Orazi'
- Sara Lucatello
- Maria Lugaro
- 'Raffaele G. Gratton'
- George Angelou
- Angela Bragaglia
- Eugenio Carretta
- 'Alan Alves-Brito'
- 'Inese I. Ivans'
- Thomas Masseron
- Alessio Mucciarelli
nocite:
- '[@marino11b]'
- '[@kur93]'
- '[@abia11]'
- '[@abrito12]'
- '[@roederer11]'
- '[@lugaro12]'
title: |
Fluorine variations in the globular cluster NGC 6656 (M22):\
implications for internal enrichment timescales[^1]
---
Introduction
============
Although Galactic globular clusters (GCs) display a distribution in their global parameters (e.g., mass, metallicity, concentration, horizontal-branch morphology), the internal variation of elements affected by proton captures (hereafter $p$-capture elements) appears a ubiquitous feature (@carretta09a,b). It is clear that GC stars exhibit large changes in the C, N, O, Na, Mg, and Al abundances, whereas (in archetypical systems at least) internal spreads in iron-peak, heavy $\alpha$- (Ca, Ti) and $slow$ neutron-capture ($s$-process) elements all remain within observational uncertainties (@carretta09c, 2010a; @james04; @smith08; @dorazi10). The changes in the $p$-capture elements give rise to a clear chemical pattern: depletion in C, O, and Mg abundances always correspond to enhancements in N, Na, and Al (the so-called light-element anticorrelations). This behavior bears the evidence of H burning at high temperature and points to the presence of multiple stellar generations. It is argued that the ejecta from a fraction of first generation of stars (initially C-O-Mg rich, sharing the same chemical composition of field stars at the same metallicity) mix with primordial gas, providing a medium from which the second generation stars (C-O-Mg poor and N-Na-Al rich) formed. We refer the reader to [@gratton12] for a recent, extensive review on this topic. In this scenario, the H-burning abundance patterns from the first generation stars are imprinted in the second generation and are present from birth. The nature of the stars that enriched the intercluster gas remains uncertain but possible candidates include intermediate-mass asymptotic giant branch (AGB) stars undergoing hot bottom burning (HBB, e.g., @dantona83; @ventura01), fast rotating massive stars (e.g., @decressin07), massive binaries (@demink09), and novae (@smith96; @maccarone12).
Interestingly, this already complex picture is further obfuscated by the presence of some peculiar clusters, such as $\omega$ Centauri (@jp10; @marino11a), NGC 1851 (@yong08; @carretta10a), Terzan 5 (@ferraro09), NGC 6715 (M54, @carretta10b), and NGC 2419 (@cohen11, see however @mucciarelli12 for a different view). In these GCs, along with changes in $p$-capture elements, internal variations in the heavy element abundances have been detected. Species ranging from the iron-peak (e.g., Fe) to the $s$-process elements (Ba, La) vary stochastically from cluster to cluster beyond what is expected from observational errors.
The metal-poor GC NGC 6656 (M22, \[Fe/H\][^2]=$-$1.70 @harris96 -updated in 2010) belongs to this class of GCs, and due to its peculiar nature has received extensive attention (@pilachowski82; @norris83; @bw92; @kayser08; @marino09; @dacosta09; @dacosta10; @abrito12). Recently, Marino et al. (2011b, hereafter MSK11) presented results from their high-resolution spectroscopic study of 35 giant stars, deriving abundances for iron-peak, $\alpha$, $p$-capture, and neutron-capture elements. This detailed abundance analysis provided a unique opportunity to investigate the chemical enrichment history of M22.
This GC is comprised of two distinct groups of stars, characterised by an offset in metallicity and in $s$-process element content. The first group displays a metallicity of $<$\[Fe/H\]$>$=$-$1.82$\pm$0.02 with $<$\[$s$/Fe\]$>$=$-$0.01$\pm$0.01 and the second group has $<$\[Fe/H\]$>$=$-$1.67$\pm$0.01 with $<$\[$s$/Fe\]$>$=+0.35$\pm$0.02. Note that the \[$s$/Fe\] ratios were computed by averaging the abundances of Y, Zr, Ba, La, Nd (see MSK11 for details). Each of these two sub-groups exhibits the classical Na-O and C-N anticorrelations shown by the archetypical GCs. Given that the \[Eu/Fe\] ratio serves as a $rapid$ neutron-capture ($r$-process) tracer[^3] and that enhancements in the $s$-process elements are not accompanied by a similar trend in the \[Eu/Fe\] ratio, it can be inferred that the dichotomy in the $n$-capture abundances may be due to a first generation of polluters that produced the $s$-process only. One possible scenario, which was advocated by MSK11, is that the $weak$ $s$-process component activated in stars with masses larger than $\sim$25 M$_{\odot}$ during core He-burning and C-shell phases (@raiteri93; @pignatari10), may have contributed to the observed abundance patterns. However, in a complimentary study, @roederer11 focused on the heavy element content (from Y to Th) of six stars across the two stellar sub-groups, and ruled out the massive star origin. They concluded that the gas from which the second stellar population formed was enriched in $s$-process material from a class of relatively massive AGB stars (M $\approx$ 5 M$_{\odot}$). In these stars, the production of $s$-process elements is due to the activation of the $^{22}$Ne($\alpha$,n)$^{25}$Mg neutron source, whereas in their lower-mass counterparts the main neutron source is the $^{13}$C($\alpha$,n)$^{16}$O reaction (@busso99).
Neither of the proposed scenarios provides a comprehensive explanation for all the observed chemical features and we are left with numerous unsolved issues. For example, [@roederer11] question why the $s$-process dichotomy is only present in M22: if massive AGB stars are the cause of the GC Na-O anticorrelation, then all clusters should present the $s$-process elements correlated with Na and anti-correlated with O, which is not observed (e.g., @dorazi10).
In this paper we turn to an alternative diagnostic. We present fluorine abundances for a sample of six cool giant stars in M22, carefully selected from both sub-stellar groups as defined by MSK11. Fluorine abundances are a powerful tracer of the polluter mass range in M22 because the F production is highly dependent on the stellar mass.
Theoretical models of AGB stars (e.g., @jorissen92) predict that F is produced due to the activation of the chain of reactions $^{18}$O(p,$\alpha$)$^{15}$N($\alpha$,$\gamma$)$^{19}$F in the He intershell during the recurrent thermal pulses associated with He burning. During the early phases of each thermal pulse, the H-burning ashes are ingested in the convective region developing in the He intershell. These ashes are rich in $^{13}$C and $^{14}$N and their ingestion in the He-burning layer results in production of $^{18}$O due to $\alpha$ captures on $^{14}$N. At the same time protons are released by the $^{14}$N(n,p)$^{14}$C reaction, with neutrons coming from $^{13}$C($\alpha$,n)$^{16}$O. After the quenching of each thermal pulse the envelope may sink in mass deep in the He intershell and carry $^{19}$F to the convective envelope via a process known as the “third dredge-up” (TDU). The peak of F production in AGB stars is reached for stars of initial masses $\sim$ 2 [M$_{\odot}$]{} [@lugaro04]. If the mass of the stars is higher than roughly 5 [M$_{\odot}$]{}, and depending on the metallicity, fluorine is destroyed both via $\alpha$ captures in the He intershell, and via proton captures at the base of the convective envelope due to HBB. AGB stars that experience HBB can also destroy O and Mg and produce Na and Al; as a consequence, according to the multiple population scenario, we should expect the abundances of F to be correlated with O (and Mg) and anti-correlated with those of Na (and Al). This prediction was observationally confirmed by [@smith05] in the intermediate-metallicity GC M4 and by [@yong+08] in NGC 6712.
Other sites have also been suggested for F production: the $\nu$-process in core-collapse supernovae (@woosley90), and core He burning in Wolf-Rayet stars (@meynet00; @palacios05). However, low-mass AGB stars are the only site observationally confirmed (@jorissen92; @abia10).
This paper is organized as follows: observations are described in Section \[sec:obs\], while details on abundance analyses are given in Section \[sec:analysis\]. Our results are then presented in Section \[sec:results\] and discussed in Section \[sec:discussion\]. A summary closes the manuscript (Section \[sec:summary\]).
Observations {#sec:obs}
============
Our sample includes six giant stars, for which stellar parameters (T$_{\rm eff}$, log$g$, \[Fe/H\] and microturbulence $\xi$) along with $p$-capture and $s$-process element abundances were derived by MSK11. We selected three stars belonging to the metal-poor component (MP, also $s$-process poor) and three stars from the metal-rich (MR, $s$-process rich) one. Within each of these sub-groups we also selected both O-rich (Na-poor) and O-poor (Na-rich) stars, spanning a range from \[O/Fe\]=+0.11 to \[O/Fe\]=+0.48 dex.
The main objective of our investigation was the determination of the fluorine abundances. Observationally, F (whose only stable isotope is $^{19}$F) is difficult to detect spectroscopically; the only atomic lines (the ground-state transitions of F [i]{}) that might be revealed lie in the far UV. On the other hand, HF molecular transitions are easily observable in the near-infrared (around $\sim$23000 Å). Our analysis focuses on the HF(1$-$0) R9 transition, located at $\lambda$23358.3 Å. Although not being the strongest feature in this wavelength range, this line is considered to be one of the best abundance indicator for F, because it is free of blends (@abia09; @lucatello11).
-------- ------------- --------------- -------- ------- ----------- -----
Star RA Dec V K Exposures S/N
(hh:mm:ss) ($^{o}$:’:”) (mag) (mag) (s)
IV-97 18:36:41.06 $-$23:58:18.9 11.043 6.759 10x90s 300
III-14 18:36:15.10 $-$23:54:54.6 11.134 6.743 17x120s 400
III-15 18:36:15.61 $-$23:55:01.2 11.362 7.138 8x180s 300
C 18:36:10.21 $-$23:48:44.0 11.309 6.737 6x180s 300
III-52 18:36:10.18 $-$23:54:21.8 11.526 7.459 10x180s 350
V-2 18:36:28.02 $-$23:55:01.6 11.498 7.276 14x120s 350
-------- ------------- --------------- -------- ------- ----------- -----
High-resolution, near-infrared spectroscopic observations were carried out in service mode with CRIRES (CRyogenic high-resolution InfraRed Echelle Spectrograph, @kaeufl04) located at VLT UT1 on 2011 April, July and August (program: 087.0319(A), PI: VD). In Table \[t:log\], we list information on target stars, reporting identifications, magnitudes (see MSK11 for details), exposure times, and S/N ratios per pixel around the HF feature. Note that our selection was limited to the cooler stars in the MSK11’s sample due to our imposed requirement of relatively strong HF lines. We also observed several (hot) early-type stars before and/or after of each target observations, in order to remove telluric contamination from our scientific frames.
We employed the 0.4$^{''}$ slit and the grating order \#24, achieving a resolution of $R$$\sim$50,000 and a wavelength coverage from $\lambda$22948.5 Å to $\lambda$23410.3 Å. This allowed us to include the HF(1$-$0) R9 line, numerous $^{12}$C$^{16}$O vibration/rotation lines (used to derive C abundances) and the Na [i]{} line at $\lambda$23379 Å.
Data reduction was accomplished by means of the CRIRES pipeline (version 2.4), running under the [gasgano]{} environment[^4], which provides one-dimensional, wavelength calibrated spectra. Telluric feature subtraction, rest-frame translation and continuum normalization were then carried out within IRAF[^5]. An example of our spectra is shown in Figure \[f:spectra\] for star III-52; HF, CO and Na [i]{} lines are marked.
![Example of a spectrum for star III-52 []{data-label="f:spectra"}](spectrumIII52.eps){width="8cm"}
Abundance analysis {#sec:analysis}
==================
Fluorine abundances were determined through spectral synthesis using the MOOG code (@sneden73, 2011 version) and the Kurucz (1993) set of stellar atmosphere models (with no overshooting) as in MSK11’s analysis, from which we retrieved stellar parameters as well as O and $s$-process element abundances. However, had we instead adopted the MARCS grid (@gustafsson08), the difference on the resulting abundances would have been less than 0.04 dex.
Concerning the HF feature, we took as excitation potential (EP) $\chi$=0.227 eV (@decin00) and a log$gf$=$-$3.971 (@lucatello11). This last value is very close to that used in previous studies focusing on F abundance determination in GCs (e.g., @smith05; @yong+08; @abrito12), log$gf$=$-$3.955 (from @jorissen92). On the other hand, our EP is about 0.25 eV lower compared to those works: this difference implies an offset between our abundances and those previously published in the literature of roughly 0.30 dex (see also the discussion in Section \[sec:alan\]). Furthermore, we derived C abundances. For this purpose, we assumed O values from the optical range given by MSK11, since our spectra did not cover any suitable OH line, whose stronger transitions extend in the $H$ band (around $\sim$15000Å). The CO line lists come from B. Plez (private communication). Finally, for the Na [i]{} line at $\lambda$23379, atomic parameters ($\chi$=3.750 eV; log$gf$=0.530) were taken from VALD[^6].
As a first step, we checked our line list on the infra-red atlas of the Arcturus spectrum (@hinkle, available at ftp://ftp.noao.edu/catalogs/arcturusatlas/). Assuming a T$_{\rm eff}$=4286 K, log$g$=1.67, $\xi$=1.74 kms$^{-1}$ and \[Fe/H\]=$-$0.52 (following @ramirez11) we obtained an \[F/Fe\]=$-$0.15[^7], to be compared to the value given by [@abia09] of \[F/Fe\]=0.10 dex. The difference is completely explained by the higher EP adopted in that study.
Moreover, we inferred a C abundance of A(C)=8.01 (under the assumption that A(O)=8.81, that is \[O/Fe\]=0.40, with a solar abundance of A(O)$_{\odot}$=8.93), which is in very good agreement with values of [@abia09] (i.e., A(C)=8.06) and [@ryde10] (A(C)=8.08). As for C solar abundance we adopted the value of A(C)$_{\odot}$=8.56, leading to \[C/Fe\]=$-$0.03.
Finally, from the Na [i]{} feature at 23379Å, we derived A(Na)=6.01 which results in \[Na/Fe\]=+0.2 dex (setting A(Na)$_{\odot}$=6.33).
Comparison between synthetic and observed spectra were carried out in a similar way for our sample stars; an example of spectral synthesis is given in Figures \[f:synth1\] and \[f:synth2\] for star C.
![Synthesis of the HF feature for star C.[]{data-label="f:synth1"}](synth1.eps){width="8cm"}
![Same as Figure \[f:synth1\] but for CO (upper panel) and Na (lower panel)[]{data-label="f:synth2"}](synth2.eps "fig:"){width="8cm"}\
![Same as Figure \[f:synth1\] but for CO (upper panel) and Na (lower panel)[]{data-label="f:synth2"}](synth3.eps "fig:"){width="8cm"}
The sensitivity of the F abundance to input stellar parameters was evaluated by separately changing effective temperature, surface gravity and microturbulence values. The intensity of the synthetic line of the HF is particularly sensitive to the adopted T$_{\rm eff}$, the other parameters affecting it at a lower degree (see also @abia09; 2011). A change of $\Delta$(T$_{\rm eff}$)=+70 K, $\Delta$(log$g$)=+0.15, and $\Delta$($\xi$)=+0.13 km s$^{-1}$ (conforming to error estimates given in Table 4 of MSK11) results in a difference in A(F) of +0.10, 0.02 and $-$0.02 dex, respectively. The variation of the input metallicity in the model atmosphere has instead a negligible effect. These are the typical uncertainties that we then summed in quadrature providing a total error, due to stellar parameters, of 0.11 dex in our \[F/H\] ratios. Errors due to the best-fit determination (related to the S/N of the spectra and including uncertainties due to the continuum placement) are instead $\pm$0.07 dex. However, we caution the reader that this value should be treated as a lower limit, since the impact of the telluric correction on this considerably weak feature is significant (see Section \[sec:results\]).
The total internal error in \[F/H\] is then obtained adding in quadrature both uncertainties, resulting in 0.13 dex.
Finally, as far as C and Na are concerned, the typical uncertainties are 0.12 and 0.10 dex, respectively.
Results {#sec:results}
=======
[lccccccccr]{} Star & T$_{\rm eff}$$^{*}$ & $\log g$$^{*}$ & \[Fe/H\]$^{*}$ & $\xi$$^{*}$ & \[O/Fe\]$^{*}$ & $s$-rich$^{*}$ & \[C/Fe\] & \[F/Fe\] & \[Na/Fe\]\
& (K) & (cms$^{-2}$) & & (kms$^{-1}$) & & & & &\
& & & & & & & & &\
IV-97 & 4000 & 0.05 & $-$1.94 & 2.00 & 0.40 & no & $-$1.10 & $-$0.70 & $-$0.40\
III-14 & 4030 & 0.35 & $-$1.82 & 2.15 & 0.48 & no & $-$1.10 & $-$0.80 & $-$0.30\
III-15 & 4070 & 0.40 & $-$1.82 & 1.85 & 0.11 & no & $-$1.15 & $-$1.00 & 0.34\
& & & & & & & & &\
C & 3960 & 0.30 & $-$1.69 & 2.25 & 0.25 & yes & $-$0.60 & $-$0.80 & 0.30\
III-52 & 4075 & 0.60 & $-$1.63 & 1.75 & 0.45 & yes & $-$0.10 & $-$0.60 & 0.05\
V-2 & 4130 & 0.65 & $-$1.57 & 1.75 & 0.15 & yes & $-$0.40 & $-$0.90 & 0.31\
& & & & & & & & &\
Our results are shown in Table \[t:results\], where we report stellar parameters and abundances from MSK11 along with our estimates for F, C, and Na. Even within our quite limited sample (six stars), we found that [*the F abundance shows a large star-to-star variation*]{}, ranging from \[F/H\]=$-$2.82 dex to \[F/H\]=$-$2.23 dex (i.e., a factor of $\sim$4). The average abundance is $<$\[F/H\]$>$=$-$2.55$\pm$0.08 (rms=0.20), implying that the amplitude of this change is beyond the measurement uncertainties (see Section \[sec:analysis\]). Moreover, taking into account the typical errors for O and F abundances, our study suggests that the F variation is comparable with that of O, as also found by [@yong+08] in the GC NGC 6712.
![Fluorine abundances (\[F/H\]) as a function of \[O/H\].[]{data-label="f:OF"}](OF.eps){width="8cm"}
![\[F/H\] versus \[Na/H\] from MSK11 (left panel) and from this study (right panel). Symbols are as in Figure \[f:OF\].[]{data-label="f:NaF"}](NaF.eps){width="8cm"}
In Figures \[f:OF\] and \[f:NaF\] we plot \[F/H\] ratios as a function of \[O/H\] and \[Na/H\], respectively. As one can see, F abundances are positively correlated with O: considering the whole sample, the Pearson correlation coefficient results in $r$=0.89, with a probability to be random smaller than 2%. Focusing on the F-Na diagram (Figure \[f:NaF\]), there is the hint for a F-Na anticorrelation, but in our small sample the linear correlation coefficient is not statistically meaningful. However, this does not prove the lack of an anticorrelation, because we are dealing with small numbers (only six points), heavily reducing the power of statistical tests. In addition, there is no a priori reason why we should combine the two sub-groups as far as the F-Na plane is concerned, because we do not expect that they must behave in the same way. If we look at each component separately the presence of a F-Na anticorrelation is indeed much more evident (given that we have only three points for each group, it is not meaningful to perform a statistical test separately).
The observed chemical pattern can be explained as the evidence of H-burning at high temperature, via the CNO cycle, which causes the destruction of F, in conjunction with O depletion and Na enhancement. More interestingly, those (anti)correlations are revealed in each of the M22 sub-components (the $s$-rich and $s$-poor groups); the implications of this finding are discussed in detail in Section \[sec:discussion\].
Regarding Na, we show both our estimate from near-infrared spectroscopy as well as LTE abundances from the optical range by MSK11 (right and left-hand panels of Figure \[f:NaF\]). A difference of $\Delta$(\[Na/Fe\])=0.31$\pm$0.04 (rms=0.09) dex is found between the two estimates (see Figure \[f:NaIROpt\]) where we compare the two measurements; non-LTE effects can totally account for such a discrepancy (e.g., @lind11). The total average Na abundance is $<$\[Na/Fe\]$>$=0.05$\pm$0.13 (rms=0.33); considering separately the two groups we obtain instead a $<$\[Na/Fe\]$>$$_{s-\rm poor}$=$-$0.12$\pm$0.23 and $<$\[Na/Fe\]$>$$_{s-\rm rich}$=+0.22$\pm$0.08, which implies $\Delta^{\rm rich}_{\rm poor}$\[Na/Fe\]=0.34$\pm$0.17 dex. This value is in good agreement with that derived by MSK11, based on the whole sample of 35 giants, being $\Delta^{\rm rich}_{\rm poor}$\[Na/Fe\]=0.23$\pm$0.07.
![\[Na/Fe\] from the optical range by MSK11 and from this study[]{data-label="f:NaIROpt"}](NaIR_Opt.eps){width="8cm"}
Finally, our sample displays an average C abundance of $<$\[C/Fe\]$>$=$-$0.74$\pm$0.18, with $<$\[C/Fe\]$>$$_{s-\rm poor}$=$-$1.12$\pm$0.02 and $<$\[C/Fe\]$>$$_{s-\rm rich}$=$-$0.37$\pm$0.14. Thus, on average, $s$-rich stars exhibit larger C abundances, which qualitatively agrees with previous works (e.g., @brown90). However, while the difference between the two groups from the optical CH bands derived by MSK11 is $\Delta^{\rm rich}_{\rm poor}$\[C/Fe\]=0.35$\pm$0.13 dex, we achieved a much larger value of $\Delta^{\rm rich}_{\rm poor}$\[C/Fe\]=0.75$\pm$0.10 dex. The same conclusion was reached by [@abrito12] who found a variation of $\Delta^{\rm rich}_{\rm poor}$\[C/Fe\]=0.78$\pm$0.15 dex, from high-resolution, near-infrared spectroscopy of nine cool giants (see also Section \[sec:alan\]). The reason of such a difference in C abundance from the optical and from the near-infrared is not clear and no obvious trends with stellar parameters (e.g., temperature, gravity, microturbulence, and/or metallicity) seem to be present. Further investigations are needed to explore this issue.
Comparison with [@abrito12] {#sec:alan}
---------------------------
In a recent paper, [@abrito12] carried out high-resolution ($R$=50,000), near-infrared (both $H$ and $K$ bands) spectroscopic observations with Phoenix@Gemini-South of nine red giant branch (RGB) stars in M22. They investigated their F content, presenting also C, N, O, Na and Fe abundances.
Four of our six stars are in common with that study, namely, III-14, III-15, III-52 and IV-97. For stars III-14 and III-52, those authors inferred \[F/Fe\]=$-$0.40 and \[F/Fe\]=$-$0.20 dex, respectively, while we obtained \[F/Fe\]=$-$0.80 and \[F/Fe\]=$-$0.60 dex. The adopted EP value can easily justify this divergence, accounting for about 0.30 dex (see Section \[sec:analysis\]); the source of the remaining $\sim$0.1 dex can be ascribed to the continuum placement, which is critical in determining abundances from such a weak line. On the other hand, for stars III-15 and IV-97 [@abrito12] obtained \[F/Fe\]=0.28 and \[F/Fe\]=0.25 dex, entailing discrepancies with our estimates larger than a factor 10. Note that stellar parameters (T$_{\rm eff}$, log$g$, and $\xi$) are the same in both works, as they come from the analysis of MSK11; for the input metallicity, Alves-Brito et al. used instead their own values coming from the IR spectroscopy and showing an offset of +0.13 dex compared to the optical ones. We investigated the nature of this substantial discordance, and attributed it to the telluric feature subtraction. In the upper panel of Figure \[f:comp\_spectra\], we directly compare our spectrum for star III-15 (solid line) with that used by Alves-Brito et al. (dotted line). Their spectrum presents stronger features in the vicinity of HF line, features expected from telluric contribution. To completely remove the contamination, Alves-Brito et al. realized that they needed early-type star targets before and/or after each scientific frame but the logistics of their run made this very difficult. We instead could observe such targets: the correction to our data, in turn affects the placing of the continuum and removes many strong features (as shown in the lower panel of Figure \[f:comp\_spectra\]). As expected, such an effect is significant for the HF feature, due to its intrinsic weakness, but only marginally affects the C and Na abundance determinations (due to the strength of their lines), and hence most of the conclusions of that paper. This is shown in Figure \[f:comp\], where we plot our \[X/Fe\] ratio as a function of those from [@abrito12] for the four stars in common: C and Na are comparable between the two studies, with differences of $\Delta$\[C/Fe\]=+0.20$\pm$0.17 dex and $\Delta$\[Na/Fe\]=+0.18$\pm$0.11 dex (in the sense Alves-Brito’s study minus our values); if we take into account the offsets in \[Fe/H\] and in the adopted solar abundances (they assumed A(C)$_{\odot}$=8.42 and A(Na)$_{\odot}$=6.17), those values become $\Delta$\[C/Fe\]=+0.19$\pm$0.17 and $\Delta$\[Na/Fe\]=+0.15$\pm$0.11 dex. On the other hand, discrepancies in F are significant and cannot be recovered from the different EP and/or solar abundances, being for the whole sample of $\Delta$\[F/Fe\]=+0.75$\pm$0.19 dex.
![Upper panel: comparison of our spectrum for star III-15 (solid line) with the one by [@abrito12] (dotted line). Lower panel: superimposition of our spectra for the same sample star with and without the telluric line subtraction (dot-dashed and solid lines, respectively).[]{data-label="f:comp_spectra"}](telluric.eps){width="9cm"}
![Comparison for the four stars in common with Alves-Brito et al. (2012, here labeled as AB12). Symbols are as in Figures \[f:OF\], \[f:NaF\], \[f:NaIROpt\].[]{data-label="f:comp"}](ALAN_comp.eps){width="9cm"}
Discussion {#sec:discussion}
==========
Our main result is the detection of F variations across our sample significantly larger than the observational uncertainties. As shown in Figures \[f:OF\] and \[f:NaF\], the changes in the F abundances are correlated with O and anticorrelated with Na. This chemical pattern qualitatively matches the predictions from the multiple population scenario, according to which the stellar ejecta from which second generation stars formed carry the signature of hot H burning causing enhancements in Na (N and Al) and depletions in O and F (C and Mg). The F-O diagram presented in Figure \[f:gcs\] demonstrates that M22 shares a similar behaviour as M4 and NGC6712, the other two GCs for which F has been explored[^8].
Furthermore, the F-O-Na (anti) correlations can be marked separately within each of the two sub-groups enclosed in M22 as clearly illustrated in Figures \[f:OF\] and \[f:NaF\], where the $s$-process poor and the $s$-process rich stars are labeled with empty and filled symbols, respectively. The same conclusion was drawn by MSK11 when considering the Na-O and C-N planes.
![\[F/Fe\] versus \[O/Fe\] for M22 (this study), M4 (@smith05), and NGC 6712 (@yong+08).[]{data-label="f:gcs"}](GCs.eps){width="8cm"}
![\[F/H\] versus \[La/H\]. Symbols as in Figures \[f:OF\], \[f:NaF\], \[f:NaIROpt\], and \[f:comp\]. The dashed line is a least-squares fit to data points.[]{data-label="f:LaF"}](LaF.eps){width="8cm"}
Very interestingly, beyond the internal spread in F characterizing each sub-component, we measured an increase in the F content between the two different stellar generations in M22. [*The $s$-process-rich group has, on average, larger F abundances than the $s$-process-poor group.*]{} This is shown in Figure \[f:LaF\], where we plot our F abundances (\[F/H\]) as a function of \[La/H\] from MSK11. There is a positive correlation between the two ratios, suggesting that the polluters responsible for the $s$-process production must account for a simultaneous F production.
There are two classes of objects producing both $s$-process elements and fluorine. The first one is very massive stars (mass roughly $>$ 40[M$_{\odot}$]{}). These produce F in the initial phases of core He burning and expel it in the interstellar medium via winds during the Wolf-Rayet phase (@meynet00). They also produce $s$-process elements during core He and shell C burning (e.g., @pignatari10). Production of both F and $s$-process elements in these massive stars depends on the initial CNO abundances and thus decreases with the stellar metallicity. Inclusion of stellar rotation enhances the $s$-process production at low metallicity (@pignatari08; @chiappini11); however, it appears to decrease the production of fluorine (@palacios05). One problem already stressed by [@roederer11] when considering these stars is that there is no reason why the SNe that enriched the $s$-process rich group host the $weak$ component and those that polluted the $s$-process poor do not (see Roederer et al. for details).
The second class of objects producing both F and $s$-process elements are AGB stars (@forestini92; @jorissen92; @mowlavi98; @karakas03; @lugaro04, 2012). To get deeper insights into the nature of the candidate AGB stars possibly responsible for the observed abundance trends in M22 we compare our results with the recent set of models by [@lugaro12]. They presented AGB models for masses 0.9$-$6.0 [M$_{\odot}$]{} and metallicity three times lower than that of the cluster under consideration (i.e., \[Fe/H\]=$-$2.3 dex).
From our data we infer that there is an increase of $\Delta$\[F/H\]$^{s-rich}_{s-poor}$=+0.40$\pm$0.15 dex in the fluorine content between the two groups. This estimate was done by taking into account the F content of the O-rich stars only, because they do not show any depletion due to the HBB. We averaged the F abundances in stars IV-97 and III-14, both belonging to the $s$-poor group, finding a mean value $<$\[F/H\]$>$=$-$2.63$\pm$0.01 dex. Since we have only one O-rich star in the $s$-rich group, III-52, we chose it as representative of the F abundance for the group, that is \[F/H\]=$-$2.23$\pm$0.15 dex. The F increase of +0.40 dex is accompanied by a corresponding enhancement in La of $\Delta$\[La/H\]$^{s-rich}_{s-poor}$=0.56 $\pm$0.18 dex, since the values are $<$\[La/H\]$>$=$-$1.92$\pm$0.15 dex and \[La/H\]=$-$1.36$\pm$0.10 dex, respectively, for the two groups.
Comparing these values with the model predictions by [@lugaro12] we found that AGB stars with masses of $\approx$ 4$-$5 [M$_{\odot}$]{} can well reproduce the observed pattern. Lower-mass AGB models do not fit our observational requirements because they over-produce fluorine. This is true even if we consider that these AGB model predictions are roughly one dex too high to match the observation of carbon-enhanced metal-poor (CEMP) stars by [@lucatello11]. Production of F in the $\approx$ 4$-$5 [M$_{\odot}$]{} mass range depends on the delicate balance between the operation of the TDU and of HBB. These stars suffer HBB and destroy F during the early phases of their AGB evolution, however, toward the end of the evolution, as the mass of the convective envelope decreases, HBB ceases while the TDU is still active resulting in mild F enhancements at the stellar surface. This explains why while the most prolific AGB stars in terms of fluorine production have initial masses around 2 [M$_{\odot}$]{}, F production still occurs at slightly higher masses. On the other hand, more massive AGB models ($>$5 [M$_{\odot}$]{}) experience hotter HBB and thus more efficient F destruction as well as higher temperatures in the thermal pulses activating also $^{19}$F($\alpha$,p)$^{22}$Ne reactions. This, combined with fewer final TDU episodes when HBB has ceased means that they do not replenish F at the stellar surface.
Interestingly, the same conclusion is drawn by [@roederer11] by exploring the heavy-element ratios, \[$hs$/$ls$\] and \[Pb/$hs$\]. Comparing their abundances with models by [@roederer10], these authors deduced that the low mass AGBs ($\leq$3[M$_{\odot}$]{}) cannot account for the observed trend. More importantly, they concluded that a match to the $s$-process element abundances is provided by the 5 [M$_{\odot}$]{} AGB model, and this is confirmed by checking the models of [@lugaro12]. Furthermore, and very interestingly, these models predict Na and C production in agreement with the observations.
[*Our result provides a further, independent confirmation to this previous hint: indications from both light (fluorine here) and heavy elements converge toward the AGB stars of the same mass as the best candidate polluters, indicating that the age difference between the two sub-groups in M22 cannot be larger than a few hundreds Myr.*]{} It should be mentioned that by analyzing the double sub-giant branch (SGB) of this cluster, [@marino12] concluded that the age spread can be [*at most*]{} $\sim$300 Myr.
The fact that three independent studies, involving different and complementary techniques/approaches, produce the same result is encouraging. However, a comprehensive understanding of the whole picture is still missing. As also stressed by [@roederer11], if relatively massive AGBs (4-5 [M$_{\odot}$]{}) produced the $s$-process elements, and if these stars are also responsible for the observed $p$-capture element anticorrelations, then it is not explained that an $s$-process enrichment is present in M22 but is not associated with O and Na abundance anomalies, nor it is seen in any other GCs where the Na-O anticorrelations are observed.
On the other hand, it is clear from the observed F abundance trends that we must select slightly more massive AGBs, i.e., $\geq$ 6[M$_{\odot}$]{} if we wish to explain the light element variations in GCs, since this is the AGB mass range where F and O can be destroyed by HBB resulting in the observed F-O correlation. We might tentatively suggest that perhaps the production of the $s$-process elements in AGB models with initial mass $\geq$ 6[M$_{\odot}$]{} is less efficient than currently predicted. This could be the result of a stronger mass-loss rate or a less efficient TDU in this mass range than those employed in the models by [@lugaro12]. This possibility is well within model uncertainties and need to be investigated; in a forthcoming paper (D’Orazi et al., in preparation) we will attack these issues, presenting new observations and AGB models and discussing their strength/weakness in reproducing the observed abundance trends in GCs.
Alternatively, we may conclude that massive AGBs are not the inter-cluster polluters; however several lines of evidence point to those stars, such as the need for a Li production between first and second generation stars (as in the case of M4, @dm10; @mucciarelli11; @monaco12). Further efforts, from both observational and theoretical perspectives, are needed; in particular the lack of a complete set of models for AGB stars and massive rotating stars with different mass and metallicity, following the whole nucleosynthetic path from Li to Pb, still hampers a quantitatively robust comparison between theory and observations.
Summary and concluding remarks {#sec:summary}
==============================
We presented fluorine abundances for a sample of six RGB stars belonging to the metal-poor globular cluster M22. The sample was selected to include $s$-process-rich and $s$-process-poor stars, as defined in MSK11. In addition, within each of these cluster sub-components, we targeted both O-rich (Na-poor) and O-poor (Na-rich) stars.
We gathered evidence of the presence of a F-O correlation and of an F-Na anticorrelation. Such chemical pattern, notably revealed in each cluster sub-group, is in agreement with F destruction during the hot H burning: fluorine follows the same trend defined by O, Na, C, N, Mg, Al, as predicted by the multiple population scenario.
Most interestingly, we found that the $s$-process- (metal-) rich component is also characterized by a larger F content than the $s$-process (metal-) poor component. The comparison between our observations and AGB models points to stars with masses around 4-5 [M$_{\odot}$]{} as responsible for such a trend, corroborating previous hints by [@roederer11] and [@marino12], and confirming that the age spread across the two different stellar generations in M22 cannot be larger than a few hundreds Myr.
The publication made extensive use of the NASA ADS and SIMBAD databases. We kindly acknowledge B. Plez for having provided his unpublished line lists. This work was partially funded by PRIN INAF 2009 (“Formation and Early Evolution of Massive Star Clusers”) and PRIN INAF 2011 (“Multiple populations in globular clusters: their role in the Galaxy assembly”). V.D. thanks Kais Hamza for very useful discussions.
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[^1]: Based on observations taken with ESO telescopes under program 087.0319(A)
[^2]: We use the notation \[X/H\]=A(X)$-$A(X)$_{\odot}$, where A(X)=log($\frac{N_{X}}{N_H}$)+12
[^3]: In the Solar System, $\approx$ 97% of Eu was synthesized via the $r$-process (@burris00, and references therein).
[^4]: http://www.eso.org/sci/software/gasgano/
[^5]: IRAF is the Image Reduction and Analysis Facility, a general purpose software system for the reduction and analysis of astronomical data. IRAF is written and supported by National Optical Astronomy Observatories (NOAO) in Tucson, Arizona.
[^6]: Vienna Atomic Line Database (www.astro.uu.se/ vald/php/vald.php)
[^7]: We used as solar fluorine abundances A(F)$_{\odot}$=4.56 (@ag89; @asplund05).
[^8]: [@cunha03] presented F abundances for two giants in $\omega$ Cen. However, they provide an F measurement only for star ROA219, giving an upper limit for star ROA 324. Discussion related to the internal F variation in this peculiar GC is still not possible with the currently available measurements. For this reason we acquired CRIRES spectra of 12 $\omega$ Cen giants; results will be presented in a forthcoming paper (S. Lucatello et al., in preparation).
| {
"pile_set_name": "ArXiv"
} |
1.0 cm
\
.1cm [*$^{(a)}$Instituto de Física, Universidade de São Paulo,\
C. Postal 66318, 05314-970 São Paulo, SP, Brazil*]{}\
.3cm [*$^{(b)}$S. N. Bose National Centre for Basic Sciences,\
Block JD, Sector III, Salt Lake, Kolkata$-$700098, India*]{}\
.1cm [E-mails: [email protected]; [email protected]]{}
1.0 cm
[**Abstract:**]{} We derive the complete set of off-shell nilpotent ($s^2_{(a)b} = 0$) and absolutely anticommuting ($s_b s_{ab} + s_{ab} s_b = 0$) Becchi-Rouet-Stora-Tyutin (BRST) ($s_b$) as well as anti-BRST symmetry transformations ($s_{ab}$) corresponding to the combined Yang-Mills and non-Yang-Mills symmetries of the $(2 + 1)$-dimensional Jackiw-Pi model within the framework of augmented superfield formalism. The absolute anticommutativity of the (anti-)BRST symmetries is ensured by the existence of [*two*]{} sets of Curci-Ferrari (CF) type of conditions which emerge naturally in this formalism. The presence of CF conditions enables us to derive the coupled but equivalent Lagrangian densities. We also capture the (anti-)BRST invariance of the coupled Lagrangian densities in the superfield formalism. The derivation of the (anti-)BRST transformations of the auxiliary field $\rho$ is one of the key findings which can neither be generated by the nilpotent (anti-)BRST charges nor by the requirements of the nilpotency and/or absolute anticommutativity of the (anti-)BRST transformations. Finally, we provide a bird’s-eye view on the role of auxiliary field for various massive models and point out few striking similarities and some glaring differences among them.\
[ PACS numbers:]{} 11.15.-q, 03.70.+k, 11.10Kk, 12.90.+b\
[*Keywords*]{}: Jackiw-Pi model; augmented superfield formalism; Curci-Ferrari conditions; (anti-)BRST symmetry transformations; nilpotency and absolute anticommutativity\
Introduction
============
The co-existence of mass and gauge invariance [*together*]{} is still one of the main issues connected with the gauge theories, in spite of the astonishing success of the standard model of particle physics which is based on (non-)Abelian 1-form gauge theories. However, it is worthwhile to mention that, in the case of sufficiently strong vector couplings, the gauge invariance does not entail the masslessness of gauge particles [@Schwinger:1962tn; @Schwinger:1962tp]. Thus, it is needless to say that the mass generation in gauge theories is a crucial issue which has attracted a great deal of interest [@Deser:1981wh; @Deser:1982vy].
In the recent past, many models for the mass generation have been studied in the diverse dimensions of spacetime. In this context, mention can be made of about 4D topologically massive (non-)Abelian gauge theories, with $(B \wedge F)$ term, where 1-form gauge field acquires a mass in a natural fashion [@Freedman:1980us; @Allen:1990gb; @Harikumar:2001eb]. One of the key features associated with such models is that the 1-form gauge field gets a mass without taking any recourse to the Higgs mechanism. We have thoroughly investigated these models within the framework of Becchi-Rouet-Stora-Tyutin (BRST) as well as superfield formalism [@Gupta:2008he; @Gupta:2010xh; @Gupta:2009up; @Kumar:2011zi; @Krishna:2010dc; @Malik:2011pm]. It is interesting to point out that the main issues connected with the 4D Abelian topologically massive models are that they suffer from the problems connected with renormalizability when straightforwardly generalized to the non-Abelian case [@Henneaux:1997mf]. However, this issue can be circumvented by the introduction of extra field (see, e.g. [@Lahiri:1996dm; @Lahiri:1999uc]).
At this juncture, it is worth mentioning about the lower dimensional non-Abelian massive models, such as $(2 + 1)$-dimensional Jackiw-Pi (JP) model [@Jackiw:1997jga], which are free from the above mentioned issues. The silent features of JP model are as follows. First, it is parity conserving model due to the introduction of a 1-form vector field having odd parity. Second, mass and gauge invariance are respected together. Third, it is endowed with the two independent sets of local continuous symmetries, namely; the usual Yang-Mills (YM) symmetries and non-Yang-Mills (NYM) symmetries. Finally, it is free from the problems connected with the 4D topologically massive models. These features make JP model attractive and worth studying in detail.
The JP model has been explored in many different prospects such as constraint analysis and Hamiltonian formalism [@Dayi:1997in], establishment of Slavnov-Taylor identities and BRST symmetries [@DelCima:2011bx]. Furthermore, this model is also shown to be ultraviolet finite and renormalizable [@DelCima:2012bm]. We have applied superfield formalism and derived the full set of off-shell nilpotent and absolutely anticommuting BRST as well as anti-BRST symmetry transformations corresponding to the both YM and NYM symmetries of JP model [@Gupta:2011cta; @Gupta:2012ur]. Within the superfield formalism, we have been able to derive the [*proper*]{} (anti-)BRST transformations for the auxiliary field $\rho$ which can neither be deduced by the conventional means of nilpotency and/or absolute anticommutativity of (anti-)BRST symmetries nor generated by the conserved (anti-)BRST charges. At this stage, we would like to point out that the derivation of the proper anti-BRST symmetries have utmost importance because they play a fundamental role in the BRST formalism (see, e.g. [@Curci:1976ar; @Ojima:1980da; @Hwang:1989mn] for details). In fact, both the symmetries (i.e. BRST and anti-BRST) have been formulated in an independent way [@Hwang:1983sm].
Recently, the (anti-)BRST symmetries for perturbative quantum gravity in curved as well as complex spacetime, in linear as well as in non-linear gauges have been found [@mir1; @mir2] and a superspace formulation of higher derivative theories [@mir3], Chern-Simons and Yang-Mills theories on deformed superspace [@mir4; @mir5] within BV formalism have also been established. Moreover, the study of massless and massive fields with totally symmetric arbitrary spin in AdS space has been carried out in the framework of BRST formalism [@mets].
The main motivations behind our present investigation are as follows. First, the derivation of off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetry transformations corresponding to the combined YM and NYM symmetries of JP model. As, in our recent works (cf. [@Gupta:2011cta; @Gupta:2012ur]), we have already established the corresponding proper (anti-) BRST symmetry transformations, individually, for both the YM and NYM cases, within the framework of superfield formalism. Second, to establish the Curci-Ferrari (CF) conditions in the case of combined symmetries. These CF conditions are hallmark of any non-Abelian 1-form gauge theories [@Curci:1976ar] and have a close connection with gerbes [@Bonora:2007hw], within the framework of BRST formalism. Third, to procure appropriate coupled Lagrangian densities which respect the (anti-)BRST symmetries derived from augmented superfield approach. Finally, to point out the role of auxiliary field $\rho$, which is very special to this model (cf. [@Dayi:1997in; @Gupta:2011cta] for details).
This paper is organized in the following manner. In Section 2, we recapitulate the underlying symmetries of 3D JP model. We derive the off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetries corresponding to the combined YM and NYM symmetries of JP model, within the framework of superfield formalism, in Section 3. Section 4 contains the derivation of coupled Lagrangian densities that respect the preceding (anti-) BRST symmetries. The conservation of (anti-)BRST charges is shown in Section 5. We also discuss about the novel observations of our present study in this section. Section 6 is devoted for the discussions of ghost symmetries and BRST algebra. In Section 7, we provide a bird’s-eye view on the role of auxiliary field in the context of various massive models. Finally, in Section 8, we make some concluding remarks.
In Appendix A, we show the nilpotency and absolute anticommutativity of the (anti-) BRST charges within the framework of augmented superfield formalism. We also capture (anti-)BRST invariance of coupled Lagrangian densities in the superfield framework.
[*Conventions and notation:*]{} We adopt here the conventions and notation such that the 3D flat Minkowski metric $\eta_{\mu\nu} =$ diag $(+ 1, - 1, -1)$ and the 3D totally antisymmetric Levi-Civita tensor $\varepsilon_{\mu\nu\eta}$ satisfies $\varepsilon_{\mu\nu\eta}\,\varepsilon^{\mu\nu\eta}= - 3!,\;
\varepsilon_{\mu\nu\eta}\,\varepsilon^{\mu\nu\kappa}$ $= - 2! \delta^\kappa_\eta$, etc. with $\varepsilon_{012} = - \varepsilon^{012} = +1$. The Greek indices $\mu, \nu, \eta, ... = 0, 1, 2$ correspond to the 3D spacetime directions and Latin indices $i, j, k,... = 1,2$ correspond to the space directions only. The dot and cross product between two non-null vectors $P$ and $Q$ in the $SU(N)$ Lie algebraic space are defined as $P \cdot Q = P^a Q^a,\; P \times Q = f^{abc}\, P^a Q^b T^c$. The $SU(N)$ generators $T^a$ (with $a, b, c,... = N^2 - 1$) follow the commutation relation $[T^a,\, T^b] = i f^{abc}\, T^c$ where the structure constants $f^{abc}$ are chosen to be totally antisymmetric in $a,b,c$ for the semi-simple $SU(N)$ Lie algebra [@Wein].
Preliminaries: Jackiw-Pi model
==============================
We start off with the massive, non-Abelian, gauge invariant Jackiw-Pi model in $(2+ 1)$-dimensions of spacetime. The Lagrangian density of this model is given by [@Jackiw:1997jga; @Gupta:2011cta] $$\begin{aligned}
{\cal L}_0 &=& - \frac{1}{4}\, F_{\mu\nu}\cdot F^{\mu\nu}
- \frac{1}{4}\, \big(G_{\mu\nu} + g F_{\mu\nu} \times \rho\big) \cdot \big(G^{\mu\nu} + g F^{\mu\nu} \times \rho\big) \nonumber\\
&+& \frac{m}{2}\, \varepsilon^{\mu\nu\eta}\,F_{\mu\nu}\cdot \phi_\eta, \label{2.1}\end{aligned}$$ where the 2-form $F^{(2)} = d A^{(1)} + i g \big(A^{(1)} \wedge A^{(1)} \big)
= \frac{1}{2!}\,\big(dx^\mu \wedge dx^\nu \big)F_{\mu\nu}\cdot T$ defines the curvature tensor $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu - g(A_\mu \times A_\nu)$ for the non-Abelian 1-form \[$A^{(1)} = dx_\mu A^\mu \cdot T$\] gauge field $A_\mu = A_\mu \cdot T$ where $d = dx^\mu \partial_\mu$ is the exterior derivative (with $d^2 = 0$). Similarly, another 2-form $G^{(2)} = d\phi^{(1)} + i g \big(A^{(1)} \wedge \phi^{(1)}\big) + ig \big(\phi^{(1)} \wedge A^{(1)}\big)
= \frac{1}{2!}\,\big(dx^\mu \wedge dx^\nu\big)\,G_{\mu\nu} \cdot T$ defines the curvature tensor $G_{\mu\nu} = D_\mu \phi_\nu - D_\nu \phi_\mu$ corresponding[^1] to 1-form $[\phi^{(1)} = dx^\mu\phi_\mu \cdot T]$ vector field $\phi_\mu = \phi_\mu \cdot T$. In the above, the vector fields $A_\mu$ and $\phi_\mu$ have opposite parity thus the JP model becomes parity invariant, $\rho$ is an auxiliary field, $g$ is the coupling constant and $m$ defines the mass parameter.
Local gauge symmetries: YM and NYM
----------------------------------
The above Lagrangian density respects two sets of local and continuous gauge symmetry transformations, namely; YM gauge transformations $(\delta_1)$ and NYM gauge transformations $(\delta_2)$. These symmetry transformations are [@Gupta:2011cta; @Gupta:2012ur] $$\begin{aligned}
&&\delta_1 A_\mu = D_\mu \Lambda, \quad \delta_1 \phi_\mu = - g(\phi_\mu \times \Lambda),
\quad \delta_1 \rho = - g(\rho \times \Lambda),\nonumber\\
&& \delta_1 F_{\mu\nu} = - g(F_{\mu\nu} \times \Lambda),
\quad \delta_1 G_{\mu\nu} = - g(G_{\mu\nu} \times \Lambda), \label{2.2}\end{aligned}$$ $$\begin{aligned}
\delta_2 A_\mu = 0, \quad \delta_2 \phi_\mu = D_\mu \Sigma, \quad \delta_2 \rho = + \Sigma, \quad
\delta_2 F_{\mu\nu} = 0, \quad \delta_2 G_{\mu\nu} = - g(F_{\mu\nu} \times \Sigma), \label{2.3}\end{aligned}$$ where $\Lambda \equiv \Lambda \cdot T $ and $\Sigma \equiv \Sigma \cdot T$ are the $SU(N)$ valued local gauge parameters corresponding to the YM and NYM gauge transformations, respectively. Under the above local and infinitesimal gauge transformations the Lagrangian density (\[2.1\]) transforms as $$\begin{aligned}
&&\delta_1 {\cal L}_0 = 0, \qquad \delta_2 {\cal L}_0 = \partial_\mu \Big[\frac{m}{2}\, \varepsilon^{\mu\nu\eta}\, F_{\nu\eta}\cdot \Sigma\Big]. \end{aligned}$$ As a consequence, the action integral $S = \int d^3x {\cal L}_0$ remains invariant under both the gauge transformations ($\delta_1$ and $\delta_2$) for the physically well-defined fields which vanish off rapidly at infinity. We would like to point out that in order to maintain the NYM symmetry, we have to have the auxiliary field $\rho$ in the theory (cf. Section 7 for details).
Combined gauge symmetry
-----------------------
In the above, we have seen that both the YM and NYM transformations are the symmetries of the theory. Thus, the combination of the above symmetries \[i.e. $(\delta = \delta_1 + \delta_2$)\] would also be the symmetry of theory. Under the combined gauge transformation $\delta$, namely; $$\begin{aligned}
&&\delta A_\mu = D_\mu \Lambda, \qquad \delta \phi_\mu = D_\mu \Sigma - g(\phi_\mu \times \Lambda), \qquad
\delta \rho = \Sigma - g(\rho \times \Lambda),\nonumber\\
&& \delta F_{\mu\nu} = - g(F_{\mu\nu} \times \Lambda), \qquad \delta G_{\mu\nu} = - g(G_{\mu\nu} \times \Lambda)
-g (F_{\mu\nu} \times \Sigma), \label{2.5}\end{aligned}$$ the Lagrangian density (\[2.1\]) remains quasi-invariant. To be more specific, the Lagrangian density transforms to a total spacetime derivative $$\begin{aligned}
\delta {\cal L}_0 = \partial_\mu \Big[\frac{m}{2}\, \varepsilon^{\mu\nu\eta}\, F_{\nu\eta}\cdot \Sigma\Big]. \end{aligned}$$ Thus, the action integral remains invariant (i.e. $\delta S = \delta \int d^3x {\cal L}_0 = 0$) under the combined symmetry ($\delta$), too.
(Augmented) superfield approach
===============================
We apply Bonora-Tonin’s (BT) superfield approach to the BRST formalism [@Bonora:1980pt; @Bonora:1980ar], to derive the off-shell nilpotent and absolutely anticommuting (anti-) BRST symmetry transformations for the 1-form gauge field $A_\mu$ and corresponding (anti-)ghost fields $(\bar C)C$.
(Anti-)BRST symmetries: Gauge and (anti-)ghost fields
-----------------------------------------------------
For this purpose, we generalize 1-form connection $A^{(1)}$ (and corresponding 2-form curvature $F^{(2)}$) and exterior derivative $d$ onto the $(3,2)$-dimensional supermanifold, as $$\begin{aligned}
d \to \tilde d &=& dZ^M\partial_M = dx^\mu\,\partial_\mu + d\theta\, \partial_\theta + d\bar\theta \,\partial_{\bar\theta},
\qquad \tilde d^2 = 0,\nonumber\\
A^{(1)} \to \tilde{\cal A}^{(1)} &=& dZ^M \tilde {\cal A}_M = dx^\mu\,\tilde {\cal A}_\mu(x, \theta, \bar\theta)
+ d\theta\, {\tilde {\bar{ \cal F}}} (x, \theta, \bar\theta)
+ d \bar\theta\, {\tilde{{\cal F}}} (x, \theta, \bar\theta), \nonumber\\
F^{(2)} \to \tilde {\cal F}^{(2)} &=& \frac{1}{2!}\,(dx^M \wedge dx^N)\,\tilde {\cal F}_{MN} = \tilde d \tilde {\cal A}^{(1)}
+ i g \big(\tilde {\cal A}^{(1)} \wedge \tilde {\cal A}^{(1)} \big) ,\end{aligned}$$ where $Z^M = (x^\mu, \theta, \bar \theta)$ are superspace coordinates characterizing the $(3, 2)$-dimensional supermanifold. In the above expression, $\theta$ and $\bar \theta$ are the Grassmannian variables (with $\theta^2 = \bar \theta^2 = 0,
\theta \bar \theta + \bar \theta \theta = 0$) and $\partial_\theta, \partial_{\bar\theta}$ are corresponding Grassmannian derivatives. We also generalize 3D gauge field \[$A_\mu (x)$\] and (anti-)ghost fields $[(\bar C)C(x)]$ of the theory to their corresponding superfields onto the $(3,2)$-dimensional supermanifold.
Now, these superfields can be expanded along the Grassmannian directions, in terms of the basic fields ($A_\mu, C, \bar C$) and secondary fields ($R_\mu$, $\bar R_\mu$, $S_\mu$, $B_1$, $B_2$, $\bar B_1$, $\bar B_2$, $s$, $\bar s$), in the following manner, $$\begin{aligned}
\tilde {\cal A}_{\mu} (x, \theta, \bar\theta) &=& A_\mu (x) + \theta\, \bar R_\mu (x) + \bar \theta\, R_\mu (x)
+ i \,\theta \,\bar\theta \, S_\mu (x), \nonumber\\
\tilde {\cal F} (x, \theta, \bar\theta) &=& C (x) + i\,\theta\, \bar B_1 (x) + i\,\bar \theta\, B_1 (x)
+ i \,\theta\, \bar\theta \, s (x), \nonumber\\
{\tilde {\bar {\cal F}}} (x, \theta, \bar\theta) &=& \bar C (x) + i\,\theta\, \bar B_2 (x) + i\,\bar \theta\;
B_2 (x) + i \,\theta \,\bar\theta \; \bar s (x). \label{3.2}\end{aligned}$$ Here $ \tilde {\cal A}_{\mu} (x, \theta, \bar\theta), \tilde {\cal F} (x, \theta, \bar\theta), {\tilde{\bar {\cal F}}} (x, \theta, \bar\theta) $ are superfields corresponding to the basic fields $A_\mu (x),$ $C (x)$ and $\bar C (x)$, respectively. Now these secondary fields, in the above expression, can be determined in terms of the basic and auxiliary fields of the underlying theory through the application of horizontality condition (HC) (cf. [@Bonora:1980pt; @Bonora:1980ar] for details). This HC can be mathematically expressed in the following fashion $$\begin{aligned}
d\,A^{(1)} + i \,g \big(A^{(1)} \wedge A^{(1)}\big)= \tilde d \,\tilde{\cal A}^{(1)}
+ i \,g\big(\tilde{\cal A}^{(1)} \wedge \tilde{\cal A}^{(1)}\big) \Longleftrightarrow F^{(2)} = \tilde {\cal F}^{(2)}. \label{3.3}\end{aligned}$$ Exploiting the above HC, we obtain the following relationships among the basic, auxiliary and secondary fields of the theory $$\begin{aligned}
&& R_\mu = D_\mu C, \quad \bar R_\mu = D_\mu \bar C, \quad B_1 = - \frac{i}{2}\,g\,(C \times C),\quad
\bar B_2 = - \frac{i}{2}\, g\,(\bar C \times \bar C), \nonumber\\
&& B + \bar B = -\,i\, g\,(C \times \bar C),\quad
s = -\,g\,(\bar B \times C), \quad \bar s = +\, g\,(B \times \bar C), \nonumber\\
&& S_\mu = D_\mu B \,+ \,i\, g \,(D_\mu C \times \bar C) \equiv - \,D_\mu \bar B - \,i\, g\,(D_\mu \bar C \times C), \label{3.4}\end{aligned}$$ where we have chosen $\bar B_1 = \bar B$ and $B_2 = B$.
Substituting the relationships (\[3.4\]) into the super-expansion of superfields in (\[3.2\]), we procure following explicit expansions $$\begin{aligned}
\tilde {\cal A}^{(h)}_{\mu} (x, \theta, \bar\theta) &=& A_\mu (x) + \theta D_\mu \bar C (x)
+ \bar \theta D_\mu C (x)+ \theta \,\bar\theta \, \big[i D_\mu B - g(D_\mu C \times \bar C)\big](x) \nonumber\\
&\equiv& A_\mu (x) + \theta \big(s_{ab}\, A_\mu (x)\big) + \bar \theta \big(s_b \,A_\mu (x)\big)
+ \theta \bar\theta \big(s_b\, s_{ab} \,A_\mu (x)\big), \nonumber\\
\tilde {\cal F}^{(h)} (x, \theta, \bar\theta) &=& C (x) + \theta \big(i \bar B (x)\big) + \bar \theta \Bigl
[\frac{g}{2}\, (C \times C) \Bigr](x) + \theta \bar\theta \big[-i g\big(\bar B \times C)\big](x) \nonumber\\
&\equiv& C (x) + \theta \big(s_{ab} C (x)\big) + \bar \theta \big(s_b C (x)\big)
+ \theta \bar\theta \big(s_b s_{ab} \,C (x)\big),\nonumber\\
{\tilde {\bar {\cal F}}}^{(h)} (x, \theta, \bar\theta) &=& \bar C (x) +
\theta \Bigl [\frac{g}{2} \,(\bar C \times \bar C)\Bigr](x) + \bar \theta \big(i B (x)\big)
+ \theta \bar\theta \big[i g(B \times \bar C)\big](x) \nonumber\\
&\equiv& \bar C (x) + \theta \big(s_{ab} \bar C (x)\big) + \bar \theta\, \big(s_b \bar C (x)\big)
+ \theta \bar\theta \big(s_b \,s_{ab}\, \bar C (x)\big). \label{3.5}\end{aligned}$$ In the above, the superscript $(h)$ on the superfields denotes the super-expansion of the superfields obtained after the application of HC (\[3.3\]). Thus, from the above expressions, we can easily identify the (anti-)BRST symmetry transformations corresponding to the gauge field $A_\mu$ and (anti-)ghost fields $(\bar C)C$. These transformations are explicitly listed below $$\begin{aligned}
&& s_b A_\mu = D_\mu C, \qquad s_b C = \frac{g}{2}\, \big(C \times C\big),
\qquad s_b \bar B = -\, g\,\big(\bar B \times C\big),\nonumber\\
&& s_b \bar C = i\, B,\qquad s_b B = 0, \label{3.6}\end{aligned}$$ $$\begin{aligned}
&&s_{ab} A_\mu = D_\mu \bar C, \qquad s_{ab} \bar C = \frac {g}{2} \,\big(\bar C \times \bar C\big),
\qquad s_{ab} B = - g\,\big(B \times \bar C\big), \nonumber\\
&& s_{ab} C = i \,\bar B,\qquad s_{ab} \bar B = 0. \label{3.7}\end{aligned}$$ We point out that, the (anti-)BRST symmetry transformations for the Nakanishi-Lautrup auxiliary fields $B$ and $\bar B$ have been derived with the help of absolute anticommutativity and nilpotency properties of the above (anti-)BRST symmetries.
(Anti-)BRST symmetries for $\phi_\mu$, $\beta$ and $\bar \beta$
---------------------------------------------------------------
In the previous subsection, we applied BT superfield approach to derive the off-shell nilpotent and absolutely anti-commuting (anti-)BRST symmetry transformations for the gauge field $(A_\mu)$ and corresponding (anti-)ghost fields $(\bar C)C$. Now, in order to derive the proper (anti-)BRST symmetries for the vector field $(\phi_\mu)$, corresponding (anti-)ghost fields $[(\bar \beta) \beta] $ and auxiliary field $(\rho)$, we have to go beyond the BT approach. For this purpose, we have exploited the power and strength of augmented superfield approach.
To derive the (anti-)BRST symmetries for the vector field $(\phi_\mu)$ and corresponding (anti-) ghost fields $[(\bar \beta) \beta]$, we invoke the following HC $$\begin{aligned}
\tilde{\cal G}^{(2)} + \tilde {\mathscr{F}}^{(2)} \equiv G^{(2)} + {\mathscr{F}}^{(2)}, \label{hc}\end{aligned}$$ where $G^{(2)}$, ${\mathscr{F}}^{(2)}$ are define in the following fashion $$\begin{aligned}
G^{(2)} &=&d\phi^{(1)} + i g\big(A^{(1)} \wedge \phi^{(1)}\big) + ig \big(\phi^{(1)} \wedge A^{(1)}\big)
= \frac{1}{2!}\,\big(dx^\mu \wedge dx^\nu\big)\,G_{\mu\nu},\nonumber\\
\mathscr{F}^{(2)} &=& -ig\big(F^{(2)} \wedge \rho^{(0)}\big) + ig\big(\rho^{(0)} \wedge F^{(2)}\big)
= \frac{g}{2!}\,\big(dx^\mu \wedge dx^\nu\big)(F_{\mu\nu} \times \rho),\end{aligned}$$ and $\tilde{\cal G}^{(2)}$, $\tilde {\mathscr{F}}^{(2)}$ are the generalizations of $G^{(2)}$, ${\mathscr{F}}^{(2)}$ onto the superspace, respectively, which can be explicitly represented in the following manner $$\begin{aligned}
\tilde{\cal G}^{(2)} &=& \tilde d \tilde \Phi^{(1)}
+ i\, g\, \big(\tilde {\cal A}^{(1)}_{(h)} \wedge \tilde \Phi^{(1)}\big)
+ i\, g\, \big(\tilde \Phi^{(1)} \wedge \tilde {\cal A}^{(1)}_{(h)}\big), \nonumber\\
\tilde {\mathscr{F}}^{(2)} &=& - i\, g\, \big(\tilde {\cal F}^{(2)}_{(h)} \wedge \tilde \rho^{(0)} \big)
+ i\, g\, \big(\tilde \rho^{(0)} \wedge \tilde {\cal F}^{(2)}_{(h)}\big).\end{aligned}$$ In the above expression, the quantities $\tilde {\cal A}^{(1)}_{(h)}, \tilde \Phi^{(1)}$ and $\tilde \rho^{(0)}$ are given as $$\begin{aligned}
\tilde {\cal A}^{(1)}_{(h)} (x, \theta, \bar \theta) &=& dx^\mu \,\tilde {\cal A}^{(h)}_\mu(x, \theta, \bar \theta)
+ d\theta \,{\tilde {\bar {\cal F}}}^{(h)}(x, \theta, \bar \theta)
+ d \bar \theta\, \tilde {\cal F}^{(h)}(x, \theta, \bar \theta), \nonumber\\
\tilde \Phi^{(1)} (x, \theta, \bar \theta) &=& dx^\mu\, \tilde\Phi_\mu(x, \theta, \bar\theta)
+ d \theta \; \tilde {\bar \beta}(x, \theta, \bar\theta)
+ d \bar \theta \; \tilde\beta(x, \theta, \bar\theta),\nonumber\\
\tilde \rho^{(0)} (x, \theta, \bar \theta) &=& \tilde \rho(x, \theta, \bar\theta), \end{aligned}$$ where the sub/super script $(h)$ denotes the quantities obtained after the application of HC. The superfields in the above expression, corresponding to the basic fields $\phi_\mu, \beta, \bar\beta$ and $\rho$ of the theory, can be expanded in terms of the secondary fields, as follows $$\begin{aligned}
\tilde \Phi_\mu(x, \theta, \bar\theta) &=& \phi_\mu(x) + \theta\, \bar P_\mu(x) + \bar \theta\, P_\mu(x)
+ i\,\theta\,\bar\theta\, Q_\mu(x),\nonumber\\
\tilde \beta(x, \theta, \bar\theta) &=& \beta(x) + i\, \theta\, \bar R_1(x) + i\, \bar \theta\, R_1(x)
+ i\,\theta\,\bar\theta\, s_1(x),\nonumber\\
\tilde{\bar \beta}(x, \theta, \bar\theta) &=& \bar \beta(x) + i\, \theta\, \bar R_2(x) + i\, \bar \theta\, R_2(x)
+ i\,\theta\,\bar\theta\, s_2(x),\nonumber\\
\tilde \rho (x, \theta, \bar\theta) &=& \rho(x) + \theta\, \bar b(x) + \bar \theta\, b(x)
+ i\,\theta\,\bar\theta\, q(x), \label{4.5}\end{aligned}$$ where $P_\mu, \bar P_\mu, b, \bar b, s_1, s_2$ are fermionic secondary fields and $R_1, \bar R_1, R_2, \bar R_2, Q_\mu, q$ are bosonic in nature.
Exploiting the above HC (\[hc\]) which demands that the coefficients of wedge products $(dx^\mu \wedge d \theta), \, (dx^\mu \wedge d \bar\theta),\, (d \theta \wedge d \theta),\,
(d \bar \theta \wedge d \bar \theta),\, (d \theta \wedge d \bar \theta)$ set equal to zero. We get following expressions: $$\begin{aligned}
&& \tilde {\cal D}_\mu \tilde {\bar \beta} - \partial_\theta \tilde \Phi_\mu
- g\,\Big(\tilde \Phi_\mu \times {\tilde {\bar {\cal F}}}^{(h)}\Big) = 0, \qquad
\partial_\theta \tilde {\bar \beta} - g\,\Big({\tilde {\bar {\cal F}}}^{(h)} \times \tilde {\bar \beta}\Big) = 0,\nonumber\\
&&\tilde {\cal D}_\mu \tilde \beta - \partial_{\bar \theta} \tilde \Phi_\mu
- g\,\Big(\tilde \Phi_\mu \times {\tilde {\cal F}}^{(h)}\Big) = 0,\qquad
\partial_{\bar \theta} \tilde \beta - g\,\Big({\tilde {\cal F}}^{(h)} \times \tilde \beta\Big) = 0,\nonumber\\
&& \partial_\theta \tilde \beta + \partial_{\bar \theta} \tilde {\bar \beta}
- g\,\Big({\tilde {\bar {\cal F}}}^{(h)} \times \tilde \beta\Big)
- g\,\Big(\tilde{\cal F}^{(h)} \times \tilde {\bar \beta}\Big) =0, \label{4.6}\end{aligned}$$ where $\tilde {\cal D}_\mu \bullet = \partial_\mu \bullet - g\big(\tilde {\cal A}^{(h)}_\mu \times \bullet \big)$. Using the expansion (\[4.5\]) in (\[4.6\]), we get following relationships amongst the basic and secondary fields of the theory, namely; $$\begin{aligned}
R_1 &=& - i\,g (C \times \beta), \quad \bar R_2 = - i\,g (\bar C \times \bar \beta), \quad
s_1 = - g (\bar B \times \beta) + g (C \times \bar R),\nonumber\\
s_2 &=& g\,(B \times \bar \beta) - g\,(\bar C \times R),
\quad R + \bar R + i\, g(C \times \bar \beta) + i \,g (\bar C \times \beta)= 0, \nonumber\\
P_\mu &=& D_\mu \beta - g (\phi_\mu \times C), \quad
D_\mu \bar R_2 + i\, g (D_\mu \bar C \times \bar \beta) + i\, g (D_\mu \bar \beta \times \bar C) = 0, \nonumber\\
\bar P_\mu &=& D_\mu \bar \beta - g (\phi_\mu \times \bar C),
\quad D_\mu R_1 + i \,g (D_\mu C \times \beta) + i \,g (D_\mu \beta \times C) = 0,\nonumber\\
Q_\mu &=& D_\mu R + g (B \times \phi_\mu) + i\, g(D_\mu C \times \bar \beta)
+ i \,g[D_\mu \beta \times \bar C - g(\phi_\mu \times C)\times \bar C]\nonumber\\
&\equiv& - D_\mu \bar R - g (\bar B \times \phi_\mu) - i\, g(D_\mu \bar C \times \beta)
- i \,g[D_\mu \bar \beta \times C - g(\phi_\mu \times \bar C)\times C], \nonumber\\ \label{3.14}\end{aligned}$$ where we have chosen $\bar R_1 = \bar R$, $R_2 = R$. Substituting, these values of secondary fields in (\[4.5\]), we have following form of superfield expansions $$\begin{aligned}
{\tilde \Phi_\mu}^{(h)} (x, \theta, \bar \theta) &=& \phi_\mu(x) + \theta \big[D_\mu \bar \beta - g\,(\phi_\mu \times \bar C)\big](x)
+ \bar \theta \big[D_\mu \beta - g\,(\phi_\mu \times C)\big](x)\nonumber\\
&+& \theta\, \bar \theta\,\big[i \,D_\mu R - i\, g\, (\phi_\mu \times B) - g\,(D_\mu C \times \bar \beta) - g\,(D_\mu \beta \times \bar C) \nonumber\\
&+& g^2\,(\phi_\mu \times C)\times \bar C \big](x) \nonumber\\
&\equiv& \phi(x) + \theta \,\big(s_b \,\phi(x)\big) + \bar \theta \,\big(s_{ab} \,\phi (x)\big)
+ \theta\,\bar\theta \,\big(s_b \,s_{ab}\,\phi(x)\big),\nonumber\\
{\tilde \beta}^{(h)} (x, \theta, \bar \theta) &=& \beta (x) + \theta \,\big[i\, \bar R (x)\big]
+ \bar \theta\, \big[g\, (C \times \beta)\big](x) \nonumber\\
&+& \theta\, \bar \theta\,\big[- i\, g\,(\bar B \times \beta) - i\, g\, (\bar R \times C) \big](x) \nonumber\\
&\equiv& \beta(x) + \theta \,\big(s_b \,\beta(x)\big) + \bar \theta \,\big(s_{ab} \,\beta (x)\big)
+ \theta\,\bar\theta \,\big(s_b \,s_{ab}\,\beta(x)\big),\nonumber\\
{\tilde {\bar \beta}}^{(h)} (x, \theta, \bar \theta) &=& \bar \beta (x) + \theta\, \big[g \,(\bar C \times \bar \beta)\big](x)
+ \bar \theta \,\big[i\, R(x)\big] \nonumber\\
&+& \theta \,\bar \theta\,\big[i \,g \,(B \times \bar \beta) + i\, g \,(R \times \bar C)\big](x) \nonumber\\
&\equiv& \bar \beta(x) + \theta \,\big(s_b \,\bar \beta(x)\big) + \bar \theta \,\big(s_{ab} \,\bar \beta (x)\big)
+ \theta\,\bar\theta \,\big(s_b \,s_{ab}\,\bar \beta(x)\big), \label{3.15}\end{aligned}$$ here $(h)$ on the superscript of superfields represents the respective quantities obtained after the application of HC (\[hc\]). Therefore, (anti-)BRST symmetry transformations for vector field $(\phi_\mu)$ and (anti-)ghost fields $[(\bar \beta)\beta]$ are obvious from the above super-expansions.
(Anti-)BRST symmetries for auxiliary field $\rho$
-------------------------------------------------
In order to derive the proper (anti-)BRST symmetry transformations for the auxiliary field $\rho$, we look for a quantity which remains invariant (or should transform covariantly) under the combined gauge transformations (\[2.5\]). Such gauge invariant quantity will serve a purpose of ‘physical quantity’ (in some sense) which could be generalized onto the $(3,2)$-dimensional supermanifold. Furthermore, being a ‘physical quantity’ it should remain unaffected by the presence of Grassmannian variables when the former is generalized onto the supermanifold. Thus, keeping above in mind, we note that under the combined gauge transformations (\[2.5\]), the quantity $(D_\mu \rho - \phi_\mu)$ transforms covariantly (as the quantities $F_{\mu\nu}$ and $G_{\mu\nu} + g (F_{\mu\nu} \times \rho)$ do). This can be explicitly checked as follows $$\begin{aligned}
\delta(D_\mu \rho - \phi_\mu) = - g \,(D_\mu \rho - \phi_\mu)\times \Lambda.\end{aligned}$$ Therefore, the above quantity serves our purpose and it can also be expressed in the language of differential forms as follows $$\begin{aligned}
d \rho^{(0)} + i\, g\, \big(A^{(1)} \wedge \rho^{(0)}\big) - i\, g\, \big(\rho^{(0)} \wedge A^{(1)}\big) - \phi^{(1)}
&=& dx^\mu \big(D_\mu \rho - \phi_\mu \big), \end{aligned}$$ which is clearly a 1-form object. Now, we generalize this 1-form object onto the $(3, 2)$-dimensional supermanifold and demand that it should remain unaffected by the presence of Grassmannian variables. This, in turn, produces the following HC $$\begin{aligned}
d \rho^{(0)} + i\, g \big(A^{(1)} \wedge \rho^{(0)}\big) - i\, g \big(\rho^{(0)} \wedge A^{(1)}\big) - \phi^{(1)} &\equiv&
\tilde d \tilde \rho^{(0)} + i\, g \big(\tilde {\cal A}^{(1)}_{(h)} \wedge \tilde \rho^{(0)}\big) \nonumber\\
&-& i \,g \big(\tilde \rho^{(0)} \wedge \tilde {\cal A}^{(1)}_{(h)}\big) - \tilde \Phi^{(1)}_{(h)}. \qquad\label{gir}\end{aligned}$$ This HC can also be derived from the integrability of (\[hc\]) (see e.g., [@ThierryMieg:1982un] for details on the topic). The r.h.s. of the above HC can be simplified as $$\begin{aligned}
&& \tilde d \tilde \rho^{(0)} + i g \big(\tilde {\cal A}^{(1)}_{(h)} \wedge \tilde \rho^{(0)}\big)
- i g \big(\tilde \rho^{(0)} \wedge \tilde {\cal A}^{(1)}_{(h)} \big) - \tilde \Phi^{(1)}_{(h)} = \nonumber\\
&& dx^\mu\Big[{\tilde {\cal D}_\mu} \tilde \rho - \tilde \Phi_\mu^{(h)}\Big]
+ d\theta \Big[\partial_\theta \tilde \rho - \tilde{\bar \beta}^{(h)}
- g \Big({\tilde {\bar {\cal F}}}^{(h)} \times \tilde \rho\Big)\Big]
+ d\bar \theta \Big[\partial_{\bar \theta} \tilde \rho - \tilde \beta^{(h)}
- g \Big(\tilde {\cal F}^{(h)} \times \tilde \rho\Big)\Big]. \qquad \;\end{aligned}$$ Exploiting (\[gir\]), and set the coefficients of $d\theta, d\bar \theta$ equal to zero, we have the following relationships, namely; $$\begin{aligned}
\partial_\theta \tilde \rho - \tilde{\bar \beta}^{(h)} - g\,\Big({\tilde {\bar {\cal F}}}^{(h)} \times \tilde \rho\Big) =0,\qquad
\partial_{\bar \theta} \tilde \rho - \tilde \beta^{(h)} - g\,\Big(\tilde {\cal F}^{(h)} \times \tilde \rho\Big) =0.\end{aligned}$$ Plugging the values of superfield expansions from (\[3.5\]), (\[4.5\]) and (\[3.15\]) into the above expressions, we get the following relationships amongst the basic and secondary fields $$\begin{aligned}
b &=& \beta - g(\rho \times C), \qquad \bar b = \bar \beta - g(\rho \times \bar C),\nonumber\\
q &=& R + g(B \times \rho) + i g (\bar C \times \beta) - i g^2(\rho \times C)\times \bar C\nonumber\\
&\equiv& - \bar R - i g (C \times \bar \beta) - g (\bar B \times \rho) + i g^2 (\rho \times \bar C)\times C.\end{aligned}$$ We point out that, however, there also exist other relationships but they are same as quoted in equation (\[3.14\]).
Finally, substituting these values of secondary fields into (\[4.5\]), we obtain the following superfield expansion for the super-auxiliary field $\tilde \rho (x, \theta, \bar\theta)$ $$\begin{aligned}
\tilde \rho^{(h)} (x, \theta, \bar \theta) &=& \rho (x) + \theta \,\big[\bar \beta - g\,(\rho \times \bar C)\big](x)
+ \bar \theta \,\big[\beta - g\,(\rho \times C)\big](x) \nonumber\\
&+ & i \theta\, \bar \theta\, \big[R + g\,(B \times \rho) + i\, g\, (\bar C \times \beta)
- i \,g^2\,(\rho \times C)\times \bar C\big](x), \nonumber\\
&\equiv& \rho(x) + \theta \,\big(s_b \,\rho(x)\big) + \bar \theta \,\big(s_{ab} \,\rho (x)\big)
+ \theta\,\bar\theta \,\big(s_b \,s_{ab}\,\rho(x)\big), \label{3.22}\end{aligned}$$ where $(h)$ as the superscript on the generic superfield denotes the corresponding superfield expansion obtained after the application of HC (\[gir\]). The (anti-) BRST symmetry transformations for the auxiliary field $\rho$ can be easily deduced from the above expansion. Thus, we have derived the proper (anti-)BRST symmetry transformations for the vector field $(\phi_\mu)$, corresponding (anti-)ghost fields $[(\bar \beta)\beta]$ and auxiliary field $(\rho)$ within the framework of augmented superfield formalism. Moreover, the (anti-)BRST symmetry transformations for the Nakanishi-Lautrup auxiliary fields $R$ and $\bar R$ have been derived with the help of anticommutativity and nilpotency properties of the (anti-)BRST symmetries. These symmetry transformations are listed below $$\begin{aligned}
&& s_b \phi_\mu = D_\mu \beta - g \big(\phi_\mu \times C\big), \quad s_b \beta = g \big(C \times \beta\big),
\quad s_b \rho = \beta -\, g \big(\rho \times C\big), \nonumber\\
&& s_b \bar \beta = i\, R, \quad s_b R = 0, \quad s_b \bar R = - g \big(\bar R \times C\big)
- g \big(\bar B \times \beta\big), \label{3.23}\end{aligned}$$ $$\begin{aligned}
&&s_{ab} \phi_\mu = D_\mu \bar \beta - g \big(\phi_\mu \times \bar C\big),
\quad s_{ab} \bar \beta = g \big(\bar C \times \bar \beta\big),
\quad s_{ab} \rho = \bar \beta - g \big(\rho \times \bar C\big),\nonumber\\
&&s_{ab} \beta = i \,\bar R, \quad s_{ab} \bar R = 0, \quad
\quad s_{ab} R = - g \big(R \times \bar C\big) - g \big(B \times \bar \beta\big). \label{3.24}\end{aligned}$$ These (anti-)BRST symmetry transformations as well as the transformations listed in (\[3.6\]) and (\[3.7\]) are off-shell nilpotent $(s_{(a)b}^2 \Psi = 0)$ and absolutely anticommuting $[(s_b s_{ab} + s_{ab} s_b) \Psi = 0]$ in nature. Here $\Psi$ represents any generic field of the theory. These properties (i.e. nilpotency and anticommutativity) are two key ingredients of the BRST formalism. The anticommutativity property for the vector fields ($\phi_\mu$ and $A_\mu$) and auxiliary field ($\rho$) is satisfied only on the constrained surface parametrized by the CF conditions (cf. (\[3.26\]) below). For instance, one can check that $$\begin{aligned}
&&\{s_b,\, s_{ab}\} A_\mu = iD_\mu \big[B + \bar B + i (C \times \bar C)],\nonumber\\
&& \{s_b, \,s_{ab}\} \phi_\mu = iD_\mu\big[R + \bar R + i g(C \times \bar \beta) + i g (\bar C \times \beta) \big]
+ i g \big[B + \bar B + i g (C \times \bar C)\big] \times \phi_\mu, \nonumber\\
&& \{s_b, \,s_{ab}\} \rho = i\big[R + \bar R + i g(C \times \bar \beta) + i g (\bar C \times \beta) \big]
+ i g \big[B + \bar B + i g (C \times \bar C)\big] \times \rho, \end{aligned}$$ whereas, for all the [*rest*]{} of the fields (of our present 3D JP model), the absolute anticommutativity property (i.e. $ \{s_b, s_{ab} \} \Psi = 0$) is valid [*without*]{} invoking the CF type conditions.
Before, we wrap up this section, some crucial points are in order. First and foremost, a very careful look at (\[3.4\]) and (\[3.14\]) reveals, respectively, the existence of two sets of Curci-Ferrari (CF) type conditions, namely; $$\begin{aligned}
&(i)& B + \bar B + i g (C \times \bar C) = 0, \nonumber\\
&(ii)& R + \bar R + i \,g\,\big(C \times \bar \beta\big) + i\, g\,\big(\bar C \times \beta\big) = 0. \label{3.26}\end{aligned}$$ These conditions are key signatures of any $p$-form gauge theory when the latter is discussed within the framework of BRST formalism. In our case, the above mentioned CF conditions emerge very naturally within the framework of (augmented) superfield formalism. In fact, CF conditions $(i)$ and $(ii)$ emerge from the HC (\[3.3\]) and (\[hc\]), respectively, when we set the coefficients of $(d\theta \wedge d \bar \theta)$ equal to zero. Second, the absolute anticommutativity of (anti-)BRST symmetries is ensured by these CF type conditions. Third, these CF type conditions are (anti-)BRST invariant. Finally, these CF type conditions play a crucial role in the derivation of the coupled (but equivalent) Lagrangian densities. We have discussed this aspect, in detail, in our next section.
Coupled Lagrangian densities
============================
In this section, we construct the coupled (but equivalent) Lagrangian densities which respect nilpotent as well as anticommuting (anti-)BRST symmetry transformations derived in the previous section (cf. Section 3). In order to proceed further, a few important points are in order. First, the mass dimensions (in natural units $c = \hbar =1$) of the various fields in our present 3D theory are: $[A_\mu] = [\phi_\mu] = [C] = [\bar C] = [\beta] = [\beta]
= [M]^{\frac{1}{2}}, \; [B] = [\bar B] = [R] = [\bar R] = [M]^{\frac{3}{2}},\;
[\rho] = [M]^{-\frac{1}{2}}, \;$ and the coupling constant $g$ has the mass dimension $[g] = [M]^{\frac{1}{2}}$. Second, the fermionic (anti-)ghost fields $(\bar C) C$ and $(\bar \beta) \beta$ carry ghost numbers $(\mp1)$, respectively whereas rest of the (bosonic) fields carry ghost number equal to zero. Third, the nilpotent (anti-)BRST transformations increase the mass dimension by one unit when they operate on any generic field of the theory. In other words, we can say that the (anti-)BRST transformations carry mass dimension equal one (in natural units). Fourth, the (anti-)BRST transformations (decrease)increase the ghost number by one unit when they act on any field of the theory. This means that (anti-)BRST transformations carry ghost number $(\mp 1)$, respectively. These points are very important in constructing the (anti-)BRST invariant coupled Lagrangian densities.
Exploiting the basic tenets of the BRST formalism, the most appropriate (anti-)BRST invariant Lagrangian densities that can be written in terms of nilpotent and absolutely anticommuting (anti-)BRST symmetry transformations are as follows [@ThierryMieg:1982un] $$\begin{aligned}
{\cal L}_b &=& {\cal L}_0 + s_b\,s_{ab}\bigg[\frac{i}{2} A_\mu \cdot A^\mu + C \cdot \bar C
+ \frac{i}{2} \phi_\mu \cdot \phi^\mu + \frac{1}{2}\, \beta \cdot \bar \beta\bigg],\nonumber\\
&&\nonumber\\
{\cal L}_{\bar b} &=& {\cal L}_0 - s_{ab}\,s_b\bigg[\frac{i}{2} A_\mu \cdot A^\mu + C \cdot \bar C
+ \frac{i}{2} \phi_\mu \cdot \phi^\mu + \frac{1}{2}\,\beta \cdot \bar \beta\bigg],\end{aligned}$$ where ${\cal L}_0$ is our starting gauge invariant Lagrangian density (\[2.1\]). We would like to emphasize that each term in the square brackets is Lorentz scalar and chosen in such a way that they have ghost number zero and mass dimension one (in natural units). Moreover, the (constant) factors in front of each term are picked for the algebraic convenience. Utilizing the off-shell nilpotent (anti-) BRST transformations from (\[3.6\]), (\[3.7\]), (\[3.23\]) and (\[3.24\]), we obtain the following explicit Lagrangian densities, namely; $$\begin{aligned}
{\cal L}_b &=& - \frac{1}{4}\, F_{\mu\nu}\cdot F^{\mu\nu}
- \frac{1}{4}\, \big(G_{\mu\nu} + g F_{\mu\nu} \times \rho\big) \cdot \big(G^{\mu\nu} + g F^{\mu\nu} \times \rho\big)
+ \frac{m}{2}\, \varepsilon^{\mu\nu\eta}\,F_{\mu\nu}\cdot \phi_\eta\nonumber\\
&+& \frac{1}{2}\,\big[B \cdot B + \bar B \cdot \bar B\big] + B\cdot \big(\partial^\mu A_\mu\big)
+ \frac{1}{2}\,\big[R + i g (C \times \bar \beta)\big]\cdot \big[R + i g (C \times \bar \beta)\big]\nonumber\\
&+& \big[R + i g (C \times \bar \beta)\big]\cdot \big(D^\mu \phi_\mu\big)
- i \partial_\mu \bar C \cdot D^\mu C - i D_\mu \bar \beta \cdot D^\mu \beta,\nonumber\\
&&\nonumber\\
{\cal L}_{\bar b} &=& - \frac{1}{4}\, F_{\mu\nu}\cdot F^{\mu\nu}
- \frac{1}{4}\, \big(G_{\mu\nu} + g F_{\mu\nu} \times \rho\big) \cdot \big(G^{\mu\nu} + g F^{\mu\nu} \times \rho\big)
+ \frac{m}{2}\, \varepsilon^{\mu\nu\eta}\,F_{\mu\nu}\cdot \phi_\eta\nonumber\\
&+& \frac{1}{2}\,\big[B \cdot B + \bar B \cdot \bar B\big] - \bar B\cdot \big(\partial^\mu A_\mu\big)
+ \frac{1}{2}\,\big[\bar R + i g (\bar C \times \beta)\big]\cdot \big[\bar R + i g (\bar C \times \beta)\big]\nonumber\\
&-& \big[\bar R + i g (\bar C \times \beta)\big]\cdot \big(D^\mu \phi_\mu\big)
- i D_\mu \bar C \cdot \partial^\mu C - i D_\mu \bar \beta \cdot D^\mu \beta, \label{4.2}\end{aligned}$$ where $B, \bar B$ and $R, \bar R$ are the Nakanishi-Lautrup type auxiliary fields. These Lagrangian densities are coupled because these Nakanishi-Lautrup auxiliary fields $B, \bar B$ and $R, \bar R$ are related through the CF conditions (\[3.26\]).
It can be checked that the (anti-)BRST transformations ($s_{(a)b}$) leave the above Lagrangian densities quasi-invariant. To be more specific, under the operations of nilpotent (anti-)BRST transformations, the Lagrangian densities $({\cal L}_{\bar b}) {\cal L}_b$ transform to a total spacetime derivative, in the following fashion, respectively $$\begin{aligned}
s_{ab} {\cal L}_{\bar b} &=& \partial_\mu \bigg[\frac{m}{2}\, \varepsilon^{\mu\nu\eta} F_{\nu\eta} \cdot \bar \beta
- \bar B \cdot (D^\mu \bar C) - \bar R\cdot D^\mu \bar \beta - i g \big(\bar C \times \beta\big)\cdot D^\mu \bar \beta\bigg], \nonumber\\
&&\nonumber\\
s_b {\cal L}_b &=& \partial_\mu \bigg[\frac{m}{2}\, \varepsilon^{\mu\nu\eta} F_{\nu\eta} \cdot \beta + B \cdot (D^\mu C)
+ R\cdot D^\mu \beta + i g \big(C \times \bar \beta\big)\cdot D^\mu \beta\bigg]. \quad\end{aligned}$$ Thus, the action integral corresponding to the above Lagrangian densities remain invariant under ($s_{(a)b}$). Furthermore, it is interesting to note that the following variations are true: $$\begin{aligned}
s_{ab} {\cal L}_b &=& \partial_\mu \Big[\frac{m}{2}\, \varepsilon^{\mu\nu\eta} F_{\nu\eta} \cdot \bar \beta
+ B \cdot \partial^\mu \bar C + \big(R + i g C \times \bar \beta \big)\cdot D^\mu \bar \beta \Big]\nonumber\\
&-& \Big[D_\mu\big(B + \bar B + i g C \times \bar C\big)\Big]\cdot \partial^\mu \bar C
- \Big[D_\mu\big(R + \bar R + ig C \times \bar \beta + i g \bar C \times \beta \big)\Big] \cdot D^\mu\bar \beta\nonumber\\
&-& g \Big[R + ig \big(C \times \bar \beta\big)
+ D^\mu \phi_\mu \Big]\cdot \Big[\big(B + \bar B + i g C \times \bar C\big)\times \bar \beta\Big], \nonumber\\
&&\nonumber\\
s_b {\cal L}_{\bar b} &=& \partial_\mu \Big[\frac{m}{2}\, \varepsilon^{\mu\nu\eta} F_{\nu\eta} \cdot \beta
- \bar B \cdot \partial^\mu C - \big(\bar R + i g \bar C \times \beta \big)\cdot D^\mu \beta \Big]\nonumber\\
&+& \Big[D_\mu\big(B + \bar B + i g C \times \bar C\big)\Big]\cdot \partial^\mu C
+ \Big[D_\mu\big(R + \bar R + ig C \times \bar \beta + i g \bar C \times \beta \big)\Big] \cdot D^\mu \beta \nonumber\\
&-& g \big[\bar R + ig \big(\bar C \times \beta \big)
- D^\mu \phi_\mu\Big]\cdot \Big[\big(B + \bar B + i g C \times \bar C\big)\times \beta\Big].\end{aligned}$$ Therefore, it is evident from the above variations that the Lagrangian densities ${\cal L}_b$ and ${\cal L}_{\bar b}$ also respect the anti-BRST ($s_{ab}$) and BRST ($s_b$) transformations, respectively only on the constrained hypersurface defined by the CF conditions (\[3.26\]). As a result, both the Lagrangian densities are equivalent and they respect BRST as well as anti-BRST symmetries on the constrained hypersurface spanned by CF conditions \[cf. (\[3.26\])\].
Conserved charges: Novel observations
=====================================
In our previous section, we have seen that the coupled Lagrangian densities (and corresponding actions) respect the off-shell nilpotent and continuous (anti-)BRST symmetry transformations. As a consequence, according to Noether’s theorem, the invariance of the actions under the continuous (anti-) BRST transformations lead to the following conserved (anti-)BRST currents ($J^\mu_{(a)b}$), namely; $$\begin{aligned}
J^\mu_{ab} &=& - (D_\nu \bar C) \cdot \Big[F^{\mu\nu} - g \big(G^{\mu\nu} + g F^{\mu\nu} \times \rho\big)\times \rho
- m \,\varepsilon^{\mu\nu\eta} \phi_\eta\Big] - \bar B \cdot (D^\mu \bar C) \nonumber\\
&-& \frac{i}{2}\,g \big(\bar C \times \bar C\big) \cdot \partial^\mu C
- (D_\nu \bar \beta)\cdot \Big(G^{\mu\nu} + g F^{\mu\nu} \times \rho \Big)
+ g (\phi_\nu \times \bar C) \cdot \Big(G^{\mu\nu} + g F^{\mu\nu} \times \rho \Big) \nonumber\\
&+& g (\phi^\mu \times \bar C)\cdot \big(\bar R + i g \bar C\times \beta\big)
- \bar R\cdot D^\mu \bar \beta - i g \big(\bar C \times \bar \beta \big)\cdot D^\mu \beta
- \frac{m}{2}\,\varepsilon^{\mu\nu\eta} F_{\nu\eta} \cdot \bar \beta,\nonumber\\
&&\nonumber\\
J^\mu_b &=& - (D_\nu C) \cdot \Big[F^{\mu\nu} - g \big(G^{\mu\nu} + g F^{\mu\nu} \times \rho\big)\times \rho
- m \,\varepsilon^{\mu\nu\eta} \phi_\eta\Big] + B \cdot (D^\mu C)\nonumber\\
&+& \frac{i}{2}\,g \big(C \times C\big) \cdot \partial^\mu \bar C - (D_\nu \beta)\cdot \Big(G^{\mu\nu} + g F^{\mu\nu} \times \rho \Big)
+ g (\phi_\nu \times C) \cdot \Big(G^{\mu\nu} + g F^{\mu\nu} \times \rho \Big)\nonumber\\
&-& g (\phi^\mu \times C)\cdot \big(R + i g C\times \bar \beta\big)
+ R\cdot D^\mu \beta + i g \big(C \times \beta \big)\cdot D^\mu \bar \beta
- \frac{m}{2}\,\varepsilon^{\mu\nu\eta} F_{\nu\eta} \cdot \beta.\end{aligned}$$ One can check that the conservation (i.e. $\partial_\mu J^\mu_b = 0$) of BRST current ($J^\mu_b$) can be proven by exploiting the Euler-Lagrange (E-L) equations of motion that are derived from the Lagrangian density ${\cal L}_b$. These E-L equations are as listed below: $$\begin{aligned}
&& D_\mu F^{\mu\nu} - g\,D_\mu\big[\big(G^{\mu\nu} + g\, F^{\mu\nu} \times \rho \big)\times \rho\big]
+ g \,\big(G^{\mu\nu} + g\, F^{\mu\nu} \times \rho \big) \times \phi_\mu - m \,\varepsilon^{\mu\nu\eta}\,(D_\mu \phi_\eta)
\nonumber\\
&& - \; \partial^\nu B - i g \big(\partial^\nu \bar C \times C\big)
+ g \big(R + i g C \times \bar \beta\big)\times \phi^\nu
+ i\, g \big(\bar \beta \times D^\nu \beta\big) - i\,g \big(\beta \times D^\nu \bar \beta\big) = 0, \nonumber\\
&& D_\mu \big[G^{\mu\nu} + g (F^{\mu\nu} \times \rho) \big] - D^\nu \big[R + i g(C \times \bar \beta)\big]
- \frac{m}{2}\, \varepsilon^{\mu\nu\kappa}\, F_{\mu\kappa} = 0, \nonumber\\
&& \big[G^{\mu\nu} + g (F^{\mu\nu} \times \rho) \big] \times F_{\mu\nu} = 0, \qquad R + ig (C \times \bar \beta) + D_\mu \phi^\mu = 0,
\qquad B = - \big(\partial_\mu A^\mu\big), \nonumber\\
&& \partial_\mu (D^\mu C) = 0, \qquad D_\mu (\partial^\mu \bar C) = 0, \qquad D_\mu (D^\mu \beta) = 0.
\qquad D_\mu (D^\mu \bar \beta) = 0, \label{5.2}\end{aligned}$$ $$\begin{aligned}
&& D_\mu F^{\mu\nu} - g\,D_\mu\big[\big(G^{\mu\nu} + g\, F^{\mu\nu} \times \rho \big)\times \rho\big]
+ g \,\big(G^{\mu\nu} + g\, F^{\mu\nu} \times \rho \big) \times \phi_\mu - m \,\varepsilon^{\mu\nu\eta}\,(D_\mu \phi_\eta)
\nonumber\\
&& +\; \partial^\nu \bar B + i g \big(\partial^\nu C \times \bar C\big)
- g \big(\bar R + i g \bar C \times \beta\big)\times \phi^\nu
+ i\, g \big(\bar \beta \times D^\nu \beta\big) - i\,g \big(\beta \times D^\nu \bar \beta\big) = 0, \nonumber\\
&& D_\mu \big[G^{\mu\nu} + g (F^{\mu\nu} \times \rho) \big] + D^\nu \big[\bar R + i g(\bar C \times \beta)\big]
- \frac{m}{2}\, \varepsilon^{\mu\nu\kappa}\, F_{\mu\kappa} = 0, \nonumber\\
&& \big[G^{\mu\nu} + g (F^{\mu\nu} \times \rho) \big] \times F_{\mu\nu} = 0, \qquad \bar R + ig (\bar C \times \beta) - D_\mu \phi^\mu = 0,
\qquad \bar B = \big(\partial_\mu A^\mu\big), \nonumber\\
&& D_\mu (\partial^\mu C) = 0, \qquad \partial_\mu (D^\mu \bar C) = 0, \qquad D_\mu (D^\mu \beta) = 0,
\qquad D_\mu (D^\mu \bar \beta) = 0, \label{5.3}\end{aligned}$$ which emerge from the Lagrangian density ${\cal L}_{\bar b}$.
Exploiting the above E-L equations of motion (cf. (\[5.2\]) and (\[5.3\])), the conserved currents $J^\mu_{(a)b}$ can be written in simpler forms as: $$\begin{aligned}
J^\mu_{ab} &=& - \partial_\nu \Big(\big[F^{\mu\nu} - g \big(G^{\mu\nu} + g F^{\mu\nu} \times \rho\big)\times \rho
- m \,\varepsilon^{\mu\nu\eta} \phi_\eta\big]\cdot \bar C + \big[G^{\mu\nu} + g F^{\mu\nu} \times \rho\big]\cdot \bar \beta\Big)\nonumber\\
&+& (\partial^\mu \bar B) \cdot \bar C - \bar B \cdot (D^\mu \bar C)
+ \frac{i}{2}\,g \big(\bar C \times \bar C\big) \cdot \partial^\mu C
- (\bar R + i g \bar C \times \beta \big)\cdot (D^\mu \bar \beta) \nonumber\\
&+& D^\mu\big(\bar R + i g \bar C \times \beta\big) \cdot \bar \beta,\nonumber\\
&&\nonumber\\
J^\mu_b &=& - \partial_\nu \Big(\big[F^{\mu\nu} - g \big(G^{\mu\nu} + g F^{\mu\nu} \times \rho\big)\times \rho
- m \,\varepsilon^{\mu\nu\eta} \phi_\eta\big]\cdot C + \big[G^{\mu\nu} + g F^{\mu\nu} \times \rho\big]\cdot \beta\Big)\nonumber\\
&+& B \cdot (D^\mu C) - (\partial^\mu B) \cdot C - \frac{i}{2}\,g \big(C \times C\big) \cdot \partial^\mu \bar C
+ (R + i g C \times \bar \beta \big)\cdot (D^\mu \beta)\nonumber\\
&-& D^\mu\big(R + i g C \times \bar \beta\big)\cdot \beta. \end{aligned}$$ Now, the proof of conservation laws ($\partial_\mu J^\mu_{(a)b} = 0$) is quite straightforward. The temporal components (i.e. $\int d^2x J^0_{(a)b} = Q_{(a)b}$) of the above conserved currents ($J^\mu_{(a)b}$) lead to the following conserved (i.e. ${\dot Q}_{(a)b} =0$) (anti-)BRST charges ($Q_{(a)b}$), namely; $$\begin{aligned}
Q_{ab} &=& - \int d^2x \bigg[\bar B \cdot (D^0 \bar C) - (\partial^0 \bar B) \cdot \bar C
- \frac{i}{2}\,g \big(\bar C \times \bar C\big) \cdot \partial^0 C
+ (\bar R + i g \bar C \times \beta \big)\cdot (D^0 \bar \beta)\nonumber\\
&-& D^0\big(\bar R + i g \bar C \times \beta\big)\cdot \bar \beta \bigg], \nonumber\\
&&\nonumber\\
Q_b &=& \int d^2x \bigg[ B \cdot (D^0 C) - (\partial^0 B) \cdot C
- \frac{i}{2}\,g \big(C \times C\big) \cdot \partial^0 \bar C
+ (R + i g C \times \bar \beta \big)\cdot (D^0 \beta) \nonumber\\
&-& D^0\big(R + i g C \times \bar \beta\big)\cdot \beta \bigg].\label{5.5}\end{aligned}$$ It turns out that the conserved, nilpotent ($Q^2_{(a)b} = 0$, see below) and anticommuting ($Q_b\,Q_{ab} + Q_{ab}\, Q_b = 0$, see below) (anti-)BRST charges are the generators of the (anti-)BRST symmetry transformations, respectively. For the sake of brevity, these transformations can be obtained by exploiting the following symmetry properties: $$\begin{aligned}
s_b \Psi = - i \big[\Psi, \; Q_b\big]_{\pm}, \qquad s_{ab} \Psi = - i \big[\Psi,\; Q_{ab} \big]_{\pm},\qquad
\Psi = A_\mu, \phi_\mu, C, \bar C, \beta, \bar \beta\end{aligned}$$ The $(\pm)$ signs as the subscript on the square brackets represent (anti)commutators corresponding to the generic field $\Psi$ being (fermionic)bosonic in nature (see, e.g. [@Gupta:2009bu] for details). The (anti-)BRST transformations of the Nakanishi-Lautrup auxiliary fields $B, \bar B, R, \bar R$ have been derived from the basic requirements (i.e. nilpotency and/or absolute anticommutativity properties) of the (anti-)BRST symmetry transformations.
It is worthwhile to mention that, even though, the (anti-)BRST charges $(Q_{(a)b})$ are conserved, nilpotent as well as anticommuting in nature (see below), they are unable to generate the proper (anti-)BRST transformations (i.e. $s_b \rho = \beta - g (\rho \times C)$ and $s_{ab} \rho = \bar \beta - g (\rho \times \bar C)$) of the auxiliary field $\rho$. Furthermore, the nilpotency and absolute anticommutativity properties of the (anti-)BRST transformations also fail to produce the transformations of $\rho$. This is one of the novel observations of our present endeavor. Although, we have derived these transformations by exploiting the power and strength of the augmented superfield formalism which produces the off-shell nilpotent ($s^2_{(a)b} = 0$) as well as absolutely anticommuting ($s_b \,s_{ab} + s_{ab}\, s_b = 0$) (anti-)BRST symmetry transformations for [*all*]{} the basic and auxiliary fields of the theory.
The nilpotency ($Q^2_{(a)b} =0$) of the (anti-)BRST charges reflects the fermionic nature whereas the anticommutativity ($Q_b\,Q_{ab} + Q_{ab}\, Q_b = 0$) shows that the (anti-)BRST charges are linearly independent of each other. These properties can be verified in the following straightforward manner: $$\begin{aligned}
s_b Q_b &=& - i \{Q_b,\; Q_b\} = 0 \Rightarrow Q^2_b = 0,\nonumber\\
s_{ab} Q_{ab} &=& - i \{Q_{ab},\; Q_{ab}\} = 0 \Rightarrow Q^2_{ab} = 0,\nonumber\\
s_b Q_{ab} &=& - i \{Q_{ab},\; Q_b\} = 0 \Rightarrow Q_{ab}\,Q_{b} + Q_{b}\, Q_{ab} = 0,\nonumber\\
s_{ab} Q_b &=& - i \{Q_b,\; Q_{ab}\} = 0 \Rightarrow Q_b\,Q_{ab} + Q_{ab}\, Q_b = 0.\end{aligned}$$ We point out that in proving the anticommutativity property ($Q_b\,Q_{ab} + Q_{ab}\, Q_b = 0$) of the (anti-)BRST charges we have used the CF conditions (\[3.26\]). For the sake of brevity, one can check $$\begin{aligned}
s_b Q_{ab} &=& - i \int d^2x\,\Big[\bar B \cdot \partial^0\Big(B + \bar B + i g C \times \bar C\Big)\Big] \nonumber\\
&+& \int d^2x\,\Big[g \Big(\big(B + \bar B + i g C \times \bar C\big) \times \beta \Big) \cdot D^0\beta
- g D^0\Big(\big(B + \bar B + i g C \times \bar C\big) \times \beta \Big) \cdot \bar \beta\Big] \nonumber\\
&-& i\int d^2x \Big[\big(R + \bar R + i g C \times \bar \beta
+ i g \bar C \times \beta\big) \cdot D^0\big(R + i g C \times \bar \beta\big)\Big],\nonumber\\\end{aligned}$$ $$\begin{aligned}
s_{ab} Q_b &=& i \int d^2x\,\Big[B \cdot \partial^0\Big(B + \bar B + i g C \times \bar C\Big)\Big] \nonumber\\
&-& \int d^2x\,\Big[g \Big(\big(B + \bar B + i g C \times \bar C\big) \times \bar \beta \Big) \cdot D^0 \beta
- g D^0\Big(\big(B + \bar B + i g C \times \bar C\big) \times \bar \beta \Big) \cdot \beta\Big] \nonumber\\
&+& i\int d^2x \Big[\big(R + \bar R + i g C \times \bar \beta
+ i g \bar C \times \beta\big) \cdot D^0\big(\bar R + i g \bar C \times \beta\big)\Big].\end{aligned}$$ It is clear from the above expressions that $s_b Q_{ab} = 0$ and $s_{ab} Q_b = 0$ if and only if CF conditions (\[3.26\]) are satisfied. As a consequence, the (anti-)BRST charges are anticommuting only on the constrained hypersurface defined by the CF conditions (\[3.26\]).
Ghost scale symmetry and BRST algebra
=====================================
The Lagrangian densities (\[4.2\]), in addition to the (anti-)BRST symmetry transformations, also respect the continuous ghost scale symmetry $(s_g)$. These symmetry transformations are given as follows $$\begin{aligned}
&&C \to e^{+\Omega}\,C, \qquad \bar C \to e^{-\Omega}\,\bar C, \qquad \beta \to e^{+\Omega}\,\beta,
\qquad \bar \beta \to e^{-\Omega}\,\bar \beta, \nonumber\\
&& \big(A_\mu, \phi_\mu, \rho, B, \bar B, R, \bar R\big) \to e^0 \big(A_\mu, \phi_\mu, \rho, B, \bar B, R, \bar R\big)\end{aligned}$$ where $\Omega$ is the global scale parameter. The numbers ($\pm1, 0$) in the exponential of the above transformations stand for ghost numbers of the corresponding fields. For instance, the ghost fields $(C, \beta)$ carry ghost number $(+1)$ and anti-ghost fields $(\bar C, \bar \beta)$ have ghost number ($-1$). The rest (bosonic) fields have ghost number zero. The infinitesimal version of the above continuous transformation is given by $$\begin{aligned}
&& s_g C = + \Omega \,C, \qquad s_g \bar C = - \Omega \,\bar C, \qquad s_g \beta = + \Omega \,\beta,
\qquad s_g \bar \beta = - \Omega \bar \beta, \nonumber\\
&& s_g\big(A_\mu, \phi_\mu, \rho, B, \bar B, R, \bar R\big) =0. \label{6.2}\end{aligned}$$ It is straightforward to check that under the above continuous ghost scale symmetry transformations (\[6.2\]) both the Lagrangian densities remain invariant (i.e. $s_g{\cal L}_b = s_g {\cal L}_{\bar b} = 0$). As a consequence, the existence of ghost scale symmetry leads to the following Noether’s conserved current ($J^\mu_g$) and charge ($Q_g$): $$\begin{aligned}
J^\mu_g &=& i \Big[\bar C \cdot (D^\mu C) - (\partial^\mu \bar C) \cdot C
+ \bar \beta \cdot (D^\mu \beta) - (D^\mu \bar \beta) \cdot \beta \Big], \nonumber\\
Q_g &=& i \int d^2x \,\Big[\bar C \cdot (D^0 C) - (\partial^0 \bar C) \cdot C
+ \bar \beta \cdot (D^0 \beta) - (D^0 \bar \beta) \cdot \beta \Big].\end{aligned}$$ The conservation law $(\partial_\mu J^\mu_g = 0)$ can be proven by exploiting the E-L equations of motion (\[5.2\]). The ghost charge $Q_g$ also turns out to be the generator of the ghost scale symmetry transformations (\[6.2\]). For instance, one can check that $s_g C = - i [C, \, \Omega\,Q_g] = +\Omega\, C.$
The above ghost charge $Q_g$ together with the nilpotent (anti-)BRST charges $Q_{(a)b}$ obey a standard BRST algebra. In operator form, this algebra can be given as follow $$\begin{aligned}
&& Q^2_b = 0, \qquad Q^2_{ab} = 0, \qquad \big\{Q_b,\; Q_{ab}\big\} = Q_b\,Q_{ab} + Q_{ab}\, Q_b = 0,\nonumber\\
&& i \big[Q_g, \,Q_b\big] = + Q_b, \qquad i \big[Q_g, Q_{ab}\big] = - Q_{ab}, \qquad Q^2_g \ne 0. \label{6.4}\end{aligned}$$
Let us consider a state $|\psi \rangle _n$, in the quantum Hilbert space of states, such that the ghost number of the state is defined in the following manner $$\begin{aligned}
i Q_g |\psi \rangle _n = n |\psi \rangle _n ,\end{aligned}$$ where $n$ is the ghost number of the state $|\psi \rangle _n$. Now, it is easy to check, with the help of above algebra (\[6.4\]), that following relationships holds $$\begin{aligned}
&& i Q_g Q_b |\psi \rangle _n = (n + 1) Q_b |\psi \rangle _n , \nonumber\\
&& i Q_g Q_{ab} |\psi \rangle _n = (n - 1) Q_{ab} |\psi \rangle _n , \end{aligned}$$ which shows that the BRST charge $Q_b$ increases the ghost number by one unit when it operates on a quantum state whereas the anti-BRST charge $Q_{ab}$ decreases it by one unit. In other words, we can say that the (anti-)BRST charge carry the ghost numbers $(\mp 1)$, respectively. A careful look at the expressions of the (anti-)BRST and ghost charges, where the ghost numbers of the fields are concerned, also reveal the same observations.
Role of auxiliary field: A bird’s-eye view
==========================================
In this section we provide a brief synopsis about few striking similarities and some glaring differences among the 3D non-Abelian JP model, 4D topologically massive non-Abelian 2-form gauge theory [@Malik:2010gu; @Kumar:2010kd] and the 4D modified gauge invariant Proca theory in the realm of well-known St[ü]{}ckelberg formalism (see, e.g. [@Ruegg:2003ps] for details).
Jackiw-Pi model
---------------
It is interesting to note that, if we make the following substitution $$\begin{aligned}
\phi_\mu \;\longrightarrow \; \phi_\mu + D_\mu \rho, \label{7.1}\end{aligned}$$ in our starting Lagrangian density (\[2.1\]), the 2-form $G_{\mu\nu}$ and mass term re-defined as $$\begin{aligned}
G_{\mu\nu}& \longrightarrow & G_{\mu\nu} - g \big(F_{\mu\nu} \times \rho \big),\nonumber\\
\frac{m}{2}\,\varepsilon^{\mu\nu\eta}\,F_{\mu\nu} \cdot \phi_\eta & \longrightarrow &
\frac{m}{2}\,\varepsilon^{\mu\nu\eta}\,F_{\mu\nu} \cdot \phi_\eta
+ \partial_\eta \Big[\frac{m}{2}\, \varepsilon^{\mu\nu\eta}\,F_{\mu\nu} \cdot \rho \Big]
- \frac{m}{2}\,\varepsilon^{\mu\nu\eta}\,\big(D_\eta F_{\mu\nu}\big) \cdot \rho. \qquad\end{aligned}$$ In the above, the term $\frac{m}{2}\,\varepsilon^{\mu\nu\eta}\,\big(D_\eta F_{\mu\nu}\big) \cdot \rho$ is zero due to the validity of the well-known Bianchi identity $(D_\mu F_{\nu\eta} + D_\nu F_{\eta\mu} + D_\eta F_{\mu\nu} = 0)$. Therefore, the mass term remains invariant, modulo a total spacetime derivative, under the re-definition (\[7.1\]). As a consequence, the modified Lagrangian density, modulo a total spacetime derivative, is given by $$\begin{aligned}
\tilde{\cal L}_0 = - \frac{1}{4}\, F_{\mu\nu}\cdot F^{\mu\nu}
- \frac{1}{4}\, G_{\mu\nu}\cdot G^{\mu\nu} + \frac{m}{2}\, \varepsilon^{\mu\nu\eta}\,F_{\mu\nu}\cdot \phi_\eta. \label{7.3}\end{aligned}$$ It is clear that the auxiliary field $\rho$ is completely eliminated from the above Lagrangian density. We point out that, even though, Lagrangian density (\[7.3\]) respects the YM gauge transformations (\[2.2\]) but it fails to respect the NYM gauge transformations (\[2.3\]). The similar observation can also be seen in the case of 4D topologically massive non-Abelain 2-form gauge theory as well as in the 4D modified gauge invariant version of Proca theory.
4D massive non-Abelian 2-form gauge theory
------------------------------------------
The Lagrangian density for the 4D massive non-Abelian 2-form gauge theory is given by (see, for details [@Kumar:2011zi; @Malik:2010gu; @Kumar:2010kd]) $$\begin{aligned}
{\cal L} = - \frac{1}{4}\, F_{\mu\nu}\cdot F^{\mu\nu}
+ \frac{1}{12}\, H_{\mu\nu\eta}\cdot H^{\mu\nu\eta} + \frac{m}{4}\, \varepsilon^{\mu\nu\eta\kappa}\,B_{\mu\nu}\cdot F_{\eta\kappa}, \label{7.4}\end{aligned}$$ where 3-form $H_{\mu\nu\eta} = D_\mu B_{\nu\eta} + D_\nu B_{\eta\mu} + D_\eta B_{\mu\nu} + g(F_{\mu\nu} \times K_\eta)
+ g(F_{\nu\eta} \times K_\mu)+ g (F_{\eta\mu} \times K_\nu)$ is the field strength tensor corresponding to the 2-form gauge field $B_{\mu\nu}$ and the 2-form field strength tensor $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu - g (A_\mu \times A_\nu)$ corresponds to the 1-form gauge field $A_\mu$. The coupling constant is represented by $g$ and $D_\mu$ is the covariant derivative. The auxiliary field $K_\mu$ is the compensating field. This Lagrangian density respects the two types of gauge transformations – the scalar gauge transformation $(\tilde \delta_1)$ and vector gauge transformation $(\tilde \delta_2)$, namely; [@Kumar:2011zi; @Malik:2010gu; @Kumar:2010kd] $$\begin{aligned}
&&\tilde \delta_1 A_\mu = D_\mu \Omega, \qquad \tilde \delta_1 B_{\mu\nu} = - g(B_{\mu\nu} \times \Omega),
\qquad \tilde \delta_1 K_\mu = - g(K_\mu \times \Omega), \nonumber\\
&& \tilde \delta_2 A_\mu =0, \qquad \tilde \delta_2 B_{\mu\nu} = - (D_\mu \Lambda_\nu - D_\nu \Lambda_\mu),
\qquad \tilde \delta_2 K_\mu = - \Lambda_\mu, \label{7.5}\end{aligned}$$ where $\Omega (x)$ and $\Lambda_\mu (x)$ are the local scalar and vector gauge parameters, respectively. We note that if we re-define the $B_{\mu\nu}$ field as $$\begin{aligned}
B_{\mu\nu} \;\longrightarrow \; B_{\mu\nu} + (D_\mu K_\nu - D_\nu K_\mu),\end{aligned}$$ the 3-form field strength tensor $H_{\mu\nu\eta}$ and the mass term modify as follows $$\begin{aligned}
H_{\mu\nu\eta} & \longrightarrow& \tilde H_{\mu\nu\eta}\;=\; D_\mu B_{\nu\eta} + D_\nu B_{\eta\mu} + D_\eta B_{\mu\nu}, \nonumber\\
\frac{m}{4}\,\varepsilon^{\mu\nu\eta\kappa}\,B_{\mu\nu} \cdot F_{\eta\kappa} &\longrightarrow&
\frac{m}{4}\,\varepsilon^{\mu\nu\eta\kappa}\,B_{\mu\nu} \cdot F_{\eta\kappa}
+ \partial_\mu \Big[\frac{m}{2}\, \varepsilon^{\mu\nu\eta\kappa}\, K_\nu \cdot F_{\eta\kappa} \Big] \nonumber\\
&& \qquad \qquad \qquad \qquad -\; \frac{m}{2}\,\varepsilon^{\mu\nu\eta\kappa}\,K_\nu \cdot \big(D_\mu F_{\eta\kappa}\big). \end{aligned}$$ and the compensating auxiliary vector field $K_\mu$ disappears from the Lagrangian density (\[7.4\]). Furthermore, the mass term $\displaystyle \frac{m}{4}\,\varepsilon^{\mu\nu\eta\kappa}\,B_{\mu\nu} \cdot F_{\eta\kappa}$ remains intact modulo a total spacetime derivative. Thus, the modified Lagrangian density can be given in the following manner (modulo a total spacetime derivative) $$\begin{aligned}
\tilde {\cal L} = - \frac{1}{4}\, F_{\mu\nu}\cdot F^{\mu\nu}
+ \frac{1}{12}\, {\tilde H}_{\mu\nu\eta}\cdot \tilde{H}^{\mu\nu\eta}
+ \frac{m}{4}\, \varepsilon^{\mu\nu\eta\kappa}\,B_{\mu\nu}\cdot F_{\eta\kappa}. \label{58}\end{aligned}$$ Clearly, the above Lagrangian density is no longer invariant under the vector gauge transformation even though it respects the scalar gauge transformations \[cf. (\[7.5\])\].
It is clear form the above discussions that both the above models (i.e. JP model and 4D massive non-Abelian 2-form gauge theory) are very similar to each other in the sense that under the re-definitions of the fields $\phi_\mu$ and $B_{\mu\nu}$ the auxiliary fields $\rho$ and $K_\mu$ are eliminated from their respective models. As a result, the modified Lagrangian densities (\[7.3\]) and (\[58\]) do not respect the symmetry transformations $(\delta_2)$ and $(\tilde \delta_2)$, respectively. Thus, the auxiliary fields $\rho$ and $K_\mu$ are required in their respective models so that these models respect both the gauge symmetry transformations \[cf. (\[2.2\]), (\[2.3\]) and (\[7.5\])\].
Modified version of Abelian Proca theory
----------------------------------------
The above key observations can also be seen in the case of modified gauge invariant Abelian Proca theory. The gauge invariant Lagrangian density of this model is as follows [@Ruegg:2003ps] $$\begin{aligned}
{\cal L}_s = - \frac{1}{4}\, F_{\mu\nu}\,F^{\mu\nu} + \frac{m^2}{2}\, A_\mu \,A^\mu
+ \frac{1}{2}\,\partial_\mu \phi\,\partial^\mu \phi + m A_\mu\, \partial^\mu \phi, \label{7.8}\end{aligned}$$ where $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ is the field strength tensor corresponding to $A_\mu$, $\phi$ is the St[ü]{}ckelberg field and $m$ represents the mass of the photon field $A_\mu$. Under the following local gauge transformations $$\begin{aligned}
\delta_{(gt)}A_\mu = \partial_\mu \chi(x), \qquad \delta_{(gt)} \phi = - m\,\chi(x), \label{7.9}\end{aligned}$$ the Lagrangian density (\[7.8\]) remains invariant. Here $\chi(x)$ is the local gauge transformation parameter. It can be checked that under the following re-definition $$\begin{aligned}
A_\mu \;\longrightarrow\; A_\mu - \frac{1}{m}\,\partial_\mu \phi,\end{aligned}$$ the St[ü]{}ckelberg field $\phi$ completely disappears from the Lagrangian density (\[7.8\]). As a consequence, the resulting Lagrangian density does not respect the above gauge transformations (\[7.9\]).
The above observation is very similar to the JP model and the massive non-Abelian 2-form gauge theory. As a consequence, the field $\rho$ (in JP model) and $K_\mu$ (in massive non-Abelian 2-form theory) are like the St[ü]{}ckelberg field. However, the key difference is that these St[ü]{}ckelberg like fields (i.e. $\rho$ and $K_\mu$) are auxiliary fields in their respective models whereas, in the modified gauge invariant Proca theory, the St[ü]{}ckelberg field $\phi$ is dynamical in nature.
Conclusions
===========
In our present investigation, we have derived the off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetry transformations corresponding to the combined YM and NYM symmetries of the JP model. For this purpose, we have utilized the power and strength of augmented superfield approach. The derivation of proper (anti-)BRST symmetries for the auxiliary field $\rho$ is one of the main findings of our present endeavor. These (anti-)BRST symmetry transformations corresponding to the auxiliary field $\rho$ can neither be generated from the conserved (anti-)BRST charges nor deduced by the requirement of nilpotency and/or absolute anticommutativity of the (anti-)BRST symmetry transformations.
One of the main features of the superfield formalism is the derivation of CF conditions which, in turn, ensure the absolutely anticommutativity of (anti-) BRST symmetry transformations. The CF conditions, a hallmark of any non-Abelian 1-form gauge theories [@Curci:1976ar], appear naturally within the framework of superfield formalism and also have connections with gerbes [@Bonora:2007hw]. In our present case of combined YM and NYM symmetries of JP model, there exist [*two*]{} CF conditions (cf. (\[3.26\])). This is in contrast to the YM symmetries case where there exist only [*one*]{} CF condition [@Gupta:2011cta] and in NYM symmetries case, [*no*]{} CF condition was observed [@Gupta:2012ur]. Moreover, these CF conditions have played a central role in the derivation of coupled Lagrangian densities (cf. Section 4).
Furthermore, we have obtained a set of coupled Lagrangian densities which respect the above mentioned (anti-)BRST symmetry transformations. The ghost sector of these coupled Lagrangian densities is also endowed with another continuous symmetry - the ghost symmetry. We have exploited this symmetry to derive the conserved ghost charge. Moreover, we have pointed out the standard BRST algebra obeyed by all the conserved charges of the underlying theory.
At the end, we have provided a bird’s-eye view on the role of auxiliary field in the context of various massive models. For this purpose, we have taken three different cases of 3D JP model, 4D massive non-Abelian 2-form gauge theory and the 4D modified version of Abelian Proca theory. We have shown that the field $\rho$ (in JP model) and $K_\mu$ (in massive non-Abelian 2-form theory) are like St[ü]{}ckelberg field ($\phi$) of Abelian Proca model. However, $\rho$ and $K_\mu$ are auxiliary fields whereas $\phi$ is dynamical, in their respective models. Finally, we capture the (anti-)BRST invariance of the coupled Lagrangian densities (cf. (\[4.2\])), nilpotency and absolute anticommutativity of (anti-)BRST charges (cf. (\[5.5\])) within the framework of superfield approach.
Acknowledgments {#acknowledgments .unnumbered}
===============
The research work of SG is supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) grant 151112/2014-2.
(Anti-)BRST invariance, nilpotency and anticommutativity: Superfield approach
=============================================================================
It is interesting to point out that the super expansions (\[3.5\]), (\[3.15\]) and (\[3.22\]) can be expressed in terms of the translations of the corresponding superfields along the Grassmannian directions of the $(3,2)$-dimensional supermanifold, as $$\begin{aligned}
&& s_b \Psi (x) = \frac {\partial}{\partial \bar \theta} \, \tilde \Psi^{(h)}(x,\theta,\bar\theta)\Big|_{\theta = 0}, \qquad \quad
s_{ab} \Psi (x) = \frac {\partial}{\partial \theta} \, \tilde \Psi^{(h)}(x,\theta,\bar\theta)\Big|_{\bar \theta = 0},\nonumber\\
&&s_b\, s_{ab} \Psi (x) = \frac {\partial}{\partial \bar \theta} \, \frac {\partial}{\partial \theta}
\, \tilde \Psi^{(h)} (x,\theta,\bar\theta), \label{b1}\end{aligned}$$ where $\Psi (x)$ is any generic field of the underlying 3D theory and $\Psi^{(h)} (x,\theta,\bar\theta)$ is the corresponding superfield obtained after the application of HC. The above expression captures the off-shell nilpotency of the (anti-)BRST symmetries because of the properties of Grassmannian derivatives, i.e. $\partial_\theta^2 = \partial_{\bar \theta}^2 = 0$. Moreover, the anticommutativity property of the (anti-)BRST symmetry transformations is also clear from the expansions (\[3.5\]), (\[3.15\]) and (\[3.22\]), in the following manner $$\begin{aligned}
\Big(\frac {\partial}{\partial \theta} \; \frac {\partial}{\partial \bar \theta} +
\frac {\partial}{\partial \bar \theta} \; \frac {\partial}{\partial \theta} \Big) \; \tilde \Psi^{(h)}
(x, \theta, \bar \theta ) = 0. \label{b2}\end{aligned}$$ Thus, the expressions (\[b1\]) and (\[b2\]) provide the geometrical interpretations for the (anti-) BRST symmetry transformations in terms of the translational generators $(\partial_\theta, \partial_{\bar\theta})$ along the Grassmannian directions of the $(3, 2)$-dimensional supermanifold.
Furthermore, the nilpotency of the (anti-)BRST charges can also be realized, within the framework of superfield formalism, in the following manner $$\begin{aligned}
Q_b &=& \frac {\partial}{\partial \bar \theta} \, \int d^2 x \,
\Big[B(x) \cdot \tilde {\cal A}_0^{(h)} (x, \theta, \bar\theta) + i \, {\dot{\tilde {\bar {\cal F}}}}^{(h)}(x, \theta, \bar\theta)
\cdot \tilde {\cal F}^{(h)} (x, \theta, \bar\theta)\nonumber\\
&+& \Big(R(x)+ ig \tilde {\cal F}^{(h)}(x, \theta, \bar\theta) \times {\tilde {\bar \beta}}^{(h)}(x, \theta, \bar\theta) \Big)
\cdot \tilde\Phi^{(h)}_0 (x, \theta, \bar\theta) \nonumber\\
&+& i {\tilde {\cal D}}_0{\tilde {\bar \beta}}^{(h)}
(x, \theta, \bar\theta) \cdot \tilde \beta^{(h)} (x, \theta, \bar\theta)\Big] \bigg|_{\theta = 0} \nonumber\\
&\equiv& \int d^2 x \int d \bar\theta \;
\Big[B(x) \cdot \tilde {\cal A}_0^{(h)} (x, \theta, \bar\theta) + i \; {\dot{\tilde {\bar {\cal F}}}}^{(h)}(x, \theta, \bar\theta)
\cdot \tilde {\cal F}^{(h)} (x, \theta, \bar\theta ) \nonumber\\
&+& \Big(R(x)+ ig \tilde {\cal F}^{(h)}(x, \theta, \bar\theta) \times {\tilde {\bar \beta}}^{(h)}(x, \theta, \bar\theta) \Big)
\cdot \tilde\Phi^{(h)}_0 (x, \theta, \bar\theta) \nonumber\\
&+& i {\tilde {\cal D}}_0{\tilde {\bar \beta}}^{(h)}
(x, \theta, \bar\theta) \cdot \tilde \beta^{(h)} (x, \theta, \bar\theta)\Big] \bigg|_{\theta = 0}. \end{aligned}$$ This, in turn, implies $$\begin{aligned}
\frac {\partial}{\partial \bar\theta} \; Q_b \bigg|_{\theta = 0} = 0
\quad \Longrightarrow \quad Q_b^2 = 0,\end{aligned}$$ because of the nilpotency property of the Grassmannian derivative (i.e. $\partial_{\bar \theta}^2 = 0$). It is interesting to point out that the nilpotency of above BRST charge $(Q_b)$, when written in ordinary 3D spacetime, $$\begin{aligned}
Q_b &=& \int d^2 x\, s_b \ \Big[B(x) \cdot A_0 (x) + i \; \dot{\bar C} (x) \cdot C(x)
+ \Big(R(x) + i g C(x) \times \bar \beta(x)\Big)\cdot \phi_0(x) \nonumber\\
&+& i D_0 \bar \beta(x) \cdot \beta(x) \Big],\end{aligned}$$ is straightforward and encoded in the nilpotency property $(s_b^2 = 0)$ of the BRST transformations $(s_b)$. In other words, $s_b Q_b = -i \{Q_b, Q_b\} = 0$ is true due to above mentioned reason. Moreover, using the CF-conditions, there is yet another way to express the above BRST charge where nilpotency is quite clear, as can be seen from the following expression $$\begin{aligned}
Q_b &=& i \frac {\partial} {\partial \bar \theta} \, \frac {\partial} {\partial \theta} \int
d^2 x \Big[ \tilde {\cal A}_0^{(h)} (x, \theta, \bar \theta) \cdot \tilde {\cal F}^{(h)}
(x, \theta, \bar \theta) + \tilde \Phi_0^{(h)} (x, \theta, \bar \theta) \cdot \tilde \beta^{(h)}
(x, \theta, \bar \theta)\Big] \nonumber\\
&\equiv& i \int d^2 x \, s_b\, s_{ab} \, \Big[\, A_0 (x) \cdot C(x) + \phi_0(x) \cdot \beta(x)\Big].\end{aligned}$$ This is true only on the constrained surface spanned by CF conditions. Similarly, we can express the anti-BRST charge $(Q_{ab})$ in the following two different ways: $$\begin{aligned}
Q_{ab} &= & - \frac {\partial}{\partial \theta} \, \int d^2 x \,
\Big[\bar B(x) \cdot \tilde {\cal A}_0^{(h)} (x, \theta, \bar\theta) + i \, \dot{\tilde{{\cal F}}}^{(h)}(x, \theta, \bar\theta)
\cdot {\tilde {\bar {\cal F}}}^{(h)} (x, \theta, \bar\theta)\nonumber\\
&-& \Big(\bar R(x)+ ig {\tilde {\bar {\cal F}}}^{(h)}(x, \theta, \bar\theta) \times {\tilde {\beta}}^{(h)}(x, \theta, \bar\theta) \Big)
\cdot \tilde\Phi^{(h)}_0(x, \theta, \bar\theta)\nonumber\\
&-& i {\tilde {\cal D}}_0{\tilde {\beta}}^{(h)}
(x, \theta, \bar\theta) \cdot {\tilde {\bar \beta}}^{(h)} (x, \theta, \bar\theta)\Big] \bigg|_{\theta = 0} \nonumber\\
& \equiv & i \frac {\partial}{\partial \theta} \frac {\partial}{\partial \bar \theta}
\int d^2 x \Big[ \tilde {\cal A}_0^{(h)} (x, \theta, \bar\theta) \cdot
{\tilde {\bar {\cal F}}}^{(h)} (x, \theta, \bar\theta)
+ \tilde \Phi_0^{(h)} (x, \theta, \bar \theta) \cdot {\tilde {\bar \beta}}^{(h)} (x, \theta, \bar \theta)\Big]. \label{A7}\end{aligned}$$ In the above, the second expression is valid on the constrained hypersurface parametrized by the CF conditions. The nilpotency of anti-BRST charge (i.e. $Q_{ab}^2 = 0$) is assured by the nilpotency $(\partial_\theta^2 = 0)$ of the Grassmannian derivative $\partial_\theta$, as described below $$\begin{aligned}
\frac {\partial}{\partial \theta} \; Q_{ab} \bigg|_{\bar \theta = 0} = 0
\quad \Longrightarrow \quad Q_{ab}^2 = 0.\end{aligned}$$ In 3D ordinary space, the above expression (\[A7\]) can be written in the following fashion $$\begin{aligned}
Q_{ab} &=& - \int d^2 x \; s_{ab}\; \Big [ \bar B(x) \cdot A_0(x) + i \; \dot C (x) \cdot \bar C (x)
+ \Big(\bar R(x) + i g \bar C(x) \times \beta(x)\Big)\cdot \phi_0(x) \nonumber\\
&+& i D_0 \beta(x) \cdot \bar \beta(x)\Big] \nonumber\\
&\equiv& - i\, \int s_{ab} s_b \,\Big[\, A_0 (x) \cdot \bar C(x) + \phi_0(x) \cdot \bar \beta(x)\Big].\end{aligned}$$ Here, the nilpotency of the anti-BRST charge lies in the equation $s_{ab} Q_{ab} = - i \{Q_{ab}, Q_{ab} \} = 0$ because of the fact that $s_{ab}^2 = 0$.
Furthermore, in order to prove the (anti-)BRST invariance of the coupled Lagrangian densities, within the framework of superfield formalism, we first generalize our starting Lagrangian density (${\cal L}_0$) onto the $(3,2)$-dimensional supermanifold, as follows $$\begin{aligned}
{\cal L}_0 \to \tilde {\cal L}_0 &=& - \frac{1}{4}\, \tilde {\cal F}^{\mu\nu (h)}
\cdot \tilde {\cal F}_{\mu\nu}^{(h)} - \;\frac{1}{4}\, \Big[\tilde {\cal G}^{\mu\nu (h)}
+ g \, \tilde {\cal F}^{\mu\nu (h)} \times \tilde \rho^{(h)} \Big]
\cdot \Big[\tilde {\cal G}_{\mu\nu}^{(h)} + g \, \tilde {\cal F}_{\mu\nu}^{(h)} \times \tilde \rho^{(h)} \Big] \nonumber\\
&+& \frac {m}{2}\,\varepsilon^{\mu\nu\eta} \, \tilde {\cal F}_{\mu\nu}^{(h)} \cdot \tilde \Phi_\eta^{(h)}.\end{aligned}$$ This Lagrangian density ($\tilde {\cal L}_0$) is free from the Grassmannian variables (cf. Section 3, for details). Therefore, the followings are true $$\begin{aligned}
\frac {\partial}{\partial \bar \theta} \; \tilde {\cal L}_0\bigg|_{\theta = 0} = 0, \qquad
\frac {\partial}{\partial \theta} \; \tilde {\cal L}_0\bigg|_{\bar \theta = 0} = 0, \end{aligned}$$ which captures the (anti-)BRST invariance of the starting Lagrangian density ${\cal L}_0$. Similarly, we can also generalize the coupled Lagrangian densities (\[4.2\]) onto the $(3,2)$-dimensional supermanifold in the following manner $$\begin{aligned}
{\cal L}_{\bar b} \longrightarrow \tilde {\cal L}_{\bar b} & =& \tilde {\cal L}_0 - \frac {\partial}
{\partial \theta}\, \frac {\partial} {\partial \bar \theta}\, \Big[ \frac {i} {2} \, \tilde {\cal A}_\mu^{(h)}
\cdot \tilde {\cal A}^{\mu (h)} + \tilde {\cal F}^{(h)} \cdot {\tilde {\bar {\cal F}}}^{(h)}
+ \frac {i} {2} \, \tilde \Phi_\mu^{(h)} \cdot \tilde \Phi^{\mu (h)}
+ \frac{1}{2}\, \tilde \beta^{(h)} \cdot {\tilde {\bar \beta}}^{(h)} \Big],\nonumber\\
&&\nonumber\\
{\cal L}_{b} \longrightarrow \tilde {\cal L}_{b} & = & \tilde {\cal L}_0 + \frac {\partial}
{\partial \bar \theta}\, \frac {\partial} {\partial \theta}\, \Big [ \frac {i} {2} \, \tilde {\cal A}_\mu^{(h)}
\cdot \tilde {\cal A}^{\mu (h)} + \tilde {\cal F}^{(h)} \cdot {\tilde {\bar {\cal F}}}^{(h)}
+ \frac {i}{2} \, \tilde \Phi_\mu^{(h)} \cdot \tilde \Phi^{\mu (h)}
+ \frac{1}{2}\, \tilde \beta^{(h)} \cdot {\tilde {\bar \beta}}^{(h)} \Big].\nonumber\\\end{aligned}$$ Now, the (anti-)BRST invariance of the above coupled Lagrangian densities is straightforward because of the fact $(\partial_\theta^2 = \partial_{\bar \theta}^2 = 0)$. Thus, we have $$\begin{aligned}
\frac {\partial}{\partial \theta}\; \tilde{\cal L}_{\bar b}\bigg|_{\bar \theta = 0} = 0,\qquad
\frac {\partial}{\partial \bar \theta}\; \tilde{\cal L}_b \bigg|_{\theta = 0} = 0,\end{aligned}$$ which imply the (anti-)BRST invariance of the coupled Lagrangian densities within the framework of superfield formalism.
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[^1]: The covariant derivative is defined as $D_\mu * = \partial_\mu * - g (A_\mu \times * )$.
| {
"pile_set_name": "ArXiv"
} |
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abstract: |
The Hopf algebra structure underlying Feynman diagrams which governs the process of renormalization in perturbative quantum field theory is reviewed. Recent progress is briefly summarized with an emphasis on further directions of research.\
---
\[section\] \[theorem\][Proposition]{} \[theorem\][Definition]{} \[theorem\][Lemma]{} \[theorem\][Corollary]{} \[section\] \[section\] \[theorem\][Remark]{} \[theorem\][Example]{} \[section\]
v
Ł ¶
ł
\
KURUSCH EBRAHIMI-FARD[^1]\
\
DIRK KREIMER[^2]\
\
December 6, 2005\
[`J. Phys. A: Math. Gen., 38, (2005), R385-R406.`]{}
\
Introduction and Overview
=========================
Quantum field theory (QFT) by now has a long and outstandingly successful history in all theories of physics. Merging the two major revolutionary achievements of early 20th century physics, quantum mechanics and special relativity, the founding fathers of QFT were setting out for an unified description of elementary particles phenomena. Its ideas and techniques found far reaching applications in different and very distinct areas of theoretical physics, and pure and applied mathematics.
Several approaches to QFT have been developed so far. Wightman’s early axiomatic [@StWh] setting leading to constructive QFT, together with Haag’s mathematically elegant and rigorous algebraic formulation of QFT in terms of von Neumann algebras [@Haag], best describes the nowadays common believe of what should be the general physical principles underlying any QFT. Still, despite the enormous and mathematically rigorous progress which has been made using these formulations, both approaches have several problems to make fruitful contact with experimental results, whilst they give a crucial insight into the structure of free quantum fields.
The perturbative approach to quantum field theory is the most successful. Theoretical predictions of physical quantities made by using their expansion in terms of –renormalized– Feynman graphs match experimental results with a vertiginous high precision. Nevertheless, in most, if not all, of the interesting and relevant 4-dimensional quantum field theories, performing even simple perturbative calculations one cannot avoid facing ill-defined integrals. The removal of these divergences in a sound way is the process of renormalization, better known by the illustrative description of ”sweeping them under the carpet". The basic idea of perturbative renormalization in QFT goes back to Kramers [@Brown], and was successfully applied for the first time in a 1947 seminal paper by Bethe [@Bethe], dealing with the concrete problem of the self energy contribution for the Lamb shift in perturbative quantum electrodynamics (QED). The latter can nowadays be regarded as one of the best tested physics theories. Its modern extension to the standard model of elementary particles represents one of the cornerstones of our present understanding of the physical world. Here again the perturbative treatment together with renormalization is the bread-and-butter of the practitioner in high energy physics.
Maintaining the physical principles of locality, unitarity, and Lorentz invariance, renormalization theory may be summed up by the statement that to all orders in perturbation theory the (ultraviolet) divergencies can be absorbed in a redefinition of the parameters defining the QFT. Here two distinct concepts enter, that of renormalizability, and the process of renormalization. The former distinguishes those theories with only a finite number of parameters, lending them considerably more predictive power. The process of renormalization instead works indifferently of the number of parameters.
Soon after Bethe’s paper on perturbative QED, there have been several approaches to establish that quantum field theories are renormalizable in general. Dyson [@Dyson1; @Dyson2] was the first to do so, using integral equations and skeleton expansions for Green’s functions. His work was then continued by Salam and Weinberg. Unfortunately, this attempt failed in the first instance, due to a problem related to a particular 14th order QED graph, but could be cured later. The second approach, based on earlier work by Stückelberg and Green, was taken by Bogoliubov and Parasiuk [@BP; @BS], using a recursive subtraction method, known as Bogoliubov’s $\mathrm{\bar{R}}$-map. Also their proof contained a loophole, but eventually found its final and satisfying form with the work of Hepp [@Hepp] and later Zimmermann [@Zimmermann]. This standard result is nowadays well-known under the name Bogoliubov–Parasiuk–Hepp–Zimmermann (BPHZ) renormalization prescription. Later, Epstein and Glaser [@EpGl] presented a rigorous proof of renormalizability situated in the realm of the axiomatic treatment of QFT. A fourth approach was taken by Blaer and Young [@BY], using the renormalization group equations, going back to a suggestion by Callan. At this point we refer the interested reader to consult the work by Caswell and Kennedy [@CaswellK], Brown [@Brown], Delamotte [@Delamotte], Collins [@Collins], and Wightman [@Wightman] for more references and details.
Notwithstanding its somewhat notorious reputation, renormalization theory, together with the gauge principle, forms the backbone of the perturbative approach to physically relevant quantum field theories. These days, the modern point of view, represented by the concept of Wilson’s renormalization group, elevates it even to a fundamental structure in the understanding of high energy physics.
Unfortunately, despite its accomplishments, renormalization theory was stigmatized, especially for its lack of a firm mathematical underpinning. Indeed, examining the current introductory and advanced literature on renormalization, as it is used in everyday applications in many branches of physics, one feels the need for a more conceptual picture unifying mathematical and computational aspects. A possible reason for this situation might have been the fact that its building blocks, the (one-particle irreducible) Feynman graphs in itself appeared to be unrelated to a sound mathematical structure that may underlie the renormalization prescription in perturbative QFT.
Almost five decades after Bethe’s work, this changed to a great extend with the original paper by Kreimer [@Kreimer1] introducing the notion of Hopf algebra. The ensuing work by Kreimer [@Kreimer2; @Kreimer6] and collaborators, especially those of Broadhurst and Kreimer [@BK1; @BK2; @BK3; @BK4], and Connes and Kreimer [@CK1; @CK2; @CK3] explored this new approach both in terms of its mathematical and physical content, as well as its computational aspects. The Hopf algebraic setting captures the combinatorial and algebraic aspects of the process for renormalization by organizing the Feynman graphs into a combinatorial Hopf algebra, $\mathcal{H}_F$, which is a connected graded commutative bialgebra, essentially characterized by its non-cocommutative coproduct structure map. The formulation of renormalization using Hopf algebras was completed in the work of Connes and Kreimer. It gives rise to an elegant and useful disentanglement of analytic and algebraic aspects of perturbative renormalization in general QFT, affirming the remark that ”Few physicists object nowadays to the idea that diagrams contain more truth than the underlying formalism\[...\]" made by Veltman and ’t Hooft in [@tHV].
In this review we will focus on an elementary introduction to the Hopf algebra structure on Feynman graphs combined with the description of a completely algebraic formulation of renormalization in terms of a factorization problem valid for any renormalization scheme, and based on a theorem by Atkinson and Spitzer’s identity [@EGK1; @EGK2].
Let us continue with some more details. The restricted dual of the Hopf algebra of Feynman graphs, denoted by $\mathcal{H}^*_{F}$, contains the group $\mathcal{G}:=char(\mathcal{H}_F,\mathbb{C})$ of characters, that is, algebra homomorphisms from $\mathcal{H}_F$ to the underlying base field $\mathbb{C}$. Feynman rules are understood as such linear and multiplicative maps, associating to each Feynman graph, seen as a collection of vertices and edges, its corresponding Feynman integral. This group of characters possesses a corresponding Lie algebra of derivations, or infinitesimal characters, $\mathcal{L}:=\partial
char(\mathcal{H}_F,\mathbb{C})$, which comes from a fundamental pre-Lie algebra structure on Feynman graphs.
The ill-defined Feynman integrals are plagued with ultraviolet divergences in general, and demand for a regularization prescription, where we replace the base field $\mathbb{C}$ by a (commutative and unital) algebra $A$ of Feynman amplitudes. Alternatively, we might consider Taylor expansions on the level of the integrands. Whichever way, this leads us to consider the space of $A$-valued, or regularized, linear maps ${\mathrm{Hom}}(\mathcal{H}_{F},A)$, which contains $\mathcal{G}_A:=char(\mathcal{H}_{F},A)$, the group of regularized characters, respectively its associated Lie algebra $\mathcal{L}_A:=\partial char(\mathcal{H}_{F},A)$. As a principal example serves dimensional regularization, where $A:=\mathbb{C}[\varepsilon^{-1},\varepsilon]]$, the field of Laurent series. In this context perturbative renormalization finds a compact formulation as a factorization problem in the group $\mathcal{G}_A$, to wit, the algebraic Birkhoff decomposition of Feynman rules [@CK1; @CK2]. The initial proof of the Connes–Kreimer factorization of regularized Feynman rules uses the property that Laurent series actually form a commutative Rota–Baxter algebra [@Baxter; @Rota1] with the pole part projection, $R:=R_{ms}$, as linear Rota–Baxter operator (minimal subtraction scheme map) fulfilling the Rota–Baxter relation (of weight 1) $$R(x)R(y)+R(xy)=R\big(R(x)y+xR(y)\big), \; \forall x,y \in A.$$ The linearity of $R$ permits to define a unital, but now non-commutative complete filtered Rota–Baxter algebra structure on the space ${\mathrm{Hom}}(\mathcal{H}_{F},A)$, with convolution as associative product. One of the fundamental results in the realm of commutative Rota–Baxter algebras is Spitzer’s classical identity [@EGK1; @EGK2; @RotaSmith; @Spitzer], and using its generalization to non-commutative Rota–Baxter algebras, together with Atkinson’s factorization theorem [@Atkinson] for Rota–Baxter algebras, one can show that the multiplicative factorization of Connes–Kreimer follows from an additive decomposition through the exponential map [@EGK1; @EGK2]. Hereby we realize Bogoliubov’s $\mathrm{\bar{R}}$-map as a special case of Spitzer’s identity.
We hope that this brief review will guide the reader to crucial aspects of the recent developments related to the Hopf algebraic description of renormalization theory. The long list of references is meant to indicate the rich spectrum of research directions triggered by this approach. The modest mathematical style, i.e. we do not strive for a rigorous theorem-proof presentation might help the interested and novice reader to get a glimpse of the new aspects which opened with the Hopf algebra point of view on perturbative renormalization. Some of the remarks made during the expository writing indicate points to be further developed. But, we should underline that this article is neither meant to be an introduction to (perturbative) quantum field theory nor to renormalization theory in general. Rather, we would like to focus on the by now well-understood and established combinatorial-algebraic picture that makes renormalization theory in perturbative QFT such a challenging and venerable subject from both, mathematical and physical perspectives. After reading this article and going back to the vast existing physics literature on renormalization theory in all its facets, the reader may get an idea of the interesting open questions related to its Hopf algebraic description.
In the following we will comment on assorted references with respect to their research directions in this field, in the hope to facilitate access to this developing subject. Due to the review character of this work and limited space none of these topics could be treated in full detail. We start by mentioning two recent papers [@FG3; @Manchon] devoted in great detail to the general Hopf algebra structure in renormalization theory. A more mathematical, but shorter summary was given in [@Boutet].
The initial discovery of the Hopf algebra structure grew out of a study of the number-theoretic properties of graphs with many subdivergences, see [@Kreimer9], which provides an overview of some of the results up to the year 1999, including the link between knot theory, Feynman graphs and number theory [@BK5].
The original work of Kreimer, and Connes and Kreimer [@CK1; @CK2; @CK3; @CK4; @Kreimer1; @Kreimer2] explores and settles the Hopf algebraic formulation of renormalization for general perturbative QFT, and links it to non-commutative geometry. It thereby establishes the Birkhoff decomposition for Feynman rules giving rise to an unexpected correspondence with the Riemann–Hilbert problem. In [@CORV; @FG1; @FG2; @GKM1; @Kastler1; @KW; @Kreimer4; @RV] further details were given. Malyshev’s work [@Malyshev1] shows the general character of Connes’ and Kreimer’s combinatorial Hopf algebra, applying it to Riemann surfaces in the context of ribbon Feynman graphs.
The link to Connes’ non-commutative geometry becomes evident in terms of a Hopf algebra of non-planar rooted trees [@CK1], solving a universal problem in Hochschild cohomology, and forming the role model for the Hopf algebraic structure of renormalization. This work renewed considerably the interest in Hopf algebraic aspects of combinatorics such as rooted tree Hopf algebras. In this context one must point out the work of G.-C. Rota [@Rota4] and his school, especially Rota’s and A. Joni’s seminal work [@JoniRota] from the late 1970ies, forming the starting point for the theory of incidence Hopf algebras, further developed in [@Ehrenborg; @Schmitt], see also [@FG3]. Holtkamp [@Holtkamp][^3] showed that the non-commutative version of Connes–Kreimer’s Hopf algebra of rooted trees is isomorphic to Loday and Ronco’s [@LodayRonco]. Aguiar and collaborators explored in more detail rooted tree Hopf algebras [@AguBS; @AguSot1; @AguSot2]. Hoffman in [@Hoffman] improved a result of Panaite [@Panaite], showing the isomorphism between the dual of Connes–Kreimer’s Hopf algebra and Grossmann–Larson’s [@G-L] rooted tree Hopf algebra (see also [@Foissy]). Brouder [@Brouder] explored the relation to Butcher’s seminal work on Runge–Kutta integration methods [@Butcher]. Turaev in [@Turaev1; @Turaev2] extended some combinatorial aspects of the Connes–Kreimer results, especially with respect to the notion of pre-Lie coalgebras, and thereby also gave a neat description of Connes–Kreimer’s rooted tree Hopf algebra.
Chapoton and Livernet [@ChapotonLivernet] described free pre-Lie algebras in terms of rooted tree operads. Mencattini and Kreimer [@KM1; @KM2] further analyzed the insertion and elimination Lie and pre-Lie algebraic structures of Feynman graphs [@CK4] in terms of infinite matrix representations.
In [@EGK1; @EGK2] the meaning of the Rota–Baxter relation in the context of Connes–Kreimer’s Birkhoff decomposition is investigated in detail, pointing out some parallels to the theory of classical integrable systems [@BBT; @STS1]. It thereby provides the algebraic underpinning for the factorization in terms of complete filtered Rota–Baxter algebras, Spitzer’s identity and Atkinson’s multiplicative decomposition theorem for Rota–Baxter algebras. This work was further extended in [@EG2; @EGGV] describing the combinatorics of renormalization in terms of unipotent triangular matrix representations, and their factorization capturing the process of renormalization analogously to the Birkhoff decomposition of Connes–Kreimer. An interesting application of renormalization techniques and Rota–Baxter algebras as described here to the iteration of symbols of pseudodifferential operators can be found in [@MP].
The work of Broadhurst and Kreimer [@BK1; @BK2; @BK3; @BK4] develops many computational and physics aspects. They show how to use the coproduct structure of the Hopf algebra to efficiently compute the forest formula and use the Hochschild cohomology of the algebra to resum the perturbative series. The latter two of the aforementioned references hence form the starting point for the latest work of Kreimer [@Kreimer7; @KreimerDS], and Bergbauer and Kreimer [@BergbauerKreimer2], putting emphasis on Hochschild cohomology of Hopf algebras as a source of locality, the Dyson–Schwinger equations, and even the Slavnov–Taylor identities for the couplings in generic gauge theories.
Finally, the authors in [@GKM2; @Malyshev2; @Sakakibara] started to analyze some aspects of renormalization group calculations in the Hopf algebra context.
The work [@BKK; @BW] (see also [@MUW]) is of more computational character, indicating the efficiency of the use of Hopf algebras in perturbative renormalization.
Several people [@BergbauerKreimer1; @Gudrun] investigated the link between the Hopf algebra of renormalization to the most rigorous approach to renormalization in perturbative QFT, provided by the Epstein–Glaser prescription.
Recently, progress has been made in the mathematical context of number theory, and motivic structures of Feynman integrals. The notion of equisingular connections was used to explore Tannakian categories and Galois symmetries in the spirit of differential Galois theory in [@CM1; @CM2; @CM3]. Underlying the notion of an equisingular connection is the locality of counterterms, which itself results from Hochschild cohomology. The resulting Dyson–Schwinger equation allows for gradings similar to the weight- and Hodge filtrations for the polylogarithm [@Kreimer3; @Kreimer8]. More concretely, the motivic nature of primitive graphs has been established very recently by Bloch, Esnault and Kreimer [@BlochEK].
Let us briefly outline the organization of the paper. Section \[section2\] introduces briefly the basic Lie and Hopf algebra structures in perturbative renormalization, including the pre-Lie composition of Feynman graphs and Bogoliubov’s $\bar{\mathrm{R}}$-operation. The next section uses perturbative QED as a simple example to manifest the aforementioned notions. In Section \[section4\] we formulate the process of renormalization in perturbative QFT as a factorization problem in purely algebraic terms. Emphasis is put on the freedom in choosing a particular regularization prescription, captured via the notion of commutative unital Rota–Baxter algebra. Section \[section5\] outlines the use of the Birkhoff decomposition introduced in the former section on the level of diffeomorphisms of coupling constants, in the realm of dimensional regularization together with the minimal subtraction scheme as a particular useful renormalization prescription. The review ends with a brief section on the role of Hochschild cohomology in perturbative renormalization. In an appendix we collect some general facts about Rota–Baxter algebras as they form the main ingredient for Section \[section4\].
From the Lie and Hopf algebras of graphs to Bogoliubov’s formula {#section2}
================================================================
In this section we describe the elementary Lie and Hopf algebra structures underlying perturbation theory. The reader looking for a mathematical rigorous and detailed presentation of Hopf algebras and related aspects is referred to the standard texts such as [@Abe; @FGV; @Kassel; @Sw] (see also [@FG3; @Manchon]).
Let $\mathbb{K}$ be a field of characteristic zero. All $\mathbb{K}$-algebras, denoted by a triple $(A,m,\eta)$, where $A$ is a $\mathbb{K}$-vector space with a product $m: A \otimes A \to
A$ and a unit map $\eta: \mathbb{K} \to A$, are supposed to be associative. Similarly for coalgebras, denoted by the triple $(C,\Delta, \bar{e})$, where the coproduct map $\Delta: C \to C
\otimes C$ fulfills coassociativity, i.e. $(\Delta \otimes {\mathrm{id}})
\circ \Delta=({\mathrm{id}}\otimes\Delta) \circ \Delta$, and $\bar{e}: C \to
\mathbb{K}$ is the counit map. The identity will be denoted by $\One$. All algebra homomorphisms are supposed to be unital. A bialgebra, denoted by a quintuple $\left( H =
\bigoplus_{i=0}^\infty H_{(i)}, m, \eta, \Delta, \bar{e} \right)$ consists of an algebra and coalgebra structure in a compatible way. Here, $\Delta: H \to H \otimes H$ is the coproduct, $m: H
\otimes H \to H$ the product. These maps together with the counit $\bar{e}: H\to \mathbb{K}$ fulfil the standard bialgebra axioms. See the above general references for details. It is called connected graded if $H_{(i)} H_{(j)} \subset H_{(i+j)}$ and $\Delta(H_{(i)}) \subset \bigoplus_{j+k=i} H_{(j)} \otimes
H_{(k)},$ and if $\Delta(\One)=\One\otimes\One$ and $H_{(0)}=\mathbb{K}\One$. The counit simply is $\bar{e}(\One) =
1\in\mathbb{K}$ and $\bar{e}=0$ on $\bigoplus_{i=1}^\infty
H_{(i)}.$
We call $\ker\bar{e}$ the augmentation ideal of $H$ and denote by $P$ the projection $H \rightarrow \ker\bar{e}$ onto the augmentation ideal, $P = {\mathrm{id}}- \eta\bar{e}.$ Furthermore, we use Sweedler’s notation $\Delta(h)=\sum h^\prime \otimes
h^{\prime\prime}$ for the coproduct. Let us define $${\rm Aug}^{(k)}= \big(\underbrace{P\otimes\cdots \otimes P}_{k\;{\rm times}}\big)\,
\Delta^{k-1},\;\; H \to \{\ker\bar{e}\}^{\otimes k},$$ as a map into the $k$-fold tensor product of the augmentation ideal. Here, $\Delta^{k-1}$ is defined inductively by $\Delta^0:={\mathrm{id}}$, and $\Delta^n:=(\Delta^{n-1}\otimes
{\mathrm{id}})\circ\Delta$ for $n>0$. We let $${\sl H}^{(k)}=\ker{\rm Aug}^{(k+1)}/\ker{\rm Aug}^{(k)},$$ $\forall k \geq 1$. All bialgebras considered here are bigraded in the sense that $$H=\bigoplus_{i=0}^\infty H_{(i)}=\bigoplus_{k=0}^\infty {\sl H}^{(k)},$$ where $H_{(k)} \subset \oplus_{j=1}^k {\sl H}^{(j)}$ for all $k
\geq 1$. $H_{(0)}\simeq {\sl H}^{(0)}\simeq \mathbb{K}$.
While these algebraic notions may seem rather abstract, they indeed govern the structure of quantum field theory. To understand how, we have first to study the pre-Lie algebra structure of one-particle irreducible Feynman graphs.
The Pre-Lie Structure of Feynman Graphs {#subsection1}
---------------------------------------
For each quantum field theory, we have an underlying free theory which provides propagators and hence Feynman rules for edges in Feynman graphs[^4]. The request for local interactions and a renormalizable theory then gives us Feynman rules for interactions. With those graphs come the sets of one-particle irreducible (1PI) graphs contributing to a chosen amplitude. The amplitudes are distinguished by the external fields, asymptotically free fields represented by external edges in the graphs. We call this an external leg structure, denoted by $\underline{r}$. For a renormalizable theory, there is a finite number of such external leg structures, one for each monomial in the Lagrangian.
For each such Feynman graph we hence have vertices as well as internal and external edges. External edges are edges which have an open end not connected to a vertex. They indicate the particles participating in the scattering amplitude under consideration and each such edge carries the quantum numbers of the corresponding free field. The internal edges and vertices form a graph in their own right. For an internal edge, both ends of the edge are connected to a vertex. For a graph $\Gamma$ we denote by $\Gamma^{[0]}$ its set of vertices and by $\Gamma^{[1]}:=\Gamma^{[1]}_{\rm int} \cup \Gamma^{[1]}_{\rm ext}$ its set of internal and external edges. Furthermore, $\omega_{\underline{r}}$ is the number of space-time derivatives appearing in the corresponding monomial in the Lagrangian.
We are considering 1PI Feynman graphs. By definition a graph $\Gamma$ is 1PI if and only if all graphs, obtained by removal of any one of its internal edges, are still connected. Such 1PI graphs are naturally graded by their number of independent loops, the rank of their first homology group $H_{[1]}(\Gamma,\mathbb{Z})$. We write $|\Gamma|$ for this degree of a graph $\Gamma$. Note that $|{\bf res}(\Gamma)|=0$, where we let ${\bf res}(\Gamma)$, called the residue of $\Gamma$, be the graph obtained when all edges in $\Gamma^{[1]}_{\rm int}$ shrink to a point. The graph we obtain in this manner consists of a single vertex, to which the edges $\Gamma^{[1]}_{\rm ext}$ are attached. In case the initial graph was a self-energy graph, we regard its residue as a single edge. We denote the set of all external leg structures $\underline{r}$ by $\mathfrak{R}$. For a renormalizable QFT it consists of the edges and vertices corresponding to the monomials in the Lagrangian.
Having specified free quantum fields and local interaction terms between them, one immediately obtains the set of 1PI graphs, and can consider for a given external leg structure $\underline{r}$ the set $M_{\underline{r}}$ of graphs with that external leg structure. The Green’s function for the corresponding amplitude is then obtained as the evaluation under the Feynman rules of the formal sum $$\Gamma^{\underline{r}}:= 1 + \sum_{{\bf res}(\Gamma)={\underline{r}}}
\alpha^{|\Gamma|}\frac{\Gamma}{{\rm sym}(\Gamma)},$$ where we divide by the symmetry factor ${{\rm sym}(\Gamma)}$ and $\alpha$ is a small parameter like (the square of) a coupling constant. These sums exhibit rich structure thanks to the algebraic structures of the single graphs [@KreimerDS] to be discussed now.
For a renormalizable theory, we can define a superficial degree of divergence $$\omega= \sum_{\underline{r}\in \Gamma^{[1]}_{\rm int}\cup
\Gamma^{[0]}}\omega_{\underline{r}}-4|H_{[1]}(\Gamma,\mathbb{Z})|,$$ for each such external leg structure: $\omega(\Gamma) =
\omega(\Gamma^\prime)$ if ${\bf res}(\Gamma) = {\bf
res}(\Gamma^\prime)$, all graphs with the same external leg structure have the same superficial degree of divergence. Only for a finite number of distinct external leg structures $\underline{r}
\in \mathfrak{R}$ will this degree indeed signify a divergence. Our first observation is that there is a natural pre-Lie algebra structure on 1PI graphs.
To see this, we define a bilinear operation on graphs $$\Gamma_1 * \Gamma_2 = \sum_\Gamma n(\Gamma_1,\Gamma_2;\Gamma)\Gamma,$$ where the sum is over all 1PI graphs $\Gamma$. Here, $n(\Gamma_1,\Gamma_2;\Gamma)$ is a section coefficient which counts the number of ways a subgraph $\Gamma_2$ in $\Gamma$ can be reduced to a point such that $\Gamma_1$ is obtained. The above sum is evidently finite as long as $\Gamma_1$ and $\Gamma_2$ are finite graphs, and the graphs which contribute necessarily fulfill $|\Gamma| = |\Gamma_1| + |\Gamma_2|$ and ${\bf res}(\Gamma) = {\bf
res}(\Gamma_1)$.
One then has:
The operation $\ast$ is pre-Lie: $$[\Gamma_1\ast\Gamma_2]\ast \Gamma_3 - \Gamma_1\ast[\Gamma_2\ast \Gamma_3] =
[\Gamma_1\ast \Gamma_3]\ast \Gamma_2 - \Gamma_1\ast[\Gamma_3\ast \Gamma_2].$$
This is evident when one rewrites the $*$ product in suitable gluing operations, using the dichotomy of inserting in nested or disjoint manner. See [@EMK; @Kreimer5; @Kreimer10; @Kreimer11] for more details.
Note that the equation claims that the lack of associativity in the bilinear operation $\ast$ is invariant under permutation of the elements indexed by $2,3$. This suffices to show that the anti-symmetrization of this map fulfils the Jacobi identity. Hence we get a Lie algebra ${\cal L}$ by anti-symmetrizing this operation: $$[\Gamma_1,\Gamma_2] = \Gamma_1\ast\Gamma_2-\Gamma_2\ast\Gamma_1.
\label{Lie}$$ This Lie algebra is graded and of finite dimension in each degree. Let us look at a couple of examples for pre-Lie products. We take graphs from quantum electrodynamics (QED) as a rather self-evident example. For the graphs $\gg$ and $\epemg$ with residues ${\bf
res}(\!\gg\!) = \!\!\begin{array}{c}\\[-.6cm] \scalebox{0.5}{\BOSONprop}\end{array}\!$ respectively ${\bf res}(\! \epemg
\!)=\!\!\begin{array}{c}\\[-.8cm] \!\scalebox{0.5}{\FERMprop}{}\end{array}\!$, we find $$\begin{aligned}
\gg\ast \epemg & = & 2\ggv.
\label{pre-LieExam}
\end{aligned}$$ Together with ${\cal L}$ one is led to consider the dual of its universal enveloping algebra ${\cal U}({\cal L})$ using the theorem of Milnor and Moore [@MilnorMoore]. For this we use the above grading by the loop number.
This universal enveloping algebra ${\cal U}({\cal L})$ is build from the tensor algebra $${\bf T}=\bigoplus_k T^k,\;T^k=\underbrace{{\cal L}\otimes\cdots\otimes {\cal L}}_{k\; {\rm times}},$$ by dividing out the ideal generated by the relations $$a\otimes b - b \otimes a = [a,b] \in {\cal L}.$$ Note that in ${\cal U}({\cal L})$ we have a natural concatenation product $m_*$. Even more, ${\cal U}({\cal L})$ carries a natural Hopf algebra structure with this product. For that, the Lie algebra ${\cal L}$ furnishes the primitive elements: $$\Delta_*(a)=a\otimes 1+1\otimes a,\;\forall a\in {\cal L}.$$ It is by construction a connected finitely graded Hopf algebra which is cocommutative but not commutative.
We can then consider its graded dual which will be a Hopf algebra ${\cal H}_{F}(m,\eta,\Delta,\bar{e})$ which is commutative but not cocommutative. One finds the coproduct $\Delta$ upon using a Kronecker pairing $$<Z_\Gamma,\delta_{\Gamma^\prime}>= \begin{cases} 1,\;\Gamma=\Gamma^\prime \\
{0,\; {\rm
else}}.\end{cases}$$ From there, one determines all other structure maps with ease, demanding that $$\langle Z_{[\Gamma_2,\Gamma_1]},\delta_\Gamma \rangle =
\langle Z_{\Gamma_1}\otimes Z_{\Gamma_2} - Z_{\Gamma_2}\otimes Z_{\Gamma_1},\Delta(\delta_\Gamma)\rangle.$$ In the above, we distinguished carefully between graphs $\Gamma$ as generators of the Lie algebra, denoted by $Z_\Gamma$, and graphs $\Gamma$ as generators of the Hopf algebra, denoted by $\delta_\Gamma$. The Lie algebra of graphs exponentiates to the character group of the Hopf algebra as explained below, eventually leading to Birkhoff factorization in that group.
The space of primitives of ${\cal U}({\cal L})$ is in one-to-one correspondence with the set ${\rm Indec}({\cal H}_{F})$ of indecomposables of ${\cal H}_{F}$, which is the linear span of its generators.
${\cal H}_{F}$ is a connected graded commutative Hopf algebra which describes renormalization theory. It operates on the superficially divergent 1PI Feynman graphs of the theory. The residues of these graphs are in one-to-one correspondence with the terms in the Lagrangian of a given theory. Often it is the case that several terms in a Lagrangian correspond to graphs with the same number and type of external legs, but match to different form-factor projections of the graph. In such cases, the above approach can be easily adopted. Below in Section \[section3\] we give an example for QED, incorporating its form-factor decomposition into our approach.
Bogoliubov’s recursive subtraction formula {#subsection2}
------------------------------------------
The above algebra structures are available once one has decided on the set of 1PI graphs of interest. Those one-particle irreducible graphs $\Gamma$ provide the generators $\delta_\Gamma$ of the Hopf algebra ${\cal H}_{F}=\oplus_{i=0}^\infty \H_{(i)}$, where ${\cal
H}_{ F, {\rm{lin}}}:={\rm span}(\delta_\Gamma)$, with their disjoint union providing the commutative product, which we denote by juxtaposition.
Let $\Gamma$ be a 1PI graph. The Hopf algebra ${\cal H}_{F}$ described above comes out to have a coproduct $\Delta:{\cal H}_{F}
\to {\cal H}_{F} \otimes {\cal H}_{F}$: $$\Delta(\Gamma) = \Gamma \otimes \One + \One \otimes\Gamma +
\sum_{\gamma{\subset}\Gamma}\gamma\otimes\Gamma/\gamma,$$ where the sum is over all unions of 1PI superficially divergent proper subgraphs, and we extend this definition to products of graphs, $\Delta(\Gamma_1\Gamma_2)=\Delta(\Gamma_1)\Delta(\Gamma_2)$, so that we get a bialgebra.
While the pre-Lie product respectively the Lie bracket inserted graphs into each other, dually the coproduct disentangles them. This is precisely what we make use of in renormalization theory: we have to render each subgraph finite before we can construct a local counterterm. Having a coproduct, two further structure maps of ${\cal H}_{F}$ are immediate, the counit and the antipode. The counit $\bar{e}$ vanishes on any non-trivial Hopf algebra element, $\bar{e}(X)=0$, $X \neq \One$, but $\bar{e}(\One )=1$. The antipode $S:{\cal H}_{F} \to {\cal H}_{F}$ is given by $$S(\Gamma) = -\Gamma- \sum_{\gamma{\subset}
\Gamma}S(\gamma)\Gamma/\gamma,$$ for $\Gamma \in \ker \bar{e}$, and $S(\One)=\One$. We can work out examples for the coproduct of a graph: $$\begin{aligned}
\Delta\big( \!\! \ggv \!\!\big) & = &\!\! \ggv\!\!\otimes \One + \One \otimes\!\! \ggv + 2\epemg\otimes \gg.
\end{aligned}$$ And an antipode: $$S\big(\!\!\!\! \begin{array}{l} \\[-0.4cm]
\epemfg \end{array} \!\!\!\!\big) = -\!\!\begin{array}{l} \\[-0.4cm]
\epemfg \end{array} \!\!+ \gg\epem.$$ We note in passing that the gluing operation underlying the pre-Lie insertion of graphs relies on gluing data which can be reconstructed from the subgraphs $\gamma$ and cographs $\Gamma/\gamma$ in the above coproduct. This is crucial in the proof of locality of counterterms upon studying the Hochschild cohomology of this Hopf algebra [@Kreimer10].
We have by now obtained a Hopf algebra generated by combinatorial elements, 1PI Feynman graphs. Its existence is indeed automatic once one has chosen interactions and free fields.
As disjoint scattering processes give rise to independent (divergent) amplitudes one is led to the study of regularized characters of the Hopf algebra, to say $A$-valued maps $\phi:
{\cal H}_{F} \to A$ such that $\phi \circ m=m_A \circ (\phi
\otimes \phi)$. Here $A$ denotes a unital commutative algebra.
Usually, Hopf algebra characters, i.e., linear multiplicative maps, assign to any element in the Hopf algebra an element in the base field, and form a group under convolution, denoted by $\mathcal{G}$. Motivated by the need for regularizing our theory, due to ultraviolet (UV) divergencies showing up in higher loop calculations, we take here a slightly more general point of view, replacing the base field as target space, say $\mathbb{C}$, by a suitable commutative and unital algebra, $A$, of -regularized- Feynman amplitudes. The group of regularized, or $A$-valued, Hopf algebra characters is denoted by $\mathcal{G}_A$, and the group law is given by the convolution product $$\phi_1 \star \phi_2 := m_A \circ (\phi_1 \otimes \phi_2) \circ \Delta,
\label{def:convol}$$ so that the coproduct, counit and coinverse (the antipode) give the product, unit and inverse of this group, as befits a Hopf algebra.
The study of tree-level amplitudes in lowest order perturbation theory justifies to assign to each edge a propagator and to each elementary scattering process a vertex which define the Feynman rules $\phi({\rm \bf res}(\Gamma))$ and the underlying Lagrangian, on the level of residues of these very graphs. With the Feynman rules providing a canonical character $\phi$, we will have to make one further choice: a renormalization scheme. The need for such a choice is no surprise: after all we are eliminating short-distance singularities in the graphs which renders their remaining finite part ambiguous, albeit in a most interesting manner.
We choose a $\mathbb{K}$-linear map $R:A \to A$, from which we obviously demand that it does not modify the UV-singular structure, i.e., $R^2=R$, and furthermore that it obeys $$R(x)R(y) + R(xy) = R\big(R(x)y) + R(xR(y)\big),
\label{RB}$$ an equation which guarantees the multiplicativity of renormalization and lies at the heart of the Birkhoff decomposition which emerges below: it tells us that elements in $A$ split into two parallel subalgebras given by the image and kernel of $R$. Algebras for which such a map exists fall into the class of Rota–Baxter algebras, well-known in mathematics. See below in Section \[section4\] (and Appendix \[appendix:RotaBaxter\]) for more details .
Let us take a shortcut for the moment and see how all the above structure comes together in renormalization theory. Starting with a regularized Feynman rules character $\phi$, we define a further character $S_ R^\phi$ which deforms $\phi \circ S$, the inverse of the $\phi$, slightly and delivers the counterterm for $\Gamma$ in the renormalization scheme $R$: $$\label{counterterm}
S_R^\phi(\Gamma)=-R\big[m_A(S_R^\phi\otimes\phi\circ P)\Delta(\Gamma)\big]=
-R[\phi(\Gamma)]-R\left[\sum_{\gamma{\subset}\Gamma}
S_R^\phi(\gamma)\phi(\Gamma/\gamma)\right],$$ for $\Gamma$ in $\ker\bar{e}$. Comparing with the undeformed inverse of $\phi$ $$\phi \circ S(\Gamma) = m_A(\phi \circ S \otimes\phi \circ P)\Delta(\Gamma)=
-\phi(\Gamma)-\sum_{\gamma{\subset} \Gamma}\phi\circ S (\gamma)\phi(\Gamma/\gamma)$$ allows to easily understand finiteness of renormalized quantities, thanks to the independence of counterterms on kinematical variables. Later, in Section \[section4\] we will fully derive Equation (\[counterterm\]) and the results below from a more mathematical point of view from the fact that $R$ is a Rota–Baxter operator.
We conclude that $S_R^\phi$ is an element of the group of regularized characters, ${\cal G}_{A}$, of the Hopf algebra, $S_R^\phi\in {\rm Spec}({\cal G}_{A})$. We now have determined the renormalized Lagrangian: $$Z^{\underline{r}}=S_R^\phi(\Gamma^{\underline{r}}).$$ The standard results of renormalization theory follow immediately using the group of regularized characters: the renormalization of a graph $\Gamma$ is obtained by the application of a renormalized character, $S_R^\phi \star \phi$ $$S_R^\phi\star\phi(\Gamma)=m_A(S_R^\phi\otimes\phi)\Delta(\Gamma)$$ for $\Gamma \in \ker \bar{e}$, and Bogoliubov’s $\bar{\mathrm{R}}$-operation is obtained as $$\label{BogosBARmap}
\bar{\mathrm{R}}(\Gamma)=m_A(S_R^\phi\otimes\phi)({\rm id}\otimes P)\Delta(\Gamma)=
\phi(\Gamma)+ \sum_{\gamma{\subset}\Gamma}
S_R^\phi(\gamma)\phi(\Gamma/\gamma).$$ In the following we write $\bar{R}(\Gamma)=:\bar{\phi}(\Gamma)$, so that we have $$S_R^\phi\star\phi(\Gamma)=\bar{\phi}(\Gamma) + S_R^\phi(\Gamma).$$ $S_R^\phi\star\phi$ is an element in the group of regularized characters, $\mathcal{G}_A$, of the Hopf algebra. This Lie group has indeed the previous Lie algebra ${\cal L}$ of graph insertions as its Lie algebra: ${\cal L}$ exponentiates to ${\cal G}_A$.
What we have achieved at this moment is a local renormalization of quantum field theory. Let $m_{\underline{r}}$ be a monomial in the Lagrangian $L$ of degree $\omega_{\underline{r}}$, $$m_{\underline{r}}=D_{\underline{r}}\{\varphi\},$$ for some suitable derivation $D_{\underline{r}}$ on the fields $\varphi$. Then one can prove using the Hochschild cohomology and induction over the augmentation degree in ${\cal H}_F$:
(locality) $$Z^{\underline{r}}D_{\underline{r}}\{\varphi\}=D_{\underline{r}}Z^{\underline{r}}\{\varphi\},$$ renormalization commutes with infinitesimal space-time variations of the fields.
Let us finally give the renormalization of a Feynman graph, say $\Gamma=\!\!\ggv$.
Example: QED {#section3}
============
The QED Lagrangian (density) for an electron coupled to the electromagnetic field in coordinate space reads (we skip the $1/2(\partial \cdot A)^2$ term) $$\label{def:QED}
L_{QED}(\alpha,m) = i\bar{\psi} \partial\!\!\!/ \psi
+ \bar{\psi} eA\!\!\!/ \psi
+ m \bar{\psi} \psi
+ \frac{1}{4} F^2,$$ with the electromagnetic field tensor $F_{\mu\nu}:=\partial_\mu
A_\nu - \partial_\nu A_\mu$ and $F^2:=F_{\mu\nu}F^{\mu\nu}$. The Euler-Lagrange equations for this term give the Maxwell equations. For the first and third term they give the Dirac equation. We use units defined by $c=\hbar=1$, for which the elementary charge squared, $e^2=4\pi \alpha_{QED}$. It is a dimensionless quantity depending on the fine-structure constant $\alpha:=\alpha_{QED}
\simeq \frac{1}{137}$. The first term in (\[def:QED\]) describes the kinetic energy of the electron, and together with the mass term, $m \bar{\psi} \psi$, it constitutes the free Lagrangian density for an electron. The second term in (\[def:QED\]) describes the minimal coupling of the electron with the electromagnetic field. The first two terms are dictated by local gauge invariance of the QED Lagrangian, i.e., invariance with respect to multiplying the electron field $\psi$ by a position dependent phase factor. Let us introduce for every term in $L_{QED}$ a 1PI Green’s function $G^{r}(p^2,\alpha,m,\mu)$, with $$\label{def:QEDmonomials}
r \in \left\{ i\bar{\psi} \partial\!\!\!/\psi ,\
\bar{\psi}e A\!\!\!/ \psi,\
m \bar{\psi} \psi ,\
\frac{1}{4}F^2
\right\},$$ all of which transform as scalars under the Lorentz group.
Usually, 1PI Green’s functions of QED are given in standard notation by the expressions
1. $\Gamma_{\nu}(p_1,p_2,m,\alpha,\mu)$ the vertex function;
2. $S_{\rm F}^{-1}(p,m,\alpha,\mu)$ the inverse fermion propagator;
3. $P^{-1}_{\nu\tau}(p,m,\alpha,\mu)$ the inverse photon propagator,
all depending on the bare parameters mass $m$ and coupling constant $\alpha$, and ’t Hooft’s parameter $\mu$ which sets the scale for the one-parameter groups of automorphisms of the Lie algebra which run the renormalization group [@CK2]. An extra parameter, such as ’t Hooft’s unit mass $\mu$, enters naturally in the context of regularization, and is needed for dimensional reasons [@Collins]. From symmetry considerations we find the following form factor decompositions, where we now assume the vertex at zero momentum transfer for simplicity:
1. $
\Gamma_{\nu}(p,p,m,\alpha,\mu)
= e G^{\bar{\psi} A\!\!\!/ \psi}(p^2,m,\alpha,\mu)\gamma_{\nu}
+ eH^{\bar{\psi} A\!\!\!/ \psi}(p^2,m,\alpha,\mu)\frac{p\!\!/ p_{\nu}}{p^2}
$;
2. $
S^{-1}(p,m,\alpha,\mu)=G^{\bar{\psi} p\!\!/ \psi}(p^2,m,\alpha,\mu)p\!\!/
+ G^{ m\bar{\psi} \psi}(p^2,m,\alpha,\mu)m\mathbb{I}
$;
3. $
P^{-1}_{\nu\tau}(p,m,\alpha,\mu)
= \Pi_{\rm tr}^{\frac{1}{4} F^2}(p^2,m,\alpha,\mu)\left\{g_{\nu\tau}p^2 - p_\nu p_\tau \right\}
+ \Pi_{\rm long}^{\frac{1}{2}(\partial\cdot A)^2}(p^2,m,\alpha,\mu)p_\nu p_\tau
$ ,
reintroducing a longitudinal term for completeness. Let us introduce a graphical notation for the monomials of the QED Lagrangian (\[def:QED\]), which will form the building blocks of our graphical Hopf algebra of Feynman graphs. $$\bar{\psi} \partial\!\!\!/ \psi \
\longleftrightarrow
\begin{array}{ccc}
\FERMprop\\[-0.1cm]
{\tiny{\partial}}
\end{array}
\qquad\quad
m \bar{\psi} \psi \
\longleftrightarrow
\begin{array}{c}
\FERMprop\\[-0.3cm]
{\tiny{m}}
\end{array}
\qquad\quad
\begin{array}{cc}
&\\
\bar{\psi} eA\!\!\!/ \psi \ \longleftrightarrow & \\ [-0.9cm]
& \QEDvertex
\end{array}$$ The first two graphs on the left represent the electron propagator parts, corresponding to the derivation and mass contribution in (\[def:QED\]), respectively. The third graph is the QED vertex, representing the interaction of the electromagnetic field with fermions in (\[def:QED\]). To ease the notation we suppressed spinorial indices.
Next we have the transversal respectively longitudinal parts of the bosonic photon propagator, $$ \frac{1}{4} F^2\
\longleftrightarrow
\begin{array}{c}
\BOSONprop\\[-0.3cm]
{\tiny{ t}}
\end{array}
\qquad \quad
1/2(\partial \cdot A)^2\
\longleftrightarrow
\begin{array}{c}
\BOSONprop\\[-0.3cm]
{\tiny{\rm long}}
\end{array}$$ In the following, we will work with a transversal photon propagator for concreteness. Once we have the graphical notation we define the -coordinate space- QED Feynman rules $\widehat{\phi}$ such that for any $\underline{r} \in
\mathfrak{R}_{QED}$ $$\label{def:QEDgraphs}
\mathfrak{R}_{QED}:=\left\{
\begin{array}{cccc}
\begin{array}{ccc}
\FERMprop\\[-0.1cm]
{\tiny{\partial}}
\end{array},
&
\begin{array}{c}
\FERMprop\\[-0.3cm]
{\tiny{m}}
\end{array},
&
&,\
\begin{array}{c}
\\[-0.2cm]
\BOSONprop\\[-0.3cm]
{\tiny{t}}
\end{array} \\[-1.3cm]
& & \QEDvertex\!\! &
\end{array}\right\}$$ we get back the corresponding coordinate space QED Lagrange monomial $r$ in (\[def:QEDmonomials\]) $$\widehat{\phi}(\underline{r}) = r.$$ We write the QED Lagrangian (\[def:QED\]) pictorially $$\label{def:graphQED}
L_{QED}(\alpha,m) = \sum_{\underline{r}\in \mathfrak{R}_{QED} }
\widehat{\phi}(\underline{r}).$$ With the QED free propagators and vertex in $\mathfrak{R}_{QED}$ at hand we have available the one-particle irreducible Feynman diagrams which provide amplitudes corresponding to these propagations and interaction, as before. We can now introduce partitions of unity for the form-factor decomposition of any Green’s function we are interested in. For example, if we do want to decompose the self-energy of the fermion into its kinetic energy and mass part We can easily incorporate this by promoting our Hopf algebra to pairs $(\Gamma,\sigma)$ [@CK1], where $\sigma$ indicates the desired form-factor obtained by composing the Feynman rules with a suitable projector as above.
The sum over all projectors defines a partition of unity $${\rm id}=\sum_\sigma P_\sigma.$$ This structure can be easily incorporated on the level of Hopf algebras, generalizing the study of external structures by setting for the pairs $(\Gamma,\sigma)$ $$\Delta(\Gamma,\sigma)=\sum (\Gamma^\prime,1)\otimes (\Gamma^{\prime\prime},\sigma).$$ Note that should we wish we can partition the unity on the left hand side above, $$(\Gamma^\prime,1)\to
(\Gamma^\prime,\sigma_R)$$ if we want to use information that only particular form-factors $\sigma_R$ need renormalization. Under the Feynman rules these pair of graphs then evaluate to the amplitudes corresponding to the structure functions defined by the projectors signified by the indicated external leg structures. The resulting Hopf algebras for such pairs $(\Gamma,\sigma)$ are decorated versions of the ones for graphs only, and define graph-like structures very similar to the tree-like structures of Turaev for the Hopf algebra of rooted trees [@Turaev1]. Examples can be found in [@Delbourgo-Kreimer; @BK1].
As an example, we might wish to renormalize the mass part of $\epemfg$, using the knowledge that the photon self-energy is transversal. We hence work out the coproduct $$\Delta\big((\!\!\!\!\begin{array}{l} \\[-0.4cm]
\epemfg \end{array}\!\!\!\!,\sigma_m)\big)= (\!\!\!\!\begin{array}{l} \\[-0.4cm]
\epemfg \end{array}\!\!\!\!,\sigma_m)\otimes \One+
\One\otimes(\!\!\!\!\begin{array}{l} \\[-0.4cm]
\epemfg \end{array}\!\!\!\!,\sigma_m)+(\gg,\sigma_{\rm trans})\otimes (\epem,\sigma_m).$$ Under the Feynman rules, we evaluate using the corresponding projectors and obtain the expected Lorentz scalar structure functions and counterterms following the routine as outlined in Eqs. (\[renorm\]).
Renormalization as a factorization problem {#section4}
==========================================
As we have seen the notion of connected graded commutative Hopf algebra appears naturally in the context of perturbative renormalization of 1PI Feynman graphs. Both, composing Feynman graphs in terms of the pre-Lie insertion product, where we replace vertices by Feynman graphs with compatible external leg structure, as well as their decomposition by eliminating subgraphs, i.e., replacing non-trivial 1PI subgraphs by their residues, look very familiar when inspecting the subtraction procedure encoded in the original BPHZ prescription [@BP; @BS; @CaswellK; @Collins]. The later was invented to extract the finite part of the Feynman integral corresponding to a Feynman graph via a regularized Feynman rules character, while maintaining fundamental physical principles, such as locality, unitarity, and Lorentz invariance.
The commutative Hopf algebra of Feynman graphs, $\mathcal{H}_{F}$, and its graded dual, $\mathcal{H}^*_{F}=\mathrm{Hom}(\mathcal{H}_{F},\mathbb{C})$ are intimately related by the Milnor–Moore theorem. The space $\mathcal{H}^*_{F}$ together with the convolution product and the counit map $\bar{e}: \mathcal{H}_{F} \to \mathbb{C}$ as unit forms a unital, associative and non-commutative $\mathbb{C}$-algebra, which contains the group of characters, $\mathcal{G}:=char(\mathcal{H}_{F},\mathbb{C})$, i.e., linear functionals $\phi \in \mathcal{H}^*_{F}$ from $\mathcal{H}_{F}$ to $\mathbb{C}$ respecting multiplication, $\phi(\Gamma_1
\Gamma_2)=\phi(\Gamma_1)\phi(\Gamma_2)$, $\Gamma_1$, $\Gamma_2 \in
\mathcal{H}_{F}$. This group of multiplicative maps possesses a corresponding Lie algebra, $\mathcal{L}=\partial
char(\mathcal{H}_{F},\mathbb{C}) \subset \mathcal{H}^*_{F}$, of derivations, or infinitesimal characters, i.e., linear maps $Z \in
\mathcal{H}^*_{F}$, satisfying Leibniz’ rule $$Z(\Gamma_1 \Gamma_2)= Z(\Gamma_1)\bar{e}(\Gamma_2)
+\bar{e}(\Gamma_1) Z(\Gamma_2)$$ for all $\Gamma_1$, $\Gamma_2 \in \mathcal{H}_{F}$. The grading of $\mathcal{H}_{F}$ implies a decreasing filtration on $\mathcal{H}^*_{F}$, which allows us to introduce a metric, and therefore a distance map. $\mathcal{H}^*_{F}$ is complete with respect to the induced topology. The exponential map $\mathrm{exp}^{\star}$ gives a bijection between the Lie algebra $\mathcal{L}$ and its corresponding group $\mathcal{G}$.
Using QED as an example we have seen that in general Feynman rules for any perturbative QFT form a subclass of characters. Also, we had to face the severe problem that the associated Feynman integrals for graphs beyond the tree level suffer from ultraviolet divergencies in the limit of large momenta, or equivalently small distances. Therefore, one is forced to invoke a regularization of such integrals, or more generally the Feynman rules themselves. Actually, there is no specific selection rule for such a regularization, indeed one must assure that the final physical result is independent of such an unphysical intermediate step. At the same time it is of vital importance that the regularization prescription used in calculations respects as many physical properties of the underlying theory as possible, such as gauge symmetries. We will ignore such subtleties and take the following stance. In the above Hopf algebraic setting, the regularization of our theory is achieved by replacing the base field $\mathbb{C}$ as target space of maps in $\mathcal{H}^*_{F}$ by an unital algebra $A$, of which we demand commutativity, and the existence of a linear map $R$ satisfying the Rota–Baxter relation $$\label{eq:RBrel}
R(x)R(y)+R(xy)=R\big(R(x)y\big)+\big(xR(y)\big)$$ for all $x,y \in A$. For $R$ being such a Rota–Baxter map, $\tilde{R}:={\mathrm{id}}_A - R$ also satisfies relation (\[eq:RBrel\]). Such algebras are well-known in mathematics under the name Rota–Baxter algebra (see Appendix \[appendix:RotaBaxter\] for more details). As a principal example we mention here dimensional regularization, where the image of Feynman rules lives in the field of Laurent series, $A=\mathbb{C}[\varepsilon^{-1},\varepsilon]]$, with the pole part projection $R:=R_{ms}$ as Rota–Baxter map. In the examples one encounters in QED renormalization calculations, where the regularization could have been for instance a simple cut-off of the momentum integrals, the map $R$ is given in terms of an evaluation at a specific fixed momentum (on-shell scheme), $R_{q}(f)(p):=f(q)$, and trivially satisfies relation (\[eq:RBrel\]), as it is an idempotent algebra homomorphism, i.e., the zeroth order in the Taylor expansion of the map $f$ at point $q$.
The space $\A:=\mathrm{Hom}(\mathcal{H}_{F},(A,R))$ of $A$-valued linear functionals, together with the convolution product (\[def:convol\]) and unit $e:=u_A \circ \bar{e}$, $e(\One)=1_A$, forms an associative non-commutative algebra, containing the group of regularized characters, $\mathcal{G}_A :=
char(\mathcal{H}_{F},A)$, and its corresponding Lie algebra, $\mathcal{L}_A := \partial char(\mathcal{H}_{F},A)$, of regularized derivations. The linearity of the Rota–Baxter map $R$ on the regularization target space $A$, gives rise to a Rota–Baxter algebra structure on $\mathrm{Hom}(\mathcal{H}_{F},(A,R))$, induced in terms of the linear map $\R$, which is defined for any $f \in \A$, by $\R(f):=R
\circ f \in \A$. As before, we can equip $\A$ with a decreasing filtration of Rota–Baxter ideals $$\A=\A_0 \supset \A_1 \supset \dots \supset \A_n \supset \dots$$ making it a complete filtered non-commutative Rota–Baxter algebra with convolution product as composition, $(\A,\R,\{\A_n\}_{n\geq
1})$, since $\R(\A_n) \subset \A_n$ for all $n$. Here we have $\mathcal{L}_A$ as a Lie subalgebra of $\A_1$, and $\mathcal{G}_A$ is a subgroup of $\widehat{\mathcal{G}}:=e +\A_1$, such that $$\begin{aligned}
\exp^{\star}: & \A_1 \to e+\A_1,\quad \exp^{\star}(Z):=
\sum_{n=0}^\infty\frac{Z^{\star n}}{n!}, \label{eq:exp} \\
\log^{\star}: & e+\A_1 \to \A_1,\quad \log^{\star}(e+Z):=
-\sum_{n=1}^\infty \frac{(-Z)^{\star n}}{n} \label{eq:log}\end{aligned}$$ are well-defined with respect to convolution and inverse to each other. Furthermore $\exp^{\star}$ restricts to a bijection between $\mathcal{L}_A$ and $\mathcal{G}_A$.
Atkinson’s [@Atkinson] (see Appendix \[appendix:RotaBaxter\]) factorization theorem for associative Rota–Baxter algebras implies in the above setting, that for a fixed $\phi = e + Z \in \mathcal{G}_A$ the solutions $X
\in e+\mathcal{R}(\A_1)$, $Y \in e+\tilde{\mathcal{R}}(\A_1)$ of the equations $$\label{eq:At1}
X = e - \R(X \star Z) \ \;\mathrm{ resp.}\ \; Y = e - \tilde{\R}(Z \star Y)$$ solve the factorization problem $$\label{eq:At2}
e + Z = \phi = X^{-1} \star Y^{-1},$$ which can be easily checked. If the Rota–Baxter map $R$ is idempotent, the decomposition in (\[eq:At2\]) is unique. In the following we denote $\phi_{-}:=X$ and $\phi_{+}=:Y^{-1}$. Spitzer’s classical identity [@Spitzer] for commutative Rota–Baxter algebras can be generalized to non-commutative Rota–Baxter algebras, thereby implying one of the main results of the Hopf algebraic approach to renormalization in QFT [@CK1], to wit the algebraic Birkhoff decomposition of Connes and Kreimer, which we formulate as a theorem.
[[[@EGK2]]{}]{} Let $\H_{F}$ be a connected graded Hopf algebra of Feynman graphs associated with a perturbatively treated renormalizable QFT. Let $A$ be a commutative unital Rota–Baxter algebra with an idempotent Rota–Baxter operator $R$. Let $\A$ be the complete filtered algebra ${\mathrm{Hom}}(\H_{F},A)$. \[thm:algBi\]
Equation (\[special2\]) follows by general arguments for Rota–Baxter algebras, as we can write for $\phi_{+}$ the following equation $$\phi_{+}=e + \tilde{\R}(\phi_{+} \star (e-\phi^{-1}))
=e + \tilde{\R}(\phi_{-} \star (\phi-e)).$$ The unique map $\chi: \A_1 \to \A_1$ in (\[eq:Spi\]) is the key-result for the generalization of Spitzer’s identity to non-commutative Rota–Baxter algebras, and satisfies the equation $$\chi(Z)= Z - BCH\Big(\R\big(\chi(Z)\big),\tilde{\R}\big(\chi(Z)\big)\Big),
\label{BCH-recur}$$ where $BCH(x,y)$ denotes the Baker–Campbell–Hausdorff relation $$\exp(x)\exp(y)=\exp\big(x+y+BCH(x,y)\big).$$ The non-linear map $\chi$ was introduced in [@EGK1; @EGK2], and was called the $BCH$-recursion. The reader may find it helpful to consult [@EG2] for more details, and [@Manchon] for a more conceptual proof in the context of Lie algebras.
As a proposition to this theorem, we mention without giving further details the fact that Bogoliubov’s $\bar{\mathrm{R}}$-operation (\[BogosBARmap\]) can be written as an exponential using the double Rota–Baxter convolution product, $\star_{\R}$, on $(\A,\R,\{\A_n\}_{n\geq 1})$ (see Eq. (\[def:doubleProd\]) in Appendix \[appendix:RotaBaxter\]): $$\bar{\mathrm{R}}(\Gamma)=\bar{\phi}(\Gamma)= \phi_{-}\star
(\phi-e)(\Gamma)=
-\exp^{\star_{\R}}\big(-\chi(Z) \big)(\Gamma)$$ for $\Gamma \in \ker \bar{e}$. Finally let us mention that the above notion of complete Rota–Baxter algebra and Theorem \[thm:algBi\] becomes very transparent for uni- and nilpotent upper (or lower) triangular matrices with entries in a commutative Rota–Baxter algebra [@EG2; @EGGV].
The above theorem presents a purely algebraic setting for the formulation of renormalization as a factorization problem in the group of regularized Hopf algebra characters, situated in the theory of non-commutative Rota–Baxter algebras with idempotent Rota–Baxter map. The formulae for the counterterm (\[special1\]) and renormalized character (\[special2\]) are completely dictated by a general decomposition structure, which characterizes Rota–Baxter algebras [@Atkinson]. The additional property of $R$ being a projector implies a direct decomposition of the algebra, hence the uniqueness of the factorization in (\[Birkhoff\]). We would like to emphasis the necessary freedom in the choice of the regularization prescription, encoded in the particular structure of the commutative Rota–Baxter algebra $A$ as target space of linear Hopf algebra functionals in ${\mathrm{Hom}}(\H_{F},A)$.
Specializing the target space Rota–Baxter algebra $A$ in the above theorem to the field of Laurent series, i.e., using dimensional regularization, we recover the original setting in [@CK1], opening a hitherto hidden geometric viewpoint on perturbative renormalization in terms of a correspondence to the Riemann–Hilbert problem. This approach was further extended in [@CM1; @CM2; @CM3].
Diffeomorphisms of physical parameters {#section5}
======================================
In the above, we obtained a unique Birkhoff decomposition of Feynman rules $\phi \in Spec({\cal G}_A)$ into two characters $\phi_- =: S_R^\phi \in Spec({\cal G}_A)$ and $\phi_+ = S_R^\phi
\star \phi \in Spec({\cal G}_A)$, for any idempotent Rota–Baxter map $R$. Thanks to Atkinson’s theorem this is possible for any renormalization scheme $R$. For the minimal subtraction scheme it amounts to the decomposition of the Laurent series $\phi(\Gamma)(\varepsilon)$, which has poles of finite order in the regulator $\varepsilon$, into a part holomorphic at the origin and a part holomorphic at complex infinity. This has a geometric interpretation upon considering the Birkhoff decomposition of a loop around the origin, providing the clutching data for the two half-spheres defined by that very loop, which is central in the work of Connes and Kreimer [@CK2; @CK3]. The geometric interpretation leads to motivic Galois theory upon studying the equisingularity of the corresponding connection in the Riemann–Hilbert correspondence [@CM1; @CM2], itself a result of the Hochschild cohomology of these Hopf algebras [@BergbauerKreimer1; @BergbauerKreimer2].
Our understanding of each term in the perturbative expansion and its renormalization have found hence satisfying mathematical interpretations. The character group ${\cal G}_A$ is a poorly understood object though, it is far too big. Fortunately renormalization can be captured by the study of diffeomorphisms of physical parameters, as by the very definition the range of allowed modifications in renormalization theory is the variation of the coefficients of monomials $\hat{\phi}(\underline{r})$ of the underlying Lagrangian $$L=\sum_{\underline{r}\in \mathfrak{R}}Z^{\underline{r}}\, \hat{\phi}(\underline{r}).$$ We can now eliminate the use of ${\cal G}_A$ as one can regain the Birkhoff decomposition at the level of diffeomorphisms of the coupling constants.
One proceeds by using that renormalized couplings provide a formal diffeomorphism $$g_{\rm new}=g_{\rm old}\;Z^g,$$ where $$Z^g= \frac{Z^v}{\prod_{e \in {\bf res}(v)^{[1]}_{\rm ext}}\sqrt{Z^e}},$$ for some vertex $v$, which obtains the new coupling in terms of a diffeomorphism of the old. This formula provides indeed a Hopf algebra homomorphism from the Hopf algebra of diffeomorphisms to the Hopf algebra of Feynman graphs, regarding $Z^g$, a series over counterterms for all 1PI graphs with the external leg structure corresponding to the coupling $g$, in two different ways: it is at the same time a formal diffeomorphism in the coupling constant $g_{\rm old}$ and a formal series in Feynman graphs. As a consequence, there are two competing coproducts acting on $Z^g$. Their consistency defines the required homomorphism, which transposes to a homomorphism from the largely unknown group of regularized characters of ${\cal H}_F$ to the one-dimensional diffeomorphisms of this coupling. Hence one concludes [@CK3]:
Let the unrenormalized effective coupling constant $g_{\rm eff}
(\varepsilon)$ viewed as a formal power series in $g$ be considered as a loop of formal diffeomorphisms and let $g_{\rm
eff} (\varepsilon)= (g_{{\rm eff}_-})^{-1}(\varepsilon) \, g_{{\rm
eff}_+}(\varepsilon)$ be its Birkhoff decomposition in the group of formal diffeomorphisms. Then the loop $g_{{\rm eff}_-}
(\varepsilon)$ is the bare coupling constant and $g_{{\rm eff}_+}
(0)$ is the renormalized effective coupling.
The role of Hochschild cohomology {#section6}
=================================
The Hochschild cohomology of the Hopf algebras of 1PI graphs illuminates the structure of 1PI Green’s functions in various ways:
- it gives a coherent proof of locality of counterterms –the very fact that $$[Z^{\underline{r}},D_{\underline{r}}]=0,$$ the coefficients in the Lagrangian remain independent of momenta, and hence the Lagrangian a polynomial expression in fields and their derivatives; [@BergbauerKreimer2; @Kreimer10; @KreimerDS]
- the quantum equation of motions take a very succinct form identifying the Dyson kernels with the primitives of the Hopf algebra [@BergbauerKreimer2; @Kreimer10; @KreimerDS], and hence replacing a sum over all graphs by a sum over all primitive graphs;
- sub-Hopf algebras emerge from the study of the Hochschild cohomology which connect the representation theory of these Hopf algebras to the structure of theories with internal symmetries leading to the Slavnov–Taylor identities for the couplings [@KreimerDS];
- these Hopf algebras are intimately connected to the structure of transcendental functions like the generalized polylogarithms which play a prominent role these days ranging from applied particle physics to recent developments in mathematics, in particular the primitive graphs which provide the Dyson kernels allow for a motivic interpretation [@BlochEK].
For more information, we refer the reader to the literature indicated.
Basic facts about general Rota–Baxter algebras {#appendix:RotaBaxter}
==============================================
For the reader’s convenience we collect some basic notions of more mathematical nature concerning mainly Rota–Baxter operators, in the hope that from the above presentation they become redundant. For more details we refer the reader to the standard literature, e.g. [@Atkinson; @Guo1; @RotaSmith]. Rota–Baxter operators (also known as Baxter operators in older mathematical references) were an active field of mathematical research in the late 1960s and early 1970s. After an almost three decades long period of dormancy they reappeared, as if on cue, in the mathematical literature in the context of dendriform algebras [@Aguiar; @K1; @JLL], number theory [@Guo2], generalizations of shuffle products [@EG1; @GuoKeigher], and Hopf algebras [@AndrewsGuo], as well as in theoretical physics in the seminal work of Kreimer and collaborators on the Hopf algebra of renormalization [@CK1; @EGK1; @EGK2; @Kreimer2].
In the following $\mathbb{K}$ denotes the base field of characteristics zero, over which all algebraic structures are defined. In general an algebra always means an associative unital $\KK$-algebra, not necessarily commutative. The algebra unit is simply denoted by $1$.
In Section \[section4\] we encountered Rota–Baxter operators respectively the Rota–Baxter relation in the context of renormalization schemes, i.e. subtraction operators for the BPHZ method. Let $A$ be an algebra together with a linear endomorphism $R: A \to A$. We call the tuple $(A,R)$ a Rota–Baxter algebra of weight $\theta \in \KK$, if the map $R$ fulfills the Rota–Baxter relation (of weight $\theta$) $$\label{eq:RBrelation}
R(x)R(y)+\theta R(xy)=R\big(R(x)y + xR(y)\big)$$ for all $x,y \in A$. Without proof we state the fact that the operator $\tilde{R}:=\theta {\mathrm{id}}_A - R$ is a Rota–Baxter operator of weight $\theta$, too, such that the mixed relation $$\label{eq:mixedRBrel}
R(x) \tilde{R}(y)=\tilde{R}\big(R(x)y\big) + R\big(x\tilde{R}(y)\big)$$ is satisfied for all $x,y \in A$. The map $B:=\theta{\mathrm{id}}_A-2R$ satisfies the modified Rota–Baxter relation $$B(x)B(y)+\theta^2 xy=B\big(B(x)y + xB(y)\big).$$ For $\theta \neq 0$, the normalized map $\theta^{-1} R$ is a Rota–Baxter operator of weight one. Therefore without lost of generality we may suppose in the following the canonical weight one case. A Rota–Baxter (left-) right-ideal $I$ is a (left-) right-ideal $I$ of $A$ such that $R(I) \subseteq I$. A Rota–Baxter ideal is a Rota–Baxter left- and right-ideal.
The American mathematician Glen Baxter introduced this relation 1960 in his probability studies in fluctuation theory [@Baxter]. Later, the Italian born American mathematician Gian-Carlo Rota [@RotaSmith; @Rota1; @Rota2; @Rota3], and others [@Cartier; @Kingman], notably F.V. Atkinson [@Atkinson], explored in detail Baxter’s work from different perspectives in analysis, algebra and combinatorics. The case $\theta = 0$ corresponds to the integration by parts property of the usual Riemann integral $I: \F\to\F$, $I[f](x) := \int_0^x
f(t)\,dt$ in the algebra $\F$ of continues functions on $\RR$, to wit, $$I[f_1]\,I[f_2] = I\bigl[ I[f_1] f_2 + f_1 I[f_2] \bigr],
\label{eq:ach-so}$$ for $f_1,f_2 \in \F$. We already encountered the pole part projection $R_{ms}$ in dimensional regularization as an example of an idempotent Rota–Baxter map of weight one. The images of $R$ as well as $\tilde{R}$ form subalgebras in $A$. Let $R$ be a projector on $A$. For $R$ to satisfy the Rota–Baxter relation is equivalent to a direct decomposition of $A=R(A)\oplus
\tilde{R}(A)$. This is just the special case of Atkinson’s additive decomposition theorem [@Atkinson], characterizing a general Rota–Baxter algebra $(A,R)$ as a subdirect difference of the images of $R$ and $\tilde{R}$.
The Lie algebra associated to $(A,R)$, with standard commutator bracket forms a Rota–Baxter Lie algebra, $(\L_A,R)$, with $R$ fulfilling $$\label{eq:RBLieRelation}
[R(x),R(y)]+\theta R([x,y])=R\big([R(x),y] + [x,R(y)]\big),$$ better known as (operator) classical Yang–Baxter equation. Let us mention here that it was rediscovered in this form in the early 1980ies by some Russian physicists in the context of classical integrable systems (see e.g. [@BBT; @STS1] for references and more details). This curious coincidence of Baxter and Baxter just happens to reveal the connections of Rota–Baxter operators with many areas of mathematics and physics.
The vector space underlying $A$, equipped with the new product $$\label{def:doubleProd}
x \star^{(1)}_R y:=xR(y) + R(x)y - xy$$ is again a Rota–Baxter algebra with Rota–Baxter map $R$, which we denote $(A_1,R)$. Hence, all Rota–Baxter algebras $(A,R)$, associative or non-associative, come with a whole hierarchy of so-called double Rota–Baxter algebras, $(A_n,R)$, $n\in\NN$. Relation (\[eq:RBrelation\]) naturally implies that $R(x
\star^{(1)}_R y)=R(x)R(y)$, i.e. $R$ is an algebra homomorphism from $A_1$ to $A$, or more generally from $A_n$ to $A_{n-1}$.
Baxter’s original motivation for his work was to prove that for a commutative algebra $A$ together with a linear map $R$ satisfying relation (\[eq:RBrelation\]), that later bore his name, the following identity for fixed $a \in A$ holds in the formal power series ring $\A:=A[[t]]$, with $t$ being a commuting parameter. This famous relation is called Spitzer’s classical identity, and appeared in Frank Spitzer’s 1956 paper [@Spitzer]. In [@EGK1] this was generalized to non-commutative Rota–Baxter algebras, based on a BCH-type recursion formula, the key result we used earlier, see Eq. (\[BCH-recur\]). A more general setting in which the above factorization may be regarded, and of which the algebra $A[[t]]$ is just a special case, is that of complete filtered Rota–Baxter algebras [@EGK2; @EG2; @EGGV].
The right hand side of (\[eq:SpitzerId\]) is the unique solution of the recursive equation $$\label{eq:Atkinson1}
X= 1 + tR(Xa).$$ This is a natural generalization of the recursion $f=1+I[gf]$ corresponding to the differential equation $f'=gf$, $f(0)=1$, solved by $\exp(I[g])$, in $(\F,I)$ where $I$ is the Riemann integral. It was again Atkinson [@Atkinson] in 1963, who observed that for any Rota–Baxter algebra, not necessarily commutative, a solution to Eq. (\[eq:Atkinson1\]), and its companion equation for $\tilde{R}$ $$\label{eq:Atkinson2}
Y= 1 + t\tilde{R}(aY)$$ in $A[[t]]$, solve the multiplicative decomposition problem $$\label{eq:AtkinsonFact}
(1-\theta a) = X^{-1}Y^{-1},$$ for any element $a \in A$. $R$ being a projector implies a unique decomposition of the element $(1-\theta a) \in A$. This genuine factorization property of Rota–Baxter algebras is further analyzed in [@EG2].
: The first author is very thankful for a PhD grant provided by the Ev. Studienwerk Villigst, as well as for an extra travel grant which allowed him to visit BIRS (Canada). Also, he would like to thank warmly the Theory Department at the Physics Institute of Bonn University for unlimited support and encouragement. Both authors would like to thank the Fields Institute, where this work was finished, for warm hospitality. The collaboration with Prof. L. Guo is acknowledged.
[0]{}
[^1]: [email protected]; currently visiting the Fields Institute, Toronto, Canada.
[^2]: [email protected] and [email protected], Center for Math.Phys., Boston University.
[^3]: See also L. Foissy’s PhD-thesis.
[^4]: We assume that the reader has already seen a Feynman graph, otherwise he might take a brief look on Eq. (\[pre-LieExam\]).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study the notions of weak rational ergodicity and rational weak mixing as defined by Jon Aaronson. We prove that various families of infinite measure-preserving rank-one transformations possess (or do not posses) these properties, and consider their relation to other notions of mixing in infinite measure.'
address:
- |
Harvard College\
University Hall\
Cambridge, MA 02138, USA
- |
University of Minnesota\
Minneapolis, MN 55455-0213, USA
- 'University of California, Los Angeles, CA 90095-1555, US '
- |
Department of Mathematics\
Williams College\
Williamstown, MA 01267, USA
author:
- Irving Dai
- Xavier Garcia
- Tudor Pădurariu
- 'Cesar E. Silva'
bibliography:
- 'References.bib'
title: 'On Rationally Ergodic and Rationally Weakly Mixing Rank-One Transformations'
---
Definitions and Preliminaries
=============================
Let $(X,\mathcal{B},\mu)$ be a standard Borel measure space with a $\sigma$-finite nonatomic measure $\mu$. In most cases, we will assume that $\mu$ is infinite. A transformation $T: X \rightarrow X$ is [**measurable**]{} if $T^{-1}A\in\mathcal B$ for all $A\in\mathcal B$. A measurable transformation $T$ is [**measure-preserving**]{} if $\mu(A)=\mu(T^{-1}A)$ for all $A\in\mathcal B$. We say that $T$ is [**ergodic**]{} if every $T$-invariant set (i.e, $T^{-1}A=A $ mod $\mu$) is null ($\mu(A)=0$) or full $(\mu(X\setminus A)=0$). We say that $T$ is [**conservative**]{} if for every measurable set $A$ of positive measure, there exists a positive integer $n$ such that $\mu(A \cap T^{-n}A)>0$. It follows that $T$ is conservative and ergodic if and only if for every set $A$ of positive measure, $\bigcup_{n=0}^\infty T^{-n}A=X$ mod $\mu$. An [**invertible measurable transformation**]{} is a measurable transformation whose inverse is also measurable. Throughout this paper, we will assume that $T$ is an invertible, conservative ergodic, measure-preserving transformation on $(X,\mathcal{B},\mu)$, and we will typically use the forward images $T^nA$ instead of $T^{-n}A$.\
\
When $T$ is a measure-preserving transformation on a probability space $X$, the Birkhoff ergodic theorem states that ergodicity is equivalent to having the convergence $$\begin{aligned}
\label{birk}
\dfrac{1}{n} \displaystyle \sum_{k = 0}^{n-1} \mu(A \cap T^k B) \rightarrow \mu(A)\mu(B)\end{aligned}$$ for all measurable $A, B \subset X$. This gives a quantitative estimate for the average number of visits of one set to another. When $X$ has infinite measure, however, the Birkhoff ergodic theorem implies that the Cesaro averages of converge to $0$ for all pairs $A, B$ of finite measure. Moreover, in [@Aa77] Aaronson proved that there exists no sequence of normalizing constants for which the averages of converge to $\mu(A)\mu(B)$, and he proposed in turn the definitions of rational ergodicity and weak rational ergodicity.\
For any measurable set $F \subset X$ of finite positive measure, define the $\mathbf{intrinsic}$ $\mathbf{weight}$ $\mathbf{sequence}$ of $F$ to be
$$u_n(F) = \dfrac{\mu(F \cap T^nF)}{\mu(F)^2}$$ and write
$$a_n(F) = \displaystyle \sum_{k = 0}^{n-1} u_k(F).$$
A transformation $T$ is said to be [**weakly rationally ergodic**]{} (see [@Aa77]) if there exists a measurable set $F \subset X$ of positive finite measure such that for all measurable $A$, $B$ $\subset F$, we have
$$\label{e:raterg}
\dfrac{1}{a_n(F)} \displaystyle \sum_{k = 0}^{n-1} \mu(A \cap T^kB) \rightarrow \mu(A)\mu(B)$$
as $n \rightarrow \infty$. If this convergence happens only along a subsequence $\{n_i\}$ of $\mathbb{N}$, we say that $T$ is $\mathbf{subsequence}$ $\mathbf{weakly}$ $\mathbf{rationally}$ $\mathbf{ergodic}$. To emphasize the set $F$, we will sometimes say $T$ is $\mathbf{weakly}$ $\mathbf{rationally}$ $\mathbf{ergodic}$ $\mathbf{on \ F}$. Note that any measure-preserving ergodic transformation on a probability space is trivially weakly rationally ergodic, by taking $F$ to be the whole space itself. Then $a_n(F) = n$, so reduces to the Cesaro sum definition of ergodicity.\
A transformation $T$ is said to be $\mathbf{(spectrally)}$ $\mathbf{weakly}$ $\mathbf{mixing}$ if whenever $f\in L^\infty(X,\mu)$ and $f\circ T= z f $ for some $z\in \mathbb{C}$, then $f$ is constant a.e. When $X$ is a probability space, this is equivalent to ergodicity of the Cartesian square and also to the strong Cesaro convergence $$\dfrac{1}{n} \displaystyle \sum_{k = 0}^{n-1} |\mu(A \cap T^k B) - \mu(A)\mu(B)| \rightarrow 0$$ for all measurable $A, B \subset X$. In [@AaLiWe79], it was shown that for infinite measure-preserving transformations, (spectral) weak mixing is strictly weaker than ergodicity of the Cartesian square.\
\
Another property we consider that is equivalent to weak mixing in the finite measure-preserving case is double ergodicity. This property was introduced by Furstenberg in [@Fu81] and was shown to be equivalent to weak mixing for probability-preserving transformations, but was not given a specific name. A transformation $T$ is said to be [**doubly ergodic**]{} if for every pair of sets $A$ and $B$ with positive measure, there exists a positive integer $n$ for which $\mu(A \cap T^nA)$ and $\mu(B \cap T^nA)$ are simultaneously nonzero. In the infinite measure-preserving case, double ergodicity is strictly stronger than spectral weak mixing and is properly implied by ergodic Cartesian square [@BoFiMaSi01].\
\
More recently, Aaronson introduced another notion of weak mixing for infinite measure that generalizes rational ergodicity. A transformation $T$ is said be [**rationally weakly mixing**]{} (see [@Aa12]) if there exists a measurable set $F \subset X$ of positive finite measure such that for all measurable $A$, $B$ $\subset F$, we have $$\label{e:rwm}
\dfrac{1}{a_n(F)} \displaystyle \sum_{k = 0}^{n-1} |\mu(A \cap T^kB) - \mu(A)\mu(B)u_k(F)| \rightarrow 0$$
as $n \rightarrow \infty$. Again, it is clear that rational weak mixing reduces to the usual definition of weak mixing in the finite measure-preserving case.\
\
We now describe our main results. In Section \[S:ratergodicity\] we prove that a large class of rank-one transformations are weakly rationally ergodic and discuss the notions of rational ergodicity and bounded rational ergodicity in this context. In Section \[S:ratweakmix\] we construct a class of rank-one transformations that are not rationally weakly mixing; in particular, we obtain a transformation which is rationally ergodic and spectrally weakly mixing but not rationally weakly mixing. This negatively answers a question of Aaronson’s. (After this work was completed, we learned that Aaronson had also independently answered this question [@Aa12b].) Section \[S:doubleerg\] shows that rational weak mixing implies double ergodicity and constructs a transformation that is not rationally weakly mixing and which we conjecture to be doubly ergodic. Section \[S:zerotype\] proves that the notion of zero-type for infinite measure-preserving transformations (whose spectral definition is similar to the mixing condition in the case of probability-preserving transformations) is independent of rational weak mixing. Finally, in Section \[S:ratweakmixex\] we present a class of rank-one transformations that are rationally weakly mixing. As remarked in [@Aa12], all the examples of rationally weakly mixing transformations constructed in [@Aa12] are of the type $T\times S$, where $T$ is an infinite measure-preserving $K$-automorphism and $S$ is a mildly mixing probability-preserving transformation. These examples have countable Lebesgue spectrum and are of a different nature than our rank-one constructions.
Acknowledgements
----------------
This paper was based on research done by the Ergodic Theory group of the 2012 SMALL Undergraduate Research Project at Williams College. Support for this project was provided by the National Science Foundation REU Grant DMS - 0353634 and the Bronfman Science Center of Williams College. We are indebted to Jon Aaronson for conversations and suggestions during discussions of our work at the 2012 Williams Ergodic Theory Conference. We would also like to acknowledge the other members of the 2012 Ergodic Theory group: Shelby Heinecke, Emily Wickstrom, and Evangelie Zachos. We would like to thank the referee for comments that improved the paper.
Rank-One Transformations (Basics)
---------------------------------
We briefly review (rank-one) cutting-and-stacking transformations (see e.g. [@Si08]). A $\mathbf{column}$ is an ordered collection of pairwise disjoint intervals (called $\mathbf{levels}$) in $\mathbb{R}$, each of the same measure. We think of the levels in a column as being stacked on top of each other, so that the $(j+1)$-st level is directly above the $j$-th level. Every column $C = \{J_j\}$ is associated with a natural column map $T_C$ sending each point in $J_j$ to the point directly above it in $J_{j+1}$. (Note that $T_C$ is undefined on the top level of $C$.) A $(\mathbf{rank}$-$\mathbf{one})$ $\mathbf{cutting}$-$\mathbf{and}$-$\mathbf{stacking}$ construction for $T$ consists of a sequence of columns $C_n$ such that:
(a) The first column $C_0$ is the unit interval.
(b) Each column $C_{n+1}$ is obtained from $C_n$ by cutting $C_n$ into $r_n\geq 2$ subcolumns of equal width, adding any number of new levels (called $\mathbf{spacers}$) above each subcolumn, and stacking every subcolumn under the subcolumn to its right. In this way, $C_{n+1}$ consists of $r_n$ copies of $C_n$, possibly separated by spacers.
(c) The collection of levels $\displaystyle \bigcup_n C_n$ forms a generating semiring for $\mathcal{B}$.
Observing that $T_{C_{n+1}}$ agrees with $T_{C_n}$ everywhere where $T_{C_n}$ is defined, we then take $T$ to be the limit of $T_{C_n}$ as $n \rightarrow \infty$.
Rank-One Transformations (Notation)
-----------------------------------
Let $T$ be a rank-one transformation, and fix any column $C_n$ of $T$. We denote the number of levels in $C_n$ by $h_n$ and write $w_n$ for the width of each level. We denote the height of any level $J$ in $C_n$ by $h(J)$, with the convention that $0 \leq h(J) < h_n$. For each $0 \leq k < r_n$, let $s_{n,k}$ be the number of spacers added above the $k$-th subcolumn of $C_n$, and denote the number of levels in the $k$-th subcolumn (after adding spacers) by $h_{n, k} = h_n + s_{n, k}$.\
\
Define $T$ to be $\mathbf{normal}$ if $s_{n, r_n-1} > 0$ for infinitely many values of $n$. (This means that at least one spacer is added above the rightmost subcolumn infinitely many times.) In addition, we say that $T$ has a $\mathbf{bounded \ number \ of}$ $\mathbf{cuts}$ if $\sup \{r_n\} < \infty$; this implies that $T$ is partially rigid and of infinite conservative index [@AdFrSi97].\
\
Given any level $J$ from $C_n$ and any column $C_m$ of $T$ with $m \geq n$, we define the $\mathbf{descendants}$ of $J$ in $C_m$ to be the collection of levels in $C_m$ whose disjoint union is $J$. We denote this set by $D(J, m)$. Occasionally, we will also use $D(J, m)$ to refer to the heights of the descendants of $J$ in $C_m$. In the case when $J$ is the unit interval $I$, for each $m \in \mathbb{N}$ it will be convenient to define $M_m = \max(D(I, m))$. (That is, $M_m$ is the height of the uppermost descendant of $I$ in $C_m$.)\
\
We say that $T$ $\mathbf{grows \ exponentially}$ if $2s_{n,r_n-1} \geq h_{n+1}$ for every $n$. Intuitively, this means that the upper half of every column $C_n$ consists of spacers added during the $(n-1)$-st stage of construction. In particular, the descendants of any level $J$ from an earlier column must lie in the lower half of $C_n$. Note that any $T$ which grows exponentially is clearly normal.
Rational Ergodicity {#S:ratergodicity}
===================
In this section, we establish some introductory ideas and prove that a large class of rank-one transformations are rationally ergodic.\
\
We begin with a computational lemma. Suppose that $T$ is a normal rank-one transformation. Then we claim that the partial sums $a_n(J)$ for any level $J$ can be computed from the descendant heights $D(J,N)$ for $N$ sufficiently large. More precisely,
\[L:lem2.1\] Let $T$ be a normal rank-one transformation. Fix any level $J$ and $n \in \mathbb{N}$. Then for every $N$ sufficiently large, we have $$\mu(J \cap T^kJ) = w_N\cdot |D(J, N) \cap (k + D(J, N))|$$ for all $0 \leq k < n$. Consequently, $$\displaystyle \sum_{k = 0}^{n-1} \mu(J \cap T^kJ) = w_N\cdot \sum_{k = 0}^{n-1} |D(J, N) \cap (k + D(J, N))|.$$
Fix any level $J$, and let $n \in \mathbb{N}$ be arbitrary. Since $T$ is normal, we can find some column $C_N$ in which all the heights $D(J, N)$ are at most $h_N - n$. For any $0 \leq k < n$ and level $J_i \in D(J, N)$, the image $T^k(J_i)$ is then the level in $C_N$ of height $h(J_i) + k$. The conclusion follows immediately.
We will sometimes need to compute $\mu(J \cap T^kJ)$ for $k < 0$. For this, simply observe that $$\mu(J \cap T^kJ) = \mu(T^{-k}J \cap J)$$ and $$|D(J, N) \cap (k + D(J, N))| = |(-k + D(J, N)) \cap D(J, N)|,$$ so in fact Lemma \[L:lem2.1\] holds for all $-n < k < n$.\
\
We thus calculate $D(J, N)$. Suppose that $J$ is a level in $C_j$ of height $h(J)$. Then $J$ splits into $r_j$ levels in $C_{j+1}$ of heights $$\{h(J)\} \cup \{h(J) + \displaystyle \sum_{k = 0}^i h_{j, k}: 0 \leq i < r_j - 1\}.$$ Letting $$H_j = \{0\} \cup \left\{\sum_{k = 0}^i h_{j, k} : 0 \leq i < r_j - 1\right\},$$ it follows inductively that $$D(J, N) = h(J) + H_j \oplus H_{j+1} \oplus \cdots \oplus H_{N-1}.$$
\
We now show that every normal rank-one transformation satisfies condition for $A$, $B$ finite unions of levels and $F$ the unit interval. In this context, we note that Aaronson [@Aa77 Theorem 6.1] has shown every set of finite measure $F$ contains a dense algebra of sets satisfying , but at the same time it is never true that is satisfied for all measurable sets in every set $F$ of finite positive measure [@Aa77 Theorem 6.2].
\[T:normal\] Let $T$ be a normal rank-one transformation. Then $T$ satisfies condition for $A$, $B$ finite unions of levels and $F$ the unit interval.
Let $F = I$ denote the unit interval. We begin by proving for $A = B = J$, where $J$ is the bottom level of any column $C_j$. We need to show that $$\dfrac{1}{a_n(I)} \displaystyle \sum_{k = 0}^{n-1} \mu(J \cap T^{k}J) \rightarrow \mu(J)^2$$ as $n \rightarrow \infty$. For $N$ sufficiently large (as a function of $n$), we have $$\displaystyle \sum_{k = 0}^{n-1} \mu(J \cap T^{k}J) = w_N \left( \sum\limits_{k = 0}^{n-1}|D(J,N)\cap (k + D(J,N))| \right)$$ by Lemma \[L:lem2.1\]. Now, writing $$D(I, N) = H_0 \oplus H_1 \oplus \cdots \oplus H_{N-1}$$ and $$D(J, N) = H_j \oplus H_{j+1} \oplus \cdots \oplus H_{N-1},$$ we may express $D(I, N) = A \oplus B$ and $D(J, N) = B$ with $A = H_0 \oplus H_1 \oplus \cdots \oplus H_{j-1}$. Noting that $\mu(J) = 1/|D(I, j)| = 1/|A|$, we thus wish to show $$\displaystyle \dfrac{w_N}{a_n(I)} \left(|A|^2\sum\limits_{k = 0}^{n-1}|B \cap (k + B)|\right) \rightarrow 1.$$ We give the term inside the parentheses a combinatorial interpretation. Let $P(n)$ denote the number of ordered quadruplets $(a, a', b, b')$ with $a, a' \in A$ and $b, b' \in B$ for which $0 \leq b - b' < n$. Then the above quotient is precisely $w_NP(n) / a_n(I)$, since $|B \cap (k + B)|$ counts the number of pairs $b, b' \in B$ with $k = b - b'$.\
\
Now let $M$ be the maximum value of $A - A$. We claim that the following inequality holds: $$\displaystyle \sum_{k = M}^{n - 1 - M} |(A \oplus B) \cap (k + A \oplus B)| \leq P(n) \leq \sum_{k = -M}^{n - 1 + M} |(A \oplus B) \cap (k + A \oplus B)|.$$ Indeed, the sum on the left counts the number of quadruplets $(a, a', b, b')$ with $M \leq a - a' + b - b' < n - M$; the sum on the right counts the number of quadruplets with $-M \leq a - a' + b - b' < n + M$. Clearly, any quadruplet with $M \leq a - a' + b - b' < n - M$ has $0 \leq b - b' < n$. Similarly, any quadruplet with $0 \leq b - b' < n$ has $-M \leq a - a' + b - b' < n + M$. Recalling that $A \oplus B = D(I, N)$, it thus follows that $$\dfrac{1}{a_n(I)} \sum_{k = M}^{n - 1 - M} \mu(I \cap T^kI) \leq \dfrac{w_N}{a_n(I)} (P(n)) \leq \dfrac{1}{a_n(I)} \sum_{k = -M}^{n - 1 + M} \mu(I \cap T^kI).$$ Now, $M$ is a fixed constant, independent of $n$. Furthermore, the sequence $\mu(I \cap T^k(I))$ is bounded above by 1 but has divergent sum. Hence both sides of the above inequality tend to 1 as $n \rightarrow \infty$, showing that $w_NP(n) / a_n(I) \rightarrow 1$, as desired. This proves for $A = B = J$ , where $J$ is the bottom level of any column.\
\
We now prove for $J$ and $J'$ any two levels in the same column. By applying $T^{-1}$ and using the fact that $T$ is measure-preservin, we may assume that one of the two levels (say $J$) is actually the bottom level of the column. Letting $J' = T^d(J)$ for some $d$, we wish to show $$\dfrac{1}{a_n(I)} \sum_{k = 0}^{n-1} \mu(J \cap T^{k + d}J) \rightarrow \mu(J)^2.$$ Now, we have from before that $$\dfrac{1}{a_n(I)} \sum_{k = 0}^{n-1} \mu(J \cap T^{k}J) \rightarrow \mu(J)^2.$$ Since $\mu(J \cap T^{k}J)$ is bounded and $a_n(I) \rightarrow \infty$, the conclusion follows immediately.\
\
Finally, we extend to finite unions of levels. Without loss of generality, we may assume that $J$ and $J'$ are both disjoint unions of images of the same level $K$. The desired statement then follows from summing together the limits for each pair of images.
We now show that under certain conditions, we can extend the results of Theorem \[T:normal\] to all sets $A$ and $B$ (thus proving weak rational ergodicity).
\[T:wraterg\] Let $T$ be an exponentially growing rank-one transformation with a bounded number of cuts. Then $T$ is weakly rationally ergodic.
We show that for $T$ satisfying the above hypotheses, it suffices to prove for finite unions of levels (as in Theorem \[T:normal\]). Indeed, given arbitrary measurable sets $A, B \subset I$, choose $D \subset I$ a finite union of levels for which $\mu(D \triangle B) < \varepsilon$. We claim that there is some constant $c$ such that $$\left| \dfrac{1}{a_n(I)} \sum\limits_{k = 0}^{n-1}\mu(A\cap T^kB) - \dfrac{1}{a_n(I)}\sum_{k = 0}^{n-1}\mu(A\cap T^kD)\right| \leq c\varepsilon$$\
for every $n$. Indeed, let $B_0 = B \cap D$, and write $B = B_0 \cup B_1$ and $D = B_0 \cup D_1$. Then the above difference reduces to $$\left| \dfrac{1}{a_n(I)} \sum\limits_{k = 0}^{n-1}\mu(A\cap T^kB_1) - \dfrac{1}{a_n(I)}\sum_{k = 0}^{n-1}\mu(A\cap T^kD_1)\right|.$$ Now, we claim that we can bound $$\displaystyle \dfrac{1}{a_n(I)} \sum\limits_{k = 0}^{n-1}\mu(A\cap T^kB_1) \leq c \mu(B_1)$$ for some $c$ independent of $n, A,$ and $B_1$. Applying this bound with $D_1$ in place of $B_1$ will bound (4) by $c(\mu(B_1)+\mu(D_1)) \leq 2c\varepsilon$, as desired.\
\
Recall that $M_m = \max(D(I, m))$, for $m\in\mathbb{N}$. Clearly, $\{M_m\}$ is an increasing sequence. For any fixed $n$, if we choose $m$ such that $M_{m-1} \leq n-1 < M_m$, we have $$\displaystyle \sum_{k = 0}^{n-1} \mu(A \cap T^kB_1) \leq \sum_{k = 0}^{n-1} \mu(I \cap T^kB_1) \leq \sum_{k = 0}^{M_m} \mu(I \cap T^kB_1)$$ and $$\displaystyle \sum_{k = 0}^{M_{m-1}} \mu(I \cap T^kI) \leq \sum_{k = 0}^{n-1} \mu(I \cap T^kI) = a_n(I).$$ To prove (5), it thus suffices to find some $c$ such that $$\sum_{k = 0}^{M_m} \mu(I \cap T^kB_1) \leq c\mu(B_1) \sum_{k = 0}^{M_{m-1}} \mu(I \cap T^kI)$$ for every $m$.\
\
Now observe that the sets $T^kI$ with $-M_m \leq k \leq M_m$ cover each point of $I$ exactly $|D(I, m)|$ times. Indeed, consider the column $C_m$ and fix any $x \in I$. Let $x$ be contained in $J$, where $J$ is some level from $D(I, m)$. For any level $J'$ in $D(I, m)$, we claim that there is exactly one value of $k$ between $-M_m$ and $M_m$ for which $T^kJ' \cap J \neq \emptyset$. Indeed, suppose $0 \leq k \leq M_m$ and $T^kJ' \cap J \neq \emptyset$. Any forward image $T^kJ'$ with $0 \leq k \leq M_m$ is just a translation upwards by $k$ levels, since $h_m \geq 2M_m$. (This is implied by our hypothesis that $T$ is exponentially growing.) Hence in this case $k$ must equal $h(J) - h(J')$. On the other hand, suppose $-M_m \leq k < 0$ and $T^kJ' \cap J \neq \emptyset$. Then $J' \cap T^{-k}J \neq \emptyset$, and exactly the same argument shows that $-k = h(J') - h(J)$ (i.e., $k = h(J) - h(J')$). The claim is then immediate.\
\
We thus have $$\sum_{k = -M_m}^{M_m} \mu(I \cap T^kB_1) = \sum_{k = -M_m}^{M_m} \mu(T^kI \cap B_1) = |D(I, m)| \mu(B_1)$$ and $$\sum_{k = -M_{m-1}}^{M_{m-1}} \mu(I \cap T^kI) = |D(I, m-1)|.$$ Hence $$\sum_{k = -M_m}^{M_m} \mu(I \cap T^kB_1) = \left(\dfrac{|D(I,m)|}{|D(I,m-1)|}\right)\mu(B_1)\left(\sum_{k = -M_{m-1}}^{M_{m-1}}\mu(I \cap T^kI)\right)$$ and so $$\sum_{k = 0}^{M_m} \mu(I \cap T^kB_1) \leq \left(\dfrac{|D(I,m)|}{|D(I,m-1)|}\right)\mu(B_1)\left(2\left(\sum_{k = 0}^{M_{m-1}}\mu(I \cap T^kI)\right) - 1\right).$$\
But $|D(I,m)|/|D(I, m-1)| = r_{m-1}$, and $T$ has a bounded number of cuts. We thus easily obtain (6). Hence (4) holds, and we can approximate $B$ with $D$ a finite union of levels. Applying a similar argument to $A$ shows that it suffices to prove for all $A$, $B$ finite unions of levels, which is the content of Theorem \[T:normal\].
We now consider some alternate notions of rational ergodicity, also due to Aaronson [@Aa77]. For any measurable function $f$, recall the notation $$S_n(f) = \displaystyle \sum_{k=0}^{n-1} f \circ T^k.$$ We say that $T$ is $\mathbf{rationally \ ergodic}$ if there exists a set $F$ of positive finite measure which satisfies a $\mathbf{Renyi}$ $\mathbf{inequality}$; i.e., there is some constant $M$ such that $$\label{e:renyi}
\displaystyle \int_F (S_n(1_F))^2 dm \leq M \left( \int_F S_n(1_F) dm \right)^2$$ for every $n \in \mathbb{N}$. If this inequality holds only on a subset $\{n_i\} \subset \mathbb{N}$, we say that $T$ is $\mathbf{subsequence}$ $\mathbf{rationally}$ $\mathbf{ergodic}$. Some authors adopt this as the definition of rational ergodicity instead (see e.g. [@Ci11]). It was shown in [@Aa77] that rational ergodicity implies weak rational ergodicity. It is not currently known whether these notions are equivalent.\
\
We say that $T$ is $\mathbf{boundedly \ rationally \ ergodic}$ (see [@Aa79]) if there exists a set $F$ of positive finite measure such that $$\displaystyle \sup_{n \geq 1} \left\| \dfrac{1}{a_n(F)} S_n(1_F) \right\|_{\infty} < \infty.$$ In [@Aa79], it was shown that bounded rational ergodicity is a strictly stronger property than rational ergodicity. It is not difficult to see that the proof of Theorem \[T:wraterg\] (in particular, the establishment of (5) for all $B_1$) yields bounded rational ergodicity for the transformations in question. Indeed, set $A = I$ in (5). Then there is a constant $c$ such that for any $n$ and $B_1 \subset I$, $$\int_{B_1} \dfrac{1}{a_n(I)} S_n(1_I) dm = \dfrac{1}{a_n(I)} \sum_{k = 0}^{n-1} \mu(I \cap T^kB_1) \leq c \mu(B_1).$$ This means that the average value of $S_n(1_I)/a_n(I)$ on $B_1$ is bounded above by $c$. Since this holds for every $B_1$, the essential supremum of $S_n(1_I)/a_n(I)$ must also be bounded above by $c$. Hence $T$ is boundedly rationally ergodic.\
\
Aaronson proved in [@Aa79] that every dyadic tower over the adding machine is boundedly rationally ergodic; Theorem \[T:wraterg\] extends this result to a larger class of transformations and uses a different approach. Some interesting examples of exponentially growing rank-one transformations with a bounded number of cuts include:
(a) Hajian-Kakutani skyscraper-type constructions [@HaKa70]: $r_n = 2$, $\{s_{n, 0} = 0, s_{n, 1} \geq 2h_n\}.$
(When $s_{n, 1} = 2h_n+1$ the transformation is spectral weakly mixing, see [@AdFrSi97]).
(b) Chacón-like constructions: $r_n = 3$, $\{s_{n, 0} = 0, s_{n, 1} = 1, s_{n, 2} \geq 3h_n + 1\}$.
(When $s_{n, 2} = 3h_n + 1$ the transformation has infinite ergodic index, see [@AdFrSi97], but is not power weakly mixing, see [@GHPSW03].)
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We now prove a slightly different version of Theorem \[T:wraterg\] without the hypothesis of a bounded number of cuts but obtain the conclusion only a a subsequence.
\[T:renyi\] Let $T$ be an exponentially growing rank-one transformation. Then $T$ is subsequence rationally ergodic on $F = I = (0, 1)$ along the sequence $\{n_m = M_m + 1\}$.
We verify the Renyi inequality with $n = M_m+1$ and $M = 2$. Let $D(I, m) = \{I_j\}$ be the descendants of $I$ in column $C_m$, and set $N = |D(I, m)|$. Order $\{I_j\}$ by height of appearance in $C_m$ so that $I_1$ is the lowermost level of $\{I_j\}$ in $C_m$ and $I_N$ is the uppermost. Denote the heights of $\{I_j\}$ in $C_m$ by $\{h(I_j)\}$.\
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Now, $S_n(1_I)(x)$ is equal to the number of $k$ with $0 \leq k \leq n-1$ such that $T^k(x) \in I$. Since $T$ is exponentially growing, this implies that $S_n(1_I)$ is constant on each $I_j$ and that the value of $S_n(1_I)$ on any fixed $I_l$ is the cardinality of the intersection $(h(I_l) + \{0, 1, \cdots, n-1\}) \cap \{h(I_j)\}$. (The relevant forward images $T^kI_l$ are simply upward translations.) On the other hand, it is obvious that $h(I_j) \in (h(I_l) + \{0, 1, \cdots, n-1\})$ exactly when $j \geq l$. Hence $S_n(1_I)$ takes the value $N + 1 - l$ on $I_l$. Restricting the domain of $S_n(1_I)$ to $I$, we thus have $$S_n(1_I) = \displaystyle \sum_{l=1}^{N} (N + 1 - l) 1_{I_l}.$$ Proving (7) is thus equivalent to showing $$\displaystyle \sum_{l = 1}^N (N+1-l)^2w_m \leq 2 \left( \sum_{l=1}^N (N + 1 -l)w_m \right)^2.$$ Now, $w_m = 1/|D(I, m)| = 1/N$. Multiplying through by $N^2$ and reindexing yields the equivalent inequality $$\displaystyle N\left( \sum_{l = 1}^N l^2 \right) \leq 2 \left( \sum_{l=1}^Nl \right)^2.$$ The result then follows from the formulas for power sums.
Rational Weak Mixing {#S:ratweakmix}
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In this section, we present a large class of transformations that are $not$ rationally weakly mixing. We obtain as a corollary the existence of transformations which are rationally ergodic and spectrally weakly mixing, but not rationally weakly mixing.\
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We begin with an example of a rank-one transformation which is subsequence rationally weakly mixing.\
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First, consider the Chacón rank-one transformation $T$ constructed by starting with the unit interval, cutting each column in half, and adding a single spacer on top of the right subcolumn at every step [@Ch69]. This transformation is finite measure-preserving and weakly mixing; thus, it is rationally weakly mixing. We claim that (in particular) $T$ is rationally weakly mixing on the unit interval $I = (0, 1)$.\
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It is clear from the definition of weak rational ergodicity that if $T$ is weakly rationally ergodic on $F$, then $T$ is weakly rationally ergodic on any subset of $F$. Moreover, it was shown in [@Aa12] that for $T$ rationally weakly mixing, the class of sets $F$ satisfying is the same as the class of sets $F$ satisfying . This establishes the claim.\
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Now let $$\phi_n(A, B) = \dfrac{1}{a_n(I)} \displaystyle \sum_{k = 0}^{n-1} |\mu(A \cap T^{k}B) - \mu(A)\mu(B)u_k(I)|$$ be the quotient from , and let $D_m$ denote the collection of dyadic intervals of the form $(i/2^m, (i+1)/2^m)$ for $0 \leq i < 2^m$. Since $D_1$ is a finite collection and $T$ is rationally weakly mixing, there exists some natural number $m_1$ such that for all $A, B \in D_1$ we have $\phi_{m_1}(A ,B) < 1/2$. We claim that in fact this inequality is true for every rank-one transformation $\tilde{T}$ which shares its first $m_1$ stages of construction with $T$ (i.e., $\tilde{C}_n = C_n$ for all $n < m_1$). Indeed, for $A, B \subset I$, the value of $\phi_{m_1}(A ,B)$ depends only on the first $m_1$ stages of the construction of $T$, since the heights $D(I, m_1)$ are all less than $h_{m_1} - m_1$.\
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We now define our desired transformation. We begin by following the construction of the transformation $T$ as described above, until we reach $C_{m_1}$. Then, at the $m_1$-th iteration, we add $2h_{m_1}$ spacers above the right subcolumn. Now, adding one spacer at each subsequent iteration gives another finite measure-preserving transformation, which is also weakly mixing. Hence, there is some $m_2 > m_1$ such that $\phi_{m_2}(A, B) < 1/4$ for all $A,B \in D_{1} \cup D_{2}$.\
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We thus continue adding a single spacer at each step until we reach $C_{m_2}$, at which point we add $2h_{m_2}$ spacers. Proceeding inductively in this manner, we obtain a cutting-and-stacking transformation $T$ and a sequence $\{m_i\}$ such that for each $i$, $\phi_{m_i}(A, B) < 1/2^i$ for all $A, B \in D_1 \cup D_2 \cup \cdots \cup D_i$. The result is an invertible, infinite measure-preserving transformation which is rationally weakly mixing along $\{m_i\}$ for dyadic intervals.\
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In order to extend to all subsets of $I$, we use the following result due to Aaronson [@Aa12]:
Let $T$ be an invertible measure-preserving transformation on a Polish space $X$, and assume that $T$ is rationally ergodic on some open set $F$. Suppose there is a countable base $\mathcal{C}$ for the topology of $F$ such that for every finite subcollection $\{C_i\} \subset \mathcal{C}$, there exists a finite subcollection $\{D_i\} \subset \mathcal{C}$ which is disjoint and has the same union. Then to establish rational weak mixing, it suffices to prove condition for elements of $\mathcal{C}$.
Lemma 3.1 also holds for establishing subsequence rational weak mixing, so long as rational ergodicity is known along the same subsequence. Since the transformation $T$ above may expressed as a dyadic tower over the adding machine, $T$ is rationally ergodic [@Aa79]. It follows from Lemma 3.1 that $T$ is subsequence rationally weakly mixing.\
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We now present a large class of examples that are not rationally weakly mixing. It will be convenient to write $$u_k(A, B) = \dfrac{\mu(A \cap T^kB)}{\mu(A)\mu(B)}$$ so that for $A$, $B$ of positive measure, we can divide by $\mu(A)\mu(B)$ to obtain $$\dfrac{1}{a_n(F)} \displaystyle \sum_{k = 0}^{n-1} |u_k(A,B) - u_k(F)| \rightarrow 0.$$ It is not difficult to see that in this case we must have $a_n(F)/a_n(A,B) \rightarrow 1$ [@Aa12]. This yields the following theorem:
\[T:notrwm\] Let $T$ be a rank-one transformation constructed by cutting $C_n$ in half and adding at least $c_n \geq 2h_n$ spacers on top of the right subcolumn at every step. Then $T$ is not rationally weakly mixing.
We prove by contradiction. Suppose that $T$ is rationally weakly mixing on some set $F$. Choose a level $J$ which is at least $(3/4)$-full of $F$, and let $J_1$ and $J_2$ be the left and right halves of $J$. By applying $T^{-1}$ to $F$, we may assume that $J$ is the bottom level of some column $C_j$. Now, both $J_1$ and $J_2$ intersect $F$ in positive measure, so $$\displaystyle \dfrac{1}{a_n(F)} \displaystyle \sum_{k = 0}^{n-1} \left| u_k(J_1 \cap F) - u_k(F) \right| \rightarrow 0$$ and $$\displaystyle \dfrac{1}{a_n(F)} \displaystyle \sum_{k = 0}^{n-1} \left| u_k(J_1 \cap F, J_2 \cap F) - u_k(F) \right| \rightarrow 0.$$ Moreover, $a_n(F)/a_n(J_1 \cap F) \rightarrow 1$. Multiplying through by this limit and using the triangle inequality, we obtain $$\displaystyle \dfrac{1}{a_n(J_1 \cap F)} \displaystyle \sum_{k = 0}^{n-1} \left| u_k(J_1 \cap F, J_2 \cap F) - u_k(J_1 \cap F) \right| \rightarrow 0.$$ Now, fix $k$ and suppose that $u_k(J_1 \cap F) > 0$. Then $\mu(J_1 \cap T^kJ_1) > 0$, so for sufficiently large $N$ we have $k \in D(J_1, N) - D(J_1, N)$. Similarly, if $u_k(J_1 \cap F, J_2 \cap F) > 0$, then $\mu(J_1 \cap T^{k + h_j}J_1) = \mu(J_1 \cap T^kJ_2) > 0$, which implies that $k + h_j \in D(J_1, N) - D(J_1, N)$. Hence we cannot have both $u_k(J_1 \cap F)$ and $u_k(J_1 \cap F, J_2 \cap F)$ nonzero, since then we would have $h_j \in (D(J_1, N) - D(J_1, N)) - (D(J_1, N) - D(J_1, N))$. As $D(J_1, N) = \{0, h_{j+1}\} \oplus \{0, h_{j+2}\} \oplus \cdots \oplus \{0, h_{N-1}\}$, this is easily seen to be impossible (given the fact that $c_n \geq 2h_n$ for all $n$).\
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It is then immediate that $$\left| u_k(J_1 \cap F, J_2 \cap F) - u_k(J_1 \cap F) \right| \geq u_k(J_1 \cap F)$$ for every $k$. Indeed, if $u_k(J_1 \cap F) = 0$ then we are done; otherwise, $u_k(J_1 \cap F, J_2 \cap F)$ is $0$. It follows that the quotient (8) is bounded below by 1, which is a contradiction. This shows that $T$ is not rationally weakly mixing.
In particular, we obtain the following:
Let $T$ be the transformation constructed by cutting each column $C_n$ in half and adding $2h_n + 1$ spacers on top of the right subcolumn. Then $T$ is rationally ergodic and spectrally weakly mixing but not rationally weakly mixing.
This transformation is rationally ergodic by Theorem \[T:wraterg\] and the discussion following (and also by [@Aa79 Theorem 1]), and is spectrally weakly mixing by [@AdFrSi97 Proposition 1.1]. By Theorem \[T:notrwm\], however, $T$ is not rationally weakly mixing.
This negatively answers a question of Aaronson’s. (As noted in the introduction, this result was obtained independently by Aaronson in [@Aa12b 1.1].)\
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We now extend Theorem \[T:notrwm\] to other rank-one transformations. Define $$H = \displaystyle \bigcup_{j = 0}^\infty H_j \setminus \{0\}$$ and observe that the elements of $H$ are increasing when listed in the obvious order. (Begin with successive elements of $H_0 \setminus \{0\}$, followed by successive elements of $H_1 \setminus \{0\}$, and so on.) We say that a rank-one transformation is $\mathbf{steep}$ if $t_{i + 1} \geq 4 t_i$ for every pair of successive $t_i, t_{i+1} \in H$. Clearly, the transformations of Theorem \[T:notrwm\] are steep. In general, such transformations can be constructed by adding an exponentially increasing number of spacers above successive subcolumns.\
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Steep transformations satisfy several nice properties, chief among which is a linear independence condition that allows us to extend Theorem 3.2. Suppose we have a linear combination $$\displaystyle \sum_{t \in H} c_t t = 0$$ with the coefficients $c_t \in \{-2, -1, 0, 1, 2\}$. Then it is easily seen that all the $c_t$ must be 0. Similarly, we also obtain a uniqueness condition that will be useful in Section 5: every integer $k$ has at most one representation $$k = \sum_{t \in H} c_t t$$ with the $c_t \in \{-1, 0, 1\}$. Altering the definition of steepness slightly yields stronger forms of these properties; for example, requiring $t_{i + 1} \geq 5 t_i$ results in uniqueness of representation with $c_t \in \{-2, -1, 0, 1, 2\}$.
Let $T$ be a normal, steep rank-one transformation. Then $T$ is not rationally weakly mixing.
We sketch the proof and leave the details to the reader. As before, we proceed by contradiction. Suppose $T$ is rationally weakly mixing on $F$, and let $J$ be a level $(3/4)$-full of $F$. Without loss of generality, we may assume that $J$ is the bottom level of some column $C_j$. Now, there must exist at least two descendants $J_1$ and $J_2$ of $J$ in $C_{j+1}$ that have positive intersection with $F$. For these levels, we have $J_2 = T^dJ_1$ for some $d \in H_j - H_j$. It then suffices to show that $d$ cannot be contained in $(D(J_1, N) - D(J_1, N)) - (D(J_1, N) - D(J_1, N))$, which follows from the linear independence (9).
Relation to Double Ergodicity {#S:doubleerg}
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In this section we show that rational weak mixing implies double ergodicity and present an example suggesting the converse implication is false.\
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We begin by proving that rational weak mixing on $F$ implies double ergodicity for subsets of $F$.
Suppose that $T$ is rationally weakly mixing on $F$. Then $T$ is doubly ergodic for all $A,B \subset F$.
Let $A, B \subset F$, and fix $\delta > 0$ such that $$\delta < \frac{1}{2}\min(\mu(A)^2, \mu(A)\mu(B)).$$
Since $T$ is rationally weakly mixing on $F$, $$\dfrac{1}{a_n(F)} \displaystyle \sum_{k = 0}^{n-1} |\mu(A \cap T^kA) - \mu(A)^2u_k(F)| \rightarrow 0$$ and $$\dfrac{1}{a_n(F)} \displaystyle \sum_{k = 0}^{n-1} |\mu(A \cap T^kB) - \mu(A)\mu(B)u_k(F)| \rightarrow 0.$$ Summing these together, we obtain (by contradiction) that there exists a positive integer $k$ for which $u_k(F)>0$ and $$|\mu(A\cap T^kA)-\mu(A)^2u_k(F)|+|\mu(A\cap T^kB)-\mu(A)\mu(B)u_k(F)|<\delta u_k(F).$$ We thus have $$|\mu(A\cap T^kA)-\mu(A)^2u_k(F)|<\delta u_k(F)$$ and so $$\mu(A\cap T^kA)>u_k(F) (\mu(A)^2-\delta)>0$$ for this $k$. Similarly, $$\mu(A \cap T^kB)>u_k(F) (\mu(A)\mu(B)-\delta)>0.$$ By construction of $\delta$, this shows that $T$ is doubly ergodic on $F$.
We now extend this result to all of $X$. It was shown in [@Aa77] that if $T$ is weakly rationally ergodic on $F$, it is weakly rationally ergodic on any finite union $F_N = F \cup T(F) \cup \cdots \cup T^{N-1}(F)$. It follows that the analogous statement holds for rational weak mixing, giving the following theorem:
Suppose that $T$ is rationally weakly mixing. Then $T$ is doubly ergodic.
Let $T$ be rationally weakly mixing on $F$, and suppose that $T$ is not doubly ergodic. Fix $A, B \subset X$ for which the double ergodicity condition fails; i.e., choose $A$ and $B$ such that for every $n$, either $\mu(A \cap T^nA) = 0$ or $\mu(A \cap T^nB) = 0$. Since $F$ sweeps out $X$, there is some $N$ for which $F_N$ intersects both $A$ and $B$ in positive measure. Then $\tilde{A} = F_N \cap A$ and $\tilde{B} = F_N \cap B$ are sets of positive measure which fail the double ergodicity condition. But $T$ is doubly ergodic on $F_N$, a contradiction.
It is worth noting that (in general) the class of sets on which $T$ is doubly ergodic is $not$ a hereditary ring. For example, let $T$ be any doubly ergodic transformation on $X$, and define $S$ on $X \times \{0, 1\}$ by $S(x, 0) = (T(x), 1)$ and $S(x, 1) = (x, 0)$. Then $S$ is doubly ergodic on both $X \times \{0\}$ and $X \times \{1\}$, but not doubly ergodic on all of $X \times \{0, 1\}$. (Let $A = X \times \{0\}$ and $B = X \times \{1\}$.)\
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We now investigate whether rational weak mixing is strictly stronger than double ergodicity. It will be useful for us consider transformations that are “almost" steep. Recall that $T$ is steep if for any pair of successive elements $t_i, t_{i + 1}$ in $H = (H_0 \cup H_1 \cup \cdots) \setminus \{0\}$, we have $t_{i+1} \geq 4 t_i$. Now, suppose $T$ is constructed so that:
(a) Each column $C_n$ is cut into at least three subcolumns ($r_n \geq 3$).
(b) We add zero spacers above the first subcolumn and one spacer above the second ($s_{n, 0} = 0$ and $s_{n, 1} = 1$).
(c) We add a sufficient number of spacers above each subsequent subcolumn so that $$\displaystyle \sum_{k = 0}^i h_{n, k} \geq 4 \left( \sum_{k = 0}^{i-1} h_{n, k} \right)$$ for every $2 \leq i \leq r_n -1$.
Then $T$ is “almost" steep, in the sense that $t_{i+1} < 4t_i$ only when $t_i$ and $t_{i+1}$ are the first two nonzero elements of some $H_n$. For such $T$, we can still extract a (slightly technical) algebraic uniqueness condition in the spirit of (9). Indeed, let $$B_n = \{h_{n, 0}, h_{n, 0} + h_{n,1}\} \times \{h_{n, 0}, h_{n, 0} + h_{n,1}\}$$ and define $$A_n = (H_n \times H_n) \setminus (\Delta H_n \cup B_n).$$ (Here, $\Delta H_n = \{(x, x) : x \in H_n\}$.) Then for any $(a, b), (a', b') \in A_n$ and $-M_n \leq k, k' \leq M_n$, the equality $$k + a - b = k' + a' - b'$$ implies $$a = a', b = b', k = k'.$$ (The proof of this is not difficult and is left to the reader.) Before we proceed, it will be useful to establish following lemma:
Let $J$ be any level, and fix $N$ sufficiently large. Suppose $(a, b) \in A_N$ and $-M_N \leq k \leq M_N$. Then $$\mu(J\cap T^{k + a - b}J) = \dfrac{1}{r_N} \mu(J\cap T^{k}J).$$
By Lemma \[L:lem2.1\], we have $$\mu(J\cap T^{k}J) = w_N\cdot |D(J, N) \cap (k + D(J, N))|$$ and $$\mu(J\cap T^{k + a - b}J) = w_{N+1}|D(J, N+1) \cap (k + a - b + D(J, N+1))|.$$ By uniqueness of (10), every representation of $k + a - b$ as an element of $D(J, N+1) - D(J, N+1)$ corresponds to exactly one representation of $k$ as an element of $D(J, N) - D(J, N)$, and vice-versa. Hence $$\mu(J\cap T^{k + a - b}J) = \dfrac{w_{N+1}}{w_N} \mu(J_1\cap T^{k}J) = \dfrac{1}{r_N} \mu(J\cap T^{k}J),$$ as desired.
We now show that if $T$ is almost steep and $\{r_n\}$ is sufficiently large, $T$ cannot be rationally weakly mixing.
\[T:almoststeep\] Let $T$ be a rank-one transformation. Suppose that $T$ is almost steep (as described above), and that $$\displaystyle \sum_{n = 0}^{\infty} \dfrac{1}{r_n} < \infty.$$ Then $T$ is not rationally weakly mixing.
We begin by proving that $T$ is not rationally weakly mixing on levels. Let $J$ be the bottom level of any column $C_j$, and let $J_1$ and $J_2$ be any two descendants of $J$ in $C_{j+1}$. Then $J_1 = T^dJ_2$ for some $d \in H_j - H_j$. As in Theorem \[T:notrwm\], it suffices to disprove the convergence $$\displaystyle \dfrac{1}{a_n(J_1)} \displaystyle \sum_{k = 0}^{n-1} \left|u_k(J_1) - u_k(J_1, J_2) \right| \rightarrow 0.$$ To do this, define $$P_m = \sum\limits_{k = -M_m}^{M_m}|\mu(J_1\cap T^kJ_1)-\mu(J_1\cap T^{k+d}J_1)|$$ and $$Q_m = \sum\limits_{k = -M_m}^{M_m}\mu(J_1\cap T^kJ_1).$$ For $m$ sufficiently large, $R_m = P_m/ Q_m$ approximates the quotient (11), so it is enough to show that $R_m$ is bounded below by some positive constant.\
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Any choice of $(a, b) \in A_m$ and $- M_m \leq k \leq M_m$ yields a unique number $k + a - b$ between $-M_{m+1}$ and $M_{m+1}$. Hence $$\begin{aligned}
P_{m+1} &= \sum\limits_{k = -M_{m+1}}^{M_{m+1}}|\mu(J_1\cap T^kJ_1)-\mu(J_1\cap T^{k+d}J_1)| \\
&\geq \sum_{(a,b) \in A_m} \sum\limits_{k = -M_m}^{M_m}|\mu(J_1\cap T^{k + a - b}J_1)-\mu(J_1\cap T^{k+a-b+d}J_1)| \\
&= \dfrac{1}{r_m} \left(\sum_{(a,b) \in A_m}\sum\limits_{k = -M_m}^{M_m}|\mu(J_1\cap T^{k}J_1)-\mu(J_1\cap T^{k+d}J_1)|\right) \\
&= \dfrac{|A_m|}{r_m} P_m.\end{aligned}$$ Moreover, the same argument as in Theorem \[T:wraterg\] shows $$Q_m = \sum\limits_{k = -M_m}^{M_m}\mu(J_1\cap T^kJ_1) = |D(J_1, m)|\mu(J_1)$$ from which it follows that $$Q_{m+1} = r_mQ_m.$$ We thus obtain $$R_{m+1} \geq \dfrac{|A_m|}{r_m^2} R_m.$$ Now, $|A_m| = r_m^2 - r_m - 2$, so $R_m$ is bounded below by $$\displaystyle \prod_{k=0}^{\infty} \left(1 - \dfrac{1}{r_k} - \dfrac{2}{r_k^2}\right)R_0$$ which is a positive constant by the hypotheses of the theorem. This bounds (10) from below along the sequence $\{M_m + 1\}$. Since $T$ is rationally ergodic along the same sequence by Theorem \[T:renyi\], it follows that $T$ is not rationally weakly mixing.
We now show that $T$ is doubly ergodic for levels, suggesting that rational weak mixing is strictly stronger than double ergodicity.
The transformation $T$ above is doubly ergodic for levels.
We check that for any pair of levels $A$ and $B$, there exists an integer $n$ such that both $\mu(A\cap T^nA)>0$ and $\mu(B\cap T^nA)>0$. Without loss of generality, we may assume that $A$ is the bottom level of some column $C_j$ and that $B = T^dA$. It then suffices to prove there exists an $n$ such that both $n$ and $n+d$ are in $D(A,N)-D(A,N)$ (for $N$ sufficiently large). This is easy; simply choose $$n = h_{j+1,0}+\cdots+h_{j+d,0}.$$ Then $$n + d = ((2h_{j+1, 0} + 1) +\cdots + (2h_{j+d,0} + 1)) - (h_{j+1,0}+\cdots+h_{j+d,0}),$$ as desired.
Independence from Zero-type {#S:zerotype}
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We now show that (subsequence) rational weak mixing and zero-type are independent (i.e., do not imply each other). We say that $T$ is $\mathbf{zero}$-$\mathbf{type}$ if $\mu(A \cap T^nA) \rightarrow 0$ for all sets $A$ of finite measure [@HaKa64]. It is well-known that in order to show a conservative ergodic transformation is zero-type, it suffices to check this convergence for a single set $A$ of positive finite measure [@HaKa64]. We show that every steep transformation with an increasing number of cuts is zero-type.
Let $T$ be a normal, steep rank-one transformation, and suppose that $\{r_n\}$ is nondecreasing with $\sup\{r_n\} = \infty$. Then $T$ is zero-type.
Consider $I = (0, 1)$. For $N$ sufficiently large, we have $$\mu(I \cap T^k I) = \dfrac{|D(I, N) \cap (k + D(I, N))|}{|D(I, N)|}.$$ Now, $|D(I, N) \cap (k + D(I, N))|$ counts the number of representations $$k = \displaystyle \sum_{i = 0}^{N-1} (d_i - d_i')$$ with $d_i, d'_i \in H_i$. (Recall that $D(I, N) = H_0 \oplus H_1 \oplus \cdots \oplus H_{N-1}$.) If $k \notin D(I, N) - D(I, N)$, then $\mu(I \cap T^k I) = 0$, so suppose that $k \in D(I, N) - D(I, N)$. Then there is at least one representation $$k = \displaystyle \sum_{i = 0}^{N-1} (x_i - x_i')$$ with $x_i, x'_i \in H_i$. Now, fix $n$ and suppose $x_n - x'_n \neq 0$. By uniqueness of representation, any other representation (12) of $k$ must have $d_n = x_n$ and $d'_n = x'_n$. In particular, the only indices $i$ at which (12) can differ from (13) are those for which $x_i - x'_i = 0$. In these cases we must have $d_i = d'_i$, but otherwise there are no restrictions (i.e., $d_i = d'_i$ can be any element of $H_i$). Hence $$\displaystyle |D(I, N) \cap (k + D(I, N))| = \prod \limits_{x_i - x'_i = 0} |H_i|$$ with the product being taken over all $i$ for which $x_i - x'_i = 0$. Since $$\displaystyle |D(I, N)| = \prod \limits_{i = 0}^{N-1} |H_i|$$ it follows that $$\mu(I \cap T^k I) = \left( \prod \limits_{x_i - x'_i \neq 0} |H_i| \right)^{-1}.$$ Now, if $k > M_n$, then the representation (13) of $k$ must have $x_m - x'_m \neq 0$ for some $m \geq n$. This implies that $$\mu(I \cap T^k I) \leq \dfrac{1}{|H_m|} = \dfrac{1}{r_m} \leq \dfrac{1}{r_n},$$ which shows $\mu(I \cap T^k I) \rightarrow 0$ as $k \rightarrow \infty$. Hence $T$ is zero-type, as desired.
We thus have:
There exist rank-one transformations that are zero-type but not rationally weakly mixing.
In [@Aa12b], Aaronson recently constructed a zero-type transformation of the form $T\times S$, where $S$ is a Markov shift, such that $T\times S$ is not subsequence rationally weakly mixing. Our examples, however, are rank-one, so of a different nature, and were constructed independently.\
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We note that it follows from Theorem F in Aaronson [@Aa12] that there exist subsequence rationally weakly mixing transformation of positive type; a rank-one example is given by the subsequence rationally weakly mixing transformation of Section 3. (Indeed, this is partially rigid since $\mu(I \cap T^{h_i}I) \geq 1/2$ for every $i$.) Aaronson [@Aa12b] also constructed positive-type, rank-one, transformations that are not subsequence rationally ergodic.
Examples of Rational Weak Mixing {#S:ratweakmixex}
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We end with a construction of a positive-type rank-one transformation which is rationally weakly mixing. Let $T$ be a Chacón-like transformation ($r_n = 3$, $\{s_{n, 0} = 0, s_{n, 1} = 1, s_{n, 2} \geq 3h_n + 1\}$) with enough spacers added above every third subcolumn so as to have $h_{n+1} = 3^c h_n$ for some fixed integer $c \geq 2$. Then $h_n = 3^{cn}$ and $D(I, n) = \displaystyle H_0 \oplus H_1 \oplus \cdots \oplus H_{n-1}$, where $H_i = \{0,h_i,2h_i +1\}$.
\[T:ratwm\] The above transformation $T$ is rationally weakly mixing.
We prove rational weak mixing for levels. Let $j\in \mathbb N_0$, $J = J_1$ be the bottom level of $C_j$, and let $J_2 = T^dJ_1$. As in the proof of Theorem \[T:almoststeep\], it suffices to show the convergence (11). Now, for any $n$, we may choose $m$ such that $M_{m-1} \leq n-1 < M_m$. Then the quotient (11) is asymptotically bounded above by $P_m / Q_{m-1} = 3 P_m / Q_m$, so it suffices to prove $P_m / Q_m \rightarrow 0$ as $m \rightarrow \infty$.\
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By the triangle inequality, $$\begin{aligned}
P_m &= \displaystyle\sum_{k = -M_m}^{M_m}|\mu(J \cap T^k J) - m (J \cap T^{k+d}J)| \\
&\leq \displaystyle \sum_{\ell=0}^{d-1}\displaystyle\sum_{k = -M_m}^{M_m} |\mu(J \cap T^{k+\ell} J) - m (J \cap T^{k+1+\ell}J)|.\end{aligned}$$ Since each of the $d$ outer sums on the right differs from the $\ell=0$ sum by a finite number of terms, it suffices to prove the convergence with only the $\ell= 0$ sum. That is, we wish to show $$\frac{\sum\limits_{k = -M_m}^{M_m}|\mu(J\cap T^kJ)-\mu(J\cap T^{k+1}J)|}{\sum\limits_{k = -M_m}^{M_m}\mu(J\cap T^kJ)} \rightarrow 0.$$ To do this, it will be useful to introduce some auxiliary functions. Given $(d, d') \in D(J, m) \times D(J, m)$, write $$d - d' = \displaystyle \left(\sum_{i=j}^{m - 1} d_i\right) - \left(\sum_{i=j}^{m - 1} d_i'\right)$$ with each $d_i, d_i' \in \{0, h_i, 2h_i + 1\}$. Replacing each instance of $2h_i + 1$ with $2h_i$ in (14) yields a sum of the form $$\displaystyle \sum_{i = j}^{m-1} \varepsilon_i3^{ci}$$ with each $\varepsilon_i \in \{-2, -1, 0, 1, 2\}$. This defines a function $$g: D(J, m) \times D(J, m) \rightarrow \{-2, -1, 0, 1, 2\}^{m - j}$$ taking the pair $(d, d')$ to the vector $\varepsilon = (\varepsilon_i)_{i = j}^{m-1}$, with the $\varepsilon_i$ as in (15).\
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For each $\varepsilon \in \{-2, -1, 0, 1, 2\}^{m - j}$, define the “multiplicity function" $\tilde{\varepsilon}$ on $D(J, m) - D(J, m)$ by $$\tilde{\varepsilon}(k) = |g^{-1} (\varepsilon) \cap \{(d,d') | d,d' \in D(J,m) \text{ and } d-d'=k \}|.$$ That is, $\tilde{\varepsilon}(k)$ counts the number of pairs $(d,d')$ in $g^{-1}(\varepsilon)$ with $d-d'=k$. Then
The following properties hold:
(a) Fix $k \in D(J, m) - D(J, m)$. Then $$\displaystyle \sum_{\varepsilon} \tilde{\varepsilon}(k) = |D(J,m)\cap (k + D(J, m))|.$$ where the sum on the left is taken over all $\varepsilon \in \{-2, -1, 0, 1, 2\}^{m - j}$.
(b) Fix $\varepsilon = (\varepsilon_i)_{i = j}^{m-1} \in \{-2, -1, 0, 1, 2\}^{m - j}$. For each $p \in \{-2, -1, 0, 1, 2\}$, let $a_p$ be the number of $\varepsilon_i$ equal to $p$. Then $$\displaystyle \sum_k \tilde{\varepsilon}(k)=3^{a_0}2^{a_1+a_{-1}}.$$ where the sum on the left is taken over all $k \in D(J, m) - D(J, m)$.
For (a), simply observe that both the left and right-hand expressions count the number of pairs $(d, d') \in D(J, m) \times D(J, m)$ for which $d-d'=k$. For (b), observe that the sum on the left counts the number of pairs $(d,d')$ whose associated vector is $\varepsilon$. Now, if $\varepsilon_i = 0$ in (15), then we must have $d_i = d_i'$ in (14), and there are three ways that this can happen. Similarly, if $\varepsilon_i = 1$, then either $d_i = h_i$ and $d_i' = 0$, or $d_i = 2h_i + 1$ and $d_i' = h_i$. Proceeding in this manner, a counting argument yields the desired equality.
($\textit{Proof of Theorem~\ref{T:ratwm}, continued.}$)\
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Applying Lemma 6.2 (a) and Lemma 2.1, we thus need to show $$\displaystyle \frac{\sum\limits_{k = -M_m}^{M_m}\left|\sum\limits_{\varepsilon}\tilde{\varepsilon}(k)-\sum\limits_{\varepsilon}\tilde{\varepsilon}(k+1)\right|}{\sum\limits_{k = -M_m}^{M_m}\sum\limits_{\varepsilon}\tilde{\varepsilon}(k)} \rightarrow 0.$$ By the triangle inequality, it suffices to prove this convergence after exchanging the order of summation in both the numerator and denominator. To this end, we exhibit a nonincreasing function $c(t)$ which converges to $0$ such that $$\displaystyle R(\varepsilon):=\frac{\sum\limits_{k}|\tilde{\varepsilon}(k)-\tilde{\varepsilon}(k+1)|}{\sum\limits_{k} \tilde{\varepsilon}(k)}\leq c(a_1+a_{-1})$$ for each $\varepsilon$. Once we have such a function, we obtain the following bound for large enough $N$:
$$\begin{aligned}
\displaystyle \sum_{\varepsilon}\sum\limits_{k = -M_m}^{M_m}|\tilde{\varepsilon}(k)-\tilde{\varepsilon}(k+1)| &\leq \sum_{\varepsilon}\sum\limits_{k = -M_m}^{M_m} \tilde{\varepsilon}(k) c(a_1 + a_{-1}) \\
&\leq c(N) \sum_{\substack{{\varepsilon} \text{ with} \\ {a_1 + a_{-1} \geq N}}}\sum\limits_{k = -M_m}^{M_m} \tilde{\varepsilon}(k) \\
&+ c(0)\sum_{\substack{{\varepsilon} \text{ with} \\ {a_1 + a_{-1} < N}}}\sum\limits_{k = -M_m}^{M_m} \tilde{\varepsilon}(k).\end{aligned}$$
Dividing this by the denominator of (16) yields $$\displaystyle \frac{\sum\limits_{\varepsilon}\sum\limits_{k = -M_m}^{M_m}|\tilde{\varepsilon}(k)-\tilde{\varepsilon}(k+1)|}{\sum\limits_{\varepsilon}\sum\limits_{k = -M_m}^{M_m}\tilde{\varepsilon}(k)} \leq c(N) + c(0) d(N,m)$$ where $$\displaystyle d(N, m) = \dfrac{\sum\limits_{\substack{{\varepsilon} \text{ with} \\ {a_1 + a_{-1} < N}}} \sum\limits_{k = -M_m}^{M_m} \tilde{\varepsilon}(k) }{\sum\limits_{\varepsilon}\sum\limits_{k = -M_m}^{M_m}\tilde{\varepsilon}(k)}.$$ We claim that $d(N, m) \rightarrow 0$ as $m \rightarrow \infty$. Combined with the fact (still to be proven) that $c(N) \rightarrow 0$ as $N \rightarrow \infty$, this will imply the convergence of (16) to zero.\
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By Lemma 6.2 (b), $$\begin{aligned}
d(N, m) &= \left( \sum\limits_{\substack{{\varepsilon} \text{ with} \\ {a_1 + a_{-1} < N}}} 3^{a_0}2^{a_1+a_{-1}} \right) / \left( \sum\limits_{\varepsilon}3^{a_0}2^{a_1+a_{-1}} \right) \\
&\leq 2^N \left( \sum\limits_{\varepsilon} 3^{a_0} \right) / \left( \sum\limits_{\varepsilon}3^{a_0}2^{a_1+a_{-1}} \right)\end{aligned}$$ with the sums taken over all $\varepsilon \in \{-2, -1, 0, 1, 2\}^{m - j}$. Now, $\sum_{p=-2}^{2} a_p = m - j$ for each such $\varepsilon$, so we can view the sums over $\varepsilon$ as sums over 5-tuples $(a_0,a_{-1},a_{1},a_{-2},a_{2})$ of non-negative integers summing to $m-j$. That is, the above expression is equal to\
$$2^N \left(\sum \binom{m-j}{a_0, a_{-1}, a_{1}, a_{-2}, a_{2}} 3^{a_0} \right) / \left( \sum \binom{m-j}{a_0, a_{-1}, a_{1}, a_{-2}, a_{2}} 3^{a_0}2^{a_1}2^{a_{-1}} \right)$$\
with the summation as described above and $\binom{m-j}{a_0, a_{-1}, a_{1}, a_{-2}, a_{2}}$ the multinomial coefficient “$m-j$ choose $a_0, \cdots, a_2$". By the identity $$(x_1 + x_2 + x_3 + x_4 + x_5)^n = \sum_{e_1 + e_2 + e_3 + e_4 + e_5 = n} \binom{n}{e_1, e_2, e_3, e_4, e_5} x_1^{e_1} \cdots x_5^{e_5},$$ this is equal to $$2^N (3 + 1 + 1 + 1 + 1)^{m-j}/(3 + 2 + 2 + 1 + 1)^{m-j}.$$ Hence $D(N, m) \leq 2^N (7/9)^{m-j}$, which clearly goes to zero as $m \rightarrow \infty$.\
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Next, we will show that $$s(n):=\sup_{\varepsilon} \left\lbrace R(\varepsilon): a_1+a_{-1}=n\right\rbrace$$ converges to $0$. (See (17) for a definition of $R(\varepsilon)$.) Setting $$c(t):=\sup\left\lbrace s(n): n\geq t\right\rbrace$$ then produces a nonincreasing function with the desired properties and completes the proof.\
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Fix $\varepsilon$, and let $a$ be the minimum element of $D(J, m) - D(J, m)$ for which $\tilde{\varepsilon}(a) > 0$. Any $k \in D(J,m) - D(J,m)$ is expressible as $k = \sum \varepsilon_i 3^{ci} + \sum (+1) + \sum (-1)$, with the $+1$’s and $-1$’s coming from choosing $2h_i + 1$ for $d_i$ and $d_i'$ in (14). We now ask: how many $+1$’s and $-1$’s do we have for $k = a$? We have only one way of obtaining $\varepsilon_i = 2$ in (14): namely, $(2h_i + 1) - 0$. Similarly, we only have one way of obtaining $\varepsilon_i = -2$: namely, $0 - (2h_i +1)$. This introduces $a_2$ number of $+1$’s and $a_{-2}$ number of $-1$’s. We have three ways of obtaining $\varepsilon_i = 0$, none of which introduce a net number of $+1$’s or $-1$’s. For $\varepsilon_i = 1$, we have two possibilities: either $h_i - 0$ or $(2h_i + 1) - h_i$. Since we want to minimize $a$, we choose the former. Similarly, for $\varepsilon_i = -1$, we must have either $0 - h_i$ or $h_i - (2h_i + 1)$, and to minimize $a$ we choose the latter. It thus follows that $a$ has $a_2$ number of $+1$’s and $a_{-2} + a_{-1}$ number of $-1$’s; moreover, $\tilde{\varepsilon}(a) = 3^{a_0}$. It is then not difficult to see that $$\tilde{\varepsilon}(a + k) = 3^{a_0} \binom{a_1 + a_{-1}}{k}$$ for all $0 \leq k \leq a_1 + a_{-1}$, and is $0$ otherwise.\
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Letting $n = a_1 + a_{-1}$, we thus have $$R(\varepsilon)=\dfrac{1}{2^n} \left(\sum_{k = 0}^{n-1} \left|\binom{n}{k}-\binom{n}{k+1}\right| + 2\right).$$ Suppose $n = 2l -1$. (The case when $n$ is even is dealt with similarly.) Since $$\left|\binom{n}{k}-\binom{n}{k-1}\right|=\binom{n + 1}{k}\left|\frac{(n + 1)-2k}{n+1}\right|,$$ the above expression yields $$\begin{aligned}
R(\varepsilon) &= \dfrac{1}{2^n} \left( \sum_{k=1}^{n}\binom{n + 1}{k}\left|\frac{(n + 1)-2k}{n+1}\right| + 2\right) \\
&= \dfrac{1}{2^n} \left(\sum_{k=1}^{2l - 1} \binom{2l}{k} \left| \dfrac{l - k}{l} \right| + 2\right) \\
&\leq \dfrac{1}{2^n} \left(2 \sum_{k=0}^{l} \binom{2l}{k} \left(\dfrac{l - k}{l} \right)\right).\end{aligned}$$ Using the combinatorial identity $$\sum_{k=0}^{l} \binom{2l}{k} \left( \frac{l-k}{l} \right) = \frac{l+1}{2l}\binom{2l}{l+1},$$ we obtain $$R(\varepsilon) \leq \dfrac{1}{2^{2l}} \binom{2l}{l+1}.$$ It is not difficult to see that this goes to $0$ as a function of $l$, thus proving that $T$ is rationally weakly mixing for levels. By Theorem \[T:wraterg\] and Lemma 3.1, it follows that $T$ is rationally weakly mixing.
| {
"pile_set_name": "ArXiv"
} |
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abstract: 'We develop a complete theoretical description of photoassociative Stimulated Raman Adiabatic Passage (STIRAP) near a Feshbach resonance in a thermal atomic gas. We show that it is possible to use *low intensity* laser pulses to directly excite the continuum at a Feshbach resonance and transfer nearly the entire atomic population to the lowest rovibrational level in the molecular ground state. In case of a broad resonance, commonly found in several diatomic alkali molecules, our model predicts a transfer efficiency $\eta$ up to 97% for a given atom pair, and up to 70% when averaged over an atomic ensemble. The laser intensities and pulse durations needed for optimal transfer are $10^2-10^3$ W/cm$^2$ and several $\mu$s. Such efficiency compares to or surpasses currently available techniques for creating stable diatomic molecules, and the versatility of this approach simplifies its potential use for many molecular species.'
address:
- 'Department of Physics, University of Connecticut, Storrs, CT 06269'
- 'ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138'
- 'Department of Physics, University of Connecticut, Storrs, CT 06269'
author:
- 'Elena Kuznetsova and Susanne F. Yelin'
- 'Marko Gacesa, Philippe Pellegrini, and Robin Côté'
title: Efficient formation of ground state ultracold molecules via STIRAP from the continuum at a Feshbach resonance
---
Introduction
============
The realization of rovibrationally stable dense samples of ultracold diatomic molecules remains one of the major goals in the field of atomic and molecular physics. While cooling diatomic alkali molecules was seen as a logical next step following the optical cooling of atoms, many of the possible applications currently under investigation extend beyond atomic and molecular physics. Testing fundamental symmetries based on high-precision spectroscopy of ultracold molecules [@DeMille-EDM; @Hudson; @weak-inter] or the attempts to detect the time variation of fundamental constants [@fine-structure] are examples of such applications. Another one is ultracold chemistry, where the interacting species and products are in a coherent quantum superposition state and could be realized by controlling reactive collisional processes [@quant-chem]. Important insights about new phases of matter could be gained from strong anisotropic dipole-dipole interaction between ultracold dipolar molecules [@Baranov]. Finally, ultracold polar molecules could also represent an attractive platform for quantum computation [@DeMille; @Susanne]. Many of those applications require dense samples of ultracold polar molecules in the lowest rovibrational state that makes them collisionally stable and long-lived.
Translationally ultracold (100 nK - 1 mK) molecules are produced from an ultracold atomic gas by photoassociation (PA) [@photoass] or magnetoassociation (MA) [@Feshbach]. In a typical PA scheme, a pair of colliding atoms is photoassociated into a bound electronically excited molecular state that spontaneously decays, forming molecules in the electronic ground state. In magnetoassociation, a magnetic field is adiabatically swept across a Feshbach resonance, converting two atoms in a matching scattering state into a molecule. Both techniques produce weakly bound molecules in highly excited vibrational states of the ground electronic potential. Such molecules have to be rapidly transferred to deeply bound vibrational states before they are lost from the trap due to inelastic collisions.
Stimulated Raman Adiabatic Passage (STIRAP) [@STIRAP] has recently attracted significant interest as an efficient way to produce deeply bound molecules, starting from Feshbach molecules [@STIRAP-Fesh; @STIRAP-Fesh1]. It allows to realize high transfer efficiency and preserve the high phase-space density of an initial atomic gas. In STIRAP, the laser pulses, coupling an initial and a final state to an intermediate excited state, are applied in a counter-intuitive sequence where a pump pulse is preceeded by a Stokes pulse. During the transfer, the system stays in a “dark” state, [*i.e.*]{}, a coherent superposition of initial and final states, preventing any losses that would otherwise occur from the excited state. By adiabatically changing amplitudes of the laser pulses, the “dark” state evolves from the initial to the final state, resulting in nearly 100% transfer efficiency [@STIRAP].
Efficient adiabatic passage from the continuum requires laser pulses shorter than the coherence time of the continuum [@Vardi; @Vardi-OE; @Ye-photoass]. The adiabaticity condition of STIRAP, $\Omega \tau_{\mathrm{tr}} \gg 1$, where $\tau_{\mathrm{tr}}$ is the transfer time, therefore implies a large effective Rabi frequency $\Omega$ for the pulses. In addition, dipole matrix elements between the continuum and the bound state are usually small, and so the pump pulse that couples the continuum and the excited state would require a very high intensity, which proves impractical. Thus the previous STIRAP experiments [@STIRAP-Fesh], being restricted by the very short coherence time of the continuum, used a Feshbach molecular state as an initial state.
The small continuum-bound dipole matrix elements can be dramatically increased by photoassociating atoms in the vicinity of a Feshbach resonance. It has been shown, both theoretically and experimentally, that the photoassociation rate increases in the presence of a Feshbach resonance by several orders of magnitude [@PA-Fesh; @PA-Fesh1; @PA-Fesh2; @FOPA]. This can be explained by considering that delocalized scattering states acquire some bound-state character due to admixture of a bound level associated with a closed channel, resulting in a large increase of the Franck-Condon factor between the initial scattering state and the final excited state. The recently proposed Feshbach Optimized Photoassociation (FOPA) technique [@FOPA] relies on this enhancement to directly reach deeply bound ground state vibrational levels from the scattering continuum. Consequently, photoassociation in the vicinity of a Feshbach resonance is expected to increase molecular formation rate up to $10^{6}$ molecules/s [@FOPA].
In the present work, we combine the approach used in FOPA with STIRAP for reducing the required pulse intensity. We predict highly efficient transfer of an entire atomic ensemble into the lowest rovibrational level in the molecular ground state.
The paper is organized as follows. In Section II, we derive a theoretical model of a combined atomic and molecular system. Fano theory is used to describe the interaction of a bound molecular state with the scattering continuum, represented as closed and open channel, respectively. The resulting continuum states are coupled by two laser fields to the vibrational target state in the ground state via the intermediate excited molecular electronic vibrational state. In Section III, we present the results of numerical solutions of the model for several alkali dimers. We find optimal Rabi frequencies and profiles of STIRAP pulses for those systems. Finally, we conclude in Section IV.
Model
=====
We consider a three level system as represented in Figure \[Levels\]. The ground level labeled ${\left| 1 \right\rangle}$ is the final product state to which a maximun of population must be transfered. Typically, this level will be the lowest virational level ($v''=0,J''=0$) of a ground molecular potential. This ground level is coupled to an excited bound level ${\left| 2 \right\rangle}$ of an excited molecular potential via a ”Stokes” pulse depicted by the blue down-arrow in Figure \[Levels\]. This level ${\left| 2 \right\rangle}$ is itself coupled via a pump pulse (red up-arrow) to an initial continuum of unbound scattering states ${\left| \Psi_{\epsilon} \right\rangle}$ of energies $\epsilon$ (grey area in Figure \[Levels\]). If we denote $C_{1}$, $C_{2}$ and $C(\epsilon)$ the time dependent amplitudes associated to the final, intermediate, and initial states ${\left| 1 \right\rangle}$, ${\left| 2 \right\rangle}$, and ${\left| \Psi_{\epsilon} \right\rangle}$, respectively, then the total wave function ${\left| \Phi \right\rangle}$ of the system is given by:
$${\left| \Phi \right\rangle}=C_{1}{\left| 1 \right\rangle}+C_{2}{\left| 2 \right\rangle}+\int \; d\epsilon \; C(\epsilon){\left| \Psi_{\epsilon} \right\rangle}.$$
![Schematics: a population from the initial state ${\left| \Psi_{\epsilon} \right\rangle}$ is transferred to a final target state ${\left| 1 \right\rangle}$ via an intermediate state ${\left| 2 \right\rangle}$. Both ${\left| \Psi_{\epsilon} \right\rangle}$ and ${\left| 1 \right\rangle}$ are coupled to ${\left| 2 \right\rangle}$ by a pump and a Stokes pulse, respectively labeled $\Omega_P$ and $\Omega_S$. A bound level ${\left| b \right\rangle}$ corresponding to a closed channel can be imbedded in the continuum.[]{data-label="Levels"}](Levels.eps){width="70.00000%"}
No restriction applies to the definition of the continuum state ${\left| \Psi_{\epsilon} \right\rangle}$ as it can be associated to either a single-channel or a multi-channel scattering state. In this work, we consider the multi-channel case in which a bound level ${\left| b \right\rangle}$ associated to a closed channel is embedded in the continuum of scattering states ${\left| \epsilon' \right\rangle}$ of an open channel. When the energy of ${\left| \epsilon' \right\rangle}$ coincides with that of ${\left| b \right\rangle}$, a so-called Feshbach resonance [@Fesh] occurs. These are common in binary collisions of alkali atoms due to hyperfine mixing and the tuning of the Zeeman interaction by an external magnetic field, hence the possibility to control interatomic interactions with a magnetic field. Following the Fano theory presented in Ref. [@Fano], the scattering state ${\left| \Psi_{\epsilon} \right\rangle}$ can be expressed as:
$${\left| \Psi_{\epsilon} \right\rangle}=a(\epsilon){\left| b \right\rangle}+\int d\epsilon' \; b(\epsilon,\epsilon'){\left| \epsilon' \right\rangle}\;,$$
with $$a(\epsilon)=\sqrt{\frac{2}{\pi\Gamma(\epsilon)}}\sin\Delta \;,$$ and $$b(\epsilon,\epsilon')=\frac{1}{\pi}\sqrt{\frac{\Gamma(\epsilon')}{\Gamma(\epsilon)}}\frac{\sin\Delta}{\epsilon-\epsilon'}-\cos\Delta \; \delta(\epsilon-\epsilon') \;.$$ Here, $\Delta =-\arctan(\frac{\Gamma}{2(\epsilon-\epsilon_{F})})$ is the phase shift due to the interaction between ${\left| b \right\rangle}$ and the scattering state ${\left| \epsilon \right\rangle}$ of the open channel. We assume $\Delta \in [-\pi/2,\pi/2]$. The width of the Feshbach resonance, $\Gamma=2\pi|V(\epsilon)|^{2}$, is weakly dependent on the energy, while $V(\epsilon)$ is the interaction strength between the open and closed channels. The position of the resonance, $\epsilon_{F}=E_{b}+P\int\frac{|V(\epsilon')|^{2}d\epsilon'}{\epsilon -\epsilon'}$, includes an interaction induced shift from the energy of the bound state $E_{b}$.
If we label $E_i$ the energy of the state ${\left| i \right\rangle}$, the total Hamiltonian $H$ is given by: $$H=\sum_{i=1,2} E_{i} |i\rangle \langle i| +
\int d\epsilon \; \epsilon |\Psi_{\epsilon}\rangle \langle \Psi_{\epsilon}|
+V_{\mathrm{light}} \;.$$ The light-matter interaction Hamiltonian $V_{\mathrm{light}}$ takes the form: $$V_{\mathrm{light}}=-\vec{\mu}_{21}\cdot\vec{{\cal E}}_{S}
|2\rangle \langle 1|
-\int d\epsilon\; \vec{\mu}_{2\Psi_{\epsilon}}\cdot\vec{{\cal E}}_{p}{\left| 2 \right\rangle}
\langle\Psi_{\epsilon}| + \mathrm{H.c.}\; ,$$ where $\vec{{\cal E}}_{p,S}=\hat{\vec{e}}_{p,S}{\cal E}_{p,S}\exp(-i\omega_{p,S}t)+\mathrm{c.c.}$ are the pump and Stokes laser fields of polarization $\hat{\vec{e}}_{p,S}$, respectively, while $\vec{\mu}_{21}$ and $\vec{\mu}_{2\Psi_{\epsilon}}$ are the dipole transition moments between the states ${\left| 2 \right\rangle}$ and ${\left| 1 \right\rangle}$, and ${\left| 2 \right\rangle}$ and ${\left| \Psi_{\epsilon} \right\rangle}$, respectively. In this form the Hamiltonian already takes into account mixing between the bound state of the closed channel and scattering states of the open channel. The Schrödinger equation describing STIRAP conversion of two atoms into a molecule is: $$\begin{aligned}
i\hbar \frac{\partial C_{1}}{\partial t} & = & E_{1}\;C_{1}-\vec{\mu}^{*}_{21}\cdot\vec{{\cal E}}^{*}_{S}\;C_{2},\\
i\hbar \frac{\partial C_{2}}{\partial t} & = & E_{2}\;C_{2}-\vec{\mu}_{21}\cdot\vec{{\cal E}}_{S}\;C_{1}-\int_{\epsilon_{th}}^{\infty}d\epsilon\; \vec{\mu}_{2\Psi_{\epsilon}}\cdot\vec{{\cal E}}_{p}\;C(\epsilon),\\
i\hbar \frac{\partial C(\epsilon)}{\partial t} & = & \epsilon \;C(\epsilon)-\vec{\mu}^{*}_{2\Psi_{\epsilon}}\cdot\vec{{\cal E}}^{*}_{p}\;C_{2}.\end{aligned}$$
For simplicity, we set the origin of the energy to be the position of the ground state ${\left| 1 \right\rangle}$, and use the rotating wave approximation with $C_1=c_1$, $C_2=c_2e^{-i\omega_S t}$, and $C(\epsilon)=c(\epsilon)e^{-i(\omega_S-\omega_P)t}$. The Schrödinger equation becomes: $$\begin{aligned}
\label{eq:c1}
i \frac{\partial c_{1}}{\partial t} & = & -\Omega_{S}c_{2},\\
\label{eq:c2}
i\frac{\partial c_{2}}{\partial t} & = & \delta c_{2}-\Omega_{S}c_{1}-\int_{\epsilon_{th}}^{\infty} d\epsilon\; \Omega_{\epsilon}c(\epsilon),\\
\label{eq:cont-ampl}
i\frac{\partial c(\epsilon)}{\partial t} & = & \Delta_{\epsilon} c(\epsilon)-\Omega^{*}_{\epsilon}c_{2},\end{aligned}$$ where $\delta=E_{2}/\hbar -\omega_{S}$, $\Delta_{\epsilon}=\epsilon/\hbar -(\omega_{S}-\omega_{p})$, and $\epsilon_{th}$ is the dissociation energy of the ground electronic potential with respect to the state ${\left| 1 \right\rangle}$. The Rabi frequencies of the fields are $\Omega_{S}=\vec{\mu}_{21}\cdot\hat{\vec{e}}_{S}{\cal E}_{S}/\hbar$ (assumed real), $\Omega_{\epsilon}=\vec{\mu}_{2\Psi_{\epsilon}}\cdot\hat{\vec{e}}_{p}{\cal E}_{p}/\hbar$.
The previous system of three equations can be reduced into a two-equation system by eliminating the continuum amplitude $c(\epsilon)$ in Eq.(\[eq:cont-ampl\]). Introducing a solution in the form of $c(\epsilon)=s(\epsilon)\exp{(-i\Delta_{\epsilon}t)}$ into Eq.(\[eq:cont-ampl\]), we get $$s=i\int^{t}_{0}dt'\;\Omega^{*}_{\epsilon}(t')c_{2}(t')e^{i\Delta_{\epsilon}t'}+s_{\epsilon}(t=0) ,$$ where $t=0$ is some moment before the collision of the two atoms. The resulting continuum amplitude is $$c=i\int^{t}_{0}dt'\;\Omega^{*}_{\epsilon}(t')c_{2}(t')e^{i\Delta_{\epsilon}(t'-t)}+s_{\epsilon}(t=0)e^{-i\Delta_{\epsilon}t}.$$
Inserting this result into Eq. (\[eq:c2\]), we obtain a final system of equations for the amplitudes of the bound states: $$\begin{aligned}
\label{eq:c1-new}
i \frac{\partial c_{1}}{\partial t} &=& -\Omega_{S}c_{2} ,\\
\label{eq:c2-new}
i\frac{\partial c_{2}}{\partial t}
&=& \delta c_{2}-\Omega_{S}c_{1}+i\int_{\epsilon_{th}}^{\infty}d\epsilon\;\Omega_{\epsilon}(t)\int^{t}_{0}dt'\;\Omega_{\epsilon}(t')^{*}c_{2}(t')e^{i\Delta_{\epsilon}(t'-t)} \nonumber \\
&&-\int_{\epsilon_{th}}^{\infty}d\epsilon\; \Omega_{\epsilon}(t)s_{\epsilon}(t=0)e^{-i\Delta_{\epsilon}t} , \\
&\equiv& \delta c_{2}-\Omega_{S}c_{1}+T-S .\nonumber\end{aligned}$$\
The third term of Eq. (\[eq:c2-new\]), labelled $T$, corresponds to the back-stimulation term, whereas the last term, labelled $S$, corresponds to the source function. In this source term, the initial amplitude of the continuum wave function $s_{\epsilon}(t=0)$ describing a collision at $t_{0}$ of two atoms with relative energy $\epsilon_{0}$ has been discussed in various contributions [@Vardi; @Vardi-OE; @Ye-photoass]. A Gaussian wavepacket provides the most classical description of a two-atom collision characterized by a minimal uncertainty relation between the energy bandwidth $\delta_{\epsilon}$ of the wavepacket and the duration of the collision: $$\label{s_epsiton_t0}
s_{\epsilon}(t=0)=\frac{1}{(\pi \delta^{2}_{\epsilon})^{1/4}}e^{-\frac{(\epsilon -\epsilon_{0})^{2}}{2\delta^{2}_{\epsilon}}+\frac{i}{\hbar}(\epsilon -\epsilon_{0})t_{0}}.$$
Futhermore, the Rabi frequency of the field coupling continuum states ${\left| \Psi_{\epsilon} \right\rangle}$ to the state ${\left| 2 \right\rangle}$ is given by [@Fano] $$\label{eq:Rabi-fr}
\Omega_{\epsilon}=\frac{\vec{\mu}_{2\epsilon}\cdot\hat{\vec{e}}_{p}{\cal E}_{p}}{\hbar}\frac{q\Gamma/2+\epsilon -\epsilon_{F}}{\sqrt{(\Gamma/2)^{2}+(\epsilon - \epsilon_{F})^{2}}}\mathrm{sgn}(\epsilon-\epsilon_{F}),$$ where $\vec{\mu}_{2\epsilon}$ is the dipole matrix element between an unperturbed scattering state ${\left| \epsilon \right\rangle}$ and the state ${\left| 2 \right\rangle}$, and $q$ is the Fano parameter, expressed as: $$\label{eq:q}
q=\frac{(\vec{\mu}_{2b}\cdot \hat{\vec{e}}_{p})+P\int\frac{V(\epsilon')(\vec{\mu}_{2\epsilon'}\cdot \hat{\vec{e}}_{p})d\epsilon'}{\epsilon-\epsilon'}}{\pi V^{*}(\epsilon)(\vec{\mu}_{2\epsilon}\cdot \hat{\vec{e}}_{p})},$$ where $\hat{\vec{e}}_{p}$ is the polarization vector of the pump field, and $\vec{\mu}_{2b}$ is the dipole matrix element between bound states ${\left| 2 \right\rangle}$ and ${\left| b \right\rangle}$. The $q$ factor is essentially the ratio of the dipole matrix elements from the state ${\left| 2 \right\rangle}$ to the bound state ${\left| b \right\rangle}$ (modified by the continuum) and to an unperturbed continuum state ${\left| \epsilon \right\rangle}$. This factor can be made much larger than unity, and as will be shown below, the total dipole matrix element from the continuum can be enhanced by this factor in the presence of the resonance. The magnitude of $q$ can be controlled by the choice of the vibrational state ${\left| 2 \right\rangle}$. Selecting a tightly bound excited vibrational state will increase the bound-bound and decrease the continuum-bound dipole matrix elements, resulting in larger $q$. On the contrary, choosing a highly excited state close to a dissociation threshold decreases $q$.
Using the expressions given in Eqs.(\[s\_epsiton\_t0\]), (\[eq:Rabi-fr\]), and (\[eq:q\]) for the initial amplitude of the continuum wave function, the Rabi frequency between the continuum state ${\left| \Psi_{\epsilon} \right\rangle}$ and the excited bound state ${\left| 2 \right\rangle}$, and the Fano parameter, respectively, we obtain the following complete expression for the source term: $$\label{eq:S}
S=S_{0}\int_{\epsilon_{th}}^{\infty}d\epsilon \;g(q,\epsilon)\; \mathrm{sgn}(\epsilon-\epsilon_{F})
e^{-\frac{(\epsilon -\epsilon_{0})^{2}}{2\delta^{2}_{\epsilon}}+\frac{i(\epsilon -\epsilon_{0})t_{0}}{\hbar}}e^{-i\Delta_{\epsilon}t},$$ with $S_{0}=\vec{\mu}_{2\epsilon}\cdot\hat{\vec{e}}_{p}{\cal E}_{p}/\hbar(\pi \delta_{\epsilon})^{1/4}$, and where the function $g(q,\epsilon)$ is defined as $$g(q,\epsilon)\equiv \frac{q+\frac{2}{\Gamma}(\epsilon-\epsilon_{F})}{\sqrt{1+\frac{4}{\Gamma^2}(\epsilon-\epsilon_{F})^{2}}} .
\label{eq:g}$$ We assume that the unperturbed continuum is structureless and the coresponding Rabi frequency $\vec{\mu}_{2\epsilon}\cdot\hat{\vec{e}}_{p}{\cal E}_{p}/\hbar$ depends only weakly on the energy. We also extend $\epsilon_{th}$ to $-\infty$ to have the initial continuum wavefunction normalized to unity: $\int_{-\infty}^{\infty}d\epsilon \; |C(\epsilon)|^{2}=1$.
We can as well obtain a complete expression for the back-stimulation term $T$. We have: $$T=\left|\frac{\vec{\mu}_{2\epsilon}\hat{\vec{e}}_{p}}{\hbar}\right|^{2}{\cal E}_{p}(t)\int_{\epsilon_{th}}^{\infty}d\epsilon \; g^2(q,\epsilon)
\int_{0}^{t} dt'\;c_{2}(t'){\cal E}_{p}(t')e^{i\Delta_{\epsilon}(t'-t)} .$$ Extending the lower integration limit allows for an analytical solution for the integrals over energy and time, leading to the following expression for the back-stimulation term: $$\begin{aligned}
T&=&\label{eq:back-stimulation}
\left|\frac{\vec{\mu}_{2\epsilon}\hat{\vec{e}}_{p}}{\hbar}\right|^{2}
\left [\pi \hbar {\cal E}_{p}^{2}(t)c_{2}(t)+\frac{\pi \Gamma}{2}(q-i)^{2}{\cal E}_{p}(t) \right. \nonumber\\
&& \left. \times\int_{0}^{t}dt'\; c_{2}(t')
{\cal E}_{p}(t')e^{[\Gamma/2\hbar+i(\epsilon_{F}/\hbar-\omega_{S}+\omega_{p})]
(t'-t)} \right] .\end{aligned}$$
Results
=======
In this work, we consider two different cases: first, when $\Gamma \gg \delta_{\epsilon}$, [*i.e.*]{}, when the width $\Gamma$ of the Feshbach resonance is much larger than the thermal energy spread $\delta_{\epsilon}$ of the colliding atoms, and second when $\Gamma \ll \delta_{\epsilon}$. By considering these two limiting cases of broad and narrow resonances, more practical expressions for both the source term $S$ and the back stimulation term $T$ can be found. The derivation of the final system of equations is given in \[appendixA\]. Here, we only describe the solutions of these systems for both broad and narrow resonances.
Using the parameters of the Stokes and pump photoassociating pulses listed in Table \[table:res-par\] for a broad ($\Gamma = 1$ mK) and a narrow ($\Gamma = 1$ $\mu$K) Feshbach resonance, we obtain the results depicted in Fig. \[resonances1\], with the left column corresponding to the broad resonance, and the right column to the narrow resonance. The top row shows the variation of the Rabi frequencies over the time period required for the population transfer calculated using Eqs.(\[eq:c1-new\]) and (\[eq:c2-new\]) along with population in the intermediate state ${\left| 2 \right\rangle}$ (middle row) and final state ${\left| 1 \right\rangle}$ (bottom row).
![Time-dependence of the Stokes and pump pulses (top row) and population in state $|2\rangle$ (middle row) and target state $|1\rangle$ (bottom row) for the STIRAP sequence. The left column is for a broad Feshbach resonance, while the right column is for a narrow resonance (see Table \[table:res-par\] for values of parameters used). The dashed blue lines in the left column are the results obtained without resonance, when the parameters are adjusted to obtain the same overall transfer fraction as for the broad resonance. Stokes Rabi frequency is in units of $10^{6}$ s$^{-1}$, while the pump Rabi frequency is in dimensionless units $\left(16\pi/\delta_{\epsilon}\right)^{1/4}\vec{\mu }_{2\epsilon}\vec{e}_{p}{\cal E}_{p}$ in the broad resonance limit and $\left(2\pi/\Gamma\right)^{1/2}\vec{\mu}_{2\epsilon}\vec{e}_{p}{\cal E}_{p}$ in the narrow resonance limit. Note that the scale for the Rabi frequencies in the narrow resonance case is 40 times the scale for the broad resonance, and the magnitude of the pump Rabi frequency is enlarged 10 times for better visibility.[]{data-label="resonances1"}](resonances1.eps){width="\textwidth"}
For the broad case, we considered a Feshbach resonance with a width $\Gamma=1$ mK, which is a typical value for broad resonances (see examples in Appendix A.1), and a thermal atomic ensemble with an energy bandwidth $\delta_{\epsilon}=10$ $\mu$K. We see that the transfer can reach $\sim 97$% of the continuum state into the target state $|1\rangle$ (see Fig. \[resonances1\] c). The parameters of the Gaussian laser pulses we used (optimized Rabi frequencies, durations and delays of laser pulses) are given in Table \[table:res-par\]: the peak intensities of the Stokes and pump fields were calculated from Rabi frequencies as $I_{S}=c{\cal E}^{2}_{S}/8\pi=c(\Omega_{S}^{0}\hbar)^{2}/8\pi \mu^{2}_{21}$ and $I_{p}=c{\cal E}^{2}_{p}/8\pi=c(\Omega_{p}^{0})^{2}\delta_{\epsilon}/32\pi^{3/2}\mu_{2\epsilon}^{2}$, where we use Eq.(\[eq:q\]) to estimate the continuum-bound dipole matrix element $\mu_{2\epsilon}\approx \mu_{2b}/q \pi V(\epsilon)=\sqrt{2}\mu_{2b}/q\sqrt{\pi \Gamma}$, resulting in $I_{p}=q^{2}c (\Omega_{p}^{0})^{2}\delta_{\varepsilon}\Gamma/64\sqrt{\pi}\mu^{2}_{2b}$.
------------ --------------------- ---------- ------------------- ------------ ---------------- --------- --------- ------------ ------------
[Reso-]{} $\delta_{\epsilon}$ $\Gamma$ $\Omega^0_{S}$ $I_{S}$ $I_{p}$ $T_{S}$ $T_{p}$ $\tau_{S}$ $\tau_{p}$
[nance]{} $\mu$K $\mu$K $10^{8}$ s$^{-1}$ W/cm$^{2}$ W/cm$^{2}$ $\mu$s $\mu$s $\mu$s $\mu$s
[None]{} 10 — 0.72 62 $4\times 10^5$ 1.5 3 0.75 1.0
[Broad]{} 10 1000 0.74 65 4000 1.4 3.4 0.65 1.0
[Narrow]{} 100 1 2.24 600 400 0.157 0.3 0.1 0.207
------------ --------------------- ---------- ------------------- ------------ ---------------- --------- --------- ------------ ------------
: Parameters of the Stokes and pump photoassociating pulses providing optimal population transfer shown in Fig.\[resonances1\]. We use $q=10$, $\gamma = 10^8$ s$^{-1}$, and $\mu _{2b}=\mu _{21}=0.1$ D (1 D=$10^{-18}$ esu cm = 0.3934 $ea_0$). Rabi frequencies are modeled by Gaussians $\Omega_{S,p}=\Omega_{S,p}^{0}\exp{(-(t-t_{0}\pm \tau_{S,p}))/T_{S,p}^{2}}$, where $\pm$ refer to the Stokes and pump pulse, respectively.
\[table:res-par\]
When comparing the results for a broad resonance to the unperturbed continuum ([*i.e.*]{}, far from the resonance), we find that the source term $S$ is enhanced by the factor $g(q,\epsilon_{0})$ (see Eq. (\[eq:S-wide\]) in \[appendixA\]): $$g(q,\epsilon_0)= \frac{q+\frac{2}{\Gamma}(\epsilon_0-\epsilon_{F})}{\sqrt{1+\frac{4}{\Gamma^2}(\epsilon_0-\epsilon_{F})^{2}}} .$$ This factor has a maximum at $2(\epsilon_{0}-\epsilon_{F})/\Gamma=1/q$, with the corresponding maximum value $\sqrt{1+q^2}\approx q$ for $q\gg 1$: hence, the source amplitude is enhanced $q$ times. In this limit, all populated continuum states experience the same transition dipole matrix element enhancement factor to the state ${\left| 2 \right\rangle}$, so that the system essentially reduces to the case of a flat continuum with an uniformly enhanced transition dipole matrix element. One thus expects that in this limit, the adiabatic passage should be efficient, requiring less pump laser intensity when compared to the unperturbed ([*i.e.*]{} without resonance) scattering continuum. This is clearly demonstrated in Fig. \[resonances1\] (left column, dashed lines): to reach the same $\sim 97$% transfer efficiency achieved with the broad resonance, a very large pump laser intensity is required if there is no resonance in the continuum (Fig. \[resonances1\] a), while the Stoke laser intensity is basically the same. So, the comparison of Rabi frequencies for the broad resonance and no resonance cases shows that, to achieve the same transfer efficiencies, the required peak pump pulse intensity is about $100$ times larger without resonance. Condering the intensity used in this particular example, this would lead to intensities in the range of $5\times 10^5$ W/cm$^2$, making STIRAP from the continuum technically impossible to achieve without a resonance. This is consistent with the analysis of photoassociative adiabatic passage from an unstructured continuum [@Ye-photoass], and the above prediction that in the presence of a wide resonance the required pump laser intensity is reduced by a factor of $\sim 1/q^{2}$.
Results of adiabatic passage in a narrow resonance limit are shown in Fig. \[resonances1\] (right column). We considered a typical value of $\Gamma=1$ $\mu$K for a narrow resonance (see examples in Appendix A.2) and the ensemble energy bandwidth $\delta_{\epsilon}=100$ $\mu$K. Again, we give the parameters providing the optimal transfer in Table \[table:res-par\]. In this limit, the transfer efficiency is lower: in the specific case analyzed here, it does not exceed 47%. The reason for this lower efficiency is destructive quantum interference which leads to electromagnetically induced transparency [@EIT] in the transition from the continuum to the excited state. It can be explained using the following argument (see Fig. \[fig:interference\]). The limit of a narrow Feshbach resonance corresponds to a weak coupling between the bound Feshbach state and the scattering continuum, and thus can be neglected in this simplified explanation. The system then can be viewed as consisting of bound and continuum states ${\left| b \right\rangle}$ and ${\left| c \right\rangle}$ having the same energy, which are coupled by the pump field to a molecular state ${\left| 2 \right\rangle}$, itself coupled to the state ${\left| 1 \right\rangle}$ by the Stokes field. Assuming that initially all the population is in the state ${\left| c \right\rangle}$, due to the small interaction strength between ${\left| b \right\rangle}$ and ${\left| c \right\rangle}$, we can eliminate the state ${\left| b \right\rangle}$, taking into account its coupling to ${\left| 2 \right\rangle}$ by the pump laser as the formation of “dressed" states ${\left| \pm \right\rangle}=({\left| 2 \right\rangle}\pm {\left| b \right\rangle})/\sqrt{2}$. If the dipole matrix element of the ${\left| b \right\rangle}\rightarrow {\left| 2 \right\rangle}$ transition is much larger than that of the ${\left| c \right\rangle}\rightarrow {\left| 2 \right\rangle}$ transition, the detuning of the “dressed" states $|\Delta_{\pm}|=\Omega_{p}^{2b}\gg \Omega_{p}^{2c},\;\Omega_{S}$. As a result, the one-photon coupling of ${\left| c \right\rangle}$ to the excited state, as well as two-photon coupling to ${\left| 1 \right\rangle}$ vanishes, preventing the adiabatic transfer. This mechanism is similar to the Fano interference effect, the difference is that the continuum is initially populated. One can therefore view it as an inverse Fano effect. The effective dipole matrix element of the ${\left| c \right\rangle}\rightarrow {\left| 2 \right\rangle}$ transition is $\mu_{2c}\sim \mu_{2b}/q\sqrt{\xi}$. In the case we analyzed, $q=10$, $\xi=\Gamma/\sqrt{2}\delta_{\epsilon}=0.007$, and $\mu_{2c}\approx \mu_{2b}$, which gives $\sim 50\%$ transfer efficiency.
The transfer efficiency increases if the Feshbach state is far detuned from the populated continuum. Our calculations show that for a Feshbach state detuning $\gg |\Omega_{2b}|^{2}/\gamma$, the transfer efficiency reaches $70\%$ using the laser pulse parameters in Table \[table:res-par\]. We note that the smaller intensity of the pump pulse used for the narrow resonance, as compared to the broad resonance, is due to the fact that we used the same $q=10$ and assumed $\mu_{2b}=0.1$ D for both resonances. From the definition of $q$, it means that the continuum-bound dipole matrix element $\mu_{2\epsilon}$ is higher in the narrow than in the broad resonance case we considered. This explains the smaller resulting pump pulse intensity. The overall conclusion for a narrow resonance is that, as opposed to a broad resonance, the presence of the Feshbach resonance prevents one from realizing high transfer efficiencies. It should be noted, however, that the destructive quantum interference effect is based on negligible interaction between the Feshbach and continuum states during the transfer time, since $\tau_{\mathrm{tr}} < \delta_{\epsilon}^{-1} \ll \Gamma^{-1}$. This argument shows that already for $\Gamma \ge \delta_{\epsilon}$, there is enough interaction to neutralize the effect of destructive interference. Therefore, we expect that the broad resonance limit can be extended down to $\Gamma \sim \delta_{\epsilon}$, making it applicable to a wide variety of atomic species.
Applications to the conversion of an entire atomic ensemble into a ground rovibrational molecule gas
====================================================================================================
The results of Fig. \[resonances1\] are for a pair of atoms having a specific mean collision energy $\epsilon_{0}=\hbar(\omega_{S}-\omega_{p})$. Such situation could be realized in very tight traps, [*e.g.*]{}, in tight optical lattices. For a system with a wider energy distribution, one would like to find an ensemble averaged transfer efficiency, and thus one needs to calculate the transfer probability $P(\epsilon_{0})=\left|c_{1}\right|^{2}$ for all $\epsilon_{0}$ within the thermal spread of energies, and perform the averaging as $$P_{\rm avg}=\frac{2}{\sqrt{\pi}(k_{B}T)^{3/2}}\int_{0}^{\infty}e^{-\epsilon_{0}/k_{B}T}\sqrt{\epsilon_{0}}P(\epsilon_{0})d\epsilon_{0},$$ where we assume a Maxwell-Boltzmann energy distribution, the pump laser resonant with the center of the distribution at $\langle \epsilon \rangle=3/2k_{B}T$, and set the bandwidth of the distribution at $\delta_{\epsilon}=\sqrt{\langle (\Delta \epsilon)^{2} \rangle}=\sqrt{3/2}k_{B}T$. The results are shown in Fig.\[resonances2\]. In this case, while the maximal transfer efficiency in the broad resonance case is $\sim 70\%$, it can be achieved with lower laser intensities than in the case of a pair of atoms of Fig. \[resonances1\].
![Same as Fig. \[resonances1\], but for the energy averaged transfer. The parameters are listed in Table \[table:res-par-avg\].[]{data-label="resonances2"}](resonances2.eps){width="\textwidth"}
------------ --------------------- ---------- ------------------- ------------ ------------------ --------- --------- ------------ ------------
[Reso-]{} $\delta_{\epsilon}$ $\Gamma$ $\Omega^0_{S}$ $I_{S}$ $I_{p}$ $T_{S}$ $T_{p}$ $\tau_{S}$ $\tau_{p}$
[nance]{} $\mu$K $\mu$K $10^{8}$ s$^{-1}$ W/cm$^{2}$ W/cm$^{2}$ $\mu$s $\mu$s $\mu$s $\mu$s
[None]{} 10 — 0.50 30 $1.7\times 10^5$ 1.5 3.3 0.75 1.3
[Broad]{} 10 1000 0.60 40 2500 1.3 3.2 0.7 1.25
[Narrow]{} 100 1 2.24 600 400 0.157 0.3 0.1 0.207
------------ --------------------- ---------- ------------------- ------------ ------------------ --------- --------- ------------ ------------
: Parameters of the Stokes and pump photoassociating pulses providing optimal population transfer shown in Fig.\[resonances2\] for averaging over a Maxwell-Boltzmann distribution of energies. We use $q=10$, $\gamma = 10^8$ s$^{-1}$, and $\mu _{2b}=\mu _{21}=0.1$ D (1 D=$10^{-18}$ esu cm = 0.3934 $ea_0$).
\[table:res-par-avg\]
Given the adiabatic photoassociation probability $P(\epsilon)$ for two colliding atoms with relative energy $\epsilon$, we can calculate the number of atoms photoassociated during the time overlap $\tau_{\mathrm{tr}}$ of the Stokes and pump pulses. During this time, the atom with the energy $\epsilon=\mu v^{2}/2$, where $\mu$ is the reduced mass, will collide with atoms in the volume $\pi b^{2}v \tau_{\mathrm{tr}}$, where $\pi b^{2}$ is the collision cross-section. The impact parameter for the collision corresponding to a partial wave with angular momentum $J$ is $b=(J+1/2)\hbar/p=(J+1/2)\hbar/\sqrt{2\mu\epsilon}$. The number of collisions that atoms with a relative energy in the interval $\left(\epsilon,\epsilon+d\epsilon\right)$ will experience during the transfer time is therefore $N(\epsilon)d\epsilon=\pi b^{2} v \tau_{\mathrm{tr}}\rho(\epsilon)d\epsilon$, where $\rho(\epsilon)=2\rho \exp{(-\epsilon/k_{B}T)}\sqrt{\epsilon}/\sqrt{\pi}(k_{B}T)^{3/2}$ is the spectral density of the atoms ($\rho$ is the density of the sample). Finally, $J=0$ for ultracold $s$-wave collisions, and the fraction of atoms in the energy interval $\left(\epsilon,\epsilon+d\epsilon\right)$ photoassociated by the two pulses is $f(\epsilon)=P(\epsilon)N(\epsilon)$, or $$f(\epsilon)=\frac{\sqrt{2\pi} \hbar^{2}}{4(\mu k_{B}T)^{3/2}}\tau_{\mathrm{tr}}\rho P(\epsilon)\exp{(-\epsilon/k_{B}T)}.$$ The total fraction of atoms photoassociated by a pair of pulses is $f=\int_{0}^{\infty} d\epsilon\; f(\epsilon) \approx \langle P_{\rm avg} \rangle \rho \sqrt{2\pi} \tau_{\mathrm{tr}}\hbar^{2}/4\mu^{3/2}\sqrt{k_{B}T}$, where we assumed that $P(\epsilon)$ does not significantly vary within the ensemble, and approximated it by the averaged value. Considering as an example $^{6}$Li atoms at T=100 $\mu$K with an atomic density $\rho=10^{12}$ cm$^{-3}$, an overlap time $\tau_{\mathrm{tr}}\sim 1$ $\mu$s, and assuming $P_{\rm avg}=0.7$, the fraction of atoms photoassociated by the Stokes and pump pulses is $f \sim 2.5\times 10^{-4}$: for heavier atoms $f \sim 10^{-6}-10^{-5}$. It will therefore require $\sim 10^{4}-10^{6}$ pairs of pulses to convert an entire atomic ensemble into deeply bound molecules.
Since only a small fraction of atoms can be transferred to ${\left| 1 \right\rangle}$ by a pair of STIRAP pulses, a train of pulse pairs can be applied to photoassociate the entire atomic ensemble. To prevent excitation of molecules in ${\left| 1 \right\rangle}$ by subsequent pulses, they have to be removed before the next pair of pulses is applied. This could be realized by applying, after each pair of Stokes and pump pulses, a relatively long pulse resonant to a transition from ${\left| 1 \right\rangle}$ to some other vibrational level in the excited electronic potential which decays spontaneously to a deep vibrational state in the ground electronic potential. This long pulse would optically pump molecules out of the state ${\left| 1 \right\rangle}$ to deeper vibrational states in the ground electronic potential. It therefore has to be longer than the spontaneous decay time of the excited state. Care has to be taken that the excited state does not decay back into the scattering continuum. This would empty the ${\left| 1 \right\rangle}$ state and deposit molecules into ground potential vibrational states according to Franck-Condon factors before the next pair of pulses arrives. Finally, after all atoms have been converted into molecules the recently demonstrated optical pumping for molecules method [@opt-pump] can be applied, which would transfer molecules from all populated vibrational states into the ground level $v=0$.
The optimal strategy is to actually choose an excited state that decays mostly to the $v=0$ level. This would allow one to avoid storing molecules in unstable vibrational states and using the optical pumping method. If such a state cannot be directly reached from ${\left| 1 \right\rangle}$, a four-photon STIRAP transfer can be applied [@Our-PRA], which provides efficient transfer to deeply bound molecular states. It allows one to choose the final state ${\left| 1 \right\rangle}$, from which the excited state decaying predominantly to $v=0$ can be easily reached. In this case rotational selectivity can also be preserved, since only $v=0,J=0$ and $v=0,J=2$ states will be populated.
The total time required to photoassociate the whole atomic ensemble and transfer it to the $v=0$ level can be estimated as follows. As the numerical results show, adiabatic passage requires $\sim 5$ $\mu$s, the follow-up pulse emptying state ${\left| 1 \right\rangle}$ can have a $\sim 100$ ns duration, if the excited state lifetime is tens of ns, resulting in the whole sequence $\sim 6$ $\mu$s. Then the train of $10^{4}-10^{6}$ pulse pairs will take $\sim 0.1-10$ s. The final step, optical pumping to the $v=0$ level, requires $\sim$ hundred $\mu$s, so the overall formation time is $\sim 0.1-10$ s. Given an illuminated volume $\sim 1$ mm$^{3}$ and an atomic density $\rho \sim 10^{12}$ cm$^{-3}$ the resulting production rate is expected to be $10^{8}-10^{10}$ molecules/s. This compares well with the recent experiment on STIRAP production of ground state KRb molecules starting from the Feshbach state, where the entire cycle including creation of Feshbach molecules takes $\sim 10$ s [@STIRAP-Fesh].
Conclusion
==========
Combining photoassociation and coherent optical transfer to molecular ground vibrational states can allow one to convert an entire atomic ensemble into deeply bound molecules, and to produce a high phase-space density ultracold molecular gas. We have analyzed photoassociative adiabatic passage in a thermal ultracold atomic gas near a Feshbach resonance. The presence of a bound state imbedded in and resonant with scattering continuum states strongly enhances the continuum-bound transition dipole matrix element to an excited electronic state, thus requiring less laser intensity for efficient transfer. In the limit of a wide resonance when compared to the thermal spread of collision energies, the dipole matrix element is enhanced by the Fano parameter $q$. Choosing a tightly bound excited vibrational state, $q$ can be made much larger than unity, resulting in the intensity of the pump pulse required for efficient adiabatic passage to be $\sim 1/q^{2}$ times smaller than in the absence of the resonance. We modeled the adiabatic passage using typical parameters of alkali dimers and found intensities and durations of STIRAP pulses providing optimal transfer. Intensities of the pump pulse, coupling the continuum to an excited state, were found to be a few kW/cm$^{2}$, which is $\sim 100$ times smaller than without resonance. Optimal pulse durations are several $\mu$s, resulting in energies per pulse $\sim 10$ $\mu$J for a focus area of $1$ mm$^{2}$.
If the Feshbach resonance is narrow compared to the thermal energy spread of colliding atoms, adiabatic passage is hindered by destructive quantum interference. The reason is that electromagnetically induced transparency significantly reduces the transition dipole matrix element from the scattering continuum to an excited state in the presence of the bound Feshbach state. In the narrow resonance limit, photoassociative adiabatic passage is therefore more efficient if the resonance is far-detuned.
Due to low atomic collision rates at ultracold temperatures, only a small fraction of atoms can be converted into molecules by a pair of photoassociative pulses. To convert an entire atomic ensemble, a train of pulse pairs can be applied. We estimate that $10^{4}-10^{6}$ pulse pairs will associate an atomic gas of alkali dimers with a density $10^{12}$ cm$^{-3}$ in an illuminated volume of $1$ mm$^{3}$ in $0.1-10$ s, resulting in extremely high production rates of $10^{8}-10^{10}$ molecules/s. High transfer efficiencies combined with low intensities of adiabatic photoassociative pulses also make the broad resonance limit attractive for quantum computation. For example, a scheme proposed in [@Ostrovskaya] can be realized, where qubit states are encoded into a scattering and a bound molecular states of polar molecules. To perform one and two-qubit operations, this scheme requires a high degree of control over the system, which our model readily offers.
Finally, marrying FOPA and STIRAP is a very promising avenue to produce large amounts of molecules, for a variety of molecular species. In fact, although we described here examples based on magnetically induced Feshbach resonances, such resonances are extremely common, and can be induced by several interactions, such as external electric fields or optical fields. Even in the absence of hyperfine interactions, other interactions can provide the necessary coupling, such as in the case of the magnetic dipole-dipole interaction in $^{52}$Cr [@pfau; @pavlovic].
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was partially founded by the National Science Foundation, Army Research Office, and the U.S. Department of Energy, Office of Basic Energy Sciences.
Adiabatic passage in the limits of broad and narrow Feshbach resonances {#appendixA}
=======================================================================
In this appendix, we discuss Eqs.(\[eq:c1-new\]) and (\[eq:c2-new\]) for various relative widths of the Feshbach resonance $\Gamma$ with respect to the thermal energy spread $\delta_{\epsilon}$ of the colliding atoms. We first describe the case of a broad resonance, [*i.e.*]{}, when the width of the Feshbach resonance greatly exceeds the thermal energy spread ($\Gamma\gg\delta_{\epsilon}$), and second consider the opposite situation of a narrow resonance ($\Gamma\ll\delta_{\epsilon}$). Finally, we briefly present the case where there is no resonance.
Limit of a broad Feshbach resonance $\Gamma\gg \delta_{\epsilon}$
-----------------------------------------------------------------
The typical thermal energy spread for colliding atoms in photoassociation experiments with non-degenerate gases is $\delta_{\epsilon} \sim 10-100$ $\mu$K. The broad resonance case occurs for resonances having a width of several Gauss ($\sim 1$ mK), for which we have $\Gamma/\delta_{\epsilon}\sim 10 -100$. A wide variety of systems exhibit broad resonances. For instance, they can be found in collision of $^{6}$Li atoms at 834 G for the $|f=1/2,m_{f}=1/2\rangle$ channel ($\Gamma = 302$ G= 40 mK) and in $^{7}$Li at 736 G for the $|f=1,m_{f}=1\rangle$ channel ($\Gamma = 145$ G = 19 mK). We note here that these values of $\Gamma$ are slightly different than the “magnetic" width $\Delta B$ usually given and based on the modelling of the scattering length.
The source function can be readily calculated from Eq.(\[eq:S\]) by noticing that the Rabi frequency term can be set at $\epsilon=\epsilon_{0}$ corresponding to the maximum of the Gaussian function in the integrand. Using the function $g(q,\epsilon)$ defined in Eq.(\[eq:g\]), the result takes the form $$\begin{aligned}
\label{eq:S-wide}
S_{w}&=&S_{0}\sqrt{2\pi}\delta_{\epsilon} g(q, \epsilon_{0})\mathrm{sgn}(\epsilon_{0}-\epsilon_{F}) e^{-(t-t_{0})^{2}\delta^{2}_{\epsilon}/2\hbar^{2}-i(\epsilon_{0}/\hbar-(\omega_{S}-\omega_{p}))t} \nonumber \\
&=&S_{\mathrm{no-res}}g(q, \epsilon_{0})\mathrm{sgn}(\epsilon_{0}-\epsilon_{F}),\end{aligned}$$ where $S_{\mathrm{no-res}}$ is the source function without a resonance given below in Eq.(\[eq:sorce-nr\]). Strictly speaking, this expression is valid for $|\epsilon_{F}-\epsilon_{0}|\ge \delta_{\epsilon}$, but since $\Gamma \gg \delta_{\epsilon}$ Eq.(\[eq:S-wide\]) is a good approximation for a wide range of detunings $\epsilon_{F}-\epsilon_{0}$.
The back-stimulation term (\[eq:back-stimulation\]) can be further simplified in the limit of a broad resonance. In this case, both $c_{2}(t)$ and ${\cal E}_{p}(t)$ change on a time scale $\sim 1/\delta_{\epsilon}$, [*i.e.*]{}, slowly compared to the decay time $\sim \hbar/\Gamma$ of the exponent. Therefore, we can rewrite (\[eq:back-stimulation\]) as: $$\left|\frac{\vec{\mu}_{2\epsilon}\hat{\vec{e}}_{p}}{\hbar}\right|^{2}\pi \hbar \left[1+\frac{(q-i)^{2}}{1+2i(\epsilon_{F}-\hbar(\omega_{S}-\omega_{p}))/\Gamma}\right]c_{2}(t){\cal E}^{2}_{p}(t).$$ The system (\[eq:c1-new\])-(\[eq:c2-new\]) in the case of a broad resonance becomes: $$\begin{aligned}
\label{eq:c1-new1}
i \frac{\partial c_{1}}{\partial t} & = & -\Omega_{S}c_{2},\\
\label{eq:c2-new1}
i\frac{\partial c_{2}}{\partial t} & = & -\Omega_{S}c_{1} - S_{w}+(\delta -i\gamma)c_{2} \nonumber\\
& & -i\pi \hbar|\Omega_{\mathrm{no-res}}(t)|^{2}\left[1+\frac{(q-i)^{2}}{1+2i(\epsilon_{F}-\hbar(\omega_{S}-\omega_{p}))/\Gamma}\right]c_{2},\end{aligned}$$ where $\Omega_{\mathrm{no-res}}=\vec{\mu}_{2\epsilon}\hat{\vec{e}}_{p}{\cal E}_{p}/\hbar$ is the continuum-bound Rabi frequency in the absence of resonance. We also added a spontaneous decay term $\gamma c_{2}$, assuming that the excited molecules dissociate into high energy continuum states and the resulting atoms leave a trap. From Eq.(\[eq:S-wide\]), one can see that in a broad resonance case, the source amplitude is enhanced by the factor $g(q,\epsilon_{0})=(q+2(\epsilon_{0}-\epsilon_{F})/\Gamma)/\sqrt{1+4(\epsilon_{0}-\epsilon_{F})^{2}/\Gamma^{2}}$ when compared to the unperturbed continuum case. This factor has a maximum at $2(\epsilon_{0}-\epsilon_{F})/\Gamma=1/q$, with the corresponding maximum value $g_{\rm max}\sim q$ for $q\gg 1$.
Limit of a narrow Feshbach resonance $\Gamma \ll \delta_{\epsilon}$
-------------------------------------------------------------------
This situation occurs when the width of the resonance is of the order of a few micro-Gauss or less. Examples of narrow resonances include $^{6}$Li$^{23}$Na at 746 G for the $|f_{1}=1/2,m_{f1}=1/2\rangle |f_{2}=1,m_{f2}=1\rangle$ channel ($\Gamma=7.8$ mG = 1 $\mu$K) [@LiNa], or $^{6}$Li$^{87}$Rb at 882 G for the $|f_{1}=1/2,m_{f1}=1/2\rangle |f_{2}=1,m_{f2}=1\rangle$ channel (p-wave, $\Gamma =10$ mG = 1.3 $\mu$K).
We note that the source term expressed in Eq.(\[eq:S\]) can be rewritten in a time representation: $$\begin{aligned}
\label{eq:source}
S&=&S_{0}\sqrt{2\pi}\delta_{\epsilon}e^{-i(\epsilon_{0}/\hbar-(\omega_{S}-\omega_{p}))t} \nonumber \\
&\times & \left[ e^{-(\tau -\tau_{0})^{2}} + \xi e^{2iD-D^{2}}\int_{-\infty}^{\infty}e^{-(\tau'-iD)^{2}}(I_{1}(\xi|\tau-\tau_{0}-\tau'|) \right. \nonumber \\
&& -L_{-1}(\xi|\tau-\tau_{0}-\tau'|)-iq(I_{0}(\xi|\tau-\tau_{0}-\tau'|) \nonumber \\
&& \left.-L_{0}(\xi|\tau-\tau_{0}-\tau'|))\mathrm{sgn}(\tau-\tau_{0}-\tau'))d\tau'\right] , \end{aligned}$$ where we introduced the dimensionless variables $\tau=t\delta_{\epsilon}/\sqrt{2}\hbar$, $D=(\epsilon_{F}-\epsilon_{0})/\sqrt{2}\delta_{\epsilon}$, $\xi=\Gamma/\sqrt{2}\delta_{\epsilon}$; $I_{0,1}$ and $L_{0,-1}$ are modified Bessel and Struve functions. One can see from this expression that the source function is a sum of the pure source function of the unperturbed continuum, given by the first term in square brackets, and of the admixed bound state, given by the integral. The coefficient $\xi=\Gamma/\sqrt{2}\delta_{\epsilon}$, which is the ratio of the Feshbach resonance width to the width of the thermal energy spread, gives the ratio of contributions from the bound state and the unperturbed continuum, respectively.
It is then easier to notice that in the limit of a narrow resonance, the Gaussian function in the integrand of Eq.(\[eq:source\]) is much narrower than the Bessel and Struve functions, which change on the time scale $\sim 1/\xi$. Therefore the source term can be aproximated as: $$\begin{aligned}
\label{eq:S-narrow-res}
S_{n}&=&S_{0}\sqrt{2\pi}\delta_{\epsilon}e^{-i(\epsilon_{0}/\hbar-(\omega_{S}-\omega_{p}))t}
[e^{-(\tau -\tau_{0})^{2}} \nonumber \\
&&+\xi \sqrt{\pi}e^{2iD-D^{2}}(I_{1}(\xi|\tau -\tau_{0}|)-L_{-1}(\xi|\tau -\tau_{0}|) \nonumber \\
&&-iq(I_{0}(\xi|\tau -\tau_{0}|)-L_{0}(\xi|\tau -\tau_{0}|))\mathrm{sgn}(\tau -\tau_{0}))]. \end{aligned}$$
Since $\xi \ll 1$, the real part of the source function is given by the first term in the square brackets, which is a pure continuum source function, while the imaginary part is due to the admixed bound state and its magnitude depends on the product $\xi q$. Using asymptotic expansions of modified Bessel and Struve functions $I_{0}(x)-L_{0}(x)\rightarrow -2/\pi x$, $I_{1}(x)-L_{-1}(x)\rightarrow -2/\pi x^{2}$, it is seen from Eq.(\[eq:S-narrow-res\]) that the contribution to the source function from the bound state decays on the time scale $|\tau -\tau_{0}|\sim 1/\xi$, while the contribution from the unperturbed continuum decays on the time scale $|\tau -\tau_{0}|\sim 1 \ll 1/\xi$.
In the limit of a narrow resonance the system (\[eq:c1-new\])-(\[eq:c2-new\]) becomes:
$$\begin{aligned}
\label{eq:c1-new2}
i \frac{\partial c_{1}}{\partial t} & = & -\Omega_{S}c_{2},\\
\label{eq:c2-new2}
i\frac{\partial c_{2}}{\partial t} & = & -\Omega_{S}c_{1}-S_{n}+ (\delta -i\gamma)c_{2} \nonumber\\
&&-i\left|\frac{\vec{\mu}_{2\epsilon}\hat{\vec{e}}_{p}}{\hbar}\right|^{2}\left[\pi \hbar{\cal E}_{p}^{2}c_{2}+\frac{\pi\Gamma}{2}(q-i)^{2}{\cal E}_{p}(t)\right. \nonumber \\
& & \left.\times\int_{0}^{t} dt'\; c_{2}(t'){\cal E}_{p}(t')e^{\Gamma(t'-t)/2\hbar
+i(\epsilon_{F}/\hbar-(\omega_{S}-\omega_{p}))(t'-t)}\right] .\end{aligned}$$
Continuum without resonance
---------------------------
Finally, let us consider the case of a continuum without resonance. In this case the continuum-bound Rabi frequency Eq.(\[eq:Rabi-fr\]) is: $$\Omega_{\epsilon}=\Omega_{\mathrm{no-res}}=\vec{\mu}_{2\epsilon}\cdot\hat{\vec{e}}_{p}\;{\cal E}_{p}/\hbar ,$$ and the source function is $$\label{eq:sorce-nr}
S_{\mathrm{no-res}}=S_{0}\sqrt{2\pi}\delta_{\epsilon}\e^{-(t-t_{0})^{2}\delta_{\epsilon}^{2}/2\hbar^{2}-i(\epsilon_{0}/\hbar -(\omega_{S}-\omega_{p}))t} .$$
The back-stimulation term (\[eq:back-stimulation\]) reduces to $$\left|\vec{\mu}_{2\epsilon}\cdot\hat{\vec{e}}_{p}/\hbar\right|^{2}\pi \hbar {\cal E}_{p}^{2}c_{2}=\pi \hbar \left|\Omega_{\mathrm{no-res}}(t)\right|^{2}c_{2} ,$$ and the system (\[eq:c1-new\])-(\[eq:c2-new\]) takes the simple form: $$\begin{aligned}
\label{eq:c1-no-res}
i \frac{\partial c_{1}}{\partial t} & = & -\Omega_{S}c_{2},\\
\label{eq:c2-no-res}
i\frac{\partial c_{2}}{\partial t} & = & -\Omega_{S}c_{1} + (\delta -i\gamma)c_{2}
-i\pi \hbar|\Omega_{\mathrm{no-res}}(t)|^{2}c_{2}-S_{\mathrm{no-res}}. \end{aligned}$$
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Networks often represent systems that do not have a long history of studies in traditional fields of physics, albeit there are some notable exceptions such as energy landscapes and quantum gravity. Here we consider networks that naturally arise in cosmology. Nodes in these networks are stationary observers uniformly distributed in an expanding open FLRW universe with any scale factor, and two observers are connected if one can causally influence the other. We show that these networks are growing Lorentz-invariant graphs with power-law distributions of node degrees. These networks encode maximum information about the observable universe available to a given observer.'
address:
- '$^1$ Departament de F[í]{}sica Fonamental, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain'
- '$^2$ Center for Complex Network Research and Department of Physics, Northeastern University, 360 Huntington Avenue, Boston, MA 02115, USA'
- '$^3$ Department of Physics, Northeastern University, 360 Huntington Avenue, Boston, MA 02115, USA'
author:
- 'Mari[á]{}n Bogu[ñ]{}[á]{}$^1$, Maksim Kitsak$^2$, and Dmitri Krioukov$^3$'
title: Cosmological networks
---
Introduction
============
Network science is intrinsically multidisciplinary because the systems it studies come from different domains of science. Complex networks are everywhere indeed—in communication technologies, social and political sciences, biology and medicine, economics, or even linguistics [@DorMen-book03; @newman03c-review; @BoLaMoChHw06]. That is why many fields of science—computer science, social sciences, biology, statistics, mathematics, and certainly physics—have contributed tremendously over the last decade to network research. Surprisingly, even though statistical physics has been applied with great success to understanding complex networks, the systems that these networks represent can rarely display a long history of broad interest and focused research in traditional fields of physics. In fact, none of the network examples above provide an exception to this general rule. Exceptions, such as energy landscape networks [@Doye2002] and networks in background-independent approaches to quantum gravity [@BoLe87; @KrKi12; @KoMa08; @RoSp10], are rare indeed.
Here we add to this relatively short list of complex physical networks, a class of networks that naturally arise in cosmology. Specifically, we consider evolving networks of causal connections among stationary (co-moving) observers, homogeneously distributed in any open Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime [@Weinberg08-book]. These networks are purely classical. Nodes can represent a dust of classical particles, or (clusters of) galaxies, or indeed imaginary observers, scattered randomly throughout the space. The horizons of all the observers expand, and for any particular observer $O$ at any given proper time $\tau$, the network consists of all other observers within $O$’s horizon, up to a certain cut-off time $\tau_\nu>0$ in the past, which can be interpreted as the time of last scattering or the red shift beyond which the observer cannot observe [@Weinberg08-book]. A directed link from observer $B$ to observer $A$ in this network exists if $B$ is within $A$’s retarded horizon. The retarded horizon of $A$ is $A$’s horizon at earlier time $\tau_r<\tau$ such that light emitted by $A$ at time $\tau_r$ reaches $O$ at time $\tau$. This means that if there are some physical processes running at each observer, then directed paths between observers $X$ and $O$ in this network represent all possible causal relations between $X$ and $O$, including indirect relations over paths longer than one hop, Fig. \[fig:1\]. Here we show that this evolving network of maximum information about the universe that any observer can collect by her proper time $\tau$, is a growing power-law graph in any open homogeneous and isotropic (FLRW) spacetime.
![[**Direct vs. indirect causal relations.**]{} Black edges show the direct causal relations between observers. In b) and c) the blue paths are indirect causal relations between observers $X$ and $O$.[]{data-label="fig:1"}](fig1.pdf){width="0.8\linewidth"}
We emphasize a critical difference between these cosmological networks and causal sets in de Sitter network cosmology considered in [@KrKi12]. The latter are discretizations of $4$-dimensional spacetime—nodes are elementary events (points in space and time), and two events are connected if they are causally related, i.e., if they lie within each other’s light cones. The resulting networks are directed acyclic graphs, and all the linking dynamics is the appearance of new links connecting new nodes to the existing nodes lying in their past light cones. No new links appear between already existing nodes, since any two events are either timelike-separated and thus connected, or spacelike separated and thus disconnected. The cosmological networks considered here are discretizations of $3$-dimensional space. Time remains continuous. Therefore the evolution of nodes in these networks represent world-lines of co-moving observers. These networks have directed cycles, and new links not only connect new nodes to existing ones, but also appear at a certain rate between existing nodes, as they do in many complex networks [@DorMen-book03; @newman03c-review; @BoLaMoChHw06].\
Overlapping horizons in the Milne universe
==========================================
The metric in an open FLRW spacetime is given by $$ds^2=-d\tau^2+R(\tau)^2\left[d\chi^2+\sinh^2{\chi} d \Omega_{d-1}^2\right],
\label{eq:1}$$ where $\tau>0$ and $\chi>0$ are the cosmic time and “radial” coordinates, $d \Omega_{d-1}^2$ is the metric on the unit $(d-1)$-dimensional sphere, and $R(\tau)$ is the scale factor of the universe given by the Friedmann equations [@Weinberg08-book]. The scale factor $R(\tau)$ is just a conformal factor in the spacial part of the metric, where coordinates $(\chi, \Omega_{d-1})$ describe the hyperbolic $d$-dimensional space $\mathbb{H}^d$ of constant curvature $K=-1$. The spacetime is thus foliated by $d$-dimensional hyperbolic spaces: for any time $\tau$, the space is the hyperbolic $d$-dimensional space of constant curvature $K=-1/R(\tau)$. To simplify the calculations, we assume that $R(\tau)=\tau$, meaning that we are considering the Milne universe—a completely empty universe without any matter or dark energy [@Mukhanov05-book]. The results presented henceforth do not depend on a particular form of scale factor $R(\tau)$. We discuss this important point at the end.
In $(2+1)$ dimensions (the generalization to $(d+1)$ with $d>2$ is straightforward), the change of coordinates $(\tau,\chi,\theta)$ to $$\label{eq:coordinates}
\begin{array}{ccl}
x&=&\tau \sinh{\chi} \cos{\theta}\\
y&=&\tau \sinh{\chi} \sin{\theta}\\
t&=&\tau \cosh{\chi}
\end{array}$$ transforms the metric in Eq. (\[eq:1\]) into the Minkowski metric $$ds^2=-dt^2+dx^2+dy^2.
\label{minkowski}$$ However, this transformation does not map the original spacetime in Eq. (\[eq:1\]) to the whole Minkowski spacetime, but only to the future light cone of the event $t=x=y=0$. Indeed, the radial Minkowski coordinate $r=\sqrt{x^2+y^2}$ of an event at coordinates $(\tau,\chi,\theta)$ is $r=t
\tanh{\chi}$. This means that a stationary observer—that is, an observer at rest in the co-moving coordinates $(\chi,\theta)$ in $\mathbb{H}^2$—is receding from the origin $x=y=0$ at constant speed $v=\tanh{\chi}\le 1$. Consistent with homogeneity and isotropy of the universe, we assume that stationary observers are also homogeneously and isotropically distributed throughout space with constant density $\delta$. These observers are therefore points distributed in the hyperbolic space $\mathbb{H}^2$ according to a Poisson point process with point density $\delta$. In the Milne cosmology, an infinite number of such observers are thus initially at the origin of coordinates (the big bang), and then they all start moving in all directions within a bubble –in the considered case this bubble is a disk in $\mathbb{R}^2$– that expands at the speed of light (see the $(x,y)$ plane in Fig. \[fig:2\]). Because the distribution of observers is uniform in $\mathbb{H}^2$, any stationary observer will “see” all other observers receding from her with the Lorentz-invariant density of speeds $v$ $$\delta(v) \propto \delta \frac{v}{(1-v^2)^{3/2}}.
\label{speed_distribution}$$
{width="\linewidth"}
Without loss of generality or breaking Lorentz invariance, in what follows we focus on the stationary observer $O$ at rest at coordinate $\chi=0$, and therefore also at rest at $x=y=0$. According to Eq. (\[eq:coordinates\]), $O$’s proper time $\tau$ is equal to the time coordinate $t$ in the Minkowski spacetime. First, we determine the horizon of $O$ at any given proper time $\tau$. This horizon is the radius of the part of the universe that $O$ can observe, up to the past cut-off time $\tau_\nu$, which can be any positive number, $0<\tau_\nu<\tau$. This radius is determined by the intersection of $O$’s past light cone with the hyperboloid at time $\tau_\nu$, Fig. \[fig:2\]. At time $\tau > \tau_{\nu}$, the farthest particle that $O$ can observe is moving at a speed such that light emitted at proper time $\tau_{\nu}$ reaches $O$ at this time $\tau$, yielding the following simple expression for the hyperbolic radius of $O$’s horizon: $$\label{eq:horizon}
\chi_h=\ln{\left(\frac{\tau}{\tau_{\nu}}\right)}.$$ The size of the network, i.e., the number is nodes in it, is in this case the number of other observers that $O$ can observe, equal to the number of points within a hyperbolic disk of radius $\chi_h$. This number grows asymptotically linearly with time $\tau$: $$N(\tau)=2 \pi \delta (\cosh{\chi_h}-1)=\pi \delta \left[ \frac{\tau}{\tau_{\nu}}+\frac{\tau_{\nu}}{\tau}-2\right] \approx \pi \delta \frac{\tau}{\tau_{\nu}}.
\label{N_tau}$$ Any two observers $A$ and $B$ in $O$’s horizon are connected by a directed link from $B$ to $A$ if $B$ lies within the retarded horizon of $A$. If $A$’s radial coordinate is $\chi$, then the retarded horizon of $A$ is defined as its horizon at time $\tau_\chi=\tau e^{-\chi}$. According to Eq. (\[eq:horizon\]), $\tau_\chi$ is such that if $A$ emits light at her proper time $\tau_\chi$, then this light reaches $O$ at time $\tau$. This means that if $A$ has some physical state (possibly causally influenced by $B$) at time $\tau_\chi$, then this state can causally influence $O$ by time $\tau$.
Figure \[fig:2\] shows observer $A$ lying within the horizon of observer $O$. Observer $B$ is connected to $A$ because $B$ lies within $A$’s retarded horizon at time $\tau_\chi$, the latest time in $A$’s history that can influence $O$ at time $\tau$. Observer $C$ is outside of this horizon and therefore is not connected to $A$. The link between $O$ and $A$ is bi-directed because they lie within each other horizons.
Mapped to the hyperbolic plane, the horizon of observer $O$ is a disk of radius $\chi_h$, whereas the horizon of observer $A$ is a disk of radius $\chi_h-\chi$ centered at $A$ who is located at radial coordinate $\chi$. This disk is tangent to $O$’s horizon as illustrated in Fig. \[fig:3\]. The expected number of direct incoming connections to observer $A$, i.e., $A$’s in-degree $\bar{k}_{in}(\chi)$, is thus given by the number of points within a disk of radius $\chi_h-\chi$: $$\bar{k}_{in}(\chi)=2 \pi \delta (\cosh{(\chi_h-\chi)}-1) \approx \pi \delta e^{-(\chi-\chi_h)}.$$ On the other hand, since observers are distributed uniformly according to the hyperbolic metric, the density of them located at radial coordinate $\chi$ is given by distribution $$\rho(\chi)=\frac{\sinh{\chi}}{\cosh{\chi_h}-1} \approx e^{\chi-\chi_h}.
\label{rho_chi}$$ We thus have a combination of two exponential dependencies: $\bar{k}_{in}(\chi)\sim e^{-\chi}$ and $\rho(\chi)\sim e^\chi$. As one can check [@newman05], if in general the expected value $\bar{k}(x)$ of some random variable $k$ decays exponentially, $\bar{k}(x) \sim e^{-\alpha x}$, $\alpha>0$, as a function of random variable $x$ whose distribution is also exponential, $\rho(x) \sim e^{\beta x}$, $\beta>0$, then the distribution of $k$ is a power law, $P(k) \sim k^{-\gamma}$, with exponent $\gamma = \beta/\alpha+1$. In our case $\alpha=\beta=1$, so that $\gamma=2$: $$P(k_{in}) \sim \frac{1}{k_{in}^2},\quad\text{if $1\ll k_{in} < \pi \delta e^{\chi_h}$}.$$ In large networks with $\chi_h \gg1$ the average in-degree scales as $\langle
k_{in}\rangle \sim \pi \delta \chi_h \approx \pi \delta \ln{(N/\pi \delta)}$. The degree distributions in many large real networks are also close to power laws with exponents close to $2$ [@DorMen-book03; @newman03c-review; @BoLaMoChHw06].
![[**Milne universe projected onto the hyperbolic plane.**]{} All moving observers in Fig. \[fig:2\] and their horizons can be mapped to the hyperbolic plane $\mathbb{H}^2$ via the change of coordinates in Eq. (\[eq:coordinates\]). After the mapping, observers become static points on $\mathbb{H}^2$, while their horizons expand with cosmic time. The blue area is the horizon of observer $O$ of hyperbolic radius $\chi_h$. The green area is $A$’s retarded horizon of radius $\chi_h-\chi$, centered at $A$ and tangent to $O$’s horizon. Nodes $B$ and $C$ are the same as in Fig. \[fig:2\]. The picture does not depend on scale factor $R(\tau)$, which determines only how horizon $\chi_h$ grows with cosmic time $\tau$. []{data-label="fig:3"}](fig3.pdf){width="0.8\linewidth"}
Next we focus on the expected number of out-going connection, i.e., out-degree, of a node located at $(\chi,0)$. It is equal to the number of points within a domain in $\mathbb{H}^2$ defined as the locus of points $(\chi',\theta)$ such that their hyperbolic distances to the point $(\chi,0)$, $x$, are smaller than the radius of their retarded horizons $\chi_h-\chi'$, that is, $$\bar{k}_{out}(\chi)=2 \delta \int_{0}^{\pi}d\theta \int_0^{\chi_h} d \chi'\,\sinh{\chi'}\,\Theta(\chi_h-\chi'-x),$$ where $\Theta(\cdot)$ is the Heaviside step function. In the limit $\chi_h\gg 1$, the integration yields $$\label{eq:out-degree}
\bar{k}_{out}(\chi) \approx
\begin{cases}
2\delta \sqrt{\displaystyle{\frac{e^{\chi_h}}{\cosh{\chi}}}}\,K\left(\tanh{\chi}\right) &\text{if $0 \le \chi<\chi_h$,}\\
0 &\text{if $\chi=\chi_h$,}
\end{cases}$$ where $K(\cdot)$ is the complete elliptic integral of the first kind. In the regime $1<\chi<\chi_h$, the average out degree is well approximated by $$\bar{k}_{out}(\chi) \approx 2 \sqrt{2} \delta \chi e^{(\chi_h-\chi)/2}.$$ For the same combination-of-exponentials reasons as in the in-degree case, this exponential scaling, combined with the one in Eq. (\[rho\_chi\]), implies that the out-degree distribution scales as $$P(k_{out}) \sim k_{out}^{-3},\quad\text{for $k_{out}\gg1$,}$$ with logarithmic corrections. We notice however that observers near (but not exactly at) the edge of the horizon have out-degrees approximately equal to $\chi_h$. Therefore, the out-degree distribution is asymptotically a power law with a lower cut-off that grows as $\chi_h$ with time.
We note that new connections appear not only between new and existing nodes, but also between pairs of already existing nodes, not previously connected. This type of linking creates directed cycles in the network. The appearance of new links between existing nodes is a simple consequence of the continuous expansion of the horizons of all observers. The resulting network dynamics is illustrated in Fig. \[fig:4\], where three snapshots of a growing network are taken. The horizon of the central observer $O$ (the blue dashed circle) grows over time, discovering an exponentially increasing number of new observers. Gray connections indicate purely directed causal relations between observers, that is, one is aware of the other. As time goes on, directed connections are reciprocated (connections in red), meaning that an increasing number of pairs of observers are getting mutually aware of each other.
{width="\linewidth"}
Finally we emphasize that our analysis is by no means limited to the Milne universe. Almost exactly the same results hold for any open FLRW universe with any scale factor $R(\tau)$. The same picture as in Fig. \[fig:3\] would apply there. The only minor difference is the rate at which new nodes join the network, defined by the radius of the observer’s horizon as a function of time. Specifically, given $R(\tau)$, this radius is $$\chi_h= \int^{t} \frac{d \tau}{R(\tau)},$$ generalizing Eq. (\[eq:horizon\]).\
Imperfect communication
=======================
Up to this point we have assumed that all observers entering the horizon of another observer are detected with probability $1$. If we assume that the probability of connection between observers decays exponentially with the hyperbolic distance $x$ between them, $$\label{eq:imperfect-p(x)}
p(x) =p e^{-\beta x},$$ then the average in-degree of an observer at coordinate $\chi$ is $$\begin{aligned}
\bar{k}_{in}(\chi)&=&2\pi \delta p \int_0^{\chi_h-\chi} \sinh{\chi'}\,e^{-\beta \chi'} d \chi'\nonumber\\
&=&2\pi \delta p\frac{1-e^{\beta(\chi-\chi_h)}[\beta \sinh{(\chi_h-\chi)}+\cosh{(\chi_h-\chi)}] }{\beta^2-1}.\end{aligned}$$ If $\beta \ge 1$ and $\chi_h\gg 1$ the average in-degree of nodes is constant and the network becomes similar to a random geometric graph. In random geometric graphs, nodes lie in a geometric space, and two nodes are connected if the distance between them in the space is below a given threshold. These graphs have Poisson distributions of node degrees [@Penrose03-book]. We can show that the in-degree distribution in our imperfect networks with $\beta\geq1$ is also Poisson. This is intuitively expected because in this case observers are connected only to other observers in their small neighborhoods. The case with $\beta<1$ is more interesting. In this case, the average in-degree of an observer located at $\chi$ is $\bar{k}_{in}(\chi) \sim e^{(1-\beta)(\chi_h-\chi)}$. As a consequence, for the same combination-of-exponentials reasons as before, the in-degree distribution scales asymptotically as a power law $P(k_{in}) \sim k_{in}^{-\gamma}$ with exponent $$\gamma=2+\frac{\beta}{1-\beta},$$ which can take any value between $2$ and $\infty$, as shown in Fig. \[fig:5\].
![Complementary cumulative in-degree distribution $P_c(k_{in})=\sum_{k_{in}' \geq k_{in}}P(k_{in}')$ in simulated Milne networks with exponents $\gamma=2.1,2.3,2.5,2.7$ grown up to $N=10^5$ nodes. The solid lines are power laws with the same exponents. Inset: degree-dependent clustering coefficient for the undirected versions of the same networks. The average clustering coefficients excluding nodes of degree $1$ are $\bar{C}=0.67, 0.47, 0.41, 0.38$ for $\gamma=2.1,2.3,2.5,2.7$, respectively. The networks are disassortative, meaning that the correlations of degrees of connected nodes (not shown) are negative, due to structural constraints imposed by the scale-free degree distribution [@Boguna:2004eh]. []{data-label="fig:5"}](fig5.pdf){width="0.8\linewidth"}
This result may have interesting cosmological implications concerning what part of the universe our observers can observe. Indeed, in the case of imperfect communication with $\beta\in(0,1)$, observer $O$ directly detects only $\sim e^{(1-\beta)\chi_h}$ other observers. Therefore by the time the number of observers within $O$’s horizon is $\sim N$, $O$ detects only $\sim N^{1-\beta}$ of those, so that the fraction of the universe that $O$ sees directly ($\sim N^{1-\beta}/N$) approaches zero as time goes on. However there are also indirect causal paths, shown in blue in Fig. \[fig:1\]. Any observer connected to $O$ via either direct or indirect causal paths can still be detected by $O$. The question of what fraction of the universe can be observed by $O$ becomes a variation of the bond percolation problem, well studied in network science. In the classical bond percolation problem, we are given a large network in which we retain or delete each link (also called “bond” for historical reasons) with probability $p$ and $1-p$. There often exists a critical value $p_c$ of this probability corresponding to the phase transition in the system: if $p>p_c$ the network is in the percolated phase, meaning that a macroscopical fraction of nodes belong to the largest connected component, while for $p<p_c$ the network decomposes into many small connected components. There is no such phase transition in random scale-free networks with power-law exponent $\gamma<3$. They are always in the percolated phase, $p_c=0$ [@SeKrBo11]. In our imperfect cosmological networks with $\beta\in(0,1)$, the given network is the perfect network with $\beta=0$ and $p=1$ in which we retain links with probability in Eq. (\[eq:imperfect-p(x)\]), and the question is now what fraction of the network is connected to $O$ via at least one causal path, direct or indirect. This problem is more involved that the standard bond percolation problem, but one may suspect that since the network is scale-free, there should exist a regime, perhaps with $\beta<1/2$, in which the network is always percolated. This would imply that a macroscopic fraction of the universe can be observed by any observer.
![Bond percolation simulations of imperfect-communication networks. The left column shows the upper and lower bounds $g_O^{u,l}(p)$ for the fraction of nodes causally connected to $O$ for different values of $\gamma$ and $N=3 \times 10^5$. The right column shows the critical values $p_c^{u,l}(N)$ for the same bounds, measured as the value of $p$ that maximizes the susceptibilities $\xi^{u,l}$ in Eq. (\[chi\]). The dashed lines are power law fits $p_c^{u,l}(N) \sim N^{-1/\nu}$ with exponents $1/\nu=0.3(7), 0.3(2)$ for $\gamma=2.1$, $1/\nu=0.2(5), 0.1(5)$ for $\gamma=2.5$, and $1/\nu=0.1(8), 0.0(7)$ for $\gamma=3$.[]{data-label="fig:6"}](fig6.pdf){width="0.8\linewidth"}
We next support these expectations in simulations. Let $g_{O}$ be the fraction of nodes within $O$’s horizon that are connected to $O$ via at least one causal path in an imperfect-communication network with the link existence probability (\[eq:imperfect-p(x)\]). From the exposition above, including Fig. \[fig:1\], the causal path is defined as a directed path $P=\{n_1,n_2,\ldots,O\}$ such that the retarded horizon $H_{n_i}$ of any node $n_i$ in the path, $i=1,2,\ldots$ (or equivalently the set of $n_i$’s neighbors in the perfect network), contains all subsequent nodes in the path: $n_j \in H_{n_i}$ for any $j>i$. The problem of finding if such a path exists between a given node $n_1$ and $O$ is likely to be an NP-hard combinatorial problem, because checking all directed paths between $n1$ and $O$ seems unavoidable. We did not attempt either to prove the NP hardness of the problem or to find its computationally admissible solution, because it is much easier to provide upper and lower bounds for $g_{O}$. An upper bound $g_{O}^u$ is just the number of nodes connected to $O$ via any directed path, not necessarily causal. As a lower bound $g_{O}^l$ we use the number of nodes connected to $O$ by at least one causal path, and located up to three hops away from $O$, which comprise a significant fraction of all nodes within $O$’s horizon.
![Percolation susceptibility of the upper bound $\xi^u$ Eq. (\[chi\]) as a function of $p$ for different network sizes and $\gamma=2.1$ (results for other values of $\gamma$ are qualitatively similar). For all values of $\gamma<3$, $\xi^u$ and $\xi^l$ show a peak that moves to the left as the system size increases. At the same time, the maximum value of $\xi^u$ and $\xi^l$ diverges as a function of $N$ as $\xi^{u,l}_{max}\sim N^{\gamma'/\nu}$. Dashed lines in the inset are power law fits with exponents $\gamma'/\nu=0.8(0)$. \[fig:7\]](fig7.pdf){width="0.8\linewidth"}
Figure \[fig:6\] shows the results for these bounds in numerical simulations of networks with up to $N=3 \times 10^5$ nodes, $\gamma=2.1,2.5,3$ and $p\in[0,1]$. The upper and lower bounds $g_{O}^u$ and $g_{O}^l$ increase monotonically as functions of $p$, suggesting that, as expected, the percolation threshold is zero. To check if it is indeed zero, we measure the susceptibilities $\xi^u$ and $\xi^l$ defined as $$\xi^{u,l}=N\frac{\langle [g_{O}^{u,l}]^2 \rangle -\langle g_{O}^{u,l} \rangle^2}{\langle g_{O}^{u,l} \rangle},
\label{chi}$$ where averages $\langle\cdot\rangle$ are taken over a large number (10000 in our case) of different bond percolation realizations for each combination of values of $N$, $\gamma$, and $p$. In continuous phase transitions, the fluctuations of a property of interest ($\xi$ in our case) diverge at a critical parameter value in the thermodynamic limit $N\to\infty$. In finite-size systems, this divergence manifests itself as a maximum of function $\xi(p)$ that becomes sharper for larger $N$, see Fig. \[fig:7\]. The value of $p=p_c$ corresponding to this maximum can be used as an estimate of the critical parameter value $p_c$ [@Newman99-book]. The right column in Fig. \[fig:6\] shows the values of thus estimated $p_c$s as functions of $N$ for bounds $\xi^{u,l}(p)$ in our networks. For $\gamma<3$ the critical points of both upper and lower bounds go to zero as power laws $p_c \sim N^{-1/\nu}$. This means that the percolation threshold is indeed zero in the thermodynamic limit ($p_c\to0$ as $N\to\infty$), and that observer $O$ can observe a finite fraction of the universe for any value of $p$. However if $\gamma=3$, then while the upper bound critical value goes to zero as $N$ goes to infinity, the critical value corresponding to the lower bound becomes nearly size independent. This implies that for $\gamma>3$, there exists a critical point $p_c$ below which our observer $O$ can observe only her local neighborhood.\
Conclusions
===========
In summary, the physical network of (indirect) causal relations between observers uniformly distributed in any open FLRW universe is a Lorentz-invariant scale-free graph with strong clustering, Fig. \[fig:5\]. This network represents maximum information about the universe that any particular observer can collect by a certain time. More precisely, paths in this network are all possible communication channels between observers. Perhaps coincidentally, in the perfect case without information loss ($\beta=0$), this network has the same statistical properties ($\gamma=2$ and strong clustering) as the maximally navigable networks [@BoKrKc08], i.e., networks that are most conductive with respect to targeted information signaling. The crucial requirement for this coincidence is that the universe must be open, Eq. (\[eq:1\]). Bubble universes are open in most inflationary cosmologies [@Kleban2011], and the current measurements of our universe do not preclude that it is open either, although it is definitely close to being flat [@Komatsu11].
These results may be interesting for both network science and cosmology. From the network science perspective, they may help to develop a “General Relativity” of networks, an analogy of the Einstein equations that would describe network dynamics within a unified framework, in which network nodes might be analogous to our observers or galaxies. Here we have considered an idealized case where nodes are massless points distributed uniformly in the space. It remains unclear how the picture would change if points have masses, perhaps distributed according to some heterogeneous distributions similar to the distribution of the masses of galaxies in the universe [@FoPo04], and if the spatial distribution of points deviates from uniform, as it does for galaxies [@LaPe010] and for real networks embedded in hyperbolic spaces [@PaBoKr11].
From the cosmology perspective, it has been suggested that measures of photons from the cosmic microwave background scattered by high energy electrons in clusters of galaxies could be used to probe the last scattering surface (LSS) at many different length scales, and thus overcome the limitations of the cosmic variance [@Kamionkowski:1997fk]. In this context the cosmological networks we have considered here may be interesting because they contain not only direct connections within causal horizons, but also all possible indirect causal connections. The galaxy-scattered photons represent the latter indirect connections between the LSS and us, albeit made of only two hops, as in Fig. \[fig:1\] b. Yet the knowledge of the density of clusters of galaxies throughout the universe, coupled with our network representation, can be used to estimate the maximum information we could ever obtain from the LSS by counting the total number of causal paths connecting such surface to us. The discussed percolation problem on these networks may be of particular interest in that respect.\
We thank Jaume Garriga for very useful comments and suggestions. This work was supported by a James S. McDonnell Foundation Scholar Award in Complex Systems; the ICREA Academia prize, funded by the [*Generalitat de Catalunya*]{}; MICINN project No. FIS2010-21781-C02-02; [*Generalitat de Catalunya*]{} grant No. 2014SGR608; DARPA grant No.HR0011-12-1-0012; NSF grants No. CNS-1344289, CNS-0964236, and CNS-1039646; and by Cisco Systems.\
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A synoptic view on the long-established theory of light propagation in crystalline dielectrics is presented, where charges, tightly bound to atoms (molecules, ions) interact with the microscopic local electromagnetic field. Applying the Helmholtz-Hodge decomposition to the current density in Maxwell’s equations, two decoupled sets of equations result determining separately the divergence-free (transversal) and curl-free (longitudinal) parts of the electromagnetic field, thus facilitating the restatement of Maxwell’s equations as equivalent field-integral equations. Employing a suitably chosen basis system of Bloch functions we present for dielectric crystals an exact solution to the inhomogenous field-integral equations determining the *local* electromagnetic field that polarizes individual atoms or ionic subunits in reaction to an external electromagnetic wave. From the solvability condition of the associated homogenous integral equation then the propagating modes and the photonic bandstructure $\omega_{n}\left(\mathbf{q}\right)$ for various crystalline symmetries $\Lambda$ are found solving a *small* sized $3M\times3M$ matrix eigenvalue problem, with [$M$]{} denoting the number of polarizable atoms (ions) in the unit cell. Identifying the *macroscopic* electric field inside the sample with the spatially low-pass filtered *microscopic* local electric field, the dielectric $3\times3$-tensor $\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)$ of crystal optics emerges, relating the accordingly low-pass filtered microscopic polarization field to the macroscopic electric field, solely with the individual microscopic polarizabilities $\alpha\left(\mathbf{R},\omega\right)$ of atoms (molecules, ions) at a site $\mathbf{R}$ and the crystalline symmetry as input into the theory. Decomposing the microscopic local electric field into longitudinal and transversal parts, an effective wave equation determining the radiative part of the macroscopic field in terms of the transverse dielectric tensor $\varepsilon_{ab}^{\left(T\right)}\left(\mathbf{q},\omega\right)$ is deduced from the exact solution to the field-integral equations. The Taylor expansion $\varepsilon_{ab}^{\left(T\right)}\left(\mathbf{q},\omega\right)=\varepsilon_{ab}^{\left(T\right)}\left(\omega\right)+i\sum_{c}\gamma_{abc}\left(\omega\right)q_{c}+\sum_{c,d}\alpha_{abcd}\left(\omega\right)q_{c}q_{d}+...$ around $\mathbf{q}=\mathbf{0}$ provides then insight into various optical phenomena connected to retardation and non-locality of the dielectric tensor, in full agreement with the phenomenological reasoning of Agranovich and Ginzburg in “Crystal Optics with Spatial Dispersion, and Excitons” (Springer Berlin Heidelberg, 1984): the eigenvalues of the tensor $\varepsilon_{ab}^{\left(T\right)}\left(\omega\right)$ describing chromatic dispersion of the index of refraction and birefringence, the first order term $\gamma_{abc}\left(\omega\right)$ specifying rotary power (natural optical activity), the second order term $\alpha_{abcd}\left(\omega\right)$ shaping the effects of a spatial-dispersion-induced birefringence, a critical parameter for the design of lenses made from $CaF_{2}$ and $BaF_{2}$ for optical lithography systems in the ultraviolet. In the *static* limit an exact expression for $\varepsilon_{\varLambda}$ is deduced, that conforms with general thermodynamic stability criteria and reduces for cubic symmetry to the Clausius-Mossotti relation. Considering various dielectric crystals comprising atoms with known polarizabilities from the literature, in all cases the calculated indices of refraction, the rotary power and the spatial-dispersion-induced birefringence coincide well with the experimental data, thus illustrating the utility of the theory. For ionic crystals, exemplarily for $CsI$ and $RbCl$, a satisfactory agreement between theory and the measured chromatic dispersion of the index of refraction is shown over a wide frequency interval, ranging from ultraviolet to far infra-red, accomplishing this with an appreciably smaller number of adjustable parameters compared to the well known Sellmeier fit.'
author:
- Marius Dommermuth
- Nils Schopohl
bibliography:
- 'References.bib'
title: On the Theory of Light Propagation in Crystalline Dielectrics
---
\[sec:Intro\]Introduction
=========================
When optical signals traverse a transparent dielectric, for example a fused quartz (silica) prism, the light travels at different speed depending on frequency $f=\frac{\omega}{2\pi}$ , so that the shape of a wave packet, say composed of mixed frequencies $\left|f-f_{c}\right|\leq\frac{\varDelta f_{c}}{2}$ around a carrier frequency $f_{c}$ in a frequency interval of width $\varDelta f_{c}$, tends to spread out. This is the well known chromatic dispersion effect resulting from the frequency dependence of the refractive index $n=n\left(\omega\right)$. Microscopic considerations based on first principles reveal, that the frequency dependence of the refractive index $n\left(\omega\right)$ is directly connected to the *retarded* response of the polarizable constituents of matter, the latter distinguishing themselves as atoms, molecules or ions. The chromatic dispersion effect is further supplemented by the effects of crystalline anisotropy and also by the effects of spatial dispersion brought about by the *non local* dependence of this response, that is a charge at point $\mathbf{r}$ recollects the action exerted on it at another position $\mathbf{r}'$ [@L.V.Keldysh2012; @Maksimov2012].
Both fundamental features of the electromagnetic response, retardation and non locality, can be described jointly by a dielectric (tensor) function $\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)$ depending on (circular) frequency $\omega$ and on wave vector $\mathbf{q}$. While the dependence of the dielectric function on $\omega$ explains the chromatic dispersion effect, and its anisotropy for $\mathbf{q}=\mathbf{0}$ describes birefringence, the dependence on wave vector $\mathbf{q}$ is directly connected to phenomena like natural optical activity and spatial dispersion induced birefringence. Furthermore, under the influence of a *static* magnetic induction field $\mathbf{B}^{\left(0\right)}$, respectively a *static* electric field $\mathbf{E}^{\left(0\right)}$, the additional dependence of the dielectric function on those static fields gives rise to many other magneto-optic and electro-optic phenomena, for instance the Faraday effect, the Pockels effect and also the Kerr effect [@Agranovich1984].
When electromagnetic waves propagate inside a dielectric material, the microscopic local electric field convoyed by those waves exerts a small supplemental force onto charge carriers inside the atoms (ions, molecules) comprising that material, pulling apart inside each atom the positively charged nucleus and the negatively charged bound electrons by a (small) shift in *opposite* direction. Of course the position of the total mass of a charge neutral atom, considering valence electrons and nucleus together, remains then unchanged. Since the atomic nucleus is certainly much heavier than the bound electrons, the resulting shift of the barycenter of the bound electrons by far surpasses the associated tiny shift of the position of the atomic nucleus. This is the effect of *induced electronic polarization.*
A fully microscopic theory of the optical properties of a material certainly requires to consider the positive charged atomic nuclei and the electrons, taking into account the laws of quantum statistical physics and low energy quantum electrodynamics, for example [@L.V.Keldysh2012; @H.Bilz1984]. In the ensuing discussion we shall accept a phenomenological (semiclassical) picture of the matter light interaction based on the fundamental fact, that matter is stable, i.e. the energy of an atom located at a site $\mathbf{R}$ is, in the absence of external fields, a minimum against any (small) displacement of its bound electrons. Accordingly, in reaction to the presence of a (weak) local time dependent electric field $\mathbf{E}\left(\mathbf{R},t\right)$ the electrons bound to an atom redistribute themselves, so that considered from outside, an atom comprising a number $Z$ of electrons, acquires an *induced* electric dipole moment. This is the basic idea of the phenomenological *classical* model of atomic polarizability due to Lorentz, who described the induced dipole moment of electrons, tied to a heavy (immobile) nucleus by a harmonic spring, solving a harmonic oscillator problem with that local electric field acting as a drive. Incidentally, the Lorentz model predicts a frequency dependent Fourier amplitude
[^1] of the induced electronic dipole moment of an atom at site $\mathbf{R}$ that coincides with a full quantum mechanical calculation, see supplemental material [@Supplementary]: $$\begin{aligned}
\tilde{d}_{a}\left(\mathbf{R},\omega\right) & = & \sum_{a'}\alpha_{a,a'}\left(\mathbf{R},\omega\right)\tilde{E}_{a'}\left(\mathbf{R},\omega\right)\label{eq: induced dipole moment of one atom}\\
a,a' & \in & \left\{ x,y,z\right\} \nonumber \end{aligned}$$ Here $\alpha_{a,a'}\left(\mathbf{R},\omega\right)$ denotes the atom-individual electric polarizability given by $$\alpha_{a,a'}\left(\mathbf{R},\omega\right)=\frac{\left|e\right|^{2}}{m}\sum_{\nu\neq0}\frac{f_{a,a'}\left(\nu;\mathbf{R}\right)}{\omega_{\nu}^{2}\left(\mathbf{R}\right)-\left(\omega+\frac{i}{2\tau_{\nu}\left(\mathbf{R}\right)}\right)^{2}}\:,\label{eq: atom polarizability}$$ the summation index $\nu$ running here over all eigenstates $\left|\Phi_{\nu}\left(\mathbf{R}\right)\right\rangle $ of the multi-electron configration of that atom except the groundstate $\left|\Phi_{0}\left(\mathbf{R}\right)\right\rangle $. Indeed, this expression looks like it was derived from an ensemble of (classical) harmonic oscillators with resonance frequencies $\omega_{\nu}\left(\mathbf{R}\right)$, each oscillator being driven by the local electric field with a factor of proportionality $f_{a,a'}\left(\nu;\mathbf{R}\right)>0$, the so called oscillator strength. However, the physical meaning of the oscillator strength is only revealed by quantum mechanics. With $\hat{d}_{a}=-\left|e\right|\sum_{n=1}^{Z}\hat{r}_{a}^{\left(n\right)}$ denoting the dipole operator of the count $Z$ of electrons tied to an atom at position $\mathbf{R}$, then according to quantum mechanics $$\begin{aligned}
f_{a,a'}\left(\nu;\mathbf{R}\right) & = & \frac{1}{\left|e\right|^{2}}\frac{2m}{\hbar}\omega_{\nu}\left(\mathbf{R}\right)\left\langle \Phi_{0}\left(\mathbf{R}\right)|\hat{d}_{a}|\Phi_{\nu}\left(\mathbf{R}\right)\right\rangle \left\langle \Phi_{\nu}\left(\mathbf{R}\right)|\hat{d}_{a'}|\Phi_{0}\left(\mathbf{R}\right)\right\rangle \:,\end{aligned}$$ explaining why the oscillator strength is a measure for how much a bound electron contributes to the electric polarizability of an atom, say under a transition from the multi-electron ground state $\left|\Phi_{0}\left(\mathbf{R}\right)\right\rangle $ of the atom Hamiltonian with eigenvalue $\hbar\Omega_{0}\left(\mathbf{R}\right)$ to an excited multi-electron eigenstate $\left|\Phi_{\nu}\left(\mathbf{R}\right)\right\rangle $ with eigenvalue $\hbar\Omega_{\nu}\left(\mathbf{R}\right)$. The differences $\omega_{\nu}\left(\mathbf{R}\right)\equiv\Omega_{\nu}\left(\mathbf{R}\right)-\Omega_{0}\left(\mathbf{R}\right)$ in (\[eq: atom polarizability\]) are the optical transition frequencies, the life-time parameter $\tau_{\nu}\left(\mathbf{R}\right)>0$ describes spontaneous emission as reasoned by quantum electrodynamics and thus being always present if the atom was excited to an eigenstate state $\nu\neq0$. Expanded details how the result (\[eq: atom polarizability\]) can be derived within first order time dependent perturbation theory in response to a weak time dependent electric field, including a discussion of the $f$-sum rule, see supplemental material [@Supplementary].
De facto, the spectrum of atoms with $Z>1$ cannot be calculated ab initio with sufficient precision, the exact taking into account of electronic correlations being (alas) an unsolved problem. In what follows we therefore conceive the optical transition frequencies $\omega_{\nu}\left(\mathbf{R}^{\left(j\right)}\right)$, the life-time parameter $\tau_{\nu}\left(\mathbf{R}^{\left(j\right)}\right)$ and the oscillator strengths $f_{a,a'}\left(\nu,\mathbf{R}^{\left(j\right)}\right)$ of each atom species positioned at a site $\mathbf{R}^{\left(j\right)}=\mathbf{R}+\boldsymbol{\eta}^{\left(j\right)}$, with $\mathbf{R}\in\Lambda$ a lattice vector and $\boldsymbol{\eta}^{\left(j\right)}$ indicating a position of an atom (ion, molecule) inside a unit cell $C_{\Lambda}$ of the crystal $\Lambda$, as fitting parameters, so that the optical properties as calculated from the dielectric tensor $\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)$ of the crystal coincide with experiment. How this objective can be accomplished, and in particular how $\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)$ depends on the individual atom polarizabilities $\alpha_{a,a'}\left(\boldsymbol{\eta}^{\left(j\right)},\omega\right)$, and thus via (\[eq: atom polarizability\]) on the atom individual multi-electron spectrum, we shall elaborate in what follows.
A time dependent external electromagnetic signal $\mathbf{E}_{ext}\left(\mathbf{r},t\right)$ incident upon a material probe polarizes the atoms inside, and for weak amplitude of the incident field the induced polarization at the atom sites $\mathbf{R}$ will then be proportional to that amplitude. However the microscopic local electric field $\mathbf{E}\left(\mathbf{r},t\right)$ polarizing an atom (ion, molecule) positioned at site $\mathbf{R}$ is not known a priory, because all atoms in the sample will get polarized and hence all act back with a (retarded) reaction in response to the primary external field $\mathbf{E}_{ext}\left(\mathbf{r},t\right)$. Then everywhere inside (and outside) of the probe there holds $$\mathbf{E}\left(\mathbf{r},t\right)=\mathbf{E}_{ext}\left(\mathbf{r},t\right)+\mathbf{E}_{ind}\left(\mathbf{r},t\right),\label{eq:total electric field I}$$ the secondary induced electric field $\mathbf{E}_{ind}\left(\mathbf{r},t\right)$ being a superposition of the individually radiated and retarded electric fields emitted by all the atoms inside the sample, that have been polarized in turn by that field $\mathbf{E}\left(\mathbf{r},t\right)$. With a suitable model of the polarizability of individual atoms thus an integral equation determining the microscopic local electric field $\mathbf{E}\left(\mathbf{r},t\right)$ emerges.
So far, everything said is well known from the original (early) literature [@Ewald1916; @Ewald1916a; @Ewald1917; @Ewald1938; @Laue1931] and from highly cited textbooks on crystal optics, for example [@Born1999; @Fluegge2013; @Agranovich1984; @Laue1960]. For a concise summary of the pioneering works on crystal optics of Ewald and v. Laue (and later authors) see [@Authier2012]. Nevertheless, we believe the approach we present in what follows truely discerns from traditional presentations of the subject. For example, nowhere do we make use of the Ewald-Oseen extinction theorem to recover the correct index of refraction $n$ of a material, it being customary practice in the so called rigorous theory of dispersion [@Born1999; @Fluegge2013] to consider the wave incident from free space to be propagated with vacuum light velocity $c$ and the signal induced in the sample to be propagated with velocity $c/n$, the Lorentz-Lorenz formula connecting the index of refraction $n$ with the polarizability of individual atoms thus emerging as a solvability condition for the field-integral equation stated (implicitely) in (\[eq:total electric field I\]).
Outline {#outline .unnumbered}
-------
In Sec. II we first establish on the basis of the fundamental Maxwell equations an exact *integral equation* for the microscopic local electric field $\mathbf{E}\left(\mathbf{r},t\right)$ in (\[eq:total electric field I\]) with an explicit formula for the integral kernel that derives directly from the atom polarizability (\[eq: atom polarizability\]). Posing boundary conditions for the components of the electromagnetic field at the boundary of a material probe is then redundant. Moreover, the frequency dependence of longitudinal and transversal parts of the electromagnetic field are treated consistently, thus making everywhere in our calculations the correct static limit $\omega\rightarrow0$ accessible.
In Sec. III we solve the field-integral equation for crystalline dielectric materials exactly making use of a set of non-standard Bloch functions, not constructed from plane waves but designed from eigenfunctions of the position operator, representing a *complete* orthonormal system of eigenfunctions of the translation operator $T_{\mathbf{R}}$ under a shift by a lattice vector $\mathbf{R}\in\Lambda$. Accordingly, instead of expanding the kernel of the field-integral equation in the well known basis of plane waves $e^{i\left(\mathbf{q}+\mathbf{G}\right)\mathbf{r}}$, thus requiring to handle for each wave vector $\mathbf{q}$ in the Brillouin zone $C_{\Lambda^{-1}}$ of a lattice $\Lambda$ then (infinite) matrices labelled by reciprocal lattice vectors $\mathbf{G},\mathbf{G}'\in\Lambda^{-1}$, our choice of eigenfunctions of the translation operator $T_{\mathbf{R}}$ sidesteps the inversion (and truncation) of such large matrices, thus easing notably the determination of the photonic bandstructure of a crystal. Also we show, if the incident electromagnetic wave was purely transversal, yet the *microscopic* local electric field features both, a transversal and a longitudinal component, see Fig.(\[fig:local\_field\_projections\]), the strength of the longitudinal component being strongly dependent on the density of polarizable atoms (ions, molecules) in the crystal.
Thereafter we present, exemplifying our calculation method, results for the photonic bandstructure of diamond ($M=2$), that have been calculated previously with other (computationally more time-consuming) methods. Our findings for the photonic bandstructures for various monoatomic Bravais lattices ($M=1$) we present and discuss in the supplemental material [@Supplementary]. In comparison to well known (phenomenological) work on photonic bandstructures [@Leung1990; @Zhang1990], within which the (macroscopic) Maxwell’s equations in a superlattice are solved assuming a spatially repetetive varying index of refraction, it turns out that in our approach based on the field-integral equations the need to eliminate unphysical “longitudinal” modes doesn’t arise.
In section \[sec:Macroscopic-Electric-Field-and-dielectric-function\] we introduce the notion of a *macroscopic* electric field $\mathcal{\boldsymbol{E}}\left(\mathbf{r},t\right)$ inside a material, conceiving it with regard to spatial variations as a low pass filtered signal, with the solution to the field-integral equation (\[eq:local field integral equation\]) as input. Relating the macroscopic polarization to that macroscopic electric field, thereafter the macroscopic dielectric $3\times3-$tensor $\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)$ of a crystalline dielectric material emerges, with chromatic and spatial dispersion fully taken into account. The exact expression for $\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)$ thus obtained is solely dependent on the symmetry of the lattice $\varLambda$ under consideration and on the polarizability $\alpha\left(\mathbf{R}^{\left(j\right)},\omega\right)$ of individual atoms (ions, molecules) positioned at their equilibrium sites $\mathbf{R}^{\left(j\right)}$ in the crystal. We also confirm, exemplarily for the ionic crystal $CsI$, that our formula for the dielectric tensor with regard to its frequency dependence indeed obeys to the Lyddane-Sachs-Teller relation. With a view to the key role of locality claimed by *macroscopic electrodynamics* [@Liu2009] we caution the reader not to discard the $\mathbf{q}$-dependence of the dielectric tensor $\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)$. As emphasized by Ginzburg and Agranovich [@Agranovich1984], a rich variety of optical effects, including rotary power and spatial dispersion induced birefringence, manifestly proof the importance of the non local nature of the dielectric response.
We next derive directly from the microscopic field-integral equations a set of *differential equations* that determine the spatial variation of the *transversal* and *longitudinal* parts of the *macroscopic* electric field $\mathcal{\boldsymbol{E}}\left(\mathbf{r},t\right)$, without prior knowledge of the microscopic local field $\mathbf{E}\left(\mathbf{r},t\right)$.
If the external field was purely transversal, this set of coupled differential equations reduces to a wave equation determining the radiative part of the macroscopic field. In this way the parts of the dielectric tensor $\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)$ are indentified that determine the *transversal* dielectric tensor $\varepsilon^{\left(T\right)}\left(\mathbf{q},\omega\right)$ comprising the optical properties of a dielectric crystal. Assuming spatial dispersion is a weak effect, which assumption applies for many transparent media, the Taylor expansion of $\varepsilon_{ab}^{\left(T\right)}\left(\mathbf{q},\omega\right)$ around $\mathbf{q}=\mathbf{0}$ then provides (yet in an implicit manner) access to various optical phenomena featuring the propagation of light in dielectric crystals [@Agranovich1984], for instance chromatic dispersion and birefringence, rotary power (natural optical activity) and also the (weak) effects of a spatial-dispersion-induced birefringence, the latter being a critical problem for the design of lens elements made from crystalline materials like $CaF_{2}$ and $BaF_{2}$ widely used in optical lithograpy systems in the ultraviolet [@Burnett2002; @Serebryakov2003]. Further we summarize in section \[sec:chromatic-dispersion,-optical activity and spatial dispersion induced birefringence\], see Table \[results\], Fig.\[Sellmeier\] and also Fig.\[fig:CaF2\_and\_BaF2\], to what large extend our theory of the dielectric tensor for crystalline dielectrics agrees with measurements over a wide range of optical frequencies for a series of well known crystalline materials, including for example Bi$_{12}$TiO$_{20}$ and also Bi$_{12}$SiO$_{20}$, both crystals featuring a large number of basis atoms ($M=66$) in the unit cell, thus demonstrating the utility of our approach.
If on the other hand the external field was purely longitudinal, the differential equations derived from the field-integral equations reduce to a Poisson type equation for a scalar potential function determining the macroscopic electric field, thus identifying the parts of the dielectric tensor $\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)$ featuring electric-field *screening,* like in electrostatics. Also we deduce an exact analytic formula for the *static* dielectric tensor $\varepsilon_{\Lambda}$, that conforms with general thermodynamic stability criteria [@Kirzhnitz2012] and applies for all $14$ monoatomic Bravais lattices ($M=1$). For the special case of cubic symmetry the long-known Clausius-Mossotti relation is recovered.
\[sec:local field\]The Field-Integral Equations
===============================================
Consider a *fixed* inertial frame, with a polarizable dielectric material at rest occupying a volume $\Omega$ in that frame. Without loss of generality let the microscopic charge density inside $\Omega$ be given the representation $$\rho\left(\mathbf{r},t\right)=\boldsymbol{-\nabla}\cdot\mathbf{P}\left(\mathbf{r},t\right),\label{eq: representation of charge density}$$ with $\mathbf{P}\left(\mathbf{r},t\right)$ denoting the vector field of electric polarization [@L.V.Keldysh2012]. We find it then convenient to decompose the associated microscopic current density $$\mathbf{j}\left(\mathbf{r},t\right)=\partial_{t}\mathbf{P}\left(\mathbf{r},t\right)\label{eq: representation of current density}$$ flowing inside $\Omega$ into *longitudinal* and *transversal* parts. To serve this purpose we introduce integral kernels $$\begin{aligned}
\Pi_{aa'}^{\left(L\right)}\left(\mathbf{r}-\mathbf{r}'\right) & = & \lim_{\kappa\rightarrow\infty}\int\frac{d^{3}q}{\left(2\pi\right)^{3}}e^{i\mathbf{q}\cdot\mathbf{\left(\mathbf{r}-\mathbf{r}'\right)}}\bar{\Pi}_{aa'}^{\left(L\right)}\left(\mathbf{q}\right)\frac{1}{1+\frac{\left|\mathbf{q}\right|^{2}}{\kappa^{2}}}\label{eq: projection operators I}\\
\Pi_{aa'}^{\left(T\right)}\left(\mathbf{r}-\mathbf{r}'\right) & = & \lim_{\kappa\rightarrow\infty}\int\frac{d^{3}q}{\left(2\pi\right)^{3}}e^{i\mathbf{q}\cdot\mathbf{\left(\mathbf{r}-\mathbf{r}'\right)}}\bar{\Pi}_{aa'}^{\left(T\right)}\left(\mathbf{q}\right)\frac{1}{1+\frac{\left|\mathbf{q}\right|^{2}}{\kappa^{2}}}\nonumber \end{aligned}$$ with labels $a,a',a''\in\left\{ x,y,z\right\} $ specifying Cartesian components, and $$\begin{aligned}
\bar{\Pi}_{aa'}^{\left(L\right)}\left(\mathbf{q}\right) & = & \frac{q_{a}q_{a'}}{\left|\mathbf{q}\right|^{2}}\;,\label{eq:Fourier transform projection operators}\\
\bar{\Pi}_{aa'}^{\left(T\right)}\left(\mathbf{q}\right) & = & \delta_{a,a'}-\frac{q_{a}q_{a'}}{\left|\mathbf{q}\right|^{2}}\;.\nonumber \end{aligned}$$ Denoting the convolution of two kernels $A_{aa'}\left(\mathbf{r},\mathbf{r}'\right)$ and $B_{aa'}\left(\mathbf{r},\mathbf{r}'\right)$ with the symbol $$\left[A\circ B\right]_{aa'}\left(\mathbf{r},\mathbf{r}'\right)=\int_{\Omega_{P}}d^{3}r''\sum_{a''}A_{aa''}\left(\mathbf{r},\mathbf{r}''\right)B_{a''a'}\left(\mathbf{r}'',\mathbf{r}'\right),\label{eq:kompact notation convolution}$$ and writing for the kernel representing unity $$\left[I\right]_{aa'}\left(\mathbf{r},\mathbf{r}'\right)=\delta_{aa'}\delta^{\left(3\right)}\left(\mathbf{r}-\mathbf{r}'\right),\label{eq: unity kernel}$$ then the validity of all the relations distinctive for projection operators are readily confirmed: $$\begin{aligned}
\Pi^{\left(L\right)}+\Pi^{\left(T\right)} & = & I\label{eq:general relations projection operator}\\
\Pi^{\left(L\right)}\circ\Pi^{\left(L\right)} & = & \Pi^{\left(L\right)}\nonumber \\
\Pi^{\left(T\right)}\circ\Pi^{\left(T\right)} & = & \Pi^{\left(T\right)}\nonumber \\
\Pi^{\left(T\right)}\circ\Pi^{\left(L\right)} & =\:0\:= & \Pi^{\left(L\right)}\circ\Pi^{\left(T\right)}\nonumber \end{aligned}$$ In position space, for $\left|\mathbf{\mathbf{r}}\right|\neq0$, the kernel of the longitudinal and the transverse projection operator both correspond to a dipole field, see [@ClaudeCohen-Tannoudji1989] for a concise derivation: $$\begin{aligned}
\Pi_{aa'}^{\left(T\right)}\left(\mathbf{r}\right) & = & \frac{2}{3}\delta_{a,a'}\delta^{\left(3\right)}\left(\mathbf{\mathbf{r}}\right)+\Theta\left(\left|\mathbf{\mathbf{r}}\right|-0^{+}\right)\nabla_{a}\nabla_{a'}\left(\frac{1}{4\pi\left|\mathbf{r}\right|}\right)\label{eq:projection operators II}\\
\Pi_{aa'}^{\left(L\right)}\left(\mathbf{r}\right) & = & \frac{1}{3}\delta_{a,a'}\delta^{\left(3\right)}\left(\mathbf{\mathbf{r}}\right)-\Theta\left(\left|\mathbf{\mathbf{r}}\right|-0^{+}\right)\nabla_{a}\nabla_{a'}\left(\frac{1}{4\pi\left|\mathbf{r}\right|}\right)\nonumber \end{aligned}$$ It follows from what has been said that the longitudinal part (L) , respectively the transversal part (T) of the current distribution $j_{a}\left(\mathbf{r},t\right)$ conforms with the convolution integrals $$j_{a}^{(L,T)}(\mathbf{r},t)=\int d^{3}r'\sum_{a'}\Pi_{aa'}^{\left(L,T\right)}\left(\mathbf{r}-\mathbf{r}'\right)j_{a'}(\mathbf{r}',t).\label{eq:longitudinal-transverse part of current density}$$ It should be underlined, this link between the original vector field $\mathbf{j}(\mathbf{r},t)$ and the associated longitudinal (transversal) part $\mathbf{j}^{\left(L,T\right)}(\mathbf{r},t)$ is *non local*. There holds by construction $$\begin{aligned}
\mathbf{j}(\mathbf{r},t) & = & \mathbf{j}^{(L)}(\mathbf{r},t)+\mathbf{j}^{(T)}(\mathbf{r},t)\label{eq: Helmholtz decomposition current density}\\
\boldsymbol{\nabla}\mathbf{\cdot j}^{(T)}(\mathbf{r},t) & = & 0\nonumber \\
\mathbf{\boldsymbol{\nabla}}\wedge\mathbf{j}^{(L)}(\mathbf{r},t) & = & \mathbf{0}.\nonumber \end{aligned}$$ According to the Helmholtz-Hodge theorem such a decomposition of a vector field $\mathbf{j}\left(\mathbf{r},t\right)$ is unique. A recent thourough discussion and compilation of the literature on the subject can be found in [@Bhatia2013].
Because Maxwell’s equations are linear, particular monochromatic solutions $$\begin{aligned}
\mathbf{E}^{(L,T)}\left(\mathbf{r},t\right) & = & \mathbf{\tilde{E}}^{(L,T)}\left(\mathbf{r},\omega\right)e^{-i\omega t}\label{eq: time harmonic fields}\\
\mathbf{B}\left(\mathbf{r},t\right) & = & \mathbf{\tilde{B}}\left(\mathbf{r},\omega\right)e^{-i\omega t},\nonumber \end{aligned}$$ with Fourier amplitudes $\mathbf{\tilde{E}}^{(L,T)}\left(\mathbf{r},\omega\right)$ and $\mathbf{\tilde{B}}\left(\mathbf{r},\omega\right)$, can be superposed to construct any wanted time dependence. Applying next the projection operators $\Pi_{aa'}^{\left(L,T\right)}$ to the in time Fourier transformed Maxwell equations, there emerge two groups of decoupled equations in the space-frequency domain for the respective Fourier amplitudes. The first group relates to the components $\tilde{E}_{a}^{\left(L\right)}(\mathbf{r},\omega)$ of the Fourier amplitudes of the longitudinal electric field, $$\begin{aligned}
\boldsymbol{\nabla}\mathbf{\cdot\tilde{E}}^{(L)}(\mathbf{r},\omega) & = & \frac{1}{\varepsilon_{0}}\tilde{\varrho}\left(\mathbf{r},\omega\right)\label{eq: longitudinal Maxwell equations}\\
-i\omega\varepsilon_{0}\tilde{E}_{a}^{\left(L\right)}(\mathbf{r},\omega)+\tilde{j}_{a}^{(L)}(\mathbf{r},\omega) & = & 0\;,\nonumber \end{aligned}$$ the second group involves the Cartesian components of the Fourier amplitudes of the transversal electromagnetic field, which obey to the following six inhomogenous *scalar* Helmholtz equations, the respective source terms being provided by the Fourier amplitudes of the transversal current density, with $c=\frac{1}{\sqrt{\varepsilon_{0}\mu_{0}}}$ the speed of light in free space: $$\begin{aligned}
\left(-\nabla^{2}-\frac{\omega^{2}}{c^{2}}\right)\tilde{E}_{a}^{(T)}\left(\mathbf{r},\omega\right) & = & \mu_{0}i\omega\tilde{j}_{a}^{(T)}\left(\mathbf{r},\omega\right)\label{eq: transversal Maxwell equations}\\
\left(-\nabla^{2}-\frac{\omega^{2}}{c^{2}}\right)\tilde{B}_{a}\left(\mathbf{r},\omega\right) & = & \mu_{0}\left[\mathbf{\boldsymbol{\nabla}}\wedge\tilde{\mathbf{j}}(\mathbf{r},\omega)\right]_{a}\nonumber \end{aligned}$$
We are interested to solve (\[eq: longitudinal Maxwell equations\]) and (\[eq: transversal Maxwell equations\]) considering now a geometry consisting of two *disjoint* material bodies at rest, see Fig.\[fig:SourceRegions\], i.e. $\Omega=\Omega_{S}\cup\Omega_{P}$ and $\Omega_{S}\cap\Omega_{P}=\emptyset$.
![\[fig:SourceRegions\]Schematic illustration of a source domain $\Omega_{\text{S}}$ and a probe volume $\Omega_{\text{P}}$ indicating non vanishing current densities inside. ](Gebiete)
Let us refer to the body $\Omega_{S}$ as the source, and to the body $\Omega_{P}$ as the probe. Accordingly, we split the current density $\tilde{j}_{a}\left(\mathbf{r},\omega\right)$ into an *externally* controlled partial current $\tilde{j}_{ext,a}\left(\mathbf{r},\omega\right)$ flowing solely inside $\Omega_{S}$, and into an *induced* partial current $\tilde{j}_{ind,a}\left(\mathbf{r},\omega\right)$ flowing solely inside the probe $\Omega_{P}$: $$\begin{aligned}
\tilde{j}_{a}\left(\mathbf{r},\omega\right) & = & \begin{cases}
\tilde{j}_{ext,a}\left(\mathbf{r},\omega\right) & \textrm{for}\:\mathbf{r}\in\Omega_{S}\\
\tilde{j}_{ind,a}\left(\mathbf{r},\omega\right) & \textrm{for}\:\mathbf{r}\in\Omega_{P}\\
0 & \textrm{for}\:\mathbf{r}\notin\Omega_{S}\cup\Omega_{P}
\end{cases}\label{eq:decomposition of current into external+induced part}\end{aligned}$$ If the source $\Omega_{S}$ was a cavity producing a laser beam, and if the surface of the probe $\Omega_{P}$ would be reflecting (parts) of the radiation incident back into that cavity, certainly there would exist a backaction from the probe to the source, leading then to the existence of an induced partial current flowing also inside the source region $\Omega_{S}$. We preclude here and in the following any such backaction effects. Provided transfer of charge between $\Omega_{S}$ and $\Omega_{P}$ is prohibited, there holds charge conservation separately (!) inside $\Omega_{S}$ and inside $\Omega_{P}$. Consequently, inside the domain $\Omega_{S}$ there holds $$i\omega\tilde{\rho}_{ext}(\mathbf{r},\omega)=\boldsymbol{\nabla}\cdot\mathbf{\tilde{j}}_{ext}(\mathbf{r},\omega).\label{eq: continuity equation}$$ Of course $\tilde{\rho}_{ext}(\mathbf{r},\omega)\equiv0$ and $\tilde{j}_{ext,a}\left(\mathbf{r},\omega\right)\equiv0$ for $\mathbf{r}\notin\Omega_{S}$.
It follows from (\[eq: longitudinal Maxwell equations\]) that the longitudinal electric field $\tilde{E}_{a}^{\left(L\right)}(\mathbf{r},\omega)$ is already determined by the longitudinal part of the current distribution (\[eq:decomposition of current into external+induced part\]), a seemingly simple result. Indeed from (\[eq: longitudinal Maxwell equations\]) and (\[eq:decomposition of current into external+induced part\]) then $$\tilde{E}_{a}^{(L)}\left(\mathbf{r},\omega\right)=\tilde{E}_{ext,a}^{(L)}\left(\mathbf{r},\omega\right)+\frac{1}{\epsilon_{0}}\frac{1}{i\omega}\tilde{j}_{ind,a}^{\left(L\right)}\left(\mathbf{r},\omega\right),\label{eq:longitudinal electric field}$$ where the *external* longitudinal field $$\tilde{E}_{ext,a}^{(L)}\left(\mathbf{r},\omega\right)=\frac{1}{\epsilon_{0}}\frac{1}{i\omega}\tilde{j}_{ext,a}^{\left(L\right)}\left(\mathbf{r},\omega\right)\equiv-\frac{\partial}{\partial r_{a}}\tilde{\phi}_{ext}(\mathbf{r},\omega)\label{eq:external longitudinal electric field}$$ is derived from a scalar potential $$\tilde{\phi}_{ext}(\mathbf{r},\omega)=\int_{\Omega_{S}}d^{3}r'\frac{\tilde{\rho}_{ext}(\mathbf{r}',\omega)}{4\pi\varepsilon_{0}\left|\mathbf{r}-\mathbf{r}'\right|},\label{eq:external scalr potential}$$ like in electrostatics. Even though $\tilde{j}_{ext,a}(\mathbf{r},\omega)$ was restricted to be non vanishing solely inside the source domain $\Omega_{S}$, its longitudinally projected part $\tilde{j}_{ext,a}^{\left(L\right)}\left(\mathbf{r},\omega\right)$ also exists outside of $\Omega_{S}$, the non locality of that projection thus becoming manifest.
The electromagnetic fields radiated by the transversal current density $\tilde{j}_{a}^{(T)}\left(\mathbf{r},\omega\right)$ in (\[eq: transversal Maxwell equations\]) can be readily determined introducing the *retarded* Green’s function $$\begin{aligned}
\tilde{g}\left(\mathbf{r}-\mathbf{r}',\omega\right) & = & \frac{\exp\left(i\frac{\omega}{c}\left|\mathbf{r}-\mathbf{r}'\right|\right)}{4\pi\left|\mathbf{r}-\mathbf{r}'\right|}\label{eq: Helmholtz propagator}\\
\mathtt{Im}\left(\omega\right) & \rightarrow & 0^{+},\nonumber \end{aligned}$$ with $\tilde{g}\left(\mathbf{r}-\mathbf{r}',\omega\right)$ the solution of the three-dimensional inhomogeneous *scalar* Helmholtz equation in free space $\mathbb{R}^{3}$ with a point source (of strength unity) at position $\mathbf{r}'$: $$\left(-\nabla^{2}-\frac{\omega^{2}}{c^{2}}\right)\tilde{g}\left(\mathbf{r}-\mathbf{r}',\omega\right)=\delta^{\left(3\right)}\left(\mathbf{r}-\mathbf{r}'\right)\label{eq: Helmholtz propagator II}$$ As the distance $\left|\mathbf{r}-\mathbf{r}'\right|$ to the source position at $\mathbf{r}'$ increases then $\tilde{g}\left(\mathbf{r}-\mathbf{r}',\omega\right)$ behaves like an *outgoing* spherical wave.
Based on Green’s identity applied to the domain $\Omega=\Omega_{S}\cup\Omega_{P}$, and again applied to the complementary domain $\mathbb{R}^{3}\setminus\Omega$, there follow directly from (\[eq: transversal Maxwell equations\]) and (\[eq: Helmholtz propagator II\]) for all points $\mathbf{r}\notin\Omega_{S}$ the integral representations $$\tilde{E}_{a}^{(T)}\left(\mathbf{r},\omega\right)=\tilde{E}_{ext,a}^{(T)}\left(\mathbf{r},\omega\right)+i\omega\mu_{0}\int d^{3}r'\tilde{g}\left(\mathbf{r}-\mathbf{r}',\omega\right)\tilde{j}_{ind,a}^{(T)}\left(\mathbf{r}',\omega\right)\label{eq:Integral representation transversal electric field}$$ and $$\tilde{B}_{a}\left(\mathbf{r},\omega\right)=\tilde{B}_{ext,a}\left(\mathbf{r},\omega\right)+\mu_{0}\int d^{3}r'\tilde{g}\left(\mathbf{r}-\mathbf{r}',\omega\right)\left[\mathbf{\boldsymbol{\nabla}'}\wedge\tilde{\mathbf{j}}_{ind}(\mathbf{r}',\omega)\right]_{a}.\label{eq:Integral equation magnetic induction field}$$ For details of the derivation of (\[eq:Integral representation transversal electric field\]) and (\[eq:Integral equation magnetic induction field\]), see supplemental material [@Supplementary].
According to (\[eq: representation of current density\]) the Fourier amplitude of the *induced* current density flowing inside the probe volume $\Omega_{P}$ is directly proportional to the Fourier amplitude of the microscopic electric polarization: $$\tilde{j}_{ind,a}\left(\mathbf{r},\omega\right)=-i\omega\tilde{P}_{a}\left(\mathbf{r},\omega\right)\label{eq: connection induced current - electric polarizabilty}$$ Combining now the respective longitudinal and transverse parts by adding the integral representations (\[eq:longitudinal electric field\]) and (\[eq:Integral representation transversal electric field\]) there follows $$\tilde{E}_{a}\left(\mathbf{r},\omega\right)=\tilde{E}_{ext,a}\left(\mathbf{r},\omega\right)+\frac{1}{\epsilon_{0}}\int d^{3}r'\sum_{a'}\mathcal{\tilde{G}}_{aa'}(\mathbf{r}-\mathbf{r}',\omega)\tilde{P}_{a'}\left(\mathbf{r}',\omega\right),\label{eq:local field in terms of polarization}$$ with $$\begin{aligned}
\mathcal{\tilde{G}}_{aa'}(\mathbf{r}-\mathbf{r}',\omega) & = & \frac{\omega^{2}}{c^{2}}\int d^{3}s\:\tilde{g}\left(\mathbf{r}-\mathbf{s},\omega\right)\Pi_{aa'}^{\left(T\right)}\left(\mathbf{s}-\mathbf{r}'\right)-\Pi_{aa'}^{\left(L\right)}\left(\mathbf{r}-\mathbf{r}'\right)\label{eq:3x3 matrix propagator}\\
& = & \lim_{\kappa\rightarrow\infty}\int\frac{d^{3}q}{\left(2\pi\right)^{3}}\frac{e^{i\mathbf{q}\cdot\left(\mathbf{r}-\mathbf{r}'\right)}}{1+\frac{\left|\mathbf{q}\right|^{2}}{\kappa^{2}}}\left[\frac{\omega^{2}}{c^{2}}\frac{\delta_{a,a'}-\frac{q_{a}q_{a'}}{\left|\mathbf{q}\right|^{2}}}{\left|\mathbf{q}\right|^{2}-\frac{\omega^{2}}{c^{2}}-i0^{+}}-\frac{q_{a}q_{a'}}{\left|\mathbf{q}\right|^{2}}\right]\nonumber \end{aligned}$$ denoting the electromagnetic kernel. The Fourier transformed kernel $\mathcal{\tilde{G}}_{aa'}(\mathbf{r},\omega)$ in the wave vector domain we denote as $$\begin{aligned}
\mathcal{\bar{G}}_{aa'}(\mathbf{q},\omega) & = & \int d^{3}re^{-i\mathbf{q}\cdot\mathbf{r}}\mathcal{\tilde{G}}_{aa'}(\mathbf{r},\omega)\label{eq: Fourier transform of 3x3 matrix propagator}\\
& = & \frac{\omega^{2}}{c^{2}}\frac{\delta_{a,a'}-\frac{q_{a}q_{a'}}{\left|\mathbf{q}\right|^{2}}}{\left|\mathbf{q}\right|^{2}-\frac{\omega^{2}}{c^{2}}-i0^{+}}-\frac{q_{a}q_{a'}}{\left|\mathbf{q}\right|^{2}}\nonumber \\
& = & \frac{\frac{\omega^{2}}{c^{2}}\delta_{a,a'}-q_{a}q_{a'}}{\left|\mathbf{q}\right|^{2}-\frac{\omega^{2}}{c^{2}}-i0^{+}}\:.\nonumber \end{aligned}$$
Assuming a small amplitude of the perturbing external field $\tilde{E}_{ext,a}\left(\mathbf{r},\omega\right)$ there results inside the probe at a position $\mathbf{r}\in\Omega_{P}$ the (total) microscopic polarization $$\tilde{P}_{a}\left(\mathbf{r},\omega\right)=\varepsilon_{0}\int_{\Omega_{P}}d^{3}r'\sum_{a'}\tilde{\chi}_{ext,aa'}\left(\mathbf{r},\mathbf{r}',\omega\right)\tilde{E}_{ext,a'}\left(\mathbf{r}',\omega\right)\:.\label{eq:polarization vs external field}$$ Within a fully microscopic approach the response kernel $\tilde{\chi}_{ext,aa'}\left(\mathbf{r},\mathbf{r}',\omega\right)$ is calculated from Kubo’s formula in reaction to the presence of the *external* field $\tilde{E}_{ext,a'}\left(\mathbf{r}',\omega\right).$ However, what we are really interested in here, is not the response kernel $\tilde{\chi}_{ext,aa'}\left(\mathbf{r},\mathbf{r}',\omega\right)$ connecting the polarization $\tilde{P}_{a}\left(\mathbf{r},\omega\right)$ inside the probe with the *external* field $\tilde{E}_{ext,a'}\left(\mathbf{r}',\omega\right)$, but the dielectric susceptibility kernel $\tilde{\chi}_{aa'}\left(\mathbf{r},\mathbf{r}',\omega\right)$ connecting $\tilde{P}_{a}\left(\mathbf{r},\omega\right)$ with the microscopic *local* electric field $\tilde{E}_{a'}\left(\mathbf{r}',\omega\right)$ that acts on each atom (ion, molecule): $$\tilde{P}_{a}\left(\mathbf{r},\omega\right)=\varepsilon_{0}\int_{\Omega_{P}}d^{3}r'\sum_{a'}\tilde{\chi}_{aa'}\left(\mathbf{r},\mathbf{r}',\omega\right)\tilde{E}_{a'}\left(\mathbf{r}',\omega\right)\label{eq:polarization vs local field}$$ As emphasized by Keldysh [@L.V.Keldysh2012], there is no need to consider dielectric and magnetic susceptibilities separately, the latter being already incorporated in the *non locality* of the dielectric kernel.
The fundamental field-integral equation determining the microscopic *local* electric field is then given by an inhomogenous integral equation comprising the dielectric kernel $\tilde{\chi}$: $$\tilde{E}_{a}\left(\mathbf{r},\omega\right)=\tilde{E}_{ext,a}\left(\mathbf{r},\omega\right)+\int_{\Omega_{P}}d^{3}r'\sum_{a'}\left[\mathcal{\tilde{G}}\circ\tilde{\chi}\right]_{aa'}\left(\mathbf{r},\mathbf{r}',\omega\right)\tilde{E}_{a'}\left(\mathbf{r}',\omega\right)\label{eq:local field integral equation}$$ The link between the dielectric susceptibility kernel $\tilde{\chi}$ and the response kernel $\tilde{\chi}_{ext}$ is readily identified [@L.V.Keldysh2012], combining (\[eq:polarization vs external field\]) with (\[eq:polarization vs local field\]): $$\tilde{\chi}=\tilde{\chi}_{ext}\circ\left[I+\mathcal{\tilde{G}}\circ\tilde{\chi}_{ext}\right]^{-1}=\left[I+\tilde{\chi}_{ext}\circ\mathcal{\tilde{G}}\right]^{-1}\circ\tilde{\chi}_{ext}\label{eq: connection dielectric kernel to response kernel}$$ For a more detailed explanation, see supplemental material [@Supplementary]. So, if $\tilde{\chi}_{ext}$ was known, say from a full microscopic calculation with Kubo’s formula, then $\tilde{\chi}$ follows from (\[eq: connection dielectric kernel to response kernel\]). In case the probe volume $\Omega_{P}$ was of finite size, solving the integral equation (\[eq: connection dielectric kernel to response kernel\]) for the microscopic dielectric susceptibility kernel $\tilde{\chi}$ poses a formidable (numerical) problem, the non-locality radius of $\tilde{\chi}_{ext}$ being substantially enhanced up to the macroscopic scale by the presence of a boundary [@L.V.Keldysh2012]. In the next section III we circumvent this problem and introduce a phenomenological model for $\tilde{\chi}$ that proves a posteriori to be appropriate to describe in detail the propagation of light in dielectric crystals.
Microscopic Local Electric Field in Crystalline Dielectrics\[sec:Microscopic-Local-Electric-Field-in crystalline-dielectrics\]
==============================================================================================================================
Crystalline order assumes each equilibrium position of atoms (ions, molecules) inside a material corresponds to a site vector $\mathbf{R}^{\left(j\right)}=\mathbf{R}+\boldsymbol{\eta}^{\left(j\right)}$, where $\mathbf{R}$ denotes a lattice vector in a Bravais lattice $\Lambda$, and $\boldsymbol{\eta}^{\left(j\right)}$ with $j\in\left\{ 1,2,...,M\right\} $ is a set of position vectors indicating equilibrium positions of the atoms (molecules, ions) inside the unit cell $C_{\Lambda}$. The entire probe volume $\Omega_{P}$ can be thought of to be filled translating a number $\left|\Lambda_{P}\right|$ of Wigner-Seitz unit cells $C_{\Lambda}$ by lattice vectors $\mathbf{R}\in\Lambda$. This subset of all such Bravais lattice vectors $\mathbf{R}\in\Lambda$ we denote as $\Lambda_{P}\subset\Lambda$. Under the action of the external electric field $\tilde{E}_{ext,a'}\left(\mathbf{r}',\omega\right)$ each atom (ion, molecule) gets polarized proportional to the *local* electric field $\tilde{E}_{a}\left(\mathbf{R}^{\left(j\right)},\omega\right)$ acting at position $\mathbf{R}^{\left(j\right)}$ of that atom. Then a simple *phenomenological* model for the microscopic dielectric susceptibility kernel in crystalline insulators emerges assembling first all individual atom contributions located at positions $\boldsymbol{\eta}^{\left(j\right)}$ inside the unit cell positioned around the origin $\mathbf{R}=\mathbf{0}$, and then sum over all such cells filling the probe volume $\Omega_{P}$ of the crystal lattice $\Lambda$ under consideration: $$\begin{aligned}
\tilde{\chi}_{aa'}\left(\mathbf{r},\mathbf{r}',\omega\right) & = & \frac{1}{\varepsilon_{0}}\sum_{\mathbf{R}\in\Lambda_{P}}\:\sum_{1\leq j,j'\leq M}\alpha_{aa'}\left(\boldsymbol{\eta}^{\left(j\right)},\boldsymbol{\eta}^{\left(j'\right)},\omega\right)\delta^{\left(3\right)}\left(\mathbf{r}-\mathbf{R}-\mathbf{\boldsymbol{\eta}}^{\left(j\right)}\right)\delta^{\left(3\right)}\left(\mathbf{r}'-\mathbf{R}-\boldsymbol{\eta}^{\left(j'\right)}\right)\label{eq: phenomenological dielectric kernel}\\
\nonumber \\
\alpha_{aa'}\left(\boldsymbol{\eta}^{\left(j\right)},\boldsymbol{\eta}^{\left(j'\right)},\omega\right) & = & \delta_{j,j'}\alpha_{aa'}^{\left(I\right)}\left(\boldsymbol{\eta}^{\left(j\right)},\omega\right)+\left(1-\delta_{j,j'}\right)\alpha_{aa'}^{\left(II\right)}\left(\boldsymbol{\eta}^{\left(j\right)},\boldsymbol{\eta}^{\left(j'\right)},\omega\right)\label{eq: model for microscopic polarizability}\end{aligned}$$ Essentially, this susceptibility kernel describes a lattice periodic arrangement of point dipoles [@Laue1931; @Ewald1938; @Laue1960]. The diagonal terms $j=j'$ refer to the afore mentioned effects of induced electronic polarization of *single* atoms (ions, molecules) at the positions $\boldsymbol{\eta}^{\left(j\right)}$ inside the unit cell $C_{\Lambda}$ around the lattice vector $\mathbf{R}=\mathbf{0}$. Based on the functional form of the frequency dependence of the individual microscopic polarizability (\[eq: atom polarizability\]) of a single atom (ion, molecule), in actual fact being equivalent to a Lorentz-oscillator model, we write now a simple phenomenological *ansatz* with fitting parameters $\alpha_{0}^{\left(j\right)}$and $\omega_{0}^{\left(j\right)}$(and possibly also including a small life-time parameter $\tau^{\left(j\right)}=1/\gamma^{\left(j\right)}>0$ representing spontaneous emission, see supplemental material [@Supplementary]): $$\alpha_{aa'}^{\left(I\right)}\left(\boldsymbol{\eta}^{\left(j\right)},\omega\right)=\delta_{a,a'}\frac{\alpha_{0}^{\left(j\right)}}{1-\left[\frac{\omega+\frac{i}{2}\gamma^{\left(j\right)}}{\omega_{0}^{\left(j\right)}}\right]^{2}}\label{eq: isotropic Lorentz polarizability diagonal}$$ If $\omega\ll\omega_{0}^{\left(j\right)}$ the atom polarizability is well approximated by its static value $\alpha_{0}^{\left(j\right)}$. For most frequencies, except if $\omega$ approaches a transition frequency $\omega_{0}^{\left(j\right)}$ near to an absorption band, we may assume $\gamma^{\left(j\right)}\rightarrow0^{+}$.
In case there are two or more basis atoms in the Wigner-Seitz cell qualitatively new effects need to be considered. Off diagonal terms $j\neq j'$ exist for $M\geq2$ and designate mutual influences of atoms (ions, molecules) positioned at different sites $\boldsymbol{\eta}^{\left(j\right)}$ and $\boldsymbol{\eta}^{\left(j'\right)}$ inside the unit cell. On one hand, in reaction to the presence of a propagating electromagnetic wave the overlap integral(s) determining the sharing of electron pairs between neighbouring atoms undergo (slight) changes. On the other hand, in crystals like $NaCl$, $CsI$, $RbCl$ etc., the formation of *ions* needs to be taken into account. Besides the effect of induced electronic polarization of *single* atoms (ions), attributed to a shift of the barycenter of the electrons bound to individual atoms (ions) under the action of the local field on-site $\mathbf{R}^{\left(j\right)}$, an additional shift of the position of the positive ions relative to the position of the negative ions concurs, thus leading to ionic *displacement* polarizability. This effect is typically noticeable in the electromagnetic response to radiation with frequency $\omega$ of order of characteristic *lattice vibration* frequencies $\omega_{ph}$, de facto being mainly of concern for radiation in the infrared.
Conceiving also the off-diagonal polarizabilities $j\neq j'$ not as input from microscopic theory, but in the phenomenological guise of a Lorentz-oscillator model for oppositely charged ion pairs [@Ashcroft1981],
$$\alpha_{aa'}^{\left(II\right)}\left(\boldsymbol{\eta}^{\left(j\right)},\boldsymbol{\eta}^{\left(j'\right)},\omega\right)=\delta_{a,a'}\frac{\alpha_{0}^{\left(j,j'\right)}}{1-\left[\frac{\omega+\frac{i}{2}\gamma^{\left(j,j'\right)}}{\omega_{0}^{\left(j,j'\right)}}\right]^{2}}\:,\label{eq: isotropic Lorentz polarizability off diagonal}$$
with fitting parameters $\alpha_{0}^{\left(j,j'\right)}$, $\omega_{0}^{\left(j,j'\right)}$ and small damping $\gamma^{\left(j,j'\right)}$, we find the well known chromatic dispersion of the index of refraction $n\left(\omega\right)$ is nicely reproduced by our theory of the dielectric tensor $\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)$ for a variety of dielectric crystals, see Table \[results\] in section \[sec:Macroscopic-Electric-Field-and-dielectric-function\]. While the electronic polarizabilities (\[eq: isotropic Lorentz polarizability diagonal\]) of single atoms are often well approximated by their static value, the ionic displacement polarizabilities (\[eq: isotropic Lorentz polarizability off diagonal\]) have a characteristic dependence on frequency in the infrared. Note, that we refrain adding the contributions of the electronic polarizabilities of single atoms to the contributions of the ionic displacement polarizabilities, there being no justification for this [@Ashcroft1981].
For example, in the case of $CsI$ with $M=2$ ions in the unit cell, our phenomenological approach in (\[eq: phenomenological dielectric kernel\]) brings into altogether $6$ parameters. These are the two transition frequencies $\omega_{0}^{\left(1\right)}$, $\omega_{0}^{\left(2\right)}$ in the optical regime to model the *induced electronic polarization* of the two ions together with two values $\alpha_{0}^{\left(1\right)}$, $\alpha_{0}^{\left(2\right)}$ for the individual *static* polarizabilities of the respective ions, and a third resonance frequency $\omega_{0}^{\left(1,2\right)}$ with associated value $\alpha_{0}^{\left(1,2\right)}$shaping the strength of the ionic displacement polarizabilty, that frequency $\omega_{0}^{\left(1,2\right)}$ being characteristic of lattice vibrational frequencies $\omega_{ph}$. Such a distinction is justifiable if the time scale for electronic polarization of individual ions is much faster than the time scale for the lattice vibrations: $\omega_{ph}\simeq\omega_{0}^{\left(1,2\right)}\ll\min\left(\omega_{0}^{\left(1\right)},\omega_{0}^{\left(2\right)}\right)$. With the model (\[eq: phenomenological dielectric kernel\]) we then successfully reproduced experimental data for the chromatic dispersion of the refraction index $n\left(\omega\right)$, exemplarily for $CsI$ and $RbCl$, over a wide frequency interval ranging from ultraviolet to infrared, utilizing only these $6$ parameters instead of $17$ parameters as required by a Sellmeier fit, see Fig.\[Sellmeier\] and Table \[fit\].
Restricting to optical frequencies well above the infrared range (but always well below atomic excitation energies), then mostly the electrons bound around individual ions (atoms) will react to the electromagnetic fields. In this case the effect of induced electric polarization is predominant and the effects of ionic polarisation, as represented by the off diagonal contributions $j\neq j'$ in (\[eq: model for microscopic polarizability\]), can be considered as small, so that $$\begin{aligned}
\omega & \gg & \omega_{ph}\nonumber \\
\alpha_{aa'}\left(\boldsymbol{\eta}^{\left(j\right)},\boldsymbol{\eta}^{\left(j'\right)},\omega\right) & = & \delta_{j,j'}\alpha_{aa'}^{\left(I\right)}\left(\boldsymbol{\eta}^{\left(j\right)},\omega\right)\:.\label{eq: high frequency limit of optical polarizability}\end{aligned}$$ It should be noted that retaining in (\[eq: high frequency limit of optical polarizability\]) the possibility of non zero off diagonal Cartesian terms $a\neq a'$ in the Lorentz-oscillator model of atomic polarizabilities $\tilde{\alpha}_{aa'}\left(\boldsymbol{\eta}^{\left(j\right)},\omega\right)$ then enables the study of crystals composed of anisotropic polarizable ionic or molecular subunits in the elementary cell. For instance, the study of the influence of an external *static* magnetic induction field $\mathbf{B}_{0}$ or *static* electric field $\mathbf{E}_{0}$ on the propagation of light in crystals also requires to retain non zero off diagonal Cartesian components $a\neq a'$ in (\[eq: high frequency limit of optical polarizability\]). As important examples we mention the *magneto-optical* Faraday effect and the *electro-optical* Pockels effect [@Agranovich1984].
Obviously, under a translation by a lattice vector $\mathbf{R}\in\Lambda$ the dielectric kernel (\[eq: phenomenological dielectric kernel\]) remains invariant: $$\tilde{\chi}_{a,a'}\left(\mathbf{r}+\mathbf{R},\mathbf{r}'+\mathbf{R},\omega\right)=\tilde{\chi}_{a,a'}\left(\mathbf{r},\mathbf{r}',\omega\right)\label{eq: periodicity of dielectric kernel}$$ The dielectric kernel considered at fixed position $\mathbf{r}\in\Omega_{P}$ as a function of $\mathbf{r}'-\mathbf{r}$ will typically undergo already discernible variations traversing a short route of order of the interatomic distance $a$. In what follows we show, that the microscopic local electric field amplitude $\tilde{\mathbf{E}}\left(\mathbf{r},\omega\right)$ then also displays discernible spatial variations on that same short length scale $a$, even though the primary incident light signal $\tilde{\mathbf{E}}_{ext}\left(\mathbf{r},\omega\right)$ was discernibly varying only on the much longer length scale set by the wavelength $\lambda\gg a$ of light in free space.
Non-Standard Bloch Functions {#non-standard-bloch-functions .unnumbered}
-----------------------------
Because of the periodicity (\[eq: periodicity of dielectric kernel\]) it is an obvious choice to expand the microscopic local electric field in a complete basis of Bloch eigenstates of the translation operator $\hat{T}_{\mathbf{\mathbf{R}}}$ shifting the argument of any function $f\left(\mathbf{r}\right)$ according to $$\hat{T}_{\mathbf{\mathbf{R}}}f\left(\mathbf{r}\right)=f\left(\mathbf{r}+\mathbf{R}\right)\label{eq: shift operator}$$ with $\mathbf{R}\in\Lambda$ a Bravais lattice vector. Routinely in problems with a lattice periodic potential an expansion provided by the orthonormal and complete basis system of plane waves constructed from eigenfunctions of the *momentum* operator is deployed $$\left\{ e^{i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}}\right\} _{\mathbf{k}\in C_{\Lambda}^{-1},\mathbf{G}\in\Lambda^{-1}}\;.\label{eq: basis system plane waves}$$ Here $\Lambda^{-1}$ denotes the reciprocal lattice conjugate to the lattice $\Lambda$ and $C_{\Lambda}^{-1}$ is the Brillouin zone. Making use of $e^{i\mathbf{G}\cdot\mathbf{R}}=1$ for $\mathbf{R}\in\Lambda$ and any reciprocal lattice vector $\mathbf{G}\in\Lambda^{-1}$ one readily confirms $$\hat{T}_{\mathbf{\mathbf{R}}}e^{i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}}=e^{i\mathbf{k}\cdot\mathbf{R}}e^{i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}}.$$ The eigenvalues $e^{i\mathbf{k}\cdot\mathbf{R}}$ associated with the eigenfunctions $e^{i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}}$ of the translation operator $\hat{T}_{\mathbf{\mathbf{R}}}$ being highly degenerate, any function of the form $$\begin{aligned}
f_{\mathbf{k}}\left(\mathbf{r}\right) & = & e^{i\mathbf{k}\cdot\mathbf{r}}u\left(\mathbf{r}\right)\\
u\left(\mathbf{r}\right) & = & \sum_{\mathbf{G}\in\Lambda^{-1}}u_{\mathbf{G}}e^{i\mathbf{G}\cdot\mathbf{r}}=u\left(\mathbf{r}+\mathbf{R}\right)\:,\end{aligned}$$ (Fourier coefficients denoted as $u_{\mathbf{G}}$), is an eigenfunction of the translation operator(s) $\hat{T}_{\mathbf{\mathbf{R}}}$: $$\hat{T}_{\mathbf{\mathbf{R}}}f_{\mathbf{k}}\left(\mathbf{r}\right)=e^{i\mathbf{k}\cdot\mathbf{R}}f_{\mathbf{k}}\left(\mathbf{r}\right)$$
Based on a rigorous theory of the microscopic electromagnetic response kernel, Dolgov and Maximov [@Maksimov2012] presented for crystalline systems a profound analysis of the dielectric function, representing the microscopic kernels in (\[eq: connection dielectric kernel to response kernel\]) as (infinite) matrices with respect to the basis system (\[eq: basis system plane waves\]). Adversely, the integral kernel $\mathcal{\tilde{G}}\circ\tilde{\chi}$ in (\[eq:local field integral equation\]), when represented in the basis of plane waves $\left\{ e^{i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}}\right\} _{\mathbf{k}\in C_{\Lambda}^{-1},\mathbf{G}\in\Lambda^{-1}}$, turns out to be a full up matrix $\left\langle \mathbf{G}+\mathbf{k}\right|\left[\mathcal{\tilde{G}}\circ\tilde{\chi}\right]_{aa'}\left|\mathbf{G}'+\mathbf{k}\right\rangle $, labeled by a wavevector $\mathbf{k}\in C_{\Lambda^{-1}}$ and an infinite set of reciprocal lattice vectors $\mathbf{G},\mathbf{G}'\in\Lambda^{-1}$. In the (numerical) calculations thus the handling of large matrices is required.
An alternative set of eigenfunctions of the translation operator(s) $\hat{T}_{\mathbf{\mathbf{R}}}$, so that the dielectric kernel when represented in the new basis appears as a *sparse* matrix, is highly desirable. Observing, that any lattice periodic function $u\left(\mathbf{r}\right)$ can be generated from a *fragment* $u^{\left(0\right)}\left(\mathbf{s}\right)$ that equals to $u\left(\mathbf{s}\right)$ inside the Wigner-Seitz cell $C_{\Lambda}$ and is zero outside, $$u\left(\mathbf{r}\right)=\sum_{\mathbf{R}'\in\Lambda}u^{\left(0\right)}\left(\mathbf{r}+\mathbf{R}'\right),\label{eq: generate lattice periodic function I-1}$$ we may as well represent any such fragment $u^{\left(0\right)}\left(\mathbf{s}\right)\equiv\left\langle \mathbf{s}|u^{\left(0\right)}\right\rangle $ as a linear combination $$\left|u^{\left(0\right)}\right\rangle =\int_{C_{\Lambda}}d^{3}s'u^{\left(0\right)}\left(\mathbf{s}'\right)\left|\mathbf{s}'\right\rangle ,$$ with the complete and orthonormal set of eigenstates $\left\{ \left|\mathbf{s}\right\rangle \right\} _{\mathbf{s}\in C_{\Lambda}}$ of the position operator $\hat{r}_{a}$ obeying to the well known relations $$\begin{aligned}
\hat{r}_{a}\left|\mathbf{s}\right\rangle & = & s_{a}\left|\mathbf{s}\right\rangle \nonumber \\
\int_{C_{\Lambda}}d^{3}s\left|\mathbf{s}\right\rangle \left\langle \mathbf{s}\right| & = & \hat{1}_{C_{\Lambda}}\nonumber \\
\left\langle \mathbf{r}|\mathbf{s}\right\rangle & = & \delta^{\left(3\right)}\left(\mathbf{s}-\mathbf{r}\right).\end{aligned}$$ So instead of expanding into the basis system of plane waves $\left\{ e^{i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}}\right\} _{\mathbf{k}\in C_{\Lambda}^{-1},\mathbf{G}\in\Lambda^{-1}}$ there emerges as an alternative expanding into the following system of eigenfunctions of the translation operator $\hat{T}_{\mathbf{\mathbf{R}}}$: $$\begin{aligned}
w\left(\mathbf{r};\mathbf{s},\mathbf{k}\right) & = & \frac{e^{i\mathbf{k}\cdot\mathbf{r}}}{\sqrt{\left|\Lambda_{P}\right|}}\sum_{\mathbf{R}'\in\Lambda}\left\langle \mathbf{r}|\mathbf{\mathbf{s}+\mathbf{R}'}\right\rangle \label{eq: basis w(r,s,k)}\end{aligned}$$ By construction: $$\hat{T}_{\mathbf{\mathbf{R}}}w\left(\mathbf{r};\mathbf{s},\mathbf{k}\right)=w\left(\mathbf{r}+\mathbf{R};\mathbf{s},\mathbf{k}\right)=e^{i\mathbf{k}\cdot\mathbf{R}}w\left(\mathbf{r};\mathbf{s},\mathbf{k}\right)$$ The set of states $\left\{ w\left(\mathbf{r};\mathbf{s},\mathbf{k}\right)\right\} _{\mathbf{k}\in C_{\Lambda^{-1}},\mathbf{\mathbf{s}}\in C_{\Lambda}}$ is for one thing labeled by a wavevector $\mathbf{k}\in C_{\Lambda^{-1}}$, that ranges through the count $\left|\Lambda_{P}\right|$ of values of wave vectors $\mathbf{k}$ in the first Brillouin zone $C_{\Lambda^{-1}}$, consistent with the Born-von Karman periodic boundary conditions, and for another thing it is labeled by position vectors $\mathbf{s}\in C_{\Lambda}$ that range within the Wigner-Seitz cell $C_{\Lambda}$ of the lattice $\Lambda$. The system $\left\{ w\left(\mathbf{r};\mathbf{s},\mathbf{k}\right)\right\} _{\mathbf{k}\in C_{\Lambda^{-1}},\,\mathbf{\mathbf{s}}\in C_{\Lambda}}$ spans a *complete* and *orthonormal* basis system of eigenfunctions of the translation operator(s) $\hat{T}_{\mathbf{\mathbf{R}}}$, for a proof see supplemental material [@Supplementary].
Solution of the Field-Integral Equations for a Dielectric Crystal {#solution-of-the-field-integral-equations-for-a-dielectric-crystal .unnumbered}
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Representing now the microscopic local electric field in the complete and orthonormal basis system $\left\{ w\left(\mathbf{r};\mathbf{s},\mathbf{k}\right)\right\} _{\mathbf{k}\in C_{\Lambda^{-1}},\,\mathbf{\mathbf{s}}\in C_{\Lambda}}$ we write $$\tilde{E}_{a}\left(\mathbf{r},\omega\right)=\sum_{\mathbf{k}\in C_{\Lambda^{-1}}}\int_{C_{\Lambda}}d^{3}s\:w\left(\mathbf{r};\mathbf{s},\mathbf{k}\right)\tilde{\mathfrak{e}}_{a}\left(\mathbf{s},\mathbf{k},\omega\right)\:.\label{eq:local electric field spanned in basis w(r,s,k)}$$ Conversely, the expansion coefficients representing that field are $$\begin{aligned}
\tilde{\mathfrak{e}}_{a}\left(\mathbf{s},\mathbf{k},\omega\right) & = & \int_{\Omega_{P}}d^{3}r'\left[w\left(\mathbf{r}';\mathbf{s},\mathbf{k}\right)\right]^{\dagger}\tilde{E}_{a}\left(\mathbf{r}',\omega\right)\:.\label{eq: expansion coefficient of local field}\end{aligned}$$ So, with $$\begin{aligned}
\tilde{\mathfrak{e}}_{ext,a}\left(\mathbf{s},\mathbf{k},\omega\right) & = & \int_{\Omega_{P}}d^{3}r'\left[w\left(\mathbf{r}';\mathbf{s},\mathbf{k}\right)\right]^{\dagger}\tilde{E}_{ext,a}\left(\mathbf{r}',\omega\right)\label{eq:expansion coefficients external field}\end{aligned}$$ the field-integral equation (\[eq:local field integral equation\]) is transformed into an equivalent integral equation determining the expansion coefficients $\tilde{\mathfrak{e}}_{a}\left(\mathbf{s},\mathbf{k},\omega\right)$: $$\tilde{\mathfrak{e}}_{a}\left(\mathbf{s},\mathbf{k},\omega\right)=\tilde{\mathfrak{e}}_{ext,a}\left(\mathbf{s},\mathbf{k},\omega\right)+\sum_{\mathbf{k}'\in C_{\Lambda^{-1}}}\int_{C_{\Lambda}}d^{3}s'\sum_{a'}\left\langle \mathbf{s},\mathbf{k}\right|\left[\mathcal{\tilde{G}}\circ\tilde{\chi}\right]_{a,a'}\left|\mathbf{s}',\mathbf{k}'\right\rangle \tilde{\mathfrak{e}}_{a'}\left(\mathbf{s}',\mathbf{k}',\omega\right)\label{eq:local field integral equation-1}$$ The matrix elements of the dielectric kernel in the basis (\[eq: basis w(r,s,k)\]) are readily evaluated, see supplemental material [@Supplementary]: $$\begin{aligned}
\left\langle \mathbf{s},\mathbf{k}\right|\left[\mathcal{\tilde{G}}\circ\tilde{\chi}\right]_{a,a'}\left|\mathbf{s}',\mathbf{k}'\right\rangle & = & \int_{\Omega_{P}}d^{3}r\int_{\Omega_{P}}d^{3}r'\left[w\left(\mathbf{r};\mathbf{s},\mathbf{k}\right)\right]^{\dagger}\left[\mathcal{\tilde{G}}\circ\tilde{\chi}\right]_{a,a'}\left(\mathbf{r},\mathbf{r}',\omega\right)w\left(\mathbf{r}';\mathbf{s}',\mathbf{k}'\right)\label{eq:matrix element dielectric kernel}\\
& = & \frac{1}{\varepsilon_{0}}\sum_{j',j''}\sum_{a''}\left[\zeta_{\varLambda}(\mathbf{s}-\boldsymbol{\eta}^{\left(j''\right)},\mathbf{k},\omega)\right]_{a,a''}\alpha_{a'',a'}^{j'',j'}\left(\mathbf{k},\omega\right)\delta^{\left(3\right)}\left(\mathbf{s}'-\boldsymbol{\eta}^{\left(j'\right)}\right)\delta_{\mathbf{k},\mathbf{k}'}\nonumber \end{aligned}$$ Here we introduced notation such that $$\alpha_{a'',a'}^{\left(j'',j'\right)}\left(\mathbf{k},\omega\right)\equiv e^{-i\mathbf{k}\cdot\mathbf{\boldsymbol{\eta}}^{\left(j''\right)}}\alpha_{a'',a'}\left(\boldsymbol{\eta}^{\left(j''\right)},\boldsymbol{\eta}^{\left(j'\right)},\omega\right)e^{i\mathbf{k}\cdot\boldsymbol{\eta}^{\left(j'\right)}},\label{eq:block matrix alpha}$$ and $\tilde{\zeta}_{\varLambda}$ denotes a $3\times3$ matrix of *lattice sums* formed with the electromagnetic kernel (\[eq:3x3 matrix propagator\]): $$\left[\zeta_{\varLambda}(\mathbf{s},\mathbf{k},\omega)\right]_{a,a'}=\begin{cases}
\sum_{\mathbf{R}'\in\varLambda}e^{-i\mathbf{k}\cdot\left(\mathbf{s}+\mathbf{R}'\right)}\mathcal{\tilde{G}}_{a,a'}\left(\mathbf{s}+\mathbf{R}',\omega\right) & \textrm{if}\:\mathbf{s}\neq\mathbf{0}\\
\sum_{\mathbf{R}'\in\varLambda\setminus\left\{ \mathbf{\mathbf{0}}\right\} }e^{-i\mathbf{k}\cdot\mathbf{R}'}\mathcal{\tilde{G}}_{a,a'}\left(\mathbf{R}',\omega\right) & \textrm{if}\:\mathbf{s}=\mathbf{0}
\end{cases}\label{eq: lattice sums zeta}$$ For any Bravais lattice vector $\mathbf{R}\in\varLambda$ there holds $$\zeta_{\varLambda}(\mathbf{s}+\mathbf{R},\mathbf{k},\omega)=\zeta_{\varLambda}(\mathbf{s},\mathbf{k},\omega).\label{eq: periodicity property of lattice sum}$$ The definition of the lattice sums (\[eq: lattice sums zeta\]) by cases is to ensure no atom can get polarized by its self-generated electromagnetic field. It is important to realize that $$\left[\zeta_{\varLambda}^{\left(0\right)}(\mathbf{k},\omega)\right]_{aa'}\equiv\left[\zeta_{\varLambda}(\mathbf{s}=\mathbf{0},\mathbf{k},\omega)\right]_{aa'}\neq\lim_{\left|\mathbf{\mathbf{s}}\right|\rightarrow0^{+}}\left[\zeta_{\varLambda}(\mathbf{s},\mathbf{k},\omega)\right]_{aa'}.\label{eq: zeta lattice sum null value}$$ Instead there holds $$\left[\zeta_{\varLambda}^{\left(0\right)}(\mathbf{k},\omega)\right]_{aa'}=\lim_{\left|\mathbf{\mathbf{s}}\right|\rightarrow0^{+}}\left[\zeta_{\varLambda}(\mathbf{s},\mathbf{k},\omega)-\mathcal{\tilde{G}}\left(\mathbf{s},\omega\right)\right]_{aa'}.\label{eq: lattice sums zeta for s=00003D0}$$ A fast and precise numerical method for the calculation of the lattice sums $\zeta_{\varLambda}(\mathbf{s},\mathbf{k},\omega)$ and $\zeta_{\varLambda}^{\left(0\right)}(\mathbf{k},\omega)$ we disclose in our supplemental material [@Supplementary].
Insertion of (\[eq:matrix element dielectric kernel\]) gives at once the field amplitudes $\tilde{\mathfrak{e}}_{a}\left(\mathbf{s},\mathbf{k},\omega\right)$ in terms of a finite number of amplitudes $\tilde{\mathfrak{e}}_{a}^{\left(j\right)}\left(\mathbf{k},\omega\right)\equiv\tilde{\mathfrak{e}}_{a}\left(\boldsymbol{\eta}^{\left(j\right)},\mathbf{k},\omega\right)$, with $j\in\left\{ 1,2,...,M\right\} $ counting the positions $\mathbf{s}=\boldsymbol{\eta}^{\left(j\right)}$ of the polarizable atoms (ions, molecules) inside the Wigner-Seitz cell $C_{\Lambda}$ and $a\in\left\{ x,y,z\right\} $ denoting Cartesian components: $$\tilde{\mathfrak{e}}_{a}\left(\mathbf{s},\mathbf{k},\omega\right)=\tilde{\mathfrak{e}}_{ext,a}\left(\mathbf{s},\mathbf{k},\omega\right)+\frac{1}{\varepsilon_{0}}\sum_{j',j''}\sum_{a',a''}\left[\zeta_{\varLambda}(\mathbf{s}-\boldsymbol{\eta}^{\left(j''\right)},\mathbf{k},\omega)\right]_{a,a''}\alpha_{a'',a'}^{\left(j'',j'\right)}\left(\mathbf{k},\omega\right)\tilde{\mathfrak{e}}_{a'}^{\left(j'\right)}\left(\mathbf{k},\omega\right)\label{eq:amplitudes of local field in basis w(r,s,k)}$$ Taking subsequently for $j=1,2,...,M$ the limit $\mathbf{s}\rightarrow\boldsymbol{\eta}^{\left(j\right)}$, then from (\[eq:amplitudes of local field in basis w(r,s,k)\]) a $3M\times3M$ system of linear equations determining those amplitudes $\tilde{\mathfrak{e}}_{a}^{\left(j\right)}\left(\mathbf{k},\omega\right)$ subject to the prescribed amplitudes $\tilde{\mathfrak{e}}_{ext,a}^{\left(j\right)}\left(\mathbf{k},\omega\right)$ of the external field is obtained: $$\tilde{\mathfrak{e}}_{a}^{\left(j\right)}\left(\mathbf{k},\omega\right)=\tilde{\mathfrak{e}}_{ext,a}^{\left(j\right)}\left(\mathbf{k},\omega\right)+\frac{1}{\varepsilon_{0}}\sum_{j',j''}\sum_{a',a''}\left[\zeta_{\varLambda}(\boldsymbol{\eta}^{\left(j\right)}-\boldsymbol{\eta}^{\left(j''\right)},\mathbf{k},\omega)\right]_{a,a''}\alpha_{a'',a'}^{\left(j'',j'\right)}\left(\mathbf{k},\omega\right)\tilde{\mathfrak{e}}_{a'}^{\left(j'\right)}\left(\mathbf{k},\omega\right)\label{eq: system of 3Mx3M equations for amplitudes of local field}$$ This result clearly brings out the advantage of the basis system $\left\{ w\left(\mathbf{r};\mathbf{s},\mathbf{k}\right)\right\} _{\mathbf{k}\in C_{\Lambda^{-1}},\mathbf{\mathbf{s}}\in C_{\Lambda}}$ over the conventional basis system of plane waves $\left\{ e^{i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}}\right\} _{\mathbf{k}\in C_{\Lambda}^{-1},\mathbf{G}\in\Lambda^{-1}}$, thus dispensing the consideration of large full up dielectric matrix kernels $\left\langle \mathbf{G}+\mathbf{k}\right|\left[\mathcal{\tilde{G}}\circ\tilde{\chi}\right]_{aa'}\left|\mathbf{G}'+\mathbf{k}\right\rangle $ labeled by an infinite number of reciprocal lattice vectors $\mathbf{G},\mathbf{G}'\in\Lambda^{-1}$. As an incidental remark see [^2].
The determination of the expansion coefficients $\tilde{\mathfrak{e}}_{a}\left(\mathbf{s},\mathbf{k},\omega\right)$ representing the local electric field amplitude via (\[eq:amplitudes of local field in basis w(r,s,k)\]) requires solving (\[eq: system of 3Mx3M equations for amplitudes of local field\]), a *small* sized $3M\times3M$ linear system of equations. Introducing the $3M\times3M$ matrices $$\Gamma_{a,a'}^{\left(j,j'\right)}\left(\mathbf{k},\omega\right)=\sum_{j''}\sum_{a''}\left[\zeta_{\varLambda}(\boldsymbol{\eta}^{\left(j\right)}-\boldsymbol{\eta}^{\left(j''\right)},\mathbf{k},\omega)\right]_{a,a''}\frac{1}{\varepsilon_{0}}\alpha_{a'',a'}^{\left(j'',j'\right)}\left(\mathbf{k},\omega\right)\label{eq: Block matrix Gamma}$$ and $$\left[I\right]_{a,a'}^{\left(j,j'\right)}=\delta_{j,j'}\delta_{a,a'}$$ the explicit solution of (\[eq: system of 3Mx3M equations for amplitudes of local field\]) reads $$\tilde{\mathfrak{e}}_{a}^{\left(j\right)}\left(\mathbf{k},\omega\right)=\sum_{j'}\sum_{a'}\left(\left[I-\Gamma\left(\mathbf{k},\omega\right)\right]^{-1}\right)_{a,a'}^{\left(j,j'\right)}\tilde{\mathfrak{e}}_{ext,a'}^{\left(j'\right)}\left(\mathbf{k},\omega\right).\label{eq: explicit amplitudes of local field}$$ Finally, substituting the expansion coefficients (\[eq: explicit amplitudes of local field\]) into (\[eq:amplitudes of local field in basis w(r,s,k)\]) we then obtain from (\[eq:local electric field spanned in basis w(r,s,k)\]) the following explicit representation for the microscopic local electric field: $$\begin{aligned}
\tilde{E}_{a}\left(\mathbf{r},\omega\right) & = & \sum_{\mathbf{k}\in C_{\Lambda^{-1}}}\int_{C_{\Lambda}}d^{3}s\:w\left(\mathbf{r};\mathbf{s},\mathbf{k}\right)\tilde{\mathfrak{e}}_{a}\left(\mathbf{s},\mathbf{k},\omega\right)\label{eq:microscopic local electric field}\\
& = & \tilde{E}_{ext,a}\left(\mathbf{r},\omega\right)+\sum_{\mathbf{k}\in C_{\Lambda^{-1}}}\frac{e^{i\mathbf{k}\cdot\mathbf{r}}}{\sqrt{\left|\Lambda_{P}\right|}}\sum_{j',j''}\sum_{a',a''}\left[\zeta_{\varLambda}(\mathbf{r}-\boldsymbol{\eta}^{\left(j''\right)},\mathbf{k},\omega)\right]_{a,a''}\frac{1}{\varepsilon_{0}}\alpha_{a'',a'}^{\left(j'',j'\right)}\left(\mathbf{k},\omega\right)\tilde{\mathfrak{e}}_{a'}^{\left(j'\right)}\left(\mathbf{k},\omega\right)\nonumber \end{aligned}$$ Incidentally, the general result (\[eq:microscopic local electric field\]) for the microscopic local electric field inside a crystal comprises also the case of multiple beams of incident signals, all with frequency $\omega$ but possibly with different wavevectors $\mathbf{k}$, in this case represented by a multitude of expansion coefficients $\tilde{\mathfrak{e}}_{ext,a'}^{\left(j'\right)}\left(\mathbf{k},\omega\right)$ carrying different wave vectors $\mathbf{k}$, for example see [@Laue1960].
It should be emphasized that the well known dispersion relation $\omega=c\left|\mathbf{q}\right|$ of light in vacuum expresses the solvability condition for the external field (\[eq:incident external signal\]) solving the *homogeneous* Maxwell equations in free space. However, in reality external signals represent solutions of the *inhomogeneous* Maxwell equations with a source current $\tilde{j}_{ext,a}\left(\mathbf{r},\omega\right)=\sum_{\mathbf{q}'}\bar{j}_{ext,a}\left(\mathbf{q}',\omega\right)e^{i\mathbf{q}'\cdot\mathbf{r}}$ flowing inside a source volume $\Omega_{S}$, the latter being usually positioned at a large (but finite) distance to the probe volume $\Omega_{P}$, see Eq. (57) in supplemental material [@Supplementary]. So a typical external signal $\tilde{E}_{ext,a}\left(\mathbf{r},\omega\right)$ at *fixed* (circular) frequency $\omega$ is composed of a bunch of mixed wave vectors $\mathbf{q}$’ forming that signal: $$\tilde{E}_{ext,a}\left(\mathbf{r},\omega\right)=\sum_{\mathbf{q}'}\mathcal{\bar{E}}_{ext,a}\left(\mathbf{\mathbf{q}}',\omega\right)e^{i\mathbf{\mathbf{q}}'\cdot\mathbf{r}}$$ This feature enables to regard $\omega$ and $\mathbf{q}$ from now on as *independent* variables. Making use of the superposition principle it suffices to specialize to a single beam. For convenience we consider now an external signal in the guise of a plane wave, with (circular) frequency $\omega$ and wavevector $\mathbf{q}$ so that $$\tilde{E}_{ext,a}\left(\mathbf{r},\omega\right)=\mathcal{\bar{E}}_{ext,a}\left(\mathbf{q},\omega\right)e^{i\mathbf{q}\cdot\mathbf{r}}\:.\label{eq:incident external signal}$$
Assuming then $\mathbf{q}\in C_{\Lambda^{-1}}$, certainly not a strong limitation for optical signals propagating in crystalline materials, it follows at once from the defining equation (\[eq:expansion coefficients external field\]) that the expansion coefficients of the external signal are independent on the value of the mode index $\mathbf{s}\in C_{\Lambda}$. All the more the coefficients $\tilde{\mathfrak{e}}_{ext,a'}^{\left(j'\right)}\left(\mathbf{k},\omega\right)$ then being independent on $\boldsymbol{\eta}^{\left(j\right)}$ there holds $$\tilde{\mathfrak{e}}_{ext,a}\left(\mathbf{s},\mathbf{k},\omega\right)=\sqrt{\left|\Lambda_{P}\right|}\delta_{\mathbf{k},\mathbf{q}}\mathcal{\bar{E}}_{ext,a}\left(\mathbf{q},\omega\right)\equiv\tilde{\mathfrak{e}}_{ext,a'}^{\left(j'\right)}\left(\mathbf{k},\omega\right)\:.\label{eq: amplitude of external fielod}$$ Insertion of (\[eq: amplitude of external fielod\]) into (\[eq: explicit amplitudes of local field\]) then reduces (\[eq:microscopic local electric field\]) to $$\begin{aligned}
\tilde{E}_{a}\left(\mathbf{r},\omega\right) & = & \mathcal{\bar{E}}_{ext,a}\left(\mathbf{q},\omega\right)e^{i\mathbf{q}\cdot\mathbf{r}}\label{eq:microscopic local field for incident plane wave with small q}\\
& & +e^{i\mathbf{q}\cdot\mathbf{r}}\sum_{j',j'',j'''}\sum_{a',a'',a'''}\left[\zeta_{\varLambda}(\mathbf{r}-\boldsymbol{\eta}^{\left(j''\right)},\mathbf{q},\omega)\right]_{a,a''}\frac{1}{\varepsilon_{0}}\alpha_{a'',a'}^{\left(j'',j'\right)}\left(\mathbf{q},\omega\right)\left(\left[I-\Gamma\left(\mathbf{q},\omega\right)\right]^{-1}\right)_{a',a'''}^{\left(j',j'''\right)}\mathcal{\bar{E}}_{ext,a'''}\left(\mathbf{q},\omega\right).\nonumber \end{aligned}$$ Exploiting next the lattice periodicity of $\zeta_{\varLambda}(\mathbf{\mathbf{s}},\mathbf{q},\omega)$, see (\[eq: periodicity property of lattice sum\]), there holds for $\mathbf{s}\neq\mathbf{0}$ the Fourier series representation $$\left[\zeta_{\varLambda}(\mathbf{s},\mathbf{q},\omega)\right]_{a,a'}=\sum_{\mathbf{G}\in\Lambda^{-1}}e^{i\mathbf{\mathbf{G}}\cdot\mathbf{s}}\left[\bar{\zeta}_{\varLambda}(\mathbf{G},\mathbf{q},\omega)\right]_{a,a'},\label{eq: Fourier series representation lattice sum}$$ with $\bar{\zeta}_{\varLambda}(\mathbf{G},\mathbf{q},\omega)$ denoting the Fourier coefficients, see supplemental material [@Supplementary]: $$\begin{aligned}
\left[\bar{\zeta}_{\varLambda}(\mathbf{G},\mathbf{q},\omega)\right]_{a,a'} & = & \frac{1}{\left|C_{\Lambda}\right|}\int_{C_{\Lambda}}d^{3}s\left[e^{-i\mathbf{\mathbf{G}}\cdot\mathbf{s}}\zeta_{\varLambda}(\mathbf{s},\mathbf{q},\omega)\right]_{a,a'}\label{eq: Fourier coefficients lattice sum}\\
& = & \frac{1}{\left|C_{\Lambda}\right|}\mathcal{\bar{G}}_{a,a'}(\mathbf{q}+\mathbf{G},\omega)\nonumber \end{aligned}$$ Insertion of the Fourier series representation (\[eq: Fourier series representation lattice sum\]) into (\[eq:microscopic local field for incident plane wave with small q\]) leads finally to the following result for the microscopic local electric field: $$\tilde{E}_{a}\left(\mathbf{r},\omega\right)=\mathcal{\bar{E}}_{ext,a}\left(\mathbf{q},\omega\right)e^{i\mathbf{q}\cdot\mathbf{r}}+\sum_{a'',a'''}\frac{1}{\left|C_{\Lambda}\right|}\sum_{\mathbf{G}\in\Lambda^{-1}}e^{i\left(\mathbf{q}+\mathbf{G}\right)\cdot\mathbf{r}}\mathcal{\bar{G}}_{a,a''}(\mathbf{q}+\mathbf{G},\omega)\left[\bar{K}_{\Lambda}\left(\mathbf{G},\mathbf{q},\omega\right)\right]_{a'',a'''}\mathcal{\bar{E}}_{ext,a'''}\left(\mathbf{q},\omega\right).\label{eq:eq:microscopic local field Fourier series representation}$$ Here we introduced the kernel $$\left[\bar{K}_{\Lambda}\left(\mathbf{G},\mathbf{q},\omega\right)\right]_{a'',a'''}\equiv\sum_{j',j'',j'''}e^{-i\mathbf{G}\cdot\boldsymbol{\eta}^{\left(j''\right)}}\sum_{a'}\frac{1}{\varepsilon_{0}}\alpha_{a'',a'}^{\left(j'',j'\right)}\left(\mathbf{q},\omega\right)\left(\left[I-\Gamma\left(\mathbf{q},\omega\right)\right]^{-1}\right)_{a',a'''}^{\left(j',j'''\right)},\label{eq: kernel K(G,q,om)}$$ a quantity being directly connected to the microscopic atom-individual polarizabilities, see (\[eq:block matrix alpha\]).
Eq. (\[eq:eq:microscopic local field Fourier series representation\]) constitutes the central result of this section. It explicitely determines the microscopic *local* electric field inside a crystal, that is the electric field that polarizes atoms (ions, molecules) at their positions $\mathbf{R}^{\left(j\right)}=\mathbf{R}+\boldsymbol{\eta}^{\left(j\right)}$ inside a unit cell of the lattice in reaction to an incident plane wave (\[eq:incident external signal\]). While the external field $\tilde{E}_{ext,a}\left(\mathbf{r},\omega\right)$ displays spatial variations on the length scale set by the wavelength $\lambda$ of the incident optical signal in free space, the contributions of the reciprocal lattice vectors $\mathbf{G}\neq\mathbf{0}$ to the Fourier series in ** (\[eq:eq:microscopic local field Fourier series representation\]) make it manifest that the *microscopic* field $\tilde{E}_{a}\left(\mathbf{r},\omega\right)$ displays on the back of the slowly varying envelope $e^{i\mathbf{q}\cdot\mathbf{r}}$ rapid spatial variations on the (atomic) scale set by the lattice constant $a_{\Lambda}\ll\lambda$, see Fig. \[fig:short\_wavelength\_field\_plot\].
![\[fig:short\_wavelength\_field\_plot\]The spatial variation of the microscopic local electric field $\tilde{\mathbf{E}}\left(\mathbf{r},\omega\right)$ representing the solution (\[eq:eq:microscopic local field Fourier series representation\]) to the inhomogenous field-integral equations (\[eq:local field integral equation\]) for an *external* electric field amplitude $\mathbf{\tilde{E}}_{ext}\left(\mathbf{r},\omega_{ext}\right)=\mathbf{e}^{\left(z\right)}\exp\left(i\mathbf{q}\cdot\mathbf{r}\right)\left[\text{\ensuremath{\frac{V}{m}}}\right]$ in the guise of a plane wave, propagating in x - direction and linearly polarized in z - direction. The plot visualizes the rapid spatial variations of the microscopic local electric field traversing a simple cubic crystal along a path $\mathbf{r}\left(x\right)=x\cdot\mathbf{e}^{\left(x\right)}+\frac{a_{\Lambda}}{2}\left(\mathbf{e}^{\left(y\right)}+\mathbf{e}^{\left(z\right)}\right)$ assuming for the external field a vacuum wavelength $\lambda=50\,nm$ in the extreme ultraviolet. A corresponding visualization that applies for visible violet light with vacuum wavelength $\lambda=400\,nm$ is presented in Fig.\[fig:Local\_Macro\_Fields\].](short_wavelength_local_field)
 The spatial variation of the microscopic local electric field $\tilde{\mathbf{E}}\left(\mathbf{r},\omega\right)$ representing the solution (\[eq:eq:microscopic local field Fourier series representation\]) to the inhomogenous field-integral equations (\[eq:local field integral equation\]) for an *external* field $\mathbf{\tilde{E}}_{ext}\left(\mathbf{r},\omega_{ext}\right)=\mathbf{e}^{\left(z\right)}\exp\left(i\mathbf{q}\cdot\mathbf{r}\right)\left[\text{\ensuremath{\frac{V}{m}}}\right]$ in the guise of a plane wave, propagating in x - direction and linearly polarized in z - direction. The plot visualizes the rapid spatial variations of the microscopic local electric field traversing a simple cubic crystal along a path $\mathbf{r}\left(x\right)=x\cdot\mathbf{e}^{\left(x\right)}+\frac{a_{\Lambda}}{2}\left(\mathbf{e}^{\left(y\right)}+\mathbf{e}^{\left(z\right)}\right)$ assuming for the external field a vacuum wavelength $\lambda=400\,nm$ corresponding to visible violet light. The applied parameters in (a) are: $a_{\Lambda}=3.5\text{\AA}$, $\frac{\alpha}{4\pi\varepsilon_{0}}=8{\text{\AA}}^{3}$, $\omega_{\text{ext}}=\frac{2\pi c}{\lambda}$ , $n=3.4256$, $\mathbf{q}=\frac{\omega_{\text{ext}}}{c}n\mathbf{e}^{\left(x\right)}$. The inset zooms to a smaller scale so that the spatial variations of the microscopic local electric field are discernible. (b) Schematic illustration of the path along which the spatial variations of the microscopic local electric field are displayed in (a). (c) Depiction of the maximal electric field strenght $\max_{x}\tilde{E}_{z}\left(\mathbf{r}\left(x\right),\omega_{ext}\right)$ along the path $\mathbf{r}\left(x\right)$ as displayed in (b), varying the modulus $Q$ of the wave vector $\mathbf{Q}=Q\,\mathbf{e}^{\left(x\right)}$ of the external field $\mathbf{\tilde{E}}_{ext}\left(\mathbf{r},\omega_{ext}\right)=\mathbf{e}^{\left(z\right)}\exp\left(i\mathbf{Q}\cdot\mathbf{r}\right)\left[\text{\ensuremath{\frac{V}{m}}}\right]$. As is clearly on view, only solutions to the field-integral equations with wave vector $\left|\mathbf{Q}-\mathbf{q}\right|<\delta Q$ meeting the conditions of tolerance set by the solvability condition of the homogeneous field-integral equations, $\omega_{ext}=\omega_{n}\left(\mathbf{Q}\right)$ as shown in the inset of (c), may propagate with sufficient intensity inside the crystal. By increasing the lattice constant to a value of $a_{\Lambda}=5\text{\AA}$, the density of polarizable atoms as well as the witdh $\delta Q$ of the distribution are decreased considerably, see (d).](total_local_field)
 The spatial variation of the microscopic local electric field $\tilde{\mathbf{E}}\left(\mathbf{r},\omega\right)$ representing the solution (\[eq:eq:microscopic local field Fourier series representation\]) to the inhomogenous field-integral equations (\[eq:local field integral equation\]) for an *external* field $\mathbf{\tilde{E}}_{ext}\left(\mathbf{r},\omega_{ext}\right)=\mathbf{e}^{\left(z\right)}\exp\left(i\mathbf{q}\cdot\mathbf{r}\right)\left[\text{\ensuremath{\frac{V}{m}}}\right]$ in the guise of a plane wave, propagating in x - direction and linearly polarized in z - direction. The plot visualizes the rapid spatial variations of the microscopic local electric field traversing a simple cubic crystal along a path $\mathbf{r}\left(x\right)=x\cdot\mathbf{e}^{\left(x\right)}+\frac{a_{\Lambda}}{2}\left(\mathbf{e}^{\left(y\right)}+\mathbf{e}^{\left(z\right)}\right)$ assuming for the external field a vacuum wavelength $\lambda=400\,nm$ corresponding to visible violet light. The applied parameters in (a) are: $a_{\Lambda}=3.5\text{\AA}$, $\frac{\alpha}{4\pi\varepsilon_{0}}=8{\text{\AA}}^{3}$, $\omega_{\text{ext}}=\frac{2\pi c}{\lambda}$ , $n=3.4256$, $\mathbf{q}=\frac{\omega_{\text{ext}}}{c}n\mathbf{e}^{\left(x\right)}$. The inset zooms to a smaller scale so that the spatial variations of the microscopic local electric field are discernible. (b) Schematic illustration of the path along which the spatial variations of the microscopic local electric field are displayed in (a). (c) Depiction of the maximal electric field strenght $\max_{x}\tilde{E}_{z}\left(\mathbf{r}\left(x\right),\omega_{ext}\right)$ along the path $\mathbf{r}\left(x\right)$ as displayed in (b), varying the modulus $Q$ of the wave vector $\mathbf{Q}=Q\,\mathbf{e}^{\left(x\right)}$ of the external field $\mathbf{\tilde{E}}_{ext}\left(\mathbf{r},\omega_{ext}\right)=\mathbf{e}^{\left(z\right)}\exp\left(i\mathbf{Q}\cdot\mathbf{r}\right)\left[\text{\ensuremath{\frac{V}{m}}}\right]$. As is clearly on view, only solutions to the field-integral equations with wave vector $\left|\mathbf{Q}-\mathbf{q}\right|<\delta Q$ meeting the conditions of tolerance set by the solvability condition of the homogeneous field-integral equations, $\omega_{ext}=\omega_{n}\left(\mathbf{Q}\right)$ as shown in the inset of (c), may propagate with sufficient intensity inside the crystal. By increasing the lattice constant to a value of $a_{\Lambda}=5\text{\AA}$, the density of polarizable atoms as well as the witdh $\delta Q$ of the distribution are decreased considerably, see (d).](Unit-Cell)
 The spatial variation of the microscopic local electric field $\tilde{\mathbf{E}}\left(\mathbf{r},\omega\right)$ representing the solution (\[eq:eq:microscopic local field Fourier series representation\]) to the inhomogenous field-integral equations (\[eq:local field integral equation\]) for an *external* field $\mathbf{\tilde{E}}_{ext}\left(\mathbf{r},\omega_{ext}\right)=\mathbf{e}^{\left(z\right)}\exp\left(i\mathbf{q}\cdot\mathbf{r}\right)\left[\text{\ensuremath{\frac{V}{m}}}\right]$ in the guise of a plane wave, propagating in x - direction and linearly polarized in z - direction. The plot visualizes the rapid spatial variations of the microscopic local electric field traversing a simple cubic crystal along a path $\mathbf{r}\left(x\right)=x\cdot\mathbf{e}^{\left(x\right)}+\frac{a_{\Lambda}}{2}\left(\mathbf{e}^{\left(y\right)}+\mathbf{e}^{\left(z\right)}\right)$ assuming for the external field a vacuum wavelength $\lambda=400\,nm$ corresponding to visible violet light. The applied parameters in (a) are: $a_{\Lambda}=3.5\text{\AA}$, $\frac{\alpha}{4\pi\varepsilon_{0}}=8{\text{\AA}}^{3}$, $\omega_{\text{ext}}=\frac{2\pi c}{\lambda}$ , $n=3.4256$, $\mathbf{q}=\frac{\omega_{\text{ext}}}{c}n\mathbf{e}^{\left(x\right)}$. The inset zooms to a smaller scale so that the spatial variations of the microscopic local electric field are discernible. (b) Schematic illustration of the path along which the spatial variations of the microscopic local electric field are displayed in (a). (c) Depiction of the maximal electric field strenght $\max_{x}\tilde{E}_{z}\left(\mathbf{r}\left(x\right),\omega_{ext}\right)$ along the path $\mathbf{r}\left(x\right)$ as displayed in (b), varying the modulus $Q$ of the wave vector $\mathbf{Q}=Q\,\mathbf{e}^{\left(x\right)}$ of the external field $\mathbf{\tilde{E}}_{ext}\left(\mathbf{r},\omega_{ext}\right)=\mathbf{e}^{\left(z\right)}\exp\left(i\mathbf{Q}\cdot\mathbf{r}\right)\left[\text{\ensuremath{\frac{V}{m}}}\right]$. As is clearly on view, only solutions to the field-integral equations with wave vector $\left|\mathbf{Q}-\mathbf{q}\right|<\delta Q$ meeting the conditions of tolerance set by the solvability condition of the homogeneous field-integral equations, $\omega_{ext}=\omega_{n}\left(\mathbf{Q}\right)$ as shown in the inset of (c), may propagate with sufficient intensity inside the crystal. By increasing the lattice constant to a value of $a_{\Lambda}=5\text{\AA}$, the density of polarizable atoms as well as the witdh $\delta Q$ of the distribution are decreased considerably, see (d).](field_resonance)
 The spatial variation of the microscopic local electric field $\tilde{\mathbf{E}}\left(\mathbf{r},\omega\right)$ representing the solution (\[eq:eq:microscopic local field Fourier series representation\]) to the inhomogenous field-integral equations (\[eq:local field integral equation\]) for an *external* field $\mathbf{\tilde{E}}_{ext}\left(\mathbf{r},\omega_{ext}\right)=\mathbf{e}^{\left(z\right)}\exp\left(i\mathbf{q}\cdot\mathbf{r}\right)\left[\text{\ensuremath{\frac{V}{m}}}\right]$ in the guise of a plane wave, propagating in x - direction and linearly polarized in z - direction. The plot visualizes the rapid spatial variations of the microscopic local electric field traversing a simple cubic crystal along a path $\mathbf{r}\left(x\right)=x\cdot\mathbf{e}^{\left(x\right)}+\frac{a_{\Lambda}}{2}\left(\mathbf{e}^{\left(y\right)}+\mathbf{e}^{\left(z\right)}\right)$ assuming for the external field a vacuum wavelength $\lambda=400\,nm$ corresponding to visible violet light. The applied parameters in (a) are: $a_{\Lambda}=3.5\text{\AA}$, $\frac{\alpha}{4\pi\varepsilon_{0}}=8{\text{\AA}}^{3}$, $\omega_{\text{ext}}=\frac{2\pi c}{\lambda}$ , $n=3.4256$, $\mathbf{q}=\frac{\omega_{\text{ext}}}{c}n\mathbf{e}^{\left(x\right)}$. The inset zooms to a smaller scale so that the spatial variations of the microscopic local electric field are discernible. (b) Schematic illustration of the path along which the spatial variations of the microscopic local electric field are displayed in (a). (c) Depiction of the maximal electric field strenght $\max_{x}\tilde{E}_{z}\left(\mathbf{r}\left(x\right),\omega_{ext}\right)$ along the path $\mathbf{r}\left(x\right)$ as displayed in (b), varying the modulus $Q$ of the wave vector $\mathbf{Q}=Q\,\mathbf{e}^{\left(x\right)}$ of the external field $\mathbf{\tilde{E}}_{ext}\left(\mathbf{r},\omega_{ext}\right)=\mathbf{e}^{\left(z\right)}\exp\left(i\mathbf{Q}\cdot\mathbf{r}\right)\left[\text{\ensuremath{\frac{V}{m}}}\right]$. As is clearly on view, only solutions to the field-integral equations with wave vector $\left|\mathbf{Q}-\mathbf{q}\right|<\delta Q$ meeting the conditions of tolerance set by the solvability condition of the homogeneous field-integral equations, $\omega_{ext}=\omega_{n}\left(\mathbf{Q}\right)$ as shown in the inset of (c), may propagate with sufficient intensity inside the crystal. By increasing the lattice constant to a value of $a_{\Lambda}=5\text{\AA}$, the density of polarizable atoms as well as the witdh $\delta Q$ of the distribution are decreased considerably, see (d).](field_resonance_thin_medium)
The result (\[eq:microscopic local field for incident plane wave with small q\]) also reveals that the strength of the microscopic local field amplitude inside the crystal strongly depends on the choice of the propagation direction $\mathbf{\hat{q}}=\mathbf{\frac{q}{\left|\mathbf{q}\right|}}$, a feature being directly connected to the photonic band structure implicitely encoded in the eigenvalues of the matrix $\Gamma\left(\mathbf{q},\omega\right)$. The huge size of the induced electric field strength, represented by the difference $\tilde{E}_{a}\left(\mathbf{r},\omega\right)-\tilde{E}_{ext,a}\left(\mathbf{r},\omega\right)$, can indeed be prorated to the predominant longitudinal character of the microscopic local electric field (\[eq:microscopic local field for incident plane wave with small q\]) inside a probe volume with a high density of polarizable atoms (ions, molecules). This is intuitively accessible in view of the quasi static electric field inside a material probe at a point $\mathbf{r}\in\Omega_{P}$ originating from nearby positioned induced atomic dipoles.
To elucidate the nature of the microscopic local electric field in (\[eq:microscopic local field for incident plane wave with small q\]) and (\[eq:eq:microscopic local field Fourier series representation\]) respectively, let us decompose it into *longitudinal* and *transversal* parts. With (\[eq:eq:microscopic local field Fourier series representation\]) we write now $$\begin{aligned}
\tilde{E}_{a}^{\left(L,T\right)}\left(\mathbf{r},\omega\right) & = & \sum_{a'}\int d^{3}r'\Pi_{a,a'}^{\left(L,T\right)}\left(\mathbf{r}'\right)\tilde{E}_{a'}\left(\mathbf{r}-\mathbf{r}',\omega\right)\\
& = & \sum_{a'}\bar{\Pi}_{a,a'}^{\left(L,T\right)}\left(\mathbf{q}\right)\mathcal{\bar{E}}_{ext,a'}\left(\mathbf{q},\omega\right)e^{i\mathbf{q}\cdot\mathbf{r}}\nonumber \\
& & +\sum_{a',a'',a'''}\frac{1}{\left|C_{\Lambda}\right|}\sum_{\mathbf{G}\in\Lambda^{-1}}e^{i\left(\mathbf{q}+\mathbf{G}\right)\cdot\mathbf{r}}\bar{\Pi}_{a,a'}^{\left(L,T\right)}\left(\mathbf{q}+\mathbf{G}\right)\mathcal{\bar{G}}_{a',a''}(\mathbf{q}+\mathbf{G},\omega)\left[\bar{K}_{\Lambda}\left(\mathbf{G},\mathbf{q},\omega\right)\right]_{a'',a'''}\mathcal{\bar{E}}_{ext,a'''}\left(\mathbf{q},\omega\right).\nonumber \end{aligned}$$ Taking into account (\[eq: Fourier transform of 3x3 matrix propagator\]) we have $$\begin{aligned}
\sum_{a'}\bar{\Pi}_{a,a'}^{\left(L\right)}\left(\mathbf{q}+\mathbf{G}\right)\mathcal{\bar{G}}_{a',a''}(\mathbf{q}+\mathbf{G},\omega) & = & -\frac{\left(\mathbf{q}+\mathbf{G}\right)_{a}\left(\mathbf{q}+\mathbf{G}\right)_{a''}}{\left|\mathbf{q}+\mathbf{G}\right|^{2}}\\
\sum_{a'}\bar{\Pi}_{a,a'}^{\left(T\right)}\left(\mathbf{q}+\mathbf{G}\right)\mathcal{\bar{G}}_{a',a''}(\mathbf{q}+\mathbf{G},\omega) & = & \frac{\omega^{2}}{c^{2}}\frac{\delta_{a,a''}-\frac{\left(\mathbf{q}+\mathbf{G}\right)_{a}\left(\mathbf{q}+\mathbf{G}\right)_{a''}}{\left|\mathbf{q}+\mathbf{G}\right|^{2}}}{\left|\mathbf{\mathbf{q}}+\mathbf{G}\right|^{2}-\frac{\omega^{2}}{c^{2}}-i0^{+}}.\end{aligned}$$ The searched-for longitudinal and transversal parts of the microscopic local electric field are thus explicitely determined: $$\begin{aligned}
\tilde{E}_{a}^{\left(L\right)}\left(\mathbf{r},\omega\right) & = & \mathcal{\bar{E}}_{ext,a}^{\left(L\right)}\left(\mathbf{q},\omega\right)e^{i\mathbf{q}\cdot\mathbf{r}}\label{eq: longitudinal microscopic field}\\
& & -\sum_{a'',a'''}\frac{1}{\left|C_{\Lambda}\right|}\sum_{\mathbf{G}\in\Lambda^{-1}}e^{i\left(\mathbf{q}+\mathbf{G}\right)\cdot\mathbf{r}}\frac{\left(\mathbf{q}+\mathbf{G}\right)_{a}\left(\mathbf{q}+\mathbf{G}\right)_{a''}}{\left|\mathbf{q}+\mathbf{G}\right|^{2}}\left[\bar{K}_{\Lambda}\left(\mathbf{G},\mathbf{q},\omega\right)\right]_{a'',a'''}\mathcal{\bar{E}}_{ext,a'''}\left(\mathbf{q},\omega\right)\nonumber \\
\tilde{E}_{a}^{\left(T\right)}\left(\mathbf{r},\omega\right) & = & \mathcal{\bar{E}}_{ext,a}^{\left(T\right)}\left(\mathbf{q},\omega\right)e^{i\mathbf{q}\cdot\mathbf{r}}\label{eq:transversal microscopic field}\\
& & +\sum_{a'',a'''}\frac{1}{\left|C_{\Lambda}\right|}\sum_{\mathbf{G}\in\Lambda^{-1}}e^{i\left(\mathbf{q}+\mathbf{G}\right)\cdot\mathbf{r}}\frac{\omega^{2}}{c^{2}}\frac{\delta_{a,a''}-\frac{\left(\mathbf{q}+\mathbf{G}\right)_{a}\left(\mathbf{q}+\mathbf{G}\right)_{a''}}{\left|\mathbf{q}+\mathbf{G}\right|^{2}}}{\left|\mathbf{\mathbf{q}}+\mathbf{G}\right|^{2}-\frac{\omega^{2}}{c^{2}}-i0^{+}}\left[\bar{K}_{\Lambda}\left(\mathbf{G},\mathbf{q},\omega\right)\right]_{a'',a'''}\mathcal{\bar{E}}_{ext,a'''}\left(\mathbf{q},\omega\right)\nonumber \end{aligned}$$ . It is important to realize that the longitudinal component $\tilde{E}_{a}^{\left(L\right)}\left(\mathbf{r},\omega\right)$ increases rapidly as the density of polarizable atoms (ions, molecules) in the probe volume increases, see Fig.\[fig:field\_contributions\_and\_index\].
 The spatial variation of the z-component $\tilde{E}_{z}^{(L)}\left(\mathbf{r},\omega\right)$ of the longitudinal part of the microscopic local electric field (magenta) and the spatial variation of the z-component $\tilde{E}_{z}^{(T)}\left(\mathbf{r},\omega\right)$ of the transversal part of the microscopic local electric field (green) corresponding to the same parameters as in Fig. \[fig:Local\_Macro\_Fields\]. The local field $\tilde{E}_{z}\left(\mathbf{r},\omega\right)=\tilde{E}_{z}^{\left(L\right)}\left(\mathbf{r},\omega\right)+\tilde{E}_{z}^{\left(T\right)}\left(\mathbf{r},\omega\right)$ is for a crystal of high particle density (high refractive index), as depicted in (b), dominated by its static longitudinal (dipole) part $\tilde{E}_{z}^{\left(L\right)}\left(\mathbf{r},\omega\right)$, while the tranversal part $\tilde{E}_{z}^{\left(T\right)}\left(\mathbf{r},\omega\right)$ is distinctly smaller. ](local_field_projections)
 The spatial variation of the z-component $\tilde{E}_{z}^{(L)}\left(\mathbf{r},\omega\right)$ of the longitudinal part of the microscopic local electric field (magenta) and the spatial variation of the z-component $\tilde{E}_{z}^{(T)}\left(\mathbf{r},\omega\right)$ of the transversal part of the microscopic local electric field (green) corresponding to the same parameters as in Fig. \[fig:Local\_Macro\_Fields\]. The local field $\tilde{E}_{z}\left(\mathbf{r},\omega\right)=\tilde{E}_{z}^{\left(L\right)}\left(\mathbf{r},\omega\right)+\tilde{E}_{z}^{\left(T\right)}\left(\mathbf{r},\omega\right)$ is for a crystal of high particle density (high refractive index), as depicted in (b), dominated by its static longitudinal (dipole) part $\tilde{E}_{z}^{\left(L\right)}\left(\mathbf{r},\omega\right)$, while the tranversal part $\tilde{E}_{z}^{\left(T\right)}\left(\mathbf{r},\omega\right)$ is distinctly smaller. ](local_field_projections_zoom)
 Plot of the ratio of maximum field strengths $\frac{\max_{x}\left|\tilde{E}_{z}^{(L,T)}\left(\mathbf{r}\left(x\right),\omega\right)\right|}{\max_{x}\left|\tilde{E}_{z}\left(\mathbf{r}\left(x\right),\omega\right)\right|}$ vs. particle density $\nu_{P}$ with parameters like in Fig.\[fig:Local\_Macro\_Fields\]. Near to the border of instability at $\nu=\nu_{P}^{\left(c\right)}$ the transversal (radiative) amplitude $\tilde{E}_{z}^{(T)}\left(\mathbf{r},\omega\right)$ is strongly suppressed while the longitudinal amplitude $\tilde{E}_{z}^{\left(L\right)}\left(\mathbf{r},\omega\right)$ becomes large. Conversely, at low particle density $\nu_{P}\rightarrow0$ the longitudinal amplitude $\tilde{E}_{z}^{\left(L\right)}\left(\mathbf{r},\omega\right)$ vanishes while the transversal amplitude essentially coincides with the field amplitude of the external field. (b) The variation of the index of refraction $n\left(\omega\right)$ vs. the density $\nu_{P}$ of polarizable atoms (ions, molecules) in a dielectric crystal. At the borderline of stability the index of refraction $n\left(\omega\right)$ displays a singularity as $\nu_{P}$ approaches (from below) the critical density $\nu_{P}^{\left(c\right)}$.](projected_field_contributions)
 Plot of the ratio of maximum field strengths $\frac{\max_{x}\left|\tilde{E}_{z}^{(L,T)}\left(\mathbf{r}\left(x\right),\omega\right)\right|}{\max_{x}\left|\tilde{E}_{z}\left(\mathbf{r}\left(x\right),\omega\right)\right|}$ vs. particle density $\nu_{P}$ with parameters like in Fig.\[fig:Local\_Macro\_Fields\]. Near to the border of instability at $\nu=\nu_{P}^{\left(c\right)}$ the transversal (radiative) amplitude $\tilde{E}_{z}^{(T)}\left(\mathbf{r},\omega\right)$ is strongly suppressed while the longitudinal amplitude $\tilde{E}_{z}^{\left(L\right)}\left(\mathbf{r},\omega\right)$ becomes large. Conversely, at low particle density $\nu_{P}\rightarrow0$ the longitudinal amplitude $\tilde{E}_{z}^{\left(L\right)}\left(\mathbf{r},\omega\right)$ vanishes while the transversal amplitude essentially coincides with the field amplitude of the external field. (b) The variation of the index of refraction $n\left(\omega\right)$ vs. the density $\nu_{P}$ of polarizable atoms (ions, molecules) in a dielectric crystal. At the borderline of stability the index of refraction $n\left(\omega\right)$ displays a singularity as $\nu_{P}$ approaches (from below) the critical density $\nu_{P}^{\left(c\right)}$.](refractive_index)
Photonic Bandstructure\[sec:Photonic-Bandstructure\] {#photonic-bandstructuresecphotonic-bandstructure .unnumbered}
----------------------------------------------------
If no external field was incident, i.e. for $\tilde{\mathfrak{e}}_{ext,a}^{\left(j\right)}\left(\mathbf{q},\omega\right)\equiv0$, a non trivial field amplitude $\tilde{\mathfrak{e}}_{a}^{\left(j\right)}\left(\mathbf{q},\omega\right)$ solving the homogenous system of equations (\[eq: system of 3Mx3M equations for amplitudes of local field\]) is obviously identical to an eigenvector $\tilde{\mathfrak{v}}_{a,n}^{\left(j\right)}\left(\mathbf{q},\omega\right)$ associated with the *special* eigenvalue $$\gamma_{n}\left(\mathbf{q},\omega\right)=1\label{eq: eigenvalue unity}$$ of the eigenvalue problem $$\sum_{j'}\sum_{a'}\Gamma_{a,a'}^{\left(j,j'\right)}\left(\mathbf{k},\omega\right)\tilde{\mathfrak{v}}_{a',n}^{\left(j'\right)}\left(\mathbf{q},\omega\right)=\gamma_{n}\left(\mathbf{q},\omega\right)\tilde{\mathfrak{v}}_{a,n}^{\left(j\right)}\left(\mathbf{q},\omega\right).\label{eq: eigenvalue problem photonic bandstructure}$$ The dispersion relation of photons, i.e. the photonic bandstructure $\omega_{n}\left(\mathbf{q}\right)$, can now be readily determined for any number $M$ of basis atoms inside the unit cell $C_{\Lambda}$ by first solving (numerically) the eigenvalue problem (\[eq: eigenvalue problem photonic bandstructure\]) for a given wave vector $\mathbf{q}\in C_{\varLambda^{-1}}$ as a function of $\omega$, thus obtaining a family of $3M$ eigenvalue curves $\gamma_{n}\left(\mathbf{q},\omega\right)$ vs. $\omega$, and then solving (numerically) for $n=1,2,3,...3M$ the implicit equations (\[eq: eigenvalue unity\]) for the unknown $\omega$ so that $$\left[\gamma_{n}\left(\mathbf{q},\omega\right)-1\right]_{\omega\rightarrow\omega_{n}}=0.\label{eq: requirement determining photonic bandstructure}$$ Varying then the wavevector $\mathbf{q}$ along (widely) different symmetry lines inside the Brillouin zone $C_{\Lambda^{-1}}$ various pieces of the photonic bandstructure $\omega_{n}\left(\mathbf{q}\right)$ emerge, as is exemplarily displayed for the diamond lattice ($M=2$) in Fig. \[fig:diamond-lattice\](a). Like electrons moving in a periodic potential also electromagnetic waves propagating in a crystal are governed by a bandstructure, for example [@Leung1990; @Zhang1990; @Soezueer1993]. While the wave function for electrons (discarding spin-orbit forces and Zeeman splitting ) is a scalar, propagating electromagnetic waves are vectorfields. Incident lightsignals composed of frequencies within an omni-directional band gap will be reflected from such a crystal irrespective of the light source being polarized or unpolarized, which is interesting for technical applications, for instance dielectric mirrors, filters or antenna-substrates.
 Photonic band structure $\omega_{n}\left(\mathbf{q}\right)$ in units of $\omega_{0}=\frac{2\pi c}{a_{\Lambda}}$ as calculated from (\[eq: eigenvalue problem photonic bandstructure\]) for a diamond lattice with wave vector $\mathbf{q}$ orientated along various symmetry lines of the Brillouin zone choosing a lattice constant $a_{\Lambda}=6\text{\AA}$ and assuming a static electronic polarizability $\frac{\alpha_{0}}{4\pi\varepsilon_{0}}=3{\text{\AA}}^{3}$. (b) Taking radiation damping into account, the meaning of the red dots being explained in the supplemental material [@Supplementary], the photonic band structure $\omega_{n}\left(\mathbf{q}\right)$ of a dielectric crystalline material ceases to make sense approaching the boundary of the Brillouin zone, where $\omega_{n}\left(\mathbf{q}\right)$ enters into the soft x-ray regime. ](PBS-diamond)
 Photonic band structure $\omega_{n}\left(\mathbf{q}\right)$ in units of $\omega_{0}=\frac{2\pi c}{a_{\Lambda}}$ as calculated from (\[eq: eigenvalue problem photonic bandstructure\]) for a diamond lattice with wave vector $\mathbf{q}$ orientated along various symmetry lines of the Brillouin zone choosing a lattice constant $a_{\Lambda}=6\text{\AA}$ and assuming a static electronic polarizability $\frac{\alpha_{0}}{4\pi\varepsilon_{0}}=3{\text{\AA}}^{3}$. (b) Taking radiation damping into account, the meaning of the red dots being explained in the supplemental material [@Supplementary], the photonic band structure $\omega_{n}\left(\mathbf{q}\right)$ of a dielectric crystalline material ceases to make sense approaching the boundary of the Brillouin zone, where $\omega_{n}\left(\mathbf{q}\right)$ enters into the soft x-ray regime. ](PBS-diamond-Damping)
Further examples of photonic band structures calculated from (\[eq: eigenvalue problem photonic bandstructure\]) and (\[eq: requirement determining photonic bandstructure\]) with our method, exemplarily for sc-, fcc- and bcc-lattices ($M=1$), are given in supplemental material [@Supplementary]. Our results comply well with calculations carried out within the frame of the Fano-Hopfield model [@Klugkist2006; @Antezza2009] in the case of weak coupling of the oscillators. Earlier calculations for super-lattices on the basis of the macroscopic Maxwell equations [@Leung1990; @Zhang1990; @Soezueer1993], carried out in a basis of plane waves $e^{i\left(\mathbf{q}+\mathbf{G}\right)\mathbf{r}}$, require for $\mathbf{q}$ fixed a large number $N$ of reciprocal lattice vectors $\mathbf{G}$ to be taken into account, for accurate computations typically $N\geq2000$ [@Hermann2001], thus leading to a huge $3N\times3N$ matrix eigenvalue problem for the modes and mode freqencies. Unfortunately, these calculations suffer from an inconsistency, as a number $N$ of spurious zero eigenvalue modes needs explicit elimination by an ad hoc transversality constraint. Because in our approach the extension of the scatterers is tiny compared to the lattice constant, our results cease agreement with theirs in certain details of the photonic bandstructure at high photon frequency, while for optical frequencies and below our results are in full agreement with theirs. Regarding the calculational cost of our approach: in the case of sc-, fcc- and bcc-lattices with one atom in the unit cell we solve (for each $\mathbf{q}$-vector) a $3\times3$ eigenvalue problem, and in the case of diamond with two atoms in the unit cell then a $6\times6$ eigenvalue problem. Parameters of interest to propagation of light pulses, for instance group velocity and effective photon mass, can be determined conveniently using $k\cdot p$-perturbation theory [@Hermann2001], a method often employed in solid state electronic band structure theory [@Ashcroft1981].
Finally, it should be considered that higher bands $n>1$ of the photonic band structure $\omega_{n}\left(\mathbf{q}\right)$ in real crystalline materials are not credibly calculable. While the concept of a photonic band structure exhibiting many band branches $\omega_{n}\left(\mathbf{q}\right)$ certainly applies for (artificial) superlattice structures with large mesoscopic lattice constant $a_{\Lambda}\gg a$, it can be accepted only with reserve for a real dielectric material. This is because for a microscopic lattice constant $a_{\Lambda}\simeq a$ photon frequencies above $\omega\geq\omega_{\Lambda}=c\times\frac{\pi}{a_{\Lambda}}$ are far and beyond the ultra-violet. In this case the effects of radiation damping, as represented by the imaginary part of the lattice sums, $$\mathtt{Im}\left[\zeta_{\varLambda}^{\left(0\right)}(\mathbf{q},\omega)\right]_{aa'}=-\frac{1}{6\pi}\left(\frac{\omega}{c}\right)^{3}\delta_{a,a'},\label{eq: Im Zeta_0}$$ can no longer be neglected. For a derivation of (\[eq: Im Zeta\_0\]) see supplemental material [@Supplementary]. In Fig.\[fig:diamond-lattice\](b) the effect of taking into account radiation damping is exemplarily shown for the diamond lattice ($M=2$). The corresponding plots revealing the influence of radiation damping for the sc-, fcc- and bcc-lattices ($M=1$) we also present in [@Supplementary]. Of course, if the wavelength of the external electromagnetic field is ultra short then the point dipole ansatz (\[eq: phenomenological dielectric kernel\]) for the dielectric susceptibility should be extended to include also magnetic dipoles and electric quadrupoles on equal footing [@Raab2005].
The Dielectric Tensor of Macroscopic Electrodynamics \[sec:Macroscopic-Electric-Field-and-dielectric-function\]
===============================================================================================================
The *macroscopic* electric field $\mathcal{\tilde{E}}_{a}\left(\mathbf{r},\omega\right)$ inside the probe we conceive as a low-pass filter applied to the Fourier-transformation $\bar{E}_{a}\left(\mathbf{q},\omega\right)$ of the spatially rapidly varying microscopic local electric field $\tilde{E}_{a}\left(\mathbf{r},\omega\right)$. Introducing a cut-off wavenumber $q_{c}$ so that $q<q_{c}$ implies $\mathbf{q}\in C_{\Lambda^{-1}}$, then $$\begin{aligned}
\mathcal{\tilde{E}}_{a}\left(\mathbf{r},\omega\right) & = & \int\frac{d^{3}q}{\left(2\pi\right)^{3}}e^{i\mathbf{q}\cdot\mathbf{r}}\Theta\left(q_{c}-\left|\mathbf{q}\right|\right)\bar{E}_{a}\left(\mathbf{q},\omega\right)\label{eq: low pass filtered microscopic electric field}\\
\bar{E}_{a}\left(\mathbf{q},\omega\right) & = & \frac{1}{\left|\Omega_{P}\right|}\int_{\Omega_{P}}d^{3}re^{-i\mathbf{q}\cdot\mathbf{r}}\tilde{E}_{a}\left(\mathbf{r},\omega\right).\nonumber \end{aligned}$$ Likewise, the *macroscopic* polarization $\mathcal{\tilde{P}}_{a}\left(\mathbf{r},\omega\right)$ inside the probe we conceive as a low-pass filter applied to the Fourier-transformation $\bar{P}_{a}\left(\mathbf{q},\omega\right)$ of the microscopic polarization $\tilde{P}_{a}\left(\mathbf{r},\omega\right)$, as defined in (\[eq:polarization vs local field\]): $$\begin{aligned}
\mathcal{\tilde{P}}_{a}\left(\mathbf{r},\omega\right) & = & \int\frac{d^{3}q}{\left(2\pi\right)^{3}}e^{i\mathbf{q\cdot}\mathbf{r}}\Theta\left(q_{c}-\left|\mathbf{q}\right|\right)\bar{P}_{a}\left(\mathbf{q},\omega\right)\label{eq: low pass filtered microscopic polarization}\\
\bar{P}_{a}\left(\mathbf{q},\omega\right) & = & \frac{1}{\left|\Omega_{P}\right|}\int_{\Omega_{P}}d^{3}re^{-i\mathbf{q}\cdot\mathbf{r}}\tilde{P}_{a}\left(\mathbf{r},\omega\right)\nonumber \end{aligned}$$ Thus in the long wavelength limit the Fourier components $\bar{\mathcal{E}}_{a}\left(\mathbf{q},\omega\right)$ of the macroscopic field $\tilde{\mathcal{E}}_{a}\left(\mathbf{r},\omega\right)$ coincide with those of the microscopic local field, and the Fourier components $\mathcal{\bar{P}}_{a}\left(\mathbf{q},\omega\right)$ of the macroscopic polarization $\mathcal{\tilde{P}}_{a}\left(\mathbf{r},\omega\right)$ coincide with those of the microscopic polarization: $$\begin{aligned}
\mathcal{\bar{E}}_{a}\left(\mathbf{q},\omega\right) & = & \Theta\left(q_{c}-\left|\mathbf{q}\right|\right)\bar{E}_{a}\left(\mathbf{q},\omega\right)\\
\mathcal{\bar{P}}_{a}\left(\mathbf{q},\omega\right) & = & \Theta\left(q_{c}-\left|\mathbf{q}\right|\right)\bar{P}_{a}\left(\mathbf{q},\omega\right)\end{aligned}$$ Restricting to $\left|\mathbf{q}\right|<q_{c}$ we now define the dielectric $3\times3$ tensor $\left[\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)\right]_{aa'}$ of a crystalline dielectric by requiring the macroscopic polarization being proportional to the macroscopic electric field: $$\mathcal{\bar{P}}_{a}\left(\mathbf{q},\omega\right)=\varepsilon_{0}\sum_{a'\in\left\{ x,y,z\right\} }\left[\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)-I\right]_{aa'}\mathcal{\bar{E}}_{a'}\left(\mathbf{q},\omega\right)\label{eq: implicit definition dielectric function}$$ Insertion of (\[eq:eq:microscopic local field Fourier series representation\]) gives an explicit expression determining the Fourier amplitude of the microscopic local electric field: $$\begin{aligned}
\bar{E}_{a}\left(\mathbf{k},\omega\right) & = & \frac{1}{\left|\Omega_{P}\right|}\int_{\Omega_{P}}d^{3}re^{-i\mathbf{k}\cdot\mathbf{r}}\tilde{E}_{a}\left(\mathbf{r},\omega\right)\label{eq: Fourier transform microscopic local field}\\
& = & \mathcal{\bar{E}}_{ext,a}\left(\mathbf{q},\omega\right)\delta_{\mathbf{k},\mathbf{q}}+\sum_{a'',a'''}\frac{1}{\left|C_{\Lambda}\right|}\sum_{\mathbf{G}'\in\Lambda^{-1}}\delta_{\mathbf{k},\mathbf{q}+\mathbf{G}'}\mathcal{\bar{G}}_{a,a''}(\mathbf{q}+\mathbf{G}',\omega)\left[\bar{K}_{\Lambda}\left(\mathbf{G}',\mathbf{q},\omega\right)\right]_{a'',a'''}\mathcal{\bar{E}}_{ext,a'''}\left(\mathbf{q},\omega\right)\nonumber \end{aligned}$$ Decomposing $\mathbf{k}=\mathbf{k}_{0}+\mathbf{G}$ with $\mathbf{k}_{0}\in C_{\Lambda^{-1}}$ and $\mathbf{G}\in\Lambda^{-1}$ there holds for $\mathbf{q}\in C_{\Lambda^{-1}}$ and $\mathbf{G}'\in\Lambda^{-1}$ $$\delta_{\mathbf{k},\mathbf{q}+\mathbf{G}'}=\delta_{\mathbf{k}_{0},\mathbf{q}}\delta_{\mathbf{G},\mathbf{G}'}.$$ Therefore $$\begin{aligned}
\bar{E}_{a}\left(\mathbf{k},\omega\right) & = & \delta_{\mathbf{k}_{0},\mathbf{q}}\bar{E}_{a}\left(\mathbf{q}+\mathbf{G},\omega\right)\\
\bar{E}_{a}\left(\mathbf{q}+\mathbf{G},\omega\right) & = & \mathcal{\bar{E}}_{ext,a}\left(\mathbf{q},\omega\right)\delta_{\mathbf{G},\mathbf{0}}+\sum_{a'',a'''}\frac{1}{\left|C_{\Lambda}\right|}\mathcal{\bar{G}}_{a,a''}(\mathbf{q}+\mathbf{G},\omega)\left[\bar{K}_{\Lambda}\left(\mathbf{G},\mathbf{q},\omega\right)\right]_{a'',a'''}\mathcal{\bar{E}}_{ext,a'''}\left(\mathbf{q},\omega\right)\:.\label{eq:Fourier transform microscopic local field II}\end{aligned}$$ Let us abbreviate for $\mathbf{G}=\mathbf{0}$: $$\left[\bar{K}_{\Lambda}(\mathbf{q},\omega)\right]_{a,a''}\equiv\left[\bar{K}_{\Lambda}(\mathbf{0},\mathbf{q},\omega)\right]_{a,a''}=\sum_{1\leq j,j''\leq M}\left(\frac{1}{\varepsilon_{0}}\alpha\left(\mathbf{q},\omega\right)\circ\left[I-\Gamma\left(\mathbf{q},\omega\right)\right]^{-1}\right)_{a,a''}^{\left(j,j''\right)}\label{eq: kernel K(G=00003D0,q,om)}$$ Then for $\mathbf{q}\in C_{\Lambda^{-1}}$: $$\begin{aligned}
\bar{E}_{a}\left(\mathbf{q},\omega\right) & = & \sum_{a'}\left[I+\frac{1}{\left|C_{\Lambda}\right|}\bar{\mathcal{G}}(\mathbf{q},\omega)\circ\bar{K}_{\Lambda}(\mathbf{q},\omega)\right]_{a,a'}\mathcal{\bar{E}}_{ext,a'}\left(\mathbf{q},\omega\right)\label{eq: Fourier transform microscopic E- field}\end{aligned}$$ On the other hand there holds keeping in mind the restriction $\mathbf{q}\in C_{\Lambda^{-1}}$: $$\begin{aligned}
\bar{P}_{a}\left(\mathbf{q},\omega\right) & = & \varepsilon_{0}\frac{1}{\left|\Omega_{P}\right|}\int_{\Omega_{P}}d^{3}re^{-i\mathbf{q}\cdot\mathbf{r}}\int_{\Omega_{P}}d^{3}r'\sum_{a'}\tilde{\chi}_{aa'}\left(\mathbf{r},\mathbf{r}',\omega\right)\tilde{E}_{a'}\left(\mathbf{r}',\omega\right)\label{eq: Fourier transform microscopic polarization}\\
& = & \frac{\varepsilon_{0}}{\left|C_{\Lambda}\right|}\sum_{a'}\left[\bar{K}_{\Lambda}(\mathbf{q},\omega)\right]_{a,a'}\mathcal{\bar{E}}_{ext,a'}\left(\mathbf{q},\omega\right)\nonumber \end{aligned}$$ Restricting to the long wavelength limit $\left|\mathbf{q}\right|<q_{c}$ and thus identifying $\mathcal{\bar{P}}_{a}\left(\mathbf{q},\omega\right)=\bar{P}_{a}\left(\mathbf{q},\omega\right)$ and $\mathcal{\bar{E}}_{a}\left(\mathbf{q},\omega\right)=\bar{E}_{a}\left(\mathbf{q},\omega\right)$, and then combining (\[eq: Fourier transform microscopic E- field\]) and (\[eq: implicit definition dielectric function\]), a conditional equation determining the macroscopic dielectric tensor $\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)$ is found $$\begin{aligned}
\bar{\mathcal{P}}_{a}\left(\mathbf{q},\omega\right) & = & \varepsilon_{0}\sum_{a'}\left[\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)-I\right]_{aa'}\mathcal{\bar{E}}_{a'}\left(\mathbf{q},\omega\right)\\
& = & \varepsilon_{0}\sum_{a',a''}\left[\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)-I\right]_{aa'}\left[I+\frac{1}{\left|C_{\Lambda}\right|}\bar{\mathcal{G}}(\mathbf{q},\omega)\circ\bar{K}_{\Lambda}(\mathbf{q},\omega)\right]_{a',a''}\mathcal{\bar{E}}_{ext,a''}\left(\mathbf{q},\omega\right)\\
& \stackrel{!}{=} & \frac{\varepsilon_{0}}{\left|C_{\Lambda}\right|}\sum_{a'}\left[\bar{K}_{\Lambda}(\mathbf{q},\omega)\right]_{a,a'}\mathcal{\bar{E}}_{ext,a'}\left(\mathbf{q},\omega\right).\end{aligned}$$ Insisting both lines should be identical for any external field amplitude $\mathcal{\bar{E}}_{ext,a}\left(\mathbf{q},\omega\right)$ immediately leads (with help of elementary matrix algebra) to the identification $$\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)-I=\frac{1}{\left|C_{\Lambda}\right|}\bar{K}_{\Lambda}(\mathbf{q},\omega)\circ\left[I+\frac{1}{\left|C_{\Lambda}\right|}\bar{\mathcal{G}}(\mathbf{q},\omega)\circ\bar{K}_{\Lambda}(\mathbf{q},\omega)\right]^{-1}.\label{eq: macroscopic dielectric function arbitrary M}$$ This is a central result. The macroscopic dielectric tensor $\left[\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)\right]_{aa'}$ is solely determined by the lattice sums $\tilde{\zeta}_{\varLambda}(\mathbf{s},\mathbf{q},\omega)$ of the Bravais lattice $\Lambda$ under consideration and the individual polarizations $\alpha_{a'',a'}\left(\boldsymbol{\eta}^{\left(j''\right)},\boldsymbol{\eta}^{\left(j'\right)},\omega\right)$ of the atoms (ions) inside the unit cell. As a test of the analytic structure of the dielectric function $\varepsilon_{\Lambda}\left(\mathbf{q}=\mathbf{0},\omega\right)$ in the complex frequency domain we checked the Lyddane-Sachs-Teller relation, see Fig. \[fig:Lyddane-Sachs-Teller\]. In agreement with general considerations under $\omega\rightarrow-\omega$ the real part of $\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)$ is an even function of $\omega$ and the imaginary part is an odd one.
![\[fig:Lyddane-Sachs-Teller\]Plot of real and imaginary parts of dielectric function $\varepsilon_{\Lambda}\left(\mathbf{q}=\mathbf{0},\omega\right)\equiv\varepsilon_{\Lambda}\left(\omega\right)$ as calculated from (\[eq: macroscopic dielectric function arbitrary M\]) for $CsI$ with microscopic polarizabilities with the parameters from Table \[fit\]. The ionic polarizability with frequency dependence given by (\[eq: isotropic Lorentz polarizability off diagonal\]) also includes a small damping parameter $\gamma>0$. The roots of $\text{Re}\left[\varepsilon_{\Lambda}\left(\omega\right)\right]$ are: $\omega_{T}=16.08$THz and $\omega_{L}=19.53$THz, corresponding to frequencies of the transversel and longitudinal optical modes respectively. The zeros of $\text{Re}\left[\varepsilon_{\Lambda}\left(\omega\right)\right]$ obey to the Lyddane-Sachs-Teller relation [@Lyddane1941], connecting the value of the static dielectric function $\varepsilon_{\Lambda}^{(0)}=4.45$ to its value $\varepsilon_{\Lambda}^{(\infty)}\equiv\varepsilon_{\Lambda}\left(\omega_{T}\cdot10^{2}\right)=3.05$ above and beyond all optical phonon frequencies: $\frac{\omega_{L}^{2}}{\omega_{T}^{2}}=\frac{\varepsilon_{\Lambda}^{(0)}}{\varepsilon_{\Lambda}^{(\infty)}}$ .](Lyddane-Sachs-Teller)
Macroscopic Electric Field {#macroscopic-electric-field .unnumbered}
--------------------------
It follows from what has beeen said that the macroscopic electric field amplitude is determined directly from the Fourier series representation (\[eq:eq:microscopic local field Fourier series representation\]) discarding all contributions of reciprocal lattice vectors $\mathbf{G}\neq\mathbf{0}$: $$\begin{aligned}
\mathcal{\tilde{E}}_{a}\left(\mathbf{r},\omega\right) & = & \sum_{a'}\left(\delta_{a,a'}+\sum_{a''}\frac{1}{\left|C_{\Lambda}\right|}\mathcal{\bar{G}}_{a,a''}(\mathbf{q},\omega)\left[\bar{K}_{\Lambda}\left(\mathbf{q},\omega\right)\right]_{a'',a'}\right)\mathcal{\bar{E}}_{ext,a'}\left(\mathbf{q},\omega\right)e^{i\mathbf{q}\cdot\mathbf{r}}\label{eq:microscopic local field for incident plane wave with small q-1}\end{aligned}$$ In position space then the longitudinal (L) and transversal (T) *macroscopic* electric field amplitudes read $$\begin{aligned}
\tilde{\mathcal{E}}_{a}^{\left(L\right)}\left(\mathbf{r},\omega\right) & = & \mathcal{\bar{E}}_{ext,a}^{\left(L\right)}\left(\mathbf{q},\omega\right)e^{i\mathbf{q}\cdot\mathbf{r}}-\frac{1}{\left|C_{\Lambda}\right|}\sum_{a',a''}\frac{q_{a}q_{a''}}{\left|\mathbf{q}\right|^{2}}\left[\bar{K}_{\Lambda}\left(\mathbf{q},\omega\right)\right]_{a'',a'}\mathcal{\bar{E}}_{ext,a'}\left(\mathbf{q},\omega\right)e^{i\mathbf{q}\cdot\mathbf{r}}\label{eq: longitudinal macroscopic field}\\
\tilde{\mathcal{E}}_{a}^{\left(T\right)}\left(\mathbf{r},\omega\right) & = & \mathcal{\bar{E}}_{ext,a}^{\left(T\right)}\left(\mathbf{q},\omega\right)e^{i\mathbf{q}\cdot\mathbf{r}}+\frac{1}{\left|C_{\Lambda}\right|}\sum_{a',a''}\frac{\omega^{2}}{c^{2}}\frac{\delta_{a,a''}-\frac{q_{a}q_{a''}}{\left|\mathbf{q}\right|^{2}}}{\left|\mathbf{\mathbf{q}}\right|^{2}-\frac{\omega^{2}}{c^{2}}-i0^{+}}\left[\bar{K}_{\Lambda}\left(\mathbf{q},\omega\right)\right]_{a'',a'}\mathcal{\bar{E}}_{ext,a'}\left(\mathbf{q},\omega\right)e^{i\mathbf{q}\cdot\mathbf{r}}\:.\label{eq:transversal macroscopic field}\end{aligned}$$ Comparing now with (\[eq: longitudinal microscopic field\]) and (\[eq:transversal microscopic field\]) we see at once that $$\begin{aligned}
\tilde{E}_{a}^{\left(L\right)}\left(\mathbf{r},\omega\right) & = & \tilde{\mathcal{E}}_{a}^{\left(L\right)}\left(\mathbf{r},\omega\right)+\delta\tilde{E}_{a}^{\left(L\right)}\left(\mathbf{r},\omega\right)\\
\delta\tilde{E}_{a}^{\left(L\right)}\left(\mathbf{r},\omega\right) & = & -\sum_{a'',a'''}\frac{1}{\left|C_{\Lambda}\right|}\sum_{\mathbf{G}\neq\mathbf{0}}e^{i\left(\mathbf{q}+\mathbf{G}\right)\cdot\mathbf{r}}\frac{\left(\mathbf{q}+\mathbf{G}\right)_{a}\left(\mathbf{q}+\mathbf{G}\right)_{a''}}{\left|\mathbf{q}+\mathbf{G}\right|^{2}}\left[\bar{K}_{\Lambda}\left(\mathbf{G},\mathbf{q},\omega\right)\right]_{a'',a'''}\mathcal{\bar{E}}_{ext,a'''}\left(\mathbf{q},\omega\right)\nonumber \\
\nonumber \\
\tilde{E}_{a}^{\left(T\right)}\left(\mathbf{r},\omega\right) & = & \tilde{\mathcal{E}}_{a}^{\left(T\right)}\left(\mathbf{r},\omega\right)+\delta\tilde{E}_{a}^{\left(T\right)}\left(\mathbf{r},\omega\right)\\
\delta\tilde{E}_{a}^{\left(T\right)}\left(\mathbf{r},\omega\right) & = & \sum_{a'',a'''}\frac{1}{\left|C_{\Lambda}\right|}\sum_{\mathbf{G}\neq\mathbf{0}}e^{i\left(\mathbf{q}+\mathbf{G}\right)\cdot\mathbf{r}}\frac{\omega^{2}}{c^{2}}\frac{\delta_{a,a''}-\frac{\left(\mathbf{q}+\mathbf{G}\right)_{a}\left(\mathbf{q}+\mathbf{G}\right)_{a''}}{\left|\mathbf{q}+\mathbf{G}\right|^{2}}}{\left|\mathbf{\mathbf{q}}+\mathbf{G}\right|^{2}-\frac{\omega^{2}}{c^{2}}-i0^{+}}\left[\bar{K}_{\Lambda}\left(\mathbf{G},\mathbf{q},\omega\right)\right]_{a'',a'''}\mathcal{\bar{E}}_{ext,a'''}\left(\mathbf{q},\omega\right)\:.\nonumber \end{aligned}$$ Accordingly the microscopic local electric field and the macroscopic electric field differ by the contributions of the sums over all reciprocal lattice vectors $\mathbf{G}\neq\mathbf{0}$: $$\begin{aligned}
\tilde{E}_{a}\left(\mathbf{r},\omega\right) & = & \tilde{\mathcal{E}}_{a}\left(\mathbf{r},\omega\right)+\delta\tilde{E}_{a}\left(\mathbf{r},\omega\right)\label{eq: microscopic local electric field II}\\
\delta\tilde{E}_{a}\left(\mathbf{r},\omega\right) & = & \sum_{a'',a'}\frac{1}{\left|C_{\Lambda}\right|}\sum_{\mathbf{G}\neq\mathbf{0}}e^{i\left(\mathbf{q}+\mathbf{G}\right)\cdot\mathbf{r}}\mathcal{\bar{G}}_{a,a''}(\mathbf{q}+\mathbf{G},\omega)\left[\bar{K}_{\Lambda}\left(\mathbf{G},\mathbf{q},\omega\right)\right]_{a'',a'}\mathcal{\bar{E}}_{ext,a'}\left(\mathbf{q},\omega\right)\nonumber \end{aligned}$$ In Fig. \[fig:local\_vs\_macro\_fields\] we compare the spatial variation of the transversal *macroscopic* electric field amplitude (\[eq:transversal macroscopic field\]) with the spatial variation of the transversal *microscopic* local electric field (\[eq:transversal microscopic field\]) along a path as shown in Fig.\[fig:Local\_Macro\_Fields\], assuming the *external* electric field was purely transversal, i.e. $\tilde{\mathcal{E}}_{ext,a}^{\left(L\right)}\left(\mathbf{r},\omega\right)=0$. The residue $\delta\tilde{E}_{z}^{\left(T\right)}\left(\mathbf{r},\omega\right)$ turns out to be smaller by a factor $10^{-5}$ compared to the size of the original amplitudes $\tilde{E}_{z}^{(T)}\left(\mathbf{r},\omega\right)$.
 Plot of the transversal part $\tilde{E}_{z}^{(T)}\left(\mathbf{r},\omega\right)$ of the microscopic local electric field (green) and plot of amplitude $\mathcal{\tilde{E}}_{z}^{\left(T\right)}\left(\mathbf{r},\omega\right)$ of the transversal macroscopic field (red dots) along a path $\mathbf{r}\left(x\right)=x\cdot\mathbf{e}^{\left(x\right)}+\frac{a_{\Lambda}}{2}\left(\mathbf{e}^{\left(y\right)}+\mathbf{e}^{\left(z\right)}\right)$ with parameters as in Fig.\[fig:Local\_Macro\_Fields\], revealing the transversal macroscopic field $\mathcal{\tilde{E}}_{z}^{\left(T\right)}\left(\mathbf{r},\omega\right)$ essentially coincides with the transversal part $E_{z}^{(T)}\left(\mathbf{r},\omega\right)$ of the microscopic local electric field. (b) Plot of the residue $\delta\tilde{E}_{z}^{(T)}\left(\mathbf{r},\omega\right)$ along the same path $\mathbf{r}\left(x\right)$.](local_vs_macro_field)
 Plot of the transversal part $\tilde{E}_{z}^{(T)}\left(\mathbf{r},\omega\right)$ of the microscopic local electric field (green) and plot of amplitude $\mathcal{\tilde{E}}_{z}^{\left(T\right)}\left(\mathbf{r},\omega\right)$ of the transversal macroscopic field (red dots) along a path $\mathbf{r}\left(x\right)=x\cdot\mathbf{e}^{\left(x\right)}+\frac{a_{\Lambda}}{2}\left(\mathbf{e}^{\left(y\right)}+\mathbf{e}^{\left(z\right)}\right)$ with parameters as in Fig.\[fig:Local\_Macro\_Fields\], revealing the transversal macroscopic field $\mathcal{\tilde{E}}_{z}^{\left(T\right)}\left(\mathbf{r},\omega\right)$ essentially coincides with the transversal part $E_{z}^{(T)}\left(\mathbf{r},\omega\right)$ of the microscopic local electric field. (b) Plot of the residue $\delta\tilde{E}_{z}^{(T)}\left(\mathbf{r},\omega\right)$ along the same path $\mathbf{r}\left(x\right)$.](local_macro_difference)
Macroscopic Magnetic Induction Field {#macroscopic-magnetic-induction-field .unnumbered}
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The amplitude of the microscopic local magnetic induction field $\tilde{\mathbf{B}}\left(\mathbf{r},\omega\right)$ is of course directly connected to the amplitude of the local microscopic electric field $\tilde{\mathbf{E}}\left(\mathbf{r},\omega\right)$ via Faraday’s law: $$\begin{aligned}
\tilde{\mathbf{B}}\left(\mathbf{r},\omega\right) & = & \frac{1}{i\omega}\mathbf{\boldsymbol{\nabla}}\wedge\tilde{\mathbf{E}}\left(\mathbf{r},\omega\right)\end{aligned}$$ Insertion of the representation (\[eq: microscopic local electric field II\]) for the microscopic local electric field amplitude, $\tilde{E}_{a}\left(\mathbf{r},\omega\right)=\tilde{\mathcal{E}}_{a}\left(\mathbf{r},\omega\right)+\delta\tilde{E}_{a}\left(\mathbf{r},\omega\right)$, leads in this way immediately to $$\begin{aligned}
\tilde{B}_{c}\left(\mathbf{r},\omega\right) & = & \sum_{b,a\in\left\{ x,y,z\right\} }\frac{1}{i\omega}\epsilon_{cba}\frac{\partial}{\partial r_{b}}\left\{ \tilde{\mathcal{E}}_{a}\left(\mathbf{r},\omega\right)+\delta\tilde{E}_{a}\left(\mathbf{r},\omega\right)\right\} \:.\label{eq: magnetic induction field}\end{aligned}$$ Identifying now the macroscopic magnetic induction field amplitude via $$\tilde{\mathcal{B}}_{c}\left(\mathbf{r},\omega\right)=\sum_{b,a\in\left\{ x,y,z\right\} }\frac{1}{i\omega}\epsilon_{cba}\frac{\partial}{\partial r_{b}}\tilde{\mathcal{E}}_{a}\left(\mathbf{r},\omega\right)$$ then $$\tilde{B}_{c}\left(\mathbf{r},\omega\right)=\tilde{\mathcal{B}}_{c}\left(\mathbf{r},\omega\right)+\delta\tilde{B}_{c}\left(\mathbf{r},\omega\right)\:,$$ where the correction term representing the difference to the microscopic magnetic induction field amplitude is $$\begin{aligned}
\delta\tilde{B}_{c}\left(\mathbf{r},\omega\right) & = & \sum_{b,a\in\left\{ x,y,z\right\} }\frac{1}{i\omega}\epsilon_{cba}\frac{\partial}{\partial r_{b}}\delta\tilde{E}_{a}\left(\mathbf{r},\omega\right)\nonumber \\
& = & \frac{\omega}{c^{2}}\sum_{c',c'',c'''}\frac{1}{\left|C_{\Lambda}\right|}\sum_{\mathbf{G}\neq\mathbf{0}}e^{i\left(\mathbf{q}+\mathbf{G}\right)\cdot\mathbf{r}}\epsilon_{cc'c''}\frac{q_{c'}+G_{c'}}{\left|\mathbf{\mathbf{q}+\mathbf{G}}\right|^{2}-\frac{\omega^{2}}{c^{2}}-i0^{+}}\left[\bar{K}_{\Lambda}\left(\mathbf{G},\mathbf{q},\omega\right)\right]_{c'',c'''}\mathcal{\bar{E}}_{ext,c'''}\left(\mathbf{q},\omega\right)\:.\end{aligned}$$ Like in the electric field case, the *macroscopic* magnetic induction field amplitude $\tilde{\mathcal{B}}_{c}\left(\mathbf{r},\omega\right)$ represents the low pass filtered *microscopic* local magnetic induction field amplitude $\tilde{B}_{c}\left(\mathbf{r},\omega\right)$. The plot of the residue $\delta\tilde{B}_{y}\left(\mathbf{r},\omega\right)=\tilde{B}_{y}\left(\mathbf{r},\omega\right)-\tilde{\mathcal{B}}_{y}\left(\mathbf{r},\omega\right)$ is displayed in Fig.\[fig:B-Field\]. Clearly, $\tilde{B}_{y}\left(\mathbf{r},\omega\right)$ and $\tilde{\mathcal{B}}_{y}\left(\mathbf{r},\omega\right)$ essentially coincide. Note that in the electric field case the relative size of the residue $\delta\tilde{E}_{z}^{\left(T\right)}\left(\mathbf{r},\omega\right)$ along the same path turned out to be even smaller, see Fig.\[fig:local\_vs\_macro\_fields\].
 Plot of component $\tilde{B}_{y}\left(\mathbf{r},\omega\right)$ of the *microscopic* local magnetic induction field amplitude (blue) and plot of component $\mathcal{\tilde{B}}_{y}\left(\mathbf{r},\omega\right)$ of the *macroscopic* magnetic induction field amplitude (red dots) along a path $\mathbf{r}\left(x\right)=x\cdot\mathbf{e}^{\left(x\right)}+\frac{a_{\Lambda}}{2}\left(\mathbf{e}^{\left(y\right)}+\mathbf{e}^{\left(z\right)}\right)$ with parameters as in Fig.\[fig:Local\_Macro\_Fields\], revealing that $\mathcal{\tilde{B}}_{y}\left(\mathbf{r},\omega\right)$ essentially coincides with $\tilde{B}_{y}\left(\mathbf{r},\omega\right)$. (b) Plot of the residue $\delta\tilde{B}_{y}\left(\mathbf{r},\omega\right)$ along the same path $\mathbf{r}\left(x\right)$.](BLocalMacro)
 Plot of component $\tilde{B}_{y}\left(\mathbf{r},\omega\right)$ of the *microscopic* local magnetic induction field amplitude (blue) and plot of component $\mathcal{\tilde{B}}_{y}\left(\mathbf{r},\omega\right)$ of the *macroscopic* magnetic induction field amplitude (red dots) along a path $\mathbf{r}\left(x\right)=x\cdot\mathbf{e}^{\left(x\right)}+\frac{a_{\Lambda}}{2}\left(\mathbf{e}^{\left(y\right)}+\mathbf{e}^{\left(z\right)}\right)$ with parameters as in Fig.\[fig:Local\_Macro\_Fields\], revealing that $\mathcal{\tilde{B}}_{y}\left(\mathbf{r},\omega\right)$ essentially coincides with $\tilde{B}_{y}\left(\mathbf{r},\omega\right)$. (b) Plot of the residue $\delta\tilde{B}_{y}\left(\mathbf{r},\omega\right)$ along the same path $\mathbf{r}\left(x\right)$.](BDifference)
Deducing the Differential Equations of Macroscopic Electrodynamics\[sec:Deducing-the-Differential\] {#deducing-the-differential-equations-of-macroscopic-electrodynamicssecdeducing-the-differential .unnumbered}
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Restricting to long wavelengths so that $\left|\mathbf{q}\right|<q_{c}$ let us rewrite (\[eq: Fourier transform microscopic E- field\]) in the guise $$\begin{aligned}
\bar{\mathcal{E}}_{a}\left(\mathbf{q},\omega\right)-\mathcal{\bar{E}}_{ext,a}\left(\mathbf{q},\omega\right) & = & \frac{1}{\left|C_{\Lambda}\right|}\sum_{a',a''}\bar{\mathcal{G}}_{a,a'}(\mathbf{q},\omega)\circ\left[\bar{K}_{\Lambda}(\mathbf{q},\omega)\right]_{a',a''}\mathcal{\bar{E}}_{ext,a''}\left(\mathbf{q},\omega\right)\label{eq: low pass filtered field integral equation}\\
& = & \sum_{a',a''}\bar{\mathcal{G}}_{a,a'}(\mathbf{q},\omega)\circ\left[\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)-I\right]_{a',a''}\mathcal{\bar{E}}_{a''}\left(\mathbf{q},\omega\right)\:.\nonumber \end{aligned}$$ Multiplication on both sides with the inverse of the kernel (\[eq: Fourier transform of 3x3 matrix propagator\]) gives $$\sum_{a'}\left[\bar{\mathcal{G}}^{-1}(\mathbf{q},\omega)\right]_{a,a'}\left[\bar{\mathcal{E}}_{a'}\left(\mathbf{q},\omega\right)-\bar{\mathcal{E}}_{ext,a'}\left(\mathbf{q},\omega\right)\right]=\sum_{a'}\left[\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)-I\right]_{a,a'}\mathcal{\bar{E}}_{a'}\left(\mathbf{q},\omega\right)\:.\label{eq: intermediate I}$$ In terms of the projection operators (\[eq:Fourier transform projection operators\]) then $$\frac{\omega^{2}}{c^{2}}\left[\bar{\mathcal{G}}^{-1}(\mathbf{q},\omega)\right]_{a,a'}=\left(\left|\mathbf{q}\right|^{2}-\frac{\omega^{2}}{c^{2}}\right)\bar{\Pi}_{a,a'}^{\left(T\right)}\left(\mathbf{q}\right)-\frac{\omega^{2}}{c^{2}}\bar{\Pi}_{a,a'}^{\left(L\right)}\left(\mathbf{q}\right)\:,\label{eq: inverse 3x3 propagator}$$ so that (\[eq: intermediate I\]) leads to $$\sum_{a'}\left[\left|\mathbf{q}\right|^{2}\bar{\Pi}^{\left(T\right)}\left(\mathbf{q}\right)-\frac{\omega^{2}}{c^{2}}\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)\right]_{a,a'}\bar{\mathcal{E}}_{a'}\left(\mathbf{q},\omega\right)=\sum_{a'}\left[\left(\left|\mathbf{q}\right|^{2}-\frac{\omega^{2}}{c^{2}}\right)\bar{\Pi}_{a,a'}^{\left(T\right)}\left(\mathbf{q}\right)-\frac{\omega^{2}}{c^{2}}\bar{\Pi}_{a,a'}^{\left(L\right)}\left(\mathbf{q}\right)\right]\bar{\mathcal{E}}_{ext,a'}\left(\mathbf{q},\omega\right)\:.\label{eq:intermediate II}$$
Identifying transversal and longitudinal components of the Fourier amplitudes of the electric field, $$\begin{aligned}
\sum_{a'}\bar{\Pi}_{a,a'}^{\left(L,T\right)}\left(\mathbf{q}\right)\bar{\mathcal{E}}_{ext,a'}\left(\mathbf{q},\omega\right) & = & \bar{\mathcal{E}}_{ext,a}^{\left(L,T\right)}\left(\mathbf{q},\omega\right)\label{eq: longitudinal and transversal projections of macroscopic electric field}\\
\sum_{a'}\bar{\Pi}_{a,a'}^{\left(L,T\right)}\left(\mathbf{q}\right)\bar{\mathcal{E}}_{a'}\left(\mathbf{q},\omega\right) & = & \bar{\mathcal{E}}_{a}^{\left(L,T\right)}\left(\mathbf{q},\omega\right)\nonumber \end{aligned}$$ let us reexpress the respective Fourier amplitudes of the *external* electric field in terms of the original sources inside the source domain $\Omega_{S}$, namely the transversal external current distribution $\bar{j}_{ext,a}^{(T)}\left(\mathbf{q},\omega\right)$ and the external charge distribution $\bar{\varrho}_{ext}\left(\mathbf{q},\omega\right)$: $$\begin{aligned}
\left(\left|\mathbf{q}\right|^{2}-\frac{\omega^{2}}{c^{2}}\right)\bar{\mathcal{E}}_{ext,a}^{\left(T\right)}\left(\mathbf{q},\omega\right) & = & \mu_{0}i\omega\bar{j}_{ext,a}^{(T)}\left(\mathbf{q},\omega\right)\label{eq: external transversal field}\\
\bar{\mathcal{E}}_{ext,a}^{\left(L\right)}\left(\mathbf{q},\omega\right) & = & -iq_{a}\bar{\phi}_{ext}\left(\mathbf{q},\omega\right)=-iq_{a}\frac{\bar{\varrho}_{ext}\left(\mathbf{q},\omega\right)}{\varepsilon_{0}\left|\mathbf{q}\right|^{2}}\label{eq: external longitudinal field}\end{aligned}$$ So the right hand side in (\[eq:intermediate II\]) reduces together with the Fourier transformed relation (\[eq:external longitudinal electric field\]) to $$\begin{aligned}
\sum_{a'}\left[\left(\left|\mathbf{q}\right|^{2}-\frac{\omega^{2}}{c^{2}}\right)\bar{\Pi}_{a,a'}^{\left(T\right)}\left(\mathbf{q}\right)-\frac{\omega^{2}}{c^{2}}\bar{\Pi}_{a,a'}^{\left(L\right)}\left(\mathbf{q}\right)\right]\bar{\mathcal{E}}_{ext,a'}\left(\mathbf{q},\omega\right) & = & \left(\left|\mathbf{q}\right|^{2}-\frac{\omega^{2}}{c^{2}}\right)\bar{\mathcal{E}}_{ext,a}^{\left(T\right)}\left(\mathbf{q},\omega\right)-\frac{\omega^{2}}{c^{2}}\bar{\mathcal{E}}_{ext,a}^{\left(L\right)}\left(\mathbf{q},\omega\right)\nonumber \\
& = & \mu_{0}i\omega\left[\bar{j}_{ext,a}^{(T)}\left(\mathbf{q},\omega\right)+\bar{j}_{ext,a}^{(L)}\left(\mathbf{q},\omega\right)\right]\\
& = & \mu_{0}i\omega\bar{j}_{ext,a}\left(\mathbf{q},\omega\right)\:.\end{aligned}$$ Consequently (\[eq:intermediate II\]) assumes the guise $$\begin{aligned}
\sum_{a'}\left[\left|\mathbf{q}\right|^{2}\bar{\Pi}^{\left(T\right)}\left(\mathbf{q}\right)-\frac{\omega^{2}}{c^{2}}\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)\right]_{a,a'}\bar{\mathcal{E}}_{a'}\left(\mathbf{q},\omega\right) & = & \mu_{0}i\omega\bar{j}_{ext,a}\left(\mathbf{q},\omega\right)\:.\label{eq:intermediate III}\end{aligned}$$
If the dependence on wavevector $\mathbf{q}$ of the dielectric tensor in (\[eq:intermediate III\]) can be ignored, we replace $\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)\rightarrow\varepsilon_{\Lambda}\left(\omega\right)$ and obtain then in position space the well known (so called) *vector* wave equation determining the *macroscopic* electric field: $$\begin{aligned}
\nabla\wedge\left[\nabla\wedge\tilde{\mathcal{\boldsymbol{E}}}\left(\mathbf{r},\omega\right)\right]-\frac{\omega^{2}}{c^{2}}\varepsilon_{\Lambda}\left(\omega\right)\tilde{\mathcal{\boldsymbol{E}}}\left(\mathbf{r},\omega\right) & = & \mu_{0}i\omega\tilde{\mathbf{j}}_{ext}\left(\mathbf{r},\omega\right)\label{eq: vector wave equation for macroscopic electric field}\end{aligned}$$ It should be noted, that here $\tilde{\mathcal{\boldsymbol{E}}}\left(\mathbf{r},\omega\right)$ still may be decomposed into divergence-free (transversal) and curl-free (longitudinal) parts, $\tilde{\mathcal{\boldsymbol{E}}}\left(\mathbf{r},\omega\right)=\tilde{\mathcal{\boldsymbol{E}}}^{\left(T\right)}\left(\mathbf{r},\omega\right)+\tilde{\mathcal{\boldsymbol{E}}}^{\left(L\right)}\left(\mathbf{r},\omega\right)$. Thus it is deceptive to interpret (\[eq: vector wave equation for macroscopic electric field\]) as a *wave equation* determining electromagnetic radiation as propagating photons with speed determined by the eigenvalues of the dielectric tensor $\varepsilon_{\Lambda}\left(\omega\right)$, unless $\tilde{\mathcal{\boldsymbol{E}}}^{\left(L\right)}\left(\mathbf{r},\omega\right)\equiv\mathbf{0}$.
To find the differential equations for the transversal and longitudinal parts of the macroscopic field amplitude let us first introduce block matrix notation specifying transversal and longitudinal projections of the dielectric tensor: $$\begin{aligned}
\varepsilon_{a,a'}^{\left(A,B\right)}\left(\mathbf{q},\omega\right) & = & \sum_{b,b'}\bar{\Pi}_{a,b}^{\left(A\right)}\left(\mathbf{q}\right)\left[\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)\right]_{b,b'}\bar{\Pi}_{b',a'}^{\left(B\right)}\left(\mathbf{q}\right)\label{eq: projections of dielectric tensor}\\
A,B & \in & \left\{ L,T\right\} \nonumber \end{aligned}$$ Then the vector wave equation (\[eq:intermediate III\]) separates into two coupled equations for the respective transversal and longitudinal Fourier amplitudes of the macroscopic field: $$\begin{aligned}
\sum_{a'}\left(\left|\mathbf{q}\right|^{2}\delta_{a,a'}-\frac{\omega^{2}}{c^{2}}\varepsilon_{a,a'}^{\left(T,T\right)}\left(\mathbf{q},\omega\right)\right)\bar{\mathcal{E}}_{a'}^{\left(T\right)}\left(\mathbf{q},\omega\right)-\frac{\omega^{2}}{c^{2}}\sum_{a'}\varepsilon_{a,a'}^{\left(T,L\right)}\left(\mathbf{q},\omega\right)\bar{\mathcal{E}}_{a'}^{\left(L\right)}\left(\mathbf{q},\omega\right) & = & \mu_{0}i\omega\bar{j}_{ext,a}^{(T)}\left(\mathbf{q},\omega\right)\label{eq: coupled field equatios}\\
\sum_{a'}\varepsilon_{a,a'}^{\left(L,T\right)}\left(\mathbf{q},\omega\right)\bar{\mathcal{E}}_{a'}^{\left(T\right)}\left(\mathbf{q},\omega\right)+\sum_{a'}\varepsilon_{a,a'}^{\left(L,L\right)}\left(\mathbf{q},\omega\right)\bar{\mathcal{E}}_{a'}^{\left(L\right)}\left(\mathbf{q},\omega\right) & = & \bar{\mathcal{E}}_{ext,a}^{\left(L\right)}\left(\mathbf{q},\omega\right)\nonumber \end{aligned}$$
Choosing Eq. (\[eq: vector wave equation for macroscopic electric field\]) as a starting point for the transport theory of radiation (light intensity) inside a (possibly disordered) material appears according to what has been said questionable, as the fluctuation contribution $\tilde{E}_{a}\left(\mathbf{r},\omega\right)-\tilde{\mathcal{E}}_{a}\left(\mathbf{r},\omega\right)\equiv\delta\tilde{E}_{a}\left(\mathbf{r},\omega\right)$, see Eq.(\[eq: microscopic local electric field II\]), is in this case not included, despite the product $\delta\tilde{E}_{a}\left(\mathbf{r},\omega\right)$$\delta\tilde{E}_{b}\left(\mathbf{r}',\omega\right)$ apparently comprising a spatially slowly varying interference contribution.
Wave Equation with Renormalized Speed of Light {#wave-equation-with-renormalized-speed-of-light .unnumbered}
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If the external field was purely transversal, i.e. $\bar{\mathcal{E}}_{ext,a}^{\left(L\right)}\left(\mathbf{q},\omega\right)\equiv0$, then the longitudinal component of the macroscopic field is readily eliminated in (\[eq: coupled field equatios\]), provided the inverse of the longitudinal block $\varepsilon_{\Lambda}^{\left(L,L\right)}\left(\mathbf{q},\omega\right)$ exists: $$\bar{\mathcal{E}}_{a}^{\left(L\right)}\left(\mathbf{q},\omega\right)=-\sum_{a'}\left[\left[\varepsilon^{\left(L,L\right)}\left(\mathbf{q},\omega\right)\right]^{-1}\circ\varepsilon^{\left(L,T\right)}\left(\mathbf{q},\omega\right)\right]_{a,a'}\bar{\mathcal{E}}_{a'}^{\left(T\right)}\left(\mathbf{q},\omega\right)$$ Insertion leads to $$\sum_{a'}\left(\left|\mathbf{q}\right|^{2}\delta_{a,a'}-\frac{\omega^{2}}{c^{2}}\left[\varepsilon^{\left(T,T\right)}\left(\mathbf{q},\omega\right)-\varepsilon^{\left(T,L\right)}\left(\mathbf{q},\omega\right)\circ\left[\varepsilon^{\left(L,L\right)}\left(\mathbf{q},\omega\right)\right]^{-1}\circ\varepsilon^{\left(L,T\right)}\left(\mathbf{q},\omega\right)\right]_{a,a'}\right)\bar{\mathcal{E}}_{a'}^{\left(T\right)}\left(\mathbf{q},\omega\right)=\mu_{0}i\omega\bar{j}_{ext,a}^{(T)}\left(\mathbf{q},\omega\right)\:.$$ Introducing as an effective *transversal* dielectric tensor $$\begin{aligned}
\varepsilon^{\left(T\right)}\left(\mathbf{q},\omega\right) & = & \varepsilon^{\left(T,T\right)}\left(\mathbf{q},\omega\right)-\varepsilon^{\left(T,L\right)}\left(\mathbf{q},\omega\right)\circ\left[\varepsilon^{\left(L,L\right)}\left(\mathbf{q},\omega\right)\right]^{-1}\circ\varepsilon^{\left(L,T\right)}\left(\mathbf{q},\omega\right)\:,\label{eq: transversal dielectric tensor}\end{aligned}$$ then $$\sum_{a'}\left(\left|\mathbf{q}\right|^{2}\delta_{a,a'}-\frac{\omega^{2}}{c^{2}}\varepsilon_{a,a'}^{\left(T\right)}\left(\mathbf{q},\omega\right)\right)\bar{\mathcal{E}}_{a'}^{\left(T\right)}\left(\mathbf{q},\omega\right)=\mu_{0}i\omega\bar{j}_{ext,a}^{(T)}\left(\mathbf{q},\omega\right)\:.\label{eq: wave equation transverse macroscopic field Fourier space}$$
If the dependence on wavevector $\mathbf{q}$ of the (transversal) dielectric function can be ignored, then $\varepsilon_{ab}^{\left(T\right)}\left(\mathbf{q},\omega\right)\rightarrow\varepsilon_{ab}^{(T)}\left(\omega\right)$, and equation (\[eq: wave equation transverse macroscopic field Fourier space\]) corresponds in position space to a (scalar) wave equation determining the Cartesian components of the transversal macroscopic electric field amplitude $\tilde{\mathcal{\boldsymbol{E}}}^{\left(T\right)}\left(\mathbf{r},\omega\right)$ propagating inside a dielectric crystal, in full agreement with the standard theory of the propagation of polarized light in transparent dielectric materials: $$\sum_{a'}\left(-\nabla^{2}\delta_{a,a'}-\frac{\omega^{2}}{c^{2}}\varepsilon_{aa'}^{(T)}\left(\omega\right)\right)\mathcal{\tilde{E}}_{a'}^{\left(T\right)}\left(\mathbf{r},\omega\right)=\mu_{0}i\omega\tilde{j}_{ext,a}^{\left(T\right)}\left(\mathbf{r},\omega\right)\label{eq: wave equation for transversal macroscopic field}$$ With a choice of a coordinate frame such that the dielectric tensor is diagonal in that frame, $\varepsilon_{aa'}^{(T)}\left(\omega\right)=\delta_{aa'}n_{a}^{2}\left(\omega\right)$, one finds for light propagating along a high symmetry axis corresponding to an eigenvector of that dielectric tensor the usual reduction of the speed of light, $c\rightarrow c/n_{a}\left(\omega\right)$ characteristic for in general birefringent crystalline dielectrics. Note that because of chromatic dispersion of the index of refraction then those eigenvectors may undergo a corresponding chromatic dispersion of axes as well, yet this effect being observable only for monoclinic and triclinic crystalline symmetry [@Born1999].
If the dependence on wavevector $\mathbf{q}$ of the (transversal) dielectric function cannot be ignored, the Taylor expansion $$\varepsilon_{ab}^{\left(T\right)}\left(\mathbf{q},\omega\right)=\varepsilon_{ab}^{\left(T\right)}\left(\omega\right)+i\sum_{c}\gamma_{abc}\left(\omega\right)q_{c}+\sum_{c,d}\alpha_{abcd}\left(\omega\right)q_{c}q_{d}+...\label{eq:Taylor expansion transversal dielectric tensor}$$ around $\mathbf{q}=\mathbf{0}$ in (\[eq: wave equation transverse macroscopic field Fourier space\]) provides then insight into various optical phenomena connected to retardation and non-locality of the dielectric tensor, in full agreement with the phenomenological reasoning of Agranovich and Ginzburg [@Agranovich1984]: the eigenvalues of the symmetric tensor $\varepsilon_{ab}^{(T)}\left(\omega\right)=\varepsilon_{ba}^{(T)}\left(\omega\right)$ describing chromatic dispersion of the index of refraction and birefringence, the antisymmetric first order term $\gamma_{abc}\left(\omega\right)$ specifying rotary power (natural optical activity), the second order terms $\alpha_{abcd}\left(\omega\right)$ shaping the intrinsic effects of a spatial-dispersion-induced-birefringence. Of course, in a centrosymmetric crystal there exists no natural optical activity: $\gamma_{abc}\left(\omega\right)\equiv0$. The tensor $\alpha_{abcd}\left(\omega\right)$ originates from the symmetry-breaking evoked by the finite $\mathbf{q}$-vector of the light [@Agranovich1984], it displays in general $3\times3\times3\times3=81$ components. But for crystals with cubic symmetry the number of independent components of that tensor reduces substantially: for symmetry group $T$ and $T_{h}$ there exist four independent components, for symmetry group $T_{d},$$O_{h}$ and $O$ there exist three independent components and for isotropic systems that number reduces to two [@Agranovich1984]. The (weak) effects of a dispersion-induced-birefringence indeed give as a matter of fact reason for concern regarding the image quality of dielectric lenses made from $CaF_{2}$ and $BaF_{2}$, a topic of prime importance designing modern lithographic optical systems in the ultraviolet [@Burnett2002; @Serebryakov2003].
The dielectric tensor $\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)$ being a functional of the microscopic polarizabilities $\alpha\left(\boldsymbol{\eta}^{\left(j\right)},\omega\right)$ of atoms (ions, molecules) located at position $\boldsymbol{\eta}^{\left(j\right)}$ in the unit cell $C_{\Lambda}$, for example see (\[eq: atom polarizability\]), undergoes variations in proportion to changes of those atom individual polarizabilities caused by external static fields. For instance, in the presence of a *static* external magnetic induction field $\mathbf{B}_{0}$, the polarizability of a single atom displays now a magnetic field induced anisotropy [@Baranova1979], even though the atom-individual polarizability in zero field $\alpha\left(\boldsymbol{\eta}^{\left(j\right)},\omega\right)$ was isotropic: $$\begin{aligned}
\alpha_{aa'}\left(\boldsymbol{\eta}^{\left(j\right)},\omega;\mathbf{B}_{0}\right) & = & \left(\left[I+i\frac{\omega}{\left|e\right|}\alpha\left(\boldsymbol{\eta}^{\left(j\right)},\omega\right)\circ b\right]^{-1}\circ\alpha\left(\boldsymbol{\eta}^{\left(j\right)},\omega\right)\right)_{aa'}\label{eq: microscopic atom polarizability in static B-field}\\
b_{aa'} & = & \sum_{a''}\epsilon_{aa'a''}B_{a''}\nonumber \end{aligned}$$ Such changes then reflect in corresponding changes of the dielectric tensor. As a result the Taylor expansion of the transversal dielectric tensor of the crystal now reads to leading order in the small quantities $\mathbf{q}$ and $\mathbf{B}_{0}$: $$\begin{aligned}
\varepsilon_{ab}^{\left(T\right)}\left(\mathbf{q},\omega;\mathbf{B}_{0}\right) & = & \varepsilon_{ab}^{\left(T\right)}\left(\omega\right)+i\sum_{c}\beta_{abc}\left(\omega\right)B_{0,c}+i\sum_{c}\gamma_{abc}\left(\omega\right)q_{c}+...\label{eq: Taylor expansion transversal dielectric tensor in static B-field}\end{aligned}$$ Note the symmetry $$\varepsilon_{ab}^{\left(T\right)}\left(-\mathbf{q},\omega;-\mathbf{B}_{0}\right)=\varepsilon_{ba}^{\left(T\right)}\left(\mathbf{q},\omega;\mathbf{B}_{0}\right)\:.$$ Correspondingly, if the dependence of the transversal dielectric tensor on wavevector $\mathbf{q}$ can be ignored, then light propagation in the presence of a static field $\mathbf{B}_{0}$ inside a dielectric crystal is governed by the wave equation (\[eq: wave equation for transversal macroscopic field\]), but with a dielectric tensor now dependent on that *static* magnetic field: $$\varepsilon_{ab}^{\left(T\right)}\left(\omega;\mathbf{B}_{0}\right)=\varepsilon_{ab}^{\left(T\right)}\left(\omega\right)+i\sum_{c}\beta_{abc}\left(\omega\right)B_{0,c}\label{eq: magnetic field dependence transversal dielectric tensor}$$ The presence of the antisymmetric tensor $\beta_{abc}\left(\omega\right)=\sum_{c'}\epsilon_{abc'}\lambda_{c'c}\left(\omega\right)$ in (\[eq: magnetic field dependence transversal dielectric tensor\]) and with that said in (\[eq: wave equation for transversal macroscopic field\]), now leads to left and right circularly polarized waves propagating at slightly different speeds, thus giving rise to a magnetic field induced *circular* birefringence. If $\mathbf{B}_{0}\wedge\mathbf{q}=\mathbf{0}$, i.e. if $\mathbf{B}_{0}$ is orientated parallel or anti-parallel to the direction $\mathbf{\hat{q}}$ of light propagation, this is the well known Faraday rotation effect of a light waves linear polarization in a static magnetic field.
Electric-Field Screening {#electric-field-screening .unnumbered}
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Conversely, if the external current source was purely longitudinal, i.e. $\bar{j}_{ext,a}^{(T)}\left(\mathbf{q},\omega\right)\equiv0$, there follows directly from (\[eq: coupled field equatios\]) upon elimination of the transversal part $\bar{\mathcal{E}}_{a}^{\left(T\right)}\left(\mathbf{q},\omega\right)$ in favour of the field amplitude $\bar{\mathcal{E}}_{a}^{\left(L\right)}\left(\mathbf{q},\omega\right)$ now the relation $$\bar{\mathcal{E}}_{a}^{\left(L\right)}\left(\mathbf{q},\omega\right)=\sum_{a'}\left(\left[\varepsilon^{\left(L\right)}\left(\mathbf{q},\omega\right)\right]^{-1}\right)_{a,a'}\bar{\mathcal{E}}_{ext,a'}^{\left(L\right)}\left(\mathbf{q},\omega\right)\:,\label{eq:Longitudinal macroscopic field Fourier space}$$ with the *longitudinal* dielectric tensor $$\varepsilon^{\left(L\right)}\left(\mathbf{q},\omega\right)=\varepsilon^{\left(L,T\right)}\left(\mathbf{q},\omega\right)\circ\frac{\frac{\omega^{2}}{c^{2}}}{\left|\mathbf{q}\right|^{2}I-\frac{\omega^{2}}{c^{2}}\varepsilon^{\left(T,T\right)}\left(\mathbf{q},\omega\right)}\circ\varepsilon^{\left(T,L\right)}\left(\mathbf{q},\omega\right)+\varepsilon^{\left(L,L\right)}\left(\mathbf{q},\omega\right)\label{eq:longitudinal dielectric tensor}$$ describing *electric-field screening*. From equation (\[eq:Longitudinal macroscopic field Fourier space\]) we readily infer $$\sum_{a,a'}iq_{a}\varepsilon_{aa'}^{\left(L\right)}\left(\mathbf{q},\omega\right)\bar{\mathcal{E}}_{a'}^{\left(L\right)}\left(\mathbf{q},\omega\right)=\sum_{a}iq_{a}\bar{\mathcal{E}}_{ext,a}^{\left(L\right)}\left(\mathbf{q},\omega\right)=\frac{\bar{\varrho}_{ext}\left(\mathbf{q},\omega\right)}{\varepsilon_{0}}\:.$$ For optical frequencies and below it is (obviously) adequate to approximate $\varepsilon^{\left(L\right)}\left(\mathbf{q},\omega\right)\simeq\varepsilon^{\left(L,L\right)}\left(\mathbf{q},\omega\right)$, provided $\det\left[\left|\mathbf{q}\right|^{2}I-\frac{\omega^{2}}{c^{2}}\varepsilon^{\left(T,T\right)}\left(\mathbf{q},\omega\right)\right]\neq0$. Then $$\begin{aligned}
\sum_{a,a'}iq_{a}\varepsilon_{aa'}^{\left(L\right)}\left(\mathbf{q},\omega\right)\bar{\mathcal{E}}_{a'}^{\left(L\right)}\left(\mathbf{q},\omega\right) & = & \sum_{a,a'}iq_{a}\varepsilon_{aa'}^{\left(L,L\right)}\left(\mathbf{q},\omega\right)\bar{\mathcal{E}}_{a'}^{\left(L\right)}\left(\mathbf{q},\omega\right)\nonumber \\
& = & \sum_{a,a',b,b'}iq_{a}\Pi_{ab}^{\left(L\right)}\left(\mathbf{q}\right)\left[\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)\right]_{bb'}\Pi_{b'a'}^{\left(L\right)}\left(\mathbf{q}\right)\bar{\mathcal{E}}_{a'}^{\left(L\right)}\left(\mathbf{q},\omega\right)\nonumber \\
& = & \sum_{b,b'}iq_{b}\left[\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)\right]_{bb'}\bar{\mathcal{E}}_{b'}^{\left(L\right)}\left(\mathbf{q},\omega\right)\:.\end{aligned}$$ In position space, and representing the longitudinal electric field $\mathcal{\tilde{E}}^{\left(L\right)}\left(\mathbf{r},\omega\right)=-\nabla\tilde{\phi}\left(\mathbf{r},\omega\right)$, this conforms in the long wavelength limit $\mathbf{\left|q\right|}\rightarrow0$ to the usual Poisson type equation of electrostatics determining the scalar potential $\tilde{\phi}\left(\mathbf{r},\omega\right)$ of a given charge distribution $\tilde{\varrho}_{ext}\left(\mathbf{r},\omega\right)$: $$\boldsymbol{-\nabla}\boldsymbol{\mathbf{\cdot}}\left[\varepsilon_{\Lambda}\left(\omega\right)\nabla\tilde{\phi}\left(\mathbf{r},\omega\right)\right]=\frac{\tilde{\varrho}_{ext}\left(\mathbf{r},\omega\right)}{\varepsilon_{0}}\label{eq: Poisson equation for longitudinal macroscopic field}$$ Now it is manifest that it is the dielectric $3\times3$ tensor $$\varepsilon_{\Lambda}\left(\omega\right)\equiv\lim_{\left|\mathbf{q}\right|\rightarrow0}\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)$$ that describes electric-field screening, like in electrostatics.
Results of Calculations for the Chromatic Dispersion, Natural Optical Activity and Spatial-Dispersion-Induced Birefringence in Various Dielectric Crystals\[sec:chromatic-dispersion,-optical activity and spatial dispersion induced birefringence\] {#results-of-calculations-for-the-chromatic-dispersion-natural-optical-activity-and-spatial-dispersion-induced-birefringence-in-various-dielectric-crystalssecchromatic-dispersion-optical-activity-and-spatial-dispersion-induced-birefringence .unnumbered}
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Conceiving the polarizability of atoms or ions not as given input from microscopic theory, but as a fit function of the type (\[eq: model for microscopic polarizability\]), the diagonal parts depending on two parameters $\omega_{0}^{\left(j\right)}$ and $\alpha_{0}^{\left(j\right)}$ for each atom or ion species $j\in\left\{ 1,2,...M\right\} $ and also the off diagonal parts depending on two parameters $\omega_{0}^{\left(j,j'\right)}$ and $\alpha_{0}^{\left(j,j'\right)}$ for each pair $\left(j,j'\right)$ of ions $j,j'\in\left\{ 1,2,...M\right\} $ in the unit cell $C_{\Lambda}$, we find the well known Sellmeier fit [@Brueckner2011] of the frequency dependence of the index of refraction $n\left(\omega\right)$ is nicely reproduced from the eigenvalues of the dielectric tensor $\varepsilon_{ab}^{\left(T\right)}\left(\mathbf{q}=\mathbf{0},\omega\right)$ for a variety of ionic compounds, see exemplarily our results for $CsI$ and $RbCl$ in Fig.\[Sellmeier\] and $CaF_{2}$ and $BaF_{2}$ in Fig.\[fig:CaF2\_and\_BaF2\]. The relative error of our calculations compared to a fit of experimental data for $n\left(\omega\right)$ with the Sellmeier formula for the mentioned crystals is less than 1%.
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 Plot of index of refraction $n\left(\omega\right)$ vs. free space wavelength $\lambda=\frac{2\pi c}{\omega}$ for $CaF_{2}$ and $BaF_{2}$. Displayed are values (dots) calculated from the macroscopic dielectric function (\[eq: macroscopic dielectric function arbitrary M\]) for $\mathbf{q}=\mathbf{0}$, solely with the lattice symmetry and the model for electronic polarizabilities (\[eq: isotropic Lorentz polarizability diagonal\]) as input, the respective values of the parameters $\alpha_{0}^{\left(j\right)}$ and $\omega_{0}^{\left(j\right)}$ as in Table \[fit\]. The relative error compared to a fit of experimental data for $n\left(\omega\right)$ with the multi-parameter Sellmeier formula [@Li1980] (solid lines) is less than 1%. The spatial dispersion induced birefringence $\Delta\text{n}$ as calculated from the $\mathbf{q}$-dependence of the macroscopic dielectric function (\[eq: macroscopic dielectric function arbitrary M\]) is displayed (red) in (b) for $CaF_{2}$ and (c) for $BaF_{2}$. To compare with experimental data [@Burnett2001] (blue dots) the orientation of the $\mathbf{q}-$vector was chosen along the diagonal of the x-y plane. ](refractive_index_CaF2_and_BaF2)
 Plot of index of refraction $n\left(\omega\right)$ vs. free space wavelength $\lambda=\frac{2\pi c}{\omega}$ for $CaF_{2}$ and $BaF_{2}$. Displayed are values (dots) calculated from the macroscopic dielectric function (\[eq: macroscopic dielectric function arbitrary M\]) for $\mathbf{q}=\mathbf{0}$, solely with the lattice symmetry and the model for electronic polarizabilities (\[eq: isotropic Lorentz polarizability diagonal\]) as input, the respective values of the parameters $\alpha_{0}^{\left(j\right)}$ and $\omega_{0}^{\left(j\right)}$ as in Table \[fit\]. The relative error compared to a fit of experimental data for $n\left(\omega\right)$ with the multi-parameter Sellmeier formula [@Li1980] (solid lines) is less than 1%. The spatial dispersion induced birefringence $\Delta\text{n}$ as calculated from the $\mathbf{q}$-dependence of the macroscopic dielectric function (\[eq: macroscopic dielectric function arbitrary M\]) is displayed (red) in (b) for $CaF_{2}$ and (c) for $BaF_{2}$. To compare with experimental data [@Burnett2001] (blue dots) the orientation of the $\mathbf{q}-$vector was chosen along the diagonal of the x-y plane. ](induced_birefringence_CaF2)
 Plot of index of refraction $n\left(\omega\right)$ vs. free space wavelength $\lambda=\frac{2\pi c}{\omega}$ for $CaF_{2}$ and $BaF_{2}$. Displayed are values (dots) calculated from the macroscopic dielectric function (\[eq: macroscopic dielectric function arbitrary M\]) for $\mathbf{q}=\mathbf{0}$, solely with the lattice symmetry and the model for electronic polarizabilities (\[eq: isotropic Lorentz polarizability diagonal\]) as input, the respective values of the parameters $\alpha_{0}^{\left(j\right)}$ and $\omega_{0}^{\left(j\right)}$ as in Table \[fit\]. The relative error compared to a fit of experimental data for $n\left(\omega\right)$ with the multi-parameter Sellmeier formula [@Li1980] (solid lines) is less than 1%. The spatial dispersion induced birefringence $\Delta\text{n}$ as calculated from the $\mathbf{q}$-dependence of the macroscopic dielectric function (\[eq: macroscopic dielectric function arbitrary M\]) is displayed (red) in (b) for $CaF_{2}$ and (c) for $BaF_{2}$. To compare with experimental data [@Burnett2001] (blue dots) the orientation of the $\mathbf{q}-$vector was chosen along the diagonal of the x-y plane. ](induced_birefringence_BaF2)
Let us emphasize our approach warrants notably fewer fit parameters compared to a Sellmeier fit. For example, to reproduce the experimentally observed chromatic dispersion of $CsI$ over a wide frequency intervall, ranging from the ultraviolet to far-infrared, a satisfactory fit to the experimental data within our approach needs only two functions of the type (\[eq: isotropic Lorentz polarizability diagonal\]) to model the *induced electronic polarization* of individual ions and a third fit function of the type (\[eq: isotropic Lorentz polarizability off diagonal\]) to model the *ionic polarization* effect. So our fit relies only on $6$ parameters modelling the microscopic polarizabilities of atoms (ions, molecules, ion pairs) compared, for example, to the $17$ parameters required by the Sellmeier fit [@Li1976], describing chromatic dispersion of the refractive index of $CsI$.
Furthermore, for (ultraviolet) light propagating along the diagonal of the x-y plane, i.e. $\mathbf{q}=\frac{\left|\mathbf{q}\right|}{\sqrt{2}}\left(\mathbf{e}^{\left(x\right)}+\mathbf{e}^{\left(y\right)}\right)$, the dielectric tensor $\varepsilon_{ab}^{\left(T\right)}\left(\mathbf{q},\omega\right)$ reveals two transversal modes capable to propagate with slightly different speeds inside the crystals mentioned above, thus causing an intrinsic birefringence $\Delta n\left(\omega\right)$ induced by spatial dispersion. Our calculation of $\Delta n\left(\omega\right)$ for the afore mentioned ionic crystals and a comparison with experimental data can be found in Fig.\[Sellmeier\] and Fig.\[fig:CaF2\_and\_BaF2\]. The applied fit parameters entering the calculations of $n\left(\omega\right)$ and $\Delta n\left(\omega\right)$ are listed in Table \[fit\].
Having thus determined the model polarizabilities (\[eq: isotropic Lorentz polarizability diagonal\]) and (\[eq: isotropic Lorentz polarizability off diagonal\]) for each (different) atom species and each ion pair, the dependence of the dielectric function on wave vector $\mathbf{q}$ is in our approach already fixed by the crystalline structure of the material under consideration, i.e. the rotary power $\gamma_{abc}\left(\omega\right)$ and the dispersion induced anisotropy $\alpha_{abcd}\left(\omega\right)$ are already implicitely encoded in the $\mathbf{q}-$dependence of the transversal dielectric tensor $\varepsilon_{ab}^{\left(T\right)}\left(\mathbf{q},\omega\right)$. To what large extend our calculations agree with published experimental data over a wide range of optical frequencies for a series of quite different crystalline materials we summarize in Table \[results\], and in particular in Fig. \[Sellmeier\] and Fig.\[fig:CaF2\_and\_BaF2\].
While the refractive indices are deduced from the square root of the (real) eigenvalues of the transversal dielectric function for $\mathbf{q}=\mathbf{0}$, the rotary power is determined from the imaginary part of the off-diagonal elements of $\varepsilon_{ab}^{\left(T\right)}\left(\mathbf{q},\omega\right)$ for wave propagation along the crystals’ (optical) z-axis, i.e. $\mathbf{q}=\left|\mathbf{q}\right|\mathbf{e}^{\left(z\right)}$. The examples of crystal structures listed in Table \[results\] cover the cubic crystal system as well as all uniaxial crystal systems, where the number M of ions comprising the unit cell $C_{\Lambda}$ varies between $M=4$ (for e.g. hexagonal BeO) and $M=66$ (for e.g. cubic $Bi_{12}SiO_{20}$ and $Bi_{12}TiO_{20}$). It should be pointed out, that in contrast to the results shown in Fig.\[Sellmeier\] and \[fig:CaF2\_and\_BaF2\], our calculations for the refractive index as well as for the rotary power, both presented in Table \[results\], solely rest on published data of (anisotropic) electronic polarizabilities being reported in the particular cited references.
As a side remark let us point out, that the described principal effects of non locality, the optical activity $\gamma_{abc}\left(\omega\right)$ and/or the dispersion induced anisotropy $\alpha_{abcd}\left(\omega\right)$, at first sight being small effects compared to $n_{a}^{2}-1$ with $n_{a}^{2}$ representing the eigenvalues of the tensor $\varepsilon_{ab}^{\left(T\right)}\left(\mathbf{q},\omega\right)$ in ordinary crystalline materials, could well be comparable to $n_{a}^{2}-1$ in artificial periodic structures choosing appropriately taylored super lattices, see [@Gorlach2016].
crystal space group $\lambda$ (nm) $\alpha'=\frac{\alpha}{4\pi\varepsilon_{0}}$ ([Å]{}$^{3}$) $n^{(\text{exp})}$ $n^{(\text{calc})}$ $\rho^{(\text{exp})}$ ($\frac{\text{degree}}{\text{mm}}$) $\rho^{(\text{calc})}$ ($\frac{\text{degree}}{\text{mm}}$) references
--------------------- -------------- ---------------- ---------------------------------------------------------------- ---------------------- ---------------------- ----------------------------------------------------------- ------------------------------------------------------------ ---------------------------------------------------------------------
$\alpha$-SiO$_{2}$ P3$_{1}$21 508 $\alpha'_{\text{Si}}=0.207$ $n_{\text{o}}=1.548$ $n_{\text{o}}=1.543$ -29.73 -29.25 [@Gualtieri2000; @Devarajan1986; @Radhakrishnan1951; @Lowry]
$\alpha'_{\text{O}}=1.213$ $n_{\text{e}}=1.558$ $n_{\text{e}}=1.550$
$\beta$-SiO$_{2}$ P6$_{2}$22 517 $\alpha'_{\text{Si}}=0.185$ $n_{\text{o}}=1.536$ $n_{\text{o}}=1.534$ +33.6 +29.84 [@Wyckoff; @Devarajan1986; @1962; @Lowry]
$\alpha'_{\text{O}}=1.250$ $n_{\text{e}}=1.544$ $n_{\text{e}}=1.539$
TiO$_{2}$ P4$_{2}$/mnm 589 $\alpha'_{\text{Ti}}=0.1862$ $n_{\text{o}}=2.613$ $n_{\text{o}}=2.600$ / / [@Restori1987; @Parker1961; @DeVore1951]
$\left[\alpha{}_{O}^{\parallel}\right]'=2.6006$ $n_{\text{e}}=2.909$ $n_{\text{e}}=2.921$
$\left[\alpha{}_{O}^{\perp}\right]'=2.2863$
BeO P6$_{3}$mc 633 $\alpha'_{\text{Be}}=0.007$ $n_{\text{o}}=1.717$ $n_{\text{o}}=1.713$ / / [@Vidal-Valat1987; @Dimitrov2002; @Weber]
$\alpha'_{\text{O}}=1.290$ $n_{\text{e}}=1.732$ $n_{\text{e}}=1.717$
NaClO$_{3}$ P2$_{1}$3 633 $\alpha'_{\text{Na}}=0.290$ 1.514 1.526 +2.44 +3.76 [@Abrahams1977a; @Devarajan1986; @Chandrasekhar1967; @Abrahams1977]
$\alpha'_{\text{Cl}}=0.010$
$\alpha'_{\text{O}}=1.600$
SrTiO$_{3}$ Pm$\bar{3}$m 589 $\alpha'_{\text{Sr}}=1.0666$ 2.410 2.409 / / [@Abramov1995; @Chaib2004; @Weber]
$\alpha'_{\text{Ti}}=0.1859$
$\alpha'_{\text{O}}=2.3940$
Bi$_{12}$TiO$_{20}$ I23 633 $\alpha'_{\text{Bi}}=0.0625$ 2.562 2.553 -5.9 -6.12 [@Swindells1988; @Weber; @Feldman1970]
$\alpha'_{\text{Ti}}=0.272$
$\alpha'_{\text{O}}=3.725$
Bi$_{12}$SiO$_{20}$ I23 650 $\alpha'_{\text{Bi}}=0.150$ 2.52 2.50 -20.5 -19.35 [@Abrahams1979a; @Devarajan1986; @Weber; @Abrahams1979]
$\alpha'_{\text{Si}}=0.001$
$\alpha'_{\text{O}}=3.540$
$\alpha$-AlPO$_{4}$ P3$_{1}$21 633 $\alpha'_{\text{Al}}=0.050$ $n_{\text{o}}=1.524$ $n_{\text{o}}=1.541$ +14.6 +11.23 [@Thong1979; @Devarajan1986; @Weber; @Malgrange2014]
$\alpha'_{\text{P}}=0.050$ $n_{\text{e}}=1.533$ $n_{\text{e}}=1.545$
$\alpha'_{\text{O}}=1.370$
BaTiO$_{3}$ P4mm 589 $\alpha'_{\text{Ba}}=1.9460$ $n_{\text{o}}=2.426$ $n_{\text{o}}=2.400$ / / [@Xiao2008; @Chaib2004; @Wemple1968]
$\alpha'_{\text{Ti}}=0.1859$ $n_{\text{e}}=2.380$ $n_{\text{e}}=2.380$
$\alpha'_{\text{O}}=2.3940$
CaCO$_{3}$ R$\bar{3}$cH 589 $\alpha'_{\text{Ca}}=0.792$ $n_{\text{o}}=1.658$ $n_{\text{o}}=1.626$ / / [@Maslen1993; @Lawless1964; @Ghosh1999]
$\alpha'_{\text{C}}=0.000$ $n_{\text{e}}=1.486$ $n_{\text{e}}=1.513$
$\left[\alpha{}_{\text{O}}^{\left(||\right)}\right]'=1.384$
$\left[\alpha{}_{\text{O}}^{\left(\perp\right)}\right]'=1.328$
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ion/binding $\frac{\alpha_{0}}{4\pi\varepsilon_{0}}\;\left[{{\ifmmode\begingroup\def\b@ld{bold} $\omega_{0}\;\left[\text{eV}\right]$
\text{\ifx\math@version\b@ld\bfseries\fi\AA}\endgroup\else\AA\fi}}^{3}\right]$
---------------------------------- ------------------------------------------------------------------------------------- --------------------------------------
$\text{Ba}^{2+}$ 1.577 16.353
$\text{Ca}^{2+}$ 0.759 27.484
$\text{Cl}^{-}$ 4.500 12.959
$\text{Cs}^{+}$ 2.884 33.220
$\text{F}^{-}$ (in $BaF_{2}$) 1.165 15.789
$\text{F}^{-}$ (in $CaF_{2}$) 0.866 15.860
$\text{I}^{-}$ 6.241 8.253
$\text{Rb}^{+}$ 0.285 7.359
$\text{Cs}^{+}$— $\text{I}^{-}$ 1.519 0.012
$\text{Rb}^{+}$— $\text{Cl}^{-}$ 2.214 0.019
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: \[fit\] Estimated fit parameters applied to the electronic and ionic Lorentz oscillator models (\[eq: isotropic Lorentz polarizability diagonal\]) and (\[eq: isotropic Lorentz polarizability off diagonal\]), respectively, regarding our calculations of the refractive index $n\left(\omega\right)$ as well as the spatial-dispersion-induced birefringence $\Delta n\left(\omega\right)$, for ionic crystals $BaF_{2}$, $CaF_{2}$, $CsI$ and $RbCl$.
Static Limit of the Dielectric Function for Monoatomic Bravais Lattices {#static-limit-of-the-dielectric-function-for-monoatomic-bravais-lattices .unnumbered}
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Assuming for simplicity $M=1$, then there is no loss of generality setting $\boldsymbol{\eta}^{\left(1\right)}=\mathbf{0}$. Identifying then, see (\[eq:block matrix alpha\]), $$\begin{aligned}
\alpha_{a''a'}\left(\omega\right) & \equiv & \alpha_{a'',a'}^{\left(1,1\right)}\left(\mathbf{k},\omega\right)\\
I_{aa'} & = & \delta_{a,a'},\nonumber \end{aligned}$$ one readily infers from (\[eq: kernel K(G=00003D0,q,om)\]) the explicit representation $$\left[\bar{K}_{\Lambda}(\mathbf{q},\omega)\right]_{a',a''}=\left(\frac{\alpha\left(\omega\right)}{\varepsilon_{0}}\circ\left[I-\zeta_{\varLambda}^{\left(0\right)}(\mathbf{q},\omega)\circ\frac{\alpha\left(\omega\right)}{\varepsilon_{0}}\right]^{-1}\right)_{a',a''}.\label{eq: kernel K for M=00003D1}$$ Elementary matrix algebra leads then to the result $$\begin{aligned}
& & \varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)-I\nonumber \\
& = & \frac{1}{\left|C_{\Lambda}\right|}\frac{\alpha\left(\omega\right)}{\varepsilon_{0}}\circ\left[I-\zeta_{\varLambda}^{\left(0\right)}(\mathbf{q},\omega)\circ\frac{\alpha\left(\omega\right)}{\varepsilon_{0}}\right]^{-1}\circ\left[I+\frac{1}{\left|C_{\Lambda}\right|}\bar{\mathcal{G}}(\mathbf{q},\omega)\circ\frac{\alpha\left(\omega\right)}{\varepsilon_{0}}\circ\left[I-\zeta_{\varLambda}^{\left(0\right)}(\mathbf{q},\omega)\circ\frac{\alpha\left(\omega\right)}{\varepsilon_{0}}\right]^{-1}\right]^{-1}\nonumber \\
& = & \frac{1}{\left|C_{\Lambda}\right|}\frac{\alpha\left(\omega\right)}{\varepsilon_{0}}\circ\left[I-\left(\zeta_{\varLambda}^{\left(0\right)}(\mathbf{q},\omega)-\frac{1}{\left|C_{\Lambda}\right|}\bar{\mathcal{G}}(\mathbf{q},\omega)\right)\circ\frac{\alpha\left(\omega\right)}{\varepsilon_{0}}\right]^{-1}.\label{eq: dielectric function M=00003D1}\end{aligned}$$ In the supplemental material [@Supplementary] it is shown that $$\begin{aligned}
& & \left(\zeta_{\varLambda}^{\left(0\right)}(\mathbf{q},\omega)-\frac{1}{\left|C_{\Lambda}\right|}\bar{\mathcal{G}}(\mathbf{q},\omega)\right)_{a,a'}\label{eq: sum over reciprocal lattice vectors for dielectric function}\\
& = & \lim_{\left|\mathbf{s}\right|\rightarrow0^{+}}\left(\frac{1}{\left|C_{\Lambda}\right|}\sum_{\mathbf{G}\in\Lambda^{-1}\setminus\left\{ \mathbf{0}\right\} }e^{i\mathbf{\mathbf{G}}\cdot\mathbf{s}}\frac{\frac{\omega^{2}}{c^{2}}\delta_{a,a'}-\left(\mathbf{G}+\mathbf{q}\right)_{a}\left(\mathbf{G}+\mathbf{q}\right)_{a'}}{\left|\mathbf{\mathbf{\mathbf{G}+\mathbf{q}}}\right|^{2}-\frac{\omega^{2}}{c^{2}}-i0^{+}}-\int\frac{d^{3}k}{\left(2\pi\right)^{3}}e^{i\mathbf{k}\cdot\mathbf{s}}\frac{\frac{\omega^{2}}{c^{2}}\delta_{a,a'}-k_{a}k_{a'}}{\left|\mathbf{k}\right|^{2}-\frac{\omega^{2}}{c^{2}}-i0^{+}}\right).\nonumber \end{aligned}$$ The lattice sum (\[eq: sum over reciprocal lattice vectors for dielectric function\]) is conveniently evaluated along the lines indicated by Ewald [@Ewald1916; @Ewald1938], splitting the sum into two absolutely converging sums, one converging rapidly in the Fourier domain and the other converging rapidly in the spatial domain. For details and a discussion of our (modified) splitting method, see supplemental material [@Supplementary]. For simple cubic lattices the expression (\[eq: sum over reciprocal lattice vectors for dielectric function\]) can also be evaluated employing Jacobi theta functions [@Draine1993; @Borwein2013].
In the static limit, first $\left|\mathbf{q}\right|\rightarrow0$ and then $\omega\rightarrow0$, we write $$\lim_{\omega\rightarrow0}\lim_{\left|\mathbf{q}\right|\rightarrow0}\left[\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)-I\right]_{a,a'}=\left[\varepsilon_{\Lambda}-I\right]_{a,a'}.$$ A comprehensive analysis of the general properties of electromagnetic response functions, quite apart from a particular model description of a material and only based on general principles such as causality and thermodynamic stability, has been given by Kirzhnitz [@Kirzhnitz2012], who derived for the isotropic case $\left[\varepsilon_{\Lambda}\right]_{a,a'}=\varepsilon I_{a,a'}$ as a lower bound of allowed *static* values of the dielectric function: $$\begin{aligned}
\varepsilon & \geq & 1\label{eq: stabilty criterion Kirzhnitz}\end{aligned}$$
Introducing the (dimensionless) Lorentz factors $$\mathcal{L}_{a,a'}=\left|C_{\Lambda}\right|\lim_{\omega\rightarrow0}\lim_{\left|\mathbf{q}\right|\rightarrow0}\left[\zeta_{\varLambda}^{\left(0\right)}(\mathbf{q},\omega)-\frac{1}{\left|C_{\Lambda}\right|}\bar{\mathcal{G}}(\mathbf{q},\omega)\right]_{a,a'},\label{eq:Lorentz factor}$$ solely dependent on the lattice geometry, and identifying $\frac{1}{\left|C_{\Lambda}\right|}=\frac{\left|\Lambda_{P}\right|}{\left|\Omega_{P}\right|}=\nu_{P}$ with the *density* of polarizable atoms in the probe volume, there follows from (\[eq: dielectric function M=00003D1\]) for mono-atomic Bravais lattices $(M=1)$ the following exact formula for the static dielectric tensor: $$\left[\varepsilon_{\Lambda}\right]_{a,a'}=\left[\frac{I+(I-\mathcal{L})\frac{\alpha}{\varepsilon_{0}}\nu_{P}}{I-\mathcal{L}\frac{\alpha}{\varepsilon_{0}}\nu_{P}}\right]_{a,a'}\label{eq:static dielectric tensor}$$ While the polarizability $\alpha$ refers to individual atomic (ionic, molecular) properties, the dielectric tensor $\varepsilon_{\Lambda}$ also depends via the $3\times3$ matrix of Lorentz factors $\mathcal{L}$ on how the atoms are assembled to build the crystal $\Lambda$. Note that the particle density $\nu_{P}$ of a dielectric crystalline probe volume is bounded from above by the a critical value $\nu_{P}^{\left(c\right)}$, $$\begin{aligned}
\nu_{P} & < & \nu_{P}^{\left(c\right)}\equiv\frac{1}{\mathcal{L}_{max}}\frac{\varepsilon_{0}}{\alpha}\;,\end{aligned}$$ with $\mathcal{L}_{max}$ denoting the largest positive eigenvalue of the matrix $\mathcal{L}$. This condition indeed implies the matrix $\varepsilon_{\Lambda}-I$ being *positive definit*, thus generalizing the stability criterion (\[eq: stabilty criterion Kirzhnitz\]) to the anisotropic case.
It should be noted that the trace of the matrix $\mathcal{L}$ as defined in (\[eq:Lorentz factor\]) is normalized to unity: $$\textrm{tr}\left(\mathcal{L}\right)=1\label{eq: trace identity Lorentz factors}$$ For a proof see supplemental material [@Supplementary]. Numerical values for the Lorentz factors are readily calculated for all $14$ monoatomic Bravais lattices $\Lambda$ along the lines indicated in the supplemental material [@Supplementary].
In the special case of a lattice with tetragonal or hexagonal symmetry the matrix $\mathcal{L}$ becomes diagonal, $\mathcal{L}_{a,a'}=\delta_{a,a'}\mathcal{L}_{a}$. In this case the trace identity (\[eq: trace identity Lorentz factors\]) leads to $\mathcal{L}_{x}=\mathcal{L}_{y}=\frac{1}{2}\left(1-\mathcal{L}_{z}\right)$, with $\mathcal{L}_{z}$ being a universal function, solely dependent on the ratio $\frac{a_{z}}{a_{x}}$ of the lattice constants, $a_{z}$ parallel and $a_{x}$ perpendicular to the crystalline $z-$axis. In Fig. \[fig:Lorentz\_factors\] that function $\mathcal{L}_{z}$ is plotted vs. the ratio of lattice constants $\frac{a_{z}}{a_{x}}$ for simple tetragonal (st), body-centered tetragonal (bct) and also hexagonal (hex) lattice symmetry. Our results for the Lorentz-factors obtained with Eq. (\[eq:Lorentz factor\]) coincide with previous works, for example [@Colpa1971a; @Colpa1971; @Vanzo2014].
![\[fig:Lorentz\_factors\] Plot of Lorentz factor $\mathcal{L}_{z}$ for simple tetragonal (blue), body-centered tetragonal (red) and hexagonal (green) lattice symmetries vs. the ratio $\frac{a_{z}}{a_{x}}$ of lattice constants. The inset shows a zoom into the region where $\mathcal{L}_{z}$ assumes a value around $\frac{1}{3}$, characteristic for isotropic systems. ](Lorentz_factors)
### Clausius-Mossotti Relation {#clausius-mossotti-relation .unnumbered}
For cubic symmetry there holds $\mathcal{L}_{a,a'}=\frac{1}{3}\delta_{a,a'}$ and the static dielectric tensor (\[eq:static dielectric tensor\]) reduces to the well known Clausius-Mossotti relation for *isotropic* systems, see supplemental material [@Supplementary]: $$\begin{aligned}
\left[\varepsilon_{\Lambda}\right]_{a,a'} & = & \delta_{a,a'}\frac{1+\frac{2}{3}\frac{\alpha}{\varepsilon_{0}}\nu_{P}}{1-\frac{1}{3}\frac{\alpha}{\varepsilon_{0}}\nu_{P}}\label{eq: Clausius-Mossotti}\end{aligned}$$ Incidentally, the relation (\[eq: Clausius-Mossotti\]) applies for a wide class of (isotropic, non polar) materials, including dielectric liquids and gases. On a final note: our derivation of (\[eq: Clausius-Mossotti\]) completely avoids the usual trick of introducing the Lorentz sphere, where the medium outside of this sphere is considered as a continuum. For an in-depth explanation of that trick see for example [@Ashcroft1981].
Conclusions
===========
The field-integral equation approach presented in this article differs from traditional presentations of crystal optics, for example [@Authier2012; @Born1999; @Fluegge2013]. Based on the Helmholtz-Hodge theorem the source term in the microscopic Maxwell equations representing the current density has been decomposed into longitudinal and transversal parts, thus establishing the formulation of equivalent (inhomogenous) field-integral equations with a kernel modelling the induced microscopic polarization $\mathbf{\tilde{P}}\left(\mathbf{r},\omega\right)$ inside a dielectric crystal as a convolution integral of the dielectric susceptibility $\chi_{aa'}\left(\mathbf{r},\mathbf{r}',\omega\right)$ (\[eq: phenomenological dielectric kernel\]) and the Fourier amplitude $\mathbf{\tilde{E}}\left(\mathbf{r},\omega\right)$ of the microscopic local electric field, see (\[eq:polarization vs local field\]). Exploiting the lattice periodicity of the dielectric susceptibility tensor $\chi_{aa'}\left(\mathbf{r}+\mathbf{R},\mathbf{r}'+\mathbf{R},\omega\right)=\chi_{aa'}\left(\mathbf{r},\mathbf{r}',\omega\right)$ it is then natural to expand the field amplitude $\mathbf{\tilde{E}}\left(\mathbf{r},\omega\right)$ into a complete and orthonormal basis of eigenfunctions of the operator $T_{\mathbf{R}}$ generating translations by a lattice vector $\mathbf{R}\in\Lambda$, see (\[eq: shift operator\]). But instead of expanding the solution to the field-integral equations in the well known basis of plane waves $e^{i\left(\mathbf{q}+\mathbf{G}\right)\mathbf{r}}$, constructed from eigenfunctions of the *momentum* operator, thus requiring to handle for each wave vector $\mathbf{q}$ in the Brillouin zone $C_{\Lambda^{-1}}$ of the lattice $\Lambda$ then (infinite dimensional) matrices labelled by reciprocal lattice vectors $\mathbf{G},\mathbf{G}'\in\Lambda^{-1}$, we use non-standard Bloch functions representing a complete and orthonormal system of eigenfunctions of the translation operator $T_{\mathbf{R}}$ constructed from eigenfunctions of the *position* operator, see (\[eq: basis w(r,s,k)\]). Our choice of basis functions indeed enables to sidesteps the inversion (and truncation) of large matrices with regard to the reciprocal lattice vectors $\mathbf{G},\mathbf{G}'\in\Lambda^{-1}$, thus easing significantly the numerical effort.
Considering the *homogenous* field-integral equations the electromagnetic modes and the photonic band structure $\omega_{n}\left(\mathbf{q}\right)$ of a dielectric crystal have been identified solving a *small* sized $3M\times3M$ matrix eigenvalue problem, with $M$ denoting the number of polarizable atoms (ions, molecules) in the elementary cell $C_{\Lambda}$ of the lattice. A radiation damping term $\propto\omega^{3}$, that originates *not* from the damping terms in the atom-individual polarizabilities (\[eq: atom polarizability\]), but from the retarded Helmholtz propagator evaluating the lattice sum without the self field term, see (\[eq: Im Zeta\_0\]), has been taken into account in all our calculations. Exemplarily we then presented the photonic band structure of diamond ($M=2$). An overview of our results for the photonic bandstructure of (primitive) sc-, fcc- and bcc lattices we present in the supplemental material [@Supplementary]. In all cases we find quantitative agreement with previously published work on photonic band structures calculated with other methods.
Considering the *inhomogenous* field-integral equations, choosing a (circular) frequency $\omega$ and a wavevector $\mathbf{q}$ obeying to $\omega=\omega_{n}\left(\boldsymbol{q}\right)$, the microscopic local electric field amplitude $\mathbf{\tilde{E}}\left(\mathbf{r},\omega\right)$ in the presence of a slowly varying time harmonic external field with amplitude $\mathbf{\tilde{E}}_{ext}\left(\mathbf{r},\omega\right)$ is found to display in general a radiative transversal part (T) and also a longitudinal part (L), the size of the longitudinal amplitude $\mathbf{\tilde{E}}^{\left(L\right)}\left(\mathbf{r},\omega\right)$, see (\[eq: longitudinal microscopic field\]), compared to the size of the transversal amplitude $\mathbf{\tilde{E}}^{\left(T\right)}\left(\mathbf{r},\omega\right)$, see (\[eq:transversal microscopic field\]), being strongly depend on the density of polarizable atoms (ions, molecules) in the crystal, see Fig.\[fig:field\_contributions\_and\_index\]. In a sufficiently dense packed dielectric crystal we find the longitudinal amplitude $\mathbf{\tilde{E}}^{\left(L\right)}\left(\mathbf{r},\omega\right)$ being substantially larger compared to the transversal amplitude $\mathbf{\tilde{E}}^{\left(T\right)}\left(\mathbf{r},\omega\right)$, but in a dilute (artificial) superlattice with polarizable subunits positioned at the lattice sites the longitudinal field amplitude becomes small compared to the transversal one, see Fig.\[fig:field\_contributions\_and\_index\]. While the microscopic local electric field amplitude $\mathbf{\tilde{E}}\left(\mathbf{r},\omega\right)$ displays inside a densely packed lattice rapid spatial variations with large amplitude traversing a distance set by the microscopic inter-particle distance $a$, it turns out that the transversal (radiative) part $\mathbf{\tilde{E}}^{\left(T\right)}\left(\mathbf{r},\omega\right)$ essentially coincides with the slowly varying *macroscopic* electric field amplitude $\mathcal{\boldsymbol{\tilde{E}}}\left(\mathbf{r},\omega\right)$, that field $\mathbf{\tilde{\mathcal{\boldsymbol{E}}}}\left(\mathbf{r},\omega\right)$ being conceived as the low pass filtered microscopic local field amplitude, see (\[eq: low pass filtered microscopic electric field\]). If the external field $\mathbf{\tilde{E}}_{ext}\left(\mathbf{r},\omega\right)$ was purely transversal and slowly varying, then the amplitude of the residue $\delta\mathbf{\tilde{E}}^{\left(T\right)}\left(\mathbf{r},\omega\right)=\mathbf{\tilde{E}}^{\left(T\right)}\left(\mathbf{r},\omega\right)-\mathbf{\tilde{\mathcal{\boldsymbol{E}}}}^{\left(T\right)}\left(\mathbf{r},\omega\right)$ is tiny, see Fig.\[fig:local\_vs\_macro\_fields\].
Conceiving correspondingly the *macroscopic* Polarization $\tilde{\mathcal{\boldsymbol{P}}}\left(\mathbf{r},\omega\right)$ as the low pass filtered *microscopic* polarization $\mathbf{\tilde{P}}\left(\mathbf{r},\omega\right)$, see (\[eq: low pass filtered microscopic polarization\]), then the dielectric $3\times3$ tensor $\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)$ of macroscopic electrodynamics emerges from the requirement $\bar{\mathcal{\boldsymbol{P}}}\left(\mathbf{q},\omega\right)=\varepsilon_{0}\left[\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)-I\right]\mathcal{\bar{E}}\left(\mathbf{q},\omega\right)$, with $\bar{\mathcal{\boldsymbol{P}}}\left(\mathbf{q},\omega\right)$ and $\mathcal{\bar{E}}\left(\mathbf{q},\omega\right)$ denoting the spatial Fourier transformation of those amplitudes. The derived exact formula for the dielectric tensor $\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)$, see (\[eq: macroscopic dielectric function arbitrary M\]), then solely depends on the microscopic polarizabilities $\alpha_{aa'}\left(\boldsymbol{\eta}^{\left(j\right)},\boldsymbol{\eta}^{\left(j'\right)},\omega\right)$ of atoms (molecules, ions) together with the crystalline symmetry $\Lambda$ as input into the theory.
Regarding the propagation of light signals inside a dielectric crystal then the question was clarified, which parts of the dielectric tensor $\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)$ describe the renormalization of the speed of light and possibly birefringence, chromatic dispersion, rotary power and spatial-dispersion induced birefringence, and which parts govern electric-field screening. Accordingly we derived directly from the field-integral equations, see (\[eq: low pass filtered field integral equation\]), a set of (coupled) differential equations for the longitudinal and transversal components determining the macroscopic electric field $\mathcal{\boldsymbol{\tilde{E}}}\left(\mathbf{r},\omega\right)$ directly, without any prior knowledge of the microscopic local electric field $\mathbf{\tilde{E}}\left(\mathbf{r},\omega\right)$. If the external field was purely transversal then an effective wave equation for the transversal (radiative) macroscopic field emerged, thus identifying the pieces of $\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)$ comprising the transversal dielectric tensor $\varepsilon_{aa'}^{\left(T\right)}\left(\mathbf{q},\omega\right)$ , see (\[eq: transversal dielectric tensor\]). Conversely, if the external field was purely longitudinal, then an effective Poisson type equation for a scalar potential $\tilde{\phi}\left(\mathbf{r},\omega\right)$ turned up, so that the longitudinal macroscopic field is represented by $\mathbf{\tilde{\mathcal{\boldsymbol{E}}}}^{\left(L\right)}\left(\mathbf{r},\omega\right)=-\boldsymbol{\nabla}\tilde{\phi}\left(\mathbf{r},\omega\right)$, thus identifying the pieces of the dielectric tensor comprising the longitudinal dielectric tensor $\varepsilon_{aa'}^{\left(L\right)}\left(\mathbf{q},\omega\right)$ being liable for electric-field screening (like in electrostatics), see (\[eq:longitudinal dielectric tensor\]). In the static limit, $\mathbf{q}\rightarrow\mathbf{0}$ and $\omega\rightarrow0$, an exact expression for the dielectric tensor $\varepsilon_{\Lambda}$ is derived that applies for all $14$ mono-atomic Bravais lattices, see (\[eq:static dielectric tensor\]). Our result for $\varepsilon_{\Lambda}$ in particular conforms with general (thermodynamic) stability criteria [@Kirzhnitz2012], and for cubic symmetry the well known Clausius-Mossotti relation is recovered.
The Taylor expansion of the transversal dielectric tensor, $\varepsilon_{ab}^{\left(T\right)}\left(\mathbf{q},\omega\right)=\varepsilon_{ab}^{\left(T\right)}\left(\omega\right)+i\sum_{c}\gamma_{abc}\left(\omega\right)q_{c}+\sum_{c,d}\alpha_{abcd}\left(\omega\right)q_{c}q_{d}+..$ around $\mathbf{q}=\mathbf{0}$, provides then insight into various optical phenomena connected to retardation and non locality of the dielectric response, in full agreement with the phenomenological reasoning of Agranovich and Ginzburg [@Agranovich1984]: the eigenvalues of the tensor $\varepsilon_{ab}^{\left(T\right)}\left(\omega\right)$ describing chromatic dispersion of the index of refraction and birefringence, the first order term $\gamma_{abc}\left(\omega\right)$ specifying rotary power (natural optical activity) in crystals lacking inversion symmetry, the second order term $\alpha_{abcd}\left(\omega\right)$ shaping the (weak) effects of a spatial-dispersion induced birefringence, nowadays a critical parameter for the design of lenses made from $CaF_{2}$ and $BaF_{2}$ for optical lithograpy systems in the ultraviolet.
Considering various dielectric crystalline materials comprising atoms (molecules, ions) with known polarizabilities from the literature, in all cases the calculated indices of refractions, the rotary power and the dispersion induced birefringence have been shown to coincide well with the experimental data displayed in table \[results\], thus illustrating the utility of the theory. For ionic crystals, exemplarily for $CsI$ and $RbCl$, a satisfactory agreement between theory and the measured chromatic dispersion of the index of refraction is manifest over a wide frequency interval, ranging from ultraviolet to far infra-red, accomplishing this with an appreciably smaller number of adjustable parameters directly linked to microscopic atomindividual polarizabilities compared to the well known Sellmeier fit. Further we computed for the cubic fluorites $CaF_{2}$ and $BaF_{2}$ the frequency dependence of the intrinsic birefingence, as represented by three independent components of the tensor $\alpha_{abcd}\left(\omega\right)$, and found good agreement with published data [@Burnett2001; @Burnett2002].
Even though all our calculations are based on a phenomenological model of microscopic polarizabilities, see (\[eq: isotropic Lorentz polarizability diagonal\]) and (\[eq: isotropic Lorentz polarizability off diagonal\]), due to the conformity with the fundamental frequency dependence of atom-individual polarizabilities as predicted by quantum mechanics, see (\[eq: atom polarizability\]), our results for the dielectric tensor $\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)$ may well claim general validity in the range of optical frequencies and below. While the frequency dependence of $\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)$ is deeply anchored in the *retarded* response of polarizable atoms (ions, molecules) comprising a dielectric crystal, the dependence of $\varepsilon_{\Lambda}\left(\mathbf{q},\omega\right)$ on the wave vector $\mathbf{q}$ of the propagating light describes the effects attributed to the *non-locality* of that response, for instance optical activity and intrinsic birefringence, all theses effects being primarily dependent on details of the crystalline symmetry. Finally it should be noted, that our theory of the macroscopic electric field doesn’t make use of the notion of a displacement field $\mathbf{D}$, and therefore we avoided to mention it.
### Outlook {#outlook .unnumbered}
We are confident the presented field-integral equation approach can be extended to calculations of the microscopic local electric field near to the surface of a dielectric crystal and also to thin films. In addition we consider it possible to extend our approach to disordered systems, for instance crystals subject to substitutional site disorder, thus enabling a theory of the dielectric tensor for disordered systems within the frame of the coherent potential approximation (CPA).
The corresponding author thanks Oleg Dolgov for insightfull remarks and for bringing the references [@L.V.Keldysh2012; @Maksimov2012; @Kirzhnitz2012] to his attention, all of this a good while ago. He also thanks Mario Liu for sharing with him over the years his thoughts on macroscopic electrodynamics. Last not least we thank Oliver Eibl, Klaus-Peter Federsel, József Fortágh, Reinhold Kleiner and Claus Zimmermann for useful discussions.
[^1]: We conform to the convention, that the Fourier transformation of a function $F(t)$ of time $t$ and its Fourier inverse $\tilde{F}\left(\omega\right)$ as a function of (circular) frequency $\omega$ are defined by $$\begin{aligned}
F\left(t\right) & = & \int_{-\infty}^{\infty}\frac{d\omega}{2\pi}e^{-i\omega t}\tilde{F}\left(\omega\right)\\
\tilde{F}\left(\omega\right) & = & \int_{-\infty}^{\infty}dte^{i\omega t}F\left(t\right)\:,\end{aligned}$$ whereas the Fourier transformation of a function $f(\mathbf{r})$ of position $\mathbf{r}$ and its Fourier inverse $\bar{f}(\mathbf{q})$ as a function of wave vector $\mathbf{q}$ are defined by: $$\begin{aligned}
f\left(\mathbf{r}\right) & = & \int\frac{d^{3}q}{\left(2\pi\right)^{3}}e^{i\mathbf{q}\cdot\mathbf{r}}\bar{f}\left(\mathbf{q}\right)\\
\bar{f}\left(\mathbf{q}\right) & = & \int d^{3}re^{-i\mathbf{q}\cdot\mathbf{r}}f\left(\mathbf{r}\right)\end{aligned}$$
[^2]: The expansion basis system $\left\{ w\left(\mathbf{r};\mathbf{s},\mathbf{k}\right)\right\} _{\mathbf{k}\in C_{\Lambda^{-1}},\mathbf{\mathbf{s}}\in C_{\Lambda}}$ also could be used to advantage reducing the computational effort determining the band structure of a massive particle moving in a periodic array of *point* scatterers, rewriting the Schr[ö]{}dinger eigenvalue problem as an integral equation, akin to the KKR method, for example see [@Ashcroft1981].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Bones are always wrapped by soft tissues. As a result, bones in their X-ray images are obscured and become unclear. In this paper, we tackle this problem and propose a novel task to virtually decompose the soft tissue and bone by image processing algorithms. This task is fundamentally different from segmentation because the decomposed images share the same imaging domain. Our decomposition task is also fundamentally different from the conventional image enhancement. We propose a new mathematical model for such decomposition. Our model is ill-posed and thus it requires some priors. With proper assumptions, our model can be solved by solving a standard Laplace equation. The resulting bone image is theoretically guaranteed to have better contrast than the original input image. Therefore, the details of bones get enhanced and become clearer. Several numerical experiments confirm the effective and efficiency of our method. Our approach is important for clinical diagnosis, surgery planning, recognition, deep learning, etc.'
author:
- 'Yuanhao Gong [^1]'
bibliography:
- 'IEEEabrv.bib'
- 'IP.bib'
title: ' Decompose X-ray Images for Bone and Soft Tissue'
---
[Yuanhao: Asymmetric Kernel]{}
X-ray; bone; soft tissue; Laplace equation
Introduction {#sec:intro}
============
-ray has being frequently used in biomedical imaging and clinical diagnosis, especially for bone research and human body diagnosis. X-ray has been studied and developed since 1895. Nowadays, it has become a popular way for bone diagnosis in clinical applications.
Bones are very common in mammals. The bone skeleton provides the basic structure support for mammals such that the body can keep the rigidity when it changes its status (walking, running, or dancing). Bones are also important for the mammals’ health. This is one reason that bone research is an important topic for human.
Meanwhile, bones in mammals are usually wrapped by various soft tissues. Such soft tissue helps the growth and development of bones. It also provides some protection for the bones, reducing the possible external force and pressure.
These bones and their surrounding soft tissues might have various thickness and density. Such property can be used to distinguish them from each other, especially in their images.
In the X-ray imaging, X-rays are absorbed and scattered by the soft tissue. X-rays are also (significantly) reduced by the dense bones. When the X-rays finally reach the sensors, the images for bones and soft tissues show very different features.
Dense structure such as bones blocks more X-ray and its image region on the sensor receives less X-ray. Therefore, it is darker than other regions. In contrast, less dense and thin soft tissue does not block much X-ray and its image region on the sensor receives more X-ray.
To better visualize the bone region, modern X-ray images usually take the residual between a constant maximum value (caused by the given dose) and the original X-ray image, leading to a brighter bone region and a darker soft tissue region.
$\Longrightarrow $
$\Longrightarrow $
In such X-ray image, bone region has higher image intensity and can be seen clearly. And we can set different intensity range to visualize the bone region (Window Technique). However, such visualized bone regions are still composed by bones and soft tissues because they are overlapped from the sensor point of view.
To obtain better bone visualization, some other methods try to increase the image contrast, such as histogram equalization, Contrast Limited Adaptive Histogram Equalization (CLAHE), etc. However, they [**simultaneously**]{} enhance the bone and soft tissue. Even though the image becomes visually better, the relationship between image intensity and actual X-ray dose becomes complex (even unknown), leading to difficulties for clinical diagnosis.
To tackle these problems, we propose to estimate the soft tissue image and bone image simultaneously without losing the linear relationship between image intensity and physical property of the imaging objects. Two examples from our method is shown in Fig. \[fig:illu\]. The bone details are enhanced, which is theoretically guaranteed. The details of our method will be explained in later sections.
Scattering Light in Physics
---------------------------
As shown in the left column of Fig. \[fig:illu\], bones are usually surrounded by the soft tissue. This physical configuration is similar with many natural scenes. One example is the foggy weather, as shown in Fig. \[fig:phy\] (a). The fog can be considered as “soft tissue” (low density) and the buildings can be considered as “bone” (high density).
The physics behind this phenomena is the scattering light [@Hulst]. Scattering light is common in various scenarios, such as X-ray images in clinics, foggy weather in natural scene, and fluorescence images in biological images [@gong:gdp; @gong:phd]. The scatter light might downgrade the image quality. For example, the soft tissue in human body scatters the X-ray, making the bone details unclear.
The scattering light has been studied long time ago. In 1871, Rayleigh studies this physics when the wavelength of light is larger than the radius of particles in the medium [@Hulst]. The more general case with ball shape particles were studied by Mie and it is called Mie scattering. These physics achievements give the theoretical foundation for modern image dehazing approaches.
Scattering Light in Natural Images
----------------------------------
A typical example of scattering light in nature is the fog. As a result, the image from foggy natural scene is not clear. Image processing algorithms that virtually remove the fog are called dehazing (as shown in the top row of Fig. \[fig:phy\]).
dehaze\
$\Longrightarrow $
decompose bone and soft tissue\
$\Longrightarrow $
For natural images, the dehazing mathematical model is simplified as [@fattal:2008; @DarkPrior] $$\label{eq:model}
f(x,y)=J(x,y)t(x,y)+A(1-t(x,y))\,,$$ where $f$ is the observed image, $J$ is the unknown clear image to be estimated, $t$ is the transmission map to be estimated, and $A$ is the global atmospheric light to be estimated. In the past few years, dehazing algorithms have made a significant progress. These methods can be categorized into three types: simple contrast enhancement [@Tan2008; @fattal:2008], dark channel based methods [@DarkPrior] and deep learning methods [@Cai2016].
Early work considers the foggy images do not have enough contrast and they simply increase the contrast [@Tan2008; @fattal:2008]. Such methods are suffering from heavy computation and usually have obvious artifacts in the result.
One important achievement is the dark channel prior based dehazing methods [@DarkPrior]. Dark channel states that there must be a low value intensity in a local neighborhood region. And the resulting model can be efficiently solved by the guided image filter, which popularizes the dark channel prior.
Deep learning is another type of scattering light removal methods [@Cai2016; @Engin2018]. It assumes the clear and foggy image pairs are given and the process that maps the foggy image to its corresponding clear image can be implicitly learned by a neural network. In such methods, the foggy images are usually synthetic. The paired images in practical applications are difficult to be obtained.
Bone Suppression
----------------
Instead of interested by bones, some applications focus on the soft tissue such as pneumonia. For these applications, they try to reduce the visualization of bones. Such task is called bone suppression [@Suzuki2006; @Chen2014; @Li2020].
Such methods require strong prior information about the imaging objects, such as the rib shapes for Chest X-ray images. And they usually require to exactly find the bone boundaries (bone segmentation). Such methods are difficult to be extended from one imaging object to other imaging objects. For example, the methods developed for rib can not easily be used for feet or knees images.
Even with the accurate bone segmentation, the resulting soft tissue images may have obvious artifacts because their assumptions are not always valid for the input images.
These limitations motivate us to develop a new and generic mathematical model. Instead of suppressing bones or soft tissue, our model decompose one X-ray image into one soft tissue image and one bone image. These two images have exactly the same imaging domain. Our task is fundamentally different from bone enhancement task and bone suppression task. In fact, our method [**simultaneously**]{} does bone enhancement and bone suppression. As illustrated in Fig. \[fig:role\], our soft image can be considered as bone suppression while our bone image can be considered as bone enhancement.
Motivation and Contributions
----------------------------
The soft tissue in human body usually scatters the X-ray, severely reducing the quality of bone details in the resulting images. This fact motivates us to construct a novel mathematical model that can decompose bones and the soft tissue in X-ray images. The decomposed soft tissue image can be used for its related study such as pneumonia. The bone image can be adopted for its related research such as bone fracture.
The scattering light in X-ray images by the soft tissue shares the same physical law as the fog in natural images [@gong:gdp; @gong:phd]. Thus, the dehazing methods that have been developed for natural foggy images must be also valid on X-ray images [@Gong2019]. We adopt the transfer learning in machine learning community by applying the dehazing model onto X-ray images.
Different from the bone enhancement or suppression, we propose to decompose the input X-ray image into one bone image and one soft tissue image. Such task is named as Bone and Soft Tissue Decomposition (BSTD). We construct a new mathematical model that can effectively decompose the soft tissues in X-ray images. Our method decomposes the input X-ray image into background image (soft tissue) and bone image. Be aware the difference between our model and the bone segmentation task. Bone segmentation separates the imaging domain into bone region and background region (without overlap). However, our background and bone images share the same imaging domain (exactly overlapped with the same imaging domain). Such difference is illustrated in Fig. \[fig:seg\].
Our contributions are in following folds:
- We propose a new image processing task named as Bone and Soft Tissue Decomposition (BSTD).
- We propose a new mathematical model for BSTD. This model is based on the well known image dehazing model, but with proper assumptions for X-ray images.
- With some assumptions, the BSTD model leads to a standard Laplace equation, which can be efficiently solved.
- The resulting bone image is theoretically guaranteed to have better image contrast than the original input image.
Our Method
==========
In this section, we first show the novel mathematical model that decomposes the soft tissue and the bones. Then, we introduce some proper assumptions to make our model well-posed. With these assumptions, our model leads to a standard Laplace equation, which has an efficient solver. Finally, our model can be efficiently solved. Our method estimates one soft tissue image and one bone image, which are bone suppression and bone enhancement, respectively. Moreover, the bone image is theoretically guaranteed to have better image contrast and the original input. We will prove this property in later section.
Mathematical Model
------------------
We modify the dehazing model for natural images in Eq. \[eq:model\] to develop our model for X-ray images. First, we define the soft tissue image (background image) as $$S(x,y)=A(1-t(x,y)),\,\mathrm{where}\, A=1\,.$$ Here, we assume $A=1$. The reason is that only X-ray can reach the sensors (there is no other light resource). We further define the unknown bone image $U(x,y)$ as a linear scaling of $J(x,y)$ $$U(x,y)=\frac{1}{\alpha}J(x,y)\,,$$ where $\alpha\ge 0$ is a scalar parameter. Taking these two equations into Eq. \[eq:model\], we propose following model for X-ray images: $$\label{eq:ourmodel}
f(x,y)=\frac{1}{\alpha}U(x,y)(1-S(x,y))+S(x,y)\,,$$ where $f(x,y)\in [0,1]$ is the observed image, $U$ is the unknown bone image, $S(x,y)$ is the background image, $\alpha\ge 0$ is a scalar parameter.
Our model keeps the physical meaning of the image intensity. When $f(x,y)=S(x,y)$, it would force $U(x,y)=0$. It means that the observation only comes from the background. When $f(x,y)=\frac{1}{\alpha}U(x,y)$, it would force $S(x,y)=0$. It indicates that the observation only comes from bones. Otherwise, the observation is composed by the background and bones as the similar way in natural images.
We use $\alpha$ as global constant variable, instead of spatially varying $\alpha(x,y)$. Although $\alpha(x,y)$ could achieve better visual result, it might introduce artifacts and it would lose the relationship between actual dose and image intensity in X-ray image. But when we use spatially constant $\alpha$, such linear scaling will keep such relationship between the actual physics and the intensity in X-ray images.
In later section, we will prove that $\alpha\ge 1$ (Eq. \[eq:alpha\]), which theoretically guarantees to increase the image contrast. This property becomes clear when we set the background $B(x,y)=0$. That is $f(x,y)=\frac{1}{\alpha}U(x,y)$. It means $\nabla U(x,y)=\alpha \nabla f(x,y)$, where $\nabla$ is the standard gradient operator. Therefore, the contrast in bone image $U(x,y)$ is theoretically larger than the contrast in the input image $f(x,y)$. This theoretical property is numerically confirmed by all our experiments.
In our model, if the background image $S(x,y)$ is already known, the bone image can be easily computed. Therefore, we can solve our model by finding the $S(x,y)$ and $\alpha$. We first introduce some assumptions for our model. These assumptions will be used to estimate $S(x,y)$ in the following section.
Assumptions
-----------
Since our model (Eq. \[eq:ourmodel\]) is ill-posed, we have to make some assumptions to solve this model. First of all, we assume $S(x,y)\le f(x,y)$. This assumption makes sure that $U(x,y)\ge 0$.
Second, we assume $0\le S(x,y)<1$, which avoids the denominator to be zero. As a result, $\frac{1}{1-S(x,y)}>1$, which helps in improving the bone image contrast. Based on this assumption, we can prove $\alpha\ge 1$ in later sections (Eq. \[eq:alpha\]).
Third, background image $S(x,y)$ is smooth, especially for soft tissue region. More specifically, we assume $S(x,y)$ is second order differentiable [@GONG2019329; @gong:cf]. Such smoothness assumption is reasonable because the physical configuration of soft tissue is always smooth. As shown in following section, $S(x,y)$ can be obtained by solving a Laplace equation.
Fourth, we assume that the maximum value in $U(x,y)$ is one, to determine the value of $\alpha$. In later section, we can prove that $\alpha\ge 1$. Therefore, for most pixel locations (statistically), the contrast in the resulting bone image is always larger than the contrast in the original input image. In other words, the bone image is enhanced.
Soft Tissue Image
-----------------
Now, we have enough assumptions to find $S(x,y)$. We estimate the background image by a two-step strategy. First, we roughly estimate a mask that covers bones. Be aware that the mask only needs to cover the bones. It does not necessarily align with bones’ boundary. Therefore, there are multiple ways to generate such mask. It can be easily obtained by a simple threshold method followed by morphology operations. It can also be estimated by active contour methods. It can even be given interactively by users. In short, the way of obtaining this mask is flexible. Be aware that our mask only needs to cover the bones, but does not need to align the mask’ boundary exactly on the bones’ boundary. Thus, it is much easier than the bone segmentation task.
Second, we find $S(x,y)$ by solving a Laplace equation. Let $M(x,y)$ denote our mask. Now, we need to estimate the soft tissue intensity in this mask. This problem can be modeled as following minimization task $$\label{eq:poisson}
\min \int\int_{M}||\nabla S||^2\,,\mathrm{s.t.}\,S_{\partial M}=f_{\partial M}\,,$$ where $\partial$ denotes the boundary. The optimal solution of this energy is the standard Laplace equation $$\label{eq:poisson2}
\Delta S_M=0\,,\mathrm{s.t.} \,S_{\partial M}=f_{\partial M}$$ This equation can be efficiently solved by the convolution pyramid method [@Farbman2011], which has linear computational complexity. The estimated background image is shown in Fig. \[fig:exam\](c).
Solving Eq. \[eq:poisson2\] is numerically efficient. The running time is 0.1 seconds in MATLAB on a ThinkPad P1 laptop with Intel Xeon E2176 CPU. The image resolution is $1022\times 757$. Such performance is enough for clinical applications in practice.
Bone Image
----------
After estimating $S(x,y)$, we need to estimate $\alpha$ for bone image $U(x,y)$ estimation. As mentioned, we assume the maximum value in $U(x,y)$ is one. Therefore, we define $$\label{eq:alpha0}
\alpha\equiv\frac{1}{\max\{\frac{f(x,y)-S(x,y)}{1-S(x,y)}\}}\,.$$ Since $0\le f(x,y)\le 1$, we have $\frac{f(x,y)-S(x,y)}{1-S(x,y)}\le 1$. As a result, we can prove $$\label{eq:alpha}
\alpha\ge 1\,.$$. This parameter linearly increases the contrast in the bone image. As mentioned, such linearity can keep the physical meaning of intensity in X-ray images.
Finally, the bone image can be computed as (shown in Fig. \[fig:exam\](d) and Fig. \[fig:examB\](c)) $$\label{eq:ourbone}
U(x,y)=\alpha\frac{f(x,y)-S(x,y)}{1-S(x,y)}\,.$$
One example is shown in Fig \[fig:exam\]. And the middle line intensity profiles of original and our results are shown in Fig. \[fig:examB\]. In this case, $\alpha=1.44$ and the image contrast is enhanced. As shown in Eq. \[eq:alpha\], $\alpha\ge 1$ and the enhancement is theoretically guaranteed. All our experiments also confirm this property.
Now, let us study the gradient of the resulting bone image to show the image contrast enhancement. From Eq. \[eq:ourbone\], we can get the gradient of $U$ $$\nabla U=\alpha[\frac{\nabla f}{1-S}-\frac{1-f}{(1-S)^2}\nabla S]\,.$$ Since we assume $S(x,y)$ is smooth, we know that $\nabla S(x,y)\approx 0$ for most locations [@gong:gdp] (see the gradient statistics in Fig.5 from Ref. [@GONG2019329]). Therefore, we have $$\nabla U\approx\alpha\frac{\nabla f}{1-S}\ge \alpha\nabla f\ge \nabla f\,.$$ This result indicates that the bone image has better image contrast than the input image for most of pixels. Moreover, the larger $\alpha$, the better bone image contrast.
Model Solver Summary
--------------------
In summary, our model in Eq. \[eq:ourmodel\] can be efficiently solved by Algorithm \[algo:HW\].
input X-ray image $f(x,y)$ obtain the mask $M(x,y)$ by active contour or user input compute $S(x,y)$ by solving Eq. \[eq:poisson2\] compute $\alpha$ by Eq. \[eq:alpha0\] compute $U(x,y)$ by Eq. \[eq:ourbone\] $S(x,y)$, $U(x,y)$
Experiments
===========
We performed three experiments for our method. First, we perform our method on several X-ray images, showing our method is not restricted by specific imaging objects. Second, we compared our method with image enhancement method and dehazing method, showing that our modification of the original dehazing indeed helps in this task. Third, we perform our method on a hand X-ray image dataset, showing its effectiveness and efficiency.
Several results from our method are shown in Fig. \[fig:ex\]. The left column is the original input image. The right two columns are the soft tissue and bone image, respectively. It can be told that the soft tissue image is smooth as we assumed. Meanwhile, the bone image has better image contrast as desired. Moreover, our method can reach real-time performance on these X-ray images. The running time of our method on these images is reported in Table \[tab:1\].
image resolution time (seconds)
-------------------- ---------------- ----------------
Fig. \[fig:ex\](a) 319$\times$442 0.031
Fig. \[fig:ex\](d) 193$\times$382 0.019
Fig. \[fig:ex\](g) 514$\times$711 0.094
Fig. \[fig:ex\](j) 336$\times$471 0.041
: The running time in seconds of our algorithm
\[tab:1\]
We further compare our method with a classical image enhancement method and a dehazing method for natural images, which uses dark channel prior [@DarkPrior]. The results are shown in Fig. \[fig:exp1\] and \[fig:exp2\]. The classical image enhancement method (histogram equalization) enhances both the soft tissue and bones. And relationship between image intensity and physical X-ray is lost.
Our model is different from the dehazing model. The dehazing method for natural images can not completely remove the soft tissue in X-ray image, as shown by the red arrows in Fig. \[fig:exp1\] and \[fig:exp2\]. In contrast, our method does not have this issue. This is because we estimate a better soft tissue image. Moreover, our bone image has better image contrast, which is theoretically guaranteed as described in previous sections.
In the third experiment, we applied our method on a hand X-ray image data set (RSNA), which contains more than 10,000 hand X-ray images. And the image has high resolution (usually larger than $1514\times2044$). These images are collected from clinical applications. Therefore, we can apply our method on these practical images, showing the efficiency and effectiveness of our method on real high resolution images.
In each panel of Fig. \[fig:last\], the input image (left) is decomposed into soft tissue (middle) and bone image (right) by our method. Although we only show the first ten images from the data set, the results for the rest images are similar.
The bone images have better image contrast since the parameter $\alpha\ge 1$ is theoretically guaranteed. Such enhancement can also be directly told by radiologists. Such enhancement is good for bone diagnosis in practical applications.
Moreover, the running time of our method on such high resolution images is less than half second in the MATLAB language on a laptop. Therefore, it can achieve higher performance on a better hardware in real applications. If higher performance is required, our model can be solved by the parallel Laplace equation solver on a modern graphic process unit (GPU), which usually has thousands of cores.
We believe that such bone and soft tissue decomposition is important for X-ray images, bone study, soft tissue diagnosis, etc. And the mathematical model can be very efficiently solved by solving a Laplace equation.
Conclusion
==========
In this paper, we propose to decompose one X-ray image into a soft tissue image and a bone image. We name this task as Bone and Soft Tissue Decomposition (BSTD). For this task, we develop a novel mathematical model. Our mathematical model is inspired by the natural dehazing model, but with proper extension for X-ray images.
With several assumptions, our model leads to a Laplace equation, which can be efficiently solved. Solving the 2D Laplace equation is a classical problem. And we use the wavelet solver developed in [@Farbman2011] to solve this equation. After solving this equation, we obtain the soft tissue image.
With the soft tissue image and the original input image, we can compute the scaling parameter $\alpha$. After getting the value of $\alpha$, we can compute the bone image with a close form solution expression. The bone image is uniquely determined by the soft tissue image.
The resulting bone images are theoretically guaranteed to have better image contrast (larger gradient) because of $\alpha\ge 1$. Several numerical experiments have confirmed this property. Better image contrast is important for clinical diagnosis, such as bone fracture and surgery planning.
Our method can enhance the details on bones in X-ray images, without losing the relationship between the intensity and actual physical X-ray received on the sensor. This property is different from the conventional image enhancement methods. Our result can improve other bone related tasks, such as bone segmentation, recognition, diagnosis, surgery planning, etc.
Moreover, our method is numerically fast. It can process 0.77 Million pixels per second in MATLAB software on a ThinkPad P1 laptop with Intel Xeon E2176 CPU. For real X-ray images with resolution $2044\times1514$, our method only requires 0.35 seconds to finish the bone and soft tissue decomposition task. Our method can be applied in a large range of applications. It can be used for bone study, for example, bone fracture diagnosis. It can also be used in bone age assessment, reducing the influence of soft tissue. Our method can also be used for applications where the soft tissue is the main concern, for example, pneumonia in chest X-ray images. Our method can be used as a pre-processing approach for deep learning training data set preparation. false In the future, we plan to solve our mathematical model by modern convolution neural networks (CNN). Thanks to their excellent achievements in the past few years, CNN have been used in many different image processing and computer vision tasks. Our mathematical model is the loss function in such CNN. And our results from X-ray images can be used as training ground truth. We believe that the CNN can learn to generate the soft tissue and bone images from one input X-ray image. The network can be trained on the paired data $(f_i, S_i, U_i)$, where $f_i$ is the input image and $S_i$, $U_i$ are results from our method.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was supported in part by the National Natural Science Foundation of China under Grant 61907031.
[^1]: Manuscript received April 19, 2005; revised September 17, 2014.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We propose a generic topological insulator bilayer (TIB) system to study the excitonic condensation with self-consistent mean-field (SCMF) theory. We show that the TIB system presents the crossover behavior from the Bardeen-Cooper-Schrieffer (BCS) limit to Bose-Einstein condensation (BEC) limit. Moreover, by comparison with traditional semiconductor systems, we find that for the present system the superfluid property in the BEC phase is more sensitive to electron-hole density imbalance and the BCS phase is more robust. Applying this TIB model into Bi$_{2}$Se$_{3}$-family material, we find that the BEC phase is most probable to be observed in experiment. We also calculate the critical temperature for Bi$_{2}$Se$_{3}$-family TIB system, which is $\mathtt{\sim}100$ K. More interestingly, we can expect this relative high-temperature excitonic condensation since our calculated SCMF critical temperature is approximately equal to the Kosterlitz-Thouless transition temperature.'
author:
- Zhigang Wang
- Ningning Hao
- 'Zhen-Guo Fu'
- Ping Zhang
title: Excitonic condensation for the surface states of topological insulator bilayers
---
[^1]
Introduction
============
Recent technological advances in microfabrication bring growing interests in studying exciton condensation in different bilayer physical systems such as the semiconductor electron-hole bilayers [@Snoke; @Sahin; @Zhu] and graphene bilayers [@CHZhang; @MacDonald]. A number of novel physical phenomena are obtained in these systems, such as the BCS-BEC crossover [@Comte] as well as the subtle phase transition in the crossover region induced by the density imbalance [@Strinati], the dark and bright excitonic condensation under spin-orbit coupling [@Can], anomalous exciton condensation in high Landau levels in magnetic field [@MacDonald], room-temperature superfluidity in graphene bilayers [@Mac], etc. The conventional electron-hole bilayers are fabricated with semiconductor heterostructures such as GaAs/AlGaAs/GaAs. The character of the semiconductor electron-hole bilayers is that the electron and hole bands are quadratic ones with different effective masses, which means missing particle-hole symmetry in these kinds of systems and small superfluid density. Hence, in semiconductor electron-hole bilayers, the excitonic condensation needs very low temperature. Another better candidate for electron-hole bilayers is graphene, which has a two-dimensional (2D) massless linear Dirac-band structure in low energy limit. However, the coupling between different Dirac-cone structures in the same Brilliouin zone brings flaw to graphene to fabricate electron-hole bilayers [@Franz].
On the other hand, another growing interest in condensed matter physics is the very recent theoretical prediction [@Bernevig] and experimental verification [@Konig] of the topological insulators [@Kane] (TIs) with strong spin-orbit interaction. Several three-dimensional (3D) solids, such as Bi$_{1-x}$Sb$_{x}$ alloys, Bi$_{2}$Se$_{3}$-family crystals, have been identified [@Fu; @Hsieh; @HJZhang; @Xia; @Chen] to be strong TIs possessing anomalous band structures. The energy scale for the surface states of these 3D TIs is dominated by the $k$-linear spin-orbit interaction. Especially, the strong TIs surface has single Dirac-cone band structure which is also different from graphene. As a result, it is expected that the excitonic condensate of these topological surface states probably have new characters.
![(Color online) Left panel: Schematic structure of double-well topological insulators in $x$-$y$ plane. The external gates can independently tune the electron and hole densities. Right panel: The linear energy dispersion around the Dirac point of the electrons and holes. []{data-label="f1"}](fig1.eps){width="0.6\linewidth"}
Inspired by this expectation, in this paper we propose a topological insulator bilayer (TIB) model analogous to Ref. [@Franz1], a gated double TI layers separated by an insulating spacer. Using this TIB model, we numerically study the excitonic condensation of TI surface states. We find that the system also presents BCS-BEC crossover along with the change in carrier densities in zero temperature limit. However, there are two characters different from those of conventional excitonic condensation in semiconductor bilayer systems. The first is that the BCS phase of TIB is more robust than that of the semiconductor bilayer systems; the second is that the superfluidity of the TIB is more sensitive to the electron-hole density imbalance than that of the semiconductor bilayer systems. These two characters physically root in the $k$-linear band dispersion of the TIB. Moreover, by putting this TIB model in Bi$_{2}$Se$_{3}$-family material, we investigate the excitonic condensation and only find the BEC phase occurring due to the values of the parameters of the material. The critical temperature of excitonic condensation in Bi$_{2}$Se$_{3}$-family TIB is also calculated in the self-consistent mean-field (SCMF) approximation ($\sim$ $100$ K), which is found to be higher than that in the traditional semiconductor electron-hole bilayers. More interestingly, we can expect this relative high-temperature excitonic condensation since our calculated SCMF critical temperature is approximately equal to the Kosterlitz-Thouless (KT) transition temperature.
The TIB Model
=============
The TIB system is schematically illustrated in the left panel in Fig. \[f1\]. Two TI films are separated by an insulating spacer of thickness $d$, and the electron (hole) density can be independently tuned by the external gate voltage $V_{1}$ ($V_{2}$). The linear dispersions of the TIs around Dirac point are cartoonishly depicted in the right panel in Fig. \[f1\]. The grand-canonical Hamiltonian describing this TIB system can be written as $$\begin{aligned}
H & =-\sum_{p,\mathbf{k,}\sigma}\mu_{p}\hat{p}_{\mathbf{k}\sigma}^{\dag}\hat{p}_{\mathbf{k}\sigma}+\sum_{p,\mathbf{k}}\hslash v_{F}^{p}\left(
k_{x}-ik_{y}\right) \hat{p}_{\mathbf{k}\uparrow}^{\dag}\hat{p}_{\mathbf{k}\downarrow}+h.c.\nonumber\\
& +\frac{1}{2\Omega}\sum_{p,p^{\prime}}\sum_{\mathbf{k},\mathbf{k}^{\prime
},\mathbf{q,}\sigma,\sigma^{\prime}}V_{\mathbf{q}}^{pp^{\prime}}\hat
{p}_{\mathbf{k}+\mathbf{q}\sigma}^{\dag}\hat{p}_{\mathbf{k}^{\prime
}-\mathbf{q}\sigma^{\prime}}^{\prime\dag}\hat{p}_{\mathbf{k}^{\prime}\sigma^{\prime}}^{\prime}\hat{p}_{\mathbf{k}\sigma}. \label{formula1}$$ Here, $\mathbf{k}$, $\mathbf{k}^{\prime}$, and $\mathbf{q}$ are 2D wave vectors in the layers, $\Omega$ is the quantization volume. $\mu_{p}$ is the chemical potential for electron layer ($p$=$e$) or hole layer ($p$=$h$). $\hat{p}_{\mathbf{k}\sigma}$ indicates the annihilation operator of electron at the wave vector $\mathbf{k}$ and spin $\sigma$ (=$\uparrow,\downarrow$) for electron layer ($p$=$e$), and hole layer ($p$=$h$). Note that $v_{F}^{e}$=$v_{F}$ and $v_{F}^{h}$=$-v_{F}.$The surface states of the strong TI film have the linear dispersion: $\epsilon_{\mathbf{k}e,h}$=$\pm\hslash
v_{F}|\mathbf{k}|$. $V_{\mathbf{q}}^{pp^{\prime}}$ is the Fourier transform of the Coulomb interaction: the intralayer Coulomb repulsive interaction $V_{\mathbf{q}}^{ee}$($V_{\mathbf{q}}^{hh}$)=$2\pi e^{2}/\left(
q\varepsilon\right) $, and the interlayer Coulomb attractive interaction [@Balatsky; @Shim] $V_{\mathbf{q}}^{eh}$=$-2\pi e^{2}\exp\left( -qd\right)
/\left( q\varepsilon\right) $, which indicates that on the one hand, in the limit of $d\rightarrow0$, the interaction between electron and hole becomes that in monolayer; on the other hand, in the large thickness limit $d\rightarrow\infty$, the interactions between the electrons in upper layer and holes in lower layer should vanish. Here, $\varepsilon$ is the background dielectric constant. Furthermore, for the present TIB system, the two TI films are separated by an inulating spacer such as SiO$_{2}$, and the spin-orbit interaction in the spacer is obviously negligible. Thus that it can be expected that our model is appropriate in neglectering the interlayer hopping coupling.
In the basis $(\hat{e}_{\uparrow},\hat{e}_{\downarrow},\hat{h}_{\uparrow},\hat{h}_{\downarrow})^{T}$, the Hamiltonian (\[formula1\]) can be decoupled to $H_{MF}$ under the mean-field approximation: $\Delta_{\sigma\sigma^{\prime
}}(\mathbf{k})$=$\sum_{\mathbf{q}}V^{eh}(\mathbf{q})\langle\hat{e}_{\mathbf{k}+\mathbf{q},\sigma}^{\dag}\hat{h}_{\mathbf{k}+\mathbf{q},\sigma^{\prime}}\rangle$, $\Sigma_{\sigma\sigma^{\prime}}^{(p)}(\mathbf{k})$=$-\sum_{\mathbf{q}}V^{pp}(\mathbf{q})\langle\hat{p}_{\mathbf{k}+\mathbf{q},\sigma}^{\dag}\hat{p}_{\mathbf{k}+\mathbf{q},\sigma^{\prime}}\rangle$. Then, $H_{MF}$ can be diagonalized with a 4$\times$4 unitary matirx $U(\mathbf{k})$, $U^{\dag}(\mathbf{k})H_{MF}(\mathbf{k})U(\mathbf{k})$=$\operatorname{diag}(E_{1}(\mathbf{k}),E_{2}(\mathbf{k}),E_{3}(\mathbf{k}),E_{4}(\mathbf{k}))$. The unitary matrix $U(\mathbf{k})$ is construsted by the normalized eigenfunctions of the Hamiltonian (\[formula1\]), which can be numerically calculated by diagonalizing the Hamiltonian matrix (\[formula1\]) in the basis $(\hat{e},\hat{e},\hat
{h},\hat{h})^{T}$. Explicitly, the elements $U_{ij}(\mathbf{k})$ denotes the $i$-th component of the eigenfunction corresponding to the eigenvalue $E_{j}$. The relevant mean-field equations to be solved for the variables $\mu_{e}$, $\mu_{h}$, and the gap functions $\Delta_{\sigma\sigma^{\prime}}(\mathbf{k})$ and self energies $\Sigma_{\sigma\sigma^{\prime}}^{(p)}(\mathbf{k})$ are $$\Delta_{jl}(\mathbf{k})=-\frac{1}{\Omega}\sum_{i=1}^{4}\sum_{\mathbf{q}}V_{\mathbf{q}}^{eh}U_{ji}^{\ast}(\mathbf{k}+\mathbf{q})U_{li}(\mathbf{k}+\mathbf{q})f(E_{i}(\mathbf{k}+\mathbf{q})),\label{formula2}$$$$\begin{aligned}
\Sigma_{jl}^{(e)}(\mathbf{k}) & =\frac{1}{\Omega}\sum_{i=1}^{4}\sum_{\mathbf{q}}V_{\mathbf{q}}^{ee}U_{ji}^{\ast}(\mathbf{k}+\mathbf{q})U_{li}(\mathbf{k}+\mathbf{q})f(E_{i}(\mathbf{k}+\mathbf{q})),\label{formula3}\\
\Sigma_{jl}^{(h)}(\mathbf{k}) & =\frac{1}{\Omega}\sum_{i=1}^{4}\sum_{\mathbf{q}}V_{\mathbf{q}}^{hh}U_{ji}^{\ast}(\mathbf{k}+\mathbf{q})U_{li}(\mathbf{k}+\mathbf{q})f(E_{i}(\mathbf{k}+\mathbf{q})),\label{formula4}$$$$\begin{aligned}
n_{e} & =\frac{1}{\Omega}\sum_{i=1}^{2}\sum_{j=2}^{3}\sum_{\mathbf{k}}\left\vert U_{ij}(\mathbf{k})\right\vert ^{2}f(E_{j}(\mathbf{k})),\label{formula5}\\
n_{h} & =\frac{1}{\Omega}\sum_{i=3}^{4}\sum_{j=2}^{3}\sum_{\mathbf{k}}\left[
1-\left\vert U_{ij}(\mathbf{k})\right\vert ^{2}f(E_{j}(\mathbf{k}))\right]
,\label{formula6}$$ where $f\left( E_{i}(\mathbf{k})\right) $=$1/(1+e^{E_{i}(\mathbf{k})/k_{B}T})$ is the Fermi distribution function and $E_{i}(\mathbf{k})$ ($i=1,...,4$) are the eigen-energies of $H_{MF}(\mathbf{k})$. In Table I we give an explicit correspondence between $\Delta_{\sigma\sigma^{\prime}}(\mathbf{k})$, $\Sigma_{\sigma\sigma^{\prime}}^{(p)}(\mathbf{k})$ and $\Delta_{jl}(\mathbf{k})$, $\Sigma_{jl}^{(p)}(\mathbf{k})$. $$\overset{\text{TABLE I. The correspondence between }\Delta_{\sigma
\sigma^{\prime}}(\mathbf{k})\text{, }\Sigma_{\sigma\sigma^{\prime}}^{(p)}(\mathbf{k})\text{ and }\Delta_{jl}(\mathbf{k})\text{, }\Sigma
_{jl}^{(p)}(\mathbf{k})}{\begin{tabular}
[c]{|cc|cc|cc|}\hline\hline
$\Delta_{\sigma\sigma^{\prime}}(\mathbf{k})$ & $\Delta_{jl}(\mathbf{k})$ &
$\Sigma_{\sigma\sigma^{\prime}}^{(e)}(\mathbf{k})$ & $\Sigma_{jl}^{(e)}(\mathbf{k})$ & $\Sigma_{\sigma\sigma^{\prime}}^{(h)}(\mathbf{k})$ &
$\Sigma_{jl}^{(h)}(\mathbf{k})$\\\hline
$\sigma\sigma^{\prime}$ & $jl$ & $\sigma\sigma^{\prime}$ & $jl$ &
$\sigma\sigma^{\prime}$ & $jl$\\\hline
$\uparrow\uparrow$ & 13 & $\uparrow\uparrow$ & 11 & $\uparrow\uparrow$ & 33\\
$\uparrow\downarrow$ & 14 & $\uparrow\downarrow$ & 12 & $\uparrow\downarrow$ &
34\\
$\downarrow\uparrow$ & 23 & $\downarrow\uparrow$ & 21 & $\downarrow\uparrow$ &
43\\
$\downarrow\downarrow$ & 24 & $\downarrow\downarrow$ & 22 & $\downarrow
\downarrow$ & 44\\\hline
\end{tabular}
\ \ }$$ In addition, for the present 2D case the average interparticle spacing is given by [@Strinati] $$r_{s}=\frac{1}{\sqrt{\frac{\pi}{2}\left( n_{e}+n_{h}\right) }}.
\label{formula7}$$
Many meaningful physical quantities, including the order parameters, can be obtained by self-consistently solving four-band Eqs. (\[formula2\])-(\[formula6\]) with the confinement of the electron and hole number densities. We numerically calculate the exciton’s energy spectrum and the order parameters under different exciton number densities: $r_{s}$=$1.5$, $\alpha$=$0$ and $5.0$, $\alpha$=$0$. Here the density imbalance parameter $\alpha$ is defined as $\alpha\mathtt{\equiv}\left( n_{e}\mathtt{-}n_{h}\right) /\left( n_{e}\mathtt{+}n_{h}\right) $. The calculated results are correspondingly shown by solid lines in Fig. \[f4\](a) and the inset in Fig. \[f2\].
Because the main goal of this paper is to focus on the general properties of the order parameters and neglect the other spin-dependent physical conditions, such as the effect of the Rashba-type spin-orbit coupling by surface inversion asymmetry, we plan to simplify our TIB model, i.e., to define a single order parameter $\Delta(\mathbf{k})$, which can approximately replace the four $\Delta_{\sigma\sigma^{\prime}}(\mathbf{k})$. Similar to that in the semiconductor case [@Strinati], the corresponding simplified grand-canonical Hamiltonian describing this TIB system can be approximately written as$$\begin{aligned}
H & =\sum_{\mathbf{k},p}\left( \epsilon_{\mathbf{k}p}-\mu_{p}\right)
c_{\mathbf{k}p}^{\dag}c_{\mathbf{k}p}+\frac{1}{2\Omega}\label{formula8}\\
& \times\sum_{\substack{\mathbf{k},\mathbf{k}^{\prime},\mathbf{q}\\p,p^{\prime}}}V_{\mathbf{k}-\mathbf{k}^{\prime}}^{pp^{\prime}}c_{\mathbf{k}+\mathbf{q}/2p}^{\dag}c_{-\mathbf{k}+\mathbf{q}/2p^{\prime}}^{\dag}c_{-\mathbf{k}^{\prime}+\mathbf{q}/2p^{\prime}}c_{\mathbf{k}^{\prime
}+\mathbf{q}/2p}.\nonumber\end{aligned}$$ With the SCMF theory, Eq. (\[formula8\]) can be rewritten in a $2\mathtt{\times}2$ matrix in the basis $(e,h)^{T}$, the relevant mean-field equations to be solved for the variables $\mu_{e}$, $\mu_{h}$, and the gap function $\Delta_{\mathbf{k}}$ are $$\Delta_{\mathbf{k}}=-\frac{1}{\Omega}\sum_{\mathbf{k}^{\prime}}V_{\mathbf{k}-\mathbf{k}^{\prime}}^{eh}\frac{\Delta_{\mathbf{k}^{\prime}}}{2E_{\mathbf{k}^{\prime}}}\left[ f(E_{\mathbf{k}^{\prime}}^{+})-f(E_{\mathbf{k}^{\prime}}^{-})\right] , \label{formula9}$$$$\begin{aligned}
\Sigma_{\mathbf{k}}^{e} & =\frac{1}{\Omega}\sum_{\mathbf{k}^{\prime}}V_{\mathbf{k}-\mathbf{k}^{\prime}}^{ee}\left[ u_{\mathbf{k}}^{2}f(E_{\mathbf{k}^{\prime}}^{+})+v_{\mathbf{k}}^{2}f(E_{\mathbf{k}^{\prime}}^{-})\right] ,\label{formula10}\\
\Sigma_{\mathbf{k}}^{h} & =\frac{1}{\Omega}\sum_{\mathbf{k}^{\prime}}V_{\mathbf{k}-\mathbf{k}^{\prime}}^{hh}\left[ v_{\mathbf{k}}^{2}f(E_{\mathbf{k}^{\prime}}^{+})+u_{\mathbf{k}}^{2}f(E_{\mathbf{k}^{\prime}}^{-})\right] , \label{formula11}$$$$\begin{aligned}
n_{e} & =\frac{1}{\Omega}\sum_{\mathbf{k}}\left\{ u_{\mathbf{k}}^{2}f(E_{\mathbf{k}}^{+})+v_{\mathbf{k}}^{2}\left[ f(E_{\mathbf{k}}^{-})\right] \right\} ,\label{formula12}\\
n_{h} & =\frac{1}{\Omega}\sum_{\mathbf{k}}\left\{ u_{\mathbf{k}}^{2}[1-f(E_{\mathbf{k}}^{-})\}+v_{\mathbf{k}}^{2}\left[ 1-f(E_{\mathbf{k}}^{+})\right] \right\} , \label{formula13}$$ where $u_{\mathbf{k}}^{2}$=$1\mathtt{-}v_{\mathbf{k}}^{2}$=$\frac{1}{2}\left(
1\mathtt{+}\xi_{\mathbf{k}}/E_{\mathbf{k}}\right) $, and $E_{\mathbf{k}}^{\pm}$=$\delta\xi_{\mathbf{k}}\mathtt{\pm}E_{\mathbf{k}}$ with $\delta
\xi_{\mathbf{k}}$=$\frac{1}{2}\left( \xi_{\mathbf{k}e}\mathtt{+}\xi_{\mathbf{k}h}\right) $ and $E_{\mathbf{k}}$=$\sqrt{\xi_{\mathbf{k}}^{2}\mathtt{+}\Delta_{\mathbf{k}}^{2}}$ that are given by $\xi_{\mathbf{k}p}$=$\epsilon_{\mathbf{k}p}\mathtt{-}\mu_{p}\mathtt{+}\Sigma_{\mathbf{k}}^{p}$ ($p$=$e,h$).
![(Color online) (a) Exciton’s energy spectrum with $r_{s}$=$1$.$5$ and $\alpha$=$0$. The solid and dashed lines correspond to Eqs. (\[formula1\]) and (\[formula8\]), respectively; (b) Exciton density of the states with $r_{s}$=$5$ and $\alpha$=$0$. []{data-label="f4"}](fig2.eps){width="0.5\linewidth"}
![(Color online) Wave-vector dependence of the gap function $\Delta(\mathbf{k})$ for $\alpha$=$0$ and several values of $r_{s}$. Inset: calculated $\Delta_{\sigma\sigma^{\prime}}(\mathbf{k})$ from original Hamiltonian (\[formula1\]) at $r_{s}$=$1.5$ and $5.0$. The solid and dashed lines are corresponding to $\Delta_{\uparrow\uparrow}$ (=$\Delta
_{\downarrow\downarrow}$) and $\Delta_{\uparrow\downarrow}$ (=$\Delta
_{\downarrow\uparrow}$), respectively. Comparing with $\Delta_{\sigma
\sigma^{\prime}}(\mathbf{k})$ in the inset, we can approximately use $\Delta(\mathbf{k})$ replacing $\Delta_{\sigma\sigma^{\prime}}(\mathbf{k})$ to study the general properties of the order parameters without other spin-dependent interactions.[]{data-label="f2"}](fig3.eps){width="0.6\linewidth"}
We also self-consistently calculate the exciton’s energy spectrum from two-band Eqs. (\[formula9\])-(\[formula13\]). The result, for comparison with the original exact four-band results from Eqs. (\[formula2\])-(\[formula6\]), is plotted in Fig. \[f4\](a) with red dashed lines under the same parameters as used in four-band calculations. The corresponding density of states is shown in Fig. \[f4\](b). From Fig. \[f4\](a) one can clearly find that the exciton energy spectrum within the two-band approximation is wonderfully consistent with that within the exct four-band formalism. Another character found from Figs. \[f4\](a) and \[f4\](b) is that there is an evident stable energy gap protecting the excitonic condensation. In addition, we would like to point out that the parity of the linear dispersion relations of the particles and holes is odd, while the parity of the particle-particle and hole-hole Coulomb interaction is even. This parity asymmetry results in the energy-shift in Fig. \[f4\](a) and the corresponding DOS asymmetry in Fig. \[f4\](b) as well as the asymmetry in Fig. \[f3a\] below. In the following of this paper, all the numerical results except for those shown in the inset of Fig. 3 are calculated from two-band SCMF Eqs. (\[formula9\])-(\[formula13\]).
Numerical results and application to the Bi$_{2}$Se$_{3}$-family material
=========================================================================
First, we calculate the wave-vector dependence of $\Delta(\mathbf{k})$ for equal densities ($\alpha$=$0$) and several values of $r_{s}$. The results are shown in Fig. \[f2\]. We can find the generic feature of the BCS-BEC crossover behavior similar to that in the semiconductor bilayers. However, the striking character in the TI bilayers is that the maximum value of $\Delta(\mathbf{k})$ in the BCS limit is much larger than that in the traditional semiconductor electron-hole bilayers [@Strinati]. This prominent difference means that the BCS phase of TIB is more robust than that of the semiconductor bilayer for equal-density case. Also shown in Fig. 3 (inset) are the calculated four-band gap functions $\Delta_{\sigma
\sigma^{\prime}}(\mathbf{k})$ at $r_{s}$=1.5 and 5.0, which show the approximate coincidence in amplitude with the two-band result of $\Delta(\mathbf{k})$.
Because there are no obvious interface between BCS and BEC regimes in terms of the density, we plot in Fig. \[f33\] the calculated momentum magnitude $k$ at which the order parameter takes its maximum value $\Delta_{\max}$ versus $r_{s}$ at $\alpha$=$0$. From this figure, one can see that as $r_{s}\mathtt{\longrightarrow}0$, the number density $n_{e}$ ($n_{h}$) and $k_{\Delta_{\max}}$ tend to infinity, the exciton’s phase is in the BCS regime. On the other hand, as $r_{s}\mathtt{\longrightarrow}\infty$, the number density $n_{e}$ ($n_{h}$) and $k_{\Delta_{\max}}$ tend to $0$, and the exciton’s phase is now in the BEC regime. As $r_{s}$ takes a moderate value, the system is in a mixed regime.
![(Color online) BCS-BEC phase transform: The momentum magnitude $k$ at which the order parameter takes its maximum value $\Delta_{\max}$ versus $r_{s}$ (or number density) at $\alpha$=$0$. The dots are the calculated data, while the solid line is to guide the eyes.[]{data-label="f33"}](fig4.eps){width="0.6\linewidth"}
The effect of $\alpha$ on $\Delta_{\max}$ is shown in Fig. \[f3a\], where $\Delta_{\max}$=$\max\left\{ \Delta_{\mathbf{k}}\right\} $. It is evident to find that the density imbalance actually suppresses $\Delta_{\max}$ and it has different effects on two sides of the crossover. In the BEC regime, the main effect of the density imbalance is to reduce the number of electron-hole pairs, which results in that the superfluid properties are less sensitive to density imbalance. In the BCS regime, the density imbalance leads to the mismatch of the Fermi surfaces of electrons and holes and the finite momentum pairing, which is easier to be broken. However, comparing with that in the traditional semiconductor bilayers, we find that the superfluid property in the BEC phase in our case is more sensitive to electron-hole density imbalance. As an example, for $r_{s}$=$20$ the maximum of gap function $\Delta_{\max}$ for TIB disappears as $\alpha$ takes a value smaller than $0.5$, while it always takes finite values at $\alpha$ varies in the whole zone $\left( -1,1\right) $ for the traditional semiconductor electron-hole bilayers [@Strinati].
Now we apply this TIB model to study the condensation of electron-hole pairs for the topological surface states of the Bi$_{2}$Se$_{3}$-family material. The two TI films in the left panel of Fig. \[f1\] now are two ultrathin TI Bi$_{2}$Se$_{3}$-family films [@Hasegawa] (about $80$ Å thick). With the adopted experimental [@Nakajima] lattice constants $a$=$4.143$ Å and $c$=$28.636$ Å, we calculate the first-principles surface band structure of Bi$_{2}$Se$_{3}$-family [@Wang] by a simple supercell approach with spin-orbit coupling included and obtain the approximate Hamiltonian form describing the gapless surface states of Bi$_{2}$Se$_{3}$-family as follows:$$H(\mathbf{k})=\gamma k^{2}+\hslash v_{F}\left( k_{x}\sigma_{y}-k_{y}\sigma_{x}\right) . \label{formula14}$$ Although this Hamiltonian has the same form as that of the conventional two-dimensional electron gas (2DEG) system with Rashba spin-orbit coupling, the intrinsic difference between these two kinds of systems is that the $k$-linear spin-orbit interaction is primary to the TI surface states, while the parabolic term is dominant in the conventional 2DEG. By fitting the first-principles results, the parameters in Eq. (\[formula14\]) are given as $\gamma$=$0.21$ eV nm$^{2}$ and $\hslash v_{F}$=$0.2$ eV nm (namely, $v_{F}$=$3.04\times10^{5}$m/s). That means the energy dispersion around the Dirac point can be accurately described by $\epsilon_{\mathbf{k}}$=$\pm\hslash
v_{F}|\mathbf{k}|$ when the wave-vector $|\mathbf{k}|$ is much smaller than $1.0$ nm$^{-1}$. For numerical calculation, we choose nm as the length unit and $0.2$ eV as the energy unit in the following discussion. The dielectric constant $\varepsilon$=$1$ and the spacer width $d$=$10$ Å. In fact, the condition that the wave-vector $|\mathbf{k}|$ is much smaller than $1.0$ nm$^{-1}$ requires that only for $r_{s}\geq5$, then the TIB model is valid for Bi$_{2}$Se$_{3}$-family material. This means that the BEC phase is most possible to emerge in Bi$_{2}$Se$_{3}$-family bilayer system.
![(Color online) Maximum value $\Delta_{\max}$=$\max\left\{
\Delta_{\mathbf{k}}\right\} $ as a function of $\alpha$ for $d$=$1$ and several values of $r_{s}$. []{data-label="f3a"}](fig5.eps){width="0.6\linewidth"}
Now, let us discuss the critical temperature of this TIB system. The relation between the $\Delta_{\max}$ and temperature $T$ is respectively shown in Fig. \[f5\](a) for $d$=$1$, $\alpha$=$0$, and several values of $r_{s}$, and \[f5\](b) for $d$=$1$, $r_{s}$=$5$.$0$, and several values of $\alpha$. From Fig. \[f5\](a), we can find that the critical temperature $T_{c}$ decreases as $r_{s}$ increases (i.e., as the particle density decreases). For the Bi$_{2}$Se$_{3}$-family bilayer at $r_{s}$=$5$.$0$, the critical temperature $T_{c}$ is calculated as 0.05 in unit of 0.2 eV. That means the critical temperature $T_{c}$ is about $8\mathtt{\sim}10$ meV ($100$ K), which is much higher than that in the traditional semiconductor electron-hole bilayers. Although the Bi$_{2}$Se$_{3}$-family TIB system is in the BEC phases ($r_{s}$=$5$.$0$, $20.0$), the numerical calculated results shown in Fig. \[f5\](a) are consistent with the general relation of BCS superconductor,$$\frac{2\Delta(0)}{T_{c}}=2\pi e^{-\gamma}\approx3.53, \label{formula15}$$ where $\Delta(0)$ is the energy gap at zero temperature. The introduced electron-hole density imbalance ($\alpha\mathtt{\neq}0$) can reduce the critical temperature. This character is clearly shown in Fig. \[f5\](b): by increasing the density imbalance $\alpha$, the critical temperature $T_{c}$ decreases.
![(Color online) (a) Maximum value $\Delta_{\max}$=$\max\left\{
\Delta_{\mathbf{k}}\right\} $ vs temperature $T$ for $d$=$1$, $\alpha$=$0$, and several values of $r_{s}$. (b) $\Delta_{\max}$ as a function of the temperature $T$ at $r_{s}$=$5$ and several values of $\alpha$.[]{data-label="f5"}](fig6.eps){width="0.7\linewidth"}
As it is known that in 2D superfluids, the critical temperature is often substantially overestimated by mean-filed theory. It is ultimately limited by entropically driven vortex and antivortex proliferation at the Kosterlitz-Thouless (KT) transition temperature $T_{\text{KT}}$=$\frac{\pi}{2}\rho_{s}(T_{\text{KT}})$ with $\rho_{s}(T)$ being the superfluid density (the phase stiffness). Ref. [@Mac] gives an approximate formula to calculate the counterflow current, which is read as $$\rho_{s}(T)\approx\frac{v^{2}\hslash^{2}}{16\pi T}\int kdk\left[ \sec
\text{h}^{2}\left( \frac{\Delta^{z}}{2T}\right) -\sec\text{h}^{2}\left(
\frac{\Delta}{2T}\right) \right] , \label{formula16}$$ where $\Delta^{z}=\frac{-\mu_{e}+\mu_{h}+\Sigma_{\mathbf{k}}^{e}-\Sigma_{\mathbf{k}}^{h}}{2}$, and $\Delta=\sqrt{\left( \Delta^{z}\right)
^{2}}+\sqrt{\Delta_{\mathbf{k}}^{2}}$. We adopt this formula to calculate the superfluid density. The temperature dependence of superfluid density is shown in Fig. \[f6\] at $r_{s}$=$5$ and $\alpha$=$0$. From Fig. \[f6\], it is evident to estimate that the KT transition temperature $T_{\text{KT}}$ is about $0.05$ in unit of $0.2$ eV. Comparing with the critical temperature $T_{c}$ in Fig. \[f5\] at $r_{s}$=$5$ and $\alpha=0$, the striking conclusion is reached: $T_{c}\mathtt{\approx}$ $T_{\text{KT}}$, which means that high-temperature ($\mathtt{\sim}$100 K) excitonic condensation may occur in the Bi$_{2}$Se$_{3}$-family TIB system. On the other hand, we can estimate the KT temperature with the zero-temperature phase stiffness $\rho_{s}(T$=$0)\mathtt{\approx}E_{F}/4\pi$ which is similar to the graphene bilayers [@Mac]. Considering the case shown in Fig. \[f4\], the Fermi energy $E_{F}$ can be numerically calculated and is given to be $\mathtt{\sim}0.4$ (in unit of $0.2$ eV). Hence, the KT temperature is estimated as $T_{\text{KT}}\mathtt{\approx}E_{F}/8\mathtt{\approx}0.05$ in unit of $0.2$ eV. This means that the two estimated methods are consistent and the high-temperature excitonic condensation can emerge in the Bi$_{2}$Se$_{3}$-family TIB system.
![(Color online) The calculated $T_{KT}$ at $r_{s}$=$5$ and $\alpha
$=$0$.[]{data-label="f6"}](fig7.eps){width="0.5\linewidth"}
Summary and conclusions
=======================
In summary, we have performed a generic TIB model to study the excitonic condensation with the SCMF theory for the topological surface states. Similar to the traditional semiconductor electron-hole bilayers, the TIB system presents the crossover behavior from BCS limit to BEC limit by changing the exciton’s density. However, two prominent novel characters different from the traditional semiconductor electron-hole bilayers are found. One is that the superfluid property in the BEC phase is more sensitive to electron-hole density imbalance. The other is that the BCS phase is more robust than that of the semiconductor bilayer. Applying this TIB model to Bi$_{2}$Se$_{3}$-family material, we find that the BEC phase is most possibly observed in experiment. Moveover, we theoretically estimate the critical temperature for the Bi$_{2}$Se$_{3}$-family TIB system and find that it is much higher than that in the traditional semiconductor electron-hole bilayers. For example, at $r_{s}$=$5$ and $\alpha$=$0$, the critical temperature $T_{c}$ is obtained as about $100$ K. We have also studied the phase stiffness and find that the KT transition doesn’t suppress the critical temperature for Bi$_{2}$Se$_{3}$-family in SCMF approximation.
This work was supported by NSFC under Grants No. 90921003 and No. 10904005, and by the National Basic Research Program of China (973 Program) under Grant No. 2009CB929103.
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[^1]: Corresponding author. Email address: zhang\[email protected]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A thermodynamic theory is developed to describe the behavior of the entanglement between the coin and position degrees of freedom of the quantum walk on the line. This theory shows that, in spite of the unitary evolution, a steady state is established after a Markovian transient stage. This study suggests that if a quantum dynamics is developed in a composite Hilbert space (*i.e.* the tensor product of several sub-spaces) then the behavior of an operator that only belongs to one of the sub-spaces may camouflage the unitary character of the global evolution.'
author:
- Alejandro Romanelli
title: The quantum walk temperature
---
Introduction
============
The concept of isolated system plays a fundamental role in the formulation of the quantum mechanics. This concept is an idealization that was constructed as an aid to understand some phenomena displayed by real systems which may be regarded as approximately isolated. However, since about 50 years ago, the study of quantum decoherence has acquired a central position in the formulation of the quantum mechanics. In fact, concepts such as thermodynamic equilibrium seem impossible to coordinate with the idea of isolated system because the quantum state for such a system follows a unitary evolution and it cannot reach a final equilibrium state at $t\rightarrow\infty $.
In this context we ask ourselves if it is possible to introduce the concept of temperature for an isolated quantum system which evolves in a composite Hilbert space. In this paper the system known as the quantum walk on the line [@QW] has been chosen as a model to answer this question . The quantum walk (QW) is a natural generalization of the classical random walk in the frame of quantum computation and quantum information processing and it is receiving permanent attention [childs,Linden,Alejo3]{}. It has the property to spread over the line linearly in time as characterized by the standard deviation $\sigma
(t)\sim t$, while its classical analog spreads out as $\sigma
(t)\sim t^{1/2}$. This property, as well as quantum parallelism and quantum entanglement, could be used to increase the efficiency of quantum algorithms [@Shenvi; @Childs; @et]. Recently we have been investigating [@alejo2010; @alejo2011; @armando2011] the asymptotic behavior of the QW on the line, focusing on the probability distribution of chirality independently of position. We showed that this distribution has a stationary long-time limit that depends on the initial conditions. This result is unexpected in the context of the unitary evolution of the QW because such a behavior is usually associated to a Markovian process. In this paper we further explore the behavior of the chirality distribution and define a thermodynamic equilibrium between the degrees of freedom of position and chirality. This equilibrium allows to introduce a temperature concept for this unitary closed system. We obtain a master equation with a time-dependent population rate, that describes the transient behavior of the reduced density operator of the QW towards thermodynamic equilibrium. The QW’s reduced density operator shows an surprising behavior: Its behavior looks diffusive but however the global evolution of the system is unitary.
The paper is organized as follows. In the next section the standard QW model is developed, in the third section the entanglement temperature is defined. Then in the fourth and fifth sections the entanglement temperature is obtained for localized initial conditions and for distributed initial conditions respectively. In the six section the transient behavior towards thermal equilibrium is studied, and finally in the last section some conclusions are drafted.
QW on the line
==============
The composite Hilbert space of the QW is the tensor product $\mathcal{H}_{s}\otimes \mathcal{H}_{c}$ where $\mathcal{H}_{s}$ is the Hilbert space associated to the motion on the line and $\mathcal{H}_{c}$ is the chirality (or coin) Hilbert space. In this composite space the walker moves, at discrete time steps $t\in
\mathbb{N}$, along a one-dimensional lattice of sites $k\in
\mathbb{Z}$. The direction of motion depends on the state of the chirality, with the eigenstates $R$ and $L$. The wave vector can be expressed as the spinor $$|\Psi (t)\rangle =\sum\limits_{k=-\infty }^{\infty }\left[
\begin{array}{c}
a_{k}(t) \\
b_{k}(t)\end{array}\right] |k\rangle , \label{spinor}$$where the upper (lower) component is associated to the left (right) chirality.
Then $P_{kL}(t)=\left\vert a_{k}(t)\right\vert ^{2}$ and $P_{kR}(t)=\left\vert b_{k}(t)\right\vert ^{2}$ denote the probability of finding the walker at $\left( k,t\right) $ and the coin in state $R$ and $L$, respectively. The probability of finding the walker at $\left( k,t\right) $ is $$P(k,t)=\left\langle \Psi _{k,t}\right. \left\vert \Psi _{k,t}\right\rangle
=\left\vert a_{k}(t)\right\vert ^{2}+\left\vert b_{k}(t)\right\vert ^{2},
\label{prob}$$and $\sum_{k}P(k,t)=1$.
The QW is ruled by a unitary map whose standard form is [@Romanelli09; @Alejo2; @Alejo1; @Alejo4] $$\begin{aligned}
a_{k}(t+1)& =a_{k+1}(t)\,\cos \theta \,+b_{k+1}(t)\,\sin \theta ,\, \notag
\\
b_{k}(t+1)& =a_{k-1}(t)\,\sin \theta \,-b_{k-1}(t)\,\cos \theta .
\label{mapa}\end{aligned}$$where $\theta \in \left[ 0,\pi /2\right] $ is a parameter defining the bias of the coin toss ($\theta =\frac{\pi }{4}$ for an unbiased or Hadamard coin). The global left and right chirality probabilities is defined as $$\begin{aligned}
P_{L}(t)& \equiv \sum_{k=-\infty }^{\infty }P_{kL}(t)=\sum_{k=-\infty
}^{\infty }\left\vert a_{k}(t)\right\vert ^{2},\, \notag \\
P_{R}(t)& \equiv \sum_{k=-\infty }^{\infty }P_{kR}(t)=\sum_{k=-\infty
}^{\infty }\left\vert b_{k}(t)\right\vert ^{2}, \label{chirality}\end{aligned}$$with $P_{R}(t)+P_{L}(t)=1$. The global chirality distribution (GCD) is defined as the distribution formed by the couple $\left[
\begin{array}{c}
P_{L}(t) \\
P_{R}(t)\end{array}\right] $. It is shown in Ref. [@alejo2010] that the GCD satisfies the following map $$\begin{aligned}
{\left[
\begin{array}{c}
P_{L}(t+1) \\
P_{R}(t+1)\end{array}\right] }& ={\left(
\begin{array}{cc}
\cos ^{2}\theta & \sin ^{2}\theta \\
\sin ^{2}\theta & \cos ^{2}\theta
\end{array}\right) }\left[
\begin{array}{c}
P_{L}(t) \\
P_{R}(t)\end{array}\right] \notag \\
& +\mathrm{Re}\left[ Q(t)\right] \sin {2}\theta \left[
\begin{array}{c}
1 \\
-1\end{array}\right] , \label{master}\end{aligned}$$where $$Q(t)\equiv \sum_{k=-\infty }^{\infty }a_{k}(t)b_{k}^{\ast }(t). \label{qdet}$$The two dimensional matrix in Eq.(\[master\]) can be interpreted as a transition probability matrix for a classical two dimensional random walk as it satisfies the necessary requirements, namely, all its elements are positive and the sum over the elements of any column or row is equal to one. On the other hand, it is clear that $Q(t)$ accounts for the interferences. When $Q(t)$ vanishes the behavior of the GCD can be described as a classical Markovian process. However in the generic case $Q(t)$ together with $P_{L}(t)$ and $P_{R}(t)$ are time depend functions that have long-time limiting values [@alejo2010] which are determined by the initial conditions of Eq.(\[mapa\]). Eq.(\[master\]) can be solved in this limit. We define $$\begin{aligned}
\Pi _{L}& \equiv
\begin{array}{c}
\lim \text{ }P_{L}(t) \\
t\rightarrow \infty~~~~
\end{array},\, \notag \\
\Pi _{R}& \equiv
\begin{array}{c}
\lim \text{ }P_{R}(t) \\
t\rightarrow \infty~~~~
\end{array},\, \notag \\
Q_{0}& \equiv
\begin{array}{c}
\lim \text{ }Q(t) \\
t\rightarrow \infty~~
\end{array},\, \label{asym}\end{aligned}$$and then we obtain the asymptotic stationary solution for the GCD as $${\left[
\begin{array}{c}
\Pi _{L} \\
\Pi _{R}\end{array}\right] }=\frac{1}{2}\left[
\begin{array}{c}
1+2\mathrm{Re}(Q_{0})/\tan \theta \\
1-2\mathrm{Re}(Q_{0})/\tan \theta
\end{array}\right] . \label{estacio}$$ This interesting result for the QW shows that the long-time probability to find the system with left or right chirality has a limit. Therefore, although the dynamical evolution of the QW is unitary, the evolution of its GCD has an asymptotic limit characteristic of a diffusive behavior. This situation is further surprising if we compare our case with the case of the QW on finite graphs [@Aharonov] where it is shown that there is no converge to any stationary distribution.
Entanglement and temperature
============================
The concept of entanglement is an important element in the development of quantum communication, quantum cryptography and quantum computation. In this context several authors have investigated the relation between asymptotic entanglement and the initial conditions of the QW [Carneiro,abal,Annabestani,Omar,Pathak,Petulante,Venegas,Endrejat,Ellinas1,Ellinas2,Maloyer]{}. Other authors [@Venegas1; @Chandrashekar] have proposed to use the QW as a tool for quantum algorithm development and as an entanglement generator potentially useful to test quantum hardware.
The unitary evolution of the QW generates entanglement between the coin and position degrees of freedom. To characterize this entanglement we start with the von Neumann entropy which is quantum analogue of the Gibbs entropy $$S_{N}(\rho )=-\mathrm{tr}(\rho \log \rho ). \label{uno}$$where $\rho =|\Psi (t)\rangle \langle \Psi (t)|$ is the density matrix of the quantum system. Due to the unitary dynamics of the QW the system remains in a pure state and this entropy vanishes. However for these pure states the entanglement between the chirality and the position can be quantified by the associated von Neumann entropy for the reduced density operator [Carneiro,abal]{} that defines the entropy of entanglement $$S(\rho )=-\mathrm{tr}(\rho _{c}\log \rho _{c}), \label{dos}$$ where $$\rho _{c}=\mathrm{tr}(\rho ), \label{dos}$$ and the partial trace is taken over the positions. Using the wave function Eq.(\[spinor\]) and its normalization properties we obtain the reduced density operator [@Carneiro] $$\rho_{c} =\left(
\begin{array}{cc}
P_{L}(t) & Q(t) \\
Q(t)^{\ast } & P_{R}(t)\end{array}\right) . \label{rho}$$ The density operator $\rho_{c}$ has the eigenvalues $$\lambda _{\pm}=\frac{1}{2}\left[ 1\pm \sqrt{1-4\left(
P_L(t)\,P_R(t)-\left\vert Q(t)\right\vert ^{2}\right) }\right]. \label{lam}$$ Then the entanglement entropy Eq.(\[dos\]) is expressed through these two eigenvalues as $$S(\rho)=-\lambda_{+}\log \lambda_{+}-\lambda_{-}\log \lambda_{-}.
\label{ttres}$$In the asymptotic regime $t\rightarrow\infty$ the eigenvalues go to a stationary limit, $\lambda _{\pm}\rightarrow\Lambda _{\pm}$ and $$\Lambda _{\pm}=\frac{1}{2}\left[ 1\pm \sqrt{1-4\left(\Pi_L\,\Pi_R-\left\vert
Q_0\right\vert ^{2}\right) }\right]. \label{lam0}$$ Further from Eq.(\[estacio\]) follows the relation $$\Pi_L\,\Pi_R= \frac{1}{4}-{\left(\frac{\mathrm{Re}(Q_{0})}{\tan \theta}\right)}^{2}, \label{piqu}$$ which is substituted in Eq.(\[lam0\]), and then the asymptotic eigenvalues are expressed as $$\Lambda _{\pm}=\frac{1}{2}\pm \sqrt{\chi}, \label{lam1}$$ with $$\chi\equiv \left\vert Q_0\right\vert ^{2}+{\left({\mathrm{Re}(Q_{0})}/{\tan
\theta}\right)}^{2}. \label{defi}$$ Note that the values of the interference term $Q_{0}$ are constrained to satisfy the condition $$0<\Lambda _{+}\Lambda _{-}<1, \label{condition0}$$ and then $$0<\chi<\frac{1}{4}. \label{condition}$$
![Dimensionless thermodynamic function normalized by $\log(2)$ as a function of the dimensionless parameter $\protect\chi$. From the top to the bottom they are: in thick line $\protect\beta\protect\epsilon$, in dashed line the entropy $S_0$, in thin line the energy $\protect\beta U$ and in dashed-dot line the Helmholtz free energy $\protect\beta A$.[]{data-label="f1"}](fig1.eps)
The entanglement entropy has an asymptotic limit too $$S_0=-\Lambda _{+}\log \Lambda _{+}-\Lambda _{-}\log \Lambda _{-},
\label{s0}$$ that only depends on the initial conditions through the interference term $Q_{0}$. Therefore we are led to consider that after some transient time the QW achieves a thermodynamic equilibrium between the position and chirality degrees of freedom. In order to make a fuller description of this equilibrium it is necessary to connect the eigenvalues of $\rho_c$ with its associated Hamiltonian operator $H_c$. To obtain this connection we shall use the quantum Brownian motion model of Ref.[@Kubo]. We considered the system associated with the chirality degrees of freedom and characterized by the density matrix $\rho_c$ in thermal contact (entanglement) with the bath system associated with the position degrees of freedom, the lattice. In this context $\rho_c$ satisfies the equation $$\frac{\partial\rho_{c}}{\partial
t}=\frac{1}{i\hbar}[H_c,\rho_{c}]+\Gamma \rho_{c}, \label{scho}$$ where $[H_c,\rho_{c}]$ is the commutator and $\Gamma \rho_{c}$ represents the Brownian motion of $\rho_c$ induced by the noise (fluctuating forces) exerted on $\rho_c$ by the lattice (position degrees of freedom). In the equilibrium (stationary) situation we must have ${\partial\rho_{c}}/{\partial t}= 0$ and $\Gamma
\rho_{c}=0$ [@Kubo], that is $$[H_c,\rho_{c}]=0. \label{scho2}$$ Therefore, in the asymptotic regime, the density operator $\rho_{c}$ must be an explicit function of the Hamiltonian operator which must be time independent. Now we call $\{\Phi_{+},\Phi_{-}\}$ the eigenfunctions of the density matrix, and then the operators $H_{c}$ and $\rho_{c}$ are both diagonal in this basis. Therefore, the eigenvalues $\Lambda _{+}$ and $\Lambda _{-}$ depend of the corresponding eigenvalues of $H_c$. We take these eigenvalues to be $\{-\epsilon,\epsilon\}$ without any loss of generality, they represent the possible values of the entanglement energy. This interpretation agrees with the fact that $\Lambda _{+}$ and $\Lambda _{-}$ are the probabilities that the system is in the eigenstate $\Phi_{+}$ or $\Phi_{-}$. The precise dependence between $\Lambda _{\pm}$ and $\pm \epsilon$ is determined by the type of ensemble we construct. The main proposal of this paper is that this equilibrium corresponds to a quantum canonical ensemble. Therefore we propose that $$\Lambda _{\pm}\equiv\frac{e^{\pm\beta\epsilon}}{e^{\beta\epsilon}+e^{-\beta\epsilon}}, \label{lam2}$$ which defines the entanglement temperature $T\equiv1/\beta$. Then the probability that the state chosen at random from the ensemble $\{\Phi_{+}
$, $\Phi_{-}$}, possesses an energy $\epsilon$ is determined by the Boltzmann factor $e^{-\beta\epsilon}$. Let us call $\widetilde{\rho_{c}}$ the diagonal expression of the density operator $\rho_{c}$, then $$\widetilde{\rho_{c}}=\left(
\begin{array}{cc}
\Lambda _{+} & 0 \\
0 & \Lambda _{-}\end{array}\right) =\frac{1}{e^{\beta\epsilon}+e^{-\beta\epsilon}}\left(
\begin{array}{cc}
e^{\beta\epsilon} & 0 \\
0 & e^{-\beta\epsilon}\end{array}\right) . \label{rhoni}$$ This operator is formally the same density operator that corresponds to an electron with possesses an intrinsic spin and a magnetic moment in a external magnetic field [@pathria].
Starting from Eq.(\[rhoni\]) it is possible to build the thermodynamics for the QW entanglement. The partition function of the system is then given by $$\mathcal{Z}=e^{\beta\epsilon}+e^{-\beta\epsilon}=2\cosh({\beta\epsilon}).
\label{betaA}$$ Accordingly, and also using Eqs.(\[lam0\],\[lam2\]), the temperature is given by $$T=2\epsilon/\ln\left(\frac{1+2\sqrt{\chi}}{1-2\sqrt{\chi}}\right),
\label{betae}$$ the Helmholtz free energy by $$A=-\frac{1}{\beta}\ln[2\cosh({\beta\epsilon})]=\frac{T}{2}\ln\left(\frac{1}{4}-\chi\right), \label{betaA}$$ the internal energy by $$U=-\epsilon\tanh(\beta\epsilon)=-2\epsilon\sqrt{\chi}, \label{betaU}$$ and finally the entropy by $$S_0=\beta U-\beta A, \label{s02}$$ where this last thermodynamic definition for the entropy of course agrees with the previous Shannon expression in Eq.(\[s0\]). To finished this section in Fig. \[f1\] we present the dependence of these thermodynamic magnitudes with the interference parameter $\chi$.
Localized initial conditions
============================
![Isothermal curves as functions of the dimensionless angles $\protect\gamma$ and $\protect\varphi$. Due to the rotation symmetry in the angle $\varphi$, only four zones are distinguished: two “cold” and two “hot”. The “hot” zones (orange on line) have six isotherms; from inside to outside their temperatures are: $T/T_{0}=6.5,~3.2,~2.2,~1.6,~1.3$ and $1.1$. The “cold” zones (blue on line) have five isotherms; from outside to inside their temperatures are: $T/T_{0}=0.9,~0.8,~0.7,~0.68$ and $0.66$. The straight (green) lines corresponded to $T/T_{0}=1$, see Eq.(\[varphygamma\]).[]{data-label="f2"}](fig2.eps)
![The isothermal curves of Fig. \[f2\] shown on the Bloch sphere. The QW initial chirality determines the entanglement temperature, see Eqs.(\[betae\],\[psi0\],\[chi2\]).[]{data-label="f3"}](fig3.eps)
As seen in the previous section the thermodynamics of entanglement only depends on the interference term $Q_{0}$ which in turn only depends on the initial conditions, as shown in [@alejo2010].
In order to investigate this dependence on the initial conditions of the system we consider first the localized case. The initial state of the walker is assumed to be sharply localized at the origin with arbitrary chirality, thus $$|\Psi (0)\rangle =\left(
\begin{array}{c}
\cos ({\gamma}/{2}) \\
\exp i\varphi \text{ }\sin ({\gamma}/{2})\end{array}\right) |0\rangle , \label{psi0}$$where $\gamma \in \left[ 0,\pi \right]$ and $\varphi \in \left[ 0,2\pi \right] $ define a point on the unit three-dimensional Bloch sphere. The expression of $Q_{0}$ was obtained in Ref. [@abal], fixing the bias of the coin toss $\theta =\pi /4$, following the method developed by Nayak and Vishwanath [@nayak]$$Q_{0}=\frac{1}{2}(1-\frac{1}{\sqrt{2}})\left[ \cos \gamma \text{ }+\sin
\gamma \text{ }(\cos \varphi +i\sqrt{2}\sin \varphi )\right] .
\label{q0entengl}$$ Using this result in Eq.(\[defi\]) the dependence of $\chi$ with the initial conditions is given by $$\chi=\chi_{0}\left(1+ \cos \varphi \sin 2\gamma \right) , \label{chi2}$$ where $\chi_{0}={3}/{4}-1/\sqrt{2}$. It is useful to define a characteristic temperature (in units of $\epsilon$) $$T_{0}=2/ \left[\ln\left(\frac{1+2\sqrt{\chi_{0}}}{1-2\sqrt{\chi_{0}}}\right)\right], \label{t0}$$ in order to express any other temperature as a proportionality with $T_0=1/\beta_0$. Then from Eq.(\[betae\]) we obtain an expression for $\beta$ as a function of the angles $\gamma$ and $\varphi$ $$\cos\varphi \sin
2\gamma=\left(\frac{\tanh\beta}{\tanh\beta_0}\right)^2-1.
\label{varphygamma}$$ Figures \[f2\] and \[f3\] show the level curves (isotherms) for the entanglement temperature as a function of the QW initial position. In Fig. \[f2\] the initial position is defined through the angles $\gamma$ and $\varphi$ and in Fig. \[f3\] it is defined through the position on the Bloch sphere (see Eq.(\[psi0\])). Both figures show four regions, two of them corresponding to temperatures $T>T_0$ (orange color on line) and the other two to temperatures $T<T_0$ (blue color on line). The longest isotherms (green color on line) correspond to the temperature $T=T_{0}$ and their initial conditions are $\gamma=0,~\pi/2,~\pi$ and $\varphi=\pi/2,~3\pi/2$.
Distributed initial conditions
==============================
![Isothermal curves as functions of the dimensionless angles $\protect\gamma$ (initial conditions) and $\protect\theta$ (bias of the coin). Four curves (with different colors on line) are presented; each curve has two branches placed symmetrically on both sides of $\protect\gamma=\protect\pi/2$, where $T=\infty$. The values of $T$ are given by Eqs.(\[betae2\]). From left to right the values of $T$ are, in units of $\protect\epsilon$: 0.5 (purple), 1. (blue), 2. (green) and 5. (orange). The diagram has two discontinuities in $\theta=\protect\pi/2$ and in $\protect\gamma=\protect\pi/2$ (see Eqs.(\[betae2\],\[re2\])).[]{data-label="f4"}](fig4.eps)
In previous works [@Eugenio; @alejo2010] we have studied the QW with extended initial conditions. Now the entanglement temperature is studied in such a case. The following extended Gaussian distributions is proposed: $$a_{k}^{0}\equiv{\left[ \frac{1}{\sigma_0\sqrt{2\pi }}\exp \left( -\frac{k^{2}}{2\sigma_0^{2}}\right) \right]}^{\frac{1}{2}}\cos({\gamma}/{2}) \text{,}
\label{aes}$$$$b_{k}^{0}\equiv e^{i \varphi }{\left[ \frac{1}{\sigma_0\sqrt{2\pi }}\exp
\left( -\frac{k^{2}}{2\sigma_0^{2}}\right) \right]}^{\frac{1}{2}}\, \sin({\gamma}/{2})\text{,} \label{bes}$$where $\sigma_0$ is the initial standard deviation, $\gamma \in \left[ 0,\pi \right]$ determines the initial proportion of the left and right chirality and $\varphi\in \left[ 0,2\pi \right] $ is a global phase. Using these initial conditions, Eqs.(\[aes\], \[bes\]), the asymptotic value of $Q(t)$, see Eqs.(\[qdet\],\[asym\]), was obtained [@alejo2010] as $$Q_0=\frac{1}{2}\cos\gamma\,\tan\theta\,, \label{q1}$$ with the restrictions $$\sigma_0\gg1, \label{sig1}$$ and $$\cos\varphi=\frac{\tan\theta}{\tan\gamma}. \label{phi2}$$ Replacing Eq.(\[q1\]) in Eq.(\[defi\]) we obtain $$\chi= \left( \frac{\cos\gamma}{2\cos\theta}\right)^{2}, \label{chides}$$ and then using Eq.(\[betae\]) we have $$\beta\epsilon=\frac{1}{2}\ln\left(\frac{\left|\cos\theta\right|
+\left|\cos\gamma\right|}{\left|\cos\theta\right|-\left|\cos\gamma\right|}\right), \label{betae2}$$ where taking into account Eqs.(\[condition\],\[chides\]) the initial condition satisfies the constraint $${\left|\cos\gamma\right|}<{\left|\cos\theta\right|}. \label{re2}$$
The functions $Q_0$, $\chi$ and $\beta$ vanish for $\gamma=\pi/2$ ( see Eqs.(\[q1\],\[chides\],\[betae2\])) and simultaneously the entanglement entropy Eq.(\[ttres\]) has its maximum value $S_0
= 1$. This maximum value is achieved when the entanglement temperature is $T=\infty$. Under these conditions the system behaves as a classical Markov process [@alejo2010]. On the other hand, the initial conditions $\gamma$ and $\varphi$ are not independent (see Eq.(\[phi2\])) and for each value of $\gamma$ there is only one value of $T$, then for fixed $\theta$, it is not possible to have isotherms as functions of $\gamma$ and $\varphi$. Instead the entanglement temperature depends on $\theta$ and $\gamma$ from Eq.(\[betae2\]), *i.e-* the choice of the bias of the coin toss $\theta$ or of the initial proportion of the chirality $\gamma$ could lead to the same entanglement temperature. Fig. \[f4\] shows the isotherms as functions of $\gamma$ and $\theta$.
Transient behavior
==================
![Envelope of the probability $\protect\lambda _{+}-\Lambda
_{+}$ as a function of the dimensionless time $t$ for two different initial conditions. Each initial condition is established by the couple $(\varphi, \gamma)$ in Eq. (\[psi0\]). Their values are $(\protect\pi /8, \protect\pi /4)$ for the full black line and $(\protect\pi/4, \protect\pi /3) $ for the dashed red line. In both cases, the temperature is $T=0.79~T_{0}$.[]{data-label="f5"}](fig5.eps)
In the QW a stationary entanglement is established between the chirality and position degrees of freedom after a transient time. This fact allowed us to introduce the concept of entanglement temperature. The transient behavior of the system is studied using the original map Eq.(\[mapa\]) in a numerical code with initial conditions given by Eq.(\[psi0\]). These numerical calculations are summarized in Figs. \[f5\] and \[f6\]. Fig. \[f5\] presents the difference between the transient ($\lambda _{+}$) and the stationary ($\Lambda _{+}$) eigenvalues of the density matrix as a function of time (see Eqs.(\[lam\],\[lam0\])). The figure only presents the envelope of the curves because the real eigenvalues dynamics is very intricate; it presents quick oscillations with high density of paths where it is only possible distinguish its global contour. However, the average evolution of the system is determined by the envelope dynamics. Each envelope has two branches placed symmetrically on both sides of $\lambda _{+}-\Lambda _{+}=0$. Two pair of curves are presented in full and dashed lines with color black and red respectively. In both cases, the envelopes decay for $t\rightarrow \infty $ as a power law $1/t^{c}$ with $c=0.490$ for the dashed red line and $c=0.486$ for the full black line. The envelopes of $\lambda _{\pm}$ will be called $\widetilde{\lambda}_{\pm}$ respectively.
![The power law exponent as a function of the dimensionless angle $\protect\gamma $. The initial conditions $(\varphi, \gamma)$ correspond to the isotherms $T=1.1~T_{0}$.[]{data-label="f6"}](fig6.eps)
It was numerically verified for several initial conditions given by Eq.(\[psi0\]) that the transient behavior of $\widetilde{\lambda}_{\pm}-{\Lambda}_{\pm}$ can be adjusted by a power law function of time. Fig. \[f6\] shows the power law exponent $c$ as a function of the initial condition $\gamma $ for the same temperature. Remember that, for $T$ and $\gamma $ given, $\varphi $ is determined by Eq.(\[varphygamma\]). Therefore, the exponent $c$ has a dependence with the initial conditions, however this dependence is not determined by the asymptotic temperature value.
Additionally, the transient behavior of $\widetilde{\lambda}_{\pm}-{\Lambda}_{\pm}$ was numerically studied, using initial conditions given by Eqs.(\[aes\],\[bes\]) with $\sigma\gg1$. In these cases, the system showed a negligible transient dynamics, in concordance with the calculation developed in Ref. [@alejo2010]. Therefore, for these initial conditions, the reduced density matrix is essentially always in thermodynamic equilibrium.
With the aim to understand the transient behavior of the system we develop an analytic theory implementing a parallelism between the reduced density operator Eq.(\[rhoni\]) and the density operator of an electron in a external magnetic field. With this picture on mind we propose the following master equation for the probabilities ${\lambda} _{+}$ and ${\lambda}_{-}$ $$\begin{aligned}
\frac{d{\lambda}_{-}}{dt}& ={\lambda}_{+}\,w_{+-}\,-{\lambda}_{-}\,w_{-+},\, \notag \\
\frac{d{\lambda}_{+}}{dt}& ={\lambda}_{-}\,w_{-+}\,-{\lambda}_{+}\,w_{+-}, \label{masterfin}\end{aligned}$$ where $w_{+-}$ and $w_{-+}$ are transition probabilities per unit of time, that can be understood as population rates. $w_{+-}$ corresponds to the transition ${\lambda}_{+}\rightarrowtail{\lambda}_{-}$ and $w_{-+}$ corresponds to the transition ${\lambda}_{-}\rightarrowtail{\lambda}_{+}$. These rates are time-dependent functions, and their behaviors are known in the limit $t\rightarrow\infty$ when ${d{\lambda}_{\pm}}/{dt}\longrightarrow 0$. In this limit, the stationary solution of Eq.(\[masterfin\]) must be the couple $\Lambda _{-}$ and $\Lambda _{+}$, given by Eq.(\[lam0\]). Then the asymptotic values of the population rates satisfy $$\frac{w_{b}}{w_{a}}= \frac{\Lambda _{-}}{\Lambda _{+}}, \label{rate}$$ where $w_{a}$ and $w_{b}$ are defined by $$\begin{aligned}
w_{a}& \equiv
\begin{array}{c}
\lim \text{ }w_{-+} \\
t\rightarrow \infty~~~
\end{array},\, \label{rates} \\
w_{b}& \equiv
\begin{array}{c}
\lim \text{ }w_{+-} \\
t\rightarrow \infty~~~
\end{array}.\, \label{rates1}\end{aligned}$$Eq.(\[rate\]) expresses a condition of detailed balance which says that the rate of occurrence for any transition equals the rate for the inverse transition. Using our knowledge about the transient and asymptotic behaviors, the following population rates are proposed $$\begin{aligned}
w_{+-}=w_{b}+\xi(t),\, \label{rate2}\\
w_{-+}=w_{a}-\xi(t),\, \label{rate3}\end{aligned}$$ where $$\xi(t)=\frac{K}{t^{c}}[\omega\sin(\omega
t+\delta)+(\frac{c}{t}-w_{a}-w_{b})\cos(\omega t+\delta)],
\label{xi}$$ with $c>0$, $K$, $\omega$ and $\delta$ constants. The general solution of Eq.(\[masterfin\]) with these population rates is $$\begin{aligned}
{\lambda}_{+}=\Lambda _{+}+\frac{K}{t^{c}}\cos(\omega t+\delta)+d\,
e^{-(w_{a}+w_{b})t},\, \label{ratefin}\\
{\lambda}_{-}=\Lambda_{-}-\frac{K}{t^{c}}\cos(\omega t+\delta)-d\,
e^{-(w_{a}+w_{b})t},\, \label{ratefin1}\end{aligned}$$ where $d$ is an additional constant. Note that $e^{-(w_{a}+w_{b})t}\rightarrow0$ faster than ${1/t^{c}}$ for $t\rightarrow\infty$, then Eqs.(\[ratefin\],\[ratefin1\]) verify the asymptotic behavior obtained numerically. All the constants $K$, $c$, $\omega$, $\delta$ and $d$ depend on the initial conditions, however their values should be compatible with the positive character of the functions $\lambda_{+}$, $\lambda_{-}$, $w_{+-}$ and $w_{-+}$. In particular, to describe correctly the numerical results, $K$ takes a finite value for localized initial conditions and $K$ takes negligible value for distributed initial conditions. In summary, the Brownian motion equation for our reduced density matrix Eq.(\[scho\]), takes the form of a master equation Eq.(\[masterfin\]).
Finally it is important to point out that the asymptotic behavior found here is similar to the behavior of the simple cellular automaton known as *sandpile* [@btw]. Such behaviors are characteristic of extended dynamical systems with spatial degrees of freedom. They naturally evolve to self-organized states with correlations that decay with a power law.
Conclusion
==========
The unitary evolution of the QW in a composite Hilbert space is studied. In particular the entanglement between chirality and position degrees of freedom is investigated. After a transient time the system establishes a stationary entanglement between the coin and the position that allows to develop a thermodynamic theory. The asymptotic reduced density operator is used to introduce the entanglement thermodynamic functions in the canonical equilibrium. These thermodynamic functions characterize the asymptotic entanglement. It is shown that the QW initial condition determines the system’s temperature, as well as other thermodynamic function. A map for the isotherms is analytically built for arbitrary localized initial conditions. Additionally, it is shown that choosing appropriately the bias of the coin-toss, it is possible to obtain a predetermined entanglement temperature.
The transient dynamics of the reduced density operator outside the thermodynamic equilibrium is also studied. We show numerically that this transient behavior can be adjusted with a power law, whose exponent depends on the initial conditions. We built a master equation to describe this behavior where the population rates have a time dependence. The accuracy of the master equation solution is numerically verified and it is shown that the reduced density has a cellular automaton behavior.
The behavior of the reduced density operator looks diffusive but it has a dependence with the initial conditions, the global evolution of the system is unitary. Then, if an observer only had information related with the chirality degrees of freedom, it would be very difficult for him to recognize the unitary character of the quantum evolution. In general, from this simple model we can conclude that if the quantum system dynamics is developed in a composite Hilbert space, then the behavior of operators that only belong to one sub-space could camouflage the unitary character of the global evolution.
I acknowledge stimulating discussions with Víctor Micenmacher, Guzmán Hernández, Raúl Donangelo and Armando Pérez and the support from PEDECIBA and ANII.
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| {
"pile_set_name": "ArXiv"
} |
=8.2in =6.3in
Revised on\
[MKN Theory of Bound States]{}\
\
\
[*1900 E. Kenwood Blvd., Milwaukee, WI 53211.*]{}
[**Abstract**]{}
This paper derives the MKN (Maung-Kahana-Norbury) theory of bound states which incorporates the Lande subtraction technique to remove the singularities of the Cornell potential.
NRSE in Momentum Space
======================
Non-relativistic Schrödinger equation (NRSE) in configuration space has been solved exactly for some potentials, such as the Coulomb and simple harmonic oscilator potentials. NRSE with a linear potential can be solved analytically for the $S$-state only as we will show later. For $l>0$, we resort to numerical methods. NRSE $r$-space codes are commonly known to be conditionally unstable [@iitaka; @succi], while the momentum space codes do not have the same problem. The momentum space code has an additional advantage of being easily adaptable to relativistic equations. NRSE in momentum space takes the form $${p^{2}\over 2\mu}\,\phi({\rm\bf p}) +
\int V({\rm\bf q})\phi({\rm\bf p'})\,dp' = E\,\phi({\rm\bf p}),
\label{nrse_ms}$$ where ${\rm\bf q}={\rm\bf p}-{\rm\bf p'}$.
Proof:
NRSE in momentum space can be derived from its configuration space counterpart $$-{\hbar^{2}\over 2\mu}\triangledown^{2}\psi({\rm\bf x}) +
V({\rm\bf x})\psi({\rm\bf x}) = E\,\psi({\rm\bf x})
\label{nrse_cs}$$ by Fourier transform. First we define the following: $$\begin{aligned}
\phi({\rm\bf p})&=&\int \psi({\rm\bf x})e^{i{\rm\bf k}\cdot{\rm\bf x}}d^{3}x,
\label{ft3}\\
\psi({\rm\bf x})&=&\int \phi({\rm\bf p})e^{-i{\rm\bf k}\cdot{\rm\bf x}}
d^{3}p,\label{ft3b}\\
{\rm\bf p}&=&\hbar{\rm\bf k},\end{aligned}$$ and ignore factors of $2\pi$ in Fourier and inverse Fourier transforms. As usual, we assume periodic boundary condition or $\triangledown\phi=\phi=0$ at infinity. We Fourier-transform Eq. \[\[nrse\_cs\]\] term by term. The term on the right hand side of Eq. \[\[nrse\_cs\]\] is obtained simply by Eq. \[\[ft3\]\]. The first term involves $\triangledown^{2}\psi$ and is transformed as $$\begin{aligned}
&& \int \triangledown^{2}\psi({\rm\bf x})
\,e^{i{\rm\bf k}\cdot{\rm\bf x}}\,d^{3}x
\nonumber\\
&=& \int \triangledown\psi({\rm\bf x})\,e^{i{\rm\bf k}\cdot{\rm\bf x}}\cdot
d{\rm\bf S} - \int \triangledown\psi({\rm\bf x})\cdot\triangledown
e^{i{\rm\bf k}\cdot{\rm\bf x}}\,d^{3}x \nonumber \\
&=& -i{\rm\bf k}\cdot
\int e^{i{\rm\bf k}\cdot{\rm\bf x}}\triangledown\psi({\rm\bf x})\,d^{3}x
\nonumber\\
&=& -i{\rm\bf k}\cdot\left[ \int \psi({\rm\bf x})
e^{i{\rm\bf k}\cdot{\rm\bf x}}\,
d{\rm\bf S} - \int \psi({\rm\bf x})
\triangledown e^{i{\rm\bf k}\cdot{\rm\bf x}}\,d^{3}x \right]
\nonumber \\
&=& -i{\rm\bf k}\cdot\left[ -i{\rm\bf k}
\int \psi({\rm\bf x}) e^{i{\rm\bf k}\cdot{\rm\bf x}}\,d^{3}x \right]
\nonumber \\
&=& -k^{2}\phi({\rm\bf p}).
\label{ft1}\end{aligned}$$ The second term involves $V({\rm\bf x})\psi({\rm\bf x})$. With Eq. \[\[ft3b\]\], the second term of Eq. \[\[nrse\_cs\]\] is transformed as $$\begin{aligned}
\int V({\rm\bf x})\psi({\rm\bf x}) e^{i{\rm\bf k}\cdot{\rm\bf x}}\,d^{3}x
&=& \int V({\rm\bf x})
\left[ \int \phi({\rm\bf p'}) e^{-i{\rm\bf k'}\cdot{\rm\bf x}}\,dp' \right]
e^{i{\rm\bf k}\cdot{\rm\bf x}}\,d^{3}x \nonumber \\
&=& \int \int V({\rm\bf x})\phi({\rm\bf p'})
e^{i({\rm\bf k}-{\rm\bf k'})\cdot{\rm\bf x}}\,d^{3}x\,d^{3}p'
\nonumber \\
&=& \int d^{3}p' \,\phi({\rm\bf p'}) \int d^{3}x\,V({\rm\bf x})
e^{i({\rm\bf k}-{\rm\bf k'})\cdot{\rm\bf x}}
\nonumber \\
&=& \int V({\rm\bf p}-{\rm\bf p'})\phi({\rm\bf p'})\,d^{3}p'
\label{ft2}\end{aligned}$$ At last, Eqs. \[\[ft3\]\], \[\[ft1\]\] and \[\[ft2\]\] are all needed to Fourier transform Eq. \[\[nrse\_cs\]\] into Eq. \[\[nrse\_ms\]\]. The proof is complete.
In this paper, we attempt to re-derive the results previously obtained by Maung *et al.* [@maung93] in 1993. The power law potential in $r$ space is given by $$V^{N}(r)=\left\{
\begin{array}{l l}
0 & r < 0 \\
\lambda_{N}\lim_{\eta\to 0}r^{N}e^{-\eta r} & r\geq 0, \, \eta > 0
\end{array}
\right.$$
Let $G=\hbar=c=1$. Define ${\bf q} \equiv {\bf p}-{\bf p'}$. The momentum space potential can be obtained by Fourier transform. $$\begin{aligned}
V^{N}({\bf q})&=&\frac{1}{(2\pi)^{3}}\int^{\infty}_{-\infty} V^{N}(r)
e^{i{\bf r}\cdot{\bf q}}d^{3}r \label{pot}\\
&=&\frac{\lambda_{N}}{(2\pi)^{3}}\lim_{\eta\to 0}\int^{\infty}_{0}\int^{1}_{-1}
\int^{2\pi}_{0}
r^{N}e^{-\eta r}e^{irq\cos\theta}r^{2}drd\cos\theta d\phi\\
&=&\frac{\lambda_{N}}{4\pi^{2}}\frac{1}{iq}\lim_{\eta\to 0}
\int^{\infty}_{0}\int^{1}_{-1}
r^{N+1}e^{-\eta r}e^{irq\cos\theta}drd(irq\cos\theta)\\
&=&\frac{\lambda_{N}}{4\pi^{2}}\frac{1}{iq}\lim_{\eta\to 0}
\int^{\infty}_{0}
r^{N+1}e^{-\eta r}\left( e^{irq}-e^{-irq} \right)dr\\
&=&\frac{\lambda_{N}}{4\pi^{2}}\frac{1}{iq}\lim_{\eta\to 0}
(-1)^{N+1}\frac{\partial^{N+1}}{\partial\eta^{N+1}}\int^{\infty}_{0}
e^{-\eta r}\left( e^{irq}-e^{-irq} \right)dr\\
&=&\frac{\lambda_{N}}{4\pi^{2}}\frac{1}{iq}\lim_{\eta\to 0}
(-1)^{N+1}\frac{\partial^{N+1}}{\partial\eta^{N+1}}
\left[ \frac{1}{\eta - iq} - \frac{1}{\eta + iq} \right]\\
&=&\frac{\lambda_{N}}{4\pi^{2}}\frac{1}{iq}\lim_{\eta\to 0}
(-1)^{N+1}\frac{\partial^{N+1}}{\partial\eta^{N+1}}
\left[ \frac{2iq}{\eta^{2} + q^{2}} \right]\end{aligned}$$ The final form of the momentum space potential is $$V^{N}({\bf q})=\frac{\lambda_{N}}{2\pi^{2}}\lim_{\eta\to 0}
(-1)^{N+1}\frac{\partial^{N+1}}{\partial\eta^{N+1}}
\left[ \frac{1}{\eta^{2} + q^{2}} \right],$$ where $N=-1$ corresponds to the Coulomb potential and $N=1$ the linear potential. Together they give the Cornell potential $$V({\bf q})\equiv V^{C}({\bf q}) + V^{L}({\bf q})
=V^{N=-1}({\bf q}) + V^{N=1}({\bf q}).$$
Next we want to perform a partial wave expansion of $V^{N}$. There are 3 useful formulas, the Wigner-Eckart Theorem[@sak85] $$<E'l'm'|T|Elm>=\delta_{l'l}\delta_{m'm}T_{l}(E),$$ the addition of spherical harmonics $$\sum_{m} Y_{lm}(\Omega)Y^{\ast}_{lm}(\Omega')
=\frac{2l+1}{4\pi}P_{l}(\cos\theta),$$ and the orthogonality of spherical harmonics $$\int d\Omega Y^{\ast}_{lm}(\Omega) Y_{l'm'}(\Omega) = \delta_{l'l}\delta_{m'm},$$ which are used in deriving the following result. $$\begin{aligned}
<{\bf p}|V^{N}|{\bf p'}>&=&\sum_{lm}\sum_{l'm'}
<{\bf p}|lm><lm|V^{N}|l'm'><l'm'|{\bf p'}> \\
&=&\sum_{lm}\sum_{l'm'}<p\Omega|lm><lm|V^{N}|l'm'><l'm'|p'\Omega'> \\
&=&\sum_{lm}\sum_{l'm'}<p|<\Omega|lm><lm|V^{N}|l'm'><l'm'|\Omega'>|p'> \\
&=&\sum_{lm}<\Omega|lm><p|V^{N}_{l}|p'><lm|\Omega'> \\
&=&\sum_{lm}V^{N}_{l}(p,p')Y_{lm}(\Omega)Y^{\ast}_{lm}(\Omega') \\
&=&\sum_{l}\frac{2l+1}{4\pi}V^{N}_{l}(p,p')P_{l}(\cos\theta)
\label{eq:Vpp}\end{aligned}$$ In scattering and bound state problems, it is customary to expand the momentum space wavefunction $\phi({\bf p})$ in partial waves, such that $$\phi({\bf p})=\sum_{nlm}c_{nlm}\phi_{nl}(p)Y_{lm}(\Omega),$$ where $c_{nlm}$’s are coefficients of the expansion \[p. 396 of Ref 1 \].
The non-relativistic Schrödinger equation in momentum space is given as $$\begin{aligned}
\left( \hat{E}-\frac{{\bf p}^{2}}{2\mu} \right) \phi({\bf p})
&=&\int V^{N}({\bf q}) \phi({\bf p'}) d^{3}{\bf p'} \\
&=&\int <{\bf p}|V^{N}|{\bf p'}> \phi({\bf p'}) d^{3}{\bf p'}\end{aligned}$$
Expand NRSE in partial waves. $$\begin{aligned}
\left( \hat{E}-\frac{{\bf p}^{2}}{2\mu} \right) \sum_{nlm} c_{nlm}
\phi_{nl}(p) Y_{lm}(\Omega) &=&
\int p'^{2} dp' d\Omega' \sum_{nlm} V^{N}_{l}(p,p')
Y_{lm}(\Omega) Y^{\ast}_{lm}(\Omega') \nonumber \\
&&\quad \times\sum_{n'l'm'} c_{n'l'm'} \phi_{n'l'}(p') Y_{l'm'}(\Omega') \\
&=&\int p'^{2}dp' \sum_{nlm} V^{N}_{l}(p,p') c_{nlm} \phi_{nl}(p')
Y_{lm}(\Omega).\end{aligned}$$ The $nl$-th terms can be separated by inspection with the help of the identity $\hat{E}\,\phi_{nl}(p)=E_{nl}\,\phi_{nl}(p).$ The partial wave NRSE is $$\left( E_{nl} - \frac{p^{2}}{2\mu} \right) \phi_{nl}(p) =
\int p'^{2} dp' V^{N}_{l}(p,p') \phi_{nl}(p').$$ Use the orthogonality of Legendre polynomials, $$\int^{1}_{-1} P_{l}(x) P_{l'}(x) dx = \frac{2}{2l+1} \delta_{l'l},$$ and Eq. (\[eq:Vpp\]), we can calculate the potential matrix elements as follow. $$\begin{aligned}
&&\quad \int^{1}_{-1} <{\bf p}|V^{N}|{\bf p'}> P_{l}(\cos\theta) d\cos\theta \\
&&=\sum_{l'} \frac{2l'+1}{4\pi} V^{N}_{l'}(p,p')
\int^{1}_{-1} P_{l}(\cos\theta) P_{l'}(\cos\theta) d\cos\theta \\
&&=\sum_{l'} \frac{1}{2\pi} V^{N}_{l'}(p,p') \delta_{l'l} \\
&&=\frac{1}{2\pi} V^{N}_{l}(p,p')\end{aligned}$$ In order words, $$V^{N}_{l}(p,p')=2\pi\int^{1}_{-1} V^{N}({\bf q}) P_{l}(\cos\theta)d\cos\theta.
\label{eq:V^N_l}$$ Define $$y \equiv \frac{p^{2}+p'^{2}+\eta^{2}}{2p'p},
\label{eq:y}$$ and use the definition of the Legendre polynomial of the second kind $Q_{n}(z)$, $$Q_{n}(z)={1 \over 2}\int^{1}_{-1}\frac{1}{z-t}P_{n}(t)dt,$$ we can modify Eq. (\[eq:V\^N\_l\]) as $$\begin{aligned}
V^{N}_{l}(p,p')
&=&2\pi\int^{1}_{-1} V^{N}({\bf q}) P_{l}(\cos\theta)d\cos\theta \\
&=&\frac{\lambda_{N}}{\pi} \lim_{\eta\to 0}
\frac{\partial^{N+1}}{\partial\eta^{N+1}}
\int^{1}_{-1} \frac{1}{q^{2}+\eta^{2}} P_{l}(\cos\theta) d\cos\theta \\
&=&\frac{\lambda_{N}}{\pi} \lim_{\eta\to 0}
\frac{\partial^{N+1}}{\partial\eta^{N+1}}
\int^{1}_{-1} \frac{1}{p^{2}+p'^{2}-2p'p\cos\theta+\eta^{2}}
P_{l}(\cos\theta) d\cos\theta \\
&=&\frac{\lambda_{N}}{\pi} \lim_{\eta\to 0}
\frac{\partial^{N+1}}{\partial\eta^{N+1}}
\int^{1}_{-1} \frac{1}{2p'p(y-\cos\theta)} P_{l}(\cos\theta) d\cos\theta \\
&=&\frac{\lambda_{N}}{\pi} \lim_{\eta\to 0}
\frac{\partial^{N+1}}{\partial\eta^{N+1}} \frac{Q_{l}(y)}{p'p}.\end{aligned}$$ The coulomb potential corresponds to $N=-1$ and has the form $$V^{C}_{l}(p,p')=\frac{\lambda_{C}}{\pi} \lim_{\eta\to 0} \frac{Q_{l}(y)}{p'p}.$$ The linear potential corresponds to $N=1$ and has the form $$\begin{aligned}
V^{L}_{l} &=& \frac{\lambda_{L}}{\pi} \lim_{\eta\to 0}
\frac{\partial^{2}}{\partial\eta^{2}} \frac{Q_{l}(y)}{p'p} \\
&=& \frac{\lambda_{L}}{\pi} \lim_{\eta\to 0}
\frac{\partial}{\partial\eta} \left[ \frac{\eta}{(p'p)^{2}} Q'_{l}(y) \right]\\
&=& \frac{\lambda_{L}}{\pi} \lim_{\eta\to 0}
\left[ \frac{Q'_{l}(y)}{(p'p)^{2}} + \frac{\eta^{2}}{(p'p)^{3}} Q''_{l}(y)
\right].\end{aligned}$$ There are 3 useful relations in terms of Legendre polynomials of the 2nd kind [@stegun]: $$\begin{aligned}
Q_{0}(y) &=& {1 \over 2} \ln \left| \frac{y+1}{y-1} \right|,
\label{eq:Q} \\
Q_{l}(y) &=& P_{l}(y)Q_{0}(y)-w_{l-1}(y),
\label{eq:Ql} \\
w_{l-1}(y) &=& \sum^{l}_{m=1}{1 \over m} P_{l-m}(y)P_{m-1}(y).\end{aligned}$$
The singularities of $V^{C}_{l}(p,p')$ and $V^{L}_{l}(p,p')$ come from the singularies of $Q_{l}(y)$ and $Q''_{l}(y)$. From Eq. (\[eq:Ql\]), it is obvious that the singularities of $Q_{l}(y)$ and $Q''_{l}(y)$ again come from those of $Q_{0}(y)$, $Q'_{0}(y)$ and $Q''_{0}(y)$. In order to treat the singularities of the momentum space Cornell potential, we need to control the singularities of $Q_{0}(y)$, $Q'_{0}(y)$ and $Q''_{0}(y)$ first and foremost. Substitute Eq. (\[eq:y\]) into Eq. (\[eq:Q\]), we have $$Q_{0}(y)={1 \over 2}\ln\left[ \frac{(p+p')^{2}+\eta^{2}}{(p-p')^{2}+\eta^{2}}
\right].
\label{eq:Q0}$$ Differentiating Eq. (\[eq:Q0\]) yields $$\begin{aligned}
Q'_{0}(y)&=&{1 \over 2}\frac{\partial}{\partial y} \ln \left|
\frac{y+1}{y-1} \right| \\
&=& {1 \over 2}\left[ \frac{1}{y+1} - \frac{1}{y-1} \right] \\
&=& \frac{1}{1-y^{2}} \\
&=& p'p \left[ \frac{1}{(p+p')^{2} + \eta^{2}} -
\frac{1}{(p-p')^{2} + \eta^{2}} \right] \label{eq:Q'}\end{aligned}$$ Differentiating again gives $$\begin{aligned}
Q''_{0}(y) &=& \frac{2y}{(1-y^{2})^{2}} \\
&=& \frac{p^{2}+p'^{2}+\eta^{2}}{p'p}
\left[ p'p \left( \frac{1}{(p+p')^{2}+\eta^{2}}
-\frac{1}{(p-p')^{2}+\eta^{2}} \right) \right]^{2},\end{aligned}$$ or $$\frac{\eta^{2}}{p'p}Q''_{0}(y)=\eta^{2} \left( p^{2}+p'^{2}+\eta^{2} \right)
\left[ \frac{1}{(p+p')^{2}+\eta^{2}}
-\frac{1}{(p-p')^{2}+\eta^{2}} \right]^{2}. \label{eq:Q''}$$
There are two useful identities which we want to prove: $$\int^{\infty}_{0} {1\over p'}Q_{0}(y, \eta=0)\, dp'
= {\pi^{2}\over 2}, \label{eq:integral_1}$$ $$\int^{\infty}_{0} \left[ \frac{\eta^{2}}{p'p} Q''_{0}(y) + Q'_{0}(y) \right]
dp' = 0 \label{eq:integral_2}.$$
Proof:\
The integral in Eq. \[\[eq:integral\_1\]\] is derived as follow: $$\begin{aligned}
&& \int^{\infty}_{0} {1\over p'}Q_{0}(y,\eta=0)\,dp' \\
&=& {1\over 2} \int^{\infty}_{0} {1 \over p'} \ln \left(\frac{p+p'}{p-p'}
\right)^{2} dp'\\
&=& {1\over 2} \left[ \int^{a}_{0} {1 \over x} \ln \left( \frac{x+a}{x-a}
\right)^{2} dx + \int^{\infty}_{a} {1 \over x} \ln \left( \frac{x+a}{x-a}
\right)^{2} dx \right] \\
&=& \int^{\infty}_{a} {1 \over x} \ln \left( \frac{a+x}{a-x}\right) dx +
\int^{\infty}_{a} {1 \over x} \ln \left( \frac{x+a}{x-a} \right) dx \\
&=& - \int^{0}_{\infty} {1\over ae^{-u}} \ln \left(
\frac{a+ae^{-u}}{a-ae^{-u}} \right) ae^{-u} du +
\int^{\infty}_{0} {1\over ae^{u}} \ln \left(
\frac{ae^{u}+a}{ae^{u}-a} \right) ae^{u} du \\
&=& 2\left[ \int^{\infty}_{0} \ln \left( \frac{1+e^{-u}}{1-e^{-u}} \right) du
\right] \\
&=& 2\left[ \int^{\infty}_{0} \ln (1+e^{-u})\,du -
\int^{\infty}_{0} \ln (1-e^{-u})\,du \right] \\
&=& 2\left[ {\pi^{2}\over 12} + {\pi^{2}\over 6} \right] \label{eq:sum} \\
&=& {\pi^{2}\over 2}\end{aligned}$$ The results of Eq. \[\[eq:sum\]\] come from relations BI((256))(10) and BI((256))(11) in Gradshteyn and Ryzhik [@GR80].
Eq. \[\[eq:Q’\]\] has 4 simple poles: $\alpha = p+i\eta$, $\alpha^{\ast}$, $\beta = -p+i\eta$ and $\beta^{\ast}$. $Q'_{0}(y)$ can be rewritten as $$Q'_{0}(y) = p'p \left[ \frac{-1}{(p'-\alpha)(p'-\alpha^{\ast})} +
\frac{1}{(p'-\beta)(p'-\beta^{\ast})} \right]$$ The contour integral $\oint Q'_{0}(y) dz$ over the upper complex plane has 2 residues: $Res(\alpha)$ and $Res(\beta)$. Use the formula $Res(z_{0}) = \lim_{z \to z_{0}} (z-z_{0})f(z)$ to calculate these residues, $$\begin{aligned}
Res(\alpha) &=& \lim_{p'\to\alpha} p'p \left[ \frac{-1}{p'-\alpha^{\ast}}
+ \frac{p'-\alpha}{(p'-\beta)(p'-\beta^{\ast})} \right] \\
&=& -p(p+i\eta)\left[ {1\over 2i\eta} + 0 \right] \\
&=& -\frac{p(p+i\eta)}{2i\eta},\end{aligned}$$ $$\begin{aligned}
Res(\beta) &=& \lim_{p'\to\beta} p'p
\left[ \frac{-(p'-\beta)}{(p'-\alpha)(p'-\alpha^{\ast})} +
{1\over (p'-\beta^{\ast})} \right] \\
&=& p(-p+i\eta)\left[ 0 + {1\over 2i\eta} \right] \\
&=& \frac{p(-p+i\eta)}{2i\eta},\end{aligned}$$ $$\begin{aligned}
Res(\alpha)+Res(\beta) &=& -\frac{p(p+i\eta)}{2i\eta} +
\frac{p(-p+i\eta)}{2i\eta} \\
&=& -\frac{p^{2}}{i\eta}.\end{aligned}$$ Since the contour at infinity is zero and $Q'_{0}(y)$ along the real axis is symmetric around the origin, we obtain $$\begin{aligned}
\int^{\infty}_{0} Q'_{0}(y)\, dp' &=& {1\over 2}\oint Q'_{0}(y)\, dz \\
&=& {1\over 2}\left( 2\pi i\sum Res \right) \\
&=& -\frac{\pi p^{2}}{\eta}.\end{aligned}$$ $(\eta^{2}/p'p)Q''_{0}(y)$ has the same poles as $Q'_{0}(y)$ but of order 2. Residues of order $m$ are calculated by the formula $$Res(z_{0})=\lim_{z \to z_{0}}{1\over (m-1)!}\,
\frac{d^{m-1}}{dz^{m-1}}(z-z_{0})^{m}f(z).$$ Again we can simplify the algebra by rewriting Eq. \[\[eq:integral\_2\]\] as $$\frac{\eta^{2}}{p'p} Q''_{0}(y)=\eta^{2} \left( p^{2} + p'^{2} + \eta^{2}
\right) \left[ \frac{-1}{(p'-\alpha)(p'-\alpha^{\ast})} +
\frac{1}{(p'-\beta)(p'-\beta^{\ast})} \right]^{2}.$$ The residues are $$\begin{aligned}
Res(\alpha) &=& \lim_{p'\to\alpha}{d\over dp'}
\eta^{2} \left( p^{2} + p'^{2} + \eta^{2} \right)
\left[ \frac{-1}{(p'-\alpha^{\ast})} +
\frac{p'-\alpha}{(p'-\beta)(p'-\beta^{\ast})} \right]^{2} \\
&=& \eta^{2}\left\{ 2(p+i\eta)\left[ \frac{-1}{2i\eta} \right]^{2} +
4p(p+i\eta)\left[ \frac{-1}{2i\eta} \right]
\left[ \frac{1}{(2i\eta)^{2}} +
{1\over 2p(2p+2i\eta)} \right]\right\} \\
&=& \frac{\eta^{2}}{2}\left\{ -{p\over\eta^{2}} - {i\over\eta} +
{p^{2}\over i\eta^{3}} + {p\over\eta^{2}} - {1\over i\eta} \right\} \\
&=& {p^{2}\over 2i\eta},\end{aligned}$$ $$\begin{aligned}
Res(\beta) &=& \lim_{p'\to\beta}{d\over dp'}
\eta^{2} \left( p^{2} + p'^{2} + \eta^{2} \right)
\left[ \frac{-(p'-\beta)}{(p'-\alpha)(p'-\alpha^{\ast})} +
\frac{1}{p'-\beta^{\ast}} \right]^{2} \\
&=& \eta^{2}\left\{ 2(p-i\eta)\left[ \frac{1}{2i\eta} \right]^{2} +
4p(p-i\eta)\left[ \frac{1}{2i\eta} \right]
\left[ \frac{-1}{(2i\eta)^{2}} +
{1\over 2p(-2p+2i\eta)} \right]\right\} \\
&=& \frac{\eta^{2}}{2}\left\{ {p\over\eta^{2}} - {i\over\eta} +
{p^{2}\over i\eta^{3}} - {p\over\eta^{2}} - {1\over i\eta} \right\} \\
&=& {p^{2}\over 2i\eta}.\end{aligned}$$ Hence the sum of residues is $$\begin{aligned}
Res(\alpha)+Res(\beta) &=& {p^{2}\over 2i\eta} + {p^{2}\over 2i\eta}\\
&=& {p^{2}\over i\eta}\end{aligned}$$ Since $(\eta^{2}/p'p)Q''_{0}(y)$ is symmetric around the origin, we can integrate along the same contour as before and obtain $$\begin{aligned}
\int^{\infty}_{0} \frac{\eta^{2}}{p'p} Q''_{0}(y)\, dp' &=&
{1\over 2} \oint \frac{\eta^{2}}{pz} Q''_{0}(y)\, dz \\
&=& {1\over 2} \left( 2\pi i \sum Res \right) \\
&=& \frac{\pi p^{2}}{\eta}.\end{aligned}$$ From these results, it is obvious that Eq. \[\[eq:integral\_2\]\] is true. The proof is complete.
A simple example is the momentum space Schrödinger equation with a linear potential in the $S$-state [@maung93; @norbury91], $$\frac{p^{2}}{2\mu}\,\phi_{n0}(p)+\frac{\lambda_{L}}{\pi p^{2}}
\int^{\infty}_{0}\underbrace{\left[ {\eta^{2}\over p'p}Q''_{0}(y)+
Q'_{0}(y)\right]}_{V^{L}_{0}(p,p')}\,
\phi_{n0}(p')\, dp'
= E_{n0}\,\phi_{n0}(p),
\label{se1}$$ where $y=(p^{2}+p'^{2})/2p'p$. Lande subtraction [@maung93; @norbury91] involves subtracting a zero term $$\int^{\infty}_{0}\left[ {\eta^{2}\over p'p}Q''_{0}(y)+Q'_{0}(y)\right]\,dp'
=0$$ from Eq. \[\[se1\]\] such that $$\frac{p^{2}}{2\mu}\,\phi_{n0}(p)+\frac{\lambda_{L}}{\pi p^{2}}\int^{\infty}_{0}
\left[ {\eta^{2}\over p'p}Q''_{0}(y)+Q'_{0}(y)\right]
[\phi_{n0}(p')-\phi_{n0}(p)]\, dp'
= E_{n0}\,\phi_{n0}(p).
\label{se2}$$ Using Eqs. \[\[eq:Q’\],\[eq:Q”\]\], the integral in Eq. \[\[se2\]\] for $p>0$ in the limit of $y\to 1$ can be shown to equal $$\lim_{\eta\to 0}\,\lim_{p\to p'}\,
{\lambda_{L}\over\pi}\,\left[2\eta^{2}\left({1\over (p-p')^{2}+\eta^{2}}
\right)^{2}
-{1\over (p-p')^{2}+\eta^{2}}\right]\,(p-p')^{2}\,{d\phi_{n0}\over dp}=0.
\label{sing1}$$ The order of the limits in Eq. \[\[sing1\]\] is important. The reverse order will lead to the nonsensical result $\int Q'_{0}(y)\,dp'=0$. Next, in the limit of $p,p'\to 0$, $(p+p')^{2}=(p-p)^{2}$. By substituting this equality into Eqs. \[\[eq:Q’\],\[eq:Q”\]\], it can be shown again that the integral in Eq. \[\[se2\]\] vanishes for $p\to 0$ at $y=1$. At the end, the integral vanishes at $y=1,\;\forall\, p$. Away from the singularities, both integrands in the integral of Eq. \[\[se2\]\] are finite. By taking $\eta\to 0$, the first integrand vanishes. The final form of Eq. \[\[se2\]\] is $$\frac{p^{2}}{2\mu}\,\phi_{n0}(p)+\frac{\lambda_{L}}{\pi p^{2}}\int^{\infty}_{0}
Q'_{0}(y)\,[\phi_{n0}(p')-\phi_{n0}(p)]\, dp'
= E_{n0}\,\phi_{n0}(p),
\label{lin0}$$ where $Q'_{0}(y)=1/(1-y^{2})$.
The momentum space NRSE with a coulomb potential is given as $${p^{2}\over 2\mu}\phi_{nl}(p) + {\lambda_{C}\over \pi p}
\int^{\infty}_{0} P_{l}(y)\frac{Q_{0}(y)}{p'} \phi_{nl}(p')p'^{2}dp'
-{\lambda_{C}\over \pi p}\int^{\infty}_{0}w_{l-1}(y)\phi_{nl}(p')
p'dp'= E_{nl}\phi_{nl}(p).$$ Use Eq. \[\[eq:integral\_1\]\] to subtract out the logarithmic singularity and obtain $$\begin{aligned}
&& {p^{2}\over 2\mu}\phi_{nl}(p) + {\lambda_{C}\over \pi p}
\int^{\infty}_{0} P_{l}(y)\frac{Q_{0}(y)}{p'} \left[ p'^{2}\phi_{nl}(p')
- \frac{p^{2}\phi_{nl}(p)}{P_{l}(y)} \right] dp' +
{\lambda_{C}\over \pi p} \left[ {\pi^{2}\over 2}p^{2}\phi_{nl}(p)\right]
\nonumber \\
&& \quad -{\lambda_{C}\over \pi p}\int^{\infty}_{0}w_{l-1}(y)\phi_{nl}(p')
p'dp' = E_{nl}\phi_{nl}(p).\end{aligned}$$
Before we perform Lande subtraction on the NRSE with a confining potential, we need the identity $$P'_{l}(1)=\frac{l(l+1)}{2}. \label{eq:P'}$$ Proof:\
We use the recursion relation $xP'_{x}-P'_{l-1}(x)=lP_{l}(x)$, the equality $P_{l}(1)=1$ to obtain the following relations: $$\begin{aligned}
P'_{l}(1) - P'_{l-1}(1) &=& l \nonumber \\
P'_{l-1}(1) - P'_{l-2}(1) &=& l-1 \nonumber \\
\vdots\quad && \nonumber \\
P'_{2}(1) - P'_{1}(1) &=& 2 \nonumber \\
P'_{1}(1) - P'_{0}(1) &=& 1.\end{aligned}$$ Add these relations and use $P_{0}(x)=1$ or $P'_{0}(x)=0$, we prove Eq. \[\[eq:P’\]\]. Next we want to examine the singularities of $Q'_{l}(y)$ and $Q''_{l}(y)$. Differentiating Eq. \[\[eq:Ql\]\] once and twice, we have $$\begin{aligned}
Q'_{l} &=& P'_{l}Q_{0} + P_{l}Q'_{0} - w'_{l-1}, \\
Q''_{l} &=& P''_{l}Q_{0} + 2P'_{l}Q'_{0} + P_{l}Q''_{0} - w''_{l-1}.\end{aligned}$$ $\eta^{2}P''_{l}Q_{0}$, $\eta^{2}P'_{l}Q'_{0}$ and $\eta^{2}w''_{l-1}$ vanish in the limit of $\eta\to 0$. The momentum space confining potential is $$V^{L}_{l}(p',p)={\lambda_{L}\over \pi} \lim_{\eta\to 0} \left\{ P_{l}(y)
\left[ \frac{\eta^{2}}{(p'p)^{3}} Q''_{0}(y) + \frac{Q'_{0}(y)}{(p'p)^{2}}
\right] + \frac{P'_{l}(y)Q_{0}(y) - w'_{l-1}(y)}{(p'p)^{2}} \right\}.$$ Lastly, perform Lande subtraction and use Eq. \[\[eq:P’\]\]. Take the limit of $\eta=0$, we derive the momentum space NRSE with a confining potential as $$\begin{aligned}
&& {p^{2}\over 2\mu}\phi_{nl}(p) + {\lambda_{L}\over \pi p^{2}}
\int^{\infty}_{0} P_l(y)Q'_{0}(y)
\left[ \phi_{nl}(p') - {\phi_{nl}(p) \over P_{l}(y)} \right]dp' \nonumber \\
&& \quad + {\lambda_{L}\over \pi p^{2}} \int^{\infty}_{0} P'_{l}(y)
{Q_{0}(y)\over p'} \left[ p'\phi_{nl}(p') - {l(l+1)\over 2}
\frac{p\phi_{nl}(p)}{P'_{l}(y)} \right]dp' \nonumber \\
&& \quad + {\lambda_{L}\over \pi p^{2}} {l(l+1)\over 2} \left[ {\pi^{2}\over 2}
p\phi_{nl}(p) \right]
+ {\lambda_{L}\over \pi p^{2}} \int^{\infty}_{0} w'_{l-1}(y)
\phi_{nl}(p')\,dp' = E_{nl}\phi_{nl}(p).\end{aligned}$$
Exact Solution of NRSE with a Linear Potential
==============================================
In the next section, we are going to solve the NRSE numerically with a linear potential. In this section, we will solve the same equation exactly so that we can use the analytic results to check their numerical counterparts. The Hamiltonian equation can be written as $$\left( {d^{2}\over dr^{2}} + {2\over r}\,{d\over dr} \right)\,R -
2\mu[\lambda_{L}\,r - E]\,R = 0. \label{linear_r}$$ Let $S\equiv r\,R$, then Eq. \[\[linear\_r\]\] can be simplified as $${d^{2}\over dr^{2}}S - 2\mu[\lambda_{L}\,r - E]S = 0.$$ If we define as new variable $$x\equiv \left( {2\mu\over \lambda_{L}}^{2} \right)^{1\over 3}
[\lambda_{L}r - E], \label{linear_s}$$ Eq. \[\[linear\_s\]\] can be transformed as $$S'' - xS = 0,$$ which is the Airy equation. The solution which satisfies the boundary condition of $S\to 0$ as $x\to \infty$ is the Airy function ${\rm Ai}(x)$. Fig. \[\[ai\]\] illustrates the graph of the Airy function. By noticing that $S\equiv r\,R$ vanishes at $r=0$, we infer that $S(r=0)$ must coincide with a zero of ${\rm Ai}(x)$. It is made possible by letting the eigen-energy act as a horizontal shift which shifts the origin to the left along the $x$-axis. If $S$ is plotted against $r$ instead of $x$, $S$ will vanish at the origin if $E_{n}$ is chosen appropriately. This conclusion leads to the eigen-energy formula $$E_{n} = -x_{n}\,\left( {\lambda_{L}^{2}\over 2\mu} \right)^{1\over 3},$$ where $x_{n}$ is the $n$-th zero of the Airy function counting from $x=0$ along $-x$.
In Norbury *et. al.*’s [@norbury91] paper, the values $\lambda_{L} = 5$ and $\mu = 0.75$ are used. The eigen-energy formula is $$E_{n} = -2.554364772\,x_{n}.$$ Table \[ai\_tab\] lists the zeros of the Airy function and the corresponding exact eigen-energies.
Conclusion
==========
The $p$-space formalism shown in this paper can be applied to any arbitrary potential in principle although only the cases of $n=-1,1$ in $r^{n}$ are considered in this paper. More complicated potentials require the calculation of integrals involving other powers of $r$. The potential involving the coupling constant $\alpha(r)$ in QCD near the asymptotic freedom region is such an example. In these cases, we may need to reply on numerical integration to evaluate the integrals of the Lande subtraction terms.
[99]{}
T. Iitaka, *Physical Review E*, [**49**]{}, 4684 (1994).
S. Succi, *Physical Review E*, [**53**]{}, 1969 (1996).
K. M. Maung, D. E. Kahana and J. W. Norbury, *Physical Review D*, [**47**]{}, 1182 (1993).
J. J. Sakurai, [*Modern Quantum Mechanics*]{}, (Redwood City, CA: Addison- Wesley, 1985), 238-242.
M. Abramowitz and I. A. Stegun, [*Handbook of Mathematical Functions*]{}, (Washington, D. C.: National Bureau of Standards, 1964), 333–335.
I. S. Gradshteyn and I. M. Ryshik, *Tables of Integrals, Series, and Products*, (San Diego: Academic, 1980), 526.
J. W. Norbury, D. E. Kahana and K. M. Maung, *Canadian Journal of Physics*, [**70**]{}, 86 (1991).
W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, [*N*umerical Recipes in C]{}, (Cambridge: Cambridge, 1997).
S. Gasiorowicz, *Quantum Physics* (New York: Wiley, 1974), 197.
n $x_{n}$ $E_{n}$
---- -------------- -------------
1 -2.33810741 5.972379202
2 -4.08794944 10.44211404
3 -5.52055983 14.10152355
4 -6.78670809 17.33572806
5 -7.94413359 20.29221499
6 -9.02265085 23.04714148
7 -10.04017434 25.64626764
8 -11.00852430 28.11978666
9 -11.93601556 30.48893766
10 -12.82877675 32.76937540
: Zeros of the Airy function and the corresponding eigen-energies in GeV for $l=0$, $\mu = 0.75 \rm\, GeV$, $\lambda_{L} = 5 \rm\, GeV$.
\[ai\_tab\]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We investigate the quantum versus classical dynamics of a microwave cavity-coupled-Cooper pair transistor (CPT) system, where an applied dc bias causes the system to self-oscillate via the ac Josephson effect. Varying the dc bias allows the self-oscillation frequency to be tuned. An unusual feature of the system design is that the dc bias does not significantly affect the high quality factor of the cavity mode to which the CPT predominantly couples. The CPT-cavity mode system has a mechanical analogue involving a driven coupled pendulum-oscillator system. The corresponding, nonlinear classical dynamical equations exhibit chaotic, as well as aperiodic motions depending on the initial conditions and the nature and strengths of the damping/noise forces. The quantum master equation exhibits such phenomena as dynamical tunnelling and the generation of nonclassical states from initial classical states. Obviating the need for an external ac-drive line, which typically is harder to noise filter than a dc bias line, the self-oscillating system described here has considerable promise for demonstrating macroscopic quantum dynamical behavior.'
author:
- 'M.P. Blencowe'
- 'A.D. Armour'
- 'A.J. Rimberg'
title: 'Quantum-classical correspondence for a dc-biased cavity resonator–Cooper-pair transistor system[^1]'
---
Introduction
============
The work presented in this chapter has its origins in a seemingly mundane microwave engineering question: is it possible to apply a dc voltage (or current) bias to the center conductor of a superconducting coplanar microwave cavity, without significantly affecting the quality factor of, say, the first and second microwave modes of the cavity? Our original motivation behind this question was to devise a circuit quantum electrodynamics (QED) based scheme [@wallraffnature04] that can generate and detect quantum states of a mechanical resonator [@armournjp08; @blencowenjp08], where the dc bias is required to strongly couple a nanomechanical resonator to a superconducting qubit. However, it turns out that having such a dc bias functionality opens up possibilities for other heretofore difficult-to-realize quantum dynamical investigations, one of which we shall focus on here.
We shall in particular investigate the quantum dynamics of the device shown in Fig. \[cavityfig\], which comprises two Josephson junctions (JJ) in series with a gate electrode, and where the source electrode to one of the JJ’s contacts the center conductor of the microwave cavity, while the drain electrode from the other JJ contacts the ground plane of the microwave cavity. The following section describes how the microwave cavity design allows the application of a dc voltage bias $V_{\mathrm{dc}}$ to the center conductor, while maintaining a very large quality factor of the second microwave mode to which the JJ’s strongly couple [@chenapl11]. For not too large a $V_{\mathrm{dc}}$ bias, the JJ’s operate in the subgap region as a “Cooper pair transistor" (CPT), where the dc bias generates a tunable oscillating supercurrent through the CPT via the ac Josephson effect. The tunneling Cooper pairs will both emit into and absorb photons from the second microwave mode, and it is the resulting coupled CPT-cavity mode quantum dynamics that will be of central interest to us.
Related devices comprising one or more JJ’s embedded in a microwave cavity date back to just a few years following the discovery of the ac Josephson effect [@josephsonpl62], where classical signatures of the resonant microwave modes of the tunnel junctions themselves, interacting with the alternating tunnel currents, were observed and discussed [@werthamer; @zimmerman; @smith]. Beginning in the ‘90’s, investigations addressed the effect of a structured electromagnetic environment with resonant modes on the current-voltage characteristics of dc voltage biased JJ’s [@holstprl94; @ingoldprb94; @hofheinzprl01]. And more recently, similar investigations involving double JJ devices were carried out [@leppakangasprb08; @pashkinprb11]. However, the quality factors of the electromagnetic modes in these devices were small, typically less than 10, to be contrasted with quality factors exceeding $10^3$ for the present device design [@chenapl11] shown in Fig. \[cavityfig\]. As a consequence, emitted microwave photons will now remain in the cavity mode for many Cooper pair tunnel oscillation cycles before leaking out of the cavity mode; it does not make sense to treat the microwave cavity as an electromagnetic environment for the CPT. Instead, the cavity and CPT should be viewed as a strongly-coupled, quantum coherent system.
The CPT-cavity mode device has a mechanical analogue involving a driven coupled pendulum-oscillator system (Sec. \[mechanicalsec\]). The corresponding, nonlinear classical dynamical equations exhibit chaotic, as well as aperiodic motions depending on the initial conditions and the nature and strengths of the damping/noise forces. Thus, the device in principle allows the experimental investigation of the quantum dynamics of a system for which the corresponding classical dynamics is chaotic. There is a long tradition of using Josephson junction devices for investigating macroscopic quantum dynamics in systems with corresponding nonlinear classical equations [@srivastavapr87; @leggettcp09]. The Sussex group carried out some of the first, pioneering work in the ‘80’s [@prancehpa83], which was followed by the demonstration of quantum tunneling by the Clarke group at Berkeley [@clarkescience88], and which culminated in demonstrations over a decade later of superposition states by the Lukens [@friedmannature00] and Mooij [@chiorescuscience03] groups at Stonybrook and Delft, respectively. Subsequent, related developments have largely focussed on the realization of superconducting quantum bits for quantum computing applications [@neeleynature10; @dicarlonature10], although JJ devices still occasionally are used for exploring macroscopic quantum dynamics and the transition to classical dynamics [@fedorovprl11].
A large body of theoretical work concerning the quantum-classical correspondence for driven systems has focused on the Duffing and other anharmonic oscillators [@dykmanzetf88; @zurekprl94; @brunjpa96; @kohlerpre97; @habibprl98; @bhattacharyaprl00; @monteolivapre01; @habibprl02; @peanoprb04; @everittpre05; @marthalerpra06; @dykmanpre07; @greenbaumpre07; @serbanprl07; @katznjp08; @versopra10; @ketzmerickpre10], as well as on various rigid rotor models [@todaptps89; @casatiptps89; @foxpra90; @foxpra91; @grahampra91; @grahamprl91; @foxpre94; @latkapra94; @mirbachprl95; @gorincp97; @mouchetpre01; @mouchetpre06; @reichl04; @haake10]; many insights have been gained by investigating dynamical properties using a quantum phase space (i.e., Wigner or Husimi function [@takahashiptps89; @leepre93]) description, and by examining the Floquet states and associated quasienergy spectra. However, relevant experimental results have been few [@steckscience01; @chaudhurynature09; @vijayrsi09]. One of the key difficulties is that most experimental realisations require an [*external*]{} ac signal to drive the system, which can be one of the most significant sources of noise, preventing the system from displaying manifest quantum dynamical behavior. In contrast, the CPT-resonator system in Fig. \[cavityfig\] generates its own ac drive, i.e., it self-oscillates. As a consequence of the ac-Josephson effect, only a dc voltage bias $V_{\mathrm{dc}}$ is required, and by varying $V_{\mathrm{dc}}$, the drive frequency can be tuned. Since it is considerably easier to noise filter a dc bias line than an ac-drive line, the device described here has considerable promise for exhibiting macroscopic quantum dynamical behavior.
The outline of this chapter is as follows. In Sec. \[devicesec\], we give a description of the CPT-cavity system. The classical system equations are derived in Sec. \[classicalsec\], and solutions to these equations are discussed in Sec. \[sec:classd\]. The corresponding quantum master equation is derived and a quantum phase space representation of the system state is described in Sec. \[sec:quantumeq\]. Solutions to the quantum master equation are discussed in Sec. \[quantumdynsec\] and in Sec. \[classicallimitsec\] we investigate the classical limit of the quantum master equation. We conclude in Sec. \[sec:conclusion\].
Much of the analysis will in fact deal with a simplified system comprising the driven ‘pendulum’ part of the device. An analysis of the full CPT-resonator mode system dynamics, including results from experiment, will be published elsewhere.
The cavity-Cooper pair transistor device {#devicesec}
========================================
To introduce a dc bias into a high-$Q$ microwave cavity, we begin with a standard coplanar-waveguide-based resonator that is one wavelength $\lambda$ long at the operating frequency, illustrated schematically in Fig. \[cavityfig\](a). As is usually the case, the ends of the cavity are terminated by small capacitors (on the order of a few fF) that even at a typical operating frequency of 5 GHz have a large impedance. To a first approximation, then, we can treat these terminations as open circuits, so that the cavity voltage is a local maximum at the cavity end, and the cavity current a local minimum. At a distance $\lambda/4$ from each end of the cavity, the situation is reversed: the cavity voltage is minimal and the current maximal, so that the $\lambda/4$ points are low impedance points.
![\[cavityfig\] (a) Schematic diagram of a dc biased microwave cavity, showing the location of the inductively terminated bias lines and the sample location (black dot). (b) Illustration of a CPT embedded in a dc-biased cavity at the central voltage antinode. ](BookFig1.eps){width="9cm"}
At the $\lambda/4$ low impedance points we then introduce dc bias lines consisting of sections of waveguide terminated with an inductance $L_{b}$. These lines are chosen to have a length $\lambda/2$, so that the impedance they present to the main cavity line at the $\lambda/4$ point is the same as their terminating impedance $i \omega L_{b}$. For even a small inductance of a few nH, this impedance can be substantial at the operating frequency of the cavity.
A microwave photon approaching either dc biasing “T” junction will therefore see a short circuit (the low impedance of the main line) in parallel with a large impedance (the dc bias line) and to first order the cavity photons will be unaffected by the presence of the dc bias lines. The second (full wave) resonance of the cavity should still enjoy a very large $Q$ of up to several thousand in the presence of a dc bias. By placing a CPT at the center of the cavity (the black dot in Fig. \[cavityfig\](a)), where there is an antinode in the cavity voltage, it should be possible to strongly couple the CPT to the cavity and use an applied dc bias to produce self oscillations of the CPT/cavity system via the ac Josephson effect.
![\[imagefig\] (a) Optical micrograph of a microwave cavity with inductively terminated dc bias lines. A contact pad for the CPT gate is visible at the bottom center. (b) Electron micrograph of the center of the cavity showing the gate lead entering the cavity at lower left, the bright Ti/Au contact pads, and a CPT at the right. (c) Detailed view of the CPT. The gate lead is at left the the Josephson junctions at the top and bottom of the central island. ](BookFig2.eps){width="9cm"}
Electron and optical micrographs of a device based on these ideas are shown in Fig. \[imagefig\]. The cavity itself is fabricated out of a Nb film on an undoped Si substrate, as shown in Fig. \[imagefig\](a). Input and output lines on the left and right are coupled to the main line by small capacitors. The dc bias lines extend toward the top of the image; each is terminated by a small on-chip spiral inductor. Cavities based on this design have been shown to posses a large $Q$ of several thousand for the full wave mode at a temperature of 4 K even when a dc bias voltage or current is applied to the central conductor of the cavity [@chenapl11].
At the center of the cavity, a narrow wire to be used as a gate line for the CPT is brought through the ground plane of the waveguide, as shown in Fig. \[imagefig\](b). Two thin Ti/Au contact pads are added to the central conductor and ground plane of the cavity to the right of the entry point for the gate wire. These contact pads, which are driven superconducting by the proximity effect, allow for good metal-to-metal contact between the CPT and cavity. Finally, the CPT and its gate are added to the structure using standard electron beam lithography and shadow evaporation techniques, as in Fig. \[imagefig\](c).
Classical model of device {#classicalsec}
=========================
Closed system equations
-----------------------
The effective lumped element model description of the dc voltage $V_{\mathrm{dc}}$ biased microwave cavity-coupled Cooper pair transistor (CPT) device is illustrated in Fig. \[fig:circuit\]. It is supposed that, for the considered $V_{\mathrm{dc}}$ bias range, the CPT couples predominantly to a particular mode of the cavity. We neglect for the time being the cavity and CPT sources of dissipation, modeled by the parallel $LC$ network admittance and series $LC$ network impedance, respectively, focusing first on writing down the closed system equations of motion. For a typical device, the cavity effective capacitance $C$ is a few pF, while the Josephson junction (JJ) capacitance $C_J$ is at least a few hundred aF, and the gate bias capacitance is about $10~{\mathrm{aF}}$. Furthermore, the effective bias line inductance $L_b$ is a few nH and the cavity effective inductance $L$ is a few tenths of nH. Thus, the typical size hierarchies are $L_{b} \gg L$ and $C \gg C_{J} \gg C_{g}$. We shall make use of these to simplify by approximation the equations of motion. Using Kirchhoff’s Laws and the constitutive relations for the various lumped circuit elements, it is straightforward to obtain the equations of motion. In terms of the phase differences across the two JJs, the equations are $$\begin{aligned}
(C+C_J)\ddot{\varphi}_1 &+&C\ddot{\varphi}_2+{L}^{-1}(\varphi_1+\varphi_2-\varphi^0_1-\varphi^0_2)=\cr
&-&\frac{2\pi I_c}{\Phi_0}\sin\varphi_1+\frac{2\pi}{\Phi_0}L_b^{-1}V_{\mathrm{dc}}t
\label{phi1eq}
\end{aligned}$$ and $$\begin{aligned}
(C+C_J+C_g)\ddot{\varphi}_2 &+&C\ddot{\varphi}_1+{L}^{-1}(\varphi_1+\varphi_2-\varphi^0_1-\varphi^0_2)=\cr
&-&\frac{2\pi I_c}{\Phi_0}\sin\varphi_2
+\frac{2\pi}{\Phi_0}L_b^{-1}V_{\mathrm{dc}}t+\frac{2\pi}{\Phi_0}C_g\dot{V}_g,
\label{phi2eq}
\end{aligned}$$ where $\Phi_0=h/(2e)$ is the flux quantum, $I_c$ is the JJ critical current, $\varphi_i^0$ is an integration constant, and we have assumed $L\ll L_b$. Transforming to ‘center-of-mass’ (CoM) and relative phase coordinates $\gamma_{\pm}=(\varphi_1\pm\varphi_2)/2$, Eqs. (\[phi1eq\]) and (\[phi2eq\]) become $$C\ddot{\gamma}_+ +{L}^{-1}(\gamma_+-\gamma_+^0) =-\frac{\pi I_c}{\Phi_0}\sin\gamma_+\cos\gamma_- +\frac{\pi}{\Phi_0}L_b^{-1}V_{\mathrm{dc}}t
\label{gamma+eq}$$ and $$C_J\ddot{\gamma}_- =-\frac{2\pi I_c}{\Phi_0}\sin\gamma_-\cos\gamma_+ -\frac{\pi}{\Phi_0}C_g\dot{V}_g,
\label{gamma-eq}$$ where we have assumed $ C_g\ll C_J\ll C$.
Eqs. (\[gamma+eq\]) and (\[gamma-eq\]) follow via the Euler-Lagrange equations from the Lagrangian $$\begin{aligned}
{\mathcal{L}}&=&\frac{1}{2}\left(\frac{\Phi_0}{\pi}\right)^2 C \dot{\gamma}_+^2 +\frac{1}{4}\left(\frac{\Phi_0}{\pi}\right)^2 C_J\dot{\gamma}_-^2 +\frac{\Phi_0}{\pi}L_b^{-1}V_{\mathrm{dc}} t \gamma_+ + \frac{\Phi_0}{2\pi}C_g V_g \dot{\gamma}_-\cr
&& -\frac{1}{2} \left(\frac{\Phi_0}{\pi}\right)^2 {L}^{-1}(\gamma_+-\gamma_+^0)^2+\frac{\Phi_0}{\pi} I_c \cos\gamma_-\cos\gamma_+.
\label{lagrangianeq}\end{aligned}$$ The Hamiltonian is $$\begin{aligned}
{\mathcal{H}}&=&\left(\frac{\pi}{\Phi_0}\right)^2(2 C)^{-1}p_+^2+\left(\frac{\Phi_0}{\pi}\right)^2 (2L)^{-1} (\gamma_+-\gamma_+^0)^2 -\frac{\Phi_0}{\pi} L_b^{-1} V_{\mathrm{dc}} t\gamma_+\cr
&& +E_{C_J} (N-N_g)^2 -2 E_J \cos\gamma_+\cos\gamma_-,
\label{hamiltonianeq}\end{aligned}$$ where $N=p_-/\hbar$ is minus the number of excess Cooper pairs on the island, $N_g=C_g V_g/(2e)$ is the polarization charge induced by the applied gate voltage bias $V_g$ in units of Cooper pair charge, $E_{C_J}=(2e)^2/(2\cdot2C_J)=e^2/C_J$ is the approximate CPT charging energy (neglecting $C_g$), i.e., the electrostatic energy cost for putting one additional Cooper pair on the CPT island, and $E_J =I_c\Phi_0/(2\pi)$ is the Josephson energy of a single JJ.
It is convenient to work instead in terms of the shifted CoM coordinate: $\tilde{\gamma}_+=\gamma_+ -\gamma_+^0-\omega_d t$, where the driving frequency is $$\omega_d=\frac{L}{L_b}\frac{eV_{\mathrm{dc}}}{\hbar}.
\label{drivefreq2eq}$$ Performing this canonical transformation with the appropriate generating function, we obtain the following transformed Hamiltonian: $$\begin{aligned}
{\mathcal{H}}&=&\left(\frac{\pi}{\Phi_0}\right)^2(2 C)^{-1}p_+^2+\left(\frac{\Phi_0}{\pi}\right)^2 (2L)^{-1}\gamma_+^2 +E_{C_J} (N-N_g)^2 \cr
&& -2 E_J\cos\gamma_- \cos(\gamma_+ +\omega_d t),
\label{transformedhamiltonian3eq}\end{aligned}$$ where we have dropped the tilde on the shifted $\gamma_+$ coordinate and have set $\gamma_+^0 =0$.
The key observation to make about Hamiltonian (\[transformedhamiltonian3eq\]) is the presence of the time-dependent drive, which originates from the ac Josephson effect, and can be controlled via the externally applied $V_{\mathrm{dc}}$ bias \[Eq. (\[drivefreq2eq\])\]; the nonlinear system self-oscillates. In contrast to most other driven nonlinear system investigations, no externally applied ac drive is required, thus eliminating one of the main sources of noise that hinders the demonstration of macroscopic quantum dynamics.
Mechanical analogue model {#mechanicalsec}
-------------------------
In order to gain insights into the cavity mode-CPT dynamics, as well as motivate other parameter choices, it is useful to consider a mechanical analogue. Hamiltonian (\[transformedhamiltonian3eq\]) can be reexpressed in the following form: $${\mathcal{H}}=\frac{J_+^2}{2I_+}+\frac{1}{2}I_+\omega_+^2 \gamma_+^2 +\frac{1}{2 I_-}\left[J_- -\left(\frac{\Phi_0}{2\pi}\right)C_g V_g\right]^2-I_- \omega_-^2 \cos(\gamma_+ +\omega_d t)\cos\gamma_- ,
\label{mechanicalhamiltonianeq}$$ where $I_+ =C(\Phi_0/\pi)^2$, $I_-=\hbar^2/(2 E_{C_J})$, $\omega_+=1/\sqrt{LC}$, and $\omega_-=2\sqrt{E_J E_{C_J}}/\hbar$. From Eq. (\[mechanicalhamiltonianeq\]), we see that the cavity mode-CPT system is equivalent to a system consisting of two coupled rotors with moments of inertia $I_{\pm}$ and angular momentum $J_{\pm}$. Neglecting the rotor coupling, the ‘+’ rotor behaves as a torsional oscillator with frequency $\omega_+$. For small $\gamma_+$ angular displacements and with the drive turned off (i.e., $V_{\mathrm{dc}}=0$), the ‘-’ rotor behaves as a pendulum. For small $\gamma_-$ displacements, the pendulum oscillates approximately harmonically with frequency $\omega_-$. With the drive turned on (i.e., $V_{\mathrm{dc}}\neq 0$), the pendulum rotor’s ‘gravitational acceleration’ is sinusoidally modulated at frequency $\omega_d$, periodically switching sign as a result. The gravitational acceleration is also modulated by the torsional oscillator coordinate. The ratio of the rotors’ moments of inertia is $$\frac{I_+}{I_-}=\frac{R_K}{Z}\frac{E_{C_J}}{\hbar\omega_+}=\frac{2C}{C_J},
\label{massratioeq}$$ where $Z=\pi\sqrt{L/C}\approx 50~\Omega$ is the cavity impedance and $R_K=h/e^2 \approx 25.8~{\mathrm{k}\Omega}$ is the von Klitzing constant. The frequency ratio for small angle, undriven oscillations is $$\frac{\omega_-}{\omega_+}=2\frac{\sqrt{E_J E_{C_J}}}{\hbar\omega_+}.
\label{frequencyratioeq}$$ For typical capacitance values, CPT charging and Josephson energies of a few Kelvins (in units of $k_B$), and for a cavity mode frequency $\omega_+ = 2\pi \times 5~{\mathrm{GHz}}$ ($\hbar\omega_+/k_B =0.24~{\mathrm{K}}$), we see that the ratios in Eqs. (\[massratioeq\]) and (\[frequencyratioeq\]) are large: the moment of inertia ratio is of order $10^4$ and the frequency ratio of order $10$. Thus, the mechanical analogue corresponds to a fast pendulum with a small moment of inertia that is coupled to a slow torsional oscillator with a large moment of inertia.
A measure of the zeropoint fluctuations in the pendulum angular coordinate $\gamma_-$ is $$\Delta_{zp}^-=\sqrt{\frac{\hbar}{2 I_- \omega_-} }=\sqrt[4]{\frac{4 E_{C_J}}{E_J}}.
\label{gammamzpeq}$$ For typical CPT parameter values, we have $\Delta^-_{zp}\approx1$, i.e., the zeropoint uncertainty is comparable to the size of the $\gamma_-$ coordinate space ($=2\pi$ radians). Thus, we don’t expect the driven quantum pendulum dynamics to resemble much the dynamics of the driven classical pendulum, which can be chaotic. Recovering the classical pendulum limit requires a smaller charging energy than Josephson energy, for example a ‘transmon’-like CPT [@kochpra07]. The classical limit will be discussed in detail in Sec. \[classicallimitsec\].
How do the mechanical analogue moments of inertia compare in magnitude to those of actual mechanical systems? The hydrogen molecule has a rotational moment of inertia $\approx 5\times 10^{-48}~{\mathrm{kg m^2}}$ [@horizfp27]. For $E_{C_J}\sim 5~{\mathrm{K}}$ ($\equiv 431~\mu{\mathrm{eV}}\equiv 6.9\times 10^{-23}~{\mathrm{J}}$), we have $I_-\approx 8\times 10^{-47}~{\mathrm{kg m^2}}$. Thus, the typical CPT pendulum equivalent moment of inertia is an order of magnitude larger than that of the hydrogen molecule. For a transmon-like CPT, the moment of inertia is about two orders of magnitude larger than that of the hydrogen molecule. From Eq. (\[massratioeq\]), we see that the cavity mode torsional oscillator equivalent moment of inertia $I_+$ is about $10^5$ times larger than that of the hydrogen molecule.
Open system equations
---------------------
The cavity-CPT device is subject to several sources of dissipation and noise. Two significant electromagnetic environment sources arise from the capacitive couplings between cavity and input/output microwave lines and the capacitive coupling between the CPT island and gate voltage bias line. Referring to Fig. \[fig:circuit\] , we model the cavity noise/dissipation by an infinite parallel network of $LC$ ‘bath’ oscillators, and the gate voltage noise/dissipation by an infinite series network of $LC$ ‘bath’ oscillators [@yurke83; @devoret95; @burkard04]. The actual dissipative mechanisms can be modeled by such infinite oscillator networks by making appropriate choices for the oscillator frequency distribution spectra. In the following, we will analyze the two noise/dissipation sources independently, beginning first with the cavity noise source.
Extending Hamiltonian (\[transformedhamiltonian3eq\]) to include the infinite parallel network of $LC$ oscillators, we obtain: $$\begin{aligned}
{\mathcal{H}}&=&\left(\frac{\pi}{\Phi_0}\right)^2(2 C)^{-1}p_+^2+\left(\frac{\Phi_0}{\pi}\right)^2 (2L)^{-1}\gamma_+^2 +E_{C_J} (N-N_g)^2 \cr
&& -2 E_J\cos\gamma_- \cos(\gamma_+ +\omega_d t)\cr
&&+\left(\frac{2\pi}{\Phi_0}\right)^2\sum_i \frac{p_i^2}{2C_i}+\left(\frac{\Phi_0}{2\pi}\right)^2\sum_i \frac{1}{2L_i}(\phi_i-2\gamma_+)^2,
\label{transformeddisshamiltonian2eq}\end{aligned}$$ where $\phi_i$ is the phase coordinate across the network capacitance $C_i$. Integrating Hamilton’s equations of motion for the network oscillator coordinate $\phi_i$, we obtain: $$\phi_i (t)=\phi_i (0) \cos\omega_i t +\frac{p_i(0)}{m_i\omega_i}\sin\omega_i t +\lambda_i \int_0^t dt'\frac{\sin \omega_i(t-t')}{m_i\omega_i}\gamma_+(t'),
\label{phiisolneq}$$ where $\omega_i=1/\sqrt{L_i C_i}$, the network oscillator “masses" are $m_i =C_i \left({\Phi_0}/{(2\pi)}\right)^2$ and the system-network oscillator couplings are $\lambda_i=2\left({\Phi_0}/{(2\pi)}\right)^2 {(L_i)^{-1}}$. Following the approach of Ref. [@cortes85], we integrate (\[phiisolneq\]) by parts and substitute into the equations for $\gamma_+$ and $p_+$ to obtain the following Langevin equation: $$\ddot{\gamma}_+=-\frac{1}{LC}\gamma_+ -\left(\frac{\pi}{\Phi_0}\right)^2 \frac{ 2 E_J}{C} \cos\gamma_-\sin(\gamma_+ +\omega_d t)-\int_0^t dt' \Gamma(t-t')\dot{\gamma}_+ (t') +f_n(t),
\label{langevineq}$$ where we have assumed that the couplings $\lambda_i$ are small and we have neglected frequency renormalization terms and where $$\Gamma(t) =\left(\frac{\pi}{\Phi_0}\right)^2 \frac{1}{C} \sum_i \frac{\lambda^2_i}{m_i\omega_i^2} \cos\omega_i t=\frac{1}{C}\sum_i\frac{1}{L_i}\cos\omega_i t
\label{kerneleq}$$ is the damping kernel and $$f_n(t) =\left(\frac{\pi}{\Phi_0}\right)^2 \frac{1}{C} \sum_i \lambda_i \left(\phi_i (0) \cos\omega_i t +\frac{p_i(0)}{m_i\omega_i}\sin\omega_i t\right)
\label{forcenoiseeq}$$ is the noise force. Assuming the network oscillator initial coordinates $\phi_i(0)$, $p_i(0)$ are randomly distributed according to the Maxwell-Boltzmann thermal distribution at temperature $T$, we find for the force-force correlation function: $$\langle f_n (t) f_n (0)\rangle=\left(\frac{\pi}{\Phi_0}\right)^2 \frac{1}{C} k_B T\ \Gamma (t).
\label{correlationeq}$$ With $(\Phi_0/\pi)^2 C$ being the “mass" of the $\gamma_+$ coordinate, we see that (\[correlationeq\]) obeys the usual fluctuation-dissipation relation. With the Markovian approximation $\Gamma(t)\approx \frac{2}{RC} \delta (t)$, Eq. (\[langevineq\]) describes a dissipative cavity mode where the admittance in Fig. \[fig:circuit\] is simply replaced by a resistance $R$.
Moving on now to modelling the gate voltage noise, we insert an infinite series $LC$ network between the gate voltage source $V_g$ and the gate capacitance $C_g$. Hamiltonian (\[transformedhamiltonian3eq\]) is then modified approximately as follows: $$\begin{aligned}
{\mathcal{H}}&=&\left(\frac{\pi}{\Phi_0}\right)^2(2 C)^{-1}\left(p_+ -\sum_i\frac{C_g}{C_i}p_i\right)^2+\left(\frac{\Phi_0}{\pi}\right)^2 (2L)^{-1}\gamma_+^2\cr
&&+E_{C_J} \left(N-N_g+\frac{1}{\hbar}\sum_i\frac{C_g}{C_i}p_i\right)^2 -2 E_J\cos\gamma_- \cos(\gamma_+ +\omega_d t)\cr
&&+\left(\frac{2\pi}{\Phi_0}\right)^2\sum_i \frac{p_i^2}{2C_i}+\left(\frac{\Phi_0}{2\pi}\right)^2\sum_i \frac{1}{2L_i}\phi_i^2,
\label{transformedgatedisshamiltonian2eq}\end{aligned}$$ where $\phi_i$ is the phase coordinate across the $C_i$ network capacitance, and we assume $C_g$ is small compared to the other capacitances. In the following, we neglect the coupling between the infinite series network $p_i$ and the $p_+$ coordinates, since this results simply in adding to the dissipation due to the cavity mode loss considered above. Integrating Hamilton’s equations of motion for the network oscillator coordinate $p_i$, we obtain: $$\begin{aligned}
p_i(t)&=&-m_i\omega_i \phi_i(0) \sin\omega_i t +m_i\dot{\phi}_i (0)\cos\omega_i t \cr
&&-m_i\omega_i \lambda_i\int_0^t dt' \sin[\omega_i (t-t')] (N(t')-N_g),
\label{pisolneq}\end{aligned}$$ where the network oscillator frequencies and masses are the same as for the cavity mode, while the system-network oscillator couplings are now $\lambda_i=(2 E_{C_J}/\hbar) (C_g/C_i)$. Substituting Eq. (\[pisolneq\]) into Hamilton’s equation for $\dot{\gamma}_-$, integrating by parts and neglecting renormalization and shift terms, we obtain the following equation: $$\begin{aligned}
\dot{\gamma}_-&=&\frac{2 E_{C_J}}{\hbar} (N-N_g)+\frac{1}{\hbar}\sum_i m_i\lambda_i^2 \int_0^t d t' \cos\omega_i (t-t') \dot{N}(t')\cr
&&+\frac{1}{\hbar}\sum_i \lambda_ip_i^{(0)}\cr
&=&\frac{2 E_{C_J}}{\hbar} (N-N_g)-\frac{2 E_J}{\hbar^2}\sum_i m_i\lambda_i^2 \int_0^t d t' \cos\omega_i (t-t')\cr
&&\times \sin\gamma_-(t') \cos(\gamma_+(t')+\omega_d t')+\frac{1}{\hbar}\sum_i \lambda_ip_i^{(0)},
\label{gammadampeq}\end{aligned}$$ where $$p_i^{(0)}(t)=-m_i \omega_i \phi_i (0) \sin\omega_i t +m_i \dot{\phi}_i (0)\cos\omega_i t.
\label{bathmomentumeq}$$ Assuming the network oscillator initial coordinates $\phi_i(0)$, $\dot{\phi}_i (0)$ are randomly distributed according to the Maxwell-Boltzmann thermal distribution at temperature $T$, we find for the correlation relation: $$\begin{aligned}
&&\left\langle \left(\sum_i\lambda_i p_i^{(0)}(t)\right) \left(\sum_i\lambda_i p_i^{(0)}(0)\right)\right\rangle=k_B T \sum_i m_i\lambda_i^2 \cos\omega_i t\cr &&=k_B T \left(\frac{e C_g}{C_J}\right)^2 \sum_i \frac{1}{C_i}\cos\omega_i t.
\label{networkcorreleq}\end{aligned}$$ Now, we have: $$\sum_i\lambda_i p^{(0)}_i=\frac{e C_g}{C_J} \frac{\Phi_0}{2\pi}\sum_i\dot{\phi}^{(0)}_i=\frac{e C_g}{C_J} V^{(0)}_{\mathrm{network}},
\label{gateforcenoiseeq}$$ where $V^{(0)}_{\mathrm{network}}$ is the fluctuating voltage across the unloaded series network. But in the Markovian approximation, the voltage noise across a resistance is $$\left\langle V^{(0)}_{\mathrm{network}}(t)V^{(0)}_{\mathrm{network}}(0)\right\rangle =2 k_B T R \delta(t),
\label{networkcorrel2eq}$$ and thus $$\left\langle \left(\sum_i\lambda_i p_i^{(0)}(t)\right) \left(\sum_i\lambda_i p_i^{(0)}(0)\right)\right\rangle =2 k_B T\left(\frac{e C_g}{C_J}\right)^2 R \delta(t),
\label{gateforcecorreleq}$$ with $$\sum_i \frac{1}{C_i}\cos\omega_i t=2R\delta(t),
\label{markovapproxeq}$$ where $R$ is the effective resistance characterizing the loss associated with the gate voltage noise. Substituting Eq. (\[markovapproxeq\]) into the damping term of Eq. (\[gammadampeq\]), we obtain for the $\gamma_-$, $N$ coordinate equations in the presence of gate voltage noise and associated damping within the Markov approximation: $$\dot{\gamma}_-=\frac{2 E_{C_J}}{\hbar} (N-N_g)-\frac{2 E_J}{\hbar^2}\left(\frac{e C_g}{C_J}\right)^2 R\sin\gamma_- \cos(\gamma_+ +\omega_d t) +\frac{1}{\hbar}\sum_i \lambda_i p_i^{(0)}
\label{markovgammaeq}$$ and $$\dot{N}=-\frac{2 E_J}{\hbar} \sin\gamma_-\cos (\gamma_++\omega_d t).
\label{Ndoteq}$$
Now that we have analyzed both the cavity noise and gate voltage noise, we finally write down in dimensionless form the classical Markovian Langevin equations for the cavity-CPT system in the presence of both noise sources. In first order form, the equations of motion are: $$\begin{aligned}
\dot{\gamma}_+&=&p_+\cr
\dot{p}_+&=&-\gamma_+ +f \sin(\gamma_++\omega_d \tau)\cos\gamma_- -Q_c^{-1} p_+ + {\cal{N}}_c (\tau)\cr
\dot{\gamma}_-&=&\frac{2 E_{C_J}}{\hbar\omega_+} (N-N_g)-\frac{4\pi E_J}{\hbar\omega_+}\left(\frac{C_g}{C_J}\right)^2 \frac{R_g}{R_K}\sin\gamma_- \cos(\gamma_+ +\omega_d \tau) +{\cal{N}}_g (\tau)\cr
\dot{N}&=&-\frac{2 E_J}{\hbar\omega_+}\sin\gamma_-\cos(\gamma_++\omega_d\tau),
\label{dimlesslangevineq}\end{aligned}$$ where the dimensionless conversions are $\tau=\omega_+ t$ and $\tilde{p}_+=p_+/(I_+\omega_+)$, with $\omega_+=1/\sqrt{LC}$. The dimensionless drive force amplitude and frequency are $f=\pi L I_c/\Phi_0=4(Z/R_K) E_J/(\hbar\omega_+)$ and $\tilde{\omega}_d=(L/L_b) e V_{\mathrm{dc}}/(\hbar\omega_+)$, respectively \[with the tildes subsequently dropped in Eq. (\[dimlesslangevineq\]) and below\]. The cavity mode quality factor is $Q_c=R_c \sqrt{C/L}$ in terms of the cavity mode resistance $R_c$, while $R_g$ denotes the gate voltage resistance. The associated dimensionless cavity and gate bias noise “forces" satisfy the respective correlation relations $$\left\langle {\cal{N}}_c(\tau){\cal{N}}_c(0)\right\rangle=2\left(\frac{\pi}{\Phi_0}\right)^2 L k_B T_c Q_c^{-1}\delta(\tau)
\label{cavitynoisecorreq}$$ and $$\left\langle {\cal{N}}_g(\tau){\cal{N}}_g(0)\right\rangle=\frac{4\pi k_B T_g}{\hbar\omega_+} \left(\frac{C_g}{C_J}\right)^2\frac{R_g}{R_K} \delta(\tau),
\label{gatenoisecorreq}$$ where we distinguish the cavity mode environment and gate voltage effective noise temperatures, since they are not necessarily the same in experiment.
Classical dynamics {#sec:classd}
==================
The set of Langevin equations (\[dimlesslangevineq\]) provides a full description of the classical stochastic dynamics of the system. Numerical integration of these equations averaged over many different realizations of the noise allows one to obtain probability distributions for all of the system variables. Ultimately these distributions could then be compared with appropriately chosen quasiprobability distributions for the corresponding quantum degrees of freedom. However, this approach is rather demanding from a computational point of view, especially for the quantum dynamics. We will restrict ourselves to outlining the behavior of the simpler system consisting of the driven Cooper-pair transistor alone. In effect this corresponds to the limit of small $f$, $T_c$ and $Q_c$.
Looking at Eq. (\[dimlesslangevineq\]), it is clear that for a strongly damped and weakly driven cavity, the variable $p_+$ will remain small and hence to a good approximation it will be possible to drop the $p_+$ dependence of the $N,\gamma_-$ equations so that the latter become entirely decoupled from the evolution of the cavity variables. In this limit we are left with just the pair of equations, $$\begin{aligned}
\dot{\gamma}_-&=&\frac{2 E_{C_J}}{\hbar\omega_+} (N-N_g)-\frac{4\pi E_J}{\hbar\omega_+}\left(\frac{C_g}{C_J}\right)^2 \frac{R_g}{R_K}\sin\gamma_- \cos(\omega_d\tau) +{\cal{N}}_g (\tau)\cr
\dot{N}&=&-\frac{2 E_J}{\hbar\omega_+}\sin\gamma_-\cos(\omega_d\tau).
\label{dimlesslangevineq2}\end{aligned}$$
We start by solving Eq. (\[dimlesslangevineq2\]) in the limit where $R_g=0$. In this regime the equations are simple classical equations of motion for $N$ and $\gamma_-$. Nevertheless, they reveal a complex dynamical behavior which has already been investigated in different contexts (see e.g. [@mouchetpre06]). Depending on the initial conditions and the choice of parameters, the system typically has a mixed phase space in which the behavior is either chaotic or quasiperiodic. The phase space is visualized in a stroboscopic plot in which a point is plotted after each period of the drive, examples of which are shown in Fig. \[fig:class1\]. In the limit $E_J\rightarrow 0$ the system is integrable with natural frequencies $2E_{C_J}N$, hence for very small values of $E_J/E_{C_J}$ the phase space is perturbed around resonances [@reichl04] which occur at $N=\pm \omega_d/(2E_{C_J})$ (see Fig. \[fig:class1\]a); as $E_J/E_{C_J}$ is increased the resonances get larger and a chaotic sea forms when they overlap. Islands of stability (where the orbits remain quasiperiodic) are found near $N=\pm \omega_d/(2E_{C_J})$ even when $E_J/E_{C_J}>1$ (see Fig. \[fig:class1\]b).
We can explore the sensitivity of the system to dissipation (as opposed to noise) by setting $T_g=0$ and changing the value of $R_g$. We find that even rather low levels of dissipation can have a significant effect on the the long time behavior. For example, for the parameters used in Fig. \[fig:class1\]b with $R_g\simeq 50\Omega$, the phase space appears to contain only two attractive fixed points (one associated with each of the resonances). However, the chaotic sea is present as a transient, albeit one which can be rather long-lived: for certain initial conditions it only disappears after $>10^3$ periods of the drive.
Before examining the full behavior of Eq. (\[dimlesslangevineq2\]) with dissipation and noise, it is also worth considering the effect of averaging over an ensemble of initial conditions. In order to make a comparison with the quantum dynamics we need to consider how an initial [*distribution*]{} of $N,\gamma_-$ values evolves. Because of the chaotic behavior of the system the effects of considering a range of initial coordinates can be very dramatic even after a relatively short period of time. Starting from a Gaussian distribution of initial states centered on a point in the chaotic sea, leads to a set of trajectories that spreads out rapidly over the chaotic sea as can be seen in Fig. \[fig:class2\]a. The islands within the chaotic sea stand out (the handful of points that lie within the islands come from initial points that didn’t fall within the chaotic sea). Clearly averaging over a range of initial conditions has a dramatic effect on the dynamics of the averages of the system, this is particularly clear for the quantity $\langle \cos\gamma_-\rangle$ which very rapidly becomes a periodic oscillation with period $\tau_d=2\pi/\omega_d$ as shown in Fig. \[fig:class2\]b.
Examples of the probability distribution for the classical, noisy, evolution of the system are shown in Fig. \[fig:stoch1\]. The numerical interaction is carried out using a generalization of the Heun method used for deterministic differential equations [@breuer]. In this case an average is carried out both over realizations of the noise and the initial conditions which are chosen from a Gaussian distribution with variances $\Delta N=\Delta\gamma_-=1/\sqrt{2}$ centered on a given point in phase space. When noise is added to the system the trajectories eventually diffuse between the chaotic sea and the quasiperiodic orbits so that the difference in the probability distribution over the island and chaotic sea regions gets washed out over time. In Fig. \[fig:stoch1\] the remnants of the island can be seen at $\tau=10\tau_d$, but by $\tau=25\tau_d$ they have disappeared completely.
Quantum model of device {#sec:quantumeq}
=======================
Quantum master equation
-----------------------
The Poisson bracket relations for the classical canonical coordinates are $$\left\{\gamma_+,p_+\right\}=1;~\left\{\gamma_-,N\right\}=\hbar^{-1}
\label{poissoneq}$$ (where recall $N=p_-/\hbar$). Applying the correspondence principle, the quantum commutation relations are $$[\hat{\gamma}_+,\hat{p}_+]=i\hbar;~[\hat{\gamma}_-,\hat{N}]=i.
\label{commutatoreq}$$ However, the phase coordinates are not periodic functions of their associated system configuration spaces; the representations of the commutation relations (\[commutatoreq\]) give the unbounded eigenvalue spectrum $\mathbb{R}$ for the corresponding phase operators. While this is not a problem for the ‘torsional’ oscillator because of the strong harmonic confining potential, which limits the accessible region of configuration space, the ‘pendulum’ typically explores the whole of its unit circle ($S^1$) configuration space. A suitable pendulum configuration space function is $e^{i\gamma_-}$ with Poisson bracket relation: $$\left\{e^{i\gamma_-},N\right\}=i\hbar^{-1}e^{i\gamma_-}.
\label{poisson2eq}$$ The corresponding commutation relation is then $$[e^{i\hat{\gamma}_-},\hat{N}]=-e^{i\hat{\gamma}_-}.
\label{commutator2eq}$$ Eq. (\[commutator2eq\]) has infinitely many unitarily inequivalent representations [@kastruppra06] that can be labelled by a real parameter $0\leq\delta<1$. Each representation is spanned by a number basis $|N\rangle_{\delta}$, where $$\hat{N} |N\rangle_{\delta}=(N+\delta)|N\rangle_{\delta}, N=0,\pm1,\pm2,\dots.
\label{deltarepeq}$$ Introduce raising and lowering operators for the torsional (CoM) coordinate: $$\gamma_+=\sqrt{\frac{\hbar}{2I_+\omega_+}}(a+a^+);~p_+=i\sqrt{\frac{I_+\omega_+\hbar}{2}}(a^+-a),
\label{raiselowereq}$$ where recall $\omega_+=(LC)^{-1/2}$, the ‘moment of inertia’ is $I_+=C(\Phi_0/\pi)^2$, and we have dropped the hats on the operators for notational convenience. The CoM phase coordinate oscillator zero-point uncertainty is $$\Delta^+_{zp}=\sqrt{\frac{\hbar}{2I_+\omega_+}}=\sqrt{\frac{\pi\sqrt{L/C}}{R_K}}=\sqrt{\frac{Z}{R_K}},
\label{zpeq}$$ where recall $Z\approx 50~\Omega$ is the cavity impedance and $R_K=h/e^2 \approx 25.8~{\mathrm{k}}\Omega$ is the von Klitzing constant, so that $\Delta^+_{zp}=(50/25800)^{1/2}\approx 0.04$. The Hamiltonian operator corresponding to (\[transformedhamiltonian3eq\]) is $$\begin{aligned}
&&{\cal H}=\hbar\omega_+ a^+a +E_{C_J}\sum_{N=-\infty}^{+\infty}\left(N+\delta-N_g\right)^2|N\rangle_{\delta \delta}\langle N|\cr
&&-E_J\sum_{N=-\infty}^{+\infty}\left(|N+1\rangle_{\delta \delta}\langle N|+|N-1\rangle_{\delta \delta}\langle N|\right)\cos\left[\Delta^+_{zp} (a+a^+) +\omega_d t\right].\cr
&&\label{quantumhamiltonianeq}\end{aligned}$$ The parameter $\delta$ appearing in the Hamiltonian operator is a purely quantum signature of the nontrivial topology of the corresponding classical pendulum’s configuration space $S^1$. An interesting question concerns the particular value for $\delta$ that Nature chooses and why [@kastruppra06; @kowalski; @bahr]. However, it is likely not possible to measure $\delta$ in experiment, since from (\[quantumhamiltonianeq\]) it is clear that the effect of a nonzero $\delta$ value is indistinguishable from that due to the presence of an excess charge on the CPT island. From now on, we shall set $\delta=0$.
We now derive the open system quantum master equations within the self-consistent Born approximation (SCBA) following the approach reviewed in Ref. [@paz01]. In the following, we analyze the two noise/dissipation sources independently, beginning first with the cavity mode environment. We write the Hamiltonian (\[transformeddisshamiltonian2eq\]) as ${\cal{H}}={\cal{H}}_S +{\cal{H}}_E +{\cal{V}}$ where the Hamiltonian ${\cal{H}}_S$ describes the cavity mode-CPT system \[Eq. (\[quantumhamiltonianeq\])\], the Hamiltonian ${\cal{H}}_E$ describes the infinite parallel $LC$ network environment, and the interaction part is $${\cal{V}}=-2\left(\frac{\Phi_0}{2\pi}\right)^2\gamma_+ \sum_i \frac{1}{L_i}\phi_i=-\gamma_+\sum_i\lambda_i\phi_i,
\label{interactionparteq}$$ where $\lambda_i=2(\Phi_0/(2\pi))^2(L_i)^{-1}$. Defining $B=\sum_i\lambda_i\phi_i$, we obtain for the master equation within the SCBA: $$\begin{aligned}
\dot{\rho}(t)&=&-\frac{i}{\hbar}\left[{\cal{H}}_S ,\rho(t)\right] \cr
&&-\frac{1}{\hbar^2}\int_0^t dt' \left\{\frac{1}{2}\langle\left\{B(t),B(t')\right\}\rangle\left[\gamma_+,\left[\gamma_+(t'-t),\rho(t)\right]\right]\right.\cr
&&+\left.\frac{1}{2}\langle\left[B(t),B(t')\right]\rangle\left[\gamma_+,\left\{\gamma_+(t'-t),\rho(t)\right\}\right]\right\},
\label{detailedcavitymastereq}\end{aligned}$$ where the operators $B(t)$ and $\gamma_+ (t'-t)$ are in the interaction picture and the expectation values are performed assuming the environment (infinite parallel $LC$ network) is in a thermal state. We have $$\begin{aligned}
\frac{1}{2}\langle\left\{B(t),B(0)\right\}\rangle&=&\frac{1}{\pi}\int_0^{\infty}d\omega J(\omega) \cos\omega t \left(1+2 N(\omega)\right)\cr
\frac{1}{2}\langle\left[B(t),B(0)\right]\rangle&=&-\frac{i}{\pi} \int_0^{\infty}d\omega J(\omega) \sin\omega t
\label{bathcorrelationseq},\end{aligned}$$ where $N(\omega)=[\exp(\hbar\omega/k_B T_c)-1]^{-1}$ is the thermal occupation number of the environment at frequency $\omega$ and where the spectral function is $$J(\omega)=\sum_i \frac{\pi\hbar}{2 m_i\omega_i}\lambda_i^2\delta (\omega-\omega_i),
\label{spectralfunctioneq}$$ with $m_i=C_i (\Phi_0/(2\pi))^2$ and $\omega_i=1/\sqrt{L_iC_i}$.
In Sec. \[classicallimitsec\], we compare the quantum versus classical dynamics and establish conditions under which the former is well approximated by the latter–the so-called classical limit. A necessary condition to be in the classical limit is that the environment temperature must be sufficiently large such that we can make the approximation $N(\omega)\approx k_B T_c/(\hbar\omega)\gg 1$. This requires $k_B T_c\gg\hbar\omega_+$. The environment correlation function then becomes $$\frac{1}{2}\langle\left\{B(t),B(0)\right\}\rangle=k_B T_c \sum_i \frac{\lambda_i^2}{m_i\omega_i^2}\cos\omega_i t = \left(\frac{\Phi_0}{\pi}\right)^2 C k_B T_c\ \Gamma(t),
\label{highTcorreq}$$ where $\Gamma(t)$ is the classical damping kernel (\[kerneleq\]). If, furthermore, the spectral function upper cut-off satisfies $\Lambda\gg k_B T_c$, then we can make the Markovian approximation $\Gamma (t) \approx \frac{2}{RC}\delta(t)$ and $$\frac{1}{2}\langle\left[B(t),B(t')\right]\rangle=-i \frac{\hbar}{2}\left(\frac{\Phi_0}{\pi}\right)^2 C \frac{d}{dt'} \Gamma(t-t')\approx -i \frac{\hbar}{2}\left(\frac{\Phi_0}{\pi}\right)^2 C \cdot\frac{2}{RC}\frac{d}{dt'}\delta(t-t').
\label{dampcorreq}$$ Substituting expressions (\[highTcorreq\]) and (\[dampcorreq\]) into Eq. (\[detailedcavitymastereq\]), integrating by parts and using $p_+=\left(\Phi_0/\pi\right)^2 C\dot{\gamma}_+$, we obtain $$\dot{\rho}(t)=-\frac{i}{\hbar} \left[ {\cal{H}}_S,\rho(t)\right]-\frac{i}{2\hbar} \Gamma\left[\gamma_+,\left\{p_+,\rho(t)\right\}\right]-\frac{1}{\hbar^2}\Gamma I_+ k_B T_c \left[\gamma_+
\left[\gamma_+,\rho(t)\right]\right],
\label{standardmastereq}$$ where $\Gamma=1/(RC)$ is the damping rate. Eq. (\[standardmastereq\]) is just the standard Born-Markov master equation for a quantum Brownian particle in the high temperature limit [@paz01], where the second term on the right hand side describes damping and the third term on the right hand side describes diffusion.
While Eq. (\[standardmastereq\]) is appropriate for investigating the classical limit, under the cryogenic conditions of an actual experiment and for say an $\omega_+\gtrsim 2\pi\times 5~{\mathrm{GHz}}$ cavity mode, we expect that $k_B T_c\ll\hbar\omega_+$, so that a low temperature limit is more appropriate. Using Eq. (\[raiselowereq\]) to express the master equation in terms of raising and lowering operators, making the rotating wave approximation and the replacement $k_BT_c\rightarrow \hbar\omega_+/2$, we obtain the following ‘low temperature’ master equation: $$\dot{\rho}=-\frac{i}{\hbar}[{\cal{H}}_S,\rho]-\frac{1}{2}\Gamma \left(a^+a\rho +\rho a^+ a-2a\rho a^+\right).
\label{mastereq}$$ The non-Hermitian part of the master equation (\[mastereq\]) is of the Lindblad form, ensuring that the solution to (\[mastereq\]) for the density matrix $\rho(t)$ is consistent. However, in the above outlined derivation of this master equation, there is no apparent justification for the replacement $k_B T_c\rightarrow \hbar\omega_+/2$, since $\omega_+$ is not in general the characteristic frequency for dynamics described by the nonlinear, time-dependent Hamiltonian ${\cal{H}}_S$. Only for a harmonic oscillator Hamiltonian with classical frequency $\omega_+$ is this replacement justified, provided the damping is sufficiently weak, i.e., $\hbar \Gamma\ll k_B T_c \ll\hbar\omega_+$. Nevertheless, for better or worse, we shall follow common practise and assume that master equation (\[mastereq\]) provides an adequate model for weakly damped cavity mode-CPT quantum dynamics at low temperatures.
We move on now to derive the master equation modelling gate voltage noise, starting from the Hamiltonian (\[transformedgatedisshamiltonian2eq\]). For the master equation within the SCBA, we obtain \[c.f., Eq. (\[detailedcavitymastereq\])\] $$\begin{aligned}
\dot{\rho}(t)&=&-\frac{i}{\hbar}\left[{\cal{H}}_S ,\rho(t)\right] \cr
&&-\frac{1}{\hbar^2}\int_0^t dt' \left\{\frac{1}{2}\langle\left\{B(t),B(t')\right\}\rangle\left[N,\left[N(t'-t),\rho(t)\right]\right]\right.\cr
&&+\left.\frac{1}{2}\langle\left[B(t),B(t')\right]\rangle\left[N,\left\{N(t'-t),\rho(t)\right\}\right]\right\},
\label{detailedgatemastereq}\end{aligned}$$ where now $B=\sum_i\lambda_i p_i$, with $\lambda_i =(2 E_{C_J}/\hbar)(C_g/C_i)$. The correlation functions still take the same form as Eq. (\[bathcorrelationseq\]), but where now the spectral function is $$J (\omega)=\sum_i \frac{\pi m_i\omega_i\hbar}{2}\lambda_i^2\delta(\omega-\omega_i).
\label{gatespectralfunction}$$ In the high temperature limit, appropriate for comparing the quantum versus classical dynamics, the environment correlation functions become $$\begin{aligned}
\frac{1}{2}\langle\left\{B(t),B(0)\right\}\rangle&=&k_B T_g\left(\frac{e C_g}{C_J}\right)^2 \sum_i \frac{1}{C_i}\cos\omega_i t=2 k_B T\left(\frac{e C_g}{C_J}\right)^2 R_g \delta(t)\cr
\frac{1}{2}\langle\left[B(t),B(0)\right]\rangle&=&\frac{i\hbar}{2 k_B T} \frac{d}{dt}\left[\frac{1}{2}\langle\left\{B(t),B(0)\right\}\rangle\right]=i\hbar\left(\frac{e C_g}{C_J}\right)^2 R_g\frac{d}{dt}\delta(t),\cr
&&\label{quantumgatecorreq}\end{aligned}$$ where the last equality results from making the Markovian approximation. Substituting the correlation relations (\[quantumgatecorreq\]) into the master equation (\[detailedgatemastereq\]), integrating the damping term by parts and using also Heisenberg’s equation to solve for $\dot{N}$, we obtain: $$\begin{aligned}
\dot{\rho}(t)&=&-\frac{i}{\hbar}\left[{\cal{H}}_S ,\rho(t)\right] -\frac{1}{\hbar^2}\left(\frac{e C_g}{C_J}\right)^2 k_B T_g R_g \left[N,\left[N,\rho(t)\right]\right]\cr
&&-\frac{i}{\hbar^2}\left(\frac{e C_g}{C_J}\right)^2 E_J R_g \left[N,\left\{\sin\gamma_-\cos(\gamma_+ +\omega_d t),\rho(t)\right\}\right],
\label{markovgatemastereq}\end{aligned}$$ where the second term on the right hand side describes diffusion and the third term describes damping. Note the atypical, explicit time-dependence in the damping term.
In contrast with the more familiar quantum Brownian master equation (\[standardmastereq\]) for the cavity mode environment, there is no corresponding simple prescription for recovering from Eq. (\[markovgatemastereq\]) a consistent, low temperature master equation that is appropriate for lower gate voltage noise levels expected in an actual cryogenic experiment, where the bias lines are filtered. Given the difficulties in finding such a low temperature master equation, we shall instead in Sec. \[quantumdynsec\] ‘take the path of least resistance’ and simply compare the quantum dynamics in both the presence and absence of the environment non-Hermitian terms in (\[markovgatemastereq\]), so as to gain some understanding of the system quantum dynamics in the presence of gate voltage noise.
Wigner functions and coherent states {#wignersec}
------------------------------------
In order to make a comparison between the quantum and classical dynamics of our systems we need to identify suitable tools with which to describe the quantum dynamics. For continuous systems coherent states prove useful as initial states because they are localized in phase space and have minimum uncertainty making them the quantum states most closely connected to a classical phase space point. Furthermore, the Wigner function quasiprobability distribution provides an effective way of visualizing the quantum evolution of a continuous variable system in phase space and signals the presence of quantum interference effects by turning negative. Whilst both conventional coherent states and the Wigner function can be applied directly to the cavity degrees of freedom, the Cooper-pair transistor is different since the Cooper pair number is discrete. Nevertheless one can define appropriate versions of both coherent states and Wigner functions for the $N,\gamma_-$ degrees of freedom, but with some important differences compared to the usual continuous variable case.
Given the basic algebra of our number and phase operators, $[\hat{\gamma}_-,\hat{N}]=i$ and $[\hat{N},{\rm e}^{i\hat{\gamma}_-}]={\rm e}^{i{\gamma}_-}$, we can use a definition of coherent states first developed for angular momentum and rotation angle variables. We define the coherent states [@kowalski; @bahr] as eigenstates of the operator $\hat{X}={\rm e}^{i\hat{\gamma}_--\hat{N}}$ and hence they take the form $$|\chi\rangle=\frac{1}{n^{1/2}}\sum_{j=-\infty}^{+\infty}\chi^{-j} {\rm e}^{-j^2/2} |j\rangle,$$ where the normalization factor is $$n=\sum_{j=-\infty}^{+\infty} |\chi|^{-2j}{\rm e}^{-j^2}.$$ The complex parameter $\chi$ can be written in terms of a charge ${N}$ and phase ${\gamma}_-$, $\chi={\rm e}^{i{\gamma}_--{N}}$, which are closely related to the expectation values of the corresponding operators [@kowalski], $$\langle \chi |\hat{N}|\chi\rangle\simeq {N}$$ and $$\langle \chi |{\rm e}^{i\hat{\gamma}_-}|\chi\rangle\simeq {\rm e}^{-1/4}{\rm e}^{i{\gamma}_-}.$$
Although the properties of these states differ somewhat from those of the harmonic oscillator coherent states they nevertheless take a fairly simple and intuitive form in phase space. Again, following the approach developed for angular momentum and rotation angle variables, we can use the form of the Wigner function developed for this case [@berry; @mukunda; @bizarro]. For a system with density operator ${\rho}$ the quasiprobability distribution is given by [@bizarro], $$\begin{aligned}
W_N(\gamma_-,t)&=&\frac{1}{\pi}\int_{-\pi/2}^{+\pi/2} d\gamma'_- e^{-2iN\gamma'_-}\langle\gamma_- +\gamma'_- |\rho|\gamma_- -\gamma'_-\rangle\cr
&=&\frac{1}{2}\sum_{\mu=0,1}\sum_{N'=-\infty}^{+\infty} \frac{\sin\left[\left(N-N'-\mu/2\right)\pi\right]}{\left(N-N'-\mu/2\right)\pi} w_{N'+\mu/2} (\gamma_-,t),
\label{circlewignereq}\end{aligned}$$ where $$\begin{aligned}
w_{N+\mu/2} (\gamma_-,t)&=&\frac{1}{\pi}\sum_{N'=-\infty}^{+\infty} e^{-2i(N'+\mu_-/2) \gamma_-} \langle N-N'|\rho|N+N'+\mu\rangle\cr
&=&\frac{1}{\pi}\int_{-\pi}^{+\pi}d\gamma'_- e^{-2i(N+\mu_-/2) \gamma'_-} \langle \gamma_- -\gamma'_-|\rho|\gamma_- +\gamma'_-\rangle
\label{parityfunctioneq}
\end{aligned}$$ and $|\gamma_-+\gamma_-'\rangle$ is one of the phase states. Note that the $w$ functions have definite parity, i.e., $w_{N+\mu/2}(\gamma_- +\pi,t)=(-1)^{\mu} w_{N+\mu/2}(\gamma_- ,t)$. Thus the Wigner function takes the form of a discrete series of strips labeled by the Cooper-pair number and which are continuous in the phase.
Examples of Wigner functions for a charge state $|N=3\rangle$ and the coherent state $|\chi={\rm e}^{i\pi/2-2}\rangle$ are shown in Fig. \[fig:exwfns\]. The Wigner function for a charge state, $|N=M\rangle$, takes a very simple form: $W_N(\gamma_-)=\delta_{N,M}/(2\pi)$, i.e., it is just a uniform strip for $N=M$ and zero elsewhere. Interestingly the charge states are the only examples of pure states which give rise to Wigner functions which are positive everywhere [@rigas]. For number-phase variables the Wigner functions of coherent states do have negative regions, in contrast with the situation for continuous variable systems [@hudson]. However, if we wish to consider an initial quantum state which is analogous to a classical point in phase space the coherent states are still a very good choice as the amount of negativity which actually occurs is in fact very small in practice \[See Fig. \[fig:exwfns\]b\] and our only choice if we wished to eliminate the negativity entirely while still using a pure state is to work with charge states which are completely spread out in phase. As we can see from Fig. \[fig:exwfns\]b the Wigner function for $|\chi={\rm e}^{i\pi/2-2}\rangle$ is strongly peaked around $N=2,\gamma_-=\pi/2$ and, apart from very small regions of negativity [@rigas2008], is very reminiscent of the corresponding continuous variable case.
Quantum dynamics {#quantumdynsec}
================
In exploring the quantum dynamics we again choose to focus on just the behavior of the Cooper-pair transistor, as we did with the classical dynamics. This in effect means that we take the limit $\Delta^+_{zp}\rightarrow 0$ in the Hamiltonian \[Eq. (\[quantumhamiltonianeq\])\]. The presence in the model of a non-zero gate resistance and large effective gate voltage noise temperature ensures that the long-time behavior of the CPT system is classical for sufficiently large $E_J$, in the sense that the Wigner function will be everywhere positive and also a smoothly varying function of $N$ (see Sec. \[classicallimitsec\]). However, over short times a very different picture emerges: even very ‘classical’ choices of the initial state can evolve naturally into states with strongly non-classical features.
We start by considering how the system evolves starting from an initial state which is ‘classical’ in the sense that its Wigner function is relatively smooth as a function of $N$. The initial state we choose to use (illustrated in Fig. \[fig:qcoh\]a) is the steady-state of the system when the dc voltage is switched off (i.e. the $\omega_d=0$ limit). and hence should be easy to prepare in practice. Figure \[fig:qcoh\] shows a series of snapshots of the Wigner function at different times after we set $\omega_d=1$ at time $t=0$. The Wigner function becomes stretched, reaches the edge of the phase space, and then starts to warp around on itself. This wrapping around leads to interference and the formation of regions where the Wigner function is negative. Very similar results are obtained when the system is initially in a coherent state instead.
Over longer time-scales and for sufficiently large $E_J/E_{C_J}$ the quantum dynamics matches up fairly well with the stochastic dynamics (described in Sec. \[sec:classd\]). A good overview of the dynamics can be obtained by looking at some of the moments as a function of time, as shown in Fig. \[fig:moments\]. The behavior of $\langle N^2\rangle$ and $\langle\cos\gamma_-\rangle$ in both the quantum and stochastic dynamics is eventually periodic, a consequence of the underlying periodicity of the drive. The average charge, $\langle N\rangle$ decays rapidly to zero in both cases. The classical stochastic dynamics comes close to reproducing the behavior seen in the quantum dynamics of both $\langle N^2\rangle$ and $\langle\cos\gamma_-\rangle$ in the long time limit, although the amplitude of the stochastic oscillations is slightly smaller than the quantum ones. Going beyond the moments of the system, we can compare the full stochastic probability distribution with the Wigner function of the corresponding quantum evolution as shown in Fig. \[fig:qdist\].
Different choices of initial conditions corresponding to different regions of the classical phase space lead to rather different quantum evolutions, although again this is only a transient effect in the presence of dissipation. Analyses of similar systems [@mouchetpre01; @mouchetpre06] (without dissipation) have shown that they can display chaos-assisted quantum tunneling [@davisjcp81; @ballentine90; @tomsovic94; @latkapra94; @kohlerpre98; @steckscience01; @mouchetpre01; @mouchetpre06; @chaudhurynature09]. In the classical dynamics any point initially chosen to be within one of the stable islands is confined to either the $N>0$ or $N<0$ part of the phase space (depending on the initial conditions) and the value of $N$ oscillates quasi-periodically on a timescale $\tau_d/2$. In the quantum dynamics, the system can tunnel between these two regions of phase space. We looked at the evolution of an initial coherent state centered on points which are either within a stable island or the chaotic sea in the classical phase space (with parameters which match those of Fig. \[fig:class1\]b). The dynamics of the average charge for these two cases is compared in Fig. \[fig:qtun\]. An initial state centred on a stable island (Fig. \[fig:qtun\]a) shows fast oscillations in $\langle N\rangle$, which have the same time-scale and (initially) a similar amplitude to those seen in the classical dynamics. However, there is also an underlying much slower oscillation which takes the system to the opposite side of the phase space. In contrast, for an initial coherent state centered in the chaotic sea (Fig. \[fig:qtun\]b) the behavior is much less regular. The differences in behavior wash out very rapidly when dissipation is included, an important sign that the slow oscillations involve coherent superpositions. The slow oscillations in Fig. \[fig:qtun\]a take the system from a state localized around one of the regions of phase space corresponding to a classical island of stability, to one which is largely localized on the other stable island. The system tunnels via a state which shows strong interference effects, as can be seen from the Wigner functions calculated after a quarter and after a half of the slow oscillation period shown in Fig. \[fig:qtunw\].
Classical limit {#classicallimitsec}
===============
A Wigner function’s domain is given by the eigenvalue spectra of the canonically conjugate operators from which the function is constructed. For the number/phase Wigner function (see Sec. \[wignersec\]), the domain consists of parallel lines \[$\gamma_-=(-\pi,\pi)$ and $N=0, \pm1, \pm2,\dots$\]. Furthermore, since these eigenvalues are the possible outcomes of measurements of the associated observables, it is natural to use the Wigner function to define a quantum phase space dynamics from which the classical phase space limiting dynamics can be recovered by approximation. In the following, we again restrict ourselves to the driven pendulum subsystem dynamics only, resulting from formally setting $\Delta^+_{zp}= 0$ in the Hamiltonian (\[quantumhamiltonianeq\]).
In contrast to a nonlinear system with the configuration space topology $\mathbb{R}$, such as the commonly investigated one-dimensional Duffing oscillator, it does not appear possible to derive a closed form equation for the Wigner function $W_N(\gamma_-,t)$ starting from the master equation (\[markovgatemastereq\]). The problem lies in the $\sin\gamma_-$ potential and damping terms, which give rise formally to Wigner functions shifted by $1/2$ in their $N$-arguments. Because of the non-trivial $S^1$ configuration space topology, the $N$ coordinate domain of the Wigner function consists of the discrete integers, not half-integers. On the other hand, it is possible to write down closed form equations for the definite parity functions $w_{N+\mu/2}(\gamma_-,t)$. From (\[markovgatemastereq\]) and (\[parityfunctioneq\]), we obtain after some algebra: $$\begin{aligned}
&&\frac{\partial w_{N+\mu/2}}{\partial t}=-\frac{2 E_{C_J}}{\hbar} (N+\mu/2 -N_g) \frac{\partial w_{N+\mu/2}}{\partial \gamma_-}\cr
&&-\frac{2 E_J}{\hbar}\sin\gamma_- \cos (\omega_d t)\left(w_{N-1/2 +\mu/2}-w_{N+1/2 +\mu/2}\right)\cr
&&+\frac{1}{\hbar^2}\left(\frac{e C_g}{C_J}\right)^2 k_B T_g R_g \frac{\partial^2 w_{N+\mu/2}}{\partial \gamma_-^2}\cr
&&-\frac{1}{\hbar^2}\left(\frac{e C_g}{C_J}\right)^2 E_J R_g \frac{\partial}{\partial\gamma_-}\left[\sin\gamma_- \cos(\omega_d t)\left(w_{N-1/2 +\mu/2}+w_{N+1/2 +\mu/2}\right)\right].\cr
&&\label{paritywignermastereq}\end{aligned}$$ Using the $\gamma_-$ integral form of the Wigner function in (\[circlewignereq\]) to analytically continue the function to arbitrary real $N$, we obtain from Eqs. (\[paritywignermastereq\]) and (\[circlewignereq\]) the following equation for the Wigner function: $$\begin{aligned}
\frac{\partial W_{N}}{\partial t}&=&-\frac{2 E_{C_J}}{\hbar} (N -N_g) \frac{\partial W_{N}}{\partial \gamma_-}\cr
&& -\frac{2 E_J}{\hbar}\sin\gamma_- \cos (\omega_d t)\left(W_{N-1/2 }-W_{N+1/2}\right)\cr
&&+\frac{1}{\hbar^2}\left(\frac{e C_g}{C_J}\right)^2 k_B T_g R_g \frac{\partial^2 W_{N}}{\partial \gamma_-^2}\cr
&&-\frac{1}{\hbar^2}\left(\frac{e C_g}{C_J}\right)^2 E_J R_g \frac{\partial}{\partial\gamma_-}\left[\sin\gamma_- \cos(\omega_d t)\left(W_{N-1/2}+W_{N+1/2}\right)\right].\cr
&&\label{wignermastereq}\end{aligned}$$ Under conditions where $W$ varies slowly with $N$, we can Taylor expand the fractionally-shifted $W$ functions to first order as a good approximation and obtain the following classical master equation for the probability distribution $P(\gamma_-,N,t)$: $$\begin{aligned}
\frac{\partial P}{\partial t}&=&-\frac{2 E_{C_J}}{\hbar} (N -N_g) \frac{\partial P}{\partial \gamma_-}+\frac{2 E_J}{\hbar}\sin\gamma_- \cos (\omega_d t)\frac{\partial P}{\partial N}\cr
&&+\frac{1}{\hbar^2}\left(\frac{e C_g}{C_J}\right)^2 k_B T_g R_g \frac{\partial^2 P}{\partial \gamma_-^2}\cr
&&-\frac{2}{\hbar^2}\left(\frac{e C_g}{C_J}\right)^2 E_J R_g \frac{\partial}{\partial\gamma_-}\left[\sin\gamma_- \cos(\omega_d t)P\right].
\label{classicalmastereq}\end{aligned}$$ This master equation is equivalent to the classical pendulum Langevin equation (\[dimlesslangevineq\]), with $(\gamma_+,p_+)$ set to zero.
In contrast to the usual situation for a nonlinear system with trivial configuration space topology $\mathbb{R}$, the classical limit (\[classicalmastereq\]) of the quantum master equations (\[paritywignermastereq\]) and (\[wignermastereq\]) was not obtained by identifying and then discarding a higher derivative quantum term involving the anharmonic system potential [@zurekprl94]. Rather, the difference between the quantum and classical equations is more subtle and linked to the discreteness of the number (equivalently angular momentum) operator, which in turn is a consequence of the non-trivial configuration space topology. All that is required to recover the classical master equation to a good approximation is that the Wigner function varies by only a small amount as its argument $N$ increases or decreases by one.
Clearly, the pendulum state must have non-negligible overlap with a large number of angular momentum eigenstates $|N\rangle$ if the Wigner function is to depend smoothly on $N$; recovering the pendulum classical limit necessarily requires $E_{C_J}\ll E_J$, e.g., a ‘transmon’-like CPT [@kochpra07]. However, the latter inequality is not a sufficient condition: as shown in Fig. \[fig:qcoh\], an initially ‘classical’, i.e., smooth, positive practically everywhere Wigner function can evolve through stretching and shrinking into a Wigner function that is non-smooth in $N$, so that the classical master equation approximation (\[classicalmastereq\]) breaks down. Furthermore, because of the $S^1$ periodicity of the configuration space, the stretching pendulum wavefunction can eventually interfere with itself, resulting in an oscillatory Wigner function with significant negative regions.
Even though the initial state-dependent ‘transient’ dynamics will display quantum features, the dynamics will eventually settle into a steady state where the Wigner function is practically positive everywhere and well-approximated by the classical master equation (\[classicalmastereq\]), provided $k_BT_g\gtrsim E_J\gg E_{C_J}$. The smaller is the gate voltage resistance $R_g$ (or gate capacitance $C_g$), the longer is the duration of the transient quantum interval.
Interestingly, when the charging energy is not small, i.e., $E_{C_J}\gtrsim E_J$, the steady state Wigner distribution will still be practically positive everywhere and hence interpretable as a probability density, provided $k_BT_g\gtrsim E_{C_J}$. However, because the Wigner function is non-smooth and with non-negligible support over only a small range in $N$, the classical pendulum master equation no longer accurately describes the Wigner function dynamics. The question then arises as to whether there is an approximate classical description that is distinct from the classical pendulum equation. Such an equation must necessarily treat $N$ as a discrete coordinate and so is more appropriately interpreted in terms of the charge dynamics of the CPT. For $E_J\ll E_{C_J}$ and sufficiently large gate voltage resistance and effective noise temperature, $P(E)$ theory may provide an adequate classical description in terms of incoherently tunneling Cooper pairs [@ingold92; @ingoldprb94; @leppakangasprb06]. Otherwise, Cooper pair tunneling across the Josephson junction is an inherently quantum coherent process: even though the steady state Wigner function evolution is practically positive everywhere, the model dynamics must be interpreted as quantum in nature when the charging energy is large.
Conclusion {#sec:conclusion}
==========
In this chapter, we have investigated a strongly non-linear superconducting device consisting of a Cooper pair transistor (CPT) that is coupled to a dc voltage biased microwave cavity and driven by the dc bias via the ac Josephson effect. Our main focus has been on comparing the quantum and classical dynamics of the system – the “quantum-classical correspondence” – in particular establishing the circumstances under which the corresponding dynamics are similar.
We derived the corresponding classical Langevin and quantum master equations of motion, which describe the dynamics of the CPT-cavity system in the presence of an environment consisting of dissipative circuit elements. Although we did not investigate the dynamics of the full system, we did explore the dynamics of the driven CPT (which is the non-linear element of the device) in the limit where the cavity does not affect its behavior. The quantum-classical correspondence was elucidated by considering the Wigner function representation of the evolving CPT quantum state. Numerical simulation of the quantum and classical dynamics shows that, apart from initial state-dependent transients, the evolution becomes very similar in the limit of large Josephson energies, for which the discreteness of the Cooper pair number on the CPT island is unimportant. Interestingly, though, the transient behavior of the system can lead to highly non-classical states even when the initial state is apparently very classical.
Future work will need to explore how the full system (CPT and cavity degrees of freedom) behaves, as well as connect the dynamics of the system to quantities that are measured in experiment. On a more technical level, further analysis of the interaction between the CPT degrees of freedom and the gate impedance is needed in order to derive the correct description of the dissipative dynamics in the low temperature limit.
Acknowledgements {#acknowledgements .unnumbered}
================
M.P.B. thanks F. Nori and the Advanced Science Institute, RIKEN, for their hospitality and support, where part of this work was carried out. MPB acknowledges support from the NSF under grant number DMR-0804477, ADA acknowledges support from the EPSRC under grant number EP/E034442X/1, and AJR is supported by the NSF and AFOSR/DARPA under grant numbers DMR-0804488 and FA9550-10-1-0047, respectively.
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[^1]: To appear in [*Fluctuating Nonlinear Oscillators*]{}, Edited by Mark Dykman (Oxford University Press).
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author:
- 'G. Catanzaro[^1]'
date: 'Received 10 September 2012 / Accepted 7 December 2012 '
title: 'Spectroscopic atlas of H$\alpha$ and H$\beta$ in a sample of northern Be stars[^2]'
---
[Be stars are fast-rotating early-type emission line stars. It is generally assumed that observed emission is generated in a rotating disk-like envelope, as supported by the observed correlation between the stellar projected rotational velocity $v \sin i$ and the width of the emission lines. Then, high-resolution spectroscopic observations of Balmer lines profiles play an important role in putting constraints on Be stars modeling.]{} [We present Balmer lines spectroscopy for a sample of 48 Be stars. For most of them, H$\alpha$ and H$\beta$ have been observed more than two times, in a total period spanning almost two years between 2008 and 2009.]{} [Spectral synthesis of the H$\alpha$ profile was performed following two steps: photospheric contribution was computed by using Kurucz’s code ATLAS9 and SYNTHE, and disk emission was derived by the approach of Hummel & Vrancken (2000, A&A, 302, 751).]{} [For 26 out of 48 stars, a modeling of the total H$\alpha$ emission, i.e. photospheric absorption plus disk net emission, has been attempted. By this modeling we derived an estimation of the disk radius, as well as the inclination angle between the rotational axis with line of sight and the base density at the stellar equator. For the stars observed more than once, we also discuss the variability of H$\alpha$ and H$\beta$ for what concerns both the equivalent width and the spectral profile. We found 16 stars with variable equivalent width and 7 stars with clear signs of profile variations.]{} [For all the stars in our sample, we derive all the fundamental astrophysical quantities, such as, effective temperature, gravity, and projected rotational velocity. We found 13 stars whose equivalent width is variable with a confidence level greater than 80$\%$ and 7 object for which spectral profiles show change with time. According to the commonly used classification scheme, we classified 16 stars as belonging to class 1, 13 to class 2, 11 are shell stars, 6 objects do not show net emission, and 2 stars display transitions from class 1 and 2. For the class 1 stars, we confirm the correlation between $v \sin i$ and peak separation. Concerning the geometry of the disk, we derived the base density at the stellar equator, the radius, and the inclination angle between rotational axis and line of sight. The maximum concentration of stars occurs for disk dimensions ranging in the interval of 6 to 8 stellar radii and for inclination angles going from 23$^\circ$ to $35^\circ$.]{}
Introduction
============
Classical Be stars are early-type B-type stars whose spectra have one or more emission lines in the Balmer series. In particular, the H$\alpha$ emission line is typically the dominant feature in the spectra of such stars, and many authors have modeled H$\alpha$ line profiles to understand the Be star phenomenon better.
The emission lines observed in the spectra are explained in terms of the recombination that occurs in a flattened circumstellar disk, according to the widely accepted model first proposed by Struve ([@struve31]). The disk is a decretion disk; i.e., the source of the disk material is the central star, generated by the equatorial flow of stellar material. One of the key factors in creating the disk is supposed to be the very high value of rotational velocity. In fact, Be stars are known to have higher rotational velocities than a sample of normal B-type stars. From statistical considerations on the $v \sin i$ distribution among Be stars, Porter ([@porter96]) estimated that Be stars rotate at a equatorial velocity equal to 80 $\%$ of the critical rotation velocity:
$$v_{\rm eq}\,=\,0.8\,\sqrt{\frac{GM_*}{R_*}} .$$
The observed emission lines take a variety of shapes, which following the scheme proposed by Hanuschik ([@hanu96]), range from wine bottle profiles, singly or double-peaked profiles, to shell spectra, when the central absorption must extend below the stellar continuum flux. The various shapes are explained as a dependency of $i$, the inclination angle of the star’s rotation axis to the observer’s line of sight. In particular, shell profiles occur only when the disk is viewed equator-on ($i\,=\,90^\circ$), while the single peak and wine bottle occur only for near pole-on ($i\,=\,0^\circ$) viewings, and double-peaked profiles occur at mid-inclination angles.
Double-peaked profiles have been observed both symmetric, which are the two peaks have the same intensity, and asymmetric, the peaks have different height over the continuum level. The current theory is that asymmetry arises from one-armed density waves in the circumstellar disk, which is also known as the global disk oscillation model. In this model, a one-armed oscillation mode is superposed on an unperturbed, axisymmetric disk (Okazaki, [@okazaki97]). Another aspect of Be stars emission is their variability. For example, about one third of all double-peaked profiles exhibit changing asymmetry, with the so-called violet-to-red ratio (V/R) being cyclically variable on timescales of years to decades.
Observations in different spectral regions help astronomers probe different regions of the stellar disk and then put constraints on modeling these stars. For example, recently Meilland et al. ([@meilland07]) have used VLTI/AMBER to observe $\alpha$ Arae in the Br$_\gamma$ line, which was constrained very strongly the rotational property of its disk, concluded that its rotation is purely Keplerian.
In this paper we present an atlas of observed H$\alpha$ and H$\beta$ spectral lines in a sample of bright Be stars. We payed particular attention to modeling the H$\alpha$ profile and to the time variability of Balmer lines profiles.
HD Spec. b-y m$_1$ c$_1$ H$\beta$ T$_{\rm eff}$ (K)
-------- ---------- ---------- ------- ---------- ---------- ------------------- -- --
6811 B7Ve 0.006 0.082 0.697 2.676 12600
10516 B2Vep – – – – 25410
11415 B3III $-$0.059 0.094 0.419 2.665 15680
37202 B2IVp – – – – 21480
37490 B3III $-$0.016 0.060 0.184 2.576 20200
41335 B2Vne 0.031 0.020 0.002 2.463 20230
43285 B6Ve $-$0.055 0.104 0.481 2.670 14840
44458 B1Vpe – – – – 26600
45542 B6III $-$0.049 0.106 0.544 2.657 14080
47054 B8Ve $-$0.022 0.096 0.698 2.677 12500
50658 B8IIIe $-$0.015 0.086 0.596 2.626 13600
50820 B3IVe+ – – – – 17900
52918 B1V $-$0.072 0.063 0.021 2.591 24720
53416 B8 – – – – 12000
58050 B2Ve $-$0.087 0.081 0.169 2.495 20260
58343 B2Vne 0.011 0.073 0.264 2.566 18600
58715 B8Ve $-$0.037 0.124 0.791 2.729 11700
60855 B2/B3V $-$0.017 0.059 0.194 2.609 19984
61224 B8/9 IV 0.041 0.061 0.773 2.656 12000
65875 B2.5Ve 0.004 0.040 0.102 2.451 22540
71072 B4IV – – – – 16550
91120 B8/9IV/V 0.005 0.095 0.956 2.742 10600
109387 B6IIIpe $-$0.023 0.072 0.351 2.566 16850
138749 B6Vnne $-$0.040 0.083 0.479 2.683 14910
142926 B9pec $-$0.029 0.109 0.764 2.677 12000
142983 B8Ia/Iab $-$0.022 0.081 0.664 2.590 12860
143275 B0.2IVe $-$0.016 0.036 $-$0.020 2.605 26700
162428 A0 – – – – 12200
162732 Bpshe $-$0.045 0.096 0.786 2.669 11740
164284 B2Ve 0.057 0.022 0.001 2.495 21650
164447 B8Vne $-$0.013 0.095 0.660 2.721 12900
183362 B3Ve $-$0.009 0.056 0.184 2.528 20260
183656 B6sh 0.050 0.037 0.809 2.641 11700
183914 B8Ve $-$0.030 0.109 0.725 2.737 12270
187567 B2.5IVe 0.052 0.007 0.093 2.531 23110
189687 B3IVe $-$0.057 0.081 0.281 2.635 17980
191610 B2.5Ve $-$0.039 0.062 0.162 2.553 20600
192044 B7Ve $-$0.032 0.086 0.603 2.632 13460
193911 B8IIIne $-$0.015 0.073 0.644 2.627 13100
210129 B7Vne $-$0.024 0.101 0.476 2.643 15000
212571 B1Ve – – – – 23300
214168 B2Ve $-$0.045 0.055 0.024 2.609 24790
216057 B5Vne $-$0.013 0.088 0.470 2.671 15090
216200 B3IVne 0.100 0.040 0.331 2.634 17600
217050 B4IIIep $-$0.006 0.062 0.472 2.501 15090
217543 B3Vpe $-$0.034 0.079 0.241 2.650 18860
217675 B6Ve 0.002 0.042 0.477 2.650 15050
217891 B6Ve $-$0.050 0.109 0.476 2.630 14910
: Spectral types, luminosity classes, Strömgren photometry, and derived effective temperatures for our program stars. Spectral types and luminosity classes are taken from the SIMBAD database[]{data-label="param"}
-------- ------ ---------------- ---- --------------------- ----------- ------------- ------------- ----------------------- --------------- --------------- ------------ ----------------------- ------------- -----------
HD HR Name N Cl. v$\sin i$ M$_*$ R$_*$ $\Delta v_{\rm peak}$ v$_c$ v$_{\rm eq}$ $i$ $\rho _0$ R$_d$/R$_*$ rem
(km/s) (M$_\odot$) (R$_\odot$) (km s$^{-1}$) (km s$^{-1}$) (km s$^{-1}$) ($^\circ$) (g cm$^{-3}$)
6811 335 $\phi$ And 3 2 80 4.0 3.1 – 500 400 12 8.50$\cdot$10$^{-14}$ 6.4 3,5
10516 496 $\phi$ Per 1 2 240 17.0 10.0 180 570 450 22 4.20$\cdot$10$^{-12}$ 4.3 3,4,5
11415 542 $\epsilon$ Cas 1 abs 40 6.5 4.2 – – – – – – –
37202 1910 $\zeta$ Tau 2 2 120 10.5 6.1 290 – – – – – 1,2,3,4,5
37490 1934 $\omega$ Ori 2 1 180 8.9 5.1 190 580 460 23 8.00$\cdot$10$^{-12}$ 3.1 1,2,3,4
41335 2142 V696 Mon 2 2 310 8.9 5.1 – – – – – – 1,2,3
43285 2231 – 1 abs 230 6.0 4.0 – – – – – – –
44458 2284 FR CMa 1 2 180 18.0 10.5 150 570 460 23 3.00$\cdot$10$^{-12}$ 5.0 3
45542 2343 $\nu$ Gem 1 sh 160 5.1 3.5 175 – – – – – 2,3,4
47054 2418 – 1 1 180 4.0 3.1 140 500 400 27 3.80$\cdot$10$^{-13}$ 5.8 3
50658 2568 $\psi$09 Aur 1 sh 230 4.5 3.3 150 510 400 80 3.50$\cdot$10$^{-14}$ 8.5 3,5
50820 2577 – 1 2 150 7.4 4.6 – 550 440 20 7.20$\cdot$10$^{-14}$ 13.8 –
52918 2648 19 Mon 1 abs 240 17.0 10.0 – – – – – – –
53416 – – 1 1 190 4.0 3.5 120 470 370 30 7.60$\cdot$10$^{-14}$ 9.1 –
58050 2817 – 3 1 110 9.0 5.3 – – – – – – 3,5
58343 2825 FW CMa 1 2 50 7.5 4.7 – 550 440 6 1.10$\cdot$10$^{-13}$ 13.7 2,3,4
58715 2845 $\beta$ CMi 3 1 210 3.8 3.0 130 490 390 32 1.25$\cdot$10$^{-13}$ 7.0 2,3,4,5
60855 2921 V378 Pup 1 2 240 9.0 5.3 150 – – – – – 2,3
61224 2932 – 2 1 200 4.0 3.1 150 500 400 30 3.50$\cdot$10$^{-13}$ 6.2 3
65875 3135 V695 Mon 3 2 180 12.0 7.0 – – – – – – 2,3
71072 – – 2 abs 100 6.5 4.5 – – – – – – –
91120 4123 – 9 1 250 3.0 2.5 180 480 380 41 1.15$\cdot$10$^{-13}$ 6.8 2,3
109387 4787 $\kappa$ Dra 13 1$\leftrightarrow$2 170 7.0 4.5 140 540 430 23 4.00$\cdot$10$^{-12}$ 5.1 3,5
138749 5778 $\theta$ CrB 11 abs 310 6.0 4.0 – – – – – – 5
142926 5938 V839 Her 10 sh 275 4.0 3.1 240 500 400 44 6.00$\cdot$10$^{-12}$ 4.3 5
142983 5941 48 Lib 13 sh 390 4.1 3.2 – – – – – – 2,3
143275 5953 $\delta$ Sco 4 2 165 18.0 10.5 – – – – – – 6
162428 – – 5 1 240 4.0 3.0 180 500 400 36 9.00$\cdot$10$^{-12}$ 6.1 4
162732 6664 $\zeta$ Her 6 sh 40 3.8 3.0 260 – – – – – 3,5
164284 6712 66 Oph 6 1 240 10.5 6.1 150 570 460 32 8.80$\cdot$10$^{-14}$ 9.3 2,3
164447 6720 V974 Her 4 1 180 4.0 3.5 140 470 370 28 1.75$\cdot$10$^{-13}$ 5.8 5
183362 7403 V558 Lyr 4 2 220 9.0 5.3 – – – – – – 3
183656 7415 V923 Aql 5 sh 190 3.8 3.0 250 – – – – – 2,3
183914 7418 $\beta$ Cyg B 5 1 240 4.0 3.0 160 500 400 37 9.00$\cdot$10$^{-14}$ 8.2 3,4
187567 7554 V1339 Aql 4 1$\rightarrow$2 200 13.0 7.7 140 570 450 26 3.00$\cdot$10$^{-13}$ 7.3 –
189687 7647 25 Cyg 6 sh 230 7.5 4.7 250 550 440 31 3.05$\cdot$10$^{-13}$ 3.6 3,4
191610 7708 28 Cyg 4 1 250 9.0 5.3 270 570 455 33 2.60$\cdot$10$^{-13}$ 3.9 3,4
192044 7719 20 Vul 3 1 240 4.5 3.3 150 510 410 38 1.10$\cdot$10$^{-13}$ 10.1 –
193911 7789 25 Vul 3 1 170 4.0 3.5 100 470 370 27 2.30$\cdot$10$^{-13}$ 6.0 2,3,5
210129 8438 25 Peg 8 1 130 6.0 4.0 70 530 430 19 1.50$\cdot$10$^{-12}$ 5.6 1,2,3
212571 8539 $\pi$ Aqr 3 1 220 13.0 7.7 280 570 450 41 6.00$\cdot$10$^{-13}$ 3.9 1,2,3,4
214168 8603 8 Lac B 3 2 150 17.0 10.0 – – – – – – –
216057 8682 – 4 abs 260 6.0 4.0 – – – – – – –
216200 8690 14 Lac 3 sh 195 13.0 7.7 – – – – – – 5
217050 8731 – 1 sh 240 6.0 4.0 – – – – – – 3,4,5
217543 8758 V378 And 2 sh 305 7.5 4.7 – – – – – – –
217675 8762 $o$ And 2 sh 200 6.0 4.0 – – – – – – 4,5
217891 8773 $\beta$ Psc 6 2 75 6.0 4.0 – 535 430 10 5.20$\cdot$10$^{-13}$ 6.1 1,2,4,5
-------- ------ ---------------- ---- --------------------- ----------- ------------- ------------- ----------------------- --------------- --------------- ------------ ----------------------- ------------- -----------
Observation and data reduction
==============================
The idea that underlies this catalog is to create a homogeneous data set of stars observed with the same instrument, the telescope of the [*M.G. Fracastoro*]{} station of INAF-Catania Astrophysical Observatory. For this purpose, we queried the “Catalogue of Be stars” compiled by Jaschek & Egret ([@jaschek82]) selecting all the objects with V$\leq$7 and observable at the latitude of the observatory, which means all the stars with $\delta \ge -22^\circ$. The result of this query is the sample of 48 Be stars reported in Table \[param\]. The limiting magnitude was chosen to obtain a good compromise between the exposure time and the signal-to-noise ratio.
The present catalog is based on new spectroscopic observations of all the stars in our sample, which spectral type are distributed between B1 and A0 (according to the histograms displayed in Fig. \[spect\_type\]), and luminosities classes are III, IV, and V, as in the SIMBAD database[^3]. Some of these stars’ spectra have never been published in other catalogs similar to ours, at least to our knowledge.late
![Histogram showing the distribution of program stars as a function of spectral type. []{data-label="spect_type"}](20357fig1.ps){width="8cm" height="8cm"}
All the spectra of our program stars have been acquired with the 91-cm telescope and FRESCO, the fiber-fed REOSC echelle spectrograph that allows spectra to be obtained in the range of 4300–6800 [Å]{} with a resolution R=21000. The spectra were recorded on a thinned, back-illuminated (SITE) CCD with 1024$\times$1024 pixels of 24 $\mu$m size, whose typical readout noise is of about 8 e$^{-}$ and the gain is 2.5 e$^-$/ADU. All the spectra have been acquired during several observing runs spanning two years between 2008 and 2009.
The reduction of spectra, which included the subtraction of the bias frame, trimming, correcting for the flat-field and the scattered light, the extraction of the orders, and the wavelength calibration, was done by using the NOAO/IRAF package[^4]. The amount of scattered light correction was about 10 ADU. After dividing the extracted spectra by flat-field, the residual shape the spectrum was removed by dividing each spectral order by a Legendre function of a low order. Typical S/N of our spectra is $\sim$100. For some stars this limit has not been reached because its apparent magnitude is close to V$\approx$7, in which case the S/N was about 50.
Finally, the IRAF package [rvcorrect]{} was used to include the velocity correction due to the Earth’s motion, which moved the spectra into the heliocentric rest frame. The task [splot]{} and its facilities were used to measure the peaks separation in the H$\alpha$ profile. Errors in the pixels position were converted in errors on the separations, and were evaluated in $\approx$ 20 km s$^{-1}$.
For each spectrum we measured the equivalent widths (including underlying absorption) of both H$\alpha$ and H$\beta$, where a negative value means the corresponding line shows net emission. All the measured equivalent widths are reported in Table \[eqw\].
-------- ------------------- ------ ------ ------------------ ------ ------
HD EW(Å) F C EW(Å) F C
6811 0.87$\pm$0.28 1.84 0.65 6.45$\pm$0.09 1.99 0.67
37202 $-$15.26$\pm$1.17 1.41 0.49 2.34$\pm$0.01 1.41 0.49
37490 $-$5.96$\pm$1.09 1.41 0.49 1.83$\pm$0.18 1.41 0.49
41335 $-$31.92$\pm$0.78 1.41 0.49 $-$1.44$\pm$0.52 1.41 0.49
58050 $-$4.40$\pm$0.31 1.94 0.66 5.49$\pm$0.04 1.73 0.63
58715 $-$1.19$\pm$0.36 1.94 0.66 7.96$\pm$0.06 1.98 0.66
61224 $-$5.59$\pm$0.48 1.41 0.49 5.14$\pm$0.40 1.41 0.49
65875 $-$43.00$\pm$0.89 1.90 0.66 0.04$\pm$0.60 1.77 0.64
71072 3.14$\pm$0.08 1.41 0.49 5.22$\pm$0.23 1.41 0.49
91120 $-$0.40$\pm$0.50 3.49 0.92 8.07$\pm$0.15 3.21 0.91
109387 $-$19.17$\pm$0.44 3.58 0.94 3.04$\pm$0.06 2.85 0.90
142926 $-$1.63$\pm$0.45 3.68 0.93 7.05$\pm$0.13 3.06 0.90
142983 $-$24.66$\pm$0.86 3.31 0.93 3.09$\pm$0.14 3.85 0.95
143275 $-$10.13$\pm$2.29 2.34 0.76 $-$0.81$\pm$0.20 2.30 0.75
162428 $-$15.57$\pm$0.14 2.44 0.80 4.78$\pm$0.38 2.40 0.80
162732 $-$6.61$\pm$0.88 2.52 0.83 5.92$\pm$0.50 2.89 0.85
164284 1.37$\pm$1.57 2.71 0.84 5.18$\pm$0.08 2.64 0.84
164447 $-$0.95$\pm$1.13 2.64 0.81 6.68$\pm$0.15 2.68 0.82
183362 $-$26.39$\pm$0.15 1.88 0.65 1.25$\pm$0.15 1.79 0.64
183656 $-$7.68$\pm$0.72 2.50 0.80 5.15$\pm$0.18 2.28 0.80
183914 $-$0.94$\pm$0.22 2.78 0.82 8.10$\pm$0.07 2.53 0.80
187567 $-$21.82$\pm$0.66 2.14 0.74 0.56$\pm$0.19 2.21 0.74
189687 0.93$\pm$0.25 2.93 0.86 4.76$\pm$0.05 3.11 0.87
191610 0.12$\pm$0.24 2.03 0.72 4.69$\pm$0.15 2.04 0.72
192044 $-$9.96$\pm$0.13 1.97 0.66 5.25$\pm$0.05 1.96 0.66
193911 $-$5.45$\pm$0.13 1.95 0.66 5.10$\pm$0.05 1.89 0.65
210129 $-$14.17$\pm$0.21 3.26 0.90 5.11$\pm$0.15 3.13 0.89
212571 $-$5.40$\pm$0.04 1.99 0.67 1.98$\pm$0.14 1.98 0.66
214168 $-$14.79$\pm$0.35 2.00 0.67 1.87$\pm$0.52 1.82 0.65
216057 4.62$\pm$0.15 2.36 0.76 7.19$\pm$0.10 2.21 0.74
216200 1.15$\pm$0.43 1.98 0.66 5.08$\pm$0.21 1.97 0.66
217543 2.35$\pm$0.76 1.41 0.49 5.31$\pm$0.11 1.41 0.49
217675 3.05$\pm$0.28 1.41 0.49 5.64$\pm$0.18 1.41 0.49
217891 $-$12.94$\pm$0.64 2.70 0.84 4.70$\pm$0.19 2.70 0.84
-------- ------------------- ------ ------ ------------------ ------ ------
: Program stars for which we have more than two spectra. We reported for both lines mean equivalent widths, their F values, computed according Eq. \[Fval\], and the confidence levels of the detected variability.[]{data-label="eqw_test"}
Classification and fit of the H$\alpha$ line profiles
=====================================================
Almost all the stars in our sample were found with emission in the H$\alpha$. Then, considering the shape of their profile and according to the classification scheme proposed by Hanuschik ([@hanu88]), we classified our stars as belonging to
- class 1 when they exhibit a rather symmetrical double peak structure with V/R$\approx$1,
- class 2 when they have an asymmetric single peak or a dominant peak with a much weaker secondary peaked,
- shell stars when the central reverse is deeper than the continuum level,
- [*abs*]{} when there is not emission above the continuum level.
Our classification is reported in the fourth column of Table \[summary\].
Only for the stars belonging to class 1, plus stars from other classes but with V/R ratio close to unity, we attempt an estimation of the disk dimension. The approach we used was to minimize the difference between observed and synthetic profiles, computed in two separate steps. First of all, we calculated the photospheric H$\alpha$, then the contribution due to the net emission of the disk and then we added these two synthetic profiles obtained separately. These two steps are described in the following:
- [*Computation of the photospheric profile*]{}
We first computed the photospheric H$\alpha$ profiles for all our program stars. They were generated in three steps: [*i)*]{} first, we computed an LTE model atmosphere using the ATLAS9 code (Kurucz [@kur93]), [*ii)*]{} the stellar spectrum was then synthesized using SYNTHE (Kurucz & Avrett [@kur81]), and [*iii)*]{} the spectrum was convolved with the instrumental and rotational profiles.
First of all we had to obtain an estimation of the effective temperature for each target. Considering that the continuum energy distribution of Be stars is typical of normal early-type stars both in the visual and UV, but not in the IR, where an excess could be present because of the hot circumstellar dust (Zickgraf [@zick00]), effective temperatures were computed from Strömgren photometry (Hauck & Mermilliod [@hauck98]) using the algorithm coded by Moon ([@moon85]), with the exception of eight stars for which photometry is not available. This method is allowed because it does not involve any IR filter. For seven of them we adopted the temperatures from the literature: HD10516, HD37202, and HD44458 from Soubiran et al. ([@soubiran10]), HD58020 and HD71072 from Hohle et al. ([@hohle10]), HD162428 from Moujtahid et al. ([@mouj99]), and HD212571 from Wu et al. ([@wu11]), while for HD53416 we derived an estimation of temperature from spectral type and the calibration by Kenyon & Hartman ([@kenyon95]). Since our targets have luminosity class IV/V (as reported in the SIMBAD database), we fixed the surface gravity to $\log g $=4.0, except for HD11415, HD37490, HD45542, HD50658, HD109387, HD193911, and HD217050 (luminosity class III) for which $\log g$=3.0 has been preferred. Radii and masses were adopted following the calibration in Drilling & Landolt ([@drill99]). Assuming these atmospheric parameters, we computed the v$\sin i$ of each star by spectral synthesis of the observed Mg[i]{} $\lambda$4481 [Å]{}. This line was chosen because in the spectral range of our targets, it reaches its maximum depth and therefore it is better suited to determining the rotational velocity. Errors on the projected rotational velocities are $\approx$ 15 km s$^{-1}$.
- [*Computation of net disk emission*]{}
We have adopted the Be disk model approach of Hummel & Vrancken ([@hummel00]) that is based on models developed by Horne & Marsh ([@horne86]) and Horne ([@horne95]) for accretion disks in cataclysmic variables. The disk is assumed to be axisymmetric and centered over the equator of the underlying star, and the gas density varies as
$$\rho(R,Z)\,=\,\rho_0R^{-n} exp\left[-\frac{1}{2}\left(\frac{Z}{H(R)}\right)^2\right]$$
where R and Z are the radial and vertical cylindrical coordinates (in units of stellar radii), $\rho_0$ is the base density at the stellar equator, [*n*]{} a radial density exponent, and H(R) the disk vertical scale height. The neutral hydrogen population within the disk is found by equating the photo-ionization and recombination rates (Gies et al. [@gies07]). The disk gas is assumed to be isothermal and related to the stellar effective temperature T$_{\rm eff}$ by T$_d$=0.6T$_{\rm eff}$ (Carciofi & Bjorkman [@carciofi06]).
This approach take the contribution of the central star’s finite size on the H$\alpha$ line formation process into account, i.e. the obscuration of the disk by the central star at any given inclination. The numerical model represents the disk by a large grid of azimuthal and radial surface elements, and the equation of transfer is solved along a ray through the center of each element according to
$$I_\lambda\,=\,S_\lambda^L\,(1-e^{-\tau_\lambda})\,+\,I_\lambda^S\,e^{-\tau_\lambda}$$
where I$_\lambda$ is the derived specific intensity, S$_\lambda^L$ the source function for the disk gas (taken as the Planck function for the disk temperature T$_d$), I$_\lambda^S$ the specific intensity for the H$\alpha$ of the star, and $\tau_\lambda$ the integrated optical depth along the ray. The first term applies to all the disk area elements that are unocculted by the star, while the second term applies to all elements that correspond to the projected photospheric disk of the star. The absorption line adopted in I$ _\lambda^S$ is Doppler-shifted according to solid-body rotation for the photospheric position in a star that is rotating at 80$\%$ of the critical value.
Electron scattering is not taken into account in the line profile computation. So we do not expect to reproduce the wings of strong lines well, since for these lines the broadening of the wings due to the electron scattering can not be neglected. Disk kinematic is taken into account using a rotational velocity law written as
$$V(R)\,=\,V_{\rm rot}^*R^{-j}$$
(Hutchings [@hutch70]), where R represents the radial coordinate that has its origin at the center of the star, and $V_{\rm rot}^*$ denotes the actual rotational velocity at the stellar surface. The exponent ranges from $j\,=\,1/2$ for pure Keplerian rotation and $j\,=\,1$ corresponding to conservation of angular momentum. Likewise the value of $j$ is still matter of debate, recent studies seem to converge toward the Keplerian value (Hummel & Vrancken [@hummel00], Meilland et al. [@meilland07]). Thus, in this study we assumed the disk to be in pure Keplerian rotation.
Once we obtained and combined these two contributions, we started the minimization algorithm using as goodness-of-fit test the parameter
$$\chi^2 = \frac{1}{N} \sum (\frac{I_{\rm obs} - I_{\rm th}}{\delta I_{\rm obs}})^2$$
where $N$ is the total number of points, $I_{\rm obs}$ and $I_{\rm th}$ are the intensities of the observed and computed profiles, respectively, and $\delta I_{\rm obs}$ is the photon noise.
As initial guesses for the inclination angle $i$ and for the disk radius, we used the equations
$$v_{\rm eq} \sin i\,=\,0.8\,\sqrt{\frac{GM_*}{R_*}}\,\sin i
\label{incl}$$
where $v_{\rm eq} \sin i$ is the value measured in our spectra, and
$$\frac{\Delta v_{\rm peak}}{2\,v \sin i}\,=\,r_d^{-j}
\label{rad}$$
(equation \[6\] in Hanuschik et al. ([@hanu88])) where $\Delta v_{\rm peak}$ is the separation between violet and red peaks, as measured in each profile and reported in Table \[summary\].
Then, fine tuning was carried out using the [*amoeba*]{} minimization[^5] algorithm between observed and computed profiles. In our procedure two assumptions have been made: radial density exponent has been fixed to $n$=3 and, as stated before, the Keplerian rotation of the disk has been considered ($j$=1/2). The first hypothesis, regarding the value of the density exponent can be justified by considering the work of Grundstrom & Gies ([@grund06]). These authors computed several theoretical curves that described the dimension of the disk radius as a function of the H$\alpha$ equivalent width, for different values of the inclination angle i and different values of n. They concluded that the overall shape of those curves for different n and equal i are almost the same, since it is a small difference of $\approx 3 \%$ in correspondence of equivalent widths between -2 and -15 [Å]{} when n change from 3 to 3.5. They then suggest that the particular choice of n is not as important as the choice of the right i. Moreover, Porter & Rivinius ([@porter03]) from IR flux excess in Be stars suggested that n falls in the interval $n\,= \,2\div4$. Thus, on the basis of these results, we fixed the value of the density exponent to the middle value of n=3. Total H$\alpha$ profiles, star+disk, are presented in Figs. \[halpha1\], \[halpha2\], and \[halpha3\].
To derive an estimation of the disk radius, we used the method developed by Grundstrom & Gies ([@grund06]) to form a synthetic image of the system star+disk in the plane of sky by summing the intensity over a 2.8 nm band centered on H$\alpha$. We collapsed this image along the projected major axis to get the summed spatial intensity, and we adopted the value for which the summed intensity drops to half its maximum value as effective disk radius.
We estimated the errors on the disk dimensions and on the inclination angles to be $\pm$2R$_*$ and $\pm$3$^\circ$, respectively. These determinations have been estimated by varying in Eqs. \[incl\] and in \[rad\] the observed quantities $v \sin i$ and peak separation by their experimental errors and considering as uncertainties the semi-amplitude of this variation.
All the adopted and derived parameters are reported in Table \[summary\].
![Correlation between measured v $\sin i$ and peaks separation. The points have been grouped on the basis of the mean equivalent width: EW$\le$-20 [Å]{} (filled squares), -20[Å]{}$<$EW$\le$-15[Å]{} (filled circles), -10[Å]{}$<$EW$\le$-5[Å]{} (filled triangles), -5[Å]{}$<$EW$\le$0[Å]{} (open circles), and EW$\ge$0 [Å]{} (open squares). []{data-label="corr"}](20357fig2.ps){width="8cm"}
H$\alpha$ and H$\beta$ Variability {#variab}
==================================
Usually Be stars display variability in their equivalent width (EW) and/or in their spectral profile.
To find whether a star presents equivalent width variation, we applied to both H$\alpha$ and H$\beta$ equivalent widths the statistical method called F-test. When more than one observation was present for a given target, we calculated the amplitude of the variation, $\Delta\,EW$, and the standard deviation of the sample using
$$\sigma\,=\,\sqrt{ \frac{1}{N-1} \sum (EW_i - \overline{EW})^2 }\\$$
where N is the number of points, and the (N-1) corresponds to the degree of freedom used for the F-test. Having obtained $\Delta EW$ and $\sigma$, we calculated a simple observable to assess the variability for a given target using a ratio of the form:
$$F\,=\,\frac{\Delta\,EW}{\sigma} .
\label{Fval}$$
This simple ratio represents the number of times that the amplitude of the variation is greater than the standard deviation.
To determine whether or not this number is meaningful and whether the star shows variability, we evaluated the corresponding confidence level; see last column of Table \[eqw\_test\]. We considered all the stars for which C$\ge$80$\%$ as definitively variable (13 stars, that is 41$\%$ of the sample), while we cannot say anything for the target with C$\le$50$\%$ (22$\%$ of the sample) due to the small number of points. The 37$\%$ of the stars, are probably variable but within a confidence level ranging from 50$\%$ to 80$\%$. In any case, all the targets show the same behavior in both spectral lines.
In this atlas we show the full set of our profiles (H$\alpha$ and H$\beta$) for the 48 observed stars. In Figs. \[variab1\] to \[variab5\], we show the Balmer profiles observed for program stars with multiple measurements, In each panel we reported for each profile also the last four digits of the heliocentric julian date of the observation. In some cases to improve the visualization, profiles have been blown up according to
{width="8cm" height="7cm"} {width="8cm" height="7cm"}
$$F'_\lambda = [(F_\lambda - 1) * k] + 1$$
where F$_\lambda$ denotes the observed flux, F$'_\lambda$ the displayed flux, and k is the magnification factor.
In Fig. \[one\_shell\] we show all the stars for which we collected one spectrum only. Most of the stars have been measured several times during the 2008/2009 period, but only few stars showed significant variations in their spectral profiles. These objects are discussed separately in the following:
[*HD37202 -*]{} This star has been observed in two nights separated by 205 days. It is clearly visible an increase in flux in the violet peak in both spectral lines, even if it is more evident in H$\beta$.
[*HD41335 -*]{} As for the previous object, only two observations have been acquired for this star in a range of 207 days. The star exhibits V/R variations in both lines.
[*HD58050 -*]{} The triple-peaked structure visible in the H$\alpha$ observed in the first night is missing in the other two spectra. No features are seen in the H$\beta$.
[*HD109387 -*]{} This star was observed for 13 nights in a period spanning 110 days. The top part of the H$\alpha$ emission profiles shows irregular variability, with back and forth changing from class 2 toward class 1, but no evident sign of day-by-day variability has been observed. No variations have been detected in H$\beta$ double-peaked profile.
[*HD142926 -*]{} During 110 days this shell star was observed 10 times showing a slight V/R variability in the H$\alpha$.
[*HD143275 -*]{} In the four spectra acquired by us, $\delta$Sco shows important changes in the emission of H$\alpha$. The V/R changes over the observational period and it shows a flat core in the first spectrum.
[*HD164284 -*]{} Observed for six times in 536 days, this star exhibits equivalent width variability. The double-peaked emission of the H$\alpha$ profile decrease with time, although the V/R remains constant and $\approx$1. Also the H$\beta$ shows a change in the shape, since it is the last profile without any peak.
[*HD183362 -*]{} This star shows an increased emission level on the red side of its profile.
[*HD183656 -*]{} V/R variability has been observed, both in H$\alpha$ and in H$\beta$ profiles, in the six spectra acquired in a range of 358 days.
[*HD187567 -*]{} This star show variability in the H$\alpha$ profile, and evolves from class 1 toward class 2.
[*HD189687 -*]{} In the H$\alpha$, this star does not show any sign of variability for the first month of observations. In the last spectrum taken after 32 days from the second to last, it starts to show an increase of the flux in the red peak. No H$\beta$ variability has been detected.
[*HD191610 -*]{} This star shows a gradual increment of the flux in the red peak.
Discussion and conclusions
==========================
In this paper we presented a homogeneous sample of H$\alpha$ and H$\beta$ line profiles observed in 32 Be stars, which show emission at least in the H$\alpha$ line. According to Hanuschik ([@hanu88]), we classified our targets on the basis of the following scheme:
- 16 stars, 33$\%$ of the sample, in class 1;
- 13 stars, 27$\%$ of the sample, in class 2;
- 11 stars, 23$\%$ of the total, has been classified as shell stars;
- 6 stars, 12$\%$ of the total, do not show net emission.
In this list we do not include the two stars that show any phase transition between classes 1 and 2.
This frequency distribution shows that the majority of our sample of 48 Be stars, randomly distributed in spectral type, belongs to class 1 profiles. Regarding the 13 stars classified as class 2, seven of them are single peak, while six show structured profiles.
Two stars showed variability from one class to another. HD187567 has undergone an evolution from class 1 to class 2, while the behavior of HD109387 is more complicated. In 110 days, this star has been observed 13 times, and it showed a transition from class 2 to class 1 and back again to class 2.
In Fig. \[corr\] we compared the behavior of the H$\alpha$ peaks separation to $v \sin i$ for the stars of the class 1 (included the class 2 HD60855). A linear correlation seems to exist, as expected from the work of Hanuschik et al. ([@hanu88]), although two stars discarded from this trend, namely HD191610 and HD212571. To verify this correlation we computed the Pearson [*r*]{} coefficient, obtaining [*r*]{}=0.72 and the linear fit given by the equation:
$$y = (0.56 \pm 0.13) \cdot x + (29.94 \pm 26.80) .$$
Thus, we confirm that $v \sin i$ and H$\alpha$ peaks separation are in linear correlation, confirming the disk-like geometry of Be star envelopes and, probably this assumption is not valid for HD191610 and HD212571.
For all the stars belonging to the class 1, we attempted to model the emission with the purpose of deriving some parameters such inclination angle, base density at the stellar equator and disk radius. In Fig. \[hist\] we show the distribution of stars as a function of the disk radius (right panel) and of the inclination angle (left panel). The histograms were built considering a binning equal to the estimated errors, that is, 2R$_*$ and 3$^\circ$ in the disk dimension and inclination angle, respectively. They show as there is a major concentration of stars for disk around $6 \div 8$ R$_*$ (about 17$\%$ of our sample) and for angles around $23^\circ \div 35^\circ$ (about 28$\%$ of the sample).
Moreover, with the aim of inferring line profile variability, for most of the stars of our sample we obtained more than one spectrum in a period spanning two years between 2008 and 2009. All but seven stars, those discussed in Sect. \[variab\], do not show any evident sign of variability in both Balmer lines.
This research made use of the SIMBAD database, operated at the CDS, Strasbourg, France.
Carciofi, A. C., & Bjorkman, J. E. 2006, ApJ, 639, 1081 Drilling, J., & Landolt, A. U. 1999, in: Allen’s Astrophysical Quantities, Fourth Edition, Edited by Arthur N. Cox, Los Alamos, NM Gies, D. R., Bagnuolo, W. G. Jr., Baines, E. K., et al. 2007, ApJ, 654, 527 Grundstrom, E. D., & Gies, D. R. 2006, A&A, 651, 53 Hanuschik, R. W. 1996, A&A, 308, 170 Hanuschik, R. W., Hummel, W., Sutorius, E., Dietle, O., & Thimm, G., 1996, A&ASS, 116, 309 Hanuschik, R. W., Kozok J. R., & Kaiser, D. 1988, A&A, 189, 1988 Hauck, B., & Mermilliod, M. 1998, A&AS, 129, 431 Hohle, M. M., Neuhauser, R., & Schutz, B.F. 2010/ AN, 331, 349 Horne, K. 1995, A&A, 297, 273 Horne, K., & Marsh, T. R. 1986, MNRAS, 218, 761 Hummel, W., & Vrancken, M. 2000, A&A, 302, 751 Hutchings, J. B. 1970, MNRAS, 150, 55 Jaschek, M., & Egret, D., 1982, A catalogue of Be stars. In: Be stars, IAU Symp. 98, 261 Ed. M. Jaschek and H.-G. Groth, Munich Kurucz, R.L. 1993, A new opacity-sampling model atmosphere program for arbitrary abundances. In: Peculiar versus normal phenomena in A-type and related stars, IAU Colloquium 138, M.M. Dworetsky, F. Castelli, R. Faraggiana (eds.), ASP Conference Series Vol. 44, p.87 Kenyon, S. J., & Hartmann, L., 1995, ApJS, 101, 117 Kurucz, R. L., & Avrett, E. H. 1981, SAO Special Rep., 391 Meilland, A., Stee, Ph., Vannier, M. et al. 2007, A&A, 464, 59 Moon, T. T. 1985, Comm. from the Univ. of London Obs., 78 Moujtahid, A., Zorec, J., & Hubert, A. M. 1999, A&A, 349, 151 Nelder, J. A., & Mead, R., 1965, Computer Journal, Vol 7, pp 308-313. Okazaki, A. T. 1997, A&A, 318, 548 Porter , J. M., & Rivinius, Th., 2003, PASP, 115, 1153 Porter, J. M. 1996, MNRAS, 280, L31 Saad, S.M., Kubát J., Korcáková, D., et al., 2006, A&A, 450, 427 Silaj, J., Jones, C. E., Tycner, C., Sigut, T. A. A., & Smith, A.D., 2010, ApJS, 187, 228 Slettebak, A., Collins, G. W., Truax, R., 1992, ApJS, 81, 335 Soubiran, C., Le Campion, J.-F., Cayrel De Strobel, G., & Caillo, A. 2010, A&A, 515, 111 Struve, O. 1931, ApJ, 73, 94 Wu, Y., Singh, H. P., Prugnel, P., Gupta, R., & Koleva, M. 2011, A&A, 525, 71 Zickgraf, F.-J., 2000, The connection with B\[e\] stars. In: The Be phenomenon in Early-Type stars, IAU Colloquium 175, M. A. Smith, H. F. Henrichs, J. Fabregat (eds), ASP Conference Series Vol. 214, p. 26
Fit emission
============
$
\begin{array}{cc}
\includegraphics[width=8cm]{20357figA1a.ps} &
\includegraphics[width=8cm]{20357figA1b.ps} \\
\includegraphics[width=8cm]{20357figA1c.ps} &
\includegraphics[width=8cm]{20357figA1d.ps} \\
\includegraphics[width=8cm]{20357figA1e.ps} &
\includegraphics[width=8cm]{20357figA1f.ps}\\
\includegraphics[width=8cm]{20357figA1g.ps} &
\includegraphics[width=8cm]{20357figA1h.ps}
\end{array}$
$
\begin{array}{cc}
\includegraphics[width=8cm]{20357figA2a.ps} &
\includegraphics[width=8cm]{20357figA2b.ps} \\
\includegraphics[width=8cm]{20357figA2c.ps} &
\includegraphics[width=8cm]{20357figA2d.ps} \\
\includegraphics[width=8cm]{20357figA2e.ps} &
\includegraphics[width=8cm]{20357figA2f.ps}\\
\includegraphics[width=8cm]{20357figA2g.ps} &
\includegraphics[width=8cm]{20357figA2h.ps}
\end{array}$
$
\begin{array}{cc}
\includegraphics[width=8cm]{20357figA3a.ps} &
\includegraphics[width=8cm]{20357figA3b.ps} \\
\includegraphics[width=8cm]{20357figA3c.ps} &
\includegraphics[width=8cm]{20357figA3d.ps} \\
\includegraphics[width=8cm]{20357figA3e.ps} &
\includegraphics[width=8cm]{20357figA3f.ps}\\
\includegraphics[width=8cm]{20357figA3g.ps} &
\includegraphics[width=8cm]{20357figA3h.ps}
\end{array}$
$
\begin{array}{cc}
\includegraphics[width=8cm]{20357figA4a.ps} &
\includegraphics[width=8cm]{20357figA4b.ps}
\end{array}$
Variability
===========
-------- ------------ ------------- ------------ -------- ------------ ------------- ------------ -------- ------------ ------------- ------------
HD JD EW([Å]{}) EW([Å]{}) HD JD EW([Å]{}) EW([Å]{}) HD JD EW([Å]{}) EW([Å]{})
(2450000+) H$\alpha$ H$\beta$ (2450000+) H$\alpha$ H$\beta$ (2450000+) H$\alpha$ H$\beta$
6811 4714.6282 1.06 6.53 142926 4543.6113 $-$2.82 7.14 187567 4654.5327 $-$ 21.09 0.81
4747.5640 0.99 6.46 4554.6243 $-$1.61 7.04 4655.5890 $-$ 21.46 0.59
4749.6417 0.55 6.36 4555.6541 $-$1.68 6.84 4678.5234 $-$ 22.24 0.40
10516 4749.6602 $-$25.21 0.37 4556.6487 $-$1.40 6.80 4714.4153 $-$ 22.50 0.44
11415 4749.6753 4.25 5.83 4575.5709 $-$1.61 7.11 189687 4654.5758 0.76 4.70
37202 4544.2610 $-$16.09 2.33 4576.5771 $-$1.68 7.14 4655.5411 0.93 4.75
4749.7298 $-$14.44 2.35 4586.5501 $-$1.49 7.12 4677.5324 0.62 4.77
37490 4544.2799 $-$5.19 1.95 4588.5402 $-$1.38 7.05 4681.5430 1.01 4.76
4749.7175 $-$6.73 1.70 4616.4682 $-$1.46 7.03 4682.5777 0.89 4.76
41335 4542.3012 $-$31.37 $-$1.81 4653.3880 $-$1.18 7.20 4714.4602 1.35 4.84
4749.7436 $-$32.48 $-$1.07 142983 4543.5309 $-$26.32 3.24 191610 4712.3812 0.48 4.81
43285 4543.3009 5.06 7.57 4555.5462 $-$23.71 3.15 4713.3568 $-$0.00 4.63
44458 4544.3009 $-$31.82 $-$0.80 4556.5508 $-$24.48 3.08 4747.3386 $-$0.01 4.81
45542 4542.3231 $-$0.91 5.93 4575.5004 $-$24.74 3.03 4749.4744 $-$0.00 4.51
45725 4542.3417 9.38 11.36 4576.4808 $-$24.61 3.15 192044 4712.4081 $-$9.93 5.21
47054 4543.3275 $-$6.86 6.17 4585.4945 $-$25.81 3.07 4713.3805 $-$10.10 5.24
50658 4542.3850 1.34 5.67 4586.4283 $-$25.08 3.16 4747.3594 $-$9.84 5.30
50820 4544.3397 $-$8.32 $-$2.73 4587.4507 $-$24.20 2.75 193911 4712.4526 $-$5.42 5.04
52918 4542.4174 2.77 3.78 4588.4201 $-$23.85 3.24 4713.4085 $-$5.34 5.12
53416 4556.2808 $-$4.66 6.42 4599.4307 $-$24.62 3.03 4747.3837 $-$5.60 5.14
58050 4544.3849 4.31 5.44 4616.3757 $-$25.63 3.05 210129 4654.5979 $-$13.86 4.85
4554.2984 4.74 5.51 4653.4399 $-$23.49 3.29 4677.5640 $-$14.28 5.02
4556.3476 4.14 5.51 4654.4038 $-$24.03 2.97 4678.5672 $-$14.30 5.24
58343 4543.3636 $-$19.29 2.75 143275 4677.3348 $-$11.01 $-$0.78 4681.5694 $-$14.18 5.04
58715 4542.4392 $-$0.79 7.91 4681.3050 $-$6.94 $-$0.64 4682.6018 $-$14.14 5.11
4554.3634 $-$1.29 7.95 4682.3327 $-$10.28 $-$0.73 4714.4878 $-$14.01 5.23
4556.3986 $-$1.49 8.02 4714.2604 $-$12.30 $-$1.09 4747.4096 $-$14.54 5.33
60855 4543.3894 $-$38.59 $-$0.95 162428 4586.5791 $-$15.50 5.03 4749.5015 $-$14.04 5.04
61224 4555.2948 $-$5.93 5.42 4588.5667 $-$15.72 4.12 212571 4712.4805 $-$5.37 2.11
4556.4231 $-$5.25 4.85 4677.3624 $-$15.69 4.99 4714.5117 $-$5.44 2.00
65875 4543.4116 $-$42.00 $-$0.28 4681.3258 $-$15.58 4.80 4749.5255 $-$5.40 1.83
4544.4331 $-$43.32 0.74 5011.4665 $-$15.38 4.96 214168 4713.4584 $-$15.13 2.12
4554.4361 $-$43.69 $-$0.33 162732 4586.6250 $-$7.24 6.13 4747.4338 $-$14.44 2.22
71072 4555.3404 3.20 5.05 4616.5402 $-$6.95 5.91 4749.5512 $-$14.80 1.27
4576.3047 3.09 5.38 4654.4306 $-$6.00 6.40 216057 4713.4828 4.48 7.25
91120 4543.4858 $-$1.39 8.32 4677.4028 $-$7.04 4.94 4714.5419 4.60 7.15
4554.3985 $-$0.31 8.19 4682.3762 $-$7.31 6.07 4747.4580 4.84 7.29
4555.4708 $-$0.40 8.15 5011.5115 $-$5.09 6.04 4749.5750 4.58 7.07
4556.4743 $-$0.52 8.02 164284 4575.5993 $-$0.01 5.19 216200 4713.5068 1.07 5.04
4576.3563 $-$0.66 7.90 4576.6058 $-$0.00 5.04 4714.5693 1.61 5.31
4585.3822 $-$0.66 8.02 4616.5681 0.91 5.23 4749.6007 0.76 4.90
4586.3393 $-$0.01 7.84 4677.4774 1.43 5.12 217050 4747.5172 $-$27.42 1.82
4587.3192 $-$0.01 8.05 4682.4266 1.64 5.26 217543 4713.5577 1.81 5.38
4588.3168 0.37 8.16 5011.5620 4.25 5.23 4747.5433 2.89 5.23
109387 4543.5060 $-$20.01 3.05 164447 4575.6219 $-$2.03 6.62 217675 4713.6001 3.24 5.51
4554.4814 $-$19.83 3.01 4677.4415 $-$1.24 6.64 4747.4769 2.85 5.77
4555.5156 $-$19.34 3.07 4681.3697 $-$1.44 6.68 217891 4678.6041 $-$12.16 4.93
4556.5191 $-$19.09 2.97 4749.3442 $-$1.02 6.93 4681.6008 $-$12.46 4.86
4575.4348 $-$19.43 3.02 5011.5811 0.96 6.52 4713.6187 $-$12.85 4.76
4576.4464 $-$19.42 2.98 183362 4653.5050 $-$ 26.22 1.35 4714.6005 $-$12.80 4.66
4585.4577 $-$19.09 3.02 4682.5161 $-$ 26.45 1.08 4747.4941 $-$13.47 4.57
4586.3921 $-$19.00 2.98 4714.3334 $-$ 26.50 1.33 4749.6231 $-$13.90 4.42
4587.3898 $-$18.78 2.99 183656 4653.5999 $-$8.27 5.24
4588.3867 $-$18.77 3.07 4681.4527 $-$7.59 5.25
4599.3878 $-$18.45 3.14 4714.3630 $-$8.01 5.35
4616.3342 $-$19.11 3.13 4749.3742 $-$8.07 4.98
4653.3303 $-$18.84 3.12 5011.6055 $-$6.48 4.94
183914 4653.5454 $-$1.01 8.03
4681.4958 $-$0.91 8.09
4682.5518 $-$0.96 8.20
4714.3886 $-$0.61 8.14
4749.4339 $-$1.23 8.06
-------- ------------ ------------- ------------ -------- ------------ ------------- ------------ -------- ------------ ------------- ------------
$
\begin{array}{cc}
\includegraphics[width=8cm]{20357figB1a.ps} &
\includegraphics[width=8cm]{20357figB1b.ps} \\
\includegraphics[width=8cm]{20357figB1c.ps} &
\includegraphics[width=8cm]{20357figB1d.ps} \\
\includegraphics[width=8cm]{20357figB1e.ps} &
\includegraphics[width=8cm]{20357figB1f.ps}
\end{array}$
$
\begin{array}{cc}
\includegraphics[width=8cm]{20357figB2a.ps} &
\includegraphics[width=8cm]{20357figB2b.ps} \\
\includegraphics[width=8cm]{20357figB2c.ps} &
\includegraphics[width=8cm]{20357figB2d.ps} \\
\includegraphics[width=8cm]{20357figB2e.ps} &
\includegraphics[width=8cm]{20357figB2f.ps}
\end{array}$
$
\begin{array}{cc}
\includegraphics[width=8cm]{20357figB3a.ps} &
\includegraphics[width=8cm]{20357figB3b.ps} \\
\includegraphics[width=8cm]{20357figB3c.ps} &
\includegraphics[width=8cm]{20357figB3d.ps} \\
\includegraphics[width=8cm]{20357figB3e.ps} &
\includegraphics[width=8cm]{20357figB3f.ps}
\end{array}$
$
\begin{array}{cc}
\includegraphics[width=8cm]{20357figB4a.ps} &
\includegraphics[width=8cm]{20357figB4b.ps} \\
\includegraphics[width=8cm]{20357figB4c.ps} &
\includegraphics[width=8cm]{20357figB4d.ps} \\
\includegraphics[width=8cm]{20357figB4e.ps} &
\includegraphics[width=8cm]{20357figB4f.ps}
\end{array}$
$
\begin{array}{cc}
\includegraphics[width=8cm]{20357figB5a.ps} &
\includegraphics[width=8cm]{20357figB5b.ps} \\
\includegraphics[width=8cm]{20357figB5c.ps} &
\includegraphics[width=8cm]{20357figB5d.ps} \\
\includegraphics[width=8cm]{20357figB5e.ps} &
\includegraphics[width=8cm]{20357figB5f.ps}
\end{array}$
$
\begin{array}{cc}
\includegraphics[width=8cm]{20357figB6a.ps} &
\includegraphics[width=8cm]{20357figB6b.ps} \\
\includegraphics[width=8cm]{20357figB6c.ps} &
\includegraphics[width=8cm]{20357figB6d.ps} \\
\includegraphics[width=8cm]{20357figB6e.ps}
\end{array}$
{width="9cm"} {width="9cm"} {width="9cm"}
[^1]: I wish to dedicate this paper to my child never born, to keep track of his passage through my life.
[^2]: Table B.1 and all observed spectra are available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/
[^3]: http://simbad.u-strasbg.fr/simbad/
[^4]: IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc.
[^5]: The amoeba routine implements the simplex method of Nelder & Mead ([@nelder65]).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The reduction criterion is a well known necessary condition for separable states, and states violating this condition are entangled and also 1-distillable. In this paper we introduce a new set of necessary conditions for separability of multipartite states, obtained from a set of positive but not completely positive maps. These conditions can be thought of as generalisations of the reduction criterion to multipartite systems. We use tripartite Werner states as an example to investigate the entanglement detecting powers of some of these new conditions, and we also look at what these conditions mean in terms of distillation. Finally, we show that these maps can be used to give a partial solution to the *subsystem problem*, as described in [@Sub].'
author:
- William Hall
title: Multipartite reduction criteria for separability
---
Introduction
============
Entanglement [@Ent] is one of the most intriguing phenomena in quantum mechanics. Its inherently non-classical nature caused much controversy in the early years of its discovery [@EPR; @Bell], and in more recent years many practical uses have been found for entanglement in the developing subject of quantum information science [@EntQI; @NC]. This has led to a concerted effort to understand the nature of entanglement, and to create a consistent theory that allows us to both determine when an arbitrary mixed state is entangled, and to quantify the entanglement of that state. While the nature of bipartite pure state entanglement is well understood [@Ent1], there is still much work to be done in the case of arbitrary mixed multipartite states; indeed, we still do not have an operational criteria to determine whether an arbitrary multipartite mixed state is entangled. However, in [@EntNC], a necessary and sufficient condition for a bipartite state to be entangled was established using positive maps, and this was generalised to the multipartite case in [@EntNC_M]. We will make use of the positive map formalism heavily in this paper. A good review can be found in [@EntRev].
The *reduction criterion* [@Red; @Red2] gives a necessary condition for a state to be separable; states which violate this criterion are hence entangled, and it can also be shown that these states are 1-distillable. In this paper we will produce some further necessary criteria for states to be (semi-)separable by introducing a new set of positive but not completely positive maps, and furthermore we will use the tripartite Werner states discussed in [@W3] to investigate their entanglement detecting properties. Finally, we will also discuss what the violation of these criteria means in terms of distillation.
The criteria described in this paper are related to the subsystem compatibility problem: *Given states of all proper subsystems of a multipartite system, what are the necessary and sufficient conditions for these subsystem states to be compatible with a single state of the whole system?* In [@Sub], this problem is solved for a classical system of $n$ bits, and some of these conditions can be translated into necessary conditions for a system of $n$ qubits. We will show in this paper how our new set of positive maps can be used to derive some of these conditions in a more general setting.
The reduction and generalised reduction criteria
================================================
Let $\mathcal{H}$ be a Hilbert space, and $\mathcal{B}(\mathcal{H})$ be the set of (bounded) operators on $\mathcal{H}$. The reduction map $\Lambda: \mathcal{B}(\mathcal{H}) \to \mathcal{B}(\mathcal{H})$, which is defined by $$\Lambda(\rho) = \operatorname{Tr}(\rho) \openone - \rho$$ ($\openone$ is the identity matrix) is a positive map: If $\rho$ has positive eigenvalues $\lambda_i$ $(i = 1,\ldots,n)$ then $\operatorname{Tr}(\rho)= \sum_i \lambda_i$, and hence the eigenvalues of $\Lambda(\rho)$ are $\sum_{i \neq j} \lambda_i$ $(j = 1,\ldots,n)$, which are also positive. The map is not however completely positive: for instance, let ${| \Psi^+ \rangle} = \frac{1}{\sqrt{2}}({| 00 \rangle} + {| 11 \rangle}) \in \mathbb{C}^2 \otimes \mathbb{C}^2$, then $$I \otimes \Lambda ({| \Psi^+ \rangle \langle \Psi^+ |}) = \frac{\openone}{2} - {| \Psi^+ \rangle \langle \Psi^+ |}$$ (where $I : \mathcal{B}(\mathcal{H}) \to \mathcal{B}(\mathcal{H})$ above is the identity map i.e. $I(\rho) = \rho)$ clearly has a negative eigenvalue. Hence for a bipartite state $\rho \in \mathcal{B}(\mathcal{H}_A \otimes \mathcal{H}_B)$, $$I \otimes \Lambda (\rho) = \rho_A \otimes \openone - \rho \geq 0$$ (where $\rho_A = \operatorname{Tr}_B \rho$) is a necessary condition for a quantum state to be separable. This is known as the *reduction criterion*. Any state violating this condition is not only entangled, but in fact is 1-distillable. This was originally proved in [@Red] and shown in a more abstract fashion in [@DW].
Let us define the maps $\Lambda^{(n)} : \mathcal{B}(\mathcal{H}_1 \otimes \ldots \otimes \mathcal{H}_n) \to \mathcal{B}(\mathcal{H}_1 \otimes \ldots \otimes \mathcal{H}_n)$ by $$\Lambda^{(n)}(\rho) = \sum_{B \subseteq N} (-1)^{|B|} \rho_B \label{GRM}$$ where $N=\{1,\ldots,n\}$, $|B|$ denotes the number of elements in the set $B$ and $\rho_B$ is the reduced density matrix of $\rho$ over the subsystems given in the set $B$, padded out with identities in the other systems (e.g. if $N=\{1,2,3\}$, and $B=\{1\}$, then $\rho_B = \operatorname{Tr}_{2,3}(\rho) \otimes \openone_2 \otimes \openone_3$). For example, $$\begin{aligned}
\Lambda^{(1)}(\rho) &=& \operatorname{Tr}(\rho)\openone - \rho ;\\
\Lambda^{(2)}(\rho) &=& \operatorname{Tr}(\rho)\openone - \rho_1 - \rho_2 + \rho ;\\
\Lambda^{(3)}(\rho) &=& \operatorname{Tr}(\rho)\openone - \rho_1 - \rho_2 - \rho_3 + \rho_{12} + \rho_{13} + \rho_{23} - \rho.\end{aligned}$$ It is from these maps that we will form our necessary criteria for separability. We first note a preliminary result:
$\Lambda^{(n)}(\rho_1 \otimes \ldots \otimes \rho_n) = \bigotimes_{i=1}^n (\operatorname{Tr}(\rho_i)\openone - \rho_i)$
*Proof* By induction: Trivially true for $n=1$; furthermore, assuming the result is true for the $n-1$ case, $$\Lambda^{(n)}(\rho_1 \otimes \ldots \otimes \rho_n) = \sum_{B \subseteq N} (-1)^{|B|} \rho_B$$ $$= \sum_{B \subseteq N, 1 \notin B} (-1)^{|B|} \rho_B + \sum_{B \subseteq N, 1 \in B} (-1)^{|B|} \rho_B$$ $$= (\operatorname{Tr}(\rho_1)\openone - \rho_1) \otimes \sum_{B \subseteq N / \{1\}} (-1)^{|B|} \rho_B$$ $$= \bigotimes_{i=1}^n (\operatorname{Tr}(\rho_i)\openone - \rho_i) \ \Box$$
This result immediately shows the sense in which we can think about these maps as generalising the reduction criterion. We now prove that these maps are positive, and furthermore they satisfy a further condition known as *$2n$-decomposability*. We first recall the definition of Schmidt number for density matrices [@Schmidt]: A bipartite density matrix $\rho$ has Schmidt number $k$ if
1. For any decomposition $\rho = \sum_i p_i {| \psi_i \rangle \langle \psi_i |}$, at least one of the ${| \psi_i \rangle}$ has Schmidt rank (number of non-zero coefficients in Schmidt decomposition) at least $k$;
2. There exists a decomposition of $\rho$ with all vectors ${| \psi_i \rangle}$ of Schmidt rank no more than $k$.
A map $\Lambda: \mathcal{B}(\mathcal{H}) \to \mathcal{B}(\mathcal{H})$ is $k$-decomposable if we can write $\Lambda = \Lambda_{CP} \circ T$, where $T$ is the transpose and $\Lambda_{CP}(\rho) = \sum_i V_i \rho V_i^\dagger$ is a completely positive map such that each $V_i$ has rank $k$.
With these definitions, we are ready to state our main result:
$\Lambda^{(n)}$ is a positive map, and is $2n$-decomposable.
*Proof* We make use of the Jamiołkowski correspondence [@J]: Let $\mathcal{K} = \mathcal{H}_1 \otimes \ldots \otimes \mathcal{H}_n$ and consider the operator $A^{(n)} \in \mathcal{B}(\mathcal{K} \otimes \mathcal{K})$ defined by $$A^{(n)} = \left( I \otimes \Lambda^{(n)} \right) (P_+)$$ where $$P_+ = \sum_{i_1, \ldots, i_n, j_1, \ldots j_n} {| i_1 \ldots i_n i_1 \ldots i_n \rangle \langle j_1 \ldots j_n j_1 \ldots j_n |},$$ a multiple of the projector onto the maximally entangled state on $\mathcal{K} \otimes \mathcal{K}$, and ${| i_r \rangle}$ forms an orthonormal basis for $\mathcal{H}_r$. We number the $2n$ systems as $\mathcal{H}_1, \ldots, \mathcal{H}_{2n}$ (so that $\mathcal{H}_{n+r} \cong \mathcal{H}_r$). Then, using Lemma 1, $$\begin{aligned}
A^{(n)} &=& \sum {| i_1 \ldots i_n \rangle \langle j_1 \ldots j_n |} \otimes \Lambda^{(n)} ({| i_1 \ldots i_n \rangle \langle j_1 \ldots j_n |} )\\
&=& \sum {| i_1 \ldots i_n \rangle \langle j_1 \ldots j_n |} \otimes \left( \bigotimes_{k=1}^n (\delta_{i_k j_k} \openone - {| i_k \rangle \langle j_k |})\right) \\\end{aligned}$$ Let us now write the above as an n-fold tensor product, grouping together the Hilbert spaces $\mathcal{H}_i$ for each $i$ (i.e. systems 1 and $n+1$ etc.). We can then write $$\begin{aligned}
A^{(n)} &=& \sum_{i_1, j_1} \left( {| i_1 j_1 \rangle \langle i_1 j_1 |} - {| i_1 i_1 \rangle \langle j_1 j_1 |} \right)_{1,n+1} \otimes \\
&& \ldots \otimes \sum_{i_n, j_n} \left( {| i_n j_n \rangle \langle i_n j_n |} - {| i_n i_n \rangle \langle j_n j_n |} \right)_{n-1,2n} \\
&=& \bigotimes_{k=1}^n \left( \openone - {P_+} \right)_{k,n+k} \\\end{aligned}$$ Let us consider the partial transpose of this over the second system $\mathcal{K}$ (i.e. over systems $n+1, \ldots, 2n$). Since $$\begin{aligned}
(\openone - {P_+})_{k,n+k}^{T_{n+k}} &=& \openone_{k,n+k} - \sum_{i_k, j_k} {| i_k j_k \rangle \langle j_k i_k |}_{k,n+k} \\
&=& \openone_{k,n+k} - {V}_{k,n+k}\end{aligned}$$ where $V$ is the swap operator, we have $$(A^{(n)})^{T_B} = \bigotimes_{k=1}^n \left( \openone - {V} \right)_{k,n+k} \label{witness}$$ where $T_B$ indicates partial transpose over the systems $n+1$ to $2n$. This is a positive operator, as $V$ has eigenvalues $\pm 1$. Furthermore, $\openone - V$ has a Schmidt rank 2 decomposition: defining ${| \psi_{ij} \rangle} = {| ij \rangle} - {| ji \rangle}$, then $$\begin{aligned}
\sum_{i<j} {| \psi_{ij} \rangle \langle \psi_{ij} |} &=& \sum_{i \neq j} {| ij \rangle \langle ij |} - {| ij \rangle \langle ji |} \\
&=& \openone - V .\end{aligned}$$ Hence $(\openone-V)_{k,n+k}$ is a Schmidt rank 2 operator (the operator is clearly not separable and hence cannot have a Schmidt rank 1 decomposition). Due to the the tensor product structure of $A^{(n)}$ in (\[witness\]), it must have Schmidt rank $2n$.
It is possible to obtain $\Lambda^{(n)}$ from $A^{(n)}$ using the inversion formula [@J; @DW] $$D = \left( I \otimes \Lambda \right)(P_+) \Leftrightarrow \Lambda(\rho) = \operatorname{Tr}_A \left[ D (\rho^T \otimes \openone) \right].$$ Suppose $D = \left( {| \psi \rangle \langle \psi |} \right)^{T_B}$, with ${| \psi \rangle}$ having Schmidt decomposition $\sum_{i=1}^n c_i {| a_i, b_i \rangle}$ i.e. Schmidt rank is $n$. Some elementary algebra and the above inversion formula allows us to obtain that $\Lambda(\rho) = V \rho^T V^\dagger$, with $V=\sum_{i=1}^n c_i {| b_i^* \rangle \langle a_i |}$, an operator of rank $n$, where if ${| b_i \rangle} = \sum_m b_{im} {| m \rangle}$, then ${| b_i^* \rangle} = \sum_m b_{im}^* {| m \rangle}$. Hence by linearity, for $D^{T_B}$ being positive and having Schmidt rank $k$, $\Lambda$ is $k$-decomposable. It follows that $\Lambda^{(n)}$ is a $2n$-decomposable map. $\Box$
From these conditions we can obtain some of the necessary conditions mentioned briefly in the introduction to this paper. For odd $n$, the coefficient of $\rho$ in the expression for $\Lambda^{(n)}(\rho)$ is negative, and hence the expression $$\Lambda^{(n)}(\rho) + \rho = \sum_{B \subset N} (-1)^{|B|} \rho_B$$ is positive. Furthermore, this expression is written only in terms of the reduced density matrices of all of the proper subsystems of an $n$-partite system. Hence if the above expression is *not* positive, it follows that the given reduced density matrices are not compatible with an overall state $\rho$. This generalises part of the necessary condition in [@Sub], which is made for $n$-partite systems of qubits with $n$ odd, to the case of $n$-partite systems (still for odd $n$) where each individual system has arbitrary (finite) dimension.
Entanglement detection
======================
We note that none of these maps are completely positive; this is evident from the construction above. An example [^1] is $\mathcal{H}_i = \mathcal{H} \equiv \mathbb{C}^2$ for $i = 1,\ldots, n+1$ (i.e. $n+1$ qubits), and $\rho \in \mathcal{B}(\mathcal{H}^{\otimes (n+1)})$ defined by $\rho={| \psi \rangle \langle \psi |}$, with ${| \psi \rangle} = \frac{1}{\sqrt{2}} ({| 00 \rangle}_{12} + {| 11 \rangle}_{12}) \otimes {| 0 \rangle}_{3\ldots n+1}$. Then, using (\[GRM\]), $$I_1 \otimes \Lambda^{(n)}_{2\ldots n+1} (\rho) = \sum_{B \subset \{2,\ldots, n+1\}} (-1)^{|B|} \rho_{B \cup \{1\}}$$ $$= (\rho_1 \otimes \openone_2 - \rho_{12}) \otimes \sum_{B \subset \{3,\ldots, n+1\}} (-1)^{|B|} \rho_B$$ $$= (\rho_1 \otimes \openone_2 - \rho_{12}) \otimes \Lambda^{(n-1)}_{3\ldots n+1} ({| 0 \rangle \langle 0 |}_{3\ldots n+1})$$ $$= (\openone_{12}/2 - {| \psi_+ \rangle \langle \psi_+ |}) \otimes \Lambda^{(n-1)}_{3\ldots n+1} ({| 0 \rangle \langle 0 |}_{3\ldots n+1})$$ which has negative eigenvalues (the subscripts label which systems each map acts on). We can hence think of each of these maps as providing necessary conditions for a density matrix to be separable: for a multipartite system with $n$ parts, and $A \subseteq N=\{1,\ldots,n\}$ with $|A|=k$, $$I_{N \setminus A} \otimes \Lambda^{(k)}_A (\rho) \geq 0$$ is a necessary condition for separability, where the subscript set denotes the systems that the map operates on. For example, for a tripartite state $\rho$, we have that $$\rho_1 - \rho_{12} - \rho_{13} + \rho \geq 0$$ is a necessary condition for $\rho$ to have the semi-separable form $\rho= \sum_i p_i \rho_1^i \otimes \rho_{23}^i$.
We note that if $I_1 \otimes \Lambda^{(n-1)}_{2\ldots n}(\rho) \ngeq 0$, then $I_1 \otimes \Lambda^{(n)}_{2\ldots n+1}(\rho \otimes \sigma) = I_1 \otimes \Lambda^{(n-1)}_{2\ldots n}(\rho) \otimes (Tr(\sigma)\openone -\sigma)$, which will also have negative eigenvalues.
One issue we are concerned with is the entanglement detecting power of these maps, especially in relation to the reduction criterion. The important result we will show here is that *there are states detected by $\Lambda^{(2)}$ that are not detected by $\Lambda^{(1)}$, and vice versa*. To do this we utilise the tripartite Werner states that are introduced in [@W3] (we choose these states in particular as the entangled two-party Werner states are not detected by the reduction criterion [@Red]).
Let us consider the Hilbert space $(\mathbb{C}^d)^{\otimes 3}$, and let us define the permutation operators $$V_\pi ({| \phi_1 \rangle} {| \phi_2 \rangle} {| \phi_3 \rangle}) = {| \phi_{\pi^{-1}(1)} \rangle} {| \phi_{\pi^{-1}(2)} \rangle} {| \phi_{\pi^{-1}(3)} \rangle}$$ where $\pi \in S_3$. Then [@W3 Lemma 1] states that the tripartite Werner states are given by $\rho = \sum_\pi \mu_\pi V_\pi (\mu_\pi \in \mathbb{C})$. We can rewrite these states using the following linear combinations, which we obtain from the representation theory of the group $S_3$: $$\begin{aligned}
R_+ &=& \frac{1}{6} \left( \openone + V_{(12)} + V_{(13)} + V_{(23)} + V_{(123)} + V_{(132)} \right) \\
R_- &=& \frac{1}{6} \left( \openone - V_{(12)} - V_{(13)} - V_{(23)} + V_{(123)} + V_{(132)} \right) \\
R_0 &=& \frac{1}{3} \left( 2\openone - V_{(123)} - V_{(132)} \right) \\
R_1 &=& \frac{1}{3} \left( 2V_{(23)} - V_{(13)} - V_{(12)} \right) \\
R_2 &=& \frac{1}{\sqrt{3}} \left( V_{(12)} - V_{(13)} \right) \\
R_3 &=& \frac{i}{\sqrt{3}} \left( V_{(123)} - V_{(132)} \right)\end{aligned}$$ We note that $R_+$ and $R_-$ represent the projections onto symmetric and antisymmetric subspaces (trivial and alternating representations of $S_3$), $R_0$ the projection onto the orthogonal subspace corresponding to the two-dimensional representation of $S_3$, and that $R_i \ (i = 1,2,3)$ act as the Pauli matrices in this subspace. This leads to Lemma 2 of [@W3]:
Let $\rho = \sum_k c_k R_k \ (k \in \{+,-,0,1,2,3\})$, and define $r_k(\rho) = tr(\rho R_k)$. Then $\rho$ is a density matrix if and only if $$r_+,r_-,r_0 \geq 0, \ r_+ + r_- + r_0 = 1, \ r_1^2 + r_2^2 + r_3^2 \leq r_0^2.$$
We will consider a subset of these states parameterised by two real variables $a,b$. Let $\rho = \frac{1}{N}(a R_+ + (1-a)R_0 + bR_1)$. For positivity we require $0 \leq a \leq 1$, and $|r_1| \leq r_0$, which is equivalent to $|b|\leq a$. $N=\operatorname{Tr}(\rho)$ is the normalisation factor, and is given by $$N = \frac{1}{6}d(d+1)(3a(2-d) + 4(d-1)).$$ We will consider the eigenvalues of $I_{12} \otimes \Lambda^{(1)}_3(\rho) = \rho_{12} - \rho$ and $I_{2} \otimes \Lambda^{(1)}_{23}(\rho) = \rho_1 - \rho_{12} - \rho_{13} + \rho$. If $\rho$ is an entangled state, we are interested in the *sign* of the eigenvalues: if all the eigenvalues are positive, then any present entanglement is not detected; one negative eigenvalue detects the entanglement. We note that $N$ is independent of $a$ if $d=2$, and for $d>2$, $N<0$ if and only if $a > \frac{4}{3} \left( 1+ \frac{1}{d-2} \right) > 1$, and hence we can ignore this normalisation factor for the purposes of determining the sign of the eigenvalues. Some elementary algebra outlined in Appendix A allows us to determine the eigenvalues of $\rho_{12} - \rho$ as $$\begin{array}{c}
\frac{1}{3N}(d-1)(2-a-b), \\
\frac{1}{3N}(d+1)(2-2a+b), \\
\frac{1}{6N}\bigg(a(d+2)+(1-a)(4d-6)+2b) \pm \\
\Big\{(1/4)(a(d+2)-4(1-a)+b(12-2d))^2 +
\\ (3/4)(a(d+2)-4(1-a)-2bd)^2 \Big\}^{1/2} \bigg)
\end{array} \label{e1}$$ and the eigenvalues of $\rho_1 - \rho_{12} - \rho_{13} + \rho$ as $$\begin{array}{c}
\frac{1}{6N} ( a((d+2)(d-3)+6) + 4(1-a)(d-1)^2 \\
+ 4b(d-1) ), \\
\frac{1}{6N} ( a(d+1)(d+2) - 4(1-a)(d+1)(d-3) \\
- 4b(d+1) ), \\
\frac{1}{6N} ( ad(d+2) + 2(1-a)(2(d+1)(d-1) - 4d +1) \\
+ 2b(1-d) ) ), \\
\frac{1}{6N} ( a(d+2)(d-2) + 2(1-a)(2(d+1)(d-1) - 4d + 5) \\
+ b(2d-10) )
\end{array} \label{e2}$$ (we note that for $d=2, R_-=0$ and hence the second expression in both lists above is *not* an eigenvalue of the corresponding operator). To determine which states are detected by which maps, we need only look at when the above eigenvalues are positive/negative and hence we can ignore the normalisation $1/N$, since for $0 \leq a \leq 1$, $N>0$. Below is a figure giving an example for $d=2$. It clearly shows states detected by $\Lambda^{(2)}$ but not $\Lambda^{(1)}$, and vice-versa.
![Plot for $d=2$ illustrating which maps detect the state $\rho$ for given values of $a,b$. The white region represents states detected by both maps (i.e. have a negative eigenvalue), the lightest grey represents states detected by $\Lambda^{(2)}$ but not $\Lambda^{(1)}$; the middle grey those detected by $\Lambda^{(1)}$ but not $\Lambda^{(2)}$, and the darker grey is where neither map detects the state. The inside of the outlined box on the right of the graph represents states biseparable in the $1|23$ cut (this can be determined from [@W3]) and hence $I_{1} \otimes \Lambda^{(2)}_{23}(\rho)$ will never detect these states. ](g4)
Interestingly, for the above states for $d>2$ all states detected by $\Lambda^{(2)}$ are also detected by $\Lambda^{(1)}$. However no definite conclusions can be drawn from this since we are only looking at a small subset of symmetric states.
Distillation criteria
=====================
It is a well known result that if a bipartite state violates the reduction criterion then it is 1-distillable. This was first proved in [@Red] by giving an explicit distillation protocol and later in [@DW] using a more abstract positive map formalism. In this section we investigate the potential for the maps introduced above to be used as a distillation criterion.
Consider a bipartite state $\rho \in \mathcal{B}(\mathcal{H}_A \otimes \mathcal{H}_B)$. We recall the following theorem from [@DW]:
$\rho$ is 1-undistillable if and only if $(I \otimes \Lambda)(\rho) \geq 0$ for all 2-decomposable maps $\Lambda: \mathcal{B}(\mathcal{H}_B) \to \mathcal{B}(\mathcal{H}_B)$.
The reduction map $\Lambda^{(1)}$ is 2-decomposable, and so if $\rho$ satisfies the relation $$\rho_A \otimes \openone - \rho \ngeq 0 \label{re}$$ then $\rho$ is 1-distillable. This gives us the distillation criterion at the start of this article. Now, let us consider $\rho^{\otimes n} \in \mathcal{B}(\mathcal{H}_A^{\otimes n} \otimes \mathcal{H}_B^{\otimes n})$. Let us act $\Lambda^{(n)}$ on $\mathcal{H}_B^{\otimes n}$, and the identity map on $\mathcal{H}_A^{\otimes n}$. We wish to consider what the condition $$I_{1\ldots n} \otimes \Lambda^{(n)}_{n+1 \ldots 2n} (\rho^{\otimes n}) \ngeq 0 \label{newred}$$ means in terms of the distillation of $\rho$. We can deduce this from the following lemma:
$I_{1\ldots n} \otimes \Lambda^{(n)}_{n+1 \ldots 2n} (\rho^{\otimes n}) = (I \otimes \Lambda (\rho))^{\otimes n}$
*Proof* Again by induction. Trivial for $n=1$; assuming the $n-1$ case, $$I_{1\ldots n} \otimes \Lambda^{(n)}_{n+1 \ldots 2n} (\rho^{\otimes n})= \sum_{B \subset \{n+1, \ldots, 2n \} } (-1)^{|B|} \rho^{\otimes n}_{B \cup \{1, \ldots, n \} }$$ $$= ( {\operatorname{Tr}}_B( \rho ) \otimes \openone - \rho)_{1,n+1} \otimes \sum_{B \subset \{n+2, \ldots, 2n\}} (-1)^{|B|} \rho^{\otimes (n-1)}_{B \cup \{2, \ldots, n \}}$$ $$= (\operatorname{Tr}_B(\rho) \otimes \openone - \rho)_{1,n+1} \otimes \left( I_{2\ldots n} \otimes \Lambda^{(n)}_{n+2 \ldots 2n} (\rho^{\otimes (n-1)}) \right)$$ and hence the result holds by induction. $\Box$
From this result it is clear that $$\begin{aligned}
I_{1\ldots n} \otimes \Lambda^{(n)}_{n+1 \ldots 2n} (\rho^{\otimes n}) \ngeq 0 & \Leftrightarrow & (I \otimes \Lambda (\rho))^{\otimes n} \ngeq 0 \\
&\Leftrightarrow& (I \otimes \Lambda (\rho)) \ngeq 0 \end{aligned}$$ and so condition (\[newred\]) is equivalent to (\[re\]), the violation of the reduction criterion.
Conclusion
==========
In this paper we have given a set of positive but not completely positive maps that can be used to define a new set of separability criteria, and we have shown that they can be thought of as multi-party forms of the reduction criterion. We have analysed the entanglement detecting powers of these maps, and have shown that there are states that $\Lambda^{(2)}$ detects but not $\Lambda^{(1)}$, and vice-versa. It is highly likely that there will be similar results for the maps in general, and in theory, by using $n$-party analogues of the Werner states, we could obtain further results. It remains to be seen (although from the above results it seems unlikely) whether there is any hierarchy within these separability criteria. We have also analysed the criteria from a distillation viewpoint, and have shown that we can recover the criteria for distillation that we obtain from the reduction criterion.
The author would like to thank Paul Butterley for providing the impetus behind the above work, Matthias Christandl for a very illuminating conversation, Lieven Clarisse and Anthony Sudbery for checking the final manuscript, and finally the Engineering and Physical Sciences Research Council (UK) for supporting this work.
Details of calculations for tripartite Werner states
====================================================
In this appendix we give a few more details of the calculations required to obtain the eigenvalues of the given maps in section 3. Suppose that $\rho = \sum_k c_k R_k \ (k \in \{+,-,0,1,2,3\})$. Then defining $$\nu_+ = \frac{d}{6}(d^2 + 3d+2), \ \nu_- = \frac{d}{6}(d^2 - 3d+2), \ \nu_0 = \frac{d}{3}(d^2 - 1),$$ it can easily be shown that $r_+ = c_+ \nu_+, r_- = c_- \nu_-, r_i = 2 c_i \nu_0 \ (i \in \{0,1,2,3\})$. Furthermore, for $k=+,-,0$, the projectors $R_k$ are orthogonal, and project onto a subspace of dimension $\nu_k$, and $R_1,R_2,R_3$ act as Pauli matrices within the subspace of $R_0$. Hence the eigenvalues of $\rho$ are given by $$c_+ \textrm{ (multiplicity }\nu_+), \ c_- \textrm{ (multiplicity }\nu_-)$$ $$c_0 \pm \sqrt{c_1^2 + c_2^2 + c_3^3} \textrm{ (multiplicity }\nu_0) \label{eig}$$ This allows us to easily investigate the properties of $I_{12} \otimes \Lambda^{(1)}_3(\rho)$ and $I_{2} \otimes \Lambda^{(1)}_{23}(\rho)$. Now take $\rho = \frac{1}{N}(a R_+ + (1-a)R_0 + bR_1)$ with $N = \frac{1}{6}d(d+1)(3a(2-d) + 4(d-1))$ as defined above. We reiterate that for positivity we require $0 \leq a \leq 1$, and $|b|\leq a$. Inverting the relations for $R_+$ etc. in terms of $V_\rho$, and some tedious but elementary algebra allows us to write $$\begin{aligned}
\lefteqn{\rho_{12} - \rho =} \\
&& \frac{1}{3N}(d-1)(2-a-b)R_+ \\
&+& \frac{1}{3N}(d+1)(2-2a+b)R_- \\
&+& \frac{1}{6N} (a(d+2) + (1-a)(4d-6) + 2b)R_0 \\
&-& \frac{1}{12N} (a(d+2)-4(1-a)+b(12-2d))R_1 \\
&+& \frac{\sqrt{3}}{12N}(a(d+2)-4(1-a)-2bd)R_2 \end{aligned}$$ and $$\begin{aligned}
\lefteqn{\rho_1 - \rho_{12} - \rho_{13} + \rho =} \\
&& \frac{1}{6N} a((d+2)(d-3)+6) \\
&& + 4(1-a)(d-1)^2 + 4b(d-1) )R_+ \\
&+& \frac{1}{6N} ( a(d+1)(d+2) - 4(1-a)(d+1)(d-3) \\
&& - 4b(d+1) )R_- + \frac{1}{6N} ( a(d-1)(d+2) \\
&& + (1-a)(4(d-1)(d+1) - 8d + 6) - 4b)R_0 \\
&+& \frac{1}{6N} ( a(d+2) - 4(1-a) - b(6-2d) ) R_1 \end{aligned}$$ from which, using (\[eig\]) above also, we can easily determine the eigenvalues of $\rho_{12} - \rho$ and $\rho_1 - \rho_{12} - \rho_{13} + \rho$ given in equations (\[e1\]) and (\[e2\]) above. We finish by noting that we do not need to consider $\rho_{13} - \rho$ additionally, as it can be shown to have identical eigenvalues to $\rho_{12} - \rho$ (this follows simply from the symmetry of $\rho$ between systems 2 and 3, and the symmetry of the expressions for $V_\rho$ in terms of $R_+$ etc.).
[16]{}
E. Schrödinger, Proc. Camb. Philos. Soc. **35**, 555 (1935). A. Einstein, B. Poldosky and N. Rosen, Phys. Rev. **23**, 777 (1935). J. S. Bell, Physics **1**, 195 (1964). For example, see A. K. Ekert, Phys. Rev. Lett. **67**, 661 (1991); C. H. Bennett and S. J. Wiesner, *ibid.* **69**, 2881 (1992); C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres and W.K. Wootters, *ibid* **70**, 1895 (1993). M. Nielsen and I. Chuang, *Quantum Computation and Quantum Information* (Cambridge University Press, Cambridge, 2000). For example see C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wootters, Phys. Rev. A **54**, 3824 (1996). M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A **223**, 1 (1996). M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A **283**, 1 (2001). M. Horodecki, P. Horodecki, and R. Horodecki, in *Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments*, edited by G. Alber (Springer-Verlag, Berlin, 2001). M. Horodecki and P. Horodecki, Phys. Rev. A **59**, 4206 (1999). N.J. Cerf, C. Adami and R.M. Gingrich, Phys. Rev. A **60**, 898 (1999). L. Clarisse, Phys. Rev. A **71**, 032332 (2005). T. Eggeling and R. F. Werner, Phys. Rev. A **63**, 042111 (2001). P. Butterley, A. Sudbery and J. Szulc, Found. Phys. (to be published). A. Jamiołkowski, Rep. Math. Phys. **3**, 275 (1972). B. M. Terhal and P. Horodecki, Phys. Rev. A **61**, 040301(R) (2000).
[^1]: This was discovered in the $n=3$ case by Paul Butterley.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Transfer learning has been proven as an effective technique for neural machine translation under low-resource conditions. Existing methods require a common target language, language relatedness, or specific training tricks and regimes. We present a simple transfer learning method, where we first train a “parent” model for a high-resource language pair and then continue the training on a low-resource pair only by replacing the training corpus. This “child” model performs significantly better than the baseline trained for low-resource pair only. We are the first to show this for targeting different languages, and we observe the improvements even for unrelated languages with different alphabets.'
author:
- |
Tom Kocmi Ond[ř]{}ej Bojar\
\
Charles University, Faculty of Mathematics and Physics\
Institute of Formal and Applied Linguistics\
Malostransk[é]{} n[á]{}m[ě]{}st[í]{} 25, 118 00 Prague, Czech Republic\
[@ufal.mff.cuni.cz]{}
bibliography:
- 'emnlp2018.bib'
title: 'Trivial Transfer Learning for Low-Resource Neural Machine Translation'
---
Introduction
============
Neural machine translation (NMT) has made a big leap in performance and became the unquestionable winning approach in the past few years . The main reason behind the success of NMT in realistic conditions was the ability to handle large vocabulary and to utilize large monolingual data . However, NMT still struggles if the parallel data is insufficient (e.g. fewer than 1M parallel sentences), producing fluent output unrelated to the source and performing much worse than phrase-based machine translation .
Many strategies have been used in MT in the past for employing resources from additional languages, see e.g. , , , or . For NMT, a particularly promising approach is transfer learning or “domain adaptation” where the “domains” are the different languages.
For example, train a “parent” model in a high-resource language pair, then use some of the trained weights as the initialization for a “child” model and further train it on the low-resource language pair. In , the parent and child pairs shared the target language (English) and a number of modifications of the training process were needed to achieve an improvement in translation from Hansa, Turkish, and Uzbek into English with the help of French-English data.
explore a related scenario where the parent language pair is also low-resource but it is related to the child language pair. They improved the previous approach by using a shared vocabulary of subword units (BPE, ). Additionally, they used transliteration to improve their results.
In this paper, we contribute empirical evidence that transfer learning for NMT can be simplified even further. We leave out the restriction on relatedness of the languages and extend the experiments to parent–child pairs where the target language changes. Moreover, we do not utilize any special modifications to the training regime or data pre-preprocessing.
In contrast to previous work, we test the method with the Transformer model , instead of the recurrent approaches . As documented in e.g. and anticipated in WMT18,[^1] the Transformer model seems superior to other NMT approaches.
Method Description
==================
The proposed method is extremely simple: We train the parent language pair for a number of iterations and switch the training corpus to the child language pair for the rest of the training, without resetting any of the training (hyper)parameters.
As such, this method is similar to the transfer learning proposed by but uses the shared vocabulary as in . The novelty is that we are removing the restriction about relatedness of the language pairs, and in contrast to the previous papers, we show that this simple style of transfer learning can be used on both sides (i.e. either the source or the target language), not only with the target language common to both parent and child model. In fact, the method is effective also for fully unrelated language pairs.
Our method does not need any modification of existing NMT frameworks. The only requirement is to use a shared vocabulary of subword units (we use wordpieces, ) across both language pairs. This is achieved by learning wordpiece segmentation from the concatenated source and target sides of both the parent and child language pairs. All other parameters of the model stay the same as for the standard NMT training.
During the training we first train the NMT model for the high-resource language pair until convergence. This model is called “parent”. After that, we train the child model without any restart, i.e. only by changing the training corpora to the low-resource language pair.
Details on Shared Vocabulary {#sec:shared_vocabulary}
----------------------------
Current NMT systems use vocabularies of subword units instead of whole words. Using subword units gives a balance between the flexibility of separate characters and efficiency of whole words. It solves the out-of-vocabulary words problem and reduces the vocabulary size. The majority of NMT systems use either the byte pair encoding or wordpieces . Given a training corpus and the desired maximal vocabulary size, either method produces deterministic rules for word segmentation to achieve the fewest possible splits.
Our method requires the vocabulary shared across both the parent (translating from language XX to YY) and the child model (translating from AA to BB). This is obtained by concatenating both training corpora into one corpus of sentences in languages AA, BB, XX and YY. [^2]
Due to our focus on low-resource language pairs, we decided to generate the vocabulary in a balanced way by selecting the same amount of sentences from both language pairs. We thus use the same number of sentence pairs of the parent corpus as there are in the child corpus.
We did not experiment with any other balancing of the vocabulary. Future research could also investigate the impact of using only the child corpus for vocabulary generation or various amounts of used sentences.
We generated vocabularies aiming at 32k subword types. The exact size of the vocabulary varies from 26.1k to 34.8k. All experiments of a given language set use the same vocabulary. Vocabulary overlap in each language set is further studied in Section \[sec:vocab\_analysis\].
Model Description
=================
We use the Transformer sequence-to-sequence model as implemented in Tensor2Tensor version 1.4.2. Our models are based on the “big single GPU” configuration as defined in the paper. To fit the model to our GPUs (NVIDIA GeForce GTX 1080 Ti with 11 GB RAM), we set the batch size to 2300 tokens and limit sentence length to 100 wordpieces.
We use exponential learning rate decay with the starting learning rate of 0.2 and 32000 warm up steps and Adam optimized. In our experiments, we find that it is undesirable to reset learning rate as it leads to the loss of the performance from the parent model. Therefore the transfer learning is handled only by changing the training corpora and nothing else.
Decoding uses the beam size of 8 and the length normalization penalty is set to 1.
The models were trained for 1M steps (approx. 140 hours), which was sufficient for models to converge to the best performance. We selected the model with the best performance on the development test for the final evaluation on the testset.
Datasets
========
In our experiments, we compare low-resource and high-resource language pairs spanning two orders of magnitude of training data sizes. We consider Estonian () and Slovak () as low-resource languages compared to the Finnish () and Czech () counterparts.
The choice of languages was closely related to the languages in this year’s WMT 2018 shared tasks. In particular, Estonian and Finnish (paired with English) were suggested as the main focus for their relatedness. We added Czech and Slovak as another closely related language pair. Russian () for the parent model was chosen for two reasons: (1) written in Cyrillic, there will be hardly any intersection in the shared vocabulary with the child language pairs, and (2) previous work uses transliteration to handle Russian, which is a nice contrast to our work. Finally, we added Arabic (), French () and Spanish () for experiments with unrelated languages.
The sizes of the training datasets are in Table \[table:dataset\_sizes\].
------- -------- ------- -------- -------- --------
Lang. Sent.
pair pairs First Second First Second
ET,EN 0.8 M 14 M 20 M 631 k 220 k
FI,EN 2.8 M 44 M 64 M 1697 k 545 k
SK,EN 4.3 M 82 M 95 M 1059 k 610 k
RU,EN 12.6 M 297 M 321 M 2202 k 3161 k
CS,EN 40.1 M 491 M 563 M 6253 k 4130 k
AR,RU 10.2 M 243 M 252 M 2299 k 2099 k
FR,RU 10.0 M 295 M 238 M 1339 k 2045 k
ES,FR 10.0 M 297 M 288 M 1426 k 1323 k
ES,RU 10.0 M 300 M 235 M 1433 k 2032 k
------- -------- ------- -------- -------- --------
: Datasets sizes overview. We consider Estonian and Slovak low-resource languages in our paper. Word counts and vocabulary sizes are from the original corpus, tokenizing only at whitespace and preserving the case. []{data-label="table:dataset_sizes"}
If not specified otherwise we use training, development and test sets from WMT.[^3] Pairs with training sentences with less than 4 words or more than 75 words on either the source or the target side are removed to allow for a speedup of Transformer by capping the maximal length and allowing a bigger batch size. The reduction of training data is small and based on our experiments, it does not change the performance of the translation model.
We use the Europarl and Rapid corpora for Estonian-English. We disregard Paracrawl due to its noisiness. The development and test sets are from WMT news 2018.
The Finnish-English was prepared as in , removing Wikipedia headlines. The dev and test sets are from WMT news 2015.
For English-Czech, we use all paralel data allowed in WMT2018 except Paracrawl. The main resource is CzEng 1.7 (the filtered version, ). The devset is WMT newstest2011 and the testset is WMT newstest2017.
Slovak-English uses corpora from , detokenized by Moses.[^4] WMT newstest2011 serves as the devset and testset.
The Russian-English training set was created from News Commentary, Yandex and UN Corpus. As the devset, we use WMT newstest 2012.
The language pairs Arabic-Russian, French-Russian, Spanish-French and Spanish-Russian were selected from UN corpus , which provides over 10 million multi-parallel sentences in 6 languages.
Results
=======
In this section, we present results of our approach. Statistical significance of the winner (marked with ) is tested by paired bootstrap resampling against the baseline (child-only) setup (1000 samples, conf. level 0.05; ).
As customary, we label the models with the pair of the source and target language codes, for example the English-to-Estonian translation model is denoted by .
The vocabularies are generated as described in \[sec:shared\_vocabulary\] separately for each experimented combination of parent and child. The same vocabulary is used whenever the parent and child use the same set of languages, i.e. disregarding the translation direction and model stage (parent or child).
English as the Common Language {#sec:english-as-the-common}
------------------------------
Table \[tab:highresourceparent\] summarizes our results for various combinations of high-resource parent and low-resource child language pairs when English is shared between the child and parent either in the encoder or in the decoder.
We confirm that sharing the target language improves performance as previously shown [@zoph-EtAl:2016:EMNLP2016; @Nguyen2017]. This gains up to 2.44 BLEU absolute for ETEN with the FIEN parent. Using only the parent (FIEN) model to translate the child (ETEN) test set gives a miserable performance, confirming the need for transfer learning or “finetuning”.
----------------- ---------- ------- --------
Parent - Child Transfer Child Parent
enFI - enET 19.74 17.03 2.32
FIen - ETen 24.18 21.74 2.44
**enCS - enET** 20.41 17.03 1.42
**enRU - enET** 20.09 17.03 0.57
**RUen - ETen** 23.54 21.74 0.80
enCS - enSK 17.75 16.13 6.51
CSen - SKen 22.42 19.19 11.62
enET - enFI 20.07 19.50 1.81
ETen - FIen 23.95 24.40 1.78
enSK - enCS 22.99 23.48 6.10
SKen - CSen 28.20 29.61 4.16
----------------- ---------- ------- --------
: Transfer learning with English reused either in source (encoder) or target (decoder). The column “Transfer” is our method, baselines correspond to training on one of the corpora only. Scores (BLEU) are always for the child language pair and they are comparable only within lines or when the child language pair is the same. “Unrelated” language pairs in bold. Upper part: parent larger, lower part: child larger. (“EN” lowercased just to stand out.)[]{data-label="tab:highresourceparent"}
A novel result is that the method works also for sharing the source language, improving ENET by up to 2.71 BLEU thanks to ENFI parent.
Furthermore, the improvement is not restricted only to related languages as Estonian and Finnish as shown in previous works. Unrelated language pairs (shown in bold in Table \[tab:highresourceparent\]) like Czech and Estonian work too and in some cases even better than with the related datasets. We reach an improvement of 3.38 BLEU for ENET when parent model was ENCS, compared to improvement of 2.71 from ENFI parent. This statistically significant improvement contradicts who concluded that the more related the languages are, the better transfer learning works. We see it as an indication that the size of the parent training set is more important than relatedness of languages.
The results with Russian parent for Estonian child (both directions) show that transliteration is also not necessary. Because there is no vocabulary sharing between Russian Cyrilic and Estonian Latin (except numbers and punctuation, see Section \[sec:vocab\_analysis\] for further details), the improvement could be attributed to a better coverage of English; an effect similar to domain adaptation.
On the other hand, this transfer learning works well only when the parent has more training data than the child. As presented in the bottom part of Table \[tab:highresourceparent\], low-resource parents do not generally improve the performance of better-resourced childs and sometimes, they even (significantly) decrease it. This is another indication, that the most important is the size of the parent corpus compared to the child one.
The baselines are either models trained purely on the child parallel data or only on the parent data. The second baseline only indicates the relatedness of languages because it is only tested but never trained on the child language pair. Also, we do not add any language tag as in . This also highlights that the improvement of our method cannot be directly attributed to the relatedness of languages: e.g. Czech and Slovak are much more similar than Czech and Estonian (Parent Only BLEU of translation out of English is 6.51 compared to 1.42) and yet the gain from transfer learning is larger for Estonian (+3.38) than from Slovak (+1.62).
Child Training Sents Transfer BLEU Baseline BLEU
---------------------- --------------- ---------------
800k 19.74 17.03
400k 19.04 14.94
200k 17.95 11.96
100k 17.61 9.39
50k 15.95 5.74
10k 12.46 1.95
: Maximal score reached by ENET child for decreasing sizes of child training data, trained off an ENFI parent (all ENFI data are used and models are trained for 800k steps). The baselines use only the reduced ENET data. []{data-label="tab:simulated_lowresource"}
Simulated Very Low Resources
----------------------------
In Table \[tab:simulated\_lowresource\], we simulate very low-resource settings by downscaling the data for the child model. It is a common knowledge, that gains from transfer learning are more pronounced for smaller childs. The point of Table \[tab:simulated\_lowresource\] is to illustrate that our approach is applicable even to extremely small child setups, with as few as 10k sentence pairs. Our transfer learning (“start with a model for whatever parent pair”) may thus resolve the issue of applicability of NMT for low resource languages as pointed out by .
Parent Convergence
------------------
Figure \[fig:progress\] compares the performance of the child model when trained from various training stages of the parent model. The performance of the child clearly correlates with the performance of the parent. Therefore, it is better to use a parent model that already converged and reached its best performance.
Direction Swap in Parent and Child
----------------------------------
Parent - Child Transfer Baseline Aligned
---------------- ---------- ---------- ---------
enFI - ETen 22.75 21.74 24.18
FIen - enET 18.19 17.03 19.74
enRU - ETen 23.12 21.74 23.54
enCS - ETen 22.80 21.74 not run
RUen - enET 18.16 17.03 20.09
enET - ETen 22.04 21.74 21.74
ETen - enET 17.46 17.03 17.03
: Results of child following a parent with swapped direction. “Baseline” is child-only training. “Aligned” is the more natural setup with English appearing on the “correct” side of the parent, the numbers in this column thus correspond to those in Table \[tab:highresourceparent\]. []{data-label="tab:shared_english"}
Relaxing the setup in Section \[sec:english-as-the-common\], we now allow a mismatch in translation direction of the parent and child. The parent XX-EN is thus followed by an EN-YY child or vice versa. It is important to note that Transformer shares word embeddings for the source and target side. The gain can be thus due to better English word embeddings, but definitely not due to a better English language model. It would be interesting to study the effect of not sharing the embeddings but we leave it for some future work.
The results in Table \[tab:shared\_english\] document that an improvement can be reached even when none of the involved languages is reused on the same side. This interesting result should be studied in more detail. hinted possible gains even when both languages are distinct from the low-resource languages but in a multilingual setting. Not surprisingly, the improvements are better when the common language is aligned.
The bottom part of Table \[tab:shared\_english\] shows a particularly interesting trick: the parent is not any high-resource pair but the very same EN-ET corpus with source and target swapped. We see gains in both directions, although not always statistically significant. Future work should investigate if this performance boost is possible even for high-resource languages. Similar behavior has been shown in , where in contrast to our work they mixed the data together and added an artificial token indicating the target language.
No Language in Common
---------------------
Parent - Child Transfer Baseline
---------------- ---------- ----------
ARRU - ETEN 22.23 21.74
ESFR - ETEN 22.24 21.74
ESRU - ETEN 22.52 21.74
FRRU - ETEN 22.40 21.74
: Transfer learning with parent and child not sharing any language.[]{data-label="tab:no_shared"}
Our final set of experiments examines the performance of ETEN child trained off parents in totally unrelated language pairs. Without any common language, the gains cannot be attributed, e.g., to the shared English word embeddings. The vocabulary overlap is mostly due to short n-grams or numbers and punctuations.
We see gains from transfer learning in all cases, mostly significant. The only non-significant gain is from Arabic-Russian which does not share the script with the child Latin at all. (Sharing of punctuation and numbers is possible across all the tested scripts.) The gains are quite similar (+0.49–+0.78 BLEU), supporting our assumption that the main factor is the size of the parent (here, all have 10M sentence pairs) rather than language relatedness.
Analysis
========
Here we provide a rather initial analysis of the sources of the gains.
Vocabulary Overlap {#sec:vocab_analysis}
------------------
Out method relies on the vocabulary estimated jointly from the child and parent model. In Transformer, the vocabulary is even shared across encoder and decoder. With a large overlap, we could expect a lot of “information reuse” between the parent and the child.
Since the subword vocabulary depends on the training corpora, a little clarification is needed. We take the vocabulary of subword units as created e.g. for ENRU-ENET experiments, see Section \[sec:shared\_vocabulary\]. This vocabulary contains 28.2k subwords in total. We then process the training corpora for each of the languages with this shared vocabulary, ignore all subwords that appear less than 10 times in each of the languages (these subwords will have little to no impact on the result of the training) and break down the total 28.2k subwords into classes depending on the languages in which the particular subword was observed, see Table \[tab:enruet-breakdown\].
ET EN RU % Subwords
---- -------------- ---- ------------
29.93%
20.69%
29.03%
10.06%
1.39%
0.00%
8.89%
28.2k (100%)
41.03%
: Breakdown of subword vocabulary of experiments involving ET, EN and RU.[]{data-label="tab:enruet-breakdown"}
We see that the vocabulary is reasonably balanced, with each language having 20–30% of subwords unique to it. English and Estonian share 10% subwords not seen in Russian while Russian shares only 0–1.39% of subwords with each of the other languages. Overall 8.89% of subwords are seen in all three languages.
A particularly interesting subset is the one where parent languages help the child model, in other words subwords appearing anywhere in English and also tokens common to Estonian and Russian. For this set of languages, this amounts to 20.69+10.06+1.39+0.0+8.89 = 41.03%. We list this number on a separate line in Table \[tab:enruet-breakdown\], “From parent”. These subwords get their embeddings trained better thanks to the parent model.
Languages Unique in a Lang. In All From Parent
------------- -------------------- -------- -------------
ET-EN-FI 24.4-18.2-26.2 19.5 49.4
ET-EN-RU 29.9-20.7-29.0 8.9 41.0
ET-EN-CS 29.6-17.5-21.2 20.3 49.2
AR-RU-ET-EN 28.6-27.7-21.2-9.1 4.6 6.2
ES-FR-ET-EN 15.7-13.0-24.8-8.8 18.4 34.1
ES-RU-ET-EN 14.7-31.1-21.3-9.3 6.0 21.4
FR-RU-ET-EN 12.3-32.0-22.3-8.1 6.3 23.1
: Summary of vocabulary overlaps for the various language sets. All figures in % of the shared vocabulary.[]{data-label="tab:vocab_stats"}
Table \[tab:vocab\_stats\] summarizes this analysis for several language sets, listing what portion of subwords is unique to individual languages in the set, what portion is shared by all the languages and what portion of subwords benefits from the parent training. We see a similar picture across the board, only AR-RU-ET-EN stands out with the very low number of subwords (6.2%) available already in the parent. The parent AR-RU thus offered very little word knowledge to the child and yet lead to a gain in BLEU.
Output Analysis
---------------
Since we rely on automatic analysis, we need to prevent some potential overestimations of translation quality due to BLEU. For this, we took a closer look at the baseline ENET model (BLEU of 17.03 in Table \[tab:highresourceparent\]) and two ENET childs derived from ENCS (BLEU of 20.41) and ENRU parent (BLEU 20.09).
Table \[tab:enet-details\] confirms the improvements are not an artifact of uncased BLEU. The gains are apparent with several (now cased) automatic scores.
BLEU nPER nTER nCDER chrF3 nCharacTER
----------- ------- ------- ------- ------- ------- ------------
Base ENET 16.13 47.13 32.45 36.41 48.38 33.23
ENRU+ENET 19.10 50.87 36.10 39.77 52.12 39.39
ENCS+ENET 19.30 51.51 36.84 40.42 52.71 40.81
: Various automatic scores on ENET test set. Scores prefixed “n” reported as $(1-\text{score})$ to make higher numbers better.[]{data-label="tab:enet-details"}
As documented in Table \[tab:bleu-comps\], the improved outputs are considerably longer. In the table, we show also individual $n$-gram precisions and brevity penalty (BP) of BLEU. The longer output clearly helps to reduce the incurred BP but the improvements are also apparent in $n$-gram precisions. In other words, the observed gain cannot be attributed solely to producing longer outputs.
Length BLEU Components BP
----------- -------- -------------------- -------
Base ENET 35326 48.1/21.3/11.3/6.4 0.979
ENRU+ENET 35979 51.0/24.2/13.5/8.0 0.998
ENCS+ENET 35921 51.7/24.6/13.7/8.1 0.996
: Candidate total length, BLEU $n$-gram precisions and brevity penalty (BP). The reference length in the matching tokenization was 36062.[]{data-label="tab:bleu-comps"}
ENRU+ENET ENCS+ENET
------- ----------------- -----------------
rb 15902 (44.2 %) 15924 (44.3 %)
- 9635 (26.8 %) 9485 (26.4 %)
b 7209 (20.0 %) 7034 (19.6 %)
r 3233 (9.0 %) 3478 (9.7 %)
Total 35979 (100.0 %) 35921 (100.0 %)
: Comparison of improved outputs vs. the baseline and reference.[]{data-label="tab:token-anots"}
Table \[tab:token-anots\] explains the gains in unigram precisions by checking which tokens in the improved outputs (the parent followed by the child) were present also in the baseline (child-only, denoted “b” in Table \[tab:token-anots\]) and/or confirmed by the reference (denoted “r”). We see that about 44+20% of tokens of improved outputs can be seen as “unchanged" compared to the baseline because they appear already in the baseline output (“b”). (The 44% “rb” tokens are actually confirmed by the reference.)
The differing tokens are more interesting: “-” denotes the cases when the improved system produced something different from the baseline and also from the reference. Gains in BLEU are due to “r” tokens, i.e. tokens only in the improved outputs and the reference but not the baseline “b”. For both parent setups, there are about 9–9.7 % of such tokens. We looked at these 3.2k and 3.5k tokens and we have to conclude that these are regular *Estonian* words; no Czech or Russian leaks to the output and the gains are *not* due to simple token types common to all the languages (punctuation, numbers or named entities). We see identical BLEU gains even if we remove all such simple tokens from the candidates and references. A better explanation of the gains thus still has to be sought for.
Related Work
============
propose multi-way multi-lingual systems, with the main goal of reducing the total number of parameters needed to cater multiple source and target languages. To keep all the language pairs “active” in the model, a special training schedule is needed. Otherwise, catastrophic forgetting would remove the ability to translate among the languages trained earlier.
is another multi-lingual approach: all translation pairs are simply used at once and the desired target language is indicated with a special token at the end of the source side. The model implicitly learns translation between many languages and it can even translate among language pairs never seen together.
Lack of parallel data can be tackled by unsupervised translation . The general idea is to mix monolingual training of autoencoders for the source and target languages with translation trained on data translated by the previous iteration of the system.
When no parallel data are available, the trainset of closely related high-resource pair can be used with transliteration approach as described in .
Aside from the common back-translation , simple copying of target monolingual data back to source has been also shown to improve translation quality in low-data conditions.
Similar to transfer learning is also curriculum learning , where the training data are ordered from foreign out-of-domain to the in-domain training examples.
Conclusion
==========
We presented a simple method for transfer learning in neural machine translation based on training a parent high-resource pair followed a low-resource language pair dataset. The method works for shared source or target side as well as for language pairs that do not share any of the translation sides. We observe gains also from totally unrelated language pairs, although not always significant.
One interesting trick we propose for low-resource languages is to start training in the opposite direction and swap to the main one afterwards.
The reasons for the gains are yet to be explained in detail but our observations indicate that the key factor is the size of the parent corpus rather than e.g. vocabulary overlaps.
Acknowledgments {#acknowledgments .unnumbered}
===============
This study was supported in parts by the grants SVV 260 453, GAUK 8502/2016, and 18-24210S of the Czech Science Foundation. This work has been using language resources and tools stored and distributed by the LINDAT/CLARIN project of the Ministry of Education, Youth and Sports of the Czech Republic (projects LM2015071 and OP VVV VI CZ.02.1.01/0.0/0.0/16 013/0001781).
[^1]: <http://www.statmt.org/wmt18/translation-task.html>
[^2]: Having separate vocabularies for the parent and child and switching from the XX-YY to AA-BB vocabulary when we switch the training corpus leads on an expected drop in performance. Independent vocabularies use different IDs even for identical subwords and the network cannot rely on any of its weights from the parent training.
[^3]: <http://www.statmt.org/wmt18/>
[^4]: <https://github.com/moses-smt/mosesdecoder>
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
In this paper, we address the problem of enhancing the speech of a speaker of interest in a cocktail party scenario when visual information of the speaker of interest is available.
Contrary to most previous studies, we do not learn visual features on the typically small audio-visual datasets, but use an already available face landmark detector (trained on a separate image dataset).
The landmarks are used by LSTM-based models to generate time-frequency masks which are applied to the acoustic mixed-speech spectrogram. Results show that: *(i)* landmark motion features are very effective features for this task, *(ii)* similarly to previous work, reconstruction of the target speaker’s spectrogram mediated by masking is significantly more accurate than direct spectrogram reconstruction, and *(iii)* the best masks depend on both motion landmark features and the input mixed-speech spectrogram.
To the best of our knowledge, our proposed models are the first models trained and evaluated on the limited size GRID and TCD-TIMIT datasets, that achieve speaker-independent speech enhancement in a multi-talker setting.
address: |
$^{\star}$Department of Engineering “Enzo Ferrari”, University of Modena and Reggio Emilia, Modena, Italy\
$^{\dagger}$Istituto Italiano di Tecnologia, Ferrara, Italy
bibliography:
- 'references.bib'
title: 'Face Landmark-based Speaker-Independent Audio-Visual Speech Enhancement in Multi-Talker Environments'
---
audio-visual speech enhancement, cocktail party problem, time-frequency mask, LSTM, face landmarks
Introduction {#sec:intro}
============
In the context of speech perception, the *cocktail party effect* [@cocktail_party; @mcdermott] is the ability of the brain to recognize speech in complex and adverse listening conditions where the attended speech is mixed with competing sounds/speech.
Speech perception studies have shown that watching speaker’s face movements could dramatically improve our ability at recognizing the speech of a target speaker in a multi-talker environment [@ZionGolumbic1417; @Ma_Wei].
This work aims at extracting the speech of a target speaker from single channel audio of several people talking simultaneously. This is an ill-posed problem in that many different hypotheses about what the target speaker says are consistent with the mixture signal. Yet, it can be solved by exploiting some additional information associated to the speaker of interest and/or by leveraging some prior knowledge about speech signal properties (e.g., [@bregman]). In this work we use face movements of the target speaker as additional information.
This paper *(i)* proposes the use of face landmark’s movements, extracted using Dlib [@Kazemi_2014_CVPR; @dlib09] and *(ii)* compares different ways of mapping such visual features into time-frequency (T-F) masks, then applied to clean the acoustic mixed-speech spectrogram.
By using Dlib extracted landmarks we relieve our models from the task of learning useful visual features from raw pixels. That aspect is particularly relevant when the training audio-visual datasets are small.
The analysis of landmark-dependent masking strategies is motivated by the fact that speech enhancement mediated by an explicit masking is often more effective than mask-free enhancement [@yuxuan_wang_training_2014].
All our models were trained and evaluated on the GRID [@cooke_audio-visual_2006] and TCD-TIMIT [@harte_tcd-timit:_2015] datasets in a speaker-independent setting.
Related work
------------
Speech enhancement aims at extracting the voice of a target speaker, while speech separation refers to the problem of separating each sound source in a mixture. Recently proposed audio-only single-channel methods have achieved very promising results [@DANet17; @Isik2016SingleChannelMS; @Kolbaek17]. However the task still remains challenging. Additionally, audio-only systems need separate models in order to associate the estimated separated audio sources to each speaker, while vision easily allow that in a unified model.
Regarding audio-visual speech enhancement and separation methods an extensive review is provided in [@rivet:hal-00990000]. Here we focus on the deep-learning methods that are most related to the present work.
Our first architecture (Section \[ssec:vidland2mask\]) is inspired by [@gabbay_seeing_2017], where a pre-trained convolutional neural network (CNN) is used to generate a clean spectrogram from silent video [@ephrat2017improved]. Rather than directly computing a time-frequency (T-F) mask, the mask is computed by thresholding the estimated clean spectrogram. This approach is not very effective since the pre-trained CNN is designed for a different task (video-to-speech synthesis). In [@gabbay2018visual] a CNN is trained to directly estimate clean speech from noisy audio and input video. A similar model is used in [@hou_audio-visual_2018], where the model jointly generates clean speech and input video in a denoising-autoender architecture.
[@hou_audio-visual_2016] shows that using information about lip positions can help to improve speech enhancement. The video feature vector is obtained computing pair-wise distances between any mouth landmarks. Similarly to our approach their visual features are not learned on the audio-visual dataset but are provided by a system trained on different dataset. Contrary to our approach, [@hou_audio-visual_2016] uses position-based features while we use motion features (of the whole face) that in our experiments turned out to be much more effective than positional features.
Although the aforementioned audio-visual methods work well, they have only been evaluated in a speaker-dependent setting. Only the availability of new large and heterogeneous audio-visual datasets has allowed the training of deep neural network-based speaker-independent speech enhancement models [@ephrat_looking_2018; @afouras_conversation:_2018; @owens2018audio].
The present work shows that huge audio-visual datasets are not a necessary requirement for speaker-independent audio-visual speech enhancement. Although we have only considered datasets with simple visual scenarios (i.e., the target speaker is always facing the camera), we expect our methods to perform well in more complex scenarios thanks to the robust landmark extraction.
MODEL ARCHITECTURES {#sec:model}
===================
We experimented with the four models shown in Fig. \[fig:model\]. All models receive in input the target speaker’s landmark motion vectors and the power-law compressed spectrogram of the single-channel mixed-speech signal. All of them perform some kind of masking operation.
VL2M model {#ssec:vidland2mask}
----------
At each time frame, the video-landmark to mask (VL2M) model (Fig. \[fig:model\]a) estimates a T-F mask from visual features only (of the target speaker). Formally, given a video sequence $\textbf{v} = [\textbf{v}_1, \dots ,\textbf{v}_T], \, \textbf{v}_t \in \mathbb{R}^n$ and a target mask sequence $\textbf{m} = [\textbf{m}_1, \dots ,\textbf{m}_T], \, \textbf{m}_t \in \mathbb{R}^d$, VL2M perform a function $\mathcal{F}_{vl2m}(\textbf{v}) = \mathbf{\hat{m}}$, where $\mathbf{\hat{m}}$ is the estimated mask.
The training objective for VL2M is a Target Binary Mask (TBM) [@Anzalone2006; @Kjems2009], computed using the spectrogram of the target speaker only. This is motivated by our goal of extracting the speech of a target speaker as much as possible independently of the concurrent speakers, so that, e.g., we do not need to estimate their number. An additional motivations is that the model takes as only input the visual features of the target speaker, and a target TBM that only depends on the target speaker allows VL2M to learn a function (rather than approximating an ill-posed one-to-many mapping). Given a clean speech spectrogram of a speaker $\mathbf{s}=[\mathbf{s}_1, \dots, \mathbf{s}_T], \, \mathbf{s}_t \in \mathbb{R}^d$, the TBM is defined by comparing, at each frequency bin $f \in [1, \dots, d]$, the target speaker value $\textbf{s}_t[f]$ vs. a reference threshold $\tau[f]$. As in [@gabbay_seeing_2017], we use a function of long-term average speech spectrum (LTASS) as reference threshold. This threshold indicates if a T-F unit is generated by the speaker or refers to silence or noise. The process to compute the speaker’s TBM is as follows:
1. The mean $\pi[f]$ and the standard deviation $\sigma[f]$ are computed for all frequency bins of all seen spectrograms in speaker’s data.
2. The threshold $\tau[f]$ is defined as $\tau[f] = \pi[f] + 0.6 \cdot \sigma[f]$ where $0.6$ is a value selected by manual inspection of several spectrogram-TBM pairs.
3. The threshold is applied to every speaker’s speech spectrogram $\mathbf{s}$. $$\mathbf{m}_t[f] = \left\{
\begin{array}{ll}
1, & \text{if $\mathbf{s}_t[f] \geq \tau[f]$,} \\
0, & \text{otherwise.} \\
\end{array}
\right.$$
The mapping $\mathcal{F}_{vl2m}(\cdot)$ is carried out by a stacked bi-directional Long Short-Term Memory (BLSTM) network [@graves13]. The BLSTM outputs are then forced to lay within the $[0,1]$ range. Finally the computed TBM $\mathbf{\hat{m}}$ and the noisy spectrogram $\mathbf{y}$ are element-wise multiplied to obtain the estimated clean spectrogram $\mathbf{\hat{s}^m} = \mathbf{\hat{m}} \circ \mathbf{y}$, where $\textbf{y}=[\textbf{y}_1, \dots \textbf{y}_T], \, \textbf{y}_t \in \mathbb{R}^d$.
The model parameters are estimated to minimize the loss: $$\scalebox{0.78}[1]{$J_{vl2m} = \sum_{t=1}^T \sum_{f=1}^d -\mathbf{m}_t[f] \cdot \log(\mathbf{\hat{m}}_t[f]) - (1-\mathbf{m}_t[f]) \cdot \log(1-\mathbf{\hat{m}}_t[f])$} \nonumber$$
VL2M\_ref model {#ssec:full_model}
---------------
VL2M generates T-F masks that are independent of the acoustic context. We may want to refine the masking by including such context. This is what the novel VL2M\_ref does (Fig. \[fig:model\]b). The computed TBM $\mathbf{\hat{m}}$ and the input spectrogram $\mathbf{y}$ are the input to a function that outputs an Ideal Amplitude Mask (IAM) $\mathbf{p}$ (known as FFT-MASK in [@yuxuan_wang_training_2014]). Given the target clean spectrogram $\mathbf{s}$ and the noisy spectrogram $\mathbf{y}$, the IAM is defined as: $$\mathbf{p}_t[f] = \frac{\mathbf{s}_t[f]}{\mathbf{y}_t[f]}$$ Note that although IAM generation requires the mixed-speech spectrogram, separate spectrograms for each concurrent speakers are not required.
The target speaker’s spectrogram $\mathbf{s}$ is reconstructed by multiplying the input spectrogram with the estimated IAM. Values greater than $10$ in the IAM are clipped to $10$ in order to obtain better numerical stability as suggested in [@yuxuan_wang_training_2014].
The model performs a function $\mathcal{F}_{mr}(\textbf{v},\textbf{y}) = \mathbf{\hat{p}}$ that consists of a VL2M component plus three different BLSTMs $\mathcal{G}_m$, $\mathcal{G}_y$ and $\mathcal{H}$.
$\mathcal{G}_m(\mathcal{F}_{vl2m}(\textbf{v})) = \textbf{r}_m$ receives the VL2M mask ${\mathbf{\hat{m}}}$ as input, and $\mathcal{G}_y(\textbf{y}) = \textbf{r}_y$ is fed with the noisy spectrogram. Their output $\textbf{r}_m, \textbf{r}_y \in \mathbb{R}^z$ are fused in a joint audio-visual representation $\mathbf{h}=[\textbf{h}_1, \dots , \textbf{h}_T]$, where $\textbf{h}_t$ is a linear combination of $\textbf{r}_{m_t}$ and $\textbf{r}_{y_t}$: $\mathbf{h}_t = \textbf{W}_{hm} \cdot \textbf{r}_{m_t} + \textbf{W}_{hy} \cdot \textbf{r}_{y_t} + \textbf{b}_h$. $\mathbf{h}$ is the input of the third BLSTM $\mathcal{H}(\mathbf{h})=\mathbf{\hat{p}}$, where $\mathbf{\hat{p}}$ lays in the \[0,10\] range. The loss function is: $$J_{mr} = \sum_{t=1}^T \sum_{f=1}^d (\mathbf{\hat{p}}_t[f] \cdot \mathbf{y}_t[f] - \mathbf{s}_t[f])^2$$
Audio-Visual concat model {#ssub:concat_model}
-------------------------
The third model (Fig. \[fig:model\]c) performs early fusion of audio-visual features. This model consists of a single stacked BLSTM that computes the IAM mask $\mathbf{\hat{p}}$ from the concatenated $[\mathbf{v},\mathbf{y}]$. The training loss is the same $J_{mr}$ used to train VL2M\_ref. This model can be regarded as a simplification of VL2M\_ref, where the VL2M operation is not performed.
Audio-Visual concat-ref model {#ssub:concat_ref_model}
-----------------------------
The fourth model (Fig. \[fig:model\]d) is an improved version of the model described in section \[ssub:concat\_model\]. The only difference is the input of the stacked BLSTM that is replaced by $[\mathbf{\hat{s}^m},\mathbf{y}]$ where $\mathbf{\hat{s}^m}$ is the denoised spectrogram returned by VL2M operation.
Experimental setup
==================
Dataset
-------
All experiments were carried out using the GRID [@cooke_audio-visual_2006] and TCD-TIMIT [@harte_tcd-timit:_2015] audio-visual datasets. For each of them, we created a mixed-speech version.
Regarding the GRID corpus, for each of the $33$ speakers (one had to be discarded) we first randomly selected $200$ utterances (out of $1000$). Then, for each utterance, we created $3$ different audio-mixed samples. Each audio-mixed sample was created by mixing the chosen utterance with one utterance from a different speaker.
That resulted in $600$ audio-mixed samples per speaker.
The resulting dataset was split into disjoint sets of $25$/$4$/$4$ speakers for training/validation/testing respectively.
The TCD-TIMIT corpus consists of $59$ speakers (we excluded $3$ professionally-trained lipspeakers) and $98$ utterances per speaker. The mixed-speech version was created following the same procedure as for GRID, with one difference. Contrary to GRID, TCD-TIMIT utterances have different duration. Thus $2$ utterances were mixed only if their duration difference did not exceed $2$ seconds. For each utterance pair, we forced the non-target speaker’s utterance to match the duration of the target speaker utterance. If it was longer, the utterance was cut at its end, whereas if it was shorter, silence samples were equally added at its start and end.
The resulting dataset was split into disjoint sets of $51$/$4$/$4$ speakers for training/validation/testing respectively.
LSTM training
-------------
In all experiments, the models were trained using the Adam optimizer [@adam]. Early stopping was applied when the error on the validation set did not decrease over $5$ consecutive epochs.
VL2M, AV concat and AV concat-ref had $5$, $3$ and $3$ stacked BLSTM layers respectively. All BLSTMs had $250$ units. Hyper-parameters selection was performed by using random search with a limited number of samples, therefore all the reported results may improve through a deeper hyper-parameters validation phase.
VL2M\_ref and AV concat-ref training was performed in $2$ steps. We first pre-trained the models using the oracle TBM $\mathbf{m}$. Then we substituted the oracle masks with the VL2M component and retrained the models while freezing the parameters of the VL2M component.
Audio pre- and post-processing
------------------------------
The original waveforms were resampled to 16 kHz. Short-Time Fourier Transform (STFT) $\mathbf{x}$ was computed using FFT size of 512, Hann window of length 25 ms (400 samples), and hop length of 10 ms (160 samples). The input spectrogram was obtained taking the STFT magnitude and performing power-law compression $\mathbf{\lvert x \rvert}^p$ with $p=0.3$. Finally we applied per-speaker 0-mean 1-std normalization.
In the post-processing stage, the enhanced waveform generated by the speech enhancement models was reconstructed by applying the inverse STFT to the estimated clean spectrogram and using the phase of the noisy input signal.
Video pre-processing
--------------------
Face landmarks were extracted from video using the Dlib [@dlib09] implementation of the face landmark estimator described in [@Kazemi_2014_CVPR]. It returns 68 x-y points, for an overall 136 values. We upsampled from 25/29.97 fps (GRID/TCD-TIMIT) to $100$ fps to match the frame rate of the audio spectrogram. Upsampling was carried out through linear interpolation over time.
The final video feature vector $\mathbf{v}$ was obtained by computing the per-speaker normalized motion vector of the face landmarks by simply subtracting every frame with the previous one. The motion vector of the first frame was set to zero.
[lSSS]{} & [SDR]{} & [PESQ]{} & [ViSQOL]{}\
Noisy & -1.06 & 1.81 & 2.11\
VL2M & 3.17 & 1.51 & 1.16\
VL2M\_ref & $\; \; \mathbf{6.50}$ & $\; \; \mathbf{2.58}$ & $\; \; \mathbf{2.99}$\
AV concat & 6.31 & 2.49 & 2.83\
AV c-ref & 6.17 & $\; \; \mathbf{2.58}$ & 2.96\
[lS @ S @S|S @ S @ S]{} & &\
& [SDR]{} & [PESQ]{} & [ViSQOL]{} & [SDR]{} & [PESQ]{} & [ViSQOL]{}\
Noisy & 0.21 & 1.94 & 2.58 & -5.34 & 1.43 & 1.62\
VL2M & 3.02 & 1.81 & 1.70 & -2.03 & 1.43 & 1.25\
VL2M\_ref & 6.52 & 2.53 & 3.02 & 2.83 & 2.19 & 2.53\
AV concat & 7.37 & 2.65 & 3.03 & 3.02 & 2.24 & 2.49\
AV c-ref & $ \; \; \mathbf{8.05}$ & $ \; \; \mathbf{2.70}$ & $ \; \; \mathbf{3.07}$ & $ \; \; \mathbf{4.02}$ & $ \; \; \mathbf{2.33}$ & $ \; \; \mathbf{2.64}$\
[lS @ S @S|S @ S @ S]{} & &\
& [SDR]{} & [PESQ]{} & [ViSQOL]{} & [SDR]{} & [PESQ]{} & [ViSQOL]{}\
Noisy & 0.21 & 2.22 & 2.74 & -3.42 & 1.92 & 2.04\
VL2M & 2.88 & 2.25 & 2.62 & -0.51 & 1.99 & 1.98\
VL2M\_ref & 9.24 & 2.81 & 3.09 & 5.27 & 2.44 & 2.54\
AV concat & 9.56 & 2.80 & 3.09 & 5.15 & 2.41 & 2.52\
AV c-ref & $ \; \; \mathbf{10.55}$ & $ \; \; \mathbf{3.03}$ & $ \; \; \mathbf{3.21}$ & $ \; \; \mathbf{5.37}$ & $ \; \; \mathbf{2.45}$ & $ \; \; \mathbf{2.58}$\
Results
=======
In order to compare our models to previous works in both speech enhancement and separation, we evaluated the performance of the proposed models using both speech separation and enhancement metrics. Specifically, we measured the capability of separating the target utterance from the concurrent utterance with the source-to-distortion ratio (SDR) [@vincent_performance_2006; @raffel2014mir_eval]. While the quality of estimated target speech was measured with the perceptual PESQ [@rix_perceptual_2001] and ViSQOL [@hines_visqol:_2012] metrics. For PESQ we used the narrow band mode while for ViSQOL we used the wide band mode.
As a very first experiment we compared landmark position vs. landmark motion vectors. It turned out that landmark positions performed poorly, thus all results reported here refer to landmark motion vectors only.
We then carried out some speaker-dependent experiments to compare our models with previous studies as, to the best of our knowledge, there are no reported results of speaker-independent systems trained and tested on GRID and TCD-TIMIT to compare with. Table \[tab:grid\_spk\_dep\] reports the test-set evaluation of speaker-dependent models on the GRID corpus with landmark motion vectors. Results are comparable with previous state-of-the-art studies in an almost identical setting [@gabbay_seeing_2017; @gabbay2018visual].
Table \[tab:grid\] and \[tab:tcdtimit\] show speaker-independent test-set results on the GRID and TCD-TIMIT datasets respectively. V2ML performs significantly worse than the other three models indicating that a successful mask generation has to depend on the acoustic context. The performance of the three models in the speaker-independent setting is comparable to that in the speaker-dependent setting.
AV concat-ref outperforms V2ML\_ref and AV concat for both datasets. This supports the utility of a refinement strategy and suggests that the refinement is more effective when it directly refines the estimated clean spectrogram, rather than refining the estimated mask.
Finally, we evaluated the systems in a more challenging testing condition where the target utterance was mixed with $2$ utterances from $2$ competing speakers. Despite the model was trained with mixtures of two speakers, the decrease of performance was not dramatic.
Code and some testing examples of our models are available at <https://goo.gl/3h1NgE>.
Conclusion
==========
This paper proposes the use of face landmark motion vectors for audio-visual speech enhancement in a single-channel multi-talker scenario. Different models are tested where landmark motion vectors are used to generate time-frequency (T-F) masks that extract the target speaker’s spectrogram from the acoustic mixed-speech spectrogram.
To the best of our knowledge, some of the proposed models are the first models trained and evaluated on the limited size GRID and TCD-TIMIT datasets that accomplish speaker-independent speech enhancement in the multi-talker setting, with a quality of enhancement comparable to that achieved in a speaker-dependent setting.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The measurements of the Doppler shifts of the Fraunhofer lines, scattered by the dust grains in the solar F-corona, provides the insight on the velocity field of the dust and hence on its origin. We report on such measurements obtained during the total eclipse of March 29, 2006. We used a Fabry-Pérot interferometer with the FOV of $5.9\degr$ and the spectral resolution of about 5000 to record Fraunhofer spectral lines scattered by the dust of the F-Corona. The spectral region was centered on the $5172.69 \rmn{\AA}$ line. The measured line-on-sight velocities with the amplitude in the range from -10 to 10 km$\cdot \rmn{s^{-1}}$ show that during our observations the dust grains were on the orbit with a retrograde motion in a plane nearly perpendicular to the ecliptics. This indicates their cometary origin. Indeed, at the end of March, 2006, SOHO recorded several sungrazing comets with the orbital elements close to what was deduced from our measurements. We conclude that the contribution of comets to the dust content in the region close to the Sun can be more important albeit variable in time. We also deduce that the size of the most of the dust grains during our observations was less than 0.1 $\umu$m.'
author:
- |
L. I. Shestakova$^{1}$, A. Chalabaev$^{2}$, B. I. Demchenko$^{1}$, and F. K. Rspaev$^{1}$ [^1]\
$^{1}$Kazakh National Space Agency, Fesenkov Astrophysical Institute, Kamenskoie Plato, Observatory 23, Almaty, 050020, Kazakhstan\
$^{2}$Laboratoire d’Astrophysique de Grenoble, UMR 5571, CNRS, Université Joseph-Fourier, BP 53X, Grenoble, CEDEX 09, France
bibliography:
- 'eclipse06.bib'
date: 'Accepted xx. Received 2010 March 8; in original form 2010 xx xx'
title: 'The velocity of the dust near the Sun during the Solar Eclipse of March 29, 2006 and sungrazing comets'
---
\[firstpage\]
zodiacal dust – Sun: general – comets: general – (stars:) circumstellar matter – techniques: imaging spectroscopy
Introduction
============
The work of @hulst47 established, after an earlier suggestion by Grotrian [as cited by @hulst47], that the emission of the solar outer corona, called F-corona after the Fraunhofer lines composing its spectrum, is due to the scattering of the solar light by the dust particles.
In recent decennia it became more clear that the dust component near the Sun cannot be described as a mere extrapolation of the zodiacal light disc with the dust grains orbiting in the plane of the ecliptics and slowly drifting to the Sun under the Poynting-Robertson effect as one could accept in a first approximation.
The measurements of the Doppler shifts of the Fraunhofer lines in the spectrum of the F-corona, first done during the total eclipse of July 31, 1981 by @scheglov87, and the analysis of the derived velocities by @shest87, indicated that, although most of the dust particles are orbiting in the ecliptic plane, there are also grains orbiting at high values of inclination angle $i$, implying that the region close to the Sun receives a dust contribution from long-period comets. Also, the *in situ* measurements of the dust particle velocity distribution on board of HELIOS-1 showed the existence of two distinct components in the interplanetary dust, cometary and asteroidal [@grun80]. Among more recent results, the high frequency of the Sun-grazing comets recorded by SOHO [@macqueen91] makes also to consider the cometary contribution as important. Furthermore, studies of extrasolar planetary and dust systems showed the existence of Falling Evaporating Bodies, possibly comets [e.g. @beust98]. This observational progress was followed by a thorough theoretical modelling taking into account complex composition of the dust in the solar vicinity (see [@mann00], @koba08).
It would have been important to obtain further measurements of the line-on-sight (thereafter LOS) velocities of the dust near the Sun to have a better idea of the relative contributions of the cometary and the ecliptic disc dust components and of their possible variation in time. After the first work of 1981, these measurements, to the best of our knowledge, were successfully attempted only once, during the eclipse of July 11, 1991 [@aimanov95] confirming the first results.
Here we present new higher quality measurements of the LOS velocity of the dust in the F-corona obtained during the eclipse on March 29, 2006 [see @shest07 for a preliminary report], and discuss their implications.
{width="18cm"}
Observations
============
The site of the observations was the village of Mugalzhar in the Aktobe region of the Republic of Kazakhstan, at $\phi = 48\degr 35\arcmin$ and $\lambda = 58\degr 27\arcmin$, situated in the middle of the totality band. According to our calculations, the beginning of the total eclipse was at UT 11h 32m 40, its end at 11h 35m 30s, and the duration of the total phase was 170s. The Sun was at $27\fdg 5$ above the horizon. The weather was mostly rainy and windy during the days preceding the eclipse, but on the eclipse day, the sky was clear with no wind and excellent transparency.
We used the same optical set-up as in our previous measurements in 1991 [@scheglov87], i.e. a Fabry-Pérot (hereafter FP) spectrometer with a coronographic mask rejecting the light of the solar corona and thus reducing the background. The entrance lens of 10 cm diameter has its focal plane at the field lense, the latter is followed by a collimator. The FP etalon in series with an order separation filter is placed in the parallel beam close the exit pupil imaged by the field lense. It is followed by a photographic objective and a CCD detector. The latter was an Apogee Alta-10 CCD device with $2048^{2}$ pixels of 14 $\umu$m size. During the data reduction, the frames were binned by $2\times 2$ pixels, so that to avoid any confusion we will refer from now on to the detector format of $1024^{2}$ pixels with the 28 $\umu$m pixel size.
The solar angular radius at the time of the eclipse was $\mathcal{R}_{\sun} = 961$ arcsec, the measured on the CCD value was 46.5pix. This gives the equivalent focal length of 279.4 mm, and the platescale of 20.66 arcsec/pixel. The field of view (FOV) is $5.9\degr$, corresponding to the elongations $\epsilon < 11$, however, to reduce the background light, the central region around the Sun, $\epsilon < 2.6$, is hidden by the coronographic mask (hereafter, the elongation $\epsilon$ is given in angular solar radii $\mathcal{R}_{\sun}$).
{width="14"}
Spectro-imaging
---------------
We used a mechanically adjustable FP etalon with the free space $\Delta=70\,\umu$m. The order separation filter has the FWHM = $10\,\rmn{\rmn{\AA}}$ with the central wavelength close to that of the line at 5172.69 $\rmn{\AA}$. In the wavelength space, the set-up is such that a frame includes 4 FP orders. The highest, on-the-axis order $N=\Delta/\lambda$, is at the centre of the frame and is unseen due to the use of the coronographic mask; the orders lower than $N-4$ are outside of the FOV.
The dark subtracted frames obtained during the eclipse are shown in Fig.\[fig:fig1\]. The left frame, denoted as “D”, is one of the calibration interferograms of the daylight sky, scattered on a white screen, recorded shortly before and after the total phase of the eclipse. The use of the white screen allows a homogenous brightness distribution over the field of view without changing the spectrometer position which remains at the same position as during the total phase. The frame shows concentric rings of the Fraunhofer absorption lines scattered in the terrestrial atmosphere.
The dark circle at the centre of the frame corresponds to the coronographic mask. The dark line corresponds to a fiducial wire indicating the East-West axis, the West to the left along the wire and the North down. For convenience of interpretation, the frames are rotated so that the frames horizontal axis coincides with the ecliptic plane. The frames are dark current subtracted.
The frame “E”in the middle of the Fig.\[fig:fig1\] is our main scientific exposure, of 130s, taken on the circumsolar region during the eclipse. It was started a few seconds after the beginning of the totality and stopped well before its end, so that contamination by coronal or chromospheric lines was avoided. The concentric dark rings are the solar Fraunhofer lines scattered on the dust particles of the F-corona.
Finally, the frame “L”at the right of the Fig.\[fig:fig1\], with the exposure time of 20s was started close to the end of the total phase and lasted a few seconds beyond, so that it caught the light of the solar corona which was just appearing from behind the mask (but not yet the photosphere). On this frame, one can see, additionally to the absorption spectral features of the F-corona, the bright emission rings of the green coronal \[Fe XIV\] line at $\lambda 5302.86
\rmn{\AA}$. Its wavelength lies far from the centre of the our narrow-band filter, however the emission is so strong that its light passed in the filter transmission wings. The corresponding interference rings well visible in Fig.\[fig:fig1\] traced the location of the 3 used Fabry-Pérot orders and gave useful reference points for the data reduction. Its measured $\rmn{FWHM} =
1.2\pm 0.1\rmn{\AA}$ gives the spectral resolution of the instrument.
The daylight sky interferograms “D” provided the wavelength standard and allowed to eventually measure the Doppler shift of the dust particles of the F-corona, while the interferogram of the coronal green line $\lambda 5302.86\rmn{\AA}$ allowed to calibrate the spectral geometry of the frames, and in particular to accurately define the rings centre as it will be discussed in Section\[Section:Reduction\].
{width="12"}
Brightness calibration
----------------------
The weather conditions being excellent and stable, the Fabry-Pérot data were calibrated in brightness relative to that of the solar disc, $B_{\sun}$. Two 0.1s exposures on the Sun through a combination of a neutral and green filters were taken to the East, at $\epsilon =
4.0$, and to the West, at $\epsilon = 4.4$ from the Sun position at the total phase of the eclipse. The Sun elevation for the calibration and for the eclipse frames was nearly the same, about $30\degr$. The uncovered surface of the solar disc was 0.94 and 0.25 respectively for the two exposures. Taking this into account, the daylight sky brightness off the eclipse is estimated to has been $4\cdot
10^{-5}B_{\sun}$.
The brightness of the F-corona continuum emission, $B_{F}$, was measured to be $4.1\cdot 10^{-9}B_{\sun}$ at the elongation $\epsilon
= 4.0$, it was $3.9\cdot 10^{-9}B_{\sun}$ at $\epsilon = 4.4$, and $1.7\cdot 10^{-9}B_{\sun}$ for the $\epsilon$ range from 9 to 10. The brightness decrement in the F-corona is in agreement with the results of @koutchmy78. Compared to the daylight sky, the F-corona is fainter by a factor of $10^{4}$.
Data reduction {#Section:Reduction}
==============
The extraction of the Doppler shift of the solar absorption lines out of the Fabry-Pérot interferograms needs a thorough data reduction process. It is worthwhile to be describe here in details.
Noise and trends handling
--------------------------
After the bias and dark current subtraction, all the frames were $2\times 2$ binned, which increased the signal-to-noise ratio without changing spectral or spatial resolution. The next step was defining all regions meaningless for further reduction, which includes the central dark region corresponding to the coronographic mask, that of the fiducial wire, the part lying out of the field-of-view of the detector and the regions of strong light induced by spurious reflections. The resulting numerically masked area was about 30%.
The “hot", “cold" and “dead" pixels were cleaned out using a median filter. The visual analysis showed that the “clouds" of defective pixels did not exceed a region of 10-12 pixels a size. We used therefore a non-linear circular filter covering 31 neighbour pixels (i.e. 2n+1, where n is the maximum scale of “clouds"). The correction was applied only to the pixels with the count exceeding 20% of the median value. The number of corrected pixels was less than 4% on the eclipse frames and less than 0.5% on the calibration frames.
For convenience of interpretation, all frames were then rotated so that the horizontal axis coincides with the ecliptic plane.
The next step was the flat-fielding. Usually, it is done by a straightforward division by a averaged flat-field frame with a subsequent masking of the pixels resulting from the division by zero or close to zero values. We decided to optimize this operation by adding a small constant to the flat-field frame. This is similar in its spirit to the Tikhonov regularisation of ill-defined inverse problems. The value of the constant was defined by analysing the histogram of the counts of the flat-field frame. The aim of this operation is to keep moderate, or negligible, the change induced in the signal-to-noise ratio. Thus, the division on the flat-field does not change the structure of the frames and avoids a useless addition of a noise.
Further, we measured the two-dimensional low frequency trend of the resulting frames in order to subtract it and to keep only useful spectro-spatial variations. It was done by using iteratively a linear circular moving average filter. The best filter diameter and the number of iterations were searched by trials in such a way that the frame with the subtracted 2D trend would still keep the structures on a 10-15 pixels scale, which is that of the recorded Fraunhofer lines. The best filter had the diameter of 13 pixels, covering 137 closest pixels, and the best number of iterations was 4. Higher the number of iterations, closer the iterative filter to a linear gaussian smoothing filter, $exp[-(x{^2} + y{^2}/(2 \sigma_{g}^{2})]$, where $\sigma_{g}$ defines the degree of the smoothing. The advantage of using an iterative filter is the possibility to control the achieved smoothness by limiting the number of iterations.
We tried square and rectangular moving average filters, however they give rise to spur line-like features. We also applied the methods of high and low enveloping curves, but it did not give better results.
The FP interferogram after median filtering and subtraction of the 2D low frequency trend is shown in Fig.\[fig:fig2\].
Finally, the data were passed through the gaussian filter with $\sigma_{g} = 1.3$ pixels, or $FWHM\approx 3$ pixels which is close to the measured $FWHM$ of the point-spread function of the experiment.
---------- ------------------------ ----------------------- ---------
Radius Elongation $\epsilon$ Wavelength Element
(pixels) ($\mathcal{R}_{\sun}$) $\lambda (\rmn{\AA}$)
182 3.92 5168.91+5169.04 +
196 4.22 5167.33
220 4.72 5183.62
238 5.12 5162.28
266 5.72 5159.06
298 6.40 5172.69
328 7.06 5167.33
344 7.40 5183.62
358 7.68 5162.28
374 8.04 5159.06
402 8.64 5172.69
425 9.14 5167.33
448 9.64 5162.28
460 9.88 5159.06
485 10.42 5172.69
---------- ------------------------ ----------------------- ---------
\[tab:tab\_lines\]
Elongation range Mean $\epsilon$ Lines used
------------------ ----------------- ------------------------
3.12 - 4.09 3.66 5172.7, 5169.0
3.12 - 4.82 4.09 5169.0, 5167.3
4.09 - 4.82 4.52 5167.3, 5183.6
4.82 - 5.63 5.27 5183.6, 5167.3
5.63 - 6.37 6.02 5159.1, 5172.7
6.36 - 7.01 6.71 5172.7, 5167.3
7.01 - 7.74 7.40 5167.3, 5183.6, 5162.3
7.74 - 8.43 8.08 5162.3, 5159.1
8.43 - 9.12 8.77 5172.7, 5167.3
9.12 - 9.63 9.38 5167.3, 5162.3
9.63 - 10.25 9.98 5162.3, 5159.1, 5172.7
\[tab:tab\_range\]
Defining the interferograms centre
----------------------------------
In an ideal Fabry-Pérot interferogram, the wavelength $\lambda$ is constant on a circular ring, and varies with its radius as $r{^2}$. Let $X_{c}$ and $Y_{c}$ denote the coordinates of the centre of the interference rings. Their values depend on many optical parameters, which can vary with temperature and mechanical flexures, so that they must be carefully defined for each frame. After different trials, we found that the most reliable way to measure the centre position was to use a correlation method in the following way.
Let us denote $z=r^{2}$. We adopt a first guess of the centre coordinates $X_{0}$ and $Y_{0}$, and divide the frame on two sub-frames, left and right, symmetrically with the respect to $X_{0}$. For each sub-frame, we compute counts *vs* $z$, which gives us two functions $L(z)$ and $R(z)$ respectively for the left and the right sub-frames. We compute then the correlation of $L$ and $R$ , and vary $X_{0}$. The maximum of the correlation gives the value of $X_{0}$, which is adopted as $X_{c}$. In a similar way, we define the best value of $Y_{c}$, correlating the upper and the lower sub-frames.
Such a correlation measurement can give false and biased results if, for example, there is a strong asymmetry in the intensity of interference patterns. To have an additional check, we applied another method to smaller parts of the patterns, using only arcs of the rings. For a given ring, we take first guess values of $X_{c}$, $Y_{c}$, $r$, and the width of the ring $dr$. Varying the values of $X_{c}$, $Y_{c}$ and $r$, we find the values such that the sum of counts is the less (for absorption lines). To insure a good statistics, the value of $dr$ must be sufficiently large, but without covering neighbor ring patterns. We used values of $dr$ in the range from 2 to 12 pixels. The difference of the centre coordinates defined by this method and that of correlations did not exceed 0.3 pix, which is at the level of the expected uncertainty.
The resulting values of the center coordinates are defined with an accuracy better than 0.3 pix.
Reduced spectrograms
--------------------
Once the centre of the interference rings was found, it is convenient to transform the data presentation from cartesian to polar coordinates $(r, \phi)$. The frames were oriented so that the polar angle $\phi$ and the position angle on the sky, $PA$ are the same for all of them. The extracted spectrograms in the form of the flux integrated over all values of the position angle $PA$ in a function of the radius counted from the interferogram centre are plotted in Fig.\[fig:fig3\] for the daylight sky, the circumsolar region during the total eclipse and the frame “L" including the coronal $\mbox{Fe\,{\sc xiv}}, \lambda
5302.86 \rmn{\AA}$ line. The 3 pics of the coronal emission line trace the 3 Fabry-Pérot spectral orders. The relevant scattered $\mbox{Fe\,{\sc i}}$, $\mbox{Fe\,{\sc ii}}$ and $\mbox{Mg\,{\sc i}}$ Fraunhofer lines, in absorption, are indicated. Their wavelengths, the values of the ring radius $r$ and the corresponding value of the elongation $\epsilon$ are given in the Table\[tab:tab\_lines\]. Albeit the used Fabry-Pérot orders overlap, the spectral features, fortunately, are distinct and can be easily identified.
{width="12"}
{width="12"}
{width="12"}
The line-on-sight velocity of the dust.
----------------------------------------
The Doppler shift between the eclipse and the daylight spectrograms was measured by cross-correlation. The corresponding dust LOS velocity $V$ writes: $$V = \frac{c \Delta_{r}^{2}}{2f{^2}}
\label{eq:eq1}$$ or, substituting the numerical values: $$V= \frac{3\times 10^{6}\times 0.028{^2}\times
\Delta_{r}^2}{2\times 86.65{^2}} = 0.01566 \times \Delta_{r}^2
\label{eq:eq2}$$ where $f = 86.55$ mm is the focal distance of the camera objective, 0.028 mm is the pixel size, $\Delta_{r}^2 = r_{e}^{2} - r_{sky}^{2}$, where $r_{e}$ and $r_{sky}$ are the radii in pixels respectively for the eclipse and daylight sky frames. The pixels convert to the elongation $\epsilon$ given that $\mathcal{R}_{\sun} = 46.5$ pixels.
The sampling in elongation $\epsilon$ and in position angle $PA$ was as follows. We used intervals of $\epsilon$ as given in Tab.\[tab:tab\_range\] choosing their spanning as regular as possible in a function of the useful spectral lines. The full range of $PA$ was simply divided on 36 sectors $20\degr$ wide each.
For convenience of the analysis, we computed two kinds of average velocities, $\bar{V}_{\epsilon}$, which is the average over all values of $PA$, and $\bar{V}_{PA}$, which is the average over all values of the elongation $\epsilon$.
The value $\bar{V}_{\epsilon}$ is the function of $\epsilon$ only; it is plotted in Fig.\[fig:fig\_vel\_elong\]. If the LOS velocities of the dust grains are distributed in a central symmetry with the respect to the Sun, as it is expected for a pure circular motion, then $\bar{V}_{\epsilon}$ is 0; if not, it reflects the presence of a radial motion, an infall or outflow, at this particular value of $\epsilon$. The Fig.\[fig:fig\_vel\_elong\] shows that radial motions with velocities about $10\,\rmn{km}\cdot\rmn{s}^{-1}$ or less might be present at $\epsilon=4.5$ and $\epsilon\approx9$, and they are absent, or very small, in between these elongations.
The value $\bar{V}_{PA}$ is a function of the position angle $PA$ only, it is plotted in Fig.\[fig:fig\_vel\_pa\]. It is so particular that we give also for comparison a similar plot from the eclipse on July 31, 1981. For 2006, the triangle marks indicate measurements using two different daylight sky interferograms; their scattering provides an estimate of the uncertainty $\delta
\bar{V}_{PA}\simeq 1.6\,\rmn{km}\cdot\rmn{s}^{-1}$. The values for the $PA$ range $360\degr-450\degr$ are added for convenience, they merely repeat those for $0-90\degr$. Plotted are also the least squares $sinus$ fits in the form: $\bar{V}_{PA} = k_{1} + k_{2}\cdot
sin(k_{3}\cdot PA +k_{4})$. The fit coefficients, for 2006, are as follows: $k_{1} = 0.3\pm 0.3$, $k_{2} = 18.7\pm 0.3$, $k_{3} =
0.01773\pm 0.00016$ and $k_{4} = 1.22\pm 0.04$, and for 1981, $k_{1} =
8.8\pm 2.5$, $k_{2} = 22.0\pm 3.5$, $k_{3} = 0.019\pm 0.001$ and $k_{4} = 2.4\pm 0.3$.
Discussion {#Section:Discussion}
==========
------------ ------------- -------------- ------------ -------------- ---------------
Comet name Epoch Perihelion q Perihelion Longitude of Inclination i
in $AU$ argument the node
CK06F050 March 21.96 0.0050 82.38 4.20 144.57
CK06F060 March 23.04 0.0333 56.09 75.03 74.13
CK06F070 March 28.64 0.0050 84.83 3.42 145.72
CK06F080 March 31.10 0.0052 84.02 5.67 144.58
------------ ------------- -------------- ------------ -------------- ---------------
\[tab:tab\_comets\]
For the prograde orbits being strictly in the ecliptic plane, the minimum LOS velocity $\bar{V}_{PA}$ should be at $90\degr$, i.e.to the East from the Sun, and the maximum $\bar{V}_{PA}$ at $270\degr$, i.e.to the West. The velocity curve measured on July 31, 1981, is close to what is expected from such a motion (with, possibly, a slight difference, the minimum of $\bar{V}_{PA}$ being at $120\degr$ and the maximum at $300\degr$).
As to the $\bar{V}_{PA}$ curve on March 29, 2006, the values of $PA$ of the extrema differ dramatically from what is expected from the prograde motion in the ecliptics, namely the minimum $\bar{V}_{PA}$ is at $195\degr,$ and the maximum is at $15\degr$, meaning a retrograde motion in a plane nearly perpendicular to the ecliptics.
The dust rotating in the prograde direction in the ecliptics is also barely present showing the “jump” of $\bar{V}_{PA}$ at $PA \simeq
120\degr$ and the symmetric to it another “jump” at $PA \simeq
300\degr$ (see Fig.\[fig:fig\_vel\_pa\], upper plot).
Let us verify whether the the LOS velocity behavior agrees with what one would expect from the keplerian motion. This can be seen from the variation of the LOS velocity with the elongation $\epsilon$ at a given $PA$. We assume that the elongation $\epsilon$ of the scattering dust and its heliocentric distance are in a linear relation. We choose the values of $15\degr$ and $195\degr$ for the $PA$, which corresponds to the extrema of $\bar{V}_{PA}$ curve. First of all, the LOS velocities in the considered $PA$’s were averaged within a sector of $\pm30\degr$ wide, then we subtracted from it the $\bar{V}_{\epsilon}$ value. The resulting “orbital" LOS velocity $V_{orb}(\epsilon )$ is given in Fig.\[fig:fig\_vel\_elong\] together with the linear fit. The latter gives $\bar{V}_{orb}(\epsilon) = -2.2 + 3.1\epsilon$ with the $1\sigma$ uncertainty of $\pm2$ on the additive term and $\pm0.3$ on the factor. The success of the fit indicates that the motion is indeed keplerian.
Summarizing what is indicated by the $\bar{V}_{PA}$ curve on March 29, 2006, we conclude that the bulk of the dust grains was in a keplerian retrograde motion in a plane nearly perpendicular to the ecliptics. In the inner Solar system, the only objects having this kind of motions are comets.
It happens that indeed around the eclipse date, the SOHO spacecraft recorded a series of sungrazing comets [@soho06] of the Kreutz group (@marsden67, @sekanina07). One of them, CK06F050, fell onto the Sun a day before the eclipse, another one, CK06F070, fell a day and a half after the eclipse. For convenience of the reader, the orbital elements as given by @soho06, are reproduced in the Tab.\[tab:tab\_comets\]. It is quite possible that there were also a chain of smaller fragments in between the entities recorded by SOHO, and thus the FOV of our eclipse frame is filled by the dust brought by the Kreutz group comets moving to the Sun on retrograde orbits.
The date of our observations is 8 days past the equinox, so that the line-of-sight to the Sun projects to approximately 8$\degr$ East from that to the vernal point $\Omega$. The longitude of the node of the indicated comets is distant from the $\Omega$ point, according to Tab.\[tab:tab\_comets\], by $3\degr-6\degr$. This means that our line-of-sight is inclined with the respect to the line of nodes by only $2\degr - 5\degr$. The perihelion argument lies at $84\degr-85\degr$ from the line of nodes.
This means that our image plane is nearly perpendicular to the orbital plane of the comets, and the line-of-sight slides over it.
If the dust grains were strictly confined to the orbital plane, we would see the selected direction indicated by the extrema of the LOS velocities at $PA\approx55\degr$, which would correspond to the inclination of the parent comet’s orbit of $i = 145\degr$. But our data indicate a different axis for the extrema velocities, namely $PA\approx15\degr$, which indicates $i = 105\degr$, i.e.the orbital plane of the dust grains is turned with respect to that of the comets by $40\degr$. The orbital plane cannot change under the gravitational force. Hence, we have to assume another reasons, the effect of the magnetic field on electrically charged dust particles being the most plausible. It follows that the size of particles is quite small, which, in turn, suggests their cometary origin. According to the detailed models of the dust grains dynamics near the Sun by [@krivov98] and [@mann00], the Lorentz force dominates the gravity for the dust grains smaller than 0.1 $\umu$m i size, and, as show their numerical simulations @mann00, the orbits of such grains get randomized which is not th case for the grains with size larger than 0.1 $\umu$m. Interestingly, for grains with the size smaller than 0.01 $\umu$m, the ratio charge-to-mass may be so high that they can be accelerated outward by the interaction with the solar wind as shows their detection by STEREO experiment @meyer09.
A more detailed modelling of the presented data, taking into account the scattering geometry, dynamics, the density and size distributions of the dust grains, would be highly desirable to quantify further the dust physical properties near the Sun, but this is largely beyond the scope of the present article.
Conclusions
===========
The reported here measurements of the Doppler shifts of the Fraunhofer lines, scattered by the dust grains in the solar F-corona, show that at the date of our observations the dust grains were on the orbit with a retrograde motion in a plane at $i\approx 105\degr$, i.e. nearly perpendicular to the ecliptics. This points to their cometary origin. Indeed, at the end of March, 2006, SOHO recorded several sungrazing comets with the orbital elements close to what was deduced from our measurements. We conclude that the contribution of comets to the dust content in the region close to the Sun can be important albeit variable in time. This contribution can explain the already noticed change of the dust distribution from one in the axial symmetry far from the Sun to that in the central, or may be spherical, symmetry @mann00.
We also derive that the observed plane of the dust grains orbit is slightly different from that of the parent comet(s), which indicates that the size of grains is small, less than 0.1 $\umu$m, so that they are deviated from the initial orbit by the Lorentz force. This also means that the observed dust grains were released by the comet(s) shortly before our observations.
The importance of comets in a circumstellar environment is general, let us recall e.g. the “Falling Evaporating Bodies” recorded in the spectra of $\beta$ Pic [e.g. @beust98]. It would be interesting to investigate whether they can provide a sufficient transport of the dust grains between far and close environments of the central star, and to contribute to the dissemination of crystallized material in the recently detected exozodiacal dust discs ([@absil06], [-@absil09]). For a more detailed discussion on the possible role of comet see also [@augereau09] and references therein.
Acknowledgments
===============
We are indebted to many persons who helped to make these measurements possible. A.Dubovitskiy, G.Minasiantz, M.Bayiliev insured the transportation of the instrument and helped with its set-up at the site of the eclipse. A.Didenko provided the Apogee CCD camera, T.Hua supplied a high quality filter on the line used in this work. We benefited from useful discussions with E.le Coarer, T.Bonev, V.Golev, A.-M.Lagrange, J.-C.Augereau. The expedition to the site of the solar eclipse was funded by the Kazakh National Space Agency and Laboratoire d’Astrophysique de Grenoble (UMR 5571 of the CNRS and Université Joseph-Fourier), the work on the data analysis became possible thanks to the EGIDE administrated ECO-NET grant 18837-XG of the French Ministry of Foreign Affaires.
\[lastpage\]
[^1]: E-mail: [email protected] (LIS), [email protected] (AC)
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'With a recent claim of superluminal neutrinos shown to be in error, 2012 may not be a propitious time to consider the evidence that one or more neutrinos may indeed be tachyons. Nevertheless, there are a growing number of observations that continue to suggest this possibility – albeit with an $m_{\nu}^2<0$ having a much smaller magnitude than was implied by the original OPERA claim. One recently published non-standard analysis of SN 1987A neutrinos supports a tachyonic mass eigenstate, and here we show how it leads to 3 + 3 mirror neutrino model having an unconventional mass hierarchy. The model incorporates one superluminal active-sterile neutrino pair, and it is testable in numerous ways, including making a surprising prediction about an unpublished aspect of the SN 1987 A neutrinos. Additional supporting evidence involving earlier analyses of cosmic rays is summarized to add credence to the tachyonic neutrino hypothesis.'
address: George Mason University
author:
- Robert Ehrlich
title: Tachyonic neutrinos and the neutrino masses
---
Published as: Astropart. Phys., 41 (2013) 1–6
Introduction
============
In this paper we summarize various observations suggesting that one of the neutrinos is consistent with being a tachyon as originally defined, i.e., a particle with $m^2<0$ and $v > c$ that obeys relativistic kinematics,[@Bilaniuk] a possibility first raised by Chodos, Hauser and Kostelecky.[@Chodos] As is well known, time-of-flight measurements of neutrinos no longer show any indication of superluminality, but they do set useful upper limits at the GeV energy scale.[@Bertolucci; @Adamson] There is also the upper limit on $\delta=(v-c)/c$ at low energies (around 20 MeV) set by SN 1987A, i.e., $\delta < 2 \times 10^{-9}$.[@Kamioka] As shown in the next section, however, there are reasons to disbelieve this much more stringent upper limit.
SN 1987A neutrino data
======================
Questioning the upper limit on $\delta$
---------------------------------------
The burst of 24 neutrinos seen in the Kamioka,[@Kamioka] IMB[@IMB] and Baksan[@Baksan] detectors, arrived about 3 hours before the light was recorded from SN 1987A. This early arrival was presumably due to the delay experienced by photons emitted from the collapsing SN core, which was not the case for the emitted neutrinos. However, the value of the photon delay need not have been the entire 3 h, therefore the early neutrino arrival is normally assumed to set only an upper limit on any excess above c for their speed, $\delta < 2 \times 10^{-9}$. Here we show that one cannot rule out a third superluminal mass eigenstate that arrived long before the other 24 neutrinos. This assertion, however, does not refer to the burst of 5 events observed in the LSD detector underneath Mt. Blanc,[@Mt; @Blanc] which occurred during a 7 s interval nearly 4 hours before the 24 event burst, as we can easily show. Consider hypothetical superluminal neutrinos of some fixed $m^2$ and varying values for their energy $E$ that are assumed to have originated in a very brief burst. Relativistic kinematics under the approximation that $\sqrt{1-m^2/E^2} \approx 1-\frac{m^2}{2E^2}$ requires that the neutrino arrival time t can be expressed as
$$t = \frac{m^2t_0}{2E^2}$$
where $t_0$ denotes the light travel time from the supernova, and $t=0$ would be the arrival time of $m^2 = 0$ neutrinos – which as shown in reference 12 is probably equivalent (within $\pm 0.5 s$) to setting $t = 0$ for the earliest arriving neutrino in each detector for the 24 event burst. From Eq. 1 we therefore find that if $m^2$ is fixed that the spread in the neutrino arrival times will be related to the spread in their energies according to:
$$\frac{\Delta E}{E}=\frac{\Delta t}{2t}=\frac{7 s}{2\times 4 h}=\frac{1}{5140}$$
Eq. 2 implies that in order to be observed $t=4h$ early within a burst as short as $\Delta t =7 s$ the superluminal neutrinos would have had to be monochromatic to one part in 5140 – which is virtually inconceivable for neutrinos from an exploding supernova. Turning the argument around, we can say that superluminal neutrinos with the energy spread seen for events in the three detectors, i.e., $\frac{\Delta E}{E} \approx 1$ would have arrival times spread over many hours and would certainly not be recognized as a pulse above background (around 1 event in 8 seconds). Whatever the source of the Mt. Blanc neutrinos, they could not have been due to brief superluminal burst emitted from SN 1987A. The inability to recognize a superluminal signal as a short pulse above background would be even less possible for larger excesses above light speed, where the spread in arrival times would be even larger. Thus, the normally assumed upper limit $\delta < 2 \times 10^{-9}$ from SN 1987A data is not correct, because any real superluminal signal would have gone unnoticed for large $\delta$ if one is expecting to see a pulse above background.
Two claimed mass eigenstates
----------------------------
The neutrinos from SN 1987A have been the subject of hundreds of papers, both theoretical and phenomenological.[@Pagliaroli] Some of these papers analyze the data to infer an upper limit on the electron neutrino mass, which ranges typically from 12 to 16 $eV$,[@Bahcall; @Arnett] although one 2010 analysis has claimed a 5.8 eV upper limit,[@Pagliaroli1] and still more refined methods may allow future galactic supernova to achieve mass limits as low as 0.14 eV.[@Ellis] In marked contrast to finding upper limits, a 2012 paper has claimed evidence for the presence of two (non-superluminal) mass eigenstates for the SN 1987 A neutrinos,[@Ehrlich0] following the method of earlier similar analyses by Cowsik[@Cowsik] and Huzita.[@Huzita] The heavier mass eigenstate has $m_2=21.4 \pm 1.2 eV$, while the lighter one has $m_1=4.0 \pm0.5 eV$ – similar to the values cited by Cowsik in 1988,[@Cowsik] but with considerably smaller uncertainties.
Before considering the implications of this result for a third superluminal state having $m_3^2<0,$ and why such outlandishly heavy mass eigenstates need not conflict with well-established upper limits on neutrino masses, e.g., from cosmology, a brief summary of the basis of the claim of 4.0 and 21.4 eV mass eigenstates is in order. The analysis is based on an observed correlation between recorded neutrino energies $E_k$ and arrival times $t_k$ for the $k = 1, 2, \cdots 24$ events in the three detectors (excluding the 5 events from the Mt. Blanc detector). If the neutrinos reaching Earth were all emitted nearly simultaneously then based on Eq. 1 on a plot of $1/E^2$ versus $t$ all those neutrinos having a mass $m_1$ will lie on a line of slope $2/(t_0m_1^2)$ while those having a mass $m_2$ will lie on a line of slope $2/(t_0m_2^2).$ Fig. 1 of reference 12 clearly shows that every one of the 24 neutrinos do lie on or near one of two straight lines, and the two fits have acceptable chi square. The fact that those two straight lines also nearly pass through the origin implies that the choice of $t=0$ for the first arriving neutrino in each detector made by each experiment nearly conforms to the definition of $t=0$ used in Eq. 1. It should also be noted that the claim of two mass eigenstates is not contradicted by the fact that all arriving neutrinos are detected only as (anti)electron neutrinos, because it is only for mass not flavor eigenstates that all neutrinos having some specific energy E travel at some fixed mass-dependent speed. Moreover, supernova neutrino data is unique in that no other time of flight measurement could possibly have the time resolution to observe separate mass eigenstates, since the distance to SN 1987A is approximately $3 \times 10^{14}$ Earth diameters.
The main weakness of the claim of two mass eigenstates is that it rests on there being near-simultaneous supernova neutrino emissions (within $\pm 0.5 s$) of most of the detected SN 1987A neutrinos. Supernova core collapse models in fact do show that the burst of electron neutrinos and antineutrinos rises and falls by an order of magnitude in the first second,[@Totani; @Bruenn] while some models show it lasting only about 0.02 seconds.[@Myra] Alternatively, it is possible some of the neutrinos detected from SN 1987A were emitted over an extended period of time, but they had a strange correlation between their energy and emission time that mimicked two mass eigenstates on an plot of $1/E^2$ versus $t$. One could conceivably accommodate this correlation within the framework of a composite model consisting of the sum of two thermal spectra.[@Lamb] Ultimately, however, there is no way to know precisely what fraction of the neutrinos emitted during a supernova core collapse are emitted in the first second. While supernova modeler Thomas Janka has suggested the number is likely to be no more than half,[@Janka] the fraction of the 24 $\emph{observed}$ neutrinos emitted during the first second could be considerably greater than half, given the softer spectra of later-emitted neutrinos.[@Raffelt]
SN 1987A and superluminal neutrinos?
------------------------------------
In the remainder of this section we show that even though the two mass eigenstates claimed for SN 1987A are not superluminal their existence (if confirmed) would imply that there must be a third unobserved eigenstate that is superluminal in order to be compatible with cosmological upper limits on the sum of the masses of the three flavor states, i.e., $\sum^3_{j=1} |m_j| < 0.28 eV,$[@Thomas] and that of the electron neutrino mass, ${m_\nu}_e < 2 eV$ from tritium beta decay.[@PDG]
We can express the effective mass of the fth flavor state in terms of a sum over all the mass eigenstates $m_i$ as:
$$m^2_f=\Sigma_i|U_{f,i}|^2 m^2_i$$
Thus, in light of the large values of $m_1$ and $m_2$, only if the 3rd mass eigenstate has ${m_3}^2 <0$ could the three flavor eigenstates all be quite close to zero, having either $m_f^2>0$ or $m_f^2<0,$ depending on the values of the $U_{f,i}$. Equivalently, the known upper limits on $\sum^3_{j=1} |m_j|$ and ${m_\nu}_e$ together with the confirmed existence of $4.0 eV$ and $21.4 eV$ neutrino mass eigenstates would *require* that the third mass eigenstate be superluminal with $m_3^2<0$.
A 3+3 neutrino model and the neutrino mass hierarchy
====================================================
How many sterile neutrinos?
---------------------------
One might also object that masses as large as 4.0 and 21.4 eV are incompatible with the two well-measured neutrino oscillation results for the $\Delta m^2$: $7.6 \times 10^{-5} eV^2$ and $2.4 \times 10^{-3} eV^2$ that suggest very small values for the masses themselves. However, one could accommodate those two measured values if there were three sterile neutrinos two of which were nearly degenerate with the 4.0 eV and 21.4 eV active neutrinos, and also if the measured $\Delta m^2$ values to date are between one active and one sterile neutrino. The empirical basis for sterile neutrinos has been gaining strength, since the entire collection of neutrino oscillation experiments can be fit with one or more sterile neutrinos, and they fit best with three of them.[@Kopp; @Conrad] For example, a collective fit with one sterile neutrino has a good probability of 55% but a compatibility between data sets of only 0.043%, while the three sterile neutrino fit has a $90\%$ probability and a compatibility of $53\%$ between data sets.[@Conrad] Although one might not be especially impressed with a good fit having as many as free parameters as occurs with 6 oscillating neutrinos, as we shall see later, the number of $\emph{independent}$ free parameters is far less than what one might think.
A 3 + 3 neutrino model
----------------------
Here we discuss a 3 + 3 neutrino model assuming three active/sterile pairs, which differs significantly from earlier models,[@Berezhiani] because (a) one pair is superluminal, (b) the mass splitting of each active-sterile pair is very small, and (c) the $\Delta m^2$ seen in oscillation experiments to date are all between a sterile and an active neutrino. The 3 + 3 model mass hierarchy is depicted in Fig. 1, and it consists of three right/left handed neutrino doublets, whose mass splittings for the two $m^2>0$ (tardyonic) doublets are taken to be the values found from neutrino oscillation experiments. It is of course the right handed states that are the sterile ones – at least in the case of the two tardyonic doublets. It should be noted that if the $4.0 eV$ and $21.4 eV$ mass eigenstate result is genuine, the mass hierarchy suggested in Fig. 1 is the only one compatible with neutrino oscillation experiments, with the exception of switching the two labeled $\Delta m^2$ values and reversing the order of each R and L state.
It is extremely interesting that the two mass splittings labeled in Fig 1. when expressed as a fraction of each doublet’s $m^2$, are identical within experimental uncertainties, i.e., $\frac{\Delta {m_1}^2}{{m_1}^2} = 4.8 \times 10^{-6}$ and $\frac{\Delta {m_2}^2}{{m_2}^2} = 5.2 \times10^{-6}$. If the fractional mass splitting for the third (superluminal) mass doublet has the same value, and $\Delta {m_3}^2\approx 1 eV^2$ as suggested by short-distance accelerator and reactor neutrino experiments,[@Louis] we then find an approximate mass of the superluminal mass doublet: ${m_3}^2 =-\frac{\Delta {m_3}^2}{5 \times 10^{-6}} \approx -200,000 eV^2=-0.2 keV^2$.
Global fits to 3 + 3 models
---------------------------
In a 3 + 3 global fit to all experiments the main interest is in seeing whether any large $\Delta m^2$ values are required to get a good fit (beyond the well-measured 3 small $\Delta m^2$), and what constraints can be placed on their values. As is well known, in the standard mass hierarchy, one assumes that the three active neutrino masses have $m << 1 eV,$ and the one or more sterile neutrinos are considerably heavier. The basis of this hierarchy rests on the assumption that the two very small well-measured $\Delta m^2$ values from oscillation experiments are both between the three active neutrinos (1,2,3), while in our model they are each between the two members of an active-sterile doublet. In other words, the well-measured value of $\Delta m_{1,2}^2$ is assigned to our $\Delta m_{1,4}^2$ and that of $\Delta m_{2,3}^2$ to our $\Delta m_{2,5}^2$ – a difference that is irresolvable in a given experiment, whether one is doing a search for actual oscillations or the appearance/disappearance of a flavor state. The distinction between the two model hierarchies is of vital importance, however, when doing a 3 + 3 global fit to all experiments, since it will affect the best fit values of the three large $\Delta m^2.$ Thus, the Conrad et. al. 3 + 3 fit was based on the standard mass hierarchy in which the three active neutrinos are much lighter than the three sterile ones.[@Conrad] In this fit there are only three independent large $\Delta m^2$ values which they take to be $\Delta m_{14}^2, \Delta m_{15}^2$ and $\Delta m_{16}^2.$ Only the first of Conrad’s three fitted large $\Delta m^2$ agree with those predicted from our 3 + 3 model, which are $1.0eV^2,$ $200,000eV^2,$ and $21.4^2-4.0^2=442 eV^2$ – see Fig. 1. This disagreement is to be expected, of course, given the differences between our mass hierarchy and the conventional one Conrad et. al. used in doing their fit. Thus, in our 3 + 3 model there is only one $\Delta m^2=2.4\times 10^{-3} eV^2$ not two. In addition the three independent large $\Delta m^2$ values in our model are $\Delta m_{12}^2, \Delta m_{13}^2$ and $\Delta m_{36}^2$ (3 double arrows in Fig. 1), and the other large $\Delta m^2$ are given in terms of them. For example, these eight large $\Delta m^2$ are all approximately equal to three significant figures: $\Delta m_{13}^2, \Delta m_{34}^2, \Delta m_{46}^2, \Delta m_{16}^2, \Delta m_{23}^2, \Delta m_{26}^2, \Delta m_{35}^2, \Delta m_{56}^2,$ as are these four: $\Delta m_{12}^2, \Delta m_{15}^2, \Delta m_{24}^2, \Delta m_{45}^2.$ These equivalencies would be more obvious had all the $\Delta m^2$ spacings been drawn to scale.
A digression: Earlier work suggesting ${m^2_\nu}_e \approx - 0.16 eV^2$
=======================================================================
Given the highly controversial nature of the claims made in the previous section it is useful to summarize earlier evidence for tachyonic neutrinos before considering theoretical support for them, and how those claims can be tested – in some cases using existing data.
The shape of the high energy cosmic ray spectrum
------------------------------------------------
Chodos, Hauser and Kostelecky in 1985[@Chodos] suggested that one could test whether electron neutrinos are tachyons based on the beta decay of stable particles whose energy exceeds some threshold. In 1999, following a suggestion by Kostelecky,[@Kostelecky] Ehrlich adopted this idea to modeling the cosmic ray spectrum.[@Ehrlich1; @Ehrlich2] It is well known that the observed spectrum satisfies a power law $\frac{dN}{dE}\approx E^{-\gamma}$ where $\gamma$ changes value relatively abruptly at an energy in the vicinity of 4 PeV, which is known as the knee of the spectrum. One can interpret the presence of the knee using the Chodos et. al. idea that protons are decaying with this energy as their threshold, and they are increasingly depleted from the spectrum above this energy. For protons, the threshold energy is inversely related to the absolute value of the tachyon mass (in $eV$) through[@Ehrlich1]
$$E_{th}=\frac{1.7 PeV}{\sqrt{-m_{\nu_e}^2}}$$
A second change in the spectrum power law known as the ankle occurs around $10^4 PeV.$ In Ehrlich’s model a good fit was obtained to the high energy spectrum (including both the knee and the ankle), by assuming ${m_{\nu _e}}^2= -0.16\pm .09 eV^2.$
Neutral hadrons in the cosmic rays from Cygnus X-3
--------------------------------------------------
One important prediction of the fit to the cosmic ray spectrum was the existence of a neutron line (from proton decay) that occurred right at the knee.[@Ehrlich1] Evidence for a neutron line at the knee was subsequently reported based on cosmic rays pointing back to Cygnus X-3, an X-ray binary having a 4.79 h period.[@Ehrlich3] At PeV-scale energies cosmic rays pointing back to a particular distant source constitutes evidence that those primary cosmic rays are neutral particles, given the strength of the galactic magnetic field. Of the four groups that had reported high energy cosmic rays from Cygnus X-3 during the 1970$'$s and 1980$'$s with signal strengths at the 4-5 $\sigma$ level, only the Lloyd-Evans group had events above 1 PeV.[@r14] That data showed an excess of 28.4 events in two adjacent energy bins straddling 5 PeV, with an uncertainty in the two bin total of 5.0 events, i.e., $N=28.4\pm5.0$ ($5.7\sigma$). Thus, the energy at which the peak occurs came very near the knee of the spectrum, which was previously interpreted as the threshold for protons to beta decay into neutrons. Given that evidence also existed for the Cygnus X-3 cosmic rays being neutral hadrons, Ehrlich interpreted the 4.5 PeV peak associated with Cygnus X-3 as a confirmation of the earlier prediction of a neutron line at the knee of the cosmic ray spectrum.[@Ehrlich3] Today many cosmic ray researchers express skepticism about the reality of those early reports of cosmic rays from Cygnus X-3. The conventional wisdom is that the only primary cosmic rays pointing back to sources at PeV energies are photons or neutrinos. In fact, a more recent high statistics cosmic ray study failed to observe non-episodic cosmic rays from Cygnus X-3 at PeV energies.[@r13] However, it should be noted that this negative result need not disprove the validity of the earlier observations since Cygnus X-3 is known to be an episodic source that is especially intense at times of strong radio flares when the RF luminosity increases a thousand fold.
Theoretical support and challenges to $m^2<0$ neutrinos
=======================================================
While there is yet no commonly accepted field theory of tachyons, a number of researchers including Ciborowski, Rembielinski, and Radzikowski have made important steps towards such a theory.[@Ciborowski; @Radzikowski] There are, however, many theoretical reasons that have been cited for being skeptical of superluminal neutrinos, including the magnitude of the violation of Lorentz Invariance (VLI) that they might entail.[@Cowsik1] However, while these VLI constraints and the Cohen-Glashow (C-G) effect[@Cohen] conflict with a $\delta$ as large as the initial spurious OPERA value, they do not rule out much smaller values below the upper limits set by various experiments.[@Miniboone; @Adamson1; @Icecube] In particular, note that for the C-G effect the threshold energy for tachyonic neutrino bremsstrahlung varies as $\delta^{-15},$ so the effect is of no consequence for very small $\delta.$ Thus, neither VLI nor the C-G effect is an argument against superluminal neutrinos in general. Moreover, not only has VLI been shown to be compatible with extensions of the standard model,[@Kostelecky2] but Chodos recently has provided a theoretical rationale for tachyonic neutrinos.[@Chodos2] He shows that one can construct a Lagrangian that satifies Light Cone Reflection, a new spacetime symmetry that arises in the context of a limited form of Lorentz Invariance, in which $\pm m^2$ neutrino pairs arise naturally. Unfortunately, since Chodos’ model explicitly requires that $m^2_{tachyon} = -m^2_{tardyon},$ it is not consistent with our 3 + 3 model. One that is consistent is a theory by Jentschura and Wundt who generalize the Dirac equation, based on a pseudo-Hermitian Hamiltonian. Their theory allows for a left-handed tachyonic neutrino mass eigenstate with a free mass parameter that is compatible with our value of $m_3^2 = -200,000eV^2,$[@Jentschura] and it also leaves room for the addition of sterile neutrinos.[@Jentschura2] Moreover, the same authors note that these extensions to the Dirac equation allow tachyonic neutrinos to be a candidate for the acceleration of the universe or dark energy.[@Jentschura1]
Tests of tachyonic neutrinos and the 3 + 3 model
================================================
The first proposed test is the least intereting one, and is likely only to result in setting upper limits, while all the rest involve testing far more specific predictions discussed earlier.
Time of flight experimental searches for no specific ${m_\nu}^2<0$
------------------------------------------------------------------
Time of flight experiments involving Earthly distances should be feasible so long as a tachyon had an $-m^2$ on the order of many $keV^2.$ The lowest mass one might be able to detect using Earthly distances might be around $m^2= - 0.0019 MeV^2= - 1,900 keV^2$ based on Eq. 1 using $E=20 MeV$, $t = 50 ns$ and $t_0 = 6400 km/(3\times 10^5 km/s)$. Such a result would be many orders of magnitude smaller than the original OPERA claim.
Confirming that ${m^2_\nu}_e \approx -0.16 eV^2$
-------------------------------------------------
If the forthcoming Katrin experiment designed to have a sensitivity of ${m_\nu}_e = 0.2 eV$ or $|{m^2_\nu}_e| = 0.04 eV^2$[@Katrin] fails to see a tachyonic electron neutrino, a PeV-scale cosmic ray experiment might well do so. In particular, finding confirming evidence for cosmic ray protons decaying at or above the knee of the spectrum and giving rise to a neutron line at that energy would be very convincing evidence. Of course, PeV-scale cosmic rays from the binary star Cygnus X-3 have been widely dismissed as an aberration by most cosmic ray researchers, in light of the very high statistics later negative search which found no indication of a steady state signal from that source.[@r13] A new experiment that looked at Cygnus X-3 and other possible sources, but $\emph{only}$ at times of large flares, and in a narrow phase window based on the binary’s period might well show a signal.
Confirming the existence of mass eigenstates $m_{\nu_1}=4.0eV$ and $m_{\nu_2}=21.4eV$
-------------------------------------------------------------------------------------
An observation that confirmed the existence of these two mass eigenstates claimed based on the SN 1987A data, would by implication show the need for a third superluminal mass eigenstate in order to conform to existing mass limits on the sum of the flavor state masses. One possibile test might come from a global fit to all oscillation experiments to see if one finds three large $\Delta m^2$ values, consistent with those predicted by the 3 + 3 model masses, as discussed in an earlier section. Another test would be provided by the fortunate occurence of a supernova in a our galaxy or another one nearby, but these occur only about twice in a century. There is, however, no need to wait for another galactic supernova, since the existence of mass eigenstates $m_{\nu_1}=4.0eV$ and $m_{\nu_2}=21.4eV$ are quite within the realm of a short baseline neutrino oscillation experiment. For example, given $\Delta m^2= 21.4^2-4.0^2=442 eV^2,$ we find $\lambda=\frac{2\pi E}{1.267\Delta m^2}=11.2m$ (for E = 1 GeV). A particularly suitable neutrino source might be the Spallation Neutron Source (SNS) at the Oak Ridge National Laboratory, given its high intensity, short pulse width, and the large percentage (about 30 %) of neutrino flux that is monochromatic ($E=30 MeV$). At this neutrino energy we find that $\Delta m^2= 442 eV^2$ would yield an oscillation wavelength of about 34 cm, which could be readily observed.
Confirming that ${m_{\nu_3}}^2 \approx -0.2 keV^2$
--------------------------------------------------
Searching for a ${m_{\nu_3}}^2 \approx -0.2 keV^2$ neutrino in an oscillation experiment should be doable at very high neutrino energies. For example, the predicted oscillation wavelength between ${m_{\nu_3}}^2 \approx -0.2 keV^2$ and some much smaller mass is $\lambda=24.7 m$ (for E = 1 TeV) or $\lambda=2.47 m$ (for E = 100 GeV). However, such a test could not distinguish between ${m_{\nu_3}}^2 \approx -0.2 keV^2$ and ${m_{\nu_3}}^2 \approx +0.2 keV^2.$ One measurement that could make this sign distinction would be an unreported and possibly unexamined aspect of the existing data on neutrinos from SN 1987A. For a typical neutrino energy of 20 MeV, $-0.2 keV^2$ neutrinos would have arrived around 25 min earlier than the main neutrino pulse. Of course, based on Eq. 2 any such superluminal neutrinos would be spread over many minutes and would not be recognized as a pulse above background because of their spread in energy. Nevertheless, given the energy dependence of the background events, there is a surprisingly simple way to discern a superluminal signal – at least for the Kamioka data for which nearly all the background events in the detector have energies below 12 MeV – the height of the dashed line in Fig. 2 (lower graph).
The 17 minute time interval depicted in Fig. 2 (lower) includes the 12 event neutrino burst reported by Kamioka seen just after 7:35 UT. In order to investigate background more thoroughly Kamioka has provided similar plots for 7 other time intervals selected at random, some before the 12 event pulse and some after.[@Kamioka] If we exclude the one 17 minute interval that happens to fall in the one hour before the 12 event burst, Kamioka shows only one background event out of about 1000 that has an energy above 12 MeV in the entire 7 x 17 = 119 minutes, or about 0.5 background events per hour. Thus, selecting only events having $E > 12 MeV$ is an extremely powerful background suppressor.
Recall that on a plot of $1/E^2$ versus neutrino arrival time $t$ events corresponding to a specific neutrino mass lie on a straight line passing through the origin whose slope is inversely proportional to $m^2$. Given that 8 of the 12 actual SN 1987A events observed in Kamioka have $E >12 MeV$ (see Fig. 2), it would not be surprising that any superluminal eigenstate might have perhaps 4 neutrinos associated with it for which $E> 12 MeV.$ The four dots in Fig. 2 (upper) shows what the signature might look like for such a ${m_{\nu_3}}^2 \approx -0.2 keV^2$ signal. Those four simulated events were arbitrarily assumed to have a uniform time distribution (equivalent to $\frac{dN}{dE}\propto 1/E^3$). Note that there is no real need to do anything more realistic here, given the extremely low background – perhaps 0.5 background events in the 1 hour interval before 7:35 UT, as long as we focus only on events having $E>12 MeV.$ Thus, should the $1/E^2$ versus t plot of the Kamioka neutrino data for this one hour period show perhaps 4 real events falling $\emph{anywhere}$ on a line through the origin having the predicted approximate slope corresponding to ${m_{\nu_3}}^2 \approx -0.2 keV^2,$ this would constitute an unambiguous signature of a superluminal neutrino.
A tantalizing hint that this possibility might be due to more than the author’s feverish imagination is provided by the one real Kamioka event (the square in Fig. 2 upper graph) that falls in the only 17 min time interval falling in the one hour before the 12 event burst. This event lies quite near the predicted straight line. It is silly to provide a calculation of the probability of this occurring based on random background, given only one event, but it is probably only about 1/100. This estimate is based on the likelihood of a background event occuring in that 17 minute interval (about 1/10), and its likelihood of it lying very close to the predicted line (about 1/10). This one event proves nothing, but if it were possible for Kamioka, IMB and Baksan to reexamine their old data one might find more persuasive confirming evidence. Thus, the probability of N events falling on or near the line as a result of random background would be on the order of $p = 10^{-2N}.$
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| {
"pile_set_name": "ArXiv"
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abstract: 'In this work we introduce two experimental proposals that could shed some light upon the inertial properties of intrinsic spin. In particular we will analyze the role that the gravitomagnetic field of the Earth could have on a quantum system with spin $1/2$. We will deduce the expression for Rabi transitions, which depend, explicitly, on the coupling between the spin of the quantum system and the gravitomagnetic field of the Earth. Afterwards, the continuous measurement of the energy of the spin $1/2$ system is considered, and an expression for the emerging quantum Zeno effect is obtained. Thus, it will be proved that gravitomagnetism, in connection with spin $1/2$ systems, could induce not only Rabi transitions but also a quantum Zeno effect.'
author:
- |
A. Camacho [^1] [^2]\
Astrophysikalisches Institut Potsdam.\
An der Sternwarte 16, D–14482 Potsdam, Germany.
title: 'Quantum Zeno effect and the detection of gravitomagnetism.'
---
=7.8in =6 in = -1 cm
Introduction.
=============
In more than three–quarters of a century the theory of general relativity (GR) has achieved a great experimental triumph. Neverwithstanding, at this point it is also important to comment that all the current direct confirmations of GR are confirmations of weak field corrections to the Galilei–Newton mechanics \[1\]. We must also add that one of the most important, and yet undetected, predictions of GR is the so called gravitomagnetic field \[1\], sometimes also called Lense–Thirring effect \[2\], which is generated by mass–energy currents. Its measurement would constitute a direct experimental evidence against an absolute inertial frame of reference, and would at the same time show the basic role that local inertial frames play in nature, i.e., it would be a direct proof that local inertial frames are influenced and dragged by mass–energy currents relative to other mass.
The first efforts in the detection of this gravitomagnetic field are quite old \[3\] and have already included many interesting proposals \[4, 5, 6\].
An additional topic in connection with gravitomagnetism is related to its coupling with intrinsic spin, this issue is of fundamental interest since it comprises the inertial properties of intrinsic spin. It is noteworthy to comment that this point is under constant analysis \[7\].
In this work we introduce two experimental proposals that could lead to the detection of the coupling between intrinsic spin and the gravitomagnetic field. We analyze the role that the gravitomagnetic field of the Earth could have on a quantum system with spin $1/2$, i.e., our results could allow us to confront the effects of mass–energy density currents upon spin. In particular we deduce a Rabi formula, which depends on the coupling between the spin of the quantum system and the gravitomagnetic field of the Earth. Afterwards, the continuous measurement of the energy of the spin $1/2$ system is considered, and a Zeno effect is obtained.
Rabi transitions and the gravitomagnetic field.
===============================================
Let us consider a spin $1/2$ system immersed in the gravitational field of a rotating uncharged, idealized spherical body with mass $M$ and angular momentum $J$. In the weak field and slow motion limit the metric, in the Boyer–Lindquist coordinates, reads \[8\]
[$$\begin{aligned}
ds^2 = -c^2\left( 1 - {2GM\over c^2r}\right)dt^2 + \left( 1 - {2GM\over c^2r}\right)^{-1}dr^2 \nonumber\\
+ r^2\left(d\theta^2 + \sin^2\theta d\phi^2\right) - {4GJ\over c^2r}\sin^2\theta d\phi dt.\end{aligned}$$]{}
The gravitomagnetic field in this case is approximately \[1\]
[$$\begin{aligned}
\vec {B} = 2{G\over c^2}{\vec {J} - 3(\vec {J}\cdot\hat {x})\hat {x}\over |\vec {x}|^3}.\end{aligned}$$]{}
We will assume that the expression that describes the precession of orbital angular momentum, immersed, for instance, in the gravitational field of the Earth, can be also used for the description of the dynamics in the case of intrinsic spin. This is a natural extension of general relativity \[7\].
Let us now denote the angular momentum of our spherical body by $\vec {J} = J\hat {z}$, being $\hat {z}$ the unit vector along the direction of the angular momentum. Our quantum particle is prepared such that $\vec {S} = S_z\hat {z}$, it has vanishing small velocity and acceleration, and it is located on the $z$–axis, with coordinate $Z$.
There is a formal analogy between the weak field and slow motion of the gravitomagnetic field in general relativity and the magnetic field in electromagnetism \[1\]. Following this analogy we may write down the interaction Hamiltonian (acting in the two–dimensional spin space of our spin $1/2$ system), which gives the coupling between $\vec {B}$ and the spin, $\vec {S}$, of our particle
[$$\begin{aligned}
H = - \vec {S}\cdot\vec {B}.\end{aligned}$$]{}
Introducing expression (2) we may rewrite the interaction Hamiltonian as follows
[$$\begin{aligned}
H = 2{GJ\hbar\over c^2Z^3}\left[|+><+| - |-><-|\right].\end{aligned}$$]{}
Here $|+>$ and $|->$ represent the eigenkets of $S_z$. Clearly, the introduction of the gravitomagnetic field renders two energy states
[$$\begin{aligned}
E_{(+)} = 2{GJ\hbar\over c^2Z^3},\end{aligned}$$]{}
[$$\begin{aligned}
E_{(-)} = -2{GJ\hbar\over c^2Z^3},\end{aligned}$$]{}
where $E_{(+)}$ ($E_{(-)}$) is the energy of the spin state $+\hbar/2$ ($-\hbar/2$). Let us now define the frequency
[$$\begin{aligned}
\Omega = \left(E_{(+)} - E_{(-)}\right)/\hbar = 4{GJ\over c^2Z^3}.\end{aligned}$$]{}
The present analogy allows us to consider the emergence of Rabi transitions \[9\]. In order to do this let us now introduce a rotating magnetic field, which, at the point where the particle is located, has the following form
[$$\begin{aligned}
\vec {b} = b\left[\cos(wt)\hat {x} + \sin(wt)\hat {y}\right],\end{aligned}$$]{}
where $\hat {x}$ and $\hat {y}$ are two unit vectors perpendicular to the $z$–axis, and $b$ is a constant magnetic field.
Under these conditions the total Hamiltonian reads
[$$\begin{aligned}
H_T = 2{GJ\hbar\over c^2Z^3}\left[|+><+| - |-><-|\right] \nonumber\\
- {eb\hbar\over 2mc}\left[e^{-iwt}|+><-| + e^{iwt}|-><+|\right].\end{aligned}$$]{}
Looking for a solution in the form $|\alpha> = c_{(+)}(t)|+> +~c_{(-)}(t)|->$, we find the usual situation \[9\] (our quantum system has been initially prepared such that $c_{(-)}(0) = 1$ and $c_{(+)}(0)= 0$.)
[$$\begin{aligned}
c_{(-)}(t) = \exp\left[-i{E_{(-)}\over\hbar}t\ + {i\over 2}(w - \Omega)t\right]\left[\cos(\Gamma t) - i{(w - \Omega)\over 2\Gamma}\sin(\Gamma t)\right],\end{aligned}$$]{}
[$$\begin{aligned}
c_{(+)}(t) = i{eb\over 2mc\Gamma}\exp\left[-i{E_{(+)}\over\hbar}t\ - {i\over 2}(w - \Omega)t\right]\sin(\Gamma t).\end{aligned}$$]{}
where $\Gamma = \sqrt{({eb\over 2mc})^2 + {(w - \Omega)^2\over 4}}$.
In this way we find
[$$\begin{aligned}
{|c_{(-)}(t)|^2\over |c_{(-)}(t)|^2 + |c_{(+)}(t)|^2} =
\left[ 1 + {({eb\over 2mc\Gamma})^2\sin^2(\Gamma t)\over \cos^2(\Gamma t) +
{(w - \Omega)^2\over 4\Gamma^2}\sin^2(\Gamma t)}\right]^{-1}.\end{aligned}$$]{}
Clearly, the Rabi transitions depend upon the coupling between spin and the gravitomagnetic field.
[$$\begin{aligned}
\left(4{GJ\over c^2Z^3} - w\right)^2 = 4\left[\Gamma^2 - \left({eb\over 2mc}\right)^2\right].\end{aligned}$$]{}
Quantum Zeno effect and gravitomagnetism.
=========================================
Let us now measure, continuously, the energy of our spin $1/2$ system, such that $E$ is the measurement output, and that this experiment lasts a time $T$. This kind of measuring process can be described by the so called effective Hamiltonian formalism \[10, 11\], which is one of the models that exist in the topic of quantum measurement theory \[12\]. In our case the corresponding effective Hamiltonian reads
[$$\begin{aligned}
H_{eff} = 2{GJ\hbar\over c^2Z^3}\left[1 +
i{2\hbar\over T\Delta E^2}\left(E - {GJ\hbar\over c^2Z^3}\right)\right]|+><+|\nonumber\\
- 2{GJ\hbar\over c^2Z^3}\left[1 +
i{2\hbar\over T\Delta E^2}\left(E + {GJ\hbar\over c^2Z^3}\right)\right]|-><-|\nonumber\\
- {eb\hbar\over 2mc}\left[e^{-iwt}|+><-| + e^{iwt}|-><+|\right] -
i{E^2\hbar\over T\Delta E^2}\Pi,\end{aligned}$$]{}
where $\Pi$ is the unit operator in the spin space of our particle. Looking for solutions with the form $|\alpha> = c_{(+)}(t)|+> +~c_{(-)}(t)|->$, we deduce
[$$\begin{aligned}
c_{(-)}(t) = \exp\left[-i{E_{(-)}\over\hbar}t\ - {(E_{(-)} - E)^2\over T\Delta E^2}t
+ i\tilde{\Gamma} t\right]\nonumber\\
\times\left[c_{(-)}(0)\cos(\beta t) - i{c_{(-)}(0)\tilde{\Gamma} + (\gamma/\hbar) c_{(+)}(0)\over \beta}\sin(\beta t)\right],\end{aligned}$$]{}
[$$\begin{aligned}
c_{(+)}(t) = \exp\left[-i{E_{(+)}\over\hbar}t\ - {(E_{(+)} - E)^2\over T\Delta E^2}t
-i\tilde{\Gamma} t\right]\nonumber\\
\times\left[c_{(+)}(0)\cos(\beta t) + i{c_{(+)}(0)\tilde{\Gamma} - (\gamma/\hbar) c_{(-)}(0)\over \beta}\sin(\beta t)\right],\end{aligned}$$]{}
where $\tilde{\Gamma} = {(w - \Omega)\over 2} + {i\over 2T\Delta E^2}
\left[(E_{(+)} - E)^2 - (E_{(-)} - E)^2\right]$, $\beta^2 = (\gamma/\hbar)^2 + \tilde{\Gamma}^2$, and finally $\gamma = -{eb\hbar\over 2mc}$.
Let us now suppose that the measurement output is the energy of the ground state, $E_{(-)}$, that we have a resonant perturbation, and that initially only the lowest energy state was populated, in other words, $E = E_{(-)}$, $\hbar w = E_{(+)} - E_{(-)}$, and $c_{(-)}(0) = 1$, $c_{(+)}(0) = 0$.
Hence (15) and (16) become
[$$\begin{aligned}
c_{(-)}(t) = \exp\left[-i{E_{(-)}\over\hbar}t\ - {(E_{(+)} - E_{(-)})^2\over 2T\Delta E^2}t\right]
\left[\cos(\beta t) - i{\tilde{\Gamma}\over \beta}\sin(\beta t)\right],\end{aligned}$$]{}
[$$\begin{aligned}
c_{(+)}(t) = -i{\gamma\over \beta\hbar}\exp\left[-i{E_{(+)}\over\hbar}t\ - {(E_{(+)} - E_{(-)})^2\over 2T\Delta E^2}t\right]\sin(\beta t).\end{aligned}$$]{}
Let us now assume that ${(E_{(+)} - E_{(-)})^4\over 4T^2\Delta E^4}> \gamma^2/\hbar^2$, then
[$$\begin{aligned}
P_{(-)}(t) =
\left[1 + {\sinh^2({\gamma\over\hbar}\tilde{\Omega}t)\over
\tilde{\Omega}^2
[\cosh({\gamma\over\hbar}\tilde{\Omega}t) + {\hbar(E_{(+)} - E_{(-)})^2\over 2T\gamma\tilde{\Omega}\Delta E^2}
\sinh({\gamma\over\hbar}\tilde{\Omega}t)]^2}\right]^{-1},\end{aligned}$$]{}
where $\tilde{\Omega} = \sqrt{{\hbar^2(E_{(+)} - E_{(-)})^4\over 4T^2\gamma^2\Delta E^4} - 1}$, $\gamma = -{eb\hbar\over 2mc}$, and $P_{(-)}(t) = {|c_{(-)}(t)|^2\over |c_{(-)}(t)|^2 + |c_{(+)}(t)|^2}$.
In the case $t\rightarrow \infty$ this last expression reduces to
[$$\begin{aligned}
P_{(-)}^{(\infty)} = \left[1 + {\left({c^2Z^3\over 4GJ\hbar}\right)^2}{ebT\Delta E^2\over mc}
\left(\sqrt{1 - {({c^2Z^3\over 4GJ\hbar})^4}({ebT\Delta E^2\over mc})^2} -1\right)^{-2}\right]^{-1}.\end{aligned}$$]{}
Clearly, Rabi transitions are inhibited, and the asymptotic value that here appears depends explicitly upon the coupling between intrinsic spin and the gravitomagnetic field, i.e., $J$ emerges in expression (20).
At this point it must be commented that the behavior of spin leads, in some cases, to the emergence of a non–geometric element in gravity \[13\].
In this work Ahluwalia has considered two different classes of flavor–oscillation clocks. The first one comprises the superposition of different mass eigenstates, associated to a quantum test particle, such that all the terms of the corresponding superposition have the same spin component. The second class of flavor–oscillation clocks, contains, at least, two distinct spin projections.
If the gravitomagnetic field is absent, then both clocks redshift identically in the corresponding gravitational field. Nevertheless, if the source of the gravitational field has a nonvanishing angular momentum, then these redshifts do not coincide any more . This fact depends not only upon the gravitomagnetic component of the gravitational field, but also on the quantum mechanical features of the employed quantum test particle. In other words, here a non–geometric element appears when gravitational and quantum mechanical phenomena are considered simultaneously.
Clearly, in the present essay we have a quantum system with spin immersed in a nonvanishing gravitomagnetic field. Nevertheless, our case is an eigenstate of the spin operator $S_z$, something that in Ahluwalia’s second class of flavor–oscillation clocks does not happen. This last remark means that our quantum system is closer to Ahluwalia’s first class of flavor–oscillation clocks than to his second one.
Finally, we must add that it is now possible to test, experimentally, the quantum Zeno effect \[12\], particularly using Penning traps to analyze Rabi transitions \[14\].
The author would like to thank A. A. Cuevas–Sosa his help, and D.-E. Liebscher for the fruitful discussions on the subject. It is also a pleasure to thank R. Onofrio for bringing Refs. 6 and 11 to my attention. The hospitality of the Astrophysikalisches Institut Potsdam is also kindly acknowledged. This work was supported by CONACYT Posdoctoral Grant No. 983023.
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[^1]: email: [email protected]
[^2]: This essay received an “honorable mention” in the Annual Essay Competition of the Gravity Research Foundation for the year 2000 — Ed.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Evolutionary algorithms offer great promise for the automatic design of robot bodies, tailoring them to specific environments or tasks. Most research is done on simplified models or virtual robots in physics simulators, which do not capture the natural noise and richness of the real world. Very few of these virtual robots are built as physical robots, and the few that are will rarely be further improved in the actual environment they operate in, limiting the effectiveness of the automatic design process. We utilize our shape-shifting quadruped robot, which allows us to optimize the design in its real-world environment. The robot is able to change the length of its legs during operation, and is robust enough for complex experiments and tasks. We have co-evolved control and morphology in several different scenarios, and have seen that the algorithm is able to exploit the dynamic morphology solely through real-world experiments.'
author:
- 'T[ø]{}nnes F. Nygaard'
- David Howard
- Kyrre Glette
bibliography:
- 'bibliography.bib'
title: Real World Morphological Evolution is Feasible
---
<ccs2012> <concept> <concept\_id>10010147.10010257.10010293.10011809.10011814</concept\_id> <concept\_desc>Computing methodologies Evolutionary robotics</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003752.10003809.10003716.10011136.10011797.10011799</concept\_id> <concept\_desc>Theory of computation Evolutionary algorithms</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
Introduction
============
In this paper, we present our shape-shifting robot, and the work we have done evolving its body shape in real-world environments. We show that evolutionary optimization is an effective technique for automated design, even when relying purely on real-world experiments on a physical robot.
The field of Evolutionary Robotics (ER) uses evolutionary computation techniques to improve various aspects of robots. Most common is to optimize the control of a robot, either through higher level control policies like movement trajectories for wheeled or flying robots, or more lower level concepts like the gait of legged robots. Most work is not done on physical robots, but on simplified models in a physics simulator [@mouret201720]. These are never 100% accurate, and the discrepancy between behavior in simulation and in the real world is referred to as the reality gap [@jakobi1995noise]. It makes evolutionary methods less effective as their results are not tuned to the specific environment and task they are meant for, but a simplified and inaccurate model of it. This is considered one of the biggest challenges in the ER field [@eiben2014grand].
![Our shape-shifting robot, with retracted legs to the left, and extended legs to the right.[]{data-label="fig.lengthComparison"}](figures/dyret_small_2.jpg "fig:") ![Our shape-shifting robot, with retracted legs to the left, and extended legs to the right.[]{data-label="fig.lengthComparison"}](figures/dyret_large_2.jpg "fig:")
There are many ways to reduce the impact of the reality gap when evolving control. Firstly, the gap can be reduced significantly by using high-resolution simulators [@collins2018towards]. Another common option is to do the start of the evolutionary run in simulation, before finishing the run in the real world [@lipson2000automatic]. This serves as a local search around the solutions found during simulation, which makes it less likely to find global optima only found in the real world. Another approach is to try to improve the simulators using real-world data. This can reduce the reality gap, but physics simulators or mathematical models will always be simplifications of the real world, and will thus never be perfect [@mouret201720]. With neural controllers, the use of plasticity for gap-crossing behavioural compensation has recently been shown possible [@qiu2020crossing], and using an embodied approach through sensor feedback has also shown some promise [@nordmoen2019evolved]. Avoiding parts of the search space where the reality gap is larger is also an option [@koos2012transferability]. The problem is that the best solutions are often the ones that are able to exploit the dynamics of the system and the peculiarities of the environment. These are also the ones that are hardest to simulate and predict correctly, so this approach often removes most of the high performing individuals.
The only approach with the potential to completely bypass the reality-gap problem is doing evolution in the real world. A challenge here, is that each evaluation can take a long time, and the physical robot(s) have to be designed, built, maintained and repaired, costing considerable amounts of effort [@nygaard2019experiences]. There are many examples where this has been done successfully, but these are often done in very simple environments in the lab [@shen2012learning; @hornby2000evolving; @heijnen2017testbed]. This does bypass the reality gap, but unless the robot is subjected to the real-world environment it will operate in, there will still be a discrepancy in behavior that will affect the performance. Only a system that can continuously evolve during operation will not be subject to the challenges traditionally associated with the reality gap.
When evolving only the control of the robot, combining simulation and real-world experiments can be fairly straightforward. The robot most often has the same freedom of control in the real world as in simulation. When it comes to the design, however, very few robots are able to change their body in a meaningful way to continue optimization in the real world after it has been built. This means that a lot of work doing evolutionary design of robots only do the optimization in simulation, before optimizing control alone in the real world [@lipson2000automatic; @hornby2003generative; @rosser2019sim2real]. This is still an effective technique, but the reality gap lessens the potential efficiency and impact of the technique considerably [@nygaard_WS_ICRA18]. There has been a few examples of evolution of the body of the robot in the real world, but these require excessive time, human intervention, or the robots are very simple and limited. Broadly, the approaches seen to date have been (i) simulate some robots, then print them [@collins2018towards], or (ii) automatically combine predefined modules to create the phenotype [@brodbeck2015morphological].
Our approach
============
Our main goal has been to do morphological optimization in the real world, and we wanted to do this on a functional robot design that could be used to solve real-world tasks. The principal idea is that morphological adaptation is built into the robot, in this case enabled through extra motors built into the legs of the platform. Our choice fell on a four legged robot, which gives a good balance between power efficiency and control complexity requirements. During the design and planning phase, we focused on developing a robust platform applicable to a range of use cases in machine learning and general robotics.
There are many different ways to parameterize the morphology of a four legged mammal-inspired robot. We decided to pursue variable length legs, as this seemed technologically feasible with a good effect on behavior and performance. The femur and tibia of each leg can all change their lengths by 50mm and 100mm respectively, as seen in Figure \[fig.lengthComparison\]. We believe that changing the length of the legs allows the robot to do a mechanical gearing of the servos, which can trade speed for force, to adapt to situations where the trade-off between these vary. We did initial experimentation both inside the lab and in realistic outdoor environments to investigate this effect, and showed that shorter legs worked better in challenging environments, while simple environments favored longer legs [@tonnesfn19icra].
Doing evolution exclusively in hardware has a range of associated challenges. One of the most important, is that the evaluation budget is severely limited when compared to physics simulation. The longer the experiments lasts, the higher the chance of sudden damage to the robot, as well as gradual wear and tear affecting the results. Each evaluation can also take several seconds to minutes, and often requires human intervention throughout the experiment. Having a system that is able to be adjusted to fit the evaluation budget at hand (as well as the requirements from the task and environment) is very important. We have implemented a control system with variable complexity that can be adjusted to the evaluation budget, as well as the requirements from the environment and task [@nygaard_evostar19].
In our first evolutionary experiment, we wanted to investigate to what degree evolution would be able to exploit dynamic morphology, given a robot with adaptable body and control. Changing control parameters gives a much more direct and commanding effect on the behavior of the system, so one might assume that the algorithm would spend most effort on control, and not necessarily utilize morphology to a significant degree. We varied the voltage to our servos, resulting in two very different conditions for the robot. Each evolutionary run took about an hour, and we did 6 runs in total. Our experiments showed that evolution was able to adapt to the reduction in servo torque of about 20%, and utilized both control and morphology to achieve this [@tonnesfn_gecco18].
We also wanted to investigate whether the robot would be able to exploit the dynamic morphology while walking on different surfaces. We ran a longer evolutionary experiment on two different types of carpets, and saw that the evolutionary search yielded significantly different bodies for the two surfaces [@nygaard2020environmental]. Each full evolutionary run took just under two hours, and we did 10 runs in total. We also tested the evolved individuals on previously unseen surfaces, where the robot performed better on surfaces qualitatively similar to the one they were evolved on, implying our method could be generalized to unknown terrains as well.
![Our robot during testing in one of the outdoor environments, with ice and snow during Norwegian winter.[]{data-label="fig.outside_path"}](figures/outside_path.jpg)
Conclusion
==========
In this paper, we have demonstrated the usefulness of real-world evolution of robot morphology and control – a concept that has previously been considered unfeasible due to the obvious challenge of efficiently changing a robot’s physical body. By evolving in the real world, we show that evolution adapts both morphology and control to different real-world scenarios that would have been very hard –if not impossible– to simulate. We discuss the design considerations for our robotic platform making these kinds of experiments possible, and the level of abstraction necessary for the evolutionary optimization. We hope to encourage further exploration of morphology and control evolution on real-world robots for adapting to complex environments.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We report on the use of elliptical pump spatial modes to increase the observed brightness of spontaneous parametric downconversion in critically phase matched crystals. Simulations qualitatively predict this improvement which depends on the eccentricity and orientation of the pump ellipse. We experimentally confirm a factor of two improvement in brightness when compared to the traditional circular-symmetric pump spatial modes. These results support previous theoretical work that proposes the use of elliptical pump modes to enhance the performance of parametric processes in anisotropic materials.'
author:
- Aitor Villar
- Arian Stolk
- Alexander Lohrmann
- Alexander Ling
bibliography:
- 'bibliography.bib'
title: Enhancing SPDC brightness using elliptical pump shapes
---
Entangled photon pair sources play a crucial role in fundamental tests of nature [@bouwmeester1997experimental; @yin2017satellite] and emerging quantum applications [@poppe2004practical; @bell2013multicolor]. Spontaneous parametric downconversion[@burnham1970observation] (SPDC) is the most common method to generate these photon pairs, with a variety of designs[@kwiat1999ultrabright; @fedrizzi2007wavelength; @villar2018experimental] and materials used. In SPDC, a photon (pump, $p$) in a nonlinear crystal can downconvert into two photons (signal, $s$; idler, $i$) obeying energy and momentum conservation.
Early studies identified the influence of pump parameters on parametric brightness[@boyd1966theory]. The most comprehensive review on pump optimization was made by Boyd and Kleinman[@boyd1968parametric], who suggested the use of circular symmetric spatial modes for pumping nonlinear processes. Though used most often, assumptions about circular (isotropic) symmetry might not optimize frequency conversion given that the vast majority of parametric processes take place in anisotropic materials. A specific case of interest, where the anisotropy is important, is when using critical phase matching in birefringent materials. When the pump is extraordinarily polarized it experiences spatial walk-off, as depicted in Fig. \[fig:experiment\](a). The interplay of pump spatial modes with walk-off was first studied by Volosov[@volosov1970effect] and Kuizenga[@kuizenga1972optimum]. They showed that elliptical pump shapes could improve frequency conversion performance in second-harmonic generation (SHG) and other parametric processes. Their main observation was that the spatial walk-off limits the overlap of any parametric emission (see Fig. \[fig:experiment\](b)). This restricts the benefits of tight pump focusing in the direction of walk-off. In contrast, this restriction is absent in the non-walk-off direction, which allows for tighter focusing. Hence, it is natural to consider different focusing conditions for different directions, leading to an elliptical (astigmatic) spatial mode.
![(a) Side view of the pump spatial walk-off (blue line) within the crystal. The collinear SPDC emission is depicted by red arrows. The dotted line is the crystal’s optical axis. (b) Sketch adapted from[@kuizenga1972optimum] representing the overlap between the pump beam (solid lines) and the downconversion beams (dotted lines), depicting the importance of controlling the pump focusing in order to maximize the overlap. (c) Collinear SPDC emission profile at the exit face of the crystal when the major axis of the pump ellipse (inset) is parallel to the walk-off direction. (d) Same as (c) but the ellipse major axis is rotated by 90$\degree$. (e) Experimental setup, 1: single-mode fiber pump output, 2: collimation lens, 3-4: cylindrical lenses, 5: focusing lens, 6: BBO crystal, 7-8: excess pump removal optics, 9: single-mode fiber, 10: photon separation via a dichroic mirror followed by coincidence detection setup.[]{data-label="fig:experiment"}](pics/experiment.pdf)
Experimental studies on SHG using elliptical pump beams verified the predicted gain, but the general observation was that the additional experimental complexity involving cylindrical focusing appeared to offset any enhancement[@steinbach1996cw; @freegarde1997general]. This observation stems from the challenges of operating elliptical beams in a resonant cavity. However, single-pass parametric processes such as SPDC are exempt from these difficulties.
In this work, we study the effect of combining elliptical pump and circular-symmetric collection modes in type-I SPDC with negative birefringent materials. Our approach is outlined in Fig. \[fig:experiment\]. We distinguish between two pump orientations; parallel (Fig. \[fig:experiment\](c)) and orthogonal (Fig. \[fig:experiment\](d)) to the walk-off plane (defined by the propagation and walk-off direction).
To systematically investigate the effect of eccentricity the pump is launched from a single-mode fiber before undergoing cylindrical focusing. The eccentricity of the pump is defined via the aspect ratio $r=\frac{\omega_y}{\omega_x}$, where $\omega_y$ ($\omega_x$) is the vertical (horizontal) waist of the ellipse. In our discussion, the vertical direction is associated with the walk-off. The change in pump aspect ratio was achieved by tuning only the waist of the horizontal axis. For the ideal pump shape, Kuizenga suggests that the size of the pump in the two directions are determined separately by considering whether walk-off is present.
To optimize the pump shape in each direction, we should utilize the birefringence parameter $B$[@boyd1968parametric]: $$B=\frac{\rho}{2}(lk_{0})^{1/2},
\label{eq:b_param}$$ where $\rho$ is the walk-off angle, $l$ is the length of the crystal and $k_0$ is the pump wave vector. In the direction where there is no walk-off, $B=0$.
The experimental layout for investigating elliptical pump performance is shown in Fig. \[fig:experiment\](e). A $\beta$-Barium Borate (BBO) crystal (cut-angle $\theta = 28.76\degree$) is pumped with light to generate type-I, collinear, non-degenerate SPDC wavelengths (signal and idler wavelengths at and , respectively). Excess pump power was removed by a dichroic mirror and a long-pass filter. Finally, the SPDC photons were coupled into a single-mode fiber and sent to a detection setup.
A range of pump aspect ratios were realized with different cylindrical lenses. Once a target aspect ratio was achieved, a plano-convex lens focused down the pump ellipse.
According to the literature [@boyd1968parametric], the optimal pump waist in the walk-off direction (given our crystal properties) is , whereas the optimal pump waist in the direction with no walk-off is . This gives the optimal aspect ratio of $r=2.9$.
These optimal values, however, do not take into account experimental limitations, such as aberrations introduced by thick lenses. Empirically, we found that in our setup a pump waist of optimized the brightness and collection efficiency when the aspect ratio was 1.0, with a collection waist of .
Therefore, to investigate the effect of the pump aspect ratio we fixed the pump size in the walk-off direction at . Correspondingly, the pump size in the non-walk off direction ranged from to . Thus, the range of aspect ratios that were implemented ranged from 4.44 to 0.66. The experiment was then repeated for the case where the major axis of the pump beam was orthogonal to the walk-off direction. In this case, the experimental aspect ratio investigated ranged from 2.94 to 0.65.
To provide better insight on the interplay of elliptical pump and circular collection modes a model was developed. Within the model, the SPDC intensity is governed by the conventional phase matching conditions of the respective ($p$, $s$ and $i$) fields. The classical $p$ field is treated as a probability distribution of $\vec{k}_p$, that follows the angular and spatial intensity distributions of a Gaussian beam profile.
For a given wavelength $\lambda_p$, starting position within the crystal $\{x,y,z\}$ and propagation direction $\vec{k}_{p}$, the weighting function $A$ can be obtained: $$A= e^{-\frac{( \omega_{x} \Delta k_{x} )^2+( \omega_{y} \Delta k_{y})^2}{2}} \cdot\operatorname{sinc}\bigg(\frac{l\Delta k_{z}}{2}\bigg)^2 .
\label{eq:sinc}$$ Here $\Delta \vec{k} = \Delta k_x\hat{x} + \Delta k_y \hat{y} + \Delta k_z \hat{z}$ is the phase mismatch in Cartesian coordinates, $\omega_{x,y}$ the beam waist in horizontal (*x*) and vertical (*y*) directions, and $l$ the crystal length. This weighting function determines the probability of generating a signal/idler pair with wavelengths $\lambda_{s}$ and $\lambda_{i}$ in the propagation directions $\hat{k}_{s}$ and $\hat{k}_{i}$. The individual treatment for each $\vec{k}_p$ accounts for effects of the wavefront curvature.
Using conventional ray-tracing techniques the $\vec{k}_{s},\vec{k}_{i}$ pair was propagated through the downstream optics towards a single-mode fiber (numerical aperture of $0.1$ and core diameter ). The overlap of the SPDC rays with the collection mode of the fiber determines the in-fiber photon pair brightness. The model is available online at: <https://github.com/arianstolk/SPDC_test/>.
The simulated and experimental results are shown in Fig. \[fig:results\_1\] and Fig. \[fig:results\_2\]. When the major axis of the pump ellipse is parallel to the walk-off direction, the experimental data agrees qualitatively with the model, which predicts increased brightness for larger pump aspect ratio. The best observed improvement over the circular symmetric mode is a factor of 2. When the major axis of the pump ellipse is orthogonal to the walk-off direction, there is a dramatic difference as the brightness shows little response when the aspect ratio changes.
In the case where the major axis is aligned with the walk-off direction, increasing the aspect ratio monotonically increased the photon pair brightness that can be observed. Due to experimental difficulties, the optimal aspect ratio of 2.9 was not achieved. However, it is instructive to consider the results of the two aspect ratios that bracket this value. These results showed a strong dependence of brightness on the orientation of the pump major axis, validating Kuizenga’s proposal.
In Fig. \[fig:results\_1\](c), there is a strong linear relationship between the brightness and collection waist, which is not present in the model.This is attributed partly to the lack of precision within the model which uses the thin lens approximation for modeling the single-mode fiber photon collection and ignores the lens thickness. Furthermore, the observed $M^2$ value of the pump beam in the walk-off direction deviates from the ideal case for large aspect ratio (see Tab. \[tab:params\] in Appendix) due to lens aberrations when using thick cylindrical lenses.
Improvements in the model (e.g. improved approximations for ray tracing of the curved optics) will help to match the experimental results and provide better predictions about the overall source performance. Additionally, an extension of this work would be the use of elliptical collection modes that are tailored to the pump profile and crystal anisotropy.
When considering asymmetric effects of the phase matching conditions on the SPDC profile, the use of a correctly oriented elliptical pump mode showed improved in-fiber brightness over the traditional circular symmetric modes. This deviates from experimental results in SHG in optical parametric oscillators, where the marginal improvement in performance did not warrant the expense of increased experimental complexity. Elliptical pump modes have already improved the brightness of entangled photon pairs sources where the crystal geometry allows an elliptical pump mode to be implemented[@villar2018experimental; @lohrmann2018high].
In many modern SPDC sources the pump is typically obtained from a laser diode which inherently exhibits an elliptical profile. Taking into account crystal anisotropy and beam walk-off it might be unnecessary to correct the elliptical pump profile, simplifying the construction of photon pair sources while improving performance.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research is supported by the National Research Foundation, Prime Minister’s Office, Singapore under its Competitive Research Programme (CRP Award No. NRF-CRP12-2013-02). This program was also supported by the Ministry of Education, Singapore.
The Boyd-Kleinman optimal circular symmetric pump beam calculation
==================================================================
In order to calculate the optimal circular symmetric pump beam suggested by Boyd-Kleinman, first the $B$ parameter of the nonlinear crystal needs to be calculated. In our case, the properties of the BBO crystal: $\rho=\SI{3.83}{\degree}$, $l=\SI{5}{\milli\metre}$ and $\lambda_0 = \SI{405}{\nano\metre}$. From Eq. \[eq:b\_param\] in the main text, $B=9.3$.
Knowing the $B$ parameter, the optimal $\xi$ value can be inferred via the optimization curves shown in Fig. \[fig:bk-curves\]:
![Optimization curves for circular symmetric beams reported in [@boyd1968parametric]. For different walk-off parameters (the $B$ number is shown above each curve) $\xi$ (x-axis) and the figure of merit $h_m$ (y-axis) is shown. Resorting to this curves and knowing the walk-off angle and the thickness of the crystal, the optimal circular symmetric beam for a particular wavelength $\lambda_0$ can be determined.[]{data-label="fig:bk-curves"}](pics/BK-single.pdf)
For $B=9.3$, $\xi$ is calculated to be 0.6579. From here, the ideal beam waist can be calculated, since $\xi=\frac{l}{b}=\frac{l\cdot\lambda}{\omega^2 \cdot 2\pi}$. In our case, $\omega=\SI{22.13}{\micro\metre}$.
Pump beam parameters
====================
Pump beam parameters used in both simulation (*Sim*) and experiment (*Exp*) are depicted in Tab. \[tab:params\]. Due to experimental factors (e.g., lens aberrations) when shaping the pump mode, the experimental Rayleigh length values are lower than the ideal values used in the simulation ($M^{2}_{\perp},M^{2}_{{\mathbin{\!/\mkern-5mu/\!}}} = 1$). This explains the difference in decaying rates between simulation and experiment.
[|c| c | c | c | c | c | c | c | c |]{} & & &$M^{2}_{{\mathbin{\!/\mkern-5mu/\!}}}$ & &$M^{2}_{\perp}$\
&Sim&Exp&Sim&Exp&Exp&Sim&Exp&Exp\
& 0.67 & 0.66 & 174.5 & 145.6 &1.1& 77.6 & 69.1&1.1\
& 1.0 & 1.03 & 77.6& 65.2 &1.1 & 77.6 &70.9 &1.1\
& 2.0 &1.85 & 19.4& 20.9 &1.0 & 77.6 &49.1&1.5\
& 3.0 & 4.44 & 8.4& 5.0 &1.0& 77.6& 30.8 &1.7\
& 0.67 & 0.65 & 77.6 & 76.0 &1.0& 174.5 & 155.5&1.1\
& 1.00& 0.97 & 77.6 & 65.2&1.1 & 77.6& 70.9&1.1\
& 2.0 &2.02 & 77.6& 66.2&1.1 & 19.4 &17.3&1.2\
& 3.0 &2.94 & 77.6 &70.9&1.0 & 8.4 &8.2&1.0\
\[tab:params\]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We use the measured X-ray luminosity function (XLF) of high-mass X-ray binaries (HMXBs) in nearby star-forming galaxies to constrain the common envelope (CE) mechanisms, which play a key role in governing the binary evolution. We find that the XLF can be reproduced quite closely under both CE mechanisms usually adopted, i.e., the $\alpha_{\rm CE}$ formalism and the $\gamma$ algorithm, with a reasonable range of parameters considered. Provided that the parameter combination is the same, the $\gamma$ algorithm is likely to produce more HMXBs than the $\alpha_{\rm CE}$ formalism, by a factor of up to $\sim$ 10. In the framework of the $\alpha_{\rm CE}$ formalism, a high value of $\alpha_{\rm CE}$ is required to fit the observed XLF, though it does not significantly affect the global number of the HMXB populations. We present the detailed components of the HMXB populations under the $\gamma$ algorithm and compare them with those in Zuo et al. and observations. We suggest the distinct observational properties, as well as period distributions of HMXBs, may provide further clues to discriminate between these two types of CE mechanisms.'
author:
- 'Zhao-Yu Zuo$^{1,3}$ and Xiang-Dong Li$^{2,3}$'
title: 'Common Envelope Mechanisms: Constraints from the X-ray Luminosity Function of High Mass X-ray Binaries'
---
Introduction
============
The common envelope (CE) evolution is among the most important and least well-constrained processes in binary evolution. It is commonly thought to occur if the mass transfer is dynamically unstable. The result is that the accreting star spirals into the envelope of the donor star [see @iben93; @taam00; @webbink08; @taam10; @Ivanova13 for reviews]. The orbital energy and angular momentum of the accreting star are then transferred into the CE via an as of yet unknown mechanism. This may end with a stellar merger or, if the binary can survive, a binary with a much shorter orbital period. The CE evolution is critical in the formation of various kinds of compact binaries.
There have been extensive three dimensional hydrodynamical simulations [e.g., @rl96; @stc98; @stb00; @Fryxell00; @OShea05; @Fryer06; @Passy12; @rt12]. However the physics of CE evolution still remains poorly understood, primarily due to a mix of various kinds of physical processes operating over a large range of timescales and length scales during the CE phase. Due to the difficulties in modeling the detailed CE evolution, population synthesis simulations commonly resort to simplified and parameterized descriptions to relate the post- and pre-CE orbital parameters [@ty79]. One such parametrization dictates the CE phase in terms of a simple energy budget [known as the $\alpha_{\rm CE}$ formalism, @heuvel76; @Webbink84; @ls88; @iben93] and the other in terms of the angular momentum budget [named as the $\gamma$ algorithm, @nvyp00; @nt05]. Both approaches have the power to account for some specific classes of post-CE binaries (PCEBs), such as cataclysmic variables [CVs, @king88], subdwarf B binaries [@mr03; @han02; @han03], low-mass X-ray binaries [@cc06], and other compact objects thought to have suffered a merger, which are probably responsible for gamma-ray bursts [@fryer99; @Thone] and Type Ia supernova [SNe Ia, @it84; @bbr05; @ruiter11; @meng11; @my12; @wh12]. There is an energetic debate over the two approaches in the literature.
High-mass X-ray binaries (HMXBs) are good examples of the effect of CE evolution. Luminous HMXBs usually have experienced CE evolution so that they have close orbits, which lead to a high wind-capture rate by the compact star. Tight orbits also help the binary survive the SN kick during the formation of the compact star. However, if the initial binary orbit is not large enough, CE evolution may lead to mergers, reducing the HMXB production. Thus, the populations and the specific characteristics (for example the orbital period distribution) of HMXBs can be used to probe the physical interactions during the CE phase.
The formation of HMXBs involves several evolutionary pathways [@bv00; @Linden10 see also Tauris & van den Heuvel 2006]. Beginning with two relatively massive stars ($\gtrsim\,10\,M_{\odot}$), the more massive primary evolves and commences mass transfer to the secondary. The mass transfer can be either dynamically stable or unstable. In the latter case, CE evolution occurs that greatly shrinks the binary orbit. The resultant binary consists of the primary’s core and the secondary. The primary’s core then collapses to form a neutron star (NS) or black hole (BH). An HMXB appears when the compact star is able to accrete from the secondary by capture of the stellar wind or Roche lobe overflow (RLOF). Note that the secondary can be on the main-sequence (MS) or a (super)giant star. In some cases the second mass transfer may also lead to a CE phase, during which the envelope of the secondary is stripped, leaving a naked helium core. This leads to the formation of HMXBs with Wolf-Rayet companions.
HMXBs have some unique statistical characteristics [for catalogs, see @lph05; @lph06]. One of the most striking features is that their X-ray luminosity function (XLF) follows a universal power law form over a broad X-ray luminosity range, from $\sim 10^{35}$ to $\sim 10^{40} \rm \,erg\,s^{-1}$. This was first discovered by @ggs03, and further confirmed recently by @mineo12 [hereafter MGS12 for short]. The XLF has been shown to follow a power law with a single slope of $\sim$ 1.6, without any significant feature near the critical Eddington luminosity of an NS or a stellar mass BH. Additionally, the collective luminosity of HMXB populations scales with the star formation rate (SFR) as $L_{\rm X} (\rm erg\,s^{-1}) \approx 2.6\cdot10^{39}\times \rm SFR (M_{\odot}\,yr^{-1})$.
In the present work, we apply the updated evolutionary population synthesis (EPS) techniques to model the XLF of HMXBs, taking into account both the $\alpha_{\rm CE}$ algorithm and $\gamma$ algorithm (with different choices of $\alpha_{\rm CE}$ and $\gamma$, respectively), to describe the CE evolution. By comparing the observational sample with our theoretical expectations, we try to discriminate or constrain the effects of the two CE mechanisms.
This paper is organized as follows. In §2 we describe the population synthesis method and the input physics for X-ray binaries (XRBs) in our model. The calculated results and discussions are presented in §3. Our conclusions are in §4.
MODEL DESCRIPTION
=================
We use the EPS code developed by @Hurley00 [@Hurley02] and recently updated by @zuo14a to calculate the expected number and the X-ray luminosity of HMXBs. In the present code, the model for compact object formation has been significantly revised by taking into account the formation of NSs through electron capture supernovae [ECS, @Podsiadlowski04] and the fallback process for both delayed and direct BH formation during core collapse [@fk01]. The prescriptions for the wind mass loss rates of massive stars [@vink01 see also Belczynski et al. 2010] and the compact remnant masses [@fryer12 see also Belczynski et al. 2012] are adopted in the code. We also update the criteria for CE occurrence as described below.
The CE Phase
------------
During the binary evolution, the mass ratio ($q=M_{\rm donor}/M_{\rm accretor}$) is a crucial factor determining the stability of mass transfer. If it is larger than a critical value, $q_{\rm crit}$, the mass transfer is dynamically unstable and a CE phase follows [@p76]. The ratio $q_{\rm crit}$ varies with the evolutionary state of the donor star at the onset of RLOF and the mass loss mechanisms during the mass transfer [@hw87; @w88; @prp02; @ch08]. In this study, we adopt an updated $q_{\rm crit}$ for Hertzsprung gap donor star, recently calculated by @shao12 [also see the Appendix A in Zuo, Li & Gu 2014a for more details]. If the primary is on the first giant branch (FGB) or the asymptotic giant branch (AGB), we use $$q_{\rm crit}=[1.67-x+2(\frac{M_{\rm c1}}{M_1})^5]/2.13$$ where $M_{\rm c1}$ is the core mass of the donor star, and $x$=d ln$R_1/$d ln$M$ is the mass-radius exponent of the donor star. If the mass donor star is a naked helium giant, $q_{\rm crit}=0.784$ [see @Hurley02 for more details].
### The $\alpha_{\rm CE}$ formalism
In the energy budget approach, the CE evolution is parameterized in terms of the orbital energy and binding energy as $E_{\rm bind} \equiv \alpha_{\rm CE}
\triangle E_{\rm orb}$ [@Webbink84; @webbink08], where the parameter $\alpha_{\rm CE}$ describes the efficiency of converting the orbital energy (**$E_{\rm orb}$**) into the kinetic energy, which is used to eject the envelope, and $E_{\rm bind}$ is the binding energy of the envelope. The CE evolution is governed by the following equation [@Kiel06]: $$\alpha_{\rm CE}[\frac{GM_{\rm c}M_{2}}{2 a_{\rm f}}-\frac{GM_{\rm
c}M_{2}}{2 a_{\rm i}}]=-\frac{GM_1M_{\rm env}}{R_{L_1}\lambda},$$ which yields the ratio of final (post-CE) and initial (pre-CE) orbital separations as $$\frac{a_{\rm f}}{a_{\rm i}}= \frac{M_{\rm c}M_{2}}{M_{1}}
\frac{1}{M_{\rm c}M_2/M_1+2M_{\rm env}/(\alpha_{\rm CE} \lambda
R_{\rm L1})},$$ where $G$ is the gravitational constant, $M_{\rm c}$ the helium-core mass of the primary star (of mass $M_1$), $M_2$ the mass of the secondary star, $R_{L_1}$ the RL radius of the primary star, $M_{\rm env}$ the mass of the primary’s envelope, $a_i$ and $a_f$ denote the initial and final orbital separations, respectively, and $\lambda$ is a parameter related to the stellar mass-density distribution.
The $\lambda$ value depends on the structure and evolution of the donor star. However, in previous studies, it was usually adopted as constant ($\sim 0.5$) for simplicity [@Hurley02; @zuo08]. Here we calculate the values of $\lambda$ from detailed stellar models including the contribution from the internal (and ionization) energies within the envelope [@zuo14a also see Xu & Li 2010 and Loveridge et al. 2011].
We consider three constant, global values of $\alpha_{\rm CE}$. For our basic model, we use $\alpha_{\rm CE}=0.5$ [@zuo14a]. We also consider two other extreme values of $\alpha_{\rm CE}=1.0$ and 0.1 since $\alpha_{\rm CE}$ is expected to be no more than unity if we consider the internal energies in calculating $E_{\rm bind}$. Different CE efficiencies for the first and second CE episodes are also examined to test its effect on the XLFs. Models with different values of $\alpha_{\rm CE}$ are denoted as A01A01, A05A05, A10A10, A01A05, and A05A01, respectively, where the two digits following each letter correspond to the values of $\alpha_{\rm CE}$ during the first and second CE episodes, respectively.
Alternatively, recent studies on WD binaries show that $\alpha_{\rm CE}$ may be a function of binary parameters rather than constant [@pw07; @zgn00; @marco11; @dkk12], although the final relationship has not yet been well developed. Following @marco11, we adopt $$\alpha_{\rm CE}=0.05\times q^{1.2},$$ where $q$ is the ratio of the donor’s mass to the accretor’s mass at the time of the CE interaction, and this model is denoted as AqAq.
### The $\gamma$ algorithm
In the angular momentum budget approach, the CE interaction is parameterized in terms of $\gamma$, the ratio of the fraction of angular momentum lost, and the fraction of mass loss: $$\frac{\triangle J}{J}=\gamma \frac{M_{\rm env}}{M_1+M_2}$$ where $\triangle J$ is the change of the total angular momentum ($J$) during the CE phase. Implicitly assuming the conservation of energy, the orbital separation after the CE is then given by $$\frac{a_{\rm f}}{a_{\rm i}}=(\frac{M_1}{M_{\rm c}})^2(\frac{M_{\rm c}+M_2}{M_1+M_2})
[1-\gamma (\frac{M_{\rm env}}{M_1+M_2})]^2$$
This description was first suggested by @nvyp00 in their investigation of the formation of double WD binaries. They found that when the energy approach is applied to describe the first CE phase, a negative value of $\alpha_{\rm CE}$ is required, which is clearly unphysical. Among the possible solutions leading to the known close double WDs, @nt05 found that $0.5<\gamma<3$ for the first (putative) CE phase, and $1<\gamma<4$ for the second CE phase. They noted that a value of $\gamma$ between 1.5 and 1.75 can account for all known observed PCEBs, including double WDs, pre-CVs, and sdB plus MS binaries.
For the $\gamma$ algorithm, we consider several constant, global values of $\gamma$ from 1.7 to 1.0, as well as different $\gamma$ values for the first and second CE episodes in our calculation. These models are denoted as G17G17, G15G15, G13G13, G10G10, G10G17 and G17G10 where the two digits following each letter correspond to the values of $\gamma$ during the first and second CE episodes, respectively.
In the study, we first compare the two mechanisms under the same assumptions, as described below. The parameter combination is kept the same as in @zuo14a, where the best-fit model in the $\alpha_{\rm CE}$ formalism is achieved. In this case, only values of $\gamma$ and $\alpha_{\rm CE}$ are changed to see their effects on the XLF. Then we manage to determine the best-fit model in the $\gamma$ algorithm by varying all the key parameters, and see their effects on the XLF (see Table 1). Finally, the two mechanisms are compared under each best-fit model (i.e., model A05A05 vs. model M1).
\[tab:m7\]
Model $P(q_0)$ IMF $f$ $\eta_{\rm Edd, BH}$ $\sigma_{\rm kick}$ winds
------- ------------------- ---------- ----- ---------------------- --------------------- ------- --
M1 $\propto q_0^{0}$ Salpeter 0.5 20 110 STD
M2 $\propto q_0^{0}$ Salpeter 0.5 20 110 WEAK
M3 $\propto q_0^{0}$ Salpeter 0.8 20 110 STD
M4 $\propto q_0^{0}$ Salpeter 0.5 100 110 STD
M5 $\propto q_0^{1}$ Salpeter 0.5 20 110 STD
M6 $\propto q_0^{0}$ MT87 0.5 20 110 STD
M7 $\propto q_0^{0}$ Salpeter 0.5 20 190 STD
M8 $\propto q_0^{0}$ Salpeter 0.5 20 265 STD
: Parameters adopted for each model under the $\gamma$ algorithm. Here $q_0$ is the initial mass ratio, IMF is the initial mass function, $f$ binary fraction, $\eta_{\rm Edd, BH}$ - the factor of super-Eddington accretion rate allowed for BH XRBs, $\sigma_{\rm kick}$ is the dispersion of kick velocity, $\eta_{\rm bol, BH(NS)}$ is the bolometric correction factor for BH(NS) XRBs, STD is the standard stellar winds while WEAK represents the standard wind mass loss rate reduced to 50%, MT87 represents the IMF of @mt87. In the best-fit model of $\gamma$ algorithm (M1), the parameters are as follows: SFH=50 Myr, $\alpha=0$, $\eta_{\rm Edd, BH}$=20, $f=0.5$, $\sigma_{\rm kick} = 110 \,\rm km\,s^{-1}$, $\eta_{\rm bol, BH}$=0.2, $\eta_{\rm bol, NS}$=0.1 and Salpeter IMF.
Input Parameters
----------------
We follow the evolution of a large number of binary systems, initially consisting of two zero-age MS stars. As HMXBs in the MGS12 samples reside in nearby star-forming galaxies, we adopt a constant SFR for 50 Myr and a fixed subsolar metallicity ($0.5\,Z_{\odot}$, where $Z_{\odot}=0.02$) accordingly [see @zuo14a for details]. Since the observed average XLF has already been normalized, we choose a Salpeter initial mass function (IMF) and set the mass range as $0.1-100\,\rm M_{\odot}$ for the normalization in order to be in parallel with MGS12[^1]. We evolve $10^6$ primordial systems[^2] and set up the same grid of initial parameters (primary mass, secondary mass and orbital separation) as in @Hurley02.
For the initial secondary’s mass ($M_2$), a power law distribution of $P(q_0)\propto q_0^{\alpha}$ is assumed, where $q_0\equiv M_2/M_1$. In our basic model, a flat distribution is assumed, i.e., $\alpha=0$. We adopt a logarithmically flat distribution of initial orbital separations $\ln a$ [@Hurley02].
We assume a binary fraction $f=0.5$ and that all binaries are initially in a circular orbit. For the SN kicks imparted on an NS, we assume a Maxwellian distribution with $\sigma_{\rm kick}=
110\, \rm km\,s^{-1}$ [@zuo14d]. For compact objects formed with partial mass fallback, the natal kicks are decreased by a factor of (1-$f_{\rm b}$) where $f_{\rm b}$ is the fraction of the stellar envelope that falls back after the SN explosion.
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X-ray luminosity and source type
--------------------------------
We adopt the same procedures to compute the $0.5-8$ keV X-ray luminosity for MS/super-giant (SG) HMXBs and Be-XRBs as in @zuo14a. For wind accretion, we use the classical @Bondi44 formula to calculate the mass transfer rate to the compact star. In the case of RLOF, we discriminate transient and persistent sources using the criteria in @l01 [i.e., Eq. 36 therein] for MS and red giant (RG) donor. The corresponding X-ray luminosity is calculated as follows: $$\begin{aligned}
&&L_{\rm X, 0.5-8 keV}\nonumber\\
&&=\left\{
\begin{array} { ll}
\eta_{\rm bol}\eta_{\rm out}L_{\rm Edd}&\ \rm transients\ in\ outbursts, \\
\eta_{\rm bol}\min(L_{\rm bol},\eta_{\rm Edd}L_{\rm Edd})&\ \rm persistent\
systems,
\end{array}
\right.\end{aligned}$$ where $\eta_{\rm bol}$ is the bolometric correction factor converting the bolometric luminosity ($L_{\rm bol}$) to the $0.5-8$ keV X-ray luminosity, ranging between $\sim 0.1$ and $\sim 0.8$ [@bel08]; $L_{\rm bol} \simeq 0.1\dot{M}_{\rm acc}c^2$ where $\dot{M}_{\rm acc}$ is the average mass accretion rate and $c$ is the velocity of light. The critical Eddington luminosity $L_{\rm Edd} \simeq 4\pi GMm_{\rm
p}c/\sigma_{T}=1.3 \times 10^{38}m$ergs$^{-1}$ (where $\sigma_{T}$ is the Thomson cross section, $m_{\rm p}$ the proton mass, and $m$ the accretor mass in the units of solar mass). We introduce the ‘Begelman’ factor $\eta_{\rm Edd}$ to allow super-Eddington luminosities. We fix $\eta_{\rm Edd, NS}=5$ for NS XRBs [@zuo14a]; for BH XRBs, $\eta_{\rm Edd, BH}$ is set as a free parameter in the study. For transient sources, the outburst luminosity is taken as a fraction ($\eta_{\rm out}$) of the critical Eddington luminosity. We take $\eta_{\rm out}=0.1$ and 1 for NS(BH) transients with orbital period $P_{\rm orb}$ less and longer than 1 day (10 hr), respectively [@chen97; @Garcia03; @bel08]. For Be-XRBs we employ a phenomenological definition as in @zuo14a [also see Belczynski & Ziolkowski, 2009]. Technically, we randomly select 25% [$f_{\rm Be}=0.25$, @s88; @z02; @mg05] of NS binaries hosting a ($3.0\,M_{\odot}-20.0\,M_{\odot}$) B/O star to be Be-XRBs, and estimate their numbers. The X-ray luminosity of a Be-XRB is calculated using the empirical relation (Eq. 11) in @dll06, which is based on the data compiled by @rp05. Considering the duration of type I outbursts in Be-XRBs [$\sim 0.2-0.3 P_{\rm orb}$, @reig11], we adopt an upper value of the duty cycle $DC_{\rm max}=0.3$ to calculate the source numbers.
Results
=======
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We first compare the results in the $\alpha_{\rm CE}$ formalism and $\gamma$ algorithm under the same input parameters: SFH=50 Myr, $\alpha=0$, $\eta_{\rm Edd, BH}$=100, $f=0.5$, $\sigma_{\rm kick} = 110 \,\rm km\,s^{-1}$, $\eta_{\rm bol, BH}$=0.6, $\eta_{\rm bol, NS}$=0.3 and Salpeter IMF [@zuo14a]. For each CE episode, models are designed by changing only one parameter each time to test its effect. Figure 1 compares the simulated XLFs with different treatments of the CE phase. Clearly, under the same parameter combination the $\gamma$ algorithm can produce more (up to one order of magnitude) HMXBs than the $\alpha_{\rm CE}$ formalism. In the framework of the $\alpha_{\rm CE}$ formalism, though all models can fit the observed XLF quite closely in most of the luminosity range (i.e., $10^{35}-\sim 10^{39}\,\rm
ergs\,s^{-1}$), a high value of $\alpha_{\rm CE}$ seems more preferable. This is mainly due to the sparseness of short period RLOF HMXBs in the case of smaller $\alpha_{\rm CE}$ [compare with the right panel of Figure 1 in @zuo14a], the progenitors of which coalesce during the binary evolution, especially in the first CE phase (see models A05A01 and A01A01). In the case of $\gamma$ algorithm, the normalization of the simulated XLFs is rather sensitive to the value of $\gamma$, especially in the first CE phase (see models G10G17 and G17G17 or models G10G10 and G17G10). Smaller values of $\gamma$ give a better fit to the observed XLF, not only in the normalization, but also in the overall shape. Considering that many parameters may considerably influence the XLF [@zuo14a], further thorough parameter studies are needed to determine the best-fit model in the $\gamma$ algorithm.
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The key parameters we vary include: the binary fraction $f$, the super-Eddington factor $\eta_{\rm Edd}$, the bolometric correction factor $\eta_{\rm bol}$, the mass ratio, the IMF, the natal kick distribution, the wind mass loss rates and the value of $\gamma$. Some parameters affect only the normalization, such as $f$ and $\eta_{\rm bol}$; some affect only the shape, for example, $\eta_{\rm Edd}$; while others affect both. We perform a suite of EPS models and find that the best-fit model in the $\gamma$ algorithm can be achieved when parameters are adopted as follows: SFH=50 Myr, $\alpha=0$, $\eta_{\rm Edd, BH}$=20, $f=0.5$, $\sigma_{\rm kick} = 110 \,\rm km\,s^{-1}$, $\eta_{\rm bol, BH}$=0.2, $\eta_{\rm bol, NS}$=0.1, Salpeter IMF and $\gamma=1.0$ (i.e., model M1). We also examine other values of $\gamma$ , and find that the results are not better than in the case of $\gamma=1.0$ (especially when $\gamma \gtrsim\,1.5$). In order to show the dependences of the XLF on the parameters, we also change these parameters one by one. The details are listed in Table \[tab:m7\].
Figure 2 clearly shows that the parameters act in different ways. Several parameters have only minor effects, i.e., the wind mass loss rate (model M2) and the initial mass ratio distribution of the secondary star (model M5). Some (e.g. models M3 and M6) mainly increase the number of HMXB populations. An increase of the binary fraction (model M3) gives more XRBs, hence an overall shift of the XLF. A flatter IMF (model M6) reflects more massive stars, hence more compact objects that may result in XRBs. An increase of the dispersion velocity $\sigma_{\rm kick}$ (models M7 and M8) means that the natal kicks of higher magnitude are chosen more frequently from the Maxwellian distribution, hence more disruptions of binaries during the SN explosions. This decreases the number of potential HMXBs, and meanwhile changes the shape of the XLF. We note the large uncertainties in $\sigma_{\rm kick}$, $f$, and $\eta_{\rm bol}$ make it difficult to tightly constrain the value of $\gamma$. The apparent luminosity ‘knee’ of XLFs is weakened if we restrict the super-Eddington factor to 20 (compare model M4 with others), implying that the maximum super-Eddington luminosity allowed is likely $\sim 20$ in the case of $\gamma$ algorithm. To sum up, in the framework of the $\gamma$ algorithm, the observed XLF can also be reconstructed within the reasonable range of the parameters adopted.
In order to explore the nature of HMXBs in the case of $\gamma$ algorithm, we also examine the detailed observational properties (i.e., orbital period, the current mass $M_2$ of the donor star, etc.) of the simulated HMXBs, and compare them with those in @zuo14a [i.e, $\alpha_{\rm CE}$ formalism] and observations. Shown in Figure 3 are the detailed components of the simulated XLF (left) and the accretion modes in XRBs (right) and in Figure 4 are the $P_{\rm orb}-L_{\rm X}$ (left) and $P_{\rm orb}-M_2$ (right) distributions in model M1. It is clear that under the $\gamma$ algorithm BH-He XRBs dominate in the low luminosity range (i.e., $L_{\rm X}<\sim10^{37}\rm\,erg\,s^{-1}$) of the XLF while this is not the case in the $\alpha_{\rm CE}$ formalism, where BH-MS XRBs instead dominate [@zuo14a]. Unfortunately, due to the limited instrument capabilities available, most of the extragalactic X-ray sources remain unresolved. We still do not clearly know their nature (for example, the spectral type of the donor star and the type of the compact star), especially the sources in low luminosities. We suggest further check with higher-precision observations is still needed in the future. The orbital period distribution is also distinct from that in @zuo14a, with a much larger population of relatively short period (less than several tens of days) systems . This is more clearly revealed in Figure 5 for the normalized orbital period $P_{\rm orb}$ distribution in models A05A05 (left) and M1 (right). We can see that short period HMXB population keeps growing under the $\gamma$ algorithm, while most HMXBs under the $\alpha_{\rm CE}$ formalism are produced within the first 20 Myrs. These distinct observational properties of HMXBs, as well as different period distributions may provide further clues to discriminate between the two models.
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We note the discrepancy in the BH-He HMXB population between models is solely a result of different treatments on CE, in which the $\gamma$ algorithm predicts a survival, while the $\alpha_{\rm CE}$ formalism predicts, a merger instead. The progenitors of BH-He HMXBs always have the following features. First, the primary stars are very massive, $\sim 30-80\,M_{\odot}$, so they can form BHs in a mild (with low/no kicks) way, which will not disrupt the system. Second, the companion stars are relatively less massive, i.e., $\sim 10-35\,M_{\odot}$. The orbits of the binaries are not too wide, of the order of tens to hundreds of $R_{\odot}$. These conditions guarantee that the primary can overfill its RL rapidly and transfer mass in a dynamically stable way (because the mass ratio is not too extreme[^3]). After that, the primary evolves to a BH, and the rejuvenated secondary star expands and fills its RL. However, due to the large mass ratio, the mass transfer this time is unstable, and a CE is triggered. The $\alpha_{\rm CE}$ formalism always leads to binary mergers due to the huge amount of binding energy in the giant envelope. In the case of $\gamma$ algorithm, the orbital evolution is determined by the mass ratio $q=M_{\rm donor}/M_{\rm accretor}$ and the core mass fraction $\mu=M_{\rm c}/M_{\rm donor}$. From Eq. 6 in @nt05, it is easy to deduce that the binary orbit not only shrinks (but still different from that in the $\alpha_{\rm CE}$ formalism), but also expands (see also Figure 3 therein). This expansion of the orbit not only avoids binary mergers, but also delays the XRB formation significantly. This is also why, under the same assumptions the $\gamma$ algorithm can produce more HMXBs than the $\alpha_{\rm CE}$ formalism and the HMXBs can keep emerging after 20 Myr in the case of the $\gamma$ algorithm.
To illustrate the formation and evolution of a typical BH-He HMXB, we present one example evolutionary sequence for $M_1$, $M_2$, $P_{\rm orb}$, and $L_{\rm X}$ under the $\gamma$ algorithm in Figure 6. We consider a primordial binary system in a $\sim 91.44\,R_{\odot}$ circular orbit. The initial stellar masses are 35.493 and 12.532 $\,M_{\odot}$ for the primary and secondary, respectively. The primary evolves first, and fills its RL on the HG (at 5.5483 Myr). The mass transfer proceeds rapidly as it evolves across the HG until the end of CHeB, at which point (5.5598$\,$Myr) it becomes an 11.069$\,M_{\odot}$ HeMS star with a 34.451$\,M_{\odot}$ (rejuvenated) MS star in a $109.626\,R_{\odot}$ orbit. Shortly after that, the naked helium star evolves across the HeHG and collapses at 6.2418 Myr, leaving a 7.617$\,M_{\odot}$ BH with an MS companion in a 167.93 $\,R_{\odot}$ orbit. Subsequently, the MS star evolves to expand and fills its RL on the HG, and then the binary enters into the CE stage (10.5754 Myr). At this time, the mass ratio is $q\sim4.3$ and the core mass fraction $\mu\sim0.3$, and the orbit shrinks slightly from 134.38$\,R_{\odot}$ to 118.55$\,R_{\odot}$, as calculated from Eq. 6 in @nt05. At the end of the CE, the envelope of the giant star is expelled, leaving a 10.58$\,M_{\odot}$ HeMS star. The stellar wind from the HeMS star is then accreted by the BH, resulting in a BH-HeMS XRB. At last, the HeMS evolves to explode as an SN (11.28 Myr), which results in a 7.348$\,M_{\odot}$ BH and disrupts the binary system.
Our findings are generally consistent with other previous studies concerning the CE evolution. For example, in the case of the $\alpha_{\rm CE}$ formalism, a high value of $\alpha_{\rm CE}$ is required to account for the observed WDMS PCEBs [$\alpha_{\rm CE}\gtrsim 0.1$, @dkw10], the shape of the delay-time distribution and the birth rate of SNe Ia for the double-degenerate systems [@my12], and the displacements of HMXBs [$\alpha_{\rm CE}\sim 0.8-1.0$, @zuo14b], while a lower value of $\alpha_{\rm CE}$ may be excluded [@my12; @zuo14b]. An exception is from @f13 where a low value of $\alpha_{\rm CE} \sim 0.1$ is preferred, most likely due to the oversimplified treatments for the binding energy parameter, where $\lambda$ is adopted as one overall, while this is not the case for massive stars [$\lambda \sim 0.1$, @xu10]. It is interesting to note that to create double WDs, the standard $\alpha_{\rm CE}$ formalism is also possible if the first mass transfer between an RG and an MS star can be stable and non-conservative. This leads to a modest widening of the orbit, with an effect similar to the $\gamma$ algorithm [@woods12]. In the framework of the $\alpha_{\rm CE}$ formalism, our simulations are also comparable to previous studies concerning HMXB populations [@pv96; @tts98; @Linden10]. The major formation pathways of HMXBs in @zuo14a are consistent with the results obtained by @Linden10. The predicted observational properties of HMXBs (such as the orbital period distributions) are also similar. The number of HMXBs is also found to be not very sensitive to the value of $\alpha_{\rm CE}$. However, it seems that neither the $\alpha_{\rm CE}$ formalism nor the $\gamma$ algorithm can account for all the specific classes of observed PCEBs [@meng11; @my12]. Moreover, even within the framework of the $\alpha_{\rm CE}$ formalism, different studies often give controversial results on the possible range of $\alpha_{\rm CE}$ and its dependence on other parameters [see @zgn00; @marco11; @davis12; @tn13 also Ivanova et al. 2013 and references therein]. Our work suggests that in the case of HMXBs, both the $\alpha_{\rm CE}$ formalism and the $\gamma$ algorithm are possible to reproduce the observed XLF. In the framework of $\alpha_{\rm CE}$ formalism, a high value of $\alpha_{\rm CE}$ is needed, although a constant value is not required. We also show the distinct observational properties, such as the period distribution of HMXBs, that may serve as possible keys to understanding the CE evolution and to discriminate between different CE models.
SUMMARY
=======
We have used an EPS code to model the XLF of HMXBs with a range of theoretical models describing the CE phase. Our study shows that the observed XLF can be reproduced quite closely under both CE mechanisms. Provided that the same parameter combination is chosen, the $\gamma$ algorithm seems to produce more HMXBs than the $\alpha_{\rm CE}$ formalism, by a factor of up to $\sim 10$. Additionally, in the framework of the $\alpha_{\rm CE}$ formalism, a high value of $\alpha_{\rm CE}$ around $\sim 0.5-1.0$ better fits the observed XLF. We present the detailed properties of HMXB populations under the $\gamma$ algorithm, and find that the simulated HMXBs have a much larger population of short period (less than about several tens of days) BH-He systems than in the $\alpha_{\rm CE}$ formalism, which may serve as clues to discriminate between the two kinds of models. Our work motivates further high-resolution X-ray and optical observations of HMXB populations in nearby star-forming galaxies.
We thank the anonymous referee for helpful suggestions that enabled us to improve the manuscript. This work was supported by the National Natural Science Foundation (under grant numbers 11103014, 11003005, 11133001, 11333004, and 10873008), the Research Fund for the Doctoral Program of Higher Education of China (under grant number 20110201120034), the National Basic Research Program of China (973 Program 2009CB824800), the Strategic Priority Research Program of CAS under grant No. XDB09010200, and the Fundamental Research Funds for the Central Universities.
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[^1]: We have improperly chosen an IMF of @Kroupa01 and set the mass range as $5.0\,M_{\odot} - +\infty$ for the normalization in @zuo14a. The predicted HMXBs were overestimated by a factor of $\sim 3$, but the results and basic conclusions of the paper remain largely unchanged [see @zuo14d for details].
[^2]: We also change the number of the binary systems by a factor of eight, and find no significant difference in the final results.
[^3]: If the mass ratio is extreme, a CE is triggered, followed by a second CE between the newly formed BH and an expanding giant, instead resulting in much tighter BH-HeMS XRBs instead. However, their population is relatively minor in this case.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- '[Antonio Garijo]{}'
- '[Xavier Jarque]{}'
- '[Mónica Moreno Rocha[^1]]{}'
title: |
Non-landing hairs in Sierpiński curve\
Julia sets of transcendental entire maps\
(Revised Version)
---
[**Keywords:**]{} Transcendental entire maps, Julia set, non-landing hairs, indecomposable continua.
[**Mathematics Subject Classification (2000):**]{} 37F10, 37F20.
Introduction {#section:intro}
============
Let $f:{{\Bbb C}}\to {{\Bbb C}}$ be a transcendental entire map. The *Fatou set* ${\cal F}(f)$ is the largest open set where iterates of $f$ form a normal family. Its complement in ${{\Bbb C}}$ is the *Julia set* ${\cal J}(f)$ and it is a non-empty and unbounded subset of the plane. When the set of singular values is bounded, we say $f$ is of *bounded singular type* and denote this class of maps by $\cal B$. It has been shown in [@Ba] and [@R] that the Julia set of a hyperbolic map in $\cal B$ contains uncountably many unbounded curves, usually known as *hairs*, [@DT]. A hair is said to *land* if it is homeomorphic to the half-closed ray $[0,+\infty)$. The point corresponding to $t=0$ is known as the *endpoint* of the hair. In contrast, if its accumulation set is a non-trivial continuum, we obtain a *non-landing* hair.
In this paper we study a particular class of non-landing hairs in the Julia set of transcendental entire maps given by $$\label{eq:themap}
f_a(z)=a (z-(1-a)) \exp(z+a),$$ when $a$ is a real parameter. For all complex values of $a$, the map $f_a$ has a superattracting fixed point at $z=-a$ and an asymptotic value at the origin whose dynamics depends on the parameter $a$. If the orbit of the asymptotic value escapes to $+\infty$, we say $a$ is an [*escaping parameter*]{}. For example, when $a>1$, the orbit of the asymptotic value escapes to $+\infty$ along the positive real axis.
To our knowledge, the family $f_{a}$ was first introduced by Morosawa, [@Moro], as an example of a transcendental entire map whose Julia set is homeomorphic to the [*Sierpiński curve*]{} continuum when $a>1$. Any planar set that is compact, connected, locally connected, nowhere dense, and has the property that any two complementary domains are bounded by disjoint simple closed curves is homeomorphic to the Sierpiński curve continuum (Whyburn, [@Why]). It is also a *universal* continuum, in the sense that it contains a homeomorphic copy of every one-dimensional plane continuum (Kuratowski, [@K]). We take advantage of this property to combinatorially construct subsets of ${{\mathcal J}}(f_a), \ a >1$, that in turn are *indecomposable continua*. An [*indecomposable continuum*]{} is a compact, connected set that cannot be written as the union of two proper connected and closed subsets. Observe that a landing hair together with the point at infinity is in fact a decomposable continuum.
Every known example in the literature of indecomposable subsets of Julia sets arises from a single family of maps, namely the exponential family $E_{\lambda}(z)=\lambda \exp(z)$. The first example was given by Devaney [@D] when $\lambda=1$ so the asymptotic value escapes to infinity and the Julia set is the whole plane. Under the assumption that either the asymptotic value escapes to infinity or has a preperiodic orbit (thus holding again ${\cal J}(E_\lambda)= {{\Bbb C}}$), several authors have been able to construct topologically distinct indecomposable continua embedded in ${{\mathcal J}}(E_\lambda)$ (see among other works, [@DJ1], [@DJM], and [@R1], where a generalization of previous results for a large set of $\lambda$-parameters can be found).
Our work provides examples of indecomposable subcontinua of Julia sets outside the exponential family and without the assumption that ${{\mathcal J}}(f_a)$ equals ${{\Bbb C}}$, since $f_a$ has a superattracting fixed point for all $a\in\mathbb C$. Denote by ${{\mathcal A}}(-a)$ the *basin of attraction* of $-a$, that is, the set of points with forward orbits converging to $-a$. Also denote by $\mathcal{A}^*(-a)$ the *immediate basin of attraction* of $-a$ which is the connected component of $\mathcal{A}(-a)$ containing $-a$. In [@Moro] Morosawa showed that all connected components of ${{\mathcal A}}(-a)$ are bounded Jordan domains. Moreover, whenever $a>1$, the orbit of the free asymptotic value escapes to infinity. Since there are no other singular values, ${{\mathcal F}}(f_a)$ cannot contain another attracting basin, or a parabolic basin, or a Siegel disk as these components must be associated with a non-escaping singular value. Maps with a finite number of singular values do not exhibit neither wandering domains ([@EL2; @GK]) nor Baker domains ([@EL2]). Hence $\mathcal{F}(f_{a})=\mathcal{A}(-a)$. In Figure \[fig:julia\_set\], we display the dynamical plane of $f_a$ for different values of $a>1$. The basin of attraction of $-a$ is shown in black, while points in the Julia set are shown in white.
Let us summarize our main results. Since ${{\mathcal J}}(f_a)$ is homeomorphic to the Sierpiński universal curve, it must contain embedded copies of planar indecomposable continua, so we obtain some of them in terms of its combinatorics. To do so, we first characterize the topology and dynamics of the boundary of $\mathcal{A}^*(-a)$ by a polynomial-like construction (Proposition \[proposition:pol\_like\]). Then, using general results of transcendental entire maps, we obtain curves in the Julia set contained in the far right plane and with specific combinatorics (Proposition \[prop:ConjugTails\]). By a controlled process of consecutive pullbacks of some of these curves, we extend them into non-landing hairs that limit upon themselves at every point (Theorem \[theorem:indecom\]). Using a result due to Curry, [@C], we show the closure of such hairs are indecomposable continua (Theorem \[thm:DoesNotSeparate\]). Finally, we study the relation between each indecomposable continuum and the boundary of ${{\mathcal A}}^*(-a)$ showing that the intersection between these two sets reduces to a unique point (Theorem \[theorem:relation\_inde\_basin\]). As a consequence of these results, we show the existence of a dense set of points in $\partial \mathcal{A}^*(-a)$ that are landing points of a unique hair (in particular there are no *pinchings* that arise as in other maps in class $\cal B$ having a superattracting basin), while there is a residual set of points in $\partial \mathcal{A}^*(-a)$ that, even though they belong to the accumulation set of a certain ray, they are not landing points of hairs.
The outline of this paper is as follows: in §\[section:dyn\_plane\] we describe the dynamical plane of $f_a$ for $a\geq 3$. §\[section:targets\] contains most of our technical results while in §\[section:indecom\] we provide the proofs of our main results.
\[\]\[\][$\zeta_1$]{} \[\]\[\][$\eta_1$]{} \[\]\[\][$\zeta_{-1}$]{} \[\]\[\][$\eta_{-1}$]{} \[\]\[\][$T_1$]{} \[\]\[\][$T_0$]{} \[\]\[\][$R$]{} \[\]\[\][$H_R$]{}
![[]{data-label="fig:sigmas"}](./figures/Moro_3punto1_sigmas1.eps){width="50.00000%"}
*Notation and terminology.*
$B_\varepsilon(x)=\{z\in {{\Bbb C}}~|~|z-x|<\varepsilon\}$.
$\overline{U}$ denotes the closure of a set $U$.
Connected components will be refered to as components.
A curve $\gamma$ *cuts across* a
1. line $L$ if the intersection $\gamma\cap L$ is not tangential,
2. rectangle $R$ if $\gamma$ cuts across both vertical boundaries of $R$ so $\gamma\cap R$ contains a component with endpoints joining those sides,
3. semi-annular region $A$ if $\gamma$ cuts across the inner and outer semicircular boundaries of $A$ so $\gamma\cap A$ contains a connected component with endpoints joining those boundaries.
Dynamical plane for escaping real parameters {#section:dyn_plane}
============================================
Consider escaping parameters of the form $a>1$ for the family of transcendental entire maps $$f_a(z) = a(z-(1-a))\exp(z+a),$$ which have a unique asymptotic value at $z=0$ and a superattracting fixed point at $z=-a$. For $a>1$, the asymptotic value escapes to infinity along the positive real line and the Fatou set reduces to the basin of attraction of $-a$, ${{\mathcal A}}(-a)$. Our first aim in this section is to provide a partition of the complex plane that will allow us to combinatorially analyze the dynamics of points in the Julia set.\
We start by taking preimages of the forward invariant set ${{\mathbb R}}^+$. Any point $z=x+iy$ in the complex plane whose image under $f_a$ is a real positive number must satisfy $$\label{eq:def_sigmas}
\begin{split}
& \left( x- (1-a) \right)\cos y - y\sin y > 0, \\
& \left( x- (1-a) \right)\sin y + y\cos y = 0.
\end{split}$$
From these conditions, the preimages of ${{\mathbb R}}^+$ are infinitely many analytic curves parametrized by $(x,\zeta_j(x))$, with $j\in {{\Bbb Z}}$. For $j=0$, $\zeta_0(x)=0$ and is defined for all $x \in (1-a,+\infty)$ while the rest of the $\zeta_j$’s are strictly monotonic functions of $x$ defined for all $x \in {{\mathbb R}}$. When $j\neq 0$, each $\zeta_j$ has two horizontal asymptotes, given by
$$\lim\limits_{x\to -\infty} \zeta_j(x) = \text{sign}(j) (2|j|-1)\pi i \quad {\rm and} \quad \lim\limits_{x\to +\infty} \zeta_j(x) =2 j \pi i.$$
For our purposes, we need to consider the preimage of the interval $(-\infty,-a)$ inside the region bounded by $\zeta_1$ and $\zeta_{-1}$ (see Figure \[fig:sigmas\]). We obtain two strictly monotonic curves $\eta_1(x)$ and $\eta_{-1}(x)$ defined for $x \in [-a,+\infty)$, satisfying $$\lim\limits_{x\to -a^+} \eta_{\pm 1}(x) = 0, \quad \quad \lim\limits_{x\to +\infty} \eta_{1}(x) = \pi i \quad {\rm and} \quad \lim\limits_{x\to +\infty} \eta_{-1}(x) = -\pi i .$$
Let $T_{0}$ denote the open and connected set containing $z=0$ and bounded by $\eta_1 \cup \eta_{-1}$. Similarly, let $T_{1}$ be the open and connected set bounded by $\zeta_{-1}\cup \eta_{-1} \cup \eta_1 \cup \zeta_{1}$. Far to the right, $T_{1}$ consists of two unbounded and disjoint strips, one above and one below the positive real line. Since most of our results involve the dynamics of points in $T_0\cup T_1$, we construct a refinement of this region. For $j=0,1$, denote by $T_{j_1}$ and $T_{j_2}$ the proper and disjoint domains in $T_j\setminus {{\Bbb R}}$ with negative and positive imaginary part, respectively. Finally, for each $j\in {{\Bbb Z}}, j\neq 0,1$, denote by $T_j$ the open and connected strip bounded by the curves $\zeta_{j-1}$ and $\zeta_j$ as Im$(z)$ increases. Then $\{T_j~|~j\in {{\Bbb Z}}\}$ defines the partition of the complex plane sought, while $\{T_{j_i}~|~j=0,1,~i=1,2\}$ defines a refinement of the region $T_0\cup T_1$.\
It is straightforward to verify that $$\begin{split}
& f_a : T_{0} \to {{\mathbb C}}\setminus \left(-\infty,-a \right], \quad {\rm and}\\
& f_a : T_{1} \to {{\mathbb C}}\setminus \left( (-\infty,-a] \cup [0,+\infty ) \right),
\end{split}$$ are one-to-one maps. Define $g_a^0=f_a^{-1}|T_0$ and $g_a^1=f_a^{-1}|T_1$ the corresponding inverse branches of $f_a$ taking values in $T_0$ and $T_1$, respectively. As for the refinement of $T_0\cup T_1$, we denote by $g_a^{j_1}$ and $g_a^{j_2}$ the appropiate restrictions of $g_a^j$ mapping into $T_{j_1}$ and $T_{j_2}$, respectively.
Assume $z$ is a point of the Julia set whose orbit is entirely contained in $\cup_{j\in {{\Bbb Z}}} T_j$. We can naturally associate to $z$ the *itinerary* $s(z)=\left(s_{0},s_{1},\ldots \right)$, with $s_{j}\in {{\Bbb Z}}$, if and only if $f_{a}^j(z)\in T_{s_j}$. Let us concentrate on the space $\Sigma_B=\{0,1\}^{{\Bbb N}}$ of [*binary sequences*]{} (in what follows *$B$-sequences*). With respect to the refinement of $T_0\cup T_1$, consider the space of [*extended sequences*]{} given by $\Sigma_E=\{0_{1},0_{2},1_{1},1_{2}\}^{{\Bbb N}}$. Since $f_a$ is a one-to-one map in $T_{0}$ and $T_{1}$, not all extended sequences are [*allowable*]{}, that is, $f_a$ behaves as a subshift of finite type over the set of points with full orbits inside $T_{0_1}\cup T_{0_2}\cup T_{1_1}\cup T_{1_2}$. Its transition matrix is given by $$A=\left(\begin{array}{cccc}1 & 0 & 1 & 0 \\0 & 1 & 0 &1 \\0 & 1 & 0 & 1 \\1 & 0 & 1 & 0 \end{array}\right),$$ and determines the space of *allowable extended sequences* (in what follows *$A$-sequences*) given by $$\Sigma_A=\{(s_0,s_1,\ldots)\in \Sigma_E ~|~s_i\in \{0_1,0_2,1_1,1_2\},a_{s_i s_{i+1}}=1, \forall i\}.$$
Denote by $\pi:\Sigma_A\to \Sigma_B$ the *projection map* that transforms an $A$-sequence into a $B$-sequence by erasing all subscripts. The form of the matrix $A$ makes evident that $\pi$ is a 2-to-1 map. For a given $B$-sequence $t$, denote by $t^1$ and $t^2$ the unique $A$-sequences so that $\pi(t^j)=t$. Observe that by interchanging all subscripts in $t^1$ we obtain $t^2$, and conversely.
\[rem:undefinedAseq\] It is important to observe that points on the real line (or on any of its preimages), do not have well defined $A$-sequences. However, since ${{\Bbb R}}$ is forward invariant under $f_a$, its dynamics and combinatorics are completely understood. Based on the next result, from now on we will only consider $B$-sequences (and its two associated $A$-sequences) that do not end in all zeros.
\[lem:ends-in-zeros\] Assume $a> 1$ and let $w\in {{\mathcal J}}(f_a)$ such that $f^k_a(w)\in T_0\cup T_1$ for all $k\geq 0$. Then, $w\in {{\Bbb R}}\cap T_j$ if and only if $s(w)=(j,0,0,\ldots)$, for $j=0,1$.
The first implication follows easily by analyzing the action of $f_a$ in ${{\Bbb R}}$. Whenever $a>1$, the set ${{\mathcal A}}^*(-a)$ intersects the real line in an open interval $(q_a,p_a)$, where $p_a$ is a repelling fixed point and $q_a$ is its only preimage in ${{\Bbb R}}$. Moreover $(-\infty, q_a)\cup (p_a,+\infty)$ consists of points that escape to +$\infty$ along ${{\Bbb R}}^+$ and hence, belong to ${{\mathcal J}}(f_a)$. Since $f_a$ sends $(-\infty, q_a]$ onto $[p_a,+\infty)$ and this second interval is fixed by $f_a$, then $w$ has a well defined itinerary given by $s(w)=(1,0,0,\ldots)$, if $w\in (-\infty, q_a]\subset T_1$, or $s(w)=(0,0,\ldots)$ if $w\in [p_a,+\infty)\subset T_0$.
To see the second implication, it is enough to show the interval $[p_a,+\infty)$ represents the only set of points in the Julia set that remain inside $T_0$ for all positive iterates. To do so, we analyze the preimages of $\eta_1$ inside $T_{0_2}$ (the case $\eta_{-1}$ and $T_{0_1}$ is analogous). Since $f_a$ maps $T_{0_2}$ onto the upper half plane Im$(z)>0$, then for each $k\geq 1$, the $k^{\rm th}$ preimage of $\eta_1$ in $T_{0_2}$, namely $\eta_1^k=(g_a^{0_2})^k(\eta_1)$, lies completely inside $T_{0_2}$ (except for its endpoint in $z=-a$) and extends towards infinity into the right half plane. In particular, it lies in the strip bounded by $[-a,+\infty)$ and $\eta_1^{k-1}$ (from bottom to top). Also, note $\eta_1^k$ and $\eta_1^j$ meet only at $z=-a$ whenever $k\neq j$. We claim that $\eta_1^k$ accumulates onto $[p_a,+\infty)$ as $k\to \infty$. For otherwise, we can find a point $x\in [-a,+\infty)$ and $\varepsilon>0$ so $B_\varepsilon(x)\cap \eta_1^k =\emptyset$ for all $k\geq 1$. Nevertheless, since $x$ belongs to the Julia set, it follows from Montel’s Theorem the existence of an integer $N>0$ for which $f_a^N(B_\varepsilon(x))\cap \eta_1\neq \emptyset$. Hence, $B_\varepsilon(x) \cap \eta_1^N\neq \emptyset$, a contradiction.
Finally, for any given point $w\in T_{0_2}$ such that $s(w)=(0,0,\dots)$, there exists an integer $m>0$ for which, either $w\in \eta_1^m$ or $w$ lies in the interior of the strip bounded by $\eta_1^{m+1}$ and $\eta_1^{m}$ (from bottom to top). In both situations, $f_a^{m+1}(w)$ lies outside $T_0$. This finishes the proof.
The rest of this section is devoted to a combinatorial description of the dynamics of points with forward orbits contained in $T_0\cup T_1$ using $A$- and $B$-sequences. First, we focus our study on the set ${{\mathcal A}}^*(-a)$ and then analyze points in the Julia set that lie far to the right in $T_0\cup T_1$. Using known results in complex dynamics we will prove that points with forward orbits completely contained in a given right hand plane are organized into continuous curves and their combinatorics are governed by the transition matrix $A$. For future reference, we compute the image of a vertical segment bounded above and below by $\zeta_1$ and $\zeta_{-1}$, respectively.
\[lemma:vertical\_segment\] Let $x\in {{\Bbb R}}$ be fixed and consider the vertical segment $L[x]=\{x+iy \, | \, \zeta_{-1}(x)\leq y \leq \zeta_1(x)\}$. Then $f_a(L[x])$ lies inside the closed round annulus $f_a(x) \leq |z| \leq f_a(x+\zeta_1(x))$.
From the definition of the map $f_a$ we have $$|f_a(x+iy)|= a \exp(x+a) \sqrt{(x-(1-a))^2 + y^2}.$$ Evidently, when restricted to $L[x]$ for a fixed $x$, the above expression reaches its minimum value when $y=0$ while its maximum value is reached whenever $y=\zeta_1(x)=\zeta_{-1}(x)$.
Dynamics near $z=-a$ {#subsection:pol_like}
--------------------
In [@Moro] it was shown that for $a>1$, each Fatou domain of $f_a$ is a bounded, connected component of $\mathcal{A}(-a)$ whose boundary is a Jordan curve. Here we show that $\overline {\mathcal{A}^*(-a)}$ is in fact a quasiconformal image of the closed unit disk. Precisely, we describe a set of points with bounded orbits inside $T_{0}\cup T_{1}$ through a polynomial-like construction (see [@DH]) around the unique and simple critical point $z=-a$. For technical reasons, we restrict to parameters $a\geq 3$ from now on.
\[proposition:pol\_like\] For any $a\geq3$, there exist open, bounded and simply connected domains $U_a$ and $V_a$ with $ -a \in \overline{U}_a \subset V_a$, such that $(f_{a},U_{a},V_{a})$ is a quadratic-like mapping. Furthermore, the filled Julia set of $(f_{a},U_{a},V_{a})$ is the image under a quasiconformal mapping of the closed unit disk and coincides with $\overline{\mathcal{A}^*(-a)}$.
\[\]\[\][$\zeta_1$]{} \[\]\[\][$\eta_1$]{} \[\]\[\][$\zeta_{-1}$]{} \[\]\[\][$\eta_{-1}$]{} \[\]\[\][$f_a$]{} \[\]\[\][$U_a$]{} \[\]\[\][$V_a$]{} \[\]\[\][$L$]{} \[\]\[\][$R$]{}
![[]{data-label="fig:polynomial"}](./figures/Moro_3punto1_polynomial.eps){width="50.00000%"}
Define $V_a$ as the open, simply connected pseudo-rectangle given by $$V_{a}=\{z \in {{\mathbb C}}\ | \ -a-6\ln a < {\rm Re}(z) < \frac{1-a}{2}, \ \zeta_{-1}({\rm Re}(z)) < {\rm Im}(z) < \zeta_{1}({\rm Re}(z)) \}.$$ First, we show that $V_{a}$ maps outside itself. Indeed, the top and bottom boundaries of $V_a$ map into a segment lying on the positive real line, thus outside $V_a$ as $(1-a)/2\leq -1$. Also, note that $V_a$ lies in the interior of the annulus $$\frac{|1-a|}{2}\leq |z| \leq |-a-6 \ln a+ i2\pi|.$$ Following the notation in Lemma \[lemma:vertical\_segment\], $L=L[-a-6\ln a]$ and $R=L[(1-a)/2]$ are the left and right hand boundaries of $V_a$. We show next the images of $L$ and $R$ lie in the complementary components of the annulus. First, for $z\in L$ we have $$|f_{a}(z)|=\frac{1}{a^5}\sqrt{\left( 1+6\ln a \right)^2+y^2} < \frac{1}{a^5}\sqrt{\left( 1+6\ln a \right)^2+4\pi^2}<\frac{1}{2} < \frac{|1-a|}{2}.$$
Similarly, if $z\in R$ $$|f_{a}(z)|=a e^{\frac{a+1}{2}}\sqrt{\left(\frac{1-a}{2} \right)^2 + y^2}>a e^{\frac{a+1}{2}} \frac{|1-a|}{2} >\sqrt{(a+6 \ln a)^2 + 4\pi^2}$$ for all $a\geq 3$, proving thus that $V_a$ is mapped outside itself under $f_a$.
Now, we define $U_a$ to be the connected component of $f^{-1}_a(V_a)$ containing $-a$. Since $-a$ is a superattracting fixed point with multiplicity one, and there are no other critical points, it follows that $\overline{U_a}\subset V_a$ and the map $f_a:U_a \to V_a$ sends $\partial U_a$ to $\partial V_a$ with degree 2, as $-a$ is a simple critical point. We conclude that $(f_a,U_a,V_a)$ is a quadratic-like mapping.
What is left to verify is that the filled Julia set of $(f_a,U_a,V_a)$ is a quasi-disk. Recall that the filled Julia set of a polynomial-like mapping is defined as the set $\{z\in U_a~|~f_a^n(z)\in U_a~\text{for~all}~n\geq 0\}$. Being $(f_a,U_a,V_a)$ a quadratic-like mapping, there exists a quasiconformal conjugacy with a polynomial of degree two that has a superattracting fixed point. Thus, the polynomial must be $z\mapsto z^2$ after a holomorphic change of variables, if necessary. So the filled Julia set of $(f_a,U_a,V_a)$ is the image under a quasiconformal mapping of the closed unit disk.
\[prop:consequences\_pol\_like\] Let $a\geq3$. The following statements hold.
1. The map $f_{a}$ restricted to the boundary of $\mathcal A^{*}(-a)$ is conjugate to the map $\theta \mapsto 2\theta$ in the unit circle.
2. Let $t\in \Sigma_E$ be an extended sequence that does not end in $0_i$’s. Then, $t$ is an $A$-sequence if and only if there exists a unique point $z \in \partial \mathcal A^{*}(-a)$ that realizes $t$ as its itinerary.
Statement (a) is a direct consequence of the previous proposition since the map $f_a$ is conjugate in $\partial \mathcal A^*(-a)$ to $z \mapsto z^2$ acting on the unit circle.
We prove statement (b) by defining a partition of the boundary of $\mathcal A^*(-a)$ that coincides with the refinement of the partition $T_0 \cup T_1$ discussed before. For simplicity, angles are measured by $[0,1]$. Denote by $z(0)$, $z(1/4)$, $z(1/2)$ and $z(3/4)$ the points in $\partial \mathcal A^*(-a)$ corresponding under the conjugacy between $f_a$ and $z \mapsto z^2$ to points in $S^1$ of angle $\theta=0,1/4,1/2 \,$ and $\,3/4$. Now label points in $\partial \mathcal A^*(-a)$ in the following way: traveling along $\partial \mathcal A^*(-a)$ in a clockwise direction, associate the symbol $0_1$ to the arc joining $z(0)$ and $z(3/4)$, the symbol $1_1$ to the arc joining $z(3/4)$ and $z(1/2)$, $1_2$ to the arc joining $z(1/2)$ and $z(1/4)$, and $0_2$ to the arc joining $z(1/4)$ and $z(0)$. We leave it to the reader to verify the transition matrix for this partition under the action of $f_a$ is exactly $A$ and the labeling is consistent with the one defined by the $T_{j_i}$’s.
\[thm:BdOrbit\] Let $z$ be a point such that $f_a^n(z)\in \overline{T_0\cup T_1}$ for all $n\geq 0$. If $z$ belongs to the Julia set and has bounded orbit, then $z\in \partial {{\mathcal A}}^*(-a)$.
If $z\in {{\mathcal J}}(f_a)$ satisfies the hypotheses, we can find $m<0<M$ so that $m< \text{Re}(f_a^n(z)) < M$ for all $n\geq 0$.
Let $\varepsilon>0$ small enough and denote by $B_\varepsilon=\overline{B_\varepsilon(0)}$. Since the orbit of the origin escapes monotonically along the positive real line, redefining $M$ if necessary, there exists an integer $N=N(M)>0$ for which $B_\varepsilon, f_a(B_\varepsilon),\ldots, f^{N}_a(B_\varepsilon)$ are pairwise disjoint compact domains, such that for all $0\leq j \leq N-1$, $$\begin{array}{ll}
& f^j_a(B_{\varepsilon}) \subset T_0 \cap \{ z \, |\, {\rm Re}(z)<M\}, \hbox{ and }\\
& f^N_a(B_{\varepsilon}) \subset T_0 \cap \{ z \, | \, {\rm Re}(z)> M\}.
\end{array}$$ Moreover, we may choose $\varepsilon$ small enough so $B_\varepsilon, f_a(B_\varepsilon),\ldots, f^{N}_a(B_\varepsilon)$ are all compact domains contained in $T_0$. Finally, select $m'\leq m<0$ so the subset $\{z\in T_1~|~\text{Re}(z)\leq m'\}$ maps completely inside $B_\varepsilon$.
Denote by $\Phi_a:{{\Bbb D}}\to {{\mathcal A}}^*(-a)$ the Böttcher coordinates tangent to the identity at the origin. For $0<r<1$ let $\Delta_r=\Phi_a(B_{r}(0))$. Clearly $\Delta_r\subset {{\mathcal A}}^*(-a)$ and maps compactly into its own interior. Moreover, we can choose $r$ small enough so for $j=0,1$, $T_j \setminus \Delta_r$ is a connected set and the intersection of $\Delta_r$ and ${{\Bbb R}}$ is an open interval $(c,d)$, since $\Phi_a$ has been chosen to be tangent to the identity at the origin. We can now define the set $E$, illustrated in Figure \[fig:periodic\], as follows $$E=\{z\in {{\Bbb C}}~|~m'< \text{Re}(z)< M,~\zeta_{-1}(\text{Re}(z))< \text{Im}(z)< \zeta_1(\text{Re}(z))\} \setminus \Omega$$ where $\Omega =\bigcup_{k=0}^{N-1} f^k_a(B_{\varepsilon})\cup \Delta_r \cup [c,M]$. It is easy to verify that $E$ is an open, bounded, connected and simply connected set and $\partial {{\mathcal A}}^*(-a) \subset E$. Moreover, $E\subset f_a(E)$ although some boundary components map into $\partial E$. Indeed, $f_a(\partial E)\cap \partial E$ consists of segments along the real line and $f^j_a(\partial B_{\varepsilon})$, for $j=1,2,\ldots,N-1$. So after $N$ iterations, the only boundary points mapping into $\partial E$ are points over the real line, thus having $B$-itineraries $(1,0,\ldots)$ or $(0, 0,\ldots)$. We show next that for $\ell>0$ sufficiently large, $f_a^{-\ell}|E$ becomes an strict contraction.
\[\]\[\][$\partial A^*(-a)$]{} \[\]\[\][$-a$]{} \[\]\[\][$\Delta_r$]{} \[\]\[\][$M$]{} \[\]\[\][$-a$]{} \[\]\[\][$B_{\varepsilon}(0)$]{} \[\]\[\][$E$]{} \[\]\[\][$m'$]{}
![[]{data-label="fig:periodic"}](./figures/periodic.eps){width="50.00000%"}
Let $s=(s_0, s_1, s_2,\ldots)$ be the $A$-sequence associated to $z$. If $s$ has finitely many $1_i$’s, then its $B$-sequence ends with $0$’s and since $z$ has bounded orbit inside $\overline{T_0\cup T_1}$, then $z\in \partial {{\mathcal A}}^*(-a)$ by Lemma \[lem:ends-in-zeros\]. If $s$ has infinitely many $1_i$’s there exists a first integer $n>N$ for which $s_{n}\in\{1_1,1_2\}$, and thus $\overline{E\cap T_{s_{n}}} \subset f^{n}(E)$, as points in $f^{n}(\partial E)$ mapping into $\partial E$ have by now itinerary $(0,0,\ldots)$.
For each $k\in \{0_1, 0_2, 1_1, 1_2\}$, the set $E\cap T_k$ is an open, connected and simply connected set with a Riemann mapping given by $\psi_k:{{\Bbb D}}\to E\cap T_k$. Consider the mapping $\Psi_{\ell}:{{\Bbb D}}\to {{\Bbb D}}$ with $\ell>n$, given by $$\Psi_{\ell} = \psi_{s_0}^{-1}\circ( g_a^{s_0}\circ \ldots \circ g_a^{s_{\ell-1}} )\circ \psi_{s_\ell}.$$ It follows that $\Psi_\ell({{\Bbb D}})$ is compactly contained in ${{\Bbb D}}$, that is, for $\ell>n$, $\Psi_\ell$ is an strict contraction with respect to the Poincaré metric on the unit disk. Consequently, the sets $\overline{\Psi_\ell({{\Bbb D}})}$ form a nested sequence of compact sets with diameters converging to zero as $\ell\to \infty$. This implies that for any $w\in {{\Bbb D}}$, $\lim_{\ell\to \infty} \Psi_\ell(w)$ exists and is independent of the point $w$. Therefore, by construction, $\psi_{s_0}(\lim_{\ell\to \infty} \Psi_\ell(0))=z$ is the unique point in $E$ with itinerary $s$. From Proposition \[prop:consequences\_pol\_like\](b) we conclude $z\in \partial {{\mathcal A}}^*(-a)$.
Dynamics near infinity {#subsection:tails_to_the_right}
----------------------
Our first aim is to prove that for $R>0$ sufficiently large and the region $$H_R = \{z\in T_0\cup T_1~|~ {\rm Re}(z)\geq R\},$$ there exist continuous curves in ${{\mathcal J}}(f_a)\cap H_R$ consisting of points whose orbits escape to $+\infty$ with increasing real part. These curves are usually known as *tails*. The existence of tails as disjoint components of the Julia set were first observed by Devaney and Tangerman [@DT] for certain entire transcendental maps and by Schleicher and Zimmer [@SZ] for the exponential family $E_\lambda(z)=\lambda \exp(z)$ and all $\lambda\in {{\Bbb C}}$. In greater generality, Barański [@Ba] and Rempe [@R] have shown the existence of tails for hyperbolic maps belonging to the class $\cal B$.
For completeness, we analyze in detail some of their results in the setting of our work to obtain tails in ${{\mathcal J}}(f_a)\cap H_R$. Once each tail has been assigned an $A$-sequence, we describe a pullback process to compute the full set of points in $T_0\cup T_1$ associated to such $A$-sequence. In the final section, we study the topological properties of that set.
Consider an entire transcendental map $f$ in the class $\cal B$. The *escaping set of $f$*, denoted as $I(f)$, is the set of points whose orbits under $f$ tend to infinity. For an entire transcendental map, Emerenko [@E] has shown the Julia set coincides with the boundary of $I(f)$. We say that two maps $f,g\in \cal B$ are *quasiconformally equivalent near infinity* if there exist quasiconformal maps $\phi_1, \phi_2:{{\Bbb C}}\to {{\Bbb C}}$ that satisfy $\phi_1 \circ f=g\circ \phi_2$ in a neighborhood of infinity.
\[theorem:lasse\] Let $f,g \in \mathcal B$ be two entire transcendental maps which are quasiconformally equivalent near infinity. Then there exist $\rho>0$ and a quasiconformal map $\theta:{{\mathbb C}}\to {{\mathbb C}}$ such that $\theta \circ f = g \circ \theta$ on $$A_{\rho}=\{ z\in{{\mathbb C}}\ | \ |f^n(z)|>\rho, \ \forall n \geq 1 \}.$$ Furthermore, the complex dilatation of $\theta$ on $I(f)\cap A_{\rho}$ is zero.
A straightforward computation shows that $f_{a}(z)$ is conjugate to the function $\tilde{f}_{a}(z)=az\exp(z+1)-(1-a)$ under the conformal isomorphism $\varphi_a(z)=z-(1-a)$. Since $\tilde{f_a}(z)= \varphi_a \circ f_a \circ \varphi_a^{-1}(z)$, it is easily verified that $\tilde{f}_{a}$ has a free asymptotical value at $z=a-1$ and a fixed critical point at $z=-1$. In turn, $\tilde{f}_{a}$ is (globally) conformally equivalent to $g_{b}(z)=bz\exp(z)$ via $\phi_{1}(z)=\alpha z+\alpha(1-a)$ and $\phi_{2}(z)=z$. Indeed, it is easy to see that $\phi_{1}\circ \tilde{f}_{a} = g_{b}\circ \phi_{2}$, where $b = e a\alpha$.
For small values of $b$, the Fatou set of $g_{b}$ consists solely of the completely invariant basin of attraction of the fixed point (and asymptotic value) $z=0$. Thus, we can describe the Julia set of $g_b$ by applying the following result found in [@Ba].
\[theo:baranski\] Let $g$ be an entire transcendental function of finite order so that all critical and asymptotic values are contained in a compact subset of a completely invariant attracting basin of a fixed point. Then ${\cal J}(g)$ consists of disjoint curves (hairs) homeomorphic to the half-line $[0,+\infty)$. Moreover, the hairs without endpoints are contained in the escaping set $I(g)$.
These disjoint curves are usually known as hairs or *dynamic rays*. Several consequences are derived from the above theorem. Firstly, if $\gamma$ denotes a hair, it can be parametrized by a continuous function $h(t), \ t\in [0,+\infty)$, such that $\gamma=h([0,+\infty))$. The point $h(0)$ is called the *endpoint* of the hair. Secondly, each hair is a curve that extends to infinity and, for $r>0$, we say that $\omega=h((r,+\infty))$ is the [*tail of the hair*]{}. Moreover, all points in a given hair share the same symbolic itinerary defined by a dynamical partition of the plane with respect to $f$, and for every point $z\in \gamma$ that is not the endpoint, we have $f^n(z)\to \infty$ as $n\to \infty$. Finally, if $f$ and $g$ are as in Theorem \[theorem:lasse\] and in addition, $g$ satisfies hypotheses in Theorem \[theo:baranski\], then near infinity the topological structure of the escaping set of $f$ is also given by disjoint curves extending to infinity. We refer to [@BJR] for a topological description of the Julia set in terms of what is known as Cantor bouquets. Furthermore, the dynamics of $f$ in those curves is quasiconformally conjugate to the dynamics of $g$ in the corresponding curves near infinity. We deduce the following result based on the previous theorems and the specific expression of $f_a$.
\[prop:ConjugTails\] Let $f_a(z) = a(z-(1-a))\exp(z+a)$, $g_b(z)=bz\exp(z)$, $a\geq 3$, and $b$ a complex parameter.
1. If $|b|$ is small enough, the Julia set of $g_{b}$ is given by the union of disjoint hairs. Each hair lands at a distinguished endpoint. Moreover, hairs without endpoints are contained in $I(g_b)$.
2. Let $R>0$ large enough. The set of points with forward $f_a$-orbits that are always contained in $H_R$ are given by the union of disjoint curves extending to infinity to the right. All points in those curves belong to $I(f_a)$. Precisely, these curves are quasiconformal copies of connected components of hairs described in (a).
3. To each curve in (b) that does not coincide with ${{\Bbb R}}$ or one of its preimages, we can assign a unique sequence $t$ in $\Sigma_A$. We denote this curve by $\omega_t$. All points in $\omega_t$ escape to infinity under the action of $f_a$ following the itinerary $t$.
4. For any $t\in \Sigma_A$, if $R>0$ is large enough, there exists a unique curve $\omega_t$ in $H_R$. Moreover, for each $r \geq R$, $\omega_t \cap \{z~|~\text{Re}(z)=r\}$ is a unique point. In particular $\omega_t$ is the graph of a function.
5. $\omega_t$ is a tail, i.e., a quasiconformal copy of a tail in (a).
Statement (a) follows directly from Theorem \[theo:baranski\]. To see statement (b) note that for any value of $a\neq 0$ and $b$, there exists $\alpha=b/(ea)$ so $f_a$ and $g_b$ are (globally) conformally equivalent. By Theorem \[theorem:lasse\], there exists $\rho>0$ such that $f_a$ and $g_b$ are conjugate on the set $A_\rho=\{z~|~|f_a^n(z)|>\rho, \forall n \geq 1\}$. Let $R>\rho$. Denote by $S$ the set of points in $H_R$ with forward $f_a$-orbits contained in $H_R$. Clearly, each point in $S$ must belong to ${{\mathcal J}}(f_a)$ (since a point in the Fatou set eventually maps into ${{\mathcal A}}^*(-a)$) and in particular, $S\subset {{\mathcal J}}(f_a)\cap A_R$. Moreover, far enough to the right, a point $z\in H_R$ whose forward orbit remains forever in $H_R$ must satisfy Re$\left(f_a^{k+1}(z)\right) > {\rm Re}\left(f_a^k(z)\right)$ for all $k>0$ (see Lemma \[lemma:vertical\_segment\]). Hence from Theorems \[theorem:lasse\] and \[theo:baranski\] we know that $S$ is the union of disjoint curves extending to infinity (that is, quasiconformal copies of components of the hairs in (a)) belonging to the escaping set.
To prove statement (c) we start by assigning to each of these curves, $\omega$, a unique sequence in $\Sigma_A$. Let $z_0$ be a point in $\omega$ and let $s(z_0)$ be its itinerary in $\Sigma_A$ in terms of the partition $T_{0_1}, T_{0_2}, T_{1_1}$ and $T_{1_2}$. By assumption this itinerary is well defined since $\omega$ is not ${{\Bbb R}}$ or one of its preimages. Let $z_1$ be another point in $\omega$. We claim that $s(z_1)=s(z_0)$ and proceed by contradiction. Let $\mathcal C$ be the connected component of $\omega$ joining $z_0$ and $z_1$. If $s(z_1)\neq s(z_0)$, there exists an integer $k\geq 0$ for which the $k^{\text th}$ entries in both itineraries are the first ones to differ. Hence there is a point $q\in \mathcal C$ such that $f_a^k(q)$ belongs to either $[R,\infty)$, $\eta_1$ or $\eta_{-1}$. Clearly $f_a^k(q)$ cannot belong to $\eta_{\pm 1}$, since otherwise $f_a^{k+1}(q)\in {{\Bbb R}}^-$ and by hypothesis the forward orbit of $q$ belongs to $H_R$. On the other hand $f_a^k(q)$ cannot belong to ${{\Bbb R}}$ since by item (b), $f_a^k\left(\omega\right)$ and ${{\Bbb R}}$ are disjoint curves of the escaping set.
Finally, there cannot be two curves having the same itinerary as this will imply the existence of an open set of points following the same itinerary, which is impossible. Thus, we may now denote by $\omega_t$ the unique curve in $H_R$ formed by escaping points with itinerary $t$.
To prove statement (d) fix $t=(t_0,t_1,\ldots, t_n,\ldots) \in \Sigma_A$ and $R$ large enough. Let $r\geq R$ and let $L[r]=\{r+iy \, | \, \zeta_{-1}(r)\leq y \leq \zeta_1(r)\}$. We denote by $I_{t_0}$ the set of points in $L[r] \cap \overline{T_{s_0}}$. We know that the image of the vertical segment $L[r]$ cuts across $H_R$ in two almost vertical lines (see Lemma \[lemma:vertical\_segment\]). Recall that when $f_a$ is restricted over the set of points with forward orbits in $H_r$, it behaves as a subshift of finite type governed by the matrix $A$, and when restricted to $T_0$ or $T_1$ it is a one-to-one map. Using these facts, we can find a unique subinterval $I_{t_0t_1}\subset I_{t_0}$ formed by points with forward orbits inside $H_r$ and itineraries starting as $(t_0,t_1,\ldots)$. Inductively, for each $n>0$, $I_{t_0t_1\ldots t_n}$ is the unique subinterval of $I_{t_0t_1\ldots t_{n-1}}$ formed by points with forward orbits inside $H_r$ and itineraries starting with $(t_0,t_1,\ldots, t_n,\ldots )$. Clearly, $$I_{t_0t_1,\ldots t_n} \subset I_{t_0t_1\ldots t_{n-1}} \subset \ldots \subset I_{t_0t_1} \subset I_{t_0}.$$ Due to the expansivity of $f_a$ in $H_R$ we obtain $\cap_{n\geq 0} I_{t_0t_1,\ldots t_n} =\{q\}$, and by construction $q$ must have itinerary $t$. Moreover $f^k(q)\to \infty$ as $k\to \infty$, so $q$ must belong to the unique curve $\omega_t$ described in (c). The above arguments imply that $\omega_t$ intersects Re$(z)=r$ at a unique point. Therefore we can parametrize $\omega_t$ on the interval $[r,\infty)$.
Finally to see statement (e) we observe that there are no endpoints associated to bounded itineraries in $H_R$, implying that each $w_t$ is a quasiconformal copy of the tail of some hair in (a). On one hand, the only points in $T_0 \cup T_1$ with bounded orbit belong to $\partial A^*(-a)$, so they are not in $H_R$. On the other hand if there were and endpoint [*of a hair*]{} with orbit escaping to infinity in $H_R$, it should have an itinerary, say, $t$. But from statement (d) there is a (unique) curve $w_t$ going from Re$(z)=R$ to infinity with such itinerary $t$, a contradiction.
Given any $A$-sequence $t$, each curve described in Proposition \[prop:ConjugTails\] will be denote by $\omega_{t}=\omega_{t}(R)$ and it will be called the *tail with itinerary $t$* contained in the half plane Re$(z)\geq R$.
The component of $\omega_t$ that cuts across $$F_R=\{z\in H_R~|~|z|<f_a(R+i\zeta_1(R))\}$$ is called the *base* of the tail, and it will be denote by $\alpha_t=\alpha_t(R)$.
See Figure \[fig:tails\].
\[\]\[\][$\zeta_1$]{} \[\]\[\][$\zeta_{-1}$]{} \[\]\[\][$\eta_{-1}$]{} \[\]\[\][$\eta_{1}$]{} \[\]\[\][$(z)=R$]{} \[\]\[\][$w_t$]{} \[\]\[\][$\alpha_t$]{} \[\]\[\][$|z|=f_a(R+i\zeta_1(R))$]{}
![[]{data-label="fig:tails"}](tails1.eps){width="40.00000%"}
We now describe a pullback construction to extend the tail $\omega_t$ into a longer curve. Recall $g_a^{j_i}=f_a^{-1}|T_{j_i}, j_i\in\{0_1,0_2,1_1,1_2\}$ and consider the shift map $\sigma: \Sigma_A\to \Sigma_A$ acting on the space of $A$-sequences. Let $t=(t_0,t_1,\ldots)\in \Sigma_A$. By the conjugacy of $f_a | \bigcup_{t\in \Sigma_A} \omega_t$ with $\sigma|\Sigma_A$, it follows that $\omega_{\sigma (t)}$ properly contains $f_{a}(\omega_t)$ since this curve lies in $H_R\setminus F_R$ (see Proposition \[prop:ConjugTails\]). Hence $f_a(\omega_t)$ is a curve that misses the base $\alpha_{\sigma(t)}$. Consequently, $g_{a}^{t_0}(\omega_{\sigma(t)})$ is a continuous curve that lies in $T_{t_0}$ and extends $\omega_t$ to the left of Re$(z)=R$. Clearly, any point in the extended curve $g_{a}^{t_0}(\omega_{\sigma (t)})$ has itinerary $t$. Inductively, consider $$g_{a}^{t_0}\circ \ldots \circ g_{a}^{t_{n-1}}\left(\omega_{\sigma^n(t)}\right).$$ This pullback iteration is always defined as long as the extended curve does not meet $z=0$, which is impossible since ${{\mathbb R}}$ is forward invariant. We thus obtain a curve of points with itinerary $t$, and each pullback iteration extends its predecessor.
Let $$\label{eq:pullback-hair}
\gamma (t)=\bigcup_{n=0}^\infty g_{a}^{t_0}\circ \ldots \circ g_{a}^{t_{n-1}}\left(\omega_{\sigma^n(t)}\right).$$ We call $\gamma (t)$ the [*hair*]{} associated to $t$.
Note that the pullback process described before may or may not produce an endpoint. In Section 4 we show that in some cases, $\gamma(t)$ is a hair with an endpoint in $\partial {{\mathcal A}}^*(-a)$, and in some other cases $\gamma(t)$ is a non-landing hair and accumulates everywhere upon itself. By the following theorem found in [@C], we will conclude that $\overline{\gamma(t)}$ is an indecomposable continuum.
\[theo:curry\] Suppose that $X$ is a one-dimensional nonseparating plane continuum which is the closure of a ray that limits on itself. Then $X$ is either an indecomposable continuum or the union of two indecomposable continua.
A *ray* is defined as the image of $[0,+\infty)$ under a continuous, one to one map. Giving any positive number $\alpha$, the image of $[\alpha,+\infty)$ under the same map is known as a *final segment* of the ray. Then, the ray *limits on itself* if it is contained in the closure of any final segment of itself.
Targets in $H_{R}$ {#section:targets}
==================
The main result in this section will be Theorem \[theorem:indecom\], where we construct $B$-sequences so their associated $A$-sequences produce hairs that accumulate everywhere on themselves.\
We now set up *targets* around the $n^{\rm th}$ *image of the base* $\alpha_{t}$ of each tail $\omega_t$. The construction is very similar to the one presented in Devaney and Jarque [@DJ2] and Devaney, Jarque and Moreno Rocha in [@DJM], although the existence of a critical point in our present case requires some modifications.\
Let $t$ be a given $A$-sequence. We first enlarge the base $\alpha_t$ inductively along $\omega_t$. Set $\alpha_{t,0}=\alpha_t$ and consider the two bases $\alpha_{\sigma^{-1}(t)}$. The set $f_{a}(\alpha_{\sigma^{-1}(t)})$ (taking the two possible bases) is a subset of $\omega_t$. Thus, the set $$\alpha_{t,1} = \alpha_{t,0} \cup f_{a}(\alpha_{\sigma^{-1}(t)}),$$ is an extension of $\alpha_{t,0}$ along $\omega_t$. Inductively, define the $n^{\rm th}$ image of the base $\alpha_t$ as $$\alpha_{t,n} = \alpha_{t,n-1} \cup f_{a}^n(\alpha_{\sigma^{-n}(t)}).$$ It is easy to verify that $\{\alpha_{t,n}\}_{n\in {{\mathbb N}}}$ is a sequence of curves satisfying the following three conditions:
1. $\alpha_{t,0} = \alpha_t$,
2. $\alpha_{t,n} \subset \alpha_{t,n+1}$, and
3. $\bigcup\limits_{n\geq 0}\alpha_{t,n} = \omega_t$.
In order to define a target around each $\alpha_{t,n}$, consider $\xi,\eta\in {{\Bbb R}}^+$ and let $$V(\xi,\eta) = \{ z \in H_R~|~\xi - 1 <{\rm Re}(z) < \eta+1 \}.$$ By definition $V( \xi, \eta)$ is a rectangular region bounded above and below by components of $\zeta_{-1}$ and $\zeta_{1}$, respectively.
\[lemma:macrolemma\] Let $R>0$ be large enough. For all $n \geq 0$ there exist positive real numbers $\xi_n$ and $\eta_n$ such that the following statements hold.
1. For every $t\in \Sigma_A$, the $n^{\rm th}$ iterate of $\alpha_t$ belongs to the interior of $V(\xi_n,\eta_n)$.
2. For every $\ell \geq 0$, $V(\xi_{n+1},\eta_{n+\ell+1})$ is compactly contained inside $f_a(V(\xi_n,\eta_{n+\ell}))$.
Set $\eta_0 = f_a\left(R+i\zeta_{1}(R)\right)$, $\eta_{n+1}=f_a\left(\eta_n+i \zeta_{1}\left(\eta_{n}\right)\right)$ and $\xi_n=f_a^n(R)$ for $n\geq 0$. It remains to verify that these values satisfy statements (a) and (b). Observe that for $R$ large enough, the image of every vertical segment $L[R]$ cuts across $T_{0}\cup T_{1}$ in two almost vertical lines: one [*near*]{} Re$(z)=f_{a}(R)$ and the other [*near*]{} Re$(z)=f_{a}\left(R+i\zeta_{1}(R)\right)$ (see Lemma \[lemma:vertical\_segment\]).
Statement (a) follows directly from the definition of $\xi_n$ and $\eta_n$, since at each step we choose $\xi_n$ and $\eta_{n}$ to be respectively the smallest and largest possible values of Re$(f^n_a(\alpha_t))$ for all $t\in \Sigma_A$, and moreover it is easy to check that $\xi_{n}<\eta_{n-1}$.
We prove statement (b) when $\ell =0$. The case $\ell >0$ follows similarly. The proof of the statement proceeds in two steps. The first one is to verify the inequality $$f_a\left(\xi_n-1+i\zeta_{1}\left(\xi_n-1\right) \right) < \xi_{n+1}-1 \,.$$ From the definition of $f_a$ we obtain that $$\begin{aligned}
f_a\left(\xi_n-1+i\zeta_{1}\left(\xi_n-1\right) \right) & < &a e^{a+\xi_n-1} \sqrt{(\xi_n-1 -(1-a))^2+4\pi^2}\\
& < & a e^{a+\xi_n} (\xi_n-(1-a))-1\\
& = & \xi_{n+1}-1,\end{aligned}$$ where the second inequality is satified if $R$ is large enough. The second step is to verify the inequality $$f_a(\eta_n+1) > \eta_{n+1}+1 = f_a(\eta_n+i\zeta_1(\eta_n))+1.$$ By evaluating both sides we obtain $$a e^{a+\eta_n+1} (\eta_n+a) > a e^{a+\eta_n}\sqrt{(\eta_n+a-1)^2+\left(\zeta_1(\eta_{n})\right)^2} +1.$$ Consequently the image of $V(\xi_n,\eta_n)$ contains $\overline{V(\xi_{n+1},\eta_{n+1})}$ as desired.
The set $V(\xi_n,\eta_n)$ will be called the $n^{\rm th}$ target of $f_a$.
Targets provide a useful tool in the proof of Theorem \[theorem:indecom\]. Before getting into the details, we briefly explain why targets are so important in our construction. For large values of $n$, an $n^{\rm th}$ target corresponds to a rectangular region with arbitrarily large real part and imaginary part bounded, in absolute value, by $2\pi$. Then, we may pullback the $n^{\text th}$ target using suitable branches of $f_a^{-1}$ to obtain two sequences of nested subsets inside $V(\xi_0,\eta_0)$. We intend to prove each nested sequence contains not only a base $\alpha_{t^{i}}, i=1,2,$ but there are also other components of $\gamma(t^i)$ accumulating into the base.\
As observed before, each target $V(\xi_n,\eta_{n+\ell})$ intersects the fundamental domains $T_{0}$ and $T_{1}$ for any $n\in {{\mathbb Z}}^+$. Denote by $W_{n,\ell}^{0_{i}}$ and $W_{n,\ell}^{1_{i}}$ the domains given by $V(\xi_n,\eta_{n+ \ell})\cap T_{0_{i}}$ and $V(\xi_n,\eta_{n+\ell})\cap T_{1_{i}}$, respectively. The next step is to show that, by considering appropriate preimages of these $W$-sets, we obtain a nested sequence of neighborhoods around two particular bases of tails.
In what follows, we will work solely with $B$-sequences and its respective $A$-sequences. We may also assume that every $B$-sequence has infinitely many 1’s to avoid taking a preimage of the positive real line. The next result is a suitable restatement of Lemma 4.3 in [@DJM] following our notation.
\[lemma:two-nested\] Let $\ell \ge 0$ and let $t=(\tau_0,1,\tau_1,1,\tau_2,1,\ldots)$ be a $B$-sequence, where each $\tau_k=\left(\tau_k^1,\ldots, \tau_k^{n_k}\right)$ denotes a finite block of binary symbols $\{0,1\}$ of length $n_k$. Let $m_j=j+1+n_0+n_1+\ldots + n_j, \ j\geq 0$. Then for each $m_{j}$ the sets $$\label{eq:W-preimage-usingB}
g_{a}^{\tau_{0}^1}\circ\cdots\circ
g_{a}^{\tau_{j}^{n_j}} \ \left(W_{m_{j},\ell}^{1_i} \right), \ i=1,2,$$ form two nested sequences of subsets of $V(\xi_0,\eta_\ell)$. Moreover, if we denote by $t^{1}$ and $t^{2}$ the two allowable $A$-sequences satisfying $\pi\left(t^{1}\right)=\pi \left(t^{2}\right)=t$, then $\alpha_{t^1,\ell}$ and $\alpha_{t^2,\ell}$ are contained each in a nested sequence of subsets of $V(\xi_0,\eta_\ell)$ given by (\[eq:W-preimage-usingB\]).
Due to Lemma \[lemma:macrolemma\], for each $m_j$ and $i=1,2$, the sets $W_{m_{j},\ell}^{1_i} \subset V(\xi_{m_j},\eta_{m_j+\ell})$, can be pulled back following $(\tau_0,1,\tau_1,1,\ldots, \tau_j)$. These preimages yield two nested sequences of subsets in $V(\xi_0,\eta_\ell)$, one sequence corresponds to preimages of $W_{m_{j},\ell}^{1_1}$ and the other to preimages of $W_{m_{j},\ell}^{1_2}$. Notice there is a unique way of pulling back each $W_{m_{j},\ell}^{1_i},\ i=1,2$ following the $B$-sequence $t$, since ${f_{a}}|{T_k},\ k=0,1$ are one-to-one maps. Also, at each step of the construction, the points belonging to these nested subsets are points of $V(\xi_0,\eta_\ell)$ with $B$-itinerary $t=(\tau_0,1,\tau_1,1,\ldots, \tau_j,\ldots)$. So, $\alpha_{t^1,\ell}$ and $\alpha_{t^2,\ell}$ must be inside all of them.
The next step in the construction is to show that in each of the nested subsets of $V(\xi_0,\eta_\ell)$ given by the previous lemma we will have not only the bases $\alpha_{t^1,\ell}$ and $\alpha_{t^2,\ell}$, but also other components of $\gamma(t)$. We split this step into two lemmas. The first lemma shows that for suitable choices of $A$-sequences the extended tails cut twice across the line Re$(z)=-\mu$ for $\mu>0$ arbitrarily large.
A finite block of $0$’s (respectively $0_{1}$’s or $0_{2}$’s) of length $k$ will be denoted by $0^k$ (respectively $0^k_1$ or $0^k_{2}$).
\[lemma:extended\_tails\] Let $t$ be any $B$-sequence, and let $t^1, t^2\in \Sigma_A$ so that $\pi(t^1)=\pi(t^2)=t$. Assume also that $0_{1}t^1$ and $0_2t^2$ are allowable. Given any $\mu>0$, there exists $K>0$ such that for all $k>K$, there exist two continuous curves, denoted by $\tilde{\omega}_{1_20_1^k t^1}$ and $\tilde{\omega}_{1_10_2^k t^2}$, that extend to infinity to the right and satisfy:
1. $\tilde{\omega}_{1_20_1^kt^1}$ and $\tilde{\omega}_{1_10_2^kt^2}$ are enlargements (to the left) of the tails with itineraries $1_20_1^kt^1$ and $1_10_2^kt^2$, respectively; and
2. $\tilde{\omega}_{1_20_1^kt^1}$ and $\tilde{\omega}_{1_10_2^kt^2}$ cuts twice across Re$\,z=-\mu$.
Consider any $A$-sequence of the form $0_{1}^mt^1$ with $m>0$ large enough so the unique tail $\omega_{0_1^mt^1}$ is parametrized by $z=(x,h(x))$, $x > R$ (see Proposition \[prop:ConjugTails\]). Lemma \[lem:ends-in-zeros\] and its proof imply that $\omega_{0_1^mt^1}$ is $\varepsilon$-close to $z=(x,0), \ x > R$ (that is, $|h(x)|<\varepsilon$ for all $x>R$).
By pulling back $\omega_{0_1^mt^1}$ using $g_{a}^{0}$, we obtain a new curve which is precisely the extended tail $\tilde{\omega}_{0_1^{m+1}t^1}$. Moreover, since the positive real line is repelling, we observe that $\tilde{\omega}_{0_1^{m+1}t^1}$ is also $\varepsilon$-close to $z=(x,0)$ with $x > g_{a}^0(R)$. Successive pullbacks via $g_{a}^{0}$ allow us to find a first positive integer $r$ such that the extended tail $\tilde{\omega}_{0_1^{m+r}t^1}$ will be $\varepsilon$-close to $z=(x,0), \ x > p_a$ where $p_a$ is the real fixed point in the boundary of $\mathcal A^*(-a)$.
At this stage we pullback again $\tilde{\omega}_{0_1^{m+r}t^1}$ using $g_{a}^{1}$. The obtained curve must coincide with the extended tail $\tilde{\omega}_{1_{2}0_{1}^{m+r}t^1}$. By construction it is a curve extending to infinity to the right, extends far into the left half plane (since $\tilde{\omega}_{0_1^{m+r}t^1}$ is close to $z=0$) and lies close to $q_a$ (the preimage of $p_a$ in $\partial \mathcal A^*(-a)$) since $\tilde{\omega}_{0_1^{m+r}t^1}$ gets arbitrarily close to $z=p_a$. For a given $\mu$, if we choose $m$ large enough, this pullback construction guarantees that $\tilde{\omega}_{1_{2}0_{1}^{m+r}t^1}$ cuts across Re$(z)=-\mu$ twice, as desired.
Analogously, if we start the construction with $0_{2}^m t^2, \ m>0$, we get a similar result for the extended tail $\tilde{\omega}_{1_{1}0_{2}^{m+r}t^2}$.
\[prop:passes\] Fix $\ell \geq 0$. Let $t$ be any $B$-sequence with $t^1$ and $t^2$ its associated $A$-sequences. Let $\tau$ be any finite block of binary symbols of length $n$ and denote by $s=\tau 110^kt$, and by $s^1,s^2$ its associated $A$-sequences. Then there exists $K>0$ such that for all $k>K$, the following statements hold.
1. The forward image of $W_{n,\ell}^{1_{1}}$ cuts three times (twice far to the left and once far to the right) across the extended tail $\tilde{\omega}_{1_20_1^kt^1}$. In other words, the hair $\gamma(1_{1}1_20_1^kt^1)$ cuts three times across $W_{n,\ell}^{1_{1}}$.
2. The hair $\gamma(s^1)$ cuts three times across $$g_{a}^{\tau_{0}}\circ\cdots\circ g_{a}^{\tau_{n-1}}\left(W_{n,\ell}^{1_{1}}\right)\ .$$ Moreover one of the connected components of $\gamma(s^1)$ in the above expression contains $\alpha_{s^1,l}$.
Analogous results corresponding for $s^2$ also apply.
The set $f_{a}\left(W_{n,\ell}^{1_{1}}\right)$ is a large semi-annulus in the upper half plane and intersects $T_0 \cup T_1$ far to the right and far to the left. From Lemma \[lemma:extended\_tails\] we can choose $K$ such that for all $k>K$, the extended tail $\tilde{\omega}_{1_{2}0_{1}^{k}t^1}$ cuts across the semi-annulus $f_{a}\left(W_{n,\ell}^{1_{1}}\right)$ three times: twice far to the left and one far to the right. Consequently there must be three components of $\gamma(1_{1}1_20_1^kt^1)$ in $W_{n,\ell}^{1_{1}}$ that map into three components of the extended tail $\tilde{\omega}_{1_{2}0_{1}^{k}t^1}$ in $f_{a}\left(W_{n,\ell}^{1_{1}}\right)$.
By taking suitable pullbacks of $W_{n,\ell}^{1_{1}}$ that follow the string of symbols $\tau=(\tau_0,\ldots,\tau_{n-1})$ and applying Lemma \[lemma:two-nested\], we can now conclude statement (b) of the present proposition.
\[theorem:indecom\] Let $\tau$ be a finite block of binary symbols of length $n$. There exists an increasing sequence of integers $k_j,\ j\geq 1$ so for the $B$-sequence $${{\mathfrak T}}= \tau110^{k_1}110^{k_2}110^{k_3}\ldots, \label{eq:IndSeq}$$ its associated $A$-sequences ${{\mathfrak T}}^1$ and ${{\mathfrak T}}^2$ determine two distinct hairs $\gamma({{\mathfrak T}}^1)$ and $\gamma({{\mathfrak T}}^2)$ that limit upon themselves, thus becoming non-landing hairs.
Let $\ell>0$, $p_0=n+1$ and for each $i\geq 1$, set $$p_i = n+1 + \sum\limits_{j=1}^{i} k_{j} + \ 2i.$$ The symbol at the $p_i^{\text{th}}$ position in (yet to be constructed) ${{\mathfrak T}}$ is equal to $1$. In general, the symbol at position $p_i$ in the $A$-sequence ${{\mathfrak T}}^j,\ j=1,2$ can be either $1_1$ or $1_2$, depending on the previous entry in ${{\mathfrak T}}^j$. In what follows we assume that all symbols at the $p_i^{\rm th}$ position in ${{\mathfrak T}}^1$ are equal to $1_{1}$ (hence, all symbols at position $p_{i}^{\rm th}$ in ${{\mathfrak T}}^2$ are $1_{2}$). It will become clear from the proof that this assumption is without lost of generality. Moreover, we shall prove only that $\gamma({{\mathfrak T}}^1)$ limits upon itself since the case of $\gamma({{\mathfrak T}}^2)$ follows in the same way.
Let $s$ be any $B$-sequence not ending in all $0$’s, and let $s^1$ and $s^2$ be its associated $A$-sequences. For each $j\geq 1$ we aim to construct inductively $B$-sequences of the form $$\begin{aligned}
\label{eq:sigma-j}
u_j &=& \tau 110^{k_1}110^{k_2}\ldots110^{k_j}\sigma^{p_j}(s),
\end{aligned}$$ so that, by carefully selecting longer blocks of zeros, the $u_j$ will converge to the $B$-sequence ${{\mathfrak T}}$ with the desired properties. Note that for all $j$, $u_j$ will be a concatenation of the first $p_j^{\rm th}$ symbols in ${{\mathfrak T}}=(\mathfrak{t}_0, \mathfrak{t}_1, \ldots)$ with $\sigma^{p_j}(s)$. In other words, $$\begin{aligned}
\label{eq:terms}
u_j &=& (\mathfrak{t}_0, \mathfrak{t}_1, \ldots, \mathfrak{t}_{p_j-1}, s_{p_j}, s_{p_j+1},\ldots).\end{aligned}$$ As before, $u_{j}^1$ and $u_{j}^2$ will denote the $A$-sequences that project into $u_j$.
We first show how to define $u_1$. Applying Proposition \[prop:passes\](a), we can choose an integer $k_1>0$ so the forward image of $W_{p_0-1,\ell}^{1_{1}}$ is cut across by the extended tail $\tilde{\omega}_{1_20_1^{k_{1}}\sigma^{p_1}(s^1)}$ three times (twice far to the left and once far to the right). Thus the hair $\gamma(1_{1}1_20_1^{k_{1}}\sigma^{p_1}(s^1))$ cuts three times across $W_{p_0-1,\ell}^{1_{1}}$. For the given block $\tau$, denote by $\tau_1$ its corresponding $A$-block that makes $u_1^1=\tau_1 1_1 1_2 0_1^{k_1}\sigma^{p_1}(s^1)$ an allowable $A$-sequence. Thus Proposition \[prop:passes\](b) implies that $$g_{a}^{\mathfrak{t}_{0}}\circ\cdots\circ g_{a}^{\mathfrak{t}_{p_0-2}}\left(W_{p_0-1,\ell}^{1_{1}}\right)$$ is a subset of $V\left(\xi_{0},\eta_{\ell}\right)$ that contains $\alpha_{u_{1}^1,\ell}$ and (at least) two other components of the hair $\gamma(u_{1}^1)$.
For $j>1$ we apply again Proposition \[prop:passes\](a) to the sequence $$\tilde{\tau} 110^{k_j}\sigma^{p_j}(s),$$ where $\tilde{\tau}=\tau110^{k_1}110^{k_2}\ldots 110^{k_{j-1}}$ and with an integer $k_{j}>0$ large enough so that the forward image of $W_{p_{j}-1,\ell}^{1_{1}}$ is cut across by the extended tail $\tilde{\omega}_{1_20_1^{k_{j}}\sigma^{p_j}(s^1)}$ three times (twice far to the left and once far to the right). Thus the hair $\gamma(1_{1}1_20_1^{k_{j}}\sigma^{p_j}(s^1))$ cuts three times across $W_{p_{j}-1,\ell}^{1_{1}}$. Pulling back $W_{p_{j}-1,\ell}^{1_{1}}$ through the finite block of binary symbols given by $\tilde{\tau}$, we get from Proposition \[prop:passes\](b) that $$g_{a}^{\mathfrak{t}_{0}}\circ\cdots\circ g_{a}^{\mathfrak{t}_{p_j-2}}\left(W_{p_{j}-1,\ell}^{1_{1}}\right)$$ is a subset of $V\left(\xi_{0},\eta_{\ell}\right)$ that contains $\alpha_{u_{j}^1,\ell}$ and (at least) two other components of the hair $\gamma(u_{j}^1)$.
As $j$ tends to infinity, the sequence $u_{j}$ converges to the desired sequence ${{\mathfrak T}}$. Indeed, from Proposition \[prop:passes\](b), the sets $$g_{a}^{\mathfrak{t}_{0}}\circ\cdots\circ g_{a}^{\mathfrak{t}_{p_j-2}}\left(W_{p_{j}-1,\ell}^{1_{1}}\right)$$ form a nested sequence of subsets of $V(\xi_{0},\eta_{\ell})$ that contains $\alpha_{u_{j},\ell}$ and two further components of the hair $\gamma(u_{j})$. Since $\ell\geq 0$ was selected in an arbitrary manner, as $j\to \infty$ we obtain $V\left(\xi_{0},\infty \right)$ contains $\omega_{{{\mathfrak T}}^1}$ and infinitely many distinct components of the hair $\gamma({{\mathfrak T}}^1)$ accumulating on it. To see this, let $z\in \omega_{{{\mathfrak T}}^1}$ be given and select $\ell_z>0$ large enough so $z$ lies in $\alpha_{{{\mathfrak T}}^1,\ell_z}$. As before, this base lies in the target $V(\xi_0,\eta_{\ell_z})$. Since ${{\mathfrak T}}$ has been already constructed, we may now apply Proposition \[prop:passes\] to $\gamma({{\mathfrak T}}^1)$ itself. There exists an integer $N>0$ sufficiently large so for all $n>N$, $\gamma(\sigma^{p_n+1}({{\mathfrak T}}^1))$ cuts across $W_{p_n-1,\ell_z}^{1_1}$ in (at least) three components, say $A^n_1, A^n_2$ and $A^n_3$, with one of them being the component of the tail $\omega_{\sigma^{p_n+1}({{\mathfrak T}}^1)}$.
As $n$ increases, the diameter of $g_a^{\mathfrak{t}_0}\circ\ldots \circ g_a^{\mathfrak{t}_{p_n}}(W_{p_n-1,\ell_z}^{1_1})$ decreases, nonetheless, Proposition \[prop:passes\] implies that for each $j=1,2,3$, $$g_a^{\mathfrak{t}_0}\circ\ldots \circ g_a^{\mathfrak{t}_{p_n}}(A_j^n)~~\text{cuts across}~~ g_a^{\mathfrak{t}_0}\circ\ldots \circ g_a^{\mathfrak{t}_{p_n}}(W_{p_n-1,\ell_z}^{1_1}).$$ Since $g_a^{\mathfrak{t}_0}\circ\ldots \circ g_a^{\mathfrak{t}_{p_n}}(A_j^n)=\alpha_{{{\mathfrak T}}^1,\ell_z}$ for some $j$, the pullback images of the remaining $A_j^n$’s accumulate lenghtwise over $\alpha_{{{\mathfrak T}}^1,\ell_z}$ (and thus along $z$) for each $n>N$.
To see that $\gamma({{\mathfrak T}}^1)$ accumulates on each of its points and not only on its tail portion, we may perform the same construction for the sequences $$110^{k_i}110^{k_{i+1}}\ldots,$$ for $i\geq 1$. Then we may pullback the corresponding hairs and their accumulations by the appropriate inverse branches of $f_{a}$ to show that $\gamma({{\mathfrak T}}^1)$ must accumulate on any point in the hair $\gamma({{\mathfrak T}}^1)$.
Geometry of hairs {#section:indecom}
=================
This last section will focus on the geometry of sets $\overline{\gamma(t)}$ when $t$ is a $B$-sequence that is either periodic or a sequence as constructed in Theorem \[theorem:indecom\]. We show that in the former case, the closure of the hair has a landing point in $\partial {{\mathcal A}}^*(-a)$ while, in the latter case, $\overline{\gamma(t^i)}$ is an indecomposable continuum for each $i=1,2$.
\[prop:PerSeqLand\] Let $t$ be a periodic $B$-sequence and $t^1, t^2$ their periodic associated $A$-sequences. Then, the hairs $\gamma(t^1)$ and $\gamma(t^2)$ land at two (repelling) periodic points $p_1,p_2\in \partial{{\mathcal A}}^*(-a)$.
Let $t=\overline{t_0\ldots t_{n-1}}$, so every block of $0$’s in $t$ has bounded length. We work only with $t^1$ since the other case follows similarly.
There exists a value $\delta>0$ and a closed ball $\overline{B_{\delta}(0)}$ so the orbit of $\overline{\gamma(t^1)}$ stays always outside $\overline{B_{\delta}(0)}$. Analogously, we can find a real value $m$ so the orbit of $\overline{\gamma(t^1)}$ does not intersect the half plane Re$(z)< m<0$.
Let $R>0$ as in Proposition \[prop:ConjugTails\]. Then, there exists a value $M\geq R$ for which the intersection of the tail $\omega_{t^1}$ with the line Re$(z)=M$ is a single point. Since the orbit of the origin escapes along the positive real line, there exists an integer $N=N(M)>0$ for which $f_a^{N-1}(0)<M\leq f_a^{N}(0)$. Select $0<\epsilon\leq \delta$ small enough so for $B_\epsilon=\overline{B_\varepsilon(0)}$ and $f_a(B_\epsilon),\ldots, f^{N}_a(B_\epsilon)$, they are compact domains contained in $T_0$. Finally, select $m'\leq m<0$ so the left half plane Re$(z)\leq m'$ maps completely inside $B_\epsilon$.
With these constants we can define an open, connected, simply connected region $E$ as in the proof of Theorem \[thm:BdOrbit\]. Then, the pullback process that defines the hair $\gamma(t^1)$ together with the contracting map $\Psi_\ell= \psi_{t_0^1}^{-1}\circ(g_a^{t_0^1}\circ \cdots \circ g_a^{t_{n-1}^1})^\ell \circ \psi_{t_0^1}$ (where $(g_a^{t_0^1}\circ \cdots \circ g_a^{t_{n-1}^1})^\ell$ denotes the $\ell$-fold composition with itself), shows that $\overline{\gamma(t^1)}\setminus \gamma(t^1)$ is a unique periodic point in $E$ with itinerary $t^1$. By Theorem \[thm:BdOrbit\], this periodic point lies in $\partial A^*(-a)$.
\[thm:DoesNotSeparate\] Let ${{\mathfrak T}}$ be a $B$-sequence as in Theorem \[theorem:indecom\], ${{\mathfrak T}}^1, {{\mathfrak T}}^2$ be its associated $A$-sequences that project onto ${{\mathfrak T}}$ and $\gamma({{\mathfrak T}}^1),\gamma({{\mathfrak T}}^2)$ their associated hairs. Then, the closure of each of these hairs is an indecomposable continuum.
As before, we restrict the proof to the $A$-sequence ${{\mathfrak T}}^1=\tau_1 1_1 1_2 0_1^{k_1}1_1 1_2 0_1^{k_2}\ldots$. Let $\Gamma^1$ be the closure of the hair $\gamma({{\mathfrak T}}^1)$. In order to apply Theorem \[theo:curry\], we must verify first that $\Gamma^1$ does not separate the plane, as from Theorem \[theorem:indecom\] $\gamma({{\mathfrak T}}^1)$ is a curve that accumulates upon itself.
First observe that $\Gamma^1$ is a set with bounded negative real part. Indeed, the first block of $0_i$’s in ${{\mathfrak T}}^1$ has finite length and this implies $\gamma({{\mathfrak T}}^1)$ (and thus $\Gamma^1$) lies to the right of the line Re$(z)=m$ for some $m<0$. For simplicity, denote by $k_0\in\{0_1,0_2,1_1,1_2\}$ the first entry in $\tau_1$, so $\Gamma^1$ lies inside the set $T_{k_0}$ and to the right of the line Re$(z)=m$. Hence $${{\Bbb C}}\setminus (T_{k_0}\cap \{z~|~\text{Re}(z)>m\})\subset {{\Bbb C}}\setminus \Gamma^1,$$ and since the bounderies of $T_0$ and $T_1$ are the graphs of strictly monotonic functions, the set in the left hand side is in fact a single component with unbounded imaginary part. This implies the existence of a single complementary component $U\subset {{\Bbb C}}\setminus \Gamma^1$ with unbounded imaginary part, while all other complementary components must have bounded imaginary parts. Also, note that $-a\in U$ and since $-a$ is a fixed point then $-a\in f_a^k(U)$ for all $k>0$.
Let $V \neq \emptyset $ be a component in ${{\Bbb C}}\setminus \Gamma^1$ with bounded imaginary part. Firstly, we assume that $V\cap {{\mathcal J}}(f_a)\neq \emptyset$. Then, by Montel’s Theorem, there exists $N>0$ such that $-a\notin f^k(V)$ for $0\leq k<N$ while $-a\in f^N_a(V)$, since $-a$ is not an exceptional value (indeed, $-a$ has infinitely many preimages). But this implies that $f_a^{N-1}(V)$ is a component with bounded imaginary part inside $T_0\cup T_1$ that must contain a point in $f_a^{-1}(-a)$, a contradiction since in $T_0 \cup T_1$ there are no preimages of $-a$ different from $-a$ itself. Secondly, we assume that $V$ does not contain points in ${{\mathcal J}}(f_a)$ and thus is a Fatou component. Since the Fatou set coincides with the basin of attraction of $-a$, there exists an integer $N>0$ so that $f^N_a(U)={{\mathcal A}}^*(-a)$ and thus $\partial {{\mathcal A}}^*(-a)\subset \partial U$, which is a contradiction with Proposition \[prop:consequences\_pol\_like\](b), as only one point in $\partial {{\mathcal A}}^*(-a)$ has itinerary ${{\mathfrak T}}^1$.
We conclude that $\Gamma^1$ does not separate the plane and by Theorem \[theo:curry\], it is either an indecomposable continuum or the union of two indecomposable continua. Nevertheless, $\gamma({{\mathfrak T}}^1)$ has a unique tail extending to infinity so it cannot be the union of two indecomposable continua.
The above two results describe an interesting relationship between the combinatorics of an itinerary $s$ and the landing properties of the curve $\gamma(s)$. Indeed, if $s$ is an $A$-sequence whose blocks of $0_i$’s have bounded length, the same arguments as in Proposition \[prop:PerSeqLand\] can be applied to show that the accumulation set of the hair is no other than the unique point $p(s)\in \partial {{\mathcal A}}^*(-a)$ that follows the given itinerary. The key step is to ensure that pullbacks of the tail are always bounded away from the postcritical and asymptotical orbits (see, for instance, Proposition 3.6 in [@Fag]), and this is always the case for $A$-sequences whose blocks of $0_i$’s have bounded length.
In contrast, whenever $s={{\mathfrak T}}$ is a $B$-sequence as in Theorem \[theorem:indecom\], then the corresponding point $p({{\mathfrak T}}^1)$ in $\partial {{\mathcal A}}^*(-a)$ is an accumulation point of $\gamma({{\mathfrak T}}^1)$, nevertheless, is not its endpoint.
\[theorem:relation\_inde\_basin\] Consider ${{\mathfrak T}}$ and $\gamma({{\mathfrak T}}^1)$ as in Theorem \[theorem:indecom\], set $\Gamma^1= \overline{\gamma({{\mathfrak T}}^1)}$. Then, there exists a unique point $p_1\in \partial {{\mathcal A}}^*(-a)$ such that $p_1\in \Gamma^1$ and $p_1$ has itinerary ${{\mathfrak T}}^1$.
Recall that ${{\mathfrak T}}=\tau110^{k_1}110^{k_2}\ldots$. For each $n\geq1$ define two preperiodic sequences given by $$\begin{aligned}
s_n&=&\tau\overline{110^{k_1}\ldots 110^{k_n}0}, \\
r_n&=&\tau\overline{110^{k_1}\ldots 110^{k_n}1}.\end{aligned}$$ Clearly, their associated $A$-sequences satisfy $s^1_n < {{\mathfrak T}}^1 < r^1_n$ for all $n\geq 1$ with respect to the distance induced by the usual order in $\Sigma_B$. Moreover, if $p(s^1_n)$ and $q(r^1_n)$ denote endpoints of the corresponding preperiodic hairs associated to $s^1_n$ and $r^1_n$, then it follows by Proposition \[prop:PerSeqLand\] that $p(s^1_n), q(r^1_n)\in \partial {{\mathcal A}}^*(-a)$. Let $p({{\mathfrak T}}^1)$ be the unique point in $\partial {{\mathcal A}}^*(-a)$ following the itinerary ${{\mathfrak T}}^1$ under the action of $f_a|\partial{{\mathcal A}}^*(-a)$.\
In the Euclidean distance restricted to $\partial {{\mathcal A}}^*(-a)$ we obtain $|p(s^1_n)-q(r^1_n)|\to 0$ as $n\to +\infty$ and clearly $p(s^1_n)<p({{\mathfrak T}}^1)<q(r^1_n)$ with the order inherited by $S^1$ under the continuous extension of the Böttcher mapping $\Phi_a:{{\Bbb D}}\to {{\mathcal A}}^*(-a)$.
For each $n$, $\gamma({{\mathfrak T}}^1)$ belongs to the region bounded above and below by $\gamma(r^1_n)$ and $\gamma(s^1_n)$, and the arc $[p(s^1_n),q(r^1_n)]\subset\partial {{\mathcal A}}^*(-a)$ containing $p({{\mathfrak T}}^1)$. Thus, if $\gamma({{\mathfrak T}}^1)$ accumulates on $\partial {{\mathcal A}}^*(-a)$, then it must accumulate at the point $p({{\mathfrak T}}^1)$.
In order to show that $\overline{\gamma({{\mathfrak T}}^1)}$ accumulates on the boundary of ${{\mathcal A}}^*(-a)$, we employ the polynomial-like construction and the symbolics of ${{\mathfrak T}}^1$. Let $m>0$ and consider the hair associated to the itinerary $$t_m=0_{k_m}11 0_{k_{m+1}}11 \ldots.$$
Note that $t_m$ is the image of ${{\mathfrak T}}^1$ under some iterates of the shift $\sigma|{\Sigma_A}$. By Proposition \[prop:passes\], the hair $\gamma(t_{m})$ is clearly close to the origin and also the hair $\gamma(1t_m)$ intersects a left half plane. Hence, a portion of $\gamma(1t_m)$ must intersect $V_a$ and its preimages under $f_a|V_a$ following $\tau110^{k_1}\ldots 0^{k_{m-1}}1$ must accumulate in $\partial {{\mathcal A}}^*(-a)$ since by Proposition \[proposition:pol\_like\], $$\bigcap_{n\geq 0} f_a^{-n}(V_a)=\partial{{\mathcal A}}^*(-a),$$ and $m>0$ has been taken arbitrarily large.
We end this section by briefly discussing the set of points in the Julia set that follow non-binary sequences. Note first that Proposition \[prop:ConjugTails\] can be easily modified to show the existence of tails with itineraries corresponding to the partition $\cup_{j\in {{\Bbb Z}}} T_j$. If $s=(s_0,s_1, \ldots)$ is a sequence that contains infinitely many non-binary symbols and $T_{s_j}\subset f_a(T_{s_{j+1}})$ for all $j\geq 0$. Let $\omega_s$ be the tail associated to $s$ that lies in the right half plane Re$(z)>R$. If in addition $s$ has (if any) blocks of $0$’s with bounded length, then the pullbacks of $\omega_s$ are always bounded away from the postcritical and asymptotic orbits, hence $\gamma(s)$ is a landing hair with an endpoint $p(s)$. By Proposition \[prop:consequences\_pol\_like\], $p(s)$ (and in fact $\overline{ \gamma(s)}$) do not lie in the boundary of any Fatou component, as otherwise, $s$ will have to end in a binary sequence. This establishes
If $s=(s_0, s_1,\ldots)\in {{\Bbb Z}}^{{\Bbb N}}$ is realizable by $f_a$, contains infinitely many non-binary symbols and its blocks of $0$’s have bounded lenght, then $\gamma(s)$ is a landing hair and a buried component of ${{\mathcal J}}(f_a)$.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The first and second author are both partially supported by the European network 035651-2-CODY, by MEC and CIRIT through the grants MTM2008Ð01486 and 2009SGR-792, respectively. The second author is also partially supported by MEC through the grant MTM2006-05849/Consolider (including a FEDER contribution). The third author is supported by CONACyT grant 59183, CB-2006-01. She would also like to express her gratitude to Universitat de Barcelona and Universitat Rovira i Virgili for their hospitality in the final stages of this article.
[Abc]{}
K. Barański, *Trees and hairs for some hyperbolic entire maps of finite order*. Math. Z. 257, 33-59 (2007). K. Barański, X. Jarque and L. Rempe *Brushing the hairs of transcendental entire functions*. Preprint arXiv:1101.4209. W. Bergweiler, *Iteration of meromorphic functions.* Bull. Amer. Math. Soc. (N.S.) 29, no. 2, 151Ð188 (1993). S. Curry, *One-dimensional nonseparating plane continua with disjoint $\epsilon$-dense subcontinua*. Topol. and its Appl. 39, 145-151 (1991). R. L. Devaney, *Knaster-like continua and complex dynamics*. Erg. Th. and Dyn. Sys. 13, 627-634 (1993). R. L. Devaney and M. Krych, *Dynamics of [exp]{}$(z)$*. Erg. Th. and Dyn. Sys. 4, 35-52 (1984). R. L. Devaney and X. Jarque, *Misiurewicz points for complex exponentials*. Int. J. Bifurc. and Chaos. 7, 1599-1616 (1997). R. L. Devaney and X. Jarque, *Indecomposable continua in complex dynamics*. Conform. Geom. Dyn. (electronic), 1-12 (2002). R. L. Devaney, X. Jarque and M. Moreno Rocha, *Indecomposable continua and Misiurewicz points in exponential dynamics*. Int. J. Bifurc. and Chaos. 15(10), 3281-3294 (2005). R. L. Devaney and F. Tangerman, *Dynamics of entire functions near the essential singularity*. Ergod. Th. and Dynam. Sys. 6, 489-503 (1986). A. Douady and J. H. Hubbard,*On the dynamics of polynomial-like mappings*. Ann. Sci. École Norm. Sup. 4, no. 2, 287-343 (1985). A. È. Erëmenko, *On the iteration of entire functions*. In Dynamical systems and ergodic theory (Warsaw, 1986), Banach Center Publ., Warsaw, 339-345 (1989). A. È. Erëmenko and M. Yu. Lyubich, *Dynamical properties of some classes of entire functions*. Ann. Inst. Fourier (Grenoble) 42 no. 4, 989-1020 (1992). N. Fagella, *Dynamics of the complex standard family*. J. Math. Anal. Appl. 229 no. 1, 1-31 (1999). P. Fatou, *Sur l’Itération des fonctions transcendentes entières*. Acta Math. 47, 337-370 (1926). L. Goldberg and L. Keen, *A finitness theorem for a dynamical class of entire functions*. Erg. Th. and Dyn. Syst. 6, 183-192 (1986). C. Kuratowski, *Topologie*. Volume [II]{}. Państwowe Wydawnictwo Naukowe, Warzawa (1961). M. Moreno Rocha, *Existence of indecomposable continua for unstable exponentials*. Top. Proc. 27, 233-244 (2002). S. Morosawa, *Local connectedness of Julia sets for transcendental entire functions*. In Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis, pp. 266-273. World Scientific, (1999). S. Morosawa, Y. Nishimura, M. Taniguchi, and T. Ueda, *Holomorphic dynamics*. Cambridge University Press, Cambridge (2000). S. B. Nadler Jr., *Continuum theory*. Marcel Dekker Inc., New York (1992). L. Rempe, *Rigidity of escaping dynamics for transcendental entire maps*. Acta Math. 203, 235-267 (2009). L. Rempe, *On nonlanding dynamic rays of exponential maps*. Ann. Acad. Sci. Fenn. Math. 32, 353-369 (2007). L. Rempe, *Siegel disks and periodic rays of entire functions*. J. Reine Angew. Math. 624, 81-102 (2008). D. Schleicher and J. Zimmer, *Escaping Points of Exponential Maps*. J. London Math. Soc. 67 (2), 380-400 (2003). G. T. Whyburn, *Topological characterization of the Sierpiński curve*. Fund. Math. 45, 320-324 (1958).
[Antonio Garijo & Xavier Jarque]{}\
[Dept. d’Enginyeria Informàtica i Matemàtiques]{}\
[Universitat Rovira i Virgili]{}\
[Av. Països Catalans 26]{}\
[Tarragona 43007, Spain]{}\
[Mónica Moreno Rocha]{}\
[Centro de Investigación en Matemáticas]{}\
[Callejón Jalisco s/n]{}\
[Guanajuato 36240, Mexico]{}
[^1]: Corresponding author. E-mail: `[email protected]`.
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abstract: 'Model selection aims to determine which theoretical models are most plausible given some data, without necessarily asking about the preferred values of the model parameters. A common model selection question is to ask when new data require introduction of an additional parameter, describing a newly-discovered physical effect. We review several model selection statistics, and then focus on use of the Bayesian evidence, which implements the usual Bayesian analysis framework at the level of models rather than parameters. We describe our *CosmoNest* code, which is the first computationally-efficient implementation of Bayesian model selection in a cosmological context. We apply it to recent WMAP satellite data, examining the need for a perturbation spectral index differing from the scale-invariant (Harrison–Zel’dovich) case.'
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[COSMOLOGICAL MODEL SELECTION]{}
\
Astronomy Centre, University of Sussex, Brighton BN1 9QH
Introduction
============
Cosmologists are becoming very good at determining the parameters of the Universe. Within the last few years observational results, exemplified by the microwave background anisotropy measurements by the Wilkinson Microwave Anisotropy Probe (WMAP), have introduced precision into cosmological modelling. Considerable sophistication is now required both in deriving theoretical predictions from models and in carrying out data analysis procedures able to squeeze the best from the data.
Within a cosmological model, the parameters indicate the importance of different effects. For instance, they describe the relative amounts of different types of material in the Universe, the geometry and expansion rate of the Universe, and the properties of the initial irregularities in the Universe which led to the formation of structure. Such parameters are not predicted by fundamental theories, but rather must be fit from data in order to decide which combination, if any, is capable of describing our Universe. A variety of cosmological data are currently well fit by a model of the Universe that is homogeneous, isotropic and spatially flat, contains cold dark matter in greater proportions than ordinary baryonic matter, and in which tiny initial perturbations evolved under Einstein’s theory of General Relativity into the structures of today. Current data constrain the parameters of this model rather well, many at the 10% level or better.
However, the presence of good data leads to a different problem, one of knowing when to stop fitting. Two different and competing models of the Universe may explain the data equally well, so how do we choose between them? The solution is one proposed by William of Occam, that the simpler model should be preferred. This is known as *Occam’s razor*. So a complicated model that explains the data slightly better than a simple one should be penalized for the extra parameters it introduces, because the extra parameters bring with them a lack of predictability. On the other hand, if a model is too simple, and cannot fit certain data well, then it can be discarded. This is a rather common type of statistical problem, both in cosmology and in other fields of astrophysics: each available parameter within a model describes some piece of physics that might be relevant to our Universe, but until measurements are made we don’t know which.
Cosmological *model selection* refers to comparing different model descriptions of the data. It doesn’t care particularly about the actual values of parameters, but rather aims to determine which [*set*]{} of parameters gives the preferred fit to observational data.
Why model select?
=================
Model selection is an extremely widespread challenge throughout science; how do you fit to data when you are unsure about the set of parameters that you should be deploying. You cannot just include every parameter you can think of in a fit to data, because inclusion of extra unnecessary parameters worsens the determination of those that are essential, so that very soon you can end up learning nothing about anything. Moreover, you can’t just use goodness of fit to the data, because typically inclusion of a new parameter will improve the goodness of fit even if that parameter has absolutely no actual relevance to the Universe. Typical attempts to avoid these problems involve [*ad hoc*]{} criteria such as ‘chi-squared per degrees of freedom’ arguments or the ‘likelihood ratio test’, in which arbitrary thresholds have to be invoked to decide which way the verdict is supposed to go. Model selection aims to put this practice on a firmer footing.
Model selection problems are ones in which the parameter set necessary to describe a given dataset is unknown, the question typically being whether new data justifies inclusion of a new physical parameter. Many of the most pressing questions in astrophysics are of this form. Cosmological examples would include whether the spatial curvature is non-zero, whether the dark energy density evolves, and whether the initial perturbation spectrum has an amplitude which varies with length scale.
Model selection statistics
==========================
The generic purpose of a model selection statistic is to set up a tension between the predictiveness of a model (for instance indicated by the number of free parameters) and its ability to fit observational data. Oversimplistic models offering a poor fit should of course be thrown out, but so should more complex models which offer poor predictive power.
There are two main types of model selection statistic that have been used in the literature so far. [**Information criteria**]{} look at the best-fitting parameter values and attach a penalty for the number of parameters; they are essentially a technical formulation of ‘chi-squared per degrees of freedom’ arguments. By contrast, the [**Bayesian evidence**]{} applies the same type of likelihood analysis familiar from parameter estimation, but at the level of models rather than parameters. It depends on goodness of fit across the entire model parameter space.
Here we discuss three possible statistics. In each case, the statistic is a single number that is a property of the model, and having computed it the models can be placed in a rank-ordered list.
Akaike Information Criterion (AIC):
: This was derived by Hirotugu Akaike in 1974, and takes the form $${\rm AIC} = -2\ln {\cal L}_{\rm max} + 2k \,,$$ where ${\cal L}$ is the likelihood ($-2\ln {\cal L}$ is often called $\chi^2$ though it generalizes it to non-gaussian distributions) and $k$ is the number of parameters in the model. The subscript ‘max’ indicates that one should find the parameter values yielding the highest possible likelihood within the model. It is obvious that this second term acts as a kind of ‘Occam factor’; initially as parameters are added the fit to data improves rapidly until a reasonable fit is achieved, but further parameters then add little and the penalty term $2k$ takes over. The generic shape of the AIC as a function of number of parameters is therefore a rapid fall, a minimum, and then a rise. The preferred model sits at the minimum.
The AIC was derived from information-theoretic considerations, specifically an approximate minimization of the Kullback–Leibler information entropy which measures the distance between two probability distributions.
Bayesian Information Criterion (BIC):
: This was derived by Gideon Schwarz in 1978, and strongly resembles the AIC. It is given by $${\rm BIC} = -2\ln {\cal L}_{\rm max} + k \ln N \,,$$ where $N$ is the number of datapoints. Since a typical dataset will have $\ln N >2$, the BIC imposes a stricter penalty against extra parameters than the AIC.
It was derived as an approximation to the Bayesian evidence, to be discussed next, but the assumptions required are very restrictive and unlikely to hold in practice, rendering the approximation quite crude.
Bayesian evidence:
: The Bayesian evidence looks rather different, being defined as $$E = \int {\cal L}(\theta) {\rm Pr}(\theta) d\theta \,.$$ Here $\theta$ is the vector of parameters of the model, and ${\rm
Pr}(\theta)$ is the prior distribution of those parameters [*before*]{} the data were obtained. The prior is an essential part of the definition of a model, upon which the evidence will ultimately depend, and might for instance be a set of ranges within which parameters are assumed to be uniformly distributed.
The evidence of a model is thus the average likelihood of the model in the prior. Unlike the statistics above, it does not focus on the best-fitting parameters of the model, but rather asks “of all the parameter values you thought were viable before the data came along, how well on average did they fit the data?”. Literally, it is the likelihood of the model given the data. Given Bayes’ theorem $$P(M|D)=\frac{P(D|M)P(M)}{P(D)} \,.$$ (here M is the model, D is the data, and the vertical bar is read as ‘given’), the evidence updates the prior model probability $P(M)$ to the posterior model probability $P(M|D)$, i.e. the probability of the model given the data.
The evidence rewards predictability of models, provided they give a good fit to the data, and hence gives an axiomatic realization of Occam’s razor. A model with little parameter freedom is likely to fit data over much of its parameter space, whereas a model which could match pretty much any data that might have cropped up will give a better fit to the actual data but only in a small region of its larger parameter space, pulling the average likelihood down.
The evidence is also known as the marginalized likelihood or, more accurately, the model likelihood. The ratio of evidences for two models is known as the Bayes factor.
Of these statistics, we would advocate using, wherever possible, the Bayesian evidence which is a full implementation of Bayesian inference and can be directly interpreted in terms of model probabilities. It is computationally challenging to compute, being a highly-peaked multi-dimensional integral, but recent algorithm development has made it feasible in cosmological contexts. We discuss it further in the next section.
If the Bayesian evidence cannot be computed, the BIC can be deployed as a substitute. It is much simpler to compute as one need only find the point of maximum likelihood for each model. However interpreting it can be difficult. Its main usefulness is as an approximation to the evidence, but this holds only for gaussian likelihoods and provided the datapoints are independent and identically distributed. The latter condition holds poorly for the current global cosmological dataset, though it can potentially be improved by binning of the data hence decreasing the $N$ in the penalty term.
The AIC has been widely used outside astrophysics, but is of debatable utility. Sometime after it was first derived, it was shown to be ‘dimensionally inconsistent’, a statistical term meaning that it is not guaranteed to give the right result even in the limit of infinite unbiased data. It may however be useful for checking the robustness of conclusions drawn using the BIC. The evidence and BIC are dimensionally consistent.
Computing and interpreting the evidence
=======================================
Computing the evidence in realistic problems is challenging, particularly in cosmology where evaluating theoretical predictions at just a single parameter point requires several seconds of CPU time with state-of-the-art codes such as `cmbfast` or `camb`. Markov chain Monte Carlo (MCMC) methods are now commonplace in cosmological parameter estimation, and efficiently trace the posterior probability distribution of the parameters of a model in the vicinity of the best-fit region. However a different sampling strategy is needed to evaluate the evidence. It can receive a large contribution from the tails of the posterior distribution of the parameters, because even though the likelihoods there are small, this region occupies a large volume of the prior probability space. Therefore the sampling strategy must effectively sample the entire prior volume to evaluate the integral (Eqn 3) accurately. Until recently, the best available strategy for evidence calculation, known as thermodynamic integration or simulated annealing, required around $10^7$ likelihood evaluations for an accurate answer for a five-parameter cosmological model, placing the problem at the limit of current supercomputer power.
Fortunately, a powerful new algorithm for evidence evaluation, known as [**nested sampling**]{}, was recently invented by John Skilling (2004). At Sussex we have implemented this algorithm for cosmology in a code named `CosmoNest`, which we recently made publically available. It has proven to be one to two orders of magnitude more efficient than thermodynamic integration, meaning that evidence calculations can now be run on a small computing cluster.
To set up the algorithm, the evidence integral is first recast as a one-dimensional integral in terms of the prior mass $X$, where $dX =
\mathrm{Pr}(\theta)\,d\theta$ with $X$ running from 0 to 1. \[A mental image to accompany this is to consider the prior parameter space as a cube, and to smash it with a large hammer. The fragments are then arranged in a line in order of increasing likelihood.\] The algorithm samples the prior a large number of times, assigning a ‘prior mass’ to each sample. The samples are ordered by likelihood, and the integration follows as the sum of the sequence, $$E = \int L(X)dX = \sum_{j=1}^m E_j\,, \quad
E_j=\frac{L_j}{2}(X_{j-1}-X_{j+1}) \,.$$ This is shown in Figure \[fig:nested\].
In order to compute the integral accurately the prior mass is logarithmically sampled. We start by randomly placing a set of $N$ points within the prior parameter space, where in a typical cosmological application $N \simeq 300$. We then iteratively discard the lowest likelihood point $L_j$, replacing it with a new point uniformly sampled from the remaining prior mass (i.e. with likelihood greater than $L_j$). Each time a point is discarded the prior mass remaining, $X_j$, shrinks by a factor that is known probabilistically, and the evidence is incremented accordingly. In this way the algorithm works its way towards the higher likelihood regions. The process is illustrated in Figure \[fig:timeseries\]. Additional details of the algorithm are in Mukherjee, Parkinson & Liddle (2006a). The algorithm is simple, works accurately even in high dimensions, and should be generally applicable in a number of areas even outside of astrophysics.
Although the evidence gives a rank-ordered list of models, it is still necessary to decide how big a difference in evidence is needed to be significant. If the prior probabilities of the models are assumed equal, the difference in log(evidence) can be directly interpreted as the relative probabilities of the models after the data. Even if people disagree on the relative prior probabilities, they will all agree on the direction in which the data, represented by the evidence, has shifted the balance. The usual interpretational scale employed is due to Sir Harold Jeffreys (from his classic 1961 book ‘Theory of Probability’), which, given a difference $\Delta \ln E$ between the evidences $E$ of two models, states that
-------------------------- -------------------------------------
$\Delta \ln E < 1$ Not worth more than a bare mention.
$1 < \Delta \ln E < 2.5$ Significant.
$2.5 < \Delta \ln E < 5$ Strong to very strong.
$5 < \Delta \ln E$ Decisive.
-------------------------- -------------------------------------
In practice we find the divisions at 2.5 (corresponding to posterior odds of about 13:1) and 5 (corresponding to posterior odds of about 150:1) the most useful.
When should model selection be deployed? If the data indicates something strongly enough, it doesn’t really matter how the statistical analysis is done. The main zone of interest is where a new parameter is ‘detected’ at between two and four ‘sigmas’ via parameter estimation techniques. These overestimate the significance of a detection because they ignore model dimensionality, and there is a well-known (in the statistics literature anyway) phenomenon called Lindley’s paradox, whereby model selection considerations can overturn an analysis based on a ‘number of sigmas’ argument. A nice discussion of Lindley’s paradox is given in Trotta (2005).
One of the bugbears of Bayesian methods is the requirement to specify priors explicitly, with the evidence depending on the choice of priors. If the data have low informative content (technically defined via the ratio of prior and posterior parameter volumes), this can be a serious issue, but it becomes less so if the data are constraining so that the posterior is well localized within any conceivable prior. In that case the evidence becomes proportional to the prior volume, and quite a substantial change in volume is needed to move models significantly around the Jeffreys scale.
Applications of model selection
===============================
There are several areas of application of model selection techniques, the main two being as follows:
Application to data:
: With real data, one can assess the viability of different models under consideration. In this case one simply computes the evidence for each model of interest and ranks them.
Model selection forecasting:
: This application aims to compare the power of different experiments before they are carried out. Many proposed experiments seek to answer model selection questions, but their capabilities are often quantified using parameter estimation projections, such as Fisher matrix forecasting. For instance, a dark energy experiment may be advertized as able to measure the equation of state parameter $w$ with an uncertainty of $\pm 0.05$, the aim being to detect deviations of $w$ from $-1$, which characterizes the cosmological constant or vacuum zero-point energy. One can instead forecast experiments’ ability to carry out model selection tests. In this case data must be simulated for a range of different assumed models, in order to investigate where in the available parameter space a given experiment can make a strong or decisive model comparison between a dynamical dark energy model and the cosmological constant. This gives a powerful tool for comparing the statistical power of competing experiments. It should also be possible to extend this concept to survey optimization, whereby one tunes survey parameters to optimize the ability to carry out model selection tests, but it is less clear that this will be fruitful.
We have extensively discussed the philosophy of model selection forecasting, with specific application to dark energy experiments, in Mukherjee et al. (2006b). In Pahud et al. (2006) we applied these ideas to determination of the nature of the primordial power spectrum of density perturbations, focussing on the ability of the Planck Satellite mission to perform model selection of this type. In this article we will focus on applications to real data.
A toy model
-----------
To help understand what is going on, we can carry out a simple toy model investigation into the spatial curvature of the Universe. According to the three-year data from WMAP (henceforth WMAP3), the total density, in units of the critical density, is $\Omega = 1.003 \pm 0.015$ (where we took the liberty of symmetrizing the uncertainty and where the Hubble Key project determination of the Hubble parameter is also used). Given this, how likely is it that the Universe is flat? For simplicity we’ll assume a gaussian likelihood corresponding to this measurement, and ignore dependence on other parameters, so we have $${\cal L} = {\cal L}_0 \exp \left[ - \frac{\left(\Omega -
1.003\right)^2}{2 \times 0.015^2} \right]$$ We also have to choose a prior range for $\Omega$; let’s say $0.1<\Omega < 2$ representing some plausible range people might have considered long before precision data emerged. Now the calculation, remembering that the evidence is just the average likelihood over the prior.
Flat model:
: We just have to evaluate the likelihood at $\Omega
= 1$. It is $E({\rm flat}) = 0.98 {\cal L}_0$.
Curved model:
: Now we have to integrate the likelihood over the prior, being sure to normalize the prior properly. This gives $E({\rm curved}) = 0.02 {\cal L}_0$.
The conclusion is that, under these assumptions, the flat model is preferred at odds of approximately 50:1.
That example was pretty boring, since $\Omega=1$ lies almost in the middle of the measured range. But suppose the result had been $\Omega
= 1.045 \pm 0.015$, a putative three-sigma detection of spatial curvature. The evidence for the curved model is unchanged (it doesn’t care what the measured value is provided it is well within the prior) while that of the flat model shrinks. Nevertheless, the end result is an odds ratio of only 2:1 in favour of the curved model. In this case, three-sigma is nowhere near enough to convincingly indicate that space is curved. Physically, the evidence is allowing for it being [*a priori*]{} very unlikely that $\Omega$ could be so close to one as to give such a low-confidence ‘detection’, yet still not be equal to one. Put another way, the $\Delta \ln E$ would be 0.7 which according to Jeffreys is hardly worth mentioning.
Real cosmology
--------------
Now we turn to a real cosmological example. The new three-year data from WMAP (see Figures \[fig:ilc\] and \[fig:spectrum\]) is for the most part uncontroversial from a model selection point of view, with parameters either being definitely required or clearly unnecessary. The exception is the scale dependence of primordial density perturbations, defined by the spectral index $n_{{\rm S}}$. These perturbations are usually considered to have been generated by inflation, a period of rapid acceleration in the early Universe. As well as solving some of the problems with the traditional hot big bang model, inflation also generically predicts the kind of observations that we now see. The many models of inflation predict a wide range of possible values for $n_{{\rm S}}$, which one should then try and fit from the data.
However, a decade before inflation was invented, Harrison and Zel’dovich independently proposed that $n_{{\rm S}}$ should be precisely one, corresponding to perturbations whose amplitude is independent of scale. At least until this year’s publication of the three-year WMAP data, the Harrison–Zel’dovich spectrum always gave a good fit to existing data. From a model selection perspective it benefits from having one parameter less than a model where $n_{{\rm
S}}$ varies, and indeed we showed in a paper last year predating WMAP3 that the Harrison–Zel’dovich model had the highest evidence, though other models including varying $n_{{\rm S}}$ were not strongly excluded.
WMAP3 gave, for the first time, indications that $n_{{\rm S}}$ might be less than one, with their main paper quoting the results (from WMAP3 data alone) $n_{{\rm S}}= 0.951^{+0.015}_{-0.019}$, which thus appears to be over 3-sigma away from unity. A similar result is found when WMAP data is combined with other independent datasets, such as the power spectrum of the large-scale distribution of galaxies and the redshift–luminosity relation of distant type Ia supernovae. So far, parameter estimation analyses performed on available data taken together seem to indicate that $n_{{\rm S}} \neq 1$ at about 3 to 4-sigma.
This significance level is exactly where Lindley’s paradox is at its strongest, making the use of model selection techniques imperative, as acknowledged in the WMAP3 papers. We have carried out such an analysis. We chose a prior on $n_{{\rm S}}$ uniform between 0.8 between 1.2; most inflationary models give $n_{{\rm S}}$ in this range and this is what was believed to be the possible range for it before the data came along. Evidences were computed using `CosmoNest`, with the calculations taking a few days on a multi-processor cluster.
According to our model selection analysis, the evidence for the $n_{{\rm S}}$ varying model is significant, but not strong or decisive. WMAP3 data on its own gives a Bayes factor of only $0.34\pm
0.26$, indicating that this data alone is unable to distinguish the two models. When WMAP3 data are used together with external data sets we estimate a $\Delta \ln E$ of $1.99\pm 0.26$, corresponding to an odds ratio of 8 to 1 in favour of the $n_{{\rm S}}$ varying model. Adding the external datasets improves the constraining power on $n_{{\rm S}}$, as they significantly extend the scales over which the primordial power spectrum affects the data. Nevertheless, the support for varying $n_{{\rm S}}$ is clearly tentative rather than compelling.
There is additional reason for some caution at present because there may be residual systematics in the data that could affect our conclusions regarding $n_{{\rm S}}$; the evidence calculation concerns statistical uncertainties only. For example, the effect of varying $n_{{\rm S}}$ in determining the power spectra shown in Figure \[fig:spectrum\] is somewhat degenerate with the signature of the relatively recent reionization of the Universe, which is mainly inferred from polarization data which is difficult to handle. There are also uncertainties associated with the modelling of the instrument beam profiles, and in whether one should attempt to model out a possible contribution to the CMB anisotropies from the Sunyaev–Zel’dovich effect. The situation will be improved with higher signal-to-noise data from additional years of WMAP observations and future experiments.
Conclusion
==========
Many of the most interesting cosmological questions are ones of model selection, not parameter estimation. With the growing precision of cosmological data, it is imperative to deploy proper model selection techniques to extract the best robust conclusions from data.
Application to the post-WMAP3 cosmological data compilation continues to indicate that the data can be well fit by quite minimal cosmological models. Five fundamental parameters are definitely required, and WMAP3 has provided suggestive indications that a sixth, the density perturbation spectral index, needs to be added to the set. According to the Bayesian evidence, however, the case for inclusion of the spectral index has yet to become compelling.
As the data improve in sensitivity we expect new model selection based questions to be both raised and answered in the next decade. These may be about the nature of dark energy, the model for reionization, the nature of inflation, the case for primordial gravitational waves, or the nature of cosmic topology. Model selection, of course, will have further applications in astrophysics and beyond.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors were supported by PPARC. We thank Pier Stefano Corasaniti, Mike Hobson, Andrew Jaffe, Martin Kunz, Cédric Pahud, John Peacock, John Skilling, and Roberto Trotta for discussion relating to these ideas.
`CosmoNest` is available for download at [http://www.cosmonest.org]{}.
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============
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} |
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abstract: 'In the case of using the higher derivative regularization we construct the subtraction scheme which gives the NSVZ-like relation for the anomalous dimension of the photino mass in softly broken ${\cal N}=1$ SQED with $N_f$ flavors in all loops. The corresponding renormalization prescription is determined by simple boundary conditions imposed on the renormalization constants. It allows fixing an arbitrariness of choosing finite counterterms in every order of the perturbation theory in such a way that the renormalization group functions defined in terms of the renormalized coupling constant satisfy the NSVZ-like relation.'
author:
- |
I.V.Nartsev, K.V.Stepanyantz\
,\
title: '**NSVZ-like scheme for the photino mass in softly broken ${\cal N}=1$ SQED regularized by higher derivatives**'
---
Introduction
============
The NSVZ $\beta$-function is an important equation relating the $\beta$-function and the anomalous dimension of the chiral matter superfields in ${\cal N}=1$ supersymmetric theories [@Novikov:1983uc; @Jones:1983ip; @Novikov:1985rd; @Shifman:1986zi]. It gives the exact $\beta$-function in the form of the geometric series for the pure ${\cal N}=1$ supersymmetric Yang–Mills (SYM) theory and can be used for proving the finiteness of ${\cal N}=2$ SYM beyond the one-loop approximation [@Shifman:1999mv; @Buchbinder:2014wra; @Buchbinder:2015eva]. Therefore, it is closely connected to the non-renormalization theorems in theories with extended supersymmetry. For the ${\cal N}=1$ supersymmetric quantum electrodynamics (SQED) with $N_f$ flavors the NSVZ relation has the form [@Vainshtein:1986ja; @Shifman:1985fi]
$$\label{NSVZ_SQED_Beta_Function}
\widetilde\beta(\alpha) = \frac{\alpha^2 N_f}{\pi}\Big(1-\widetilde\gamma(\alpha)\Big),$$
where $\alpha$ denotes the renormalized coupling constant and $\widetilde\gamma(\alpha)$ is the anomalous dimension of the chiral matter superfields. The tildes point that the renormalization group functions are defined in terms of the renormalized coupling constant (see Eq. (\[RG\_Renormalized\]) below). The NSVZ-like equation can be also written for theories with softly broken supersymmetry [@Hisano:1997ua; @Jack:1997pa; @Avdeev:1997vx]. It relates the renormalization of the gaugino mass to the renormalization of the rigid theory by the equation [@Hisano:1997ua]
$$\label{Hisano_Equation}
\frac{\alpha m}{\widetilde\beta(\alpha)} = \mbox{RGI},$$
where $m$ is the gaugino mass and RGI means that this expression is the renormalization group (RG) invariant. In this paper we will consider softly broken ${\cal N}=1$ SQED with $N_f$ flavors. In this case, by the help of Eq. (\[NSVZ\_SQED\_Beta\_Function\]), it is possible to present Eq. (\[Hisano\_Equation\]) in the form of the relation between the anomalous dimension of the photino mass and the anomalous dimension of the matter superfields,
$$\label{Gamma_M}
\widetilde\gamma_m(\alpha) = \frac{\alpha N_f}{\pi}\Big[1- \frac{d}{d\alpha}\Big(\alpha\widetilde\gamma(\alpha)\Big)\Big].$$
where $\widetilde\gamma_m(\alpha)$ is the photino mass anomalous dimension. Again using Eq. (\[NSVZ\_SQED\_Beta\_Function\]), this relation can be equivalently rewritten as
$$\label{NSVZ_Photino}
\frac{d}{d\ln\mu}\Big(\frac{m}{\alpha}\Big) = -\frac{m\alpha N_f}{\pi}\cdot \frac{d\widetilde\gamma(\alpha)}{d\alpha}.$$
This result can be combined with Eq. (\[NSVZ\_SQED\_Beta\_Function\]) into the NSVZ-like equation
$$\label{NSVZ_General}
\frac{d}{d\ln\mu}\Big(\frac{1}{(1+2m\theta^2)\alpha}\Big) = -\frac{N_f}{\pi}\Big[1-\widetilde\gamma\Big((1+2m\theta^2)\alpha\Big)\Big].$$
This implies that Eqs. (\[NSVZ\_SQED\_Beta\_Function\]) and (\[NSVZ\_Photino\]) have a similar nature, which can be revealed using the spurion technique [@Girardello:1981wz; @HelayelNeto:1984iv; @Feruglio:1984aq; @Scholl:1984hj; @Yamada:1994id]. For example, Eq. (\[NSVZ\_General\]) can be obtained by help of the Statement presented in [@Avdeev:1997vx], which have been derived by this method.
Although the NSVZ and NSVZ-like equations are well-known, to obtain them by explicit summing supergraphs is a complicated problem. At present, it has been solved only in the Abelian case by using the higher derivative regularization [@Slavnov:1971aw; @Slavnov:1972sq] in the supersymmetric version [@Krivoshchekov:1978xg; @West:1985jx].[^1] With this regularization the NSVZ relation in the Abelian case is obtained for the RG functions defined in terms of the bare coupling constant in all orders [@Stepanyantz:2011jy; @Stepanyantz:2014ima]. Note that this result is valid in an arbitrary subtraction scheme, because the RG functions defined in terms of the bare coupling constant are scheme-independent for a fixed regularization [@Kataev:2013eta]. NSVZ-like relations for such RG functions have also been proved in all orders for the Adler $D$-function [@Adler:1974gd] in ${\cal N}=1$ SQCD [@Shifman:2014cya; @Shifman:2015doa] and for the anomalous dimension of the photino mass in the softly broken ${\cal N}=1$ SQED [@Nartsev:2016nym]. All these NSVZ and NSVZ-like relations appear due to factorization of loop integrals into integrals of (double) total derivatives in the limit of the vanishing external momentum [@Soloshenko:2003nc; @Smilga:2004zr].[^2] For various non-Abelian theories the factorization into double total derivatives has been verified in the lowest loops [@Pimenov:2009hv; @Stepanyantz_MIAN; @Stepanyantz:2011bz; @Aleshin:2016yvj; @Buchbinder:2014wra; @Buchbinder:2015eva], but has yet not been derived in all orders. However, in this paper we consider the Abelian case, for which the relations
$$\label{Bare_Relations}
\beta(\alpha_0) = \frac{\alpha_0^2 N_f}{\pi}\Big(1-\gamma(\alpha_0)\Big);\qquad \frac{d}{d\ln\Lambda}\Big(\frac{m_0}{\alpha_0}\Big) = -\frac{m_0\alpha_0 N_f}{\pi}\cdot \frac{d\gamma(\alpha_0)}{d\alpha_0},$$
where $\alpha_0$ is the bare coupling constant, have been rigorously proved in all orders for softly broken ${\cal N}=1$ SQED with $N_f$ flavors, regularized by higher derivatives. These scheme-independent equations relate the RG functions defined in terms of the bare coupling constant. Note that the first equation has been verified by an explicit three-loop calculation in [@Kazantsev:2014yna].
However, the RG functions are standardly defined in terms of the renormalized coupling constant. In this case they depend on the subtraction scheme. In particular, it is possible to verify that the NSVZ relation is also scheme dependent[^3]. If (softy broken) supersymmetric theories are regularized by the dimensional reduction, the NSVZ (or NSVZ-like) scheme can be obtained from the $\overline{\mbox{DR}}$-scheme by a finite renormalization, which should be tuned in every order [@Jack:1996vg; @Jack:1996cn; @Jack:1998uj; @Jack:1997eh; @Jack:1998iy; @Jack:1999aj; @Harlander:2006xq; @Mihaila:2013wma]. For Abelian rigid theories regularized by higher derivatives the NSVZ scheme is obtained by imposing the simple boundary conditions
$$\label{Rigid_Boundary_Conditions}
Z_3(\alpha,x_0)=1;\qquad Z(\alpha,x_0)=1$$
on the renormalization constants of the charge and of the matter superfields, respectively [@Kataev:2013eta; @Kataev:2013csa; @Kataev:2014gxa]. In these equations $x_0$ is a fixed value of $x\equiv \ln \Lambda/\mu$, where $\Lambda$ is a dimensionful parameter of the regularized theory (which acts as an ultraviolet cut-off) and $\mu$ is a normalization point. (The similar conditions for the non-Abelian case are discussed in [@Stepanyantz:2016gtk].) Under the conditions (\[Rigid\_Boundary\_Conditions\]) the $\beta$-function and the anomalous dimension of the matter superfields satisfy Eq. (\[NSVZ\_SQED\_Beta\_Function\]) in all orders.
In this paper we construct the boundary conditions similar to Eq. (\[Rigid\_Boundary\_Conditions\]) under which Eq. (\[Gamma\_M\]) (or, equivalently, Eq. (\[NSVZ\_Photino\]) or Eq. (\[NSVZ\_General\])) is valid in all loops for softly broken ${\cal N}=1$ SQED with $N_f$ flavors.
NSVZ-like scheme for softly broken ${\cal N}=1$ SQED
====================================================
For rigid ${\cal N}=1$ SQED the RG functions are defined in terms of the bare coupling constant according to the prescription
$$\label{Bare_Beta_Gamma}
\beta(\alpha_0) \equiv \frac{d\alpha_0}{d\ln\Lambda}\Big|_{\alpha=\mbox{\scriptsize const}};\qquad \gamma(\alpha_0) \equiv -\frac{d\ln Z}{d\ln\Lambda}\Big|_{\alpha=\mbox{\scriptsize const}},$$
where $Z$ is the renormalization constant for the matter superfields, $\phi = \sqrt{Z} \phi_R$. The $\beta$-function can be also expressed in terms of the renormalization constant $Z_3 \equiv \alpha/\alpha_0$,
$$\beta(\alpha_0) = - \alpha_0 \frac{d\ln Z_3}{d\ln\Lambda}\Big|_{\alpha=\mbox{\scriptsize const}}.$$
In the softly broken theory we also define the anomalous dimension of the photino mass
$$\label{Bare_Gamma_M_Definition}
\gamma_m(\alpha_0) \equiv \frac{d\ln m_0}{d\ln\Lambda}\Big|_{\alpha,m=\mbox{\scriptsize const}} = - \frac{d\ln Z_m}{d\ln\Lambda}\Big|_{\alpha=\mbox{\scriptsize const}} = \frac{\alpha_0}{m_0} \frac{d}{d\ln\Lambda}\Big(\frac{m_0}{\alpha_0}\Big) + \frac{\beta(\alpha)}{\alpha_0},$$
where $Z_m\equiv m/m_0$ is the photino mass renormalization constant. To prove its scheme independence, we consider a part of the two-point Green function of the gauge superfield $\bm{V}$ corresponding to the photino mass. In the limit when the external momentum $p$ is mich larger than all masses it can be written as
$$\frac{m_0}{128\pi} \int \frac{d^4p}{(2\pi)^4} d^4\theta\, \Big(\theta^2 D^a \bm{V}(-p,\theta)\,\bar D^2 D_a \bm{V}(p,\theta) + \bar\theta^2 \bar D^{\dot a} \bm{V}(-p,\theta)\, D^2 \bar D_{\dot a} \bm{V}(p,\theta)\Big) d_m^{-1}(\alpha_0,\Lambda/p),$$
where $d_m$ is a dimensionless function. Then, according to the definition of the renormalization constant $Z_m$, the product $Z_m d_m(\alpha_0(\alpha,\Lambda/\mu),\Lambda/p)$ should be finite in the limit $\Lambda\to \infty$. Differentiating the logarithm of this expression with respect to $\ln\Lambda$, we express $\gamma_m(\alpha_0)$ via the function $d_m(\alpha_0,\Lambda/p)$,
$$\label{Gamma_Vs_Dm}
\gamma_m(\alpha_0) = \lim\limits_{p\to 0} \Big(\frac{\partial \ln d_m(\alpha_0,\Lambda/p)}{\partial\alpha_0}\cdot \beta(\alpha_0) - \frac{\partial\ln d_m(\alpha_0,\Lambda/p)}{\partial \ln p} \Big).$$
Note that the limit $p\to 0$ is needed in order to get rid of the terms proportional to $(p/\Lambda)^{n}$, where $n$ is a positive integer. The function $d_m(\alpha_0,\Lambda/p)$ is evidently scheme-independent, because it is obtained by calculating the effective action before the renormalization. The scheme-independence of the function $\beta(\alpha_0)$ has been proved in [@Kataev:2013eta]. Therefore, the right hand side of Eq. (\[Gamma\_Vs\_Dm\]) is also scheme-independent. This implies that the anomalous dimension $\gamma_m(\alpha_0)$ (defined in terms of the bare coupling constant) does not depend on the renormalization prescription. However, it depends on the regularization.
The RG functions (\[Bare\_Beta\_Gamma\]) and (\[Bare\_Gamma\_M\_Definition\]) should be distinguished from the corresponding RG functions (standardly) defined in terms of the renormalized coupling constant [@Bogolyubov:1980nc],
$$\label{RG_Renormalized}
\widetilde\beta(\alpha) \equiv \frac{d\alpha}{d\ln\mu}\Big|_{\alpha_0=\mbox{\scriptsize const}};\qquad \widetilde\gamma(\alpha) \equiv \frac{d\ln Z}{d\ln\mu}\Big|_{\alpha_0=\mbox{\scriptsize const}};\qquad \widetilde\gamma_m(\alpha) \equiv \frac{d\ln m}{d\ln\mu}\Big|_{\alpha_0,m_0=\mbox{\scriptsize const}}.$$
The main observation that allowed constructing the NSVZ scheme for the rigid ${\cal N}=1$ SQED regularized by higher derivatives is that the $\beta$-function and the anomalous dimension defined in terms of the bare coupling constant and the ones defined in terms of the renormalized coupling constant coincide, if the boundary conditions (\[Rigid\_Boundary\_Conditions\]) are imposed on the renormalization constants [@Kataev:2013eta]. Really, according to [@Stepanyantz:2011jy; @Stepanyantz:2014ima], the former RG functions satisfy the NSVZ relation independently of the subtraction scheme in all orders in the case of using the higher derivative regularization. Therefore, the latter RG functions also satisfy it under the conditions (\[Rigid\_Boundary\_Conditions\]).
Now, let us consider the softly broken theory. In this case we also impose the boundary conditions (\[Rigid\_Boundary\_Conditions\]), because the anomalous dimension of the matter superfields enter Eq. (\[Gamma\_M\]). Then the NSVZ relation (\[NSVZ\_SQED\_Beta\_Function\]) is valid for the RG functions defined in terms of the renormalized coupling constant and
$$\label{RG_Equality}
\widetilde\beta(\alpha) = \beta(\alpha_0)\Big|_{\alpha_0 = \alpha};\qquad \widetilde\gamma(\alpha) = \gamma(\alpha_0)\Big|_{\alpha_0 = \alpha}.$$
According to Ref. [@Nartsev:2016nym], the NSVZ-like relation
$$\label{Bare_Gamma_M}
\gamma_m(\alpha_0) = \frac{\alpha_0 N_f}{\pi}\Big[1- \frac{d}{d\alpha_0}\Big(\alpha_0 \gamma(\alpha_0)\Big)\Big]$$
is valid in all loops for the theory regularized by higher derivatives. Therefore, we need to find the boundary conditions under which
$$\label{Gamma_M_Equality}
\widetilde\gamma_m(\alpha) = \gamma_m(\alpha_0)\Big|_{\alpha_0 = \alpha}.$$
This can be done by repeating the argumentation of Ref. [@Kataev:2013eta]. Let us, in addition to Eq. (\[Rigid\_Boundary\_Conditions\]), impose the condition
$$\label{M_Bondary_Condition}
m(\alpha,x_0)=m_0.$$
Equivalently, we fix a value $x_0 = \ln\Lambda/\mu$ and require that the renormalization constants satisfy the equations
$$\label{Soft_Boundary_Conditions}
Z_3(\alpha,x_0)=1;\qquad Z(\alpha,x_0)=1;\qquad Z_m(\alpha,x_0)=1.$$
Then, the anomalous dimension of the photino mass defined in terms of the renormalized coupling constant can be presented as
$$\label{Calculation_Of_Gamma_M}
\widetilde\gamma_m\left(\alpha(\alpha_0,x)\right) = - \frac{d}{dx}\ln Z_m\left(\alpha(\alpha_0,x),x\right) = - \frac{\partial \ln Z_m(\alpha,x)}{\partial\alpha} \cdot \frac{\partial \alpha(\alpha_0,x)}{\partial x} - \frac{\partial \ln Z_m(\alpha,x)}{\partial x}.$$
In this equation $d/dx$ denotes the total derivative with respect to $x=\ln\Lambda/\mu$ which acts both on the explicitly written $x$ and on $x$ inside $\alpha$, unlike the partial derivative $\partial/\partial x$ which does not act on $x$ inside $\alpha$. Let us consider Eq. (\[Calculation\_Of\_Gamma\_M\]) in the point $x_0$. Then, due to the boundary conditions (\[Soft\_Boundary\_Conditions\]), we obtain
$$\frac{\partial \ln Z_m(\alpha,x)}{\partial\alpha}\Big|_{x=x_0} = \frac{\partial \ln(1)}{\partial\alpha} = 0,$$
so that the first term in Eq. (\[Calculation\_Of\_Gamma\_M\]) vanishes. In the second term $\ln Z_m$ is differentiated with respect to $\ln\Lambda/\mu$ at a fixed value of the renormalized coupling constant $\alpha$, exactly as in Eq. (\[Bare\_Gamma\_M\_Definition\]) which defines the anomalous dimension $\gamma_m(\alpha_0)$. Moreover, due to the first boundary condition in Eq. (\[Soft\_Boundary\_Conditions\]), $\alpha(\alpha_0,x_0)=\alpha_0$. This implies that both definitions of the photino mass anomalous dimension coincide, see Eq. (\[Gamma\_M\_Equality\]).
Therefore, if the boundary conditions (\[Soft\_Boundary\_Conditions\]) are imposed on the renormalization constants of softly broken ${\cal N}=1$ SQED with $N_f$ flavors regularized by higher derivatives, then the NSVZ-like equation (\[Gamma\_M\]) (and, consequently, Eqs. (\[NSVZ\_Photino\]) and (\[NSVZ\_General\])) is satisfied in all orders. Thus, we have constructed the prescription which gives the NSVZ scheme in all orders of the perturbation theory.
Conclusion
==========
In this paper we have constructed the scheme in which the anomalous dimension of the photino mass in softly broken ${\cal N}=1$ SQED with $N_f$ flavors satisfies the NSVZ-like relation (\[Gamma\_M\]) in all loops. In this relation all RG functions are defined in the standard way in terms of the renormalized coupling constant and are, therefore, scheme-dependent. That is why the considered NSVZ-like relation is valid only in a certain subtraction scheme. An important ingredient needed for its constructing is the higher derivative regularization. The matter is that with this regularization the RG functions defined in terms of the bare coupling constant satisfy the NSVZ relation and the NSVZ-like relation for the photino mass in all orders independently of the subtraction scheme. In this paper we proved that, if the boundary conditions (\[Soft\_Boundary\_Conditions\]) are imposed to the renormalization constants, the RG functions defined in terms of the bare coupling constant coincide with the RG functions defined in terms of the renormalized coupling constant, see Eqs. (\[RG\_Equality\]) and (\[Gamma\_M\_Equality\]). Consequently, the latter RG functions satisfy the NSVZ-like relations in all orders for the higher derivative regularization supplemented by the renormalization prescription (\[Soft\_Boundary\_Conditions\]). This construction is very similar to the one presented in [@Kataev:2013eta; @Kataev:2013csa; @Kataev:2014gxa] for the rigid theory. However, in the softly broken theory it is also necessary to impose one more boundary condition on the renormalization constant of the photino mass. Finally, let us again emphasize that this construction is valid only in the case of using the higher derivative regularization.
Acknowledgments {#acknowledgments .unnumbered}
===============
K.S. is very grateful to A.L.Kataev for valuable discussions.
The work of K.S. was supported by the Russian Foundation for Basic Research, grant No. 14-01-00695.
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[^1]: This regularization also includes inserting the Pauli–Villars determinants for regularizing the one-loop divergences [@Slavnov:1977zf; @Faddeev:1980be].
[^2]: In the case of using the dimensional reduction [@Siegel:1979wq] this limit is not well-defined. Therefore, the factorization into double total derivatives does not take place in this case [@Aleshin:2015qqc], and the RG functions defined in terms of the bare coupling constant do not satisfy the NSVZ relation [@Aleshin:2016rrr].
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| {
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**An Analysis of a Simple Local Search Algorithm for Graph Colouring**
[David Chalupa]{}
[ Computer Science\
School of Engineering and Computer Science\
University of Hull\
Cottingham Road\
Hull HU6 7RX, United Kingdom\
Tel.: +441482463069\
Email: `[email protected]` ]{}\
*Acknowledgement:* This is a preprint version of an article, which is currently in review. This text will be substituted later with an acknowledgement and a link to the final version of the article.
#### Abstract.
Vertex Descent is a local search algorithm which forms the basis of a wide spectrum of tabu search, simulated annealing and hybrid evolutionary algorithms for graph colouring. These algorithms are usually treated as experimental and provide strong results on established benchmarks. As a step towards studying these heuristics analytically, an analysis of the behaviour of Vertex Descent is provided. It is shown that Vertex Descent is able to find feasible colourings for several types of instances in expected polynomial time. This includes $2$-colouring of paths and $3$-colouring of graphs with maximum degree $3$. The same also holds for $3$-colouring of a subset of $3$-colourable graphs with maximum degree $4$. As a consequence, Vertex Descent finds a $3$-colouring in expected polynomial time for the smallest graph for which Brélaz’s heuristic DSATUR needs $4$ colours. On the other hand, Vertex Descent may fail for forests with maximum degree $3$ with high probability.
#### Keywords.
Vertex Descent, local search, graph colouring, algorithm analysis.
Introduction
============
*Local search* forms the basis of some of the most popular metaheuristics [@hoosstutzlesls], including tabu search [@glover1; @glover2; @tabusearch] and simulated annealing [@simulatedannealingcerny; @simulatedannealing]. These algorithms find their applications in a large spectrum of combinatorial optimisation problems. One of them, the *graph colouring problem* (GCP), is among the most widely studied problems in a body of experimental and theoretical literature. It is one of the most well-known NP-hard problems [@introalg; @karp] with applications in *multiprocessor task scheduling* [@multiprocessor], *timetabling* [@timetabling] or *frequency assignment in mobile radio networks* [@frequency].
Formally, let $G = [V,E]$ be an undirected graph. Then, the *aim of GCP* is to find $k$ disjoint colour classes $V_1$, $V_2$, ..., $V_k$ for which $\cup_{i=1}^{k} V_i = V$ and $\forall ~ \{v,w\} \in E ~ [\exists ~ i,j ~ (v \in V_i \wedge w \in V_j \wedge i \neq j)]$. A *conflict* is defined as a pair of equally coloured adjacent vertices. A colouring with no conflicts is called *feasible*. Otherwise it is *infeasible*. Minimum $k$ for which there is a feasible colouring is called *chromatic number* and is denoted by $\chi$.
Vertex Descent is a local search algorithm which represents the basic component of a large number of successful graph colouring heuristics. It tackles GCP by searching in the space of infeasible colourings and by minimising the number of conflicts between equally coloured vertices. This approach, sometimes called *k-fixed penalty strategy*, is one of the most widespread and successful approaches to solve GCP [@localsearch].
Algorithms which use Vertex Descent as a basis include the tabu search algorithm TabuCol [@tabucol], learning-based local search algorithms [@porumbel::cartography], the simulated annealing graph colouring algorithm [@johnson::annealing], the quantum annealing algorithm QACol [@quantumann; @qatuning; @distributedquantum] and a number of hybrid evolutionary algorithms that use local search (mostly TabuCol) as an intensification subroutine [@dorne::genetic; @hybrid; @ie2col; @lu::memetic; @headcoloring; @porumbel::evocoljour; @extracol].
While these are successful experimental algorithms, their theoretical understanding still seems to be very limited. On the other hand, analysis of randomised search heuristics has progressed rapidly over the last years [@searchheurtheory; @eacomplexity]. In this paper, we provide the first analytical results on the behaviour of Vertex Descent. To the best of our knowledge, this is the first analytical study of a graph colouring heuristic based on the widely used k-fixed penalty strategy.
#### Contributions
We show that the behaviour of Vertex Descent may be modelled using the fitness levels method [@lehrelevels; @eacomplexity; @sudholtlower] and fair random walks [@covertime] can be used to model its behaviour on plateaus.
For Vertex Descent, we obtain that it is able to find feasible colourings in expected polynomial time for several types of instances. These instances include:
- $(\Delta+1)$-colouring for graphs with maximum degree $\Delta$;
- $\Delta$-colouring for graphs with maximum degree $\Delta$ which are neither complete graphs $K_n$ nor odd rings and;
- $3$-colourings for $3$-colourable graphs with maximum degree $4$ such that the neighbours of each vertex with degree $4$ induce $K_1 \cup P_3$, i.e. an isolated vertex and a path of length $3$.
The last of the instance types above includes the smallest hard-to-colour graph for Brélaz’s heuristic DSATUR [@smallhardbrelaz]. On the other hand, we also obtain results on instances which are hard-to-colour for Vertex Descent. For forests with maximum degree $3$, Vertex Descent with $k = 2$ may not produce a feasible $2$-colouring with probability $1 - o(1)$. We also present a connected $3$-colourable graph for which Vertex Descent with $k = 3$ will not be able to find a feasible $3$-colouring with probability $1 - o(1)$.
The paper is structured as follows. In Section 2 we present an overview of the related work. In Section 3 we introduce the Vertex Descent algorithm. In Section 4 we present the general analytical results. The main results for Vertex Descent are presented in Section 5. Finally, in Section 6 we provide the conclusions and identify the problems which remain open.
Related Work
============
Most of the modern graph colouring heuristics are *hybrid algorithms*. Currently, the most successful algorithms include a *distributed quantum annealing algorithm* QACol [@quantumann; @qatuning], an algorithm IE$^2$Col, based on *extraction and expansion* of independent sets [@ie2col] and a *memetic algorithm* HEAD with a very small population of two colourings [@headcoloring]. All three of these algorithms use Vertex Descent combined with several different ideas. These include solution populations, tabu lists, thermal and quantum fluctuations, partition crossovers and preprocessing. However, Vertex Descent remains at the core of all these approaches.
One of the most well-known hybrid evolutionary algorithms for GCP was introduced by Galinier and Hao [@hybrid]. It used the tabu search algorithm TabuCol [@tabucol] as an intensification subroutine. TabuCol practically represents the Vertex Descent algorithm enhanced by a tabu list to prevent it from cycling. Glass and Prügel-Bennett investigated an adaptation of the hybrid algorithm by Galinier and Hao obtained by substituting TabuCol with Vertex Descent [@hybridanalysis]. They concluded that this version can perform comparably to the original variant but requires a larger population. Vertex Descent has also been used as a technique of search space sampling in an evaluation of different objective functions for the problem [@evaluationfunctionscoloring].
This indicates that an analysis of Vertex Descent is highly relevant for a better understanding of these algorithms. Most of the theoretical results were previously proven for constructive graph colouring heuristics. These results usually focus on the number of colours used in the worst case [@worstcasegreedy; @worstcasebrelaz]. The most popular constructive algorithms for GCP include Brélaz’s heuristic DSATUR [@Brelaz] and the recursive largest first heuristic [@leighton::scheduling]. Brélaz’s heuristic is known to use $n$ colours for a $3$-colourable graph on $\mathcal{O}(n)$ vertices if an unfavourable sequence of vertices is chosen [@worstcasebrelaz]. There is also a $3$-colourable graph on $8$ vertices for which Brélaz’s heuristic will always use $4$ colours [@smallhardbrelaz].
Sudholt and Zarges [@ilscoloringanalysis] analysed an iterated local search algorithm for graph colouring. This algorithm operated in the space of feasible colourings and used moves called Kempe chains and colour eliminations to minimise the number of colours used. This study provided several interesting insights into the behaviour of local search for graph colouring. However, the most popular experimental approach based on the $k$-fixed penalty strategy still seems to be overlooked in the theoretical studies.
It is worth noting that an alternative $k$-fixed approach is based on searching in the space of partial feasible colourings and minimizing the number of uncoloured vertices [@blochliger::partialcol; @morgenstern::dncs]. These approaches can also be combined [@vss]. Order-based and distributed algorithms have also been studied [@culberson; @morgenstern::dncs].
Numerous analytical results have also been obtained for other combinatorial optimisation problems. For the maximum matching problem, behaviours of local search and evolutionary algorithms for paths and worst-case approximations were investigated [@maxmatchanalysis]. For the vertex cover problem, behaviour of hybrid algorithms was analysed for specific types of graphs [@vertexcoverhybrid] and an iterated local search algorithm was investigated in the context of sparse random graphs [@vertexcoverrandom]. Fixed-parameter evolutionary algorithms were also analysed [@vertexcoverfpt]. Other studied problems include makespan scheduling [@makespandiscrep], the Euclidean travelling salesperson problem [@tspanalysis], the Eulerian cycle problem [@eulerian], the minimum spanning tree problem [@minimumspanning] or the minimum cut problem [@minimumcuts].
The Vertex Descent Algorithm
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In this section, we briefly review the Vertex Descent algorithm. Algorithm 1 presents the pseudocode. As an input, we have a graph $G = [V,E]$ and the number of colours $k$. The output is the best colouring found in the search process.
In step 1 the vertices are assigned uniformly random colours to create an initial candidate solution $S$. In step 2 the currently best solution $S_{best}$ and its objective value $f_{best}$ are initialised. Let $s(v)$ be the colour of $v$ in $S$. Then we have that $f(S) = |\left\{\{v,w\} \in E: s(v) = s(w)\right\}|$.
Algorithm 1. Vertex Descent local search algorithm for graph colouring\
--- -------------------------------------------------------------------------
Input: graph $G = [V,E]$, the number of colours $k$
Output: solution $S = \{V_1,V_2,...,V_k\}$
1 $\forall ~ v \in V$ let $v$ have a random colour in $S$
2 $S_{best} = S$, $f_{best} = f(S)$
3 while stopping criteria are not met
4 let $N(S)$ be the neighbourhood of $S$
5 $[v,c] = \arg\min_{[v,c], S' \in N(S)} f(S')$
6 let $S$ be constructed by applying move $[v,c]$ to $S$
7 if $f_{best} > f(S_{best})$ then $f_{best} = f(S')$, $S_{best} = S$
8 return $S_{best}$
--- -------------------------------------------------------------------------
An iterative procedure follows. In step 4 we scan the neighbourhood $N(S)$ of solution $S$. The set of conflicting vertices in $S$ is $C(S) = \{v \in V: \exists ~ \{v,w\} \in E ~ [s(v) = s(w)]\}$. Then, the neighbourhood $N(S)$ is a set of all solutions obtained from $S$ by recolouring any of the vertices in $C(S)$ by any of the $k-1$ other colours which the vertex can have. Therefore, there are $|N(S)| = (k-1)|C(S)|$ possible moves. In step 5 we choose the move which leads to the lowest number of conflicts. In step 6 we recolour $v$ by its new chosen colour $c$. In step 7 the best solution $S_{best}$ and its objective function value $f_{best}$ are updated, if needed. This process is repeated until a feasible colouring is found.
It is worth mentioning that the testing of the new objective values for candidate solutions $S' \in N(S)$ requires $\mathcal{O}(1)$ time if Vertex Descent is implemented using an auxiliary matrix. Therefore, step 5 requires $\mathcal{O}(kn)$ time in the worst case, since there are always at most $n$ conflicting vertices and $k-1$ possible colours.
General Results
===============
At this point, we begin with our general analytical results. In the following, Lemma 1 and Lemma 2 provide results on the total number of conflicts and its reduction by recolouring a vertex. Next, Theorem 1 provides the general result on the number of conflicts reached by Vertex Descent in expected polynomial time for an arbitrary graph.
#### Definition 1.
Let $S$ be a colouring of a graph $G = [V,E]$ and let $s(v)$ be the colour of $v$ in $S$. Then, the function $$\Gamma_S(v,c) = \left|\{ v' \in V: \{v,v'\} \in E \wedge s(v') = c\}\right|$$ denotes the number of neighbours of a vertex $v$ with colour $c$.
#### Lemma 1.
Let $S$ be a colouring of an arbitrary graph $G = [V,E]$ and let $s(v)$ be the colour of $v$ in $S$. Let $C(S)$ be the set of conflicting vertices in $S$. Then, the total number of conflicts $confl(S)$ in $S$ is: $$confl(S) = \frac{1}{2} \sum_{v \in V} \Gamma_S(v, s(v)) = \frac{1}{2} \sum_{v \in C(S)} \Gamma_S(v, s(v)).$$
#### Proof.
Consider the subgraph $G'$ induced by the set of conflicting vertices $C(S)$, containing only edges between these vertices. In such a subgraph, each conflicting vertex $v$ has degree $\Gamma(v, s(v))$. We obtain that $\sum_{v \in C(S)} deg_{G'}(v) = \sum_{v \in C(S)} \Gamma_S(v, s(v)) = 2 confl(S)$. Add the non-conflicting vertices of $G$ as isolated vertices to our induced subgraph $G'$ to form a new graph $G''$. We now have that $\sum_{v \in V} deg_{G''}(v) = \sum_{v \in V} \Gamma_S(v, s(v)) = 2 confl(S)$. $\blacksquare$
#### Lemma 2.
Let $G = [V,E]$ be an arbitrary graph a let $S$ be a $k$-colouring of $G$. If there is a vertex $v$ such that $\Gamma_S(v,c) > \left\lfloor\frac{deg(v)}{k}\right\rfloor$, then there is a move recolouring $v$ by colour $c'$ such that $\Gamma_S(v,c') \leq \left\lfloor\frac{deg(v)}{k}\right\rfloor$. Additionally, it is a move which always leads to a drop in the number of conflicts.
#### Proof.
Consider the move recolouring $v$ with colour $c'$ in a colouring $S$ such that $\Gamma_S(v,c')$ is minimal. After such a move, the number of conflicts cannot become higher. Hence, the move will be accepted. We have that $\Gamma_S(v,c') \leq \left\lfloor\frac{deg(v)}{k}\right\rfloor$. This is implied by the fact that the worst case involves evenly distributed colours between the neighbours of $v$. Since $\Gamma_S(v,c') < \Gamma_S(v,c)$, the move leads to a drop in the number of conflicts by at least one. $\blacksquare$
To analyse the behaviour of Vertex Descent in the phase of decreasing number of conflicts, we will use the method of *fitness levels* [@lehrelevels; @eacomplexity; @sudholtlower]. In this method, the search space is partitioned into levels such that each level contains colourings with an equal number of conflicts. Then, the expected time to obtain a colouring with a certain number of conflicts is derived as a sum of expected waiting times for improvements from the current fitness level to a better one. Theorem 1 provides a result on the fitness level reached by Vertex Descent in expected polynomial time.
#### Theorem 1.
Let $G = [V,E]$ be an arbitrary graph on $n$ vertices and $m$ edges. Then, after $\mathcal{O}(knm)$ time in expectation, Vertex Descent for an instance of GCP with $k$ colours will arrive at a solution $S$ with each vertex $v$ involved in at most $\left\lfloor\frac{deg(v)}{k}\right\rfloor$ conflicts. The total number of conflicts in $S$ will be: $$confl(S) \leq \frac{1}{2} \sum_{v \in V} \left\lfloor\frac{deg(v)}{k}\right\rfloor.$$
#### Proof.
Let us have a $k$-colouring $S$ of $G$. If there is a vertex $v$ involved in more than $\left\lfloor\frac{deg(v)}{k}\right\rfloor$ conflicts in $S$, then Lemma 2 implies that $v$ can be recoloured to obtain an improvement by at least one conflict. We now investigate the expected time until Vertex Descent arrives at a colouring for which each vertex is involved in at most $\left\lfloor\frac{deg(v)}{k}\right\rfloor$ conflicts. From Lemma 1, we have that there are at most $\frac{1}{2} \sum_{v \in V} \left\lfloor\frac{deg(v)}{k}\right\rfloor$ conflicts in such a colouring.
Let us pessimistically assume that the initial solution contains the maximum number of conflicts possible which is $m$. Then, for each move, we have that Vertex Descent obtains an improvement by at least one conflict. Hence, the number of moves needed to obtain our target solution is:
$$m - \frac{1}{2} \sum_{v \in V} \left\lfloor\frac{deg(v)}{k}\right\rfloor \leq m.$$
This simplified upper bound already allows us to see that $\mathcal{O}(m)$ moves are needed, with $\mathcal{O}(kn)$ being the cost of a neighbourhood scan. $\blacksquare$
Theorem 1 will further be used to upper bound the number of conflicts obtained in the initial phase of the search. However, it also implies that Vertex Descent finds $(\Delta+1)$-colourings for graphs with maximum degree $\Delta$ in expected polynomial time, including $3$-colourings for odd rings.
#### Corollary 1.
For a graph on $n$ vertices with maximum degree $\Delta$ and $k = \Delta+1$, the expected time for Vertex Descent to find a feasible $(\Delta+1)$-colouring is upper bounded by $\mathcal{O}(\Delta nm)$.
#### Corollary 2.
For an odd ring on $n$ vertices and $k = 3$, the expected time for Vertex Descent to find a feasible $3$-colouring is upper bounded by $\mathcal{O}(n^2)$.
Main Results for Vertex Descent
===============================
Theorem 1 can be used to analyse the behaviour of Vertex Descent for certain types of graphs. However, in cannot be used directly if the algorithm needs to perform an exploration of a plateau, i.e. a set of solutions with an equal number of conflicts. This is the first scenario we consider in our further investigations.
Next, we focus on $3$-colourable graphs with maximum degree $4$ for which the neighbours of all vertices with degree $4$ induce $K_1 \cup P_3$. For these graphs, we show that Vertex Descent finds a feasible $3$-colouring in expected polynomial time.
In the last part of the analysis, we present a forest with maximum degree $3$ for which Vertex Descent does not produce a feasible $2$-colouring with probability $1-o(1)$. This idea is then generalised to connected $3$-colourable graphs. We find a $3$-colourable graph for which Vertex Descent does not find a feasible $3$-colouring with probability $1-o(1)$.
$\Delta$-colouring of Graphs with Maximum Degree $\Delta$ Other than $K_n$ and Odd Rings
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Vertex Descent often has to search on plateaus of colourings with an equal number of conflicts. The most straightforward example for this is represented by $2$-colouring of paths for which the plateaus are formed by conflicts located at different positions on the path. Exploration of these plateaus by Vertex Descent can be modelled as a random walk of a conflict on the path.
Theorem 2 provides a more general result. For graphs with maximum degree $\Delta$ which are neither complete graphs $K_n$ nor odd rings, we show that Vertex Descent will produce a feasible $\Delta$-colouring in expected polynomial time.
It is also worth noting that the upper bounds in the following results are based on rather pessimistic assumptions. The aim is to show that Vertex Descent solves the instance in expected polynomial time. In practice, the algorithm usually tends to be much faster.
#### Theorem 2.
Let $G$ be a connected graph on $n$ vertices and $m$ edges with maximum degree $\Delta$ which is neither $K_n$ nor an odd ring. Then, the expected time for Vertex Descent with $k = \Delta$ to find a feasible $\Delta$-colouring for $G$ is upper bounded by $\mathcal{O}(\Delta n^4m)$.
#### Proof.
Brooks’ theorem states that for a connected graph on $n$ vertices with maximum degree $\Delta$, it holds that it is $\Delta$-colourable if and only if it is neither $K_n$ nor an odd ring [@Brooks]. This already establishes that there is a feasible $\Delta$-colouring of $G$.
Based on Theorem 1, we have that after $\mathcal{O}(\Delta nm)$ time in expectation, vertices with degree at most $\Delta-1$ will be involved in no conflicts and vertices with degree $\Delta$ will be involved in at most one conflict. Figure 1 illustrates an example of such a vertex $v$ (in the middle) with degree $\Delta$. All $\Delta$ colours must be used between the neighbours of vertex $v$, since otherwise one colour would be left for use and the conflict could be resolved in one step.
In Vertex Descent, the choice of a new colour $c'$ for $v$ is uniformly distributed, unless one of the possible colours leads to a resolution of the conflict. There are $\Delta-1$ possible choices of this new colour, each occurring with probability $1 / (\Delta-1)$. Note that there are two conflicting vertices for each conflict which must have degree $\Delta$ and both of them must be involved only in this one conflict.
After recolouring $v$ from $c$ to $c'$, the conflict “moves” in the sense that it gets resolved and possibly replaced by a conflict with another neighbour of $v$, as illustrated by Figure 1. This process leads to a random walk of each of these conflicting vertices. This random walk is fair which is implied by the probability arguments described above.
The cover time of such a random walk is $\mathcal{O}(nm)$. After this number of steps of Vertex Descent affecting a fixed conflict in expectation, this conflict will visit a sequence of positions such that each vertex was involved in the conflict at least once. Since $G$ is $\Delta$-colourable, there must be a vertex $v$ such that if $v$ becomes conflicting in a colouring $S$, then there is a colour $c$ such that $\Gamma_S(v,c) = 0$. If there was no such vertex, then for all vertices all $\Delta$ colours would be used to colour their neighbours, leading to $G$ not being $\Delta$-colourable. When such a vertex becomes conflicting, the conflict can be resolved in one step of Vertex Descent.
For each of the $\mathcal{O}(n)$ remaining fitness levels, we now consider one specific conflict and assume that other moves will not lead to an improvement. In each fitness level, it takes $\mathcal{O}(n)$ neighbourhood scans in expectation to pick a move affecting our fixed conflict and $\mathcal{O}(\Delta n)$ is the complexity of such a neighbourhood scan. The cover time of the random walk analysed above is $\mathcal{O}(nm)$ which proves the theorem. $\blacksquare$
The previous result implies that Vertex Descent finds feasible $2$-colourings for paths and even rings and feasible $3$-colourings for graphs with maximum degree $3$ in expected polynomial time.
#### Corollary 3.
For both paths and even rings on $n$ vertices and $k = 2$, the expected time for Vertex Descent to find a feasible $2$-colouring is upper bounded by $\mathcal{O}(n^5)$.
#### Corollary 4.
Let $G$ be a 3-colourable graph on $n$ vertices and $m$ edges a let its maximum degree be $\Delta \leq 3$. Then, the expected time for Vertex Descent with $k = 3$ to find a feasible $3$-colouring for $G$ is upper bounded by $\mathcal{O}(n^4m)$.
$3$-colouring of a Subset of $3$-colourable Graphs with Maximum Degree $4$
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Kochol et al. have explored the $3$-colorability problem for $3$-colorable graphs with maximum degree $4$ [@colorability3maxdegree4]. They have shown that these instances can be partitioned into easy and hard, depending on the subgraphs induced by the neighbours of vertices with degree $4$. One of the patterns leading to solvability in polynomial time is when the neighbours of each vertex with degree $4$ induce $K_1 \cup P_3$, i.e. a subgraph with an isolated vertex and a path on $3$ vertices. One of their results was that a $3$-colorability problem for a graph which only contains patterns leading to polynomial-time solvability can be transformed into a $3$-colorability problem for a graph with neighbours of each vertex with degree $4$ inducing $K_1 \cup P_3$.
In the following, we show that Vertex Descent will find a feasible $3$-colouring for such a graph in expected polynomial time.
#### Theorem 3.
Let $G$ be a 3-colourable graph on $n$ vertices and $m$ edges a let its maximum degree be $\Delta \leq 4$. Furthermore, let the neighbours of each vertex with degree $4$ induce $K_1 \cup P_3$ as a subgraph. Then, the expected time for Vertex Descent with $k = 3$ to find a feasible $3$-colouring for $G$ is upper bounded by $\mathcal{O}(n^4m)$.
#### Proof.
Based on Theorem 1, we have that after $\mathcal{O}(nm)$ time in expectation, vertices with degree at most $2$ will be involved in no conflicts and vertices with degree $3$ or $4$ will be involved in at most one conflict. For vertices with degree $3$, the arguments used in the proof of Theorem 2 can be applied to show that conflicting vertices with degree $3$ perform a fair random walk on the graph.
Consider a vertex $v$ with degree $4$ which is involved in a conflict. Figure 2 illustrates a vertex $v$ and its neighbours in $G$ and possible scenarios in the search. Note that the neighbours of our vertex $v$ with degree $4$ induce the subgraph $K_1 \cup P_3$, with edges drawn in full lines. The other edges are drawn using dashed lines. There are $4$ neighbours of $v$ and $3$ colours. Therefore, one colour must be used twice between the neighbours of $v$ and the other two only once.
Let $c_1$ be the colour which is used twice. This implies that $v$ must be coloured by $c_2$ or $c_3$. Without loss of generality, fix $c_2$ as the current colour of $v$. Let the vertices with degree $2$ in Figure 2 be called *pick vertices* and let them be denoted by $v_p^1$ and $v_p^2$. Furthermore, let $c(v)$ denote the current colour of $v$. We now use enumeration based on colours of the pick vertices.
*Case 1.* Let $c(v) = c(v_p^1) \vee c(v) = c(v_p^2)$. This implies that $v$ is in a conflict with a vertex with degree $2$. This conflict can be resolved in one step of Vertex Descent.
*Case 2.* Let $c(v) \neq c(v_p^1) \wedge c(v) \neq c(v_p^2)$. In this situation, we have to further consider whether $v_p^1$ and $v_p^2$ are coloured differently or equally.
*Case 2a.* Let $c(v_p^1) \neq c(v_p^2)$. One of the pick vertices must be coloured by $c_1$ and the other one by $c_3$. Let the vertex in the bottom right corner be denoted by $v_o$. Then, $v_o$ cannot be coloured by $c_3$, since that colour can be used only once among the neighbours of $v$.
If $v_o$ was coloured by $c_1$, it would mean that the pick vertex coloured by $c_1$ can be recoloured by $c_3$. Since the uppermost neighbour will be coloured $c_2$, this scenario leads to Case 2b (with the role of $c_1$ and $c_3$ being switched).
Let $v_o$ be coloured by $c_2$. Consequently, the uppermost neighbour of $v$ must be coloured by $c_1$, as shown in Figure 2 (on the left-hand side). This situation leads Vertex Descent to choose $c_3$ as the next colour for $v$ which leads to Case 1 and a resolution of the conflict in one step of Vertex Descent.
*Case 2b.* Let $c(v_p^1) = c(v_p^2)$. The pick vertices must be coloured by $c_1$ in this case. Figure 3 illustrates that this situation leads to a free choice of $c_2$ and $c_3$ among the other two neighbours of $v$. Hence, we are facing a fair random walk of the conflicting vertices on the subgraph obtained by pruning the pick vertices. The cover time argument can now be applied to this situation.
Let us now assume that Case 2b happens at all times when a conflict visits a position involving a vertex with degree $4$, since the other scenarios lead to a resolution of the conflict in $\mathcal{O}(1)$ steps. Since the graph obtained by pruning all pick vertices has maximum degree at most $3$, we can now use the same arguments with cover time of random walks as in Theorem 2 which concludes the proof. $\blacksquare$
At this point, it is interesting to confront this result to a result obtained by a constructive algorithm. Figure 3 illustrates graph $G_1$ which is the smallest hard-to-colour graph for Brélaz’s heuristic DSATUR [@smallhardbrelaz]. It will use $4$ colours to colour $G_1$ for all of its possible randomised runs. On the other hand, Theorem 3 implies that Vertex Descent with $k = 3$ will find a feasible $3$-colouring for $G_1$ in expected polynomial time. Therefore, it is possible to find graphs for which Vertex Descent will outperform a well-known constructive algorithm.
#### Corollary 5.
For graph $G_1$, the expected time for Vertex Descent with $k = 3$ to obtain a feasible $3$-colouring is upper bounded by $\mathcal{O}(1)$.
![An illustration of the smallest hard-to-colour graph for Brélaz’s heuristic. We denote this graph by $G_1$ and it is the smallest $3$-colourable graph, for which Brélaz’s heuristic will always use $4$ colours [@smallhardbrelaz].](Brelaz_smallest_bad.pdf)
Hard-to-colour Instances for Vertex Descent
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The previous results provide a somewhat optimistic view on the behaviour of Vertex Descent. In this section, we focus on the limitations of this algorithm.
In Theorem 4, we prove that Vertex Descent may fail to provide a feasible $2$-colouring for a forest with maximum degree $3$ with high probability. Such a forest is depicted in Figure 4 and will be denoted by $G_{2,c}$, where $c$ represents the number of identical trees in the forest.
#### Theorem 4.
For graph $G_{2,c}$ on $n = 14c$ vertices, Vertex Descent with $k = 2$ will not produce a feasible $2$-colouring with probability lower bounded by $1-(32/31)^{-\Omega(n)}$.
#### Proof.
Let us consider a single tree of $G_{2,c}$ and its vertices $A$ and $B$. Suppose that in the initial colouring, they are equally coloured. This occurs with probability $1/2$. Then, let vertices $C$, $D$, $E$ and $F$ have a different colour than $A$ and $B$. This occurs with probability $1/16$, i.e. this initial configuration is generated with probability $1/32$.
At this point, each leaf of the tree will be recoloured so that conflicts on leaves are resolved. This leaves Vertex Descent with a single conflict between $A$ and $B$. By recolouring any of these two vertices, a colouring with $2$ conflicts is reached. Next, the best move in the neighbourhood is to recolour the previous vertex back to get $1$ conflict. This leads to an infinite loop between two suboptima.
The probability that this occurs for a single tree is lower bounded by $1/32$. Therefore, the probability that this occurs for at least one tree in $G_{2,c}$ is lower bounded by $1-(31/32)^{n/14} = 1-(32/31)^{-\Omega(n)}$. $\blacksquare$
This confirms that Vertex Descent may fail to produce a feasible $2$-colouring for forests with maximum degree $3$. This is a somewhat surprising disadvantage compared to Brélaz’s heuristic which guarantees to construct a $2$-colouring for a bipartite graph [@Brelaz].
In the next analysis, we will further extend on this result and show how Vertex Descent may fail in $3$-colouring of a connected graph. Figure 5 illustrates the $3$-colourable graph $G_{3,\mathcal{L}}$. This graph consists of $\mathcal{L}$ “legs” for which Vertex Descent will be able to produce a colouring with one conflict in each “leg”. However, the algorithm will then keep cycling with high probability. We formalise this result in Theorem 5.
#### Theorem 5.
For graph $G_{3,\mathcal{L}}$ on $n = 5\mathcal{L} + 1$ vertices, Vertex Descent with $k = 3$ will not be able to find a feasible $3$-colouring with probability $1 - o(1)$.
#### Proof.
Let the vertex of $G_{3,\mathcal{L}}$ with maximum degree be called the central vertex and let it be denoted by $v_c$. We also recall that vertices with degree $2$ will be called pick vertices and vertices with degree $1$ will be referred to as leaves.
Theorem 1 implies that after $\mathcal{O}(n^3)$ time in expectation, vertices with degree $1$ or $2$ will be involved in no conflicts, vertices with degree $4$ will be involved in at most one conflict and $v_c$ will be involved in at most $\left\lfloor\frac{2\mathcal{L}}{3}\right\rfloor$ conflicts. Within a specific leg, the neighbours of $v_c$ are adjacent to each other. Therefore, if both neighbours have the same colour as $v_c$, then they are involved in $2$ conflicts which can be reduced to $1$, according to Lemma 2. Hence, at this point, there is at most $1$ conflict per each leg.
Consider the scenario illustrated by Figure 6. Let the neighbours of $v_c$ be coloured by $c_1$ or $c_2$ and let their other neighbours in the leg be coloured by $c_3$. In another leg, let the neighbours of $v_c$ be coloured by $c_1$ or $c_3$ and let their other neighbours in the leg be coloured by $c_2$.
For one of the legs, the vertices with degree $4$ cannot be recoloured by $c_3$ and for the other leg, they cannot be recoloured by $c_2$ without a worsening. This is because of the colours of pick vertices and leaves which cannot be changed, since that would require them to become conflicting. To obtain that, a worsening would have to occur first. Therefore, only the colours of vertices with degree $4$ and $v_c$ will be changed. To obtain a feasible colouring of one of the legs without recolouring of the pick vertices or leaves, $v_c$ would have to be coloured by $c_2$, and for the other leg, it would have to be coloured by $c_3$ at the same time which is in a contradiction.
The probability of choosing $c_1$ or $c_2$ for the neighbours of $v_c$ and $c_3$ for their other three neighbours in a particular leg is at least $4 / 3^5 = 4 / 243$. The same argument holds for the case of the second leg with $c_2$ and $c_3$ being reversed. Therefore, the probability of this scenario happening in at least one leg with $c_2$ being the colour of vertices with degree $4$ and at least one other leg with $c_3$ being the colour of these vertices, is lower bounded by:
$$\left[ 1 - \left(\frac{239}{243}\right)^{\mathcal{L}} \right] \left[ 1 - \left(\frac{239}{243}\right)^{\mathcal{L}-1} \right] =
1 - o(1). ~ \blacksquare$$
Although $G_{2,c}$ and $G_{3,\mathcal{L}}$ can be optimally coloured by other methods, their importance lies in their hardness for Vertex Descent. Theorem 4 and Theorem 5 clearly indicate that to colour $G_{2,c}$ and $G_{3,\mathcal{L}}$ optimally, a local search algorithm must use an additional component such as thermal fluctuations or a tabu list. Behaviour of simulated annealing [@johnson::annealing] and tabu search [@tabucol] algorithms for these instances will be very interesting to investigate. Their behaviour for hard 3-colourable instances will also be interesting to explore [@hardthreecoloring]. We believe that our results may pave the way to better understand the interplay between different components in hybrid graph colouring heuristics and lead to theoretically underlaid parameter tuning techniques.
Conclusions
===========
We have analysed the *Vertex Descent* local search algorithm for graph colouring. The behaviour of this algorithm can be modelled using fitness levels method and its search on plateaus can be modelled using fair random walks.
It has been shown that Vertex Descent finds $(\Delta+1)$-colourings for graphs with maximum degree $\Delta$ in expected polynomial time. It also obtains feasible $\Delta$-colourings in expected polynomial time for connected graphs with maximum degree $\Delta$ which are neither complete graphs $K_n$ nor odd rings. A similar polynomial-time result has been obtained for $3$-colourable graphs with maximum degree $4$ for which neighbours of each vertex with degree $4$ induce $K_1 \cup P_3$, i.e. an isolated vertex and a path of length $3$.
However, Vertex Descent may fail for $2$-colouring of a forest with maximum degree $3$ with high probability. We have also demonstrated how Vertex Descent can get stuck in an infeasible colouring region for a $3$-colouring instance.
Vertex Descent is the basis for other local search algorithms for graph colouring, including simulated annealing [@johnson::annealing] and tabu search [@tabucol]. The most successful state-of-the-art algorithms for the problem [@ie2col; @headcoloring; @quantumann] also use Vertex Descent as one of their components.
Based on the previous experimental studies [@computationalcomparisoncoloring], it has long been known that different colouring methods work for different instances. To support a well-informed design of modern experimental graph colouring heuristics, further analytical investigations of these algorithms will be needed. Relating these analyses to experimental studies will be especially useful. We believe that our results have laid down the basis for understanding of the strengths and limitations of the approaches based on local search.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The Lyman-$\alpha$ forest is a powerful probe for cosmology, but it is also strongly impacted by galaxy evolution and baryonic processes such as Active Galactic Nuclei (AGN) feedback, which can redistribute mass and energy on large scales. We constrain the signatures of AGN feedback on the 1D power spectrum of the Lyman-$\alpha$ forest using a series of eight hydro-cosmological simulations performed with the Adaptative Mesh Refinement code RAMSES. This series starts from the Horizon-AGN simulation and varies the sub-grid parameters for AGN feeding, feedback and stochasticity. These simulations cover the whole plausible range of feedback and feeding parameters according to the resulting galaxy properties. AGNs globally suppress the Lyman-$\alpha$ power at all scales. On large scales, the energy injection and ionization dominate over the supply of gas mass from AGN-driven galactic winds, thus suppressing power. On small scales, faster cooling of denser gas mitigates the suppression. This effect increases with decreasing redshift. We provide lower and upper limits of this signature at nine redshifts between $z=4.25$ and $z=2.0$, making it possible to account for it at post-processing stage in future work given that running simulations without AGN feedback can save considerable amounts of computing resources. Ignoring AGN feedback in cosmological inference analyses leads to strong biases with 2% shift on $\sigma_8$ and 1% shift on $n_s$, which represents twice the standards deviation of the current constraints on $n_s$.'
author:
- |
Solène Chabanier$^{1,2}$[^1], Frédéric Bournaud$^{1,2}$, Yohan Dubois$^{3}$, Nathalie Palanque-Delabrouille$^{1}$, Christophe Yèche$^{1}$, Eric Armengaud$^{1}$, Sébastien Peirani$^{3,4}$, Ricarda Beckmann$^{3}$\
$^{1}$IRFU, CEA, Université Paris-Saclay, F91191 Gif-sur-Yvette, France\
$^{2}$ AIM, CEA, CNRS, Université Paris-Saclay, Université Paris Diderot, Sorbonne Paris Cité, F91191 Gif-sur-Yvette, France\
$^{3}$CNRS and UMPC Université Paris 06, UMR 7095, Institut d’Astrophysique de Paris, 98 bis boulevard Arago, Paris 75014, France\
$^{4}$ Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, Nice, France\
bibliography:
- 'paper.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: 'The impact of AGN feedback on the 1D power spectra from the Ly$\alpha$ forest using the Horizon-AGN suite of simulations'
---
\[firstpage\]
(galaxies:) quasars: supermassive black holes – (galaxies:)intergalactic medium – (galaxies:) quasars: absorption lines
Introduction {#sec:intro}
============
The simulations set {#sec:simu}
===================
The 1D flux power spectrum {#sec:p1d}
==========================
Results {#sec:results}
=======
Discussion {#sec:discussion}
==========
Acknowledgements
================
This work has been carried out thanks to the support of the ANR 3DGasFlows (ANR-17-CE31-0017). This work was granted access to the HPC resources of CINES and TGCC under the allocations 2019-A0070402192 and 2019-A0070410560 made by GENCI.
\[lastpage\]
[^1]: E-mail: [email protected]
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Hua-Xing <span style="font-variant:small-caps;">Chen</span>$^{1,2,}$[^1], Atsushi <span style="font-variant:small-caps;">Hosaka</span>$^{1,}$[^2] and Shi-Lin <span style="font-variant:small-caps;">Zhu</span>$^{2,}$[^3]'
title: Light Scalar Mesons in the QCD Sum Rule
---
Introduction
============
The nature of light scalar mesons of $up$, $down$ and $strange$ quarks is not fully understood [@scalar; @Yao:2006px]. The expected members are $\sigma(600)$, $\kappa(800)$, $f_0(980)$ and $a_0(980)$ forming a nonet of flavor SU(3). Because they have the same spin and parity as the vacuum, $J^P = 0^+$, they reflect the bulk properties of the non-perturbative QCD vacuum. So far, several different pictures for the scalar mesons have been proposed. In the conventional quark model, they have a $\bar q q$ configuration of $^3P_0$ whose masses are expected to be larger than 1 GeV due to the $p$-wave orbital excitation. Furthermore, the mass ordering in a naive quark mass counting of $m_u \sim m_d < m_s$ implies $m_\sigma
\sim m_{a_0} < m_\kappa < m_{f_0}$. In chiral models, they are regarded as chiral partners of the Nambu-Goldstone bosons ($\pi, K,
\eta, \eta^\prime)$ [@Hatsuda:1994pi]. Due to the collective nature, their masses are expected to be lower than those of the quark model. Yet another interesting picture is that they are tetraquark states [@Jaffe:1976ig; @Lee:2006vk; @Brito:2004tv; @Zhang:2006xp]. In contrast with the $\bar q q$ states, their masses are expected to be around 0.6 – 1 GeV with the ordering of $m_\sigma < m_\kappa <
m_{f_0, a_0}$, consistent with the recent experimental observation [@scalar; @Yao:2006px; @experiment]. If such tetraquarks survive, they may be added to members of exotic multiquark states.
In this contribution, we would like to report the results of a systematic study of the masses of the tetraquark scalar mesons in the QCD sum rule. We find that the QCD sum rule analysis with tetraquark currents implies the masses of scalar mesons in the region of 600 – 1000 MeV with the ordering, $m_\sigma < m_\kappa <
m_{f_0, a_0}$, while the conventional $\bar q q$ currents imply masses around 1.5 GeV.
Independent Currents
====================
Let us start with currents for the scalar tetraquark, which we consider only local currents. Using the antisymmetric combination for diquark flavor structure, we arrive at the following five independent currents [@Chen:2006hy] $$\begin{aligned}
\nonumber\label{define_udud_current} S^\sigma_3 &=& (u_a^T C
\gamma_5 d_b)(\bar{u}_a \gamma_5 C \bar{d}_b^T - \bar{u}_b \gamma_5
C \bar{d}_a^T)\, ,
\\ \nonumber
V^\sigma_3 &=& (u_a^T C \gamma_{\mu} \gamma_5 d_b)(\bar{u}_a
\gamma^{\mu}\gamma_5 C \bar{d}_b^T - \bar{u}_b \gamma^{\mu}\gamma_5
C \bar{d}_a^T)\, ,
\\
T^\sigma_6 &=& (u_a^T C \sigma_{\mu\nu} d_b)(\bar{u}_a
\sigma^{\mu\nu} C \bar{d}_b^T + \bar{u}_b \sigma^{\mu\nu} C
\bar{d}_a^T)\, ,
\\ \nonumber
A^\sigma_6 &=& (u_a^T C \gamma_{\mu} d_b)(\bar{u}_a \gamma^{\mu} C
\bar{d}_b^T + \bar{u}_b \gamma^{\mu} C \bar{d}_a^T)\, ,
\\ \nonumber
P^\sigma_3 &=& (u_a^T C d_b)(\bar{u}_a C \bar{d}_b^T - \bar{u}_b C
\bar{d}_a^T)\, ,\end{aligned}$$ where the sum over repeated indices ($\mu$, $\nu, \cdots$ for Dirac, and $a, b, \cdots$ for color indices) is taken. Either plus or minus sign in the second parentheses ensures that the diquarks form the antisymmetric combination in the flavor space. The currents $S$, $V$, $T$, $A$ and $P$ are constructed by scalar, vector, tensor, axial-vector, pseudoscalar diquark and antidiquark fields, respectively. The subscripts $3$ and $6$ show that the diquarks (antidiquark) are combined into the color representation $\mathbf{\bar 3_c}$ and $\mathbf{6_c}$ ($\mathbf{3_c}$ or $\mathbf{\bar 6_c}$), respectively. The currents for other members are formed similarly. We can also use a symmetric combination for diquark flavor structure. However, they are related to the antisymmetric ones by the axial U(1) transformation [@Umekawa:2004js].
QCD Sum Rule Analysis
=====================
For the past decades QCD sum rule has proven to be a very powerful and successful non-perturbative method [@Shifman:1978bx; @Reinders:1984sr]. In sum rule analyses, we consider two-point correlation functions: $$\Pi(q^2)\,\equiv\,i\int d^4x e^{iqx}
\langle0|T\eta(x){\eta^\dagger}(0)|0\rangle \, , \label{eq_pidefine}$$ where $\eta$ is an interpolating current for the tetraquark. We compute $\Pi(q^2)$ in the operator product expansion (OPE) of QCD up to certain order in the expansion, which is then matched with a hadronic parametrization to extract information of hadron properties. At the hadron level, we express the correlation function in the form of the dispersion relation with a spectral function: $$\Pi(p)=\int^\infty_0\frac{\rho(s)}{s-p^2-i\varepsilon}ds \, ,
\label{eq_disper}$$ where $$\begin{aligned}
\rho(s) & \equiv & \sum_n\delta(s-M^2_n)\langle
0|\eta|n\rangle\langle n|{\eta^\dagger}|0\rangle \ \nonumber\\ &=&
f^2_X\delta(s-M^2_X)+ \rm{higher\,\,states}\, . \label{eq_rho}\end{aligned}$$ For the second equation, as usual, we adopt a parametrization of one pole dominance for the ground state $X$ and a continuum contribution. The mass of the state $X$ can be obtained $$M^2_X=\frac{\int^{s_0}_0 e^{-s/M_B^2}s\rho(s)ds}{\int^{s_0}_0
e^{-s/M_B^2}\rho(s)ds}\, . \label{eq_LSR}$$ We performed the sum rule analysis using all currents and their various linear combinations, and found a good sum rule by a linear combination of $A_6^\sigma$ and $V_3^\sigma$ $$\begin{aligned}
\eta^\sigma_1 = \cos\theta A^\sigma_6 + \sin\theta V^\sigma_3\, ,\end{aligned}$$ where the best choice of the mixing angle turns out to be $\cot\theta = 1 / \sqrt{2}$. For $\kappa$, $f_0$ and $a_0$, we have also found that similar linear combinations give better sum rules. The results of OPE can be found in Ref. [@Chen:2006zh]
Numerical Analysis
==================
For numerical calculations, we use the following values of condensates [@Yang:1993bp; @Ioffe:2002be; @Gimenez:2005nt]: $\langle\bar qq \rangle=-(0.240 \mbox{ GeV})^3$, $\langle\bar
ss\rangle=-(0.8\pm 0.1)\times(0.240 \mbox{ GeV})^3$,$\langle
g_s^2GG\rangle =(0.48\pm 0.14) \mbox{ GeV}^4$, $ m_u = 5.3 \mbox{
MeV}$, $m_d = 9.4 \mbox{ MeV}$, $m_s(1\mbox{ GeV})=125 \pm 20 \mbox{
MeV}$, $\langle g_s\bar q\sigma G q\rangle=-M_0^2\times\langle\bar
qq\rangle$, $M_0^2=(0.8\pm0.2)\mbox{ GeV}^2$.
The sum rules are written as power series of the Borel mass $M_B$. Since the Borel transformation suppresses the contributions from $s
> M_B$, smaller values are preferred to suppress the continuum contributions also. However, for smaller $M_B$ convergence of the OPE becomes worse. Therefore, we should find an optimal $M_B$ preferably in a small value region. We have found that the minima of such a region are 0.4 GeV for $\sigma$, 0.5 GeV for $\kappa$ and 0.8 GeV for $f_0$ and $a_0$, where the pole contributions reach around 50 % for all cases [@Chen:2006zh]. As $M_B$ is increased, the pole contributions decrease, but the resulting tetraquark masses are stable as shown in Fig. \[pic\_tetra\].
After careful test of the sum rule for a wide range of parameter values of $M_B$ and $s_0$, we have found reliable sum rules, with which we find the masses $ m_\sigma = (0.6 \pm 0.1) \; {\rm GeV}$, $
m_\kappa = (0.8 \pm 0.1) \; {\rm GeV}$, $m_{f_0,a_0} = (1 \pm 0.1)
\; {\rm GeV}\; ,$ which are consistent with the experimental results [@Yao:2006px].
For comparison, we have also performed the QCD sum rule analysis using the $\bar q q$ current within the present framework. The stable (weak $M_B$) behavior is obtained with the masses of all four mesons around 1.5 GeV. Here again we have tested various values of $M_B$ and $s_0$, and confirmed that the result shown is optimal.
Conclusions
===========
We have performed the QCD sum rule analysis with tetraquark currents, which implies the masses of scalar mesons in the region of 600 – 1000 MeV with the ordering, $m_\sigma < m_\kappa < m_{f_0,
a_0}$. We have also performed the QCD sum rule analysis with the conventional $\bar q q$ currents, which implies masses around 1.5 GeV. We have tested all possible independent tetraquark currents as well as their linear combinations. Our observation supports a tetraquark structure for low-lying scalar mesons. To test the validity of the tetraquark structure, it is also important to study decay properties, which is often sensitive to the structure of wave functions. Such a tetraquark structure will open an alternative path toward the understanding exotic multiquark dynamics which one does not experience in the conventional hadrons.
Acknowledgements {#acknowledgements .unnumbered}
================
H. X. C and A. H. thank the Yukawa Institute for Theoretical Physics at Kyoto University for hospitality during the YKIS2006 on “New Frontiers on QCD”. H.X.C. is grateful to the Monkasho fellowship for supporting his stay at RCNP, Osaka University. A.H. is supported in part by the Grant for Scientific Research ((C) No.16540252) from the Ministry of Education, Culture, Science and Technology, Japan. S.L.Z. was supported by the National Natural Science Foundation of China under Grants 10375003 and 10421503, Ministry of Education of China, FANEDD, Key Grant Project of Chinese Ministry of Education (NO 305001) and SRF for ROCS, SEM.
[10]{}
E. M. Aitala et al., Phys. Rev. Lett. 86, 770 (2001); M. Ablikim et al., Phys. Lett. B 598, 149 (2004). W. M. Yao [*et al.*]{} \[Particle Data Group\], J. Phys. G [**33**]{}, 1 (2006). T. Hatsuda and T. Kunihiro, Phys. Rept. [**247**]{}, 221 (1994). R. L. Jaffe, Phys. Rev. D [**15**]{}, 267 (1977). H. J. Lee and N. I. Kochelev, Phys. Lett. B [**642**]{}, 358 (2006). T. V. Brito, F. S. Navarra, M. Nielsen and M. E. Bracco, Phys. Lett. B [**608**]{}, 69 (2005). A. Zhang, T. Huang and T. G. Steele, arXiv:hep-ph/0612146. E. M. Aitala et al., Phys. Rev. Lett. 89, 121801 (2002); M. Ablikim et al., Phys. Lett. B 633, 681 (2006). H. X. Chen, A. Hosaka and S. L. Zhu, Phys. Rev. D 74, 054001 (2006). T. Umekawa, K. Naito, M. Oka and M. Takizawa, Phys. Rev. C [**70**]{}, 055205 (2004).
M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B [**147**]{}, 385 (1979). L. J. Reinders, H. Rubinstein and S. Yazaki, Phys. Rept. [**127**]{}, 1 (1985). K. C. Yang, W. Y. P. Hwang, E. M. Henley and L. S. Kisslinger, Phys. Rev. D [**47**]{}, 3001 (1993). B. L. Ioffe and K. N. Zyablyuk, Eur. Phys. J. C [**27**]{}, 229 (2003). V. Gimenez, V. Lubicz, F. Mescia, V. Porretti and J. Reyes, Eur. Phys. J. C [**41**]{}, 535 (2005). H. X. Chen, A. Hosaka and S. L. Zhu, arXiv:hep-ph/0609163.
[^1]: e-mail address: [email protected]
[^2]: e-mail address: [email protected]
[^3]: e-mail address: [email protected]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We calculate the asymptotic high-energy amplitude for electrons scattering at one ion as well as at two colliding ions, respectively, by means of perturbation theory. We show that the interaction with one ion [*eikonalizes*]{} and that the interaction with two ions [*causally decouples*]{}. We are able to put previous results on perturbative grounds and propose further applications for the obtained rules for interactions on the light cone. The formalism will be of use for the calculation of Coulomb corrections to electron-positron pair creation in heavy ion collisions. Finally we discuss the results and inherent dangers of the employed approximations.'
author:
- 'U. Eichmann${}^a$, J. Reinhardt${}^a$, S. Schramm${}^b$, W. Greiner${}^a$'
date: |
${}^a$Institut für Theoretische Physik,\
Johann Wolfgang Goethe-Universität, Frankfurt am Main, Germany\
${}^b$Gesellschaft für Schwerionenforschung mbH, Darmstadt, Germany
title: Electron Propagation in the Field of Colliding Nuclei at Ultrarelativistic Energies
---
Introduction
============
At ultrarelativistic energies, the theoretical treatment of scattering processes is extremely facilitated. On the one hand, the relevant equations themselves simplify, when terms of order ${\cal O}(1/\gamma^2)$ become negligible, on the other hand, the interactions simplify due to causality. In that way, high energy scattering becomes analytically accessible.
Eikonal approximations or optical models usually are formulated for the scattering of a highly energetic particle at a slow or even static center [@Abarbanel][@Torgerson]. We present a simple transformation of the covariant derivatives that is used to easily solve the opposite case. The transformation of the equations of motion for particles scattered by fast moving charge centers immediately generates the scattered wave describing the particle. Our results coincide with previous calculations performed in this reference frame [@Jackiw][@tHooft].
The summation of ladder graphs is shown to [*eikonalize*]{} as well [@Chang-Ma]. This was elegantly derived within the method of kinematically decoupling the components of the scattering process, and Lorentz transforming into the respective rest frames [@Chang-Fishbane] which inherently contains the advantages of a fast external potential.
Following a different approach we will exploit the same advantages. We perform a perturbative approach and directly approximate the external potential by its asymptotic high-energy limit which amounts to saying, that the longitudinal components of the exchanged photons can be discarded.
In doing so, one can directly rederive the amplitude for the scattering at one center and even put the recent result of Segev and Wells [@Segev] for the scattering amplitude for an electron moving in the field of two ultrarelativistic colliding ions on perturbative grounds. Moreover, one is allowed to go beyond their calculations and is provided deeper insight.
The derivations in this paper are formulated for electron scattering, but they can be immediately extended to cover the physically more relevant process of electron-positron pair production. The search for exact analytic expressions describing electron-positron pair production in heavy-ion collisions is motivated by the question whether Coulomb effects only play an inferior role at high energies. Such a conclusion might be drawn from a comparison between second-order perturbation theory results [@Bottcher] and calculations employing Furry-Sommerfeld-Maue wave functions [@Ionescu]. It should be mentioned, however, that the Coulomb distortions considered in these calculations only account for one ion, whereas the second ion enters as a perturbation.
Scattering of an electron off a fast moving source
==================================================
Transformation of the Dirac equation {#1iontrafo}
------------------------------------
We are searching for the asymptotic scattering solution of a Dirac particle from a fast moving Coulomb potential in the limit of very large collision energy. In the Lorentz gauge the Li$\acute{\rm e}$nard-Wiechert potentials for a point charge moving moving with uniform velocity $\beta$ in $+z$ direction read $$\begin{aligned}
\label{lwpot}
A_0&=&-\frac{Z\alpha \gamma}{\sqrt{\gamma^2(z-\beta t)^2 +
\vec{x}_\perp^2}}\\
\label{a0a3}
A_3&=&\beta A_0\end{aligned}$$ The equation of motion for the scattered particle becomes $$\label{dg}
\left[ \hat{\gamma_0}(i\partial_t -A_0)+\hat{\gamma_3}(i\partial_z +A_3)+
\hat{\vec{\gamma}}_\perp \cdot i\vec{\nabla}_\perp -
m\right] \psi =0$$ We set $c=\hbar=1$. The charge $e$ of the electron was absorbed into the definition of the potential. We make use of the external field approximation, i.e. we assume that the source is not influenced by the scattered particle and moves on a straight line. This treatment will be justified if the mass of the source particle is very large. To simplify the Dirac equation (\[dg\]) we use the operator identity [@Miura] $$(i\partial_x \mp
i\partial_x\ln \phi)^n=\phi^{\pm 1}(i\partial_x)^n\phi^{\mp 1}
\label{kovabl}$$ to rewrite the covariant derivatives. We must introduce two fields $\phi'$ and $\phi$ for the space and the time component of the vector-potential $A^\mu$ $$\begin{aligned}
A_0&=&i\partial_t \ln \phi\nonumber\\
A_3&=&i\partial_z \ln \phi'\end{aligned}$$ The field $\phi$ is determined to be $$\label{phidef}
\phi=e^{-i\int_{-\infty}^t dt' A_0}$$ The thus transformed Dirac equation reads (see Appendix \[adiractrans\]) $$\label{dtrans}
\left[\hat{\gamma}_0i\partial_t +\hat{\gamma}_3
(i\partial_z-\frac{1}{\beta^2\gamma^2}A_3)+
\hat{\vec{\gamma}}_\perp \cdot (i\vec{\nabla}_\perp +
i{\rm grad}_\perp
\ln \phi )- m\right]\tilde{\psi}=0$$ where $\tilde{\psi}=\psi/\phi = e^{i\int_{-\infty}^t dt' A_0}\psi$. The operator identity, together with the field $\phi$ defined in (\[phidef\]) has led to the elimination of the scalar part of the vector potential, i.e. to the temporal gauge, $A'_0=0$. For very large $\gamma$ one nearly has a purely transverse vector potential $\vec{A'}_\perp = i{\rm grad}_\perp
\ln \phi$ which is the negative time integral of the transverse electric field. From classical electrodynamics one knows, that the time integral of the transverse electric field is given by $$\label{etransint}
\int_{-\infty}^{\infty}\vec{E}_\perp=-2Z\alpha\vec{x}_\perp /
(\beta x_\perp^2)$$ This implies that $$\int_{-\infty}^\infty dt' A_0 = +\frac{1}{\beta}
Z\alpha \ln x_\perp^2 + {\cal C}$$ which reproduces (\[etransint\]) if the transverse gradient operator is applied. ${\cal C}$ is an infinite quantity which expresses the divergence of the phases in Coulomb scattering. Furthermore using (\[phidef\]) and (\[lwpot\]) it is easy to show, that the transverse vector potential exhibits a Heaviside step function dependence $\sim \theta (t -z)\vec{x}_\perp/x^2_\perp$ in the limit of very large $\gamma$. Now, since $t$ and $\gamma$ enter symmetrically in the integral, the limit $\gamma \to \infty$ corresponds to sending the upper bound of the integral to infinity. Therefore, all of the above is applicable and we find $$\label{potlimc}
\lim_{\gamma \to \infty} A_0 = +\delta (z-t) Z \alpha \ln x_\perp^2 + {\cal
C}'$$ The Coulomb phase ${\cal C}'$ in general will depend on $z$ and $t$. It can be removed by a gauge transformation, as is easily seen $$\label{cgtrafo}
\tilde{\psi}'=e^{-i\int_{-\infty}^{t}dt'{\cal C}'}\tilde{\psi}=
e^{+iZ\alpha \theta (t-z)
\ln x_\perp^2}\psi$$ This gauge transformation was first applied in [@Aichelburg]. The removal of the Coulomb phase yields a short range potential allowing for asymptotic plane wave solutions (see Appendix \[apotlim\]).
For $t\neq z$ the $t$ and $z$ dependence in both the transverse vector potential and the transformed spinor $\tilde{\psi}$ vanish in the limit $\gamma\to \infty$. By inverse transformation we find that $\psi$ solves a free Dirac equation on either side of the light front $t=z$ and can only differ by a phase.
The transformed wave function $\tilde{\psi}$ has the advantage of being continuous on the surface defined by $t=z$. In contrast, the wave function $\psi$ exhibits a discontinuous behaviour at the light front. There is a jump in that component of $\psi$ which couples to $\hat{\gamma}_-
=\hat{\gamma}_0-\hat{\gamma}_3$, the matrix structure of the interaction in the limit $\gamma \to \infty$. Using this property one directly finds for $\hat{\gamma}_-\psi$ at the discontinuity $$\label{psiout}
\hat{\gamma}_-\psi(t-z=0^+)=e^{-iZ\alpha \ln x_\perp ^2 }\hat{\gamma}_-\psi
(t-z=0^-)$$ where we ignored the irrelevant quantity ${\cal C}$.[^1] The complement $\hat{\gamma}_+\psi$ of these spinor components, where $\hat{\gamma}_+ = 2\hat{\gamma}_0
-\hat{\gamma}_-= \hat{\gamma}_0+\hat{\gamma}_+$ is continuous at $t=z$. Both parts of the spinor are coupled via the free Dirac equation on either side of the discontinuity.
By application of the LSZ-reduction formula one finds in general, that at very large scattering energies the $S$ matrix is determined by $\phi$, in which we recognize the well known eikonal form [@Fried]. Because of the identity (\[kovabl\]) this result holds independently of the power of the momentum entering in the respective wave equation. For that reason the expressions for the $S$ matrices for e.g. spinor or scalar particles only differ by an overall factor.
We first consider the properties of the previous result. The LHS of (\[psiout\]) can be expanded in plane waves. Since we consider scattering at the negative light front, we must substitute $d^3x \to dx_+d^2x_\perp$ [@Meggiolaro] and accordingly $d^3p \to dp_-d^2p_\perp$. The expansion coefficient corresponds to the $S$ matrix in momentum space, which is easily found to be $$\begin{aligned}
\label{smom}
S(p',p)&=&2\pi\delta
(p'_--p_-)
\Bigg[\left(\frac{4}{(\vec{p'}_\perp-\vec{p}_\perp)^2}
\right)^{1-iZ\alpha}\Gamma^2(1-iZ\alpha)\sin(\pi
iZ\alpha)\nonumber\\
&&+(2\pi)^2\delta(\vec{p'}_\perp-\vec{p}_\perp)\Bigg]
\overline{u}(p')\hat{\gamma}_-u(p)\end{aligned}$$ Here $u$ denotes the electron unit spinor. If the trajectory of the ion is shifted by the impact parameter $\vec{b}$, this result is simply multiplied by the factor $e^{i(\vec{p'}_\perp-\vec{p}_\perp)\cdot \vec{b}}$. $p$ and $p'$ are the incoming and outgoing momenta. We note that the negative light cone momentum $p_-=p_0-p_3$ is conserved in the scattering. The positive light cone momentum is fixed by the mass shell condition. The first term in the square brackets in (\[smom\]) corresponds to the $T$ matrix. Eq. (\[smom\]) represents a well known result which was previously derived in e.g. [@Abarbanel; @tHooft; @Jackiw; @Segev].
Perturbative Approach {#pertapp}
---------------------
In this section we want to derive the eikonal form of the $T$ matrix via perturbation theory. Several approximations are necessary to obtain the eikonal form, namely the neglect of the longitudinal components of the photon momentum, the conservation of the photon light cone momentum, as well as the simplification of the matrix structure of the interaction [@Chang-Ma]. The calculation shows that these approximations are the counterparts of the requirement of a vanishing longitudinal vector potential and the step function dependence of the transverse vector potential. Having this in mind we directly use the asymptotic high-energy expression of the potential. We then evaluate the terms of the perturbation series for the external-field scattering problem depicted by the Feynman graphs of Fig. \[fig01\].
The potential entering into our calculations is of the form $$\label{pot}
V_0(x)=V_3(x)=\delta (z-t)V_\perp(\vec{x}_\perp)$$ In the following calculations it will not be necessary to specify the explicit form of $V_\perp(\vec{x}_\perp)$. Problems related to the logarithmic potential obtained in the last section will be discussed in section \[discussion\]. We use the light-cone variables $$\left(\begin{array}{c}p_-\\ p_+\end{array}\right)=\left(\begin{array}{rr}
1&-1\\ 1&1\end{array}\right)\left(\begin{array}{c}p_0\\ p_3\end{array}\right)$$ The Feynman propagator describing the internal electron lines reads $$S_F(p)=\frac{1}{\hat{\gamma}_0p_0 -\hat{\vec{\gamma}}\cdot\vec{p}-
m+i\epsilon}=
\frac{\frac{1}{2}(\hat{\gamma}_-p_++\hat{\gamma}_+p_-)-
\hat{\vec{\gamma}}_\perp
\cdot \vec{p}_\perp +m}{p_+p_--p_\perp^2-m^2+i\epsilon}$$ The following products of gamma matrices in the light-cone representation are needed: $$\label{gamprod}
\begin{array}{ll}
\displaystyle
\hat{\gamma}_\pm \hat{\gamma}_\mp \hat{\gamma}_\pm &
\displaystyle
=4\hat{\gamma}_\pm\\
\displaystyle
\hat{\gamma}_\pm \hat{\vec{\gamma}}_\perp \hat{\gamma}_\mp&\displaystyle
= 2\hat{\vec{\alpha}}_\perp \hat{\gamma}_\mp\\
\displaystyle
\hat{\gamma}_\pm \hat{\vec{\gamma}}_\perp \hat{\gamma}_\pm
&\displaystyle =0\\
\displaystyle
\hat{\gamma}_\pm \hat{\gamma}_\mp&\displaystyle
=2\hat{\gamma}_0\hat{\gamma}_\mp\\
\displaystyle
\hat{\gamma}_\pm \hat{\gamma}_\pm&\displaystyle
=0\\
\end{array}$$ The amplitude for electron scattering in first order perturbation theory is $$\label{1order}
A^{(1)}_{p'p}=(2\pi) (-i) \delta (p'_--p_-)
F_{p'p}(V_\perp)\overline{u}(p')\hat{\gamma}_-u(p)$$ $F_{p'p}(\;\;)$ denotes the Fourier transform with respect to the transverse coordinates taken at the momentum $(\vec{p'}_\perp-\vec{p}_\perp)$ $$F_{p'p}(V_\perp)=\int d^2 x_\perp e^{-i\vec{x}_\perp \cdot
(\vec{p'}_\perp-\vec{p}_\perp)}V_\perp (\vec{x}_\perp)$$ In second order the amplitude reads $$\begin{aligned}
A^{(2)}_{p'p}&=&\int \frac{dk_+dk_-d^2k_\perp}{(2\pi)^4} (2\pi)^{2}
(-i)^2 i \delta (k_--p_-)\delta
(p'_--k_-)\frac{k_-}{k_-k_+-k_\perp^2 -m^2
+i\epsilon}\nonumber \\
\label{k+int}
&&F_{kp}(V_\perp)F_{p'k}(V_\perp)\overline{u}(p')\hat{\gamma}_-u(p)\\
&=&(2\pi) (-i)^2 \delta (p'_--p_-)
\frac{1}{2}F_{p'p}(V_\perp^2)\overline{u}(p')\hat{\gamma}_-u(p)
\nonumber\end{aligned}$$ The $k_+$ integral in (\[k+int\]) drops out using the symbolic substitution $$\label{Feynsubst}
1/(x+i\epsilon)\to P(1/x)-i\pi\delta (x)$$ since the principal value integral $P$ vanishes. It is interesting to note that the simple structure of the results (\[1order\]) and (\[k+int\]) is retained if one goes to higher orders of perturbation theory. The $n^{th}$ order amplitude factorizes into $n-1$ integrals of the form (\[k+int\]) which leads to $$\label{norder}
A^{(n)}_{p'p}=(2\pi) (-i)^n \delta (p'_--p_-)
\frac{1}{n!}F_{p'p}(V_\perp^n)\overline{u}(p')
\hat{\gamma}_-u(p)$$
This result is obtained by symmetrizing the $n-1$ integrals over the positive light cone momenta in (\[Feynsubst\]) yielding the expression $(-i2\pi)^{n-1}/n!\prod_i \delta (k_+^i)$ [@Chang-Ma]. This corresponds to reconsidering the different time orderings and finally dividing by $n!$ to prevent double counting. This symmetrization procedure directly shows that the principal value terms in (\[Feynsubst\]) do not contribute.
Clearly, with (\[norder\]) the perturbation series can be summed up to yield the result $$\label{tmatr1}
A_{p'p}=2\pi\delta (p'_--p_-){\cal
T}(\vec{p'}_\perp-\vec{p}_\perp)\overline{u}(p')\hat{\gamma}_-u(p)$$ Here we defined the momentum transfer function $${\cal
T}(\vec{p'}_\perp-\vec{p}_\perp)=F_{p'p}(e^{-iV_\perp(\vec{x}_\perp)}-1)$$ with $$V_\perp(\vec{x}_\perp)=\int_{-\infty}^{+\infty} dt V_0(x)$$ This result reproduces the eikonal form.
Solution in the field of several ions
=====================================
The case of two colliding ions
------------------------------
In the c.m. frame, the field of two ultrarelativistic colliding ions $A$ and $B$, cf. Figure \[fig02\], reads $$\label{pot2ion}
V_{0/3}(x)=\delta (z-t) V^A_\perp(\vec{x}_\perp)
\pm \delta (z+t)V^B_\perp(\vec{x}_\perp)$$
The identity (\[kovabl\]) can also be applied to potentials given by a superposition as is easily verified $$\label{mtrafomi}
(i\partial_x \mp i\sum \partial_x\ln \phi_i)^n=\left(\prod \phi_i\right)^
{\pm 1}
(i\partial_x)^n
\left(\prod \phi_i\right)^{\mp 1}$$ Since in the case of (\[pot2ion\]) we have two discontinuities, the asymptotic solution is not obtained as easily as in section \[1iontrafo\]. The explicit calculation is shown in Appendix \[asolu2ion\].
It is found, that the two ions couple to distinct components of the electron spinor. We show in this section, how this behaviour follows from perturbation theory and how it can be interpreted consistently.
We have to consider several new diagrams describing the alternate interaction of the electron with both ions. Using (\[gamprod\]) we find, that the contribution to the $T$ matrix of an arbitrary number of interactions with one ion that are sandwiched between interactions with the other ion (see Fig. \[fig03\]) , vanishes. The reason is, that we end up with an integral of the form $$A\sim\int \frac{dk_\pm}{(k_\pm p_\mp -k_\perp^2 -m^2 +i\epsilon)
(k_\pm p'_\mp -{k'}^2_\perp -m^2 +i\epsilon)}=0$$ The vanishing of this integral is immediately seen from Cauchy’s theorem since the contour can be closed in the upper half plane, where the integrand is analytic.
In the ultrarelativistic limit the electron will therefore interact with the ions separately, see Figure \[fig04\]. The separate interactions of the electron with the two ions $A$ and $B$ are linked in the following way $$\begin{aligned}
A^{tot}_{p'p}&=&\int \frac{d^2k_\perp}{(2\pi)^2}
{\cal T}_A(-\vec{p}_\perp+\vec{k}_\perp){\cal T}_B(\vec{p'}_\perp -
\vec{k}_\perp)\nonumber \\
&&\overline{u}(p')\frac{-\hat{\vec{\alpha}}_\perp
\cdot \vec{k}_\perp +
\gamma_0
m}{p'_+ p_- - {k}_\perp^2 -m^2 +i\epsilon}
\hat{\gamma}_+u(p)
\nonumber\\
&&+\int \frac{d^2k_\perp}{(2\pi)^2}
{\cal T}_A(-\vec{p}_\perp+\vec{k}_\perp){\cal T}_B(\vec{p'}_\perp -
\vec{k}_\perp)\nonumber \\
\label{Ttot}
&&\overline{u}(p')\frac{-\hat{\vec{\alpha}}_\perp
\cdot (\vec{p}_\perp +\vec{p'}_\perp -\vec{k}_\perp) +
\gamma_0
m}{p'_- p_+ -
(\vec{p}_\perp +\vec{p'}_\perp -{k}_\perp)^2 -m^2 +i\epsilon}
\hat{\gamma}_-u(p)\end{aligned}$$ Here we have already added both possible time orderings. ${\cal T}_A$ and ${\cal T}_B$ are the momentum transfer functions ${\cal T}$, defined in (\[tmatr1\]) for the interactions with ion $A$ and $B$, respectively. This result is equivalently obtained by using the discontinuous behaviour at the light fronts (see Appendix \[asolu2ion\]) and corresponds to the result of Segev and Wells [@Segev].
To understand the decoupling property, one has to consider the matrix structure of the potential. To this end we write down the Dirac equation in the following form $$\left[ i\partial_t+
\hat{\vec{\alpha}} \cdot i \vec{\nabla} - \hat{\gamma}_0m
-(1\pm \beta\hat{\alpha_z})A_0 \right] \psi =0$$ where the sign depends upon the direction of motion and $A_0$ is given by (\[lwpot\]). In the limit $\beta\to 1$ the operators $1/2(1\pm \beta\hat{\alpha_z})$ become orthogonal projection operators [@Segev]. The action of these operators can be understood if one recalls the standard form of Lorentz transformations [@Itzykson] in spinor space $$\label{ltrafo}
\psi'(x')=e^{-(i/4)\sigma_{\alpha \beta}\omega^{\alpha\beta}}\psi (x)$$ Here $\sigma_{\alpha \beta}=i/2[\gamma_\alpha,\gamma_\beta]$ and the exponent represents the product of the rapidity vector $\vec{\omega}$ times the generators of the Lorentz transformation. For a boost in $+z$ direction (\[ltrafo\]) simplifies to $$\begin{aligned}
\psi'(x')&=&e^{-\frac{\omega}{2}\hat{\alpha}_z}\psi (x)\nonumber \\
&=&{\rm cosh}\left(\frac{\omega}{2}\right)\left(1-{\rm
tanh}\left(\frac{\omega}{2}\right)\hat{\alpha}_z\right)\psi (x)\end{aligned}$$ Therefore, (see (\[lwpot\])), a Lorentz-transformed vector acting in spinor space $$(1\pm \beta\hat{\alpha_z})A_0=\gamma(1\pm
\beta\hat{\alpha_z})\frac{-Z\alpha}{r'}$$ can directly be obtained by a Lorentz transformation (\[ltrafo\]) accounting for the vectorial nature of the transformed object with a factor 2 in the exponent. The operators $(1\pm\hat{\alpha}_z)$ are $1/\gamma$ times a Lorentz transformation with effectively infinite rapidity. These operators project the Dirac spinors onto causally disconnected subspaces of the Hilbert space. Therefore it is simply causally impossible for the Dirac spinor to communicate alternately with both ions.
Therefore, even the exact expressions for the interaction of an electron with two colliding ultrarelativistic ions maintains the structure of the two-photon graph. We can interpret (\[Ttot\]) as the interaction of an electron in lowest order with a ”dressed“ potential of the form [@Baltz2] $$\label{modpot}
\tilde{V_0}(x)=\tilde{V_3}(x)
\sim\delta (z-t)\left(\left(\frac{1}{x_\perp}\right)^{2iZ\alpha}-1\right)$$ An inspection of (\[Ttot\]) reveals that the scattering amplitude is represented by a divergent integral. There are infrared divergencies caused by the poles of the momentum transfer functions ${\cal T}_A$ and ${\cal
T}_B$, see Appendix \[approp\]. It is interesting to note, that an explicit introduction of a photon mass describing a screened Coulomb potential does [*not*]{} yield a regularized expression for the functions ${\cal T}$. On the other hand, if the modified potential (\[modpot\]) is screened with a damping factor $e^{-\epsilon x_\perp}$, this leads in momentum space to $$\label{tscreened}
\tilde{V}(k)\sim
\frac{1}{(\epsilon^2+k_\perp^2)^{(1-i\alpha Z)}}\Gamma(2(1-i\alpha
Z))P_{1-2i\alpha Z}(\epsilon(\epsilon^2+ k_\perp^2)^{-\frac{1}{2}})$$ which resembles the propagator of a photon with mass $\epsilon$. $P_{1-2i\alpha Z}(\;\;)$ denotes a Legendre function.
We should stress however, that such artificial regularization procedures are not needed. There is a natural cut off since the condition for the applicability of the used approximations requires (see Appendix \[apotlim\]) $$\gamma \gg \frac{x_\perp}{|z-\beta t|}$$ which in momentum space translates into the condition $$k_\perp \gg \frac{\omega}{\gamma}$$ This lower bound for the transverse momentum corresponds to the cut off inherent in the Fourier transform of the potential, cf eq. (\[ftpot\]) The introduction of this cut off has the important property to restore the energy dependence of the amplitude which was lost when taking the limit $\gamma \to \infty$.
Solution in the field of channeled ions {#crystal}
---------------------------------------
Here we want to sketch briefly an extension of the formalism discussed so far to the case of more that two colliding charges. The causal decoupling of interactions with sources moving on the positive and negative light-cone, respectively, and the above interpretation of the interaction can be used to calculate the scattering amplitude of electrons (or more realistically electron-positron pair production) for a field configuration which corresponds to the channeling of an ion in a crystal.
We use the equal speed system, the crystal is moving in $-z$ direction. The crystal layers have a spatial distance $a\vec{e}_z$. In the ultrarelativistic case, the electron again interacts with the ion and the crystal layers separately and we get simple time orderings of the interaction. For $n$ crystal layers we have $n+1$ possibilities. For the sake of simplicity we formulate the perturbative description of the successive interactions of the electron with both the ion and the crystal layers directly with modified potentials of the form (\[modpot\]). One then obtains for the interaction with two neighbouring crystal layers the integral $$\begin{aligned}
A&=&-\delta (p'_+-p_+)e^{ip'_-\frac{a}{2}}\nonumber \\
&&\int\frac{dk_-d^2k_\perp}{(2\pi)^2}
\frac{\overline{u}(p')\hat{\gamma}_+u(p)}{-ik_-+
i\frac{k_\perp^2+m^2}{k_+}+\frac{\epsilon}{k_+}}e^{-ik_-\frac{a}{2}}{\cal
T}_{C_i}(\vec{k}_\perp-\vec{p}_\perp){\cal
T}_{C_{i+1}}(\vec{p'}_\perp-\vec{k}_\perp)\nonumber \\
&=&2\pi \delta (p'_+-p_+)
\int\frac{d^2k_\perp}{(2\pi)^2}e^{i\left(-\frac{k_\perp^2+m^2}{2p'_+}+
\frac{p'_-}{2}\right)a}\nonumber \\
&&{\cal T}_{C_i}(\vec{k}_\perp-\vec{p}_\perp)
{\cal T}_{C_{i+1}}(\vec{p'}_\perp-\vec{k}_\perp)
\overline{u}(p')\hat{\gamma}_+u(p)
\;\;\;\;a>0,\;\epsilon \to 0\end{aligned}$$ The subscripts $C_i$ and $C_{i+1}$ denote the scattering amplitudes from the interaction of the electron with the $i$th and the $(i+1)$th crystal layer. For $a<0$ (reverse direction of electron motion) the integral vanishes, which expresses, that the electron can not interact alternately with neighbouring crystal layers, due to causality. The derivation of this functional connection using (\[mtrafomi\]) is shown in Appendix \[asolu2ion\].
If the electron interacts with the ion between interacting with two distinct crystal layers, we get $$\begin{aligned}
A&=& \int \frac{d^2k_\perp d^2k'_\perp}{(2\pi)^4}
\frac{e^{ip'_-\frac{a}{2}}
\left(e^{-i ({k}_\perp^2
+m^2-i\epsilon )\frac{a}{2p_+}} -e^{-i ({k'}_\perp^2
+m^2-i\epsilon )\frac{a}{2p'_+}}\right)}{(p'_+({k}_\perp^2+m^2)-
p_+({k'}_\perp^2+m^2)-i \epsilon(p'_+-p_+))}
\nonumber \\
&&{\cal T}_{C_i}(\vec{k}_\perp-\vec{p}_\perp)
{\cal T}_A(\vec{k'}_\perp-\vec{k}_\perp)
{\cal T}_{C_{i+1}}(\vec{p'}_\perp-\vec{k'}_\perp) \nonumber \\
&&\overline{u}(p')
(-\hat{\vec{\alpha}}_\perp \cdot \vec{k}_\perp +\hat{\gamma}_0m)
(-\hat{\vec{\alpha}}_\perp \cdot
\vec{k'}_\perp +\hat{\gamma}_0m) \hat{\gamma}_+u(p)\end{aligned}$$ Successive interactions with different crystal layers factorize and any scattering process including intermediate interaction with the channeled ion gives the same amplitude. Further studies will have to show how these considerations can be put to use for the calculation of pair creation in channeling.
Discussion
==========
In the previous sections the potential of a fast moving charge has been substituted by its asymptotic high-energy expression. From a mathematical (and also a physical) point of view this is a problematic limit, since the required transformation is not an element of the Lorentz group. Furthermore, the potential (\[lwpot\]), a bounded operator in Hilbert space, gets transformed into an unbounded operator, and finally the number of spatial dimensions gets reduced from three to two.
The ansatz directly reflects the approximations made by Chang and Ma [@Chang-Ma] who neglected the longitudinal components of the photon momentum, giving the $\delta$-functions for the respective conserved light cone momenta. The above mentioned problems emerge here in the fact that the longitudinal components of the photon momentum never really vanish.
All approximations allow the well known conclusion, that the eikonal expression can be regarded as the contribution of all ladder diagrams in the high-energy limit and that it is completely compatible with a perturbative calculation.
In the case of two ions we mainly profited from the causal decoupling of the interactions implied by the presence of the factors $(1\pm \alpha_3)$. The matrix structure of the true interaction is given by $(1\pm \beta\hat{\alpha}_3)\approx
(1\pm \hat{\alpha}_3) \mp
\hat{\alpha}_3/2\gamma^2$, so that the leading corrections to this behaviour are suppressed with $1/\gamma^2$.
However, the considered calculations have inherent dangers. Fortunately we had to specify neither the transverse part of the potential nor its Fourier transform throughout our perturbative calculation. The first point may serve to generalize the validity of the result to any function $V_\perp(\vec{x}_\perp)$. However, a naive comparison even with the first order Born approximation would have failed due to the difficulties concerning the Fourier transform of the logarithm, whereas the logarithm in the argument of the exponential function is meaningful and correct. On the one hand the integration of the potential eliminates one dimension, which is finally recovered in the overall $\delta$ function for the light cone momenta. On the other hand the calculation of the scattering matrix between asymptotic states $(t\to \pm \infty)$ corresponds to the (unphysical) limit $\gamma \to \infty$. The detour via the Fourier transform of the ungauged potential gives a logarithm as well (see Appendix \[approp\]), depending, however, strongly upon a regularization mass $\mu$. This has its root in the fact, that $1/(k_\perp^2 + \mu^2)$ is not the correct two-dimensional photon propagator [@Grignani]. The divergent term $\lim_{\mu\to 0}\ln \mu^2$ is the term ${\cal C}$ in section \[1iontrafo\].
Now it is well known, that two-dimensional fields in the limit of vanishing mass are rather ill-defined objects, whereas the exponential of these fields is not. The Fourier transform of this exponential expression is elementary (see (\[smom\])). It can further be expanded into a Taylor series. Although it is not justified to identify the different terms with the Fourier transforms of the powers of the logarithm, the first term corresponds to the high energy limit of the Fourier transform of the retarded potential, which is rather accidental.
Nevertheless, the correct Fourier transform of the logarithm in two dimensions is obtained by Taylor expansion of ${\cal T}(k)$, but the limit $Z\alpha \to 0$ has to be taken [*after*]{} having integrated the expression with a test function [@Grignani]. $$\label{lnft}
\int d^2x_\perp e^{-i\vec{k}_\perp\cdot \vec{x}_\perp}\ln x_\perp^2
=\lim_{iZ\alpha\to 0}\frac{d}{d(iZ\alpha)}\left(\pi \frac{\Gamma(1-i\alpha
Z)}{\Gamma(i\alpha
Z)}\left(\frac{4}{k_\perp^2}\right)^{1-i\alpha Z}
\right)$$ This peculiarity is related to the fact, that the linearity of the Fourier transform is not strictly defined for the action on infinite series, resulting in the non-commutability of limiting procedures as in (\[lnft\]). Another way is to rewrite the logarithm by the $t$ integral of the gauged potential and expressing the integrand by means of its Fourier transform. One then finds for the Fourier transform of the logarithm in two dimensions [@Ferrari]: $$-\int d^2x_\perp e^{-i\vec{k}_\perp\cdot \vec{x}_\perp}
\ln x_\perp ^2 = \lim_{\lambda\to 0} 4 \pi \left(
\frac{1}{k_\perp^2 +\lambda^2} + \pi \delta^2(k_\perp) \ln \left(\frac{
\lambda^2}{\mu^2}\right)\right)$$ with $\lambda =\omega/\gamma$, $\mu = 2/e^C$. The condition $\lambda \to 0$ coincides with the limit $\gamma\to \infty$ (see Appendix \[approp\]). The correct treatment of this result again requires the limit to be taken after integrating the expression with a test function. Inverse transformation shows the required independence of the result of the regularzation parameter $\lambda$.
In view of the previous discussion one may infer that it is not justified to identify the eikonal expression using a gauged potential with first-order perturbation theory using the original potential, since this ignores the gauge transformation applied to the potential.
In any case the above considerations show, that the limit $\gamma \to \infty$ is pathological and that its implications should be studied with care.
Conclusion
==========
The transformation presented in section \[1iontrafo\] directly yields the scattering amplitude for (arbitrary) particles scattered at fast charge centers. Due to Lorentz invariance the expression for the amplitude holds even for the case of static scattering centers, which gives the classical form of the eikonal approximation. The essential ingredients in the static case – the vanishing of the spin current and the assumption that $\phi$ was a slowly varying function – have been replaced by the discontinuous behaviour at the light fronts in the presented case of fast scattering centers.
The gauge transformed high energy limit of the potential directly contains the necessary approximations mentioned in section \[pertapp\], and perturbative calculations using this potential can be done without further assumptions.
The results obtained by eikonal approximations or by application of the transformation (\[mtrafomi\]) are shown to be equivalent to the sum of all ladder graphs. It should therefore not be surprizing that it is possible to regain perturbative results from the eikonal expression. We showed however, that the obtained results must be studied with care, and that a wrong treatment only accidentally leads to correct results.
Finally, the interaction of the electron with several ions moving along the light cones was shown to decouple due to causality.
The high-energy scattering amplitude of electrons in the field of two colliding ions has the same structure as the second-order perturbative result [@Bottcher]. It can therefore be considered as a two-photon process with a modified potential of the form (\[modpot\]). The restrictions for the exchanged momenta imposed by the considered approximations are compatible with those encountered in the Weizsäcker–Williams method of virtual quanta. This method corresponds to a first-order Born approximation in the temporal gauge, considering only the transvere part of the interaction (the longitudinal part is suppressed by $1/\gamma^2$). The analogy of the obtained scattering amplitude to the two-photon process allows to adopt the approximations made in [@Bottcher], aiming to express the result in the Weizsäcker-Williams form. The cross section of the scattering process can therefore be obtained from the Klein–Nishina formula for Compton-scattering and photon distributions obtained from (\[modpot\]) in the temporal gauge [@Eichmann]. The simplified structure of the scattering amplitude allows for a study of the high-energy behaviour of electron-positron pair production, that accounts correctly for the Coulomb effects of both ions.
Acknowledgements {#acknowledgements .unnumbered}
================
U.E. would like to thank S.J. Chang for helpful comments, and D. Schwarz and F. Constantinescu for stimulating discussions. This work was supported by [*Deutsche Forschungsgemeinschaft*]{} DFG within the project Gr-243/44-2.
Transformation of the Dirac equation {#adiractrans}
====================================
According to (\[kovabl\]) we set $$\begin{aligned}
A_0&=& i\partial_t \ln \phi\\
A_3&=& i\partial_z \ln \phi'\end{aligned}$$ Since $A_3=\beta A_0$ and $\partial_t =-\beta \partial_z$ due to the fact, that the $t$ and $z$ dependence enters only via the combination ($z-\beta t$), we find $$\begin{aligned}
&&\hphantom{\Rightarrow\;\;}\frac{\partial_z\phi'}{\phi'}=-\beta^2
\frac{\partial_z\phi}{\phi}\\
&&\Rightarrow\;\; \phi'=\phi^{-\beta^2}\end{aligned}$$ i.e. $\phi=1/\phi'$ for $\beta \to 1$ as expected (see (\[kovabl\]),(\[a0a3\])). Inserting this into the Dirac equation, we find $$\begin{aligned}
&&\left[ \phi \hat{\gamma}_0i\partial_t \frac{1}{\phi} + \phi
\phi^{-\frac{1}{\gamma^2}}\hat{\gamma}_3i\partial_z
\frac{1}{\phi\phi^{-\frac{1}{\gamma^2}}} +\hat{\vec{\gamma}}_\perp
i\vec{\nabla}_\perp -m\right]\psi\\
&&=\left[\phi\left( \hat{\gamma}_0i\partial_t + \phi^{-\frac{1}{\gamma^2}}
\hat{\gamma}_3
i\partial_z \frac{1}{\phi^{-\frac{1}{\gamma^2}}}\right)\frac{1}{\phi}
+\hat{\vec{\gamma}}_\perp
i\vec{\nabla}_\perp -m\right]\psi\\
&&=\phi\left[ \hat{\gamma}_0i\partial_t + \hat{\gamma}_3\left(
i\partial_z -\frac{1}{\beta^2
\gamma^2}A_3\right) + \hat{\vec{\gamma}}_\perp (i \vec{\nabla}_\perp + i {\rm
grad}_\perp \ln \phi )-m \right]\tilde{\psi}\\
&&=0\end{aligned}$$ In the last step we introduced $\tilde{\psi}=\psi/\phi$.
In the ultrarelativistic limit terms of the order $1/\gamma^2$ are neglected. We end up with a Dirac equation coupled to a purely transverse vector potential exhibiting a Heaviside step function dependence $\sim \theta
(t-z)$. These properties are essential for our considerations.
Ultrarelativistic limit of the potential {#apotlim}
========================================
In this section we want to discuss the limit $\beta \to 1$ of the potential (\[lwpot\]). From section (\[1iontrafo\]) we expect the asymptotic form of the potential to be $$\lim_{\beta \to 1}\frac{-1}{\sqrt{(z-t)^2+\frac{x_\perp^2}{\gamma^2}}} =
\delta (z-t) \ln (x_\perp^2) + {\cal C}'$$ $t$ integration of (\[lwpot\]) determines ${\cal C}'$ $$\label{cprime}
{\cal C}'=\frac{-1}{|z-t|}- \delta (z-t)\ln (\gamma^2)$$ where we had to require $\gamma \gg x_\perp/|z-\beta t|$.\
An attempt to derive the limit by means of Fourier transformation of (\[lwpot\]) with respect to $z$ was presented in [@Jackiw]. The Fourier transform reads $$\begin{aligned}
\int dz e^{i\omega z} \frac{1}{\sqrt{(z-\beta t)^2+\frac{x_\perp^2}{\gamma^2}}}
&=& e^{i\omega \beta t}\int dz e^{i\omega z}
\frac{1}{\sqrt{z^2+\frac{x_\perp^2}{\gamma^2}}}\nonumber \\
&=& 2e^{i\omega \beta t} K_0\left(\frac{\omega
x_\perp}{\gamma}\right)\nonumber \\
\label{FTlim}
&\stackrel{\gamma\to
\infty}{\longrightarrow}& -2e^{i\omega t}\ln
\left(\frac{\omega x_\perp e^C}{2 \gamma}\right)\end{aligned}$$ The quantity $C$ here denotes Euler’s constant. The inverse Fourier transformation of this expression yields $$\label{arbilim}
\lim_{\beta \to 1}\frac{1}{\sqrt{(z-t)^2+\frac{x_\perp^2}{\gamma^2}}}
=\frac{1}{|z-t|} + g(x_\perp)\delta (z-t)$$ The coefficient $g(x_\perp)$ of the delta distribution in this result is not uniquely specified. Naive application of the textbook formula [@Lighthill] $$\label{lighthillf}
\int \frac{dk}{2\pi} \ln |k| e^{ikx} = -\frac{1}{2|x|}$$ would give $g(x_\perp)=2[\ln(x_\perp/2\gamma)+C]$, but (\[lighthillf\]) is valid only up to arbitrary multiples of $\delta (x)$. The validity of (\[FTlim\]), however, as well demands the condition $\gamma \gg \omega x_\perp$.
It is possible to find a gauge transformation that removes both, the long-range potential $1/|z-t|$ as well as $\delta (z-t)\ln (\gamma^2)$ (\[cprime\]). This is fulfilled with the gauge transformation [@Aichelburg] $$\label{gaugeob2}
\psi'=e^{iZ\alpha \ln\left(\gamma (z-t) +
\sqrt{1+\gamma^2(z-t)^2}\right)}\psi$$ The gauge-transformed potential reads $$\label{gpot}
A_0'=-\frac{Z\alpha\gamma}{\sqrt{\gamma^2(z-\beta t)^2 + x_\perp^2}} +
\frac{Z\alpha\gamma}{\sqrt{\gamma^2(z-\beta t)^2 + 1}}$$ and has the ultrarelativistic limit $$\label{potlim}
\lim_{\beta\to 1}A_0'=+\delta (z-t) \ln (x_\perp ^2)$$
The appearance of the logarithm follows immediately from the inhomogeneous Maxwell equations in the Lorentz gauge that reduces to a two dimensional Poisson equation in the limit $\beta\to 1$.
This gauge transformation has the advantage to yield a short-range potential that allows for asymptotic plane wave solutions. For this reason it was used to obtain a faster convergence in coupled channel calculations [@Eichler].
Solution of the Dirac equation with two discontinuities {#asolu2ion}
=======================================================
The Dirac equation for an electron moving in the field of two ultrarelativistic colliding ions $A$ and $B$ reads $$\label{de2ion}
\left[\frac{1}{2}\left(\hat{\gamma}_-i\partial_{\tau_-}+
\hat{\gamma}_+i\partial_{\tau_+}\right) +i\hat{\vec{\gamma}}_\perp
\cdot \vec{\nabla}_\perp -m -\frac{1}{2}\hat{\gamma}_-\delta (\tau_-)
V^A_\perp -\frac{1}{2}\hat{\gamma}_+\delta (\tau_+) V^B_\perp \right] \psi=0$$ where we used light cone variables $\tau_\pm=(t\pm z)/2$. One directly finds that $\hat{\gamma}_- \psi$ is discontinuous at $\tau_-$ through the action of ion $A$ and $\hat{\gamma}_+\psi$ is discontinuous at $\tau_+=0$ through the action of ion $B$, respectively.
We introduce $\psi_\pm=(1\pm \hat{\alpha}_z)\psi$ and use $2\psi=\psi_-+\psi_+$ to formulate the problem as follows $$\left(i\partial_{\tau_+} +i\hat{\vec{\alpha}}_\perp\cdot \vec{\nabla}_\perp
-\gamma_0m -\delta (\tau_+) V^B_\perp
\right)\psi_+ +\left(i\partial_{\tau_-}
+i\hat{\vec{\alpha}}_\perp\cdot \vec{\nabla}_\perp
-\gamma_0m -\delta (\tau_-) V^A_\perp\right)\psi_-=0$$ where (\[de2ion\]) has been multiplied by $2\hat{\gamma}_0$. This we rewrite as $$\begin{aligned}
&&\left(i\partial_{\tau_+} -\delta (\tau_+) V^B_\perp
\right)\psi_+ +\left(i\hat{\vec{\alpha}}_\perp\cdot \vec{\nabla}_\perp
-\gamma_0m\right)\psi_-\nonumber \\
\label{delambda}
&&=-\left(i\partial_{\tau_-}-\delta (\tau_-)
V^A_\perp\right)\psi_- -\left(i\hat{\vec{\alpha}}_\perp\cdot
\vec{\nabla}_\perp
-\gamma_0m\right)\psi_+\end{aligned}$$ By using the standard representation of Dirac matrices and simply rearranging the four equations (\[delambda\]) one obtains $$\begin{aligned}
\nonumber
&&\left[\left(\begin{array}{rr}1\!{\rm l}&0\\0&0\end{array}\right)
\left(i\partial_{\tau_+}
-\delta (\tau_+) V^B_\perp
\right)
+\left(\begin{array}{rr}0&0\\0&1\!{\rm l}\end{array}\right)
\left(i\partial_{\tau_-}-\delta (\tau_-)
V^A_\perp\right)-
m\left(\begin{array}{rr}0&1\!{\rm l}\\1\!{\rm l}&0\end{array}\right)\right.\\
&&\left.+i\partial_x
\left(\begin{array}{rr}0&-\sigma_y\\ \sigma_y&0\end{array}\right)
+i\partial_y\left(\begin{array}{rr}0&-i\sigma_x\\i\sigma_x&0\end{array}\right)
\right]\tilde{\psi}=0\\
\nonumber
{\rm where}
&&\tilde{\psi}=\left(\begin{array}{c}\psi_1+\psi_3\\ \psi_2-\psi_4\\
\psi_1-\psi_3\\ \psi_2+\psi_4\end{array}\right)\end{aligned}$$ corresponding to an isomorphic linear transformation [@Segev] with the matrix $$\label{segevtrafo}
\Lambda=\left(\begin{array}{rr}
1\!{\rm l}&\sigma_z\\1\!{\rm l}&-\sigma_z\end{array}\right)$$ Since $\Lambda$ is a bijection, each side of (\[delambda\]) has to be zero. Off the light fronts we therefore have the two equations $$\begin{aligned}
\label{psi+}
i\partial_{\tau_+}\psi_+&=&(i\hat{\vec{\alpha}}_\perp\cdot \vec{\nabla}_\perp
-\gamma_0m)\psi_-\\
i\partial_{\tau_-}\psi_-&=&(i\hat{\vec{\alpha}}_\perp\cdot \vec{\nabla}_\perp
-\gamma_0m)\psi_+\end{aligned}$$ According to (\[psiout\]) the discontinuities at the light fronts are described by $$\psi_-(\tau_-=0^+)=\phi^A(x_\perp)\psi_-(\tau_-=0^-)\;\;,\;\;\;
\psi_-(\tau_+=0^+)=\phi^B(x_\perp)\psi_+(\tau_-=0^-)$$ $\phi^A$ and $\phi^B$ are defined by (\[phidef\]) using the scalar parts of the potentials of the ions $A$ and $B$. Let us study the spinor $\psi_+$, evaluated at the surface $\tau_+=0^+$: $$\label{taupgr0}
\psi_+(\tau_+=0^+)=\phi^B \frac{i\hat{\vec{\alpha}}_\perp\cdot
\vec{\nabla}_\perp
-\gamma_0m}{p_-}\psi_-(\tau_+=0^-)$$ In the region $\tau_->0$ the electron already has interacted with ion $A$ and we can write $$\label{taupmgr0}
\psi_+(\tau_+=0^+,\tau_->0)
=\phi^B \frac{i\hat{\vec{\alpha}}_\perp\cdot
\vec{\nabla}_\perp
-\gamma_0m}{p_-}\phi^A (1-\alpha_z)\psi_p$$ where $\psi_p$ is the incoming plane wave at momentum $p$. This relation also can be obtained imeediately from (\[mtrafomi\]) and (\[psi+\]) for $\tau_+$ and $\tau_->0$. The operator $i\partial_+$ in (\[psi+\]) has been replaced by its eigenvalue $p_-$, the incoming negative light cone momentum. This is possible since $p_-$ is conserved in the interaction with ion $A$. The expansion of $\psi_+(\tau_+=0^+)$ in the plane-waves basis reads $$\psi_+(\tau_+=0^+,\tau_->0)
=\int\frac{dp'_+d^2p'_\perp}{(2\pi)^3}B(p',p)e^{-ip'_+\tau_-
+i\vec{p'}_\perp\cdot \vec{x}_\perp}u(p')$$ where we substituted $d^3p' \to dp'_+d^2p'_\perp$ [@Meggiolaro]. According to (\[taupgr0\]) the expansion coefficients are $$B(p',p)=\int_0^\infty d\tau_-\int d^2x_\perp e^{ip'_+\tau_-
-i\vec{p'}_\perp\cdot \vec{x}_\perp}\phi^B
\overline{u}(p')\frac{i\hat{\vec{\alpha}}_\perp\cdot \vec{\nabla}_\perp
-\gamma_0m}{p_-}\psi_-(\tau_+=0^-,\tau_->0)$$ In the region $\tau_->0$, $\tau_+<0$ the wave function $\psi_-$ is a freely propagating wave packet with a fixed light cone momentum $p_-$ and a superposition of transverse momenta $p_\perp$. The mass shell condition requires $p_-p_+=\vec{q}^2_\perp +m^2$. In this way $\psi_-(\tau_+<0,\tau_->0)$ can be obtained from $\psi_-(\tau_+<0,\tau_-=0^+)$. We have $$\psi_-(\tau_->0)=
\int\frac{d^2q_\perp}{(2\pi)^2}
e^{-i\frac{q_\perp^2+m^2}{p_-}\tau_--ip_-\tau_++
i\vec{q}_\perp\cdot \vec{x}_\perp}
\int d^2x'_\perp e^{i\vec{x'}_\perp\cdot
(\vec{p}_\perp-\vec{q}_\perp)}\phi^A
(1-\hat{\alpha}_z)u(p)$$ which leads to $$\begin{aligned}
B(p',p)&=&i\int \frac{d^2q_\perp}{(2\pi)^2} \int d^2x'_\perp
e^{i\vec{x'}_\perp\cdot (\vec{q}_\perp-\vec{p'}_\perp)}\phi^B\int d^2x_\perp
e^{i\vec{x}_\perp\cdot (\vec{p}_\perp-\vec{q}_\perp)}\phi^A\nonumber \\
\label{bexp}
&&\overline{u}(p')
\frac{\hat{\vec{\alpha}}_\perp\vec{q}_\perp -\gamma_0m}
{p'_+p_--q_\perp^2-m^2+i\epsilon}(1-\hat{\alpha}_z)u(p)\end{aligned}$$ Note, that the lower bound of the $\tau_-$ integration is 0, since we inserted the expression of $\psi_-$ for $\tau_->0$.
Together with the corresponding term for the reverse order of interactions with the two ions, (\[bexp\]) is the $S$ matrix for an electron scattered at the light fronts, first derived by Segev and Wells [@Segev] in an elegant way using the transformation (\[segevtrafo\]).
If both ions $A$ and $B$ move on positive light cones separated by the spatial distance $a\vec{e}_z$ (see section \[crystal\]), we obtain with (\[mtrafomi\]) for the interacting part of the spinor $\psi$ $$\begin{aligned}
\psi_+(\tau_+-a/2=0^+)&=&\phi^A\psi_+(\tau_+-a/2=0^-)\nonumber \\
&=&\phi^A\int\frac{d^2q_\perp}{(2\pi)^2}
e^{-i\frac{q_\perp^2+m^2}{p_+}\frac{a}{2}-ip_+\tau_-
+i\vec{q}_\perp\cdot \vec{x}_\perp}\nonumber \\
&&\int d^2x'_\perp e^{i\vec{x'}_\perp\cdot
(\vec{p}_\perp-\vec{q}_\perp)}\phi^B
(1+\hat{\alpha}_z)u(p)\end{aligned}$$ The expansion of $\psi_+$ in plane waves at the point $\tau_+=a/2+0^+$ yields the $S$ matrix of this process in momentum space $$\begin{aligned}
S(p',p)&=&2\pi \delta(p'_+-p_+)i\int \frac{d^2q_\perp}{(2\pi)^2}
e^{i\left(-\frac{q_\perp^2+m^2}{2p'_+}+\frac{p'_-}{2}\right)a}\nonumber \\
&& \int d^2x_\perp e^{i\vec{x}_\perp\cdot (\vec{q}_\perp-\vec{p'}_\perp)}
\phi^A \int d^2x_\perp
e^{i\vec{x}_\perp\cdot (\vec{p}_\perp-\vec{q}_\perp)}\phi^B \overline{u}(p')
(1+\hat{\alpha}_z)u(p)\end{aligned}$$ in accordance with section \[crystal\].
The photon propagator at high collision energies {#approp}
================================================
The four-dimensional Fourier transform of the potential (\[lwpot\]) reads $$\label{ftpot}
\int d^4x e^{ikx}\frac{-Z\alpha \gamma}{\sqrt{\gamma^2(z-\beta t)^2 +
\vec{x}^2_\perp}}=-(2\pi)^2 Z \alpha \delta (k_0-\beta k_3)
\frac{2}{\left(\frac{k_3}{\gamma}\right)^2 + k_\perp^2}$$ which has the following low and high-velocity limits $$\begin{aligned}
\lim_{\beta \to 0}&=&-(2\pi)^2 Z \alpha \delta (k_0)\frac{2}{|\vec{k}|^2}\\
\label{helimft}
\lim_{\beta \to 1}&=&-(2\pi)^2 Z \alpha \delta (k_0-k_3)\frac{2}{k^2_\perp}\end{aligned}$$
The last expression reflects the observation, that in the high-energy limit the longitudinal components $k_-$ and $k_+$ of the photon momentum can be dropped.
After having performed the gauge transformation (\[gaugeob2\]) and taken the limit $\gamma \to \infty$, the potential to be transformed is expression (\[potlim\]). Grignani and Mintchev [@Grignani] have shown, that it is wrong to identify the Fourier transform of (\[potlim\]) with (\[helimft\]) or with the regulated expression $1/(k_\perp^2 + \mu^2)$ with a regulating mass inserted by hand.
Calculating the time integral of $A_0$ in the eikonal expression and using (\[ftpot\]) one finds $$\int^\infty_{-\infty}dt A_0 =Z\alpha \ln (x_\perp^2) +Z\alpha
\lim_{\mu\to 0} \ln \mu^2$$ with $1/2e^C$ absorbed in $\mu$ as in [@Jackiw]. The term $\lim_{\mu\to 0} Z\alpha\ln \mu^2$ is the term ${\cal C}$ in section \[1iontrafo\] and is completely different from $\mu$ in eq (7) of [@Jackiw]!
One may attempt to calculate the two-dimensional Fourier transform of the logarithm from a Taylor expansion in powers of $iZ\alpha$ of the Fourier transform of the $T$ matrix (i.e. its transverse part) which is given by the following closed expression $$\begin{aligned}
{\cal T}(k_\perp)&=&
\left(\frac{4}{k^2_\perp}\right)^{1-i\alpha Z}\Gamma^2(1-i\alpha Z)
\sin(\pi i\alpha Z)\\
&=&\pi \frac{\Gamma(1-i\alpha Z)}{\Gamma(i\alpha
Z)}\left(\frac{4}{k^2_\perp}\right)^{1-i\alpha Z} \end{aligned}$$ The first terms of the Taylor expansion read $$\begin{aligned}
{\cal T}(k_\perp)&\approx& +4\pi i\alpha Z
\frac{1}{k^2_\perp} + 4\pi (i\alpha Z)^2 \frac{\ln
(k^2_\perp/4)+C}{k^2_\perp}\nonumber \\
\label{ttaylor}
&&+ 2\pi
(i\alpha Z)^3 \frac{\ln^2(k^2_\perp/4)+ 4C\ln(k^2_\perp/4)+4C^2}
{k^2_\perp} + \dots\end{aligned}$$ The second term would then correspond to the desired Fourier transform (times $(iZ\alpha)$), the third term correspondingly to $(iZ\alpha)^2$ times the Fourier transform of the square of the transverse part of the potential (\[potlim\]) that has to be compared with the result of Torgerson [@Torgerson].
This is, however, not justified, since the linearity of the Fourier transform is only guaranteed for finite sums and causes problems when applied to infinite series like the Taylor expansion of the exponential function. To get the correct result for the exact two dimensional euclidean photon propagator, the limit $iZ\alpha \to 0$ in $$\label{2dimpott}
\int d^2x_\perp e^{-i\vec{k}_\perp\vec{x}_\perp}\ln x_\perp^2
=\lim_{iZ\alpha\to 0}\frac{d}{d(iZ\alpha)}\left(\pi \frac{\Gamma(1-i\alpha
Z)}{\Gamma(i\alpha
Z)}\left(\frac{4}{k_\perp^2}\right)^{1-i\alpha Z}
\right)$$ has to be taken after having integrated the result with a test function. Performing the limit without this precaution gives the wrong result (\[helimft\]).
Another form of the correct Fourier transform was derived in [@Ferrari]. We obtain the equivalent form from the gauged potential $A'_0$ in (\[gpot\]). Since $$\label{lnpotdef}
-\ln x_\perp^2=\lim_{\gamma\to \infty}\int_{-\epsilon}^\epsilon dt \left(
\frac{\gamma}{\sqrt{\gamma^2t^2+x_\perp^2}}-
\frac{\gamma}{\sqrt{\gamma^2t^2+1}} \right)$$ ($\epsilon$ is arbitrary but finite) and $$\int dtd^2x_\perp e^{i\omega t-i\vec{k}_\perp\cdot\vec{x}_\perp} \left(
\frac{\gamma}{\sqrt{\gamma^2t^2+x_\perp^2}}-
\frac{\gamma}{\sqrt{\gamma^2t^2+1}} \right)=4\pi\left(
\frac{1}{\left(\frac{\omega}{\gamma}\right)^2+k_\perp^2}-2\pi
K_0\left(\frac{\omega}{\gamma}\right)\right)$$ we find by direct substitution $$-\int d^2x_\perp e^{-i\vec{k}_\perp\vec{x}_\perp}\ln x_\perp^2
=\lim_{\lambda \to 0}4\pi\left(\frac{1}{k_\perp^2+ \lambda^2}+\pi\delta^2
(k_\perp)\ln\left(\frac{\lambda^2}{\mu^2}\right)\right)$$ with $\lambda =
\omega/\gamma$, $\mu=2/e^C$. The limit has to be treated in the same way as in (\[2dimpott\]).
When naively taking the limit $iZ\alpha \to 0$ immediately, by chance one obtains the high energy limit of the ungauged potential.
[99]{} H.D.I. Abarbanel, C. Itzykson, Phys. Rev. Lett. 23 (1969) 53 R. Torgerson, Phys. Rev. 143 (1966) 1194 R. Jackiw, D. Kabat, M. Ortiz, Phys. Lett. B 277 (1992) 148 G. ’t Hooft, Phys. Lett. B 198 (1987) 61 S.J. Chang, S.K. Ma, Phys. Rev. 188 (1969) 2385; S.J. Chang, S.K. Ma, Phys. Rev. Lett. 22 (1969) 1336 S.J. Chang, P.M. Fishbane, Phys. Rev. D 2 (1970) 1104 B. Segev, J.C. Wells, Phys. Rev. A 57 (1998) 1849 C. Bottcher, M.R. Strayer, Phys. Rev. D 39 (1989) 1330 D.C. Ionescu, J. Eichler, Phys. Rev. A 48 (1993) 1176 R.M. Miura, J. Math. Phys. 9 (1968) 1202 P.C. Aichelburg, R.U. Sexl, Gen. Rel. Grav. 2 (1971) 303 H.M. Fried, Functional Methods and Models in Quantum Field Theory, MIT Press (1972) E. Meggiolaro, Phys. Rev. D 53 (1996) 3835 C. Itzykson, J.B. Zuber, Quantum Field Theory, McGraw-Hill (1985) A.J. Baltz, Phys. Rev. Lett. 78 (1997) 1231 G. Grignani, M. Mintchev, Phys. Rev. D 38 (1988) 3163 R. Ferrari, Nuovo Cimento A 19 (1974) 204 U. Eichmann, J. Reinhardt, W. Greiner, manuscript in preparation M.J. Lighthill, Introduction to Fourier analysis and generalized functions, Cambridge Univ. Press (1959) N. Toshima, J. Eichler, Phys. Rev. A 42 (1990) 3896
[^1]: The effect of the potential (\[potlimc\]) also can be described within the Aichelburg-Sexl metric. Two field-free regions of space-time meet at $z=t$, such that (the superscripts $<$ and $>$ denote $t>z$ and $t<z$, respectively) $$\begin{aligned}
\label{xtrgr}x_\perp^>&=&x_\perp^<\\
\label{zgr}z^>&=& z^<-Z\alpha \ln x_\perp^2\\
\label{tgr}t^>&=& t^<-Z\alpha \ln x_\perp^2\end{aligned}$$ The result (\[psiout\]) is then easily obtained by simply substituting (\[xtrgr\])-(\[tgr\]) into the plane wave at $t>z$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present an anti-ferromagnetically ordered ground state of Na$_{2}$IrO$_{3}$ based on density-functional-theory calculations including both spin-orbit coupling and on-site Coulomb interaction $U$. We show that the splitting of $e_{g}''$ doublet states by the strong spin-orbit coupling is mainly responsible for the intriguing nature of its insulating gap and magnetic ground state. Due to its proximity to the spin-orbit insulator phase, the magnetic ordering as obtained with finite $U$ is found to exhibit a strong in-plane anisotropy. The phase diagram of Na$_{2}$IrO$_{3}$ suggests a possible interplay between spin-orbit insulator and Mott anti-ferromagnetic insulator phases.'
author:
- Hosub
- Heungsik
- Hogyun
- 'Choong H.'
- Jaejun
title: 'Mott Insulating Ground State and its Proximity to Spin-Orbit Insulators in Na$_{2}$IrO$_{3}$'
---
Recently, the role of spin-orbit coupling (SOC) has attracted great attention in many fields of condensed matter physics. In multiferroic materials, for example, SOC combined with a large electron-lattice interaction has been suggested to be responsible for the multiferroic behavior which exhibit both non-collinear magnetic ordering and lattice polarization [@Kimura03; @Hur04]. SOC is also indispensable to anomalous Hall and spin Hall effects where Hall and spin Hall currents are generated by an external electric field, respectively [@PhysRev.95.1154; @ShuichiMurakami09052003; @PhysRevLett.92.126603]. In particular, the quantum spin Hall effect has led to the notion of topological insulators, new states of quantum matter [@zhang09; @xia09]. While they have bulk energy gaps generated by the SOC, topological insulators are characterized by the presence of gapless surface states which are protected by time-reversal symmetry [@kane:146802].
Another manifestation of strong SOC combined with on-site Coulomb interactions is the $j_{\mathrm{eff}}$=1/2 Mott insulator discovered in Sr$_{2}$IrO$_{4}$, one of the 5$d$ transition-metal oxides [@kim:076402; @moon:226402]. The novel spin-orbit integrated state with $j_{\mathrm{eff}}$=1/2 arises from the combined action of both strong SOC and intermediate on-site Coulomb interactions within the Ir 5$d$ $t_{2g}$ manifold. In addition, there has been a theoretical proposal on the room temperature quantum spin Hall effect in Na$_2$IrO$_3$ based on the $j_{\mathrm{eff}}$=1/2 physics [@shitade-2008], where the honeycomb lattice consisting of edge-shared IrO$_6$ octahedra in each Ir-O layer was considered to be an ideal realization of the Kane-Mele model, where hopping integrals between the $j_{\mathrm{eff}}$=1/2 states at the Fermi level was assumed to be an essential ingredient for the quantum spin Hall effect [@kane:146802; @PhysRevLett.95.226801]. Since the crystal structure and local environment of Ir atoms in Na$_2$IrO$_3$ are different from those of Sr$_{2}$IrO$_{4}$, however, it is necessary to clarify the electronic and magnetic structures of the Ir 5$d$ manifold in this Na$_2$IrO$_3$ compound with hexagonal lattice.
In this paper, we present novel electronic structure and magnetic properties of Na$_2$IrO$_3$ by carrying out density-functional-theory (DFT) calculations including both SOC $\lambda_{\mathrm{SO}}$ and on-site Coulomb interaction. We observe that a new form of spin-orbit coupled states emerges from the $e_{g}'$ doublet states near the Fermi level ($E_{\mathrm{F}}$) and determines the intriguing nature of its insulating gap. With an effective on-site Coulomb interaction parameter $U=2.0$ eV, the ground state of Na$_2$IrO$_3$ is found to be an antiferromagnetic (AFM) insulator with the ordered moments lying down within the honeycomb lattice of Ir atoms. The large splitting of the $e_{g}'$ doublet by the strong SOC is related to the strong in-plane anisotropy of magnetic ordering. Considering the role of SOC, we propose a phase diagram in the $\lambda_{\textrm{SO}}$–$U$ parameter space which features a phase boundary between AFM Mott insulators and SO insulators. By estimating the exchange couplings between neighboring Ir atoms, we suggest a possible frustration of magnetic ordering in its ground state, which is consistent with a recent experiment [@Takagi].
In order to examine the effects of both SOC and on-site Coulomb interaction on the electronic structure of Na$_2$IrO$_3$, it is necessary to treat both SOC and $U$ on an equal footing in the description of Ir 5$d$ states. To identify the role of each contribution as well as the interplay between them, we carried out DFT calculations within the local-density approximation (LDA), LDA including SOC (LDA+SO), and LDA+$U$ including SOC (LDA+$U$+SO) respectively. For the calculations, we used the DFT code, OpenMX [@openmx], based on the linear-combination-of-pseudo-atomic-orbitals method [@PhysRevB.67.155108], where both the LDA+$U$ method [@han:045110] and the SOC contribution were included via a relativistic $j$-dependent pseudo-potential scheme in the non-collinear DFT formalism. Double valence and single polarization orbitals were used as basis sets, which were generated by a confinement potential scheme with cutoff radii of 7.0, 7.0 and 5.0 a.u. for Na, Ir, and O atoms respectively. We used a (14$\times$14$\times$14) **k**-point grid for the k-space integration.
Up to our knowledge there is no crystal structure data for Na$_2$IrO$_3$ published yet. Thanks to the preliminary information provided by Takagi[@Takagi], we were able to construct a minimal unit-cell containing two formula units based on the hexagonal structure of Na$_{2}$RuO$_{3}$ [@Kailash04], a sibling compound of Na$_2$IrO$_3$. The crystal structure of Na$_2$IrO$_3$ can be viewed as an alternate stacking of (Ir$_{2/3}$Na$_{1/3}$)O$_2$ and Na layers. Edge-shared IrO$_6$ octahedra form a honeycomb lattice of Ir atoms. Na atoms are placed at the center of each hexagon. Upper and lower triangle oxygens are rotated by 3.5$^{\circ}$ to shorten the Ir-O distance. The positions of atoms in the unit cell were determined through the full structural optimization by the LDA calculations with 0.5$\times 10^{-3}$ Hatree/[Å]{} of force criterion. There is a possible stacking disorder in the types of the Na-layers relative to the (Ir$_{2/3}$Na$_{1/3}$)O$_2$ layers. We have checked the effect of different stacking sequences and observed a negligible change in the energy dispersions. Since the basic electronic structure is dominated by the in-plane Ir-O hybridization and remains intact regardless of the stacking sequence, we will focus on the electronic structure without structural disorder hereafter.
We investigated the electronic and magnetic structures of Na$_2$IrO$_3$ by performing LDA, LDA+SO, and LDA+$U$+SO calculations. Calculated electronic band structure near $E_{\mathrm{F}}$ are shown in Fig. \[fig:1\]. The LDA band structure in Fig. \[fig:1\](a) features the Ir 5$d$ bands of $e_{g}$ and $t_{2g}$ components separated by a large cubic crystal field $\Delta_{\textrm{cubic}}\sim$ 4 eV. While narrow $e_{g}$ bands are located at 3 eV above $E_{\mathrm{F}}$, the top of $t_{2g}$ bands are pinned at $E_{\mathrm{F}}$ and spread out to -2.0 eV below $E_{\mathrm{F}}$. Due to the extended nature of Ir 5$d$ orbitals, there are large contributions to the band structure from both the indirect hopping via the Ir 5$d$-O 2$p$ hybridization and the direct hopping between the neighboring Ir 5$d$ orbitals. From the tight-binding analysis [@Choong], even the next-nearest-neighbor hopping terms through oxygen and sodium atoms make significant contributions to the LDA band structure.
The trigonal crystal field ($\Delta_{\textrm{trigonal}}$) splits the $t_{2g}$ bands into $a_{1g}$ and $e_{g}'$ states. In addition, there is a strong hybridization between neighboring Ir 5$d$ orbitals which gives rise to the bonding and anti-bonding of $e_{g}'$ orbitals. The bonding and anti-bonding doublet states consist of $e_{g}'$ orbital pairs of two Ir atoms per unit cell. At the $\Gamma$ point of the LDA band structure, the $e_{g}'$ anti-bonding states, to be called by $e_{AB}$, are close to $E_{\mathrm{F}}$ while the $e_{g}'$ bonding states, to be called by $e_{B}$, are at about $-$0.8 eV. The $a_{1g}$ bands located at $-$1 eV have a negligible effect of the hybridization between neighboring Ir atoms but show a relatively large $c$-axis dispersion, which may be derived from the character of $a_{1g}$ orbitals pointing toward the Na atoms in the next layers. Here it is noted that the appearance of the $e_{AB}$ doublet at $E_{\mathrm{F}}$ in the LDA band structure of Na$_{2}$IrO$_{3}$ is in contrast to the presence of almost degenerate $t_{2g}$ state in Sr$_{2}$IrO$_{4}$ which serves as a basis for the $j_{\mathrm{eff}}$=1/2 state when SOC is introduced [@kim:076402].
![(Color online) Electronic band structures of Na$_2$IrO$_3$ within (a) LDA, (b) LDA+SO, and (c) LDA+$U$+SO schemes. Green, red, and blue colored energy dispersions in (a) are indicating $e_g$, $e_g'$, and $a_{1g}$ bands respectively, induced by the largest cubic and the next largest trigonal crystal fields.[]{data-label="fig:1"}](Fig1.pdf){width="8cm"}
In the LDA band structure, the doubly degenerate $e_{AB}$ states form a narrow band and cross $E_{\mathrm{F}}$. The introduction of SOC breaks the degeneracy of $e_{AB}$ by preserving the time-reversal symmetry so that the $e_{AB}$ bands split off over the whole Brillouin zone (BZ) as shown in Fig. \[fig:1\](b). Despite the split of $e_{AB}$ bands, the LDA+SO band structure is still metallic with a small electron pocket at the $A$ point and hole pockets off the $k_{c}=0$ plane near $M$. From the tight-binding analysis of the Na$_{2}$IrO$_{3}$ band structure [@Choong], we obtained $\Delta_{\textrm{trigonal}}\sim$ 0.6 eV, which is larger than the SOC parameter $\lambda_{\textrm{SO}}\sim$ 0.4 eV [@kim:076402; @PhysRevB.13.2433]. Thus the band structure of Na$_{2}$IrO$_{3}$ near $E_{\mathrm{F}}$ is characterized by the bonding $e_{B}$ and anti-bonding $e_{AB}$ states with $\Delta_{\textrm{cubic}}>
\Delta_{\textrm{trigonal}}>\lambda_{\textrm{SO}}$. Since $\Delta_{\textrm{trigonal}} > \lambda_{\textrm{SO}}$, however, the $e_{AB}$ character of the bands are maintained. Contrary to the layered perovskite Sr$_2$IrO$_4$ system, where the SOC entangles almost degenerate $t_{2g}$ orbitals with spin states and produces the spin-orbit integrated $j_{\mathrm{eff}}$=1/2, the strong trigonal field in Na$_{2}$IrO$_{3}$ suppresses the mixing of $a_{1g}$ and $e_{g}'$ states. Instead, the SOC acting on the $e_{g}'$ subspace plays a role of effective Zeeman coupling, the details of which will be discussed later. The presence of the effective Zeeman coupling is manifested in the parallel splitting of $e_{AB}$ and $e_{B}$ bands.
Similarly to the case of Sr$_{2}$IrO$_{4}$, both the on-site Coulomb interaction and the SOC are expected to be important in the description of Ir 5$d$ states. The LDA+$U$+SO band structure shown in Fig. \[fig:1\](c) was calculated with an effective $U=2.0$ eV, which was found to be consistent with angle-resolved photoemission and optical spectroscopy experiments [@kim:076402]. As a result of the combined action of both on-site Coulomb interaction and SOC, a small band gap arises between the SO-split $e_{AB}$ bands. Two $e_{AB}$ bands form valence and conduction bands with nearly the same dispersion above and below $E_{\mathrm{F}}$, respectively. Contrary to the non-magnetic metallic solution of the LDA and LDA+SO calculations, the LDA+$U$+SO solution predicts an AFM ordering with local magnetic moments lying within the $ab$ plane. The magnitude of total moment is 0.47 $\mu_B$ per each Ir atom, which is decomposed into the spin moment of 0.12 $\mu_B$ and the orbital moment of 0.35 $\mu_B$.
![Electronic band structures from LDA+SO calculations with the scaling factors of SOC strength $\lambda_{\textrm{SO}}/\lambda_0$ are (a) 0.5, (b) 1.0, (c) 1.5, and (d) 2.0, where $\lambda_0$ is the SOI magnitude of a real Ir atom. Gap opens when $\lambda_{\textrm{SO}}/\lambda_0$ is increasing from 1.0 to 1.5.[]{data-label="fig:2"}](Fig2.pdf){width="8cm"}
Despite that the importance of both $U$ and $\lambda_{\mathrm{SO}}$, the nature of the insulating ground state of Na$_{2}$IrO$_{3}$ is quite distinct from that of Sr$_{2}$IrO$_{4}$. In Sr$_{2}$IrO$_{4}$, the $j_{\mathrm{eff}}$=1/2 degeneracy can not be lifted by the SOC and the Mott-Hubbard gap can be attained only when the on-site $U$ is introduced. Thus breaking the time-reversal symmetry is essential to get the insulating ground state of Sr$_{2}$IrO$_{4}$. In the case of Na$_{2}$IrO$_{3}$, however, the broken time-reversal symmetry is not required to acquire the insulating state. As shown in Fig. \[fig:1\](b), the SO-split $e_{AB}$ bands are separated over the whole BZ so that the increase of the SOC strength can enlarge the already present gap between two $e_{AB}$ bands. To probe this idea, we carried out DFT calculations by controlling the SOC strength, which can be achieved by changing the scaling factor when generating the $j$-dependent pseudo-potential [@openmx]. Calculated results for the scale factors $\lambda_{\mathrm{SO}}/\lambda_{0}$= 0.5, 1.0, 1.5, and 2.0 are shown in Fig. \[fig:2\]. Taking the original SOC in the real Ir atom as $\lambda_{0}$ as a reference, $\lambda_{\mathrm{SO}}/\lambda_{0}$= 1.5 was found to be enough to open a full insulating gap. We call these insulating ground states as spin-orbit (SO) insulators, which have energy gaps generated by the SOC. SO insulators have no local moment and preserve the time-reversal symmetry and thus are distinct from the Mott-Hubbard insulator.
To understand the origin of SO insulators, we consider the SOC matrix elements within the $e_g'$ subspace. Since the degenerate $e_g'$ states can be written by $$\begin{aligned}
\mid e_1'\rangle=\frac{1}{\sqrt3}(\mid d_{xy}\rangle+e^{\imath\theta}\mid d_{yz}\rangle
+e^{-\imath\theta}\mid d_{zx}\rangle) \nonumber \\
\mid e_2'\rangle=\frac{1}{\sqrt3}(\mid d_{xy}\rangle+e^{-\imath\theta}\mid d_{yz}\rangle
+e^{\imath\theta}\mid d_{zx}\rangle)\end{aligned}$$ where $\theta=2\pi/3$, the on-site SOC term becomes $$\langle \mathcal{H}_{\textrm{SO}}\rangle_{e_g'}=
\langle \lambda_{\textrm{SO}}\mathbf{L}\cdot\mathbf{S}\rangle_{e_g'}=\frac{\lambda_{\textrm{SO}}}{2}
\left(\begin{array}{c|c}
\hat{n}\cdot \vec{\sigma} & \\ \hline
& \; -\hat{n}\cdot \vec{\sigma}
\end{array}\right)$$ where the basis sets are $\mid e_g'\rangle\otimes\mid
S=\frac{1}{2}\rangle=\{ \mid e_1'\alpha\rangle ,\mid e_1'\beta\rangle,\mid
e_2'\alpha\rangle,\mid e_2'\beta\rangle\}$ and $\hat{n}$ is the unit vector along the $c$-axis, i.e., the \[111\] direction in the local coordinate of IrO$_6$ octahedron. This block-diagonal form comes from the fact that $\langle \mathbf{L} \rangle$ is simultaneously diagonalized within $e_g'$ manifold and its eigenvalues are $\hat{n}$ and $-\hat{n}$, respectively. Here the SOC terms in $e_g'$ act as an internal magnetic field perpendicular to the $ab$-plane. The internal field gives rise to an effective Zeeman splitting, but the field direction in the $e_{1}'$ component is opposite to that in the $e_{2}'$ component. Thus, the effective Zeeman coupling does not break the time-reversal symmetry and $|e_1'\alpha\rangle$–$|e_2'\beta\rangle$ and $|e_1'\beta\rangle$–$|e_2'\alpha\rangle$ remain as time-reversal partners. Since the $e_{AB}$ states are the anti-bonding combination of the $e_{g}'$ orbitals of neighboring Ir atoms, the splitting of $e_{AB}$ bands by the effective Zeeman coupling is proportional to the SOC strength as shown in Fig. \[fig:2\], especially at the $\Gamma$ point.
The ground states of Na$_{2}$IrO$_{3}$ with the large SOC strength are SO insulators. The band gaps are induced by the effective Zeeman coupling of the SOC within the $e_{g}'$ subspace. Their characters are different from other types of band insulators such as covalent or ionic ones. The Fermi level is placed between bonding and anti-bonding bands in covalent solids and between different ionic configurations in ionic solids. In SO insulators, the gap is not driven by bonding characters, but mainly related to the symmetry of the states at $E_{\mathrm{F}}$. In a sense that their band gaps are generated by the SOC, SO insulators share the same ground with topological insulators though it is necessary to prove the non-trivial topology of its ground state.
One important consequence of the SO insulating phase is the proximity of the AFM ground state to the SO insulator state. In the LDA+$U$+SO calculation, the AFM ordered local moments are aligned in the $ab$-plane. Due to the huge internal field along \[111\] direction, it is hard to break the time-reversal symmetry and to develop local magnetic moments along that direction. Thus, transverse magnetic moments which are perpendicular to the internal field can be easily developed. Strong magnetic anisotropy originated from the internal magnetic fields might be seen in magnetic susceptibility measurements.
![Phase diagram in the $\lambda_{\textrm{SO}}$–$U$ parameter space depicting four different phases from LDA+$U$+SO calculations with varying $U$ and $\lambda_{\textrm{SO}}$ values. Paramagnetic metallic phase appears near the origin, Mott insulator in the region of $U > 1$, and SO insulator in the region of $\lambda_{\textrm{SO}}>1$ and $U <
1$. The real ground state is located inside Mott insulating territory.[]{data-label="fig:3"}](Fig3.pdf){width="8cm"}
To elucidate the relation between SO insulator and AFM Mott insulator phases, we explored a possible phase diagram of Na$_{2}$IrO$_{3}$ in an extended $\lambda_{\textrm{SO}}$–$U$ parameter space and present the result in Fig. \[fig:3\]. When $U$ is small and $\lambda_{\textrm{SO}}/\lambda_{0}$ is less than 1.5, the ground state remains as a paramagnetic metal. When there is no SOC, i.e., $\lambda_{\mathrm{SO}} = 0$, a ferromagentic metallic phase develops in a narrow range of the parameter space with $\lambda_{\mathrm{SO}} = 0$ upto $U=5.0$ eV. This ferromagnetic state becomes unstable in the presence of the SOC. On the other hand, for the value of $U$ smaller than about 1 eV, the SO insulator phase emerges as a non-magnetic insulator. Since the band gap is induced by the effective Zeeman coupling of the SOC within the $e_{g}'$ subspace, the Kramers degeneracy of the valence states holds up and the time-reversal symmetry remains unbroken. For the finite $\lambda_{\mathrm{SO}}$, Mott insulating AFM states develop as $U$ becomes larger than about 1.0 eV. The difference between two insulating phases, i.e., the criterion for the boundary is the existence of local magnetic moments. The Mott insulating phase has AFM ordering where on-site Coulomb repulsion breaks the symmetry developing local moments during the correlation gap opens. Our LDA+$U$+SO calculation predicts that the real ground state of Na$_2$IrO$_3$ is located in the Mott AFM region with $U=2.0 \sim 3.0$ eV and $\lambda_{\textrm{SO}}/\lambda_0=1$. However, the strongly anisotropic nature of its AFM ordering originates from its proximity to the SO insulator phase.
![(Color online) Schematic drawing of the (Ir$_{2/3}$Na$_{1/3}$)O$_2$ plane and magnetic configuration of the AFM insulating ground state of Na$_2$IrO$_3$. Magnetic moments are ordered anti-ferromagnetically lying on the $ab$-plane due to the strong internal field along the $c$-axis. Not only the NN exchange $J$ (dotted arrow) but the NNN exchange $J'$ (dashed arrow) are significant and may give rise to magnetic frustration.[]{data-label="fig:4"}](Fig4.pdf){width="8cm"}
Another important aspect in Na$_2$IrO$_3$ is magnetic frustration indicated in large $\theta_{\textrm{CW}}/T_{\textrm{N}}$ ratio from susceptibility measurements [@Takagi]. To reveal the origin of frustration, we have estimated exchange interactions $J$ and $J'$ between nearest-neighbor (NN) and next-nearest-neighbor (NNN) Ir atoms respectively.(Fig. \[fig:4\]) Calculation scheme is based on the perturbation formalism. $$J_{ij}=\frac{1}{2\pi}\int^{\epsilon_{F}}d\epsilon \left[
\hat{G}^{\uparrow}_{ij} \hat{V}_j \hat{G}^{\downarrow}_{ji} \hat{V}_i \right],$$ where $\hat{G}$ is the one-particle Green’s function and $\hat{V}$ is on-site exchange interaction potential [@PhysRevB.70.184421]. The result is $J'/J=0.47$, which means that NN and NNN exchange coupling strength are comparable and they might be a source of frustration. Above result is mainly attributed to the extended nature of Ir 5$d$ orbitals. Large direct overlap between NN Ir atoms gives FM direct exchange interaction, competing with AFM superexchange from oxygen mediated hopping channels and finally reducing AFM exchange $J$. On the other hand, the NNN hopping integrals are not negligible that the NNN AFM interaction $J'$ can be comparable and frustrate long range AFM ordering.
In conclusion we have shown that the spin-orbit entangled $e_g'$ states under the strong internal Zeeman field driven by the SOC lead to an unusual band gap. The predicted AFM ground state is in close proximity to the SO insulator phase where the AFM ordering in Na$_2$IrO$_3$ becomes strongly anisotropic with quenched moments along the $c$-axis. The highly anisotropic AFM state in Na$_2$IrO$_3$ may serve as a model system for the two-dimensional XY model with frustrated exchange interactions. One may be able to drive a crossover between AFM and SO insulators through the modulation of structural parameters or chemical substitution, though we need more study on the role of SOC in the Mott AFM phase in connection with the topological nature of SO insulators.
We are grateful to H. Takagi for sharing information prior to publication. This work was supported by the KOSEF through the ARP (R17-2008-033-01000-0). We also acknowledge the support from KISTI under the Supercomputing Application Support Program.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The properties of underluminous type Ia supernovae (SNe Ia) of the 91bg subclass have yet to be theoretically understood. Here, we take a closer look at the structure of the dim SN Ia 2005bl. We infer the abundance and density profiles needed to reproduce the observed spectral evolution between $-$$6$d and $+$$12.9$d with respect to $B$ maximum. Initially, we assume the density structure of the standard explosion model W7; then we test whether better fits to the observed spectra can be obtained using modified density profiles with different total masses and kinetic energies. Compared to normal SNe Ia, we find a lack of burning products especially in the rapidly-expanding outer layers ($v$$\gtrsim$$15000$[km s$^{-1}$]{}). The zone between $\sim$$8500$ and $15000$[km s$^{-1}$]{} is dominated by oxygen and includes some amount of intermediate mass elements. At lower velocities, intermediate mass elements dominate. This holds down to the lowest zones investigated in this work. This fact, together with negligible-to-moderate abundances of Fe-group elements, indicates large-scale incomplete Si burning or explosive O burning, possibly in a detonation at low densities. Consistently with the reduced nucleosynthesis, we find hints of a kinetic energy lower than that of a canonical SN Ia: The spectra strongly favour reduced densities at $\gtrsim$$13000$[km s$^{-1}$]{} compared to W7, and are very well fitted using a rescaled W7 model with original mass ($1.38$${M_{\odot}}$), but a kinetic energy reduced by $\sim$$30\%$ (i.e. from $1.33$$\cdot$$10^{51}$erg to $0.93$$\cdot$$10^{51}$erg).'
author:
- |
Stephan Hachinger$^{1}$, Paolo A. Mazzali$^{1,2,3}$, Stefan Taubenberger$^{1}$,\
[Rüdiger Pakmor$^{1}$, Wolfgang Hillebrandt$^{1}$]{}\
$^1$Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85748 Garching, Germany\
$^2$Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy\
$^3$Istituto Nazionale di Astrofisica-OAPd, vicolo dell’Osservatorio 5, 35122 Padova, Italy
date: 'arXiv v2, 2009-11-02. The definitive version is available at [www.blackwell-synergy.com](http://www.blackwell-synergy.com) (MNRAS 399, 1238).'
title: |
Spectral analysis of the 91bg-like Type Ia SN 2005bl:\
Low luminosity, low velocities, incomplete burning.
---
supernovae: general – techniques: spectroscopic – radiative transfer
Introduction {#sec:introduction}
============
Type Ia supernovae (SNe Ia) play a key role in modern astrophysics. They are invaluable as distance indicators for cosmology (e.g. @per97 [@per99; @rie98; @ast06; @woo07]) because of the high accuracy with which the absolute luminosity of most SNe Ia can be inferred. The luminosity varies among different objects, but the variations correlate with distance-independent light-curve parameters such as the decline in magnitudes in the $B$-band within 15 days after $B$ maximum [@phi93]. Luminosity calibration techniques exploiting this fact are mostly applied to “normal” SNe Ia [@bra93] not showing poorly-understood peculiarities. These SNe supposedly emerge from a homogeneous sample of progenitors, which are thought to be C-O white dwarfs (WDs) accreting matter from a non-degenerate companion star (single-degenerate scenario).
In the single-degenerate paradigm, the smooth variations among “normal” SNe Ia [@bra93; @nug95] can be explained within a delayed-detonation scenario [@kho91]: an initially subsonic explosion (deflagration) undergoes a deflagration-detonation transition (DDT) and proceeds as a supersonic detonation afterwards. The efficiency and extent of burning in the initial deflagration may then vary from object to object, which affects the nucleosynthesis and causes the observed variability [@maz07].
Extremely sub- or superluminous SNe Ia [e.g. @fil92bg; @phi92; @lei93; @how06], on the other hand, are more difficult to explain. Here, progenitors deviating from the Chandrasekhar mass may play a role, or some explosions might result from a merger of two WDs (double-degenerate scenario). Progenitor systems producing peculiar SNe Ia might also produce some rather “normal” explosions, contaminating the sample of homogeneous explosions used for distance determination. Clarifying which explosion scenarios lead to SNe Ia at which rates is therefore important for supernova cosmology, but it will also be of value for other fields. Studies concerned with the binary progenitors and population synthesis [e.g. @rui09], observed supernova rates [@gre08] or the impact of supernovae on their surroundings [e.g. @sat07] will profit from understanding the origin of peculiar supernovae.
Thus motivated, we analyse the 91bg-like SN 2005bl [@tau08]. SNe of the 91bg subclass are dim and decline rapidly [e.g. @fil92bg; @lei93; @tur96; @gar04]. They were used, with other SNe, to infer the slope of the relation between luminosity and decline rate of SNe Ia [@phi93], but later it became clear that dim SNe decline even more rapidly than expected from a linear luminosity$-$decline-rate relation among normal SNe [@phi99; @tau08]. Spectroscopically, 91bg-like SNe show characteristic peculiarities, such as low line velocities around $B$ maximum [e.g. @fil92bg] and clear spectral signatures of [Ti [ii]{}]{}, indicating lower ionisation [@maz97bg]. All these properties together are consistent with a low mass of newly-synthesised [$^{56}$Ni]{}. To date, no elaborate explosion models have convincingly reproduced 91bg-like SNe Ia. Pure deflagration models show even lower expansion velocities than observed in these objects, especially when little [$^{56}$Ni]{} is produced [cf. @sah08]. Delayed-detonation models might explain 91bg-like objects within a unified scenario for SNe Ia [@maz07]. Yet, there are hints of qualitative differences. One example are the improved fits to spectra of SN 1991bg of @maz97bg, enabled by a reduction in ejecta mass and kinetic energy with respect to canonical values. Ultimately, only refined analyses of photometric and spectroscopic properties can constrain explosion models.
We use a spectral synthesis code to analyse the structure and abundance stratification of SN 2005bl, reproducing its observed spectral evolution. The “abundance tomography“ method [@ste05], which we use, exploits the fact that the optically thick region of the ejecta becomes smaller as time progresses. Thus, deeper and deeper layers contribute to spectrum formation. Modelling a time series of spectra, we infer the abundance profile from the outer envelope to as deep a layer as possible. We then test whether variations in mass or explosion energy are needed to explain the differences between spectra of normal and dim SNe Ia. This is done performing abundance tomography with various density profiles, and assessing the quality of the resulting spectral fits. The range in masses and energies sampled by the modified models starts at $0.5$$\cdot$$M_{\textrm{Ch}}$ / $\sim$5$\cdot$$10^{50}$erg and extends to $1.45$$\cdot$$M_{\textrm{Ch}}$ / $\sim$2$\cdot$$10^{51}$erg ($M_{\textrm{Ch}}$: Chandrasekhar mass, $1.4{M_{\odot}}$). This choice has been motivated by parameters inferred for observed extreme SNe Ia of all kinds [@maz97bg; @how06].
The paper is structured as follows: First, we give a short introduction to the methods employed (Sec. \[sec:method\]). We then present the models for SN 2005bl (Sec. \[sec:models\]), discuss and assess them (Sec. \[sec:discussion\]), and finally draw conclusions (Sec. \[sec:conclusions\]).
Method {#sec:method}
======
The radiative transfer code we use and the abundance tomography method have already been described [@ste05; @maz08eo]. Thus, we focus on aspects necessary for an understanding of the present study.
Radiative transfer
------------------
We use a 1D Monte Carlo (MC) radiative transfer code (@abb85, @maz93a, @luc99, @maz00 and @ste05) to compute SN spectra from a given density and abundance profile. The aim is to infer the chemical structure adjusting the abundances within the envelope until an optimal fit to the observed spectra is obtained.
The code computes the radiative transfer through the SN ejecta above an assumed photosphere. The densities within the envelope are calculated from an initial density profile describing the state of the ejecta after homologous expansion has set in, which is a few seconds after the explosion (e.g. @roe05). The ejecta expand radially with $r=v\cdot t$, where $r$ is the distance from the centre, $t$ the time from explosion (see beginning of Sec. \[sec:models\]), and $v$ the velocity. Radius and velocity can be used interchangeably as coordinates.
From the photosphere, which is located at an adjustable $v_{\textrm{ph}}$, thermal radiation \[$I_{\nu}^{+}=B_{\nu}(T_{\textrm{ph}})$\] is assumed to be emitted into the atmosphere. This is of course quite a crude approximation to the pseudo-continuous radiation field deep in the ejecta [@sau06]. Notable deviations mainly appear in the red and infrared, where a departure of the flux level from that of the observed spectra sometimes cannot be avoided.
The radiation is simulated as ”photon packets“, which undergo Thomson scattering as well as line excitation-deexcitation processes, treated in the Sobolev approximation. The process of photon branching is included, which implies that the transitions for excitation and deexcitation can be different. In a branching event, the photon packet is not split up. Instead, it is emitted as a whole with a new frequency corresponding to a possible downward transition. This ”indivisible packet“ approach [@luc99] enforces radiative equilibrium. The downward transition is randomly selected, taking into account effective emission probabilities. Thus, if a large number of packets are simulated, the distribution of decays reflects the actual one.
A modified nebular approximation, which mimics effects of non-local thermodynamic equilibrium (NLTE), is used to calculate the action of the radiation field onto the gas. For each of the $30$ zones into which the envelope is discretised here, a radiation temperature $T_R$ and an equivalent dilution factor $W$ are calculated. These quantities mostly determine the excitation and ionisation state [@abb85]. Only the variables describing the state of the gas are discretised; for the paths and redshifts of photons, and for the positions of interaction surfaces of lines, continuous values are allowed.
The code iterates the radiation field and the gas conditions. Furthermore, $T_\textrm{ph}$ is automatically modified so as to match a given output luminosity $L$, taking backscattering into account. After convergence, the emerging spectrum is obtained from a formal integral solution of the transfer equation [@luc99].
Model $\!\!\!\!$ $\!\!\!\!$ $E'_{\textrm{k}} / E_{\textrm{k},W7}$ $\!\!\!\!$ $\!\!\!\!$ $M'\!/ M_{W7}$ $\!\!\!\!$ $\!\!\!\!$ $E'_{\textrm{k}}$ \[$10^{51}$ erg\] $\!\!\!\!$ $\!\!\!\!$ $M'$ \[${M_{\odot}}$\] $\!\!\!\!$ $\!\!\!\!$ $\frac{E'_\textrm{k}}{M'}$ / $\left(\frac{E_{\textrm{k}}}{M}\right)_{\!W7\!\!\!\!\!}$
------------------ ------------------------------------------------------------- -------------------------------------- ----------------------------------------------------------- ---------------------------------------------- --------------------------------------------------------------------------------------------------
w7e0.35 0.35 1.00 0.47 1.38 0.35
w7e0.5m0.5 0.50 0.50 0.66 0.69 1.0
w7e0.5m0.7 0.50 0.70 0.66 0.97 0.7
w7e0.5 0.50 1.00 0.66 1.38 0.5
w7e0.5m1.25 0.50 1.25 0.66 1.73 0.4
w7e0.7m0.7 0.70 0.70 0.93 0.97 1.0
w7e0.7 0.70 1.00 0.93 1.38 0.7
w7e0.7m1.25 0.70 1.25 0.93 1.73 0.6
w7e0.7m1.45 0.70 1.45 0.93 2.00 0.5
w7m0.7 1.00 0.70 1.33 0.97 1.4
w7 1.00 1.00 1.33 1.38 1.0
w7m1.25 1.00 1.25 1.33 1.73 0.8
w7m1.45 1.00 1.45 1.33 2.00 0.7
w7e1.45m1.45 1.45 1.45 1.93 2.00 1.0
: Density models used in this work, and their total kinetic energy $E'_\textrm{k}$ and mass $M'$. The models are named according to the scaling factors for kinetic energy and mass with respect to W7 $\left(\frac{E'_\textrm{k}}{E_{\textrm{k},W7}}\textrm{ and }\frac{M'}{M_{W7}}\right)$, and sorted according to their kinetic energy $E'_\textrm{k}$.[]{data-label="tab:scaledmodels"}
Density profiles {#sec:densityprofile}
----------------
As a first step, we adopt the density structure of the standard explosion model W7 [@nom84] as a basis for our calculations. We then repeat the abundance tomography with modified density profiles, changing the total mass and kinetic energy of the explosion. To achieve this, the values for each grid point in the W7 velocity-density structure are scaled uniformly (i.e. all velocities by one scaling factor, and all densities by another one) according to: $$\begin{aligned}
\rho'& = &\rho_{W7}\cdot\left(\frac{E'_\textrm{k}}{E_{\textrm{k},W7}}\right)^{-3/2}\cdot\left(\frac{M'}{M_{W7}}\right)^{5/2}\\
v' & = & v_{W7}\cdot\left(\frac{E'_\textrm{k}}{E_{\textrm{k},W7}}\right)^{1/2}\cdot\left(\frac{M'}{M_{W7}}\right)^{-1/2}.\end{aligned}$$
Here, $\rho'$ and $v'$ are the density and velocity coordinates of each grid point after the scaling. $E'_\textrm{k}$ and $M'$ are the new total kinetic energy and mass.
The scaled density models used in this work are listed in Table \[tab:scaledmodels\], which gives an overview of the respective $\frac{E'_\textrm{k}}{E_\textrm{k}}$ and $\frac{M'}{M}$ ratios. We have not implemented every possible energy-mass combination within the limits given in Section \[sec:introduction\]. Instead, we first constrained ourselves to a few test cases. Then, we sampled the $E'_\textrm{k}$$-$$M'$ plane more densely in the region where models of acceptable quality emerged (see Sec. \[sec:assessment\]).
W7 naturally shows some differences with respect to more recent and realistic hydrodynamical simulations, and the scaled density profiles can also be expected to do so. However, it is possible to obtain good fits to spectra of “normal“ SNe Ia like SN 2002bo [@ste05] using the W7 density structure. The results for the scaled profiles should therefore bring out possible differences between dim SNe Ia and normal ones.
Abundance tomography
--------------------
The abundance tomography method [@ste05] uses a series of photospheric spectra to establish the abundance distribution within a supernova. The idea is that the opaque core of the expanding ejecta shrinks with time. Thus, a time series of spectra carries information about the abundances in the ejecta at different depths. In the picture adopted in our code, involving an approximate photosphere, the photosphere recedes to lower velocities with time. Deeper and deeper layers become visible, leaving their imprint on the spectra.
The earliest spectrum available can be used to obtain the photospheric velocity at that time and the abundances in the outer envelope. To this aim, we optimise the code input parameters to match that spectrum, as in a one-zone spectral model (e.g. @maz97bg). The subsequent spectrum will carry the imprint of the material in the outer envelope and additionally that of the layers inside which the photosphere has receded. Because the abundances in the outer zone are already known, the abundances of the layers which have become visible can now be inferred, together with the new velocity of the photosphere. This procedure is then continued with later spectra.
The optimum parameters inferred from a spectral model are subject to some uncertainty (see also the discussion in @maz08eo). One important reason for this can be degeneracy, which makes the spectra appear similar for different parameter sets. The composition adopted for an outer layer in an early-epoch model may therefore be in conflict with a later spectrum, if the later spectrum is still influenced by the outer layers. In such cases, we revised the parameters for the outer layers so as to optimise the earlier and later spectra at the same time.
Models {#sec:models}
======
{width="14.5cm"}
We analyse five spectra of SN 2005bl, taken at , , , and with respect to $B$ maximum. Observational data and one-zone spectral models have already been presented in @tau08. As in that paper, we assume a total reddening of $E(B-V)$$=$$0.20$ and a $B$-band rise time of 17d to calculate the time $t$ from the onset of the explosion.
Later spectra were not modelled, as the photosphere has already receded to $v_\textrm{ph}$$<$$3500$[km s$^{-1}$]{} at \[for comparison, @maz08eo found $v_\textrm{ph}$$=$$4700$[km s$^{-1}$]{} at and $v_\textrm{ph}$$=$$2800$[km s$^{-1}$]{} at in SN 2004eo\]. As the photosphere reaches the [$^{56}$Ni]{}-rich zone, some energy deposition should realistically take place above the photosphere itself. This is is not taken into account in our code. Thus, to explore the innermost layers one would need to model nebular spectra (which are not available for SN 2005bl) at least as a consistency check.
The outermost ejecta of SNe Ia may partly consist of unburned material [cf. @maz07]. As the one-zone models for SN 2005bl [@tau08] showed too much absorption by burned material at high velocities, we introduced a zone with strongly reduced abundances of burning products above $v$$\gtrsim$$15000$[km s$^{-1}$]{}. This was done limiting the mass fractions of burning products in this zone to $\lesssim$$\frac{1}{10}$ their value at the photosphere at . The unburned material at $v$$\gtrsim$$15000$[km s$^{-1}$]{}, which then constitutes $\gtrsim$$98$$\%$ by mass at these velocities, is assumed to consist of carbon and oxygen in a $\sim$$1$:$1$ ratio. In a preliminary stratified-abundance model, this was found clearly to improve the synthetic spectra, also with respect to the one-zone models (cf. Sec. \[sec:taubenbergercomparison\]). Consequently, we implemented such a zone in all our stratified-abundance models (except when using the w7e0.35 density profile, which has negligible densities in the outer layers).
Below, we first discuss an abundance tomography experiment based on the original W7 density structure. We compare our synthetic spectra with the observed ones and with the one-zone model spectra of @tau08. After discussing the abundance profile, we then present models with different total mass and kinetic energy. Parameters (abundances, photospheric velocities, etc.) of all models are compiled in Appendix \[app:modelparameters\].
Abundance tomography based on W7
--------------------------------
The spectral models discussed here are shown in Fig. \[fig:sequence-w7\], where the most important spectral features are marked.
### 2005 April 16: -6d, $v_\textrm{ph}$$=$$8400\mathrm{km}\;\mathrm{s^{-1}}$ {#sec:firstspectrum-w7}
At this epoch, the supernova shows a spectrum dominated by singly-ionised species. In normal SNe, usually also doubly-ionised species are detected at such early epochs (e.g. @maz08eo). The zone between $8400$ and $15000\textrm{{km~s$^{-1}$}}$ is dominated by oxygen. The absence of the [Si [ii]{}]{} $\lambda6355$ emission peak suggests absorption by [C [ii]{}]{} $\lambda6580$. However, the mass fraction of C between $8400$ and $15000\textrm{{km~s$^{-1}$}}$ must be $<$$10\%$; otherwise the [C [ii]{}]{} feature would become too deep. Burned material (oxygen as a burning product excluded) makes up for no more than $\sim$$15\%$ in mass according to the observed line depths.
While numerous lines of intermediate-mass elements (IME) are visible, there are no absorptions that can unambiguously be attributed to Fe. We determined an upper limit to the Fe abundance of $0.01\%$, avoiding the appearance of a spurious [Fe [ii]{}]{} feature at $\sim$$4950\textrm{\AA}$. Yet, some burning products heavier than Si and S are seen in the spectra: some per mille of Ti and Cr are necessary to model the absorption trough at $\sim$$4100\textrm{\AA}$ and the feature at $\sim$$4700\textrm{\AA}$, respectively. These elements also contribute significantly to line blocking in the UV [@sau08].
### 2005 April 17: -5d, $v_\textrm{ph}$$=$$8100\mathrm{km}\;\mathrm{s^{-1}}$
The April 17 spectrum is very similar to the previous one. As the material directly above the photosphere is highly ionised, many features in this spectrum depend strongly on the abundances above $8400$[km s$^{-1}$]{}.
At $\sim$$4950\textrm{\AA}$, the stratified model has an absorption trough too deep. This is mostly due to the [Si [ii]{}]{} $\lambda5049$ line, whose strength largely depends on the Si abundance above $v$$=$$8400$[km s$^{-1}$]{}. We chose this abundance so as to match the [Si [ii]{}]{} $\lambda5972$ line of this and the previous spectrum, and a simultaneous match of the [Si [ii]{}]{} $\lambda5049$ line was not possible. Apart from this and some flux mismatch in the red, the observations are fitted well.
### 2005 April 19: -3d, $v_\textrm{ph}$$=$$7500\mathrm{km}\;\mathrm{s^{-1}}$
This model again matches the observed spectrum nicely in most regions. The Ti-dominated trough at $\sim$$4100\textrm{\AA}$ is now deeper than in the earlier spectra, and relatively hard to fit. A good model requires Ti abundances of the order of a few percent at the photosphere, and relatively large Ti abundances in the zones above. Thus, we set the Ti mass fraction to $1\%$ between $8100$ and $8400$[km s$^{-1}$]{}. At larger velocities, the abundances are sharply constrained to some per mille by the features in the spectrum.
There is still no evidence for significant amounts of Fe in the spectrum. Fe mass fractions of a few per cent in the layers between $7500$ and $8400$[km s$^{-1}$]{} are compatible with the observations, but not strictly required.
### 2005 April 26: +4.8d, $v_\textrm{ph}$$=$$6600\mathrm{km}\;\mathrm{s^{-1}}$ {#sec:05blw7-p48}
In order to fit this spectrum with its low flux in the UV and blue, the model atmosphere must contain sufficient amounts of Ti, Cr and Fe. The layers at $v$$>$$8400$[km s$^{-1}$]{} contain relatively small amounts of these elements, as dictated by the pre-maximum spectral features and UV flux. To compensate for this, large amounts are needed close to the photosphere. While the flux-blocking in the UV is quite sensitive to the abundances close to the photosphere, the depth of individual features (such as [Si [ii]{}]{} $\lambda5972$) is still more strongly influenced by the composition at $>$$8400$[km s$^{-1}$]{}.
The most notable deviation the model from the observed spectrum occurs in the blue wing of [O [i]{}]{} $\lambda 7773$, where there is too much absorption. In the outermost zone, O could only be replaced by C, but we already have a $\sim$1$:$1 C-O mixture there. If we wanted to reduce the [O [i]{}]{} absorption strength by a factor of 2 in these layers, we would have to postulate a $\sim$3$:$1 C-O mixture, which would seem quite ad-hoc. In the layers between $8400$ and $15000\textrm{{km~s$^{-1}$}}$, the amount of oxygen cannot be reduced (cf. Sec. \[sec:firstspectrum-w7\]). In Sec. \[sec:w7e0.7spectra\], we will show that a reduction of the density in the outer layers can cure this problem.
There is some mismatch around $5700$Å, which seems to be caused by a low pseudo-continuum. This impression is however also due to [Na [i]{}]{}$\;\!$D absorption at the peak between the [S [ii]{}]{} trough and the [Si [ii]{}]{} $\lambda5972$ feature. We introduced a small amount of Na above $8100$[km s$^{-1}$]{} to obtain at least some [Na [i]{}]{}$\;\!$D absorption at . The spurious absorption appearing at then indicates inaccuracies in the Na ionisation profile and its evolution with time, a common issue with synthetic spectra [@maz97bg].
### 2005 May 04: $+12.9\mathrm{d}$, $v_\textrm{ph}$$=$$3250\mathrm{km}\;\mathrm{s^{-1}}$
This model carries some conceptual uncertainty, as a possible energy deposition by [$^{56}$Ni]{} above the photosphere is not simulated in our code. Yet, the overall fit is satisfactory. Some incompatibilities with the abundances inferred for the outer layers could not be resolved. It was, for example, impossible to get rid of the absorptions at $\sim$$6500$ and $\sim$$7700\textrm{\AA}$, which are due to [Ti [ii]{}]{} $\lambda\lambda6680,6718,6785$ and [Si [ii]{}]{} $\lambda7849$, respectively. These lines were not visible in the earlier spectra.
The photosphere is now deep inside the Si-dominated zone. The extended red wing of the observed feature at $\sim$$9000$Å, caused mostly by [Si [ii]{}]{} $\lambda 9242$, indicates a large Si mass fraction. On the other hand, the small flux in the blue and UV already demands a larger fraction of Fe-group elements. While the exact amounts of Fe, Co and Ni are somewhat uncertain, their sum can be estimated to be $\sim$$30\%$. The exact number depends on the abundances of other elements blocking UV flux (mostly Ti and Cr) between $3250$ and $6600$[km s$^{-1}$]{}, which are somewhat uncertain.
![Abundances of W7 nucleosynthesis calculations [@iwa99 top panel] vs. abundance tomography of SN 2005bl, based on the original W7 density profile ( model, bottom panel).[]{data-label="fig:abundances-w7"}](abundances-w7-orig.eps "fig:"){width="8.0cm"}\
![Abundances of W7 nucleosynthesis calculations [@iwa99 top panel] vs. abundance tomography of SN 2005bl, based on the original W7 density profile ( model, bottom panel).[]{data-label="fig:abundances-w7"}](abund-models-05bl-w7.eps "fig:"){width="8.0cm"}
### Comparison to one-zone models {#sec:taubenbergercomparison}
Compared to one-zone models \[@tau08, shown in Fig. \[fig:sequence-w7\] as the magenta line\], the stratified model sequence clearly constitutes an improvement in fitting the observations. The main reason for this is the C/O-dominated shell introduced at $v$$>$$15000\textrm{{km~s$^{-1}$}}$, which makes spectral lines of burned material narrower. The changes with respect to the one-zone model are especially apparent in the pre-maximum spectra: the [Ca [ii]{}]{} H&K and [Si [ii]{}]{} $\lambda6355$ lines absorb less at high velocities, so that in the blue wings of the features only small mismatches are left. The [Ti [ii]{}]{}-dominated trough around $4100$Å now has more structure. Some deviations, even a bit more apparent then in the one-zone models, remain in the red wing of [Si [ii]{}]{} $\lambda6355$ in the earliest spectra. This is largely due to re-emission in this wavelength range, caused by elements such as Ti and Cr which block and redistribute UV flux. These elements are, however, necessary to model the spectral features (see Sec. \[sec:firstspectrum-w7\]).
### Abundance profile
In Fig. \[fig:abundances-w7\], we compare the abundance profile derived in our tomography experiment to the nucleosynthesis in W7 [@iwa99], which approximately represents a normally-luminous SN Ia [@nom84]. In our models, unburned material (counting in all of the oxygen) constitutes a much larger fraction of the ejecta, almost the outer $\sim$$0.7{M_{\odot}}$. Our analysis of the outer layers is still a bit coarse. A better-resolved analysis, yielding more exact results e.g. for the amount of IME between $8400$ and $15000$[km s$^{-1}$]{}, would be possible if spectra at earlier epochs were available (see Sec. \[sec:earlierspectra\]).
Below the outer $\sim$$0.7{M_{\odot}}$, the ejecta of SN 2005bl are dominated by IME. The transition happens in the zone between $6400$ and $8400$[km s$^{-1}$]{}. The exact transition velocity is difficult to infer, as the post-maximum spectra show only a limited sensitivity to the Si abundances below $8400$[km s$^{-1}$]{}. @maz97bg have conducted a fine analysis of the [O [i]{}]{} $\lambda7773$ line profile in SN 1991bg, and found a lower cut-off velocity of $8600$[km s$^{-1}$]{} for O. We thus implemented a relatively sharp decrease of the O abundance in favour of Si below the photosphere.
The layers between $6400$ and $8400$[km s$^{-1}$]{} already consist of $\sim$$100\%$ [$^{56}$Ni]{} in W7. We, in contrast, find (besides IME) comparatively large abundances of Ti and Cr as products of incomplete burning at these velocities (peak values in the order of some per cent). These elements contribute to the formation of the observed trough around $\sim$$4200$Å which is characteristic of 91bg-like objects past maximum, but also to the line blocking in the UV. To some extent, their effects can also be mimicked by Fe, Co and Ni. With overly large amounts of Fe, however, individual lines in the optical may show up, and the flux distribution in the UV and blue may deviate from what is observed. Large abundances of [$^{56}$Ni]{} or its decay product [$^{56}$Co]{} outside the centre would be in conflict with the nebular spectra of dim SNe Ia, which show very narrow lines [@maz97bg].
The deepest zones that we reach with our analysis are still dominated by IME. However, there are signs of a transition to the NSE-burning zone: the large amount of line blocking and flux redistribution needed to fit the spectrum clearly points towards Fe-group abundances of several $10\%$.
Compared to the one-zone models of @tau08 epoch by epoch, the abundances of burning products at the respective photospheres are larger. In a model with homogeneous composition, the inferred abundances will always be some average between those at the photosphere and those further outwards, where less burning products are present.
Models with modified density profiles {#sec:modelcomparison}
-------------------------------------
{width="16cm"}
{width="16cm"}
We now show some representative spectral models based on modified density profiles (Sec. \[sec:densityprofile\]). The reader interested in the abundances is referred to Section \[sec:w7e0.7discussion\] and Appendix \[app:modelparameters\]. Here, we focus on the differences in the spectra with respect to the W7-based models. To facilitate the understanding of these differences, we first discuss the properties of the scaled density models.
Our scaled density models span a range of masses and kinetic energies (see Table \[tab:scaledmodels\]). Scaling the total mass and energy, the amplitude and/or form of the W7 density structure is changed. What exactly happens depends on the $\frac{E'_\textrm{k}}{M'}$ ratio of the final profile with respect to $\frac{E_{W7}}{M_{W7}}$. Here, we distinguish the following three cases, which result in three classes of scaled density profiles:
- $\!\!\!\!\!\!\!\!\!\!\frac{E'_\textrm{k}}{M'}\! = \! \frac{E_{W7}}{M_{W7}}$: In this case, the scaled velocity-density profile is obtained from W7 by reducing the density at each velocity by a uniform factor. The form of the density profile in velocity space is thus left unchanged.
- $\!\!\!\!\!\!\!\!\!\!\frac{E'_\textrm{k}}{M'}\! < \! \frac{E_{W7}}{M_{W7}}$: Here, the energy per unit mass is reduced. This means that mass elements are ”shifted“ towards lower velocities. The density profile becomes steeper in velocity space, and the relative amount of mass at high velocities is smaller.
- $\!\!\!\!\!\!\!\!\!\!\frac{E'_\textrm{k}}{M'}\! > \! \frac{E_{W7}}{M_{W7}}$: Increasing the energy per unit mass ”shifts“ material outwards, opposite to the case before. As the spectra of 91bg-like SNe Ia lack absorption at high velocities in *all* lines, this is generally disfavoured. Thus, we calculated only one model sequence with such a density profile (w7m0.7).
The models we discuss below are exemplary for these three scaling types. They are named after the underlying density models (e.g. is based on w7m0.7).
### Reduced mass and energy, $\frac{E'_\textrm{k}}{M'}\! = \! \frac{E_{W7}}{M_{W7}}$:
In these models, the density is decreased at all radii. This leads to a slight improvement of the spectra (Fig. \[fig:sequence-w7e0.7m0.7\]), as the photospheres are deeper inside the ejecta, and the absorption velocities tend to be lower. Owing to the lower densities, larger mass fractions of burned material are necessary to fit the line depths. At high velocities, however, oxygen still dominates and the high-velocity absorption in the [O [i]{}]{} $\lambda 7773$ line only becomes a bit weaker.
### Reduced energy, $\frac{E'_\textrm{k}}{M'}\! < \! \frac{E_{W7}}{M_{W7}}$: {#sec:w7e0.7spectra}
In the w7e0.7 density profile (Fig. \[fig:sequence-w7e0.7\]), the densities are significantly increased below $\sim$$6500$[km s$^{-1}$]{} and decreased above $\sim$$13000$[km s$^{-1}$]{}. Thus, the spectral features become narrower compared to the W7-based sequence. Line widths and positions now generally fit the structure of the observed spectra better.
Owing to the lower densities in the outer part, there is less line blocking by heavy elements. This decreases the flux redistribution, so that the flux level in the red wing of [Si [ii]{}]{} $\lambda6355$ and redwards is matched better, especially at early times.
At the same time, the spurious high-velocity absorption is practically gone in Ca H&K and [Si [ii]{}]{} $\lambda6355$, but even more importantly in [O [i]{}]{} $\lambda7773$. The reason for this is again the decreased density in the outer layers. As the abundances in the outer layers, especially of oxygen, are not fundamentally changed with the density modification, the decrease in density translates into weaker absorption at high velocities.
### Increased mass, $\frac{E'_\textrm{k}}{M'}\! < \! \frac{E_{W7}}{M_{W7}}$:
{width="16cm"}
Despite the larger mass, the spectra show a somewhat improved quality compared to W7 (Fig. \[fig:sequence-w7m1.25\], see also line velocity measurements in Sec. \[sec:assessment\]). This illustrates that a super-Chandrasekhar total mass is not necessarily incompatible with the spectra of SN 2005bl. Remarkably, the improvement over the W7-based sequence is due to *decreased* densities in the outermost layers ($v$$\gtrsim$$15000$[km s$^{-1}$]{}) of the warped density profile.
### Reduced mass, $\frac{E'_\textrm{k}}{M'}\! > \! \frac{E_{W7}}{M_{W7}}$:
{width="16cm"}
Here, the densities in the outermost layers are increased with respect to W7. This can directly be seen in the spectra (Fig. \[fig:sequence-w7m0.7\]): all the problems which are reduced in (compared to the original W7-based model sequence) are now exacerbated.
Discussion {#sec:discussion}
==========
Assessment of the models based on different density profiles – mass and kinetic energy of dim SNe Ia. {#sec:assessment}
-----------------------------------------------------------------------------------------------------
Having discussed some representative cases in Section \[sec:modelcomparison\], we now systematically compare all models calculated on the basis of different density profiles. Our aim is to judge the quality of each model sequence in a simple and meaningful manner. To achieve this, we introduce three quality criteria:
1. *Consistence of spectra.* The main motivation to test modifications of the density were mismatches in the line velocities or widths remaining in the W7-based model sequence, especially in [O [i]{}]{} $\lambda7773$ and [Si [ii]{}]{} $\lambda6355$. Other lines of the spectrum did not show deviations as apparent, apart from [Ca [ii]{}]{} H&K, which behaves quite similar to [Si [ii]{}]{} $\lambda6355$[^1]. To assess if the lines are better fitted using different density profiles, we measured the velocities of [O [i]{}]{} $\lambda7773$ and [Si [ii]{}]{} $\lambda6355$ in each synthetic and observed spectrum at , and . Then we calculated, for each model sequence and line, the velocity difference between the observed and the synthetic spectra, averaged over the epochs.
2. *Consistence of kinetic energy.* We calculated a hypothetical kinetic energy ($E_{\textrm{k,hyp}}$) for each of the abundance profiles inferred. This is the nuclear energy release (assuming a pre-explosion composition of equal amounts C and O) minus the binding energy $|E_\textrm{bind}|$ of the WD (gravitational energy[^2] minus thermal and, in case of rotation, rotational energy). To judge the quality of a model sequence, we then compared the kinetic energy assumed in the density scaling ($E'_\textrm{k}$) to $E_{\textrm{k,hyp}}$. The calculation of $E_{\textrm{k,hyp}}$ depends on some assumptions, the first of which is that the mass fraction of IME in the obscured core below the photosphere is $\frac{1}{2}$ of that above the photosphere. Actually, this mass fraction may be between zero and the IME mass fraction above the photosphere. The possible error due to this is given below. The binding energy $|E_\textrm{bind}|$ of the progenitors (except for the $0.69$${M_{\odot}}$ ones) was calculated following @yoo05, who assume a white dwarf rotation profile resulting from binary evolution. We used their ”$BE(M;\rho_c)$“ relation (eq. 33), assuming a central density $\rho_c$ of $2.0$$\cdot$$10^9$$\textrm{g}/\textrm{cm}^3$ (which is typical for WD ignition) for $M'$$\geq$$M_\textrm{Ch}$. For sub-Chandrasekhar WDs, the central densities are lower even in the absence of rotation. We assumed negligible rotation for these cases, and obtained the central density for a given mass inverting formula (22) of @yoo05[^3].
3. *Expected light-curve width.* For models with good consistence based on the first two criteria, we additionally can check whether the density and abundance structure implies a width of the bolometric light curve ($\tau_\textrm{LC}$) compatible with that of dim SNe Ia. We calculated an expected light curve width for each model sequence, following @maz07, from the respective kinetic energy $E_{\mathrm{k}}$, ejecta mass $M'$, and total masses of IME and NSE material $M_{\textrm{IME}},M_{\textrm{NSE}}$ as: $$\tau_{\textrm{LC}}=\mathcal{N}\cdot \tilde{\kappa}^{\frac{1}{2}} E_{\mathrm{k}}^{-\frac{1}{4}} M'^{\frac{3}{2}}.$$ Here, $\tilde{\kappa}\!=\!(0.1M_{\textrm{IME}}+M_{\textrm{NSE}})/M'$ is proportional to the opacity estimate of @maz07, and $\mathcal{N}$ is a normalisation factor chosen so as to agree with their estimates of light-curve widths. In order to calculate $\tilde{\kappa}$, we can assume different burning efficiencies in the core, as above; additionally, we may adopt as $E_{\mathrm{k}}$ either the hypothetical value $E_{\textrm{k,hyp}}$ or the value $E'_{\textrm{k}}$ from the density scaling. We thus calculated again an average $\tau_{\textrm{LC}}$ and an estimate of the error introduced by these degrees of freedom. In order to judge the models, the values $\tau_{\textrm{LC}}$ were compared to $\tau_{\textrm{LC,dim}}$$=$$13.9$d, which is the average expected light curve width for the similarly dim SNe 1991bg and 1999by [@maz07].
We now discuss the quality of the models in terms of the three criteria.
### Line velocities – consistence of spectra
The differences in Doppler velocity of the [O [i]{}]{} $\lambda7773$ and [Si [ii]{}]{} $\lambda6355$ lines between observed and synthetic spectra are shown in Table \[tab:velocities\]. In this table, the models are ranked according to the absolute value of the ”mean velocity difference“, which is the average over both lines and all epochs.
Model $\!\!\!\!$ $\!\!\!\!$ $E'_{\textrm{k}} / E_{\textrm{k},W7} $ $\!\!\!$ $\!\!\!\!$ $M' / M_{W7}$ $\!\!\!\!$ $\langle\Delta{}v(\textrm{{Si~{\sc ii}}\ }\lambda 6355)\rangle$ $\!\!\!\!\!\!\!\!$ $\langle\Delta{}v(\textrm{{O~{\sc i}}\ }\lambda 7773)\rangle$ $\!\!\!\!\!\!\!\!$ $\!\!\!\!\!\!$ $\langle\Delta{}v\rangle$ $\!\!$
------------------------------ ------------------------------------------------------------ ------------------------------------- ------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------- -------------------------------------------------
05bl-w7e0.7 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ 1.00 $\!\!\!\!$ $\!\!\!\!$ 64.1 $\quad\ $ $\!\!\!\!$ -54.7 $\quad\ $ $\!\!\!\!$ 4.7
05bl-w7m1.45 $\!\!\!\!$ $\!\!\!\!$ 1.00 $\!\!\!\!$ $\!\!\!\!$ 1.45 $\!\!\!\!$ $\!\!\!\!$ 206.0 $\quad\ $ $\!\!\!\!$ 86.6 $\quad\ $ $\!\!\!\!$ 146.3
05bl-w7e0.5m0.7 $\!\!\!\!$ $\!\!\!\!$ 0.50 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ -26.5 $\quad\ $ $\!\!\!\!$ -283.8 $\quad\ $ $\!\!\!\!$ -155.1
05bl-w7e0.7m1.25 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ 1.25 $\!\!\!\!$ $\!\!\!\!$ -117.2 $\quad\ $ $\!\!\!\!$ -386.0 $\quad\ $ $\!\!\!\!$ -251.6
05bl-w7e0.5m0.5 $\!\!\!\!$ $\!\!\!\!$ 0.50 $\!\!\!\!$ $\!\!\!\!$ 0.50 $\!\!\!\!$ $\!\!\!\!$ 292.2 $\quad\ $ $\!\!\!\!$ 222.2 $\quad\ $ $\!\!\!\!$ 257.2
05bl-w7m1.25 $\!\!\!\!$ $\!\!\!\!$ 1.00 $\!\!\!\!$ $\!\!\!\!$ 1.25 $\!\!\!\!$ $\!\!\!\!$ 368.1 $\quad\ $ $\!\!\!\!$ 359.5 $\quad\ $ $\!\!\!\!$ 363.8
05bl-w7e0.7m0.7 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ 380.4 $\quad\ $ $\!\!\!\!$ 454.9 $\quad\ $ $\!\!\!\!$ 417.7
05bl-w7e0.5 $\!\!\!\!$ $\!\!\!\!$ 0.50 $\!\!\!\!$ $\!\!\!\!$ 1.00 $\!\!\!\!$ $\!\!\!\!$ -325.5 $\quad\ $ $\!\!\!\!$ -618.0 $\quad\ $ $\!\!\!\!$ -471.7
05bl-w7e0.7m1.45 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ 1.45 $\!\!\!\!$ $\!\!\!\!$ -223.4 $\quad\ $ $\!\!\!\!$ -798.2 $\quad\ $ $\!\!\!\!$ -510.8
05bl-w7 $\!\!\!\!$ $\!\!\!\!$ 1.00 $\!\!\!\!$ $\!\!\!\!$ 1.00 $\!\!\!\!$ $\!\!\!\!$ 494.7 $\quad\ $ $\!\!\!\!$ 1012.4 $\quad\ $ $\!\!\!\!$ 753.5
05bl-w7e1.45m1.45 $\!\!\!\!$ $\!\!\!\!$ 1.45 $\!\!\!\!$ $\!\!\!\!$ 1.45 $\!\!\!\!$ $\!\!\!\!$ 594.9 $\quad\ $ $\!\!\!\!$ 942.8 $\quad\ $ $\!\!\!\!$ 768.8
05bl-w7e0.5m1.25 $\!\!\!\!$ $\!\!\!\!$ 0.50 $\!\!\!\!$ $\!\!\!\!$ 1.25 $\!\!\!\!$ $\!\!\!\!$ -592.6 $\quad\ $ $\!\!\!\!$ -976.9 $\quad\ $ $\!\!\!\!$ -784.7
05bl-w7e0.35 $\!\!\!\!$ $\!\!\!\!$ 0.35 $\!\!\!\!$ $\!\!\!\!$ 1.00 $\!\!\!\!$ $\!\!\!\!$ -846.7 $\quad\ $ $\!\!\!\!$ -1131.7 $\quad\ $ $\!\!\!\!$ -989.2
05bl-w7m0.7 $\!\!\!\!$ $\!\!\!\!$ 1.00 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ 457.9 $\quad\ $ $\!\!\!\!$ 1815.7 $\quad\ $ $\!\!\!\!$ 1136.8
A decent match of line velocity is obtained especially for the model, but also, for example, for some super-Chandrasekhar mass models with $E'_\textrm{k}/M'$ lower than W7. This shows that a reduced density in the outer layers is the key to a better fit in the lines. To fit the observed lines well, models near the Chandrasekhar mass need a $E'_\textrm{k}/M'$ smaller by $\sim$$30$$-$$40\%$ with respect to W7. With too large a reduction in energy, line velocities become too low (see e.g. negative velocity differences for the model). At low masses, generally a smaller reduction in $E'_\textrm{k}/M'$ suffices: ($M'$$=$$0.69{M_{\odot}}$) as an extreme model still gives a satisfactory fit with $E'_\textrm{k}/M'$$=$$\left(E_\textrm{k}/M\right)_{W7}$.
Remarkably, for all mass values probed in this work, a reasonably good model can be obtained (judged by the line velocities). The kinetic energy $E'_\textrm{k}$ of all well-fitting models, however, is lower than $E_{\textrm{k},W7}$.
### Energetic consistence
In Table \[tab:energetics\] we show our hypothetical kinetic energy values, as well as the quantities from which they were calculated. We then judge the models by the ratio of $E_{\textrm{k,hyp}}$ to the kinetic energy assumed in the density scaling ($E'_\textrm{k}$). Ideally, this ratio should be equal to one; the larger the deviation, the lower the rank of a model.
------------------------------ ----------------------------------------------------------- ----------------------------------- ---------------------------------------- ------------------------------------------- ------------------------------------------- ------------------------------------------------------- ---------------------------------------------------------------------------
Model $\!\!\!\!$ $\!\!\!\!$ $E'_{\textrm{k}}/E_{\textrm{k},W7}$ $\!\!\!\!$ $\!\!\!\!$ $M'/M_{W7}$ $\!\!\!\!$ $\!\!\!\!$ $E'_\textrm{k}$ $\!\!\!\!$ $\!\!\!\!$ $E_{\textrm{nucl}}$ $\!\!\!\!$ $\!\!\!\!$ $E_{\textrm{bind}}$ $\!\!\!\!$ $\!\!\!\!$ $E_\textrm{k,hyp}$$^\textrm{a}$ $\!\!\!\!$ $\!\!\!\!$ $\!\!\!\!\frac{E_\textrm{k,hyp}}{E'_\textrm{k}}\!-\!1\!\!\!\!$
$\!\!\!\!$ $\!\!\!\!$ $\!\!\!\!$ $\!\!\!\!$ $\!\!\!\!$ $\!\!\!\!$ \[$10^{51}$erg\] $\!\!\!\!$ $\!\!\!\!$ \[$10^{51}$erg\] $\!\!\!\!$ $\!\!\!\!$ \[$10^{51}$erg\] $\!\!\!\!$ $\!\!\!\!$ \[$10^{51}$erg\] $\!\!\!\!$ $\!\!\!\!$
05bl-w7e0.7m1.45 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ 1.45 $\!\!\!\!$ $\!\!\!\!$ 0.93 $\!\!\!\!$ $\!\!\!\!$ 2.10 $\!\!\!\!$ $\!\!\!\!$ 1.15 $\!\!\!\!$ $\!\!\!\!$ 0.96$\,\pm\,$0.10 $\!\!\!\!$ $\!\!\!\!$ 0.03 $\ $
05bl-w7e0.7m1.25 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ 1.25 $\!\!\!\!$ $\!\!\!\!$ 0.93 $\!\!\!\!$ $\!\!\!\!$ 1.73 $\!\!\!\!$ $\!\!\!\!$ 0.85 $\!\!\!\!$ $\!\!\!\!$ 0.88$\,\pm\,$0.06 $\!\!\!\!$ $\!\!\!\!$ -0.05 $\ $
05bl-w7e0.5m0.7 $\!\!\!\!$ $\!\!\!\!$ 0.50 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ 0.66 $\!\!\!\!$ $\!\!\!\!$ 0.87 $\!\!\!\!$ $\!\!\!\!$ 0.13 $\!\!\!\!$ $\!\!\!\!$ 0.74$\,\pm\,$0.02 $\!\!\!\!$ $\!\!\!\!$ 0.11 $\ $
05bl-w7e0.7 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ 1.00 $\!\!\!\!$ $\!\!\!\!$ 0.93 $\!\!\!\!$ $\!\!\!\!$ 1.26 $\!\!\!\!$ $\!\!\!\!$ 0.49 $\!\!\!\!$ $\!\!\!\!$ 0.77$\,\pm\,$0.03 $\!\!\!\!$ $\!\!\!\!$ -0.17 $\ $
05bl-w7e0.5m0.5 $\!\!\!\!$ $\!\!\!\!$ 0.50 $\!\!\!\!$ $\!\!\!\!$ 0.50 $\!\!\!\!$ $\!\!\!\!$ 0.66 $\!\!\!\!$ $\!\!\!\!$ 0.60 $\!\!\!\!$ $\!\!\!\!$ 0.06 $\!\!\!\!$ $\!\!\!\!$ 0.53$\,\pm\,$0.00 $\!\!\!\!$ $\!\!\!\!$ -0.20 $\ $
05bl-w7e0.5m1.25 $\!\!\!\!$ $\!\!\!\!$ 0.50 $\!\!\!\!$ $\!\!\!\!$ 1.25 $\!\!\!\!$ $\!\!\!\!$ 0.66 $\!\!\!\!$ $\!\!\!\!$ 1.89 $\!\!\!\!$ $\!\!\!\!$ 0.85 $\!\!\!\!$ $\!\!\!\!$ 1.04$\,\pm\,$0.07 $\!\!\!\!$ $\!\!\!\!$ 0.23 $\ $
05bl-w7e0.7m0.7 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ 0.93 $\!\!\!\!$ $\!\!\!\!$ 0.80 $\!\!\!\!$ $\!\!\!\!$ 0.13 $\!\!\!\!$ $\!\!\!\!$ 0.68$\,\pm\,$0.01 $\!\!\!\!$ $\!\!\!\!$ -0.27 $\ $
05bl-w7m1.45 $\!\!\!\!$ $\!\!\!\!$ 1.00 $\!\!\!\!$ $\!\!\!\!$ 1.45 $\!\!\!\!$ $\!\!\!\!$ 1.33 $\!\!\!\!$ $\!\!\!\!$ 1.90 $\!\!\!\!$ $\!\!\!\!$ 1.15 $\!\!\!\!$ $\!\!\!\!$ 0.75$\,\pm\,$0.06 $\!\!\!\!$ $\!\!\!\!$ -0.43 $\ $
05bl-w7e0.5 $\!\!\!\!$ $\!\!\!\!$ 0.50 $\!\!\!\!$ $\!\!\!\!$ 1.00 $\!\!\!\!$ $\!\!\!\!$ 0.66 $\!\!\!\!$ $\!\!\!\!$ 1.44 $\!\!\!\!$ $\!\!\!\!$ 0.49 $\!\!\!\!$ $\!\!\!\!$ 0.95$\,\pm\,$0.05 $\!\!\!\!$ $\!\!\!\!$ 0.43 $\ $
05bl-w7m1.25 $\!\!\!\!$ $\!\!\!\!$ 1.00 $\!\!\!\!$ $\!\!\!\!$ 1.25 $\!\!\!\!$ $\!\!\!\!$ 1.33 $\!\!\!\!$ $\!\!\!\!$ 1.56 $\!\!\!\!$ $\!\!\!\!$ 0.85 $\!\!\!\!$ $\!\!\!\!$ 0.71$\,\pm\,$0.04 $\!\!\!\!$ $\!\!\!\!$ -0.47 $\ $
05bl-w7 $\!\!\!\!$ $\!\!\!\!$ 1.00 $\!\!\!\!$ $\!\!\!\!$ 1.00 $\!\!\!\!$ $\!\!\!\!$ 1.33 $\!\!\!\!$ $\!\!\!\!$ 1.14 $\!\!\!\!$ $\!\!\!\!$ 0.49 $\!\!\!\!$ $\!\!\!\!$ 0.65$\,\pm\,$0.02 $\!\!\!\!$ $\!\!\!\!$ -0.51 $\ $
05bl-w7m0.7 $\!\!\!\!$ $\!\!\!\!$ 1.00 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ 1.33 $\!\!\!\!$ $\!\!\!\!$ 0.65 $\!\!\!\!$ $\!\!\!\!$ 0.13 $\!\!\!\!$ $\!\!\!\!$ 0.53$\,\pm\,$0.01 $\!\!\!\!$ $\!\!\!\!$ -0.60 $\ $
05bl-w7e1.45m1.45 $\!\!\!\!$ $\!\!\!\!$ 1.45 $\!\!\!\!$ $\!\!\!\!$ 1.45 $\!\!\!\!$ $\!\!\!\!$ 1.93 $\!\!\!\!$ $\!\!\!\!$ 1.69 $\!\!\!\!$ $\!\!\!\!$ 1.15 $\!\!\!\!$ $\!\!\!\!$ 0.54$\,\pm\,$0.04 $\!\!\!\!$ $\!\!\!\!$ -0.72 $\ $
05bl-w7e0.35 $\!\!\!\!$ $\!\!\!\!$ 0.35 $\!\!\!\!$ $\!\!\!\!$ 1.00 $\!\!\!\!$ $\!\!\!\!$ 0.46 $\!\!\!\!$ $\!\!\!\!$ 1.55 $\!\!\!\!$ $\!\!\!\!$ 0.49 $\!\!\!\!$ $\!\!\!\!$ 1.07$\,\pm\,$0.06 $\!\!\!\!$ $\!\!\!\!$ 1.29 $\ $
------------------------------ ----------------------------------------------------------- ----------------------------------- ---------------------------------------- ------------------------------------------- ------------------------------------------- ------------------------------------------------------- ---------------------------------------------------------------------------
$^\textrm{a}$ The error estimate only reflects the error due to the unknown composition below the photosphere at .
For density profiles with the same mass, but different kinetic energy ${E'_\textrm{k}}$, the hypothetical kinetic energy ${E_{\textrm{k,hyp}}}$ usually varies systematically. In density models with smaller ${E'_\textrm{k}}$, densities are reduced in the high-velocity layers (see Sec. \[sec:w7e0.7spectra\]), which contain mostly unburned material. At the same time, densities are increased in lower layers, where the material is mostly burned. The velocity at which the transition (between unburned and burned material) happens does not vary much from model to model as it is constrained by spectral features. Therefore, the change in the density profile results in a larger ratio of burned to unburned material and a larger ${E_{\textrm{k,hyp}}}$. Similarly, when ${E'_\textrm{k}}$ is increased, ${E_{\textrm{k,hyp}}}$ decreases. Equality, i.e. consistence, between $E_{\textrm{k,hyp}}$ and $E'_\textrm{k}$ is usually reached at a reduced value of $E'_\textrm{k}/M'$ with respect to W7. The required reduction varies with the mass of the models (see Sec. \[sec:modeldiagram\]).
In Table \[tab:energetics\], two supermassive models (, ) rank top. However, it should be noted that the energetic quality criterion again does not single out a certain mass, but sets a point of energetic consistence for each mass. All models with larger ${E'_\textrm{k}}$ will then feature too little nucleosynthesis to explain the assumed kinetic energy. The opposite holds for models with lower ${E'_\textrm{k}}$.
### Expected light-curve width
We calculated estimates of the width of the bolometric light curve for the models ranking best in spectroscopic and energetic consistence at each mass $M'$. The resulting values, and those of the quantities needed for the calculation, are given in Table \[tab:lcwidth\].
----------------------------- ----------------------------------------------------------- ----------------------------------- ---------------------------------------- ----------------------------------------- -------------------------------------------- ----------------------------------------------------- --------------------------------------------------------- ----------------------------------------------------------------------------------
Model $\!\!\!\!$ $\!\!\!\!$ $E'_{\textrm{k}}/E_{\textrm{k},W7}$ $\!\!\!\!$ $\!\!\!\!$ $M'/M_{W7}$ $\!\!\!\!$ $\!\!\!\!$ $E'_\textrm{k}$ $\!\!\!\!$ $\!\!\!\!$ $M'$ $\!\!\!\!$ $\!\!\!\!$ $E_{\textrm{k,hyp}}$ $\!\!\!\!$ $\!\!\!\!$ $\tilde{\kappa}$$^\textrm{a}$ $\!\!\!\!$ $\!\!\!\!$ $\tau_{\textrm{LC}}$$^\textrm{a}$ $\!\!\!\!$ $\!\!\!\!$ $\!\!\frac{\tau_{\textrm{LC}}}{\tau_{\textrm{LC,dim}}}\!-\!1\!\!\!\!$
$\!\!\!\!$ $\!\!\!\!$ $\!\!\!\!$ $\!\!\!\!$ $\!\!\!\!$ $\!\!\!\!$ \[$10^{51}$erg\] $\!\!\!\!$ $\!\!\!\!$ \[${M_{\odot}}$\] $\!\!\!\!$ $\!\!\!\!$ \[$10^{51}$erg\] $\!\!\!\!$ $\!\!\!\!$ $\!\!\!\!$ $\!\!\!\!$ \[d\] $\!\!\!\!$ $\!\!\!\!$ $\!\!\!\!$
05bl-w7e0.7 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ 1.00 $\!\!\!\!$ $\!\!\!\!$ 0.93 $\!\!\!\!$ $\!\!\!\!$ 1.31 $\!\!\!\!$ $\!\!\!\!$ 0.77 $\!\!\!\!$ $\!\!\!\!$ 0.31$\,\pm\,$0.10 $\!\!\!\!$ $\!\!\!\!$ 13.8$\,\pm\,$2.5 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\quad\ $
05bl-w7e0.5m0.7 $\!\!\!\!$ $\!\!\!\!$ 0.50 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ 0.66 $\!\!\!\!$ $\!\!\!\!$ 0.97 $\!\!\!\!$ $\!\!\!\!$ 0.74 $\!\!\!\!$ $\!\!\!\!$ 0.34$\,\pm\,$0.10 $\!\!\!\!$ $\!\!\!\!$ 11.7$\,\pm\,$1.9 $\!\!\!\!$ $\!\!\!\!$ -0.16 $\quad\ $
05bl-w7e0.7m1.25 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ 1.25 $\!\!\!\!$ $\!\!\!\!$ 0.93 $\!\!\!\!$ $\!\!\!\!$ 1.73 $\!\!\!\!$ $\!\!\!\!$ 0.88 $\!\!\!\!$ $\!\!\!\!$ 0.32$\,\pm\,$0.12 $\!\!\!\!$ $\!\!\!\!$ 16.1$\,\pm\,$3.4 $\!\!\!\!$ $\!\!\!\!$ 0.16 $\quad\ $
05bl-w7e0.5m0.5 $\!\!\!\!$ $\!\!\!\!$ 0.50 $\!\!\!\!$ $\!\!\!\!$ 0.50 $\!\!\!\!$ $\!\!\!\!$ 0.66 $\!\!\!\!$ $\!\!\!\!$ 0.69 $\!\!\!\!$ $\!\!\!\!$ 0.53 $\!\!\!\!$ $\!\!\!\!$ 0.47$\,\pm\,$0.07 $\!\!\!\!$ $\!\!\!\!$ 11.2$\,\pm\,$1.2 $\!\!\!\!$ $\!\!\!\!$ -0.19 $\quad\ $
05bl-w7m1.45 $\!\!\!\!$ $\!\!\!\!$ 1.00 $\!\!\!\!$ $\!\!\!\!$ 1.45 $\!\!\!\!$ $\!\!\!\!$ 1.33 $\!\!\!\!$ $\!\!\!\!$ 2.00 $\!\!\!\!$ $\!\!\!\!$ 0.75 $\!\!\!\!$ $\!\!\!\!$ 0.28$\,\pm\,$0.11 $\!\!\!\!$ $\!\!\!\!$ 16.9$\,\pm\,$4.6 $\!\!\!\!$ $\!\!\!\!$ 0.21 $\quad\ $
05bl-w7e0.7m1.45 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ 1.45 $\!\!\!\!$ $\!\!\!\!$ 0.93 $\!\!\!\!$ $\!\!\!\!$ 2.00 $\!\!\!\!$ $\!\!\!\!$ 0.96 $\!\!\!\!$ $\!\!\!\!$ 0.35$\,\pm\,$0.16 $\!\!\!\!$ $\!\!\!\!$ 18.5$\,\pm\,$4.5 $\!\!\!\!$ $\!\!\!\!$ 0.34 $\quad\ $
----------------------------- ----------------------------------------------------------- ----------------------------------- ---------------------------------------- ----------------------------------------- -------------------------------------------- ----------------------------------------------------- --------------------------------------------------------- ----------------------------------------------------------------------------------
$^\textrm{a}$ The error estimate reflects the errors due to the unknown composition below the photosphere at , and due to the uncertainties in $E_\textrm{k}$.
The deviation of $\tau_{\textrm{LC}}$ from $\tau_{\textrm{LC,dim}}$$=$$13.9$d strongly depends on the mass $M'$. Models with larger mass clearly tend to have a larger light-curve width, although they often have lower values of $\tilde{\kappa}$, as relatively small abundances of burning products are needed to match the observed line strengths with the synthetic spectra.
Although our expected light-curve widths are quite rough estimates, one can clearly state that the criterion disfavours masses largely deviating from the Chandrasekhar mass. The least massive model, with a mass of $M'$$=$$0.5$$M_\textrm{Ch}$, presumably will not produce a broad enough light curve. Likewise, the models at $M'$$=$$1.45$$M_\textrm{Ch}$ will probably exhibit too broad a light curve, although these models are not strictly incompatible with $\tau_{\textrm{LC,dim}}$, as a large inaccuracy in $\tau_{\textrm{LC,dim}}$ results from the large mass in the obscured core.
Location of consistent models in the $E'_\textrm{k}$$-$$M'$ plane {#sec:modeldiagram}
-----------------------------------------------------------------
{width="12.5cm"}
Fig. \[fig:modeldiagram\] gives an overview of all models in an $E'_\textrm{k}$$-$$M'$ plane. According to their quality in spectroscopic terms, the models are marked with different colours; the energetic consistence is indicated by hatches.
In the figure, we also indicate where spectroscopically and energetically consistent models can generally be expected in the plane: a black line is drawn approximately where the transition between too large and too small line velocities occurs. This line is straight and runs from massive models with low $E'_\textrm{k}/M'$ to submassive models with $E'_\textrm{k}/M'$$\approx$$\left(E_\textrm{k}/M\right)_{W7}$. A green line approximately divides the regions of too large and too small nucleosynthetic energy yields. It lies in the same region as the line of spectroscopic consistence, but is curved because the WD binding energy shows a disproportionately strong increase with WD mass. For models up to $M'$$\approx$$0.7 M_\textrm{Ch}$, the binding energy is negligible compared to the nuclear energy release, whereas at higher masses it is considerable, forcing the line of consistence towards smaller $E'_\textrm{k}$ and larger nucleosynthesis yields.
The two lines of consistence are especially close to one another for masses $M'$$\lesssim$$M_\textrm{Ch}$. Models at $M'$$=$$1.45$$M_\textrm{Ch}$ are either spectroscopically or energetically inconsistent, at least under the assumptions we made in this work. Additionally, the light-curve criterion indicates that the width of the light curve is too large for these most massive models. Our least massive models with $M'$$=$$0.5$$M_\textrm{Ch}$ are also disfavoured in this respect, as they would probably show too rapid a light-curve evolution.
Criteria like those used here could give stricter limits still, if the chemical composition in the inner layers was known. This requires studies of nebular spectra of dim SNe Ia.
as a reference model {#sec:w7e0.7discussion}
---------------------
![w7e0.7 compared to the standard W7 and the w7m1.25e0.7 density profiles.[]{data-label="fig:density-reference"}](rho-w7e0.7.eps){width="8.0cm"}
As discussed above, our models give no clear indication for a deviation from the Chandrasekhar mass. The simplest modification leading to better spectral fits and roughly consistent energetics is simply a moderate downscaling of the energy, as in the model. Thus, we consider the model a “reference”. In Figures \[fig:density-reference\] and \[fig:abundances-reference\], we show the density and abundance profiles of the model. Other spectroscopically consistent models show similar densities in the outer layers, and thus also similar abundances in that zone. This can be verified in Figures \[fig:density-reference\] and \[fig:abundances-reference\], where the model is also plotted for comparison.
The model features $0.46$${M_{\odot}}$ of unburned material (including all oxygen; C constitutes $0.04$${M_{\odot}}$). IME are dominant, with a total abundance of $0.55$${M_{\odot}}$ above $3350$[km s$^{-1}$]{}, the velocity of the photosphere at . Stable Fe is present in significant amounts ($0.05$${M_{\odot}}$). The mass of [$^{56}$Ni]{} (including decay products) above $3350$[km s$^{-1}$]{} is $0.06$${M_{\odot}}$, which is a bit higher than the $0.016$$-$$0.026{M_{\odot}}$ found in @maz97bg above $3500$[km s$^{-1}$]{} for SN 1991bg. However, some of the [$^{56}$Ni]{} could be replaced by other UV-blocking elements without changing the quality of the fit. Some $0.23$${M_{\odot}}$ of material are still hidden below the photosphere, where the IME abundances may still be significant (Si of the order of several $10\%$).
{width="8.0cm"} {width="8.0cm"}\
{width="8.0cm"} {width="8.0cm"}\
Alternative spectroscopically consistent models show similar patterns in the abundance profile in velocity space, but the exact densities and abundances below $\sim$$10000$[km s$^{-1}$]{} are somewhat different. In , as an example, the densities in the inner zones are larger. Thus, the abundances of Fe, Ti and Cr must be lower too keep UV opacities reasonable. For Si, moderate changes in the number density do not cause big changes in the spectra. Therefore, the smaller Fe, Ti and Cr abundances can be balanced by slightly larger Si abundances.
The need for early time and nebular spectra of dim SNe Ia {#sec:earlierspectra}
---------------------------------------------------------
The analysis presented here could still be refined for the outermost and innermost layers. The exact abundance stratification in the outer envelope cannot be inferred from the spectrum at , whose photospheric velocity is already quite low. For the inner layers, especially the density structure and thus the abundance of Si is somewhat uncertain (see Sec. \[sec:w7e0.7discussion\]). In order to make a more precise study of dim SNe Ia possible, additional spectra in the very early and in the nebular phase are needed.
The potential of an analysis of the nebular spectra has already been shown in @maz97bg. Here, we would like to illustrate the benefit of early time spectra, showing their sensitivity to the abundances in the outer envelope. We checked the influence of the abundances between $v$$\approx$$11000$[km s$^{-1}$]{} and $15000$[km s$^{-1}$]{} on the spectrum, and found that these abundances have some effects difficult to distinguish from those of the chemical composition at lower velocities. Moreover, the (small) abundances of burned material at $\gtrsim$$15000$[km s$^{-1}$]{} cannot be exactly determined, as these only affect the extreme blue wings of the spectral features.
To explore the effect of the abundances in the outer envelope on early-time spectra, we calculated model spectra at and (Fig. \[fig:earlytimespectra\]). The luminosities at these epochs were crudely estimated from the luminosity at under the assumption of a quadratic light curve rise [cf. @rie99]. We first calculated spectra assuming the density and abundance structure. For each of the two epochs, the photospheric position was shifted from its value at to higher velocities, until the backscattering was reasonably reduced. This resulted in photospheric velocities of $11750$ and $15200$[km s$^{-1}$]{}, respectively.
After calculating these initial models, we explored the effect of changes in the chemical composition, performing three additional code runs for each epoch. In the first two runs, we reduced IME and heavier elements to $20\%$ of their original abundances, respectively. For the model, these changes were applied to the whole atmosphere. In the model, we kept the original composition at velocities $>$$15200$[km s$^{-1}$]{}constant, in order to show the sensitivity to the abundances in the zone not probed by the spectrum. In the third code run, finally, we removed oxygen in favour of carbon (so that the mass fraction $X(\textrm{C})$$=$$80\%$). This change was applied to the whole atmosphere, also at -10d, as otherwise an inverted composition (larger C abundances further inwards) would have resulted.
In Fig. \[fig:earlytimespectra\], we show the resulting spectra and give line identifications to clarify the effect of the modified abundances. Moreover, it is indicated which lines do and which do not change significantly with the modifications. At , the synthetic spectrum looks vastly different from the spectrum of a normal SN Ia. We illustrate this in the upper panel of Fig. \[fig:earlytimespectra\] by additionally plotting the earliest SN Ia spectrum ever observed (SN 1990N at , @lei91). Compared to spectra of normal SNe Ia, but also to the spectrum, lines of less ionised species appear owing to the low temperatures, which result from the low luminosity. [Si [ii]{}]{} and [S [ii]{}]{} lines, which normally characterise SNe Ia, are absent. The spectrum is especially sensitive to the abundances of Na, Ca and Fe-group elements (with Na, uncertainties in the ionisation remain a caveat, see Sec. \[sec:05blw7-p48\]). Furthermore, [C [i]{}]{} features are present around $6800$Å and $8700$Å. As the C abundance is already quite large in the outermost layers of , these features do not react strongly to a further increase of $X($C$)$. However, if much less carbon was present, they should gradually disappear.
At , the structure of the spectrum resembles somewhat more that at . Yet, the spectrum has little in common with that of the moderately subluminous, spectroscopically rather normal SN 2004eo at (@pas07; plotted in the lower panel of Fig. \[fig:earlytimespectra\]). Compared to , the spectrum still shows hints of lower temperatures: because of the scarce population of excited levels, the [S [ii]{}]{} “W-trough” does not show up. For the same reason, the [C [ii]{}]{} $\lambda 6580$ feature is weak. In the model with a larger C mass fraction, however, some of the strongest lines of [C [i]{}]{} begin to absorb at $\sim8700$Å. Furthermore, there are absorptions due to [O [i]{}]{}, [Na [i]{}]{}, [Si [ii]{}]{}, [Ti [ii]{}]{}, [Cr [ii]{}]{} and [Fe [ii]{}]{}, which should allow for an analysis of the abundances in the outer layers as soon as observations are available.
The amount of extra information which can be inferred from early-time spectra will, of course, also depend on the actual luminosity of the SN at these epochs. Larger luminosities mean higher temperatures, making lines of different ions appear. However, our results already suggest that there are interesting possibilities to infer the chemical composition of the outermost ejecta.
Conclusions {#sec:conclusions}
===========
{width="11.5cm"} {width="11.5cm"}
We conducted an abundance tomography of SN 2005bl and confirmed that nuclear burning in dim, 91bg-like SNe Ia stops at less advanced stages compared to normal SNe Ia. The spectra indicate that the abundance of burned material above $\sim$$8500$[km s$^{-1}$]{} is much lower than even in moderately-luminous objects [@maz08eo]. From $\sim$$8500$[km s$^{-1}$]{} down to $\sim$$3300$[km s$^{-1}$]{}, IME dominate the ejecta. This points towards large-scale incomplete Si-burning or explosive O burning [e.g. @woo73]. A detonation at low densities, as it proceeds in the outer layers of delayed-detonation models [@kho91], may be responsible for the abundance pattern we find. Assuming this, we need to understand how low densities could prevail in such a large fraction of the envelope at the onset of the detonation. Up to now, all explosion models which pre-expand the star by a deflagration and then detonate (e.g. @hil00, @bad03, @gam04, @roe07, @bra09) produce larger amounts of [$^{56}$Ni]{}. This indicates that either the pre-expansion is too weak or the amount of [$^{56}$Ni]{} produced in the deflagration stage is already too large. As it is uncertain whether a suitable single-degenerate model can be found, the possibility of a double-degenerate origin of dim SNe Ia deserves attention.
Besides the abundances, we have obtained information about the density profile of SN 2005bl. We showed that the spectra are incompatible with the presence of significant amounts of oxygen at $v$$\gtrsim$$13000$[km s$^{-1}$]{}. Together with the low abundances of burning products, this indicates a general lack of material at high velocities, albeit less extreme than in objects like SN 2005hk [@sah08]. We tested whether a good fit to the observed spectra is possible using a modified W7 model, scaled to a different total mass and/or energy. Indeed, a reduction of $\sim$$30\%$ in total kinetic energy yielded a spectroscopically, and also energetically consistent Chandrasekhar-mass model (). Such consistence can also be reached with somewhat super- or sub-Chandrasekhar mass density profiles, provided that they are similar to w7e0.7 at $v$$\gtrsim$$13000$[km s$^{-1}$]{}. Deviations of $\gtrsim$$30\%$ from the Chandrasekhar mass seem disfavoured. With our most massive models ($1.45$$M_\textrm{Ch}$), it proved impossible to obtain spectroscopic and energetic consistence at the same time. In addition, these models as well as the least massive ones ($0.5$$M_\textrm{Ch}$) most likely would yield a light curve not matching that of a dim SN Ia.
Sharper constraints on density models, as well as on the abundance structure in the innermost and outermost layers may be obtained from very-early-epoch and nebular spectra of dim SNe Ia. More extensive observations are needed in order to complete our picture of these objects, and of SNe Ia in general.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
This work was supported in part by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00303, ‘The Physics of Type Ia Supernovae’, and by the DFG-TCRC 33 “The Dark Universe”. We would like to thank the anonymous referee for constructive comments.
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Parameters of the models {#app:modelparameters}
========================
Table \[tab:modelparameters\] shows the code input parameters of all spectral models mentioned in the main paper. Apart from the abundances, the code takes as input the photospheric velocity $v_\textrm{ph}$, the time from explosion $t$ (see main text) and the bolometric luminosity $L_\textrm{bol}$. For different models of a given spectrum, these luminosities can differ a bit, depending on the model spectral energy distribution. In addition to the input values, Table \[tab:modelparameters\] also gives the calculated temperature of the photospheric black body emission, $T_\textrm{BB}$, for each model.
------------------------------ ------------------------------ ------------------------------------------------------------------------------------ ------------------------------------------- ------------------------------------- ------------------------------ ------------------------------ ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------ ------------------------------- ------------------------------- ------------------------------- ----------------------------------------------------- ----------------------------------------------------- --
Model $\!\!\!\!$ $\!\!$ epochs $\!\!\!\!\!\!$ $\!$ $\mathrm{lg}\!\left(\frac{L_\textrm{bol}}{{L_{\odot}}}\right)$ $\!\!\!\!\!\!$ $\!\!\!\!$ $v_{\textrm{ph}}$ $\!\!\!\!$ $\!\!\!\!$ $T_\textrm{BB}$ $\!\!\!$
$\!\!\!\!$ $\!\!\!\!$ \[d\] $\!\!$ $\!\!\!\!$ $\!\!\!\!$ $\!\!\!\!$ \[[km s$^{-1}$]{}\] $\!\!\!\!$ $\!\!\!\!$ \[K\] $\!$ $\!\!\!\!$ $X$(C) $\!\!\!\!$ $\!\!\!\!$ $X$(O) $\!\!\!\!$ $\!\!\!\!$ $X$(Na) $\!\!\!\!$ $\!\!\!\!$ $X$(Mg) $\!\!\!\!$ $\!\!\!\!$ $X$(Al) $\!\!\!\!$ $\!\!\!\!$ $X$(Si) $\!\!\!\!$ $\!\!\!\!$ $X$(S) $\!\!\!\!$ $\!\!\!\!$ $X$(Ca) $\!\!\!\!$ $\!\!\!\!$ $X$(Ti) $\!\!\!\!$ $\!\!\!\!$ $X$(Cr) $\!\!\!\!$ $\!\!\!\!$ $X$(Fe)$_{0}{}^{\textrm{a)}}$ $\!\!\!\!$ $\!\!\!\!$ $X$([$^{56}$Ni]{})$_{0}{}^{\textrm{a)}}$
05bl-w7e0.35 $\!\!\!\!$ $\!\!\!\!$ -6 $\!\!\!\!$ $\!\!\!\!$ 8.520 $\!\!\!\!$ $\!\!\!\!$ 8200 $\!\!\!\!$ $\!\!\!\!$ 9574.5 $\!\!\!\!$ $\!\!\!\!$ 0.05 $\!\!\!\!$ $\!\!\!\!$ 0.87 $\!\!\!\!$ $\!\!\!\!$ 0.0080 $\!\!\!\!$ $\!\!\!\!$ 0.03 $\!\!\!\!$ $\!\!\!\!$ 0.0025 $\!\!\!\!$ $\!\!\!\!$ 0.04 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0009 $\!\!\!\!$ $\!\!\!\!$ 0.0016 $\!\!\!\!$ $\!\!\!\!$ 0.0008 $\!\!\!\!$ $\!\!\!\!$ 0.0017 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -5 $\!\!\!\!$ $\!\!\!\!$ 8.622 $\!\!\!\!$ $\!\!\!\!$ 7975 $\!\!\!\!$ $\!\!\!\!$ 10024.7 $\!\!\!\!$ $\!\!\!\!$ 0.02 $\!\!\!\!$ $\!\!\!\!$ 0.20 $\!\!\!\!$ $\!\!\!\!$ 0.0050 $\!\!\!\!$ $\!\!\!\!$ 0.03 $\!\!\!\!$ $\!\!\!\!$ 0.0025 $\!\!\!\!$ $\!\!\!\!$ 0.60 $\!\!\!\!$ $\!\!\!\!$ 0.06 $\!\!\!\!$ $\!\!\!\!$ 0.0009 $\!\!\!\!$ $\!\!\!\!$ 0.0070 $\!\!\!\!$ $\!\!\!\!$ 0.0035 $\!\!\!\!$ $\!\!\!\!$ 0.0750 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -3 $\!\!\!\!$ $\!\!\!\!$ 8.753 $\!\!\!\!$ $\!\!\!\!$ 7600 $\!\!\!\!$ $\!\!\!\!$ 10377.4 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.15 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.04 $\!\!\!\!$ $\!\!\!\!$ 0.0015 $\!\!\!\!$ $\!\!\!\!$ 0.50 $\!\!\!\!$ $\!\!\!\!$ 0.13 $\!\!\!\!$ $\!\!\!\!$ 0.0009 $\!\!\!\!$ $\!\!\!\!$ 0.0500 $\!\!\!\!$ $\!\!\!\!$ 0.0250 $\!\!\!\!$ $\!\!\!\!$ 0.1000 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ 4.8 $\!\!\!\!$ $\!\!\!\!$ 8.870 $\!\!\!\!$ $\!\!\!\!$ 7050 $\!\!\!\!$ $\!\!\!\!$ 8658.6 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.11 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.55 $\!\!\!\!$ $\!\!\!\!$ 0.04 $\!\!\!\!$ $\!\!\!\!$ 0.0009 $\!\!\!\!$ $\!\!\!\!$ 0.0850 $\!\!\!\!$ $\!\!\!\!$ 0.0850 $\!\!\!\!$ $\!\!\!\!$ 0.1100 $\!\!\!\!$ $\!\!\!\!$ 0.0100
$\!\!\!\!$ $\!\!\!\!$ 12.9 $\!\!\!\!$ $\!\!\!\!$ 8.609 $\!\!\!\!$ $\!\!\!\!$ 3350 $\!\!\!\!$ $\!\!\!\!$ 10611.0 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.08 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.65 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0009 $\!\!\!\!$ $\!\!\!\!$ 0.0111 $\!\!\!\!$ $\!\!\!\!$ 0.0111 $\!\!\!\!$ $\!\!\!\!$ 0.0800 $\!\!\!\!$ $\!\!\!\!$ 0.1600
05bl-w7e0.5m0.5 $\!\!\!\!$ $\!\!\!\!$ -6 $\!\!\!\!$ $\!\!\!\!$ 8.510 $\!\!\!\!$ $\!\!\!\!$ 7600 $\!\!\!\!$ $\!\!\!\!$ 9626.0 $\!\!\!\!$ $\!\!\!\!$ 0.06 $\!\!\!\!$ $\!\!\!\!$ 0.72 $\!\!\!\!$ $\!\!\!\!$ 0.0080 $\!\!\!\!$ $\!\!\!\!$ 0.13 $\!\!\!\!$ $\!\!\!\!$ 0.0070 $\!\!\!\!$ $\!\!\!\!$ 0.06 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0010 $\!\!\!\!$ $\!\!\!\!$ 0.0005 $\!\!\!\!$ $\!\!\!\!$ 0.0007 $\!\!\!\!$ $\!\!\!\!$ 0.0003 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -5 $\!\!\!\!$ $\!\!\!\!$ 8.618 $\!\!\!\!$ $\!\!\!\!$ 7275 $\!\!\!\!$ $\!\!\!\!$ 10220.3 $\!\!\!\!$ $\!\!\!\!$ 0.03 $\!\!\!\!$ $\!\!\!\!$ 0.17 $\!\!\!\!$ $\!\!\!\!$ 0.0040 $\!\!\!\!$ $\!\!\!\!$ 0.08 $\!\!\!\!$ $\!\!\!\!$ 0.0070 $\!\!\!\!$ $\!\!\!\!$ 0.60 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0010 $\!\!\!\!$ $\!\!\!\!$ 0.0300 $\!\!\!\!$ $\!\!\!\!$ 0.0300 $\!\!\!\!$ $\!\!\!\!$ 0.0400 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -3 $\!\!\!\!$ $\!\!\!\!$ 8.736 $\!\!\!\!$ $\!\!\!\!$ 7000 $\!\!\!\!$ $\!\!\!\!$ 10242.6 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0030 $\!\!\!\!$ $\!\!\!\!$ 0.52 $\!\!\!\!$ $\!\!\!\!$ 0.12 $\!\!\!\!$ $\!\!\!\!$ 0.0010 $\!\!\!\!$ $\!\!\!\!$ 0.1100 $\!\!\!\!$ $\!\!\!\!$ 0.1100 $\!\!\!\!$ $\!\!\!\!$ 0.1250 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ 4.8 $\!\!\!\!$ $\!\!\!\!$ 8.837 $\!\!\!\!$ $\!\!\!\!$ 6400 $\!\!\!\!$ $\!\!\!\!$ 8613.5 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.17 $\!\!\!\!$ $\!\!\!\!$ 0.04 $\!\!\!\!$ $\!\!\!\!$ 0.0010 $\!\!\!\!$ $\!\!\!\!$ 0.2550 $\!\!\!\!$ $\!\!\!\!$ 0.2550 $\!\!\!\!$ $\!\!\!\!$ 0.2550 $\!\!\!\!$ $\!\!\!\!$ 0.0100
$\!\!\!\!$ $\!\!\!\!$ 12.9 $\!\!\!\!$ $\!\!\!\!$ 8.570 $\!\!\!\!$ $\!\!\!\!$ 3250 $\!\!\!\!$ $\!\!\!\!$ 9195.7 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.17 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0010 $\!\!\!\!$ $\!\!\!\!$ 0.2150 $\!\!\!\!$ $\!\!\!\!$ 0.2150 $\!\!\!\!$ $\!\!\!\!$ 0.1300 $\!\!\!\!$ $\!\!\!\!$ 0.2600
05bl-w7e0.5m0.7 $\!\!\!\!$ $\!\!\!\!$ -6 $\!\!\!\!$ $\!\!\!\!$ 8.520 $\!\!\!\!$ $\!\!\!\!$ 7900 $\!\!\!\!$ $\!\!\!\!$ 9734.1 $\!\!\!\!$ $\!\!\!\!$ 0.06 $\!\!\!\!$ $\!\!\!\!$ 0.83 $\!\!\!\!$ $\!\!\!\!$ 0.0060 $\!\!\!\!$ $\!\!\!\!$ 0.06 $\!\!\!\!$ $\!\!\!\!$ 0.0025 $\!\!\!\!$ $\!\!\!\!$ 0.03 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0008 $\!\!\!\!$ $\!\!\!\!$ 0.0005 $\!\!\!\!$ $\!\!\!\!$ 0.0004 $\!\!\!\!$ $\!\!\!\!$ 0.0004 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -5 $\!\!\!\!$ $\!\!\!\!$ 8.618 $\!\!\!\!$ $\!\!\!\!$ 7600 $\!\!\!\!$ $\!\!\!\!$ 10242.2 $\!\!\!\!$ $\!\!\!\!$ 0.04 $\!\!\!\!$ $\!\!\!\!$ 0.12 $\!\!\!\!$ $\!\!\!\!$ 0.0030 $\!\!\!\!$ $\!\!\!\!$ 0.04 $\!\!\!\!$ $\!\!\!\!$ 0.0025 $\!\!\!\!$ $\!\!\!\!$ 0.62 $\!\!\!\!$ $\!\!\!\!$ 0.10 $\!\!\!\!$ $\!\!\!\!$ 0.0008 $\!\!\!\!$ $\!\!\!\!$ 0.0150 $\!\!\!\!$ $\!\!\!\!$ 0.0150 $\!\!\!\!$ $\!\!\!\!$ 0.0400 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -3 $\!\!\!\!$ $\!\!\!\!$ 8.740 $\!\!\!\!$ $\!\!\!\!$ 7100 $\!\!\!\!$ $\!\!\!\!$ 10727.9 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.03 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0017 $\!\!\!\!$ $\!\!\!\!$ 0.55 $\!\!\!\!$ $\!\!\!\!$ 0.15 $\!\!\!\!$ $\!\!\!\!$ 0.0008 $\!\!\!\!$ $\!\!\!\!$ 0.0750 $\!\!\!\!$ $\!\!\!\!$ 0.0750 $\!\!\!\!$ $\!\!\!\!$ 0.1200 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ 4.8 $\!\!\!\!$ $\!\!\!\!$ 8.845 $\!\!\!\!$ $\!\!\!\!$ 6550 $\!\!\!\!$ $\!\!\!\!$ 8764.9 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.53 $\!\!\!\!$ $\!\!\!\!$ 0.05 $\!\!\!\!$ $\!\!\!\!$ 0.0008 $\!\!\!\!$ $\!\!\!\!$ 0.1225 $\!\!\!\!$ $\!\!\!\!$ 0.1225 $\!\!\!\!$ $\!\!\!\!$ 0.1650 $\!\!\!\!$ $\!\!\!\!$ 0.0100
$\!\!\!\!$ $\!\!\!\!$ 12.9 $\!\!\!\!$ $\!\!\!\!$ 8.590 $\!\!\!\!$ $\!\!\!\!$ 3350 $\!\!\!\!$ $\!\!\!\!$ 9582.3 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.68 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0008 $\!\!\!\!$ $\!\!\!\!$ 0.0400 $\!\!\!\!$ $\!\!\!\!$ 0.0400 $\!\!\!\!$ $\!\!\!\!$ 0.0800 $\!\!\!\!$ $\!\!\!\!$ 0.1550
05bl-w7e0.5 $\!\!\!\!$ $\!\!\!\!$ -6 $\!\!\!\!$ $\!\!\!\!$ 8.525 $\!\!\!\!$ $\!\!\!\!$ 8300 $\!\!\!\!$ $\!\!\!\!$ 9705.2 $\!\!\!\!$ $\!\!\!\!$ 0.05 $\!\!\!\!$ $\!\!\!\!$ 0.88 $\!\!\!\!$ $\!\!\!\!$ 0.0070 $\!\!\!\!$ $\!\!\!\!$ 0.03 $\!\!\!\!$ $\!\!\!\!$ 0.0020 $\!\!\!\!$ $\!\!\!\!$ 0.03 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0004 $\!\!\!\!$ $\!\!\!\!$ 0.0008 $\!\!\!\!$ $\!\!\!\!$ 0.0004 $\!\!\!\!$ $\!\!\!\!$ 0.0008 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -5 $\!\!\!\!$ $\!\!\!\!$ 8.625 $\!\!\!\!$ $\!\!\!\!$ 8050 $\!\!\!\!$ $\!\!\!\!$ 10185.4 $\!\!\!\!$ $\!\!\!\!$ 0.02 $\!\!\!\!$ $\!\!\!\!$ 0.12 $\!\!\!\!$ $\!\!\!\!$ 0.0035 $\!\!\!\!$ $\!\!\!\!$ 0.02 $\!\!\!\!$ $\!\!\!\!$ 0.0020 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ 0.07 $\!\!\!\!$ $\!\!\!\!$ 0.0004 $\!\!\!\!$ $\!\!\!\!$ 0.0300 $\!\!\!\!$ $\!\!\!\!$ 0.0250 $\!\!\!\!$ $\!\!\!\!$ 0.0100 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -3 $\!\!\!\!$ $\!\!\!\!$ 8.748 $\!\!\!\!$ $\!\!\!\!$ 7550 $\!\!\!\!$ $\!\!\!\!$ 10745.4 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0010 $\!\!\!\!$ $\!\!\!\!$ 0.62 $\!\!\!\!$ $\!\!\!\!$ 0.14 $\!\!\!\!$ $\!\!\!\!$ 0.0004 $\!\!\!\!$ $\!\!\!\!$ 0.0625 $\!\!\!\!$ $\!\!\!\!$ 0.0625 $\!\!\!\!$ $\!\!\!\!$ 0.1000 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ 4.8 $\!\!\!\!$ $\!\!\!\!$ 8.866 $\!\!\!\!$ $\!\!\!\!$ 6900 $\!\!\!\!$ $\!\!\!\!$ 8959.2 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.62 $\!\!\!\!$ $\!\!\!\!$ 0.06 $\!\!\!\!$ $\!\!\!\!$ 0.0004 $\!\!\!\!$ $\!\!\!\!$ 0.0675 $\!\!\!\!$ $\!\!\!\!$ 0.0675 $\!\!\!\!$ $\!\!\!\!$ 0.1600 $\!\!\!\!$ $\!\!\!\!$ 0.0100
$\!\!\!\!$ $\!\!\!\!$ 12.9 $\!\!\!\!$ $\!\!\!\!$ 8.602 $\!\!\!\!$ $\!\!\!\!$ 3350 $\!\!\!\!$ $\!\!\!\!$ 10432.1 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.78 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0004 $\!\!\!\!$ $\!\!\!\!$ 0.0190 $\!\!\!\!$ $\!\!\!\!$ 0.0190 $\!\!\!\!$ $\!\!\!\!$ 0.0600 $\!\!\!\!$ $\!\!\!\!$ 0.1200
05bl-w7e0.5m1.25 $\!\!\!\!$ $\!\!\!\!$ -6 $\!\!\!\!$ $\!\!\!\!$ 8.517 $\!\!\!\!$ $\!\!\!\!$ 8250 $\!\!\!\!$ $\!\!\!\!$ 10038.1 $\!\!\!\!$ $\!\!\!\!$ 0.05 $\!\!\!\!$ $\!\!\!\!$ 0.87 $\!\!\!\!$ $\!\!\!\!$ 0.0045 $\!\!\!\!$ $\!\!\!\!$ 0.04 $\!\!\!\!$ $\!\!\!\!$ 0.0018 $\!\!\!\!$ $\!\!\!\!$ 0.03 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0006 $\!\!\!\!$ $\!\!\!\!$ 0.0010 $\!\!\!\!$ $\!\!\!\!$ 0.0008 $\!\!\!\!$ $\!\!\!\!$ 0.0013 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -5 $\!\!\!\!$ $\!\!\!\!$ 8.615 $\!\!\!\!$ $\!\!\!\!$ 8000 $\!\!\!\!$ $\!\!\!\!$ 10511.8 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.13 $\!\!\!\!$ $\!\!\!\!$ 0.0045 $\!\!\!\!$ $\!\!\!\!$ 0.04 $\!\!\!\!$ $\!\!\!\!$ 0.0018 $\!\!\!\!$ $\!\!\!\!$ 0.62 $\!\!\!\!$ $\!\!\!\!$ 0.12 $\!\!\!\!$ $\!\!\!\!$ 0.0006 $\!\!\!\!$ $\!\!\!\!$ 0.0220 $\!\!\!\!$ $\!\!\!\!$ 0.0220 $\!\!\!\!$ $\!\!\!\!$ 0.0250 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -3 $\!\!\!\!$ $\!\!\!\!$ 8.748 $\!\!\!\!$ $\!\!\!\!$ 7650 $\!\!\!\!$ $\!\!\!\!$ 10658.7 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.02 $\!\!\!\!$ $\!\!\!\!$ 0.0020 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0010 $\!\!\!\!$ $\!\!\!\!$ 0.64 $\!\!\!\!$ $\!\!\!\!$ 0.16 $\!\!\!\!$ $\!\!\!\!$ 0.0006 $\!\!\!\!$ $\!\!\!\!$ 0.0600 $\!\!\!\!$ $\!\!\!\!$ 0.0600 $\!\!\!\!$ $\!\!\!\!$ 0.0600 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ 4.8 $\!\!\!\!$ $\!\!\!\!$ 8.861 $\!\!\!\!$ $\!\!\!\!$ 7250 $\!\!\!\!$ $\!\!\!\!$ 8628.6 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.65 $\!\!\!\!$ $\!\!\!\!$ 0.07 $\!\!\!\!$ $\!\!\!\!$ 0.0006 $\!\!\!\!$ $\!\!\!\!$ 0.0900 $\!\!\!\!$ $\!\!\!\!$ 0.0900 $\!\!\!\!$ $\!\!\!\!$ 0.0850 $\!\!\!\!$ $\!\!\!\!$ 0.0085
$\!\!\!\!$ $\!\!\!\!$ 12.9 $\!\!\!\!$ $\!\!\!\!$ 8.595 $\!\!\!\!$ $\!\!\!\!$ 3350 $\!\!\!\!$ $\!\!\!\!$ 10835.7 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.82 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0006 $\!\!\!\!$ $\!\!\!\!$ 0.0070 $\!\!\!\!$ $\!\!\!\!$ 0.0070 $\!\!\!\!$ $\!\!\!\!$ 0.0500 $\!\!\!\!$ $\!\!\!\!$ 0.1100
05bl-w7e0.7m0.7 $\!\!\!\!$ $\!\!\!\!$ -6 $\!\!\!\!$ $\!\!\!\!$ 8.514 $\!\!\!\!$ $\!\!\!\!$ 7950 $\!\!\!\!$ $\!\!\!\!$ 9749.2 $\!\!\!\!$ $\!\!\!\!$ 0.07 $\!\!\!\!$ $\!\!\!\!$ 0.75 $\!\!\!\!$ $\!\!\!\!$ 0.0050 $\!\!\!\!$ $\!\!\!\!$ 0.10 $\!\!\!\!$ $\!\!\!\!$ 0.0035 $\!\!\!\!$ $\!\!\!\!$ 0.04 $\!\!\!\!$ $\!\!\!\!$ 0.03 $\!\!\!\!$ $\!\!\!\!$ 0.0009 $\!\!\!\!$ $\!\!\!\!$ 0.0004 $\!\!\!\!$ $\!\!\!\!$ 0.0004 $\!\!\!\!$ $\!\!\!\!$ 0.0003 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -5 $\!\!\!\!$ $\!\!\!\!$ 8.619 $\!\!\!\!$ $\!\!\!\!$ 7600 $\!\!\!\!$ $\!\!\!\!$ 10346.2 $\!\!\!\!$ $\!\!\!\!$ 0.02 $\!\!\!\!$ $\!\!\!\!$ 0.10 $\!\!\!\!$ $\!\!\!\!$ 0.0025 $\!\!\!\!$ $\!\!\!\!$ 0.03 $\!\!\!\!$ $\!\!\!\!$ 0.0035 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ 0.07 $\!\!\!\!$ $\!\!\!\!$ 0.0009 $\!\!\!\!$ $\!\!\!\!$ 0.0200 $\!\!\!\!$ $\!\!\!\!$ 0.0200 $\!\!\!\!$ $\!\!\!\!$ 0.0300 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -3 $\!\!\!\!$ $\!\!\!\!$ 8.744 $\!\!\!\!$ $\!\!\!\!$ 7100 $\!\!\!\!$ $\!\!\!\!$ 10760.9 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.02 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0018 $\!\!\!\!$ $\!\!\!\!$ 0.60 $\!\!\!\!$ $\!\!\!\!$ 0.09 $\!\!\!\!$ $\!\!\!\!$ 0.0009 $\!\!\!\!$ $\!\!\!\!$ 0.0700 $\!\!\!\!$ $\!\!\!\!$ 0.0700 $\!\!\!\!$ $\!\!\!\!$ 0.1500 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ 4.8 $\!\!\!\!$ $\!\!\!\!$ 8.843 $\!\!\!\!$ $\!\!\!\!$ 6550 $\!\!\!\!$ $\!\!\!\!$ 8730.7 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.51 $\!\!\!\!$ $\!\!\!\!$ 0.07 $\!\!\!\!$ $\!\!\!\!$ 0.0009 $\!\!\!\!$ $\!\!\!\!$ 0.1200 $\!\!\!\!$ $\!\!\!\!$ 0.1200 $\!\!\!\!$ $\!\!\!\!$ 0.1600 $\!\!\!\!$ $\!\!\!\!$ 0.0100
$\!\!\!\!$ $\!\!\!\!$ 12.9 $\!\!\!\!$ $\!\!\!\!$ 8.587 $\!\!\!\!$ $\!\!\!\!$ 3250 $\!\!\!\!$ $\!\!\!\!$ 9539.3 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.50 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0009 $\!\!\!\!$ $\!\!\!\!$ 0.0590 $\!\!\!\!$ $\!\!\!\!$ 0.0590 $\!\!\!\!$ $\!\!\!\!$ 0.1250 $\!\!\!\!$ $\!\!\!\!$ 0.2500
05bl-w7e0.7 $\!\!\!\!$ $\!\!\!\!$ -6 $\!\!\!\!$ $\!\!\!\!$ 8.520 $\!\!\!\!$ $\!\!\!\!$ 8350 $\!\!\!\!$ $\!\!\!\!$ 9764.8 $\!\!\!\!$ $\!\!\!\!$ 0.06 $\!\!\!\!$ $\!\!\!\!$ 0.86 $\!\!\!\!$ $\!\!\!\!$ 0.0060 $\!\!\!\!$ $\!\!\!\!$ 0.04 $\!\!\!\!$ $\!\!\!\!$ 0.0025 $\!\!\!\!$ $\!\!\!\!$ 0.02 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0004 $\!\!\!\!$ $\!\!\!\!$ 0.0004 $\!\!\!\!$ $\!\!\!\!$ 0.0003 $\!\!\!\!$ $\!\!\!\!$ 0.0003 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -5 $\!\!\!\!$ $\!\!\!\!$ 8.617 $\!\!\!\!$ $\!\!\!\!$ 8100 $\!\!\!\!$ $\!\!\!\!$ 10135.5 $\!\!\!\!$ $\!\!\!\!$ 0.03 $\!\!\!\!$ $\!\!\!\!$ 0.13 $\!\!\!\!$ $\!\!\!\!$ 0.0030 $\!\!\!\!$ $\!\!\!\!$ 0.03 $\!\!\!\!$ $\!\!\!\!$ 0.0025 $\!\!\!\!$ $\!\!\!\!$ 0.68 $\!\!\!\!$ $\!\!\!\!$ 0.10 $\!\!\!\!$ $\!\!\!\!$ 0.0004 $\!\!\!\!$ $\!\!\!\!$ 0.0100 $\!\!\!\!$ $\!\!\!\!$ 0.0070 $\!\!\!\!$ $\!\!\!\!$ 0.0150 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -3 $\!\!\!\!$ $\!\!\!\!$ 8.745 $\!\!\!\!$ $\!\!\!\!$ 7600 $\!\!\!\!$ $\!\!\!\!$ 10632.8 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0015 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ 0.11 $\!\!\!\!$ $\!\!\!\!$ 0.0004 $\!\!\!\!$ $\!\!\!\!$ 0.0533 $\!\!\!\!$ $\!\!\!\!$ 0.0367 $\!\!\!\!$ $\!\!\!\!$ 0.0900 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ 4.8 $\!\!\!\!$ $\!\!\!\!$ 8.861 $\!\!\!\!$ $\!\!\!\!$ 6800 $\!\!\!\!$ $\!\!\!\!$ 8930.6 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.71 $\!\!\!\!$ $\!\!\!\!$ 0.07 $\!\!\!\!$ $\!\!\!\!$ 0.0004 $\!\!\!\!$ $\!\!\!\!$ 0.0550 $\!\!\!\!$ $\!\!\!\!$ 0.0400 $\!\!\!\!$ $\!\!\!\!$ 0.1150 $\!\!\!\!$ $\!\!\!\!$ 0.0100
$\!\!\!\!$ $\!\!\!\!$ 12.9 $\!\!\!\!$ $\!\!\!\!$ 8.594 $\!\!\!\!$ $\!\!\!\!$ 3350 $\!\!\!\!$ $\!\!\!\!$ 10071.5 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.77 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0005 $\!\!\!\!$ $\!\!\!\!$ 0.0167 $\!\!\!\!$ $\!\!\!\!$ 0.0167 $\!\!\!\!$ $\!\!\!\!$ 0.0650 $\!\!\!\!$ $\!\!\!\!$ 0.1300
05bl-w7e0.7m1.25 $\!\!\!\!$ $\!\!\!\!$ -6 $\!\!\!\!$ $\!\!\!\!$ 8.523 $\!\!\!\!$ $\!\!\!\!$ 8700 $\!\!\!\!$ $\!\!\!\!$ 9667.9 $\!\!\!\!$ $\!\!\!\!$ 0.06 $\!\!\!\!$ $\!\!\!\!$ 0.86 $\!\!\!\!$ $\!\!\!\!$ 0.0045 $\!\!\!\!$ $\!\!\!\!$ 0.03 $\!\!\!\!$ $\!\!\!\!$ 0.0020 $\!\!\!\!$ $\!\!\!\!$ 0.03 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0006 $\!\!\!\!$ $\!\!\!\!$ 0.0005 $\!\!\!\!$ $\!\!\!\!$ 0.0005 $\!\!\!\!$ $\!\!\!\!$ 0.0006 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -5 $\!\!\!\!$ $\!\!\!\!$ 8.623 $\!\!\!\!$ $\!\!\!\!$ 8400 $\!\!\!\!$ $\!\!\!\!$ 10101.1 $\!\!\!\!$ $\!\!\!\!$ 0.03 $\!\!\!\!$ $\!\!\!\!$ 0.13 $\!\!\!\!$ $\!\!\!\!$ 0.0020 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0020 $\!\!\!\!$ $\!\!\!\!$ 0.70 $\!\!\!\!$ $\!\!\!\!$ 0.10 $\!\!\!\!$ $\!\!\!\!$ 0.0008 $\!\!\!\!$ $\!\!\!\!$ 0.0100 $\!\!\!\!$ $\!\!\!\!$ 0.0075 $\!\!\!\!$ $\!\!\!\!$ 0.0100 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -3 $\!\!\!\!$ $\!\!\!\!$ 8.750 $\!\!\!\!$ $\!\!\!\!$ 7950 $\!\!\!\!$ $\!\!\!\!$ 10346.8 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0010 $\!\!\!\!$ $\!\!\!\!$ 0.80 $\!\!\!\!$ $\!\!\!\!$ 0.14 $\!\!\!\!$ $\!\!\!\!$ 0.0008 $\!\!\!\!$ $\!\!\!\!$ 0.0110 $\!\!\!\!$ $\!\!\!\!$ 0.0080 $\!\!\!\!$ $\!\!\!\!$ 0.0400 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ 4.8 $\!\!\!\!$ $\!\!\!\!$ 8.858 $\!\!\!\!$ $\!\!\!\!$ 7150 $\!\!\!\!$ $\!\!\!\!$ 8750.8 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.76 $\!\!\!\!$ $\!\!\!\!$ 0.08 $\!\!\!\!$ $\!\!\!\!$ 0.0008 $\!\!\!\!$ $\!\!\!\!$ 0.0290 $\!\!\!\!$ $\!\!\!\!$ 0.0240 $\!\!\!\!$ $\!\!\!\!$ 0.0900 $\!\!\!\!$ $\!\!\!\!$ 0.0100
$\!\!\!\!$ $\!\!\!\!$ 12.9 $\!\!\!\!$ $\!\!\!\!$ 8.594 $\!\!\!\!$ $\!\!\!\!$ 3350 $\!\!\!\!$ $\!\!\!\!$ 10598.5 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.84 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0008 $\!\!\!\!$ $\!\!\!\!$ 0.0140 $\!\!\!\!$ $\!\!\!\!$ 0.0140 $\!\!\!\!$ $\!\!\!\!$ 0.0400 $\!\!\!\!$ $\!\!\!\!$ 0.0800
05bl-w7e0.7m1.45 $\!\!\!\!$ $\!\!\!\!$ -6 $\!\!\!\!$ $\!\!\!\!$ 8.519 $\!\!\!\!$ $\!\!\!\!$ 8700 $\!\!\!\!$ $\!\!\!\!$ 9896.0 $\!\!\!\!$ $\!\!\!\!$ 0.09 $\!\!\!\!$ $\!\!\!\!$ 0.82 $\!\!\!\!$ $\!\!\!\!$ 0.0055 $\!\!\!\!$ $\!\!\!\!$ 0.04 $\!\!\!\!$ $\!\!\!\!$ 0.0020 $\!\!\!\!$ $\!\!\!\!$ 0.03 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0006 $\!\!\!\!$ $\!\!\!\!$ 0.0006 $\!\!\!\!$ $\!\!\!\!$ 0.0006 $\!\!\!\!$ $\!\!\!\!$ 0.0009 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -5 $\!\!\!\!$ $\!\!\!\!$ 8.620 $\!\!\!\!$ $\!\!\!\!$ 8450 $\!\!\!\!$ $\!\!\!\!$ 10296.5 $\!\!\!\!$ $\!\!\!\!$ 0.07 $\!\!\!\!$ $\!\!\!\!$ 0.13 $\!\!\!\!$ $\!\!\!\!$ 0.0030 $\!\!\!\!$ $\!\!\!\!$ 0.04 $\!\!\!\!$ $\!\!\!\!$ 0.0020 $\!\!\!\!$ $\!\!\!\!$ 0.60 $\!\!\!\!$ $\!\!\!\!$ 0.10 $\!\!\!\!$ $\!\!\!\!$ 0.0006 $\!\!\!\!$ $\!\!\!\!$ 0.0220 $\!\!\!\!$ $\!\!\!\!$ 0.0220 $\!\!\!\!$ $\!\!\!\!$ 0.0150 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -3 $\!\!\!\!$ $\!\!\!\!$ 8.752 $\!\!\!\!$ $\!\!\!\!$ 8200 $\!\!\!\!$ $\!\!\!\!$ 10222.8 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0010 $\!\!\!\!$ $\!\!\!\!$ 0.76 $\!\!\!\!$ $\!\!\!\!$ 0.12 $\!\!\!\!$ $\!\!\!\!$ 0.0013 $\!\!\!\!$ $\!\!\!\!$ 0.0275 $\!\!\!\!$ $\!\!\!\!$ 0.0275 $\!\!\!\!$ $\!\!\!\!$ 0.0450 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ 4.8 $\!\!\!\!$ $\!\!\!\!$ 8.864 $\!\!\!\!$ $\!\!\!\!$ 7350 $\!\!\!\!$ $\!\!\!\!$ 8761.1 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.78 $\!\!\!\!$ $\!\!\!\!$ 0.07 $\!\!\!\!$ $\!\!\!\!$ 0.0006 $\!\!\!\!$ $\!\!\!\!$ 0.0333 $\!\!\!\!$ $\!\!\!\!$ 0.0333 $\!\!\!\!$ $\!\!\!\!$ 0.0700 $\!\!\!\!$ $\!\!\!\!$ 0.0070
$\!\!\!\!$ $\!\!\!\!$ 12.9 $\!\!\!\!$ $\!\!\!\!$ 8.605 $\!\!\!\!$ $\!\!\!\!$ 3525 $\!\!\!\!$ $\!\!\!\!$ 10682.2 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.87 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0006 $\!\!\!\!$ $\!\!\!\!$ 0.0170 $\!\!\!\!$ $\!\!\!\!$ 0.0170 $\!\!\!\!$ $\!\!\!\!$ 0.0250 $\!\!\!\!$ $\!\!\!\!$ 0.0650
05bl-w7m0.7 $\!\!\!\!$ $\!\!\!\!$ -6 $\!\!\!\!$ $\!\!\!\!$ 8.520 $\!\!\!\!$ $\!\!\!\!$ 8150 $\!\!\!\!$ $\!\!\!\!$ 9593.4 $\!\!\!\!$ $\!\!\!\!$ 0.06 $\!\!\!\!$ $\!\!\!\!$ 0.80 $\!\!\!\!$ $\!\!\!\!$ 0.0060 $\!\!\!\!$ $\!\!\!\!$ 0.05 $\!\!\!\!$ $\!\!\!\!$ 0.0060 $\!\!\!\!$ $\!\!\!\!$ 0.05 $\!\!\!\!$ $\!\!\!\!$ 0.02 $\!\!\!\!$ $\!\!\!\!$ 0.0010 $\!\!\!\!$ $\!\!\!\!$ 0.0004 $\!\!\!\!$ $\!\!\!\!$ 0.0008 $\!\!\!\!$ $\!\!\!\!$ 0.0003 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -5 $\!\!\!\!$ $\!\!\!\!$ 8.630 $\!\!\!\!$ $\!\!\!\!$ 7850 $\!\!\!\!$ $\!\!\!\!$ 9927.4 $\!\!\!\!$ $\!\!\!\!$ 0.03 $\!\!\!\!$ $\!\!\!\!$ 0.11 $\!\!\!\!$ $\!\!\!\!$ 0.0020 $\!\!\!\!$ $\!\!\!\!$ 0.04 $\!\!\!\!$ $\!\!\!\!$ 0.0050 $\!\!\!\!$ $\!\!\!\!$ 0.72 $\!\!\!\!$ $\!\!\!\!$ 0.07 $\!\!\!\!$ $\!\!\!\!$ 0.0010 $\!\!\!\!$ $\!\!\!\!$ 0.0050 $\!\!\!\!$ $\!\!\!\!$ 0.0050 $\!\!\!\!$ $\!\!\!\!$ 0.0050 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -3 $\!\!\!\!$ $\!\!\!\!$ 8.745 $\!\!\!\!$ $\!\!\!\!$ 7475 $\!\!\!\!$ $\!\!\!\!$ 9998.2 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0020 $\!\!\!\!$ $\!\!\!\!$ 0.72 $\!\!\!\!$ $\!\!\!\!$ 0.15 $\!\!\!\!$ $\!\!\!\!$ 0.0010 $\!\!\!\!$ $\!\!\!\!$ 0.0375 $\!\!\!\!$ $\!\!\!\!$ 0.0375 $\!\!\!\!$ $\!\!\!\!$ 0.0475 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ 4.8 $\!\!\!\!$ $\!\!\!\!$ 8.844 $\!\!\!\!$ $\!\!\!\!$ 6550 $\!\!\!\!$ $\!\!\!\!$ 8654.3 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.60 $\!\!\!\!$ $\!\!\!\!$ 0.05 $\!\!\!\!$ $\!\!\!\!$ 0.0010 $\!\!\!\!$ $\!\!\!\!$ 0.0900 $\!\!\!\!$ $\!\!\!\!$ 0.0900 $\!\!\!\!$ $\!\!\!\!$ 0.1500 $\!\!\!\!$ $\!\!\!\!$ 0.0100
$\!\!\!\!$ $\!\!\!\!$ 12.9 $\!\!\!\!$ $\!\!\!\!$ 8.566 $\!\!\!\!$ $\!\!\!\!$ 3250 $\!\!\!\!$ $\!\!\!\!$ 9133.8 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.46 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0005 $\!\!\!\!$ $\!\!\!\!$ 0.0800 $\!\!\!\!$ $\!\!\!\!$ 0.0800 $\!\!\!\!$ $\!\!\!\!$ 0.1250 $\!\!\!\!$ $\!\!\!\!$ 0.2500
05bl-w7 $\!\!\!\!$ $\!\!\!\!$ -6 $\!\!\!\!$ $\!\!\!\!$ 8.525 $\!\!\!\!$ $\!\!\!\!$ 8400 $\!\!\!\!$ $\!\!\!\!$ 9845.6 $\!\!\!\!$ $\!\!\!\!$ 0.05 $\!\!\!\!$ $\!\!\!\!$ 0.84 $\!\!\!\!$ $\!\!\!\!$ 0.0025 $\!\!\!\!$ $\!\!\!\!$ 0.03 $\!\!\!\!$ $\!\!\!\!$ 0.0032 $\!\!\!\!$ $\!\!\!\!$ 0.05 $\!\!\!\!$ $\!\!\!\!$ 0.02 $\!\!\!\!$ $\!\!\!\!$ 0.0003 $\!\!\!\!$ $\!\!\!\!$ 0.0004 $\!\!\!\!$ $\!\!\!\!$ 0.0004 $\!\!\!\!$ $\!\!\!\!$ 0.0001 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -5 $\!\!\!\!$ $\!\!\!\!$ 8.627 $\!\!\!\!$ $\!\!\!\!$ 8100 $\!\!\!\!$ $\!\!\!\!$ 10291.1 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.17 $\!\!\!\!$ $\!\!\!\!$ 0.0013 $\!\!\!\!$ $\!\!\!\!$ 0.04 $\!\!\!\!$ $\!\!\!\!$ 0.0032 $\!\!\!\!$ $\!\!\!\!$ 0.65 $\!\!\!\!$ $\!\!\!\!$ 0.10 $\!\!\!\!$ $\!\!\!\!$ 0.0003 $\!\!\!\!$ $\!\!\!\!$ 0.0100 $\!\!\!\!$ $\!\!\!\!$ 0.0100 $\!\!\!\!$ $\!\!\!\!$ 0.0175 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -3 $\!\!\!\!$ $\!\!\!\!$ 8.754 $\!\!\!\!$ $\!\!\!\!$ 7500 $\!\!\!\!$ $\!\!\!\!$ 10859.7 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0011 $\!\!\!\!$ $\!\!\!\!$ 0.69 $\!\!\!\!$ $\!\!\!\!$ 0.12 $\!\!\!\!$ $\!\!\!\!$ 0.0003 $\!\!\!\!$ $\!\!\!\!$ 0.0500 $\!\!\!\!$ $\!\!\!\!$ 0.0500 $\!\!\!\!$ $\!\!\!\!$ 0.0800 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ 4.8 $\!\!\!\!$ $\!\!\!\!$ 8.859 $\!\!\!\!$ $\!\!\!\!$ 6600 $\!\!\!\!$ $\!\!\!\!$ 9110.3 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.69 $\!\!\!\!$ $\!\!\!\!$ 0.06 $\!\!\!\!$ $\!\!\!\!$ 0.0003 $\!\!\!\!$ $\!\!\!\!$ 0.0675 $\!\!\!\!$ $\!\!\!\!$ 0.0675 $\!\!\!\!$ $\!\!\!\!$ 0.1000 $\!\!\!\!$ $\!\!\!\!$ 0.0100
$\!\!\!\!$ $\!\!\!\!$ 12.9 $\!\!\!\!$ $\!\!\!\!$ 8.594 $\!\!\!\!$ $\!\!\!\!$ 3300 $\!\!\!\!$ $\!\!\!\!$ 9957.4 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.64 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0003 $\!\!\!\!$ $\!\!\!\!$ 0.0350 $\!\!\!\!$ $\!\!\!\!$ 0.0350 $\!\!\!\!$ $\!\!\!\!$ 0.0950 $\!\!\!\!$ $\!\!\!\!$ 0.1900
05bl-w7m1.25 $\!\!\!\!$ $\!\!\!\!$ -6 $\!\!\!\!$ $\!\!\!\!$ 8.515 $\!\!\!\!$ $\!\!\!\!$ 8800 $\!\!\!\!$ $\!\!\!\!$ 9685.6 $\!\!\!\!$ $\!\!\!\!$ 0.08 $\!\!\!\!$ $\!\!\!\!$ 0.81 $\!\!\!\!$ $\!\!\!\!$ 0.0040 $\!\!\!\!$ $\!\!\!\!$ 0.05 $\!\!\!\!$ $\!\!\!\!$ 0.0023 $\!\!\!\!$ $\!\!\!\!$ 0.04 $\!\!\!\!$ $\!\!\!\!$ 0.02 $\!\!\!\!$ $\!\!\!\!$ 0.0003 $\!\!\!\!$ $\!\!\!\!$ 0.0002 $\!\!\!\!$ $\!\!\!\!$ 0.0002 $\!\!\!\!$ $\!\!\!\!$ 0.0002 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -5 $\!\!\!\!$ $\!\!\!\!$ 8.622 $\!\!\!\!$ $\!\!\!\!$ 8500 $\!\!\!\!$ $\!\!\!\!$ 10207.5 $\!\!\!\!$ $\!\!\!\!$ 0.05 $\!\!\!\!$ $\!\!\!\!$ 0.12 $\!\!\!\!$ $\!\!\!\!$ 0.0040 $\!\!\!\!$ $\!\!\!\!$ 0.05 $\!\!\!\!$ $\!\!\!\!$ 0.0023 $\!\!\!\!$ $\!\!\!\!$ 0.65 $\!\!\!\!$ $\!\!\!\!$ 0.08 $\!\!\!\!$ $\!\!\!\!$ 0.0003 $\!\!\!\!$ $\!\!\!\!$ 0.0160 $\!\!\!\!$ $\!\!\!\!$ 0.0120 $\!\!\!\!$ $\!\!\!\!$ 0.0190 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -3 $\!\!\!\!$ $\!\!\!\!$ 8.756 $\!\!\!\!$ $\!\!\!\!$ 7950 $\!\!\!\!$ $\!\!\!\!$ 10729.3 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0010 $\!\!\!\!$ $\!\!\!\!$ 0.72 $\!\!\!\!$ $\!\!\!\!$ 0.15 $\!\!\!\!$ $\!\!\!\!$ 0.0003 $\!\!\!\!$ $\!\!\!\!$ 0.0425 $\!\!\!\!$ $\!\!\!\!$ 0.0400 $\!\!\!\!$ $\!\!\!\!$ 0.0400 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ 4.8 $\!\!\!\!$ $\!\!\!\!$ 8.862 $\!\!\!\!$ $\!\!\!\!$ 7000 $\!\!\!\!$ $\!\!\!\!$ 8996.5 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.78 $\!\!\!\!$ $\!\!\!\!$ 0.09 $\!\!\!\!$ $\!\!\!\!$ 0.0003 $\!\!\!\!$ $\!\!\!\!$ 0.0450 $\!\!\!\!$ $\!\!\!\!$ 0.0450 $\!\!\!\!$ $\!\!\!\!$ 0.0400 $\!\!\!\!$ $\!\!\!\!$ 0.0050
$\!\!\!\!$ $\!\!\!\!$ 12.9 $\!\!\!\!$ $\!\!\!\!$ 8.598 $\!\!\!\!$ $\!\!\!\!$ 3325 $\!\!\!\!$ $\!\!\!\!$ 10378.3 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.80 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0003 $\!\!\!\!$ $\!\!\!\!$ 0.0135 $\!\!\!\!$ $\!\!\!\!$ 0.0135 $\!\!\!\!$ $\!\!\!\!$ 0.0600 $\!\!\!\!$ $\!\!\!\!$ 0.1100
05bl-w7m1.45 $\!\!\!\!$ $\!\!\!\!$ -6 $\!\!\!\!$ $\!\!\!\!$ 8.520 $\!\!\!\!$ $\!\!\!\!$ 8800 $\!\!\!\!$ $\!\!\!\!$ 9990.6 $\!\!\!\!$ $\!\!\!\!$ 0.06 $\!\!\!\!$ $\!\!\!\!$ 0.83 $\!\!\!\!$ $\!\!\!\!$ 0.0040 $\!\!\!\!$ $\!\!\!\!$ 0.06 $\!\!\!\!$ $\!\!\!\!$ 0.0020 $\!\!\!\!$ $\!\!\!\!$ 0.04 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0005 $\!\!\!\!$ $\!\!\!\!$ 0.0003 $\!\!\!\!$ $\!\!\!\!$ 0.0003 $\!\!\!\!$ $\!\!\!\!$ 0.0006 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -5 $\!\!\!\!$ $\!\!\!\!$ 8.628 $\!\!\!\!$ $\!\!\!\!$ 8550 $\!\!\!\!$ $\!\!\!\!$ 10483.5 $\!\!\!\!$ $\!\!\!\!$ 0.03 $\!\!\!\!$ $\!\!\!\!$ 0.10 $\!\!\!\!$ $\!\!\!\!$ 0.0020 $\!\!\!\!$ $\!\!\!\!$ 0.04 $\!\!\!\!$ $\!\!\!\!$ 0.0020 $\!\!\!\!$ $\!\!\!\!$ 0.65 $\!\!\!\!$ $\!\!\!\!$ 0.12 $\!\!\!\!$ $\!\!\!\!$ 0.0005 $\!\!\!\!$ $\!\!\!\!$ 0.0150 $\!\!\!\!$ $\!\!\!\!$ 0.0150 $\!\!\!\!$ $\!\!\!\!$ 0.0200 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -3 $\!\!\!\!$ $\!\!\!\!$ 8.758 $\!\!\!\!$ $\!\!\!\!$ 8200 $\!\!\!\!$ $\!\!\!\!$ 10562.5 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0010 $\!\!\!\!$ $\!\!\!\!$ 0.78 $\!\!\!\!$ $\!\!\!\!$ 0.10 $\!\!\!\!$ $\!\!\!\!$ 0.0005 $\!\!\!\!$ $\!\!\!\!$ 0.0275 $\!\!\!\!$ $\!\!\!\!$ 0.0275 $\!\!\!\!$ $\!\!\!\!$ 0.0525 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ 4.8 $\!\!\!\!$ $\!\!\!\!$ 8.861 $\!\!\!\!$ $\!\!\!\!$ 7175 $\!\!\!\!$ $\!\!\!\!$ 9017.1 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.78 $\!\!\!\!$ $\!\!\!\!$ 0.08 $\!\!\!\!$ $\!\!\!\!$ 0.0005 $\!\!\!\!$ $\!\!\!\!$ 0.0300 $\!\!\!\!$ $\!\!\!\!$ 0.0300 $\!\!\!\!$ $\!\!\!\!$ 0.0700 $\!\!\!\!$ $\!\!\!\!$ 0.0070
$\!\!\!\!$ $\!\!\!\!$ 12.9 $\!\!\!\!$ $\!\!\!\!$ 8.605 $\!\!\!\!$ $\!\!\!\!$ 3475 $\!\!\!\!$ $\!\!\!\!$ 10418.1 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.87 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0005 $\!\!\!\!$ $\!\!\!\!$ 0.0097 $\!\!\!\!$ $\!\!\!\!$ 0.0097 $\!\!\!\!$ $\!\!\!\!$ 0.0350 $\!\!\!\!$ $\!\!\!\!$ 0.0700
05bl-w7e1.45m1.45 $\!\!\!\!$ $\!\!\!\!$ -6 $\!\!\!\!$ $\!\!\!\!$ 8.524 $\!\!\!\!$ $\!\!\!\!$ 9000 $\!\!\!\!$ $\!\!\!\!$ 10051.5 $\!\!\!\!$ $\!\!\!\!$ 0.08 $\!\!\!\!$ $\!\!\!\!$ 0.71 $\!\!\!\!$ $\!\!\!\!$ 0.0043 $\!\!\!\!$ $\!\!\!\!$ 0.10 $\!\!\!\!$ $\!\!\!\!$ 0.0035 $\!\!\!\!$ $\!\!\!\!$ 0.07 $\!\!\!\!$ $\!\!\!\!$ 0.03 $\!\!\!\!$ $\!\!\!\!$ 0.0002 $\!\!\!\!$ $\!\!\!\!$ 0.0002 $\!\!\!\!$ $\!\!\!\!$ 0.0003 $\!\!\!\!$ $\!\!\!\!$ 0.0005 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -5 $\!\!\!\!$ $\!\!\!\!$ 8.623 $\!\!\!\!$ $\!\!\!\!$ 8700 $\!\!\!\!$ $\!\!\!\!$ 10534.9 $\!\!\!\!$ $\!\!\!\!$ 0.09 $\!\!\!\!$ $\!\!\!\!$ 0.12 $\!\!\!\!$ $\!\!\!\!$ 0.0043 $\!\!\!\!$ $\!\!\!\!$ 0.07 $\!\!\!\!$ $\!\!\!\!$ 0.0035 $\!\!\!\!$ $\!\!\!\!$ 0.55 $\!\!\!\!$ $\!\!\!\!$ 0.10 $\!\!\!\!$ $\!\!\!\!$ 0.0002 $\!\!\!\!$ $\!\!\!\!$ 0.0225 $\!\!\!\!$ $\!\!\!\!$ 0.0225 $\!\!\!\!$ $\!\!\!\!$ 0.0200 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ -3 $\!\!\!\!$ $\!\!\!\!$ 8.760 $\!\!\!\!$ $\!\!\!\!$ 8100 $\!\!\!\!$ $\!\!\!\!$ 11060.7 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0015 $\!\!\!\!$ $\!\!\!\!$ 0.68 $\!\!\!\!$ $\!\!\!\!$ 0.14 $\!\!\!\!$ $\!\!\!\!$ 0.0002 $\!\!\!\!$ $\!\!\!\!$ 0.0500 $\!\!\!\!$ $\!\!\!\!$ 0.0500 $\!\!\!\!$ $\!\!\!\!$ 0.0750 $\!\!\!\!$ $\!\!\!\!$ 0.0000
$\!\!\!\!$ $\!\!\!\!$ 4.8 $\!\!\!\!$ $\!\!\!\!$ 8.868 $\!\!\!\!$ $\!\!\!\!$ 7000 $\!\!\!\!$ $\!\!\!\!$ 9313.8 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.01 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.72 $\!\!\!\!$ $\!\!\!\!$ 0.07 $\!\!\!\!$ $\!\!\!\!$ 0.0002 $\!\!\!\!$ $\!\!\!\!$ 0.0650 $\!\!\!\!$ $\!\!\!\!$ 0.0650 $\!\!\!\!$ $\!\!\!\!$ 0.0675 $\!\!\!\!$ $\!\!\!\!$ 0.0067
$\!\!\!\!$ $\!\!\!\!$ 12.9 $\!\!\!\!$ $\!\!\!\!$ 8.607 $\!\!\!\!$ $\!\!\!\!$ 3450 $\!\!\!\!$ $\!\!\!\!$ 10342.2 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0000 $\!\!\!\!$ $\!\!\!\!$ 0.84 $\!\!\!\!$ $\!\!\!\!$ 0.00 $\!\!\!\!$ $\!\!\!\!$ 0.0002 $\!\!\!\!$ $\!\!\!\!$ 0.0350 $\!\!\!\!$ $\!\!\!\!$ 0.0350 $\!\!\!\!$ $\!\!\!\!$ 0.0250 $\!\!\!\!$ $\!\!\!\!$ 0.0600
------------------------------ ------------------------------ ------------------------------------------------------------------------------------ ------------------------------------------- ------------------------------------- ------------------------------ ------------------------------ ------------------------------- ------------------------------- ------------------------------- ------------------------------- ------------------------------ ------------------------------- ------------------------------- ------------------------------- ----------------------------------------------------- ----------------------------------------------------- --
${}^{\textrm{a)}}$ The abundances of Fe, Co and Ni in our models are assumed to be the sum of [$^{56}$Ni]{} and its decay chain products ([$^{56}$Co]{} and [$^{56}$Fe]{}) on the one hand, and directly synthesised / progenitor Fe on the other hand. Thus, they are conveniently given in terms of the [$^{56}$Ni]{} mass fraction at $t=0$ \[$X($[$^{56}$Ni]{}$)_0$\], the Fe abundance at $t=0$ \[$X(\textrm{Fe})_0$\], and the time from explosion onset $t$.
[^1]: [Ca [ii]{}]{} H&K, [Si [ii]{}]{} $\lambda6355$ and [O [i]{}]{} $\lambda7773$ are usually the strongest lines in our spectra, which therefore have the highest probability of developing high-velocity absorptions.
[^2]: By ”binding energy“ and ”gravitational energy“ we always mean the absolute values here, i.e. we treat them as positive numbers.
[^3]: This formula cannot be applied for our lowest-mass models. Therefore, we inferred the binding energy of a $0.69$${M_{\odot}}$ progenitor from a WD model with constant temperature, which uses the Timmes equation of state [@tim99]. This equation of state takes into account a variable degree of electron degeneracy.
| {
"pile_set_name": "ArXiv"
} |
---
bibliography:
- '../arm.bib'
---
[**The auxiliary region method: A hybrid method for coupling PDE- and Brownian-based dynamics for reaction-diffusion systems** ]{}\
Cameron A. Smith$^{1,\ast}$, Christian A. Yates$^{1,\ast\ast}$\
$^1$**Centre for Mathematical Biology, Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, United Kingdom\
$\ast$ E-mail: [email protected]\
$\ast\ast$ E-mail: [email protected]**
Key words: hybrid modelling, stochastic reaction-diffusion, multiscale modelling, auxiliary region, partial differential equation, Brownian dynamics
Abstract {#abstract .unnumbered}
========
Reaction-diffusion systems are used to represent many biological and physical phenomena. They model the random motion of particles (diffusion) and interactions between them (reactions). Such systems can be modelled at multiple scales with varying degrees of accuracy and computational efficiency. When representing genuinely multiscale phenomena, fine-scale models can be prohibitively expensive, whereas coarser models, although cheaper, often lack sufficient detail to accurately represent the phenomenon at hand. Spatial hybrid methods couple two or more of these representations in order to improve efficiency without compromising accuracy.
In this paper, we present a novel spatial hybrid method, which we call the auxiliary region method (ARM), which couples PDE and Brownian-based representations of reaction-diffusion systems. Numerical PDE solutions on one side of an interface are coupled to Brownian-based dynamics on the other side using compartment-based “auxiliary regions”. We demonstrate that the hybrid method is able to simulate reaction-diffusion dynamics for a number of different test problems with high accuracy. Further, we undertake error analysis on the ARM which demonstrates that it is robust to changes in the free parameters in the model, where previous coupling algorithms are not. In particular, we envisage that the method will be applicable for a wide range of spatial multi-scales problems including, filopodial dynamics, intracellular signalling, embryogenesis and travelling wave phenomena.
Introduction {#sect:Introduction}
============
Reaction-diffusion models are important mathematical tools that are used to represent and understand complex biological and physical behaviours. They model the random movement of the particles (diffusion) and the interactions between particles (reactions), giving them a wide array of applications across multiple spatial scales. These applications range from the large-scale representation of striped vegetation patterns in semi-arid landscapes [@sherratt2005avs] and the spread of epidemics [@volpert2009rdw] to smaller-scale studies of pattern formation during embryogenesis [@turing1952cbm; @mort2016rdm] and, at even smaller scales, to the study of actin dynamics inside a cell’s filopodia [@erban2014msr] and intracellular dynamics [@khan2011scd; @andasari2012iid; @zhuge2000dsb]. \[xr:Refs\]
Reaction-diffusion models can be specified at different levels of detail depending on the temporal, spatial and concentration scales involved in the application (see Table \[tab:Comparison\]). At the finest scale that we will consider are microscopic dynamics. These models and methods \[xr:GFRD\] (which include Brownian motion for purely diffusive systems and Smoluchowski dynamics [@andrews2004ssc; @smoluchowski1917vem] \[xr:Andrews\] or Green’s function reaction dynamics (GFRD) [@van2005gfr] for reaction-diffusion systems) are amongst the more detailed representations of such systems, but consequently are relatively computationally expensive[^1]. \[xr:GFRD\_Smoldyn\] They require not only the knowledge of the location of all particles at all times, but in the case of second- and higher-order reactions, the pairwise distances between particles, which requires large memory, and are expensive to calculate for many time-steps. In the case of diffusion-limited reactions, time-steps must be taken to be extremely small to ensure that reactive particles do not jump past each other and that the attendant reaction events are not missed. All update steps also require the production of a normally distributed random number for each co-ordinate of each particle which can be computationally expensive depending on the reaction system that is being modelled. However, some of these expensive steps can be accelerated by considering event-driven algorithms or employing approximate algorithms with longer time-steps. GFRD is an event-driven algorithm differs from the standard method for simulating Brownian motion. It uses a maximum time-step so that only single particles, or pairs of particles, need to be considered. It then utilises the exact solution to the Smoluchowski equation in order to combine movement of, and interactions between, particles. If particles are far apart, the event-based time-steps are large. Smoldyn uses relatively long time-steps, and accounts for the error that this causes (due to possible reactant pairs passing by one another without the possibility of reacting) by making the effective particle sizes larger. \[xr:Ind\_Diff\] Micro-scale modelling is particularly useful when fine scale detail is required, for example, when considering the binding of particles to receptors [@dobramysl2015pbd; @english2004bad; @moy2000tct]. An even finer scale representation, in which atomistic dynamics can be represented, is available, if required. Typically, modelling at this scale is known as molecular dynamics, and we direct the reader to @holley1971mhp and @durr1981mmb for more information about modelling at this scale. \[xr:Even\_Finer\_Scale\]
At a coarser scale we have compartment-based or mesoscopic models. Like the fine-scale, microscopic models, these also account for stochastic variation, however particles are now considered to belong to compartments rather than having their exact locations tracked. Particles can either react with one another within a compartment, or can jump between adjacent compartments with given rates, simulating diffusion. Compartment-based models can be simulated using either exact but computationally expensive [@gillespie1977ess; @gibson2000ees; @elf2004ssb] or inexact but computationally cheaper [@gillespie2001aas] stochastic simulation algorithms (SSAs). The exact methods (so-called because they produce sample paths consistent with the associated chemical master equation) effectively assign exponential waiting times to every possible event (diffusive jump or reaction) and then choose the event with the shortest waiting time to enact. In general, they are faster than the microscopic methods, since pairwise reaction distances do not need to be calculated for bi-molecular reactions and individual particle identities are not tracked, but are less accurate, since they only record a particle’s location up to the accuracy of the compartment size, and generally particles are only allowed to react with others in the same compartment [@isaacson2013crd].
Finally, at the coarsest scale lie continuum or macroscopic models. The most commonly employed macroscopic models for reaction-diffusion systems comprise partial differential equations (PDEs)[^2]. These methods are generally only valid for high particle numbers. The stochastic variations, which are considered small enough to be neglected at high copy numbers, play a pivotal role in the dynamics at low copy numbers, leading the PDE solutions to diverge from the true underlying dynamics. There is a wealth of well established numerical methods that can quickly simulate an approximate solution to a PDE. These include finite-difference methods, finite-volume methods and finite-element methods (see for example, [@smith1985nsp; @morton2005nsp; @eymard2000fvm; @brenner2004fem]).
**Scale** **Advantages** **Disadvantages**
----------- -------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------
Micro Most accurate representation.Can be used for low copy numbers. Slow to compute reactions.Impractical for large numbers of particles.
Meso Fast for low particle numbers. Represents individual-level behaviour. Can be slow for large copy numbers.Does not retain precise location or particle identity.
Macro Fast to compute solutions.Suitable for high copy numbers.Often amenable to analytical solutions. Inaccurate for low copy numbers.Mean-field models diverge from individual dynamics for higher-order reactions.
: A comparison of the advantages and disadvantages of the three most prominent scales at which reaction-diffusion processes are modelled.[]{data-label="tab:Comparison"}
Often though, important biological and physical phenomena are genuinely multiscale [@markevich2004ssb; @black2012sfe; @gillespie2013psa; @robinson2014atr]. In spatial reaction-diffusion systems, concentration may vary over orders of magnitude. In regions of low concentration it is often important to employ detailed individual-based models in order to correctly represent the dynamics. If these models were to be employed indiscriminately throughout the domain, however, the regions of high concentration, in which there are many individual particles to be evolved, might render the system computationally intractable. In these regions, it might be acceptable to employ a coarser and less computationally expensive model. A canonical example of this phenomenon is the stochastic Fisher wave [@breuer1995mls; @breuer1994few]. The wave speed is determined by the stochastic activity at the pulled front, so it is important to employ an accurate individual-based representation of the dynamics in this region. Conversely, behind the wave front, the detailed dynamics are of little importance. It is possible, therefore, to employ a coarser, cheaper representation of the dynamics in this region.
Spatially coupled hybrid methods have been developed for precisely this purpose: to simulate spatially inhomogeneous domains both accurately and efficiently. In general, such methods are designed to accelerate expensive computations whilst maintaining reasonable levels of accuracy. The majority of spatially coupled hybrid methods divide the computational domain into distinct regions using interfaces. The dynamics of adjacent regions are represented using different methods. Regions in which detailed representations of the dynamics are required for accuracy are simulated using a fine-scale method, whereas regions in which less detail is required are modelled with a coarser, less computationally expensive method. There can be two reasons for this. The first is in order to resolve a particular region of the spatial domain in more detail, such as when looking at the behaviour of ions around gated channels [@dobramysl2015pbd], or when building a model for the energy in a liquid crystal [@robinson2017mcm]. Both of these examples have a prohibitively slow but accurate model that is required in certain regions of space, but which is too computationally expensive be used everywhere. The second reason is to simply segregate a region of the domain in which there are very few particle numbers. In these regions a coarse method (for example a continuum model) may be too inaccurate. \[xr:Usefulness\]
There exist hybrid methods that couple each of the different scales described above to one another (and indeed many more, see @smith2018seh for a comprehensive review of such methods). Macroscopic-to-mesoscopic methods have been proposed which employ averaged fluxes in order to calculate appropriate boundary conditions for each regime at the interface(s) [@wagner2004hcf; @moro2004hms; @harrison2016hac], as well as using an extra compartment within the macroscopic region [@yates2015pcm]. Mesoscopic-to-microscopic methods, which also employ extra compartments, this time in the microscopic regime, have been developed [@flegg2015cmc], and a class of methods using adapted rates of diffusion from the mesoscopic to the microscopic domains have been proposed and successfully applied to represent biological processes [@flegg2012trm; @robinson2014atr; @flegg2014atr; @dobramysl2015pbd; @erban2014msr]. There are fewer macroscopic-to-microscopic hybrid methods in the literature. Macro-to-micro methods that allow mass to flow over the interface in both directions in order to initialise particles [@franz2012mrd] or that average solutions on either side of the interface to find a flux [@alexander2002ars] can be found in the literature. For a more detailed review of spatially extended hybrid methods, see [@smith2018seh].
Two of the above-mentioned hybrid methods are of particular relevance for the purposes of this paper. The pseudo-compartment method, presented by @yates2015pcm, is a macroscopic-to-mesoscopic (specifically PDE-to-compartment) method in which the coupling is achieved using an extra compartment, known as the “pseudo-compartment”, adjacent to the interface within the macroscopic domain. In this compartment, mass is represented using both the PDE solution and the compartment-based method (with particle numbers found by direct integration of the PDE over this region). Particles are then allowed to cross the interface in both directions using the compartment-based method. We give a schematic representation of this method in Figure \[fig:Plots\_yates\_flegg\] .
The ghost cell method proposed by @flegg2015cmc is a mesoscopic-to-microscopic method which uses an extra compartment in the microscopic domain. The number of particles in this “ghost cell” is simply the number of Brownian particles which reside in this region. Again, particles are allowed to jump across the interface using the compartment-based mesoscopic method. A schematic representation of the method is given in Figure \[fig:Plots\_yates\_flegg\] .
In this paper, we employ these two methods in order to couple a macroscopic PDE description for reaction-diffusion systems to a corresponding microscopic Brownian dynamics representation through the use of “auxiliary regions”. These regions are compartments, which lie either side of the interface, and allow mass to pass between the two regimes via a mesoscopic jump process (see Figure \[fig:Schematic\] on page for a schematic representation). Within the auxiliary regions, mass is simultaneously represented using both the description for the region in which they reside (i.e. PDE or Brownian) and the mesoscopic description. Changes (i.e. reactions or diffusion events) implemented under one modelling paradigm (e.g. the compartment-based representation of the auxiliary region) are simultaneously implemented in the other (e.g. the PDE or Brownian representations in these regions). The interface, which divides the two modelling paradigms, can either be static, in which case it remains in its initial position, or adaptive, in which case it moves with the density profile in order to ensure that regions of space with few particles are simulated using the finest scale. Through a series of test cases, we demonstrate our algorithm to be more accurate and more robust to model parameters than previous PDE-to-Brownian coupling algorithms.
The paper is organised as follows. In Section \[sect:Previous\], a previous attempt at hybridising a Brownian dynamics model to its corresponding mean-field PDE description is evaluated in more detail [@franz2012mrd]. A description of our novel auxiliary region method (ARM) is presented in Section \[sect:ARM\] alongside the relevant justifications and pseudocode. Numerical results, verifying the accuracy of our hybrid method, are presented in Section \[sect:Results\]. Numerical error analysis is conducted in Section \[sect:Error\], where we also discuss restrictions on the model parameters for the effective functioning of the coupling algorithm. We conclude with a discussion of the effectiveness of our new hybrid method and suggest avenues for further exploration in Section \[sect:Discussion\].
An existing PDE-to-Brownian coupling {#sect:Previous}
====================================
In this section we summarise the pioneering work of @franz2012mrd, who were among the first to couple PDE and Brownian dynamics representations of reaction-diffusion. By replicating their results, we demonstrate that their “PDE-assisted Brownian dynamics” algorithm is not robust to simulation parameter choice, even for simple diffusive processes. This motivates the need for a more robust coupling method, which we provide in the form of the ARM in Section \[sect:ARM\].
PDE-assisted Brownian dynamics {#section:_PDE_assisted_brownian_dyanamics}
------------------------------
Hybrid methods that couple the PDE description of a reaction-diffusion system to its corresponding Brownian dynamics representation have been relatively poorly investigated in comparison to PDE-to-compartment-based and compartment-based-to-Brownian couplings. In part, this is a result of the fact that such hybrid algorithms neglect meso-scale representations of particle dynamics, meaning that they must bridge a greater scale separation than either of the other two hybrid paradigms. Mainly though, the absence of many examples of PDE-to-Brownian hybrid methods is due to the inherent difficulty when converting PDE mass to individual particles (and vice-versa) when coupling Brownian dynamics models to continuum PDE representations. Below, we describe two algorithms proposed by @franz2012mrd, but focus on the first, a method with an interfacial coupling. We choose to focus on this coupling because our ARM coupling method, described in Section \[sect:ARM\], also utilises an interface. \[xr:Franz\_Interface\]
@franz2012mrd present two related algorithms. In the first, the non-overlapping PDE and Brownian domains are separated by an interface (see Figure \[fig:franz2012mrd\]). Both PDE and Brownian representations are updated using a time-driven algorithm, with the PDE time-step much smaller than the Brownian time-step. The discretised PDE is evolved (until the time reaches the next Brownian time-step) using a centred finite-difference scheme with implicit Euler time-stepping, and PDE mass is allowed to cross the interface between the two regimes. Provided that the Brownian time-step is sufficiently small, the amount of mass that crosses the interface between Brownian time-steps gives the probability that a new particle is placed within the Brownian domain. A uniformly distributed random number is used to determine whether a particle is initialised in the Brownian regime or not. If it is, this particle’s position is randomly initialised according to the normalised density profile of the PDE mass that crossed the interface in the previous Brownian time-step. If a Brownian particle crosses into the PDE domain, a particle’s worth of mass is added to the PDE solution at its new location as a $\delta$-function and the individual particle is removed. We have illustrated this method schematically in Figure \[fig:franz2012mrd\].
@franz2012mrd found the variance in particle numbers in the Brownian region of the hybrid domain to be altered in comparison to the variance that would be expected in a fully Brownian simulation. In order to counteract this problem, they introduced a second algorithm, in which an overlap region replaces the interface. Within the overlap region, mass can be simulated as either Brownian particles or as part of the PDE. The coupling works in the same way as in the interfacing algorithm, however the Brownian particles are subsumed into the PDE only once they have crossed the boundary of the overlap region closest to the fully-PDE domain. Similarly, PDE mass can only be converted to Brownian particles once it has flowed over the overlap boundary adjacent to the fully-Brownian domain. \[xr:Franz\_Overlap\]
The Brownian time-step in the algorithm is required to be small, in order that the total probability of initialising a particle in the Brownian regime is less than one. However, the algorithm runs into difficulties if the time-step is chosen to be too small. Specifically, the amount of mass that flows over the interface between updates of the Brownian dynamics is too small in comparison to that which would be predicted theoretically using the exact diffusion kernel. This gives rise to inaccuracies in the algorithm, particularly if long simulation times are required. This sensitivity to the choice of Brownian time-step restricts the physical scenarios to which the algorithm can be applied.
In figure \[fig:Plots\_franz\] we present three snapshots of the evolution of the first version of the algorithm (interface rather than overlap region) which illustrate this problem. By time $t=2$, in Figure \[fig:Plots\_franz\] , there is a clear disparity between the hybrid method and the mean field solution (black dotted line). Disparities of this nature are not acceptable when modelling real reaction-diffusion systems, irrespective of the computational savings the algorithm is able to produce.
The auxiliary region method {#sect:ARM}
===========================
In this section we present our novel “auxiliary region method” (ARM) for coupling PDE and Brownian representations of reaction-diffusion. For simplicity we will present a version of the method with a single interface separating two regimes. However, the method can be easily generalised to multiple interfaces which separate alternating PDE and Brownian regions. Sequentially, we describe the composition of the domain and the models we employ in each region; the nature of the auxiliary regions; the implementation of movement of mass across the boundary; the implementation of reactions; and finally the specific details required for the simulation of the algorithm, including pseudocode for its implementation. All code, which has been written in MATLAB, can be found in the electronic supplementary material online.
The domain composition {#sect:ARM_description}
----------------------
Recall that, for our coupling method, space is partitioned into two regions within which we use different modelling paradigms (PDE and Brownian dynamics) to simulate the underlying reaction-diffusion system. Separating the two regions is a point interface, over which particles can jump according to a compartment-based method.
Consider a one-dimensional domain[^3] $\Omega=(L_1,L_2)\subseteq\mathbb{R}$ for some $L_1<0<L_2$. We split $\Omega$ into two regions, ${\Omega_{\text{{\tiny P}}}}=(L_1,0)$ and ${\Omega_{\text{{\tiny B}}}}=(0,L_2)$ (separated by an interface $I$ at position $0$), within which the evolution of the system will be represented using a PDE description and Brownian dynamics, respectively.
The auxiliary regions
---------------------
Particles can move between the two domains (${\Omega_{\text{{\tiny P}}}}$ and ${\Omega_{\text{{\tiny B}}}}$) via the auxiliary regions ${\Omega_{\text{{\tiny PA}}}}$ and ${\Omega_{\text{{\tiny BA}}}}$; subsets of ${\Omega_{\text{{\tiny P}}}}$ and ${\Omega_{\text{{\tiny B}}}}$ respectively, each of width $h_a>0$. Within these regions, mass/particles are simultaneously represented according to the default methodology for their domain (either PDE in ${\Omega_{\text{{\tiny P}}}}$ or Brownian dynamics in ${\Omega_{\text{{\tiny B}}}}$), but also as well-mixed particles in their respective auxiliary regions ${\Omega_{\text{{\tiny PA}}}}$ and ${\Omega_{\text{{\tiny BA}}}}$. These auxiliary regions act as a bridge between the fine- and coarse-scale descriptions. A schematic representation of domain’s composition is given in Figure \[fig:Schematic\].
We justify the use of the Brownian auxiliary region by following the methodology set out in @flegg2015cmc. The entire Brownian domain can be simulated using a mesoscopic compartment-based regime, and equivalently using a microscopic simulation. In the absence of reactions, if the particles in the microscopic simulation are “binned” into the same compartments as the mesoscopic simulation, the expected numbers in each compartment for each simulation would be the same. At this scale, the two methods are equivalent ways of simulating the same diffusive process [@flegg2015cmc].
To justify the use of the PDE auxiliary region, we appeal to the arguments of @yates2015pcm. We note that the PDE density can be thought of as the probability of finding a particle at a particular position and time, scaled by the number of particles within the PDE domain. Provided that the auxiliary region is sufficiently narrow, the PDE density within the auxiliary region can be thought of as being approximately uniformly distributed across the region with the appropriate number of particles. This is precisely the interpretation of the contents of a compartment within the mesoscopic, compartment-based framework.
The PDE regime, ${\Omega_{\text{{\tiny P}}}}$ {#sect:ARM_PDE}
---------------------------------------------
Within $\Omega_{\text{P}}$, we represent the mass of particles using: $$\begin{aligned}
\text{PDE}&:\quad {\frac{\partial {\boldsymbol{c}}}{\partial t}}(x,t)=D{\frac{\partial^2 {\boldsymbol{c}}}{\partial x ^2}}(x,t) + {\boldsymbol{f}}({\boldsymbol{c}}(x,t));\quad x\in\Omega_{\text{P}};\quad t\in(0,T),\label{eqn:PDE}\\
\text{BCs}&:\quad {\frac{\partial {\boldsymbol{c}}}{\partial x}}(x,t) = 0; \quad x\in\partial{\Omega_{\text{{\tiny P}}}}; \quad t\in(0,T), \notag \\
\text{IC}&:\quad {\boldsymbol{c}}(x,0) = {\boldsymbol{c}}_0(x); \quad x\in {\Omega_{\text{{\tiny P}}}}. \notag\end{aligned}$$ Here, ${\boldsymbol{c}}(x,t)=(c_1(x,t),...,{c_{\text{{\tiny K}}}}(x,t))^T$, denotes the density of species $k={1,\dots,K}$ at position $x$ and time $t$, $D$ is a diagonal matrix containing the Fickian diffusion constants for each species, and ${\boldsymbol{f}}$ is a function that encapsulates the effect of any reactions on each species. We also use the notation $\partial{\Omega_{\text{{\tiny P}}}}$ to represent the boundary of ${\Omega_{\text{{\tiny P}}}}$, and ${\boldsymbol{c}}_0(x)$ is the initial condition. For all the simulations presented in this paper we employ the finite-difference $\theta$-method (a general family of finite-difference methods)[^4]. Although the Crank-Nicolson method ($\theta=0.5$) is second-order accurate and unconditionally stable, we use $\theta=0.51$ since the Crank-Nicolson method can give rise to spurious oscillations when implemented on step-function initial conditions of the sort we will consider [@smith1985nsp]. \[xr:Instabilities\]
The Brownian regime, ${\Omega_{\text{{\tiny B}}}}$
--------------------------------------------------
Within ${\Omega_{\text{{\tiny B}}}}$, all particles are tracked and their positions updated according to the following stochastic differential equation (SDE) which simulates Brownian motion: $$y_i^k(t+{\Delta t}) = y_i^k(t) + \sqrt{2D_k{\Delta t}}\, \xi_i^k; \quad \xi_i^k\sim N(0,1); \quad \text{ for } \quad i\in\{1,...,{N_{\text{{\tiny HB}}}}^k(t)\} \quad\text{ and } \quad \ k\in\{1,...,K\},
\label{eqn:BD}$$ where $y_i^k(t)$ denotes the location of particle $i$ of species $k$ within ${\Omega_{\text{{\tiny B}}}}$, ${\Delta t}$ is the time-step for both the PDE and Brownian dynamics simulators[^5] and ${N_{\text{{\tiny HB}}}}^k(t)$ is the number of particles of species $k$ in ${\Omega_{\text{{\tiny B}}}}$ at time $t$. Once again, we set reflective boundary conditions at both ends of ${\Omega_{\text{{\tiny B}}}}$ to ensure that no particles can leave this domain via a Brownian diffusion event. The zero-flux boundary conditions at the interface for both PDE and Brownian regimes ensure that mass can only cross the interface according to the compartment-based method.
Movement across the interface
-----------------------------
Since both domains, ${\Omega_{\text{{\tiny P}}}}$ and ${\Omega_{\text{{\tiny B}}}}$, have zero-flux boundaries at the interface, particles can only cross over the interface via the auxiliary regions. In effect, these regions comprise a two-compartment reaction-diffusion master equation (RDME) model. Each particle in each auxiliary region jumps to its neighbouring region on the other side of the interface with a rate $d_k$ (for species $k$), which is related to the macroscopic diffusion coefficient (for species $k$), $D_k$, via$$d_k = \frac{D_k}{h_a^2}.
\label{eqn:link_d_D}$$ Here, $h_a$ is the width of each auxiliary region, which is assumed to be the same for both the Brownian and PDE auxiliary regions. In order to implement jumps (or reactions, where necessary) according to the RDME, we require particle numbers.
Borrowing terminology from @yates2015pcm, the number of “pseudo-particles” of species $k$ within the PDE auxiliary region, ${\Omega_{\text{{\tiny PA}}}}$, at time $t$, denoted ${N_{\text{{\tiny PA}}}}^k(t)$, is calculated as $${N_{\text{{\tiny PA}}}}^k(t) = \displaystyle\int_{{\Omega_{\text{{\tiny PA}}}}}{c_k(x,t)\ dx}.
\label{eqn:particle_numbers_PDE_AR}$$ The number of particles of species $k$ in the Brownian auxiliary region, ${\Omega_{\text{{\tiny BA}}}}$, is given by $${N_{\text{{\tiny BA}}}}^k(t) = \left|{\left\{j:y_j^k(t)\in {\Omega_{\text{{\tiny BA}}}}\right\}}\right|.
\label{eqn:particle_numbers_BD_AR}$$
These particle numbers allow us to define propensity functions corresponding to diffusive jumps between, or reactions within, the auxiliary regions. For diffusive jumps between the two auxiliary regions, the propensity functions for species $k$ within the PDE and Brownian auxiliary regions are (respectively): $$\begin{aligned}
{\alpha_{\text{{\tiny P}}}}^k(t) &= d_k{N_{\text{{\tiny PA}}}}^k(t)\quad\text{ for }\quad{\Omega_{\text{{\tiny PA}}}}, \label{eqn:propensity_functions_PDE}\\
{\alpha_{\text{{\tiny B}}}}^k(t) &= d_k{N_{\text{{\tiny BA}}}}^k(t)\quad\text{ for }\quad{\Omega_{\text{{\tiny BA}}}}. \label{eqn:propensity_functions_Brown}\end{aligned}$$ We note here that if ${N_{\text{{\tiny PA}}}}^k(t) < 1$, we set ${\alpha_{\text{{\tiny P}}}}^k(t) = 0$ to prevent the possibility of negative density. While it may be a problem if this scenario occurs persistently, practically speaking, we should choose the position of the interface such that density is always large enough that this does not happen. An adaptive interface will allow us to satisfy this criteria (see Section \[sect:Results\_P4\_Interface\]), and hence this problem would not occur when using such an interface. \[xr:Negative\_Particles\]
When a particle jumps from ${\Omega_{\text{{\tiny BA}}}}$ to ${\Omega_{\text{{\tiny PA}}}}$, a particle within the Brownian auxiliary region is chosen uniformly at random to be removed, and a particle’s worth of mass is added to the PDE solution uniformly across ${\Omega_{\text{{\tiny PA}}}}$ for the species, $k$, which has changed: $$c_k(x,t) = c_k(x,t) + \frac{1}{h_a}\mathds{1}_{[x\in{\Omega_{\text{{\tiny PA}}}}]},
\label{eqn:add_particle_PDE}$$ where $\mathds{1}_{[x\in A]}$ is the indicator function for $x\in A$. \[xr:Indicator\] Similarly, if a jump is enacted in the opposite direction, from ${\Omega_{\text{{\tiny PA}}}}$ to ${\Omega_{\text{{\tiny BA}}}}$, we first remove a particle’s worth of mass uniformly from ${\Omega_{\text{{\tiny PA}}}}$ for the appropriate species $k$: $$c_k(x,t) = c_k(x,t) - \frac{1}{h_a}\mathds{1}_{[x\in{\Omega_{\text{{\tiny PA}}}}]},
\label{eqn:remove_particle_PDE}$$ and a new particle is initialised within the Brownian auxiliary region, ${\Omega_{\text{{\tiny BA}}}}$, with position chosen uniformly at random.
Reaction implementation
-----------------------
Throughout ${\Omega_{\text{{\tiny P}}}}$, all reactions are implemented using the reaction operator ${\boldsymbol{f}}({\boldsymbol{c}})$. The method we employ to implement reactions within ${\Omega_{\text{{\tiny B}}}}$ depends on the location of the reactant particles. Let $\mathcal{R}$ denote the set of reaction pathways (with $|\mathcal{R}|=R$). Define the subset of reactions $\mathcal{R}^*(t)$ at time $t$ as follows:
$$\mathcal{R}^*(t)=\{\text{all reactions for which at least one set of reactant particles lies exclusively within }{\Omega_{\text{{\tiny BA}}}}\}.$$
Reactions between molecules for which at least one of the reactive molecules lies within ${\Omega_{\text{{\tiny B}}}}\backslash{\Omega_{\text{{\tiny BA}}}}$ are implemented using an appropriate microscopic approach, such as the $\lambda$-$\rho$ method [@erban2009smr; @lipkova2011abd]. However, if at least one set of participating particles lie in ${\Omega_{\text{{\tiny BA}}}}$ (i.e. $r\in \mathcal{R}^*$), care needs to be taken over the interaction of such particles and the mass on the other side of the interface in ${\Omega_{\text{{\tiny P}}}}$. As explained below we will implement the reactions $r\in \mathcal{R}^*$ for these reactant particles using the compartment-based method. For illustrative purposes, consider a reversible second-order reaction involving species $A$, $B$ and $C$: $$A +B \xrightleftharpoons[\kappa_2]{\kappa_1} C.
\label{eqn:second_order_reaction}$$ Under the $\lambda-\rho$ method [@erban2009smr] and its later modification [@lipkova2011abd], for the forward reaction, a particle of species $A$ and a particle of species $B$ are required to be within a distance $\rho$ of one another in order to react. They then react with a rate $\lambda$, where $\lambda$ is a function of both the reaction radius $\rho$ and the reaction rate $\kappa_1$. Imagine that an $A$ particle (without loss of generality) in ${\Omega_{\text{{\tiny B}}}}$ is close enough to the interface that the reaction radius $\rho$ is larger than the distance between itself and the interface. For consistency with the Brownian representation, the $A$ particle should be allowed to react with a $B$ particle in the PDE region. The implementation of such reactions would be extremely difficult. Instead, by ensuring bimolecular reactions within the auxiliary region are implemented according to the mesoscopic compartment-based method, we avoid such issues (provided that the width of the auxiliary region is chosen to be larger than the interaction radius $\rho$).
According to the backwards reaction, two particles are created after the reaction has occurred. These particles are placed a certain distance away from each other (called the dissociation radius) in order to achieve a specified probability of geminate recombination (a recombination of any pair of $A$ and $B$ particle that were initialised from the same $C$ particle). \[xr:Geminate\] If this radius intersects with the PDE regime, then there is the potential for individual particles to be initialised within ${\Omega_{\text{{\tiny P}}}}$. By again employing the mesoscopic representations for reactions we resolve this issue. All product particles are assumed to be placed uniformly throughout the Brownian auxiliary region. Particles that are products of the backwards dissociation reaction in ${\Omega_{\text{{\tiny B}}}}\backslash{\Omega_{\text{{\tiny BA}}}}$ are extremely unlikely to be placed in ${\Omega_{\text{{\tiny P}}}}$ (again, providing that the auxiliary region is larger than the dissociation radius).
For these reasons, all of the reactions $r\in \mathcal{R}^*$ (for which at least one set of participating particles lie in ${\Omega_{\text{{\tiny BA}}}}$) are implemented using the compartment-based method, in which reactions are incorporated as events in the associated Markov chain, according to the RDME. We can write the following propensity functions for reactions within ${\Omega_{\text{{\tiny BA}}}}$: $$\alpha_{r}(t)=g_r({{\boldsymbol{N}}_{\text{{\tiny BA}}}}(t))\kappa_r h_a^{1-\nu},
\label{eqn:propensity_functions_reactions}$$ for any reaction channel $r\in\mathcal{R}^*(t)$ of order $\nu$ and corresponding reaction rate $\kappa_r$, where ${{\boldsymbol{N}}_{\text{{\tiny BA}}}}(t)=({N_{\text{{\tiny BA}}}}^1(t),...,{N_{\text{{\tiny BA}}}}^K(t))^T$ and $g_r$ is the appropriate number of possible combinations of the reactants for reaction $r$ from the particles that lie within ${\Omega_{\text{{\tiny BA}}}}$. Recall, however, that in ${\Omega_{\text{{\tiny B}}}}\backslash{\Omega_{\text{{\tiny BA}}}}$, any such reactions are implemented according to the chosen microscopic reaction method [@erban2009smr; @lipkova2011abd; @doi1976std].
Simulation specifics
--------------------
The Gillespie SSA [@gillespie1977ess] is used to simulate the above-described reactions in ${\Omega_{\text{{\tiny BA}}}}$, as well as the diffusive fluxes over the interface. The SSA requires the computation of an exponential random variable which gives the time, $\tau$, until the next event, and can be found by transforming a uniform random variable $u_1\sim \text{Unif}(0,1)$ via the following equation $$\tau = \frac{1}{\alpha^0(t)}\ln\left(\frac{1}{u_1}\right).
\label{eqn:next_reaction_time}$$ Here, $\alpha^0(t)$ is the sum of all of the propensity functions: $$\alpha^0(t) = {\alpha_{\text{{\tiny P}}}}^0(t)+{\alpha_{\text{{\tiny B}}}}^0(t) + \sum_{r\in\mathcal{R}^*(t)}{\alpha_r(t)},
\label{eqn:a0}$$ where $${\alpha_{\text{{\tiny P}}}}^0(t)= \sum_{k=1}^K{{\alpha_{\text{{\tiny P}}}}^k(t)},\label{eqn:sum_PDE_props}$$ and $${\alpha_{\text{{\tiny B}}}}^0(t)=\sum_{k=1}^K{{\alpha_{\text{{\tiny B}}}}^k(t)}.\label{eqn:sum_Brownian_props}$$
The PDE solutions and Brownian dynamics are implemented using the same discrete time-step, ${\Delta t}$, and the diffusive jumps across the interface (and any required reactions, $r\in \mathcal{R}^*$) are implemented in an event-driven manner, according to the Gillespie SSA. Event-driven time-steps are implemented until the putative time for the next event passes the next Brownian/PDE update time, at which point the PDE and Brownian dynamics are updated. Pseudocode for the ARM is given in Algorithm \[alg:ARM\].
[Auxiliary region method (ARM)]{}\[alg:ARM\]
Initialise time $t=0$, set final time $T$, PDE/Brownian update time-step, ${\Delta t}$, the PDE discretisation grid size, $\Delta x$, and the auxiliary region spatial step, $h_a$. Initialise particles in both ${\Omega_{\text{{\tiny P}}}}$ and ${\Omega_{\text{{\tiny B}}}}$ as required. Calculate the time until the next PDE and Brownian update step $t_\Delta = {\Delta t}$.
Calculate the number of particles ${N_{\text{{\tiny PA}}}}^k$ and ${N_{\text{{\tiny BA}}}}^k$ in the auxiliary regions, for each species $k\in\{1,2,...,K\}$, using formulae and . Consequently, calculate the corresponding propensity functions, ${\alpha_{\text{{\tiny P}}}}^k(t)$ and ${\alpha_{\text{{\tiny B}}}}^k(t)$ as per equations and , and their sums according to equations and . Calculate $\alpha_{r}(t)$, for $r\in\mathcal{R}^*$, using equation and finally compute $\alpha^0(t)$ according to equation . \[item:while\_loop\_arm\]
Calculate the time, $\tau$, until the next auxiliary region event according to equation (\[eqn:next\_reaction\_time\]). Update the auxiliary region time $t_a = t+\tau$.
If $t_a<t_\Delta$
(i) Draw three random numbers $u_1,u_2,u_3\sim \text{Unif}(0,1)$.
(ii) If $u_1\alpha^0(t) < {\alpha_{\text{{\tiny P}}}}^0(t)$ (corresponding to a jump from ${\Omega_{\text{{\tiny PA}}}}$ to ${\Omega_{\text{{\tiny BA}}}}$):
- Use $u_2$ to determine the species, $k$, which the jump affects, with each species selected with probability proportional to its propensity function.
- Remove a particle from the PDE auxiliary region for species $k$ via equation .
- Initialise a new particle of species $k$ within ${\Omega_{\text{{\tiny BA}}}}$ at position $y^* =u_3h_a + I$.
Else if $u_1\alpha^0(t) < {\alpha_{\text{{\tiny P}}}}^0(t)+{\alpha_{\text{{\tiny B}}}}^0(t)$ (corresponding to a jump from ${\Omega_{\text{{\tiny BA}}}}$ to ${\Omega_{\text{{\tiny PA}}}}$):
- Use $u_2$ to determine the species, $k$, which the jump affects, with each species selected with probability proportional to its propensity function.
- Choose a particle of species $k$ uniformly at random from within the Brownian auxiliary region and remove it from the system. We do this by selecting an index $q$ such that $q=\lceil u_3 {N_{\text{{\tiny BA}}}}^k \rceil$, where $\lceil x \rceil$ denotes the smallest integer larger than $x$.
- Add a new particle into the PDE auxiliary region for species $k$ via equation .
Else (corresponding to a reaction in ${\Omega_{\text{{\tiny BA}}}}$)
- Use $u_2$ to choose the reaction $r\in \mathcal{R}^*(t)$ to be implemented with probability proportional to its propensity function.
- Update particle numbers (and initialise positions, if appropriate) in the Brownian representation accordingly.
(iii) Set $t = t_a$.
Else
(i) Update the PDE system using an appropriate numerical method (see Section \[sect:ARM\_description\]).
(ii) Update the positions of the Brownian particles according to equation .
(iii) Implement any reactions using an appropriate method (see Section \[sect:ARM\_description\]). Note that production reactions should be implemented after any degradation reactions in order to prevent particles being created and destroyed in the same time-step.
(iv) Set $t = t_\Delta$, update $t_{\Delta} = t + {\Delta t}$.
If $t<T$, return to \[item:while\_loop\_arm\], otherwise stop.
Results {#sect:Results}
=======
Within this section, we present four test problems which are used to demonstrate that the ARM correctly simulates reaction-diffusion systems. Two of these problems are models of pure diffusion with different initial conditions and will demonstrate that the fluxes over the interface are consistent with the expected behaviour of the fully Brownian simulations. The third problem is the formation of a morphogen gradient, which demonstrates the successful implementation of reactions in the ARM. Despite the fact that our method is valid for higher-order reactions, the first three test problems consider reactions up to first order. For such systems, no moment closure assumptions are required in deriving the mean-field reaction-diffusion PDE and hence its behaviour agrees with the mean behaviour of the individual-based models. This allows us to efficiently verify accuracy by comparing the mean behaviour of our hybrid method to the known mean-field behaviour. Finally, in test problem four, we implement a second-order reaction system in higher dimensions, indicating the applicability of the method to more complicated examples.
For each of the first three test problems, we use ${\Omega_{\text{{\tiny P}}}} = (-1,0)$ and ${\Omega_{\text{{\tiny B}}}}=(0,1)$, meaning that the interface is the single point at $0$. We take the value of the fixed PDE and Brownian update steps to be $\Delta t=0.02$, the auxiliary regions have width $h=0.05$ and the diffusion constant is $D = 0.025$. We will quantify the qualitative comparisons, presented here through density comparison snapshots, in Section \[sect:Error\]. All simulations will comprise only a single species, so henceforth, all sub- or super-scripts, $k$, pertaining to species will be removed.
Test problem 1: maintaining equilibrium {#sect:Results_P1}
---------------------------------------
For the first test problem, we simulate pure diffusion in the form of a simple Brownian motion with reflecting boundary conditions, which has Fokker-Planck equation given by the diffusion PDE and corresponding boundary conditions: $$\begin{aligned}
\text{PDE}&:\quad {\frac{\partial p}{\partial t}}=D{\frac{\partial^2 p}{\partial x ^2}};\quad x\in(-1,1);\quad t\in(0,T),\label{eqn:pure_diffusion_mean_field_model}\\
\text{BCs}&:\quad {\frac{\partial p}{\partial x}}(x,t) = 0; \quad x=-1,1; \quad t\in(0,T), \label{eqn:pure_diffusion_mean_field_model_BC} \\
\text{IC}&:\quad p(x,0) = p_0(x); \quad x\in [-1,1], \label{eqn:pure_diffusion_mean_field_model_IC}\end{aligned}$$ where $p_0(x)$ denotes the initial condition. Note that $p(x,t)$ here represents the mean-field solution across the whole domain, whereas $c(x,t)$ represents the PDE solution in ${\Omega_{\text{{\tiny P}}}}$ in the hybrid method. We initialise particles uniformly across the computational domain, so that $p_0(x) \equiv N/2$, where $N$ is the (constant) number of particles in the system.
Figure \[fig:Plots\_Unif\] shows that the ARM passes the most basic test by maintaining the steady state without causing an accumulation of mass on either side of the interface. For this test problem, we also include a plot which displays the variance in the density of particles (Figure \[fig:Vars\]). In order to calculate this variance, we have binned the spatial domain onto a mesh of size $h_a$ (the same as the auxiliary region width) and calculated the variance of the density in each bin over a number of identically initialised (up to random allocation of particles in ${\Omega_{\text{{\tiny B}}}}$) repeats. This demonstrates a problem that occurs with all hybrid methods which contain an interface coupling a stochastic to a deterministic region. The variance is damped close to the interface in the stochastic part of the domain, due to the deterministic nature of the solver on the opposite side. Specifically, the PDE effectively has a stochastic boundary condition at the interface, caused by the diffusive jumps between the auxiliary regions. This causes a higher level of variance than would be expected if it was a purely deterministic regime. However, when a particle jumps from the PDE to the Brownian dynamics auxiliary region, since the PDE region is mostly deterministic, it contributes less variance than would be expected than if the stochastic method was employed across the entire domain. There are methods that can be used in order to fix this problem, such as the use of an overlap region (e.g. [@harrison2016hac]) and replacing the PDE with an appropriate SPDE (e.g. [@alexander2002ars]). This is explored in more detail in the discussion (Section \[sect:Discussion\]).
![The plot of the variance in the density of particles at time $t = 100$ for the parameter values used to produce Figure \[fig:Plots\_Unif\]. The spatial domain is partitioned into a series of bins of width $h_a$, and the particle density variance is calculated in each bin over $S = 1000$ repeats. The blue line is the variance from the hybrid method, the black dashed line is the expected variance from the fully Brownian model, and the red line is the position of the interface. The variance can be seen to be damped in the stochastic domain close to the interface, as discussed in the text.[]{data-label="fig:Vars"}](Uniform_Var.eps){width="50.00000%"}
Test problem 2: flux over the interface {#sect:Results_P2}
---------------------------------------
The second test problem is a stress test for the interfacial flux. For the PDE part of the hybrid method we solve the same diffusion equation - as in Section \[sect:Results\_P1\]. However this time we initialise by placing all particles uniformly within the PDE domain, ${\Omega_{\text{{\tiny P}}}}$, which results in $$p_0(x) = \left\{\begin{array}{ll}
N & x\in{\Omega_{\text{{\tiny P}}}} \\
0 & x\in{\Omega_{\text{{\tiny B}}}}
\end{array} \right.$$ The results from this simulation are displayed in Figure \[fig:Plots\_Left\].
As with the uniform initial condition in test problem 1, we see from Figure \[fig:Plots\_Left\] that the hybrid method agrees with the solution of the mean-field model, indicating that the method simulates flux over the interface accurately. We have also tested our hybrid method with all the mass initialised uniformly across $\Omega_{\text{B}}$ and found a similarly good agreement between the hybrid method and the mean-field solution (figures not shown).
Test problem 3: morphogen gradient
----------------------------------
For the third test problem, we investigate the formation of a morphogen gradient from a uniform initial condition. The gradient is formed by allowing particles to diffuse throughout the domain as well as to degrade at a rate $\mu$. We also have particles entering at the left-hand boundary, $x=-1$, at rate $D\lambda$, and a zero-flux condition at $x=1$. Thus, the PDE half of the hybrid domain is governed by the mean-field model representing the expected behaviour of the fully Brownian dynamics: $$\begin{aligned}
\text{PDE}&:\quad {\frac{\partial c}{\partial t}}=D{\frac{\partial^2 c}{\partial x ^2}} - \mu c;\quad x\in(-1,0);\quad t\in(0,T),\label{eqn:morphogen_gradient_mean_field_model}\\
\text{BCs}&:\quad {\frac{\partial c}{\partial x}}(-1,t) = -\lambda;\quad {\frac{\partial c}{\partial x}}(0,t) = 0; \quad t\in(0,T), \notag \\
\text{IC}&:\quad c(x,0) = c_0(x); \quad x\in [-1,0]. \notag\end{aligned}$$ For the corresponding microscopic dynamics we implement Brownian motion for the diffusion of particles and a time-based method in order to enact the degradation reactions. We note that production of particles is not implemented within the microscopic domain since it occurs at $x=-1$. $N=500$ particles are initialised uniformly across the domain.
As demonstrated in Figure \[fig:Plots\_Morph\] the solution of the hybrid method matches that of the corresponding mean-field model, as with the previous two test problems.
Test problem 4: Higher-order systems {#sect:Results_P4}
------------------------------------
For our final test problem, we look at the reaction system: $$2A \xrightarrow{\kappa_1} \emptyset,\quad \emptyset \xrightarrow{\kappa_2} A, \label{eqn:TP4_System}$$ which takes place in a three-dimensional cuboid $\Omega\subseteq\mathbb{R}^3$ of volume $V$, where $\Omega=(x_0,x_1)\times(y_0,y_1)\times(z_0,z_1)$. We further split this domain by firstly defining the position of the adaptive planar interface $I(t) \in (x_0,x_1)$ which is to be implemented for this test problem (see Section \[sect:Results\_P4\_Interface\]). In an analogous way to in the one-dimensional case, we then define the time-dependent PDE and individual-based subdomains, ${\Omega_{\text{{\tiny P}}}}(t)$ and ${\Omega_{\text{{\tiny B}}}}(t)$, with volumes ${V_{\text{{\tiny P}}}}(t)$ and ${V_{\text{{\tiny B}}}}(t)$ respectively. These subdomains and volumes depend on $t$ due to the adaptive interface position. The interface will move according to the local density profile within the PDE and Brownian dynamics auxiliary regions ${\Omega_{\text{{\tiny PA}}}}(t)$ and ${\Omega_{\text{{\tiny BA}}}}(t)$, which are explicitly defined to be:$$\begin{aligned}
{\Omega_{\text{{\tiny PA}}}}(t) &= (I(t)-h_a,I(t))\times(y_0,y_1)\times(z_0,z_1),\\
{\Omega_{\text{{\tiny BA}}}}(t) &= (I(t),I(t)+h_a)\times(y_0,y_1)\times(z_0,z_1).\end{aligned}$$
We will firstly find a PDE in one dimension that will form the deterministic part of our domain. We do this by considering the reaction system and forming an ODE to simulate this system in three dimensions. We then include isotropic diffusion to obtain a three-dimensional PDE, and finally impose a constraint on the initial condition to simplify this to a one-dimensional PDE. We then briefly describe the process we use to evolve the individual-level behaviour, before introducing an adaptive interface. We will finish this subsection with the results of some simulations of this system. Note that from now on, we will drop the dependence on $t$ for any of the subdomains, their volumes and the interface position for brevity, unless they are explicitly needed.
### PDE model {#sect:Results_P4_PDE}
We will use the chemical master equation (CME) for the reaction system in order to derive a PDE that approximates the system in ${\Omega_{\text{{\tiny P}}}}$. Let $p_n(t)=\mathbb{P}(A(t)=n)$, where $A(t)$ is the number of particles at time $t$. Then the CME for the evolution of this probability is given by: $${\frac{\text{d} p_n}{\text{d} t}} = \frac{\kappa_1}{{V_{\text{{\tiny P}}}}}\left[(n+2)(n+1)p_{n+2} - n(n-1)p_n\right] + \kappa_2{V_{\text{{\tiny P}}}}\left[p_{n-1}-p_n\right].$$ If we now define the $k^\text{th}$ central moment $\langle A^k\rangle := \sum_{n=0}^\infty n^kp_n$, we can multiply the CME by $n$ and sum over all $n\in\mathbb{N}_0$ to yield the mean equation: $${\frac{\text{d} \langle A \rangle}{\text{d} t}} = \frac{2\kappa_1}{{V_{\text{{\tiny P}}}}}\left[\langle A\rangle - \langle A^2 \rangle\right] + \kappa_2{V_{\text{{\tiny P}}}}. \label{eqn:TP4_ODE_Exact}$$ The ODE is currently exact, but depends on the second moment of $A$. Furthermore, the ODE for every moment of $A$ depends on higher moments still — the system is not closed. In order to close the system, we follow @erban2009smr and apply Poisson moment closure, which implies: $${\mathbb{V}\text{ar}}(A) = \mathbb{E}[A] \implies \langle A^2 \rangle = \langle A \rangle + \langle A \rangle ^2. \label{eqn:Moment_Closure}$$ Applying the moment closure to the ODE , and setting $c = \langle A \rangle/{V_{\text{{\tiny P}}}}$ gives us the closed ODE $${\frac{\text{d} c}{\text{d} t}} = \kappa_2 - \kappa_1c^2.$$ Finally, including isotropic diffusion through the usual Laplace operator yields the three-dimensional PDE: $${\frac{\partial c}{\partial t}} = D\nabla^2c - \kappa_1c^2 + \kappa_2;\quad (x,y,z)\in \Omega;\quad t\in(0,T). \label{eqn:TP4_PDE_3D}$$ We will enforce an initial condition which is translationally invariant in both the $y$ and $z$ co-ordinates, which means that the dynamics will remain translationally invariant for all time. As such, $c$ is simply a function of $x$ and $t$, and the dynamics can be represented by a one-dimensional equivalent of this PDE by implementing zero-flux boundaries on all boundaries and using the transformation: $$\bar{C}(x,t) = \int_{z_0}^{z_1}{\int_{y_0}^{y_1}{c(x,t)\ dy}\ dz} = L_yL_zc(x,t),$$where $L_y=y_1-y_0$ and similarly for $L_z=z_1-z_0$. This gives: $$\begin{aligned}
\text{PDE}&:\quad {\frac{\partial \bar{C}}{\partial t}}=D{\frac{\partial^2 \bar{C}}{\partial x ^2}}-\frac{\kappa_1}{L_yL_z}\bar{C}^2 + \kappa_2L_yL_z;\quad x\in(x_0,I);\quad t\in(0,T),\label{eqn:TP4_PDE}\\
\text{BCs}&:\quad {\frac{\partial \bar{C}}{\partial x}}(x,t) = 0; \quad x=x_0,I; \quad t\in(0,T), \notag \\
\text{IC}&:\quad \bar{C}(x,0) = \bar{C}_0(x); \quad x\in [x_0,I]. \notag\end{aligned}$$
### Individual-based formulation {#sect:Results_P4_Individual}
We now turn our attention to the individual-based system. In order to simulate the three-dimensional individual-based model, we will follow the $\lambda$-$\rho$ method [@erban2009smr]. In the context of this system, whenever two particles are within the reaction radius $\rho$, they react with a probability $P_\lambda$, which is a function of the kinetic rate $\kappa_1$, the time-step $\Delta t$, and the diffusion coefficient $D$. For more information on how $P_\lambda$ is chosen, we refer the reader to @erban2009smr. The zeroth-order reaction is completed by initialising a particle uniformly throughout the individual-based domain ${\Omega_{\text{{\tiny B}}}}$ with probability $\kappa_2\Delta t{V_{\text{{\tiny B}}}}$, which we ensure is below 1 by choosing $\Delta t$ to be sufficiently small.
### Adaptive interface {#sect:Results_P4_Interface}
Test problems 1–3 have been simulated using a static interface. However, this requires *a priori* knowledge of where the interface should be for all time. When the finer scale modelling regime is required in order to resolve a specific area of space in more detail (for example, the region around ion channels [@dobramysl2015pbd]), the interface position will be known. However, if the purpose of the interface is to split regions of space in which there are high and low particle numbers, a different approach is required. In this case, the interface (or interfaces) need to move with the density of particles to maintain the computational savings they are designed to provide. We now describe a method, adapted from @robinson2014atr which allows the interface to move adaptively.
The interface at time $t$, which we shall denote by $I(t)$, moves according to local particle numbers in the auxiliary regions around it. We set two thresholds $\beta_u > \beta_l$, and move the interface towards the PDE subdomain if ${N_{\text{{\tiny PA}}}}(t) < \beta_l$ and towards the individual-based subdomain if ${N_{\text{{\tiny BA}}}}(t) > \beta_u$ (borrowing the notation from Section \[sect:ARM\]). The two threshold values are designed to prevent the interface from rapidly oscillating between two values, which is a possibility when $\beta_u=\beta_l$ due to the stochastic nature of the system. We enforce that when the interface moves, it moves a distance $h_a$, the width of the auxiliary region, in the chosen direction.
If the interface moves towards the PDE subdomain (i.e. ${N_{\text{{\tiny PA}}}}(t) < \beta_l$), we convert the PDE auxiliary region into particles, initialising each one uniformly. As ${N_{\text{{\tiny PA}}}}(t)$ is not necessarily an integer, we treat the fractional part (${N_{\text{{\tiny PA}}}}(t)\ \text{mod}\ 1$) to be the probability of initialising one extra particle within the newly formed individual-based region. We then scale the rest of the PDE subdomain to ensure that we conserve mass. During an interface movement towards the individual-based subdomain (i.e. ${N_{\text{{\tiny BA}}}}(t) > \beta_u$), the Brownian auxiliary region is converted to PDE mass by initialising a density of ${N_{\text{{\tiny BA}}}}(t)/h_a$ uniformly across the new PDE mesh points created by moving the interface. For a more detailed description of a similar method, we direct the interested reader to @robinson2014atr.
### Results {#sect:Results_P4_Results}
We consider $N$ particles initialised throughout $\Omega$ with a constant negative gradient so that the density of particles at position $x_1$ is equal to zero. This ensures that the interface will move as the dynamics progress. The results can be seen in Figure \[fig:Plots\_Second\], in which the hybrid method has been averaged over $S=1000$ repeats. The hybrid density in the case of the moving interface is represented as yellow bars throughout the domain. This is because the interface position changes with each repeat, and so very few regions of space are solely represented by one or the other modelling paradigm over all repeats.
------- ---- ------------ ------ ------------ ------
$x_0$ 0 $D$ 0.2 $\Delta t$ 0.01
$x_1$ 10 $\kappa_1$ 0.01 $h_p$ 0.1
$y_0$ 0 $\kappa_2$ 0.5 $h_a$ 0.5
$y_1$ 2 $\rho$ 0.1 $I_0$ 0.5
$z_0$ 0 $N$ 200
$z_1$ 2 $T$ 5
$V$ 40 $\beta_u$ 9.5
$\beta_l$ 4
------- ---- ------------ ------ ------------ ------
: Table of parameter values for test problem 4.[]{data-label="tab:Params"}
We can see good agreement between the hybrid method and the fully individual-based method throughout the domain, with the only discrepancy close to the left hand boundary at 0 caused by the difference between the PDE and individual-based methods due to moment closure. We compare our hybrid method to the fully individual-based method here, in contrast to the PDE solution used in test problems 1-3, due to the inaccuracy introduced in the PDE by the moment closure required for the second order reaction.
Error analysis {#sect:Error}
==============
We have seen in Section \[sect:Results\] that the solutions provided by the hybrid method visually match the mean-field solution. Within this section we quantify the difference between the solutions of these test problems. We compare the mass in the PDE and Brownian regions of the domain between the two methods. Separately we compare the density profile across the whole domain using the histogram distance error (HDE). We then proceed to investigate the dependence of the accuracy of the hybrid method on the two free simulation parameters (${\Delta t}$ and $h_a$).
Quantitative comparisons
------------------------
In order to evaluate the accuracy of the ARM for test problems 1,2 and 3, we compare its mean behaviour (averaged over $S=1000$ repeat simulations) to the mean-field model for which we compute the analytical solution across the entire domain $\Omega$, for each of our test problems. Figure \[fig:Errors\] contains nine plots which demonstrate the error for the first three test problems above; - are for test problem 1, - are for test problem 2 and - are for test problem 3. The first and second columns show particle number comparisons between the hybrid and analytical solutions. Specifically, in the first column we compare $${N_{\text{{\tiny MP}}}}(t)=\int_{-1}^0p(x,t){\mbox{d}}x,$$ the expected number of particles in ${\Omega_{\text{{\tiny P}}}}$ in the mean-field model to $${N_{\text{{\tiny HP}}}}(t)=\frac{1}{S}\displaystyle\sum_{s=1}^S\int_{-1}^0 c^s(x,t){\mbox{d}}x,$$ the expected number of particles in ${\Omega_{\text{{\tiny P}}}}$ in the hybrid method. Here, as before, $p(x,t)$ represents the mean-field PDE solution at position $x$ at time $t$ and $c^s(x,t)$ represents the PDE part of the solution in the hybrid method for repeat $s$ of $S$. Explicitly, we plot $({N_{\text{{\tiny HP}}}}-{N_{\text{{\tiny MP}}}})/{N_{\text{{\tiny MP}}}}$, which shows no bias around zero for any of the three test problems. For completeness, in the second column we also compare $${N_{\text{{\tiny MB}}}}(t)=\int_{0}^1p(x,t)\ {\mbox{d}}x,$$ the expected number of particles in ${\Omega_{\text{{\tiny B}}}}$ in the mean-field model to $${N_{\text{{\tiny HB}}}}(t)= \frac{1}{S}\displaystyle\sum_{s=1}^S {N_{\text{{\tiny HB}}}}^s(t),$$ the expected number of particles in ${\Omega_{\text{{\tiny B}}}}$ in the hybrid methods. Here, ${N_{\text{{\tiny HB}}}}^s(t)$ is the number of particles in the Brownian region of the hybrid method at time $t$ for repeat $s$ of $S$. Explicitly, we plot $({N_{\text{{\tiny HB}}}}-{N_{\text{{\tiny MB}}}})/{N_{\text{{\tiny MB}}}}$, which again shows no bias around zero for any of the three test problems.
The last column of Figure \[fig:Errors\] contains the histogram distance error (HDE), which is defined by $$\text{HDE}(t) = \frac{1}{2}\sum_{\ell=1}^{L}{\left|c_\ell^{H}(t)-c_\ell^{P}(t)\right|},
\label{eqn:HDE}$$where $\ell$ indexes a common mesh on which the solutions are compared. $c_\ell^{H}(t)$ is the normalised solution of the hybrid method at mesh point $\ell$ and time $t$, and $c_\ell^{P}(t)$ is the normalised solution of the mean-field model at the same common mesh point and time, where $$\sum_{\ell=1}^L{c_\ell^{H}(t)}=\sum_{\ell=1}^L{c_\ell^{P}(t)}=1\quad\forall t\geq 0.$$ This ensures a value of the HDE between 0 and 1. Here, 0 means that the two solutions are exactly the same, and 1 corresponds to the two solutions having non-overlapping supports. All figures were produced using the same number of repeats ($S=1000$).
In all cases, the relative errors between the mean-field and hybrid methods, in Figure \[fig:Errors\], are low with no discernible bias about zero. Similarly, all HDE plots in Figure \[fig:Errors\] are low for the majority of the simulations. This demonstrates numerically that the hybrid scheme presented in this paper is correctly reproducing the behaviour of the Brownian model in the mean-field. These error plots confirm the visual concurrence shown in Figures \[fig:Plots\_Unif\]–\[fig:Plots\_Morph\].
For the fourth test problem, we use a different error measurement due to the disparity between the mean-field PDE and individual-based systems. Consequently, we choose to compare the number of particles in the final compartment for both the hybrid method and the individual-based method. We motivate this in two ways. Firstly, using this measure of error, we are able to minimise the influence of the extra error caused by the difference between the mean-field PDE and the individual-based method. Secondly, several biological systems require detailed knowledge of the particle concentrations at the end of the domain. Apical growth of filamentous cells such as fungi [@goriely2008mmh] is such an example. If we define ${N_{\text{{\tiny H}}}}(t)$ to be the average number of particles in the region $(x_1-h_a,x_1)\times(y_0,y_1)\times(z_0,z_1)$ (where we recall that $\Omega = (x_0,x_1)\times(y_0,y_1)\times(z_0,z_1)$), when simulating the hybrid method at time $t$, and the quantity ${N_{\text{{\tiny M}}}}(t)$ to be the same for the fully microscopic simulation, we can obtain a measurement of error given by $${E_{\text{{\tiny Rel}}}}(t) = \frac{{N_{\text{{\tiny M}}}}(t)-{N_{\text{{\tiny H}}}}(t)}{{N_{\text{{\tiny M}}}}(t)}. \label{eqn:Error_Second}$$
![The error measurement given in equation for the system simulated in Figure \[fig:Plots\_Second\].[]{data-label="fig:Error_Second"}](Err_Second.eps){width="80.00000%"}
The relative error shows no long-term bias in either direction, and oscillates around zero, indicating a close agreement between our hybrid method and the ground truth individual-based method. The hybrid method completed its 1000 repeats in 485.5 seconds, while the fully individual-based method took 1047.4 seconds.
Parameter choice {#sect:Error_Parameter}
----------------
Within the ARM, there are two free parameters that need to be chosen – the width of the auxiliary regions $h_a$ and the time-step for the PDE and Brownian updates ${\Delta t}$. These need to be chosen so that the quantity $D{\Delta t}/h_a^2$ remains small enough that the particle numbers in the auxiliary regions do not become overly equilibrated between PDE/Brownian update steps. That is to say, if there is a gradient across the interface, $\Delta t$ should be small enough that the closed system of the two auxiliary regions should not reach steady state between PDE/Brownian update steps.
In order to demonstrate why $D{\Delta t}/h_a^2$ must be small, we consider the evolution of particle numbers in the two auxiliary regions between PDE/Brownian update steps. We form an ODE for particle numbers in one of these boxes (using the fact that particle numbers are conserved between PDE/Brownian updates).
Let $\nu_0$ be the (constant) number of particles in the two auxiliary regions combined, ${M_{\text{{\tiny P}}}}(t),\ {M_{\text{{\tiny B}}}}(t)$ be the mean number of particles in the PDE and Brownian auxiliary regions respectively at time $t$, and ${\mu_{\text{{\tiny P}}}},\ {\mu_{\text{{\tiny B}}}}$ be the number in the PDE and Brownian auxiliary regions respectively at time 0, which will represent the beginning of a time-step. Then, the equation for the mean number of particles in the PDE auxiliary region can be calculated from a simple probability master equation as $${\frac{\text{d} {M_{\text{{\tiny P}}}}}{\text{d} t}} = d{M_{\text{{\tiny B}}}} - d{M_{\text{{\tiny P}}}} = d(\nu_0-{M_{\text{{\tiny P}}}})-d{M_{\text{{\tiny P}}}} = d\nu_0 -2d{M_{\text{{\tiny P}}}},$$ where we recall that $d$ is the jumping rate between the two auxiliary regions and is linked to the diffusion constant, $D$, via equation . Solving this ODE gives $${M_{\text{{\tiny P}}}}(t) = \frac{1}{2}\left[\nu_0-(\nu_0-2{\mu_{\text{{\tiny P}}}})e^{-2dt}\right].
\label{eqn:ODE_MP_Sol}$$ Assuming a small time-step, ${\Delta t}$, we can approximate ${M_{\text{{\tiny P}}}}({\Delta t})$, the number of particles after a time-step has occurred, by Taylor expanding equation to first order: $$\begin{aligned}
{M_{\text{{\tiny P}}}}({\Delta t}) &= {M_{\text{{\tiny P}}}}(0) + {\Delta t}{M_{\text{{\tiny P}}}}'(0) + o({\Delta t})\\
&\approx\frac{1}{2}\left[\nu_0-(\nu_0-2{\mu_{\text{{\tiny P}}}})\right] + \frac{{\Delta t}}{2}\left[2d(\nu_0-2{\mu_{\text{{\tiny P}}}})\right]\\
&=(1-2d{\Delta t}){\mu_{\text{{\tiny P}}}} + d{\Delta t}\nu_0.\end{aligned}$$ Fixing the value of $D$ and using equation , we find that $${M_{\text{{\tiny P}}}}({\Delta t})\approx\left(1-\frac{2D{\Delta t}}{h_a^2}\right){\mu_{\text{{\tiny P}}}}+\frac{D{\Delta t}}{h_a^2}\nu_0.$$ We require the change in the number of particles over the small time-step to be small, and so would like ${M_{\text{{\tiny P}}}}({\Delta t}) \approx \mu_P$. Thus we need to choose our parameters such that the quantity $D{\Delta t}/h_a^2$ small. This elucidates an important relationships between the fixed and free parameters of the model. If the diffusion coefficient is large then we must choose a small update time-step or a larger auxiliary region length to compensate.
![A contour plot for the HDE at time $T=10$ for a series of simulations initialised with all particles uniformly distributed within the PDE region of the domain. Dark colours indicate low error. The red line is the representative contour $D{\Delta t}/h_a^2=1/2$. Here, $D=0.05$ and all simulations are averaged over $100$ repeats.[]{data-label="fig:HDE_Error"}](ARM_Error.eps){width="60.00000%"}
Figure \[fig:HDE\_Error\] shows that a large region of the ${\Delta t}-h_a$ space has a very low histogram distance error, meaning that our method is robust to parameter change, and only breaks down once the value of $D{\Delta t}/h_a^2$ becomes very large. The plot also shows that, given any choice of the width of the auxiliary regions, $h_a$, there is a value for the time-step, ${\Delta t}$, which will give a low level of error. Also, depending on our choice of $\Delta t$, we can adjust $h_a$ to make the simulation more accurate.
Discussion {#sect:Discussion}
==========
We have presented a new spatially coupled hybrid method for coupling a Brownian dynamics representation of a reaction-diffusion system to its corresponding mean-field PDE description. By bridging the gap in spatial scales with intermediate auxiliary regions, we have produced an algorithm that is not only accurate, but is also robust to the choice of the free parameters within the problem, namely the width of the auxiliary regions, $h_a$, and the fixed time-step, $\Delta t$ used to update both the PDE and Brownian dynamics. This is in direct contrast to a previously presented PDE-to-Brownian hybrid, which we demonstrated to be extremely parameter-sensitive. In order to make the ARM even more robust, applicable and efficient, we now discuss several areas for possible extension, which will be addressed in future works.
In the interests of completeness we should point out that, as with the pseudo-compartment method of @yates2015pcm, the auxiliary region method requires that the mass in the PDE auxiliary region $\Omega_\text{PA}$ be sufficient for a step function, corresponding to the mass of a particle, to be removed uniformly from across the auxiliary region. \[xr:Negative\_Particles\_Prior\] This will lead to difficulties in situations in which particle numbers are low around the interface. Arguably though, we should not employ such hybrid methods in situations for which particle density is low around the interface as the PDE will be a poor model of the true stochastic, microscopic dynamics in these regions. A possible solution to this inconvenience, is the incorporation of an adaptive interface, which we have employed in test problem 4. Such interfaces evolve with the simulation dynamics, ensuring the appropriate model is used for the corresponding particle density [@robinson2014atr].
A related issue is that of multiple interfaces. Multiple interfaces will allow the efficient simulation of stochastic reaction diffusion systems in which multiple regions of high and low concentration are expected. Such patterns will require interfaces to be dynamic in number and transient in nature. Although we have not implemented such interfaces in this work we expect it to be a relatively straightforward extension. While we have presented an example in which the system is simulated in a cuboid with a planar interface (test problem 4), non-planar interfaces, such as those which have corners or are curved, and complex domain geometries, present deeper challenges that we hope to address in a future publication.
Failing to maintain stochastic variation is a problem which is common amongst many spatially coupled hybrid methods. As a result of the deterministic nature of the PDE, the noise in the Brownian dynamics region of the domain is damped in comparison to the fully microscopic model (see Figure \[fig:Vars\]). In the literature, two approaches have been used in order to rectify this. The first is an overlap region, which has been employed in several papers [@harrison2016hac; @franz2012mrd; @flekkoy2001cpf]. These methods introduce a region of space which lies in the intersection of the two domains. In these regions, mass is simultaneously represented using both scales of description. The second is to replace the deterministic PDE with an appropriately chosen stochastic partial differential equation (SPDE). @alexander2002ars consider such a coupling and demonstrate they can indeed fix the discrepancy by using an SPDE as their continuum macro-scale model. We will address both the use of SPDEs and overlap regions (in which the region between the PDE and the Brownian dynamics regions is simulated using a purely compartment-based method) in forthcoming work.
The auxiliary region method provides a simple yet accurate method to couple an individual Brownian dynamics representation of a reaction-diffusion system to a corresponding PDE representation. Our hybrid algorithm will be of particular interest to researchers modelling reaction-diffusion systems whose concentrations vary significantly across the spatial domain. By reducing the computational expense of simulations, the ARM will facilitate the investigation of stochastic effects in such systems, in some cases, making the difference between being able to interrogate the system and not. In particular, we suggest that our method will be useful for the investigation of stochastic Turing patterns [@flegg2016srk], Fisher waves [@breuer1995mls; @breuer1994few], oscillatory dynamics [@hoffmann2014omp] and excitatory dynamics [@gerisch2013mar] with applications in embryogenesis [@mort2016rdm], intracellular dynamics [@khan2011scd] and pattern formation [@flegg2016srk] amongst others. It may also be worthwhile to interface the methods presented here with commonly used Brownian dynamics simulation software packages such as Smoldyn [@andrews2004ssc].
Acknowledgements {#acknowledgements .unnumbered}
================
Cameron Smith is supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), under the project EP/L015684/1. Dr Christian Yates would like to thank the CMB/CNCB preprint club for constructive and helpful comments on a preprint of this paper.
Comparing to PDE-assisted Brownian dynamics {#sect:Appendix}
===========================================
Within this section, we apply the same parameter values as used in Section \[sect:Previous\] in order to demonstrate that the ARM can accurately simulate the problem that PDE assisted Brownian dynamics could not (see Figure \[fig:Plots\_franz\]). Recall, that we use ${\Omega_{\text{{\tiny P}}}}=(-1,0)$, ${\Omega_{\text{{\tiny B}}}} = (0,1)$ with the interface placed at $I=0$. The only additional parameter that is to be defined is the auxiliary region width, which we set here to be $h_a=0.1$. The results can be seen in Figure \[fig:Plots\_Comp\].
As can be seen from this figure, the agreement between the mean-field and hybrid solutions is much closer than that of the PDE assisted Brownian dynamics [@franz2012mrd]. This indicates an improvement over the previous method. We also present the error plots which are described in Section \[sect:Error\] — namely the relative errors in particle numbers and the histogram distance errors.
Once again, the relative error plots (Figures \[fig:Errors\_Comp\]-) appear to show no long-term bias in either direction and the histogram distance error (Figure \[fig:Errors\_Comp\_HDE\]) is small.
[^1]: Throughout this paper, regardless of whether we have interactions between particles or not, we shall refer to models at this microscopic scale as “Brownian dynamics”.
[^2]: However, with the increasing awareness of the importance of randomness, stochastic partial differential equations (SPDEs) are also becoming popular macroscopic representations.
[^3]: Note that the method can be extended to higher dimensions with (hyper-)planar interfaces in a straight-forward manner.
[^4]: Note that this PDE can be simulated using any appropriate numerical solver, including the finite-element method or finite-volume method.
[^5]: Note that there is no requirement for the PDE and Brownian time steps to be the same. In many situation it may be useful to have a significantly finer Brownian time-step than PDE time step in order to accurately resolve the individual-based dynamics. We employ the same time-step in our simulations for simplicity.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
In the present study, a numerical method, perturbation-iteration algorithm (shortly PIA), have been employed to give approximate solutions of nonlinear fractional-integro differential equations (FIDEs). Comparing with the exact solution, the PIA produces reliable and accurate results for FIDEs.
**Keywords:** Fractional-integro differential equations, Caputo fractional derivative, Initial value problems, Perturbation-Iteration Algorithm.
author:
- |
Mehmet ŞENOL and İ. Timuçin DOLAPCİ\
Nevşehir Haci Bektaş Veli University, Department of Mathematics, Nevşehir, Turkey\
Celal Bayar University, Department of Mechanical Engineering,\
Manisa, Turkey\
e-mail:[email protected], [email protected]
title: 'On the Numerical Solution of Nonlinear Fractional-Integro Differential Equations'
---
Introduction
============
Scientists has been interested in fractional order calculus as long as it has been in classical integer order analysis. However, for many years it could not find practical applications in physical sciences. Recently, fractional calculus has been used in applied mathematics, viscoelasticity [@1], control [@2], electrochemistry [@3], electromagnetic [@4].
Developments in symbolic computation capabilities is one of the driving forces behind this rise. Different multidisciplinary problems can be handled with fractional derivatives and integrals.
[@5] and [@6] are studies that describe the fundamentals of fractional calculus give some applications. Existence and uniqueness of the solutions are also studied in [@7].
Similar to the studies in physical sciences, fractional order integro differential equations (FIDEs) also gave scientists the opportunity of describing and modeling many important and useful physical problems.
In this manner, a remarkable effort has been expended to propose numerical methods for solving FIDEs, in recent years. Fractional variational iteration method [@8; @9], homotopy analysis method [10,11]{}, Adomian decomposition method [@12; @13] and fractional differential transform method [@14; @15; @16] are among these methods.
In our study, we use the previously developed method PIA, to obtain approximate solutions of some FIDEs. This method can be applied to a wide range of problems without requiring any special assumptions and restrictions.
A few fractional derivative definitions of an arbitrary order exists in the literature. Two most used of them are the Riemann-Liouville and Caputo fractional derivatives. The two definitions are quite similar but have different order of evaluation of derivation.
The Riemann-Liouville fractional integral of order $\alpha $ is described by:$$J^{\alpha }u(x)=\frac{1}{\Gamma (\alpha )}\int_{0}^{x}(x-t)^{\alpha
-1}u(t)dt,\quad \alpha >0,\quad x>0. \label{1}$$
The Riemann-Liouville and Caputo fractional derivatives of an arbitrary order are defined as the following, respectively$$D^{\alpha }u(x)=\frac{d^{m}}{dx^{m}}\left( J^{m-\alpha }u(x)\right)
\label{2}$$$$D_{\ast }^{\alpha }u(x)=J^{m-\alpha }\left( \frac{d^{m}}{dx^{m}}u(x)\right) .
\label{3}$$where $m-1<\alpha \leqslant m$ and $m\in
\mathbb{N}
.$
Due to the appropriateness of the initial conditions, fractional definition of Caputo is often used in recent years.
The Caputo fractional derivative of a function $u(x)$ is defined as$$D_{\ast }^{\alpha }u(x)=\left\{
\begin{array}{cc}
\frac{1}{\Gamma (m-\alpha )}\int_{0}^{x}(x-t)^{m-\alpha -1}u^{(m)}(t)dt, &
m-1<\alpha \leqslant m \\
\frac{d^{m}u(x)}{dx^{m}} & \alpha =m\end{array}\right. \label{4}$$for $m-1<\alpha \leqslant m,$ $m\in
\mathbb{N}
,$ $x>0,$ $u\in C_{-1}^{m}.$
Following lemma gives the two main properties of Caputo fractional derivative.
For $m-1<\alpha \leqslant m,$ $u\in C_{\mu }^{m},$ $\mu \geqslant -1$ and $m\in
\mathbb{N}
$ then $$D_{\ast }^{\alpha }J^{\alpha }u(x)=u(x) \label{5}$$and $$J^{\alpha }D_{\ast }^{\alpha }u(x)=u(x)-\sum_{k=0}^{m-1}u^{(k)}(0^{+})\frac{x^{k}}{k!},\quad x>0. \label{6}$$
After this introductory section, Section 2 is reserved to a brief review of the Perturbation-Iteration Algorithm PIA, in Section 3 some examples are illustrated to show the simplicity and effectiveness of the algorithm. Finally the paper ends with a conclusion in Section 4.
Analysis of the PIA
===================
Differential equations are naturally used to describe problems in engineering and other applied sciences. Searching approximate solutions for complicated equations has always attracted attention. Many different methods and frameworks exist for this purpose and perturbation techniques [@17; @18; @19] are among them. These techniques can be used to find approximate solutions for both ordinary and partial differential equations.
Requirement of a small parameter in the equation that is sometimes artificially inserted is a major limitation of perturbation techniques that renders them valid only in a limited range. Therefore, to overcome the disadvantages come with the perturbation techniques, several methods have been proposed by authors [@20; @21; @22; @23; @24; @25; @26; @27; @28; @29].
Parallel to these attempts, a perturbation-iteration method has been proposed by Aksoy, Pakdemirli and their co-workers [@33; @34; @35] previously. A primary effort of producing root finding algorithms for algebraic equations [@30; @31; @32], finally guided to obtain formulae for differential equations also [@33; @34; @35]. In the new technique, an iterative algorithm is constructed on the perturbation expansion. The present method has been tested on Bratu-type differential equations [@33] and first order differential equations [@34] with success. Then the algorithms were applied to nonlinear heat equations also [@35]. Finally, the solutions of the Volterra and Fredholm type integral equations [@36] and ordinary differential equation systems [@37] have been presented by the developed method.
This new algorithm have not been used for any fractional integro differential equations yet. To obtain the approximate solutions of FIDEs, the most basic perturbation-iteration algorithm PIA(1,1) is employed by taking one correction term in the perturbation expansion and correction terms of only first derivatives in the Taylor series expansion. [@33; @34; @35].
Take the fractional-integro differential equation. $$F\left( u^{(\alpha )},u,\int_{0}^{t}{g\left( t,s,u(s)\right) ds},\varepsilon
\right) =0 \label{7}$$
where $u=u(t)$ and $\varepsilon $ is a small parameter. The perturbation expansions with only one correction term is
$$\begin{aligned}
u_{n+1} &=&u_{n}+\varepsilon {\left( u_{c}\right) }_{n}\ \notag \\
u_{n+1}^{\prime } &=&u_{n}^{\prime }+\varepsilon {\left( u_{c}^{\prime
}\right) }_{n}\ \label{8}\end{aligned}$$
Replacing Eq.$(\ref{8})$ into Eq.$(\ref{7})$ and writing in the Taylor series expansion for only first order derivatives gives
$$\begin{aligned}
&&F\left( u_{n}^{\left( \alpha \right) },u_{n},\int_{0}^{t}{g\left(
t,s,u_{n}(s)\right) ds},0\right) \notag \\
&&+F_{u}\left( u_{n}^{\left( \alpha \right) },u_{n},\int_{0}^{t}{g\left(
t,s,u_{n}(s)\right) ds},0\right) \varepsilon {\left( u_{c}\right) }_{n}
\notag \\
&&+F_{u^{\left( \alpha \right) }}\left( u_{n}^{\left( \alpha \right)
},u_{n},\int_{0}^{t}{g\left( t,s,u_{n}(s)\right) ds},0\right) \varepsilon {\left( u_{c}^{(\alpha )}\right) }_{n} \notag \\
&&+F_{\int {u}}\left( u_{n}^{\left( \alpha \right) },u_{n},\int_{0}^{t}{g\left( t,s,u_{n}(s)\right) ds},0\right) \varepsilon \int {{\left(
u_{c}\right) }_{n}} \notag \\
&&+F_{\varepsilon }\left( u_{n}^{\left( \alpha \right) },u_{n},\int_{0}^{t}{g\left( t,s,u_{n}(s)\right) ds},0\right) \varepsilon =0 \label{9}\end{aligned}$$
or
$${\left( u_{c}^{(\alpha )}\right) }_{n}\frac{\partial F}{\partial u^{(\alpha
)}}+{\left( u_{c}\right) }_{n}\frac{\partial F}{\partial u}+\left( \int {{\left( u_{c}\right) }_{n}}\right) \frac{\partial F}{\partial (\int {u})}+\frac{\partial F}{\partial \varepsilon }+\frac{F}{\varepsilon }=0 \label{10}$$
Here $(.)^{\prime }$ represents the derivative according to the independent variable and $$F_{\varepsilon }=\frac{\partial F}{\partial \varepsilon },~F_{u}=\frac{\partial F}{\partial u},~F_{u^{\prime }}=\frac{\partial F}{\partial
u^{\prime }},\ldots \label{11}$$ The derivatives in the expansion are evaluated at $\varepsilon =0$. Beginning with an initial function $u_{0}(t)$, first ${\left( u_{c}\right) }_{0}(t)$ is calculated by the help of $(\ref{10})$ and then substituted into Eq.$(\ref{8})$ to calculate $u_{1}(t)$. Iteration procedure is continued using $(\ref{10}) $ and $(\ref{8})$ until obtaining a reasonable solution.
Applications
============
Consider the following nonlinear fractional-integro differential equation [@38]:
$$\frac{d^{\alpha }u(t)}{{dt}^{\alpha }}-\int_{0}^{1}{ts({u(s))}^{2}ds}=1-\frac{t}{4},\ \ \ t>0,\ \ \ 0\leq t<1,\ \ \ 0<\alpha \leq 1 \label{12}$$
with the initial condition $u\left( 0\right) =0$ and the known exact solution for $\alpha =1$ is
$$u\left( t\right) =t \label{13}$$
Before iteration process rewriting Eq.$(\ref{12})$ with adding and subtracting $u^{\prime }(t)$ to the equation gives
$$\varepsilon \frac{d^{\alpha }u(t)}{{dt}^{\alpha }}-u^{^{{\prime }}}\left(
t\right) +{\varepsilon u}^{^{{\prime }}}\left( t\right) -{\varepsilon }\int_{0}^{1}{ts({u(s))}^{2}ds}-1+\frac{t}{4}=0 \label{14}$$
In this case for
$$F\left( u^{^{{\prime }}},u,\varepsilon \right) =\frac{1}{\Gamma (1-\alpha )}\varepsilon \int_{0}^{t}{\frac{u^{\prime }(s)}{{(t-s)}^{\alpha }}ds-u_{n}^{^{{\prime }}}\left( t\right) +\varepsilon u_{n}^{^{{\prime }}}\left( t\right)
-\varepsilon \int_{0}^{1}{ts({u_{n}(s))}^{2}ds}-1+\frac{t}{4}} \label{15}$$
and the iteration formula$$u^{^{{\prime }}}(t)+\frac{F_{u}}{F_{u^{\prime }}}u\left( t\right) =-\frac{F_{\varepsilon }+\frac{F}{\varepsilon }}{F_{u^{\prime }}} \label{16}$$
the terms that will be replaced in, are
$$\begin{aligned}
F &=&{u_{n}^{^{{\prime }}}\left( t\right) }-1+\frac{t}{4} \notag \\
F_{u} &=&0 \notag \\
F_{u^{\prime }} &=&1 \notag \\
F_{\varepsilon } &=&-{u_{n}^{^{{\prime }}}\left( t\right) }+\frac{1}{\Gamma
(1-\alpha )}\int_{0}^{t}{\frac{u^{\prime }(s)}{{(t-s)}^{\alpha }}ds}-\int_{0}^{1}{ts({u(s))}^{2}ds} \label{17}\end{aligned}$$
After substitution the differential equation for this problem, Eq.$(\ref{10})$ becomes
$$\frac{\int_{0}^{t}{{\left( -s+t\right) }^{-\alpha }{u_{n}}^{^{{\prime }}}(s)ds}}{\Gamma (1-\alpha )}+{\left( {u}_{c}^{\prime }(t)\right) }_{n}=\int_{0}^{1}{st{\left( u_{n}\left( s\right) \right) }^{2}ds}+\frac{4-t+4\left( -1+\varepsilon \right) {u}_{n}^{\prime }\left( t\right) }{4\varepsilon } \label{18}$$
Appropriate to the initial conditions, chosen $u_{0}\left( t\right) =0$ and, solving Eq.$(\ref{18})$ for $n=0$ gives $${{(u}_{c}(t))}_{0}=t-\frac{t^{2}}{8}+C_{1} \label{19}$$
This expression written in
$$u_{1}=u_{0}+\varepsilon {{(u}_{c}(t))}_{0} \label{20}$$
gives
$$u_{1}\left( x,t\right) =u_{0}\left( x,t\right) +\varepsilon (t-\frac{t^{2}}{8}+C_{1}) \label{21}$$
or
$$u_{1}\left( x,t\right) =\varepsilon (t-\frac{t^{2}}{8}+C_{1}) \label{22}$$
Solving this equation for
$$u_{1}\left( 0\right) =0 \label{23}$$
we obtain
$$C_{1}=0 \label{24}$$
For this value and $\varepsilon =1$ reorganizing $u_{1}(t)$
$$u_{1}\left( t\right) =t-\frac{t^{2}}{8} \label{25}$$
gives the first iteration result. If the iteration procedure is continued in a similar way, we obtain the following iterations.
$$u_{2}(t)=2t-\frac{571t^{2}}{3840}+\frac{t^{2-\alpha }(t+4(-3+\alpha ))}{4\Gamma (4-\alpha )}\ \label{26}$$
$$\begin{aligned}
u_{3}\left( t\right) &=&3t+\frac{29844889t^{2}}{176947200}-\frac{t^{3-2\alpha }\left( t+8\left( -2+\alpha \right) \right) }{4\Gamma \left(
5-2\alpha \right) } \notag \\
&&+\frac{t^{2}\left( 3379230+8t^{-\alpha }\left( 1051t+5760\left( -3+\alpha
\right) \right) \left( -7+\alpha \right) \left( -6+\alpha \right) \left(
-5+\alpha \right) \right) }{15360\left( -7+\alpha \right) \left( -6+\alpha
\right) \left( -5+\alpha \right) \Gamma \left( 4-\alpha \right) } \notag \\
&&-\frac{2240277\alpha +\left( 450151-28436\alpha \right) \alpha ^{2}}{15360\left( -7+\alpha \right) \left( -6+\alpha \right) \left( -5+\alpha
\right) \Gamma \left( 4-\alpha \right) } \notag \\
&&-\frac{t^{2}\left( -4+\alpha \right) \left( -1159+2\alpha \left(
529+16\left( -10+\alpha \right) \alpha \right) \right) }{64\left( -7+2\alpha
\right) {\Gamma \left( 5-\alpha \right) }^{2}} \label{27}\end{aligned}$$
The other iterations contain large inputs and are not given. A computational software program could help to calculate the other iterations up to any order. In Table 1. some of the PIA iteration results are compared with the exact solution. The results express that the present method gives highly approximate solutions. Also in Figure 1. the obtained results are illustrated graphically.
------- ---------- ---------- ---------- ---------- ---------------- ----------------
t $u_{2}$ $u_{3}$ $u_{4}$ $u_{5}$ Exact Solution Absolute Error
$0.0$ 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
$0.1$ 0.099763 0.099953 0.099990 0.099981 0.100000 1.872712E-6
$0.2$ 0.199052 0.199812 0.199962 0.199992 0.200000 7.490848E-6
$0.3$ 0.297867 0.299577 0.299915 0.299983 0.300000 1.685440E-5
$0.4$ 0.396208 0.399249 0.399850 0.399970 0.400000 2.996339E-5
$0.5$ 0.494075 0.498826 0.499765 0.499953 0.500000 4.681780E-5
$0.6$ 0.591468 0.598310 0.599662 0.599932 0.600000 6.741763E-5
$0.7$ 0.688388 0.697700 0.699541 0.699908 0.700000 9.176289E-5
$0.8$ 0.784833 0.796996 0.799400 0.799880 0.800000 1.198535E-4
$0.9$ 0.880804 0.896198 0.899241 0.899848 0.900000 1.516896E-4
$1.0$ 0.976302 0.995307 0.999063 0.999812 1.000000 1.872712E-4
------- ---------- ---------- ---------- ---------- ---------------- ----------------
: Numerical results of Example 3.1. for different $u$ values when $\protect\alpha =1$
{width="3.50in"}
Consider the following system of nonlinear fractional-integro differential equations [@39]:
$$\begin{aligned}
\frac{d^{\alpha _{1}}u(t)}{{dt}^{\alpha _{1}}} &=&1-\frac{1}{2}{\left( k^{^{{\prime }}}\left( t\right) \right) }^{2}+\int_{0}^{t}{\left( \left(
t-s\right) k\left( s\right) +u\left( s\right) k\left( s\right) \right) ds}
\notag \\
\frac{d^{\alpha _{2}}k(t)}{{dt}^{2}} &=&2t+\int_{0}^{t}{\left( \left(
t-s\right) u\left( s\right) -k^{2}\left( s\right) +u^{2}\left( s\right)
\right) ds}\ \ \ \ 0<\alpha _{1},\alpha _{2}\leq 1 \label{28}\end{aligned}$$
Given with $u\left( 0\right) =0,\ \ k(0)=1$ as initial conditions. The exact solution for $\alpha _{1} =\alpha _{2} =1$ is
$$\begin{aligned}
u\left( t\right) &=&sinht \notag \\
k(t) &=&cosht \label{29}\end{aligned}$$
Rewriting Eq.$(\ref{28})$ in the following for with adding and subtracting $u^{\prime }(t)$ and $k^{\prime }(t)$ to the equation respectively gives $$\begin{aligned}
&&\varepsilon \frac{d^{\alpha _{1}}u(t)}{{dt}^{\alpha _{1}}}+u^{\prime
}\left( t\right) -\varepsilon u^{\prime }(t)-1+\frac{1}{2}{\left( k^{\prime
}\left( t\right) \right) }^{2}-\varepsilon \int_{0}^{t}{\left( \left(
t-s\right) k\left( s\right) -u\left( s\right) k\left( s\right) \right) ds\ }
\notag \\
&&\varepsilon \frac{d^{\alpha _{2}}u(t)}{{dt}^{\alpha _{2}}}+k^{\prime
}\left( t\right) -\varepsilon k^{\prime }\left( t\right) -2t-\varepsilon
\int_{0}^{t}{\left( \left( t-s\right) u\left( s\right) +{k}^{2}\left(
s\right) -{u}^{2}\left( s\right) \right) ds} \label{30}\end{aligned}$$In this case for $$\begin{aligned}
F\left( u^{\prime },u,\varepsilon \right) &=&\frac{1}{\Gamma (1-\alpha _{1})}\varepsilon \int_{0}^{t}{\frac{u^{\prime }(s)}{{(t-s)}^{\alpha _{1}}}ds-\varepsilon \int_{0}^{t}{\left( \left( t-s\right) k\left( s\right)
+u\left( s\right) k\left( s\right) \right) ds}-1+\frac{1}{2}{\left(
k^{\prime }\left( t\right) \right) }^{2}} \notag \\
F\left( k^{\prime },k,\varepsilon \right) &=&\frac{1}{\Gamma (1-\alpha _{2})}\varepsilon \int_{0}^{t}{\frac{u^{\prime }(s)}{{(t-s)}^{\alpha _{2}}}ds-\varepsilon \int_{0}^{t}{\left( \left( t-s\right) u\left( s\right)
-k^{2}\left( s\right) +u^{2}\left( s\right) \right) ds}}-2t \label{31}\end{aligned}$$
and the iteration formula
$$u^{\prime }\left( t\right) +\frac{Fu}{Fu^{\prime }}u\left( t\right) =-\frac{F_{\varepsilon }+\frac{F}{\varepsilon }}{Fu^{\prime }} \label{32}$$
the terms that will be replaced in, are
$$\begin{aligned}
F &=&u_{n}^{\prime }(t)-1+\frac{{k_{n}^{\prime }(t)}^{2}}{2} \notag \\
F_{u} &=&0 \notag \\
F_{u^{\prime }} &=&1 \notag \\
F_{\varepsilon } &=&-u_{n}^{\prime }(t)+\frac{1}{\Gamma (1-\alpha _{1})}\int_{0}^{t}{\frac{u_{n}^{\prime }(s)}{{(t-s)}^{\alpha _{1}}}ds}-\int_{0}^{t}{((t-s)k_{n}(s)+u_{n}(s)k_{n}(s))ds} \label{33}\end{aligned}$$
and the iteration formula$$k^{\prime }\left( t\right) +\frac{F_{k}}{F_{k^{\prime }}}k\left( t\right) =-\frac{F_{\varepsilon }+\frac{F}{\varepsilon }}{F_{k^{\prime }}} \label{34}$$
the terms that will be replaced in, are
$$\begin{aligned}
F &=&k_{n}^{\prime }\left( t\right) -2t\ \notag \\
F_{k} &=&0 \notag \\
F_{k^{\prime }} &=&1 \notag \\
F_{\varepsilon } &=&-k_{n}^{\prime }(t)+\frac{1}{\Gamma (1-\alpha _{2})}\int_{0}^{t}{\frac{k_{n}^{\prime }(s)}{{(t-s)}^{\alpha _{2}}}ds}-\int_{0}^{t}{((t-s)u_{n}(s)-{k_{n}(s)}^{2}+{u_{n}(s)}^{2})ds} \label{35}\end{aligned}$$
After substitution, the system of differential equations for this problem become
$$\frac{1}{\Gamma (1-\alpha _{1})}\int_{0}^{t}{{(-s+t)}^{-\alpha _{1}}{u}}^{\prime }{{_{n}}(s)ds}+{\left( {u^{\prime }}_{c}(t)\right) }_{n}+\frac{-1+\frac{1}{2}{k}^{\prime }{{_{n}}(t)}^{2}+{u}^{\prime }{_{n}}(t)}{\varepsilon }=\int_{0}^{t}{k_{n}(s)(-s+t+u_{n}(s))ds}+{u}^{\prime }{_{n}}(t)$$
$$\frac{1}{\Gamma (1-\alpha _{2})}\int_{0}^{t}{{(-s+t)}^{-\alpha _{2}}{k}}^{\prime }{{_{n}}(s)ds}+{\left( {k^{\prime }}_{c}(t)\right) }_{n}=\int_{0}^{t}{(-{k_{n}(s)}^{2}+u_{n}(s)(-s+t+u_{n}(s)))ds}+\frac{2t+(-1+\varepsilon ){k}_{n}^{\prime }(t)}{\varepsilon } \label{36}$$
Appropriate to the initial conditions, chosen $u_{0}\left( t\right) =0$ and $k_{0}\left( t\right) =1$ and solving Eq.$(\ref{36})$ for $n=0,1,2,3,...$ the successive iterations are
$$u_{1}(t)=\frac{1}{6}(6t+t^{3}) \label{37}$$
$$k_{1}(t)=1+\frac{t^{2}}{2} \label{38}$$
$$u_{2}\left( t\right) =\frac{1}{504}t\left(
1008+168t^{2}+21t^{4}+t^{6}\right) -\frac{t^{2-\alpha _{1}}\left(
12+t^{2}+\left( -7+\alpha _{1}\right) \alpha _{1}\right) }{\Gamma (5-\alpha
_{1})} \label{39}$$
$$k_{2}\left( t\right) =1+t^{2}+\frac{t^{4}}{24}+\frac{t^{6}}{240}+\frac{t^{8}}{2016}-\frac{t^{3-\alpha _{2}}}{\Gamma (4-\alpha _{2})} \label{40}$$
Following in this manner the third iteration results, $u_{3}(t)$ and $k_{3}(t),$ are calculated. Again Table 2, Figure 2 and Figure 3 prove that PIA give remarkably approximate results. We can say that the higher iterations would give closer results.
------- --------------- ---------------- ---------------- --------------- ---------------- ----------------
t PIA $(u_{3})$ Exact Solution Absolute Error PIA $(k_{3})$ Exact Solution Absolute Error
$0.0$ 0.000000 0.000000 0.000000 1.000000 1000000. 0.000000
$0.1$ 0.100166 0.100166 1.591577E-10 1.005004 1.005004 1.191735E-11
$0.2$ 0.201335 0.201336 2.053723E-8 1.020066 1.020066 3.060393E-9
$0.3$ 0.304519 0.304520 3.556439E-7 1.045338 1.045338 7.884730E-8
$0.4$ 0.410749 0.410752 2.714842E-6 1.081073 1.081072 7.934216E-7
$0.5$ 0.521082 0.521095 1.326132E-5 1.127630 1.127625 4.774578E-6
$0.6$ 0.636604 0.636653 4.893639E-5 1.185485 1.185465 2.077300E-5
$0.7$ 0.758434 0.758583 1.490491E-4 1.255241 1.255169 7.230620E-5
$0.8$ 0.887710 0.888105 3.950285E-4 1.337648 1.337434 2.139083E-4
$0.9$ 0.025574 0.026516 9.426045E-4 1.433645 1.433086 5.592545E-4
$1.0$ 0.173128 0.175201 2.072716E-3 1.544407 1.543080 1.327116E-3
------- --------------- ---------------- ---------------- --------------- ---------------- ----------------
: Numerical results of Example 3.2. for $u_{3}$ and $k_{3}$ values when $\alpha _{1} =\alpha _{2} =1$
{width="3.50in"}
{width="3.50in"}
Conclusion
==========
In this study, Perturbation-Iteration Algorithm was introduced for some Factional Differential Equations. It is clear that the method is very simple and reliable perturbation-iteration technique and producing very approximate results. We expect that the present method could used to calculate the approximate solutions of other types of fractional differential equations.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Let $G$ be a finite group, and let $R$ be a discrete valuation ring with residue field $k$ and fraction field $K$. We say that $G$ is weakly tame at a prime $p$ if it has no non-trivial normal $p$-subgroups. By convention, every finite group is weakly tame at $0$. Using this definition, we show that if $G$ is weakly tame at ${\operatorname{char}}(k)$, then ${\operatorname{ed}}_K(G) \geqslant {\operatorname{ed}}_k(G)$. Here ${\operatorname{ed}}_F(G)$ denotes the essential dimension of $G$ over the field $F$. We also prove a more general statement of this type, for a class of étale gerbes ${{\mathscr}{X}}$ over $R$.
As a corollary, we show that if $G$ is weakly tame at $p$, then ${\operatorname{ed}}_{L} G \geqslant {\operatorname{ed}}_{k} G$ for any field $L$ of characteristic $0$ and any field $k$ of characteristic $p$, provided that $k$ contains $\overline{{\mathbb{F}}}_{p}$. We also show that a conjecture of A. Ledet, asserting that ${\operatorname{ed}}_k({\mathbb{Z}}/p^n {\mathbb{Z}}) = n$ for a field $k$ of characteristic $p > 0$ implies that ${\operatorname{ed}}_{{\mathbb{C}}}(G) \geqslant n$ for any finite group $G$ which is weakly tame at $p$ and contains an element of order $p^n$. We give a number of examples, where an unconditional proof of the last inequality is out of the reach of all presently known techniques.
address:
- |
Department of Mathematics\
1301 Mathematics Building\
University of Maryland\
College Park, MD 20742-4015\
USA
- |
Department of Mathematics\
1984 Mathematics Road\
University of British Columbia\
Vancouver, BC V6T 1Z2\
Canada
- |
Scuola Normale Superiore\
Piazza dei Cavalieri 7\
56126 Pisa\
Italy
author:
- Patrick Brosnan
- Zinovy Reichstein
- Angelo Vistoli
bibliography:
- 'ed-weakly-tame.bib'
title: |
Essential dimension\
in mixed characteristic
---
[^1]
[^2]
[^3] [^4]
Introduction
============
Let $R$ be a discrete valuation ring with residue field $k$ and fraction field $K$, and let $G$ be a finite group. In this paper we will compare ${\operatorname{ed}}_K(G)$ and ${\operatorname{ed}}_k(G)$. More generally, we will compare ${\operatorname{ed}}_K({{\mathscr}{X}})$ to ${\operatorname{ed}}_k({{\mathscr}{X}})$ for an étale gerbe ${{\mathscr}{X}}$ over $R$. For an overview of the theory of essential dimension, we refer the reader to [@brosnan-reichstein-vistoli3; @merkurjev-survey; @reichstein-icm].
To state our main result, we will need some definitions.
Suppose $S$ is a scheme. By an *étale gerbe* ${{\mathscr}{X}}{\ifinner\to\else\longrightarrow\fi}S$ we mean an algebraic stack that is a gerbe in the étale topology on $S$. Furthermore, we will always assume that there exists an étale covering $\{S_{i} {\ifinner\to\else\longrightarrow\fi}S\}$, such that the pullback ${{\mathscr}{X}}_{S_{i}}$ is of the form ${{\mathscr}{B}}_{S_{i}}G_{i}$, where $G_{i} {\ifinner\to\else\longrightarrow\fi}S_{i}$ is a finite étale group scheme.
We say that a finite group $G$ is *tame* (resp. *weakly tame*) at a prime number $p$ if $p\nmid |G|$ (resp. $G$ contains no non-trivial normal $p$-subgroup). Equivalently, $G$ is tame at $p$ if the trivial group is the (unique) $p$-Sylow subgroup of $G$, and $G$ is weakly tame at $p$ if the intersection of all $p$-Sylow subgroups of $G$ is trivial. By convention we say that every finite group is both tame and weakly tame at $0$. [^5]
By a geometric point of $S$, we mean a morphism ${\operatorname{Spec}}\Omega {\ifinner\to\else\longrightarrow\fi}S$ with $\Omega$ an algebraically closed field. We say that a finite étale group scheme $G$ over $S$ is *tame* (resp. *weakly tame*) if, for every geometric point ${\operatorname{Spec}}\Omega\to S$, the group $G(\Omega)$ is tame (resp. weakly tame) at ${\operatorname{char}}\Omega$. Similarly, we say that an étale gerbe ${{\mathscr}{X}}{\ifinner\to\else\longrightarrow\fi}S$ is *tame* (resp. *weakly tame*) if, for every object $\xi$ over a geometric point ${\operatorname{Spec}}\Omega {\ifinner\to\else\longrightarrow\fi}S$, the automorphism group ${\operatorname{Aut}}_{\Omega}\xi$ is tame (resp. weakly tame) at ${\operatorname{char}}\Omega$.
A key result of [@brosnan-reichstein-vistoli3] is the so called *Genericity Theorem* for tame Deligne–Mumford stacks, [@brosnan-reichstein-vistoli3 Theorem 6.1]. The proof of this result in [@brosnan-reichstein-vistoli3] was based on the following.
\[thmbrv3\] Let $R$ be a discrete valuation ring (DVR) with residue field $k$ and fraction field $K$, and let $${{\mathscr}{X}}{\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}R$$ be a tame étale gerbe. Then ${\operatorname{ed}}_{K}{{\mathscr}{X}}_K \geqslant {\operatorname{ed}}_{k}{{\mathscr}{X}}_{k}$.
Here ${{\mathscr}{X}}_{K}$ and ${{\mathscr}{X}}_{k}$ are respectively the generic fiber and the special fiber of ${{\mathscr}{X}}{\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}R$.
Unfortunately, the proof of [@brosnan-reichstein-vistoli3 Theorem 5.11] contains an error in the case when ${\operatorname{char}}K = 0$ and ${\operatorname{char}}k > 0$. This was noticed by Amit Hogadi, to whom we are very grateful. (See Remark \[remex\] for an explanation of the error.) For the applications in [@brosnan-reichstein-vistoli3] only the equicharacteristic case was needed, so this mistake in the proof of Theorem \[thmbrv3\] does not affect any other results in [@brosnan-reichstein-vistoli3] (the genericity theorem, in particular). However, the assertion of Theorem \[thmbrv3\] in the mixed characteristic case remained of interest to us, as a way of relating essential dimension in positive characteristic to essential dimension in characteristic $0$. In this paper, our main result is the following strengthened version of Theorem \[thmbrv3\].
\[thm.main\] Let $R$ be a DVR with residue field $k$ and fraction field $K$, and let $${{\mathscr}{X}}{\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}R$$ be a weakly tame étale gerbe. Then ${\operatorname{ed}}_{K}{{\mathscr}{X}}_K \geqslant {\operatorname{ed}}_{k}{{\mathscr}{X}}_{k}$.
In particular, [@brosnan-reichstein-vistoli3 Theorem 5.11] is valid as stated. Moreover, our new proof is considerably shorter than the one in [@brosnan-reichstein-vistoli3]. And in Sections \[sect.cors\]-\[sect.psl2\] we will deduce some rather surprising consequences.
We will give two proofs of our main result, one for gerbes of the form where ${{\mathscr}{X}}= {{\mathscr}{B}}_{R}G$, where $G$ is a (constant) finite group (Theorem \[thm.main0\]) and the other for the general case. The ideas in these two proofs are closely related; the proof of Theorem \[thm.main0\] allows us to introduce these ideas in the elementary setting of classical valuation theory. A separate proof of Theorem \[thm.main0\] also makes the applications in Sections \[sect.cors\]-\[sect.psl2\] accessible to those readers who are not familiar with, or don’t care for, the language of gerbes.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We are grateful to the referee for a thorough reading and constructive suggestions. We would also like to thank Alexander Duncan and Najmuddin Fakhruddin for helpful comments on an earlier version of this paper.
Proof of Theorem \[thm.main\] in the constant case {#sect.const}
==================================================
In this section we will prove a special case of Theorem \[thm.main\], where ${{\mathscr}{X}}= {{\mathscr}{B}}_{R}G$ for $G$ a finite group (viewed as a constant group scheme over ${\operatorname{Spec}}R$); see Theorem \[thm.main0\].
Throughout this section we will assume that $L$ is a field equipped with a (surjective) discrete valuation $\nu \colon L^* \to {\mathbb{Z}}$ and $K$ is a subfield of $L$ such that $\nu(K^*) = {\mathbb{Z}}$. We will denote the residue fields of $L$ and $K$ by $l$ and $k$, respectively. Similarly, we will denote the valuation rings by ${\mathscr}{O}_L$ and ${\mathscr}{O}_K$.
The following lemma is a special case of the Corollary to Theorem 1.20 in [@vaquie]. For the convenience of the reader, we supply a short proof.
\[lem.valuation\] ${\operatorname{trdeg}}_k(l) \leqslant {\operatorname{trdeg}}_K(L)$.
Let $u_{1}, \ldots, u_{m} \in l$ be algebraically independent over $k$. Lift each $u_i$ to $v_i \in {{\mathscr}{O}}_{L} \subseteq L$. It now suffices to show that $v_1, \dots, v_m$ are algebraically independent over $K$. Assume the contrary: $f(v_{1}, \dots , v_{m}) = 0$ for some polynomial $0 \neq f(x_1, \dots, x_m) \in K[x_{1}, \dots, x_{m}]$. After clearing denominators we may assume that every coefficient of $f$ lies in ${{\mathscr}{O}}_{K}$, and at least one of the coefficients has valuation $0$. If $f_{0}$ is the image of $f$ in $k[x_{1}, \dots, x_{m}]$ then $f_{0} \neq 0$ and $f_{0}(u_{1}, \dots, u_{m}) = 0$. This contradicts our assumption that $u_1, \dots, u_m$ are algebraically independent over $k$.
Let $L_m = \nu^{-1}(m) \cup \{ 0 \}$ and $L_{\geqslant m} = \bigcup_{j \geqslant m} \, L_j$. Note that, by definition, $L_{\geqslant 0} = {{\mathscr}{O}}_L$ is the valuation ring of $\nu$, $L_{\geqslant 1}$ is the maximal ideal, and $L_{\geqslant 0}/L_{\geqslant 1} = l$ is the residue field.
\[lem.bur5.1\] Assume that $g$ is an automorphism of $L$ of finite order $d \geqslant 1$, preserving the valuation $\nu$. Let $p = {\operatorname{char}}(l) \geqslant 0$. If $g$ induces a trivial automorphism on both $L_{\geqslant 0}/L_{\geqslant 1}$ and $L_{\geqslant 1}/L_{\geqslant 2}$, then
\(a) $d = 1$ (i.e., $g = {\mathrm{id}}$ is the identity automorphism) if $p = 0$, and
\(b) $d$ is a power of $p$, if $p > 0$.
Part (a) is proved in [@bur Lemma 5.1]; a minor variant of the same argument also proves (b). Alternatively, with some additional work, Lemma \[lem.bur5.1\] can be deduced from [@zs1 Theorem 25, p. 295]. For the reader’s convenience we will give a short self-contained proof below.
In case (b), write $d = mp^{r}$, where $m$ is not divisible by $p$. After replacing $g$ by $g^{p^{r}}$, we may assume that $d$ is prime to $p$. In both parts we need to conclude that $g$ is the identity.
Let $G$ be the cyclic group generated by $g$; then $G$ is linearly reductive. Since the action of $G$ on $l$ is trivial, the induced action on $L_{{\geqslant}i}/L_{{\geqslant}i+1}$ is $l$-linear. Furthermore, let $t \in L_1$ be a uniformizing parameter. By our assumption $g(t) = t \pmod{L_{\geqslant 2}}$. Thus multiplication by $t^{i-1}$ induces the $l$-linear $G$-equivariant isomorphism $(L_{\geqslant 1}/L_{\geqslant 2})^{\otimes i}
\simeq L_{{\geqslant}i}/L_{{\geqslant}i+1}$. Consequently, $G$ acts trivially on $L_{{\geqslant}i}/L_{{\geqslant}i+1}$ for all $i {\geqslant}0$. Since $G$ is linearly reductive, from the exact sequence $$0 {\ifinner\to\else\longrightarrow\fi}L_{{\geqslant}i}/L_{{\geqslant}i+1} {\ifinner\to\else\longrightarrow\fi}L_{{\geqslant}0}/L_{{\geqslant}i+1} {\ifinner\to\else\longrightarrow\fi}L_{{\geqslant}0}/L_{{\geqslant}i} {\ifinner\to\else\longrightarrow\fi}0$$ we deduce, by induction on $i$, that $G$ acts trivially on $L_{{\geqslant}0}/L_{{\geqslant}i}$ for every $i \geqslant 1$. Since $\bigcap_{i {\geqslant}0} L_{i} = 0$, this implies that the action of $G$ on $L_{{\geqslant}0}$ is trivial. But $L_{{\geqslant}0}$ is a domain with quotient field $L$, so $G$ also acts trivially on $L$. Since $G$ acts faithfully on $L$, we conclude that $G = \{1\}$, and the lemma follows.
\[prop.faithful\] Consider a faithful action of a finite group $G$ on $L$, such that $G$ preserves $\nu$ and acts trivially on $K$. Let $\Delta$ be the kernel of the induced $G$-action on $l$. Then $\Delta = \{ 1 \}$ if ${\operatorname{char}}(k) = 0$ and $\Delta$ is a $p$-subgroup if ${\operatorname{char}}(k) = p$.
Assume the contrary. Then we can choose an element $g \in \Delta$ of prime order $q$, such that $q \neq {\operatorname{char}}(k)$. Let $M$ be the maximal ideal of the valuation ring ${{\mathscr}{O}}_L$. Since we are assuming that $\nu(K^*) = \nu(L^*) = {\mathbb{Z}}$, we can choose a uniformizing parameter $t \in K$ for $\nu$. Since $g \in \Delta$, $g$ acts trivially on both $l = {{\mathscr}{O}}_L/M$ and $M/M^2 = l \cdot t$. By Lemma \[lem.bur5.1\], $g$ acts trivially on $L$. This contradicts our assumption that $G$ acts faithfully on $L$.
We are now ready to prove the main result of this section.
\[thm.main0\] Let $(R, \nu)$ be a discrete valuation ring with residue field $k$ and fraction field $K$, and $G$ be a finite group. If $p = {\operatorname{char}}(k) > 0$, assume that $G$ is weakly tame at $p$. Then ${\operatorname{ed}}_{K}(G) \geqslant {\operatorname{ed}}_{k}(G)$.
Set $d {\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle def}} =}}}{\operatorname{ed}}_K(G)$. Let $R[G]$ be the group algebra of $G$ and let $V_R = (\mathbb{A}_R)^{|G|}$ denote the corresponding $R$-scheme equipped with the (left) regular action of $G$. By definition $d$ is the minimal transcendence degree ${\operatorname{trdeg}}_K(L)$, where $L$ ranges over $G$-invariant intermediate subfields $K \subset L \subset K(V_K)$ such that the $G$-action on $L$ is faithful; see [@bur]. Choose a $G$-invariant intermediate subfield $L$ such that ${\operatorname{trdeg}}_K(L) = d$.
We will now construct a $G$-invariant intermediate subfield $k \subset l \subset k(V_k)$, where $V_k$ is the regular representation of $G$ over $k$, as follows. Lift the given valuation $\nu \colon K^* \to {\mathbb{Z}}$ to the purely transcendental extension $K(V_K)$ of $K$ in the obvious way. That is, $\nu: K(V_K)^* \to \mathbb{Z}$ is the divisorial valuation corresponding to the fiber of $V_R$ over the closed point in ${\operatorname{Spec}}R$. The residue field of $K(V_K)$ is then $k(V_k)$. By restriction, $\nu$ is a valuation on $L$ with $\nu(L^*)=\mathbb{Z}$. Let $l$ be the residue field of $L$. Clearly $k \subset l \subset k(V_k)$ and $\nu$ is invariant under $G$. By Proposition \[prop.faithful\], $G$ acts faithfully on $l$. Moreover, by Lemma \[lem.valuation\], ${\operatorname{trdeg}}_k(l) \leqslant d$. Thus ${\operatorname{ed}}_k(G) \leqslant d = {\operatorname{ed}}_K(G)$, as desired.
Examples illustrating Theorem \[thm.main0\] and a simple application {#sect.cors}
====================================================================
\[ex.necessary1\] The following example shows that Theorem \[thm.main0\] fails if we do not assume that $G$ is weakly tame. Choose $R$ so that ${\operatorname{char}}K = 0$, ${\operatorname{char}}k = p > 0$, and $K$ contains a $p^{2}$-th root of $1$. Let $G = C_{p^2}$ be the cyclic group of order $p^{2}$. Since $K$ contains a primitive $p^2$-th root of $1$, ${\operatorname{ed}}_K (G) = {\operatorname{ed}}_{K} ({\mathrm{C}}_{p^{2}}) = 1$. On the other hand, ${\operatorname{ed}}_k(G) = {\operatorname{ed}}_{k} ({\mathrm{C}}_{p^{2}}) = 2$; this is a special (known) case of Ledet’s conjecture, see Remark \[rem.ledet\].
\[ex.necessary2\] Here is an example showing that Theorem \[thm.main0\] fails if we do not assume that $R$ is a DVR. Let $R \subseteq {\mathbb{C}}{[\mspace{-2mu}[t]\mspace{-2mu}]}$ be the subring consisting of power series in $t$ whose constant term is real. Then $R$ is a one-dimensional complete Noetherian local ring with quotient field $K = {\mathbb{C}}{(\mspace{-3mu}(t)\mspace{-3mu})}$ and residue field $k = {\mathbb{R}}$, but not a DVR. Letting $G = C_4$ be the cyclic group of order $4$, we see that in this situation ${\operatorname{ed}}_K(G) = {\operatorname{ed}}_{{\mathbb{C}}{(\mspace{-3mu}(t)\mspace{-3mu})}} ({\mathrm{C}}_{4}) = 1$, while ${\operatorname{ed}}_k(G) = {\operatorname{ed}}_{{\mathbb{R}}} ({\mathrm{C}}_{4}) = 2$; see [@bf1 Theorem 7.6].
\[ex.semicontinuity\] (cf. [@tossici Remark 4.5(ii)]) This example shows that essential dimension is not semicontinuous in any reasonable sense, even in characteristic $0$. Consider the scheme $$S {\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle def}} =}}}{\operatorname{Spec}}{\mathbb{Q}}[u, x]/(x^{2} - u)\,.$$ The embedding ${\mathbb{Q}}[u] \subseteq {\mathbb{Q}}[u, x]/(x^{2} - u)$ gives a finite map $S {\ifinner\to\else\longrightarrow\fi}{\mathbb{A}}^{1}_{{\mathbb{Q}}}$. If $p$ is an odd prime, the inverse image of the prime $(u - p) \subseteq {\mathbb{Q}}[u]$ in $S$ consists of a point $s_{p}$ with residue field $k(s_{p}) =\mathbb{Q}(\sqrt{p}) = {\mathbb{Q}}[x]/(x^{2} - p)$. Then ${\operatorname{ed}}_{{\mathbb{Q}}(\sqrt{p})}({\mathrm{C}}_{4}) = 1$ if $-1$ is a square modulo $p$, and ${\operatorname{ed}}_{{\mathbb{Q}}(\sqrt{p})}{\mathrm{C}}_{4} = 2$ if $-1$ is not a square modulo $p$; once again, see [@bf1 Theorem 7.6]. Equivalently, ${\operatorname{ed}}_{{\mathbb{Q}}(\sqrt{p})}({\mathrm{C}}_{4}) = 1$ if $p \equiv 1 \pmod{4}$, and ${\operatorname{ed}}_{\mathbb{Q}(\sqrt{p})}{\mathrm{C}}_{4} = 2$ is $p \equiv 3 \pmod{4}$. We conclude that the set of points $s \in S$ with ${\operatorname{ed}}_{k(s)}{\mathrm{C}}_{4} = 1$ is dense in $S$, and likewise for the set of points $s \in S$ with ${\operatorname{ed}}_{k(s)}{\mathrm{C}}_{4} = 2$ is also dense in $S$.
We conclude this section with an easy corollary of Theorem \[thm.main0\].
\[cor1\] Let $p$ be a prime, $G$ a finite group weakly tame at $p$. Then
[(]{}cf. [@tossici Corollary 4.2][)]{} ${\operatorname{ed}}_{{\mathbb{Q}}}G \geqslant{\operatorname{ed}}_{{\mathbb{F}}_{p}}G$.
If $K$ is a field of characteristic $0$ and $k$ a field of characteristic $p$ containing $\overline{{\mathbb{F}}}_{p}$, then ${\operatorname{ed}}_{K}G \geqslant {\operatorname{ed}}_{k}G$.
\(a) follows directly from Theorem \[thm.main0\] by taking $R$ to be the localization of the ring of integers ${\mathbb{Z}}$ at a prime ideal $p {\mathbb{Z}}$.
\(b) Let $\overline{K}$ be the algebraic closure of $K$. Since ${\operatorname{ed}}_K(G) {\geqslant}{\operatorname{ed}}_{\overline{K}}(G)$, we may replace that $K$ by $\overline{K}$ and thus assume that $K$ is algebraically closed. Note that ${\operatorname{ed}}_{K}G = {\operatorname{ed}}_{\overline{{\mathbb{Q}}}}G$ and ${\operatorname{ed}}_{k}G = {\operatorname{ed}}_{\overline{{\mathbb{F}}}_{p}}G$; see [@brv1a Proposition 2.14] or [@tossici Example 4.10].
Choose a number field $E \subseteq \overline{{\mathbb{Q}}}$ such that ${\operatorname{ed}}_{E}G = {\operatorname{ed}}_{\overline{{\mathbb{Q}}}}G$ and let ${\mathfrak{p}}\subseteq {{\mathscr}{O}}_{E}$ a prime in the ring ${{\mathscr}{O}}_{E}$ of algebraic integers in $E$ lying over $p$. Set $E_{0} {\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle def}} =}}}{{\mathscr}{O}}_{E}/{\mathfrak{p}}$. Since $k$ contains $\overline{{\mathbb{F}}}_{p}$, there is an embedding $E_{0} \subseteq k$. By Theorem \[thm.main0\], ${\operatorname{ed}}_{E}(G) \geqslant {\operatorname{ed}}_{E_{0}}(G)$ and since $E_0 \subset k$, ${\operatorname{ed}}_{E_{0}}G \geqslant {\operatorname{ed}}_{k}G$.
\[ex.equality\] A. Duncan pointed out to us that equality in Corollary \[cor1\](b) does not always hold. For example, let $G = A_5$ be the alternating group of order $60$ and $p = 2$. Note that since $A_5$ is simple, it is weakly tame at every prime. By [@bur Theorem 6.7], ${\operatorname{ed}}_{{\mathbb{C}}}(A_5) = 2$. On the other hand, $A_5 \simeq {\mathrm{SL}}_2(\mathbb{F}_4)$ admits a $2$-dimensional faithful linear representation over any field $k$ containing $\mathbb{F}_4$, that is, the representation coming from the obvious inclusion of ${\mathrm{SL}}_2({\mathbb{F}}_4)$ into ${\mathrm{SL}}_2(k)$. The natural ($A_5$-equivariant) projection ${\mathbb{A}}^2 \dasharrow {\mathbb{P}}^1$ now tells us that ${\operatorname{ed}}_k(A_5) = 1$. In summary, $$2 = {\operatorname{ed}}_{{\mathbb{C}}}(A_5) > {\operatorname{ed}}_k(A_5) = 1 .$$
\[rem.inequality\] The group $G = A_5$ in Example \[ex.equality\] is weakly tame but not tame at $2$. We do not know of any such examples with $G$ tame. We conjecture that they do not exist. That is, if $|G|$ is prime to $p$, then under the hypotheses of Corollary \[cor1\](b), ${\operatorname{ed}}_{K}G = {\operatorname{ed}}_{k}G$, provided that $K$ is algebraically closed.
Ledet’s conjecture and its consequences
=======================================
The following conjecture is due to A. Ledet [@ledet-p].
\[conj.ledet\] If $k$ is a field of characteristic $p > 0$, $n$ is a natural number, and ${\mathrm{C}}_{p^{n}}$ is a cyclic group of order $p^{n}$, then ${\operatorname{ed}}_{k}({\mathrm{C}}_{p^{n}}) = n$.
\[rem.ledet\] It is known that in characteristic $p$, ${\operatorname{ed}}({\mathrm{C}}_{p^n}) \leqslant n$ for every $n {\geqslant}1$ (see [@ledet-p]) and ${\operatorname{ed}}({\mathrm{C}}_{p^n}) \geqslant 2$ if $n \geqslant 2$ ([@ledet-ed1 Theorems 5 and 7]). Thus the conjecture holds for $n = 1$ and $n = 2$; it remains open for every $n \geqslant 3$.
Combining Conjecture \[conj.ledet\] with Theorem \[thm.main0\], we obtain the following surprising result.
\[prop2\] Assume that a finite group $G$ is weakly tame at a prime $p$ and contains an element of order $p^n$. Let $K$ be a field of characteristic $0$. If Conjecture \[conj.ledet\] holds for $C_{p^n}$, then ${\operatorname{ed}}_{K}(G) \geqslant n$.
By Corollary \[cor1\](b), with $k = \overline{{\mathbb{F}}}_p$, we have ${\operatorname{ed}}_K(G) \geqslant {\operatorname{ed}}_k(G)$. Since $G$ contains ${\mathrm{C}}_{p^n}$, ${\operatorname{ed}}_k(G) \geqslant {\operatorname{ed}}_k({\mathrm{C}}_{p^n})$, and by Conjecture \[conj.ledet\], ${\operatorname{ed}}_k({\mathrm{C}}_{p^n}) = n$.
\[cor.semidirect\] Let $p$ be a prime and $n$ a positive integer. Choose a positive integer $m$ such that $q {\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle def}} =}}}mp^{n}+1$ is a prime. (By Dirichlet’s theorem on primes in arithmetic progressions, there are infinitely many such $m$.) Let ${\mathrm{C}}_{q}$ be a cyclic group of order $q$. Then ${\operatorname{Aut}}{\mathrm{C}}_{q} = ({\mathbb{Z}}/q{\mathbb{Z}})^{*}$ is cyclic of order $mp^{n}$; let ${\mathrm{C}}_{p^{n}} \subseteq ({\mathbb{Z}}/q{\mathbb{Z}})^{*}$ denote the subgroup of order $p^{n}$. Set $G {\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle def}} =}}}{\mathrm{C}}_{p^{n}} \ltimes {\mathrm{C}}_{q}$. Then
\(a) $G$ is weakly tame at $p$, and
\(b) if Conjecture \[conj.ledet\] holds, then ${\operatorname{ed}}_K(G) \geqslant n$ for any field $K$ of characteristic $0$.
\(a) Suppose $S \subseteq G$ is a normal $p$-subgroup. Then $S$ lies in every Sylow $p$-subgroup of $G$, in particular, in $C_{p^n}$. Our goal is to show that $S = \{ 1 \}$. The cyclic group $C_q$ of prime order $q$ acts on $S$ by conjugation. Since $q > p^n \geqslant |S|$, this action is trivial. In other words, $S$ is a central subgroup of $G$. In particular, $S$ acts trivially on $C_q$ by conjugation. On the other hand, by the definition of $G$, $C_{p^n}$ acts faithfully on $C_{q}$ by conjugation. We conclude that $S = \{1\}$, as desired.
\(b) follows from Proposition \[prop2\].
\[rem.semidirect1\] The inequality of Corollary \[cor.semidirect\](b) is equivalent to $$\label{e.semidirect}
{\operatorname{ed}}_{\mathbb{C}}({\mathrm{C}}_{p^n} \ltimes {\mathrm{C}}_{q}) \geqslant n \, ,$$ where ${\mathbb{C}}$ is the field of complex numbers (once again, see [@brv1a Proposition 2.14] or [@tossici Example 4.10]). For $n = 2$ and $3$, this inequality can be proved unconditionally (i.e., without assuming Conjecture \[conj.ledet\]) by appealing to the classifications of finite groups of essential dimension $1$ and $2$ over ${\mathbb{C}}$ in [@bur Theorem 6.2] and [@duncan-ed2 Theorem 1.1] respectively.
\[rem.semidirect2\] Let $G$ be a finite group. Set $${\operatorname{ed}^{\rm loc}}_k(G) := \max \, \{ {\operatorname{ed}}(G; p) \, | \, \text{$p$ is a prime} \},$$ where ${\operatorname{ed}}_k(G; p)$ denotes essential dimension of $G$ at a prime $p$ and the superscript “loc" stands for “local". If the base field $k$ is assumed to be fixed, we will write ${\operatorname{ed}}(G; p)$ and ${\operatorname{ed}^{\rm loc}}(G)$ in place of ${\operatorname{ed}}_k(G; p)$ and ${\operatorname{ed}^{\rm loc}}_k$, respectively.
Clearly ${\operatorname{ed}}(G) \geqslant {\operatorname{ed}^{\rm loc}}(G)$. In the language of [@reichstein-icm Section 5], computing ${\operatorname{ed}^{\rm loc}}(G)$ is a Type I problem. This problem is solved, at least in principle, by the Karpenko-Merkurjev theorem [@km2]. Computing ${\operatorname{ed}}(G)$ in those case, where ${\operatorname{ed}}(G) > {\operatorname{ed}^{\rm loc}}(G)$ is a Type II problem. Such problems tend to be very hard. For more on this, see [@reichstein-icm Section 5] or the discussion after the statement of Theorem 2 in [@reichstein-cremona].
Let us now return to the setting of Corollary \[cor.semidirect\], where $G = {\mathrm{C}}_{p^n} \ltimes {\mathrm{C}}_{q}$. Since all Sylow subgroups of $G = {\mathrm{C}}_{p^n} \ltimes {\mathrm{C}}_{q}$ are cyclic, one readily sees that ${\operatorname{ed}^{\rm loc}}_{{\mathbb{C}}}(G) = 1$. Thus the inequality is a “Type 2 problem" whenever $n \geqslant 2$. An unconditional proof of this inequality is out of the reach of all currently available techniques for any $n\geqslant 3$. However, it is shown in [@reichstein-cremona] that $$\lim_{n \to \infty} {\operatorname{ed}}_{{\mathbb{C}}} ({\mathrm{C}}_{p^{n}} \ltimes {\mathrm{C}}_{q}) \longrightarrow \infty$$ for any choice of $q$.
\[rem.ed-p\] It is shown in [@rei-vi] that if $G$ is a finite group and $k$ is a field of characteristic $p$, then $$\label{e.ed-p}
{\operatorname{ed}}_k(G; p) = \begin{cases} \text{$1$, if $p$ divides $|G|$, and} \\
\text{$0$, otherwise.}
\end{cases}$$ In particular, ${\operatorname{ed}^{\rm loc}}_k(C_{p^n}) = 1$ for every $n \geqslant 1$. So, for $n \geqslant 2$, Conjecture \[conj.ledet\] is also a Type 2 problem. Thus the situation in Corollary \[cor.semidirect\] can be described as follows: we deduce one Type II assertion from another, without being able to prove either one from first principles. Another results of this type is [@duncan-reichstein Proposition 10.8]; further examples can be found in the next section.
\[rem.main0-at-p\] In view of , Corollary \[cor1\](b) continues to hold if we replace essential dimension by essential dimension at $p$, for trivial reasons. Moreover, under the assumptions of Corollary \[cor1\], (a$^{\prime}$) ${\operatorname{ed}}_{{\mathbb{Q}}}(G; p) \geqslant{\operatorname{ed}}_{{\mathbb{F}}_{p}}(G; p)$ and (b$^{\prime}$) ${\operatorname{ed}}_{K}(G; p) \geqslant {\operatorname{ed}}_{k}(G; p)$, for any finite group $G$, not necessarily weakly tame. In (b$^{\prime}$) we can also drop the requirement that $k$ should contain $\overline{{\mathbb{F}}}_p$. Note however that our proof of Theorem \[thm.main0\] breaks down if we replace essential dimension by essential dimension at $p$.
Essential dimension of ${\mathrm{PSL}}_2(q)$ {#sect.psl2}
============================================
Let $p$ be a prime, $q = p^r$ be a prime power and $\mathbb F_q$ be a field of $q$ elements. Let $G = {\mathrm{PSL}}_2(q) = {\mathrm{PSL}}(2, {\mathbb{F}}_q)$. (To avoid confusion, we remind the reader that $G$ is the quotient of ${\mathrm{SL}}(2,{\mathbb{F}}_q)$ by its subgroup $\{\pm 1\}$. In general, it is not the same thing as the group ${\mathrm{PSL}}_2({\mathbb{F}}_q)$ of ${\mathbb{F}}_q$ points of the algebraic group ${\mathrm{PSL}}_2 = {\mathrm{PGL}}_{2}$.) For $q>3$, it is well-known that $G$ is simple; see, e.g., [@Dieudonne p. 39] or [@gfg p. 419]. Hence, $G$ is weakly tame at every prime. In this section we will work over the field $k = {\mathbb{C}}$ of complex numbers and deduce lower bounds on ${\operatorname{ed}}_{{\mathbb{C}}}(G)$ from Ledet’s conjecture.
For some $q$, these lower bounds are Type II bounds, in the sense of Remark \[rem.semidirect2\], and are genuinely new. To establish this we will compute ${\operatorname{ed}^{\rm loc}}(G)$ in every case. We begin with the following well-known description of the Sylow subgroups of ${\mathrm{PSL}}_2(q)$.
\[dih\] Let $p$ and $\ell$ be prime numbers and set $q=p^r$ for some positive integer $r$. Let $G_{\ell}$ denote an $\ell$-Sylow subgroup of $G={\mathrm{PSL}}_2(q)$. Then
1. For $\ell=p$, we have $G_{\ell}\cong (C_p)^r$.
2. For $\ell\not\in\{2,p\}$, $G_{\ell}$ is cyclic.
3. For $p$ odd and $\ell=2$, $G_{\ell}$ is dihedral.
See [@gfg Lemma 1.1 on page 418].
\[prop.psl2\] Let $p$ be a prime and $q = p^r$ be a prime power.
\(a) ${\operatorname{ed}^{\rm loc}}{\mathrm{PSL}}_2(q)= \begin{cases} r, & \text{$q$ even};\\
\max(2,r), & \text{$q$ odd}.
\end{cases} $
\(b) Let $\ell$ be a prime and $s$ be a nonnegative integer such that $2 \ell^s$ divides $q^2 - 1$. If Ledet’s Conjecture [\[conj.ledet\]]{} holds for cyclic groups of order $\ell^s$ in characteristic $\ell$, then ${\operatorname{ed}}_{{\mathbb{C}}} ( {\mathrm{PSL}}_2(q) ) \geqslant s$.
Note that part (b) is conditional on Ledet’s conjecture but part (a) is not.
Set $G={\mathrm{PSL}}_2(q)$. We begin by pointing out that $$\label{Gsize}
|G|=\begin{cases}
(q-1)q(q+1)/2, & 2\nmid q;\\
(q-1)q(q+1), & 2| q.
\end{cases}$$
\(a) Recall that ${\operatorname{ed}}_{{\mathbb{C}}}(G;\ell) = {\operatorname{ed}}_{{\mathbb{C}}}(G_{\ell}; \ell)$, where $G_{\ell}$ is a Sylow $\ell$-subgroup of $G$. So we only need to consider the primes $\ell$ dividing $|G|$; otherwise $G_{\ell} = \{ 1 \}$ and ${\operatorname{ed}}_{{\mathbb{C}}}(G_{\ell}; \ell) = 0$.
If $\ell \neq 2$ or $p$, then by Lemma \[dih\] (b), $G_{\ell}$ is cyclic; hence, ${\operatorname{ed}}_{{\mathbb{C}}}(G_{\ell}) = 1$.
If $\ell = p$, then by Lemma \[dih\] (a), $G_{\ell} = G_p = (C_p)^r$, and ${\operatorname{ed}}_{{\mathbb{C}}}(G_p; p)=r$.
If $\ell = 2$ and $p$ is odd, then by Lemma \[dih\] (c), $G_{\ell}$ is a dihedral group; hence, $G_{\ell}$ has a 2-dimensional faithful linear representation over $\mathbb C$. We conclude that ${\operatorname{ed}}_{{\mathbb{C}}}(G_{2}; 2) \leqslant 2$. On the other hand, since $G_2$ is not cyclic and $|G_2|\equiv 0\pmod{4}$, ${\operatorname{ed}}_{{\mathbb{C}}} G_{\ell}{\geqslant}2$ by [@bur Theorem 6.2]. So ${\operatorname{ed}}_{\mathbb{C}} G_{\ell}=2$.
This proves part (a) for the case that $p$ is odd. The case that $p$ is even follows directly from Lemma \[dih\] by the same method.
\(b) Note that the assertion of part (b) is vacuous if $\ell=p$ or $p = 2$. So we may assume that $p$ is odd and $\ell\neq p$. Then it follows from Lemma \[dih\] that the Sylow $\ell$-subgroup of ${\mathrm{PSL}}_2(q)$ is cyclic if $\ell$ is odd and dihedral if $\ell = 2$. Thus, by , ${\mathrm{PSL}}_2(q)$ contains an element of order ${\ell}^s$, and the desired inequality follows from Proposition \[prop2\].
\[rem1.psl2\] Note that, for odd $\ell$, Proposition \[prop.psl2\](a) gives the “Type I" lower bound: ${\operatorname{ed}}_{{\mathbb{C}}}{\mathrm{PSL}}_2(q) \geqslant \max \{ 2, r \}$; cf. Remark \[rem.semidirect2\]. We also know which finite simple groups have essential dimension $1$, $2$ or $3$ from [@bur Theorem 6.2], [@duncan-ed2] and [@beauville], respectively. Thus the lower bound of Proposition \[prop.psl2\](b) is only of interest in those cases, where $$s \geqslant \max \{r + 1, 5 \}.$$ In such cases an unconditional proof of the lower bound $${\operatorname{ed}}_{{\mathbb{C}}}({\mathrm{PGL}}_2(q)) \geqslant s$$ (i.e., a proof that does not rely on Ledet’s conjecture) is not known.
\[rem2.psl2\] It follows from Proposition \[prop.psl2\](a) that ${\operatorname{ed}}_{{\mathbb{C}}}({\mathrm{PSL}}_2(q)) \geqslant {\operatorname{ed}^{\rm loc}}_{{\mathbb{C}}}({\mathrm{PSL}}_2(q) \geqslant r$ for any $q = p^r$. Hence, if we want to deduce an interesting (Type II) lower bound on ${\operatorname{ed}}_{{\mathbb{C}}}({\mathrm{PSL}}_2(q))$ from Proposition \[prop2\], we need $\ell^s$ to divide $q \pm 1 = p^r \pm 1$ for some prime $\ell$ and some integer $s \geqslant r + 1$. This can only happen if $\ell < p$. In particular, this method gives no new information about ${\operatorname{ed}}_{{\mathbb{C}}}({\mathrm{PSL}}_2(q))$ in the case, where $q$ is a power of $2$.
\[ex2.psl\] Let $p = 31$ and $q = p^2 = 961$. Then $(q-1)/2 = 960$ is divisible by $2^6$. Thus Proposition \[prop.psl2\] yields
\(a) ${\operatorname{ed}^{\rm loc}}({\mathrm{PSL}}_2(961)) =2$ but (b) ${\operatorname{ed}}_{{\mathbb{C}}}({\mathrm{PSL}}_2(961)) \geqslant 5$.
Now let $q = p = 65537$. Note that $p$ is a Fermat prime, $p = 2^{16} + 1$. Here Proposition \[prop.psl2\] yields
\(a) ${\operatorname{ed}^{\rm loc}}({\mathrm{PSL}}_2(65537)) =2$ but (b) ${\operatorname{ed}}_{{\mathbb{C}}}({\mathrm{PSL}}_2(65537)) \geqslant 15$.
In both cases the inequality (b) is conditional on Ledet’s conjecture.
\[rem3.psl2\] It follows from [@reichstein-cremona Theorem 2] that for any $d \geqslant 1$ there are only finitely many non-abelian simple finite groups $G$ such that ${\operatorname{ed}}_{{\mathbb{C}}}(G) \leqslant d$. In some ways this assertion is more satisfying than the inequality of Proposition \[prop.psl2\](b): it is unconditional (does not rely on Ledet’s conjecture), and it covers all finite simple groups, not just those of the form ${\mathrm{PSL}}_2(q)$. On the other hand, it does not give an explicit lower bound on ${\operatorname{ed}}_{{\mathbb{C}}}(G)$ for any particular finite simple group $G$.
Proof of Theorem \[thm.main\] {#sect.proof}
=============================
We begin by remarking that an étale gerbe ${{\mathscr}{X}}{\ifinner\to\else\longrightarrow\fi}S$ is weakly tame if and only if there exists an étale cover $\{S_i {\ifinner\to\else\longrightarrow\fi}S\}$ such that each ${{\mathscr}{X}}_{S_i} {\ifinner\to\else\longrightarrow\fi}S_i$ is equivalent to ${{\mathscr}{B}}_{S_i}G_i {\ifinner\to\else\longrightarrow\fi}S_i$ with $G_i$ weakly tame étale group schemes over $S_i$.
Our proof of Theorem \[thm.main\] will rely on the following Lemma \[lem.versal\]. To state it, we need the notion of *versal object* of an algebraic stack. This is standard for classifying stacks of algebraic groups, but does not seem to be in the literature in the general case, so a short discussion is in order.
Let ${{\mathscr}{X}}{\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}F$ be an algebraic stack of finite type over a field. Then ${{\mathscr}{X}}$ preserves inductive limits, in the following sense: if $\{A_{i}\}$ is an inductive system of $F$-algebras over a filtered poset, the induced functor ${\varinjlim}{{\mathscr}{X}}(A_{i}) {\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}({\varinjlim}A_{i})$ is an equivalence of categories. If $L$ is an extension of $F$ then we can view $L$ as the inductive limit of its subalgebras $R \subseteq K$ of finite type over $F$; hence, given an object $\xi \in {{\mathscr}{X}}(L)$, there exists a finitely generated subalgebra $R \subseteq K$ and an object $\xi_{R} \in {{\mathscr}{X}}(R)$ whose image in ${{\mathscr}{X}}(L)$ is isomorphic to $\xi$.
We say that an object $\xi \in {{\mathscr}{X}}(L)$ is *versal* if it satisfies the following condition, which expresses the fact that every object of ${{\mathscr}{X}}$ over an extension of $F$ can be obtained by specialization of $\xi$.
For any $R$ and $\xi_{R}$ as above, and any object $\eta \in {{\mathscr}{X}}(K)$ over an extension $K$ of $F$ that is an infinite field, there exists a homomorphism of $F$-algebras $R {\ifinner\to\else\longrightarrow\fi}K$ such that the image of $\xi_{R}$ in ${{\mathscr}{X}}(K)$ under the induced functor ${{\mathscr}{X}}(R) {\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}(K)$ is isomorphic to $\eta$.
Versal object don’t exist in general; for example, they don’t exist when ${{\mathscr}{X}}$ has positive-dimensional moduli space. When they do exist, however, they control the essential dimension, that is, $\xi \in {{\mathscr}{X}}(L)$ is versal, then the essential dimension of $\xi$ is easily seen to be the essential dimension of ${{\mathscr}{X}}$ (in other words, no object of ${{\mathscr}{X}}$ defined over a field can have essential dimension larger than that of $\xi$).
\[lem.versal\] Let ${{\mathscr}{X}}_{F} {\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}F$ be a finite étale gerbe over a field $F$. Suppose that $A$ is a non-zero finite $F$-algebra, and that the morphism ${\operatorname{Spec}}A {\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}F$ has a lifting $\phi\colon {\operatorname{Spec}}A {\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}_{F}$. Consider the locally free sheaf $\phi_{*}{{\mathscr}{O}}_{{\operatorname{Spec}}A}$ on ${{\mathscr}{X}}_{F}$; call ${{\mathscr}{V}}{\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}_{F}$ the corresponding vector bundle on ${{\mathscr}{X}}_{F}$. Then ${{\mathscr}{V}}$ has a non-empty open subscheme $U \subseteq {{\mathscr}{V}}$. Furthermore, if $k(U)$ is the field of rational functions on $U$, the composite ${\operatorname{Spec}}k(U) {\ifinner\to\else\longrightarrow\fi}U \subseteq {{\mathscr}{V}}{\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}_{F}$ gives a versal object of ${{\mathscr}{X}}_{F}\bigl(k(U)\bigr)$.
Let us show that ${{\mathscr}{V}}$ is generically a scheme. We can extend the base field $F$, so that it is algebraically closed; in this case ${{\mathscr}{X}}_{F}$ is the classifying space ${{\mathscr}{B}}_{F}G$ of a finite group $G$, and there exists a homomorphism of $F$-algebras $A {\ifinner\to\else\longrightarrow\fi}F$. The vector bundle ${{\mathscr}{V}}{\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}_{F}$ corresponds to a representation $V$ of $G$; by the semicontinuity of the degree of the stabilizer for finite group actions, it is enough to show that $V$ has a point with trivial stabilizer. The homomorphism $A {\ifinner\to\else\longrightarrow\fi}F$ gives a morphism ${\operatorname{Spec}}F {\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}A$, and the composite ${\operatorname{Spec}}F {\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}A {\ifinner\to\else\longrightarrow\fi}{{\mathscr}{B}}_{F}G$ corresponds to the trivial $G$-torsor on ${\operatorname{Spec}}F$. If we call ${{\mathscr}{W}}$ the pushforward of ${{\mathscr}{O}}_{{\operatorname{Spec}}F}$ to ${{\mathscr}{B}}_{F}G$, then ${{\mathscr}{W}}\subseteq {{\mathscr}{V}}$. On the other hand ${{\mathscr}{W}}$ corresponds to the regular representation of $G$, and so the generic stabilizer is trivial, which proves what we want.
Let us show that the composite ${\operatorname{Spec}}k(U) {\ifinner\to\else\longrightarrow\fi}U \subseteq {{\mathscr}{V}}{\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}_{F}$ is versal; the argument is standard. Suppose that $K$ is an extension of $F$ that is an infinite field, and consider a morphism ${\operatorname{Spec}}K {\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}_{F}$. It is enough to prove that for any open subscheme $U \subseteq {{\mathscr}{V}}$, the morphism ${\operatorname{Spec}}K {\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}_{F}$ factors through $U \subseteq {{\mathscr}{V}}{\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}_{F}$. The pullback $V_{K} {\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}K$ of ${{\mathscr}{V}}{\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}_{F}$ is a vector space on $K$, and the inverse image $U_{K} \subseteq V_{K}$ of $U \subseteq {{\mathscr}{V}}$ is a non-empty open subscheme; hence $U_{K}(K) \neq \emptyset$, which ends the proof.
Let $\widehat{R}$ be the completion of $R$ and $\widehat{K}$ be the fraction field of $\widehat{R}$. Then clearly $K \subset \widehat{K}$ and thus ${\operatorname{ed}}_{K}({{\mathscr}{X}}_K) \geqslant {\operatorname{ed}}_{\widehat{K}}({{\mathscr}{X}}_{\widehat{K}})$. Thus for the purpose of proving Theorem \[thm.main\], we may replace $R$ by $\widehat{R}$. In other words, we may (and will) assume that $R$ is complete.
Let $R {\ifinner\to\else\longrightarrow\fi}A$ be an étale faithfully flat algebra such that ${{\mathscr}{X}}(A) \neq \emptyset$; since $R$ is henselian, by passing to a component of ${\operatorname{Spec}}A$ we can assume that $R {\ifinner\to\else\longrightarrow\fi}A$ is finite. An object of ${{\mathscr}{X}}(A)$ gives a lifting $\phi\colon {\operatorname{Spec}}A {\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}$; this is flat and finite. Let ${{\mathscr}{V}}{\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}$ be the vector bundle corresponding to $\phi_{*}{{\mathscr}{O}}_{{\operatorname{Spec}}A}$. If $U {\ifinner\to\else\longrightarrow\fi}{{\mathscr}{V}}$ is the largest open subscheme of ${{\mathscr}{V}}$, the Lemma above implies that $U {\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}R$ is surjective. Denote by $U_{K}$ and $U_{k}$ respectively the generic and special fiber of $U {\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}R$; call $E$ and $E_{0}$ the fields of rational functions on $U_{K}$ and $U_{k}$ respectively. Again because of the Lemma, the objects $\xi\colon {\operatorname{Spec}}E {\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}_{K}$ and $\xi_{0}\colon {\operatorname{Spec}}E_{0} {\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}_{k}$ are versal.
Consider the local ring ${{\mathscr}{O}}_{E}$ of $U$ at the generic point of $U_{k}$, which is a DVR. The residue field of ${{\mathscr}{O}}_{E}$ is $E_{0}$, and we have a morphism $\Xi\colon {\operatorname{Spec}}{{\mathscr}{O}}_{E} {\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}$ whose restrictions to ${\operatorname{Spec}}K$ and ${\operatorname{Spec}}k$ are isomorphic to $\xi$ and $\xi_{0}$ respectively.
Set $m {\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle def}} =}}}{\operatorname{ed}}_{K}{{\mathscr}{X}}_{K}$; we need to show that $\xi_{0}$ has a compression of transcendence degree at most $m$.
There exists a field of definition $K \subseteq L \subseteq E$ for $\xi$ such that ${\operatorname{trdeg}}_{K}L = m$; call $\theta\colon {\operatorname{Spec}}L {\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}$ the corresponding morphism, so that we have a factorization ${\operatorname{Spec}}E {\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}L {\xrightarrow}{\theta} {{\mathscr}{X}}$ for $\xi$. Consider the intersection ${{\mathscr}{O}}_{L}{\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle def}} =}}}{{\mathscr}{O}}_{E} \cap L \subseteq E$; then ${{\mathscr}{O}}_{L}$ is a DVR with quotient field $L$. Call $L_{0}$ it residue field; we have $L_{0} \subseteq E_{0}$. By Lemma \[lem.valuation\], ${\operatorname{trdeg}}_{k}L_{0} \leqslant{\operatorname{trdeg}}_{K}L$.
Now it suffices to show that $\xi_{0}\colon {\operatorname{Spec}}E_{0} {\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}$ factors through ${\operatorname{Spec}}L_{0}$. Assume that we have proved that the morphism $\theta\colon {\operatorname{Spec}}L {\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}$ extends to a morphism $\Theta\colon {\operatorname{Spec}}{{\mathscr}{O}}_{L} {\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}$. The composite ${\operatorname{Spec}}E \subseteq {\operatorname{Spec}}{{\mathscr}{O}}_{E} {\xrightarrow}{\Xi} {{\mathscr}{X}}$ is isomorphic to the composite ${\operatorname{Spec}}E {\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}L \subseteq {\operatorname{Spec}}{{\mathscr}{O}}_{L} {\xrightarrow}{\Theta} {{\mathscr}{X}}$; since ${{\mathscr}{X}}$ is separated, it follows from the valuative criterion of separation that the composite ${\operatorname{Spec}}{{\mathscr}{O}}_{E} {\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}{{\mathscr}{O}}_{L} {\xrightarrow}{\Theta} {{\mathscr}{X}}$ is isomorphic to $\Xi\colon {\operatorname{Spec}}{{\mathscr}{O}}_{E} {\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}$. By restricting to the central fibers we deduce that $\xi_{0}\colon {\operatorname{Spec}}E_{0} {\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}$ is isomorphic to the composite ${\operatorname{Spec}}E_{0} {\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}L_{0} {\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}$, and we are done.
To prove the existence of the extension $\Theta\colon {\operatorname{Spec}}{{\mathscr}{O}}_{L} {\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}$, notice that the uniqueness of such extension implies that to prove its existence we can pass to a finite étale extension $R \subseteq R'$, where $R'$ is a DVR; it is straightforward to check that formation of ${{\mathscr}{O}}_{L}$ and ${{\mathscr}{O}}_{E}$ commutes with such a base change. Hence we can assume that ${{\mathscr}{X}}$ has a section, so that ${{\mathscr}{X}}= {{\mathscr}{B}}_{R}G$, where $G{\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}R$ is a finite étale weakly tame group scheme. By passing to a further covering we can assume that $G {\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}R$ is constant, that is, the product of ${\operatorname{Spec}}R$ with a finite group $\Gamma$. If $A$ is an $R$-algebra, an action of $G$ on ${\operatorname{Spec}}A$ corresponds to an action of $\Gamma$.
The vector bundle ${{\mathscr}{V}}{\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}$ corresponds to a vector bundle $V_{R} {\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}R$ with an $R$-linear action of $\Gamma$, such that the induced representations of $\Gamma$ on $V_{K}$ and $V_{k}$ are faithful. Call $\widetilde{E}$ the function field of $V_{K}$ and $\widetilde{E}_{0}$ the function field of $V_{k}$; then $\widetilde{E}^{\Gamma} = E$, and therefore ${{\mathscr}{O}}_{\widetilde{E}}^{\Gamma} = {{\mathscr}{O}}_{E}$. The factorization ${\operatorname{Spec}}E{\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}L {\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}$ gives a $\Gamma$-torsor ${\operatorname{Spec}}\widetilde{L} {\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}L$ whose lift to ${\operatorname{Spec}}E$ is isomorphic to ${\operatorname{Spec}}\widetilde{E} {\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}E$; then $\widetilde{L}$ is a $\Gamma$-invariant subfield of $\widetilde{E}$. Then ${{\mathscr}{O}}_{\widetilde{L}} {\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle def}} =}}}\widetilde{L} \cap {{\mathscr}{O}}_{\widetilde{E}}$ is a $\Gamma$-invariant DVR, and ${{\mathscr}{O}}_{\widetilde{L}}^{\Gamma} = \widetilde{L}^{\Gamma} \cap {{\mathscr}{O}}_{\widetilde{E}}^{\Gamma} = {{\mathscr}{O}}_{L}$.
Call ${\mathfrak{m}}_{\widetilde{L}} \subseteq {{\mathscr}{O}}_{\widetilde{L}}$ the maximal ideal, and set $\widetilde{L}_{0} {\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle def}} =}}}{{\mathscr}{O}}_{\widetilde{L}}/{\mathfrak{m}}_{\widetilde{L}}$. If $t \in R$ is the uniformizing parameter, the image of $t$ in ${{\mathscr}{O}}_{\widetilde{L}}$, which we denote again by $t$, is a uniformizing parameter; this is $\Gamma$-invariant. The action of $\Gamma$ on ${{\mathscr}{O}}_{\widetilde{L}}$ descends to an action of $\Gamma$ on $\widetilde{L}_{0}$. By Proposition \[prop.faithful\], this action is faithful.
So the action of $\Gamma$ on ${\operatorname{Spec}}L_{0}$ is free over $k$; this implies that the action of $\Gamma$ on ${\operatorname{Spec}}{{\mathscr}{O}}_{\widetilde{L}} {\ifinner\to\else\longrightarrow\fi}{\operatorname{Spec}}R$ is free, so ${\operatorname{Spec}}{{\mathscr}{O}}_{\widetilde{L}} {\ifinner\to\else\longrightarrow\fi}({\operatorname{Spec}}{{\mathscr}{O}}_{\widetilde{L}})/G = {\operatorname{Spec}}{{\mathscr}{O}}_{L}$ is a $\Gamma$-torsor. This gives the desired morphism $\Theta\colon {\operatorname{Spec}}{{\mathscr}{O}}_{L} {\ifinner\to\else\longrightarrow\fi}{{\mathscr}{X}}$, and ends the proof of the Theorem.
\[remex\] The problem with the proof of [@brosnan-reichstein-vistoli3 Theorem 5.11] was in the last sentence of the second paragraph on page 1094. We claimed there that the discrete valuation ring $R$ in the proof can be replaced with the ring called $W(k(s))$. Since the essential dimension of the generic point can go up when we make this replacement, this is, in fact, not allowable. (In effect, our mistake boils down to using an inequality in the wrong direction.)
Note also that the proof of the characteristic $0$ genericity theorem in [@brv1a] does not rely on Theorem \[thmbrv3\]. For that argument, which was different from the proof of [@brosnan-reichstein-vistoli3 Theorem 6.1], see [@brv1a Theorem 4.1].
[^1]: Patrick Brosnan was partially supported by NSF grant DMS 1361159.
[^2]: Zinovy Reichstein was partially supported by National Sciences and Engineering Research Council of Canada Discovery grant 253424-2017.
[^3]: Angelo Vistoli was partially supported by research funds from the Scuola Normale Superiore.
[^4]: The authors are grateful to the Collaborative Research Group in Geometric and Cohomological Methods in Algebra at the Pacific Institute for the Mathematical Sciences for their support of this project.
[^5]: By a theorem of T. Nakayama [@nakayama], $G$ is weakly tame at $p$ if and only if $G$ admits a faithful completely reducible representation over some (and thus every) field of characteristic $p$. The significance of this condition in the study of essential dimension of finite groups was first observed by R. Lötscher [@lotscher]. Note that Lötscher used the term “semifaithful" in place of “weakly tame".
| {
"pile_set_name": "ArXiv"
} |
UU-NF 07\#02 (February 2007)\
\
ISSN 1401-6269\
------------------------------------------------------------------------
[Energy spectra for protons]{}
[emitted in the reaction]{}
[$^{\textrm{nat}}{\textrm{Ca}}(\textrm{n},x\textrm{p})$ [at 94 MeV]{}]{}
[Pär-Anders Söderström]{}
*Department of Neutron Research, Uppsala University,*
*Box 525, SE-75120, Uppsala, Sweden*
[**Abstract**]{}
The MEDLEY setup based at The Svedberg Laboratory (TSL), Uppsala, Sweden has previously been used to measure double-differential cross sections for elastic $nd$ scattering, as well as light ion production reactions for various nuclei in the interaction with neutrons around 95 MeV. When moved to the new beam line, the first experimental campaign was on light-ion production from Ca at 94 MeV in February 2005.
These data sets have been analyzed for proton production in forward and backward angles. The $\Delta E - \Delta E - E$ technique have been used to identify protons, and a cutoff as low as 2.5 MeV is achieved. Suppression of events induced by neutrons in the low-energy tail of the neutron field is achieved by time-of-flight techniques. The data are normalized relative to elastic $np$ scattering measured in the 20-degree telescope. Results from an accepted neutron spectrum are presented and some methods to correct for events from low energy neutrons are presented and evaluated. The data are compared with calculations using the nuclear code TALYS. It was found that TALYS systematically overestimates the compound part, and underestimates the pre-equilibrium part of the cross section.
List of acronyms {#list-of-acronyms .unnumbered}
================
Introduction
============
In the last years, fast neutrons have gathered more and more interest worldwide. Lots of applications involving fast neutrons in some way are predicted to be available within 50 years, for example one can mention fusion power reactors and transmutation of spent nuclear fuel. Even today, fast neutrons play an important role in the field of electronics, where it can cause so called single event effects, doing damage to the hardware and software. Fast neutrons are also used for hadronic radiotherapy of cancer in several places in the world, for exampel at iTemba LABS in South Africa. So understanding of fast neutron interactions is a large and important part of applied nuclear physics.
In this work the interaction between 94 MeV neutrons with calcium will be examined, in particular the productions of protons at forward and backward angles. A first rough estimation of the angular distributions will also be carried out. In this introduction to the work, some general properties of the neutron, and experiments with the neutron, will be discussed, as well as a short introduction to the nucleus under study. This is followed by a few sections about what is actually measured and some possible applications and motivations for this work in the form of a biological introduction.
The neutron
-----------
The second lightest (known) baryon in the universe is the uncharged neutron, discovered in 1932 by James Chadwick [@existneutron], when he was examining mysterious radiation emitted from beryllium, boron and lithium that had been discovered a few years earlier by @1930ZPhy...66..289B. Since the neutron is the second lightest, the slight mass difference between it and the lightest, the proton, makes the neutron unstable when it is in an unbound state and it decays according to figure \[fig:neutronbetadecay\] with a lifetime of 887 seconds [@physics].
![$\beta$-decay of a free neutron into a proton, an electron and an antineutrino.[]{data-label="fig:neutronbetadecay"}](fig/decay.png){width="40.00000%"}
The neutron interacts with its surroundings through all four of the fundamental forces: the strong nuclear force[^1], the weak nuclear force[^2], the electromagnetic force[^3] and the gravitational force[^4]. In experiments with neutrons there exist a few difficulties that do occur due to its special properties. Acceleration of neutrons is impossible today since accelerators use the charge of the particles. Instead, one has to rely on a neutron beam as a secondary beam. This means that one uses a primary beam of charged particles, for example protons, and let them hit some target that can induce a neutron production nuclear reaction. Similarly the focusing of particle beams is done by large magnets, while the beam shaping for a beam of uncharged particles must be done with collimators. Dealing with neutrons also gives rise to background and radiation protection issues. This since high-energy neutrons has a lower probability than charged particles for interaction and higher possibility for passing through the shielding material.
Calcium
-------
The nucleus under study in this work builds up a quite hard, silvery white, alkaline earth metal. The name calcium origins from the Latin word *calcis* that means *lime*, referring to the group of minerals dominated by calcium-carbonates, -oxides and -hydroxides often used as building material, for example in limestone, concrete and mortar between bricks in masonry [@encyclo].
In the human body, calcium is the most common mineral where almost all of it is located in the skeleton in the form of hydroxide apatite crystals. The skeleton of a normal adult contains about 1 kg of calcium. Some of it can also be found in the soft tissue in the form of Ca$^{2+}$ ions [@biology]. There are six stable isotopes of calcium where the doubly magic[^5] isotope $^{40}$Ca is the absolutely most common one, with a natural abundance of 96.941 % [@physics].
Cross-section measurements
--------------------------
One of the most fundamental needs in nuclear physics, as well as in all other physics and even all other sciences, is theoretical models to explain nature. Of course, another fundamental need in the field of science is also the to go the other way, by obtaining experimental data to verify theoretical models.
In nuclear and particle physics the cross section is in a way the connection between everyday life and the often complicated and abstract theoretical models. It is a quantity that both can be measured in experiments, as well as calculated from simple or complex nuclear models. The unit is a non-SI, although accepted for use with the SI, areal unit named barn[^6] that can be visualized as the effective area a projectile sees when approaching a target. This makes the interpretation of a cross section as a reaction probability a little more intuitive; the larger the effective area of the target – the larger the probability of hitting it. It is important to keep in mind though that the reaction cross section is not the same as the geometric cross section.
Ionizing radiation and humans\[sec:ionizing\_humans\]
-----------------------------------------------------
The human body is constantly exposed to radioactivity in terms of ionizing radiation. The sources are partly due to human activities like rest products from atmospheric nuclear weapons tests, nuclear power accidents, dental x-rays and many other things. But there are also various natural sources of radioactivity. For example radon gas from earths crust and cosmic radiation. Even man itself contains various radioactive isotopes and is a source for radiation, as illustrated in figure \[fig:radioman\].
![In some sense, everyone of us is a Radioactive Man [@simpsons].[]{data-label="fig:radioman"}](fig/large.jpg){width="48.00000%"}
The damage that ionizing radiation does on human tissue is mainly on the DNA and a majority of the DNA damage is indirect by ionizing the water in the cell and creating very reactive free radicals. Following [@radiobio], there are three possible damages that can occur, namely base pair damage, single strand break and double strand break. Base pair damage and single strand break are the most common of these three, but causes generally no problems for the cell and are fixed fast and easy. Double strand break on the other hand can create some more severe problems.
When the DNA, in the form of chromosomes, gets exposed to a double strand break and the cell tries to repair it there are three possible outcomes. The first is that the reparation succeeds and everything is back to normal, but the reparation can also go wrong in two different ways. One way is that we get an abnormal chromosome, like a ring chromosome where the end of each arm has been lost and repaired into a ring formation. Another abnormal chromosome, that will appear in about 60 % of chromosome aberrations, is the di-centric chromosome where the chromosome has been repaired into one with two centromeres, the region where the two the two identical parts of the chromosome touch during cell division, instead of just one. Both these examples of abnormal chromosomes are, although fatal to the cell, quite harmless to humans since the damaged cell never can survive mitosis, cell division. This is illustrated in figure \[fig:dnadam\].
The third outcome, also illustrated in figure \[fig:dnadam\], of a double strand break is the worst one. Here the chromosomes undergo stable rearrangements instead of getting abnormal shapes. The stable rearrangements could consist of translocation of two of the arms or that a small portion of an arm is removed from the chromosome. This mutation does not lead to mitotic cell death. Instead the damage will multiply at the same rate as the cell, something that very well can be a part of inducing a cancer tumor. This is the reason why cancer-inducing effects are often refereed to as stochastic effects. The probability of the statistically rare event of mutation increases with increasing exposure, but the effect of an injury is independent of the received dose.
![Possible reparation errors when fixing a double strand break and the resulting DNA-damage [@radioslides].[]{data-label="fig:dnadam"}](fig/dnadamage.jpg){width="60.00000%"}
Cancer therapy
--------------
In principle there are three ways to treat cancer. One can use surgery, chemotherapy or radiation therapy. The basics behind radiation therapy of cancer are to damage the cancer cells DNA strands and in that way prevent the cell to divide. In this case one actually wants a double strand break that ends up in an abnormal chromosome and prevents the mitosis, as mentioned in section \[sec:ionizing\_humans\]. The most common type of radiation therapy is using $\gamma$-rays from $^{60}$Co, x-rays or electron beams. However some tumors, called radioresistant tumors, seem to respond quite poorly to this treatment and therefore other methods are needed.
One way is to use hadrons such as protons, neutrons, helium ions or pions instead of photons or electrons. Since hadrons have a larger , that is the stopping power in the medium, than electrons and photons, they has a higher chance to do biological damage to the cell. Actually there are three things that make a high particle more effective than a low particle. Firstly it is harder for the cells to repair the larger damage, secondly high particles is more efficient in damaging the oxygen-poor tissue in a tumor. And finally high radiation is not as sensitive to whether or not the cell is currently dividing as low radiation [@radiobio; @hadrons]. Another property of charged hadrons is the interaction depth. While photons and neutrons deposit most of their energy in the surface, charged hadrons deposit most of their energy in a well-defined depth in the patient. This known as the Bragg peak.
Using fast neutrons for cancer therapy is a little more complex method than using photons, electrons or hadrons, since neutrons do not damage the tissue via direct interactions. Instead, nuclear reactions with neutrons damage the tissue indirectly by production of various types of light ions. This means that in order to make accurate dose calculations, typically via Monte Carlo based radiation transport codes, it is fundamental that the double-differential cross sections for light ion production are well known [@tippawan].
TALYS\[sec:talys\]
------------------
TALYS is a joint project between NRG Petten in the Netherlands and CEA Bruyéres-le-Chétel in France, and is a nuclear reactions modelling program working with nuclear reactions of energies 1 keV - 200 MeV. It is written completely from scratch in Fortran77, with the only exception of the implemented coupled-channels code ECIS-97 [@2005AIPC..769.1154K].
The two main purposes of TALYS are to be used as a physics tool during experimental analysis, and as a data tool to generate nuclear data for use in various applications. The program is also very flexible, since one can get a reasonable cross section by a minimal input like the four lines:
projectile n
element Ca
mass 40
energy 94
But it also reacts to more than 150 keywords, making it possible to adjust the nuclear models and the output [@talys].
Theory
======
As mentioned in section \[sec:talys\] the theoretical predictions are made using a nuclear reactions code called TALYS. This is an outline of how TALYS does the calculations in this particular case, after a brief introduction to the basics of nuclear reactions.
One should remember that this is only a crude simplification of the calculation process valid only for incident high-energy neutrons on a spherical nucleus and emission of protons. When dealing with more complex particles like deuterons, tritons, helium isotopes and so on effects like continuum stripping, pick-up and knock-out also needs to be taken into account. Deformed nuclei also introduce a number of corrections not needed here. An example of the different reactions contribution to the total cross section can be found in figure \[fig:talplot\].
The shell model and magic nuclei\[sec:magicmodel\]
--------------------------------------------------
In nuclear physics there is not a single simple model to describe the atomic nucleus completely. Instead several different models that fit different kinds of parameters or properties for the nuclei are used. For example there are the liquid drop model and the shell model.
The shell model for nuclei is quite analogous to the shell model for atoms. In the atomic shell model the electrons populate shells outside the nucleus and when a shell is filled they start filling the next. In a nucleus the principle is the same, only here it is protons and neutrons that occupy shells and not electrons. Since the neutron and the proton have individual shells two layers of shells are filled in parallel [@krane].
Since the shell model deals with particles populating different energy levels there are some regions of extra interest and these are the nuclei with filled, or close to filled, shells. The nuclei with certain number of nucleons, corresponding to filled shells in the shell model, are found to be more tightly bound than others. These numbers of nucleons are commonly referred to as magic numbers and are: 2, 8, 20, 28, 50, 82 and 126. Nuclei containing filled shells are consequently called magic nuclei and nuclei with both protons and neutrons in filled shells are called double magic nuclei. Examples of double magic nuclei are $^{16}$O, $^{40}$Ca, $^{48}$Ca and $^{208}$Pb.
There are theories that also higher numbers of neutrons and of protons are magic numbers which would give rise to stable exotic nuclei such as $^{298}_{114}$Uuq, $^{304}_{120}$Ubn and $^{310}_{126}$Ubh in an island of stability among the otherwise highly unstable elements at the end of the periodic table [@island].
Fundamental nuclear reactions
-----------------------------
One important field of research to understand nuclear physics is the study of nuclear reactions. The general nuclear reaction is when an incoming particle, $a$, hits a target, $X$, causing the emission of a particle, $b$, and a recoil nucleus, $Y$. This can be written on the same form as a chemical reaction $$\label{eq:nuclreact}
a + X \rightarrow Y + b$$ or in a more compact form $X(a,b)Y$ where the $Y$ often is dropped. An example[^7] of a reaction could be $$\label{eq:nuclreact2}
\textrm{p} + {^7\textrm{Li}} \rightarrow {^4\textrm{He}} + \alpha$$ or correspondingly ${^7\textrm{Li}}(\textrm{p},\alpha)$. One can now divide (\[eq:nuclreact\]) into two subgroups, elastic and inelastic nuclear reactions. An elastic reaction is when $a=b$ and $X=Y$ in (\[eq:nuclreact\]) and no energy is lost due to excitation of any of the components, while in an inelastic process the target gets excited and/or $a \neq b$.
When studying high-energy reactions, another way to classify nuclear reactions is in terms of the time scales at which they occur, or equivalently in terms of the number of collisions within the nucleus. The fastest reactions, with the fewest collisions, are called direct reactions and occur on time scales of about $10^{-22}$ s. Here the incoming projectile primarily interacts with the surface of the target. At the opposite end there is the compound reaction at time scales of about $10^{-16}-10^{-18}$ s [@krane], and in between these types pre-equilibrium reactions may occur. The spectrum of the different reactions is illustrated in figure \[fig:cpdspectra\], where also the corresponding shape of the cross section is sketched.
![The different types of reactions, adapted from [@talys]: Compound reactions are indicated by C, pre-equilibrium reactions by P and direct reactions by D. The x-axis represent reaction time and particle energy and the y-axis the cross section.[]{data-label="fig:cpdspectra"}](fig/spectra.png){width="48.00000%"}
The optical model
-----------------
Before diving into details in the different nuclear reaction it is suitable to have a closer look at the basis of nuclear reaction calculations, something that is called the . An analogy to this nuclear model is a murky glass sphere with some incoming light. Some of the light will be refracted, or elastically scattered, through the sphere, while other rays will be partially absorbed. In the nuclear model, this effect is mathematically described by defining a scattering potential as $$\label{eq:genomp}
\mathcal{U}(r,E) = \mathcal{V}(r,E) + i\mathcal{W}(r,E)$$ that can be inserted into the Schrödinger equation. Following the parametrisation in [@talys; @2003NuPhA.713..231K], these potentials can in turn be described as a sum of potentials caused by different properties of the nucleus as
$$\mathcal{V}(r,E) = - \mathcal{V}_V(r,E) + \mathcal{V}_{SO}(r,E) \cdot \mathbf{L} \cdot \mathbf{\sigma} +
\mathcal{V}_C(r)\label{eq:genvop}$$
$$\mathcal{W}(r,E) = - \mathcal{W}_V(r,E) - \mathcal{W}_D(r,E) + \mathcal{W}_{SO}(r,E) \cdot \mathbf{L} \cdot \mathbf{\sigma}\label{eq:genwop}$$
where the indices denote the volume part, $V$, the spin-orbit coupling part, $SO$, the surface derivative part, $D$, and the Coulomb part, $C$. Finally, all of these can be further broken down into an energy dependent part and a radial dependent form function part as follows
$$\mathcal{V}_V(r,E) = V_V(E) f(r,R_V,a_V)\label{eq:genvv}$$
$$\mathcal{V}_{SO}(r,E) = V_{SO}(E) \left( \frac{\hbar}{m_\pi c} \right)^2\frac{1}{r}\frac{\textrm{d}}{\textrm{d}r}f(r,R_{SO},a_{SO})\label{eq:genvso}$$
$$\mathcal{V}_C(r) = \left\{ \begin{array}{cl}
\frac{Zze^2}{2R_C}\left( 3-\frac{r^2}{R_C^2} \right) & \text{for $r \leq R_C$}\\
\frac{Zze^2}{r} & \text{for $r \geq
R_C$}
\end{array}
\right. \label{eq:genvc}$$
$$\mathcal{W}_V(r,E) = W_V(E) f(r,R_V,a_V)\label{eq:genwv}$$
$$\mathcal{W}_D(r,E) = -4 a_D W_D(E)\frac{\textrm{d}}{\textrm{d}r}f(r,R_D,a_D)\label{eq:genwd}$$
$$\mathcal{W}_{SO}(r,E) = W_{SO}(E) \left( \frac{\hbar}{m_\pi c} \right)^2\frac{1}{r}\frac{\textrm{d}}{\textrm{d}r}f(r,R_{SO},a_{SO}).\label{eq:genwso}$$
The parameters $R_i = r_iA^{1/3}$ and $a_i$ are the radius and diffuseness parameters for the nucleus and the form function often is assumed to be of Woods-Saxon type $$\label{eq:wodds-saxon}
f(r,R_i,a_i) = \frac{1}{1+e^{\frac{r-R_i}{a_i}}}.$$
These parameters are listed by @2003NuPhA.713..231K for a vast amount of nuclei and for $^{40}$Ca and incoming neutrons they are given by table \[tab:ca40omp\]. The change in the potentials with energy of the incoming neutron and distance from the center of the nucleus is plotted in figure \[fig:Ca-potentials\].
------- ---------- ------- --------- ---------- ----------
$r_V$ $a_V$ $r_D$ $a_D$ $r_{SO}$ $a_{SO}$
1.206 0.676 1.295 0.543 1.01 0.60
$V_V$ $V_{SO}$ $V_C$ $W_V$ $W_D$ $W_{SO}$
26.06 $3.984$ 0 $8.216$ $1.597$ -0.9518
------- ---------- ------- --------- ---------- ----------
: Numerical values for the parameters for $^{40}$Ca with an incident neutron.[]{data-label="tab:ca40omp"}
![Optical potentials plotted for $^{40}$Ca and incoming neutrons. The left panel shows $V_V(E)$, $V_{SO}(E)$, $W_V(E)$, $W_D(E)$ and $W_{SO}(E)$ for incident neutron energies between 0 and 200 MeV while the right panel shows $\mathcal{V}(r)$ (solid line) and $\mathcal{W}(r)$ (dashed line) for an energy of 94.5 MeV. Parametrized via [@2003NuPhA.713..231K].[]{data-label="fig:Ca-potentials"}](fig/pots1.pdf){width="\textwidth"}
![Optical potentials plotted for $^{40}$Ca and incoming neutrons. The left panel shows $V_V(E)$, $V_{SO}(E)$, $W_V(E)$, $W_D(E)$ and $W_{SO}(E)$ for incident neutron energies between 0 and 200 MeV while the right panel shows $\mathcal{V}(r)$ (solid line) and $\mathcal{W}(r)$ (dashed line) for an energy of 94.5 MeV. Parametrized via [@2003NuPhA.713..231K].[]{data-label="fig:Ca-potentials"}](fig/pots2.pdf){width="\textwidth"}
Direct reactions
----------------
To calculate the cross sections from the direct formation processes TALYS has several models incorporated, like the weak coupling model or giant resonances. For near-spherical nuclides TALYS calculates the cross sections for the direct formation processes with the . This basically means that one uses first order perturbation theory on the regular Born approximation, that the incoming and outgoing wave functions are just plane waves. This will give a result where the simple plane waves of the Born approximation is disturbed by the scattering potential of Eq. (\[eq:genomp\]). This first order model for small deformations is valid since $^{40}$Ca is a double magic nucleus, and hence near spherical. For a reaction like (\[eq:nuclreact\]), the cross section is obtained from the relation $$\label{eq:dwbaxsec}
\frac{\textrm{d}\sigma^{\textrm{D}}}{\textrm{d}\Omega} = \frac{E_a E_b}{(2\pi)^2(\hbar
c)^4} \frac{p_b}{p_a}\frac{1}{(2s_a+1)(2s_X+1)}\sum_{m_a, m_X, m_b,
m_Y}\left|\left\langle f \left| H_\textrm{int} \right| i
\right\rangle \right|^2$$ where $E_i$ is the total energy, $p_i$ the momentum, $s_i$ the spin of the particle, $i$, and $m_i$ the mass. The initial and final states $\left\langle f \right|$ and $\left| i \right\rangle$ are given by the and $H_\textrm{int}$ is the interaction potential [@lecturenotes].
Pre-equilibrium reactions
-------------------------
After the direct reaction and before the nucleus reaches equilibrium some pre-equilibrium processes takes place. In TALYS the pre-equilibrium stage is calculated using a model called the two-component exciton model, following [@talys; @2004NuPhA.744...15K]. In analogy to solid state physics one can define particle-hole pairs, also called pairs of excitons, above the Fermi surface[^8] as
$$n_{\pi} = p_{\pi} + h_{\pi} \label{eq:exi1}$$
$$n_{\nu} = p_{\nu} + h_{\nu} \label{eq:exi2}$$
$$p = p_{\pi} + p_{\nu} \label{eq:exi3}$$
$$h = h_{\pi} + h_{\nu} \label{eq:exi4}$$
$$n = n_{\pi} + n_{\nu} \label{eq:exi5}$$
where $p$ is the number of particles, $h$ the number of holes and $n$ the number of excitons while $\pi$ corresponds to protons and $\nu$ to neutrons. When the incoming projectile interacts with the nucleus it can excite a particle-hole pair. On its path through the nucleus it can either excite new particle-hole pairs, annihilate existing ones or change configuration of the existing pairs as illustrated in figure \[fig:exciton\], as well as release some excitation energy by emitting a particle.
![Different possible reaction paths in the two component exciton model for an incoming proton [@2004NuPhA.744...15K]. The numbers in the boxes correspond to $p_{\pi}$, $h_{\pi}$, $p_{\nu}$ and $h_{\nu}$ of some population $\mathcal{P}\left( p_{\pi}, h_{\pi}, p_{\nu}, h_{\nu} \right)$. The thickness of the arrows is proportional to the transition rate, and as the arrows pointing backwards are really small compared to the others, they are neglected by TALYS. This is the so called never-come-back approximation.[]{data-label="fig:exciton"}](fig/exiton.PNG){width="58.00000%"}
The more interactions the particle undergoes on its way through the nucleus, the more it looses its memory of incoming direction and energy until it reaches some equilibrium state and the compound model takes over. The cross section for this pre-equilibrium state is described by $$\label{eq:exixsec}
\frac{\textrm{d}\sigma_{k}^{\textrm{P}}}{\textrm{d} E_k} =
\sigma^{\textrm{CF}}
\sum_{p_{\pi}=p_{\pi}^0}^{p_{\pi}^\textrm{max}}
\sum_{p_{\nu}=p_{\nu}^0}^{p_{\nu}^\textrm{max}}
W_k\left( p_{\pi} , h_{\pi} , p_{\nu} , h_{\nu} , E_k \right)
\tau\left( p_{\pi} , h_{\pi} , p_{\nu} , h_{\nu} \right) P\left( p_{\pi} , h_{\pi} , p_{\nu} , h_{\nu} \right)$$ where $\sigma^{\textrm{CF}}$ is the total compound formation cross section, $W_k$ is the emission rate, $\tau$ the lifetime of the particle-hole states and $P$ the states that survived and are left from the last process.
The total compound-formation cross section is just the reaction cross section, that can be calculated from the , with the sum of the cross sections for direct reactions calculated from the subtracted, $\sigma^{\textrm{CF}} =
\sigma^{\textrm{reac}}-\sigma^{\textrm{direct}}$. The emission rate depends on known parameters, the inverse reaction cross section calculated from the and the particle-hole state density. Both the lifetime and the survived population are calculated from the internal transition rates for particle-hole pair creation, conversion and annihilation [@talys].
Compound formation
------------------
For the compound formation one can separate two different processes. At low energies it is a binary reaction where the projectile is captured by the target, followed by emission of another particle. But at high incident energies the nucleus will still be left in a highly excited state after the binary reaction and other process take over. Although the two processes at first sight seem identical, there are two major differences. One of them being that the first process is non-isotropic, and the other the presence of a width fluctuation factor in the first process that correlates the incoming and outgoing waves [@2005AIPC..769.1154K]. An analogy is often made with the evaporation of a hot fluid, that works in basically the same way, where particles evaporate from the nucleus until it is cooled down.
Also for compound reactions TALYS includes several models for different situations. At high energies, as in this case, the default model used is multiple pre-equilibrium emission within the exciton model. This is an extension of the pre-equilibrium model and includes the pre-equilibrium population $\mathcal{P}^{\textrm{pre}}
\left( Z,N,p_\pi,h_\pi,p_\nu,h_\nu,E_x(i) \right)$ that needs to be included in (\[eq:exixsec\]) and summed over all possible combinations of parameters. The populations that are not used for a new $\mathcal{P}^{\textrm{pre}}$ are used to feed another multiple emission model called multiple Hauser-Feshbach decay [@talys].
Angular distribution
--------------------
Continuum angular distributions can be achieved using the systematics by @1988PhRvC..37.2350K. The systematics is based on that the angular distribution, to a first approximation, is independent of the energy of the projectile, as well as the type of the target, projectile and ejectile. Instead it depends on the emitted particles energy and the fraction of multi-step direct and multi-step compound emissions for the given energy [@1988PhRvC..37.2350K]. In this phenomenological approach the angular distribution is described as $$\label{eq:kalbach}
\frac{\textrm{d}^2 \sigma}{\textrm{d} \Omega \textrm{d} E} = \frac{1}{4 \pi} \frac{\textrm{d} \sigma}{\textrm{d}
E} \frac{a}{\sinh(a)}\left( \cosh(a\cos\theta) +
f_{\textrm{MSD}} \sinh(a\cos\theta) \right)$$ with $a=a(E)$ as a free parameter to be fitted and $f_{\textrm{MSD}}$ as the fraction of emissions that comes from multi-step direct processes. This parameter, as well as the energy-differential cross section, $\frac{d \sigma}{d E}$, is assumed to be known. This result is physically motivated by @1994PhRvC..50.2490C. The form in Eq. (\[eq:kalbach\]) is the form that TALYS uses, but for fitting purposes the generalized form
$$\label{eq:gen_kalbach}
\frac{\textrm{d}^2 \sigma}{\textrm{d} \Omega \textrm{d} E} = b e^{a\cos\theta}$$
is used. Here $a=a(E)$ is still a free parameter and another free parameter, $b=b(E)$, is introduced.
Experimental setup\[sec:setup\]
===============================
The experiment was carried out at in Uppsala in February 2005 using a setup called MEDLEY. The MEDLEY setup is designed to measure light ion reactions like (n,p), (n,d), (n,t), (n,$^3$He) and (n,$\alpha$). Besides calcium, MEDLEY has also performed cross section measurements at 96 MeV on other nuclei of biological or technical relevance like silicon [@2004PhRvC..69f4609T], oxygen [@2006PhRvC..73c4611T], carbon [@tippcomm], iron, lead and uranium [@2004PhRvC..70a4607B]. This chapter contains a brief description of , the neutron beam, and a more in depth description of the MEDLEY spectrometer setup. The last two sections are devoted to the electronics and data acquisition system involved.
The Svedberg Laboratory
-----------------------
The protons for the primary beam was produced by the Gustaf Werner isochronous cyclotron that can accelerate protons up to energies of 180 MeV and heavy ions with charge $Q$ and mass number $A$ up to 192 $Q^2/A$ MeV. It consists of two D-shaped electrodes with a perpendicular magnetic field that makes the particles go in a circular pattern, and a high-voltage field between the electrodes that make the particles accelerate in the gap between the electrodes. By using a alternating electric field the particle will get a spiral trajectory outwards until extracted. The Gustaf Werner cyclotron is operated with a of 58 ns for 98 MeV protons. However, for protons of energies higher than 100 MeV, as well as ${^3}$He at their highest energies, relativistic effects need to be compensated for and the cyclotron switches to a frequency modulated synchrocyclotron mode where the driving is varied. This switching between modes is something that makes the Gustaf Werner cyclotron unique.
From the cyclotron the proton beam is transported to the Blue Hall, as seen in the drawing in figure \[fig:beam\]. Here it hits a highly enriched target of $^7$Li, where it induces the nuclear reaction $^7$Li(p,n)$^7$Be with a $Q$ value of 1.64 MeV [@physics], resulting in a quasi-monoenergetic neutron beam in the range of 20-180 MeV. There are different thicknesses of the lithium target available, ranging from 2 to 24 mm [@2005AIPC..769..780P], in this experiment the thickness used was 8 mm. The remaining proton beam is bent away using a bending magnet, actually recycled from the former LISA spectrometer, into a well-shielded and integrated in a Faraday cup. The charge deposited in the is used as one of three independent monitors for the neutron flux.
![Drawing of the new neutron beam facility in the Blue Hall at []{data-label="fig:beam"}](fig/neutronbeam.png){width="\textwidth"}
The actual beam is defined by a collimator, consisting of 100 cm long iron cylinders of different diameter, embedded in a wall of concrete. These cylinders can give a beam of, more or less, any shape and diameter up to 30 cm, even though the standard forms available are circular beams in steps of about 5 cm. Right after the collimator is the experimental area with the MEDLEY setup, that is used for experiments concerning neutron-induced charged particle production and will be described in more detail in section \[sec:medley\]. It is centered at a distance of 3.74 m from the lithium production target. After MEDLEY there is another setup called SCANDAL [@2002NIMPA.489..282K] used for experiments on elastic neutron scattering. Running several experiments at once like this is possible since almost the entire neutron beam passes the first setup without interacting. Two more independent neutron monitors are available namely the that uses the ionization of the gas inside the detector and via an electric field collects the ion pairs created [@knoll], and the [@tfbc]. The exit area from the collimator as well as the and the can be seen in figure \[fig:monitors\]. However, in this experiment there was a problem with the so that it could not be used as an absolute monitor of the neutron flux, but instead as a relative monitor calibrated to the .
![In the left panel, the exit of the collimator with the larger cylindrical and the smaller rectangular in front. The right panel shows the neutron spectrum emerging from the collimator [@pompcomm].[]{data-label="fig:monitors"}](fig/TSL_001.jpg){width="\textwidth"}
![In the left panel, the exit of the collimator with the larger cylindrical and the smaller rectangular in front. The right panel shows the neutron spectrum emerging from the collimator [@pompcomm].[]{data-label="fig:monitors"}](fig/GFull.jpg){width="\textwidth"}
MEDLEY spectrometer\[sec:medley\]
---------------------------------
The MEDLEY spectrometer, as seen in figure \[fig:medley\], consists of a 24 cm high cylindrical vacuum chamber with 90 cm diameter. Inside this chamber there are eight telescopes mounted at even intervals covering angles from 20 to 160 degrees, four on each side, and can be changed by rotating the table inside the chamber. The distance from the target to the different telescopes can also be varied.
![The MEDLEY spectrometer. The left panel shows a schematic sketch of the scattering chamber and in the right panel is a photography of a similar setup.[]{data-label="fig:medley"}](fig/medley.jpg){width="\textwidth"}
![The MEDLEY spectrometer. The left panel shows a schematic sketch of the scattering chamber and in the right panel is a photography of a similar setup.[]{data-label="fig:medley"}](fig/TSL_014.jpg){width="\textwidth"}
To get a clean positioning of the target within the beam without getting too much background signal from the target holder, a small aluminum frame is used that is well off the beam. The target itself is strung up in the frame using thin wires. This setup can be seen in figure \[fig:medleytarget\]. There are three of these frames so one can easily switch between targets without breaking the vacuum. For calibration purposes an ${^{241}}$Am source is also available, though this was not used in this experiment.
![One of the aluminum frames and the CH$_{2}$ target stinged up inside the MEDLEY spectrometer.[]{data-label="fig:medleytarget"}](fig/TSL_006.jpg){width="48.00000%"}
Detector telescopes
-------------------
Each of the telescopes, as can be seen in figure \[fig:telescope\], in the MEDLEY spectrometer consists of three individual detectors. First there are two silicon detectors, one thin of about 50 - 60 m and one thick of about 400 - 500 m thickness respectively. Finally there is a 5 cm long crystal of . These detectors are often referred to as the $\Delta E_1$, the $\Delta E_2$, and the $E$ detector, or sometimes A, B, and C detector. The detectors are mounted in an aluminum housing. Since the configuration of the detectors in this experiment was not the usual for MEDLEY it is listed in table \[tab:decset\].
There also exists a possibility to use plastic scintillators mounted as active anti-coincidence collimators to discard the signals from particles that did not pass straight into the detector. Attempts have actually been made in earlier experiments, but due to problems with them they were not used in this experiment.
![One of the detector telescopes. The left panel shows a schematic sketch of the telescope as well as the different kinds of events mentioned in section \[sec:silicon\] and in the right panel is a photography of a similar telescope.[]{data-label="fig:telescope"}](fig/tele.jpg){width="\textwidth"}
![One of the detector telescopes. The left panel shows a schematic sketch of the telescope as well as the different kinds of events mentioned in section \[sec:silicon\] and in the right panel is a photography of a similar telescope.[]{data-label="fig:telescope"}](fig/TSL_013.jpg){width="\textwidth"}
### Silicon detectors\[sec:silicon\]
The two most common semiconductor detectors are silicon and germanium detectors, and are in principle a large diode in reverse bias. The primary way of energy loss is via Coulomb interaction in the material where the atoms are either ionized or excited. This will create electron-hole pairs that can be collected by two electrodes and measured as a small current pulse when a particle is passing through.
Another feature that is fundamental for this experiment is that the amount of energy lost per unit length is not constant. Actually it decreases with increasing energy. In figure \[fig:stopping\] the stopping power and the distance a proton can travel before it is completely stopped, the so called range, is plotted for the range of proton energies relevant in this experiment. This property will be used for particle identification in the analysis.
![Stopping power (solid line) and range (dashed line) of protons in silicon calculated using SRIM.[]{data-label="fig:stopping"}](fig/stopping.pdf){width="48.00000%"}
These detectors are standard fully depleted surface-barrier detectors from ORTEC [@2000NIMPA.452..484D] were the silicon wafer is glued to a metal ring. One side is covered with a thin layer of gold and the other with aluminum [@ortec]. The $\Delta
E_1$ detector is used as a transmission detector, that is the particles pass through the detector leaving only a fraction of its energy. This is also the case for the $\Delta E_2$ detector at energies above 10 MeV, but for low energy events the proton is stopped in this detector. Protons that stop in $\Delta E_1$ are below the cutoff and are not included in the analysis. For an illustration of the different kinds of events, see figure \[fig:telescope\].
### CsI(Tl) detectors\[sec:CsI(Ti)\]
An inorganic scintillator is used as $E$ detector in this setup. Here the incoming radiation creates electron-hole pairs in the crystal by moving an electron from the valence band to the conduction band on its way through the crystal. In the crystal, some impurities are often added. These impurities will have lower ionization energy than the crystal, which is why the holes will drift here and ionize the impurities. The ionized impurities will then pick up an electron from the conduction band and decay to the ground state and emit a scintillation photon [@knoll]. A quite popular inorganic scintillator is the cesium iodide, available with both thallium and sodium activation. The high density of the gives it large stopping power and it also has a really high light yield of about 65000 photons/MeV [@knoll]. The wavelength of the emission light, 540 nm, is quite bad suited for a photomultiplier tube so instead a Hamamatsu photodiode is used [@2000NIMPA.452..484D]. To suite this, the last 2 cm of the 4 cm diameter cylinder is tapered off to 1.8 cm to fit the surface of the diode. Another useful feature of the worth mentioning is that it has a variable decay time for different kinds of particles [@knoll] that makes it suitable to use with pulse shape analysis.
Readout and electronics\[sec:readelectr\]
-----------------------------------------
Both the $\Delta E_1$ and the $\Delta E_2$ detector can generate a trigger signal, in order to allow triggers from both low- and high-energy particles. The signal from the detector is split up in two branches, a timing branch and an energy branch. A complete scheme is found in figure \[fig:electro\].
A signal from a detector is first amplified and then, if the pulse height is enough, converted to a logical signal using a . The timing signal is split up into two branches. One of them is connected via a logic to the other detector and telescopes as a master event. The other branch is delayed and sent as a stop signal for the timing channel in the . If the computer is busy processing earlier event the current event is discarded, otherwise it, through a logic , generates an energy gate signal for the and units, a start signal for the and finally a trigger for the data acquisition.
![Scheme of the MEDLEY electronics as of week 9, 2004[]{data-label="fig:electro"}](fig/electronics_medley_w9_2004.png){width="\textwidth"}
The energy signal is recorded from all three detectors and all eight telescopes as soon as one detector triggers. It is sent into a , with the previously named master signal as gate, after amplifying and shaping. From the cyclotron one also gets a signal that indicates a pulse from the neutron beam. The is split up through a into three branches. Two of the signals, where one is delayed, are used as a stop signal for the channels in the [^9]. It is also possible to use this signal as a veto for the master signal if the cyclotron is switched off.
Data acquisition
----------------
For the data acquisition a system called SVEDAQ is used, and it is summarized here from [@svedaqdescr]. The system is taken from the EUROGAM experiment and has been modified to be suitable for use at the . It is divided into three main blocks, as can be seen in figure \[fig:electro\], connected via two independent networks. The three blocks are the event builder, the tape server and the control and monitoring workstation.
The event builder is the block that reads out the data from the units in the VME crate as well as the units and scalers in the CAMAC crate. The event builder itself is also located in the VME. It mainly consists of a Motorola CPU that runs the readout program and sends packages to the data network when the readout buffers are full. It also sends the computer busy signals for vetoing of the master event and the 100 Hz clock so that events occurring during processing of other events are discarded.
The tape server is an optional part of the acquisition system. Of course if one wants to use the data after the experiment one needs to write it to the hard drive, but if one wants to conduct different kinds of test runs it is not needed by the event builder.
To control all this a SUN workstation is used. It is via this workstation that the system is started and stopped, but since the three blocks are more or less independent of each other the event builder and tape server will continue to write data from the CAMAC and VME even if the workstation is completely shut down. This workstation is connected to the local area network at and can be remotely controlled from there. But it is also connected to the data network via a sort-spy daemon process so it can listen to the traffic without formally being a part of the network with its own IP number. This makes it possible to display a large number of one- and two-dimensional spectra from the CAMAC signals and conduct some on-line sorting and analysis.
Data analysis
=============
The analysis of the data is split up into several parts. Since the SVEDAQ system writes the events as binary data, the raw files first have to be properly decoded. After the decoding, the information needs to be calibrated so that it corresponds to correct energy and timing information, before the particle selection procedure can take place, where events corresponding to the right particle type and incoming neutron energy are selected. Finally the relevant corrections for background and other effects can be carried out before extracting the final cross section.
During the analysis, three different data sets were analyzed simultaneously. Two of them were the calcium data and the background data, while the third was a reference data set of polyethene for calibration and normalization purposes. The complete set of runs is listed in tables \[tab:runlist180\] and \[tab:runlist0\].
ROOT
----
The main tool used for the analysis is ROOT. ROOT is an object oriented data analysis framework initiated at CERN by René Brun and Fons Rademakers[^10]. The framework is freely distributed as open source, where everyone is free to further distribute and modify it. This is something that has contributed, and still is contributing, to its rapid growth. It contains a large amount of packages concerning different part of data analysis like histogramming, curve fitting, minimization, statistics tools and much more [@ROOT].
The framework of ROOT is closely bound to a C++ interpreter called CINT that is written by Masahuru Goto. This means that one actually can use C++ as a scripting language for rapid prototype development of programs. Then the same code can be compiled to take advantage of the fast running of a machine language executable, unlike if one had used a normal scripting language like Ruby, Perl or Python for the prototyping.
Decoding
--------
The raw data from SVEDAQ are written into binary data files in the EUROGAM standards with a 24 byte header followed by event-by-event records [@svedaqdescr]. A typical example of a data sequence can be found in figure \[tab:hexraw\]. The events are recorded as a series of shorts[^11], with the first one being the number of bytes that are used to register one event. Between the first short and the telescope events there are 20 scaler channels with numbers 1-13 containing information about the telescope triggers, the 100 Hz clocks and the neutron monitors. The last seven channels are left empty. Next is the telescope events, each of them written in five shorts corresponding to three energy signals, one from each detector, and timing signals from the two silicon detectors. Finally the two signals are written before the event is ended. To translate the SVEDAQ files into ROOT files a small computer code called SV2ROOT [@terucomm] was used.
Calibration
-----------
### Energy calibration
Since the data from the runfile are given in the unit of channel numbers from the an energy calibration is needed to convert these to incoming energies. The silicon detectors are assumed to have a linear response to the energy, that is the correlation looks like $$\label{eq:linear}
E=kx+m$$ where $E$ is the energy output, $x$ is the channel number, and $k$ and $m$ the constants to be fitted.
Unfortunately the is not that simple since it has a non-linear response function between light output, $L$, and incoming energy, $E$, which makes the calibration more complicated. The light output is assumed to be given by a three-parameter formula that follows $$\label{eq:scintresponse}
L = a_0 + a_1 \left( E - a_2 A z^2 \ln \left[ \frac{E + a_2 A z^2}{a_2 A z^2} \right]
\right),$$ where $a_i$ is the fitting constants, and $A$ and $z$ is the mass number and charge of the incident particle [@2000NIMPA.452..484D]. However, to conduct the calibration we need the inverse of expression (\[eq:scintresponse\]). Since this inverse is analytically complicated to calculate, the approximation according to @2000NIMPA.452..484D $$\label{eq:energulight}
E \approx a + b L + c (bL)^2$$ is used for the hydrogen isotopes. The parameter $c$ depends only on particle type and is found by @tippawan to be $c = 0.0032$ for protons, which leaves only $a$ and $b$ as constants to be fitted.
Here the calibration of the silicon detectors is done by calculating the incoming energies needed to break through each of the silicon detectors - the punch through energies. These are calculated using a program called , based on the work of @srim. The punch through values for , , and are found in table \[tab:Sidecs\].
------------------ -------------- -------------- -------------- -------------- -------------- -------------- -------------- --------------
$\Delta E_1$ $\Delta E_2$ $\Delta E_1$ $\Delta E_2$ $\Delta E_1$ $\Delta E_2$ $\Delta E_1$ $\Delta E_2$
Thickness \[m\] 64.9 549 60.5 538 52.9 424 61.6 550
p \[MeV\] 2.42 8.62 2.32 8.51 2.34 8.63 2.13 7.41
d \[MeV\] 3.12 11.5 2.99 11.4 3.02 11.5 2.74 9.87
t \[MeV\] 3.60 13.5 3.43 13.4 3.47 13.6 3.14 11.6
$^{3}$He \[MeV\] 8.60 30.6 8.23 30.2 8.32 30.6 7.56 26.3
$^{4}$He \[MeV\] 9.56 34.4 9.14 34.0 9.24 34.5 8.38 29.6
$^{6}$Li \[MeV\] 18.1 65.3 17.3 64.6 17.5 65.4 15.9 56.3
$^{7}$Li \[MeV\] 19.1 69.7 18.3 68.9 18.5 69.8 16.7 60.0
------------------ -------------- -------------- -------------- -------------- -------------- -------------- -------------- --------------
: Thickness and punch through energies for the different silicon detectors used.[]{data-label="tab:Sidecs"}
If one plots the $\Delta E_1$ versus $\Delta E_2$, these punch though points manifest themselves as turning points in the particle bands as can be seen in the left panel of figure \[fig:rawplots\]. It is now quite straightforward to compare the channel numbers of these areas to the values in table \[tab:Sidecs\] and get the coefficients to (\[eq:linear\]).
![Calibrated $\Delta E_1$-$\Delta E_2$ and $\Delta E_2$-$E$ plots for . The red lines correspond to tabulated energy loss values for protons, deutrons, tritons, $^3$He and alpha particles in ascending order. The apparent mismatch for some lines in the right panel is due to the non-linear effects in the .[]{data-label="fig:rawplots"}](fig/de1de2.png){width="\textwidth"}
![Calibrated $\Delta E_1$-$\Delta E_2$ and $\Delta E_2$-$E$ plots for . The red lines correspond to tabulated energy loss values for protons, deutrons, tritons, $^3$He and alpha particles in ascending order. The apparent mismatch for some lines in the right panel is due to the non-linear effects in the .[]{data-label="fig:rawplots"}](fig/de2de.png){width="\textwidth"}
But once again, the poses a bit of a problem. As can be seen in the right panel of figure \[fig:rawplots\], there are no turning points available for calibration. Instead, at least in forward angles, one can use the quite strong peak of elastic p(n,p) scattering in the CH$_2$ runs as one calibration point. In the lower part of the $E$ scale the proton band is quite bent, so here a couple of points are selected and the energy deposited in the $E$ detector is calculated from the energy lost in $\Delta E_2$. Numerical values obtained for the calibration constants are listed in table \[tab:fitparamet\].
### Time calibration\[sec:timing\]
For the time calibration a couple of special runs devoted were made. During these runs cables of different lengths, giving different delay times, were put between the signal and the , displacing the peak in the recorded signal. The result from these runs can be fitted with a linear calibration formula
When each of the signals are time calibrated they need to be combined into a single time line. The two signals can be plotted against each other in figure \[fig:rf1rf2\], where four areas can be identified, schematically illustrated in figure \[fig:rfareas\]. Events in area 1 and area 3 are events that occur during one of the signal pulses, and hence get start and stop signals simultaneously. So in these areas we only get signal from one of the . In area 2 and area 4 there is a signal from both 1 and 2.
To translate the two signals into a single time scale, one of the signals is chosen as base, in this case 1. In the areas where no information exists for the chosen base signal, the information from the other signal is transformed to the chosen base according to the algorithm in [@aboutRF].
![RF 1 versus RF 2. Each point in the 8-shaped trajectory correspond to a time between detection and production of the next burst. One lap around is 58 ns.[]{data-label="fig:rf1rf2"}](fig/rf2.png){width="48.00000%"}
![Scematic illustration of the two signals and the different possible events that can occur [@aboutRF]. The is measured as the from the trigger to the next signal from the cyclotron.[]{data-label="fig:rfareas"}](fig/rfareas.PNG){width="60.00000%"}
Particle identification\[sec:partid\]
-------------------------------------
As mentioned earlier, the bands that are shown in figure \[fig:rawplots\] correspond to different kinds of charged particles. This makes the particle identification procedure quite straightforward by taking some rough cuts around the structures of the desired particle type. These cuts should be very generous, since it is always easier to remove unwanted events than to recreate wanted ones[^12] and is shown for in figure \[fig:firstcuts\]. The criterion here is that an event must be in both of these to be selected.
![First graphical selection cuts for protons in . In the left panel the cut for the $\Delta E_1$-$\Delta E_2$ plot is shown and in the right panel the $\Delta E_2$-$E$ plot is shown.[]{data-label="fig:firstcuts"}](fig/firstAB.png){width="\textwidth"}
![First graphical selection cuts for protons in . In the left panel the cut for the $\Delta E_1$-$\Delta E_2$ plot is shown and in the right panel the $\Delta E_2$-$E$ plot is shown.[]{data-label="fig:firstcuts"}](fig/firstBC.png){width="\textwidth"}
When these first rough cuts are done, the particle bands are straightened for a more refined selection of events. To do this, the energy recorded in the $E$ detector is used together with to calculate the corresponding energies in the $\Delta E_{1}$ and $\Delta E_{2}$ detectors. By subtracting this value from the energy actually recorded the particle bands are straightened out around zero. The $\Delta E_{1}(\textrm{exp})-\Delta E_{1}(\textrm{tab})$ selection is mainly used to remove noise in the low energy region in the $\Delta E_1$-$\Delta E_2$ plot, and the $\Delta
E_{2}(\textrm{exp})-\Delta E_{2}(\textrm{tab})$ selection is used to get a cleaner separation between protons and deuterons. These two are shown in figure \[fig:diff2d\].
![Refined selection cuts for protons in . In the upper left panel the cut for the $\Delta E_{1}(\textrm{exp})-\Delta E_{1}(\textrm{tab})$ plot is shown and in the upper right panel the $\Delta
E_{2}(\textrm{exp})-\Delta E_{2}(\textrm{tab})$ plot is shown. The lower panels contains the final proton cuts as in figure \[fig:firstcuts\] for together with the tabulated curve from []{data-label="fig:diff2d"}](fig/diffe1.png){width="\textwidth"}
![Refined selection cuts for protons in . In the upper left panel the cut for the $\Delta E_{1}(\textrm{exp})-\Delta E_{1}(\textrm{tab})$ plot is shown and in the upper right panel the $\Delta
E_{2}(\textrm{exp})-\Delta E_{2}(\textrm{tab})$ plot is shown. The lower panels contains the final proton cuts as in figure \[fig:firstcuts\] for together with the tabulated curve from []{data-label="fig:diff2d"}](fig/diffe2.png){width="\textwidth"}
\
![Refined selection cuts for protons in . In the upper left panel the cut for the $\Delta E_{1}(\textrm{exp})-\Delta E_{1}(\textrm{tab})$ plot is shown and in the upper right panel the $\Delta
E_{2}(\textrm{exp})-\Delta E_{2}(\textrm{tab})$ plot is shown. The lower panels contains the final proton cuts as in figure \[fig:firstcuts\] for together with the tabulated curve from []{data-label="fig:diff2d"}](fig/finalAB.png){width="\textwidth"}
![Refined selection cuts for protons in . In the upper left panel the cut for the $\Delta E_{1}(\textrm{exp})-\Delta E_{1}(\textrm{tab})$ plot is shown and in the upper right panel the $\Delta
E_{2}(\textrm{exp})-\Delta E_{2}(\textrm{tab})$ plot is shown. The lower panels contains the final proton cuts as in figure \[fig:firstcuts\] for together with the tabulated curve from []{data-label="fig:diff2d"}](fig/finalBC.png){width="\textwidth"}
Time-of-Flight measurements\[sec:tof\]
--------------------------------------
When only protons are selected, the next restriction is to only select protons that are produced by neutrons of the correct energy. Since the neutron beam is not completely monoenergetic, but contains about 60 % lower energy neutrons [@2000NIMPA.452..484D], the contribution from these has to be removed. To do this the information obtained in section \[sec:timing\] is used. The , $t$, for a particle with energy $E_k$ and velocity $v$ to travel a distance $s$ can be calculated as $$\label{eq:simpletof}
t = \frac{s}{v}.$$ The velocity of the particle is given in its rest mass, $m_0$, and its momentum, $p$, as $$\label{eq:velocity}
v = \frac{p}{\sqrt{m_0^2+\frac{p^2}{c^2}}}$$ where the $p$ is given by $$\label{eq:moment}
p = \frac{1}{c}\sqrt{E_k^2 + 2 E_k m_0 c^2}.$$ Plugging (\[eq:moment\]) into (\[eq:velocity\]) into (\[eq:simpletof\]) and simplifying yields
$$t = \frac{s}{c}\sqrt{\frac{(m_0 c^2 + E_k)^2}{E_k (E_k + 2 m_0
c^2)}}\label{eq:toft}$$
$$E_k = \sqrt{m_0^2+\frac{m_0^2}{\left( t \frac{c}{s} \right)^2-1}} - m_0\label{eq:tofe}$$
where Eqs. (\[eq:toft\]) and (\[eq:tofe\]) are equivalent.
But the time scale mentioned in section \[sec:timing\] includes both the neutrons as well as the charged particles from the target to the telescope. As can be calculated from (\[eq:toft\]), if the particle is a 94.5 MeV neutron traveling a distance of 3.74 m the would be about 29.9 ns. A 5 MeV proton traveling 26.83 cm has a of 8.71 ns while a 90 MeV proton would have a of only 2.19 ns. This makes quite a large difference, especially at low energies, and is compensated for by subtracting the charged particle , calculated from the energy of $\Delta E_{1}$, $\Delta E_{2}$ and $E$ added together. The final cuts are illustrated in figure \[fig:tofplots\].
![Neutron versus proton energy for calcium in the left panel, and CH$_2$ in the right panel, together with the cut. The the low energy tail in the neutron spectra is clearly visible in the CH$_2$ plot. The extra cut at high energies is motivated in section \[sec:tofdiss\].[]{data-label="fig:tofplots"}](fig/T8CaTOF.png){width="\textwidth"}
![Neutron versus proton energy for calcium in the left panel, and CH$_2$ in the right panel, together with the cut. The the low energy tail in the neutron spectra is clearly visible in the CH$_2$ plot. The extra cut at high energies is motivated in section \[sec:tofdiss\].[]{data-label="fig:tofplots"}](fig/T8CH2TOF.png){width="\textwidth"}
To choose a suitable sized cut, the elastic p(n,p) scattering is chosen to define the width. This peak has a of 6.6 ns, quite in agreement with @2000NIMPA.452..484D where the is measured to be about 6-7 ns, and one of the large contributions is the finite time of a beam pulse from the cyclotron. The with of the cut is chosen as two standard deviations, and is a compromise between statistics, minimizing mismatching errors, and a high lower limit of the accepted neutron spectrum. For more details, see section \[sec:mismatch\].
Corrections
-----------
For each run, the live time of the data acquisition system is corrected for. As mentioned in section \[sec:readelectr\], events that occur while an earlier event is treated are discarded. The amount of discarded events is not constant, as seen in figure \[fig:cLT\], but is assumed to be. The ratio of discarded events is obtained by comparing the 100 Hz scaler and the 100 Hz computer busy scaler. The finite efficiency of the is also corrected for. This has a non-linear dependence of energy deposited, and this is simulated by both MCNPX and GEANT [@bildcomm]. The results of these simulations are found in figure \[fig:CsIeff\]. Since they are in good agreement, one of them is choosen, in this case the GEANT results.
![Efficiency of the detector. The dashed line is a simulation with MCNPX and the solid line is a simulation with GEANT [@bildcomm].[]{data-label="fig:CsIeff"}](fig/cLT.png){width="\textwidth"}
![Efficiency of the detector. The dashed line is a simulation with MCNPX and the solid line is a simulation with GEANT [@bildcomm].[]{data-label="fig:CsIeff"}](fig/effect.pdf){width="\textwidth"}
During the entire event-by-event analysis, the data sets from the runs containing an empty frame were treated in parallel to the calcium and the polyethene data. When switching to analysis of histograms instead of events, the data from the background runs kan be subtracted from the calcium and polyethene data. In that way one can get pure signal histograms. A comparision between the calcium, polyethene and background data can be found in figure \[fig:snratio\]. The background in MEDLEY has been analyzed thoroughly by @tippawan for the old facility. Its main components are particles produced by produced neutrons reacting in the beam pipe and the reaction chamber, and the and the telescope housing acting as an active target for the beam halo.
![In the top panels, the signal and background for calcium, left, and polyethene, right, at an angle of 20 degrees. The bottom panel shows the background for the four angles analyzed: 140 degrees, 160 degrees, 40 degrees and 20 degrees in increasing order.[]{data-label="fig:snratio"}](fig/cabg20.png){width="\textwidth"}
![In the top panels, the signal and background for calcium, left, and polyethene, right, at an angle of 20 degrees. The bottom panel shows the background for the four angles analyzed: 140 degrees, 160 degrees, 40 degrees and 20 degrees in increasing order.[]{data-label="fig:snratio"}](fig/chbg20.png){width="\textwidth"}
\
![In the top panels, the signal and background for calcium, left, and polyethene, right, at an angle of 20 degrees. The bottom panel shows the background for the four angles analyzed: 140 degrees, 160 degrees, 40 degrees and 20 degrees in increasing order.[]{data-label="fig:snratio"}](fig/bg1.png){width="\textwidth"}
### Target thickness
To achieve a count rate that is acceptable, some demands are put on the target. Since the neutron beam is quite low in intensity and only a small fraction of the neutrons interacts the target has to have some finite thickness. For high-energy protons this is of negligible importance. For low energy protons, on the other hand, the energy loss can be quite significant. In table \[tab:calstop\], calculations for low energies are shown to illustrate the problem. The target used in these measurements was about 230 $\mu$m thick. These effects are compensated for through a method implemented in a computer code, TCORR, by @tcorr and is based on iterative calculations of response functions. The result of the target correction procedure is found in figure \[fig:corred\].
Ion Energy \[MeV\] $\textrm{d}E/\textrm{d}x$ \[keV/$\mu$m\] Range \[$\mu$m\] Straggling \[$\mu$m\]
-------------------- ------------------------------------------ ------------------ -----------------------
3.25 11.854 164.35 7.51
3.50 11.247 185.84 8.35
3.75 10.708 208.46 9.19
4.00 10.225 232.18 10.06
4.50 9.3934 282.85 12.85
5.00 8.7004 337.80 15.54
5.50 8.1137 396.92 18.20
6.00 7.6086 460.15 20.86
6.50 7.1698 527.41 23.55
: Stopping of protons in calcium[]{data-label="tab:calstop"}
![Target correction results for calcium in the left panel and polyethene in the right panel. Red circles represent the original data, while blue triangles show the corrected data, re-binned to 1 MeV for comparison. After three iterations the Kolmogorov test gave 100% probability for the current iteration compared to the previous one.[]{data-label="fig:corred"}](fig/ca_corred.pdf){width="\textwidth"}
![Target correction results for calcium in the left panel and polyethene in the right panel. Red circles represent the original data, while blue triangles show the corrected data, re-binned to 1 MeV for comparison. After three iterations the Kolmogorov test gave 100% probability for the current iteration compared to the previous one.[]{data-label="fig:corred"}](fig/ch_corred.pdf){width="\textwidth"}
Accepted neutron spectrum\[sec:nspectrum\]
------------------------------------------
Even with the selection in section \[sec:tof\], not all low energy neutrons were rejected. As can be seen in figure \[fig:tofplots\] the finite width of the cut results in events induced by neutrons down to around 60 MeV to be accepted. In the same figure one can also see the how the low energy p(n,p) tail wraps around the time scale and allows events by about 10-14 MeV neutrons inside the cut. To be able to subtract data originating from these events the accepted neutron spectrum is deduced from the CH$_2$ data. To get a clean neutron spectrum, the contribution from carbon has to be subtracted from the CH$_2$ data. For 95 MeV neutrons, the subtracted carbon cross section comes from existing experimental data [@tippcomm]. This is not possible for lower energy neutrons, so their contribution to the carbon spectrum is taken from tabulated data in [@ICRU].
To obtain a p(n,p) spectrum, the peak at 83 MeV is normalized to a cross section obtained from the solution SP05 of [@SAID] via a procedure described in appendix \[sec:NN\][^13]. In this first normalization, the fact that also carbon contributes slightly to the peak is ignored, but the peak is re-normalized to the cross section from [@SAID] after the subtraction of experimental carbon data.
Now the remaining CH$_2$ data are divided into bins corresponding to 10 MeV neutron energy each, translated to elastically scattered proton energies as seen in table \[tab:neubins\]. The bin right below the peak is integrated and the number of events in the bin is corrected for the variation in p(n,p) cross section, also listed in table \[tab:neubins\]. The cross section is assumed constant within the bin and equal to the cross section in the center of the bin. When doing this an assumption is made that the high Q value of the C(n,p) reaction[^14] makes the contribution from the carbon to the bin content negligible. Now, when one have the fraction of accepted neutrons in that bin compared to the full energy peak, one can subtract the corresponding carbon cross section obtained from [@ICRU]. In parallel, the obtained bin content relative to the peak is used to build up an accepted neutron spectrum. This process is then iterated downwards in energy for each of the bins until a pure p(n,p) spectrum and a corresponding neutron spectrum is available, and the resulting spectrum is found in figure \[fig:nspectra\] and in table \[tab:nspectra\].
![The deduced pure np spectrum in the left panel. Grey triangles are polyethene data and black circles are carbon data from the procedure in section \[sec:nspectrum\]. In the right panel is, the fraction of accepted neutrons per bin is shown.[]{data-label="fig:nspectra"}](fig/ch_and_c.pdf){width="\textwidth"}
![The deduced pure np spectrum in the left panel. Grey triangles are polyethene data and black circles are carbon data from the procedure in section \[sec:nspectrum\]. In the right panel is, the fraction of accepted neutrons per bin is shown.[]{data-label="fig:nspectra"}](fig/nspectrum.pdf){width="\textwidth"}
Bin \[MeV\] Contribution \[%\]
------------- --------------------
5-15 3.2
15-25 0
25-35 0
35-45 0
45-55 0
55-65 6.9
65-75 10
75-85 17
85-100 61
: Accepted neutron spectrum, as seen in figure \[fig:nspectra\].[]{data-label="tab:nspectra"}
Normalization
-------------
To obtain a value for the cross section all data are normalized to elastic p(n,p) scattering at 20 degrees. This since the p(n,p) scattering cross section is well known and the peak is clear at this angle. The peak together with a gaussian fit is seen in figure \[fig:np\].
![The elastic np peak fitted with a gaussian. The data are CH$_2$ with carbon [@tippcomm] subtracted.[]{data-label="fig:np"}](fig/npeak.pdf){width="48.00000%"}
The of the peak is found to be 6.4 MeV, in agreement with the in @2000NIMPA.452..484D, with the solid angle of the telescope as the main contribution. The cross section, $\sigma$, for each bin, $i$, in telescope $\textrm{T}X$ is now calculated from $$\label{eq:xsecnorm}
\frac{\sigma_{\textrm{Ca},i,\textrm{T}X}}{N_{\textrm{Ca},i,\textrm{T}X}} =
\frac{\sigma_{\textrm{H}}}{N_{\textrm{H}}}
\frac{2 m_{\textrm{CH}_2}}{M_{\textrm{CH}_2}}
\frac{M_{\textrm{Ca}}}{m_{\textrm{Ca}}}
\frac{\Omega_{\textrm{T}1,\textrm{T}8}}{\Omega_{\textrm{T}X}}
\frac{\Phi_{\textrm{CH}_2}}{\Phi_{\textrm{Ca}}}$$ where $N$ is the number of counts, $m$ is the target mass, $M$ is the molecular mass, $\Phi$ is the relative neutron flux[^15] and $\Omega$ is the solid angle. Since the target distances for and are the same, as seen in table \[tab:decset\] the solid angles for these telescopes are assumed to be equal. The other six telescopes, as also seen in table \[tab:decset\] however needs to be corrected with a factor of 0.546, from a simple geometrical point of view. For a more in-deep analysis of the solid angle in MEDLEY, see [@flux].
Since the polyethylene is the reference target that is regularly used in MEDLEY experiments the weight, and number of protons, has been measured with high precision to be $461.55 \pm 0.01$ mg and $3.963 \cdot 10^{22}$ protons [@flux]. The weight of the calcium target used was measured to be $235.9 \pm 0.1$ mg before the runs and $237.3 \pm 0.1$ after the runs, which averages to 236.6 mg.
The cross section, $\sigma_{\textrm{H}}^\textrm{cm}$, is taken from @2001PhRvC..63d4001R where the p(n,p) cross section at a angle of 139.0 degrees and at 96 MeV is measured to be 7.735 mb/sr. In order to extrapolate this to 94.5 MeV at a angle of 139.1 degrees, data from solution SP05 are used. From [@SAID] the cross section for 94.5 MeV and 139.1 degrees is 8.041 mb/sr, while the cross section for 96 MeV and 139.0 degrees is 7.928 mb/sr. This gives an increase in cross section of about 1.43 %. Via the procedure in section \[sec:NN\] this is translated into $\sigma_{\textrm{H}}^\textrm{lab} = 30.53$ mb/sr.
Results
=======
The results given here are those of the accepted neutron spectrum obtained in section \[sec:nspectrum\] and listed in table \[tab:nspectra\], resulting in a mean value for the neutron energy of 79.7 MeV. The total cross section for this spectrum is found in table \[tab:restot\] with a cutoff energy of 2.5 MeV.
Double-differential cross sections
----------------------------------
Double-differential cross sections at four different angles have been analyzed. The result is found in figure \[fig:dEdO\_res\] and tables \[tab:res\_deg20\]-\[tab:res\_deg160\], which also include corrected data introduced in section \[sec:meth3\].
![Experimental double-differential cross sections, filled circles, for the neutron spectrum in table \[tab:nspectra\] in the Ca(n,px) reaction. The solid histogram is TALYS calculations for this neutron spectrum. Cutoff energy is 2.5 MeV.\[fig:dEdO\_res\]](fig/deg20res_all.pdf){width="\textwidth"}
![Experimental double-differential cross sections, filled circles, for the neutron spectrum in table \[tab:nspectra\] in the Ca(n,px) reaction. The solid histogram is TALYS calculations for this neutron spectrum. Cutoff energy is 2.5 MeV.\[fig:dEdO\_res\]](fig/deg40res_all.pdf){width="\textwidth"}
\
![Experimental double-differential cross sections, filled circles, for the neutron spectrum in table \[tab:nspectra\] in the Ca(n,px) reaction. The solid histogram is TALYS calculations for this neutron spectrum. Cutoff energy is 2.5 MeV.\[fig:dEdO\_res\]](fig/deg140res_all.pdf){width="\textwidth"}
![Experimental double-differential cross sections, filled circles, for the neutron spectrum in table \[tab:nspectra\] in the Ca(n,px) reaction. The solid histogram is TALYS calculations for this neutron spectrum. Cutoff energy is 2.5 MeV.\[fig:dEdO\_res\]](fig/deg160res_all.pdf){width="\textwidth"}
Single-differential cross sections
----------------------------------
### Angular-differential cross section
To get angular-differential cross sections, the double-differential spectra are summed over and weighted for the different bin widths. The result is found in table \[tab:Eint\_tot\_all\]. In figure \[fig:res\_eint\] the angular-differential cross section for bins corresponding to compound, pre-equilibrium and direct reactions is found.
![Experimental angular-differential cross sections for three different bins and the neutron spectra in table \[tab:nspectra\] in the Ca(n,px) reaction. The chosen bins are assumed to represent the three specific reaction types. The solid line results from TALYS and the dotted line is a fit to the data according to Eq. (\[eq:gen\_kalbach\]). Cutoff energy is 2.5 MeV.[]{data-label="fig:res_eint"}](fig/angC.pdf){width="\textwidth"}
![Experimental angular-differential cross sections for three different bins and the neutron spectra in table \[tab:nspectra\] in the Ca(n,px) reaction. The chosen bins are assumed to represent the three specific reaction types. The solid line results from TALYS and the dotted line is a fit to the data according to Eq. (\[eq:gen\_kalbach\]). Cutoff energy is 2.5 MeV.[]{data-label="fig:res_eint"}](fig/angPE.pdf){width="\textwidth"}
![Experimental angular-differential cross sections for three different bins and the neutron spectra in table \[tab:nspectra\] in the Ca(n,px) reaction. The chosen bins are assumed to represent the three specific reaction types. The solid line results from TALYS and the dotted line is a fit to the data according to Eq. (\[eq:gen\_kalbach\]). Cutoff energy is 2.5 MeV.[]{data-label="fig:res_eint"}](fig/angD.pdf){width="\textwidth"}
### Energy-differential cross section
To get energy-differential cross sections, each of the energy bins in the double-differential spectra are fitted with Eq. (\[eq:gen\_kalbach\]). Using the results from this fit, the data are interpolated to the angles 60, 80, 100 and 120 degrees, and also extrapolated to 5 and 175 degrees. These data points are summed over and weighted for the different bin as well as their angular contribution. The result is found in figure \[fig:res\_angint\] and in table \[tab:res\_Aint\].
![Experimental energy-differential cross sections, filled circles, for the neutron spectrum in table \[tab:nspectra\] in the Ca(n,px) reaction. The solid histogram is TALYS calculations for this neutron spectrum. Cutoff energy is 2.5 MeV.[]{data-label="fig:res_angint"}](fig/resAint_all.pdf){width="\textwidth"}
Discussion
==========
Comparison with theory
----------------------
To be able to compare the result with theoretical calculations the contribution from the low-energy neutrons need to be subtracted. To do this three different methods, listed in sections \[sec:meth1\]-\[sec:meth3\], are presented. The difference in total cross section between these methods is shown in table \[tab:en\_tot\].
### Method 1: Rescaling of experimental data\[sec:meth1\]
In the first method for subtraction of low-energy neutron data, the energy spectrum for proton emission is assumed to be independent of the incoming neutron energy. The experimental cross section is weighted with the amount of low energy neutrons in the accepted spectrum from table \[tab:nspectra\], and rescaled to the peak in the neutron spectrum. The result is found in figure \[fig:dEdO\_res\_frac\].
The main advantage of this method is that one gets rid of the model dependencies in the TALYS calculations, and the data used are pure experimental. This means that the result at low energies, up to about 40 MeV, should be quite accurate. The main disadvantage on the other hand is the obvious fact that the cross section spectrum for 94 MeV neutrons and lower energy neutrons differ a lot at high energies. This is especially relevant at energies above the low energy bin under consideration.
### Method 2: Subtraction of TALYS data\[sec:meth2\]
Another way to correct for the low-energy neutron contribution is by using data from TALYS. For the accepted neutron spectrum in table \[tab:nspectra\], TALYS is used to calculate cross sections for each bin. The bins between 15 MeV and 55 MeV are left out, since the contribution in these bins probably derive from statistics. The result is found in figure \[fig:dEdO\_res\_tal\].
The advantage of using this method is that it makes the comparison between experiment and theory straightforward. The difference one can see in figure \[fig:dEdO\_res\_tal\] between experiment and theory is directly related to the TALYS calculations. So for code verification purposes, this method works fine. But for obtaining experimental data, this method is not so good, since the resulting experimental data points will have a heavy model dependency on calculations that are known to be incorrect.
### Method 3: Subtraction of modified TALYS data\[sec:meth3\]
A third possible method is to try and modify the TALYS data to get it more correct. From previous experiment, TALYS is known to heavily overestimate the evaporation cross section in backward angles, and slightly overestimate it at forward angles. It also appears to underestimate pre-equilibrium cross sections in forward angles [@2004PhRvC..69f4609T; @2006PhRvC..73c4611T]. Here, an assumption is made that the over/under-estimations in the TALYS calculations are truly systematic and do not vary with projectile energy. From this assumption one can divide the experimental data with the TALYS data in figure \[fig:dEdO\_res\] and get a correction histogram. This correction histogram is assumed to contain the true over/under-estimations in TALYS, and by multiplying the low-energy TALYS calculations with this before subtracting them, the true[^16] cross-section spectra are obtained. The result is found in figure \[fig:dEdO\_res\_wtal\] and, together with uncorrected data, in tables \[tab:res\_deg20\]-\[tab:res\_deg160\].
By using this method one can reduce the model dependencies compared to unmodified TALYS data, by adjusting the theory to the experiment. This method is the one that probably will give the result with smallest systematic errors for the low-energy neutron correction. However, it is first when one sees the results from similar experiments at different energies that one can conclude if the assumption made is too bold or not.
Method $\frac{\textrm{d}\sigma}{\textrm{d}\Omega}(\theta=20)$ $\frac{\textrm{d}\sigma}{\textrm{d}\Omega}(\theta=40)$ $\frac{\textrm{d}\sigma}{\textrm{d}\Omega}(\theta=140)$ $\frac{\textrm{d}\sigma}{\textrm{d}\Omega}(\theta=160)$ $\sigma_{\textrm{prod}}(\textrm{n},\textrm{p}x)$
---------------- -------------------------------------------------------- -------------------------------------------------------- --------------------------------------------------------- --------------------------------------------------------- --------------------------------------------------
1\. Exp. Data $155 \pm 2.92$ $99.4 \pm 1.54$ $31.6 \pm 0.486$ $32.1 \pm 0.700$ $746 \pm 1.97$
2\. TALYS $188 \pm 4.76$ $107 \pm 2.51 $ $17.6 \pm 0.792$ $18.6 \pm 1.14$ $679 \pm 3.74$
3\. Mod. TALYS $175 \pm 6.69$ $107 \pm 4.22$ $33.7 \pm 1.73$ $35.1 \pm 2.04$ $788 \pm 5.83$
TALYS 131 106 62.4 62.4 1070
GNASH [@ICRU] 186 145 - - 1300
: Comparison of angular-differential and total cross sections with subtracted data. Differential cross sections are given in mb/sr and total cross sections in mb. The cutoff energy is 2.5 MeV, except for the GNASH calculations of $\sigma_{\textrm{prod}}(\textrm{n},\textrm{p}x)$, where no cut-off is was applied.[]{data-label="tab:en_tot"}
Background and shielding
------------------------
As can be seen in figure \[fig:snratio\] the background problem is quite heavy. In the silicon data runs from the old facility, the signal-to-background ratio was about 8 [@tippawan]. By moving the spectrometer closer to the lithium target, and at the same time reducing the shielding, the background increased dramatically. In later data sets with energies of 180 MeV, the background problem were so large that it was virtually impossible to get acceptable results from the data [@terucomm]. To come to terms with the background problems recent GEANT simulations at St. Petersburg University show that the background for a 180 MeV neutron beam can be reduced by a factor 20-50[^17], by exchanging the concrete wall to an iron wall. The background would probably be reduced even more for a 94 MeV beam. Available iron blocks from the old CELSIUS ring have actually been used when this reconstruction was undertaken in January 2007. The first data with the new shielding will probably be taken during February 2007.
Special TOF cut\[sec:tofdiss\]
------------------------------
Run 37 was itself unusable for the analysis, but still it may contain some interesting information on the phenomena. As can be seen in the right panel of figure \[fig:tofshift\] there exists a really strong shift in the at high energies that does not seem to be constant, but disappears around 60 MeV. This effect is also visible in as can be seen in the left panel of figure \[fig:tofshift\]. This clear energy dependence makes a correction for this effect quite difficult. Therefore, the cut instead has been expanded to include these effects. This had to be done for both and since the effect still was present, but weak, after the threshold adjustment in . The reasons for this effect is unfortunately still unknown, but since it was weakened by threshold adjustment[^18] it is suspected to be correlated to the pulse height in the $\Delta E_2$ detector. This suspicion is further strengthened by the fact that an analysis of emitted deutrons shows no effect like this.
![TOF shift in in the left panel and run 37 in in the right panel.[]{data-label="fig:tofshift"}](fig/run15.png){width="\textwidth"}
![TOF shift in in the left panel and run 37 in in the right panel.[]{data-label="fig:tofshift"}](fig/run37.png){width="\textwidth"}
One can also note the energy shift in the lower part of the plot, that does not seem to be dependent on the threshold adjustment. If the cause of these problems is because of instabilities in the a possibility to get rid of this problem is to use a timer based on the leading edge technique instead. This will require some extra work with the selections and mean large corrections[^19], since the timing of a leading edge discriminator is dependent on the pulse height.
Mismatch between T1 and T8\[sec:mismatch\]
------------------------------------------
As one can see in figure \[fig:mismatch\] there are some mismatches between and . The low energy mismatch has been carefully examined, unfortunately without solving the problem. The intermediate energy mismatch seen in the calcium data in figure \[fig:mismatch\] was however found to depend on the cut. The mismatch was reduced when the upper part of the cut in figure \[fig:tofplots\] was increased. But increasing the cut also increases the amount of accepted low energy neutrons that need to be subtracted from the data. So the chosen cut of two standard deviations from the peak, is a compromise between the error from the telescope matching and the error from the low-energy neutron correction.
![Mismatch between (filled circles) and (filled triangles). The left panel shows the CH$_2$ runs and the right panel shows the calcium runs.[]{data-label="fig:mismatch"}](fig/ch2T1T8.pdf){width="\textwidth"}
![Mismatch between (filled circles) and (filled triangles). The left panel shows the CH$_2$ runs and the right panel shows the calcium runs.[]{data-label="fig:mismatch"}](fig/caT1T8.pdf){width="\textwidth"}
Reaction losses
---------------
Due to nuclear interactions within the CsI(Tl) some protons may, for example, produce gamma radiation that will escape the CsI(Tl) undetected. Thus only a certain fraction of the incident protons is detected at its true energy, and the rest will form a reaction tail of apparently lower energy. This issue is currently under investigation, but so far the results show that for low energies the lost fraction is really small and correcting for these effects only gets important at higher energies, as shown in figure \[fig:CsIeff\].
Different approaches to angular distributions
---------------------------------------------
In this work, the angular distributions were obtained by fitting Eq. (\[eq:gen\_kalbach\]) to the data. This is an approach coming from experiment, and trying to fit some angular shape to the data, that can be integrated to get the cross section. Another approach one can use is to go from theory and calculate the predicted shape, $a$, of the angular distributions and fitting Eq. (\[eq:kalbach\]) to the data under assumption that $f_{\textrm{MSD}}=1$. The actual form of $f_{\textrm{MSD}}$, as calculated by TALYS, is found in figure \[fig:ang\_fmsd\]. To calculate $a$, the parametrization used [@1988PhRvC..37.2350K] is $$\label{eq:kalb_par1}
a = 0.04\frac{E_1 {e_b}'}{{e_a}'}+1.8 \cdot 10^{-6}\left(\frac{E_1
{e_b}'}{{e_a}'}\right)^{3}+6.7 \cdot 10^{-7}\left(\frac{E_3{e_b}'}{{e_a}'}\right)^{4}$$ where $E_1 = \min({e_a}',130)$, $E_3 = \min({e_a}',41)$, ${e_a}' =
E_a + S_a$ and ${e_b}' = E_b + S_b$. Here $E_a$ is the incoming neutron energy, $E_b$ is the outgoing proton energy, and $S_{a,b}$ is the separation energy between particle $a,b$ and the compound nucleus. The separation energy, $S_{a,b}$, is calculated using the Myers and Swiatecki mass formula [@talys]. As seen in figure \[fig:ang\_a\] there exist some differences in angular distributions between the theoretical and the experimental approach, especially at higher energies, where the theory predicts the forward peaking be larger. The negative values on $a$ at really high energies, that would imply backward peaking, are probably statistical artifacts from the background subtraction.
![Theoretical and experimental approaches to angular distributions and the differences in energy-differential cross sections for the two approaches, where experimental angular distributions are filled circles and calculated angular distribution are filled triangles. The data used are the data with modified TALYS calculations subtracted. The solid line is prediction by TALYS, and the dashed line is prediction by GNASH [@ICRU].[]{data-label="fig:ang_appro"}](fig/kalb_a2.pdf){width="\textwidth"}
![Theoretical and experimental approaches to angular distributions and the differences in energy-differential cross sections for the two approaches, where experimental angular distributions are filled circles and calculated angular distribution are filled triangles. The data used are the data with modified TALYS calculations subtracted. The solid line is prediction by TALYS, and the dashed line is prediction by GNASH [@ICRU].[]{data-label="fig:ang_appro"}](fig/fmsd.pdf){width="\textwidth"}
\
![Theoretical and experimental approaches to angular distributions and the differences in energy-differential cross sections for the two approaches, where experimental angular distributions are filled circles and calculated angular distribution are filled triangles. The data used are the data with modified TALYS calculations subtracted. The solid line is prediction by TALYS, and the dashed line is prediction by GNASH [@ICRU].[]{data-label="fig:ang_appro"}](fig/kalb_exam.pdf){width="\textwidth"}
As can be seen in figure \[fig:ang\_appro\], this parametrization gives a result closer to the model at intermediate energies but deviates more at high energies, while at low energies the difference between the methods are small. This is quite expected from the approximation of $f_{\textrm{MSD}}=1$. An investigation of the parametrization with the calculated form of $f_{\textrm{MSD}}$, or even experimental determination of $f_{\textrm{MSD}}$, is something I think could be interesting in the future.
Systematic uncertainties
------------------------
Throughout this work the error bars only represent the statistical uncertainties in the experiment. To give a complete picture, the systematic errors should be added to the result. These are listed in table \[tab:sysunc\], where some values are for this particular experiment, and some values are adapted from @2006PhRvC..73c4611T.
Origin Uncertainty
------------------------- -------------
Target correction 1-10 %
Solid angle 1-5 %
Beam monitoring 2-3 %
Number of Ca 1 %
efficiency 1 %
Particle identification 1 %
Dead time $<0.1$ %
Absolute cross section 5 %
: Systematic uncertainties[]{data-label="tab:sysunc"}
Conclusions
-----------
As presented in sections \[sec:meth1\]-\[sec:meth3\] and shown in figures \[fig:dEdO\_res\_frac\] - \[fig:dEdO\_res\_wtal\], comparisons between three different methods have been made. No matter which method one chooses, the results all show the same trend as the previous experiments [@2004PhRvC..69f4609T; @2006PhRvC..73c4611T]. The evaporation peak is overestimated by TALYS and predicted to be almost isotropic, while in experiment it significantly decreases with angle. The GNASH calculations overestimate the peak even more. In the pre-equilibrium region, TALYS instead underestimates the cross section, but the predictions seem to be more accurate at larger angles.
The problem in the is an issue that needs to be investigated, especially since similar effects have been noted in other data sets as well [@tippcomm]. But since one of the mismatches in section \[sec:mismatch\] was found to be dependent on the cut, a solution to the problem might shed some light on the mismatch.
Outlook
-------
The data sets analyzed in this work was the first, and at this moment the only, data set available from the upgraded neutron facility [@2005AIPC..769..780P] at . To further improve the calcium results, experimental data from the angles 60, 80, 100 and 120 degrees should be analyzed. The reactions Ca(n,d), Ca(n,t), Ca(n,[${^3}$He]{}) and Ca(n,$\alpha$) would also be interesting to have a look at.
Apart from that, and as mentioned in section \[sec:setup\], lots of work has previously been carried out at 96 MeV [@2004PhRvC..69f4609T; @2006PhRvC..73c4611T; @2004PhRvC..70a4607B] and there still exists a never-ending selection of isotopes to be measured. But since the nuclei under study in these works are especially interesting from a biological and technical point of view it is more interesting to use the capabilities of the new neutron facility and run the experiments at 180 MeV. By obtaining more data sets, one extends the relevant database for future applications. At the same time one makes it possible to do a more quantitative analysis on the nuclei and get a better systematic feedback for the development of TALYS.
[Method 1]{}
\
![Double-differential cross sections in the four top panels, and angle integrated energy-differential cross section in the bottom panel, with experimental data (method 1) subtracted for 94 MeV neutrons in the Ca(n,px) reaction. Solid lines are predictions by TALYS, and dashed lines are predictions by GNASH [@ICRU]. Cutoff energy is 2.5 MeV.\[fig:dEdO\_res\_frac\]](fig/deg20res_frac.pdf){width="\textwidth"}
![Double-differential cross sections in the four top panels, and angle integrated energy-differential cross section in the bottom panel, with experimental data (method 1) subtracted for 94 MeV neutrons in the Ca(n,px) reaction. Solid lines are predictions by TALYS, and dashed lines are predictions by GNASH [@ICRU]. Cutoff energy is 2.5 MeV.\[fig:dEdO\_res\_frac\]](fig/deg40res_frac.pdf){width="\textwidth"}
\
![Double-differential cross sections in the four top panels, and angle integrated energy-differential cross section in the bottom panel, with experimental data (method 1) subtracted for 94 MeV neutrons in the Ca(n,px) reaction. Solid lines are predictions by TALYS, and dashed lines are predictions by GNASH [@ICRU]. Cutoff energy is 2.5 MeV.\[fig:dEdO\_res\_frac\]](fig/deg140res_frac.pdf){width="\textwidth"}
![Double-differential cross sections in the four top panels, and angle integrated energy-differential cross section in the bottom panel, with experimental data (method 1) subtracted for 94 MeV neutrons in the Ca(n,px) reaction. Solid lines are predictions by TALYS, and dashed lines are predictions by GNASH [@ICRU]. Cutoff energy is 2.5 MeV.\[fig:dEdO\_res\_frac\]](fig/deg160res_frac.pdf){width="\textwidth"}
\
![Double-differential cross sections in the four top panels, and angle integrated energy-differential cross section in the bottom panel, with experimental data (method 1) subtracted for 94 MeV neutrons in the Ca(n,px) reaction. Solid lines are predictions by TALYS, and dashed lines are predictions by GNASH [@ICRU]. Cutoff energy is 2.5 MeV.\[fig:dEdO\_res\_frac\]](fig/resAint_frac.pdf){width="\textwidth"}
[Method 2]{}
\
![Double-differential cross sections in the four top panels, and angle integrated single-differential cross section in the bottom panel, with TALYS data (method 2) subtracted for 94 MeV neutrons in the Ca(n,px) reaction. Solid lines are predictions by TALYS, and dashed lines are predictions by GNASH [@ICRU]. Cutoff energy is 2.5 MeV.\[fig:dEdO\_res\_tal\]](fig/deg20res_tal.pdf){width="\textwidth"}
![Double-differential cross sections in the four top panels, and angle integrated single-differential cross section in the bottom panel, with TALYS data (method 2) subtracted for 94 MeV neutrons in the Ca(n,px) reaction. Solid lines are predictions by TALYS, and dashed lines are predictions by GNASH [@ICRU]. Cutoff energy is 2.5 MeV.\[fig:dEdO\_res\_tal\]](fig/deg40res_tal.pdf){width="\textwidth"}
\
![Double-differential cross sections in the four top panels, and angle integrated single-differential cross section in the bottom panel, with TALYS data (method 2) subtracted for 94 MeV neutrons in the Ca(n,px) reaction. Solid lines are predictions by TALYS, and dashed lines are predictions by GNASH [@ICRU]. Cutoff energy is 2.5 MeV.\[fig:dEdO\_res\_tal\]](fig/deg140res_tal.pdf){width="\textwidth"}
![Double-differential cross sections in the four top panels, and angle integrated single-differential cross section in the bottom panel, with TALYS data (method 2) subtracted for 94 MeV neutrons in the Ca(n,px) reaction. Solid lines are predictions by TALYS, and dashed lines are predictions by GNASH [@ICRU]. Cutoff energy is 2.5 MeV.\[fig:dEdO\_res\_tal\]](fig/deg160res_tal.pdf){width="\textwidth"}
\
![Double-differential cross sections in the four top panels, and angle integrated single-differential cross section in the bottom panel, with TALYS data (method 2) subtracted for 94 MeV neutrons in the Ca(n,px) reaction. Solid lines are predictions by TALYS, and dashed lines are predictions by GNASH [@ICRU]. Cutoff energy is 2.5 MeV.\[fig:dEdO\_res\_tal\]](fig/resAint_tal.pdf){width="\textwidth"}
[Method 3]{}
\
![Double-differential cross sections in the four top panels, and angle integrated energy-differential cross section in the bottom panel, with modified TALYS data (method 3) subtracted for 94 MeV neutrons in the Ca(n,px) reaction. Solid lines are predictions by TALYS, and dashed lines are predictions by GNASH [@ICRU]. Cutoff energy is 2.5 MeV.\[fig:dEdO\_res\_wtal\]](fig/deg20res_wtal.pdf){width="\textwidth"}
![Double-differential cross sections in the four top panels, and angle integrated energy-differential cross section in the bottom panel, with modified TALYS data (method 3) subtracted for 94 MeV neutrons in the Ca(n,px) reaction. Solid lines are predictions by TALYS, and dashed lines are predictions by GNASH [@ICRU]. Cutoff energy is 2.5 MeV.\[fig:dEdO\_res\_wtal\]](fig/deg40res_wtal.pdf){width="\textwidth"}
\
![Double-differential cross sections in the four top panels, and angle integrated energy-differential cross section in the bottom panel, with modified TALYS data (method 3) subtracted for 94 MeV neutrons in the Ca(n,px) reaction. Solid lines are predictions by TALYS, and dashed lines are predictions by GNASH [@ICRU]. Cutoff energy is 2.5 MeV.\[fig:dEdO\_res\_wtal\]](fig/deg140res_wtal.pdf){width="\textwidth"}
![Double-differential cross sections in the four top panels, and angle integrated energy-differential cross section in the bottom panel, with modified TALYS data (method 3) subtracted for 94 MeV neutrons in the Ca(n,px) reaction. Solid lines are predictions by TALYS, and dashed lines are predictions by GNASH [@ICRU]. Cutoff energy is 2.5 MeV.\[fig:dEdO\_res\_wtal\]](fig/deg160res_wtal.pdf){width="\textwidth"}
\
![Double-differential cross sections in the four top panels, and angle integrated energy-differential cross section in the bottom panel, with modified TALYS data (method 3) subtracted for 94 MeV neutrons in the Ca(n,px) reaction. Solid lines are predictions by TALYS, and dashed lines are predictions by GNASH [@ICRU]. Cutoff energy is 2.5 MeV.\[fig:dEdO\_res\_wtal\]](fig/resAint_wtal.pdf){width="\textwidth"}
Acknowledgements {#acknowledgements .unnumbered}
================
- *S* – Thank you for all your support and your infinite patience with me. And thank you for always being there for me in times of need.
- *Jan “Bumpen” Blomgren* – Always having something interesting to share from the world outside. And who, despite my probably very confused appearance when I first contacted him, understood what I wanted to do and introduced me to...
- *Stephan Pomp* – My guide and supervisor throughout this. Thank you for all the provided inspiration. And for always having time for me, and my questions and ideas of various quality and relevance.
- *Masateru Hayashi* – Who helped me a lot during the first part of my work, almost like an assisting supervisor. Thank you for discussing ROOT macros, C++ programming, detectors and electronics as well as life in Japan and Sweden with me.
- *Johan Vegelius* – For sharing pizza in the evenings and work during the days, and for simply being a great fellow diploma student.
I also would like to thank everyone at INF, and TSL, that has contributed to a great working environment. And with these words, my diploma thesis is completed. A special thanks to you for reading it.
\
Pär-Anders Söderström
Appendices {#appendices .unnumbered}
==========
Figures and tables
==================
![Contribution to total cross section from various reaction types, according to TALYS calculations.[]{data-label="fig:talplot"}](fig/talysplots.pdf){width="75.00000%"}
No Si1 Si2 Distance \[cm\] Angle \[deg\] $\Delta x_1$ \[$\mu$m\] $\Delta x_2$ \[$\mu$m\]
---- --------- --------- ----------------- --------------- ------------------------- -------------------------
1 38-110B 35-150B 26.83 20 64.9 549
2 37-198D 37-051B 19.83 40 60.5 538
3 38-110D 37-052A 19.83 60 63.9 533
4 37-198G 37-052C 19.83 80 61.7 526
5 31-325C 37-199A 19.83 100 50.4 439
6 31-325D 37-199E 19.83 120 50.1 432
7 38-110G 39-100B 19.83 140 61.6 550
8 31-325H 37-199G 26.83 160 52.9 424
: Detector setup[]{data-label="tab:decset"}
----- -------- ------- ------------------------- ---------- --------- --
Run Target Beam Neutron fluence Lifetime Runtime
\[A\] \[n/cm${^2}\cdot10^8$\] \[%\] \[h\]
004 Junk
005 Empty 1.2 3.1 84 0:57
006 Empty 1.1 4.2 74 1:15
007 Empty 1.2 5.2 72 1:29
008 Empty 5.5 72 1:35
009 Empty 1:27
010 Empty 5.9 72 1:36
011 Empty 1.2 3.5 72 0:57
012 Empty 5.6 72 1:33
013 Empty 6.9 72 1:55
014 CH$_2$ 1.2 6.5 71 1:45
015 CH$_2$ 5.5 71 1:33
016 CH$_2$ 4.3 71 1:09
017 Empty 1.2 4.7 72 1:22
018 Empty 1.2 4.7 73 1:26
019 Ca 1.1 1.6 74 0:27
020 Ca 5.9 74 1:50
021 Ca 7.1 74 2:02
022 Ca 1.1 7.1 74 2:11
023 Ca 5.9 74 1:47
024 Ca 5.6 74 1:37
025 Ca 4.6 73 1:19
026 CH$_2$ 5.1 72 1:28
027 CH$_2$ 4.6 72 1:19
028 Ca 7.1 73 2:01
029 Ca 6.9 73 1:56
030 Ca 5.5 73 2:00
031 Ca 3.4 73 0:52
032 Ca 5.7 73 1:37
033 Ca 6.6 73 1:55
034 Ca 7.0 73 2:01
035 Ca 3.7 74 1:06
036 Ca 1.6 74 0:35
----- -------- ------- ------------------------- ---------- --------- --
: List of runs when the table was positioned at 180 degrees, that is with in forward and in backward direction.[]{data-label="tab:runlist180"}
----- -------- ------- ------------------------- ---------- --------- -- --
Run Target Beam Neutron fluence Lifetime Runtime
\[A\] \[n/cm${^2}\cdot10^8$\] \[%\] \[h\]
001 Junk
002 Junk 1.2
003 Junk 1.2
037 CH$_2$ 1.1 1:05
038 Junk
039 CH$_2$ 1.1 0.96 71 0:15
040 CH$_2$ 5.6 71 1:43
041 CH$_2$ 1.1 5.5 71 1:35
042 CH$_2$ 1.8 71 0:30
043 CH$_2$ 6.9 71 2:00
044 Empty 5.9 72 1:44
045 Empty 9.3 72 2:44
046 Empty 7.2 72 2:07
047 Empty 6.9 72 2:00
048 Empty 7.0 72 2:01
049 Empty 6.7 72 1:59
050 Empty 6.8 72 2:00
051 Empty 5.2 72 1:31
052 Ca 1.1 4.5 71 1:15
053 Ca 1.1 5.3 72 1:37
054 Ca 1.1 5.4 72 1:35
055 Ca 5.7 72 1:45
056 Ca 3.4 72 1:00
057 Ca 4.7 72 1:25
058 Ca 6.4 71 1:55
059 Ca 6.9 71 2:02
060 Ca 6.4 71 1:56
061 Ca 6.6 71 2:01
062 Ca 4.2 72 1:17
063 Ca 1.1 5.4 72 1:39
064 Ca 1.1 6.5 72 1:59
065 Ca 6.5 72 1:59
066 Ca 3.5 72 1:02
----- -------- ------- ------------------------- ---------- --------- -- --
: List of runs when the table was positioned at 0 degrees, that is with in forward and in backward direction.[]{data-label="tab:runlist0"}
`013C` **`4264`** `04FE` `0000` `0000` **`4265`** `05B2` `0000` `0000` **`4266`**
-------- ------------ -------- ------------ ------------ ------------ -------- ------------ -------- ------------
`083E` `0000` `0000` **`4267`** `039B` `0000` `0000` **`4268`** `06B5` `0000`
`0000` **`4269`** `06F4` `0000` `0000` **`426A`** `048B` `0000` `0000` **`426B`**
`0325` `0000` `0000` **`426C`** `0A2C` `0000` `0000` **`426D`** `074E` `0000`
`0000` **`426E`** `0C53` `0000` `0000` **`426F`** `000E` `0000` `0000` **`4270`**
`02D9` `0000` `0000` **`4271`** `0000` `0000` `0000` **`4272`** `0000` `0000`
`0000` **`4273`** `0000` `0000` `0000` **`4274`** `0000` `0000` `0000` **`4275`**
`0000` `0000` `0000` **`4276`** `0000` `0000` `0000` **`4277`** `0000` `0000`
`0000` **`4501`** `0066` `0065` `004F` `0000` `01F4` **`4502`** `004B` `02CA`
`0137` `0000` `007E` **`4503`** `007A` `0068` `0040` `0000` `001C` **`4504`**
`004B` `0034` `0043` `0000` `0206` **`4505`** `0064` `005A` `0036` `0000`
`01F5` **`4506`** `0077` `0049` `003F` `0000` `01CF` **`4507`** `0062` `006D`
`00D4` `0000` `0225` **`4508`** `0083` `0065` `004D` `0000` `01D3` **`480A`**
`0060` `0068` `0067` `004D` `0069` `0047` `005A` `0062` `0000` `4C09`
`0025` `0038` `004A` `004E` `0040` `0039` `002E` `002F` `0FDC` `0FDC`
`0FDC` `0FDC` `0000` **`4300`** **`0000`** `0083` `018F` **`FFFF`**
T1 T2 T7 T8
------- --------- --------- --------- ---------
$k_1$ 0.00334 0.00332 0.00309 0.00278
$m_1$ -0.264 -0.197 -0.284 -0.256
$k_2$ 0.00893 0.00863 0.00969 0.00904
$m_2$ -0.851 -0.667 -1.04 -0.893
$a$ -2.09 -0.734 -1.57 -1.49
$b$ 0.0259 0.0199 0.0226 0.0228
$c$ 0.0032 0.0032 0.0032 0.0032
: Calibration values, as used in Eq. (\[eq:linear\]) and Eq. (\[eq:energulight\]).[]{data-label="tab:fitparamet"}
Neutron bin \[MeV\] np bin \[MeV\] $\sigma_{(\textrm{n},\textrm{p})}$ \[mb\]
--------------------- ---------------- -------------------------------------------
5-15 4-13 292.60
15-25 13-22 153.78
25-35 22-31 100.51
35-45 31-40 73.033
45-55 40-48 57.336
55-65 48-57 47.461
65-75 57-66 40.841
75-85 66-78 36.239
: Energy binning for subtraction of carbon data from the CH$_2$ runs.[]{data-label="tab:neubins"}
------- ------------------- ------------------- -------- --------------------- ---------------------
Bin Uncorrected Corrected Bin Uncorrected Corrected
2.5-3 $0.0931 \pm 0.23$ $0.0559 \pm 0.25$ 68-70 $2.67 \pm 0.30$ $2.25 \pm 0.37$
3-4 $9.61 \pm 0.75$ $5.66 \pm 1.1$ 70-72 $2.13 \pm 0.29$ $1.85 \pm 0.32$
4-5 $9.81 \pm 0.50$ $5.90 \pm 0.86$ 72-74 $1.87 \pm 0.28$ $1.67 \pm 0.30$
5-6 $12.5 \pm 0.50$ $7.71 \pm 1.00$ 74-76 $1.15 \pm 0.26$ $1.07 \pm 0.26$
6-7 $10.4 \pm 0.50$ $6.54 \pm 0.95$ 76-78 $1.69 \pm 0.25$ $1.59 \pm 0.25$
7-8 $7.29 \pm 0.45$ $4.79 \pm 0.75$ 78-80 $0.73 \pm 0.22$ $0.716 \pm 0.22$
8-9 $6.25 \pm 0.44$ $4.13 \pm 0.73$ 80-81 $2.89 \pm 0.30$ $2.88 \pm 0.30$
9-10 $5.37 \pm 0.40$ $3.61 \pm 0.68$ 81-82 $2.05 \pm 0.28$ $2.05 \pm 0.28$
10-12 $4.00 \pm 0.28$ $2.68 \pm 0.45$ 82-83 $0.739 \pm 0.26$ $0.739 \pm 0.26$
12-14 $3.93 \pm 0.30$ $2.60 \pm 0.53$ 83-84 $1.42 \pm 0.26$ $1.42 \pm 0.26$
14-16 $2.56 \pm 0.31$ $1.66 \pm 0.45$ 84-85 $0.979 \pm 0.23$ $0.979 \pm 0.23$
16-18 $2.86 \pm 0.35$ $1.83 \pm 0.53$ 85-86 $1.3 \pm 0.23$ $1.30 \pm 0.23$
18-20 $2.02 \pm 0.35$ $1.28 \pm 0.47$ 86-87 $1.07 \pm 0.21$ $1.07 \pm 0.21$
20-24 $2.66 \pm 0.28$ $1.68 \pm 0.41$ 87-88 $0.861 \pm 0.21$ $0.861 \pm 0.21$
24-28 $2.00 \pm 0.30$ $1.25 \pm 0.39$ 88-89 $0.801 \pm 0.20$ $0.801 \pm 0.20$
28-32 $1.98 \pm 0.31$ $1.23 \pm 0.39$ 89-90 $0.444 \pm 0.18$ $0.444 \pm 0.18$
32-36 $2.83 \pm 0.33$ $1.74 \pm 0.46$ 90-91 $0.524 \pm 0.17$ $0.524 \pm 0.17$
36-40 $2.61 \pm 0.33$ $1.59 \pm 0.46$ 91-92 $0.479 \pm 0.17$ $0.479 \pm 0.17$
40-44 $2.92 \pm 0.32$ $1.78 \pm 0.49$ 92-93 $0.874 \pm 0.17$ $0.874 \pm 0.17$
44-48 $2.85 \pm 0.31$ $1.75 \pm 0.47$ 93-94 $0.345 \pm 0.14$ $0.345 \pm 0.14$
48-52 $2.66 \pm 0.29$ $1.69 \pm 0.43$ 94-95 $0.862 \pm 0.13$ $0.862 \pm 0.13$
52-56 $2.72 \pm 0.28$ $1.83 \pm 0.40$ 95-96 $0.380 \pm 0.11$ $0.380 \pm 0.11$
56-60 $2.49 \pm 0.26$ $1.77 \pm 0.35$ 96-97 $0.400 \pm 0.10$ $0.400 \pm 0.10$
60-62 $2.56 \pm 0.35$ $1.92 \pm 0.45$ 97-98 $-0.0245 \pm 0.080$ $-0.0245 \pm 0.080$
62-64 $2.57 \pm 0.33$ $1.99 \pm 0.43$ 98-99 $0.00468 \pm 0.079$ $0.00468 \pm 0.079$
64-66 $2.73 \pm 0.32$ $2.18 \pm 0.41$ 99-100 $0.0227 \pm 0.13$ $0.0227 \pm 0.13$
66-68 $2.80 \pm 0.316$ $2.27 \pm 0.41$
------- ------------------- ------------------- -------- --------------------- ---------------------
: Experimental double-differential cross sections at 20 degrees in the Ca(n,px) reaction for the neutron spectrum in table \[tab:nspectra\] and for 94 MeV neutrons according to the subtraction method in section \[sec:meth3\].[]{data-label="tab:res_deg20"}
------- ------------------ ------------------ -------- -------------------- --------------------
Bin Uncorrected Corrected Bin Uncorrected Corrected
2.5-3 $7.64 \pm 1.9$ $4.59 \pm 2.2$ 70-71 $0.489 \pm 0.17$ $0.428 \pm 0.19$
3-4 $11.3 \pm 0.57$ $6.68 \pm 1.0$ 71-72 $0.950 \pm 0.18$ $0.834 \pm 0.22$
4-5 $6.43 \pm 0.24$ $3.87 \pm 0.51$ 72-73 $0.506 \pm 0.17$ $0.456 \pm 0.18$
5-6 $12.6 \pm 0.34$ $7.80 \pm 0.95$ 73-74 $0.459 \pm 0.16$ $0.414 \pm 0.17$
6-7 $9.31 \pm 0.30$ $5.84 \pm 0.78$ 74-75 $0.738 \pm 0.16$ $0.689 \pm 0.17$
7-8 $6.97 \pm 0.28$ $4.59 \pm 0.63$ 75-76 $0.355 \pm 0.15$ $0.334 \pm 0.16$
8-9 $5.80 \pm 0.27$ $3.84 \pm 0.60$ 76-77 $0.0924 \pm 0.14$ $0.0861 \pm 0.14$
9-10 $5.16 \pm 0.25$ $3.47 \pm 0.58$ 77-78 $0.700 \pm 0.15$ $0.667 \pm 0.15$
10-12 $3.87 \pm 0.16$ $2.60 \pm 0.38$ 78-79 $0.586 \pm 0.14$ $0.571 \pm 0.14$
12-14 $2.72 \pm 0.15$ $1.81 \pm 0.34$ 79-80 $0.366 \pm 0.13$ $0.362 \pm 0.13$
14-16 $2.03 \pm 0.16$ $1.33 \pm 0.31$ 80-81 $0.411 \pm 0.13$ $0.409 \pm 0.13$
16-18 $2.29 \pm 0.17$ $1.48 \pm 0.37$ 81-82 $0.0804 \pm 0.11$ $0.0804 \pm 0.11$
18-20 $1.58 \pm 0.18$ $1.01 \pm 0.31$ 82-83 $0.215 \pm 0.12$ $0.215 \pm 0.12$
20-24 $1.47 \pm 0.14$ $0.926 \pm 0.22$ 83-84 $0.274 \pm 0.11$ $0.274 \pm 0.11$
24-28 $2.04 \pm 0.15$ $1.28 \pm 0.29$ 84-85 $0.439 \pm 0.10$ $0.439 \pm 0.10$
28-32 $1.78 \pm 0.15$ $1.11 \pm 0.27$ 85-86 $0.172 \pm 0.094$ $0.172 \pm 0.094$
32-36 $1.65 \pm 0.15$ $1.02 \pm 0.27$ 86-87 $0.295 \pm 0.089$ $0.295 \pm 0.089$
36-40 $1.48 \pm 0.15$ $0.907 \pm 0.25$ 87-88 $0.0167 \pm 0.076$ $0.0167 \pm 0.076$
40-44 $1.13 \pm 0.15$ $0.694 \pm 0.22$ 88-89 $0.360 \pm 0.080$ $0.36 \pm 0.080$
44-48 $1.55 \pm 0.14$ $0.971 \pm 0.26$ 89-90 $0.325 \pm 0.10$ $0.325 \pm 0.10$
48-52 $1.22 \pm 0.13$ $0.785 \pm 0.21$ 90-91 $-0.202 \pm 0.00$ $-0.202 \pm 0.00$
52-54 $1.12 \pm 0.18$ $0.746 \pm 0.27$ 91-92 $0.076 \pm 0.074$ $0.076 \pm 0.074$
54-56 $1.04 \pm 0.18$ $0.720 \pm 0.25$ 92-93 $0.102 \pm 0.047$ $0.102 \pm 0.047$
56-58 $1.26 \pm 0.17$ $0.897 \pm 0.27$ 93-94 $0.0237 \pm 0.063$ $0.0237 \pm 0.063$
58-60 $0.966 \pm 0.16$ $0.712 \pm 0.23$ 94-95 $0.149 \pm 0.046$ $0.149 \pm 0.046$
60-62 $0.851 \pm 0.16$ $0.647 \pm 0.2$ 95-96 $0.0882 \pm 0.046$ $0.0882 \pm 0.046$
62-64 $0.773 \pm 0.15$ $0.605 \pm 0.18$ 96-97 $0.108 \pm 0.046$ $0.108 \pm 0.046$
64-66 $0.871 \pm 0.14$ $0.701 \pm 0.19$ 97-98 $0.0595 \pm 0.044$ $0.0595 \pm 0.044$
66-68 $1.07 \pm 0.14$ $0.876 \pm 0.19$ 98-99 $0.0190 \pm 0.045$ $0.0190 \pm 0.045$
68-70 $1.07 \pm 0.13$ $0.909 \pm 0.18$ 99-100 $0.0499 \pm 0.050$ $0.0499 \pm 0.050$
------- ------------------ ------------------ -------- -------------------- --------------------
: Experimental double-differential cross sections at 40 degrees in the Ca(n,px) reaction for the neutron spectrum in table \[tab:nspectra\] and for 94 MeV neutrons according to the subtraction method in section \[sec:meth3\].[]{data-label="tab:res_deg40"}
------- -------------------- -------------------- -------- ---------------------- ---------------------
Bin Uncorrected Corrected Bin Uncorrected Corrected
2.5-3 $0.0668 \pm 0.14$ $0.0401 \pm 0.15$ 30-35 $0.175 \pm 0.020$ $0.114 \pm 0.082$
3-4 $5.41 \pm 0.42$ $3.19 \pm 0.60$ 35-40 $0.0721 \pm 0.015$ $0.0472 \pm 0.045$
4-5 $3.90 \pm 0.20$ $2.35 \pm 0.35$ 40-45 $0.0183 \pm 0.011$ $0.0123 \pm 0.018$
5-6 $8.68 \pm 0.30$ $5.37 \pm 0.69$ 45-50 $0.00676 \pm 0.0082$ $0.00474 \pm 0.010$
6-7 $6.76 \pm 0.27$ $4.26 \pm 0.59$ 50-55 $-0.0278 \pm 0.0090$ $-0.0208 \pm 0.031$
7-8 $4.21 \pm 0.21$ $2.80 \pm 0.41$ 55-60 $0.0146 \pm 0.0085$ $0.0117 \pm 0.020$
8-9 $3.55 \pm 0.18$ $2.37 \pm 0.39$ 60-65 $0.0244 \pm 0.0099$ $0.0209 \pm 0.033$
9-10 $3.37 \pm 0.16$ $2.31 \pm 0.39$ 65-70 $0.0274 \pm 0.0090$ $0.0246 \pm 0.041$
10-12 $1.97 \pm 0.055$ $1.36 \pm 0.20$ 70-75 $0.00282 \pm 0.011$ $0.00267 \pm 0.011$
12-14 $1.09 \pm 0.045$ $0.762 \pm 0.15$ 75-80 $0.0165 \pm 0.011$ $0.0161 \pm 0.021$
14-16 $0.946 \pm 0.046$ $0.663 \pm 0.16$ 80-85 $0.0562 \pm 0.013$ $0.0561 \pm 0.014$
16-18 $0.709 \pm 0.052$ $0.493 \pm 0.16$ 85-90 $0.0346 \pm 0.015$ $0.0346 \pm 0.015$
18-20 $0.283 \pm 0.053$ $0.193 \pm 0.096$ 90-95 $0.0914 \pm 0.016$ $0.0914 \pm 0.016$
20-25 $0.420 \pm 0.024$ $0.280 \pm 0.11$ 95-100 $0.0274 \pm 0.019$ $0.0274 \pm 0.019$
25-30 $0.149 \pm 0.0233$ $0.0968 \pm 0.057$
------- -------------------- -------------------- -------- ---------------------- ---------------------
: Experimental double-differential cross sections at 140 degrees in the Ca(n,px) reaction for the neutron spectrum in table \[tab:nspectra\] and for 94 MeV neutrons according to the subtraction method in section \[sec:meth3\].[]{data-label="tab:res_deg140"}
------- -------------------- ------------------- -------- ---------------------- ----------------------
Bin Uncorrected Corrected Bin Un-corrected Corrected
2.5-3 $0.152 \pm 0.26$ $0.0915 \pm 0.28$ 30-35 $0.152 \pm 0.023$ $0.0984 \pm 0.073$
3-4 $6.70 \pm 0.60$ $3.95 \pm 0.81$ 35-40 $0.204 \pm 0.020$ $0.133 \pm 0.12$
4-5 $6.44 \pm 0.38$ $3.87 \pm 0.61$ 40-45 $0.137 \pm 0.017$ $0.0918 \pm 0.10$
5-6 $6.97 \pm 0.36$ $4.31 \pm 0.62$ 45-50 $0.0132 \pm 0.015$ $0.00927 \pm 0.019$
6-7 $4.68 \pm 0.32$ $2.95 \pm 0.50$ 50-55 $0.0163 \pm 0.015$ $0.0122 \pm 0.023$
7-8 $4.05 \pm 0.30$ $2.69 \pm 0.46$ 55-60 $-0.00418 \pm 0.017$ $-0.00335 \pm 0.018$
8-9 $3.63 \pm 0.25$ $2.43 \pm 0.44$ 60-65 $-0.00651 \pm 0.019$ $-0.00558 \pm 0.021$
9-10 $2.89 \pm 0.21$ $1.98 \pm 0.38$ 65-70 $-0.0545 \pm 0.019$ $-0.0491 \pm 0.081$
10-12 $1.59 \pm 0.083$ $1.10 \pm 0.17$ 70-75 $0.0539 \pm 0.021$ $0.0511 \pm 0.073$
12-14 $0.915 \pm 0.079$ $0.640 \pm 0.14$ 75-80 $0.0449 \pm 0.024$ $0.0439 \pm 0.055$
14-16 $0.647 \pm 0.079$ $0.454 \pm 0.14$ 80-85 $0.134 \pm 0.023$ $0.134 \pm 0.026$
16-18 $0.616 \pm 0.080$ $0.428 \pm 0.16$ 85-90 $0.146 \pm 0.028$ $0.146 \pm 0.028$
18-20 $0.388 \pm 0.078$ $0.266 \pm 0.13$ 90-95 $0.254 \pm 0.030$ $0.254 \pm 0.030$
20-25 $0.223 \pm 0.030$ $0.148 \pm 0.063$ 95-100 $0.229 \pm 0.036$ $0.229 \pm 0.036$
25-30 $0.184 \pm 0.0281$ $0.120 \pm 0.070$
------- -------------------- ------------------- -------- ---------------------- ----------------------
: Experimental double-differential cross sections at 160 degrees in the Ca(n,px) reaction for the neutron spectrum in table \[tab:nspectra\] and for 94 MeV neutrons according to the subtraction method in section \[sec:meth3\].[]{data-label="tab:res_deg160"}
------- ----------------- ----------------- -------- ------------------- -------------------
Bin Uncorrected Corrected Bin Uncorrected Corrected
0-2.5 - - 46-48 $6.69 \pm 0.29$ $4.15 \pm 0.41$
2.5-3 $12.1 \pm 0.32$ $7.24 \pm 0.36$ 48-50 $5.23 \pm 0.28$ $3.32 \pm 0.37$
3-4 $102 \pm 0.52$ $59.7 \pm 0.77$ 50-52 $5.04 \pm 0.35$ $3.26 \pm 0.55$
4-5 $73.7 \pm 0.26$ $45.0 \pm 0.47$ 52-54 $4.54 \pm 0.33$ $3.05 \pm 0.56$
5-6 $129 \pm 0.33$ $78.7 \pm 0.73$ 54-56 $4.46 \pm 0.33$ $3.07 \pm 0.49$
6-7 $97.3 \pm 0.30$ $60.4 \pm 0.62$ 56-58 $5.44 \pm 0.29$ $3.77 \pm 0.53$
7-8 $69.7 \pm 0.26$ $46.1 \pm 0.49$ 58-60 $4.27 \pm 0.32$ $3.12 \pm 0.44$
8-9 $59.2 \pm 0.24$ $39.5 \pm 0.47$ 60-62 $3.85 \pm 0.31$ $2.91 \pm 0.39$
9-10 $52.5 \pm 0.21$ $35.4 \pm 0.45$ 62-64 $3.68 \pm 0.30$ $2.86 \pm 0.37$
10-12 $35.2 \pm 0.11$ $23.6 \pm 0.26$ 64-66 $4.00 \pm 0.27$ $3.21 \pm 0.36$
12-14 $24.2 \pm 0.10$ $16.4 \pm 0.23$ 66-68 $4.54 \pm 0.29$ $3.70 \pm 0.37$
14-16 $18.3 \pm 0.10$ $12.0 \pm 0.22$ 68-70 $4.50 \pm 0.28$ $3.79 \pm 0.35$
16-18 $18.0 \pm 0.11$ $12.0 \pm 0.26$ 70-72 $3.25 \pm 0.24$ $2.84 \pm 0.28$
18-20 $11.0 \pm 0.13$ $7.19 \pm 0.22$ 72-74 $2.53 \pm 0.27$ $2.27 \pm 0.28$
20-22 $11.8 \pm 0.11$ $7.34 \pm 0.24$ 74-76 $2.34 \pm 0.45$ $2.15 \pm 0.32$
22-24 $12.0 \pm 0.11$ $7.35 \pm 0.27$ 76-78 $2.23 \pm 0.25$ $2.11 \pm 0.25$
24-26 $10.9 \pm 0.12$ $6.82 \pm 0.28$ 78-80 $2.42 \pm 0.12$ $2.33 \pm 0.20$
26-28 $11.3 \pm 0.14$ $6.95 \pm 0.30$ 80-82 $2.89 \pm 0.56$ $2.89 \pm 0.74$
28-30 $10.0 \pm 0.14$ $6.25 \pm 0.31$ 82-84 $1.55 \pm 0.23$ $1.55 \pm 0.19$
30-32 $9.53 \pm 0.12$ $5.99 \pm 0.31$ 84-86 $1.71 \pm 0.15$ $1.71 \pm 0.15$
32-34 $9.74 \pm 0.12$ $6.04 \pm 0.39$ 86-88 $1.29 \pm 0.21$ $1.29 \pm 0.21$
34-36 $10.7 \pm 0.13$ $6.64 \pm 0.46$ 88-90 $2.47 \pm 0.070$ $2.47 \pm 0.070$
36-38 $8.17 \pm 0.11$ $4.76 \pm 0.39$ 90-92 $0.684 \pm 0.045$ $0.684 \pm 0.041$
38-40 $9.51 \pm 0.12$ $5.42 \pm 0.61$ 92-94 $0.942 \pm 0.044$ $0.942 \pm 0.044$
40-42 $5.76 \pm 0.13$ $2.49 \pm 0.77$ 94-96 $0.949 \pm 0.060$ $0.949 \pm 0.060$
42-44 $8.05 \pm 0.21$ $3.59 \pm 0.68$ 96-98 $0.516 \pm 0.15$ $0.516 \pm 0.15$
44-46 $7.12 \pm 0.32$ $3.94 \pm 1.3$ 98-100 $0.315 \pm 0.15$ $0.315 \pm 0.15$
44-46 $7.12 \pm 0.31$ $3.94 \pm 0.41$
------- ----------------- ----------------- -------- ------------------- -------------------
: Experimental energy-differential cross sections in the Ca(n,px) reaction for the neutron spectrum in table \[tab:nspectra\] and for 94 MeV neutrons according to the subtraction method in section \[sec:meth3\].[]{data-label="tab:res_Aint"}
------- ------------------ -----------------
Angle Uncorrected Corrected
20 $253 \pm 4.76$ $175 \pm 6.69$
40 $162 \pm 2.51$ $107 \pm 4.22$
140 $51.5 \pm 0.792$ $33.7 \pm 1.73$
160 $52.4 \pm 1.14$ $35.1 \pm 2.04$
------- ------------------ -----------------
: Experimental angular-differential cross sections in the Ca(n,px) reaction for the neutron spectrum in table \[tab:nspectra\] and for 94 MeV neutrons according to the subtraction method in section \[sec:meth3\]. Cutoff energy is 2.5 MeV.[]{data-label="tab:Eint_tot_all"}
[cc]{} Uncorrected & Corrected\
\
$1220 \pm 3.21$ & $788 \pm 5.83$\
Notes about extracting information from NN databases\[sec:NN\]
==============================================================
When using databases to obtain data on elastic scattering it is not always intuitive to get what one wants from it. The problem is that databases often gives the cross section for a given neutron scattering angle, but one most often knows the proton angle. To get the cross section one wants, some transformations between different frames and calculations of ratio-factors are needed [@flux]. This is an eight-step process to get through this without (too many) tears. First some necessary kinematic relations for a general scattering reaction are given in [@lecturenotes],
$$E_1 = T_1 + m_1\label{eq:kine1}$$
$$p_1 = \sqrt{T_1^2 + 2T_1m_1}\label{eq:kine2}$$
$$s = m_1^2+m_2^2+2E_1m_2\label{eq:kine3}$$
$$E_3^\textrm{cm} = \frac{1}{2\sqrt{s}} \left( s+m_3^2-m_4^2 \right)\label{eq:kine4}$$
$$v_\textrm{cm} = \frac{p_1}{E_1+m_2}\label{eq:kine5}$$
$$p_3^\textrm{cm} = \sqrt{(E_3^\textrm{cm})^2-m_3^2}\label{eq:kine6}$$
$$\tan\theta_3 = \frac{\sin\theta_3^\textrm{cm} \sqrt{1-v_\textrm{cm}^2}}{\cos\theta_3^\textrm{cm} +
\frac{v_\textrm{cm}E_3^\textrm{cm}}{p_3^\textrm{cm}}}\label{eq:kine7}.$$
The relevant quantities are defined in figure \[fig:kinema\], where $m_i$ is the mass, $T_i$ is the kinetic energy and $p_i$ is the momentum of particle $i$. Quantities in the system are labeled cm, unlabeled quantities are in the laboratory frame.
![Scattering in the laboratory frame, adapted from [@lecturenotes].[]{data-label="fig:kinema"}](fig/kinema.png){width="60.00000%"}
For a reaction with a neutron on a proton target we switch the indices in the notations as $1,4 \rightarrow n$ and $2,3
\rightarrow p$. Known parameters are assumed to be $m_p$, $m_n$, $T_n$ and $\theta_\textrm{lab}^p$. An unlabeled $\theta$ is always the scattering angle of the neutron.
1. Using the relations (\[eq:kine1\]) to (\[eq:kine7\]) it is straightforward, although (\[eq:kine7\]) might make it analytically tricky, to calculate $\theta^\textrm{cm}_p$.
2. The easy part: $\theta^\textrm{cm}_n = 180 -
\theta^\textrm{cm}_p$.
3. Calculate $\theta^\textrm{lab}_n$. This is preferably done via some computer code for relativistic kinematics, for example the code RELKIN circulating at [@pompcomm].
4. From the database of choice, extract the cross section in the frame for the *opposite* case, that is when the neutron is scattered in the desired proton angle. That is, the cross section $\left( \frac{\textrm{d}\sigma}{\textrm{d}\Omega}
\right)^\textrm{cm}_{\theta_n =
\theta^\textrm{cm}_p}$.\[list:steppcm\]
5. In the same way as in step \[list:steppcm\], use a database to extract the cross section in the frame for the desired case. That is, the cross section $\left( \frac{\textrm{d}\sigma}{\textrm{d}\Omega}
\right)^\textrm{cm}_{\theta_n = \theta^\textrm{cm}_n}$.\[list:stepncm\]
6. Do step \[list:steppcm\], but in the laboratory frame, to get $\left( \frac{\textrm{d}\sigma}{\textrm{d}\Omega}
\right)^\textrm{lab}_{\theta_n = \theta^\textrm{lab}_p}$ from the database.\[list:stepplab\]
7. Now to calculate the ratio factor, $R$, introduced in [@flux] for transformations between lab frame cross section and frame cross section. Since the $R$ given in the reference only is valid for particles of equal masses the results from step \[list:steppcm\] and step \[list:stepplab\] is used to calculate $R$ explicitly from the relation $\left( \frac{\textrm{d}\sigma}{\textrm{d}\Omega}
\right)^\textrm{cm}_{\theta_n = \theta^\textrm{cm}_p} = R \left( \frac{\textrm{d}\sigma}{\textrm{d}\Omega}
\right)^\textrm{lab}_{\theta_n = \theta^\textrm{lab}_p}$.\[list:stepR\]
8. Using $R$ obtained in step \[list:stepR\] it is now possible to transform the cross section from step \[list:stepncm\] into the lab frame, by the relation previously used and obtained from [@flux], $\left( \frac{\textrm{d}\sigma}{\textrm{d}\Omega}
\right)^\textrm{lab}_{\theta_n = \theta^\textrm{lab}_n} = \left( \frac{\textrm{d}\sigma}{\textrm{d}\Omega}
\right)^\textrm{cm}_{\theta_n = \theta^\textrm{cm}_n}\frac{1}{R}$.
Finally we have the cross section $\left(\frac{\textrm{d}\sigma}{\textrm{d}\Omega}\right)^\textrm{lab}_{\theta
= \theta^\textrm{lab}_n}$ which is just what we wanted, since the cross section of scattering a neutron at an angle of $\theta^\textrm{lab}_n$ is equivalent to the cross section of scattering of a proton at an angle $\theta^\textrm{lab}_p$.
Let us find the cross section for np scattering at $\theta_\textrm{lab}^p = 20$ degrees with an incoming neutron energy of $T_n=50$ MeV. Using the mass values $m_n = 939.57$ MeV/c$^2$ and $m_n = 938.27$ MeV/c$^2$ the relations (\[eq:kine1\]) to (\[eq:kine7\]) will give $\theta^\textrm{cm}_p \approx 40.5$ degrees and, consequentially, $\theta^\textrm{cm}_n \approx 139.5$ degrees. Using RELKIN [@pompcomm], a code for relativistic kinematics, $\theta^\textrm{lab}_p$ is calculated to be $\theta^\textrm{lab}_p \approx 69.4$ degrees. From the SAID database [@SAID] we get
$$\left( \frac{\textrm{d}\sigma}{\textrm{d}\Omega} \right)^\textrm{cm}_{\theta_n = 40.5} \approx
14.1 \frac{\textrm{mb}}{\textrm{sr}}\label{eq:ex_xsecpcm}$$
$$\left( \frac{\textrm{d}\sigma}{\textrm{d}\Omega} \right)^\textrm{cm}_{\theta_n = 139.5} \approx
15.0 \frac{\textrm{mb}}{\textrm{sr}}\label{eq:ex_xsecncm}$$
$$\left( \frac{\textrm{d}\sigma}{\textrm{d}\Omega} \right)^\textrm{lab}_{\theta_n = 20.0} \approx
53.9. \frac{\textrm{mb}}{\textrm{sr}}\label{eq:ex_xsecplab}$$
Note the different observable keywords `DSG` for cross section in the center of mass frame, and `DSGL` for the cross section in the laboratory frame. The ratio factor, $R$, can now be calculated using the information in (\[eq:ex\_xsecpcm\]), (\[eq:ex\_xsecplab\]) and the definition in step \[list:stepR\] $$R \equiv \frac{\left( \frac{\textrm{d}\sigma}{\textrm{d}\Omega}
\right)^\textrm{cm}_{\theta_n = \theta^\textrm{cm}_p}}{ \left( \frac{\textrm{d}\sigma}{\textrm{d}\Omega}
\right)^\textrm{lab}_{\theta_n = \theta^\textrm{lab}_p}} =
\frac{14.1}{53.9} \approx 0.261.\label{eq:ex_ratio}$$ The ratio factor from (\[eq:ex\_ratio\]) and the cross section (\[eq:ex\_xsecncm\]) can now be used to obtain $$\left(\frac{\textrm{d}\sigma}{\textrm{d}\Omega}
\right)^\textrm{lab}_{\theta_n = 69.4} = \left( \frac{\textrm{d}\sigma}{\textrm{d}\Omega}
\right)^\textrm{cm}_{\theta_n = 139.5}\frac{1}{R} = 15.0 \cdot \frac{1}{0.261}
\approx 57.5 \frac{\textrm{mb}}{\textrm{sr}}$$ which is the sought cross section.
[43]{} \[1\][\#1]{} \[1\][`#1`]{} urlstyle \[1\][doi: \#1]{}
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[^1]: More or less all nuclear reactions.
[^2]: In the $\beta$-decay showed in figure \[fig:neutronbetadecay\]
[^3]: Even though the neutron is electrically neutral it still holds a magnetic dipole moment of about -1.91 nuclear magnetons, $\mu_\textrm{N}$. This is due to the fact that a baryon has a magnetic moment equal to the sums of the quarks magnetic moments, which in the case of the neutron gives $\mu_\textrm{n} = \frac{4}{3}\mu_\textrm{d} -
\frac{1}{3}\mu_\textrm{u} = -
\frac{2}{3}\frac{M_\textrm{p}}{m_{\textrm{u},\textrm{d}}}$ . The electrical dipole moment is measured to be less than $2.9
\cdot 10^{-26}$ $e\cdot$cm [@2006PhRvL..97m1801B].
[^4]: For example in neutron stars.
[^5]: See section \[sec:magicmodel\]
[^6]: Rumors claim that @barn coined the term barn in December 1942. Trying to come up with a name for the unit 10$^{-28}$ m$^2$ they rejected both bethe, oppenheimer, and manley before they decided to use barn, since they thought that the probability of a reaction in the experiment was as high as the probability of hitting a barn when firing a gun.
[^7]: This example is actually quite historically interesting since it was the first accelerator induced nuclear reaction [@krane].
[^8]: The Fermi surface is the highest energy level when all the nucleons in a nucleus, or electrons in an atom, are at their lowest lying states - the Fermi sea.
[^9]: It may sound backwards to use the registration of a particle as a start signal and the next beam pulse as stop for the , but this is simply a matter of not overloading the electronics with unnecessary start signals. The reason to store the two times where one is delayed may not be intuitively clear either, but this is explained in more detail in section \[sec:timing\], and also in [@aboutRF].
[^10]: These two have also been a part of creating other popular data analysis and simulation tools like PAW, PIAF, and GEANT.
[^11]: One short is 2 bytes
[^12]: At least concerning background events. Misidentified events from, for example, deuterons might still be tricky to get rid of if the cuts are too generous.
[^13]: One should note that this is not the same cross section as the one used for later normalization, but this should pose no problem since this is only a relative measurement.
[^14]: The Q value for the C(n,p) reaction is 12.588 MeV [@physics].
[^15]: The normalization to neutron flux has actually already been done, simultaneous to the life time correction, using the monitor. But the factor should still be included in Eq. (\[eq:xsecnorm\]) for completeness.
[^16]: Well, true under the condition that the assumption is correct.
[^17]: Even up to a factor 100.
[^18]: Before run 39 the $\Delta E_2$ threshold in T8 was changed from 450 mV to 315 mV, and the $\Delta E_2$ threshold in T7 was changed from 400 mV to 325 mV. Before run 40 the $\Delta E_2$ threshold in T8 was further changed to 295 mV, and the $\Delta E_2$ threshold in T3 was changed from 320 mV to 350 mV.
[^19]: Of the order of 10 ns.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The class of threshold functions is known to be characterizable by functional equations or, equivalently, by pairs of relations, which are called relational constraints. It was shown by Hellerstein that this class cannot be characterized by a finite number of such objects. In this paper, we investigate classes of threshold functions which arise as intersections of the class of all threshold functions with clones of Boolean functions, and provide a complete classification of such intersections in respect to whether they have finite characterizations. Moreover, we provide a characterizing set of relational constraints for each class of threshold functions arising in this way.'
address:
- |
LAMSADE – CNRS\
Université Paris-Dauphine\
Place du Maréchal de Lattre de Tassigny\
75775 Paris Cedex 16\
France
- |
University of Luxembourg\
Computer Science and Communications Research Unit\
6, rue Richard Coudenhove-Kalergi\
L–1359 Luxembourg\
Luxembourg
- |
University of Luxembourg\
Mathematics Research Unit\
6, rue Richard Coudenhove-Kalergi\
L–1359 Luxembourg\
Luxembourg
author:
- Miguel Couceiro
- Erkko Lehtonen
- Karsten Schölzel
title: A complete classification of equational classes of threshold functions included in clones
---
Introduction and preliminaries
==============================
Introduction
------------
Two approaches to characterize properties of Boolean functions have been considered recently: one in terms of functional equations [@EFHH], another in terms of relational constraints [@Pippenger]. As it turns out, these two approaches have the same expressive power in the sense that they characterize the same properties (classes) of Boolean functions, which can be described as initial segments of the so-called “minor” relation between functions: for two functions $f$ and $g$ of several variables, $f$ is said to be a minor of $g$ if $f$ can be obtained from $g$ by identifying variables, permuting variables, or adding inessential variables (see Subsection \[susec:MinorsConstraints\]). Furthermore, a class is characterizable by a finite number of functional equations if and only if it is characterizable by a finite number of relational constraints (see, e.g., [@CF; @Pippenger]). For the sake of simplifying the presentation of constructions and proofs, we will focus on the approach by relational constraints.
Several properties of functions can be charaterized by relational constraints (or, equivalently, by functional equations. In fact, uncountably many properties are expressible by such objects, even in the simplest interesting case of functions of several variables, i.e., the Boolean functions (see [@CP; @Pippenger]). Classical examples of such properties include idempotency, monotonicity and linearity. More contemporary examples include submodularity, supermodularity and the combination of the two, i.e., modularity (see, e.g., [@CouMar; @Lovasz; @Singer; @Topkis]).
Another noteworthy example is thresholdness that is the property of those Boolean functions whose true points can be separated from the false points by a hyperplane when considered as elements of the $n$-dimensional real space ${\mathbb{R}}^n$. Threshold functions have been widely studied in the literature on Boolean functions, switching theory, system reliability theory, game theory, etc.; for background see, e.g., [@Isbell; @Muroga; @Peleg1; @Peleg2; @Taylor; @Winder].
Despite being a property expressible by relational constraints, thresholdness cannot be captured by a finite set of relational constraints (see Hellerstein [@Hellerstein]). However, by imposing additional conditions such as linearity or preservation of componentwise conjunctions or disjunctions of tuples, the resulting classes of threshold functions may become characterizable by a finite number of relational constraints. In fact, these examples can be obtained from the class of threshold functions by intersecting it with certain clones, namely, those of linear functions, conjunctions and disjunctions, respectively. (Recall that a clone is a class of functions that contains all projections and is closed under functional composition.) Another noteworthy and well-known example of such an intersection is the class of “majority games”, which results as the intersection with the clone of self-dual monotone functions. The natural question is then: Is the class of majority games characterizable by a finite number of relational constraints?
In this paper we answer negatively to this question. In fact, we will determine, for each clone of Boolean functions, whether its intersection with the class of threshold functions is finitely characterizable by relational constraints. Moreover, we provide finite or infinite characterizing sets of relational constraints accordingly.
The paper is organized as follows. In the remainder of this section, we recall basic notions and results that will be needed throughout the paper. The main results are presented in Section \[sec:main\], in particular, the classification of all intersections $C \cap {T}$, where $C$ is a clone and ${T}$ is the class of all threshold functions, as well as the corresponding characterizing set of relational constraints. For the reader’s convenience, the constructions needed for the main results will be left for Section \[sec:constructions\]. One of the main tools in our proof is Taylor and Zwicker’s [@Taylor] theorem on the existence of a $k$-asummable function that is not $(k+1)$-asummable. In Section \[sec:magic\], we slightly refine Taylor and Zwicker’s result and show how the classes of functions characterizable by the relational constraints that arise in our current work are related to each other. Appendix \[App:Post\] provides a list of the clones of Boolean functions and relations characterizing them.
Boolean functions
-----------------
Throughout the paper, we denote the set $\{1, \dots, n\}$ by ${[{n}]}$ and the set $\{0, 1\}$ by ${\mathbb{B}}$. A *Boolean function* is a map $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$ for some positive integer $n$ called the *arity* of $f$. Typical examples of Boolean functions include
- the $n$-ary $i$-th *projection* ($i \in {[{n}]}$) $e^{(n)}_i \colon {\mathbb{B}}^n \to {\mathbb{B}}$, $(a_1, \dots, a_n) \mapsto a_i$;
- *negation* ${\overline{\cdot}} \colon {\mathbb{B}}\to {\mathbb{B}}$, ${\overline{0}} = 1$, ${\overline{1}} = 0$;
- *conjunction* $\wedge \colon {\mathbb{B}}^2 \to {\mathbb{B}}$, $x \wedge y = 1$ if and only if $x = y = 1$;
- *disjunction* $\vee \colon {\mathbb{B}}^2 \to {\mathbb{B}}$, $x \vee y = 0$ if and only if $x = y = 0$;
- *modulo-$2$ addition* $\oplus \colon {\mathbb{B}}^2 \to {\mathbb{B}}$, $x \oplus y = (x + y) \bmod 2$.
The set of all Boolean functions is denoted by $\Omega$ and the set of all projections is denoted by $I_c$.
The preimage ${{f}^{-1}(1)}$ of $1$ under $f$ is referred to as the set of *true points*, while ${{f}^{-1}(0)}$ is referred to as the set of *false points*.
The $i$-th variable of a Boolean function $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$ is said to be *essential* in $f$, or that $f$ depends on $x_i$, if there are $a_1, \ldots, a_{i-1},a_{i+1}, \ldots,a_n \in {\mathbb{B}}$ such that $$f(a_1, \ldots, a_{i-1},0,a_{i+1}, \ldots, a_n) \neq f(a_1, \ldots, a_{i-1}, 1, a_{i+1}, \ldots, a_n).$$
The *dual* of a Boolean function $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$ is the function ${{f}^{\mathrm{d}}} \colon {\mathbb{B}}^n \to {\mathbb{B}}$ given by $${{f}^{\mathrm{d}}}(x_1, \dots, x_n)
= {\overline{f({\overline{x}}_1, \dots, {\overline{x}}_n)}}.$$ A function $f$ is *self-dual* if $f = {{f}^{\mathrm{d}}}$.
If $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$ and $g_1, \dots, g_n \colon {\mathbb{B}}^m \to {\mathbb{B}}$, then the *composition* of $f$ with $g_1, \dots, g_n$ is the function $f(g_1, \dots, g_n) \colon {\mathbb{B}}^m \to {\mathbb{B}}$ given by $$f(g_1, \dots, g_n)(\aa) = f(g_1(\aa), \dots, g_n(\aa))$$ for all $\aa \in {\mathbb{B}}^m$. A *clone* of Boolean functions is a subset $C$ of the set $\Omega$ of all Boolean functions that satisfies the following two conditions:
- $I_c \subseteq C$, i.e., $C$ contains all projections,
- if $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$, $g_1, \dots, g_n \colon {\mathbb{B}}^m \to {\mathbb{B}}$ and $f, g_1, \dots, g_n \in C$, then $f(g_1, \dots, g_n) \in C$, i.e., $C$ is closed under composition.
The clones of Boolean functions were completely described by Post [@Post], and they are often referred to as *Post’s classes.* We provide a list of all clones of Boolean functions in Appendix \[App:Post\].
Minors and relational constraints {#susec:MinorsConstraints}
---------------------------------
We will denote tuples in ${\mathbb{B}}^m$ by boldface letters and their entries with corresponding italic letters, e.g., $\mathbf{a} = (a_1, \dots, a_m)$. Tuples $\mathbf{a} \in {\mathbb{B}}^m$ may be viewed as mappings $\mathbf{a} \colon {[{m}]} \to {\mathbb{B}}$, $i \mapsto a_i$. With this convention, given a map $\sigma \colon {[{n}]} \to {[{m}]}$, we can write the tuple $(a_{\sigma(1)}, \dots, a_{\sigma(n)})$ as $\mathbf{a} \circ \sigma$, or simply $\mathbf{a} \sigma$.
A function $f \colon {\mathbb{B}}^m \to {\mathbb{B}}$ is a *minor* of another function $g \colon {\mathbb{B}}^n \to {\mathbb{B}}$ if there exists a map $\sigma \colon {[{n}]} \to {[{m}]}$ such that $f(\mathbf{a}) = g(\mathbf{a} \sigma)$ for all $\mathbf{a} \in {\mathbb{B}}^m$; in this case we write $f \leq g$. Functions $f$ and $g$ are *equivalent,* denoted $f \equiv g$, if $f \leq g$ and $g \leq f$. In other words, $f$ is a minor of $g$ if $f$ can be obtained from $g$ by permutation of arguments, addition and deletion of inessential arguments and identification of arguments. Functions $f$ and $g$ are equivalent if each one can be obtained from the other by permutation of arguments and addition and deletion of inessential arguments.
The minor relation $\leq$ is a quasi-order (i.e., a reflexive and transitive relation) on the set of all Boolean functions, and the relation $\equiv$ is indeed an equivalence relation. For further background see, e.g., [@CL1; @CL2; @CP; @EFHH; @Pippenger] In what follows, we shall consider minors of the following special form. Let $n \geq 2$, and let $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$. For any two-element subset $I$ of ${[{n}]}$, we define the function $f_I \colon {\mathbb{B}}^{n-1} \to {\mathbb{B}}$ by the rule $f_I(\aa) = f(\aa \delta_I)$ for all $\aa \in {\mathbb{B}}^{n-1}$, where $\delta_I \colon {[{n}]} \to {[{n-1}]}$ is given by the rule $$\label{eq:deltaI}
\delta_I(i) =
\begin{cases}
i, & \text{if $i < \max I$,} \\
\min I, & \text{if $i = \max I$,} \\
i - 1, & \text{if $i > \max I$.}
\end{cases}$$ In other words, if $I = \{i, j\}$ with $i < j$, then $$f_I(a_1, \dots, a_{n-1}) = f(a_1, \dots, a_{j-1}, a_i, a_j, \dots, a_{n-1}).$$ Note that $a_i$ occurs twice on the right side of the above equality: both at the $i$-th and at the $j$-th position. The function $f_I$ will be referred to as an *identification minor* of $f$.
It was shown by Pippenger [@Pippenger] that the classes of functions closed under taking minors are characterizable by so-called relational constraints. We will briefly survey some results which we will use hereinafter. An $m$-ary *relational constraint* is a couple $(R, S)$ of $m$-ary relations $R$ (the *antecedent*) and $S$ (the *consequent*) on ${\mathbb{B}}$ (i.e., $R, S \subseteq {\mathbb{B}}^m$). We denote the antecedent and the consequent of a relational constraint $Q$ by $R(Q)$ and $S(Q)$, respectively. If both $R(Q)$ and $S(Q)$ equal the binary equality relation, then $Q$ is called the binary *equality constraint*. Furthermore, we refer to constraints with empty antecedent and empty consequent as *empty constraints*, and to constraints where the antecedent and consequent are the full relation ${\mathbb{B}}^m$, for some $m\geq 1$, as *full constraints*. The set of all relational constraints is denoted by $\Theta$.
A function $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$ *preserves* an $m$-ary relational constraint $(R, S)$, denoted $f \triangleright (R, S)$, if for every $\mathbf{a}^{1}, \dots, \mathbf{a}^{n} \in R$, we have $f(\mathbf{a}^1, \dots, \mathbf{a}^n) \in S$. (Regarding tuples $\mathbf{a}^i$ as unary maps, $f(\mathbf{a}^1, \dots, \mathbf{a}^n)$ denotes the $m$-tuple whose $i$-th entry is $f(\mathbf{a}^1, \dots, \mathbf{a}^n)(i) = f(a^1_i, \dots, a^n_i)$.)
The preservation relation gives rise to a Galois connection between functions and relational constraints that we now briefly describe; for further background, see [@C; @CP; @Pippenger]. Define $\operatorname{cPol}\colon \mathcal{P}(\Theta) \to \mathcal{P}(\Omega)$, $\operatorname{cInv}\colon \mathcal{P}(\Omega) \to \mathcal{P}(\Theta)$ by $$\begin{aligned}
\operatorname{cPol}(\mathcal{Q}) &= \{f \in \Omega : \text{$f \triangleright Q$ for every $Q \in \mathcal{Q}$}\}, \\
\operatorname{cInv}(\mathcal{F}) &= \{Q \in \Theta : \text{$f \triangleright Q$ for every $f \in \mathcal{F}$}\}.\end{aligned}$$ We say that a set $\mathcal{F}$ of functions is *characterized* by a set $\mathcal{Q}$ of relational constraints if $\mathcal{F} = \operatorname{cPol}(\mathcal{Q})$. Dually, $\mathcal{Q}$ is *characterized* by $\mathcal{F}$ if $\mathcal{Q} = \operatorname{cInv}(\mathcal{F})$. In other words, sets of functions characterizable by relational constraints are exactly the fixed points of $\operatorname{cPol}\circ \operatorname{cInv}$, and, dually, sets of relational constraints characterizable by functions are exactly the fixed points of $\operatorname{cInv}\circ \operatorname{cPol}$.
Preservation of a relational constraint generalizes the notion of preservation of a relation, as in the classical $\mathrm{Pol}$–$\mathrm{Inv}$ theory of clones and relations, which establishes that the clones on finite sets are exactly the classes of functions that are characterized by relations (see [@BKKR; @Geiger]). In this framework, a function $f$ preserves a relation $R$ if and only if $f$ preserves the relational constraint $(R, R)$. Hence, clones are exactly the classes that are characterized by relational constraints of the form $(R, R)$ for some relation $R$.
The following result reassembles various descriptions of the Galois closed sets of functions, which can be found in [@CP; @EFHH; @Pippenger].
\[thm:characterizablefunctions\] Let $\mathcal{F}$ be a set of functions. The following are equivalent.
1. \[thm:characterizablefunctions:item1\] $\mathcal{F}$ is closed under taking minors.
2. \[thm:characterizablefunctions:item2\] $\mathcal{F}$ is characterizable by relational constraints.
3. \[thm:characterizablefunctions:item3\] $\mathcal{F}$ is of the form $$\operatorname{forbid}(A) := \{f \in \Omega : \text{$g \nleq f$ for all $g \in A$}\}$$ for some antichain $A$ with respect to the minor relation $\leq$.
It follows from the equivalence of \[thm:characterizablefunctions:item1\] and \[thm:characterizablefunctions:item2\] in Theorem \[thm:characterizablefunctions\] that the union and the intersection of classes that are characterizable by relational constraints are characterizable by relational constraints.
Note that the antichain $A$ in item \[thm:characterizablefunctions:item3\] of Theorem \[thm:characterizablefunctions\] is unique up to equivalence. In fact, $A$ can be chosen among the minimal elements of $\Omega \setminus \mathcal{F}$; the elements of $A$ are called *minimal forbidden minors for $\mathcal{F}$*.
As we will see, there are classes of functions that, even though characterizable by relational constraints, are not characterized by any finite set of relational constraints. A set of functions is *finitely characterizable* if it is characterized by a finite set of relational constraints.
The following theorem is a refinement of Theorem \[thm:characterizablefunctions\] and provides a description for finitely characterizable classes.
\[thm:finitelycharacterizablefunctions\] Let $\mathcal{F}$ be a set of functions. The following are equivalent.
1. $\mathcal{F}$ is finitely characterizable.
2. $\mathcal{F}$ is of the form $\operatorname{forbid}(A)$ for some finite antichain $A$ with respect to the minor relation $\leq$.
The Galois closed sets of relational constraints were likewise described by Pippenger [@Pippenger]; this description was extended to arbitrary, possibly infinite, underlying sets in [@CF]. We shall briefly survey Pippenger’s description of the Galois closed sets of constraints.
An $m$-ary relational constraint $(R,S)$ is a *simple minor* of an $(m+p)$-ary relational constraint $(R',S')$ if there is $h \colon \{1, \ldots, n\} \to \{1, \ldots, m+p\}$ such that $$\begin{aligned}
R
\left(\begin{array}{c}
x_1\\
\vdots \\
x_m\\
\end{array}\right)
&
\quad \iff \quad
\exists x_{m+1} \ldots \exists x_{m+p} \quad
R'
\left(\begin{array}{c}
x_{h(1)}\\
\vdots \\
x_{h(n)}\\
\end{array}\right)
\intertext{and}
S
\left(\begin{array}{c}
x_1\\
\vdots \\
x_m\\
\end{array}\right)
&
\quad \iff \quad
\exists x_{m+1} \ldots \exists x_{m+p} \quad
S'
\left(\begin{array}{c}
x_{h(1)}\\
\vdots \\
x_{h(n)}\\
\end{array}\right).
\end{aligned}$$
Note that simple minors subsume the notions of permutation, diagonalization and projection of arguments; for background see [@CF; @Pippenger].
A constraint $(R,S)$ is obtained from a constraint $(R',S)$ by *restricting the antecedent* if $R\subseteq R'$. Likewise, $(R,S)$ is obtained from a constraint $(R,S')$ by *extending the consequent* if $S\supseteq S'$. A constraint $(R,S\cap S')$ is said to be obtained from $(R,S)$ and $(R, S')$ by intersecting consequents.
A set $\mathcal{Q}$ of relational constraints is said to be *minor-closed* if it contains the binary equality constraint, the unary empty constraint, and it is closed under taking simple minors, restricting antecedents, and extending and intersecting consequents.
We can now state Pippenger’s [@Pippenger] description of the Galois closed sets of relational constraints.
\[thm:characterizableconstraints\] Let $\mathcal{Q}$ be a set of relational constraints. The following are equivalent.
1. \[thm:characterizableconstraints:item1\] $\mathcal{Q}$ is characterizable by some set of functions.
2. \[thm:characterizableconstraints:item2\] $\mathcal{Q}$ is minor-closed.
The following lemma provides a noteworthy tool for showing that certain classes of threshold functions are not finitely characterizable.
\[lemma:finiteDescendingChainIffFinitelyCharacterizable\] Let $C$ and $C_i$ for all $i \geq 1$ be classes of functions that are closed under taking minors, such that $C = \bigcap_{i \geq 1} C_i$, and $C_{i+1} \subseteq C_i$ for all $i \geq 1$. If $C$ is finitely characterizable by constraints, then there exists $\ell \in {\mathbb{N}}$ such that $C_j = C_\ell$ for all $j \geq \ell$.
By Theorem \[thm:characterizablefunctions\], each minor-closed class $C_i$ is characterized by some set $\mathcal{Q}_i$ of relational constraints, i.e., $C_i = \operatorname{cPol}\mathcal{Q}_i$ for all $i \geq 1$.
Assume that $C$ is finitely characterizable. Then there is some finite set $\mathcal{P}$ of constraints with $C = \operatorname{cPol}\mathcal{P}$, and thus $\operatorname{cInv}C = \operatorname{cInv}\operatorname{cPol}\mathcal{P}$. Since $$C = \bigcap_{i \geq 1} \operatorname{cPol}\mathcal{Q}_i = \operatorname{cPol}\bigcup_{i \geq 1} \mathcal{Q}_i$$ we can construct each $P \in \mathcal{P}$ from the constraints in $\bigcup_{i \geq 1} \mathcal{Q}_i$. Since the constraints are finite, all such constructions are finite. In particular, only a finite number of constraints from $\bigcup_{i \geq 1} \mathcal{Q}_i$ are used for each $P \in \mathcal{P}$. Since $\mathcal{P}$ is finite, this implies that only a finite number of constraints are needed to construct all $P \in \mathcal{P}$. Therefore $$C = \operatorname{cPol}\mathcal{P} \supseteq \operatorname{cPol}\bigcup_{i = 1}^l \mathcal{Q}_i
= \bigcap_{i=1}^l \operatorname{cPol}\mathcal{Q}_i
= \operatorname{cPol}\mathcal{Q}_l = C_l$$ holds for some $l \in {\mathbb{N}}$. Now this implies that for any $j \geq l$, $$C \subseteq C_j \subseteq C_l \subseteq C$$ and consequently $C = C_j = C_l$ for all $j \geq l$.
Main results: classification and characterizations of Galois closed sets of threshold functions {#sec:main}
===============================================================================================
Motivation
----------
A *threshold function* is a Boolean function $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$ such that there exist *weights* $w_1, \dots, w_n \in {\mathbb{R}}$ and a *threshold* $t \in {\mathbb{R}}$ fulfilling $$f(x_1, \dots, x_n) = 1 \iff \sum_{i=1}^n w_i x_i \geq t.$$ Another, equivalent, definition is the following. An $n$-ary Boolean function $f$ is called a *threshold function* if there is a hyperplane in ${\mathbb{R}}^n$ strictly separating the true points of $f$ from the false points of $f$, considered as elements of ${\mathbb{R}}^n$. The set of all threshold functions is denoted by ${T}$.
The class of threshold functions has remarkable invariance properties. For instance, it is closed under taking negations and duals (see Lemma \[lem:dualBl\]). Moreover, the class of threshold functions is also closed under taking minors of its members; hence it is characterizable by relational constraints by Theorem \[thm:characterizablefunctions\]. However, no finite set of relational constraints suffices.
\[thm:Hell\] The class of threshold functions is not finitely characterizable.
Imposing some additional conditions on threshold functions, we may obtain proper subclasses of ${T}$ that are finitely characterizable. Easy examples arise from the intersections of ${T}$ with the clones $L$, $\Lambda$, $V$ (see Appendix \[App:Post\]). However, as we have seen, other intersections $C \cap T$ may fail to be finitely characterizable, e.g., for $C = \Omega$.
This fact gives rise to the following problem.
Which clones $C$ of Boolean functions have the property that $C \cap {T}$ is finitely characterizable?
In the following subsection we present a solution to this problem.
Classification and characterizations of intersections of the class of threshold functions with clones
-----------------------------------------------------------------------------------------------------
We start by observing that $$L \cap {T}= \Omega(1), \qquad
\Lambda \subseteq {T}, \qquad
V \subseteq {T},$$ from which it follows that the intersection $C \cap {T}$ is a clone for any clone $C$ contained in one of $L$, $V$ and $\Lambda$. Hence, the characterization of $C \cap {T}$ for any such clone $C$ is given by the relational constraint $(R, R)$, where $R$ is the relation characterizing $C \cap {T}$ given in Appendix \[App:Post\].
We proceed to characterizing the intersections $C \cap {T}$ for the remaining clones $C$; as we will see, none of these is finitely characterizable. A characterization of the class ${T}$ of all threshold functions (i.e., for $C = \Omega$) is easily obtained with the help of the notion of asummability.
For $k \geq 2$, a Boolean function $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$ is *$k$-asummable* if for any $m \in \{2, \dots, k\}$ and for all $\aa_1, \dots, \aa_m \in f^{-1}(0)$ and $\bb_1, \dots, \bb_m \in f^{-1}(1)$, it holds that $$\aa_1 + \dots + \aa_m \neq \bb_1 + \dots + \bb_m.$$ (Addition here is standard vector addition in ${\mathbb{R}}^n$.) A function is *asummable* if it is $k$-asummable for all $k \geq 2$. It is well known that asummability characterizes threshold functions; see [@Chow; @Elgot; @Muroga].
\[thm:thresholdasummable\] A Boolean function is threshold if and only if it is asummable.
Define for $n \geq 1$, the $2n$-ary relational constraint $B_n$ as $$\begin{aligned}
R(B_n) & := \{ (x_1, \dots, x_{2n}) \in {\mathbb{B}}^{2n} : \sum_{i=1}^n x_i = \sum_{i=n+1}^{2n} x_i \} \\
S(B_n) & := {\mathbb{B}}^{2n} \setminus \{ (\underbrace{0,\dots,0}_{n},\underbrace{1,\dots,1}_n),
(\underbrace{1,\dots,1}_{n},\underbrace{0,\dots,0}_n) \}. \end{aligned}$$ Note that in the definition of $R(B_n)$ we employ the usual addition of real numbers. Denoting by $w(\aa)$ the *Hamming weight* of a tuple $\aa \in {\mathbb{B}}^n$ (i.e., the number of nonzero entries in $\aa$), we can equivalently define $R(B_n)$ as $\{(x_1, \dots, x_{2n}) \in {\mathbb{B}}^{2n} : w(x_1, \dots, x_n) = w(x_{n+1}, \dots, x_{2n})\}$.
\[lem:PolBl\] Let $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$ and $\ell \geq 2$. Then $\aa_1 + \dots + \aa_\ell \neq \bb_1 + \dots + \bb_\ell$ for all $\aa_1, \dots, \aa_\ell \in f^{-1}(0)$ and $\bb_1, \dots, \bb_\ell \in f^{-1}(1)$ if and only if $f$ preserves $B_\ell$.
Assume first that $f$ does not preserve $B_\ell$. Then there exists a matrix $$M = \begin{pmatrix}
m_1^1 & m_2^1 & \dots & m_n^1 \\
m_1^2 & m_2^2 & \dots & m_n^2 \\
\vdots & \vdots & & \vdots \\
m_1^{2 \ell} & m_2^{2 \ell} & \dots & m_n^{2 \ell}
\end{pmatrix} = \begin{pmatrix}
M^1 \\
M^2 \\
\vdots \\
M^{2 \ell}
\end{pmatrix}
= (M_1, M_2, \dots, M_n),$$ i.e., $M^1, \dots, M^{2 \ell} \in {\mathbb{B}}^n$ are the rows of $M$, and $M_1, \dots, M_n \in {\mathbb{B}}^{2 \ell}$ are the columns of $M$, such that
- $M_1, \dots, M_n \in R(B_\ell)$, and
- $\zz := g(M_1, \dots, M_n) := \begin{pmatrix} g(M^1) \\ \vdots \\ g(M^{2 \ell}) \end{pmatrix} \notin S(B_\ell)$.
Thus $\zz \in \{ (\underbrace{0,\dots,0}_l,\underbrace{1,\dots,1}_l),
(\underbrace{1,\dots,1}_l,\underbrace{0,\dots,0}_l)\}$. As $B_\ell$ is invariant under swapping the first $\ell$ rows with the last $\ell$ rows, we can assume that $\zz = (\underbrace{0, \dots, 0}_\ell, \underbrace{1, \dots, 1}_\ell)$. Then $M^1, \dots, M^\ell \in f^{-1}(0)$ and $M^{\ell + 1}, \dots, M^{2 \ell} \in f^{-1}(1)$, and $M^1 + \dots + M^\ell = M^{\ell + 1} + \dots + M^{2 \ell}$ by the definition of $B_\ell$.
Assume then that there exist $\aa_1, \dots, \aa_\ell \in f^{-1}(0)$ and $\bb_1, \dots, \bb_\ell \in f^{-1}(1)$ such that $\aa_1 + \dots + \aa_\ell = \bb_1 + \dots + \bb_\ell$. Let $M$ be the $2 \ell \times n$ matrix whose rows are $\aa_1, \dots, \aa_\ell, \bb_1, \dots, \bb_\ell$. The columns of $M$ are tuples in $R(B_\ell)$, but $f(M) = (\underbrace{0, \dots, 0}_\ell, \underbrace{1, \dots, 1}_\ell) \notin S(B_\ell)$. We conclude that $f$ does not preserve $B_\ell$.
Now it is easy to define a set of relational constraints that characterizes $k$-asummable functions. For $k \geq 2$, let $\mathcal{A}_k := \{B_n : 2 \leq n \leq k\}$.
\[lem:k-asummable\] Let $k \geq 2$. A Boolean function $f$ is $k$-asummable if and only if $f \in \operatorname{cPol}(\mathcal{A}_k)$.
Follows immediately from the definition of $k$-asummability and Lemma \[lem:PolBl\].
Let $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$. The following are equivalent.
1. \[cor:thr:item:thr\] $f$ is a threshold function.
2. \[cor:thr:item:Ak\] $f \in \bigcap_{k \geq 2} \operatorname{cPol}(\mathcal{A}_k)$.
3. \[cor:thr:item:Bn\] $f \in \operatorname{cPol}(\{B_n : n \geq 2\})$.
The equivalence of \[cor:thr:item:thr\] and \[cor:thr:item:Ak\] follows immediately from Theorem \[thm:thresholdasummable\] and Lemma \[lem:k-asummable\]. Conditions \[cor:thr:item:Ak\] and \[cor:thr:item:Bn\] are equivalent, because $$\bigcap_{k \geq 2} \operatorname{cPol}(\mathcal{A}_k)
= \operatorname{cPol}\Bigl( \bigcup_{k \geq 2} \mathcal{A}_k \Bigr)
= \operatorname{cPol}\Bigl( \bigcup_{k \geq 2} \{B_n : 2 \leq n \leq k\} \Bigr)
= \operatorname{cPol}(\{B_n : n \geq 2\}).
\qedhere$$
Since $\mathcal{A}_k \subseteq \mathcal{A}_k \cup \{B_{k+1}\} = \mathcal{A}_{k+1}$, it is clear that $\operatorname{cPol}(\mathcal{A}_{k+1}) \subseteq \operatorname{cPol}(\mathcal{A}_k)$ for all $k \geq 2$. Taylor and Zwicker have shown in [@Taylor] that for every $k \geq 2$, there exist $k$-asummable functions that are not $(k+1)$-asummable. Hence these inclusions are strict for every $k$.
\[theorem:strictAkInclusions\] For all $k \geq 2$, $\operatorname{cPol}(\mathcal{A}_{k+1}) \subset \operatorname{cPol}(\mathcal{A}_k)$.
\[thm:characterization\] The set $\operatorname{cPol}(\{ B_n : n \geq 2 \})$ is the class of all threshold functions. Moreover, for every clone $C$, the subclass $C \cap {T}$ of threshold functions is characterized by the set $\{B_n : n \geq 2\} \cup \mathcal{Q}_C$, where $\mathcal{Q}_C$ is the set of relational constraints characterizing the clone $C$, as given in Appendix \[App:Post\].
From Theorems \[theorem:strictAkInclusions\] and \[thm:characterization\] it follows that $${T}= \bigcap_{k \geq 2} \operatorname{cPol}(\mathcal{A}_k) \subset \dots \subset \operatorname{cPol}(\mathcal{A}_{\ell+1}) \subset \operatorname{cPol}(\mathcal{A}_\ell) \subset \dots \subset \operatorname{cPol}(\mathcal{A}_2)$$ holds for all $\ell \geq 3$, i.e., the sets $\operatorname{cPol}(\mathcal{A}_k)$ with $k \geq 2$ form an infinite descending chain, whose intersection is the set ${T}$ of all threshold functions.
Theorem \[thm:characterization\] provides an infinite set of relational constraints characterizing the set $C \cap {T}$ for each clone $C$. As Theorem \[thm:classification\] will reveal, the characterization provided is optimal for the clones not contained in $L$, $V$ or $\Lambda$ in the sense that for such clones $C$, the set $C \cap {T}$ is not finitely characterizable by relational constraints.
In order to proceed, we need the following lemma. Its proof is somewhat technical and is deferred to Section \[sec:constructions\].
\[lem:GCf\]
Let $f$ be a Boolean function, and let $C \in \{SM, M_cU_\infty, M_cW_\infty\}$. There exists a Boolean function $G_C(f)$ that satisfies the following conditions:
1. $G_C(f) \in C$,
2. for all $n \geq 2$, $f \in \operatorname{cPol}B_n$ if and only if $G_C(f) \in \operatorname{cPol}B_n$.
This brings together Corollaries \[cor:GSM\], \[cor:GMcUinfty\] and \[cor:GMcWinfty\], which will be proved in Section \[sec:constructions\].
\[rem:OldTheorem-theorem:strictCcapAkInclusions\] Lemma \[lem:GCf\] gives rise to a noteworthy refinement of Theorem \[theorem:strictAkInclusions\]. Indeed, by Theorem \[theorem:strictAkInclusions\], there is some $f \in \operatorname{cPol}(\mathcal{A}_k) \setminus \operatorname{cPol}(\mathcal{A}_{k+1})$ and, by Lemma \[lem:GCf\], there exists a function $G_E(f) \in E \subseteq C$ satisfying $G_{E}(f) \in \operatorname{cPol}(\mathcal{A}_k) \setminus \operatorname{cPol}(\mathcal{A}_{k+1})$. This implies that $G_{E}(f) \in (C \cap \operatorname{cPol}(\mathcal{A}_k)) \setminus (C \cap \operatorname{cPol}(\mathcal{A}_{k+1}))$ and thus $C \cap \operatorname{cPol}(\mathcal{A}_{k+1}) \subset C \cap \operatorname{cPol}(\mathcal{A}_k)$ for all $k \geq 2$.
This shows that if $C$ is a clone of Boolean functions satisfying $E \subseteq C$ for some $E \in \{ SM, M_cU_\infty, M_cW_\infty \}$, then $C \cap \operatorname{cPol}(\mathcal{A}_{k+1}) \subset C \cap \operatorname{cPol}(\mathcal{A}_k)$ for all $k \geq 2$.
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/in [ T0H/T0e, T0HT/T0eT, T0HM/T0eM, T0HTM/T0eTM, T1H/T1e, T1HT/T1eT, T1HM/T1eM, T1HTM/T1eTM]{} () – ();
plot coordinates[ ($(AT0) +(-1.5,-0.2)$) ($ (L) +( 0.1, 0.4)$) ($(VT1) +( 1.5,-0.2)$) ]{};
\[thm:classification\] Let $C$ be a clone of Boolean functions. The subclass $C \cap {T}$ of threshold functions is finitely characterizable if and only if $C$ is contained in one of the clones $L$, $V$, $\Lambda$.
This theorem is illustrated by Figure \[figure:PostsLattice\].
We have already observed that $C \cap {T}$ is finitely characterizable for every subclone $C$ of $L$, $V$ or $\Lambda$.
Now we consider all the other clones. Let $C$ be a clone such that $C \not\subseteq D$ for all $D \in \{L, V, \Lambda\}$. We can read off of Post’s lattice (see Figure \[figure:PostsLattice\]) that there is some $E \in \{SM, M_cU_\infty, M_cW_\infty\}$ such that $E \subseteq C$. It follows from Theorem \[theorem:strictAkInclusions\] and Lemma \[lem:GCf\] that for every $k \geq 2$, there exists a function $f_k \in E$ such that $f_k \in \operatorname{cPol}B_\ell$ whenever $2 \leq \ell \leq k$ and $f_k \notin \operatorname{cPol}B_{k+1}$. Note that $f_k \notin C \cap {T}$.
Suppose, on the contrary that $C \cap {T}$ is finitely characterizable. By Theorem \[thm:finitelycharacterizablefunctions\], $C \cap {T}$ is of the form $\operatorname{forbid}(A)$ for some finite antichain $A$ of minimal forbidden minors. Each one of the functions $f_k$ has a minor in $A$. Since $A$ is finite, there is an element $g \in A$ and an infinite set $S \subseteq {\mathbb{N}}$ such that $g \leq f_k$ for all $k \in S$. The function $g$ is not threshold, so there exists $p \in {\mathbb{N}}$ such that $p \geq 2$ and $g \notin \operatorname{cPol}B_p$. Being infinite, the set $S$ contains an element $q$ with $p \leq q$. Then we have $g \leq f_q$ and $f_q \in \operatorname{cPol}B_p$. We also have $g \in \operatorname{cPol}B_p$, because $\operatorname{cPol}B_p$ is closed under taking minors. This yields the desired contradiction.
Alternatively, Theorem \[thm:classification\] can be proved using Lemma \[lemma:finiteDescendingChainIffFinitelyCharacterizable\] and Remark \[rem:OldTheorem-theorem:strictCcapAkInclusions\].
As before if $C$ is a subclone of $L$, $V$ or $\Lambda$, then $C \cap {T}$ is finitely characterizable. As for any other clone $C$, we know (once again reading off of Post’s lattice) that there is some $E \in \{SM, M_cU_\infty, M_cW_\infty\}$ such that $E \subseteq C$. By Remark \[rem:OldTheorem-theorem:strictCcapAkInclusions\], we have $C \cap \operatorname{cPol}(\mathcal{A}_{k+1}) \subset C \cap \operatorname{cPol}(\mathcal{A}_k)$ for all $k \geq 2$. Furthermore, $$C \cap {T}= C \cap \bigcap_{n \geq 2} \operatorname{cPol}(\mathcal{A}_k)
= \bigcap_{n \geq 2} (C \cap \operatorname{cPol}(\mathcal{A}_k)),$$ i.e., we have an infinite descending chain the intersection of which equals $C \cap {T}$. By Lemma \[lemma:finiteDescendingChainIffFinitelyCharacterizable\] we thus conclude that $C \cap {T}$ is not finitely characterizable.
Constructions {#sec:constructions}
=============
In order to prove Lemma \[lem:GCf\], we will construct from a given Boolean function $f$, for each $C \in \{S, M_c, SM, U_\infty, M_cU_\infty, M_cW_\infty \}$, a Boolean function $G_C(f)$ that satisfies the following conditions:
1. $G_C(f) \in C$,
2. for all $\ell \geq 2$, $f \in \operatorname{cPol}B_\ell$ if and only if $G_C(f) \in \operatorname{cPol}B_\ell$.
We do this step by step. We first construct functions $G_S(f)$ and $G_{M_c}(f)$ with the desired properties. Using these two constructions as building blocks, we can construct $G_{SM}$ as $G_{M_c}(G_S(f))$. Then we construct $G_{U_\infty}(f)$, and, building upon this, we finally get $G_{M_cU_\infty}(f) := G_{U_\infty}(G_{M_c}(f))$ and $G_{M_cW_\infty}(f) := {{(G_{M_cU_\infty}(f))}^{\mathrm{d}}}$.
Construction of $G_{S}(f)$ {#subsec:ST}
--------------------------
Let $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$. Then we define $G_{S}(f) \colon {\mathbb{B}}^{n+1} \to {\mathbb{B}}$ by $$G_{S}(f)(x_1, \dots, x_{n+1}) = (x_{n+1} \wedge f(x_1, \dots, x_n)) \vee ({\overline{x}}_{n+1} \wedge {{f}^{\mathrm{d}}}(x_1, \dots, x_n)).$$
\[lemma:anytoS\] For any $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$, the function $G_{S}(f)$ is self-dual.
Let $g := G_{S}(f)$. Then $$\begin{aligned}
{{g}^{\mathrm{d}}}(\mathbf{x}, x_{n+1}) &=
{\overline{({\overline{x}}_{n+1} \wedge f({\overline{\mathbf{x}}})) \vee ({\overline{{\overline{x}}}}_{n+1} \wedge {{f}^{\mathrm{d}}}({\overline{\mathbf{x}}}))}} \\
&= (x_{n+1} \vee {{f}^{\mathrm{d}}}(\mathbf{x})) \wedge ({\overline{x}}_{n+1} \vee f(\mathbf{x})) \\
&= (x_{n+1} \wedge {\overline{x}}_{n+1}) \vee (x_{n+1} \wedge f(\mathbf{x})) \vee ({{f}^{\mathrm{d}}}(\mathbf{x}) \wedge {\overline{x}}_{n+1}) \vee ({{f}^{\mathrm{d}}}(\mathbf{x}) \wedge f(\mathbf{x})) \\
&= (x_{n+1} \wedge f(\mathbf{x})) \vee ({{f}^{\mathrm{d}}}(\mathbf{x}) \wedge {\overline{x}}_{n+1}) \\
&= g(\mathbf{x}, x_{n+1}),
\end{aligned}$$ where the second last equality holds since $${{f}^{\mathrm{d}}}(\mathbf{x}) \wedge f(\mathbf{x}) \leq (x_{n+1} \wedge f(\mathbf{x})) \vee ({{f}^{\mathrm{d}}}(\mathbf{x}) \wedge {\overline{x}}_{n+1})$$ for every $x_{n+1}$.
\[lemma:ST:fNotinBlimpliesgNotinBl\] Let $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$. If $f \notin \operatorname{cPol}B_\ell$ for some $\ell \geq 2$, then $G_{S}(f) \notin \operatorname{cPol}B_\ell$.
Assume that $f \notin \operatorname{cPol}B_\ell$, and let $g := G_{S}(f)$. Then there are $\yy_1, \dots, \yy_n \in R(B_\ell)$ with $f(\yy_1, \dots, \yy_n) \notin S(B_\ell)$. Since $g(x_1, \dots, x_n, 1) = f(x_1, \dots, x_n)$, we have $$g(\yy_1,\dots,\yy_n,\11) = f(\yy_1,\dots,\yy_n) \notin S(B_\ell).$$ Since also $\11 \in R(B_\ell)$, we conclude that $g \notin \operatorname{cPol}B_\ell$.
\[lemma:ST:fInBlImpliesgInBl\] Let $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$. If $f \in \operatorname{cPol}B_\ell$ for some $\ell \geq 2$, then $G_{S}(f) \in \operatorname{cPol}B_\ell$.
Let $g := G_{S}(f)$. Suppose, on the contrary, that $g \notin \operatorname{cPol}B_\ell$. Then there is some matrix $M$ given by $$M = \begin{pmatrix}
m_1^1 & m_2^1 & \dots & m_{n+1}^1 \\
m_1^2 & m_2^2 & \dots & m_{n+1}^2 \\
\vdots & \vdots & & \vdots \\
m_1^{2\ell} & m_2^{2\ell} & \dots & m_{n+1}^{2\ell}
\end{pmatrix} = \begin{pmatrix}
M^1 \\
M^2 \\
\vdots \\
M^{2\ell}
\end{pmatrix}
= (M_1,M_2,\dots,M_{n+1}),$$ i.e., $M^1,\dots,M^{2\ell} \in {\mathbb{B}}^{n+1}$ are the rows of $M$, and $M_1,\dots,M_{n+1} \in {\mathbb{B}}^{2\ell}$ are the columns of $M$, such that
- $M_1,\dots,M_{n+1} \in R(B_\ell)$, and
- $\zz := g(M_1,\dots,M_{n+1}) := \begin{pmatrix} g(M^1) \\ \vdots \\ g(M^{2\ell}) \end{pmatrix}
\notin S(B_\ell)$.
Thus $\zz \in \{ (\underbrace{0,\dots,0}_\ell,\underbrace{1,\dots,1}_\ell),
(\underbrace{1,\dots,1}_\ell, \underbrace{0,\dots,0}_\ell)\}$. As $B_\ell$ is invariant under swapping the first $\ell$ coordinates with the last $\ell$ coordinates, we can assume that $\zz = (\underbrace{0,\dots,0}_\ell, \underbrace{1,\dots,1}_\ell)$.
We now look at the last column $M_{n+1}$ of $M$. Since $\sum_{i=1}^\ell m_{n+1}^i = \sum_{i=\ell+1}^{2\ell} m_{n+1}^i$, and since $B_\ell$ is totally symmetric on the first $\ell$ rows and on the last $\ell$ rows, we can assume that $$M_{n+1} = (\underbrace{0,\dots,0}_\alpha, \underbrace{1,\dots,1}_\beta,
\underbrace{0,\dots,0}_\alpha, \underbrace{1,\dots,1}_\beta)$$ holds for some $\alpha,\beta \geq 0$ with $\alpha+\beta = \ell$.
We will now construct a matrix $K$ with $$K = \begin{pmatrix}
k_1^1 & k_2^1 & \dots & k_n^1 \\
k_1^2 & k_2^2 & \dots & k_n^2 \\
\vdots & \vdots & & \vdots \\
k_1^{2\ell} & k_2^{2\ell} & \dots & k_n^{2\ell}
\end{pmatrix} = \begin{pmatrix}
K^1 \\
K^2 \\
\vdots \\
K^{2\ell}
\end{pmatrix}
= (K_1,K_2,\dots,K_n),$$ that satisfies $K_1, \dots, K_n \in R(B_\ell)$ and $f(K_1, \dots, K_n) \notin S(B_\ell)$. This will yield the desired contradiction since we started with the assumption that $f \in \operatorname{cPol}B_\ell$.
We define $k_j^i$ for $1 \leq i \leq 2\ell$ and $1 \leq j \leq n$ by $$\def\arraystretch{1.2}
k_j^i = \left\{\begin{array}{llr@{{} \leq i \leq {}}l}
\overline{m}_j^{i+\ell} & \text{if } & 1 & \alpha \\
m_j^i & \text{if } & \alpha+1 & \ell \\
\overline{m}_j^{i-\ell} & \text{if } & \ell+1 & \ell+\alpha \\
m_j^i & \text{if } & \ell+\alpha+1 & 2\ell
\end{array}\right.$$ In other words, matrix $K$ is obtained from $M$ by omitting the last column, negating rows $1, \dots, \alpha$ and $\ell + 1, \dots, \ell + \alpha$, and then swapping rows $1, \dots, \alpha$ with rows $\ell + 1, \dots, \ell + \alpha$.
We need to show that $K_j \in R(B_\ell)$ for all $j \in [n]$. Let $j \in [n]$ be arbitrary, and let $$a := \sum_{i=1}^\alpha m_j^i, \qquad
b := \sum_{i=\alpha+1}^\ell m_j^i, \qquad
c := \sum_{i=\ell+1}^{\ell+\alpha} m_j^i, \qquad
d := \sum_{i=\ell+\alpha+1}^{2\ell} m_j^i.$$ Since $M_j \in R(B_\ell)$ we have $$\label{equation:ST:abEqualscd}
a+b = \sum_{i=1}^\ell m_j^i = \sum_{i=\ell+1}^{2\ell} m_j^i = c + d.$$ For $K_j$ we find the following: $$\begin{aligned}
\sum_{i=1}^\alpha k_j^i &
= \sum_{i=1}^\alpha \overline{m}_j^{i+\ell}
= \sum_{i=1}^\alpha (1-m_j^{i+\ell})
= \alpha - \sum_{i=\ell+1}^{\ell+\alpha} m_j^i
= \alpha - c, \displaybreak[0]\\
\sum_{i=\alpha+1}^\ell k_j^i & = \sum_{i=\alpha+1}^\ell m_j^i = b, \displaybreak[0]\\
\sum_{i=\ell+1}^{\ell+\alpha} k_j^i &
= \sum_{i=\ell+1}^{\ell+\alpha} \overline{m}_j^{i-\ell}
= \sum_{i=\ell+1}^{\ell+\alpha} (1-m_j^{i-\ell})
= \alpha - \sum_{i=1}^{\alpha} m_j^i
= \alpha - a, \displaybreak[0]\\
\sum_{i=\ell+\alpha+1}^{2\ell} k_j^i & = \sum_{i=\ell+\alpha+1}^{2\ell} m_j^i = d.
\end{aligned}$$ From this it follows that $$\sum_{i=1}^\ell k_j^i = \alpha - c + b \overset{\eqref{equation:ST:abEqualscd}}{=}
\alpha - a + d = \sum_{i=\ell+1}^{2\ell} k_j^i,$$ and thus $K_j \in R(B_\ell)$ for all $j \in [n]$.
We now show that $f(K^i) = 0$ if $1 \leq i \leq \ell$, and $f(K^i) = 1$ if $\ell+1 \leq i \leq 2\ell$. We need to consider four different cases for $i$:
- $1 \leq i \leq \alpha$. Then $(\overline{K}^i,0) = M^{i+\ell}$, and $$\overline{f(K^i)} = {{f}^{\mathrm{d}}}(\overline{K}^i) = g(\overline{K}^i, 0) = g(M^{i+\ell}) = 1.$$ Hence $f(K^i) = 0$.
- $\alpha+1 \leq i \leq \ell$. Then $(K^i,1) = M^i$, and $$f(K^i) = g(K^i,1) = g(M^i) = 0.$$
- $\ell+1 \leq i \leq \ell+\alpha$. Then $(\overline{K}^i,0) = M^{i-\ell}$, and $$\overline{f(K^i)} = {{f}^{\mathrm{d}}}(\overline{K}^i) = g(\overline{K}^i, 0) = g(M^{i-\ell}) = 0.$$ Hence $f(K^i) = 1$.
- $\ell+\alpha+1 \leq i \leq 2\ell$. Then $(K^i,1) = M^i$, and $$f(K^i) = g(K^i,1) = g(M^i) = 1.$$
Thus we have $$f(K_1,\dots,K_n) = \begin{pmatrix} f(K^1) \\ \vdots \\ f(K^\ell) \\ f(K^{\ell+1}) \\ \vdots \\ f(K^{2\ell}) \end{pmatrix}
= \begin{pmatrix} 0 \\ \vdots \\ 0 \\ 1 \\ \vdots \\1 \end{pmatrix} \notin S(B_\ell),$$ in contradiction to $f \in \operatorname{cPol}B_\ell$. We conclude that $g \in \operatorname{cPol}B_\ell$.
\[cor:GS\] For any Boolean function $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$, $G_S(f) \in S$ and for all $\ell \geq 2$, $f \in \operatorname{cPol}B_\ell$ if and only if $G_S(f) \in \operatorname{cPol}B_\ell$.
This brings together Lemmas \[lemma:anytoS\], \[lemma:ST:fNotinBlimpliesgNotinBl\] and \[lemma:ST:fInBlImpliesgInBl\].
Construction of $G_{M_c}(f)$ and $G_{SM}(f)$ {#subsec:SMT}
--------------------------------------------
Let $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$. We define the Boolean function $G_{M_c}(f) \colon {\mathbb{B}}^{2n} \to {\mathbb{B}}$ by the following rules
- If $w(\xx) < n$, then $G_{M_c}(f)(\xx) := 0$.
- If $w(\xx) > n$, then $G_{M_c}(f)(\xx) := 1$.
- If $\xx = (\aa, \overline{\aa})$ for some $\aa \in {\mathbb{B}}^n$, then $G_{M_c}(f)(\xx) := f(\aa)$.
- If $w(\xx) = n$ and there exists $i \in [n]$ such that $x_i = x_{n+i}$ and $x_j \neq x_{n+j}$ for all $j < i$, then $G_{M_c}(f)(\xx) := x_i$.
It is easy to verify that the function $G_{M_c}(f)$ is defined on every tuple $\xx \in {\mathbb{B}}^{2n}$.
\[lemma:StoSM\] Let $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$.
1. \[lemma:StoSM:item1\] $G_{M_c}(f) \in M_c$, i.e., $G_{M_c}(f)$ is monotone and constant-preserving.
2. \[lemma:StoSM:item2\] If $f$ is self-dual, then $G_{M_c}(f)$ is self-dual.
Let $g := G_{M_c}(f)$.
Let $\xx,\yy \in {\mathbb{B}}^{2n}$ with $\xx < \yy$. Then $w(\xx) < w(\yy)$ and one of the following cases applies: $w(\xx) < n$ or $w(\yy) > n$. In the former case, we have $g(\xx) = 0 \leq g(\yy)$; in the latter case, we have $g(\xx) \leq 1 = g(\yy)$. We conclude that $g$ is monotone. Since $w(\00) = 0 < n$ and $w(\11) = 1 > n$, it holds that $f(\00) = 0$ and $f(\11) = 1$, i.e., $f$ preserves both constants.
Assume that $f$ is self-dual. Let $\xx \in {\mathbb{B}}^{2n}$.
If $w(\xx) > n$ then $w(\overline{\xx}) < n$, and thus $(g(\xx),g(\overline{\xx})) = (1,0)$. Similarly, if $w(\xx) < n$ then $w(\overline{\xx}) > n$, and thus $(g(\xx),g(\overline{\xx})) = (0,1)$.
If $\xx = (\aa, \overline{\aa})$ for some $\aa \in {\mathbb{B}}^n$, then $(g(\xx), g(\overline{\xx})) = (f(\aa), f(\overline{\aa})) \in \{ (0,1),(1,0) \}$ since $f$ is self-dual.
Otherwise, there is some $i \in [m]$ with $x_i = x_{m+i}$ and $x_j \neq x_{m+j}$ for all $j < i$. This holds also for the negation of $\xx$, and thus $(g(\xx), g(\overline{\xx})) \in \{ (0,1),(1,0) \}$.
We conclude that $g$ is self-dual.
\[lemma:SM:fnotinBlimpliesgnotinBl\] Let $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$. If $f \notin \operatorname{cPol}B_\ell$ for some $\ell \geq 2$, then $G_{M_c}(f) \notin \operatorname{cPol}B_\ell$.
Let $f \notin \operatorname{cPol}B_\ell$ and $g := G_{M_c}(f)$. Then there are $\yy_1,\dots,\yy_n \in R(B_\ell)$ with $f(\yy_1,\dots,\yy_n) \notin S(B_\ell)$. Also $\overline{\yy}_1,\dots,\overline{\yy}_n \in R(B_\ell)$ and thus $$g(\yy_1,\dots,\yy_n,\overline{\yy}_1,\dots,\overline{\yy}_n) = f(\yy_1,\dots,\yy_n) \notin S(B_\ell).$$ Therefore $g \notin \operatorname{cPol}B_\ell$.
\[lemma:SM:finBlimpliesginBl\] Let $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$ with $f \in \operatorname{cPol}B_\ell$ for some $\ell \geq 2$. Then $G_{M_c}(f) \in \operatorname{cPol}B_\ell$.
Let $g := G_{M_c}(f)$. Suppose, on the contrary, that $g \notin \operatorname{cPol}B_\ell$. Then there is some matrix $M$ given by $$M = \begin{pmatrix}
m_1^1 & m_2^1 & \dots & m_{2n}^1 \\
m_1^2 & m_2^2 & \dots & m_{2n}^2 \\
\vdots & \vdots & & \vdots \\
m_1^{2\ell} & m_2^{2\ell} & \dots & m_{2n}^{2\ell}
\end{pmatrix} = \begin{pmatrix}
M^1 \\
M^2 \\
\vdots \\
M^{2\ell}
\end{pmatrix}
= (M_1,M_2,\dots,M_{2n}),$$ i.e., $M^1,\dots,M^{2\ell} \in {\mathbb{B}}^{2n}$ are the rows of $M$, and $M_1,\dots,M_{2n} \in {\mathbb{B}}^{2\ell}$ are the columns of $M$, such that
- $M_1,\dots,M_{2n} \in R(B_\ell)$, and
- $\zz := g(M_1,\dots,M_{2n}) := \begin{pmatrix} g(M^1) \\ \vdots \\ g(M^{2\ell}) \end{pmatrix}
\notin S(B_\ell)$.
Thus $\zz \in \{ (\underbrace{0,\dots,0}_\ell,\underbrace{1,\dots,1}_\ell),
(\underbrace{1,\dots,1}_\ell,\underbrace{0,\dots,0}_\ell)\}$. As $B_\ell$ is invariant under swapping the first $\ell$ coordinates with the last $\ell$ coordinates, we can assume that $\zz = (\underbrace{0,\dots,0}_\ell,\underbrace{1,\dots,1}_\ell)$.
We have the following possibilities for $M^i$ with $1 \leq i \leq 2\ell$:
1. \[enum:finQlimpliesginQl:smallweight\] $w(M^i) \neq n$;
2. \[enum:finQlimpliesginQl:equalpairs\] $w(M^i) = n$ and there is some $b \in [n]$ with $m^i_b = m^i_{n+b}$;
3. \[enum:finQlimpliesginQl:fromf\] $w(M^i) = n$ and $m^i_b \neq m^i_{n+b}$ for all $b \in [n]$, i.e., there is some $\aa_i \in {\mathbb{B}}^n$ with $M^i = (\aa_i,\overline{\aa_i})$.
We show that case \[enum:finQlimpliesginQl:smallweight\] cannot happen, since the weight of each row $M^i$ of $M$ is exactly $n$. Since $g(M^i) = 0$ for $1 \leq i \leq \ell$, we have $w(M^i) \leq n$ for $1 \leq i \leq \ell$. Similarly, we have $w(M^i) \geq n$ for $\ell+1 \leq i \leq 2\ell$. Thus $\sum_{i=1}^\ell w(M^i) \leq n\ell$ and $\sum_{i=\ell+1}^{2\ell} w(M^i) \geq n\ell$. Because $M_j \in R(B_\ell)$ for $1 \leq j \leq 2n$, we get $$\sum_{i=1}^\ell w(M^i) = \sum_{i=1}^\ell \sum_{j=1}^{2n} m_j^i
= \sum_{j=1}^{2n} \sum_{i=1}^\ell m_j^i
= \sum_{j=1}^{2n} \sum_{i=\ell+1}^{2\ell} m_j^i
= \sum_{i=\ell+1}^{2\ell} \sum_{j=1}^{2n} m_j^i
= \sum_{i=\ell+1}^{2\ell} w(M^i)$$ Therefore $\sum_{i=1}^\ell w(M^i) = \sum_{i=\ell+1}^{2\ell} w(M^i) = n\ell$, and $w(M^i) = n$ for $1 \leq i \leq 2\ell$. Thus the case \[enum:finQlimpliesginQl:smallweight\] cannot happen for $M^i$.
We will show that case \[enum:finQlimpliesginQl:equalpairs\] is also not possible. Suppose, on the contrary, that there is some $i \in [2\ell]$ and some $b \in [n]$ such that $m^i_b = m^i_{n+b}$, and $m^i_a \neq m^i_{n+a}$ for all $a < b$. We can assume that $b$ is the smallest number with this property.
Now we consider the weights of $M_b$ and $M_{n+b}$. Because $b$ is minimal, we have that $m^{i'}_a \neq m^{i'}_{n+a}$ for all $a < b$. Thus we have $(m^{i'}_b, m^{i'}_{n+b}) \in \{ (0,0),(0,1),(1,0) \}$ for $1 \leq i' \leq \ell$, and $(m^{i'}_b, m^{i'}_{n+b}) \in \{ (0,1),(1,0),(1,1) \}$ for $\ell+1 \leq i' \leq 2\ell$. Then $$\begin{aligned}
\sum_{i' = 1}^{\ell} (m^{i'}_b + m^{i'}_{n+b}) &
\leq n, \\
\sum_{i' = \ell+1}^{2\ell} (m^{i'}_b + m^{i'}_{n+b}) &
\geq n,
\end{aligned}$$ and at least one of these inequalities holds strictly. This implies that one of the following holds: $$\begin{aligned}
\sum_{i' = 1}^\ell m^{i'}_b & < \sum_{i' = \ell+1}^{2\ell} m^{i'}_b \quad \text{or} \\
\sum_{i' = 1}^\ell m^{i'}_{n+b} & < \sum_{i' = \ell+1}^{2\ell} m^{i'}_{n+b}.
\end{aligned}$$ This means that $M_b \notin R(B_\ell)$ or $M_{n+b} \notin R(B_\ell)$, in contradiction to the assumption. Thus no such $b$ exists, and case \[enum:finQlimpliesginQl:equalpairs\] cannot happen.
Thus case \[enum:finQlimpliesginQl:fromf\] applies for all $M^i$, i.e., $M^i = (\aa_i,\overline{\aa_i})$ for some $\aa_i \in {\mathbb{B}}^n$ holds for all $i \in [2\ell]$. By the definition of $g$ and since $f \in \operatorname{cPol}B_\ell$, we obtain $$\zz = g\begin{pmatrix} M^1 \\ \vdots \\ M^{2\ell} \end{pmatrix}
= f\begin{pmatrix} \aa_1 \\ \vdots \\ \aa_{2\ell} \end{pmatrix}
= f(M_1,\dots,M_n)
\in S(B_\ell).$$ But this is a contradiction to $\zz \notin S(B_\ell)$. Thus the matrix $M$ cannot exist, and we have $g \in \operatorname{cPol}B_\ell$.
\[cor:GMc\] For any Boolean function $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$, $G_{M_c}(f) \in M_c$ and for all $\ell \geq 2$, $f \in \operatorname{cPol}B_\ell$ if and only if $G_{M_c}(f) \in \operatorname{cPol}B_\ell$.
This brings together Lemmas \[lemma:StoSM\]\[lemma:StoSM:item1\], \[lemma:SM:fnotinBlimpliesgnotinBl\] and \[lemma:SM:finBlimpliesginBl\].
Let $G_{SM}(f) := G_{M_c}(G_{S}(f))$. Then we can conclude the following corollary from the preceding lemmas.
\[cor:GSM\] For any Boolean function $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$, $G_{SM}(f) \in SM$ and for all $\ell \geq 2$, $f \in \operatorname{cPol}B_\ell$ if and only if $G_{SM}(f) \in \operatorname{cPol}B_\ell$.
By Corollary \[cor:GS\], we have $G_{S}(f) \in S$, and by Lemma \[lemma:StoSM\], we get $G_{SM}(f) = G_{M_c}(G_S(f)) \in SM$.
By Corollary \[cor:GS\], the condition $f \in \operatorname{cPol}B_\ell$ is equivalent to $G_S(f) \in \operatorname{cPol}B_\ell$, which is in turn equivalent to $G_{SM}(f) = G_{M_c}(G_S(f)) \in \operatorname{cPol}B_\ell$ by Corollary \[cor:GMc\].
Construction of $G_{U_\infty}(f)$, $G_{M_cU_\infty}(f)$ and $G_{M_cW_\infty}(f)$ {#subsec:MCT}
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Let $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$. Define $G_{U_\infty}(f) \colon {\mathbb{B}}^{n+1} \to {\mathbb{B}}$ by $$G_{U_\infty}(f)(x_1, \dots, x_{n+1}) = x_{n+1} \wedge f(x_1, \dots, x_n).$$
\[lemma:SMtoMcUinfty\] Let $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$.
1. \[lemma:SMtoMcUinfty:item1\] $G_{U_\infty}(f) \in U_\infty$.
2. \[lemma:SMtoMcUinfty:item2\] If $f$ is monotone, then $G_{U_\infty}(f)$ is monotone.
3. \[lemma:SMtoMcUinfty:item3\] If $f(\11) = 1$, then $G_{U_\infty}(f)$ preserves both constants.
4. \[lemma:SMtoMcUinfty:item4\] If $f \in M_c$, then $G_{U_\infty}(f) \in M_cU_\infty$.
Let $g := G_{U_\infty}(f)$.
\[lemma:SMtoMcUinfty:item1\] By the definition of $g$ we have that if $g(x_1,\dots,x_{n+1}) = 1$ then $x_{n+1} = 1$. Thus $g \in U_\infty$.
\[lemma:SMtoMcUinfty:item2\] Let $\xx, \yy \in {\mathbb{B}}^{n+1}$, and assume that $\xx < \yy$. If $x_{n+1} = 0$, then $g(\xx) = 0 \leq g(\yy)$. If $x_{n+1} = 1$, then also $y_{n+1} = 1$, and since $f$ is monotone, we have $$g(\xx) = f(x_1, \dots, x_n) \leq f(y_1, \dots, y_n) = g(\yy).$$ We conclude that $g$ is monotone.
\[lemma:SMtoMcUinfty:item3\] By the definition of $g$, we have $g(\00) = 0$. Furthermore, if $f(\11) = 1$, then we have $g(\11) = f(\11) = 1$.
\[lemma:SMtoMcUinfty:item4\] Follows immediately from the previous items.
\[lemma:McUinfty:fnotinBlimpliesgnotinBl\] Let $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$. If $f \notin \operatorname{cPol}B_\ell$ for some $\ell \geq 2$, then $G_{U_\infty}(f) \notin \operatorname{cPol}B_\ell$.
The proof is exactly the same as the proof of Lemma \[lemma:ST:fNotinBlimpliesgNotinBl\].
\[lemma:McUinfty:finBlimpliesginBl\] Let $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$. If $f \in \operatorname{cPol}B_\ell$ for some $\ell \geq 2$, then $G_{U_\infty}(f) \in \operatorname{cPol}B_\ell$.
Let $g := G_{U_\infty}(f)$.
Suppose, on the contrary, that $g \notin \operatorname{cPol}B_\ell$. Then there is some matrix $M$ given by $$M = \begin{pmatrix}
m_1^1 & m_2^1 & \dots & m_{n+1}^1 \\
m_1^2 & m_2^2 & \dots & m_{n+1}^2 \\
\vdots & \vdots & & \vdots \\
m_1^{2\ell} & m_2^{2\ell} & \dots & m_{n+1}^{2\ell}
\end{pmatrix} = \begin{pmatrix}
M^1 \\
M^2 \\
\vdots \\
M^{2\ell}
\end{pmatrix}
= (M_1,M_2,\dots,M_{n+1}),$$ i.e., $M^1,\dots,M^{2\ell} \in {\mathbb{B}}^{n+1}$ are the rows of $M$, and $M_1,\dots,M_{n+1} \in {\mathbb{B}}^{2\ell}$ are the columns of $M$, such that
- $M_1,\dots,M_{n+1} \in R(B_\ell)$, and
- $\zz := g(M_1,\dots,M_{n+1}) := \begin{pmatrix} g(M^1) \\ \vdots \\ g(M^{2\ell}) \end{pmatrix}
\notin S(B_\ell)$.
Thus $\zz \in \{ (\underbrace{0,\dots,0}_\ell,\underbrace{1,\dots,1}_\ell),
(\underbrace{1,\dots,1}_\ell,\underbrace{0,\dots,0}_\ell)\}$. As $B_\ell$ is invariant under swapping the first $\ell$ coordinates with the last $\ell$ coordinates, we can assume that $\zz = (\underbrace{0,\dots,0}_\ell,\underbrace{1,\dots,1}_\ell)$.
We now look at the last column $M_{n+1}$ of $M$. Since $\sum_{i=1}^\ell m_{n+1}^i = \sum_{i=\ell+1}^{2\ell} m_{n+1}^i$, and since $B_\ell$ is totally symmetric on the first $\ell$ rows and on the last $\ell$ rows, we can assume that $$M_{n+1} = (\underbrace{0,\dots,0}_\alpha, \underbrace{1,\dots,1}_\beta,
\underbrace{0,\dots,0}_\alpha, \underbrace{1,\dots,1}_\beta)$$ holds for some $\alpha,\beta \geq 0$ with $\alpha+\beta = \ell$.
If $\alpha > 0$ then $g(M^{\ell+1}) = g(m_1^{\ell+1},\dots,m_n^{\ell+1},0) = 0 \land f(m_1^{\ell+1},\dots,m_n^{\ell+1}) = 0$, in contradiction to $g(M^{\ell+1}) = 1$. Thus $\alpha = 0$, and $M_{n+1} = \11$. But then $f(M_1,\dots,M_n) = g(M_1,\dots,M_n,\11) = \zz \notin S(B_\ell)$, which implies that $f \notin \operatorname{cPol}B_\ell$. This contradicts the assumption $f \in \operatorname{cPol}B_\ell$, and we conclude that $g \in \operatorname{cPol}B_\ell$.
\[cor:GUinfty\] For any Boolean function $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$, $G_{U_\infty}(f) \in U_\infty$ and for all $\ell \geq 2$, $f \in \operatorname{cPol}B_\ell$ if and only if $G_{U_\infty}(f) \in \operatorname{cPol}B_\ell$.
This brings together Lemmas \[lemma:SMtoMcUinfty\]\[lemma:SMtoMcUinfty:item1\], \[lemma:McUinfty:fnotinBlimpliesgnotinBl\] and \[lemma:McUinfty:finBlimpliesginBl\].
Let $G_{M_cU_\infty}(f) := G_{U_\infty}(G_{M_c}(f))$ and $G_{M_cW_\infty}(f) := {{G_{M_cU_\infty}(f)}^{\mathrm{d}}}$.
\[cor:GMcUinfty\] For any Boolean function $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$, $G_{M_cU_\infty}(f) \in M_cU_\infty$ and for all $\ell \geq 2$, $f \in \operatorname{cPol}B_\ell$ if and only if $G_{M_cU_\infty}(f) \in \operatorname{cPol}B_\ell$.
By Corollary \[cor:GMc\], we have $G_{M_c}(f) \in M_c$, and by Lemma \[lemma:SMtoMcUinfty\], we get $G_{M_cU_\infty}(f) = G_{U_\infty}(G_{M_c}(f)) \in M_cU_\infty$.
By Corollary \[cor:GMc\], the condition $f \in \operatorname{cPol}B_\ell$ is equivalent to $G_{M_c}(f) \in \operatorname{cPol}B_\ell$, which in turn is equivalent to $G_{M_cU_\infty}(f) = G_{U_\infty}(G_{M_c}(f)) \in \operatorname{cPol}B_\ell$ by Corollary \[cor:GUinfty\].
Let $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$. We define the functions $\overline{f} \colon {\mathbb{B}}^n \to {\mathbb{B}}$ and $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$, for $\uu \in {\mathbb{B}}^n$, as $$\begin{aligned}
\overline{f}(\aa) &= \overline{f(\aa)}, \\
f^\uu(\aa) &= f(\aa \oplus \uu).\end{aligned}$$ Note that ${{f}^{\mathrm{d}}} = \overline{f^\11}$, where $\11 := (1, \dots, 1) \in {\mathbb{B}}^n$.
\[lem:dualBl\] Let $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$, and let $\ell \geq 2$. The following are equivalent:
1. $f \in \operatorname{cPol}B_\ell$,
2. $f^\uu \in \operatorname{cPol}B_\ell$ for any $\uu \in {\mathbb{B}}^n$,
3. $\overline{f} \in \operatorname{cPol}B_\ell$,
4. ${{f}^{\mathrm{d}}} \in \operatorname{cPol}B_\ell$.
$\text{(i)} \iff \text{(ii)}$: Let $\aa^1, \dots \aa^n \in R(B_\ell)$. Since $R(B_\ell)$ is invariant under taking negations of its members, we also have $\overline{\aa^1}, \dots \overline{\aa^n} \in R(B_\ell)$. Let $\uu \in {\mathbb{B}}^n$, and let $\bb^i := \aa^i$ if $u_i = 0$ and $\bb^i := \overline{\aa^i}$ if $u_i = 1$, for $i \in {[{n}]}$. If $f \in \operatorname{cPol}B_\ell$, then $$f^\uu(\aa^1, \dots, \aa^n) = f(\bb^1, \dots, \bb^n) \in S(B_\ell);$$ hence $f^\uu \in \operatorname{cPol}B_\ell$. The converse implication holds, since $(f^\uu)^\uu = f$.
$\text{(i)} \iff \text{(iii)}$: Assume that $f \in \operatorname{cPol}B_\ell$, and let $\aa^1, \dots \aa^n \in R(B_\ell)$. Then $f(\aa^1, \dots, \aa^n) \in S(B_\ell)$. Since $S(B_\ell)$ is invariant under taking negations of its members, we have $$\overline{f}(\aa^1, \dots, \aa^n)
= \overline{f(\aa^1, \dots, \aa^n)}
\in S(B_\ell);$$ hence $\overline{f} \in \operatorname{cPol}B_\ell$. The converse implication holds, since $\overline{\overline{f}} = f$.
$\text{(i)} \iff \text{(iv)}$: This follows immediately from the equivalence of (i), (ii) and (iii), because ${{f}^{\mathrm{d}}} = \overline{f^\11}$.
\[cor:GMcWinfty\] For any Boolean function $f \colon {\mathbb{B}}^n \to {\mathbb{B}}$, $G_{M_cW_\infty}(f) \in M_cW_\infty$ and for all $\ell \geq 2$, $f \in \operatorname{cPol}B_\ell$ if and only if $G_{M_cW_\infty}(f) \in \operatorname{cPol}B_\ell$.
Since $M_cW_\infty = \{{{f}^{\mathrm{d}}} : f \in M_cU_\infty\}$, the claim follows from Lemma \[lem:dualBl\] and Corollary \[cor:GMcUinfty\].
Simple games and magic squares revisited {#sec:magic}
========================================
In their proof of the existence of $k$-asummable functions that are not $(k+1)$-asummable (see Theorem \[theorem:strictAkInclusions\]), Taylor and Zwicker constructed a certain family of functions [@Taylor]. We recall their construction here, and then we will refine Theorem \[theorem:strictAkInclusions\] and determine how the sets $\operatorname{cPol}B_n$ are related to each other. We will also show that Taylor and Zwicker’s functions actually constitute an antichain of minimally non-threshold functions.
Fix an integer $k \geq 3$. For $p, q \in {[{k}]}$, define the $k \times k$ matrix $A^{p,q} = (a_{i,j})$ as follows: $$a_{i,j} =
\begin{cases}
k - 1, & \text{if $(i,j) = (p,q)$,} \\
1, & \text{if $i \neq p$ and $j \neq q$,} \\
0, & \text{otherwise.}
\end{cases}$$ For example, if $k = 4$, then $A^{2,3} = \begin{pmatrix} 1 & 1 & 0 & 1 \\ 0 & 0 & 3 & 0 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & 0 & 1 \end{pmatrix}$. Let $B$ be the $k \times k$ matrix all of whose entries are equal to $k - 1$.
Let $S$ be a subset of ${[{k}]} \times {[{k}]}$. We refer to $S$ as the $i$-th *row* if $S = \{(i, j) : j \in {[{k}]}\}$, and we refer to $S$ as the $j$-th *column* if $S = \{(i, j) : i \in {[{k}]}\}$.
\[lem:roworcolumn\] Let $S \subseteq {[{k}]} \times {[{k}]}$. Then $\sum_{(p,q) \in S} A^{p,q} = B$ if and only if $S$ is a row or a column.
It is clear that if $S$ is a row or a column, then $\sum_{(p,q) \in S} A^{p,q} = B$.
Assume then that $\sum_{(p,q) \in S} A^{p,q} = B$. Clearly $S$ is nonempty, so choose an element $(p,q)$ of $S$; clearly $S$ contains another element $(p',q')$. If $p \neq p'$ and $q \neq q'$, then the entry on row $p$ column $q$ in the sum $\sum_{(p,q) \in S} A^{p,q}$ is at least $k$; hence the sum cannot be equal to $B$. Thus either $p = p'$ or $q = q'$. It is easy to see that in the former case, all remaining entries of $S$ must be on the $p$-th row, and all elements of the $p$-th row must be in $S$; in the latter case, all remaining entries of $S$ must be on the $q$-th column, and all elements of the $q$-th column must be in $S$. We conclude that $S$ is either a row or a column.
We define a function $\phi \colon {[{R}]}^{k \times k} \to {\mathbb{N}}$ that maps each $k \times k$ matrix with entries in ${[{R}]}$ to an integer, where $R$ is a sufficiently large integer that will be specified below. The function $\phi$ is defined as follows: for a matrix $M$, read the entries of $M$ from left to right and from top to bottom; the resulting string is the representation of $\phi(M)$ in base $R$. For $p, q \in {[{k}]}$, denote $w^{p,q} := \phi(A^{p,q})$ and $t := \phi(B)$. For example if $k = 4$, then $w^{2,3} = 1101003011011101_R$ and $t = 3333333333333333_R$. We must choose $R$ in such a way that when we add these numbers to form the sum $\sum_{(p,q) \in S} w^{p,q}$ for any $S \subseteq {[{k}]} \times {[{k}]}$, no carry will occur. Thus, the number $(k-1)^2 + (k-1) + 1 = k^2 - k + 1$, or anything larger, would be fine.
It is easy to see that the function $\phi$ has the following preservation property: for any $S \subseteq {[{k}]} \times {[{k}]}$, $\phi(\sum_{(p,q) \in S} A^{p,q}) = \sum_{(p,q) \in S} \phi(A^{p,q})$. It thus follows from Lemma \[lem:roworcolumn\] that for all $S \subseteq {[{k}]} \times {[{k}]}$, it holds that $\sum_{(p,q) \in S} w^{p,q} = t$ if and only if $S$ is a row or a column.
Fix a bijection $\beta \colon {[{k}]} \times {[{k}]} \to {[{k^2}]}$. The *characteristic tuple* of a subset $S$ of ${[{k}]} \times {[{k}]}$ is the tuple $\ee_S \in {\mathbb{B}}^{k^2}$, whose $i$-th entry is $1$ if $i = \beta(p,q)$ for some $(p,q) \in S$ and $0$ otherwise. With no risk of confusion, we will refer to the characteristic tuples of rows and columns also as *rows* and *columns,* respectively.
Let $\ww = (w^{\beta^{-1}(1)}, \dots, w^{\beta^{-1}(k^2)})$.
For any $n$-tuples $\aa, \bb \in {\mathbb{R}}^n$, the *dot product* is defined as $$\aa \cdot \bb = \sum_{i=1}^n a_i b_i.$$
Taylor and Zwicker’s function $f_k \colon {\mathbb{B}}^{k^2} \to {\mathbb{B}}$ is defined by the following rule: $f_k(\xx) = 1$ if and only if $\xx \cdot \ww > t$ or $\xx$ is a row. Note that for all $\xx \in {\mathbb{B}}^{k^2}$, $\xx \cdot \ww = t$ if and only if $\xx$ is a row or a column.
\[lem:fkBl\] Let $k \geq 3$ and $\ell \geq 2$. Then $f_k$ preserves $B_\ell$ if and only if $k$ is not a divisor of $\ell$.
If $\ell = mk$ for some integer $m$, then let $\aa^1, \dots, \aa^\ell$ comprise $m$ occurrences of each column, and let $\bb^1, \dots, \bb^\ell$ comprise $m$ occurrences of each row. Then, the $\aa^i$ are false points of $f_k$ and the $\bb^i$ are true points, and $\aa^1 + \dots + \aa^\ell = (m, \dots, m) = \bb^1 + \dots + \bb^\ell$. Thus $f_k$ is not $\ell$-asummable. Lemma \[lem:PolBl\] implies that $f_k$ does not preserve $B_\ell$.
Assume then that $k$ is not a divisor of $\ell$. Suppose, on the contrary, that $f_k$ does not preserve $B_\ell$. By Lemma \[lem:PolBl\], there exist $\aa^1, \dots, \aa^\ell \in f^{-1}(0)$ and $\bb^1, \dots, \bb^\ell \in f^{-1}(1)$ such that $\aa^1 + \dots + \aa^\ell = \bb^1 + \dots + \bb^\ell$. Since $\xx \cdot \ww \leq t$ for any false point $\xx$ of $f_k$, and $\xx \cdot \ww \geq t$ for any true point $\xx$, we have $$\sum_{i = 1}^\ell \aa^i \cdot \ww \leq \ell t
\qquad
\text{and}
\qquad
\sum_{i = 1}^\ell \bb^i \cdot \ww \geq \ell t.$$ On the other hand, since $\aa^1 + \dots + \aa^\ell = \bb^1 + \dots + \bb^\ell$, we have $$\sum_{i = 1}^\ell \aa^i \cdot \ww
= (\aa^1 + \dots + \aa^\ell) \cdot \ww
= (\bb^1 + \dots + \bb^\ell) \cdot \ww
= \sum_{i = 1}^\ell \bb^i \cdot \ww.$$ Consequently, $\aa^i \cdot \ww = t$ and $\bb^i \cdot \ww = t$ for all $i \in {[{\ell}]}$, and we conclude that each $\aa^i$ is a column and each $\bb^i$ is a row. Since $k$ is not a divisor of $\ell$, there necessarily exist two columns that have a different number of occurrences among $\aa^1, \dots, \aa^\ell$. Then $\phi^{-1}(\aa^1 + \dots + \aa^n)$ is a matrix that is constant along each column, but there are two columns with distinct values. This contradicts the fact that the matrix $\phi^{-1}(\bb^1 + \dots + \bb^n)$ is constant along each row. This completes the proof, and we conclude that $f_k$ preserves $B_\ell$.
\[lem:mod2sum\] The modulo-$2$ addition operation ${\oplus}$ preserves $B_\ell$ if and only if $\ell$ is odd.
The false points of $\oplus$ are $(0,0)$ and $(1,1)$, while the true points are $(0,1)$ and $(1,0)$. Hence the sum of any $\ell$ false points is of the form $(m,m)$ for some $m$ with $0 \leq m \leq \ell$. The sum of any $\ell$ true points is of the form $(m, \ell - m)$ for some $m$ with $0 \leq m \leq \ell$.
If $\ell$ is odd, then $m \neq \ell - m$ for any $m$. It follows that $\aa^1 + \dots + \aa^\ell \neq \bb^1 + \dots + \bb^\ell$ for any false points $\aa^1, \dots, \aa^\ell$ and any true points $\bb^1, \dots, \bb^\ell$. By Lemma \[lem:PolBl\], $\oplus$ preserves $B_\ell$.
If $\ell$ is even, say $\ell = 2k$, then $$\begin{gathered}
\underbrace{(0,0) + \dots + (0,0)}_k + \underbrace{(1,1) + \dots + (1,1)}_k = \\
\underbrace{(0,1) + \dots + (0,1)}_k + \underbrace{(1,0) + \dots + (1,0)}_k.\end{gathered}$$ By Lemma \[lem:PolBl\], $\oplus$ does not preserve $B_\ell$.
Let $\ell, m \geq 2$. Then $\operatorname{cPol}B_\ell \subseteq \operatorname{cPol}B_m$ if and only if $m$ divides $\ell$.
Assume first that $m$ does not divide $\ell$. If $m \neq 2$, then by Lemma \[lem:fkBl\], $f_m \in \operatorname{cPol}B_\ell$ but $f_m \notin \operatorname{cPol}B_m$. If $m = 2$, then by Lemma \[lem:mod2sum\], ${\oplus} \in \operatorname{cPol}B_\ell$ but ${\oplus} \notin \operatorname{cPol}B_m$. In either case, we conclude that $\operatorname{cPol}B_\ell \not\subseteq \operatorname{cPol}B_m$.
Assume then that $\ell = km$ for some integer $k$. Let $f \in \operatorname{cPol}B_\ell$. Let $\aa^1, \dots, \aa^n \in R(B_m)$. For each $i \in \{1, \dots, n\}$, define the tuple $\bb^i \in {\mathbb{B}}^\ell$ as $$\bb^i = (\underbrace{a^i_1, \dots, a^i_1}_k, \dots, \underbrace{a^i_m, \dots, a^i_m}_k, \underbrace{a^i_{m + 1}, \dots, a^i_{m + 1}}_k, \dots, \underbrace{a^i_{2m}, \dots, a^i_{2m}}_{k}).$$ It is clear that $\bb^i \in R(B_\ell)$. Let $\zz := f(\bb^1, \dots, \bb^n)$, that is, $$\zz = (\underbrace{f(a^1_1, \dots, a^n_1), \dots, f(a^1_1, \dots, a^n_1)}_k, \dots, \underbrace{f(a^1_{2m}, \dots, a^n_{2m}), \dots, f(a^1_{2m}, \dots, a^n_{2m})}_k).$$ Since $f \in \operatorname{cPol}B_\ell$, we have $\zz \in S(R_\ell)$. Then $$\zz \in {\mathbb{B}}^\ell \setminus \{(\underbrace{0, \dots, 0}_\ell, \underbrace{1, \dots, 1}_\ell), (\underbrace{1, \dots, 1}_\ell, \underbrace{0, \dots, 0}_\ell))\}.$$ This implies that $$f(\aa^1, \dots, \aa^n) \in {\mathbb{B}}^m \setminus \{(\underbrace{0, \dots, 0}_m, \underbrace{1, \dots, 1}_m), (\underbrace{1, \dots, 1}_m, \underbrace{0, \dots, 0}_m))\}.$$ Thus $f \in \operatorname{cPol}B_m$, and we conclude that $\operatorname{cPol}B_\ell \subseteq \operatorname{cPol}B_m$.
\[prop:tzantichain\] The functions $f_k$ ($k \geq 3$) are pairwise incomparable by the minor relation.
Let $m \neq n$, and consider the comparability of $f_m$ and $f_n$. Since all variables are essential in $f_m$ and in $f_n$, and the number of essential variables cannot increase when taking minors, we have that $f_m \not\leq f_n$ whenever $m > n$. If $m < n$, then $n$ is not a divisor of $m$ but $n$ is a divisor of itself. By Lemma \[lem:fkBl\], $f_n$ preserves $B_m$ and $f_m$ does not preserve $B_m$. Since every minor of $f_n$ preserves all relational constraints $f_n$ does, we must have that $f_m \not\leq f_n$ also in this case.
\[prop:tzmonotone\] For every $k \geq 3$, the function $f_k$ is monotone.
Let $\xx, \yy \in {\mathbb{B}}^{k^2}$. If $\xx < \yy$, then, since each $w^{p,q}$ is positive, $\xx \cdot \ww < \yy \cdot \ww$. Therefore one of the following conditions holds: $\xx \cdot \ww < t$ or $\yy \cdot \ww > t$. In the former case, $f(\xx) = 0 \leq f(\yy)$. In the latter case, $f(\xx) \leq 1 = f(\yy)$.
\[prop:tzminnonthr\] For every $k \geq 3$, the function $f_k$ is minimally non-threshold.
We need to show that every identification minor of $f_k$ is threshold. Let $(p,q)$ and $(p',q')$ be distinct elements of ${[{k}]} \times {[{k}]}$, let $I = \{\beta(p,q), \beta(p',q')\}$, and assume without loss of generality that $\beta(p,q) < \beta(p',q')$. We will show that $(f_k)_I$ is $\ell$-asummable for every $\ell \geq 2$ and hence threshold by Theorem \[thm:thresholdasummable\]. Let $\ell \geq 2$, and let $\aa^1, \dots, \aa^\ell \in ((f_k)_I)^{-1}(0)$, $\bb^1, \dots, \bb^\ell \in ((f_k)_I)^{-1}(1)$. Suppose, on the contrary, that $\aa^1 + \dots + \aa^\ell = \bb^1 + \dots + \bb^\ell$. Let $\vv \in {\mathbb{B}}^{k^2 - 1}$ be the tuple obtained from $\ww$ by replacing its $\beta(p,q)$-th entry by $w_{\beta(p,q)} + w_{\beta(p',q')}$ and deleting the $\beta(p',q')$-th entry. (Before proceeding, we ask the reader to recall the definition of $\delta_I$ from in Section \[susec:MinorsConstraints\].) It clearly holds that $\xx \cdot \vv = \xx \delta_I \cdot \ww$ for all $\xx \in {\mathbb{B}}^{k^2 - 1}$. Therefore $((f_k)_I)(\xx) = f_k(\xx \delta_I) = 1$ if and only if $\xx \cdot \vv = \xx \delta_I \cdot \ww > t$ or $\xx \delta_I$ is a row. Note that if $\xx \delta_I$ is a row or a column, then $\xx \cdot \vv = \xx \delta_I \cdot \ww = t$. In a similar way as we argued in the proof of Lemma \[lem:fkBl\], we have $$\ell t
\geq \sum_{i = 1}^\ell \aa^i \cdot \vv
= (\aa^1 + \dots + \aa^\ell) \cdot \vv
= (\bb^1 + \dots + \bb^\ell) \cdot \vv
= \sum_{i = 1}^\ell \bb^i \cdot \vv
\geq \ell t.$$ Hence $\aa^i \cdot \vv = t$ and $\bb^i \cdot \vv = t$ for all $i \in {[{\ell}]}$, that is, $\aa^i \delta_I$ is a column and $\bb^i \delta_I$ is a row for all $i \in {[{\ell}]}$.
Since $(p,q) \neq (p',q')$, we have $p \neq p'$ or $q \neq q'$. If $p \neq p'$, then none of the rows $\bb^i \delta_I$ is the $p$-th row; hence $\phi^{-1}(\bb^1 \delta_I + \dots + \bb^\ell \delta_I)$ is a matrix with a row full of $0$’s, while $\phi^{-1}(\aa^1 \delta_I + \dots + \aa^\ell \delta_I)$ has no row full of $0$’s. If $q \neq q'$, then none of the columns $\aa^i \delta_I$ is the $q$-th column; hence $\phi^{-1}(\aa^1 \delta_I + \dots + \aa^\ell \delta_I)$ is a matrix with a column full of $0$’s, while $\phi^{-1}(\bb^1 \delta_I + \dots + \bb^\ell \delta_I)$ has no column full of $0$’s. On the other hand, $$\aa^1 \delta_I + \dots + \aa^\ell \delta_I
= (\aa^1 + \dots + \aa^\ell) \delta_I
= (\bb^1 + \dots + \bb^\ell) \delta_I
= \bb^1 \delta_I + \dots + \bb^\ell \delta_I.$$ We have reached a contradiction.
We conclude that $(f_k)_I$ is $\ell$-asummable for every $\ell \geq 2$ and hence threshold.
Taylor and Zwicker’s functions $f_k$ constitute an infinite antichain of monotone, minimally non-threshold functions (Propositions \[prop:tzantichain\], \[prop:tzmonotone\], \[prop:tzminnonthr\]). It should be noted here that this antichain does not, however, characterize the set of monotone threshold functions in terms of forbidden minors, i.e., $M \cap {T}\neq \operatorname{forbid}(\{f_k : k \geq 3\})$. For example, there exist self-dual monotone non-threshold functions of arity $6$ (see, e.g., [@BioIba]), which clearly fail to have any of the $f_n$ as a minor.
Post classes {#App:Post}
============
We provide a concise description of all clones of Boolean functions as well as characterizing sets of relations $R$ – or, equivalently, relational constraints $(R,R)$ – for some clones; the characterizion of the remaining clones is easily derived by noting that if $C_1 = \operatorname{cPol}(\mathcal{Q}_1)$ and $C_2 = \operatorname{cPol}(\mathcal{Q}_2)$, then $C_1 \cap C_2 = \operatorname{cPol}(\mathcal{Q}_1 \cup \mathcal{Q}_2)$. We make use of notations and terminology appearing in [@FP] and [@JGK].
$\Omega$ denotes the clone of all Boolean functions. It is characterized by the empty relation.
$T_0$ and $T_1$ denote the clones of $0$- and $1$-preserving functions, respectively, i.e., $$T_0 = \{f \in \Omega : f(0, \dots, 0) = 0\}
\quad \text{and} \quad
T_1 = \{f \in \Omega : f(1, \dots, 1) = 1\}.$$ They are characterized by the unary relations $\{0\}$ and $\{1\}$, respectively.
$T_c$ denotes the clone of constant-preserving functions, i.e., $T_c = T_0 \cap T_1$.
$M$ denotes the clone of all monotone functions, i.e., $$M = \{f \in \Omega : \text{$f(\mathbf{a}) \leq f(\mathbf{b})$ whenever $\mathbf{a} \leq \mathbf{b}$}\}.$$ It is characterized by the binary relation ${\leq} := \{(0,0), (0,1), (1,1)\}$.
$M_0 = M \cap T_0$, $M_1 = M \cap T_1$, $M_c = M \cap T_c$.
$S$ denotes the clone of all self-dual functions, i.e., $$S = \{f \in \Omega : {{f}^{\mathrm{d}}} = f\}.$$ It is characterized by the binary relation $\{(0,1),(1,0)\}$.
$S_c = S \cap T_c$, $SM = S \cap M$.
$L$ denotes the clone of all linear functions, i.e., $$L = \{f \in \Omega : f = c_0 \oplus c_1 x_1 \oplus \dots \oplus c_n x_n\}.$$ It is characterized by the quaternary relation $\{(a,b,c,d) \in {\mathbb{B}}^4 : a \oplus b \oplus c = d\}$.
$L_0 = L \cap T_0$, $L_1 = L \cap T_1$, $LS = L \cap S$, $L_c = L \cap T_c$.
Let $a \in \{0,1\}$. A set $A \subseteq \{0,1\}^n$ is said to be *$a$-separating* if there is some $i\in [n]$ such that for every $(a_1, \dotsc, a_n) \in A$ we have $a_i = a$. A function $f$ is said to be *$a$-separating* if $f^{-1}(a)$ is $a$-separating. The function $f$ is said to be *$a$-separating of rank $k \geq 2$* if every subset $A \subseteq f^{-1}(a)$ of size at most $k$ is $a$-separating.
For $m \geq 2$, $U_m$ and $W_m$ denote the clones of all $1$- and $0$-separating functions of rank $m$, respectively. They are characterized by the $m$-ary relations ${\mathbb{B}}^m \setminus \{(0, \dots, 0)\}$ and ${\mathbb{B}}^m \setminus \{(1, \dots, 1)\}$, respectively.
$U_\infty$ and $W_\infty$ denote the clones of all $1$- and $0$-separating functions, respectively, i.e., $U_\infty = \bigcap_{k \geq 2} U_k$ and $W_\infty = \bigcap_{k \geq 2} W_k$.
$T_cU_m = T_c \cap U_m$ and $T_cW_m = T_c \cap W_m$, for $m = 2, \dotsc, \infty$.
$MU_m = M \cap U_m$ and $MW_m = M \cap W_m$, for $m = 2, \dotsc, \infty$.
$M_cU_m = M_c \cap U_m$ and $M_cW_m = M_c \cap W_m$, for $m = 2, \dotsc, \infty$.
$\Lambda $ denotes the clone of all conjunctions and constants, i.e., $$\begin{gathered}
\Lambda = \{f \in \Omega : f = x_{i_1} \wedge \dotsb \wedge x_{i_n}\} \cup
\{\mathbf{0}^{(n)}: n\geq 1\} \cup \{\mathbf{1}^{(n)}: n\geq 1\}.\end{gathered}$$ It is characterized by the ternary relation $\{(a,b,c) : a \wedge b = c\}$.
$\Lambda_0 = \Lambda \cap T_0$, $\Lambda_1 = \Lambda \cap T_1$, $\Lambda_c = \Lambda \cap T_c$.
$V$ denotes the clone of all disjunctions and constants, i.e., $$\begin{gathered}
V = \{f \in \Omega : f = x_{i_1} \vee \dotsb \vee x_{i_n}\} \cup
\{\mathbf{0}^{(n)}: n\geq 1\} \cup \{\mathbf{1}^{(n)}: n\geq 1\}.\end{gathered}$$ It is characterized by the ternary relation $\{(a,b,c) : a \vee b = c\}$.
$V_0 = V \cap T_0$, $V_1 = V \cap T_1$, $V_c = V \cap T_c$.
$\Omega (1)$ denotes the clone of all projections, negations, and constants. It is characterized by the ternary relation $\{(a,b,c) : \text{$a = b$ or $b = c$}\}$.
$I^* = \Omega(1) \cap S$, $I = \Omega(1) \cap M$.
$I_0 = I \cap T_0$, $I_1 = I \cap T_1$.
$I_c$ denotes the smallest clone containing only projections, i.e., $I_c = I \cap T_c$.
[99]{}
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The case of $\gamma$-ray absorption due to photon-photon pair production of jet photons in the external photon environment like accretion disk and broad-line region radiation field of $\gamma$-ray loud active galactic nuclei (AGN) that exhibit strong emission lines is considered. I demonstrate that this ”local opacity”, if detected, will almost unavoidably be redshift-dependent in the sub-TeV range. This introduces non-negligible biases, and complicates approaches for studying the evolution of the extragalactic background light with contemporary GeV instruments like e.g. the Gamma-ray Large Area Space Telescope (GLAST), etc., where the $\gamma$-ray horizon is probed by means of statistical analysis of absorption features (e.g. Fazio-Stecker relation, etc.) in AGN spectra at various redshifts. It particularly applies to strong-line quasars where external photon fields are potentially involved in $\gamma$-ray production.'
author:
- 'A. Reimer'
title: 'The redshift-dependence of gamma-ray absorption in the environments of strong-line AGN'
---
Introduction
============
Following the unification scheme the central nucleus of an active galaxy (AGN) consists of a black hole (BH), an accretion disk, line-emitting clouds, a dust torus, and emanates prominent jets when classified as radio-loud. The properties of radio-loud AGN viewed at a small angle to the line-of-sight are in general agreement with the common blazar properties [e.g., @Urry95; @Padovani07]. Their broadband emission covers the complete electromagnetic band, from the radio up to the $\gamma$-ray band, in some cases even reaching TeV-energies, and is widely dominated by beamed non-thermal emission from a relativistic jet. The blazar class subdivides into BL Lac objects and flat-spectrum radio quasars (FSRQs). The difference dividing both subclasses is generally considered in the detection of strong emission lines in the case of FSRQs, while in BL Lac objects the equivalent width of emission lines is depressed, or lines are absent at all. The physical reason is thought to lie predominantly in the weak accretion disk radiation field of BL Lac objects, whereas BH in the nuclei of FSRQs accrete with high rates leading to luminous accretion disk photon fields. The present work deals with $\gamma$-ray loud AGN observed during an epoche of a bright accretion disk and accompagnied with the apprearance of strong emission lines. For simplicity I will refer to them as ”quasars” in the following, although some radio-loud AGN classified conventionally as BL Lac objects may occasionally fall into this category as well [e.g., @Falomo94; @Sbarufatti06], and vice versa.
Gamma-ray production mechanisms, leptonic as well as hadronic ones, in AGN jets often involve either the external radiation fields associated with the immediate AGN environment or internal jet photons. For example, contributions from interactions in the accretion disk or broad-line region (BLR) radiation fields are often mandatory to explain the overall $\gamma$-ray spectral energy distribution (SED) from FSRQs [e.g., @Ghisellini96; @Boettcher00], while the SED of low-luminosity BL Lac objects is often fitted with either ”Synchrotron-Self-Compton” models [e.g., @Ghisellini85] or their hadronic equivalent, the ”Synchrotron-Proton blazar” models [e.g., @Mannheim93; @Muecke00a; @Aharonian00; @Muecke03]. The former scenario places the $\gamma$-ray emission region rather close to the BLR clouds. This has immediate consequences: If external radiation fields in quasar environments play a non-negligible role for $\gamma$-ray production, then, at the same time they are also significant for the quasi-resonant process of photon-photon pair production with its peak cross section possessing a comparable value to the Thomson cross section. It should be noted though, that the position of the high-energy emission region is still a matter of debate, with locations proposed to be also far away from the BLR [e.g., @Lindfors05; @Sokolov05]. If this is the case, external Compton emission [e.g., @Dermer93] or pair cascade radiation initiated by ultra-high energy cosmic ray interactions on external photon fields [e.g., @Protheroe97b; @Atoyan03] in those sources does not provide an appreciable contribution to the observed high-energy emission.
The present work concerns specifically opacity features in the $>10$ GeV regime by photon absorption through e$^+$-e$^-$ pair production in radiation fields in the vicinity of the BH, but external to the jets in $\gamma$-ray loud quasars. Past works on this subject [e.g., @Protheroe97a; @Donea03; @Becker95] indicate the importance of this process for constraining AGN properties like the location of the $\gamma$-ray region above the disk, disk radiation fields, torus temperature, etc. In contrast to these works, I will be focusing on the evolution of external radiation fields in quasar environments, and its consequences for the resulting opacity features in the $\gamma$-ray band covered by current and near future instruments. This is primarily motivated by the anticipated studies of the evolution of the extragalactic background light (EBL) through the detection of absorption features in a large sample of high-redshift sources using the Large Area Telescope (LAT) onboard GLAST. Dedicated methods have been developed here to probe the evolution of the EBL via detecting the horizon of $\gamma$-rays emitted from extragalactic sources like AGN and GRBs while propagating through the EBL to Earth [e.g., @Chen04; @Kneiske04]. They involve either the determination of the ratio of absorbed to unabsorbed flux versus redshift $z$, or the detection of the e-folding cutoff energy $E(\tau_{\gamma\gamma}=1)$ versus redshift [”Fazio-Stecker relation”; @Fazio70] in a large number of sources at various redshifts in order to disentangle intrinsic blazar features from absorption caused ones during propagation in the EBL. The common underlying reasoning for this procedure is that the observation of any redshift-dependent attenuation in $\gamma$-ray AGN can only be attributed to absorption in the EBL, and no other sources of redshift-dependent opacity exist in those sources. Here I will demonstrate that opacity due to photon absorption from $\gamma\gamma$-pair production caused in external radiation fields within the AGN system (in the following called ”local absorption”, to be distinguished from “self-absorption” in the internal jet radiation fields) will most likely result in optical depth values that increase with the source redshift, and coincidentally mimic redshift-dependent EBL-caused absorption.
The outline of this paper is organized as follows: Sect. 2 and Sect. 3 describe the considered external target photon fields (accretion disk and BLR radiation field) for photon-photon pair production, and the corresponding optical depth calculation, respectively. In Sect. 5 I will apply various models of supermassive BH growth and accretion rate evolution, that are described in Sect. 4, to the $\gamma$-ray attenuation calculations. The results with particular emphasis on the possibility of a redshift-dependence of the local opacity are presented in Sect. 5. The paper closes with conclusions and a discussion in Sect. 6.
Characterization of external radiation fields in quasars
========================================================
The most relevant target radiation fields in AGN environments for photon absorption in the LAT energy range, $\sim$0.02-300 GeV[^1], are the optical/UV bands of the accretion disk photon field and the radiation field of the BLR. These will be considered in the following. I assume the radiation fields to be located azimuthally symmetric with respect to the jet axis, and to radiate persistently during $\gamma$-ray emission[^2].
Quasar accretion disk radiation fields
--------------------------------------
The accretion disk spectrum in FSRQs is assumed to follow the cool, optically thick blackbody solution of Shakura & Sunyaev (1973) with a given accretion rate $\dot M_{\rm acc}$, suitable for AGN that show strong emission lines (“strong-line AGN”). The differential photon density $n(\epsilon,\Omega)$ into a solid angle $d\Omega=2\pi d\mu$, $\mu=\cos{\xi}$ reads $$\frac{dn(\epsilon,\Omega)}{d\Omega}=\frac{dn}{2\pi d\mu}=\frac{\epsilon^2}{2\mu c^3}\left(\frac{m_e c^2}{h}\right)^3
\left[\exp\left(\frac{\epsilon}{\theta(R)}\right)-1\right]^{-1}$$ with $R=l\sqrt{\mu^{-2}-1}$, $l$ the distance of the emission region above the BH, $\epsilon$ the photon energy, and $$\theta(R)=\frac{k_{\rm B}T(R)}{m_e c^2}\simeq 1.44 \left(\frac{M_{\rm BH}}{M_\sun}\right)^{-1/2} \left(\frac{\dot M_{\rm acc}}{M_\sun \rm{yr}^{-1}}\right)^{1/4} \left(\frac{R}{R_g}\right)^{-3/4} \left(1-\sqrt{\frac{R_i}{R}}\right)^{1/4}$$ with $R_i=6GM_{\rm BH}/c^2$ for a Schwarzschild metric, $R_g=GM_{\rm BH}/c^2$ and $k_{\rm B}$ the Boltzmann constant. For the opacity calculations the disk is considered to extend from 6 to $10^5 R_g$. Typical accretion rates for strong quasars approach the Eddington accretion rate $M_{\rm edd}$, $\dot M_{\rm acc}=0.1-1M_{\rm edd}$, respectively the Eddington luminosity $L_{\rm edd}$.
The BLR radiation field in quasars
----------------------------------
The accretion disk is considered as the photo-ionizing source of the BLR material, and emission lines are produced through recombination. The BLR geometry is approximated as a spherical shell with radius $r$ filled with clouds, extending from $r_{\rm in}$ to $r_{\rm out}$. In this picture the accretion disk as located at $r=0$. In the following the shell size is fixed to $r_{\rm in}=0.01$pc to $r_{\rm out}=0.4$pc, which are typical values for quasars [e.g., @Kaspi04]. A geometrically extended shell is supported from the observations by the AGN Watch campaigns[^3]. The opacity calculations require spectral as well as spatial knowledge of the emissivity of the BLR radiation field. Information about the cloud sizes and their spatial distribution can be deduced in general from reverberation mapping. Although most details on BLR physics and geometry have been derived by the study of radio-quiet AGN, no major differences in the broad-line flux between radio-quiet and -loud sources have been found so far [e.g., @Corbin92; @Wills93]. For the present calculations the number density $n_{\rm cl}\propto r^\alpha$ and cross-section $\sigma_{\rm cl}\propto r^\beta$ of the clouds are assumed to follow a power law with exponents $\alpha=-1.5$ and $\beta=0.6$ as derived by Kaspi & Netzer (1999). A fraction $d\tau_{BLR}=n_{\rm cl}\sigma_{\rm cl} dr$ of the central source luminosity $L_{\rm disk}$ is re-processed into line radiation such that the total BLR luminosity, assumed to be optically thin, is $L_{\rm BLR}=\tau_{\rm BLR}L_{\rm disk}$, and these lines are assumed to radiate isotropically. Observational support for a very narrow range of $\tau_{\rm BLR}$, i.e. the ratio of emitted to ionizing continuum photons, is provided by the observation of a linear correlation between Balmer line and optical disk luminosity in AGN over several orders of magnitude [e.g., @Yee80; @Shuder81]. This is compatible with photoionization/recombination theory which expects the H-line brightness to be correlated with the ionizing continuum flux as the line luminosity is driven by this ionizing disk continuum. A statistical blazar study of Celotti et al. (1997) implies $\tau_{\rm BLR}\simeq 0.01$, which is used in the following. Changing the here fixed parameters (e.g. $\tau_{\rm BLR}$, $r_{\rm in}$, $r_{\rm out}$, etc.) will alter the absolute values of the $\gamma$-ray attenuation, however, any redshift-dependence remains unaffected.
The calculation of the $\gamma$-ray opacity follows the procedure outlined in Donea & Protheroe (2003) for the geometrically thick shell case, but uses a more refined BLR line spectrum here. The ”average” BLR spectrum of Francis et al. (1991) together with the H$_\alpha$ line strength reported by Gaskell et al. (1981) sums up to 35 lines, with H$_\alpha$ and Ly$_\alpha$ being the strongest lines. For the present work the BLR spectrum is approximated as a two-component spectrum, $n(\epsilon)\propto \delta(\epsilon-\epsilon_{H\alpha}) + \delta(\epsilon-\epsilon_{Ly\alpha})$, with the total luminosity of all lines at $>4000~\AA$ ($\sim 32\%$) to be radiated at the strongest optical line (H$_\alpha$) wavelength, and the total line luminosity at $<4000~\AA$ ($\sim 68\%$) emitted at the strongest UV-line, Ly$_\alpha$. A refined treatment of the BLR spectrum is straight forward, but will not alter the results on the redshift-dependence of the local opacity, nor add qualitatively new insights to the subject of the present work.
GeV-photon absorption in quasar radiation fields
================================================
The calculation of the optical depth $$\begin{aligned}
d\tau_{\gamma\gamma}(E,l) & = & dl \int_{\epsilon_0}^\infty d\epsilon \int_{\mu_{\rm min}}^{\mu_{\rm max}} d\mu (1-\mu)
\sigma_{\gamma\gamma}(s) 2\pi \frac{dn(\epsilon,l,\mu)}{d\Omega}=\\
& = & \frac{dl}{4E^2}\int_{\epsilon_0}^\infty d\epsilon \epsilon^{-2}\int_{s_{\rm min}}^{s_{\rm max}} \int_{s_{\rm min}}^{s_{\rm max}} s \sigma_{\gamma\gamma}(s) \frac{dn(\epsilon,l,\mu)}{d\mu}\end{aligned}$$ with $E$ the primary $\gamma$-ray photon energy, $\mu=\cos{\xi}$, $s=2E\epsilon(1-\mu)$, $\epsilon_0=\min(\epsilon_{\rm thr},\epsilon_{\rm min})$, $\epsilon_{\rm thr}=(2 m_e c^2)^2/(2E(1-\mu_{\rm min}))$, $\epsilon_{\rm min}=s_{\rm min}/(2E(1-\mu_{\rm min}))$, $s_{\rm min} = 2E\epsilon(1-\mu_{\rm max})$ and $s_{\rm max} = 2E\epsilon(1-\mu_{\rm min})$ takes into account the full angle-dependent cross section [e.g., @Gould67; @Jauch76]. $\mu_{\rm min}$, $\mu_{\rm max}$ are determined by the respective geometry of the target photon field. The total cross section maximizes at $x=(1-y^{-1})^{1/2}\approx 0.7$, where $y=1/2E\epsilon(1-\mu)/(m_e^2 c^4)>1$ ($\xi$=photon interaction angle) is the threshold condition of the pair production process. The very prominent peak of the cross section near threshold reaches roughly $\sigma_{\gamma\gamma,\rm max}\approx 0.26\sigma_T$, where $\sigma_T$ is the Thomson cross section. The narrowness of the pair production cross section forces over half the interactions to occur in a small target photon energy interval $\Delta\epsilon\approx(4/3\pm 2/3)\epsilon^*$ centered on $\epsilon^*\approx 0.8(E/{\rm TeV})^{-1}$eV for a smooth broadband spectrum [@Aha-book].
Fig. \[fig1\] shows the resulting opacity from different distances $l_0$ of the $\gamma$-ray production region above the black hole to $l\rightarrow\infty$, in the accretion disk (red curves) and BLR radiation field (blue curves) for typical quasar accretion rates and BH masses. The two “bumps” in the blue opacity curves are the result of absorption in the H$_\alpha$ and Ly$_\alpha$ lines of the BLR, and smooth when a detailed multi($>30$)-line spectrum is used. In typical quasar environments the strength of local absorption is strongly dependent on the location of the emission region with respect to the target photon field. If the $\gamma$-ray emission region is located not well beyond the BLR, which is mandatory for $\gamma$-ray production that involve external photon fields, local $\gamma$-ray absorption features in FSRQ spectra have to be expected at $E(1+z)\geq$ several tens of GeV.
Black hole evolution and accretion rates
========================================
The key step of this work is the application of cosmological black hole and quasar evolution to the expected pair production opacity of $\gamma$-ray photons in AGN, with direct implications for studies of the evolution of the EBL.
With the availability of large AGN data archives enormous advances in BH demographics were made. The cosmic evolution of the BH mass accretion rate has recently been studied by Netzer & Trakhtenbrot (2007) on the basis of $\sim 10^4$ SDSS type-I radio-loud and radio-quiet AGN in a large redshift range ($z\leq 0.75$). Significantly higher accretion rates at larger redshifts were derived with an Eddington ratio of the accretion luminosity $L_{\rm acc}/L_{\rm edd} \propto (1+z)^{\delta(M_{\rm BH},z)}$ with $\delta(M_{\rm BH},z)\simeq 6-9$.
The intriguing similarity observed of the time history of star formation (SF) and BH accretion rate density [e.g., @Marconi04] as well as established relations between BH mass and some bulge properties of the host galaxy lead to the widely accepted picture of a joint evolution of QSOs/BHs and their host galaxies [see also @Barger01; @Marconi04]. [^4] Moreover, the observed BH mass function is consistent with that inferred from quasar luminosities for simple assumptions of the accretion efficiency [e.g., @Soltan82; @Cavaliere88; @Marconi04]. The evolution of their luminosity functions at both, soft and hard X-rays, shows the number density of fainter AGN to peak at lower redshifts than that of the brighter ones [e.g., @Ueda03; @Hasinger05; @Barger05]. This evidence of ”downsizing” directly leads to the picture of an ”anti-hierarchical” BH growth (i.e. high-mass BHs grow faster and low-mass BHs grow preferably at lower redshift), and has meanwhile been confirmed by multiple studies: an analysis based on the fundamental plane of accreting BHs [@Merloni04]; the phenomenological approach of Marconi et al. (2004) for determining the evolution of the BH mass function using observational constraints from the local BH mass function, the evolving X-ray luminosity functions and energetics from the X-ray background; determinations of low-luminosity high-redshift quasar luminosity functions by the GOODS collaboration [e.g., @Cristiani04]; the COMBO-17 survey at higher luminosities [@Wolf03], just to name a few. The combination of these observational findings with the theoretically advocated hierarchical clustering paradigm (based on cold dark matter) led to the awareness of feedback processes working during the process of BH mass growth in AGN systems [see e.g. @Granato04; @Lapi06; @Fontanot06].
While the parameter space is still large, the advocated evolution of BH growth and thus accretion rates has severe implications for the $\gamma$-ray quasar population where local $\gamma$-ray absorption in accretion disk and BLR radiation fields is potentially important.
In the following I use three exemplary models for the evolution of the cosmic BH accretion rate (Fig. \[fig2\]): a) the Netzer & Trakhtenbrot (2007) analysis, complemented by the Lapi et al. (2006) model for redshifts $>1$ where only modest evolution is proposed there, b) the ”anti-hierarchical” BH growth picture of Marconi et al. (2004), and c) a non-evolution scenario for comparison. The evolution of the accretion rate of model (b) has been derived by using the average BH growth history as published in Marconi et al. (2004) from redshift $z=3$ (where $100\%$ source activity has been assumed as an initial condition) to $z=0$ together with their supported accretion radiation efficiency of 0.1. Model (a) and (b) both show strongly redshift-dependent BH accretion rates with higher rates at larger redshifts for BH masses of $10^8-10^9$M$_\sun$, typical for quasars. For the present work the chosen models are used plainly as means and agencies to demonstrate how any evolution of accretion rates transforms into a redshift-dependence of the local optical pair production depth in strong-line quasars.
Is the local opacity redshift-dependent?
========================================
When studying the evolution of the EBL by means of a statistical analysis of signatures of $\gamma$-ray attenuation with effective opacity $\tau_{\rm eff}=\tau_{\rm EBL}+\tau_{\rm source}$ from extragalactic objects, recognizing and disentangling absorption taking place within the source system (”local opacity”; $\tau_{\rm source}$) and in the EBL during photon propagation to Earth (”EBL-caused opacity”; $\tau_{\rm EBL}$) is a crucial task. If both opacities each depend on redshift in the same direction, but with the exact value of each being unknown, the redshift parameter alone will not be sufficient to extract the evolution of the EBL-caused opacity. In order to assess the probability of this to take place in quasar-like AGN, in the following I will apply realistic evolving, as well as non-evolving, cosmic accretion rate curves (see Fig. \[fig2\]) to the calculation of the expected opacity from $\gamma$-ray absorption in their accretion disk and BLR radiation fields, using the procedure outlined in Sect. 2. Any evolution of accretion rates translates into an evolution of the target photon fields under consideration for photon-photon interactions, and thus into a redshift-dependence of the local opacity. For this study the position of the $\gamma$-ray production is fixed to $l_0=0.01$pc. A location of the $\gamma$-ray emission region close to the BLR is particularly preferred by leptonic as well as hadronic blazar emission models that require external photon fields like accretion disk and BLR radiation as an ingredient for $\gamma$-ray production through particle-photon interactions. If $\gamma$-ray production is located well beyond the BLR, $\gamma\gamma$-pair production on external photon fields ceases to be an important process, and so will external Compton scattering above pair production threshold owing to their comparable cross section values there, or photopion production on accretion disk and BLR photons.
The goal of this excercise is to verify a possible [*redshift-dependence*]{} of the local opacity. The absolute $\tau_{\gamma\gamma}$ values depend on the details of the target radiation fields and location of the $\gamma$-ray production [see e.g. @Donea03], and thus come with uncertainties that reflect the dimension of the free parameter space.
Fig. \[fig3\] shows the resulting opacity curves in the observer frame when using the evolutionary behaviour of accretion rates following Netzer & Trakhtenbrot (2007) and Lapi et al. (2006) for the target photon fields for $\gamma\gamma$-interactions. The fast evolution of accretion rates at low redshifts is apparent. The corresponding curves for a M$_{\rm BH}=10^9$M$_\sun$ are inflated towards higher $\tau_{\gamma\gamma}(E)$ by more than an order of magnitude, and for $E\geq 100$ GeV the photon-photon collisions occur predominantly in the accretion disk radiation field. Energy redshifting is the reason for the threshold and energy of maximum interaction probability, $E^*\approx 1.2/(1+z) (\epsilon/{\rm eV})^{-1}$TeV, to decrease with redshift. Note that for most energies $<1$TeV $\gamma$-ray absorption occurs preferentially near the increasing part of the $\tau_{\gamma\gamma}(E)$ function near pair production threshold. If the distance $l_0$ of the $\gamma$-ray production site from the disk rises beyond the BLR, $\gamma$-ray attenuation tails off as indicated in Fig.\[fig1\]. An increase in BLR size leads to an opacity decrease for an unchanged BLR luminosity by an amount that corresponds to the decrease in target photon density with increasing BLR volume.
A direct view on the redshift-dependence of the local opacity opens up by slicing Fig. \[fig3\] at the energies of interest. Fig. \[fig4\] shows the resulting $\tau_{\gamma\gamma}(z)$ curves at observer energy 100 GeV (all solid lines) and 300 GeV (all dashed lines) for all three evolution pictures for the accretion rate as shown in Fig. \[fig2\] and for typical quasar BH masses (M$_{\rm BH}=10^8$M$_\sun$: all red curves, M$_{\rm BH}=10^9$M$_\sun$: all blue curves). The black curves represent the optical depths for the situation of a non-evolving high-accretion rate disk with a high-mass BH, and of a non-evolving low-accretion rate system with a lower-mass BH. A strong redshift-dependence is apparent in almost all cases. Even for the case of non-evolving accretion rates the ”local opacities” show redshift-dependence, if $\gamma\gamma$ pair production occurs predominantly close to threshold, with an increasing slope of $\tau_{\gamma\gamma}(z)$ with redshift. The reason lies basically in the prominent peak of the pair production cross section, together with cosmological energy red-shifting: The prominent peak in the cross section leads to most photon-photon collisions occuring in a rather narrow energy range $\Delta\epsilon^*$ near threshold for smooth broadband target photon spectra. If those source spectra are located at cosmological distances, the energy red-shifting into the observer frame leads to the presented redshift-dependence of $\tau_{\gamma\gamma}$.
It is straight forward to then determine the evolution of the e-folding cutoff energy $E(\tau_{\gamma\gamma}=1)(z)$ for the series of accretion rate evolution models used so far. Fig. \[fig5\] shows a Fazio-Stecker-like presentation for the local absorption and compares with the typically expected behaviour for absorption of $\gamma$-rays in the evolving EBL [e.g. @Primack99; @Kneiske04; @Stecker06]. Potential EBL probes of GLAST’s LAT are located at a redshift $z\geq 0.5$. In all cases, and this includes also non-evolving accretion rates, decreases the e-folding cutoff energy with redshift, similar to the Fazio-Stecker relation for EBL-caused absorption. If local absorption in the external radiation fields of AGN leaves measurable imprints in the $\gamma$-ray spectra, these will be almost unavoidably redshift-dependent, and remain to be distinguished from the $\gamma$-ray opacity of the EBL.
Conclusions and discussion
==========================
This work is devoted to investigate whether any redshift-dependence of the opacity $\tau_{\gamma\gamma}$ of $\gamma$-rays produced in systems of strong-line blazars can be uniquely attributed to photon absorption in the EBL. The possible existence of further sources of redshift-dependence of $\tau_{\gamma\gamma}$ other than caused during propagation in the EBL will lead to ambiguities in estimating the EBL evolution. Strong-line quasars are in general considered as high-luminosity sources in the $\gamma$-ray domain, which have a marked probability of providing sufficient photon statistics to search for absorption breaks in a spectral analysis. Their bright external photon fields (i.e. accretion disk, BLR) are used in many high-energy emission models as a necessary ingredient for $\gamma$-ray production, e.g. via inverse Compton scattering. If this, however, takes place in the Klein-Nishina regime, and therefore above the pair production threshold, $\gamma$-ray absorption due to photon-photon interactions is of comparable importance. In some hadronic blazar models the observed $\gamma$-ray output is the result of re-distributing the injected nucleon energy via pair cascades that develop in external photon fields, and thus inevitably involves photon-photon pair production there. The focus of this study is therefore directed to $\gamma\gamma$-pair production in photon environments (here specifically accretion disk and BLR radiation field) of $\gamma$-ray loud blazars where the interactions of relativistic particles in external photon fields potentially contributes significantly to the $\gamma$-ray output above $\sim 50-100$ GeV.
The search for any redshift-dependence of this ”local” $\gamma$-ray opacity leads to the following conclusions:
- If the $\gamma$-ray emission region is located not well beyond the BLR, mandatory for $\gamma$-ray production that involve external photon fields, local $\gamma$-ray absorption features in strong-line quasar spectra have to be expected at $E(1+z) >$ several tens of GeV. Local $\gamma$-ray absorption in external AGN radiation fields ceases importance if the $\gamma$-ray production site is sufficiently distant from the BLR. While then evolutionary studies of the EBL by means of $\gamma$-ray absorption signatures are not affected, it will lead to important implications for the high-energy blazar emission models.
- Following recent progress in BH demographics, BH growth and corresponding accretion rates turn out to show a redshift-dependence with higher rates at larger redshifts. Correspondingly, the critical energy $E(\tau_{\gamma\gamma}=1)$ due to local absorption in quasar disk and BLR radiation fields decreases with redshift, similar to Fazio-Stecker’s relation for EBL absorption.
- Even for the case of no evolution of quasar disk accretion rates, $E(\tau_{\gamma\gamma}=1)$ decreases with redshift for sources of a given BH mass, at $z>1$ very similar to the Fazio-Stecker presentation of EBL absorption with the current knowledge of those systems. It results from the interplay of local absorption near pair production threshold and cosmological energy red-shifting.
- Any observed redshift-dependence of absorption features in blazars, that are prone to local $\gamma$-ray absorption, can therefore [*not*]{} serve as a unique signature for absorption occurring in the EBL radiation field. This complicates approaches for estimating the evolution of the EBL using GeV-sensitive instruments, that utilize the Fazio-Stecker relation or similar methods, and $\gamma$-ray AGN whose external photon fields are considered important in $\gamma$-ray production. As a result, it seems that only ”naked $\gamma$-ray jet sources” (i.e. AGN without noticable optical/UV radiation fields close to the $\gamma$-ray emission region; “true type-2 AGN”) are unbiased probes for studies of the evolution of the EBL on the basis of the Fazio-Stecker relation and using GeV instruments like GLAST, etc.
Consequently, an obvious choice for suitable candidate sources for this task would be blazars with particular weak or absent emission lines, generally classified as BL Lac objects (although exceptions exist, see Sect. 1). Predictions for the expected number of GeV BL Lacs range from several hundred [@Dermer07] to a few thousand [@Muecke00b] above the LAT sensitivity. It remains to be seen whether the near future GeV instruments will be detecting a sufficient number of suitable sources at $z\geq 0.5$ to allow a sensible analysis on the basis of naked $\gamma$-ray jet sources only.
Though the finding of this work adds fundamentally to already recognized complications (e.g. flux and spectral source variability) in analysis aiming to probe EBL evolution through the $\gamma$-ray horizon, it also offers new options for constraining AGN physics. The relevance of $\gamma$-ray absorption for cutting off the SED at the high energy end in the different blazar types, possibly as a consequence of the location of $\gamma$-ray production, could be probed by means of a statistical study of $\tau_{\gamma\gamma}(E,z)$ as a function of source type or object parameters. If simultaneously measured emission lines and/or accretion disk signatures indicate the presence of luminous photon fields external to the jet, the position of the $\gamma$-ray production site could be constrained. Monitoring both, the time history of the external target photon fields in AGN and the jet $\gamma$-ray flux, may offer independent verification of the importance of local absorption and extern inverse Compton scattering there, and allows conclusions on the $\gamma$-ray production location and on some properties of the BLR material [@Boettcher95]. At the same time, any non-detection of absorption features with sensitive GeV instruments puts limits on the external radiation fields in those AGN.
If local opacity shapes part of the $\gamma$-ray loud AGN population, its evolution may influence luminosity function and extragalactic $\gamma$-ray background contribution of AGN above the 50-100 GeV energy range. In both cases a redshift-dependence of the local opacity of $\gamma$-ray loud quasars will have far-reaching implications.
[**Acknowledgments**]{}
I’d like to thank for valuable feedback from the GLAST-LAT AGN science working group, in particular for very useful comments from C. Dermer, G. Madejski and B. Lott. I also thank F. Stecker for providing his model curves and for interesting discussions. This work is supported by the National Aeronautics and Space Administration under contract NAS5-00147 with Stanford University.
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[^1]: <http://www-glast.slac.stanford.edu/software/IS/glast_lat_performance.htm>
[^2]: Optical continuum variations that are generally attributed to accretion disk radiation in AGN are typically observed on time scales of months to years [e.g., @Peterson93]. This is significantly longer than typical variability time scales in the $\gamma$-ray domain, and thus allows the disk emission to be approximated as a constant photon field for the purpose of the present work.
[^3]: <http://www.astronomy.ohio-state.edu/~agnwatch/>
[^4]: Since the evolution of the EBL is strongly influenced by the cosmic SF history, one may in turn even speculate about an indirect imprint of the BH accretion history into the evolution of the EBL.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Stefan Fredenhagen,'
- Olaf Krüger
- and Karapet Mkrtchyan
---
Introduction {#sec:introduction}
============
In this paper, we investigate a Lagrangian formulation of higher-spin (HS) theories in arbitrary dimensions. The aim of this work is, in particular, to obtain restrictions for all possible independent interaction vertices of order $n\geq 4$ for massless higher-spin fields, extending the three-dimensional results of [@Fredenhagen:2019hvb]. Together with the earlier results on the cubic vertices [@Bengtsson:1986kh; @Metsaev:1991mt; @Metsaev:1991nb; @Metsaev:2005ar; @Manvelyan:2010jr; @Conde:2016izb; @Mkrtchyan:2017ixk; @Kessel:2018ugi] (see also [@Sagnotti:2010at; @Fotopoulos:2010ay; @Manvelyan:2010je; @Mkrtchyan:2011uh; @Sagnotti:2013bha; @Metsaev:2012uy]), this work intends to complete the classification of [*all independent interacting deformations*]{} of free massless HS Lagrangians [@Fronsdal:1978rb; @Campoleoni:2012th] to the lowest order in the deformation parameters (coupling constants) in Minkowski space-time of arbitrary dimensions $d\geq 3$.
HS Gravities [@Vasiliev:1990en; @Prokushkin:1998bq; @Vasiliev:2003ev] (see, e.g., [@bciv; @Didenko:2014dwa] for reviews) are generalisations of Einstein’s General Relativity which involve higher-spin gauge fields. These are symmetric tensor (Fronsdal) fields[^1] $\phi_{\m_1\dots\m_s}$, described by the Fronsdal action [@Fronsdal:1978rb] at free level, describing massless particles of spin $s$ upon quantisation[^2]. A set of free HS fields can be described by a Lagrangian, which is a sum of Fronsdal Lagrangians for spin $s$ fields. However, a full non-linear Lagrangian of *interacting* Fronsdal fields is not available to date.
Such theories are strongly constrained by gauge invariance, necessary for consistency. These gauge transformations extend those of General Relativity — space-time reparametrisations, or diffeomorphisms — to larger symmetries, involving gauge parameters that are Lorentz tensors of rank $(s-1)$ for each massless spin $s$ field. This extension of symmetries can potentially resolve some problems of General Relativity (singularities, quantisation problem, etc), making HS Gravity an attractive field of investigation.
The corresponding gauge transformation for free fields reads [^3] $$\label{freegauge}
\delta^{(0)} \phi_{\mu_{1}\dots \mu_{s}}= s\, \partial_{(\mu_{1}}\epsilon_{\mu_{2}\dots \mu_{s})}\,,$$ which generalises the well known expressions for massless vector fields ($s=1$) in gauge theory and the Graviton ($s = 2$) in linearised gravity theory.
The naive intuition from lower-spin model building suggests that one can pick an arbitrary collection of fields, including massless HS fields, and the gauge symmetries will partly constrain the interactions, leaving room for a large parameter space of theories. It turns out, that the severe constraints from HS gauge invariance rule out theories with an arbitrary choice of the particle content. Therefore, one is easily led to negative results if one chooses an arbitrary starting setup for constructing a theory with massless HS spectrum. This striking difference from textbook examples makes it tempting to conclude after some attempts that such theories cannot exist.
The problem can be traced to the global symmetries of the theory (see, e.g., [@Joung:2014qya]). Building such theories, therefore, can be addressed constructively by looking for suitable global symmetry algebras, which have to satisfy the so-called admissibility condition [@Konshtein:1988yg]. This condition rules out infinitely many potential candidate algebras (see, e.g., [@Joung:2019wwf]) and was crucial in deriving the list of admissible HS algebras [@Konstein:1989ij; @Vasiliev:1989re] and constructing full non-linear HS equations [@Vasiliev:1990en; @Prokushkin:1998bq; @Vasiliev:2003ev] in the frame formulation, proving the existence of a theory with massless HS fields. This theory, however, has unusual properties: there is an infinite tower of massless higher-spin fields with $s=0,1,2,\dots$ and a necessarily non-zero cosmological constant [@Fradkin:1986qy]. The need for a non-zero cosmological constant is related to diffeomorphism transformations, as explained in [@Fradkin:1987ks; @Joung:2013nma], essential for the Fradkin-Vasiliev solution to the Aragone-Deser problem [@Aragone:1979hx]. This argument, together with the holographic conjectures (see, e.g., [@Giombi:2016ejx; @Gaberdiel:2012uj]) motivated the intense studies of HS interactions, especially in $(A)dS_d$ background [@Manvelyan:2009tf; @Bekaert:2009ud; @Mkrtchyan:2010zz; @Vasilev:2011xf; @Joung:2011ww; @Joung:2012fv; @Boulanger:2012dx; @Francia:2016weg; @Sleight:2017fpc; @Bekaert:2014cea; @Sleight:2016dba].
The frame-like formulation of HS gravities that led to successful developments (including Vasiliev’s non-linear equations [@Vasiliev:1990en; @Prokushkin:1998bq; @Vasiliev:2003ev]) registered less progress so far in understanding the corresponding Lagrangian formulation. On the other hand, the metric-like formulation [@Fronsdal:1978rb; @Campoleoni:2012th; @Francia:2016weg] is a simple suitable setup for classifying interaction vertices and deriving restrictions on interacting Lagrangians. Here, we work in the framework of the Noether-Fronsdal program (see [@Bengtsson:1983pd; @Berends:1984rq; @Bekaert:2006us; @fms1; @Fotopoulos:2008ka; @Zinoviev:2008ck; @Boulanger:2008tg; @Manvelyan:2009vy; @Manvelyan:2010wp; @Ruehl:2011tk; @Bengtsson:2016hss; @Taronna:2017wbx; @Roiban:2017iqg; @Ponomarev:2017qab] for related literature and [@Kessel:2018ugi] for a recent summary of the status of the problem) to classify independent vertices of order $n\geq 4$ in arbitrary dimensions $d\geq 3$, generalising the $d=3$ results obtained earlier in [@Fredenhagen:2019hvb].
The situation is different only in three dimensions, where the interacting HS theories can admit arbitrary Einstein backgrounds (including Minkowski) as well as a finite spectrum of massless HS fields (see, e.g., [@Campoleoni:2010zq; @Campoleoni:2012hp; @Fredenhagen:2014oua; @Gwak:2015vfb; @Campoleoni:2017xyl]). However, such massless HS fields do not correspond to propagating particles in $d=3$, while the inclusion of matter leads to a situation similar to the higher-dimensional story in many ways.
The Noether-Fronsdal program is a systematic approach to perturbatively construct a Lagrangian $\mathcal{L}$ for an arbitrary interacting HS theory order by order. In this procedure, $\mathcal{L}$ is expanded in powers of small parameters $g_n$, $$\label{eq:noether-exp}
\mathcal{L}=\mathcal{L}_{2} + \sum_{n\geq 3} g_{n} \mathcal{L}_n + O(g_{n}^2)\,.$$ Here, $\mathcal{L}_2$ denotes the free Fronsdal Lagrangian and another sum over the different kinds of $n$-point vertices $\mathcal{L}_n$ is suppressed.
The action must be gauge-invariant, hence, $\delta \mathcal{L}$ equals a total derivative, where $\delta$ is obtained by a deformation of the free gauge transformation $\delta^{(0)}$, $$\delta=\delta^{(0)}+ \sum_{k \geq 1} \delta^{(k)}\,.$$ Here, the deformation $\delta^{(k)}$ is of $k$-th order in the fields. Since our aim is to find constraints for the *independent* vertex structures (i.e. linear in the coupling constants[^4]), the $n$-point vertex must satisfy $$\label{NE}
\delta^{(0)}\mathcal{L}_{n} + \delta^{(n-2)} \mathcal{L}_{2} = 0\ \ \text{up to total derivatives}\,.$$
In this paper, we find restrictions for all independent $n$-point vertices $\mathcal{L}_n$ for massless HS fields in arbitrary dimension $d \geq 3$, such that they satisfy Eq. (\[NE\]) [^5]. From that, we deduce a simple classification of vertices. For a summary of the explicit results, see the beginning of Section \[sec:discussion\].
The paper is organised as follows: In Section \[sec:preliminaries\], we set up notations and provide the mathematical framework for our analysis. There, we discuss that we have to analyse three different cases separately: Large dimensions $d \geq 2n - 1$ (see Section \[sec:case-2n-leq\]), low dimensions $d < n$ (see Section \[sec:case-n-\]) and the intermediate case (see comments in Section \[sec:discussion\]). We mostly consider parity-even vertices, but give a generalisation to parity-odd vertices in Section \[sec:parity-odd-vertices\]. We finally conclude in Section \[sec:discussion\].
Preliminaries {#sec:preliminaries}
=============
We want to constrain the $n$-point independent vertices $\mathcal{L}_n$ that may constitute the lowest order deformations of the free Lagrangian for massless HS fields. For this purpose, it is sufficient to restrict ourselves to the traceless and transverse (TT) sector of the Lagrangian as in [@Fredenhagen:2019hvb]. Hence, we can assume that the tensors $\phi_{\mu_1\dots\mu_s}$ that describe the gauge fields, are traceless, divergence-free and the corresponding free equation of motion is given by the (massless) Klein-Gordon equation, hence $$\label{eq:phi-trace-div}
g^{\mu_1 \mu_2} \phi_{\mu_1\dots\mu_s} = 0\,, \qquad \partial^{\mu_1} \phi_{\mu_1 \cdots \mu_s} = 0\,, \qquad
\partial^\nu \partial_\nu\, \phi_{\mu_1\dots\mu_s} \big|_\text{free e.o.m.} = 0\,.$$ The relaxation of these conditions will allow to reconstruct the full off-shell counterpart of the TT vertices as in [@Manvelyan:2010jr; @Francia:2016weg].
Vertex Generating Operators {#sec:vertex-operators}
---------------------------
It is very convenient to contract the indices of the fields each with an auxiliary vector variable $a^\mu$, $$\begin{aligned}
\phi^{(s)}(x,a)=\frac1{s!}\phi_{\mu_1\dots\mu_s}(x) a^{\mu_1}\cdots a^{\mu_s}\,.\label{phi}\end{aligned}$$ This has several advantages: First, we do not have to tackle expressions with too many indices and secondly, the tensor $\phi_{\mu_1 \cdots \mu_s}$ is by construction symmetric. We will also note later on, that the complexity of index contractions will be reduced a lot. For example, using the short-hand notation $P^\mu = \partial_{x^\mu}$ and $A^\mu = \partial_{a^\mu}$, the relations in Eq. (\[eq:phi-trace-div\]) simplify to $$\label{eq:Fierz-phi}
A^2\; \phi^{(s)} = 0\,, \qquad A\cdot P \;\phi^{(s)} = 0\,, \qquad P^2\;
\phi^{(s)}\big|_\text{free e.o.m.} = 0\,.$$ We call these relations collectively Fierz equations [@Fierz].
Now, each $n$-point vertex $\mathcal{L}_n$ in Eq. (\[eq:noether-exp\]) is a product of $n$ massless bosonic fields (and possibly derivatives thereof). But it has to be a Lorentz scalar, hence, all indices of the fields (and of the derivatives) must be fully contracted. For now, let us concentrate on parity-even vertices — we consider parity-odd vertices in Section \[sec:parity-odd-vertices\]. Then, we can write $\mathcal{L}_n$ in the following, very convenient way: $$\label{eq:L^n}
\mathcal{L}_n (x) = \mathcal{V} \left( \prod_{i = 1}^n \phi_i(x_i,a_i) \right) \Bigg|_{\substack{x_i = x\\ a_i = 0}}\,.$$ This needs some explanation:
- We use the notation set up in Eq. (\[phi\]) and dropped the spin labels of the fields: $\phi_i$ is a spin $s_i$ field, $\phi_i = \phi^{(s_i)}$.
- The term in brackets represents a function of the spacetime coordinates $x_i$ and the auxiliary vector variables $a_i$. The *vertex generating operator* $\mathcal{V}$ performs the index contractions between the fields $\phi_i$ as follows: Let $P^\mu_i = \partial_{x^\mu_i}$ and $A^\mu_i = \partial_{a^\mu_i}$ as in Eq. (\[eq:Fierz-phi\]). Then, $\mathcal{V}$ must be a polynomial in the following commuting variables: $$\begin{aligned}
\label{eq:zys}
z_{ij} = A_i \cdot A_j \big|_{1 \leq i \leq j \leq n}, \qquad y_{ij} = A_i \cdot P_j \big|_{1 \leq i, j
\leq n} , \qquad s_{ij} = P_i \cdot P_j \big|_{1 \leq i \leq j \leq n}\,.
\end{aligned}$$ The operator $z_{ij}$ induces a single contraction of indices between the fields $\phi_i$ and $\phi_j$, whereas $y_{ij}$ will take one index of the field $\phi_i$ and contract it with a derivative which acts on the field $\phi_j$. Finally, the operators $s_{ij}$ will introduce extra derivatives (a derivative of $\phi_i$ is contracted with a derivative of $\phi_j$). These are called Mandelstam variables.
- Since all of the indices in $\mathcal{L}_n$ have to be contracted, we discard all terms that still contain at least one of the auxiliary variables, when $\mathcal{V}$ acted on the terms in brackets. Thus, we set $a_i = 0$ in the end, which ensures that $\mathcal{L}_n$ is Lorentz invariant. Finally, we also set $x_i = x$. The splitting of the coordinates is useful to keep track of the derivatives acting on different fields, and has no physical consequences.
All in all, we translated the problem of ‘what is the most general form of the parity-even $n$-point vertex $\mathcal{L}_n$’ to the question ‘what is the most general form of the vertex generating operator $\mathcal{V}$ in the polynomial ring $\mathbb{R}[y_{ij}, z_{ij}|_{i \leq j}, s_{ij} |_{i \leq j}]$’. The connection between Lagrangian $\mathcal{L}_n$ and operator $\mathcal{V}$ is given by Eq. (\[eq:L\^n\]). We also ensured that $\mathcal{L}_n$ is Lorentz invariant.
There are two questions arising now: First of all, there are equivalence relations for Lagrangians: e.g., two Lagrangians that differ by a total derivative lead to the same action. We call them equivalent in this case. What does this imply for the corresponding vertex generating operators? Secondly, how do we have to constrain $\mathcal{V}$, such that $\mathcal{L}_n$ is gauge invariant? We present a general answer to these questions in the next two sections and give more details in Sections \[sec:case-2n-leq\] and \[sec:case-n-\].
Equivalence Relations for Vertex Generating Operators {#sec:equiv-relat-vert}
-----------------------------------------------------
We must take into account that different Lagrangians may describe the same theory. We say that they are *equivalent* in this case and evidently, we are only interested in $\mathcal{L}_n$ *up to equivalence*. When we encode the Lagrangians via vertex generating operators, we need to introduce a notion of equivalence for operators: *vertex operators $\mathcal{V}$ and $\mathcal{V}'$ are equivalent, $\mathcal{V} \approx \mathcal{V}'$, iff the two Lagrangians $\mathcal{L}_n$ and $\mathcal{L}_n'$, constructed from them via Eq. (\[eq:L\^n\]) are also equivalent.* We are hence only interested in $\mathcal{V}$ *up to equivalence* and summarise the different kinds of equivalence relations in the following.
The first kind of equivalence relations arises from field redefinitions $\phi_i \mapsto \phi_i + \delta \phi_i$, where $\delta \phi_i$ is non-linear in the fields. These do not change the theory, but affect the Lagrangian. For example, terms in $\mathcal{L}_2$ may contribute to $\mathcal{L}_n$ when the fields are redefined non-linearly. But in this way, the $n$-point vertices only change by terms that vanish when the free equations of motion are imposed. We say that two Lagrangians are equivalent, when they are related by such field redefinitions and deduce from Eq. (\[eq:L\^n\]) that we can choose $\mathcal{V}$ to be independent of $s_{ii}$. Furthermore, we assume that $\mathcal{V}$ does not depend on $z_{ii}$ and $y_{ii}$, because the fields are traceless and divergence free.
Mathematically speaking, we impose the equivalence relations $$\label{eq:equiv1}
y_{ii} \approx 0 , \qquad z_{ii} \approx 0 , \qquad s_{ii} \approx 0$$ and deduce that each operator in the ideal $\langle y_{ii}, z_{ii}, s_{ii} \rangle \subset \mathbb{R}[y_{ij}, z_{ij}|_{i \leq j}, s_{ij} |_{i \leq j}]$ is equivalent to $0$. Hence, we can construct equivalence classes of vertex generating operators, $$[\mathcal{V}] \in \frac{\mathbb{R}[y_{ij}, z_{ij}|_{i \leq j}, s_{ij}|_{i \leq j}]}{\langle y_{ii}, z_{ii},
s_{ii}\rangle}\,.$$ But the quotient ring is isomorphic to the subring $\mathcal{R} = \mathbb{R}\big[y_{ij}|_{i \neq j}, z_{ij}|_{i < j}, s_{ij}|_{i < j}\big]$, $$\frac{\mathbb{R}[y_{ij}, z_{ij}|_{i \leq j}, s_{ij}|_{i \leq j}]}{\langle y_{ii}, z_{ii}, s_{ii}\rangle}
\simeq
\mathcal{R} \subset \mathbb{R}[y_{ij},
z_{ij}|_{i \leq j}, s_{ij}|_{i \leq j}]\,,$$ so we can choose the vertex generating operator as $\mathcal{V} \in \mathcal{R}$. In other words, we simply dropped the dependence of $\mathcal{V}$ on $y_{ii}, z_{ii}$ and $s_{ii}$.
Secondly, acting with the operator $D^\mu = \sum_{j = 1}^n P_j^\mu$ on the term in brackets in Eq. (\[eq:L\^n\]) gives a total derivative in the Lagrangian. This does not change the action and hence, does not affect the theory. Therefore, we impose the equivalence relations $$\label{eq:equiv2}
A_i \cdot D = \sum_{j = 1}^n y_{ij} \approx 0,\qquad P_i \cdot D = \sum_{j = 1}^n s_{ij} \approx 0 \,.$$ These together generate an ideal $\mathcal{I}_D \subset \mathcal{R}$ and in the following, we consider equivalence classes of vertex generating operators in the quotient ring $$[\mathcal{V}] \in \frac{\mathcal{R}}{\mathcal{I}_D}\,.$$ As for the equivalence relations in Eq. (\[eq:equiv1\]), we could choose a convenient representative $\mathcal{V}$ in $\mathcal{R}$, but it turns out to be better to keep the quotient ring structure for now.
A last equivalence stems from ‘Schouten identities’, i.e. relations following from over-antisymmetrisation of spacetime indices. These spacetime dimension-dependent identities are exact relations at the Lagrangian level. In the polynomial ring $\mathcal{R}$, however, we forgot that we work in $d$ dimensions. Therefore, we have to impose Schouten identities as equivalence relations for vertex generating operators [^6], which form an ideal $\mathcal{I}_S \subset \mathcal{R}$ as follows: Let $b = ( P_1 , \ldots , P_n, A_1, \ldots A_n )$ be a vector of derivative operators and consider the symmetric $2n \times 2n $ matrix $$\label{defofB}
\mathcal{B} =
\big(
b_K \cdot b_L
\big)\big|_{K,L \in (1,\ldots, 2n)} =
\begin{pmatrix}
\mathcal{S} & \mathcal{Y}^T \\ \mathcal{Y} & \mathcal{Z}
\end{pmatrix}\,.$$ Here, $\mathcal{S} = (s_{ij})$, $\mathcal{Y} = (y_{ij})$, $\mathcal{Z} = (z_{ij})$ are symmetric $(n\times n)$-matrices with elements in $\mathcal{R}$. With the equivalence relations in Eq. (\[eq:equiv1\]), the diagonal elements of $\mathcal{S}$, $\mathcal{Y}$ and $\mathcal{Z}$ vanish equivalently. We also keep in mind that there are further equivalence relations from Eq. (\[eq:equiv2\]) which introduce a linear relation among the first $n$ rows (and columns) of $\mathcal{B}$, but we do not apply them right now.
Then, the ideal $\mathcal{I}_S$ is generated by all $(d + 1) \times (d+1)$ minors of $\mathcal{B}$. We show this in a moment, but note first that this implies that $\mathcal{I}_S$ is trivial for $d \geq 2n - 1$. Indeed, in this case, there is only one such minor, namely when equality holds. This minor is $\det \mathcal{B}$, which is equivalent to zero due to the equivalence relations in $\mathcal{I}_D$ (the first $n$ rows add up to a total derivative). Now we show that for $d < 2n - 1$, the above statement is true. Indeed, remove $(2n - d-1)$ rows and columns from $\mathcal{B}$, such that only the rows $K_1, \ldots, K_{d+1} \in (1,\ldots , 2n)$ and the columns $L_1, \ldots, L_{d+1} \in (1,\ldots , 2n)$ remain and call the resulting $(d+1) \times (d+1)$-matrix $M$. Then, $$\begin{aligned}
\label{eq:schouten-proof}
\det M &= \delta_{\nu_1}^{\mu_1} \cdots \delta_{\nu_{d+1}}^{\mu_{d+1}}\, B^{\mu_1}_{[K_1} \cdots B^{\mu_{d+1}}_{K_{d+1}]}\,
B^{\nu_1}_{L_1} \cdots B^{\nu_{d+1}}_{L_{d+1}} \nonumber \\
&= \delta_{\nu_1 \cdots \nu_{d+1}}^{\mu_1 \cdots \mu_{d+1}}\, B^{\mu_1}_{K_1} \cdots B^{\mu_{d+1}}_{K_{d+1}}\,
B^{\nu_1}_{L_1} \cdots B^{\nu_{d+1}}_{L_{d+1}}\end{aligned}$$ and acting with it on the term in brackets in Eq. (\[eq:L\^n\]) yields a term in the Lagrangian with over-antisymetrised indices. On the other hand, each term in the Lagrangian with over-antisymmetrised indices corresponds to a vertex generating operator $\mathcal{V}$ that contains a factor of the form on the rhs of Eq. (\[eq:schouten-proof\]) for a certain set of indices $K_i,L_i \in (1,\ldots,2n)$. Hence, $\mathcal{V} \in \mathcal{I}_S$.
At this step, it is convenient to introduce the notion of the *level* of a Schouten identity. To this end, let us first define the *level* of the rows and columns of $\mathcal{B}$ as follows: The first $n$ rows and columns of $\mathcal{B}$ are of level $0$ and all others are of level $1$. Furthermore, each $(d+1)\times(d+1)$-submatrix $M$ of $\mathcal{B}$ that is obtained by removing rows and columns inherits those row and column levels from $\mathcal{B}$. Then, the sum of row and column levels of $M$ equals the power of $A_i^\mu$ operators in $\iota_d(\det M)$. This is what we call the *level* of the Schouten identity $\det M = 0$. Denote by $I(k)$ the ideal generated by all Schouten identities of level $k$, then we have $$\label{eq:grading}
\mathcal{I}_S = \sum_{k = 0}^{2d + 2} I(k)\,,$$ where again, $d$ denotes the spacetime dimension.
Now, we consider three cases:
- For large dimensions, $d \geq 2n-1$, as discussed before, there are no non-trivial Schouten identities at all (the only possible Schouten identities arise in the case $d=n+1$, but they are zero up to total derivatives, so they are already contained in $\mathcal{I}_{D}$). This case is much simpler and we treat it separately in Section \[sec:case-2n-leq\].
- For large values of $n$, $d < n$, only the subideal $I(0)$ might be trivial (namely for $d + 1 = n$, where the level $0$ Schouten identities vanish up to a total derivative and thus are already contained in $\mathcal{I}_D$). Thanks to the variety of Schouten identities available, we are able to perform a lot of simplifications. We treat this case in Section \[sec:case-n-\].
- In the intermediate case $2n-2 \geq d \geq n$ only the ideals of level $2d - 2n + 4,\ldots, 2n$ are non-trivial. We will not study this case in full detail here, but a general characterisation of the corresponding vertices is given in Section \[sec:discussion\].
All in all, we have now considered all possible equivalences for parity-even Lagrangians. Because of the freedom of field redefinitions, we consider $\mathcal{V} \in \mathcal{R}$ and we divide out the ideals generated by total derivatives ($\mathcal{I}_D$) and Schouten identities ($\mathcal{I}_S$), $$\label{eq:[V]}
[\mathcal{V}] \in \frac{\mathcal{R}}{\mathcal{I}} \,, \qquad\qquad \mathcal{I} = \mathcal{I}_S + \mathcal{I}_D\,.$$
Imposing Gauge Invariance {#sec:impos-gauge-invar}
-------------------------
Finally, we require that $\mathcal{L}$ is gauge invariant, i.e. it satifies Eq. (\[NE\]). What does this imply for the corresponding vertex generating operator $\mathcal{V}$? Note first that the second term in Eq. (\[NE\]) vanishes when the free equations of motions are imposed. In other words, the requirement of gauge invariance for the independent vertex structures reads $$\label{eq:gauge-0}
\delta_k^{(0)} \mathcal{L}_n^{\,} \approx 0\,,$$ where $\delta_k^{(0)}$ is the free gauge transformation of the field $\phi_k$ (see Eq. (\[freegauge\])).
The latter can be simplified by contracting the tensor for the gauge parameter in Eq. (\[freegauge\]) with auxiliary vector variables $a^\mu$ as well, $$\epsilon^{(s-1)}(x, a)=\frac1{(s-1)!}\epsilon_{\mu_1\dots\mu_{s-1}}(x) a^{\mu_1}\cdots a^{\mu_{s-1}}\,.$$ Again, we drop the spin index, $\epsilon_k = \epsilon^{(s_k - 1)}$, and the linearised gauge transformation of the $k$-th field $\phi_k$ in Eq. (\[freegauge\]) reads $$\delta^{(0)}_k \phi_k (x_k,a_k) = a_k\cdot P_k\;\epsilon_k (x_k,a_k), \qquad \text{(no sum)}.$$ Note that this gauge transformation must be consistent with Eqs. (\[eq:Fierz-phi\]). We therefore impose the Fierz equations also for the gauge parameter.
All in all, we can now impose the restrictions for the vertex generating operators $\mathcal{V}$ from gauge invariance, Eq. (\[eq:gauge-0\]): $$\delta_k^{(0)} \mathcal{L}_n = \mathcal{V} \, a_k\cdot P_k \left( \epsilon_k(x_k,a_k) \prod_{1\leq i \leq n}^{i \neq k}
\phi_i(x_i,a_i) \right) \Bigg|_{\substack{x_i = x\\ a_i = 0}} \approx 0\,.$$ Since all the auxiliary vector variables $a_i$ are set to zero in the end, it immediately follows that $\mathcal{L}_n$ is gauge invariant if and only if the corresponding vertex generating operator $\mathcal{V} \in \mathcal{R}$ (via Eq. (\[eq:L\^n\])) satisfies $$\label{eq:gauge-inv}
\text{for all}\ k \in \{1,\ldots,n\} \,: \quad [\mathcal{V}, a_k \cdot P_k] =: D_k \mathcal{V} \in
\mathcal{I}_S + \mathcal{I}_D\,.$$ Here, we defined the operators $D_k$ of gauge variations. These act as linear first-order differential operators on the vertex $\mathcal{V}$: $$\begin{aligned}
D_k=\sum_{j=1}^n \Big( y_{jk}\frac{\partial}{\partial z_{kj}}+s_{kj}\frac{\partial}{\partial y_{kj}}\Big)\,.\label{gvo}\end{aligned}$$
The case $2n-1 \leq d$ {#sec:case-2n-leq}
======================
We start with the case of sufficiently high space-time dimensions where the classification of vertices is the simplest because there are no Schouten identities and we only have to take into account total derivatives, hence, $\mathcal{I} = \mathcal{I}_D$.
Gauge Invariants {#sec:gauge-invariants}
----------------
To derive the $n$-th order independent vertices we first recall the constraints on the vertex generating operators $y_{ij}\,, z_{ij}\,, s_{ij}$ in Eqs. (\[eq:equiv1\]) and (\[eq:equiv2\]) and count the independent variables:
\[constraints\] $$\begin{aligned}
y_{ii}\approx 0\,,\quad \sum_{j=1}^n y_{ij}\approx 0\,, \quad n(n-2)\; \text{variables}\; y_{ij}\,,\\
z_{ij}=z_{ji}\,,\quad z_{ii}\approx 0\,,\quad \frac{n(n-1)}2\; \text{variables}\; z_{ij}\,,\\
s_{ij}=s_{ji}\,, \quad s_{ii}\approx 0\,,\quad \sum_{j=1}^n s_{ij} \approx 0\,,\quad \frac{n(n-3)}{2}\; \text{variables}\; s_{ij}\,.\end{aligned}$$
The vertex depends altogether on $2n(n-2)$ variables, and is subject to $n$ linear differential equations that stem from Eqs. (\[eq:gauge-inv\]) and (\[gvo\]) [^7]. If these differential equations are linearly independent, the solution should depend on $2n(n-2)-n=n(2n-5)$ variables.
For cubic vertices, $n=3$, this would give three invariants, while we know that the solution depends on four invariants $y_{12}\,,\; y_{23}\,,\; y_{31}$ and $ G=y_{12}\,z_{23}+y_{23}\,z_{31}+y_{31}\,z_{12}\,$. The reason is that the three equations in that case are not linearly independent: $y_{12}\, D_1+y_{23}\, D_2+y_{31}\, D_3=0$. Due to this relation, we have, e.g., the Yang-Mills cubic vertex $V_3^{YM}=G$ and the Einstein-Hilbert cubic vertex $V_3^{EH}=G^2$.
On the other hand, one can easily see from Eq. (\[gvo\]) that the operators $D_k$ are linearly independent for $n\geq 4$. Hence, the general form of the vertices should depend on $n(2n-5)$ invariants composed of $s_{ij}\,,\; y_{ij}\,,\; z_{ij}\,$.
At this point, we introduce gauge invariant operators, which are more suitable as the building blocks of $n$-th order vertices. These are given through the following variables: $$\begin{aligned}
s_{ij}=s_{ji}\,\qquad & \frac{n(n-3)}2 \; \text{variables}\,,\\
c_{ij}=y_{ij}\,y_{ji}-s_{ij}\,z_{ij}=c_{ji}\,,\qquad & \frac{n(n-1)}2\; \text{variables}\,,\label{defc2} \\
c_{i,jk}=y_{ij}\,s_{ik}-y_{ik}\,s_{ij}=-c_{i,kj}\,, \qquad & \frac{n(n-2)(n-3)}2 \; \text{variables}\,.\label{defc3}\end{aligned}$$ It is easy to show that these expressions are gauge invariant: $$\begin{aligned}
D_k\, s_{ij}=0\,,\quad D_k\, c_{ij}=0\,,\quad D_k\, c_{i,jl}=0\,.\end{aligned}$$
Counting the number of the variables $s_{ij}$ and $c_{ij}$ is straightforward. In order to count the number of $c_{i,jk}$ variables, we count separately the number of choices for $i$ and the number of choices for the antisymmetric pair $jk$ for a given $i$ and multiply them. Naively, we choose $i$ in $n$ possible ways, and the antisymmetric pair $jk$ takes values in $\{i+1\,,\dots, i-2\; (\text{mod n})\}$, therefore takes $\frac{(n-2)(n-3)}2$ values, hence the number of $c_{i,jk}$’s is given above. These variables $c_{i,jk}$ are not linearly independent though, satisfying the following relations: $$\begin{aligned}
3\,c_{i,[jk}\, s_{i|l]}\equiv c_{i,jk}\, s_{il}+c_{i,kl}\, s_{ij}+c_{i,lj}\, s_{ik}=0\,,\label{cijk redundancy}.\end{aligned}$$ These naively are $\frac{n(n-2)(n-3)(n-4)}6$ many, given by multiplying the $n$ possible choices of $i$ and $\frac{(n-2)(n-3)(n-4)}6$ choices of the antisymmetric triple $jkl$. But again, this counting is redundant, due to linear relations between equations, involving different choices of $jkl$. These relations are also given by adding another $s_{im}$ and antisymmetrising the four indices $jklm$. This chain of reducibility can be resummed to get all linearly independent variables of $c_{i,jk}$. This is done by finding the number of possible values of $jk$ antisymmetrised pairs that correspond to the independent variables, by summing up with changing signs the numbers of components of antisymmetric tensors of $gl(n)$, starting from rank two: $$\begin{aligned}
\sum_{i=2}^{n-2} (-1)^i {{n-2}\choose i}=n-3\,.\end{aligned}$$
This means that the number of independent variables $c_{i,jk}$ is $n(n-3)$. We see that the variables $c_{i,jk}$ are redundant and we choose the following set of independent variables: $$\label{defYgenerald}
Y_{i}^{j} := c_{i,i+j\, i+1}\, ,$$ where now $j=2,\dots, n-2$, taking $n-3$ possible values (indices are always meant modulo $n$). Thus, the number of variables $Y_{i}^{j}$ is altogether $n(n-3)$. It is elementary to show that any other variable $c_{i,jk}$ can be expressed through $Y_{i}^{j}$ using Eq. : $$\begin{aligned}
\label{eq:Y(c)}
c_{i,jk}=\frac{c_{i,j\, i+1 }\,s_{ik}-c_{i,k\, i+1}\,s_{ij}}{s_{i\,i+1}}=\frac{Y_{i}^{j-i}\,s_{ik}-Y_{i}^{k-i}\,s_{ij}}{s_{i\,i+1}}\, .\end{aligned}$$
Therefore altogether we have: $$\begin{aligned}
\frac{n(n-3)}2+\frac{n(n-1)}2+n(n-3)=n(2n-5)\; \text{invariants}.\end{aligned}$$ Given that the number of independent invariants $s_{ij}\,,\, c_{ij}\,,\, Y_{i}^{j}$ is the same as the number of variables that should constitute the building blocks of $n$-th order independent vertices, it is already tempting to conclude that the most general solution is an arbitrary function of these variables. We will show this now, by allowing for dividing by Mandelstam variables and making the replacements $$\begin{aligned}
z_{ij}=\frac1{s_{ij}}(y_{ij}\,y_{ji}-c_{ij})\,,\end{aligned}$$ and, consecutively, $$\begin{aligned}
y_{i i+j}=\frac1{s_{i i+1}} (y_{i i+1}\,s_{i i+j} - Y_{i}^{j})\,, \qquad \qquad j = 2,\ldots, n-2 \mod n\,,\end{aligned}$$ expressing the vertex operator in terms of the variables $s_{ij},\, c_{ij},\, Y_{i}^{j}$ and $y_{i i+1}$. Correspondingly, the gauge variation in terms of these variables is generated by the operators $$\begin{aligned}
\label{Dksinglederivative}
D_k=s_{k k+1}\frac{\partial}{\partial y_{k k+1}}\,,\end{aligned}$$ which turn into a single derivative. Therefore, the new gauge invariance equations for the vertex operator give: $$\begin{aligned}
D_k \mathcal{V}(s_{ij},c_{ij},Y_{i}^{j},y_{i i+1})=s_{k k+1}\frac{\partial}{\partial y_{k k+1}}\mathcal{V}(s_{ij},c_{ij},Y_{i}^{j},y_{i i+1})\approx 0\,.\label{gie}\end{aligned}$$ If we go to a set of independent variables, we can conclude that the $y_{ii+1}$-derivative is equal to zero, and the vertex can be solely written in terms of the gauge invariant combinations $s_{ij},\, c_{ij},\, Y_{i}^{j}$. A gauge invariant local vertex generating operator $\mathcal{V}$ in high enough dimension ($d \geq 2n - 1$) is then in one-to-one correspondence to a polynomial in $s_{ij}$, $c_{ij}$, $Y_{i}^{j}$, allowing at most those inverse powers of Mandelstam variables, such that $\mathcal{V}$ becomes polynomial in the variables $s_{ij}$, $y_{ij}$ and $z_{ij}$, when re-expressing the combinations $c_{ij}$ and $Y_i^j$.
Building blocks of vertices
---------------------------
We have just shown that any gauge-invariant vertex $\mathcal{V}$ of order $n$ for $d\geq 2n-1$ can be rewritten as a function of the invariants $c_{ij}$, $Y_{i}^{j}$ and $s_{ij}$. This function is polynomial in $c_{ij}$ and $Y_i^j$, but can contain inverse powers of the Mandelstam variables $s_{ij}$.
In this subsection we address the question: ‘what is the most general form of this function if we assume that the vertex is local?’ First of all it is clear that any polynomial of $c_{ij}$, $Y_{i}^{j}$ and $s_{ij}$ defines a local and gauge-invariant vertex. Now let us analyse the case that the vertex contains a single pole in one $s_{ij}$ when written in terms of the invariants: $$\mathcal{V} = \frac{1}{s_{ij}}Q (c_{ij},Y_{i}^{j},s_{kl})\,.$$ Here, we assume that the polynomial $Q$ does not explicitly depend on this specific $s_{ij}$. For $\mathcal{V}$ to be local, the inverse of $s_{ij}$ has to be compensated by a term proportional to $s_{ij}$ that arises when the invariants are rewritten in terms of $s_{ij},y_{ij},z_{ij}$. One can show that in this case, $\mathcal{V}$ is a linear combination of $$b_{ijk\ell}=\frac{1}{s_{ij}}\left(c_{ij}\,s_{ik}\,s_{j\ell} -c_{i,jk}c_{j,i\ell}\right) \quad \text{and}\quad \frac{1}{s_{ij}}\left(s_{ik}c_{i,j\ell}-s_{i\ell}c_{i,jk} \right)$$ multiplied by polynomials in $c_{ij}$, $Y_i^j$ and the Mandelstam variables [^8]. The second expression is simply equal to $c_{i,k\ell }$ (see Eq. ), so it is again a polynomial in $s_{kl}$ and $c$ variables. The first one can be rewritten as $$b_{ijk\ell} = \det \begin{pmatrix}
s_{ij} & s_{ik} & y_{ji}\\
s_{\ell j} & s_{\ell k}& y_{j\ell}\\
y_{ij} & y_{ik} & z_{ij}
\end{pmatrix} +s_{k\ell} c_{ij} \, .$$ Hence, up to a shift by a polynomial in Mandelstam and $c$ variables, the building block $b_{ijk\ell}$ can be written as a determinant of a $3\times 3$-submatrix of the matrix $\mathcal{B}$ (see Eq. ). This nicely fits with the observation that also the $c$ invariants are just minors of $\mathcal{B}$, $$c_{ij} = - \det \begin{pmatrix}
s_{ij} & y_{ji}\\
y_{ij} & z_{ij}
\end{pmatrix} \quad ,\quad c_{i,jk} = \det \begin{pmatrix}
s_{ik} & s_{ij}\\
y_{ik} & y_{ij}
\end{pmatrix}\, .$$ Notice that these minors as well as the $(3\times 3)$-example above have the property that each $(n+i)$-th row (column) of the second block is accompanied by the corresponding ($i$-th) row (column) of the first block. This ensures gauge invariance because the $i$-th gauge variation transforms the $(n+i)$-th row (column) into the $i$-th row (column) leading to a vanishing determinant. Translating such a building block to the fields, the resulting expression is a pure curvature term: a tensor index of a field $i$ occurs in an antisymmetric combination with an index of a derivative acting on the field.
Of course all such minors can be written as polynomials in the $c$ invariants with negative powers of Mandelstam variables allowed. This can be explicitly seen when in the determinant we add to the $(j+n)$-th column the $j$-th column multiplied by $-\frac{y_{jj+1}}{s_{jj+1}}$, and similarly we add to the $(i+n)$-th row the $i$-th row multiplied by $-\frac{y_{ii+1}}{s_{ii+1}}$. Then one arrives at $$\det \begin{pmatrix}
(s_{ij}) & (y_{ji})\\
(y_{ij}) & (z_{ij})
\end{pmatrix} = \det \begin{pmatrix}
(s_{ij}) & \Big(\frac{1}{s_{jj+1}} c_{j,ij+1} \Big)\\
\Big(\frac{1}{s_{ii+1}}c_{i,ji+1} \Big) & \Big(\frac{1}{s_{ij}s_{ii+1}s_{jj+1}} (c_{j,j+1\,i}c_{i,i+1\,j}-s_{ii+1}s_{jj+1}c_{ij}) \Big)
\end{pmatrix}\, .\label{InvariantMinor}$$ Here, the labels $i$ and $j$ only run through the values that correspond to the rows and columns present in the minor that we are considering.
There is one additional possibility due to the linear dependencies in $\mathcal{B}$: we can take the determinant of the $(2n-1)\times (2n-1)$ submatrix that is obtained by deleting, e.g., the first row and column. This is still gauge invariant because the gauge transformation with respect to the variables of the first field transforms the first row of the second block into a linear combination of the $n-1$ rows of the first block, and the determinant still vanishes. Expressed in terms of fields, such a building block corresponds to a term of the form $$\delta^{[\mu_{2}\dotsb \mu_{2n}]}_{\nu_{2}\dotsb \nu_{2n}}\phi^{(1)}_{\mu_{n+1}}{}^{\nu_{n+1}}\partial_{\mu_{2}}\partial^{\nu_{2}}\phi^{(2)}_{\mu_{n+2}}{}^{\nu_{n+2}} \dotsb \partial_{\mu_{n}}\partial^{\nu_{n}}\phi^{(n)}_{\mu_{2n}}{}^{\nu_{2n}}\, ,\label{Lovelock}$$ which is gauge invariant up to total derivatives. This Lovelock-type vertex can be generalised in a way, where one computes the determinant of the minor of $\cal B$ containing $n-1$ rows and columns from the first block and arbitrary number $m$ of rows and columns from the second block, but these do not introduce new building blocks [^9].
Note that also the Mandelstam variables $s_{ij}$ are $1\times 1$-minors. It is tempting to speculate that all gauge invariant local vertices $\mathcal{V}$ can be written as polynomials in the types of minors of $\mathcal{B}$ mentioned above. If this speculation is correct, then for a spin configuration $s_1\geq s_2\geq \dots \geq s_n$ the lowest number of derivatives in a local vertex is $s_1+s_2+\dots + s_{n-1}$.
Lower dimension: dealing with Schouten identities {#sec:lowerd}
=================================================
In the previous section we have discussed the gauge-invariant vertices when we do not have to consider Schouten identities. When we go to lower dimensions, the ideal of relations is enlarged from $\mathcal{I}_{D}$ to $\mathcal{I}_{D}+\mathcal{I}_{S}$. Gauge-invariant vertex generating operators for large dimensions still define gauge-invariant operators in lower dimensions, but a priori, enlarging the ideal could have two effects: First, inequivalent vertices become equivalent, and second, new vertices arise that are gauge-invariant only up to the now larger set of equivalence relations. We will show in the following that the latter possibility does not lead to new equivalence classes of vertices, but that for all gauge-invariant vertex generating operators there are equivalent operators[^10] which are gauge-invariant already without the use of Schouten identities.
To show this, start with a vertex generating operator $\mathcal{V}$ as a polynomial in $s_{ij},y_{ij},z_{ij}$ that in $d$ dimensions is gauge invariant, $$D_{k}\mathcal{V} \in \mathcal{I}_{D}+\mathcal{I}_{S}\, .$$ In $\mathcal{V}$ we now express the variables $z_{ij}$ and $y_{ij}$ in terms of $c_{ij}$, $Y_{i}^{j}$ and $y_{ii+1}$, $$\mathcal{V} = Q_{\mathcal{V}} (c_{ij},Y_{i}^{j},y_{ii+1})\, ,$$ where $Q_{\mathcal{V}}$ is a polynomial in the given variables. We suppressed the dependence on Mandelstam variables which can also occur with negative powers. In these variables, the gauge variation $D_{k}$ is written as a derivative with respect to $y_{kk+1}$ as in Eq. , so we have $$D_{k}\mathcal{V}= s_{kk+1}\frac{\partial}{\partial y_{kk+1}} Q_{\mathcal{V}} (c_{ij}, Y_{i}^{j},y_{ii+1}) \in \mathcal{I}_{D}+\mathcal{I}_{S}\, .$$ When we expand $Q$ in powers of $y_{12}$, $$Q_{\mathcal{V}} (c_{ij}, Y_{i}^{j},y_{ii+1}) = \sum_{n=0}^{N} q_{n} (c_{ij}, Y_{i}^{j},y_{23},\dotsc ,y_{n1}) (y_{12})^{n} \, ,$$ we apply $(D_{1})^{N}$ to the expression and obtain $$N! (s_{12})^{N} q_{N} \in \mathcal{I}_{D}+\mathcal{I}_{S}\, .$$ When we allow us to divide by Mandelstam variables, we conclude that $$q_{N} \in \frac{1}{(s_{12})^{N}} \left(\mathcal{I}_{D}+\mathcal{I}_{S} \right)\, .$$ Similar relations can be found for all other terms in the expansion in $y_{12}$ and also in the other variables $y_{ii+1}$. Hence, we find that $$\mathcal{V} - Q_{\mathcal{V}}(c_{ij}, Y_{i}^{j},y_{ii+1})\big|_{y_{ii+1}=0} \in \frac{1}{\Delta} \left(\mathcal{I}_{D}+\mathcal{I}_{S} \right)\, ,$$ where $\Delta$ is a product of powers of Mandelstam variables. Therefore, $\mathcal{V}$ is equivalent to an operator depending only on $c_{ij}$ and $Y_{i}^{j}$ which already defines a gauge invariant vertex operator without the need of Schouten identities.
We conclude that in all dimensions, vertex generating operators can be expressed in terms of the operators identified for large dimensions. The main task for lower dimension is therefore to work out explicitly the equivalences between such operators that are induced by Schouten identities. Here, the case of low dimensions, $d<n$, is special because many Schouten identities arise that reduce the independent equivalence classes considerably. This will be discussed in detail in the subsequent section. The identifications in the intermediate case will be stated in the discussion in Section \[sec:discussion\].
In the remainder of this section we give a heuristic geometric argument why generically one does not expect new vertices to appear when we lower the dimension. In the sense of algebraic geometry, the ideal $\mathcal{I} = \mathcal{I}_S + \mathcal{I}_D$ defines a variety $V (\mathcal{I})$ as the zero-set of the polynomials contained in $\mathcal{I}$. If $\mathcal{I}$ was a prime ideal, we could think of the ring $\mathcal{R}/\mathcal{I}$ as the ring of polynomial functions on this variety. The gauge variations $D_{k}$ define $n$ vector fields on this variety, and we are looking for functions on $V(\mathcal{I})$ that are constant along the vector fields. When we enlarge the ideal to $\mathcal{I}'\supset \mathcal{I}$ by going from higher to lower dimensions where new Schouten identities occur, we concentrate on a subvariety $V (\mathcal{I}')$ of $V (\mathcal{I})$. Generically, if the vector fields do not degenerate on this subvariety, functions that are constant along $D_{k}$ on $V (\mathcal{I}')$ can be lifted to constant functions on $V (\mathcal{I})$.
The above argument only gives a very rough picture, because apart from the possible degeneration of the vector fields, there are two subtleties: First, as it was said, the argument only applies to prime ideals, but the ideals that occur are usually not prime; secondly, there could be constant polynomials on $V (\mathcal{I}')$ whose lifts to $V(\mathcal{I})$ are not polynomial. Therefore, this picture can only be seen as a heuristic explanation why generically we do not expect new gauge invariant vertices to appear when we lower the dimension.
The case $n > d$ {#sec:case-n-}
================
In this chapter, we find general restrictions for gauge invariant $n$-point vertices with $n > d$. Our result is a simple characterisation of equivalence classes $[\mathcal{V}] \in \mathcal{R} \slash \mathcal{I}$ for vertex generating operators. The results are summarised in Section \[sec:restrictions\].
As discussed in Section \[sec:equiv-relat-vert\], we have the full set of Schouten identities at hand in order to find a simple representative $\mathcal{V}$ for a given vertex. This has the advantage that a lot of simplifications are possible. On the other hand, the structure of the set of Schouten identities is very complicated, and the number of independent Schouten identities in the polynomial ring is large. This problem was solved in [@Fredenhagen:2019hvb] for $d = 3$ by observing that many Schouten identities become dependent when multiplied with an appropriate product $\Delta$ of Mandelstam variables. By multiplying a given vertex $\mathcal{V}$ with $\Delta$, the remaining independent Schouten identities can be used to deduce strong constraints for the vertex $\mathcal{V}$ itself. Essentially, one can treat the Mandelstam variables in the manipulations like numbers and also divide by them. This concept can be also employed in higher dimensions.
Formally, to be able to divide by certain combinations of Mandelstam variables, we introduce the ring of fractions, $M^{-1}\mathcal{R}$. Here, $M$ is a multiplicatively closed set containing all (finite) products of *non-zero minors of the submatrix $\mathcal{S}$ of $\mathcal{B}$* (see Eq. ): these are the expressions we want to divide by. More explicitly, let ${\text{Mi}}(\mathcal{S})$ be the set of non-zero minors of $\mathcal{S}$ [^11] and let $M={\text{Mon}}[{\text{Mi}}(\mathcal{S})]$ be the set of monomials in these minors. Then, the ring of fractions consists of formal quotients, $$M^{-1}\mathcal{R} = \Big\{ \frac{r}{\Delta} \, |\, \Delta \in M,\ r\in \mathcal{R} \Big\}\, ,$$ with the obvious rules for addition and multiplication. As also $1\in M$, we can identify $\mathcal{R}$ via $r \mapsto \frac{r}{1}$ as subring of $M^{-1}\mathcal{R}$. The ideal $\mathcal{I}=\mathcal{I}_{S}+\mathcal{I}_{D}\subset \mathcal{R}$ can then be seen as a subset of $M^{-1}\mathcal{R}$ which generates an ideal $\mathcal{I}_{M}$ in $M^{-1}\mathcal{R}$. Using the embedding of $\mathcal{R}$ into $M^{-1}\mathcal{R}$, we have an induced map of the quotient rings, $$i_{M}:\frac{\mathcal{R}}{\mathcal{I}} \to \frac{M^{-1}\mathcal{R}}{\mathcal{I}_{M}}\, .$$
As we will argue below, this map is injective, and therefore we can characterise equivalence classes of vertices uniquely by equivalence classes in the ring of fractions. The crucial observation is now that in $M^{-1}\mathcal{R}$ many of the generators of the ideal become dependent, so that $\mathcal{I}_{M}$ has a simple set of generators. This section is structured as follows. In Section \[sec:ideal\] we find a simple set of generators for $\mathcal{I}_{M}$. This enables us to find a convenient representative of $[\mathcal{V}]$ in the quotient of the ring of fractions in Section \[sec:using-equiv-relat\]. We then impose gauge invariance in Section \[sec:gener-restr-from\], which leads to strong restrictions on the vertex $\mathcal{V}$. In $d=3$, these restrictions completely rule out independent vertices (as reported in [@Fredenhagen:2019hvb]), in higher dimensions the restrictions are less strict, and we discuss them in Section \[sec:restrictions\]. In order to make the structure of this paper better accessible, we collect some proofs in Section \[sec:proofs\].
Before we proceed, we want to show that $i_{M}$ is indeed injective. If $i_{M} ([\mathcal{V}]) =[0]$, this means that $\mathcal{V}\in \mathcal{R}\cap \mathcal{I}_{M}$. Then, there is some $\Delta \in M$ such that $\Delta \mathcal{V}\in \mathcal{I}$. But if $\Delta \mathcal{V}$ defines a trivial vertex, then also $\mathcal{V}$ defines a trivial vertex, which can be seen in Fourier space, where the operators $s_{ij}$ are numbers. In particular, the polynomial $\Delta$ is non-zero on the subvariety defined by $k_{i}^{2}=0$ and $\sum k_{i}=0$. Now, if $\Delta \mathcal{V}$ defines a trivial vertex, then $$\Delta \mathcal{V} \prod_i \widehat{\phi}_i(k_i,a_i) \Big|_{a_i = 0}$$ vanishes on this subvariety. But $\Delta$ is non-vanishing almost everywhere. Hence, since $\mathcal{V}$ only depends polynomially on $k_{i}^{\mu}$, $\mathcal{V}$ applied on the fields $\widehat{\phi}_{i}$ must vanish. So we conclude that $\mathcal{V}\approx 0$, hence $[\mathcal{V}] = [0]$.
A Minimal Generating Set of Schouten Identities {#sec:ideal}
-----------------------------------------------
In this section, we find a simple set of generators for the ideal $\mathcal{I}_M$ in two steps. First, any Schouten identitiy multiplied with a certain $\Delta \in M={\text{Mon}}[{\text{Mi}}(\mathcal{S})]$ is an element in the ideal generated by the equivalence relations in Eq. (\[eq:equiv2\]) and all Schouten identities up to level 2 [^12] (recall the notion of level introduced in the paragraph before Eq. (\[eq:grading\])). In other words, $$\label{eq:MAIN} \text{there exists}\ \Delta \in {\text{Mon}}[{\text{Mi}}(\mathcal{S})] \, : \quad \Delta \cdot \mathcal{I}_S \subset
\sum_{k = 0}^2 I(k) + \mathcal{I}_D \,.$$ We show this in Section \[sec:scho-ident-revis\]. This observation implies that in the ring of fractions where we are allowed to divide by $\Delta $, we need far less generators for the Schouten identities.
In order to perform the second step, we introduce some more notations: First, $$\label{eq:N_ij}
N_{ij} =
\begin{pmatrix}
s_{ij} & \cdots & s_{i j+d-1} \\
\vdots & \ddots & \vdots \\
s_{i+d-1 \, j}&\cdots & s_{i+d-1 j+d-1}
\end{pmatrix}$$ is a $d\times d$ submatrix of $\mathcal{S}$, hence, $\det N_{ij} \in {\text{Mi}}(\mathcal{S})$ and $N_{ij}$ has full rank. Secondly, let $B_1(i,j)$ with $i,j = 1,\ldots,n$ be the following $(d+1)\times(d+1)$ submatrix of $\mathcal{B}$: It contains the rows and columns $i,i+1,\ldots,i + d - 1$ (modulo $n$) as well as another row $j$ and the column $i + n$. Hence, $$\label{eq:B1}
\det B_1(i,j) = \det
\left(
\begin{array}{cccc}
&&&0 \\
&&& y_{ii+1}\\
\multicolumn{3}{c}{\smash{\raisebox{.5\normalbaselineskip}{$N_{ii}$}}} &\vdots \\
&&& y_{ii+d-1}\\
s_{ji} & \cdots & s_{j i+d-1} & y_{ij}
\end{array}
\right)
\in I(1)~\footnote{Note that this is true for all $j = 1,\ldots, n$. If for example $j = i$, then
$\det B_1(i,j) = 0 \in I(1)$.}.$$ Finally, let $B_2(i,j)$ with $i,j = 1,\ldots,n$ be the $(d+1)\times(d+1)$ submatrix of $\mathcal{B}$ containing the rows $i,i+1,\ldots,i + d - 1$ (modulo $n$) and $i+n$, as well as the columns $j,j+1,\ldots,j + d - 1$ (modulo $n$) and $j+n$. Hence, $$\det B_2(i,j) = \det
\left(
\begin{array}{cccc}
&&& y_{ij}\\
\multicolumn{3}{c}{\smash{\raisebox{.0\normalbaselineskip}{$N_{ij}$}}} &\vdots \\
&&& y_{ij+d-1}\\
y_{ij} & \cdots & y_{i j+d-1} & z_{ij}
\end{array}
\right)
\in I(2)\,.$$ With these notations, we show in Section \[sec:schouten-ideal-2\] that there exists $\Delta \in M={\text{Mon}}[{\text{Mi}}(\mathcal{S})]$ such that $$\label{eq:I-Delta}
\Delta \cdot (\mathcal{I}_S +
\mathcal{I}_D) \subset I(0) + \left\langle \sum_{k = 1}^n s_{ik}\,,\, \det B_1(i,j)\,,\, \det B_2(i,j) \; \Big| \; i,j
= 1,\ldots, n \right\rangle\,.$$ Denote the family of generators of $I (0)$ by $(\det B_0(A))$, where $A$ labels the different equivalence relations. Then, we can conclude that $\mathcal{I}_{M}$ is generated as $$\mathcal{I}_{M} = \left\langle \sum_{k = 1}^n s_{ik}\,,\,(\det B_0 (A))\, ,\, \det B_1(i,j)\,,\, \det B_2(i,j) \; \Big| \; i,j
= 1,\ldots, n \right\rangle\,.$$
The choice of Representative {#sec:using-equiv-relat}
----------------------------
Now, let us investigate the relevant ideal $\mathcal{I}_M$ in order to choose a convenient representative for $\mathcal{V}$ in its equivalence class $[\mathcal{V}] \in M^{-1}\mathcal{R} \slash \mathcal{I}_M$.
We start by considering the Schouten identities $\det B_2(i,j) \in I(2)$, with $i \neq j$. Using a Laplace expansion along the last column, they read $$\label{eq:simplify-z}
0 \approx \det B_2(i,j) = z_{ij} \det N_{ij} + \text{terms that do not contain any } z_{kl}\,.$$ Since $\det N_{ij} \in {\text{Mi}}(\mathcal{S})$, we can divide by it in $M^{-1}\mathcal{R}$, and express $z_{ij}$ by an expression independent of any $z_{kl}$. Hence, we may choose the representative of $[\mathcal{V}]$ to be independent of $z_{ij}$. In the same way, the Schouten identities $\det B_1(i,j) \in I(1)$ take the form $$0 \approx \det B_1(i,j) = y_{ij} \det N_{ii} + p(s_{ij},y_{i i+1} ,\ldots, y_{i i+d-1}) \,.$$ Here, the polynomial $p$ only depends on $y_{i i+1} ,\ldots, y_{i i+d-1}$ [^13] and the Mandelstam variables. Using these Schouten identities, we can replace all of the operators $y_{ij}$ in $\mathcal{V}$ except for $y_{ii+1},\ldots,y_{ii+d-1}$.
Finally, we perform a change of variables in $\mathcal{V}$. Similarly to Eq. we introduce the combinations $$\label{eq:Y}
Y_i^j = s_{i i+1} y_{i i+j} - s_{i i+j} y_{i i+1} \qquad \text{for } j = 2,\ldots , d - 1\,,$$ and replace all $y_{ii+2},\dotsc ,y_{ii+d-1}$ in terms of these variables and $y_{ii+1}$. This can be done, because $s_{ii+1} \in M$ and we can divide by it in $M^{-1} \mathcal{R}$. We arrive at $$\label{eq:Q}
\mathcal{V} \approx Q_\mathcal{V}(y_{ii+1}, Y_i^j,s_{ij})\,,$$ where $Q_\mathcal{V}$ is a polynomial in $y_{ii+1}$, $Y_i^j$ and the Mandelstam variables (with coefficients that can contain inverse powers of elements in ${\text{Mi}}(\mathcal{S})$). More explicitly, we can see $[\mathcal{V}]$ as an element in the quotient $$\label{eq:V-in-MR}
[\mathcal{V}] \in \frac{M^{-1}\mathbb{R}\left[y_{ii+1}, Y_i^j , s_{ij}\right]}{\left\langle (\det B_0 (A))\,,\,\sum_{j =
1}^n s_{ij}\,,\, \det B_2(i,i)\;\big|\; i = 1,\ldots, n \right\rangle}\,.$$ There are several reasons to introduce the $Y_i^j$ variables. First, they are the gauge invariant combinations of the $y_{ij}$ variables — we have discussed this already in Section \[sec:case-2n-leq\] and it will become important in Section \[sec:gener-restr-from\]. Secondly, the remaining level-2 Schouten identities $\det B_2(i,i)$ can be written solely in terms of the $Y_i^j$’s and the Mandelstam variables, and they do not depend explicitly on $y_{ii+1}$. We show this in the rest of this section: For this purpose, consider $$s_{ii+1}^2 \det B_2(i,i) = \det
\left(
\begin{array}{ccccc}
&&&&0 \\
&&&& s_{ii+1} y_{ii+1}\\
\multicolumn{4}{c}{\smash{\raisebox{.5\normalbaselineskip}{$N_{ii}$}}} &\vdots \\
&&&& s_{ii+1}y_{ii+d-1}\\
0 & s_{ii+1} y_{ii+1} & \cdots & s_{ii+1} y_{i i+d-1} & 0
\end{array}
\right)\,.$$ The determinant of the matrix does not change when $y_{ii+1}$ times the first row is subtracted from the last one and $y_{ii+1}$ times the first column is subtracted from the last one. Hence, using the definition of $Y_i^j$ in Eq. (\[eq:Y\]), we find $$s_{ii+1}^2 \det B_2(i,i) = \det
\left(
\begin{array}{cccccc}
&&&&&0 \\
&&&&& 0\\
\multicolumn{5}{c}{\smash{\raisebox{.0\normalbaselineskip}{$N_{ii}$}}} &Y_i^2 \\
&&&&& \vdots\\
&&&&& Y_i^{d-1}\\
0 & 0 & Y_i^2 & \cdots & Y_i^{d-1} & 0
\end{array}
\right)
= - \sum_{j,k = 2}^{d-1} Y_i^j \left( \text{adj}N_{ii} \right)_{jk} Y_i^k =: q_2^i(Y_i^j,s_{jk})\,.$$ Here, we used a Laplace expansion along the last row and column. The resulting polynomials $q_2^i$ are quadratic in the $Y_i^j$ variables with coefficients that still depend on the Mandelstam variables. However, the $q_2^i$’s are independent of $y_{ii+1}$. We comment on their structure in Section \[sec:restrictions\]. All in all, we can replace the generators $\det B_2(i)$ by $q_2^i$ because we are allowed to divide by Mandelstam variables. Hence, we have the following result: $$[\mathcal{V}] \in \frac{M^{-1}\mathbb{R}\left[y_{ii+1}, Y_i^j , s_{ij}\right]}{\left\langle (\det B_0 (A))\,,\,\sum_{j =
1}^n s_{ij}\,,\, q_2^i\;\big|\; i = 1,\ldots, n \right\rangle}\,.$$
General Restrictions from Gauge Invariance {#sec:gener-restr-from}
------------------------------------------
With the results of the previous sections, we now show that the polynomial $Q_\mathcal{V}$ introduced in Eq. (\[eq:Q\]) can be chosen to be independent of $y_{i i+1}$ if the operator $\mathcal{V}$ corresponds to a gauge invariant Lagrangian $\mathcal{L}_n$. From now on, we will always consider $\mathcal{V}$ as an element in the bigger ring of fractions.
Starting from Eq. (\[eq:gauge-inv\]) and using that the operators $a_k\cdot P_k$ commute with all Mandelstam variables, we find that a gauge invariant vertex $\mathcal{L}_n$ requires $$\text{for all}\ k \in \{1, \ldots n\} \,: \quad [\mathcal{V} , a_k \cdot P_k] \in \mathcal{I}_M\,,$$ where $\mathcal{L}_n$ and $\mathcal{V}$ are related via Eq. (\[eq:L\^n\]). Now, the ideal $\mathcal{I}_M$ is gauge invariant, hence, it commutes with the operators $a_k \cdot P_k$. We deduce that the polynomial in Eq. (\[eq:Q\]) satisfies $$[Q_\mathcal{V}, a_k \cdot P_k ] \in \left\langle (\det B_0 (A))\,,\,\sum_{j =
1}^n s_{ij}\,,\, q_2^i(Y_i^j)\;\big|\; i = 1,\ldots, n \right\rangle\,.$$ With $$[y_{ii+1}, a_k \cdot P_k] = \delta_{ik} s_{i i +1} \qquad \Rightarrow \qquad [Y_i^j , a_k \cdot P_k] = \delta_{ik}
\left(s_{ii+j} s_{i i + 1} - s_{ii+1} s_{i i+j} \right) = 0\,,$$ it follows immediately that $$\text{for all}\ k = 1,\ldots, n \,:\quad s_{kk+1} \partial_{y_{kk+1}} Q_\mathcal{V} \in \left\langle (\det B_0 (A))\,,\,\sum_{j =
1}^n s_{ij}\,,\, q_2^i(Y_i^j)\;\big|\; i = 1,\ldots, n \right\rangle\,.$$ But the generators of the ideal on the rhs do not depend on $y_{ii+1}$. We conclude that $Q_\mathcal{V}$ can be chosen to be independent of $y_{ii+1}$. More explicitly, $$\label{eq:result-schoutens}
\mathcal{V} \approx Q_\mathcal{V} (Y_i^j\,,\,s_{ij})\,, \qquad[\mathcal{V}] \in \frac{M^{-1}\mathbb{R}\left[Y_i^k , s_{ij}\right]}{\left\langle (\det B_0 (A))\,,\,\sum_{j = 1}^n s_{ij} \,,\, q_2^i(Y_i^j) \;|\; i = 1,\ldots , n \right\rangle}\,.$$
Restrictions for $\mathcal{V}$ {#sec:restrictions}
------------------------------
Let us summarise our results. Eq. (\[eq:result-schoutens\]) states that each gauge invariant vertex $\mathcal{V}$ is equivalent to a vertex $Q_\mathcal{V}$, which does only depend on Mandelstam variables and $Y_i^j$. In particular, translating back to the vertex in terms of $P_i^\mu$ and $A_i^\mu$ operators, we have the following relation: $$\iota_d (Y_i^j) = 2 P_i{}_\mu A_i{}_\nu
P_{[i+1}^{\mu} A_{i+j]}^{\nu} = 2 P_i{}_{[\mu} A_i{}_{\nu]}
P_{i+1}^\mu A_{i+j}^\nu\,.$$ Now, in the vertex generated by $Q_\mathcal{V}$, an index of the $i$th field is only generated by $A_i^\mu$ via a corresponding $Y_i^j$. Hence, the $i$th field enters the Lagrangian via a curvature term (each index of the field is antisymmetrised with an index of a partial derivative acting on it). We deduce that $Q_V$ generates a Lagrangian, which can be written solely in terms of curvature terms.
The drawback of this analysis is that we loose locality on the way to this result. $Q_\mathcal{V}$ might not have a local form, since it can have inverse powers of Mandelstam variables. We can only say that for each gauge invariant vertex (generated by $\mathcal{V}$), there is a $\Delta \in M$ such that $\Delta \mathcal{V}$ can be written only in terms of curvatures.
Much stricter conditions can be found in three dimensions [@Fredenhagen:2019hvb]. In that case, there is only one $Y_i^j$ and the corresponding Schouten identity is $q_2^i = - s_{ii+1}^2 (Y_i^2)^2$. Hence, $\det B_2(i,i) = (Y_i^2)^2 \approx 0$ and $Q_\mathcal{V}$ is only linear in $Y_i$. One can then deduce that $\mathcal{V}$ itself is at most linear in each of the operators $A_i^\mu$, which means that the corresponding vertex $\mathcal{L}_n$ contains no higher-spin fields at all. This argument can also be obtained from the observation that in $d = 3$, there are simply no curvature terms for higher-spin fields.
Proofs {#sec:proofs}
------
### Proof of Eq. (\[eq:MAIN\]) {#sec:scho-ident-revis}
Let $\det M = 0$ be a Schouten identity that stems from a $(d+1)\times(d+1)$-submatrix $M$ of $\mathcal{B}$ such that $\det M \not{\in} \mathcal{I}_D$. Let $r$ ($s$) be the number of level-0 rows (columns) of $M$. Furthermore, let Let $\bar r$ ($\bar s$) be the number of level-1 rows (columns) of $M$. Hence, $r + \bar r = s + \bar s = d+1$. Without loss of generality, we assume $r \geq s$ [^14]. Furthermore, let $\bar s \geq 2$, hence, the level of the Schouten identity $\det M = 0$ is $\bar r + \bar s \geq 2$. In particular, equality holds if and only if $\bar s = 2$ and $\bar r = 0$.
With the submatrix $M$ given, we construct a $(d+2)\times(d+2)$-submatrix $\widetilde{M}$ of $\mathcal{B}$ as follows:
- Removing $(2n - d - 2)$ rows and columns from $\mathcal{B}$ results in $\widetilde{M}$.
- There is a level-0 row (which we call $Row$) and a level-0 column (called $Col$) in $\widetilde{M}$, such that removing $Row$ and $Col$ in $\widetilde{M}$ yields $M$. Hence, $\widetilde{M}$ contains $(r+1)$ level-0 rows and $(s+1)$ level-0 columns.
- The construction of $\widetilde{M}$ might not be unique, but is always possible. This can be seen as follows: In order to construct $\widetilde{M}$, there must be at least one level-0 row of $\mathcal{B}$ that is not part of $M$ (otherwise, $M$ would contain all level-0 rows of $\mathcal{B}$ which means that $\det M \in \mathcal{I}_D$ which contradicts our assumption). Furthermore, there are at least two level-0 columns of $\mathcal{B}$ that are not part of $M$, because $\bar s \geq 2$ and hence, $s \leq d - 1$ [^15]. In particular, we can always choose $\widetilde{M}$ such that the intersection of $Row$ and $Col$ contains a non-zero Mandelstam variable.
The construction of the matrix $\widetilde{M}$ is visualised in Figure \[fig:M\].
\(S) \[matrix of math nodes,nodes=[outer sep=0pt,minimum width=10mm,minimum height=10mm]{}\] [|\[fill=black!20\]| & |\[fill=black!20\]| & |\[fill=black!20\]| & |\[fill=black!20\]|\
|\[fill=black!20\]| & |\[fill=black!20\]| & |\[fill=black!20\]| & |\[fill=black!20\]|\
|\[fill=black!20\]| & |\[fill=black!20\]| & |\[fill=black!20\]| & |\[fill=black!20\]|\
|\[fill=black!20\]| & |\[fill=black!20\]| & |\[fill=black!20\]| & |\[fill=black!20\]|\
]{};
(S-4-2.south east) – (S-1-2.north east);(S-4-1.north west) – (S-4-4.north east); (S-1-1.north west) – (S-1-3.north west); (S-1-4.north east) – (S-4-4.north east); (col) at (S-1-2.north) [$s$ level-0 columns]{}; (row) at (S-2-4.east) [$r$ level-0 rows]{};
(M=) at (S-3-1.north west) [$M=$]{};
\(S) \[matrix of math nodes,nodes=[outer sep=0pt,minimum width=10mm,minimum height=10mm]{}\] [|\[fill=black!20\]| & & |\[fill=black!20\]| & |\[fill=black!20\]| & |\[fill=black!20\]|\
|\[fill=black!20\]| & & |\[fill=black!20\]| & |\[fill=black!20\]| & |\[fill=black!20\]|\
& & & & \
|\[fill=black!20\]| & & |\[fill=black!20\]| & |\[fill=black!20\]| & |\[fill=black!20\]|\
|\[fill=black!20\]| & & |\[fill=black!20\]| & |\[fill=black!20\]| & |\[fill=black!20\]|\
]{};
(S-5-3.south east) – (S-1-3.north east);(S-5-1.north west) – (S-5-5.north east);(S-1-2.north west) – (S-5-2.south west); (S-1-2.north east) – (S-5-2.south east); (S-3-1.north west) – (S-3-5.north east); (S-3-1.south west) – (S-3-5.south east);
(-1.5,0.3) – (-2.5,0.3); (-0.5,0.3) – (2.5,0.3); (-0.5,0.3) – (-1.5,0.3);
(-1.3,0.5) – (-1.3,2.5); (-1.3,-0.5) – (-1.3,-2.5); (-1.3,-0.5) – (-1.3,0.5);
(S-1-1.north west) – (S-1-4.north west); (S-1-5.north east) – (S-5-5.north east); (col) at (S-1-2.north) [$(s+1)$ level-0 columns]{}; (row) at (S-2-5.south east) [$(r+1)$ level-0 rows]{}; at (1,0.05) [$Row$]{}; at (-1.05,1.4) [$Col$]{}; at (-0.9,-0.1) [$\neq 0$]{}; (M=) at (S-3-1.west) [$\widetilde{M}=$]{};
For $\widetilde{M}$, Cramers rule states that $$\label{eq:cramer}
\mathbb{I}_{(d+2)\times (d+2)} \det \widetilde{M} - \widetilde{M} \cdot C^T = 0,$$ where $C = (c_{ij})$ denotes the cofactor matrix of $\widetilde{M} = (\widetilde{m}_{ij})$. In particluar, $c_{ij}$ is (up to a factor of $\pm 1$) equal to the determinant of the $(d+1)\times(d+1)$-submatrix obtained by deleting the $i$-th row and the $j$-th column from $\widetilde{M}$. In other words, $c_{ij}$ is a $(d+1)\times(d+1)$-minor of $\mathcal{B}$, hence $c_{ij} \in \mathcal{I}_S$. In the following, we consider only part of Eq. (\[eq:cramer\]): $$\label{eq:cramer-part}
\delta_{ji} \det \widetilde{M} - \sum_{k = 1}^{s+1} \widetilde{m}_{jk} c_{ik} - \sum_{k = s+2}^{d+2}
\widetilde{m}_{jk} c_{ik} = 0 \, \qquad i = 1,\ldots,s+1,\,j \in J\,.$$ Here, $J$ is a (non-unique) subset of $s+1$ level-$0$ rows that contains $Row$. In other words, $$J \subset \{1,\ldots, r+1 \}\,,\qquad |J| = s+1\,,\qquad Row \in J\,.$$
Performing a Laplace expansion of $\det \widetilde{M}$ along the last column of $\widetilde{M}$ (which is of level 1 because of $\bar s \geq 2$), we deduce that $\det \widetilde{M}$ is a linear combination of Schouten identities of level $\bar r + \bar s - 1$ and $\bar r + \bar s - 2$. Hence, $$\det \widetilde{M} \in I(\bar r + \bar s - 1) + I(\bar r + \bar s - 2)\,.$$ Furthermore, in the third term of Eq. (\[eq:cramer-part\]), the Schouten identities $c_{ik}$ with $k > s+1$ are of level $(\bar r + \bar s - 1)$. We therefore conclude that the middle term is an element in the following ideal: $$\label{eq:sum=in-I}
\text{for all}\ i = 1, \ldots ,s+1,\,j \in J \, :\quad \left(\sum_{k = 1}^{s+1} \widetilde{m}_{jk} c_{ik} \right) \in I(\bar r + \bar s - 1)
+ I(\bar r + \bar s - 2)\,.$$
Now, denote by $N = (\widetilde{m}_{jk})$ (with $j \in J$ and $k \in \{1, \ldots ,s+1\}$) the $(s+1)\times(s+1)$-submatrix of $\widetilde{M}$ that occurs in Eq. (\[eq:sum=in-I\]). It is also a submatrix of $\mathcal{S}$ as it only consists of level-0 rows and columns. Since $s + 1 \leq d$, we deduce that $\det N \in {\text{Mi}}(\mathcal{S})$ [^16]. In particular, $\det N \neq 0$ and by inverting $N$ in Eq. (\[eq:sum=in-I\]) using Cramers rule, we find $$\text{for all}\ j \in J,\, k \in \{1, \ldots ,s+1\} \, :\quad \det N \cdot c_{jk} \in I(\bar r + \bar s - 1) + I(\bar r + \bar s - 2)\,.$$
Finally, setting $j = Row$ and $k = Col$, we have $c_{jk} = \det M$ — which corresponds to the Schouten identity of level $(\bar r + \bar s)$ we started with. It follows directly that for all $\det M \in I(\bar r +
\bar s)$, either $\det M \in \mathcal{I}_D$ or $$\label{eq:recursiveMAIN}
\text{there exists}\ \det N \in {\text{Mi}}(\mathcal{S}) \, : \quad \det N \cdot
\det M \in
I(\bar r + \bar s - 1) + I(\bar r + \bar s - 2) \,.$$ In other words, $$\text{there exists}\ \Delta \in {\text{Mon}}[{\text{Mi}}(\mathcal{S})] \, : \quad \Delta \cdot I(\bar r + \bar s) \subset I(\bar r + \bar s -
1) + I(\bar r + \bar s - 2) + \mathcal{I}_D$$ and a recursion over $\bar r$ and $\bar s$ proves the general statement in Eq. (\[eq:MAIN\]).
### Proof of Eq. (\[eq:I-Delta\]) {#sec:schouten-ideal-2}
We prove Eq. (\[eq:I-Delta\]) in three steps. It directly follows from Eq. (\[eq:MAIN\]), as well as Eqs. (\[eq:PartA\], \[eq:PartB\] and \[eq:PartC\]).
#### Part 1: {#part-1 .unnumbered}
First of all, we show that $$\label{eq:PartA}
\text{there exists}\ \Delta \in {\text{Mon}}[{\text{Mi}}(\mathcal{S})] \, : \quad \Delta I(2) \subset I(0) + I(1) + \left
\langle \det B_2(i,j) \,\big|\, i,j = 1,\ldots , n \right \rangle\,.$$ Therefore, consider an arbitrary level-2 Schouten identity $\det M \approx 0$, with $M$ being a $(d+1)\times(d+1)$-submatrix of $\mathcal{B}$. As explained in the previous section, we may assume that $M$ has either one ($\bar r = 1$, $\bar s = 1$) or two level-1 columns ($\bar r = 0$, $\bar s = 2$). In the latter case, the proof of the previous section goes through and Eq. (\[eq:recursiveMAIN\]) is satisfied. Hence, we only need to consider the other case $(\bar r = \bar s = 1)$. In particular, we show that if $M$ contains the level-1 row $i+n$ and the level-1 column $j+n$ of $\mathcal{B}$, then: $$\label{eq:proposition1}
\text{there exists}\ \Delta \in {\text{Mon}}[{\text{Mi}}(\mathcal{S})] \,:\quad \Delta \det M \in I(1) + \langle
\det B_2(i,j) \rangle\,,$$ which implies Eq. (\[eq:PartA\]).
We prove Eq. (\[eq:proposition1\]) by induction. Therefore, fix $i$ and $j$. Let $I_2(K,L)$ be the ideal generated by all level-2 Schouten identities $\det M \approx 0$, such that the $(d+1)\times(d+1)$-submatrix $M$ of $\mathcal{B}$ has the following properties:
- $M$ contains the level-1 row $i+n$ and the level-1 column $j+n$ of $\mathcal{B}$.
- $K$ rows ($L$ columns) of $M$ stem from the rows $i,\ldots, i + d - 1$ (columns $j,\ldots, j + d - 1$) (modulo $n$) of $\mathcal{B}$.
Hence, $I_2(d,d) = \langle \det B_2(i,j) \rangle$ and Eq. (\[eq:proposition1\]) is a recursive consequence of the following two propositions: $$\begin{aligned}
\label{eq:ind1}
\text{if } K < d\,,\quad \text{there exists}\ \Delta \in {\text{Mon}}[{\text{Mi}}(\mathcal{S})] \,:&\quad \Delta \cdot I_2(K,L) \subset I(1)
+ I_2(K+1,L)\,,\\
\label{eq:ind2}
\text{if } L < d\,,\quad\text{there exists}\ \Delta \in {\text{Mon}}[{\text{Mi}}(\mathcal{S})] \,:&\quad \Delta \cdot I_2(K,L) \subset I(1) +
I_2(K,L+1)\,.\end{aligned}$$
Here, we present the proof of Eq. (\[eq:ind2\]). Eq. (\[eq:ind1\]) can be shown in the same way, except that the roles of rows and columns are interchanged. Therefore, we start with a $(d+1)\times(d+1)$-submatrix $M$ of $\mathcal{B}$, such that $\det M \in I_2(K,L)$ with $L<d$. For $M$ given, we construct a $(d+2)\times(d+2)$-matrix $\widetilde{M}$ as follows:
- Removing the first row from $\widetilde{M}$ yields a $(d+1)\times(d+2)$-submatrix $\widehat{M}$ of $\mathcal{B}$.
- There is a unique $k_0 \in \{1,\ldots , d+1\}$, such that removing the $k_0$-th column from $\widehat{M}$ yields $M$. We construct $\widetilde{M}$ such that the $k_0$-th column stems from one of the columns $j,\ldots, j + d - 1$ (modulo $n$) of $\mathcal{B}$, which is possible because $L < d$.
- Note that $\widehat{M}$ has $d+1$ level-0 columns. Hence, at least one of those cannot stem from one of the columns $j,\ldots, j + d - 1$ (modulo $n$) of $\mathcal{B}$. Let us agree that at least the $l_0$-th column has this property. Obviously, $l_0 \neq k_0$.
- The first two rows of $\widetilde{M}$ coincide, hence, $\det \widetilde{M} = 0$.
Now, Cramers rule states that $\widetilde{M}C^T = 0$, where $C = (c_{kl})$ is the cofactor matrix of $\widetilde{M} = (\widetilde{m}_{kl})$. The first column of this matrix equation reads $$\sum_{l = 1}^{d+2} \widetilde{m}_{kl} c_{1l} = 0\,.$$ Note that up to a factor of $\pm 1$, $c_{1l}$ is the determinant of the $(d+1)\times(d+1)$-matrix obtained by removing the first row and the $l$th column from $\widetilde{M}$. Hence, $c_{1l} \in I(2)$ for $l \leq d+1$ and $c_{1\, d+2} \in I(1)$. In particular, $c_{1k_0} \propto \det M \in I_2(K,L)$ and $c_{1l_0} \propto \det M_{l_0 \to k_0} \in I_2(K,L+1)$, where the matrix $M_{l_0 \to k_0}$ differs from $M$ by only one column. Indeed, it contains the $k_0$-th column of $\widetilde{M}$ instead of the $l_0$-th. We deduce that $$\sum_{1 \leq l \leq d+1 }^{l \neq l_0} \widetilde{m}_{kl} c_{1l} \in I(1) + I_2(K,L+1)\,.$$
Now, consider only the rows $2,\ldots, d+1$ of that relation. The matrix $N = (\widetilde{m}_{kl})$ with $k \in \{ 2,\ldots, d+1 \}$ and $l \in \{1,\ldots , d+1\} \backslash \{l_0\}$ is a $(d\times d)$-submatrix of $\mathcal{S}$ and can hence, be inverted using Cramers rule. We find that $$\det N \cdot c_{1l} \in I(1) + I_2(K,L+1)$$ and setting $l = k_0$ finally proves Eq. (\[eq:ind2\]).
#### Part 2:
In a second step, we show that $$\label{eq:PartB}
\text{there exists}\ \Delta \in {\text{Mon}}[{\text{Mi}}(\mathcal{S})] \quad : \quad \Delta I(1) \subset I(0) + \left \langle \det B_1(i,j)
\,|\, i,j = 1,\ldots , n\right \rangle\,.$$ The proof is similar to the previous one. Let $\det M \approx 0$ be an arbitrary level-1 Schouten identity, where $M$ is a $(d+1)\times(d+1)$-submatrix of $\mathcal{B}$. As explained in the previous section, we may assume that $M$ has exactly one level-1 column ($\bar r = 0, \bar s = 1$). In particular, we show that if $M$ contains the level-1 column $i+n$ of $\mathcal{B}$, then: $$\label{eq:proposition2}
\text{there exists}\ \Delta \in {\text{Mon}}[{\text{Mi}}(\mathcal{S})] \,:\quad \Delta \det M \in I(0) + \langle
\det B_1(i,j) \,|\, j = 1,\ldots , n \rangle\,,$$ which implies Eq. (\[eq:PartB\]).
Again, we prove Eq. (\[eq:proposition2\]) by induction. Therefore, fix $i$. Let $I_1(K,L)$ be the ideal generated by all level-1 Schouten identities $\det M \approx 0$, such that the $(d+1)\times(d+1)$-submatrix $M$ of $\mathcal{B}$ has the following properties:
- $M$ contains the level-1 column $i+n$ of $\mathcal{B}$.
- $K$ rows ($L$ columns) of $M$ stem from the rows (columns) $i,\ldots, i + d - 1$ (modulo $n$) of $\mathcal{B}$.
Hence, $M(d,d) = \langle \det B_1(i,j) \,|\, j = 1,\ldots , n \rangle$ and Eq. (\[eq:proposition2\]) is a recursive consequence of the following two propositions: $$\begin{aligned}
\label{eq:ind3}
\text{if } K < d\,,\quad\text{there exists}\ \Delta \in {\text{Mon}}[{\text{Mi}}(\mathcal{S})] \,:&\quad \Delta \cdot I_1(K,L) \subset I(0)
+ I_1(K+1,L)\,,\\
\label{eq:ind4}
\text{if } L < d\,,\quad\text{there exists}\ \Delta \in {\text{Mon}}[{\text{Mi}}(\mathcal{S})] \,:&\quad \Delta \cdot I_1(K,L) \subset I(0) +
I_1(K,L+1)\,.\end{aligned}$$
Here, we give the proof of Eq. (\[eq:ind4\]), Eq. (\[eq:ind3\]) follows analogously. For a given $(d+1)\times(d+1)$-submatrix $M$ of $\mathcal{B}$, such that $\det M \in I_1(K,L)$ with $L<d$, we construct a $(d+2)\times(d+2)$-matrix $\widetilde{M}$ as follows:
- Removing the first row from $\widetilde{M}$ yields a $(d+1)\times(d+2)$-submatrix $\widehat{M}$ of $\mathcal{B}$.
- There is a unique $k_0 \in \{1,\ldots , d+1\}$, such that removing the $k_0$-th column from $\widehat{M}$ yields $M$. Again, we construct $\widetilde{M}$ such that the $k_0$-th column stems from one of the columns $i,\ldots, i + d - 1$ (modulo $n$) of $\mathcal{B}$, which is possible because $L < d$.
- Note that $\widehat{M}$ has $d+1$ level-0 columns. Hence, at least one of those (say, the $l_0$-th) cannot stem from one of the columns $i,\ldots, i + d - 1$ (modulo $n$) of $\mathcal{B}$. Again, $l_0 \neq k_0$.
- The first two rows of $\widetilde{M}$ coincide, hence, $\det \widetilde{M} = 0$.
Now, Cramers rule states that $\widetilde{M}C^T = 0$, where $C = (c_{kl})$ is the cofactor matrix of $\widetilde{M} = (\widetilde{m}_{kl})$. Considering the first column of this matrix equation, we have $$\sum_{l = 1}^{d+2} \widetilde{m}_{kl} c_{1l} = 0\,.$$ Up to a factor of $\pm 1$, $c_{1l}$ is the determinant of the $(d+1)\times(d+1)$-matrix obtained by removing the first row and the $l$th column from $\widetilde{M}$. Hence, $c_{1l} \in I(1)$ for $l \leq d+1$ and $c_{1\, d+2} \in I(0)$. In particular, $c_{1k_0} \propto \det M \in I_1(K,L)$ and $c_{1l_0} \propto \det M_{l_0 \to k_0} \in I_1(K,L+1)$, where the matrix $M_{l_0 \to k_0}$ differs from $M$ by only one column (it contains the $k_0$-th column of $\widetilde{M}$ instead of the $l_0$-th). We deduce that $$\sum_{1\leq l \leq d+1}^{l \neq l_0} \widetilde{m}_{kl} c_{1l} \in I(0) + I_1(K,L+1)\,.$$
Again, we only consider the rows $2,\ldots, d+1$ of that relation. The matrix $N = (\widetilde{m}_{kl})$ with $k \in \{ 2,\ldots, d+1 \}$ and $l \in \{1,\ldots , d+1\} \backslash \{l_0\}$ is a $(d\times d)$-submatrix of $\mathcal{S}$ and can be inverted using Cramers rule. Finally, $$\det N \cdot c_{1l} \in I(0) + I_1(K,L+1)$$ and setting $l = k_0$ proves Eq. (\[eq:ind4\]).
#### Part 3:
Finally, we prove that for any $i \in \{1,\ldots , n\}$, $$\label{eq:PartC}
\det N_{ii} \sum_{j = 1}^n y_{ij} \in \left \langle \sum_{j = 1}^n s_{ij} \right
\rangle + \left \langle \det B_1(i,j) \right
\rangle\,,$$ where $N_{ii} \in {\text{Mi}}(\mathcal{S})$ is defined in Eq. (\[eq:N\_ij\]).
Fix $i \in \{1,\ldots , n\}$. Then, for any $j \in \{1,\ldots , n\}$, let $N_{ii}(k \to j)$ be the matrix $N_{ii}$, where the $(k+1)$st row is replaced by $
\begin{pmatrix}
s_{ji} & s_{j i + 1} & \cdots & s_{j i + d - 1}
\end{pmatrix}
$. In particular, $$\label{eq:propertyN}
\sum_{j = 1}^n \det N_{ii} ( k \to j ) \in \left \langle \sum_{j = 1}^n s_{ij} \right
\rangle\,,$$ because the determinant of $N_{ii} ( k \to j )$ is linear (especially in the $(k+1)$st row).
Now, a Laplace expansion of Eq. (\[eq:B1\]) with respect to the last column results in $$\det B_1(i,j) = y_{ij} \det N_{ii} - \sum_{k = 0}^{d-1} y_{ii+1} \det N_{ii} ( k \to j )\,,$$ which holds for all $j \in \{1 ,\ldots , n)$. In particular, $$\det N_{ii} \sum_{j = 1}^n y_{ij} = \sum_{j = 1}^n \det B_1(i,j) + \sum_{k = 0}^{d-1} y_{ii+1} \sum_{j = 1}^n
\det N_{ii} ( k \to j )\,,$$ which, taking Eq. (\[eq:propertyN\]) into account, proves Eq. (\[eq:PartC\]).
Parity-Odd Vertices {#sec:parity-odd-vertices}
===================
So far, we only discussed parity-even vertices, i.e. terms in the Lagrangian which do not involve the epsilon tensor $\epsilon_{\mu_1 \cdots \mu_d}$. However, the discussion of the previous sections can simply be generalised also for parity-odd vertices.
First of all, the most general form of a parity-odd vertex is given by Eq. (\[eq:L\^n\]) but with $\mathcal{V}$ replaced by $$\label{eq:Vtilde}
\tilde{\mathcal{V}} = \sum Q_{I_1\cdots I_d} \tilde{\mathcal{V}}^{I_1\cdots I_d}\,,$$ where $\tilde{\mathcal{V}}^{I_1 \cdots I_d} \in \mathbb{R}[y_{ij}, z_{ij}|_{i \leq j}, s_{ij} |_{i \leq j}]$ contains the parity-even contractions [^17] and $$Q_{I_1 \cdots I_d} = \epsilon_{\mu_1\cdots \mu_d} b^{\mu_1}_{I_1} \cdots b^{\mu_d}_{I_d}$$ is totally antisymmetric in its indices ($I_k = 1,\ldots ,2n$). The derivative operators $b_I$ were introduced in Section \[sec:equiv-relat-vert\], right before Eq. (\[defofB\]). Note that for $i = 1,\ldots , n$, we have $b_i = P_i$ and $b_{i+n} = A_i$. The structure of the gauge-invariant parity-odd vertices depends on the dimension:
- For $d\geq 2n$, there are no parity-odd $n-$point vertex operators, because $Q_{I_1 \cdots I_d} = 0$ (the vector $b$ has only $2n-1$ independent entries up to total derivatives).
- For $d=2n-1$, there is a unique elementary parity-odd vertex operator $$\begin{aligned}
Q^{LL}_{1\cdots 2n-1} = \e_{\m_1\dots\m_{2n-1}}P_1^{\m_{1}}\dots P_{n-1}^{\m_{n-1}} \, A_1^{\m_{n}}\,\dots A_n^{\m_{2n-1}}\,,\end{aligned}$$ which is gauge invariant up to total derivatives and squares to the Lovelock operator . This covers also the case of $n=3$ and $d=5$, consistent with [@Metsaev:2005ar].
- In the case $n > d$, we make use again of the fact that we consider $[\mathcal{V}]$ in the ring of fractions. The crucial point is that the general form of an elementary building block $Q_{I_{1}\dotsb I_{d}}$ of parity-odd vertices can be highly simplified, when it is multiplied with the upper-left $d\times d$ submatrix of $\mathcal{S}$. Denote this matrix by $S_d$. Its determinant, $$\det S_d = \frac{1}{d!} \epsilon_{\mu_1 \cdots \mu_d} \epsilon_{\nu_1 \cdots \nu_d} b_1^{\mu_1} \cdots
b_d^{\mu_d} b_1^{\nu_1} \cdots b_d^{\nu_d}\,,$$ is a non-zero minor of $\mathcal{B}$, hence, $\det S_d \in {\text{Mi}}(\mathcal{S})$ and we conclude that $$\begin{aligned}
\det S_d \cdot Q_{I_1 \cdots I_d} &=
\left(
\mathcal{B}_{1I_1} \cdots \mathcal{B}_{d I_d}
\right)\big|_{[I_1 \cdots I_d]} \cdot Q_{1\cdots d}\,.\end{aligned}$$ In other words, for any parity-odd vertex in the Lagrangian given by the vertex generating operator $\tilde{\mathcal{V}}$ in Eq. (\[eq:Vtilde\]), we find $$\label{eq:Q123}
\det S_d \cdot \tilde{\mathcal{V}} = Q_{1 \cdots d} \cdot \mathcal{V}\,,$$ where $\mathcal{V} \in \mathbb{R}[y_{ij}, z_{ij}|_{i \leq j}, s_{ij} |_{i \leq j}]$ as in the parity-even case.
Now, since we work in the ring of fractions, we can divide by $\det S_d \in {\text{Mi}}(\mathcal{S})$. Furthermore, $Q_{1 \cdots d}$ is gauge invariant: $$[Q_{1\cdots d},a_k \cdot P_k] = 0\,.$$ Hence, along the same lines as in Section \[sec:case-n-\], we find that $$\label{eq:PY-general-odd}
\tilde{\mathcal{V}} \approx Q_{1\cdots d} \cdot Q_\mathcal{V} (Y_i^j , s_{ij})\,.$$
- For $n\leq d\leq 2n-2$, one has to be more careful, taking into account the Schouten identities, as was done for cubic vertices in $d=4$ [@Conde:2016izb] and $d=3$ [@Kessel:2018ugi]. The difference here is in the possibility to use negative powers of Mandelstam variables in the case of $n\geq 4$. The idea is to use Schouten identities to bring any parity-odd vertex structure to a form, where one has a gauge-invariant “square-root of a Horndeski-type operator”: $$\begin{aligned}
Q_{I_1\cdots I_d}^{(n-1)} = \e_{\m_1\dots\m_d}P_{I_1}^{\m_1}\dots P_{I_{n-1}}^{\m_{n-1}}\,A_{I_{n}}^{\m_n}\dots
A_{I_{d}}^{\m_{d}}\,,\qquad I_k \leq n\,.\label{SHO}\end{aligned}$$ This procedure can remove redundancies present due to Schouten identities, but the uniquely fixed form of the vertex may involve negative powers of the variables[^18] $s_{ij}$. This scheme was instrumental in deriving parity-odd cubic vertices in four [@Conde:2016izb] and three dimensions [@Kessel:2018ugi]. We just need to show here, that any parity-odd vertex operator that involves less than $n-1$ derivatives, can be related to another operator with more derivatives by Schouten identities. We prove this in the rest of this section.
Let us take a generic operator of this type, $$\begin{aligned}
Q^{(k)}_{I_1\dots I_{k}\,J_{1}\dots J_{d-k}}=\e_{\m_1\dots\m_d}P_{I_1}^{\m_1}\dots P_{I_k}^{\m_k}A_{J_{1}}^{\m_{k+1}}\dots A_{J_{d-k}}^{\m_d}\,,\end{aligned}$$ with $k\leq n-2$, $1\leq I_1 < I_2 < \dots < I_k\leq n$ and $1\leq J_1 < J_2 < \dots < J_{d-k+1} \leq n$. We can form a Schouten identitiy $$\begin{aligned}
0=\e_{\m_1\dots\m_d}P_{I_1}^{\m_1}\dots P_{I_k}^{\m_k}A_{J_{1}}^{[\m_{k+1}}\dots A_{J_{d-k}}^{\m_d}\, P_{I_{k+1}}^{\n_{1}}\dots P_{I_{2k+1}}^{\n_{k+1}]}\,P^{K_{1}}_{\n_1}\dots P^{K_{k+1}}_{\n_{k+1}}\,,\end{aligned}$$ or, schematically, $$\begin{aligned}
0=Q^{(k)} \det S^{(k+1)}+O(Q^{(j>k)})\,,\label{SchoutenWk}\end{aligned}$$ where $\det S^{(k+1)}\in {\text{Mi}}(\mathcal{S})$, while $O(Q^{(j > k)})$ refers to all the terms that contain parity-odd opeartors involving more than $k$ derivatives $P_I$. Using the equation Eq. , one can replace the operator $Q^{(k)}$ with expressions that contain operators $Q^{(j)}$ with $j>k$, but also inverse powers of $\det S^{(k+1)}$ (which are non-zero). This is possible as long as $k\leq n-2$, therefore the procedure saturates when all the parity-odd operators are brought to the form .
Discussion {#sec:discussion}
==========
In this work, we complete the classification of independent vertices of arbitrary order $n\geq 3$ for massless bosonic fields with arbitrary spin in arbitrary space-time dimensions $d\geq 3$.
We briefly summarise the results:
- For dimensions $d\geq 2n-1$ there are no Schouten identities. After reducing to the independent Mandelstam variables, we find that all gauge invariant operators can be expressed as polynomials in the gauge-invariant combinations $c_{ij}$ and $Y_{i}^{j}$, $$\mathcal{V} \in M_{1}^{-1}\mathbb{R}[s_{ij},c_{ij},Y_{i}^{j}]\, ,$$ where $M_{1}$ is the set of all products of Mandelstam variables $s_{ij}$ ($i\not = j$). The invariant combinations $Y_{i}^{j}$ are labelled by $i=1,\dotsc ,n$ and $j=2,\dotsc ,n-2$.
- For dimensions $d < n$ we have the full set of Schouten identities at our disposal. All gauge invariant operators are already generated by the $Y_{i}^{j}$ where $i=1,\dotsc ,n$ and $j=2,\dotsc ,d-1$. All remaining relations are generated by level-$0$ Schouten identities and specific quadratic expressions $q_{2}^{i}$ in the variables $Y_{i}^{j}$, $$[\mathcal{V}] \in \frac{M^{-1}\mathbb{R}[s_{ij},Y_{i}^{j}]}{\langle (\det B_{0} (A)), q_{2}^{i} \rangle} \, ,$$ where again we reduced to the independent Mandelstam variables.
- In the intermediate case ($2n - 1 > d \geq n$), we have Schouten identities, but because $d\geq n$ the non-trivial Schouten identities involve at least $(d-n)+2\geq 2$ level-$1$ rows and columns. By an argument analogous to the one leading to Eq. one can show that in the ring of fractions all Schouten identities are generated by those that contain $n-1$ level-$0$ rows and columns and $(d-n)+2$ rows and columns of level-$1$. Let us denote them by $\det B_{2 (d-n)+4} (A)$, where $A$ labels the possible choices of the level-$1$ rows and columns. These generators are all gauge-invariant (up to total derivatives), and hence we can express them in terms of the invariant combinations $c_{ij}$ and $Y_{i}^{j}$ as in Section \[sec:case-2n-leq\]. Then the gauge invariant vertices are classified by equivalence classes $$[\mathcal{V}] \in \frac{M^{-1}\mathbb{R}[s_{ij},c_{ij},Y_{i}^{j}]}{\langle \det B_{2 (d-n)+4} (A)\rangle}\, .$$
An interesting question is whether the higher order vertices can induce deformations of gauge transformations for the fields involved. Deformations arise when the gauge variation is non-trivial before imposing the equations of motion. Terms in the variation that contain the equations of motion have to be compensated by a non-trivial $\delta^{(n-2)}$ in Eq. . We have found that in all dimensions, as long as we are allowed to divide by Mandelstam variables, the independent gauge-invariant vertices can be expressed in terms of the combinations $c_{ij}$ and $Y_{i}^{j}=c_{i,i+j i+1}$, but these — as defined in Eq. and Eq. — are manifestly gauge-invariant without need of equations of motion. This strongly suggests that the vertex does not induce a deformation. Strictly speaking we can only conclude that $\Delta \mathcal{V}$ for an appropriate product $\Delta $ of Mandelstam variables does not induce any deformation. However, in Fourier space $\Delta$ is simply a (generically non-zero) number and should not change the general structure of deformations, hence we do not expect that $\mathcal{V}$ itself can induce a deformation.
To recapitulate, as soon as we allow for dividing by Mandelstam variables (and hence, we loose manifest locality), the independent vertices of order $n\geq 4$ can be all written in terms of linearised curvatures of HS fields. Therefore they are manifestly gauge invariant with respect to linearised gauge transformations and do not introduce deformations for the latter. On the other hand, if such deformations of the gauge transformations, induced from cubic vertices, exist in the theory, then these vertices will be completed by further non-linear terms. This is similar to higher-curvature terms in Einstein Gravity, whose non-linear structure is gauge invariant with respect to full diffeomorphisms, induced from the Einstein-Hilbert cubic vertex. Such non-linear completions may make use of non-linear generalisation of de Wit-Freedman curvatures [@deWit:1979sib], which are not known in metric-like formulation (see, however, [@Manvelyan:2010jf]). In the frame formulation, these vertices would correspond to structures that make use of Weyl tensors and their descendants (zero form sector of Vasiliev system). In the light of our findings here, the three dimensional results of [@Fredenhagen:2019hvb] can be interpreted as particular case of the general dimensional results: all the independent vertices are given through linearised curvatures, which are on-shell trivial in $d=3$.
Even though the classification is done for Minkowski spaces, we expect the vertices found here to deform smoothly to $(A)dS$ space-times as it happens for cubic vertices. Indeed, the existence of $(A)dS$ extensions for linearised de Wit-Freedman curvatures for HS fields [@Manvelyan:2007hv] allows to straightforwardly lift vertices given through curvatures to $(A)dS_d$. Same is true for the operators and , where one can simply replace derivatives with $(A)dS_d$ covariant ones.
Our results should have a direct analogue for correlation functions of conserved tensors in $d-1$ dimensional conformal field theories, which can be classified with similar methods [@Costa:2011mg]. For $n=3$ there is a precise match between independent vertices and three-point functions [@Metsaev:2005ar; @Manvelyan:2010jr; @Giombi:2011rz; @Costa:2011mg; @Joung:2011ww; @Conde:2016izb; @Francia:2016weg; @Sleight:2017fpc; @Fredenhagen:2018guf]. It would be interesting to compare our findings for $n\geq 4$ with the group theoretic results of [@Kravchuk:2016qvl].
Next, we would like to note that there is another interpretation of the equation Eq. (\[eq:gauge-0\]) which we solved here. One can think of Eq. (\[eq:gauge-0\]) as a Ward identity for an $n$-point amplitude computed in a theory of interacting HS fields. It is clear from our discussion, that the building blocks of the amplitudes are given through $c_{ij}$, $Y_{i}^{j}=c_{i,i+j i+1}$ and Mandelstam variables, including negative powers of the latter. They correspond to arbitrary tensor contractions of linearised curvatures [@deWit:1979sib] of HS gauge fields and their derivatives. These linear de Wit-Freedman curvatures (or their traceless part: the Weyl tensors) and their derivatives are the only on-shell non-zero gauge invariants with respect to the linearised gauge transformations. It is natural that the amplitudes for $n\geq 4$ should be given through gauge invariant quantities, as they are observable.
The amplitude interpretation might be less motivated in three dimensions, since there are no propagating HS massless particles in three dimensions. As proved in [@Fredenhagen:2019hvb], there are no candidate invariants for amplitudes with such fields either for $d=3$. There is one difference between amplitudes and vertices though — the latter are supposed to be local, while the former do not have to. Given that one can always multiply the candidate invariant vertices (amplitudes) by a non-vanishing function of Mandelstam variables, one can show that relaxing locality would not help to get non-zero amplitudes in $d=3$. There is an interesting conclusion to be made here: since the amplitude is a sum of exchanges[^19] and contact vertices, vanishing amplitudes imply that the exchanges and contact vertices should cancel each other. This is only possible if the non-local parts of the exchanges sum up to zero, which should be specific to three dimensions and is presumably due to the special structure of vertices and Schouten identities present only in three dimensions. We plan to study the Lagrangian formulation of metric-like non-linear HS theories with(out) matter in the near future to expose these special properties of HS gravities in $d=3$.
#### Note added
We learned from Euihun Joung and Massimo Taronna about their preprint with related results [@EJMT], which will appear on arxiv simultaneously.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are grateful to Dario Francia, Euihun Joung and Shailesh Lal for useful discussions on the subject of this work. KM is grateful to Max Planck Institute for Gravitational Physics (Albert Einstein Institute), where part of this work was done. The hospitality of the Erwin Schr[ö]{}dinger International Institute for Mathematics and Physics during the program on “Higher Spins and Holography” where this work was initiated is greatly appreciated.
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[^1]: The index $s$ is the spin of the field and $\mu_i = 0, \ldots , d - 1$ in $d$ dimensions.
[^2]: In this paper, we will restrict ourselves to integer-spin (bosonic) fields for simplicity.
[^3]: The round (square) brackets denote (anti-)symmetrisation with weight one.
[^4]: Gauge invariance provides constraints to fix the terms proportional to higher powers of coupling constants. We are interested here in the structures that parametrise the non-trivial deformations at the lowest order in the coupling constants.
[^5]: We consider a flat Minkowski space-time but comment also on $(A)dS$ backgrounds in the Section \[sec:discussion\].
[^6]: Formally, let $\iota_d$ be the map $$\begin{aligned}
\iota_{d} : \qquad \qquad \mathcal{R} \qquad &\to \quad \mathbb{R}[P_i^{\mu},A_i^{\mu}] \\
\mathcal{V}(z_{ij},y_{ij},s_{ij}) & \mapsto \mathcal{V}(A_i \cdot A_j, A_i \cdot P_j,P_i \cdot P_j)
\end{aligned}$$ that replaces the operators $z_{ij}$, $y_{ij}$ and $s_{ij}$ by their definitions in Eq. (\[eq:zys\]). $\iota_d$ therefore reintroduces the operators $P_i$ and $A_i$ and hence, spacetime indices in $d$ dimensions in the vertex generating operator $\mathcal{V}$. The kernel $\iota_{d}^{-1} (0)$ of this map is what we call the ideal of Schouten identities in $d$ dimensions.
[^7]: Notice that the operators $D_{k}$ are consistent with these constraints , which means that $D_{k}$ acting on a constraint will lead to a constraint. Therefore we can leave the operators $D_k$ in the general form stated in Eq. (\[gvo\]) and do not need to express them in terms of a set of independent variables.
[^8]: Note that $c_{i,jk}$ can be expressed as a polynomial in $Y$’s and Mandelstam variables via Eq. (\[eq:Y(c)\]).
[^9]: By adding total derivatives they can be transformed to a expression of the type where the $n-1$ rows (columns) of the first block contain the $m$ rows (columns) corresponding to those of the second block.
[^10]: as long as we can divide by Mandelstam variables
[^11]: First, non-zero minors of order one are just the Mandelstam variables $s_{ij}$ with $i \neq j$. Secondly, all minors of order $2,3,\dots, d$ are generically non-zero — even when the equivalence relations in Eq. (\[eq:equiv2\]) are applied. Finally, all minors of order greater than $d$ do vanish due to Schouten identities. Hence, ${\text{Mi}}(\mathcal{S})$ consists of all $(2\times2)\,,\; (3\times3)\,,\; \dots (d \times d)$ subdeterminants of $\mathcal{S}$ as well as the Mandelstam variables $s_{ij}$ with $i \neq j$.
[^12]: This proof relies on the fact that $n > d$.
[^13]: the indices are considered modulo $n$.
[^14]: If $r < s$, we choose $M^T$ instead of $M$, which yields the same Schouten identity $\det M^T = \det M$. $M^T$ is a submatrix of $\mathcal{B}$ as well because $\mathcal{B}$ is symmetric.
[^15]: $\mathcal{B}$ has more than $d$ level-0 columns, since $n>d$.
[^16]: In the case that $s = 0$, $N$ is just a Mandelstam variable. But within the construction of $\widetilde{M}$, we chose $Row$ and $Col$ such that its intersection (which is $N$ in that case) is non-zero. Therefore, $\det N = N \neq 0$.
[^17]: We discussed these in the previous sections, where they were called $\mathcal{V}$.
[^18]: In the special case of cubic vertices in $d=4$ [@Conde:2016izb], one even gets negative powers of $y_{ij}$’s, which, however can be removed by inverting this procedure after solving for the vertex operator.
[^19]: The exchange is again a notion that is defined when there are particles to exchange, but this should not affect our argument, given that a propagator for massless HS fields can be formally defined in three dimensions. See, e.g., [@David:2009xg; @Giombi:2013fka]. We thank Shailesh Lal for a discussion about this point.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we consider the localization of a five-dimensional gravitino field on $f(R)$ thick branes. We get the coupled chiral equations of the Kaluza-Klein (KK) modes of gravitino by choosing the gauge condition $\Psi_z=0$. It is found that the chiral equations of the gravitino KK modes are almost the same as the ones of the Dirac fermion. However, their chiralities are precisely opposite. The chiral KK modes of gravitino could be localized on some kinds of $f(R)$ thick branes if a coupling term is introduced. We investigate the localization of gravitino on three kinds of $f(R)$ thick branes through a Yukawa-like coupling term with background scalar fields. It has been shown that all the KK modes of gravitino can not be localized on the pure geometric $f(R)$ thick branes by adding a five-dimensional gravitino mass term. However, for the $f(R)$ thick branes generated by one or two background scalar fields, only the left- or right-handed zero mode could be localized on the branes and the massive KK resonant modes are the same for both left- and right-handed gravitinos, in spite of their opposite chiralities. All these results are consistent with that of the five-dimensional Dirac fermion except their chiralities, which may be an important sign to distinguish the gravitino field and the Dirac fermion field.'
author:
- 'Xiang-Nan Zhou$^{1}$[^1], Yun-Zhi Du$^{2}$[^2], Hao Yu$^{3}$[^3], and Yu-Xiao Liu$^{3}$[^4]'
title: 'Localization of Gravitino Field on $f(R)$ Thick Branes'
---
Introduction {#scheme1}
============
The extra dimensional theory has attracted more and more attention even though the visible world is a four-dimension spacetime [@Akama:1982jy; @Rubakov:1983; @ADD; @Antoniadis:1998ig; @Randall:1999ee; @Randall:1999vf; @ArkaniHamed:2000eg; @Kim:2000mc; @Nussinov:2001rb; @Pankov:2005ar; @Abulencia:2006kk; @Dey:2009xu; @Neupane:2010ey]. Some classical physical problems including the gauge hierarchy problem (the huge difference between the Planck scale and the weak scale) [@Antoniadis:1998ig; @Randall:1999ee; @Das:2007qn; @Yang:2012dd] and the cosmological problem [@Dvali:2000rv; @Starkman:2001xu; @Kim:2000mc; @Dey:2009xu; @Neupane:2010ey] could be solved via utilizing extra dimensions. In the 1920s, the Kaluza-Klein (KK) theory was proposed to unify Einstein’s gravity and electromagnetism by introducing a compact extra spacial dimension with Plank size [@Kaluza; @Klein]. Several decades later, Akama, Rubakov and Shaposhnikov proposed the idea of domain wall braneworld with an infinite extra dimension in five-dimensional flat spacetime [@Akama:1982jy; @Rubakov:1983]. In 1998, Antoniadis and Arkani-Hamed etc introduced the famous model with large extra dimension that attempts to solve the hierarchy problem [@ADD; @Antoniadis:1998ig]. One year latter, Randall and Sundrum (RS) suggested that the extra dimension with warped geometry could be [finite [ or]{} infinite, [corresponding to]{} the RSI [@Randall:1999ee] or RSII [@Randall:1999vf] thin braneworld model. In [both]{} braneworld scenarios,]{} our visible four-dimensional world is a brane [without]{} thickness along the extra dimension, [and]{} the matter fields of the Standard Model (SM) are confined on the brane [while]{} only gravity propagates in the five-dimensional [bulk]{} spacetime. [Subsequently]{}, more realistic thick branes generated dynamically by matter fields or pure gravity were introduced [@Gregory:2000jc; @Kaloper:2000jb; @Melfo:2002wd; @Bazeia:2004dh; @Cardoso:2006nh; @Liu:2011wi; @Liu:2012rc; @Liu:2012mia; @Bazeia:2015owa; @Yu:2015wma]. In these models, there exists a non-vanishing distribution of energy density along the extra dimension.
[The braneworld [scenario]{} with]{} [ warped]{} infinite extra dimensions [requires]{} a natural physical mechanism to trap the matter fields on the branes in order to not conflict with the current experiments. Thus, it is significant to investigate the localization of the matter fields on various kinds of branes [@Chang:1999nh; @Shiromizu:1999wj; @Kehagias:2000au; @Ringeval:2001cq; @Maity:2003im; @Chatterjee:2005cr; @Melfo:2006hh; @Liu:2007gk; @Davies:2007xr; @Liu:2009ve; @Guerrero:2009ac; @Fu:2012sa; @Xie:2013rka; @Liu:2013kxz; @Zhao:2014gka; @Guo:2014nja; @Du:2015pjw]. In order to rebuild the SM on the branes, the zero modes of these matter fields (the four-dimensional massless particles) should be localized on the branes. At the same time, [the localization of massive KK modes are crucial to provide a method to explore extra dimensions.]{} For example, we may observe some physical effects of these KK particles interacting with the SM particles in the Large Hadron Collider (LHC) [@Hung:2003cj; @Guo:2011qt; @Sahin:2014dua; @Bauer:2016lbe]. In some braneworld models, there are no bounded massive KK modes for some matter fields, [ while]{} there may be some resonant KK modes [quasi-localized]{} on the branes. These massive resonant KK modes may stay on the branes for a [ long]{} time and interact with other particles, which could provide us with opportunities to find the massive resonant KK modes and prove the existence of extra dimensions [@Liu:2009ve; @Xie:2013rka; @Guo:2014nja; @Almeida:2009jc; @Liu:2009mg; @Landim:2011ki; @Du:2013bx; @Zhang:2016ksq].
Gravitino is the gauge fermion supersymmetric partner of graviton in the supersymmetry theory. It has been suggested as a candidate for dark matter in cosmology [@Chun:1993vz; @Moroi:1995fs; @Steffen:2006hw; @Feng:2010ij; @Savvidy:2012qa]. It is a fermion of spin 3/2 and obeys the Rarita-Schwinger equation. The mass of [ a]{} light gravitino is always considered around 1eV [@Chun:1993vz], but there are still some challenges to its mass [@Feng:2010ij]. Its mass is widely investigated in the models of hot and cold dark matters [@Chun:1993vz; @Steffen:2006hw], and the possibility of finding light [gravitinos]{} at the LHC was discussed in Ref. [@Shirai:2009kn]. The behaviors of gravitino around a black hole also attract attentions [@Khlopov:2004tn; @Yale:2008kx; @Arnold:2013zva; @Chen:2015jga]. Besides, gravitino is a kind of matter [field]{} beyond the SM with many special properties that the SM matter fields do not possess. Therefore, the localization of a five-dimensional gravitino field on a brane will be very interesting and give us new perspective to investigate the gravitino. Compared to the matter fields of the SM such as the scalar and fermion fields, [the works on gravitino are few and not comprehensive]{} [@Liu:2007gk; @Du:2015pjw; @Bajc:1999mh; @Oda:2000dd; @Gherghetta:2000kr; @Oda:2000wa; @Hewett:2002uq; @Lee:2007ib]. The zero mode of a [ five-dimensional]{} free gravitino can be localized on a RS-like brane only when a bulk mass term is introduced [@Oda:2000dd]. In a $D$-dimensional spacetime with $D\geq5$, the zero mode of the gravitino with a coupling term can be localized on the brane, and the localization property is similar to that of the Dirac fermion [@Liu:2007gk; @Oda:2000wa]. In addition, the behavior of the gravitino KK modes with coupling terms was investigated in Ref. [@Hewett:2002uq]. Recently, the localization and mass spectrum of the gravitino KK modes on two kinds of thin branes (the RS branes and the scalar-tensor branes) were investigated in Ref. [@Du:2015pjw]. It should be noticed that most of these investigations focused on the RS-like thin branes.
In this paper, we pay our attention to the localization of a five-dimensional gravitino field on the $f(R)$ thick branes. Although general relativity is a very successful theory, its non-renormalization motives the investigation of modified gravity theories, particularly the gravity theories including higher-order curvature terms [@Sotiriou:2008rp]. $f(R)$ gravity is a kind of modified gravity whose Lagrangian is a function of the scalar curvature $R$. It always contains [higher-order curvature invariants]{}, which could make the theory to be renormalizable [@Sotiriou:2008rp]. [Furthermore]{}, the $f(R)$ gravity could be used to explain the dark energy or dark matter and answer the astrophysical and cosmological riddles. Therefore, it has been studied widely in cosmology and braneworld [@Liu:2011wi; @Yu:2015wma; @Sotiriou:2008rp; @DeFelice:2010aj; @Nojiri:2010wj; @Nojiri:2000gv; @Nojiri:2001ae; @Giovannini:2001xg; @Afonso:2007zz; @Dzhunushaliev:2009dt; @Liu:2011am; @Bazeia:2015oqa; @Zhong:2010ae; @Zhong:2012nt; @Zhong:2015pta]. In Ref. [@Yu:2015wma], the authors investigated various kinds of $f(R)$-branes and gave their general solutions. All of these solutions are also appropriate for the general relativity braneworlds, i.e., $f(R)=R$.
In this paper, we would [ like to]{} investigate the localization of a five-dimensional gravitino field on the $f(R)$ thick branes, whose solutions [have]{} been given in Ref. [@Yu:2015wma]. The conclusions of the localization of the gravitino will have some certain universality because they are also appropriate for the general relativity braneworlds. We believe it will give us some interesting results for the structure of thick branes that the thin branes do not have. Our work is organized as follows. In Sec. II, we consider the localization of a five-dimensional free massless gravitino field on a thick brane. We introduce the gauge condition [$\Psi_z=0$]{} and get the Schrödinger-like equations of [the gravitino]{} KK modes. Then we focus on the localization of a five-dimensional gravitino field with a coupling term on a thick brane in Sec. III. Three kinds of $f(R)$ thick branes are considered and the massive KK resonances are studied. Finally, discussion and conclusion are given in Sec. IV.
Localization of free gravitino field on thick branes {#sec2}
====================================================
Firstly, we consider the localization of a free massless gravitino field on a thick brane in a five-dimensional spacetime. Usually, it can be to assume the five-dimensional line-element as $$ds^2=g_{MN}dx^Mdx^N=\text{e}^{2A(y)}\hat g_{\mu\nu}(x)dx^\mu dx^\nu + dy^2.\label{Le}$$ Here, $M$ and $N$ denote the curved five-dimensional spacetime indices, $\hat{g}_{\mu\nu}$ is the metric on the brane and the warp factor $\text{e}^{2A(y)}$ is only the function of the extra dimension $y$. For convenience, the following coordinate transformation $$\begin{aligned}
dz=\text{e}^{-A(y)}dy\label{ytoz}\end{aligned}$$ could be performed to transform the metric (\[Le\]) to be $$\begin{aligned}
ds^2=\text{e}^{2A(z)}(\hat g_{\mu\nu}dx^\mu dx^\nu + dz^2).\label{LE}\end{aligned}$$
The action of a free massless gravitino field $\Psi$ in five-dimensional spacetime is given by [@Liu:2007gk; @Oda:2000wa; @Du:2015pjw] $$\begin{aligned}
S_{{\frac}{3}{2}}=\int d^5 x \sqrt{-g}~\bar\Psi_M\Gamma^{[M}\Gamma^N\Gamma^{R]}D_N\Psi_R, \label{action}\end{aligned}$$ and the corresponding equations of motion read as $$\begin{aligned}
\Gamma^{[M}\Gamma^N\Gamma^{R]}D_N\Psi_R=0.\label{motionequation}\end{aligned}$$ The Dirac gamma matrices $\Gamma^{M}$ in curved five-dimensional spacetime satisfy $\Gamma^M=e^M_{~~\bar M}\Gamma^{\bar M}$. $\Gamma^{\bar M}$ are the gamma matrices in flat five-dimensional spacetime and $\{\Gamma^{\bar M},~\Gamma^{\bar N}\}=2\eta^{\bar M\bar N}$, where $\bar M$ and $\bar N$ represent the five-dimensional local Lorentz indices. The vielbein satisfies $g_{MN}=e_M^{~~\bar M}e_N^{~~\bar N}\eta_{\bar M\bar N}$ and for the metric (\[LE\]) it is given by $$\begin{aligned}
e_M^{~~\bar M}=\left(
\begin{array}{cc}
\text{e}^{A}\hat e_\mu^{~~\bar\mu}&
0\\
0&
\text{e}^{A}
\end{array}\right),~~~~
e^M_{~~\bar M}=\left(
\begin{array}{cc}
\text{e}^{-A}\hat e^\mu_{~~\bar\mu}&
0\\
0&
\text{e}^{-A}
\end{array}\right).\end{aligned}$$ From the relations $e_{M\bar M}=g_{MN}e^{N}_{~~\bar M}$ and $e^{M\bar M}=g^{MN}e_{N}^{~~\bar M}$, we can get $$\begin{aligned}
e_{M\bar M}=\left(
\begin{array}{cc}
\text{e}^{A}\hat e_{\mu\bar\mu}&0\\
0&\text{e}^{A}
\end{array}\right),~~~~~~
e^{M\bar M}=\left(
\begin{array}{cc}
\text{e}^{-A}\hat e^{\mu\bar\mu}&0\\
0&\text{e}^{-A}
\end{array}\right).\end{aligned}$$ Thus $\Gamma^M=\text{e}^{-A}(\hat e^\mu_{~~\bar\mu}\gamma^{\bar\mu},~\gamma^5)=\text{e}^{-A}(\gamma^{\mu},~\gamma^5)$, where $\gamma^{\mu}= \hat e^\mu_{~~\bar\mu}\gamma^{\bar\mu}$, $\gamma^{\bar\mu}$ and $\gamma^5$ are the flat gamma matrices in the four-dimensional Dirac representation. In this paper, we choose the following representation for the four-dimensional flat gamma matrices: $$\begin{aligned}
\gamma^0=\left(
\begin{array}{cc}
0&-\text{i}\mathbb{I}\\
-\text{i}\mathbb{I}&0
\end{array}\right),~~
\gamma^i=\left(
\begin{array}{cc}
0&\text{i}\sigma^i\\
-\text{i}\sigma^i&0
\end{array}\right),~~
\gamma^5=\left(
\begin{array}{cc}
\mathbb{I}&0\\
0&-\mathbb{I}
\end{array}\right).\label{Gamma}\end{aligned}$$ Here $\mathbb{I}$ is a two-by-two unit matrix and $\sigma^i$ are the Pauli matrices. In this paper, we only consider flat chick branes, i.e. $\hat{g}_{\mu\nu}=\eta_{\mu\nu}$. So we have $\hat{e}^{\mu}_{~\bar{\mu}}=\delta^{\mu}_{~\bar{\mu}}$ and $\gamma^{\mu}=\gamma^{\bar{\mu}}$. In addition, the covariant derivative of a gravitino field is defined by $$\begin{aligned}
D_N\Psi_R=\partial_N\Psi_R-\Gamma^M_{~~NR}\Psi_M+\omega_N\Psi_R,\end{aligned}$$ where the spin connection $\omega_N$ is defined by $\omega_N=\frac{1}{4}\omega_{N}^{~~\bar N\bar L}
\Gamma_{\bar N}\Gamma_{\bar L}$ and $\omega_{N}^{~~\bar N\bar L}$ is given by $$\begin{aligned}
\omega_{N}^{~~\bar N\bar L}
=\frac{1}{2}e^{M\bar N}(\partial_N e_M^{~~\bar L}-\partial_M e_N^{~~\bar L})
-\frac{1}{2}e^{M\bar L}(\partial_N e_M^{~~\bar N}-\partial_M e_N^{~~\bar N})
-\frac{1}{2}e^{M\bar N}e^{P\bar L}(\partial_M e_{P\bar R}-\partial_P e_{M\bar R})e^{~\bar R}_N.\end{aligned}$$ Thus we get the non-vanishing components of $\omega_N$: $$\begin{aligned}
\omega_\mu=\frac{1}{2}(\partial_z A)\gamma_\mu\gamma_5+\hat\omega_\mu.\end{aligned}$$ Note that the four-dimensional spin connection $\hat\omega_\mu$ on a flat brane vanishes. The non-vanishing components of $D_N\Psi_R$ are $$\begin{aligned}
D_\mu\Psi_\nu
&=&\partial_\mu\Psi_\nu-\Gamma^M_{~~\mu\nu}\Psi_M+\omega_\mu\Psi_\nu\nonumber\\
&=&\hat D_\mu\Psi_\nu+(\partial_z A)\hat g_{\mu\nu}\Psi_z+\frac{1}{2}(\partial_z A)\gamma_\mu\gamma_5\Psi_\nu,\label{G1}\\
D_\mu\Psi_z
&=&\partial_\mu\Psi_z-\Gamma^M_{~~\mu z}\Psi_M+\omega_\mu\Psi_z\nonumber\\
&=&\partial_\mu\Psi_z-(\partial_z A)\Psi_\mu+\frac{1}{2}(\partial_z A)\gamma_\mu\gamma_5\Psi_z+\hat\omega_\mu\Psi_z,\label{G3}\\
D_z\Psi_\mu
&=&\partial_z\Psi_\mu-\Gamma^M_{~~z\mu}\Psi_M+\omega_z\Psi_\mu\nonumber\\
&=&\partial_z\Psi_\mu-(\partial_z A)\Psi_\mu,\label{G4}\\
D_z\Psi_z
&=&\partial_z\Psi_z-\Gamma^M_{~~zz}\Psi_M+\omega_z\Psi_z\nonumber\\
&=&\partial_z\Psi_z-(\partial_z A)\Psi_z.\end{aligned}$$ Equation (\[motionequation\]) includes five equations because $M$ runs over all five spacetime indices. There are two kinds of equations: $M=5$ and $M=\mu$. For the first case of $M=5$, the equation of motion reads as $$\begin{aligned}
\Gamma^{[5}\Gamma^{N}\Gamma^{R]}D_N\Psi_R&=&
\Gamma^{[5}\Gamma^{\mu}\Gamma^{\nu]}D_\mu\Psi_\nu\nonumber\\
&=&\big([\Gamma^{\mu},~\Gamma^{\nu}]-g^{\mu\nu}\big)
\Gamma^5\Big(\hat D_\mu\Psi_\nu+(\partial_z A)\hat g_{\mu\nu}\Psi_z+\frac{1}{2}(\partial_z A)\gamma_\mu\gamma_5\Psi_\nu\Big) \nonumber\\
&=&0.\label{EOM1}\end{aligned}$$ In this paper, for convenience we prefer to choose the gauge condition $\Psi_z=0$, with which we introduce the KK decomposition $$\begin{aligned}
\Psi_\mu=\sum\psi^{(n)}_\mu(x)\xi_n(z), \label{KKdecomposition}\end{aligned}$$ where $\psi^{(n)}_\mu(x)$ is the four-dimensional gravitino field. Then Eq. (\[EOM1\]) is reduced to $$\begin{aligned}
\big([\gamma^{\mu},~\gamma^{\nu}]-\hat g^{\mu\nu}\big) \gamma^5~
\Big(\hat D_\mu\psi_\nu^{(n)}
+\frac{1}{2} (\partial_z A)\gamma_\mu\gamma_5\psi_\nu^{(n)}
\Big)=0. \label{EqsGravitino5}\end{aligned}$$ For the four-dimensional massive gravitino field $\psi_\mu$, it should satisfy the following four equations [@Moroi:1995fs]
\[Eqs4DGravitino\] $$\begin{aligned}
\gamma^{[\lambda}\gamma^{\mu}\gamma^{\nu]}\hat{D}_{\mu}\psi_{\nu}-m_{3/2}[\gamma^{\lambda},~\gamma^{\mu}]\psi_{\mu}&=&0, \\
\gamma^{\mu}\psi_{\mu}&=&0,\\
\hat{D}^{\mu}\psi_{\mu}&=&0,\\
(\gamma^{\mu}\hat{D}_{\mu}+m_{3/2})\psi_{\nu}&=&0.\end{aligned}$$
Here, $m_{3/2}$ is the mass of a four-dimensional gravitino field $\psi_{\mu}$. Thus the left-hand side of Eq. is always vanished for a four-dimensional gravitino field $\psi_{\mu}^{(n)}$ [satisfying]{} the above equation . On the other hand, when we choose the gauge condition $\Psi_{z}=0$, the part $\Gamma^{[5}\Gamma^{N}\Gamma^{R]}D_N\Psi_R$ in the five-dimensional gravitino action has no contribution, so Eq. (\[EOM1\]) can be ignored. Then we will focus on the case of $M=\mu$, for which the equations of motion are $$\begin{aligned}
\Gamma^{[\lambda}\Gamma^{N}\Gamma^{L]}D_N\Psi_L&=&
\Gamma^{[\lambda}\Gamma^{\mu}\Gamma^{\nu]}D_\mu\Psi_\nu+
\Gamma^{[\lambda}\Gamma^{\nu}\Gamma^{5]}D_\nu\Psi_z+
\Gamma^{[\lambda}\Gamma^{5}\Gamma^{\nu]}D_z\Psi_\nu\nonumber\\
&=&\text{e}^{-3A}\gamma^{[\lambda}\gamma^{\mu}\gamma^{\nu]}\hat{D}_{\mu}\Psi_{\nu}-\text{e}^{-3A}[\gamma^\lambda,~\gamma^\nu]\gamma_5
(\partial_z A+\partial_z)\Psi_{\nu}\nonumber\\
&=&0,
\label{EOM2}\end{aligned}$$ where we have used the gauge condition $\Psi_z=0$. When we introduce the decomposition and consider the zero mode, which corresponds to the four-dimensional massless gravitino satisfying $\gamma^{[\lambda}\gamma^{\mu}\gamma^{\nu]}\hat{D}_{\mu}\psi^{(0)}_{\nu}=0$, we get the equation of motion for the extra-dimensional configuration $\xi_0(z)$: $$\begin{aligned}
&& \gamma^{[\lambda}\gamma^{\mu}\gamma^{\nu]}\hat{D}_{\mu}\psi^{0}_{\nu}(x)\xi_0(z)
-[\gamma^\lambda,~\gamma^\nu]\gamma_5\psi^{(0)}_\nu(x)
(\partial_z A+\partial_z)\xi_0(z)\nonumber\\
&=&-(\partial_z A+\partial_z)\xi_0(z)=0.\end{aligned}$$ Obviously, the solution is $$\begin{aligned}
\xi_0(z)=C\text{e}^{-A(z)},\end{aligned}$$ where $C$ is a normalization constant. By substituting the zero mode $\xi_0(z)$ into the gravitino action (\[action\]) yields $$\begin{aligned}
S_{{\frac}{3}{2}}^{(0)} =\mathcal{I}_0
\int d^4 x~\sqrt{-\hat g}~\bar\psi^{(0)}_\lambda
\gamma^{[\lambda}\gamma^\mu\gamma^{\nu]}\hat D_\mu\psi^{(0)}_\nu(x),
\end{aligned}$$ where $\mathcal{I}_0 \equiv \int dz~\text{e}^{2A}\xi^2_0(z)=C^2\int dz =C^2\int e^{-A(y)}dy$. In order to localize the spin 3/2 gravitino on a brane, the integral $\mathcal{I}_0$ must [be]{} finite. So if we consider a RS-type brane model, then only for a finite extra dimension the zero mode of a five-dimensional free massless gravitino can be localized on the brane.
For the massive modes, we need to introduce the following chiral decomposition: $$\begin{aligned}
\Psi_\mu(x,z)
&=&\sum_n\left(\psi^{(n)}_{L\mu}(x)\xi_{Ln}(z)+\psi^{(n)}_{R\mu}(x)\xi_{Rn}(z)\right) \nonumber \\
&=&\sum_n\bigg(\left[
\begin{array}{c}
0\\ \tilde{\psi}^{(n)}_{L\mu}\xi_{Ln}
\end{array}\right]+\left[
\begin{array}{c}
\tilde{\psi}^{(n)}_{R\mu}\xi_{Rn}\\0
\end{array}\right]
\bigg),\label{CD}\end{aligned}$$ where $\tilde{\psi}^{(n)}_{L\mu}$ and $\tilde{\psi}^{(n)}_{R\mu}$ are both the two-component spinors. The effect of $P_{L,R}$ ($P_{L,R}=\frac{1}{2}[I \mp \gamma^5]$) on the gravitino field $\Psi_M$ is to single out the left- and right-handed parts, respectively, which is equivalent to the following equations: $$\begin{aligned}
\gamma^5\psi^{(n)}_{L\mu}=-\psi^{(n)}_{L\mu}, ~~~~~~\gamma^5\psi^{(n)}_{R\mu}=\psi^{(n)}_{R\mu}. \label{partiyrelation}\end{aligned}$$ Thus, substituting the chiral decomposition (\[CD\]) into Eq. (\[EOM2\]), we have $$\begin{aligned}
0&=&\gamma^{[\lambda}\gamma^\mu\gamma^{\nu]}\hat D_\mu\psi^{(n)}_{L\nu}\xi_{Ln}
+\gamma^{[\lambda}\gamma^\mu\gamma^{\nu]}\hat D_\mu\psi^{(n)}_{R\nu}\xi_{Rn}
+[\gamma^{\lambda},~\gamma^{\nu}]
(\partial_z A)\psi^{(n)}_{L\nu}\xi_{L n}\nonumber\\
&&-[\gamma^{\lambda},~\gamma^{\nu}](\partial_z A)\psi^{(n)}_{R\nu}\xi_{Rn}
+[\gamma^{\lambda},~\gamma^{\nu}]\psi^{(n)}_{L\nu}\partial_z\xi_{Ln}
-[\gamma^{\lambda},~\gamma^{\nu}]\psi^{(n)}_{R\nu}\partial_z\xi_{Rn}.\end{aligned}$$ Since the product of three gamma matrices is oblique diagonal and the product of two gamma matrices is diagonal, two equations can be obtained from above equation:
$$\begin{aligned}
\gamma^{[\lambda}\gamma^\mu\gamma^{\nu]}\hat D_\mu\psi^{(n)}_{L\nu}\xi_{Ln}
-[\gamma^{\lambda},~\gamma^{\nu}](\partial_z A)\psi^{(n)}_{R\nu}\xi_{Rn}
-[\gamma^{\lambda},~\gamma^{\nu}]\psi^{(n)}_{R\nu}\partial_z\xi_{Rn}=0,\\
\gamma^{[\lambda}\gamma^\mu\gamma^{\nu]}\hat D_\mu\psi^{(n)}_{R\nu}\xi_{Rn}
+[\gamma^{\lambda},~\gamma^{\nu}](\partial_z A)\psi^{(n)}_{L\nu}\xi_{L n}
+[\gamma^{\lambda},~\gamma^{\nu}]\psi^{(n)}_{L\nu}\partial_z\xi_{Ln}=0.\end{aligned}$$
Through the method of separation of variance and defining a parameter $m_n$, we have
$$\begin{aligned}
\frac{\gamma^{[\lambda}\gamma^\mu\gamma^{\nu]}\hat D_\mu\psi^{(n)}_{L\nu}}
{[\gamma^{\lambda},~\gamma^{\alpha}]\psi^{(n)}_{R\alpha}}
=\frac{A'\xi_{Rn}+\partial_z\xi_{Rn}}{\xi_{Ln}}=m_n,\\
\frac{\gamma^{[\lambda}\gamma^\mu\gamma^{\nu]}\hat D_\mu\psi^{(n)}_{R\nu}}
{[\gamma^{\lambda},~\gamma^{\alpha}]\psi^{(n)}_{L\alpha}}
=-\frac{(\partial_z A)\xi_{Ln}+\partial_z\xi_{Ln}}{\xi_{Rn}}=m_n,\end{aligned}$$
i.e., $$\begin{aligned}
\gamma^{[\lambda}\gamma^\mu\gamma^{\nu]}\hat D_\mu\psi^{(n)}_{L\nu}
&=&m_n[\gamma^{\lambda},~\gamma^{\alpha}]\psi^{(n)}_{R\alpha},~~~~
\gamma^{[\lambda}\gamma^\mu\gamma^{\nu]}\hat D_\mu\psi^{(n)}_{R\nu}
=m_n[\gamma^{\lambda},~\gamma^{\alpha}]\psi^{(n)}_{L\alpha},
\label{MLD}\\
(\partial_z+(\partial_z A))\xi_{Rn}&=&m_n\xi_{Ln},~~~~~~~~~~~~~~~~
(\partial_z+(\partial_z A))\xi_{Ln}=-m_n\xi_{Rn}.
\label{MRD}\end{aligned}$$ Equations (\[MLD\]) are the ones that four-dimensional chiral gravitino fields satisfy and Eqs. (\[MRD\]) are the coupled ones which KK modes $\xi_{Ln}$ and $\xi_{Rn}$ satisfy. Performing the field transformations $\xi_{Rn}(z)=\chi^{R}_{n}(z)~\text{e}^{-A}$ and $\xi_{Ln}(z)=\chi^{L}_{n}(z)~\text{e}^{-A}$, we can obtain equations for the left- and right-handed KK modes of gravitino
$$\begin{aligned}
\partial_z^2\chi^{L}_n(z)=-m^2_n\chi^L_n(z),\label{chi1}\\
\partial_z^2\chi^{R}_n(z)=-m^2_n\chi^R_n(z).\label{chi2}\end{aligned}$$
When the following normalizable conditions are introduced $$\begin{aligned}
\int\chi^{L}_{m}(z)\chi^{R}_n(z)dz=\delta_{RL}\delta_{mn}, \label{normalizable condition}\end{aligned}$$ the effective action of the four-dimensional massless and massive gravitinos can be got $$\begin{aligned}
S^{m}_{\frac{3}{2}}&=&\sum_{n}\int d^4x
\bigg[\bar{\psi}^{(n)}_{L\lambda}(x)\gamma^{[\lambda}\gamma^{\mu}\gamma^{\nu]}\partial_{\mu}\psi^{(n)}_{L\nu}(x)
-m_n\bar{\psi}^{(n)}_{L\lambda}(x)[\gamma^{\lambda},~\gamma^{\mu}]\psi^{(n)}_{R\mu}(x)\nonumber\\
&&~~~~+\bar{\psi}^{(n)}_{R\lambda}(x)\gamma^{[\lambda}\gamma^{\mu}\gamma^{\nu]}\partial_{\mu}\psi^{(n)}_{R\nu}(x)
-m_n\bar{\psi}^{(n)}_{R\lambda}(x)[\gamma^{\lambda},~\gamma^{\mu}]\psi^{(n)}_{L\mu}(x)
\bigg]\nonumber\\
&=&\sum_{n}\int d^4x\left(\bar{\psi}^{(n)}_{\lambda}(x)\gamma^{[\lambda}\gamma^{\mu}\gamma^{\nu]}\partial_{\mu}\psi^{(n)}_{\nu}(x)
-m_n\bar{\psi}^{(n)}_{\lambda}(x)[\gamma^{\lambda},~\gamma^{\mu}]\psi^{(n)}_{\mu}(x)\right).\label{4Daction}\end{aligned}$$ However, obviously the solutions of Eqs. (\[chi1\]) and (\[chi2\]) are mediocre. Thus the four-dimensional massive gravitinos cannot be localized. This conclusion is the same as Dirac fermion.
Localization of gravitino field with coupling term on thick branes
==================================================================
As what we have pointed out in the previous section, the massive KK modes of a five-dimensional free massless gravitino field cannot be localized on RS-type thick branes. Therefore, it is necessary to introduce a coupling term as the case of Dirac field. In the thin brane scenario [@Du:2015pjw], one usually introduces an additional mass term which is associated with the warp factor of the thin brane. In the scenario of thick brane generated by one or multiple background scalar fields, we could introduce a coupling term between the background scalar field and gravitino field. We consider the simplest coupling, i.e., a Yukawa-like coupling, for which the action of a five-dimensional gravitino field is $$\begin{aligned}
S_{\frac{3}{2}}=\int d^5x\sqrt{-g}\Big(\bar{\Psi}_M\Gamma^{[M}\Gamma^{N}\Gamma^{R]}D_N\Psi_R-\eta F(\phi)\bar{\Psi}_M [\Gamma^M,~\Gamma^N]\Psi_N\Big).\end{aligned}$$ Here, $F(\phi)$ is a function of the background scalar field $\phi$ and $\eta$ is the coupling constant. The equations of motion derived from the above action [are]{} $$\begin{aligned}
\Gamma^{[M}\Gamma^N\Gamma^{R]}D_N\Psi_R-\eta F(\phi)[\Gamma^M,~\Gamma^N]\Psi_N=0.\end{aligned}$$ By using the gauge condition $\Psi_z=0$ and introducing the chiral decomposition $$\begin{aligned}
\Psi_\mu(x,z)=\sum_n\text{e}^{-A(z)}\left(\psi^{(n)}_{L\mu}(x)\chi^{L}_{n}(z)+\psi^{(n)}_{R\mu}(x)\chi^{R}_{n}(z)\right),\end{aligned}$$ we can obtain the following first-order coupled equations
\[FirstOrderCoupledEquations\] $$\begin{aligned}
(\partial_z-\eta\text{e}^{A}F(\phi))\chi^L_n(z)=&-&m_n\chi^R_n(z),\\
(\partial_z+\eta\text{e}^{A}F(\phi))\chi^R_n(z)=&&m_n\chi^L_n(z).\end{aligned}$$
From the above equation , the left- and right-handed KK modes of the gravitino field satisfy the following Schrödinger-like equations:
\[EoG\] $$\begin{aligned}
(-\partial^2_z+V^L(z))\chi^L_n(z)&=&m_n^2\chi^L_n(z),\label{EoL}\\
(-\partial^2_z+V^R(z))\chi^R_n(z)&=&m_n^2\chi^R_n(z),\label{EoR}\end{aligned}$$
where the effective potentials are given by
\[Vz\] $$\begin{aligned}
V^L(z)&=&(\eta\text{e}^{A}F(\phi))^2+\eta\partial_z(\text{e}^{A}F(\phi)),\\
V^R(z)&=&(\eta\text{e}^{A}F(\phi))^2-\eta\partial_z(\text{e}^{A}F(\phi)).\end{aligned}$$
For a five-dimensional free gravitino, we have obtained the effective action (\[4Daction\]) of the four-dimensional left- and right-handed gravitinos. It is interesting that the forms of these equations for the left- and right-handed KK gravitinos (\[EoL\]) and (\[EoR\]) are the same as those of the KK modes of a Dirac field, while the only difference is their chiralities. For a given background solution of a thick brane, if the function $F(\phi)$ and the coupling parameter $\eta$ are the same, it seems that the mass spectrum of the KK gravitinos will be the same as that of the Dirac field. Here, we should note the difference of chiralities, which will give an interesting result.
Next we first review some kinds of $f(R)$ thick branes [@Zhong:2015pta; @Yu:2015wma], and then investigate the localization of the five-dimensional gravitino on these branes and give their KK mass spectra.
In the five-dimensional spacetime, the action of a general $f(R)$ thick brane model reads [@Yu:2015wma] $$\begin{aligned}
S=\int d^{5}x\sqrt{-g}\left(\frac{1}{2\kappa^{2}_{5}}f(R)+L(\phi_{i},X_i)\right),\end{aligned}$$ where $\kappa^{2}_{5}\equiv8\pi G^{5}$ is the five-dimensional gravitational constant and is set to one for convenience, $f(R)$ is a function of the scalar curvature $R$ and $L(\phi_{i},X_i)$ is the Lagrangian density of the background scalar fields $\phi_i$ with the kinetic terms $
X_i=-\frac{1}{2}g^{MN}\partial_{M}\phi_{i}\partial_N\phi_{i}$. It is predictable that the spectra of the KK modes of the gravitino field on these $f(R)$ thick branes will be almost the same as the ones of the Dirac field except their chiralities. These results could give us some important reference in the future experiments about extra dimension and gravitino.
Localization of gravitino field on the pure geometric $f(R)$ thick [branes]{} without background scalar field
-------------------------------------------------------------------------------------------------------------
Firstly, we focus on the localization of the gravitino field on the pure geometric $f(R)$ thick branes. In Ref. [@Zhong:2015pta], the authors investigated the pure geometric $f(R)$ thick branes, where the Lagrangian density of the background scalar fields $L(\phi_{i},X_i)$ vanishes. For the flat pure geometric $f(R)$ thick branes, the background metric is given by (\[Le\]) with $\hat g_{\mu\nu}=\eta_{\mu\nu}$. The solution of the warp factor $A(y)$ is [@Zhong:2015pta] $$\begin{aligned}
A(y)=-n\ln(\cosh(ky)), \label{Ay_fR1}\end{aligned}$$ where $k$ is a positive real parameter that related to the curvature of the five-dimensional spacetime and $n$ is a positive integer number. The solutions of the function $f(R)$ for $n=1$ and $n=20$ are respectively [@Zhong:2015pta] $$\begin{aligned}
f(R)&=&\frac{1}{7}(6k^2+R)\cosh(a(w(R)))-\frac{2}{7}k^2\sqrt{480-\frac{36R}{k^2}-\frac{3R^2}{k^4}}\sinh(\alpha((w(R))),~~~(n=1)\\
f(R)&=&-\frac{377600}{7803}k^2+\frac{4196}{2601}R-\frac{83}{41616k^2}R^2+\frac{13}{39951360k^4}R^3,~~~(n=20)\end{aligned}$$ where $\alpha(w)=2\sqrt{3}\arctan(\tanh(\frac{w}{2}))$ and $w(R)=\pm\text{arcsech}\left[\frac{\sqrt{20n^2+R/k^2}}{\sqrt{8n+20n^2}}\right]$. For arbitrary $n$, the [ function $f(R)$ has no a unified expression]{}, and it is hard to get [ an]{} analytical $y(z)$ from the following relation of $z(y)$ calculated from the solution : $$\begin{aligned}
z(y) =-\frac{\cosh^{n+1}(k y) \sinh(k y)
~{_2 F_1}\big(1/2, \frac{n+1}{2}, \frac{n+3}{2}, \cosh^2(k y)\big)}
{(n+1) k \sqrt{-\sinh^2(k y)}}.
\end{aligned}$$ Since there is no background scalar [field]{} in the pure geometric brane model, we may try to take $\eta F$ as the five-dimensional mass $M$ of the gravitino field. Then, the effective potentials $V^L$ and $V^R$ can be expressed in terms of the extra dimension $y$ $$\begin{aligned}
V^L(z(y))&=& \text{sech}^{2n}(ky) \big(M^2-nkM\tanh(ky)\big),\label{VLforpure}\\
V^R(z(y))&=& \text{sech}^{2n}(ky) \big(M^2+nkM\tanh(ky)\big). \label{VRforoure}
\end{aligned}$$ It is easy to see that both potentials are asymmetric and their asymptotic behaviors are $$\begin{aligned}
V^L(0)&=&M^2,~~~~~~V^L(\pm\infty)=\text{e}^{2A(\pm\infty)}(M^2\mp Mkn)=0,\\
V^R(0)&=&M^2,~~~~~~V^R(\pm\infty)=\text{e}^{2A(\pm\infty)}(M^2\pm Mkn)=0,\end{aligned}$$ which indicates that there is no bound massive KK [mode]{}. The solutions for the left- and right-handed zero modes of the gravitino field are $\chi_0^{L,R} \propto e^{\pm My}$. It is clear that both zero modes are not normalizable, and hence cannot be localized on the pure geometric $f(R)$ thick branes.
Localization of gravitino field on the $f(R)$ thick [branes]{} with $L=X-V(\phi)$
---------------------------------------------------------------------------------
Now let us consider the thick $f(R)$ branes generated by one background scalar field. For the Lagrangian density $L=X-V(\phi)=-\frac{1}{2}\partial^{M}\phi\partial_{M}\phi-V(\phi)$, the solution in this model with the Sine-Gordon potential is given by [@Yu:2015wma]
$$\begin{aligned}
f({{\hat{R}}})
\!&=&\! \hat{R}
+\!\alpha \bigg\{ \frac{24b^2\!+\!2{\hat{R}}\!+ \!2b{\hat{R}}}{2\!+\!5b}
\Big[P_{K_{-}}^{{b}/{2}}(\Xi)
\!-\! {\beta}Q_{K_{-}}^{{b}/{2}}(\Xi)
\Big] \nonumber \\
&-&\!\!
4(b^2\!-\!2bK_{+}) \Xi\Big[P_{K_{+}}^{{b}/{2}}(\Xi)
-\!\! \Xi P_{K_{-}}^{{b}/{2}}(\Xi) \nonumber \\
&+&\!\!\beta \Xi\left( Q_{K_{-}}^{{b}/{2}}(\Xi)
- Q_{K_{+}}^{{b}/{2}}(\Xi)\right)\Big] \bigg\} \Theta^{b/2}, \label{fRBrane2a} \\
V(\phi) \!&=&\! \frac{3b k^2}{8}\left[(1-4b)+(1+4b)\cos \Big(\sqrt{\frac{8}{3b}}\phi\Big)\right],~~~~\label{fRBrane2b}\\
\phi(y) \!&=&\! \sqrt{6b} \arctan \Big[\tanh\Big(\frac{ky}{2}\Big)\Big],\label{fRBrane2c}\\
A(y)\!&=&\! -b\ln\Big[\cosh (ky)\Big],\label{fRBrane2d}\end{aligned}$$
\[fRBrane2\]
where $b$ and $k$ are positive parameters related to the thickness of the brane, $\alpha$ is an arbitrary constant, $\hat{R}\equiv R/k^2$, $K_{\pm}\equiv\frac12\sqrt{(b-14) b+1}\pm1/2$, $\Xi=\sqrt{1-\Theta^2}$, $\Theta\equiv\frac{\sqrt{20b^2+{\hat{R}}}}{2\sqrt{2b+5b^2}}$, $P$ and $Q$ are the first and second kinds of Legendre functions, $\beta={P_{K_{+}}^{{b}/{2}}(0)}/{Q_{K_{+}}^{{b}/{2}}(0)}$. Note that the solution - is also appropriate for the case of $f(R)=R$. Thus our following results are also appropriate for the case of general relativity thick brane. As shown in the above subsection, it is very difficult to obtain analytical $y(z)$. Therefore, in the following we will solve the equations numerically. The effective potentials $V^L$ and $V^R$ in the physical coordinate $y$ become
\[Vzy\] $$\begin{aligned}
V^L(z(y))&=&(\eta\text{e}^{A}F(\phi))^2+\eta\text{e}^{2A}\partial_y F(\phi)+\eta(\partial_yA) \text{e}^{2A} F(\phi),\\
V^R(z(y))&=&V^L(z(y))|_{\eta\rightarrow-\eta}.\end{aligned}$$\[Vforphi\]
It is obvious that [for different forms of $F(\phi)$, the potentials $V^L$ and $V^R$ have different expressions]{}, which determine the mass spectra of the KK modes. In this paper, we would like to consider one kind of Yukawa coupling, i.e., $F(\phi)=\phi^\alpha$ with positive integer $\alpha$. For a kink configuration of the scalar $\phi$, since $V^L$ and $V^R$ are demanded to be symmetrical with respect to extra dimension $y$, $\alpha$ should be odd. Next we consider two cases: the simplest case $F(\phi)=\phi$ and the case for $\alpha>1$.
-5mm
### Case I: $F(\phi)=\phi$
For the case of $F(\phi)=\phi$, the effective potentials (\[Vforphi\]) read
$$\begin{aligned}
V^L(y)&=&\frac{1}{2}\cosh(ky)^{-1-2b}
\bigg [ 12b\eta^2\arctan\Big(\tanh(\frac{ky}{2})\Big)^2\cosh(ky)\nonumber\\
&+&\eta\sqrt{6b}k(1-2b\arctan\Big(\tanh(\frac{ky}{2})\Big)\sinh(ky)) \bigg] ,
\label{Vforphi1L} \\
V^R(y)&=&V^L(y)|_{\eta\rightarrow-\eta},\label{Vforphi1R}\end{aligned}$$
\[Vforphi1\]
which are symmetrical. The values of the potentials at the original point and infinity are given by $$\begin{aligned}
&&V^R(0)=-\eta k\sqrt{\frac{3b}{2}}=-V^L(0),\\
&&V^R(\pm\infty)=0=V^L(\pm\infty).\end{aligned}$$ It is clear that both potentials have the same asymptotic behaviors as $y\rightarrow\pm\infty$, while their values at $y=0$ are opposite. Thus only the left- or right-handed gravitino zero mode (four-dimensional massless left- or right-handed gravitino) could be localized on the $f(R)$ thick brane. The shapes of the potentials (\[Vforphi1\]) are shown in Fig. \[fig:VLRCaseII1\], from which it can be seen that for any positive $b$, $k$ and $\eta$, $V^R(z(y))$ is a volcano type of potential and there may exist a localized zero mode and a continuous gapless spectrum of massive KK modes. Furthermore, the depth of the potential $V^R$ increases with values of the parameters $\eta$, $b$ and $k$. By solving Eq. (\[EoR\]) with the potential (\[Vforphi1R\]), the zero mode of the right-handed gravitino [becomes]{} $$\begin{aligned}
\chi^{R}_{0}(z)&\propto&\exp\left(-\eta\int_{0}^{z}e^{A(\bar{z})}
F(\phi)d\bar{z}\right) \nonumber \\
&=& \exp\left( -\eta\int_{0}^{y}\phi(\bar{y}) d\bar{y}\right)\nonumber\\
&=& \exp\left( -\eta\int_{0}^{y}\sqrt{6b} \arctan \left(\tanh\left(\frac{k\bar{y}}{2}\right)\right) d\bar{y}\right),
\label{fL0CaseI}\end{aligned}$$ and its normalization condition $$\begin{aligned}
\int_{-\infty}^{\infty}(\chi^R_{0}(z))^2 dz
&=& \int_{-\infty}^{\infty}(\chi^R_{0}(y))^2 e^{-A(y)} dy \nonumber \\
&\propto& \int_{-\infty}^{\infty} \exp\left( -A(y)-2\eta\int_{0}^{y} \phi(\bar{y}) d\bar{y}\right)dy \nonumber \\
&=& \int_{-\infty}^{\infty} \exp\left( b
\text{ln}(\cosh(ky))
-2\eta\int_{0}^{y}\sqrt{6b} \arctan \left(\tanh\left(\frac{k\bar{y}}{2}\right)\right) d\bar{y}
\right)dy
<\infty
\end{aligned}$$ is equivalent to $$\begin{aligned}
\int_{0}^{\infty} \exp
\left( kby - \frac{\pi\eta}{2}\sqrt{6b}y \right)dy
<\infty
\label{NormalizationConditionfR0CaseI1}\end{aligned}$$ since $-A(y)\rightarrow kby$ and $\arctan(\tanh(\frac{ky}{2}))={\pi}/{4}$ as $y\rightarrow\infty$. The above normalization condition (\[NormalizationConditionfR0CaseI1\]) requires $$\begin{aligned}
\eta>\eta_0\equiv\frac{k}{\pi}\sqrt{\frac{2b}{3}}.
\label{NormalizationConditionfR0CaseI1Foreta}\end{aligned}$$ Thus, if the coupling constant is strong enough ($\eta>\eta_0$), the right-handed zero mode can be localized on the brane. It is not difficult to check that the left-handed zero mode can not be localized on the brane under the condition .
On the other hand, the potential $V^L(z(y))$ for positive $\eta$ is always positive and vanishes far away from the brane. This type of potential cannot trap any bound state, and hence there is no left-handed gravitino zero mode. The structure of the potential $V^L$ is determined by the parameters $k$, $b$ and $\eta$. For given $k$ and $b$, the potential $V^L$ has a barrier for a small $\eta$. When $\eta$ increases, there will be a quasi-potential well and the depth of the well will increase with the value of $\eta$. However, for given $\eta$ and $k$ (or $b$), the height of the potential $V^L$ increases with $b$ (or $k$) and the quasi-potential well changes into a barrier as the growth of $b$ (or $k$). The behavior of $V^L$ around the point $y=0$ is similar to that of the function $y^4$ and there will be three extreme points if a quasi-potential well exists around the point $y=0$. Doing third-order Taylor series expansion of $\partial_y V^L$ near the point $y=0$, we will get $$\begin{aligned}
\partial_y V^L &=&
\frac{1}{2} k^2 \eta \big[6b \eta-\sqrt{6b}k(1 + 4b)\big] y \nonumber \\
&+&\frac{1}{12} k^4 \eta \big[\sqrt{6b}k(1 + 2b) (5 + 18b) -24b\eta (1 + 3b) \big] y^3 + \mathcal{O}(z^5) .\end{aligned}$$ For $k=1$ and $b>\frac{1}{2\sqrt{3}}$, the above function has three roots and there is a quasi-potential well when $\eta>\frac{1}{6}\sqrt{\frac{6+48b+96b^2}{b}}$ (it equals 2.04124 when $b=1$).
For the case that there is a quasi-potential well for $V^L$, we could find resonance states of the gravitino, which are the massive four-dimensional gravitinos with finite lifetimes on the brane. To investigate the gravitino resonant modes, we give the definition of the relative probability by following Ref. [@Liu:2009ve]: $$\begin{aligned}
P_{L,R}(m^{2})=\frac{\int^{z_{b}}_{-z_{b}}|\chi^{L,R}(z)|^{2}dz}{\int^{z_{max}}_{-z_{max}}|\chi^{L,R}(z)|^{2}dz},\end{aligned}$$ where $2z_b$ is approximately the width of the brane, and $z_{max}=10z_b$. The left- and right-handed wavefunctions $\chi^{L,R}(z)$ are the solutions of Eqs. (\[EoG\]). The above definition could be explained that $|\chi^{L,R}(z)|^{2}$ is the probability density [@Liu:2009ve; @Almeida:2009jc]. There exists a resonant mode with mass $m_n$, if the relative probability $P(m^2)$ has a peak around $m=m_n$. These peaks should have full width at half maximum and the number of these peaks is the same as the number of the resonant modes. In order to get the solutions of Eqs. (\[EoG\]), we always need additional two types of initial conditions
$$\begin{aligned}
\label{even}
\chi^{L,R}_{\rm{even}}(0)\!\!&=&\!\!1, ~~~\partial_{z}\chi^{L,R}_{\rm{even}}(0)=0;\\
\label{odd}
\chi^{L,R}_{\rm{odd}}(0)\!\!&=&\!\!0, ~~~~\partial_{z}\chi^{L,R}_{\rm{odd}}(0)=1,\end{aligned}$$
\[EvenOddConditions\]
where $\chi^{L,R}_{\rm{even}}$ and $\chi^{L,R}_{\rm{odd}}$ correspond to the even and odd parity modes of $\chi^{L,R}(z)$, respectively.
![The probabilities $P_{L,R}$ (as a function of $m^{2}$) for finding massive resonant KK modes of the left- and right-handed gravitinos with mass $m^{2}$ on the thick brane for the coupling $F(\phi)=\phi$. Solid lines and dashed lines are plotted for the even-parity and odd-parity massive gravitinos, respectively. The parameters are set to $b=1$, $k=1$, $\eta=10$, and $z_{max} = 20$.[]{data-label="Pforphi"}](PphiforL.eps "fig:"){width="7cm" height="5cm"} ![The probabilities $P_{L,R}$ (as a function of $m^{2}$) for finding massive resonant KK modes of the left- and right-handed gravitinos with mass $m^{2}$ on the thick brane for the coupling $F(\phi)=\phi$. Solid lines and dashed lines are plotted for the even-parity and odd-parity massive gravitinos, respectively. The parameters are set to $b=1$, $k=1$, $\eta=10$, and $z_{max} = 20$.[]{data-label="Pforphi"}](PphiforR.eps "fig:"){width="7cm" height="5cm"}
-5mm
Our results are shown in Figs. \[Pforphi\], \[fig\_\_Eigenvalue1\] and Tab. \[Tableonephi\]. It is obvious that the mass spectra of the left- and right-handed gravitino resonant modes are almost the same while their parities are opposite. The first resonant mode of the left-handed gravitino is even and its shape around $z=0$ looks like a ground state. On the other hand, the first resonant mode of the right-handed gravitino is odd and it seems to be the first excited state. These results are reasonable because the effective potentials $V^L$ and $V^R$ are supersymmetric partners, which give the same spectra of the resonant modes. In fact, fermion resonances on branes have similar properties because Eqs. (\[EoG\]) of the KK modes of a gravitino are almost the same as the ones of a fermion. However, there is a difference between them, which will be explained as follows. For a five-dimensional Dirac fermion field with a coupling term, if we use the representation of the gamma matrices (\[Gamma\]) and parity relation (\[partiyrelation\]), the equations of motion of the left- and right-handed fermion KK modes $f^{L,R}$ are given by
\[Scheq\] $$\begin{aligned}
(-\partial_{z}^{2}+V^{L}(z))f^{L}&=&m^{2}f^{L},\label{ScheqLeft}
\\
(-\partial_{z}^{2}+V^{R}(z))f^{R}&=&m^{2}f^{R},
\label{ScheqRight}\end{aligned}$$
with the effective potentials
\[Vz\] $$\begin{aligned}
V^{L}(z)=\eta^{2} e^{2A}F^{2}(\phi)-\eta
e^{A}\partial_{z}F(\phi)-\eta
e^{A}(\partial_{z}A)F(\phi)\,, \label{VzL} \\
V^{R}(z)=\eta^{2} e^{2A}F^{2}(\phi)+\eta
e^{A}\partial_{z}F(\phi)+\eta
e^{A}(\partial_{z}A)F(\phi)\,. \label{VzR}\end{aligned}$$
It is obvious that the Schrödinger-like equation of the left-handed gravitino KK modes (\[EoL\]) is the one of the right-handed fermion KK modes (\[ScheqRight\]), and the Schrödinger-like equation for the right-handed gravitino KK modes (\[EoR\]) is the one of the left-handed fermion KK modes (\[ScheqLeft\]). Therefore, for a five-dimensional Dirac fermion, only the zero mode of the left-handed fermion can be localized on the $f(R)$ brane with the coupling $F(\phi)=\phi$, and the first resonant mode of the right-handed fermion is even. This difference between the fermion and gravitino KK modes comes from the difference of their field equations. For a five-dimensional Dirac fermion field with the Yukawa coupling, the field equation reads $$\left[\gamma^{\mu}\partial_{\mu}+\gamma^5(\partial_z+2\partial_zA)-\eta \text{e}^{A}F(\phi)\right]\Psi=0.$$ It should be noticed that the sign in front of $\gamma^5$ is plus. While for a bulk gravitino, Eq. (\[EOM2\]) tells us that the sign in front of $\gamma^5$ is minus, which leads to the swap of the above results. This difference is very meaningful and it could be a symbol of the distinction between Dirac fermion and gravitino fields.
In addition, the number of the resonant modes for the gravitino field increases with the coupling constant $\eta$ but decreases with the parameter $b$. The relative probability $P$ decreases when the mass of the resonant mode approaches the maximum of the potentials. Furthermore, the resonant modes become closer and closer as $m^2$ approaches the maximum of the potentials. These results are consistent with that of the Dirac fermion.
-5mm
$b$ $\eta$ $\mathcal{C}$ $\mathcal {P}$ $m^2$ $m$ $P$
-------------- -------------- --------------------- ---------------------- ---------------------- -------------------- -------------------
even 21.9171 4.68157 0.988172
$\mathcal{L}$ odd 38.3348 6.19151 0.986425
10 even 47.9467 6.92436 0.423099
odd 21.9169 4.68155 0.999673
$\mathcal{R}$ even 38.3311 6.19121 0.981257
odd 47.9328 6.92335 0.423111
even 34.2006 5.84813 0.999212
odd 63.1603 7.94735 0.999998
\[0pt\][1]{} $\mathcal{L}$ even 86.4700 9.29892 0.993796
odd 102.7540 10.13680 0.632409
15 even 112.2329 10.59400 0.238222
odd 34.1964 5.84777 0.999959
even 63.1822 7.94872 0.992303
$\mathcal{R}$ odd 86.4490 9.29780 0.998145
even 102.7740 10.13780 0.631364
odd 112.0613 10.58590 0.232899
$\mathcal{L}$ even 35.3730 5.94752 0.982136
10 odd 54.7429 7.39884 0.287870
$\mathcal{R}$ odd 35.3712 5.94737 0.981886
even 54.4129 7.37651 0.285428
even 56.8585 7.54046 0.999979
$\mathcal{L}$ odd 98.8432 9.94199 0.922543
\[0pt\][3]{} 15 even 122.4728 11.06670 0.276112
odd 56.8515 7.53999 0.999923
$\mathcal{R}$ even 98.9253 9.94612 0.902362
odd 122.6000 11.07250 0.277316
: The eigenvalue $m^2$, mass $m$ and the relative probability of the left- and right-handed gravitinos with odd-parity and even-parity solutions for the coupling $F(\phi)=\phi$. In all tables of this paper, $\mathcal {C}$ and $\mathcal {P}$ stand for chirality and parity, $\mathcal {L}$ and $\mathcal {R}$ mean left- and right-handed, respectively. The parameter $k$ is set to $k=1$.[]{data-label="Tableonephi"}
\
### Case II: $F(\phi)=\phi^\alpha$ with $\alpha>1$
Next, we consider a natural generalization of the Yukawa coupling $F(\phi)=\phi^\alpha$ with $\alpha=3,~5,~7,~\cdots$. Note that $\phi^\alpha$ becomes a double-kink for $\alpha\geq3$ since the scalar field $\phi$ is a kink. For this case, the effective potentials (\[Vforphi\]) become
$$\begin{aligned}
V^L(y)&=& \frac{1}{2} 3^{\frac{\alpha}{2}} k\eta b^{\frac{\alpha}{2}-1}
\arctan^{\alpha-1}\left(\tanh\left(ky/2\right)\right)
\text{sech}^{2b+1}(k y)\left[\alpha-2b\arctan\left(\tanh(ky/2)\right)\sinh(ky)\right] \nonumber\\
&+& 6^\alpha\eta^2\left(\sqrt{b}\arctan\left(\tanh(ky/2)\right)\right)^{2\alpha}
\text{sech}^{2b}(ky),\\
V^R(y)&=&V^L(y)|_{\eta\rightarrow-\eta}.\end{aligned}$$
\[Vforphi2\]
It is obvious that both the potentials are symmetry and vanish at $y=0$ and $y\rightarrow\pm\infty$ and they are depicted in Fig. \[fig:VLRCaseII2\] for different values of $b$ and $\alpha$. There always exists a quasi-potential well for the left-handed potential $V^L$ and a double-potential well for the right-handed one. These wells for both potentials are deeper and deeper with the [increases]{} of the parameters $b$, $\eta$, and $\alpha$, which means that there are more and more resonances with the increases of $b$, $\eta$, and $\alpha$. Since the coupling function $\phi^\alpha$ trends to a constant as $y\rightarrow\pm\infty$, the zero mode of the right-handed gravitino $$\begin{aligned}
\chi^{R}_{0}\propto\exp\left(-\eta\int_{0}^{z}e^{A(z)}
\phi^{\alpha}d z\right)
=\exp\left( -\eta\int_{0}^{y}\phi^{\alpha}d y\right)
\label{fL0CaseI}\end{aligned}$$ is equivalent to $ \exp\left(-\eta(\frac{\pi}{4}\sqrt{6b})^\alpha|y| \right)$ since $\phi^\alpha=\pm(\frac{\pi}{4}\sqrt{6b})^\alpha$ as $y\rightarrow\pm\infty$. It is not difficult to check that the normalization condition can be satisfied for any positive coupling constant $\eta$. Thus, the right-handed zero mode can be localized on the brane for any positive coupling constant [$\eta$]{}, and at the same time the [left-handed]{} one can not.
As for the massive modes, we consider the resonance states. As what we have done in the previous subsection, we solve the Schrödinger equations (\[EoG\]) numerically by using the two types of initial conditions (\[EvenOddConditions\]). The mass spectrum of the resonances is shown in Tab. \[Tableonephi3\]. It is clear that in this table the masses of the resonant modes of the left- and right-handed gravitinos are still almost same, while their parities are opposite. The number of the resonances increases with the increases of the parameters $b$, $\alpha$, and $\eta$. These resonances are closer to each other as $m^2$ increasing, which is the same as the conclusion of the case of $\alpha=1$.
-5mm
$\alpha$ $\mathcal{C}$ $\mathcal {P}$ $m^2$ $m$
---------------- --------------------- ---------------------- ---------------------- --------------------
$\mathcal{L}$ even 1.07851 1.03851
\[0pt\][3]{} $\mathcal{R}$ odd 1.07771 1.03813
even 1.34176 1.15834
odd 3.84231 1.96018
$\mathcal{L}$ even 5.85206 2.41910
odd 7.30702 2.70315
even 8.79401 2.96547
\[0pt\][5]{} odd 1.32835 1.15254
even 3.83457 1.95821
$\mathcal{R}$ odd 5.84765 2.41819
even 7.31179 2.70403
odd 8.80562 2.96743
: The eigenvalue $m^2$ and mass $m$ of the left-and right-handed gravitinos with odd-parity and even-parity solutions for the coupling $F(\phi)=\phi^{\alpha}$. The parameters are set to $k=1$, $\eta$=1, and $b=1$.[]{data-label="Tableonephi3"}
\
Localization of the gravitino field on the $f(R)$ thick [branes]{} with $L=X_1+X_2-V(\phi_1,\phi_2)$
----------------------------------------------------------------------------------------------------
In the previous subsection, the [$f(R)$-branes are]{} generated by a single canonical scalar field. In this subsection, we will analysis the localization of a bulk gravitino in the Bloch-$f(R)$ brane model, where the Lagrangian density of the scalar fields is given by $$\begin{aligned}
L=-\frac{1}{2}\partial^M\phi\partial_M\phi-\frac{1}{2}\partial^M\xi\partial_M\xi-V(\phi,\xi).\end{aligned}$$ The scalar fields $\phi$ and $\xi$ interact through the scalar potential $V(\phi,\xi)$. In the following, we consider the solution given in Ref. [@Yu:2015wma]:
$$\begin{aligned}
\label{BlochBrane1Phiy}
\phi(y) \!\!&=&\!\! v \tanh (2dvy),\\
\label{BlochBrane1Chiy}
\xi(y) \!\!&=&\!\!v \sqrt{\frac{\tilde{b}\!-\!2d}{d}}~\text{sech} (2dvy),\\
\label{BlochBrane1Ay}
A(y) \!\!&=&\!\! \frac{v^2}{9d}
\left[ (\tilde{b}\!-\!3d)\tanh^2 (2dvy)
\!-\!2\tilde{b} \ln \cosh(2dvy) \right],~~~~~~\end{aligned}$$
\[BlochBrane1\]
where $\tilde{b}>2d>0$, and the scalar potential is $$\begin{aligned}
V(\phi,\xi) = \frac12 \left[\left(\tilde{b} v^2-\tilde{b}\phi^2-d\xi^2\right)^2+4d^2 \phi^2 \xi^2 \right]-\frac43\left(\tilde{b}\phi v^2-\frac13 \tilde{b}\phi^3-d\phi\xi^2\right)^2.\end{aligned}$$ For certain given values of the parameters $v$ and $\tilde{b}$, the function $f(R)$ could have analytical expression. For example, when $v=\sqrt{3/2}$ and $\tilde{b}=3d$, we have $$\begin{aligned}
f(R)=R+\frac{2\gamma}{7}\Big[\sqrt{3(R-48 d^2)(R+120 d^2)}\sin\mathcal{Y}(R)+2\left(R+36 d^2\right)\cos\mathcal{Y}(R)\Big],\end{aligned}$$ where $\gamma$ is a parameter and $\mathcal{Y}(R)=\sqrt{3}\ln \left(\frac{\sqrt{R-48 d^2}+\sqrt{R+120 d^2}}{2 \sqrt{42}d}\right)$.
Next, we investigate the localization of a bulk gravitino with the coupling [function]{} $F(\phi)=\phi^{p}\xi^{q}$ with $p=1,3,5,\cdots$ and $q$ any integer. Such coupling was also used to localize the Dirac fermion in Refs. [@Almeida:2009jc; @Liu:2009mg; @Xie2015].
### Case I: $F(\phi)=\phi^{p}\xi^{q}$ with $q>0$
Firstly, we consider the case of $F(\phi)=\phi^{p}\xi^{q}$ with $q>0$. For convenience we let $q=1$. The most simplest one is the Yukawa coupling between the two scalar fields and the gravitino, i.e., $-\eta\phi\xi \bar{\Psi}_M [\Gamma^M,~\Gamma^N]\Psi_N$. We also assume without loss of generality that the coupling constant $\eta$ is positive. The asymptotic behaviors of the potentials (\[Vforphi\]) in this case are similar to those in the last subsection. As $z$ (or $y$)$\rightarrow\infty$, both the potentials $V^L$ and $V^R$ vanish and their values are opposite at $z=0$: $$\begin{aligned}
V^L(0)=-V^R(0)=2\eta v^3\sqrt{(\tilde{b}-2d){d}},\end{aligned}$$ which shows that there is a potential well around $z=0$ for $V^R$. Thus, it seems that the left-handed zero mode of the gravitino can not be localized on the brane, while the right-handed zero mode can be localized. However, when substituting the solution of the right-handed zero mode $$\begin{aligned}
\chi^{R}_{0}&\propto&\exp\bigg(-\eta\int_{0}^{z}d\bar{z}e^{A(\bar{z})}
\phi(\bar{z})\xi(\bar{z})\bigg) \nonumber \\
&=& \exp\bigg( -\eta\int_{0}^{y} d\bar{y}
\phi(\bar{y})\xi(\bar{y})\bigg)
= \exp\bigg(\frac{\eta v}{2d}\sqrt{\frac{\tilde{b}-2d}{d}}\text{sech}(2dv\bar{y})|^{y}_{0}\bigg)\nonumber\\
&\propto& \exp\bigg(\frac{\eta v}{2d}\sqrt{\frac{\tilde{b}-2d}{d}}\text{sech}(2dvy) \bigg)\label{CRight0}\end{aligned}$$ into the normalization condition (\[normalizable condition\]), we find the integral $$\begin{aligned}
\int_{-\infty}^{\infty}(\chi^R_{0}(z))^2 dz
&=& \int_{-\infty}^{\infty}(\chi^R_{0}(y))^2 e^{-A(y)} dy \nonumber \\
&\propto& \int_{-\infty}^{\infty} \exp\bigg( -A(y)-2\eta\int_{0}^{y}
\phi(\bar{y})\xi(\bar{y})d\bar{y} \bigg)dy \nonumber \\
&=& \mathcal{C}^2\int_{-\infty}^{\infty}\cosh(2dvy)^{\frac{2v^2\tilde{b}}{9d}} \exp\bigg(\frac{2\eta v}{2d}\sqrt{\frac{\tilde{b}-2d}{d}}\text{sech}(2dvy)-\frac{v^2}{2d}(\tilde{b}-3d)\tanh^2(2dvy) \bigg)dy
\label{fL0CaseI}\end{aligned}$$ is divergent, which means that the right-handed zero mode can not be confined on the brane. Although the potential of the right-handed gravitino is a volcanic-type one, there still does not exist the zero mode on the brane. In fact, for any $q>0$ and $p=1,3,5\cdots$, the right-handed zero mode will be a constant as $y\rightarrow\infty$ since $F(\phi)=\phi^{p}\xi^{q}\varpropto\tanh^p(2dvy)\text{sech}^q(2dvy)\rightarrow 0$. Obviously, this kind of zero mode can not satisfy the normalization condition (\[normalizable condition\]). Thus, for any $q>0$, there [exists]{} no bounded zero mode of the gravitino on the brane (the left-handed zero mode can also not be localized). Since there is no localized zero mode on the brane, we turn to the case of $q<0$.
-5mm
### Case II: $F(\phi)=\phi^{p}\xi^{q}$ with $q<0$ (or $q=-1$)
We let $q=-1$ to represent the case of $q<0$ for convenience. The potentials (\[Vforphi\]) in this case are displayed in Fig. \[fig:VLRCaseIII\]. Both the potentials $V^L$ and $V^R$ have infinite wells. For the simplest case $p=1$, both two potentials vanish as $z$ (or $y$)$\rightarrow\infty$ and their values are opposite at $z=0$: $V^L(0)=-V^R(0)=\frac{2\eta dv}{\sqrt{(\tilde{b}-2d)/d}}$. The left-handed zero mode still can not be localized on the brane since it is divergent as $z\rightarrow\infty$. While the right-handed one $$\begin{aligned}
\chi^{R}_{0}&\propto& \exp\bigg( -\eta\int_{0}^{y} d\bar{y}
\phi(\bar{y})\xi^{-1}(\bar{y})\bigg)\nonumber\\
&=& \exp\bigg(-\frac{\eta}{2}\left(\sqrt{{(\tilde{b}-2d)}{d}}~v\right)^{-1}\cosh(2dv\bar{y})|^{y}_{0}\bigg)\nonumber\\
&\propto& \exp\bigg(-\frac{\eta}{2}\left(\sqrt{{(\tilde{b}-2d)}{d}}~v\right)^{-1}\cosh(2dv\bar{y}) \bigg)\end{aligned}$$ will vanish as $y\rightarrow\infty$ for any $\eta>0$. It is not difficult to check that for any $\eta>0$ this right-handed zero mode can be localized on the brane under the condition . And for any $q<0$ and $p=1,3,5\cdots$ the right-handed zero mode will be localized. For other case of $p\geqslant3$, both potentials vanish at $z=0$: $V^L(0)=V^R(0)=0$, and the left-handed potential $V^L$ is always non-negative while $V^R$ have a double-well. Therefore, only the right-handed zero mode could be localized on the brane.
There are infinite bounded massive KK modes in this case because both the effective potentials are infinite ones. Some of our results are shown in Tab \[Tabletwophi\]. It is obvious that the mass spectra of the left- and right-handed gravitino massive bounded KK modes are almost the same while their parities are opposite as shown in the previous section. When $p=1$, the mass of the first bounded state of the left-handed gravitino (or the mass of the first excited state of the right-handed one) increases with the value of $\eta$ because the minimum of the left-handed potential $V^L$ increases with $\eta$. On the other hand, the relative width of the effective potentials decreases with the value of $\eta$ and increases with the value of $m^2$. Thus, the gaps between the bounded states will extend with the growth of $\eta$ and become narrower and narrower as $m^2$ increases. When $p\geq3$, the mass of the first bounded state of the left-handed gravitino still increases with the growth of the $\eta$, even though the minimum of the left-handed potential $V^L$ is always zero. Other conclusions are the same with the case of $p=1$.
[||c|c|c|c|c|c||]{} $p$ & $\eta$ & $\mathcal{C}$ & $\mathcal {P}$ & $m_n^2$ & $m_n$ \
& & & even& 2.4489 & 1.5649\
& & & odd & 3.6790 & 1.9181\
& & $\mathcal{L}$ & even& 4.8069 & 2.1925\
& & & odd& 5.7380 & 2.3954\
& & & $\vdots$& $\vdots$ & $\vdots$\
& 1 & & even & 0 & 0\
& & & odd & 2.4490 & 1.5649\
& & $\mathcal{R}$ & even & 3.6790 & 1.9181\
& & &odd & 4.8070 & 2.1925\
& & &even & 5.7381 & 2.3954\
& & &$\vdots$ & $\vdots$ & $\vdots$\
\[0pt\][1]{} & & &even & 5.8846 & 2.4258\
& & &odd & 9.3857 & 3.0636\
& & $\mathcal{L}$ &even & 12.2642 & 3.5020\
& & &odd & 14.7436 & 3.8397\
& & &$\vdots$ & $\vdots$ & $\vdots$\
& 2 & &even & 0 & 0\
& & &odd & 5.8849 & 2.4259\
& & $\mathcal{R}$ &even & 9.3860 & 3.0637\
& & &odd & 12.2640 & 3.5020\
& & &even & 14.7437 & 3.8398\
& & &$\vdots$ & $\vdots$ & $\vdots$\
& & & even& 1.8861 & 1.3734\
& & & odd & 3.5178 & 1.8756\
& & $\mathcal{L}$ & even& 4.5436 & 2.1316\
& & & odd& 5.6248 & 2.3717\
& & & $\vdots$& $\vdots$ & $\vdots$\
& 1 & & even & 0 & 0\
& & & odd& 1.8860 & 1.3733\
& & $\mathcal{R}$ &even & 3.5177 & 1.8756\
& & &odd & 4.5436 & 2.1316\
& & &even & 5.6248 & 2.3717\
& & &$\vdots$ & $\vdots$ & $\vdots$\
\[0pt\][3]{} & & &even & 3.6985 & 1.9232\
& & &odd & 8.1171 & 2.8491\
& & $\mathcal{L}$ &even & 10.9629 & 3.3110\
& & &odd & 13.8026 & 3.7152\
& & &$\vdots$ & $\vdots$ & $\vdots$\
& 2 & &even & 0 & 0\
& & &odd & 3.6981 & 1.9230\
& & $\mathcal{R}$ &even & 8.1170 & 2.8490\
& & &odd & 10.9624 & 3.3110\
& & &even & 13.8025 & 3.7152\
& & &$\vdots$ & $\vdots$ & $\vdots$\
\
Discussion and conclusion {#secConclusion}
=========================
In this manuscript, we investigated the localization and resonant modes of [ a]{} five-dimensional gravitino field on the $f(R)$ thick [branes]{}, and gave the Schrödinger equations for [ the gravitino KK modes]{} with the gauge condition $\Psi_z=0$. [As same as a]{} [five-dimensional free and massless Dirac fermion field]{}, the zero mode of a free massless five-dimensional gravitino field could be localized on a brane only for a compactification extra dimension and its massive KK modes can not realize the localization. Therefore we introduced the coupling term $-\eta F(\phi)\bar{\Psi}_M[\Gamma^M,~\Gamma^N]\Psi_N$ to investigate the localization of gravitino [on three kinds of $f(R)$ thick branes]{}. The relativity probability method has been applied to study the resonances of gravitino [on these $f(R)$ thick [branes]{}]{}. It has been shown that the localization and KK spectra of the five-dimensional gravitino field with the Yukawa coupling term $-\eta F(\phi)\bar{\Psi}_M[\Gamma^M,~\Gamma^N]\Psi_N$ are very similar to [those]{} of the Dirac fermion, but their chiralities are opposite. This difference may become a symbol of the distinction between a five-dimensional Dirac fermion field and gravitino field.
Firstly, we considered the localization of gravitino on the pure geometric $f(R)$ thick branes, whose the Lagrangian density $L(\phi_{i},X_i)$ of the background scalar fields is zero. With the addition of the five-dimensional mass term $\eta F(\phi)=M$, we found that in this system the KK modes of gravitino, both the zero mode and massive ones, could not be localized on the pure geometric $f(R)$ thick branes.
Then the $f(R)$ thick branes, which are generated by a single canonical background scalar field $\phi$, were considered. We introduced the Yukawa coupling function, $F(\phi)=\phi^\alpha$ with $\alpha=1,~3,~5,~7,~\cdots$ to study the localization of the gravitino field in the $f(R)$ thick brane model. There are [two types of coupling functions]{} $F(\phi)$, i.e., $\alpha=1$ and $\alpha\geq3$. For the case of $\alpha=1$, there could exist localized left- or right-handed zero mode on the brane as the coupling parameter $\eta$ satisfies $\eta>\frac{k}{\pi}\sqrt{\frac{2b}{3}}$. Furthermore, for $k=1$ and $b>\frac{1}{2\sqrt{3}}$, we could obtain massive resonances of gravitino on the brane with the condition $\eta>\frac{1}{6}\sqrt{\frac{6+48b+96b^2}{b}}$. The results [indicate]{} that the left- and right-handed gravitinos almost have the same resonant spectra, while their parities are opposite. With relation (\[partiyrelation\]), the first resonance of the left-handed gravitino is even and the one of the right-handed gravitino is odd. There is only the right-handed zero mode of gravitino confined on the brane. These results are appropriate to other cases in this paper, while they are just opposite to the Dirac fermion. For a five-dimensional Dirac fermion field, only the left-handed zero mode of Dirac fermion could be localized on the $f(R)$ thick branes and the first resonance of the left-handed Dirac fermion is odd. The difference results between the gravitino field and the Dirac fermion field come from the different sign in front of $\gamma^5$ in their dynamic equations, which may be a symbol to distinguish the Dirac fermion field and the gravitino field as they have the same coupling function $F$ and parameter $\eta$. In addition, the number of KK resonant modes for gravitino [in this braneworld system]{} increases with the increase of the coupling parameter $\eta$ while decreases with the model parameter $b$. For other case ($\alpha\geq3$), there are no bounded zero modes for both left- and right-handed gravitinos, and the number of KK resonant modes increases with the growths of the parameters $b$, $\alpha$, and the coupling parameter $\eta$.
Finally, we focused on the [Bloch-$f(R)$]{} branes which are generated by two interacted real scalar fields. The coupling function $F(\phi)=\phi^{p}\xi^{q}$ with $p=1,3,5,\cdots$ and $q$ any integer was considered in this model. For the case of $q>0$, there exist no bounded zero modes. For the case of $q<0$, the right-handed zero mode could be localized on the brane for any $\eta>0$, and there exist infinite bounded massive KK modes for both the left- and right-handed gravitinos because both the effective potentials are infinite potential wells. The gaps between the bounded states extend with the growth of $\eta$ and become narrower and narrower as $m^2$ increases.
There are still some issues. As we showed in this paper, the spectra of the KK modes of a bulk gravitino are all most the same as the one of a bulk Dirac fermion except for chiralities. Thus all the results of localization of Dirac fermion on branes could be appropriate to gravitino by interchanging the chiralities. But for some kinds of branes, we found the localized KK modes of Dirac fermion by introducing a new kind of coupling term [@Liu:2013kxz; @Zhang:2016ksq]. It is not clear whether this kind of coupling term applies to gravitino and it will be our work in the future. In addition, we just consider Minkowski branes in this paper. Localization of gravitino on dS/AdS branes is also interesting.
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[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
[^4]: [email protected], corresponding author
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Numerous authors have considered the problem of determining the Lebesgue space mapping properties of the operator $\mathcal{A}$ given by convolution with affine arc-length measure on some polynomial curve in Euclidean space. Essentially, $\mathcal{A}$ takes weighted averages over translates of the curve. In this paper a variant of this problem is discussed where averages over both translates and dilates of a fixed curve are considered. The sharp range of estimates for the resulting operator is obtained in all dimensions, except for an endpoint. The techniques used are redolent of those previously applied in the study of $\mathcal{A}$. In particular, the arguments are based upon the refinement method of Christ, although a significant adaptation of this method is required to fully understand the additional smoothing afforded by averaging over dilates.'
address: 'Jonathan Hickman, Room 5409, James Clerk Maxwell Building, University of Edinburgh, Peter Guthrie Tait Road, Edinburgh, EH9 3FD.'
author:
- Jonathan Hickman
bibliography:
- 'Reference.bib'
title: 'Uniform $L_x^p - L^q_{x,r}$ Improving for Dilated Averages over Polynomial Curves'
---
Introduction and statement of results
=====================================
Let $\gamma: I \rightarrow {\mathbb{R}}^d$ denote a (parametrisation of a) smooth curve in ${\mathbb{R}}^d$ where $I \subseteq {\mathbb{R}}$ is an interval. Consider the operator $A$ defined, at least initially, on the space of for all test functions $f$ on ${\mathbb{R}}^d$ by $$\label{operator}
Af(x,r) := \int_I f(x-r \gamma (t)) \alpha(t)\,{\mathrm{d}}t \qquad \textrm{for all $(x,r) \in {\mathbb{R}}^d \times [1,2]$.}$$ Here $\alpha$ denotes some density which is assumed to be smooth and non-negative. Thus $A$ takes averages over translates of dilates of $\gamma$. A natural problem is to establish the range of $(p,q_1,q_2)$ for which there is an a priori mixed norm estimate either of the form $$\label{mixed1}
\|Af\|_{L^{q_2}_tL^{q_1}_x({\mathbb{R}}^d \times [1,2])} \leq C_{d, \gamma} \|f\|_{L^p({\mathbb{R}}^d)}$$ or $$\label{mixed2}
\|Af\|_{L^{q_1}_xL^{q_2}_t({\mathbb{R}}^d \times [1,2])} \leq C_{d, \gamma} \|f\|_{L^p({\mathbb{R}}^d)}.$$ This question subsumes the study of the $L^p$ mapping properties of both single averages and maximal functions associated to space curves. The archetypical case to consider is when $\gamma := h \colon [0,1] \to {\mathbb{R}}^d$ is the so-called *moment curve* given by $$h(t) := (t, t^2, \dots, t^d)$$ with constant density $\alpha \equiv 1$. With this choice of curve and density the following results are known:
- Taking $q_2 = \infty$ in reduces matters to determining the set of $(p,q)$ for which the single averages $$\label{single}
\mathcal{A}_rf(x) := \int_0^1 f(x - rh(t))\,{\mathrm{d}}t$$ are type $(p,q)$ uniformly for $r \in [1,2]$. The $L^p-L^q$ mapping properties of $\mathcal{A}:= \mathcal{A}_1$ were investigated in low dimensions in a number of papers [@Littman1973; @Oberlin1987; @Oberlin1997; @Greenleaf1999; @Oberlin1999] before being completely determined in all dimensions by Stovall [@Stovall2009] using powerful new methods developed by Christ [@Christb; @Christ1998].
- On the other hand, if one sets $q_2 = \infty$ in the situation is very different. In particular, one now wishes to understand the $L^p-L^q$ mapping properties of the maximal function $\mathcal{M}$ associated to $h$, defined by $$\mathcal{M}f(x) := \sup_{1 \leq r \leq 2} |\mathcal{A}_rf(x)|.$$ A celebrated theorem of Bourgain [@Bourgain1986] established $L^p-L^p$ mapping properties for $d=2$; this result was extended by Schlag [@Schlag1997a] who proved an almost-sharp range of $L^p-L^q$ estimates.[^1] However, the problem of determining even the $L^p-L^p$ range remains open in all other dimensions. Some partial results in this direction are given in [@Pramanik2007].[^2]
In this paper the special case of and where $q_1 = q_2$ is considered. In particular, Theorem \[momentthm\] below almost completely determines the set of $(p,q)$ for which $A$ is bounded from $L^p_x({\mathbb{R}}^d)$ to $L^q_{x,r}({\mathbb{R}}^d \times [1,2])$ when $\gamma(t) := h(t)$ is the moment curve. Testing the inequality on some simple examples (see Section \[necessity\]) shows such a bound is possible only if $(1/p, 1/q)$ lies in the trapezium $\mathcal{T}_d$ given by the closed, convex hull of the set $$\{(0, 0), (1,1), (1/p_1, 1/q_1), (1/p_2, 1/q_2)\}$$ where $$\left(\frac{1}{p_1}, \frac{1}{q_1} \right):= \left(\frac{1}{d}, \frac{d-1}{d(d+1)}\right) \quad \textrm{and} \quad \left(\frac{1}{p_2}, \frac{1}{q_2} \right):= \left( \frac{d^2 - d+2}{d(d+1)}, \frac{d-1}{d+1} \right).$$
This condition is shown to be sufficient, at least up to an endpoint.
\[momentthm\] For $d \geq 2$ the operator $Af(x,r) := \mathcal{A}_rf(x)$ is bounded from $L^p_x({\mathbb{R}}^d)$ to $L^q_{x,r}({\mathbb{R}}^d\times [1,2])$ for all $(1/p,1/q) \in \mathcal{T}_d\setminus \{(1/p_1, 1/q_1)\}$. If $(1/p,1/q) \notin \mathcal{T}_d$, then $A$ is not restricted weak-type $(p,q)$.
The proof of Theorem \[momentthm\] proceeds by establishing a restricted weak-type inequality at the endpoint $(p_1, q_1)$. Therefore, except for the question of whether this weak-type endpoint inequality can be strengthened the theorem completely determines the $L^p$ mapping of $A$ for the given choice of curve.
The $d=2$ case of Theorem \[momentthm\] (when the curve is also a hypersurface) is already known to hold with a strong-type inequality at the endpoint. This result essentially appears, for example, in the work of Strichartz [@Strichartz1977] and Schlag and Sogge [@Schlag1997]. Furthermore, in [@Strichartz1977; @Schlag1997] it is observed that the critical $L^2_x-L^6_{x,r}$ inequality for dilated averages over circles is equivalent to a Stein-Tomas Fourier restriction theorem for a conic surface and connections between this theory and estimates for certain evolution equations are also discussed. In addition, the $d=2$ case follows from more recent work of Gressman [@Gressman2006; @Gressman2013] utilising methods which are rather combinatorial in nature. The combinatorial techniques found in [@Gressman2006] are akin to the arguments found in the present article; both are based on earlier work of Christ [@Christ1998], discussed later in the introduction.
For $d \geq 3$ the results appear to be new and, indeed, no previous (non-trivial) partial results are known to the author. It is remarked that the connection between the theory of dilated averages, Fourier restriction and analysis of PDE appears to be confined to the hypersurface setting but nevertheless Theorem \[momentthm\] is arguably of interest in its own right.
Theorem \[momentthm\] is in fact a special case of a more general result, Theorem \[bigthm\], described below. Indeed, rather than restricting attention to $h$, this paper considers $A$ defined with respect to any polynomial curve. In this setting the statement of the results requires some preliminary motivation and definitions.
Given an arbitrary curve $\gamma$, when investigating the mapping properties of the key consideration is curvature. One would expect $A$ is non-degenerate (in the sense that the largest possible range of estimates hold for $A$) if and only if the $d-1$ curvature functions associated to $\gamma$ are non-vanishing in the support of the density $\alpha$. This kind of phenomenon is well-known in the context of single averages and maximal functions[^3] (the latter over curves in ${\mathbb{R}}^2$) and, indeed, many other operators whose definition depends on some submanifold (or family of submanifolds) of ${\mathbb{R}}^d$ (see, for instance, [@Christ1999; @Tao2003]). One method for quantifying the relationship between the curvature of $\gamma$ and the boundedness of $A$ is to introduce a specific choice of weight $\lambda_{\gamma}$ in the definition of the operator. In particular, $\lambda_{\gamma}$ is carefully chosen to vanish at the flat points of the curve so as to ameliorate the effect of the degeneracies. One can then hope to achieve $L^p_x - L^q_{x,r}$ boundedness for the full range of exponents corresponding to the non-degenerate case under mild hypotheses on the curve. This strategy follows the example of numerous authors (notably Drury [@Drury1990], Oberlin [@Oberlin2002], Dendrinos, Laghi and Wright [@Dendrinos2009], Stovall [@Stovall2014; @Stovall2010] and Dendrinos and Stovall [@Dendrinos]) who, in considering averages defined with respect to degenerate curves, have chosen the underlying measure in the definition of the operator to be the so-called *affine arc-length* measure, described below. This measure has the desired effect of dampening any degeneracies of the curve or surface and also makes the problem both affine and parametrisation invariant.
To make this discussion precise, define the torsion $L_{\gamma}$ of the curve to be the function $$L_{\gamma}(t) := \det (\gamma^{(1)}(t) \dots \gamma^{(d)}(t) )$$ where $\gamma^{(i)}$ denotes the $i$th derivative of $\gamma$, viewed as a column vector. This function vanishes precisely when any of the $d-1$ curvature functions associated to $\gamma$ vanish. The affine arc-length measure ${\mathrm{d}}\mu_{\gamma}$ on $\gamma$ is then defined by $$\int f \,{\mathrm{d}}\mu_{\gamma} := \int_{{\mathbb{R}}} f(\gamma(t)) \lambda_{\gamma}(t)\,{\mathrm{d}}t$$ whenever $f \in C_c({\mathbb{R}}^d)$, say, where $\lambda_{\gamma}(t) := |L_{\gamma}(t)|^{2/d(d+1)}$. In this paper the operator $A_{\gamma}$ given by convolution with dilates of this measure is studied. Explicitly, $$\label{Poperator}
A_{\gamma}f(x,r) := \int_{{\mathbb{R}}} f(x-r \gamma(t)) \lambda_{\gamma}(t)\,{\mathrm{d}}t$$ for all test functions $f$ on ${\mathbb{R}}^d$.
The choice of weight $\lambda_{\gamma}$ is further motivated by the fact that the resulting measure exhibits both parametrisation and affine invariance (for a detailed discussion see, for example, [@Christc]). Consequently, for all exponents $1 \leq p,q < \infty$ satisfying the relation $1/q = 1/p - 2/d(d+1)$ any $L^p_x - L^q_{x,r}$ inequality for $A_{\gamma}$ is affine invariant in the sense that if $\gamma$ is replaced with $X \circ \gamma$ for some invertible linear transformation $X$, then the estimate still holds with the same constant. One can therefore hope to achieve estimates for $A_{\gamma}$ which are uniform over all $\gamma$ belonging to a large class of curves. A counter-example due to Sjölin [@Sjolin1974] demonstrates such uniformity is impossible if the class includes curves which exhibit an arbitrarily large number of oscillations (see also [@Dendrinos]). It is therefore natural to postulate uniform estimates are possible over all polynomial curves of some fixed degree, since the degree controls the number of oscillations.
\[bigthm\] Let $d \geq 2$ and $P \colon {\mathbb{R}}\to {\mathbb{R}}^d$ be a polynomial curve whose components have maximum degree $n$. Then $$\|A_Pf\|_{L^q_{x,r}({\mathbb{R}}^d \times [1,2])} \leq C_{n,d} \|f\|_{L^p_x({\mathbb{R}}^d)}$$ whenever $(1/p, 1/q)$ lies in $\mathcal{T}_d \setminus\{(1/p_1, 1/q_1)\}$ and satisfies $1/q = 1/p - 2/d(d+1)$. Here the constant $C_{n,d}$ depends only on $p$, $q$, the dimension $d$ and the degree $n$ of the polynomial mapping.
Again, the theorem will follow by demonstrating a uniform restricted weak-type $(p_1, q_1)$ inequality holds. In addition, Theorem \[bigthm\] is almost-sharp in the sense that if $(1/p, 1/q)$ does not lie in the intersection of $\mathcal{T}_d$ with the line $1/q = 1/p - 2/d(d+1)$, then such estimates do not hold with a finite constant.
It is remarked that the result of Theorem \[bigthm\] is new for dimensions $d \geq 3$ whilst the $d=2$ case follows from a very general theorem due to Gressman [@Gressman2013]. Indeed, Gressman’s theorem, *inter alia*, establishes the hypersurface analogue of Theorem \[bigthm\] in all dimensions, up to and including all the relevant endpoints.
If ${\mathbb{R}}$ is replaced with a bounded interval $I$ in the definition of $A_P$, then the resulting operator (which is denoted by $A_P^{\mathrm{c}}$) is trivially seen to be bounded from $L^p_{x}$ to $L^p_{x,r}$ for all $1 \leq p \leq \infty$. Real interpolation therefore yields the following corollary.
The operator $A_P^{\mathrm{c}}$ is bounded from $L^p_{x,r}$ to $L^q_x$ for all $(1/p,1/q) \in \mathcal{T}_d\setminus \{(1/p_1, 1/q_1)\}$. If $(1/p,1/q) \notin \mathcal{T}_d$ and $L_P \not\equiv 0$, then $A_P$ is not restricted weak-type $(p,q)$.
Notice, when $P = \gamma$ is the moment curve defined above, the torsion function $L_{\gamma}$ is constant and thus Theorem \[momentthm\] is indeed a special case of Theorem \[bigthm\].
It is natural to ask whether the restricted weak-type $(p_1, q_1)$ endpoint can be strengthened to a strong-type estimate. This is certainly the case in dimension $d=2$ where the inequality is a consequence of the aforementioned theorem due to Gressman [@Gressman2013]. Furthermore, one may recover the strong-type bound for $d=2$ by combining the analysis contained within the present article with an extrapolation method due to Christ [@Christb] (see also [@Stovall2009]). It is possible that the argument can be adapted to the case where $d$ belongs to a certain congruence class modulo $3$ to (potentially) establish the strong-type bound in this situation. A more detailed discussion of the validity of the strong-type endpoint appears below in Remark \[strong type remark\].
Theorems \[momentthm\] and \[bigthm\] belong to a growing body of works which have applied variants of the geometric and combinatorial arguments due to Christ [@Christ1998] to the study of operators collectively known as generalised Radon transforms, of which $A$ is an example. Essentially these operators are defined for any point $y$ belonging to $\Sigma$ an $n$-dimensional manifold by integration over a $k$-dimensional manifold $M_y$ which depends on $y$, where $k < n$ is referred to here as the *dimension of the associated family*. The techniques of [@Christ1998] have fruitfully been applied and developed in, for instance, [@Christ; @Christa; @Christ2002; @Christ2008; @Dendrinos2009; @Gressman2004; @Gressman2009; @Stovall2014; @Stovall2009; @Stovall2010; @Tao2003] to study the Lebesgue mapping properties of one-dimensional generalised Radon transforms $R$ which are, roughly, operators $R$ for which $R$ and its adjoint $R^*$ are both generalised Radon transforms given by integration over some family of curves. The approach has been less successful when considering $R$ which are *unbalanced* in the sense that $R$ and $R^*$ are both generalised Radon transforms but the dimensions of the associated families are not equal, although it has still produced results in some specific cases, for example [@Erdogan2010; @Gressman2006; @Gressman2013]. The dilated averaging operator fits into this framework by setting $\Sigma := {\mathbb{R}}^d \times (1,2)$ and for each $(x,r) \in \Sigma$ defining $M_{(x,r)}$ to be the curve parametrised by $t \mapsto x - r \gamma(t)$. Observe that although $A$ is defined by integration over curves, the adjoint of $A$ is defined by integration over $2$-surfaces and hence the operator is unbalanced.
The structure of the paper is as follows. In the following section the necessary conditions on $(p,q)$ for $A$ to be restricted weak type $(p,q)$ are discussed. In Section \[overview of refinement method\] standard methods together with estimates for single averages are combined to reduce the proof of Theorem \[bigthm\] to proving a single restricted weak-type inequality. Christ’s method of refinements is also reviewed and used to establish the simple case of Theorem \[momentthm\] when $d=3$. The remaining sections develop this method to be applicable in the general situation.
A word of explanation concerning notation is in order: throughout the paper $C$ and $c$ will be used to denote various positive constants whose value may change from line to line but will always depend only on the dimension $d$ and degree $\deg P$ of some fixed polynomial. If $X, Y \geq 0$, then the notation $X \lesssim Y$ or $Y \gtrsim X$ signifies $X \leq C Y$ and this situation is also described by “$X$ is $O(Y)$”. In addition, $X \sim Y$ indicates $X \lesssim Y \lesssim X$. Finally, the cardinality of any finite set $B$ will be denoted by $\# B$.
Acknowledgement {#acknowledgement .unnumbered}
---------------
The author wishes acknowledge his PhD supervisor, Prof. Jim Wright, for all his kind and patient guidance relating to this work. He would also like to thank both Marco Vitturi and Betsy Stovall for elucidating discussions regarding some of the references.
Necessary conditions {#necessity}
====================
Suppose the operator $A$ from Theorem \[momentthm\] satisfies a restricted weak-type $(p,q)$ inequality for some $1 \leq p,q < \infty$. Here it is shown that the exponents $p,q$ must satisfy four conditions, each corresponding to an edge of the trapezium $\mathcal{T}_d$. The first three conditions also appear in the study of the averaging operator $\mathcal{A}$ defined in the introduction and are deduced by the same reasoning. The remaining condition does not appear in the theory of single averages and here the dilation parameter plays a non-trivial rôle, although the arguments are only marginally different from those used to examine $\mathcal{A}$.
(3.1,0) -| (0,3.1) ; at (1.5, 0) [$1/p$]{}; at (0, 1.5) [$1/q$]{}; (0,0) coordinate (es) – (3.0,3.0) coordinate (ee); (0,0) coordinate (es) – (1.0,0.5) coordinate (ee); (2.0,1.5) coordinate (es) – (3.0,3.0) coordinate (ee); (1.0,0.5) coordinate (es) – (2.0,1.5) coordinate (ee); (0,0) coordinate (es) – (1.5,1.0) coordinate (ee); (0,3) coordinate (es) – (3,3) coordinate (ee); (3,0) coordinate (es) – (3,3) coordinate (ee);
(1.0,0.5) node\[dot\] (int2) ; at (1.0,0.5) [$ \big(\frac{1}{p_1}, \frac{1}{q_1}\big)$]{}; (2.0,1.5) node\[dot\] (int2) ; at (2.0,1.5) [$ \big(\frac{1}{p_2}, \frac{1}{q_2}\big)$]{}; (1.5,1) node\[dot\] (int2) ; at (1.5,1) [$ \big(\frac{1}{q_2'}, \frac{1}{p_2'}\big)$]{}; (0,0) node\[dot,label=below:[$(0,0)$]{}\] (int2) ; (3,3) node\[dot,label=right:[$(1,1)$]{}\] (int2) ;
To begin, a slight modification of a general theorem of Hörmander [@Hormander1960] implies $p \leq q$.
For the second condition, let $R(\delta) : = \prod_{j=1}^d [-\delta^j,\delta^j]$ and note that $$A \chi_{R(\delta)}(x,r) = |\{ t \in [0,1] : x - r\gamma (t) \in R(\delta) \}|.$$ If $x \in (1/2) R(\delta)$, then whenever $t \in [0, \delta/4]$ it follows that $$|x_j -rt^j| \leq \delta^j/2 + 2 (\delta/4)^j \leq \delta^j \qquad \textrm{for $j = 1, \dots, d$}$$ and therefore $$A \chi_{R(\delta)}(x,r) \geq \frac{\delta}{4} \chi_{(1/2)R(\delta)}(x).$$ Consequently, applying the hypothesised restricted weak-type estimate, $$|R(\delta)| \lesssim \left|\left\{ (x,r) \in {\mathbb{R}}^d \times [1,2] : A \chi_{R(\delta)}(x,r) > \delta/8 \right\}\right| \lesssim \left( \frac{1}{\delta}|R(\delta)|^{1/p}\right)^{q}.$$ Observe $|R(\delta)| \sim_d \delta^{d(d+1)/2}$ and so the preceding inequality implies $$\delta^{d(d+1)/(2q)} \lesssim \delta^{d(d+1)/(2p) - 1} \qquad \textrm{for all $0 < \delta < 1$.}$$ The exponents $(p,q)$ must therefore satisfy the relation $$\frac{1}{q} \geq \frac{1}{p} - \frac{2}{d(d+1)}.$$ The third condition is established by testing $A$ on $\chi_{B(\delta)}$, the characteristic function of a ball $B(\delta) \subset {\mathbb{R}}^d$ of radius $0 < \delta < 1$, centred at the origin. It is easy to see $$\label{nec2}
A \chi_{B(\delta)}(x,r) \gtrsim \delta \chi_{\mathcal{N}_r(\delta)}(x)$$ where $\mathcal{N}_r(\delta)$ is a $\delta/3$-neighbourhood of the $r$-dilate of the moment curve; that is, the set of all points $x \in {\mathbb{R}}^d$ for which $|x- r\gamma(t_0)| < \delta/3$ for some $t_0 \in [0,1]$. The hypothesised restricted weak-type estimate together with imply $$\begin{aligned}
\left|\left\{(x,r) \in {\mathbb{R}}^d \times [1,2] : x \in \mathcal{N}_r(\delta)\right\}\right| &\leq& \left|\left\{ (x,r) \in {\mathbb{R}}^d \times [1,2] : A \chi_{B(\delta)}(x,r) > C \delta \right\}\right| \\
&\lesssim& \left( \frac{1}{\delta}|B(\delta)|^{1/p}\right)^{q}.\end{aligned}$$ Observe $|B(\delta)| \sim_d \delta^d$ whilst $|\mathcal{N}_r(\delta)| \gtrsim \delta^{d-1}$ for all $r \in [1,2]$ and so the preceding inequality implies $$\delta^{(d-1)/q} \lesssim \delta^{d/p - 1}\qquad \textrm{for all $0 < \delta < 1$.}$$ Thus the exponents must satisfy the relation $$\frac{1}{q} \geq \frac{d}{d-1} \frac{1}{p} - \frac{1}{d-1}.$$
The final condition on $(1/p, 1/q)$ is deduced by considering the adjoint $A^*$ of $A$. A simple computation yields $$A^*g(x) = \int_1^2 \int_0^1 g(x + r\gamma(t), r)\,{\mathrm{d}}t {\mathrm{d}}r$$ for suitable functions $g$ defined on ${\mathbb{R}}^d \times [1,2]$. The hypothesis on $(p,q)$ is equivalent to the assumption that $A^*$ is restricted weak-type $(q',p')$. For $B(\delta)$ as above, let $F(\delta)$ denote the set $B(\delta) \times [1, 1+c\delta]$ for some small constant $c$. Observe $$A^* \chi_{F(\delta)}(x) \gtrsim \delta^2 \chi_{\mathcal{N}_1(\delta)}(-x)$$ where $\mathcal{N}_1(\delta)$ is as defined above. Therefore, $$|\mathcal{N}_1(\delta)| \leq \left|\left\{ x \in {\mathbb{R}}^d : A^* \chi_{F(\delta)}(x) \gtrsim \delta^2 \right\}\right| \lesssim \left( \frac{1}{\delta^{2}}|F(\delta)|^{1/q'}\right)^{p'}.$$ Finally, $|F(\delta)| \sim_d \delta^{d+1}$ whilst $|\mathcal{N}_1(\delta)| \gtrsim \delta^{d-1}$ and so the preceding inequality implies $$\delta^{(d-1)/p'} \lesssim \delta^{(d+1)/q' - 2} \qquad \textrm{for all $0 < \delta < 1$.}$$ It follows that the exponents must satisfy the relation $(d+1)/q' - 2 \leq (d-1)/p'$ which can be rewritten as $$\frac{1}{q} \geq \frac{d-1}{d+1} \frac{1}{p}.$$
An overview of the refinement method {#overview of refinement method}
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It remains to show the conditions on $(p,q)$ described in Theorem \[bigthm\] are sufficient to ensure $A_P$ satisfies a type $(p,q)$ inequality with the desired uniformity. Real interpolation immediately reduces matters to establishing a uniform restricted weak-type $(p_1, q_1)$ and strong type $(p_2, q_2)$ estimate for $A_P$. The latter is easily dealt with by appealing to the existing literature. Indeed, a theorem of Stovall [@Stovall2010] implies the estimate $$\label{stovallest}
\|A_Pf( \,\cdot\, , r) \|_{L^{q_2}_x({\mathbb{R}}^d)} \lesssim \|f\|_{L^{p_2}_x({\mathbb{R}}^d)}$$ holds for all $r \in [1,2]$. Taking $L^{q_2}_r([1,2])$-norms of both sides of yields the uniform type $(p_2, q_2)$ inequality for $A_P$ and the proof of Theorem \[bigthm\] is therefore reduced to establishing the following Proposition.
\[weakthm\] For $d \geq 2$ the inequality $$\label{weak}
\langle A_P\chi_E, \chi_F \rangle \lesssim |E|^{1/d}|F|^{(d^2+1)/d(d+1)}$$ is valid for all pairs of Borel sets $E \subset {\mathbb{R}}^d$ and $F \subset {\mathbb{R}}^d \times [1,2]$ of finite Lebesgue measure.
The proof of Proposition \[weakthm\] will utilise the geometric and combinatorial techniques introduced by Christ in [@Christ1998], which were briefly discussed in the introduction. Collectively these techniques are referred to as the method of refinements. In this section the rudiments of the method are reviewed. It is instructive to consider the proof of the analogue of Proposition \[weakthm\] in three dimensions ($d=3$) for the operator $A$ from the statement of Theorem \[momentthm\]. In this situation the arguments are extremely simple and only a crude version of the refinement procedure is required.
Let $E$ and $F$ denote fixed sets satisfying the hypotheses of Proposition \[weakthm\] for $d=3$. Assume, without loss of generality, that $\langle A\chi_E, \chi_F \rangle \neq 0$ where $A$ is the operator from Theorem \[momentthm\]. One wishes to establish the inequality $$\langle A\chi_E, \chi_F \rangle \lesssim |E|^{1/3}|F|^{5/6},$$ from which Theorem \[momentthm\] follows for the case $d=3$. Defining constants $\alpha$ and $\beta$ by the equation $\langle A\chi_E, \chi_F \rangle = \alpha|F| = \beta|E|$, one may rewrite the preceding inequality as a lower bound on the measure of $E$; explicitly, $$\label{momentequivweake}
|E| \gtrsim \alpha^{6} (\beta/\alpha).$$
The basic idea behind Christ’s method is to attempt to prove by using iterates of $A$ and $A^*$ to construct a natural parameter set $\Omega \subset {\mathbb{R}}^3$ and parametrising function $\Phi : \Omega \rightarrow E$ with a number of special properties. First of all, $\Phi$ must have bounded multiplicity so, by applying the change of variables formula, $$|E| \gtrsim \int_{\Omega} |J_{\Phi}(t)| \,{\mathrm{d}}t$$ where $J_{\Phi}$ denotes the Jacobian of $\Phi$. It then remains to bounded this integral from below by some expression in terms of $\alpha$ and $\beta$, which is possible provided that the parametrisation has been carefully constructed.
Following [@Christ1998], define $$\begin{aligned}
F_1 &:=& \big\{ (x,r) \in F : A \chi_E(x,r) > \alpha / 2 \big\}, \\
E_1 &:=& \big\{ y \in E : A^* \chi_{F_1}(y) > \beta / 4 \big\}.\end{aligned}$$ It is not difficult to see the assumptions on $E$ and $F$ imply $\langle A\chi_{E_1}, \chi_{F_1} \rangle \neq 0$ and therefore $E_1$ is non-empty. Fix $y_0 \in E_1$ and define a map $\Phi_1 \colon [1,2] \times [0,1] \to {\mathbb{R}}^3 \times [1,2]$ by $$\label{moment Phi1}
\Phi_1(r_1,t_1) := \left(\begin{array}{c}
y_0 + r_1h(t_1) \\
r_1
\end{array} \right).$$ Note that the set $$\Omega_1:= \left\{ (r_1, t_1) \in [1,2] \times [0,1] : \Phi_1(r_1,t_1) \in F_1 \right\}$$ satisfies $|\Omega_1| > \beta / 4$. Similarly, define a map $\Phi_2 \colon [1,2] \times [0,1]^2 \to {\mathbb{R}}^3$ by $$\label{moment Phi2}
\Phi_2(r_1, t_1, t_2) := y_0 + r_1h(t_1) - r_1h(t_2)$$ and observe for each $(r_1, t_1) \in \Omega_1$ the set $$\Omega_2(r_1, t_1):= \left\{ t_2 \in [0,1] : \Phi_2(r_1, t_1, t_2) \in E \right\}$$ satisfies $|\Omega_2(r_1, t_1)| > \alpha / 2$. Finally, define the structured set $$\Omega_2 := \big\{(r_1, t_1, t_2) \in [1,2] \times [0,1]^2 : (r_1, t_1) \in \Omega_1 \textrm{ and } t_2 \in \Omega_2(r_1, t_1) \big\}.$$ Now, $\Omega := \Omega_2 \subset {\mathbb{R}}^3$ is the parameter set alluded to above and $\Phi := \Phi_2 |_{\Omega} : \Omega \rightarrow E$ the parametrising function. Observe $\Phi$ is well-defined by the preceding observations and the polynomial nature of this map ensures it has almost everywhere bounded multiplicity.[^4] The absolute value of the Jacobian $J_{\Phi}(r_1, t_1, t_2)$ of $\Phi$ may be expressed as $$r_1^2 \left| \det \left(\begin{array}{ccc}
1 & 1 & t_2 - t_1 \\
2t_1 & 2t_2 & t_2^2 - t_1^2 \\
3t_1^2 & 3t_2^2 & t_2^3 - t_1^3 \end{array}\right) \right| = 6r_1^2 \bigg| \int_{t_1}^{t_2} V(t_1, t_2, x) \,{\mathrm{d}}x \bigg|$$ where $V(x_1, \dots, x_m) := \prod_{1\leq i<j \leq m} (x_j - x_i)$ denotes, and will always denote, the $m$-variable Vandermonde polynomial. The sign of $V(t_1, t_2, x)$ does not change as $x$ varies between $t_1$ and $t_2$ and so modulus signs can be placed inside the integral in the above expression. Thus, $$|J_{\Phi}(r_1, t_1, t_2)| \gtrsim |t_1 - t_2| \int_{t_1}^{t_2} |x-t_1||t_2-x| \,{\mathrm{d}}x \gtrsim |t_1 - t_2|^4,$$ where the last inequality may easily be deduced by removing a $|t_1-t_2|/8$-neighbourhood of the endpoints $\{t_1,t_2\}$ from the domain of integration. Consequently, by applying the change of variables formula, $$|E| \gtrsim \int_{\Omega} |J_{\Phi}(r_1, t_1, t_2)| \, {\mathrm{d}}t_2 {\mathrm{d}}r_1 {\mathrm{d}}t_1 \gtrsim \int_{\Omega_1}\int_{\Omega_2(r_1, t_1)} |t_1 - t_2|^4 \, {\mathrm{d}}t_2 {\mathrm{d}}r_1 {\mathrm{d}}t_1.$$ For each $(t_1,r_1) \in \Omega_1$ define $\widetilde{\Omega}_2(r_1, t_1) := \Omega_2(r_1, t_1)\setminus (t_1 + c\alpha, t_1 - c\alpha)$ for a suitably small constant $c$, chosen so that $|\widetilde{\Omega}_2(r_1, t_1)| \gtrsim \alpha$. Hence, $$|E| \gtrsim \int_{\Omega_1}\int_{\widetilde{\Omega}_2(r_1, t_1)} |t_1 - t_2|^4 \, {\mathrm{d}}t_2 {\mathrm{d}}r_1 {\mathrm{d}}t_1 \gtrsim \alpha^4 \int_{\Omega_1}\int_{\widetilde{\Omega}_2(r_1, t_1)} \, {\mathrm{d}}t_2 {\mathrm{d}}r_1 {\mathrm{d}}t_1 \gtrsim \alpha^5 \beta,$$ and this concludes the proof of and thereby establishes Theorem \[momentthm\] for $d=3$.
The remainder of the paper will develop this elementary argument in order to prove Proposition \[weakthm\] in any dimension $d$ and for any polynomial curve $P$.
The polynomial decomposition theorem of Dendrinos and Wright {#decomposition 1}
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The refinement method essentially reduces the problem of establishing the restricted weak-type inequality from Proposition \[weakthm\] to estimating a Jacobian determinant associated with a certain naturally arising change of variables. In the case of the moment curve this Jacobian takes a particularly simple form involving a Vandermonde polynomial $V(t)$. For a general polynomial curve $P \colon {\mathbb{R}}\to {\mathbb{R}}^d$ one is led to consider expressions of the form $$\label{JP}
J_P(t) := \det(P'(t_1) \dots P'(t_d))$$ for $t = (t_1, \dots, t_d) \in {\mathbb{R}}^d$.[^5] The multivariate polynomial $J_P$ can be effectively estimated by comparing it with the Vandermonde polynomial and a certain geometric quantity expressed in terms of the torsion function (whose definition is recalled below). This leads to what is referred to here (and in [@Dendrinos2010]) as a *geometric inequality* for $J_P$. It is often the case that such a comparison is not possible globally; however, an important theorem due to Dendrinos and Wright [@Dendrinos2010] demonstrates the existence of a decomposition of the real line into a bounded number of intervals, ${\mathbb{R}}= \bigcup_{m=1}^C \overline{I_m}$, such that such a geometric inequality holds on each constituent interval $I_m$. Furthermore, the torsion function has a particularly simple form when restricted to an $I_m$: it is comparable to a centred monomial. Restricting the analysis to an interval arising from the Dendrinos-Wright decomposition therefore significantly simplifies the situation and allows for an effective estimation of the Jacobian $J_P$.
In order to state the decomposition lemma, recall the torsion of the curve $P$ is defined to be the polynomial function $$L_P(t) := \det(P^{(1)}(t) \dots P^{(d)}(t))$$ where $P^{(i)}$ denotes the $i$th derivative of $P$.
\[Dendrinos Wright theorem\] Let $P \colon {\mathbb{R}}\to {\mathbb{R}}^d$ be a polynomial curve of degree $n$ such that $L_P \not\equiv 0$. There exists an integer $C = C_{d,n}$ and a decomposition ${\mathbb{R}}= \bigcup_{m=1}^{C} \overline{I_m}$ where the $I_m$ are pairwise disjoint open intervals with the following properties:
1) Whenever $\mathbf{t} = (t_1, \dots, t_d) \in I_m^d$ the geometric inequality $$|J_P(\mathbf{t})| \gtrsim \prod_{i=1}^d|L_P(t_i)|^{1/d} |V(\mathbf{t})|$$ holds.
2) For every $1 \leq m \leq C$ there exists a positive constant $D_{m}$, a non-negative integer $K_{m} \lesssim 1$ and a real number $b_{m} \in {\mathbb{R}}\setminus I_m$ such that $$|L_{P}(t)| \sim D_{m}|t - b_m|^{K_{m}} \qquad \textrm{for all $t \in I_m$.}$$
Theorem \[Dendrinos Wright theorem\] originally appeared in [@Dendrinos2010] where it was used to study Fourier restriction operators associated to polynomial curves (see [@Stovall] for further developments in this direction). Concurrently, Dendrinos, Laghi and Wright [@Dendrinos2009] applied the decomposition to establish uniform estimates for convolution with affine arc-length on polynomial curves in low dimensions; their results were subsequently extended to all dimensions by Stovall [@Stovall2010]. Many of the methods of this paper are based on those found in [@Dendrinos2009; @Stovall2010].
Fixing a polynomial $P$ for which $L_P \not\equiv 0$, to prove Proposition \[weakthm\] it suffices to establish the analogous uniform restricted weak-type inequalities for the local operators $$A^c_P(x,r) := \int_I f(x - r P(t)) \, \lambda_P(t){\mathrm{d}}t$$ where $I$ is any bounded interval. Furthermore, one may assume $I$ lies completely within one of the intervals $I_m$ produced by the decomposition (indeed, $A^c_P$ can always be expressed as a sum of a bounded number of operators of the same form for which this property holds). Observing the translation, reflection and scaling invariance of the problem, one may assume $D_{m} = 1$, $b_m = 0$ and $I \subset (0, \infty)$ with $|I| = 1$ without any loss of generality. Similar reductions were made in [@Stovall2010] where further details can be found. Notice under these hypotheses, $|L_P(t)| \sim t^K$ uniformly on $I$ for some non-negative integer $K \lesssim 1$.
Henceforth $A$ will denote the operator defined by $$\label{reduced operator}
Af(x, r) := \int_I f(x - r P(t)) \,{\mathrm{d}}\mu_P(t)$$ where $\mu_P$ is now the weighted measure ${\mathrm{d}}\mu_P(t) := \lambda_P(t){\mathrm{d}}t$; $\lambda_P$ is redefined as $\lambda_P(t):= t^{2K/d(d+1)}$ and the integer $K$ and interval $I$ satisfy the above properties. It remains to prove the analogue of the restricted weak-type inequality from Proposition \[weakthm\] for this operator.
To close this section it is remarked that Stovall [@Stovall2010] established an upper bound for certain derivatives of $J_P$ on the set $I^d$ in terms of $J_P$ itself. This estimate will be of use in the forthcoming analysis and is recorded presently for the reader’s convenience.
\[Stovall’s observation\] Let $S \subseteq \{1, \dots, d\}$ be a non-empty set of indices. Whenever $t = (t_1, \dots, t_d) \in I^d$, one has the estimate $$\bigg| \prod_{j \in S} \frac{\partial}{\partial t_j} J_P(t) \bigg| \lesssim \sum_{T \subseteq S} \sum_{u, \epsilon} \bigg(\prod_{{j} \in S\setminus T} t_{j}^{-1}\bigg)\bigg( \prod_{{j} \in T} t_{j}^{-\epsilon(j)}|t_{j} - t_{u({j})}|^{\epsilon(j) - 1}\bigg)|J_P(t)|$$ where the outer sum is over all subsets $T$ of $S$ and the inner sum is over all functions $u \colon T \to \{1, \dots, d\}$ with the property $u(j) \neq j$ for all $j \in T$ and all $\epsilon \colon T \to \{0,1\}$.
Parameter towers
================
Having made the reductions of the previous section, fix Borel sets $E \subseteq {\mathbb{R}}^d$ and $F \subseteq {\mathbb{R}}^d \times [1,2]$ of finite Lebesgue measure such that $\langle A \chi_E\,,\, \chi_F\rangle \neq 0$ where $A$ is of the special form described in . As in Section \[overview of refinement method\], the quantities $$\alpha := \frac{1}{|F|} \langle A\chi_E, \chi_F \rangle \quad \textrm{and} \quad \beta := \frac{1}{|E|} \langle \chi_E, A^*\chi_F \rangle$$ play a dominant rôle in the analysis. Indeed, by some simple algebra the inequality can be restated in terms of $\alpha$ and $\beta$ as either $$\label{equivweake}
|E| \gtrsim \alpha^{d(d+1)/2} (\beta/\alpha)^{(d-1)/2}$$ or $$\label{equivweakf}
|F| \gtrsim \alpha^{d(d+1)/2} (\beta/\alpha)^{(d+1)/2}.$$ The proof will proceed by attempting to establish either one of these estimates by applying a variant of the refinement procedure described earlier. In view of the $L^{p_2}_x - L^{q_2}_{x,r}$ estimate established in Section \[overview of refinement method\], henceforth it is assumed without loss of generality that $\alpha > \beta$. Indeed, the restricted weak-type $(p_2, q_2)$ inequality implies $$|E| \gtrsim \alpha^{d(d+1)/2} (\beta/\alpha)^{d}$$ from which follows in the case $\alpha \leq \beta$.
As in Section \[overview of refinement method\], either or will be established by constructing suitable parameter domain $\Omega$ and parametrising function $\Phi$ where $\Omega$ is some structured set. In this section the basic structure of such a domain $\Omega$ is described.
Consider a collection $\{\Omega_j\}_{j=1}^D$ of Borel measurable sets either of the form[^6] $$\label{floortower}
\Omega_j \subseteq [1,2]^{\floor{j/2}} \times I^j \qquad \textrm{for $j = 1, \dots, D$}$$ or $$\label{ceiltower}
\Omega_j \subseteq [1,2]^{\ceil{j/2}} \times I^j \qquad \textrm{for $j = 1, \dots, D$}.$$ In order to be concise it is useful to let $\bracket{x}$ ambiguously denote either $\ceil{x}$ or $\floor{x}$ for any $x \in {\mathbb{R}}$, where it is understood the notation is consistent within any given equation. Thus and are considered simultaneously by writing $$\Omega_j \subseteq [1,2]^{\bracket{j/2}} \times I^j \qquad \textrm{for $j = 1, \dots, D$.}$$ Assume each $\Omega_j$ has positive $(j + \bracket{j/2})$-dimensional measure. The following definitions, which borrow terminology from [@Christ; @Christa], are fundamental in what follows.
i) A collection $\{\Omega_j\}_{j=1}^D$ of the above form is a (parameter) tower of height $D \in {\mathbb{N}}$ if for any $1 < j \leq D$ and $r_1, \dots, r_{\bracket{j/2}} \in [1,2]$ and $t_1, \dots, t_j \in I$ the following holds: $$(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j \Rightarrow (\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \Omega_{j-1}$$ where $\mathbf{r}_k := (r_1, \dots, r_{\bracket{k/2}})$ and $\mathbf{t}_k := (t_1, \dots, t_k)$ for $k = j-1, j$.
ii) If a tower is described as “type 1” (respectively, “type 2”) this indicates the constituent sets are of the form described in (respectively, ). Thus, when considering type 1 (respectively, type 2) towers the symbol $\bracket{x}$ is interpreted as $\floor{x}$ (respectively, $\ceil{x}$) for any $x \in {\mathbb{R}}$.
iii) Given a type 1 (respectively, type 2) tower $\{\Omega_j\}_{j=1}^D$, fix $1< j \leq D$. For each $(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \Omega_{j-1}$ define the associated fibre $\Omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})$ to be the set $$\left\{\begin{array}{ll}
\big\{ t_j \in I : (\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j \big\}
& \textrm{if $j$ is odd (respectively even)}\\
&\\
\left\{ (r_{\bracket{j/2}}, t_j) \in [1,2] \times I : (\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j \right\} & \textrm{if $j$ is even (respectively odd).}
\end{array}\right.$$
A type $j$ tower is characterised by the property that the initial set $\Omega_1$ is $j$-dimensional, for $j=1,2$.
For example, the collection $\{\Omega_j\}_{j=1}^2$ defined in Section \[overview of refinement method\] constitutes a type 2 tower. In what follows, type 1 towers will be of primary interest. The elements of the various levels of a type 1 tower are typically denoted using the following notation: $$\mathbf{t}_1 = t_1 \in \Omega_1, \quad (\mathbf{r}_2, \mathbf{t}_2) = (r_1, t_1,t_2) \in \Omega_2, \quad (\mathbf{r}_3, \mathbf{t}_3) = (r_1, t_1, t_2, t_3) \in \Omega_3,$$ $$(\mathbf{r}_4, \mathbf{t}_4) = (r_1, r_2, t_1, t_2, t_3, t_4) \in \Omega_4,\quad (\mathbf{r}_5, \mathbf{t}_5) = (r_1, r_2, t_1, t_2, t_3, t_4, t_5) \in \Omega_5, \dots .$$
Recall that certain mappings $\Phi_1$ and $\Phi_2$, defined in and , were associated to the tower constructed in Section \[overview of refinement method\]. Presently the analogues of these mappings in the general situation are discussed. First of all one associates to every $(x_0, r_0) \in {\mathbb{R}}^d \times [1,2]$ and $y_0 \in {\mathbb{R}}^d$ a family of functions.
i) Given $(x_0, r_0) \in {\mathbb{R}}^d \times [1,2]$ define the functions $\Psi_j(x_0, r_0;\,\cdot\,) : [1,2]^{\floor{j/2}} \times I^j \to {\mathbb{R}}^d$ by $$\label{parf}
\Psi_j(x_0, r_0;\mathbf{r}_j, \mathbf{t}_j) = x_0 + \sum_{k = 1}^j (-1)^{k} r_{\floor{k/2}} P(t_k).$$ for all $\mathbf{r}_j = (r_1, \dots, r_{\floor{j/2}}) \in [1,2]^{\floor{j/2}}$ and $\mathbf{t}_j = (t_1, \dots, t_j) \in I^j$.
ii) Given $y_0 \in {\mathbb{R}}^d$ define the functions $\Psi_j(y_0;\,\cdot\,) : [1,2]^{\ceil{j/2}} \times I^j \to {\mathbb{R}}^d$ by $$\label{pare}
\Psi_j(y_0; \mathbf{r}_j, \mathbf{t}_j) = y_0 + \sum_{k = 1}^j (-1)^{k+1} r_{\ceil{k/2}} P(t_k)$$ for all $\mathbf{r}_j = (r_1, \dots, r_{\ceil{j/2}}) \in [1,2]^{\ceil{j/2}}$ and $\mathbf{t}_j = (t_1, \dots, t_j) \in I^j$.
To any tower one associates a family of mappings on the constituent sets, defined in terms of the $\Psi_j$ functions.
\[associated mappings\] Suppose $\{\Omega_j\}_{j=1}^D$ is a type 1 (respectively, type 2) tower and fix some $z_0 = (x_0, r_0) \in {\mathbb{R}}^d \times [1,2]$ (respectively, $z_0 = y_0 \in {\mathbb{R}}^d$). The family of mappings $\{\Phi_j\}_{j=1}^D$ associated to these objects is defined as follows:
i) For $1 \leq j \leq D$ odd (respectively, even) let $\Phi_j : \Omega_j \rightarrow {\mathbb{R}}^d$ denote the map $$\Phi_j(\mathbf{r}_j, \mathbf{t}_j) := \Psi_j(z_0; \mathbf{r}_j, \mathbf{t}_j).$$
ii) For $1 \leq j \leq D$ even (respectively, odd) let $\Phi_j : \Omega_j \rightarrow {\mathbb{R}}^d \times [1,2]$ denote the map $$\Phi_j(\mathbf{r}_j, \mathbf{t}_j) := \left( \begin{array}{c}
\Psi_k(z_0, \mathbf{r}_j, \mathbf{t}_j)\\
r_{\bracket{j/2}} \end{array}\right).$$
Referring back to the simple case discussed earlier, (appropriate restrictions of) the functions defined in and are easily seen to constitute the family associated to the point $y_0$ and tower $\{\Omega_j\}_{j=1}^2$ constructed in Section \[overview of refinement method\].
For notational convenience define the following quantity $$\label{kappa}
\kappa := \frac{d(d+1)}{2K + d(d+1)}.$$ Recalling the definition of $\mu_P$ from , it is also useful to let $\nu_P$ denote the measure given by the product of Lebesgue measure on $[1,2]$ with $\mu_P$. Hence, for any Borel set $R \subseteq [1,2] \times I$, $$\nu_P(R) = \int_1^2 \int_I \chi_R(r,t)\,\lambda_P(t){\mathrm{d}}t {\mathrm{d}}r.$$ Initially the following lemma is used to construct a suitable parameter tower.
\[dendrinosstovall\] There exists a point $(x_0, r_0) \in F$ and a type 1 tower $\{\Omega_j\}_{j = 1}^{d+1}$ with the following properties:
1) Whenever $(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j$ it follows that $$\alpha^{\kappa} = \max\{\alpha, \beta\}^{\kappa} \lesssim t_1 < t_2 < \dots < t_j.$$
2) For $1 \leq j \leq d+1$ odd:
i) $\Phi_j(\Omega_j) \subseteq E$;
ii) $\mu_P(\Omega_1) \gtrsim \alpha$ and if $j > 1$, then $\mu_P\left(\Omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\right) \gtrsim \alpha$ whenever $(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \Omega_{j-1}$;
iii) If $j > 1$ and $(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j$, then $$\int_{t_{j-1}}^{t_j} \,\lambda_P(t){\mathrm{d}}t \gtrsim \alpha.$$
3) For $1 < j \leq d+1$ even:
i) $\Phi_j(\Omega_j) \subseteq F$;
ii) $\nu_P \left(\Omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\right) \gtrsim \beta$ whenever $(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \Omega_{j-1}$;
iii) If $(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j$, then $$\int_{t_{j-1}}^{t_j} \,\lambda_P(t){\mathrm{d}}t \gtrsim \beta.$$
Lemma \[dendrinosstovall\] is a slight modification of a recent result due to Dendrinos and Stovall [@Dendrinos], based on a fundamental construction due to Christ [@Christ1998]. Rather than present a proof of Lemma \[dendrinosstovall\] a stronger statement, Lemma \[towerlem\], is established below.
To conclude this section it is noted that a tower admitting all the properties described in the previous lemma automatically satisfies a certain separation condition. This observation was also used in [@Dendrinos].
\[separation\] Let $\{\Omega_j\}_{j=1}^{d+1}$ be a tower with all the properties described in Lemma \[dendrinosstovall\].
i) Suppose $1 < j \leq d+1$ is odd. Then, for all $(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j$ it follows that $$t_j - t_i \gtrsim \alpha t_i^{-2K/d(d+1)} \qquad \textrm{for $1 \leq i \leq j-1$.}$$
ii) Suppose $1 < j \leq d+1$ is even. Then, for all $(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j$ it follows that $$t_j - t_{j-1} \gtrsim \beta t_{j-1}^{-2K/d(d+1)}; \qquad
t_j - t_{i} \gtrsim \alpha t_i^{-2K/d(d+1)} \quad \textrm{for $1 \leq i \leq j-2$.}$$
Let $1 < j \leq n+1$ be either odd or even and $1 \leq i \leq j-1$. If $j$ is even, then further suppose $i\leq j-2$. For $(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j$, properties 1) and 2) iii) of the construction ensure there exists some $t_{i} < s_{i} < t_{j}$ for which $$\int_{t_{i}}^{s_{i}} \lambda_P(t)\,{\mathrm{d}}t \sim \alpha.$$ Consequently, $$s_{i}^{1/\kappa} \sim \int_0^{s_{i}} \lambda_P(t)\,{\mathrm{d}}t = \int_0^{t_{i}} \lambda_P(t)\,{\mathrm{d}}t + \int_{t_{i}}^{s_{i}} \lambda_P(t)\,{\mathrm{d}}t \sim t_{i}^{1/\kappa} + \alpha$$ and, since $\alpha \lesssim t_{i}^{1/\kappa}$ holds by property 1), one concludes that $s_{i} \sim t_{i}$. Whence, $$|t_j - t_i| \geq |s_{i} - t_{i}| \gtrsim \bigg(\int_{t_{i}}^{s_{i}} \lambda_P(t)\,{\mathrm{d}}t\bigg)t_{i}^{-2K/d(d+1)} \gtrsim \alpha t_{i}^{-2K/d(d+1)}.$$ The remaining case when $j$ is even and $i = j-1$ can be dealt with in a similar fashion, applying property 3) iii).
Improved parameter towers
=========================
The properties detailed in Lemma \[dendrinosstovall\], though useful, are insufficient for the present purpose. Observe that although the even fibres of the tower constructed in Lemma \[dendrinosstovall\] are two-dimensional sets, consisting of points $(r_{j/2}, t_j) \in [1,2] \times I$, all the bounds are decidedly one-dimensional in the sense that they are in terms of the $t_j$ variables and there is little reference to the dilation parameters. An additional refinement is necessary to take advantage the higher dimensionality of the even fibres.
\[towerlem\] Fix $0 < \delta \ll 1$ a small parameter. There exists a point $(x_0, r_0) \in F$ and a tower $\{\Omega_k\}_{k = 1}^{d+1}$ satisfying all the properties of Lemma \[dendrinosstovall\] with the additional property that for each even $1 < j \leq d+1$ either $$|t_j - t_{j-1}| \geq \delta (\alpha\beta)^{1/2} t_{j-1}^{-2K/d(d+1)}$$ holds for all $(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j$, or $$|t_j - t_{j-1}| < \delta (\alpha\beta)^{1/2} t_{j-1}^{-2K/d(d+1)} \quad \textrm{ and } \quad |r_{j/2} - r_{j/2-1}| \gtrsim (\beta/\alpha)^{1/2}$$ both hold for all $(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_j$. If the former case the index $j$ is designated “pre-red”, whilst in the latter $j$ is designated “pre-blue”. The odd vertices are “pre-achromatic”; that is, they are not assigned a “pre-colour”.
The partition of the indices into the sets of odd, pre-red and pre-blue indices plays a very similar rôle to the construction of the band structure in [@Christ1998] and in particular the “slicing method” of [@Christ; @Christ1998] will be utilised.
As the prefix “pre-” suggests, later in the argument it will be convenient to relabel the indices. In particular, in the following section an updated labelling will be introduced which designates the indices either “red”, “blue” or “achromatic”.
The result is essentially established as follows. By applying the argument of Dendrinos and Stovall [@Dendrinos] one obtains an initial tower with the properties stated in Lemma \[dendrinosstovall\]. To ensure the additional property described in Lemma \[towerlem\] holds one further refines the tower, appealing to the following elementary (but notationally-involved) result.
\[refinecor\] Let $\{ \widetilde{\Omega}_j\}_{j=1}^D$ be a tower of height $D$ and for each even $1 < k \leq D$ let $$\big\{A_{k}(\mathbf{r}_{k-1}, \mathbf{t}_{k-1}) : (\mathbf{r}_{k-1}, \mathbf{t}_{k-1}) \in \widetilde{\Omega}_{k-1} \big\}$$ a collection of measurable subsets of $[1,2] \times I$. Then there exists a tower[^7] $\{\Omega_j\}_{j=1}^D$ satisfying:
a) $\Omega_1 \subseteq \widetilde{\Omega}_1$ and $\Omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \subseteq \widetilde{\Omega}_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})$ for all $(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \Omega_{j-1}$ and all $1 < j \leq D$;
b) The following estimates hold for the fibres:
i) $\mu_P(\Omega_1) \geq 2^{-\floor{D/2}} \mu_P(\widetilde{\Omega}_1)$.
ii) Whenever $1 < j \leq D$ is odd, $$\mu_P\big(\Omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\big) \geq 2^{-\floor{D/2}} \mu_P\big(\widetilde{\Omega}_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\big)$$ holds for all $(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \Omega_{j-1}$.
iii) Whenever $1 < j \leq D$ is even, $$\nu_P\big(\Omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\big) \geq 2^{-\floor{D/2}} \nu_P\big(\widetilde{\Omega}_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\big)$$ holds for all $(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \Omega_{j-1}$.
c) For each even $1 < k \leq D$ precisely one of the following holds:
i) $\Omega_{k}(\mathbf{r}_{k-1}, \mathbf{t}_{k-1}) \subseteq A_k(\mathbf{r}_{k-1}, \mathbf{t}_{k-1})$ for all $(\mathbf{r}_{k-1}, \mathbf{t}_{k-1}) \in \Omega_{k-1}$;
ii) $\Omega_{k}(\mathbf{r}_{k-1}, \mathbf{t}_{k-1}) \cap A_k(\mathbf{r}_{k-1}, \mathbf{t}_{k-1}) = \emptyset$ for all $(\mathbf{r}_{k-1}, \mathbf{t}_{k-1}) \in \Omega_{k-1}$.
\[measure remark\] Strictly speaking, the proof below will not address the issue of whether the $\Omega_j$ are measurable (as required in the definition of a tower), although the fibres certainly will be (and therefore b) i) and b) ii) make sense). In practice, when the lemma is applied below it will be clear from the choice of sets $A_{k}(\mathbf{r}_{k-1}, \mathbf{t}_{k-1})$ that the resulting $\Omega_j$ are measurable and so this omission is unimportant for the present purpose.
Proceed by induction on $D$, the case $D = 1$ being vacuous. Let $1 < D$ and fix a tower $\{\widetilde{\Omega}_j\}_{j=1}^D$. Apply the induction hypothesis to $\{\widetilde{\Omega}_j\}_{j=1}^{D-1}$ to obtain a tower $\{\widehat{\Omega}_j\}_{j=1}^{D-1}$ satisfying the properties a), b) and c) of the corollary, with $D$ replaced by $D-1$. For each $(\mathbf{r}_{D-1}, \mathbf{t}_{D-1}) \in \widehat{\Omega}_{D-1}$ define $\widehat{\Omega}_{D}(\mathbf{r}_{D-1}, \mathbf{t}_{D-1}) := \widetilde{\Omega}_{D}(\mathbf{r}_{D-1}, \mathbf{t}_{D-1}) $. If $D$ is odd define $\widehat{\Omega}_{D}$ to be $$\big\{ (\mathbf{r}_{D}, \mathbf{t}_{D}) \in \widetilde{\Omega}_D : t_D \in \omega_{D}(\mathbf{r}_{D-1}, \mathbf{t}_{D-1})\textrm{ and } (\mathbf{r}_{D-1}, \mathbf{t}_{D-1}) \in \widehat{\Omega}_{D-1} \big\}$$ where $\mathbf{r}_{D} = \mathbf{r}_{D-1}$ and $\mathbf{t}_{D} = (\mathbf{t}_{D-1}, t_{D})$; throughout this article, similar notation will be used for elements belonging to levels of various parameter towers without further comment. Similarly, if $D$ is even, then define $\widehat{\Omega}_{D}$ to be $$\big\{ (\mathbf{r}_{D}, \mathbf{t}_{D}) \in \widetilde{\Omega}_D : (r_{D/2}, t_D) \in \omega_{D}(\mathbf{r}_{D-1}, \mathbf{t}_{D-1})\textrm{ and } (\mathbf{r}_{D-1}, \mathbf{t}_{D-1}) \in \widehat{\Omega}_{D-1} \big\}.$$
If $D$ is odd, then the proof is immediately completed by letting $\Omega_j := \widehat{\Omega}_j$ for $j=1,\dots, D$. It remains to consider the case when $D$ is even, which is more involved. Define a sequence of sets $\omega_{D-k} \subseteq \widehat{\Omega}_{D-k}$ for $1 \leq k \leq D-1$ recursively as follows. For all $(\mathbf{r}_{D-1}, \mathbf{t}_{D-1}) \in \widehat{\Omega}_{D-1}$ let $$\omega_D(\mathbf{r}_{D-1}, \mathbf{t}_{D-1}) := \widehat{\Omega}_D(\mathbf{r}_{D-1}, \mathbf{t}_{D-1}) \cap A(\mathbf{r}_{D-1}, \mathbf{t}_{D-1})$$ and define $\omega_{D-1}$ to be the set $$\left\{ (\mathbf{r}_{D-1}, \mathbf{t}_{D-1}) \in \widehat{\Omega}_{D-1} : \nu_P\big(\omega_D(\mathbf{r}_{D-1}, \mathbf{t}_{D-1})\big) \geq \frac{1}{2}\nu_P\big(\widehat{\Omega}_D(\mathbf{r}_{D-1}, \mathbf{t}_{D-1})\big) \right\}.$$ Hence, $\omega_{D-1}$ is the set of points $(\mathbf{r}, \mathbf{t}) \in \widehat{\Omega}_{D-1}$ with the property that most of the associated fibre $\widehat{\Omega}_D(\mathbf{r}, \mathbf{t})$ lies in $A(\mathbf{r}, \mathbf{t})$.
Now suppose $\omega_{D-k}$ has been defined for some $1 \leq k \leq D-2$ and let $(\mathbf{r}_{D-k-1}, \mathbf{t}_{D-k-1}) \in \widehat{\Omega}_{D-k-1}$. If $k$ is odd, then $D- k$ is also odd and $\omega_{D-k}(\mathbf{r}_{D-k-1}, \mathbf{t}_{D-k-1})$ is defined to be $$\big\{ t_{D-k} \in \widehat{\Omega}_{D-k}(\mathbf{r}_{D-k-1}, \mathbf{t}_{D-k-1}) : (\mathbf{r}_{D-k}, \mathbf{t}_{D-k}) \in \omega_{D-k}\big\}.$$ Let $$\omega_{D-k-1} := \left\{ (\mathbf{r}, \mathbf{t}) \in \widehat{\Omega}_{D-k-1} : \mu_P\big(\omega_{D-k}(\mathbf{r}, \mathbf{t})\big) \geq \frac{1}{2}\mu_P\big(\widehat{\Omega}_{D-k}(\mathbf{r}, \mathbf{t})\big) \right\}$$ so that $\omega_{D - k -1}$ is the set of points $(\mathbf{r}, \mathbf{t}) \in \widehat{\Omega}_{D-k -1}$ with the property that most of the associated fibre $\widehat{\Omega}_{D-k}(\mathbf{r}, \mathbf{t})$ lies in $\omega_{D-k}(\mathbf{r}, \mathbf{t})$.
Similarly, if $k$ is even, then $D - k$ is even and $\omega_{D-k}(\mathbf{r}_{D-k-1}, \mathbf{t}_{D-k-1})$ is defined to be $$\big\{ (r_{(D-k)/2}, t_{D-k}) \in \widehat{\Omega}_{D-k}(\mathbf{r}_{D-k-1}, \mathbf{t}_{D-k-1}) : (\mathbf{r}_{D-k}, \mathbf{t}_{D-k}) \in \omega_{D-k}\big\}$$ and one completes the recursive definition by letting $$\omega_{D-k-1} := \left\{ (\mathbf{r}, \mathbf{t}) \in \widehat{\Omega}_{D-k-1} : \nu_P\big(\omega_{D-k}(\mathbf{r}, \mathbf{t})\big) \geq \frac{1}{2}\nu_P\big(\widehat{\Omega}_{D-k}(\mathbf{r}, \mathbf{t})\big) \right\}.$$ Having constructed the sequence $\omega_{D-k}$ for $1 \leq k \leq D-1$, suppose $\mu_P(\omega_1) \geq \tfrac{1}{2}\mu_P(\widehat{\Omega}_1)$. If one defines $\Omega_1 := \omega_1$ and $\Omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) := \omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})$ for $1 < j \leq D$, then one may construct a tower inductively by setting $$\label{tower recursion 1}
\Omega_j := \{(\mathbf{r}_{j}, \mathbf{t}_{j}) \in [1,2]^{(j-1)/2} \times [0,1]^{j} : (\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \Omega_{j-1} \textrm{ and } t_j \in \Omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \}$$ for $j > 1$ odd and $$\label{tower recursion 2}
\Omega_j := \{(\mathbf{r}_{j}, \mathbf{t}_{j}) \in [1,2]^{j/2} \times [0,1]^{j} : (\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \Omega_{j-1} \textrm{ and } (r_{j/2},t_j) \in \Omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \}$$ for $j$ even. It immediately follows from the definitions that the resulting tower $\{\Omega_j\}_{j-1}^{D}$ satisfies the properties stated in the lemma with c) i) holding for $k=D$.
On the other hand, if $\mu_P(\omega_1) < \tfrac{1}{2}\mu_P(\widehat{\Omega}_1)$, then define $\Omega_1 := \widehat{\Omega}_1 \setminus \omega_1$ and let $$\Omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) := \widehat{\Omega}_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \setminus\omega_j(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})$$ for $1 < j \leq D$ so that properties a) and b) i) and c) clearly hold for the resulting tower $\{\Omega_j\}_{j-1}^{D}$, which is again defined by and . To prove b) ii), suppose $1 < j \leq D$ is odd and $(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \Omega_{j-1}$. Thus, $(r_{(j-1)/2}, t_{j-1}) \in \Omega_{j-1}(\mathbf{r}_{j-2}, \mathbf{t}_{j-2})$ and, by the definition of $\omega_{j-1}(\mathbf{r}_{j-2}, \mathbf{t}_{j-2})$, it follows that $(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \widehat{\Omega}_{j-1} \setminus\omega_{j-1}$. Finally, the definition of $\omega_{j-1}$ ensures $$\begin{aligned}
\mu_P\big(\Omega_{j}(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\big) &=\mu_P\big(\widehat{\Omega}_{j}(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\big) - \mu_P\big(\omega_{j}(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\big) \\
&\geq 2^{-1}\mu_P\big(\widehat{\Omega}_{j}(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\big)\\
&\geq 2^{-\floor{(D-1)/2}+1}\mu_P\big(\widetilde{\Omega}_{j}(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})\big),\end{aligned}$$ where the induction hypothesis has been applied in the last inequality. A similar argument shows b) iii) also holds, completing the proof.
Having stated this refinement result one may proceed to prove Lemma \[towerlem\].
Observe, defining $\Sigma := {\mathbb{R}}^d \times [1,2] \times I$ it follows that $$\langle A \chi_E, \chi_F \rangle = \int_{\Sigma} \chi_{U}(x,r,t)\,\lambda_P(t) {\mathrm{d}}x {\mathrm{d}}r{\mathrm{d}}t$$ where $\lambda_P(t):= t^{2K/d(d+1)}$ and $$U := \big\{ (x,r,t) \in \Sigma : x - r P(t) \in E \textrm{ and } (x,r) \in F \big\}.$$ Writing $I := (a,b)$ where $a \geq 0$ and $b - a =1$, a method due to Dendrinos and Stovall [@Dendrinos] may be applied to produce a sequence $\{U_k\}_{k=0}^{\infty}$ of decreasing subsets of $U$ of pairwise comparable measure such that for all $k \geq 1$ either $$\label{ds1}
\int_t^b \chi_{U_{k-1}}(x , r, \tau)\, \lambda_P(\tau){\mathrm{d}}\tau \geq 4^{-(k+1)} \alpha$$ or $$\label{ds2}
\int_1^2 \int_t^b \chi_{U_{k-1}}(x - r P(t) +\rho P(\tau), \rho, \tau)\, \lambda_P(\tau){\mathrm{d}}\tau {\mathrm{d}}\rho \geq 4^{-(k+1)} \beta$$ holds for all $(x,r,t) \in U_k$. Specifically, the $\{U_k\}_{k=0}^{\infty}$ can be chosen so that $$\int_{\Sigma} \chi_{U_0}(x,r,t)\,\lambda_P(t) {\mathrm{d}}x {\mathrm{d}}r{\mathrm{d}}t \geq \frac{1}{2}\langle A \chi_E, \chi_F \rangle$$ and for all $k \geq 1$:
i) $U_k \subseteq U_{k-1}$ and $$\int_{\Sigma} \chi_{U_{k}}(x,r,t) \,\lambda_P(t) {\mathrm{d}}x{\mathrm{d}}r{\mathrm{d}}t \geq \frac{1}{4}\int_{\Sigma} \chi_{U_{k-1}}(x,r,t) \,\lambda_P(t){\mathrm{d}}x{\mathrm{d}}r{\mathrm{d}}t;$$
ii) If $k \not\equiv d \mod 2$ (respectively, $k \equiv d \mod 2$), then (respectively ) holds for all $(x,r,t) \in U_k$.
iii) Furthermore, for each $k$, if $t \in I$ is such that $(x,r,t) \in U_k$ for some $(x,r) \in {\mathbb{R}}^d \times [1,2]$, then $t \geq (\alpha/2 \kappa)^{\kappa}$ where $\kappa$ is as in .
This construction is due to Dendrinos and Stovall [@Dendrinos], however the details are appended for completeness.
Fix $(x_0, r_0, t_0) \in U_{d+1}$. The next step is to use the sets $\{U_k\}_{k=0}^{d+1}$ to construct an initial tower $\{\widetilde{\Omega}_j\}_{j=1}^{d+1}$ satisfying the properties of Lemma \[dendrinosstovall\] and such that whenever $(\mathbf{r}_j, \mathbf{t}_j) \in \widetilde{\Omega}_j$ for some $1 \leq j \leq d+1$, it follows that $$\left\{\begin{array}{ll}
\left(\Psi_j(\mathbf{r}_j, \mathbf{t}_j), r_{j/2}, t_{j}\right) \in U_{d+1-j} & \textrm{if $0 \leq j \leq d+1$ is even}\\
&\\
\left(\Psi_{j-1}(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}), r_{\floor{j/2}}, t_{j}\right) \in U_{d+1-j} & \textrm{if $1 \leq j \leq d+1$ is odd}
\end{array}\right.$$ where $\Psi_j(\mathbf{r}_j, \mathbf{t}_j) := \Psi_j(x_0, r_0;\mathbf{r}_j, \mathbf{t}_j)$ for $j \geq 1$ is as defined in and $\Psi_0(\mathbf{r}_0, \mathbf{t}_0) := x_0$. It is convenient to consider $(\mathbf{r}_0, \mathbf{t}_0)$ as some arbitrary object (say, $(\mathbf{r}_0, \mathbf{t}_0) := (r_0, t_0)$) and $\Psi_0$ as a function on the singleton set $\widetilde{\Omega}_0 := \{(\mathbf{r}_0, \mathbf{t}_0)\}$, taking the value $x_0$.
The tower $\{\widetilde{\Omega}_j\}_{j=1}^{d+1}$ is constructed recursively. To begin, define $\widetilde{\Omega}_0$ as above; suppose $\widetilde{\Omega}_{j}$ has been defined for some $0 \leq j \leq d$ and fix $(\mathbf{r}_{j}, \mathbf{t}_{j}) \in \widetilde{\Omega}_{j}$. The argument splits into two cases, depending on the parity of $j$.
Case 1: $j$ is even. {#case-1-j-is-even. .unnumbered}
--------------------
Since $(\Psi_{j}(\mathbf{r}_{j}, \mathbf{t}_{j}), r_{j/2}, t_{j}) \in U_{d+1-j}$ and $d+1-j \not\equiv d \mod 2$, one may apply to deduce $$\label{ds3}
\int_{t_{j}}^b \chi_{U_{d-j}}(\Psi_{j}(\mathbf{r}_{j}, \mathbf{t}_{j}), r_{j/2}, \tau)\, \lambda_P(\tau){\mathrm{d}}\tau \geq 4^{-(d+2-j)} \alpha.$$ It is therefore possible to choose $t_j < s_j < b$ with the property $$\label{ds4}
\int_{t_j}^{s_j} \,\lambda_P(\tau){\mathrm{d}}\tau = 4^{-(d+5/2-j)} \alpha.$$ Define the set $$\widetilde{\Omega}_{j+1}(\mathbf{r}_j, \mathbf{t}_j) := \big\{ t_{j+1} \in (s_j, b] : (\Psi_{j}(\mathbf{r}_{j}, \mathbf{t}_{j}), r_{j/2}, t_{j+1}) \in U_{d-j} \big\},$$ noting, by and , that this has measure $\mu_P(\widetilde{\Omega}_{j+1}(\mathbf{r}_j, \mathbf{t}_j)) \geq 4^{-(d+5/2-j)} \alpha$. To complete the recursive step in this case, if $j=0$ let $\widetilde{\Omega}_1 := \widetilde{\Omega}_1(\mathbf{r}_0, \mathbf{t}_0)$ whilst for $j > 0$ define $$\widetilde{\Omega}_{j+1} := \big\{ (\mathbf{r}_{j+1}, \mathbf{t}_{j+1} ) : (\mathbf{r}_{j}, \mathbf{t}_{j} ) \in \widetilde{\Omega}_j \textrm{ and } t_{j+1} \in \widetilde{\Omega}_{j+1}(\mathbf{r}_j, \mathbf{t}_j) \big\}.$$
Case 2: $j$ is odd. {#case-2-j-is-odd. .unnumbered}
-------------------
Since $(\Psi_{j-1}(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}), r_{\floor{j/2}}, t_{j}) \in U_{d+1-j}$ and $d+1-j \equiv d \mod 2$, one may apply to deduce $$\label{ds5}
\int_1^2 \int_{t_{j}}^b \chi_{U_{d-j}}(\Psi_{j}(\mathbf{r}_{j}, \mathbf{t}_{j}) +\rho P(\tau), \rho, \tau)\, \lambda_P(\tau){\mathrm{d}}\tau {\mathrm{d}}\rho \geq 4^{-(d+2-j)} \beta.$$ Here the identity $$\Psi_{j}(\mathbf{r}_{j}, \mathbf{t}_{j}) = \Psi_{j-1}(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) - r_{\floor{j/2}}P(t_j)$$ has been applied, which is a consequence of the definition . It is therefore possible to choose $t_j < s_j < b$ and $t_j < \tilde{s}_j$ with the properties $$\label{ds6}
\int_{t_j}^{s_j} \,\lambda_P(\tau){\mathrm{d}}\tau = 4^{-(d+5/2-j)} \beta, \quad \int_{t_j}^{\tilde{s}_j} \,\lambda_P(\tau){\mathrm{d}}\tau = 2^{-(d+3-j)} (\alpha\beta)^{1/2}.$$ Define $\widetilde{\Omega}_{j+1}(\mathbf{r}_j, \mathbf{t}_j)$ to be the set $$\left\{ (r_{\ceil{j/2}}, t_{j+1}) \in D(\mathbf{r}_j, \mathbf{t}_j) : \left(\Psi_{j}(\mathbf{r}_{j}, \mathbf{t}_{j}) + r_{\ceil{j/2}} P(t_{j+1}), r_{\ceil{j/2}}, t_{j+1}\right) \in U_{d-j} \right\}$$ where $D(\mathbf{r}_j, \mathbf{t}_j) := ([1,2] \times (s_j, b])\setminus R(\mathbf{r}_j, \mathbf{t}_j)$ for the rectangle $$R(\mathbf{r}_j, \mathbf{t}_j):=\big\{ (r,t) \in {\mathbb{R}}^2 : |r - r_{\floor{j/2}}| \leq 2^{-(d+4-j)} (\beta/\alpha)^{1/2} \textrm{ and } t_j \leq t \leq \tilde{s}_j \big\}.$$ Observe, by it follows that $$\int_1^2 \int_I \chi_{R(\mathbf{r}_j, \mathbf{t}_j)}(r,t)\,\lambda_P(t){\mathrm{d}}t {\mathrm{d}}r \leq 4^{-(d+3-j)}\beta.$$ Thus, by and , the set $\widetilde{\Omega}_{j+1}(\mathbf{r}_j, \mathbf{t}_j)$ has measure $\nu_P(\widetilde{\Omega}_{j+1}(\mathbf{r}_j, \mathbf{t}_j))\geq 4^{-(d+3-j)} \beta$.
Finally, to complete the recursive definition let $$\widetilde{\Omega}_{j+1} := \big\{ (\mathbf{r}_{j+1}, \mathbf{t}_{j+1} ) : (\mathbf{r}_{j}, \mathbf{t}_{j} ) \in \Omega_j \textrm{ and } (r_{\ceil{j/2}}, t_{j+1}) \in \Omega_{j+1}(\mathbf{r}_j, \mathbf{t}_j) \big\}.$$
It is easy to verify the collection $\{\widetilde{\Omega}_j\}_{j=1}^{d+1}$ forms a tower satisfying all the properties of Lemma \[dendrinosstovall\]. Finally, Lemma \[refinecor\] is applied to further refine this tower to ensure the additional property stated in Lemma \[towerlem\] holds. For each $1 < j \leq d+1$ even, define $$A(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) := \left\{ (r, t) \in [1,2] \times I : t - t_{j-1} > \delta (\alpha\beta)^{1/2}t_{j-1}^{-2K/d(d+1)} \right\}$$ for all $(\mathbf{r}_{j-1}, \mathbf{t}_{j-1}) \in \widetilde{\Omega}_{j-1}$. Letting $\{\Omega_j\}_{j=1}^{d+1}$ denote the refined tower, the existence of which is guaranteed by Lemma \[refinecor\], it is easy to see this has all the desired properties. In particular, for each $1 < j \leq d+1$ even, precisely one of the following holds:
a) For all $(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_{j}$ one has $|t_j - t_{j-1}| > \delta (\alpha\beta)^{1/2}t_{j-1}^{-2K/d(d+1)}$.
b) For all $(\mathbf{r}_j, \mathbf{t}_j) \in \Omega_{j}$ one has $|t_j - t_{j-1}| \leq \delta (\alpha\beta)^{1/2}t_{j-1}^{-2K/d(d+1)}$. In this case, observe for $t_{j-1} \leq \tau \leq t_j$ it follows that $$|\tau - t_{j-1}| \leq \delta (\alpha\beta)^{1/2}t_{j-1}^{-2K/d(d+1)} \lesssim \delta \alpha (\alpha^{\kappa})^{-2K/d(d+1)} \lesssim \delta t_{j-1}$$ by condition iii) of the sets $U_k$ described above. Thus $\tau \lesssim t_{j-1}$ and consequently $$\int_{t_{j-1}}^{t_j}\,\lambda_P(\tau){\mathrm{d}}\tau \lesssim \delta (\alpha\beta)^{1/2}.$$ If $\delta$ is chosen from the outset to be sufficiently small depending only on $d$ and $K$, then it follows that $t_{j-1} < t_j < \tilde{s}_{j-1}$ by the preceding inequality and the definition of $\tilde{s}_{j-1}$ from . Since $(r_{j/2}, t_j) \notin R(\mathbf{r}_{j-1}, \mathbf{t}_{j-1})$, one concludes that $|r_{j/2} - r_{j/2-1}| > 2^{-(d+5-j)} (\beta / \alpha)^{1/2}$.
Definition of the parameter domain {#definitionparameter}
==================================
Henceforth fix a tower $\{\Omega_j\}_{j=1}^{d+1}$ satisfying the properties stated in Lemma \[towerlem\] with a suitably small choice of $0 < \delta \ll 1$ so as to satisfy all the forthcoming requirements of the proof. It is stressed that the subsequent argument will require $\delta$ to be chosen depending only on the admissible parameters $d$ and $\deg P$; a careful examination of what follows shows such a choice of $\delta$ is always possible.
Observe that the set $\Omega_{d+1}$ is of dimension $\floor{3(d+1)/2}$; one requires a domain of either dimension $d$ or $d+1$ to effectively parametrise either the set $E$ or $F$. Two methods can be applied to remedy this excess of variables. The first is to simply consider the tower defined only to a lower level; that is, only work with $\{\Omega_j\}_{j=1}^N$ for some $N \leq d+1$. The second method is to consider the whole tower $\{\Omega_j\}_{j=1}^{d+1}$ and *freeze* a number of the variables $t_j$. What is essentially meant by this is that for some set of indices $\mathcal{I} \subset \{1, \dots, d+1\}$ and choice of $(s_i)_{i\in \mathcal{I}} \subset {\mathbb{R}}^{\# \mathcal{I}}$ each set $\Omega_j$ is replaced with $$\big\{ (r, t) \in \Omega_j : t_i = s_i \textrm{ for all }i \in \mathcal{I} \cap \{1, \dots, j\} \big\}.$$ In order to optimise the subsequent Jacobian estimates, both methods are combined below and the variables to be frozen are chosen according to the “pre-colour” of their indices. In particular only $t_i$ for $i$ a pre-blue index will be frozen. In light of this discussion define the function $\zeta: \{1, \dots, d+1\} \rightarrow \{1,2\}$ as follows: $$\zeta(j) := \left\{\begin{array}{ll}
2 & \textrm{if $j$ is pre-red} \\
1 & \textrm{otherwise}
\end{array} \right. \qquad \textrm{for $j = 1, \dots, d+1$.}$$ One can think of $\zeta(j)$ as the number of variables contributed by the fibres of the $j$th floor of the tower after the pre-blue variables have been frozen. Note that there exists a least $1 < N \leq d+1$ such that $$\label{defn N}
Z(N) := \sum_{j=1}^{N} \zeta(j) \in \{d, d+1\};$$ in particular, the number of variables contributed by the first $N$ floors corresponds to the dimension of either $E$ or $F$. At this point the proof splits into a number of different cases depending on the parity of $N$ and the value of $Z(N)$.
If $N$ is odd, then $\mathrm{Case}(1, Z(N))$ holds. If $N$ is even, then $\mathrm{Case}(2, Z(N))$ holds.
Ostensibly there are four distinct cases; however, the minimality of $N$ precludes $\mathrm{Case}(1, d+1)$ and so it suffices to consider the remaining three cases.
The preceding observations lead to the definition of a suitable parameter domain $\Omega$ and mapping $\Phi$ which will form the focus of study for the remainder of the paper. By the nature of the definitions of $\Omega$ and $\Phi$, it will be convenient to introduce a relabelling of the relevant indices as “red,” “blue” or “achromatic” to replace the established “pre-red, pre-blue, pre-achromatic” system. All these definitions depend on which $\mathrm{Case}(i, j)$ happens to hold.
Case$\mathbf{(1, d)}$ and Case$\mathbf{(2, d+1)}$: {#casemathbf1-d-and-casemathbf2-d1 .unnumbered}
--------------------------------------------------
The simplest situation corresponds to when either $\mathrm{Case}(1, d)$ or $\mathrm{Case}(2, d+1)$ holds. In both instances one defines $\Omega := \Omega_{N}$ and $\Phi := \Phi_{N}$. In the former case $\Phi : \Omega \rightarrow E$ whilst in the latter $\Phi : \Omega \rightarrow F$ by the properties 2. i) and 3. i) of Lemma \[dendrinosstovall\], respectively. Here essentially no relabelling is required: each pre-red (respectively, pre-blue) index $1 \leq j \leq N$ is designated red (respectively, blue) whilst the odd indices are achromatic (that is, they have no colour assigned to them).
Case$\mathbf{(2, d)}$: {#casemathbf2-d .unnumbered}
----------------------
This situation is slightly more complicated. Fix $t_0 \in \Omega_1$ and consider the family of sets $\{\Omega_j^*\}_{j=1}^{N}$ defined by $$\Omega_j^* := \big\{ (t_1, \dots, t_{j}) : (t_0, t_1, \dots, t_{j}) \in \Omega_{j+1} \big\} \qquad \textrm{for $j = 1, \dots, N$.}$$ It is easy to see $\{\Omega_j^*\}_{j=1}^{N}$ constitutes a type 2 tower. Let $y_0 := x_0 - r_0P(t_0)$ and let $\{\Phi^*_j\}_{j=1}^N$ denote the family of mappings associated to $\{\Omega_j^*\}_{j=1}^{N}$ and the point $y_0$, as defined in Definition \[associated mappings\]. Define $\Omega := \Omega^*_N$ and $\Phi := \Phi^*_{N}$ and observe, by property 2. i) of Lemma \[dendrinosstovall\], that $\Phi : \Omega \rightarrow E$. In this case the colouring system of the indices is redefined. In particular,
i) The index 1 is designated red;
ii) The odd indices $1 < j \leq N$ are designated red (respectively, blue) if $j+1$ was pre-red (respectively, pre-blue) in the previous scheme;
iii) The even indices are designated achromatic.
Freezing variables and families of parametrisations {#freeze}
===================================================
Rather than use a single function to parametrise $E$ or $F$, here the slicing method of Christ [@Christ; @Christ1998] is used to construct a family of maps $G_{\sigma}$. To begin some notation is introduced.
Let $M$ denote the number of non-blue indices in $\{1, \dots, N\}$ and label these indices $l_1 < l_2 < \dots < l_M$. Similarly, let $m$ (respectively, $n$) denote the number of red (respectively, blue) indices so that $M = N - n$. For the reader’s convenience the following table indicates the relationship between these parameters in the various cases.
$$\label{counting variables}
\begin{tabular}{ | c | c | c |}
\hline
& N & d \\ \hline
$\mathrm{Case}(1,d)$ & 2m +2n + 1 & 3m + 2n + 1 \\ \hline
$\mathrm{Case}(2,d+1)$ & 2m +2n & 3m + 2n - 1 \\ \hline
$\mathrm{Case}(2,d)$ & 2m +2n & 3m + 2n \\
\hline
\end{tabular}$$
These computations follow immediately from the definition of $N$ in .
Now let $ 1 \leq \mu_1 < \dots < \mu_m \leq M$ be such that $\{ l_{\mu_i}\}_{i=1}^m$ is precisely the set of red indices. Rather than the blue indices themselves, it will be useful to enumerate those indices which lie directly before a blue index. Irrespective of which case happens to hold, any blue index is at least 2 and so there are precisely $n$ indices lying directly before a blue index. In particular, let $1 \leq \nu_1 < \dots < \nu_{n} \leq M$ be such that $\{l_{\nu_j}+1\}_{j=1}^{n}$ are precisely the blue indices.
Define functions $\tau$ and $\sigma$ on $\Omega$ by $$\begin{aligned}
\tau = (\tau_1, \dots, \tau_M) &:=& ( t_{l_1}, \dots, t_{l_M}) \\
s = (s_{\nu_1}, \dots, s_{\nu_{n}}) &:=& (t_{l_{\nu_j}+1} - t_{l_{\nu_j}})_{j=1}^{n} \\
\sigma = (\sigma_{\nu_1}, \dots, \sigma_{\nu_n}) &:=& (s_{\nu_j} \tau_{\nu_j}^{2K/d(d+1)})_{j=1}^{n}.\end{aligned}$$
Finally let $\rho = (\rho_{\mu_1}, \dots, \rho_{\mu_m}, \rho_{\nu_1}, \dots, \rho_{\nu_{n}})$ where $\rho_{\mu_i}$ (respectively, $\rho_{\nu_j}$) is the dilation variable arising from the fibres of floor $l_{\mu_i}$ (respectively, $l_{\nu_j} +1$) of the tower. More precisely, $\rho_{\mu_j} := r_{\ceil{l_{\mu_i}/2}}$ for $1 \leq i \leq m$ whilst $\rho_{\nu_j} := r_{\ceil{(l_{\nu_j}+1)/2}}$ for $1 \leq j \leq n$.
Observe that the map $$\label{phi change of variables}
\varphi: (r, t) \mapsto (\rho, \tau, \sigma)$$ is a valid change of variables with Jacobian determinant satisfying $$\left|\det \frac{\partial \varphi}{\partial (r, t)}\right| = \prod_{j=1}^{n} \tau_{\nu_j}^{2K/d(d+1)}.$$ For $\sigma \in {\mathbb{R}}^n$ define the parameter set $\omega(\sigma) := \big\{ (\rho, \tau) : \varphi^{-1}(\rho, \tau, \sigma) \in \Omega \big\}$ and let $$\label{setw}
W := \big\{ \sigma \in {\mathbb{R}}^{n} : \omega(\sigma) \neq \emptyset \} \subseteq \big[0, \delta(\alpha\beta)^{1/2}\big]^n$$ where the inclusion follows from properties of the blue indices. Finally, consider the mapping $G_{\sigma}$ on $\omega(\sigma)$ by $$G_{\sigma}(\rho, \tau):= \Phi \circ \varphi^{-1}(\rho, \tau, \sigma).$$ By , it follows that in $\mathrm{Case}(1, d)$ and $\mathrm{Case}(2, d)$ the maps $G_{\sigma}$ are functions of $d$ variables and take values in $E$ whilst in $\mathrm{Case}(2, d+1)$ the $G_{\sigma}$ are functions of $d+1$ variables and take values in $F$. Hence in each case the maps $G_{\sigma}$ have the desirable property that the domain and codomain are of equal dimension.
Furthermore, the polynomial nature of maps $G_{\sigma}$ imply each has bounded multiplicity. In particular, the following well-known multiplicity lemma applies to this situation.
\[multiplicity\] Let $Q: {\mathbb{R}}^d \rightarrow {\mathbb{R}}^d$ be a polynomial mapping; that is, $Q(t) = (Q_j(t))_{j=1}^d$ for all $t \in {\mathbb{R}}^d$ where each $Q_j : {\mathbb{R}}^d \rightarrow {\mathbb{R}}$ is a polynomial in $d$ variables. Suppose the Jacobian determinant $J_Q$ of $Q$ does not vanish everywhere. Then for almost every $x \in {\mathbb{R}}^d$ the set $Q^{-1}(\{x\})$ is finite. Moreover, for almost every $x \in {\mathbb{R}}^d$ the inequality $$\label{mult1}
\# Q^{-1}(\{x\}) \leq \prod_{j=1}^d \deg (Q_j)$$ holds, where $\deg(Q_j)$ denotes the degree of $Q_j$.
The simple proof of this lemma appears in [@Christ1998]; however, it is included at the end of this section for completeness.
As a consequence of the Multiplicity Lemma, if $J_{\sigma}$ denotes the Jacobian of $G_{\sigma}$, then in $\mathrm{Case}(1, d)$ and $\mathrm{Case}(2, d)$ one concludes that the estimate $$\label{Jacobian estimate}
|E| \gtrsim \int_{\omega(\sigma)} |J_{\sigma}(\rho, \tau)| \,{\mathrm{d}}\rho{\mathrm{d}}\tau$$ holds for all $\sigma \in W$. In $\mathrm{Case}(2, d+1)$ there is a similar estimate but with $|E|$ replaced with $|F|$ on the left-hand side of the above expression. Thus, in order to establish either or in the present cases it suffices to prove a suitable estimate for the Jacobian $|J_{\sigma}(\rho, \tau)|$ on the set $\omega(\sigma)$.
This section is concluded with the proof of the Multiplicity Lemma.
Since the zero locus $Z$ of $J_Q$ is a proper algebraic subset of ${\mathbb{R}}^d$ it has measure zero. Furthermore, as $Q$ is a polynomial (and therefore locally Lipschitz) mapping the image $Q(Z)$ of $Z$ under $Q$ has measure zero. It is claimed that holds for all $x \in {\mathbb{R}}^d \setminus Q(Z)$. Indeed, fixing such an $x$ notice that $Q^{-1}(\{x\}) = \bigcap_{j=1}^d V_j^x$ where $\{V_j^x\}_{j=1}^d$ are algebraic sets given by $$V_j^x := \{ t \in {\mathbb{R}}^d : Q_j(t) - x_j = 0 \}.$$ Bezout’s theorem implies that the cardinality of this intersection is either uncountable or at most $\prod_{j=1}^d \deg (Q_j)$.[^8] It therefore suffices to show that $Q^{-1}(\{x\})$ is not uncountable; this is achieved by proving each point of the set is isolated. By the choice of $x$, whenever $t_0 \in Q^{-1}(\{x\})$ the vectors $\{ \nabla Q_j(t_0)\}_{j=1}^d$ span ${\mathbb{R}}^d$. Thus the $V_j^x$ are smooth hypersurfaces in a neighbourhood of $t_0$ which, of course, intersect at $t_0$ and are transversal at this point of intersection. It follows that $t_0$ must be an isolated point of $Q^{-1}(\{x\})$, as required.
Reduction to Jacobian estimates
===============================
Recall, in order to prove the main theorem it suffices to obtain a lower bound on either $|E|$ or $|F|$ in terms of $\alpha$ and $\beta$, as discussed in and . In the previous sections, it was shown that when either $\mathrm{Case}(1, d)$ or $\mathrm{Case}(2, d)$ holds a family of useful mappings $G_{\sigma} \colon \omega(\sigma) \to E$ can be constructed. These mappings effectively parametrise the set $E$ and, in particular, one has the estimate . A similar construction is available in $\mathrm{Case}(2, d+1)$ (this time parametrising the set $F$) and so it remains to find effective bounds for integrals such as that appearing in the right-hand side of . This will be achieved by estimating pointwise the Jacobian determinant $J_{\sigma}$ of the map $G_{\sigma}$. In order to state the main result in this direction, it is convenient to introduce the notation $$\eta := \left\{ \begin{array}{ll}
0 & \textrm{if either $\mathrm{Case}(1, d)$ or $\mathrm{Case}(2, d+1)$ holds} \\
1 & \textrm{if $\mathrm{Case}(2, d)$ holds}
\end{array}\right. .$$
\[jaclem1\] Let $\sigma \in W$, where $W$ is as defined in . Then $$|J_{\sigma}(\rho, \tau)| \gtrsim \alpha^{d(d+1)/2 - M}(\beta / \alpha )^{(m+n - \eta)/2} \prod_{l=1}^M \tau_l^{2K/d(d+1)}$$ for all $(\rho, \tau) \in \omega(\sigma)$.
Theorem \[bigthm\] is a direct consequence of Lemma \[jaclem1\].
To prove Theorem \[bigthm\] it suffices to show the estimate holds in both $\mathrm{Case}(1, d)$ and $\mathrm{Case}(2,d)$ and holds in $\mathrm{Case}(2, d+1)$. Indeed, recall both and are equivalent to the desired endpoint restricted weak-type $(p_1, q_1)$ inequality for $A$. For notational convenience, let $$|X| := \left\{ \begin{array}{ll}
|E| & \textrm{if either $\mathrm{Case}(1, d)$ or $\mathrm{Case}(2, d)$ holds} \\
|F| & \textrm{if $\mathrm{Case}(2, d+1)$ holds}
\end{array}\right. .$$
Apply Lemma \[jaclem1\] to each $\sigma \in W$ together with the Multiplicity Lemma to deduce in all cases $$\begin{aligned}
|X| &\gtrsim& \int_{\omega(\sigma)} |J_{\sigma}(\rho, \tau)| \,{\mathrm{d}}\rho{\mathrm{d}}\tau \\
&\gtrsim& \alpha^{d(d+1)/2 - M}(\beta / \alpha )^{(m+n - \eta)/2}\int_{\omega(\sigma)} \prod_{k=1}^M \tau_k^{2K/d(d+1)} {\mathrm{d}}\rho{\mathrm{d}}\tau .\end{aligned}$$ Integrating both sides of the preceding inequality over $W$, it follows that $$\label{wkpf2}
(\alpha\beta)^{n/2} |X| \gtrsim \alpha^{d(d+1)/2 - M}(\beta / \alpha )^{(m+n - \eta)/2} \int_{\varphi(\Omega)} \prod_{k=1}^M \tau_k^{2K/d(d+1)} \,{\mathrm{d}}\rho {\mathrm{d}}\tau {\mathrm{d}}\sigma$$ where $\varphi$ is the map defined in . By a change of variables, the integral on the right-hand side of can be written as $$\begin{aligned}
\int_{\varphi(\Omega)} \prod_{k=1}^M \tau_k^{2K/d(d+1)} \,{\mathrm{d}}\rho {\mathrm{d}}\tau {\mathrm{d}}\sigma &=& \int_{\Omega} \prod_{k=1}^M t_{l_k}^{2K/d(d+1)}\left| \det \frac{\partial \varphi}{\partial (r, t)}(r,t) \right| \,{\mathrm{d}}r {\mathrm{d}}t \\
&=& \int_{\Omega} \prod_{k=1}^M t_{l_k}^{2K/d(d+1)}\prod_{j=1}^{n} t_{l_{\nu_j}}^{2K/d(d+1)}\,{\mathrm{d}}r {\mathrm{d}}t.\end{aligned}$$ Arguing as in the last step of the proof of Lemma \[towerlem\], one may deduce $t_{l_{\nu_j}} \sim t_{l_{\nu_j}+1}$ for $1 \leq j \leq n$ provided that the parameter $\delta$ from Lemma \[towerlem\] is chosen to be sufficiently small (depending only on the degree of $P$ and $d$). The previous expression is therefore bounded below by a constant multiple of $$\int_{\Omega} \prod_{k=1}^N t_{k}^{2K/d(d+1)}\,{\mathrm{d}}r {\mathrm{d}}t.$$ Applying Fubini’s theorem and the estimates for the $\mu_P$-measure of the fibres of the $\Omega_j$, one may easily deduce the above integral is at least a constant multiple of $\alpha^N(\beta/\alpha)^{\floor{N/2}}$. Whence, combining these observations and multiplying both sides of by $\alpha^{-n}(\beta/\alpha)^{-n/2}$, one arrives at the estimate $$|X| \gtrsim \alpha^{d(d+1)/2 - M + N-n}(\beta / \alpha )^{(m-\eta)/2 + \floor{N/2}}.$$ Recalling $M=N-n$ and , this is easily seen to be the desired estimate.
To complete the proof of Theorem \[weakthm\] it remains to prove Lemma \[jaclem1\].
The proof of the Jacobian estimates: $\mathrm{Case}(1,d)$
=========================================================
In the previous section the proof of the main theorem was reduced to establishing the pointwise estimates for the Jacobian function described in Lemma \[jaclem1\]. Here the proof of Lemma \[jaclem1\] in $\mathrm{Case}(1,d)$ is discussed in detail. The same arguments can be adapted to treat the remaining cases, as demonstrated in the following section.
The arguments here, which are based primarily on those of [@Christ1998; @Stovall2010], are somewhat lengthy; it is convenient, therefore, to present the proof as a series of steps.
Compute the Jacobian matrix. {#compute-the-jacobian-matrix. .unnumbered}
----------------------------
Recalling the definition of the mapping $\Phi := \Phi_N$, one may use the established index notation to express $\Phi \circ \varphi^{-1}$ as $$\begin{aligned}
\Phi\circ \varphi^{-1}(\rho, \tau, \sigma) &=& x_0 -r_0P(\tau_1) + \sum_{i=1}^m \rho_{\mu_i}\big( P(\tau_{\mu_i}) - P(\tau_{\mu_i +1}) \big) \\
&& + \sum_{j=1}^n \rho_{\nu_j}\big( P(\tau_{\nu_j}+ s_{\nu_j}(\tau,\sigma)) - P(\tau_{\nu_j +1}) \big).\end{aligned}$$ One immediately deduces that $$\frac{\partial G_{\sigma}}{\partial \rho_{\mu_i}} (\rho, \tau) = P(\tau_{\mu_i}) - P(\tau_{\mu_i +1}) \qquad \textrm{for $i = 1, \dots, m$}$$ whilst $$\frac{\partial G_{\sigma}}{\partial \rho_{\nu_j}} (\rho, \tau) = P(\tau_{\nu_j} + s_{\nu_j}(\tau,\sigma)) - P(\tau_{\nu_j +1}) \qquad \textrm{for $j = 1, \dots, n$}$$ which identifies $m+n$ of the columns of the Jacobian matrix. The remaining columns correspond to differentiation with respect to the $\tau$ variables and are readily computed by expressing $\Phi\circ \varphi^{-1}$ as $$\begin{aligned}
\Phi\circ \varphi^{-1}(\rho, \tau, \sigma) &=& x_0 + \sum_{i=1}^m \big( \rho_{\mu_i}P(\tau_{\mu_i}) - \rho^*_{\mu_i-1}P(\tau_{\mu_i - 1})\big) \\
&&+ \sum_{j=1}^n \big( \rho_{\nu_j}P(\tau_{\nu_j} + s_{\nu_j}(\tau,\sigma)) - \rho^*_{\nu_j-1}P(\tau_{\nu_j})\big) - \rho^*_MP(\tau_M)\end{aligned}$$ where the $\rho^*_{\mu_i-1}$, $\rho^*_{\nu_j-1}$ and $\rho^*_M$ are defined in the obvious manner; for instance, if $\rho_{\nu_j}$ corresponds to the parameter $r_k$ via the change of variables $\varphi$, then $\rho^*_{\nu_j-1}$ is understood to correspond to the parameter $r_{k-1}$. Thus for $i=1, \dots, m$ one has $$\frac{\partial G_{\sigma}}{\partial \tau_{\mu_i}} (\rho, \tau) = \rho_{\mu_i}P'(\tau_{\mu_i}) \quad \textrm{and} \quad \frac{\partial G_{\sigma}}{\partial \tau_{\mu_i - 1}} (\rho, \tau) = - \rho^*_{\mu_i-1}P'(\tau_{\mu_i - 1})$$ whilst $$\frac{\partial G_{\sigma}}{\partial \tau_{M}} (\rho, \tau) = - \rho^*_MP'(\tau_M),$$ accounting for a further $2m+1$ columns to the Jacobian matrix. To compute the remaining $n$ columns differentiate $G_{\sigma}$ with respect to the $\tau_{\nu_j}$ to give $$\begin{aligned}
\label{taunucolumn}
\frac{\partial G_{\sigma}}{\partial \tau_{\nu_j}} (\rho, \tau) &=& \rho_{\nu_j}P'(\tau_{\nu_j}+ s_{\nu_j}(\tau, \sigma)) - \rho^*_{\nu_j-1}P'(\tau_{\nu_j}) \\
\nonumber
&& - \frac{2K}{d(d+1)} \frac{s_{\nu_j}(\tau, \sigma)}{\tau_{\nu_j}}\rho_{\nu_j}P'(\tau_{\nu_j}+ s_{\nu_j}(\tau, \sigma))\end{aligned}$$ for $j=1, \dots, n$.
Compare $J_{\sigma}$ with $J_P$. {#compare-j_sigma-with-j_p. .unnumbered}
--------------------------------
The estimation of the Jacobian $J_{\sigma}$ will be achieved by comparing it to the more tractable expression $J_P$, introduced in . Once such a comparison is established, $J_{\sigma}$ can then be bounded by means of the geometric inequality of Dendrinos and Wright (that is, Theorem \[Dendrinos Wright theorem\]). This inequality is guaranteed to hold in the appropriate setting due to the reductions made earlier in the article.
To begin, express the Jacobian determinant in the form of an integral $$J_{\sigma}(\rho, \tau)= \pm\int_{R(\tau) \times B_{\sigma}(\tau)} \wp_{\sigma}(\rho, \tau, x) \,{\mathrm{d}}x$$ where $\wp_{\sigma}(\rho, \tau, x)$ is a multi-variate polynomial and $$R(\tau) := \prod_{i=1}^{m} (\tau_{\mu_i}, \tau_{\mu_i+1}) \quad \textrm{and} \quad
B_{\sigma}(\tau) := \prod_{i=1}^{n} (\tau_{\nu_j} + s_{\nu_j}, \tau_{\nu_j+1})$$ are rectangles.[^9] The polynomial $\wp_{\sigma}(\rho, \tau, x)$ is the product of $C(\rho) = \rho_M^* \prod_{i=1}^m \rho_{\mu_i - 1}^* \rho_{\mu_i}$ and the determinant of the matrix $A_{\sigma}(\rho, \tau, x)$ obtained from original Jacobian matrix by making the following changes:
- The column $P(\tau_{\mu_i}) - P(\tau_{\mu_i +1})$ is replaced with $P'(x_i)$ for $i=1, \dots, m$.
- The column $P(\tau_{\nu_j}+ s_{\nu_j}) - P(\tau_{\nu_j +1})$ is replaced with $P'(x_{m+j})$ for $j = 1, \dots, n$.
- The columns $\rho_{\mu_i}P'(\tau_{\mu_i})$ and $- \rho^*_{\mu_i-1}P'(\tau_{\mu_i - 1})$ are replaced with $P'(\tau_{\mu_i})$ and $P'(\tau_{\mu_i - 1})$, respectively, for all $i = 1, \dots, m$. In addition, $- \rho^*_MP'(\tau_M)$ is replaced with and $P'(\tau_M)$.
- The remaining columns $n$ are unaltered; in other words, they agree with the corresponding columns of the Jacobian matrix.
Notice the unaltered columns are those corresponding to differentiation by $\tau_{\nu_j}$ and are of the form given in . Each may be expressed as the sum of three terms $$\label{taylor1}
\frac{\partial G_{\sigma}}{\partial \tau_{\nu_j}} (\rho, \tau) = \sum_{i=1}^3 T^i_{\sigma, j}(\rho,\tau)$$ where, writing $c := - 2K/d(d+1)$, $$\begin{aligned}
T^1_{\sigma, j}(\rho,\tau) &:=& (\rho_{\nu_j} - \rho^*_{\nu_j-1})P'(\tau_{\nu_j}), \\
T^2_{\sigma, j}(\rho,\tau) &:=& c \frac{s_{\nu_j}(\tau, \sigma)}{\tau_{\nu_j}}\rho_{\nu_j}P'(\tau_{\nu_j}+ s_{\nu_j}(\tau, \sigma)), \\
T^3_{\sigma, j}(\rho,\tau) &:=& \rho_{\nu_j}(P'(\tau_{\nu_j}+ s_{\nu_j}(\tau, \sigma)) - P'(\tau_{\nu_j})). \end{aligned}$$ The multi-linearity of the determinant and are now applied to express $\det A_{\sigma}(\rho, \tau, x)$ as a sum of determinants of more elementary matrices. In order to present concisely the resulting expression it is useful to introduce some notation. In particular, for $S \subseteq \mathcal{N} := \{\nu_1, \dots, \nu_n\}$, let $\Delta_S$ denote the function of $\rho$ given by $$\Delta_S(\rho) := \prod_{\nu \in S} (\rho_{\nu} - \rho^*_{\nu - 1})$$ and $R_{\sigma, S}(\tau) \subset {\mathbb{R}}^{\#S}$ the rectangle $$R_{\sigma, S}(\tau) := \prod_{\nu \in S} (\tau_{\nu}, \tau_{\nu}+ s_{\nu}).$$
With this notation $\det A_{\sigma}(\rho, \tau, x)$ equals $$\label{integrand1}
\sum_{\mathcal{S}} \Delta_{S_1}(\rho) \bigg(\prod_{\nu \in S_2}\frac{c \rho_{\nu}s_{\nu}}{\tau_{\nu}}\bigg)\bigg( \prod_{\nu \in S_3} \rho_{\nu}\bigg) \int_{R_{\sigma, S_3}(\tau)} \bigg(\prod_{\nu \in S_3}\frac{\partial}{\partial y_{\nu}}\bigg) J_P(\xi_{\mathcal{S}}(y), x)\,{\mathrm{d}}y,$$ at least up to a sign, where the sum ranges over all partitions $\mathcal{S}:=(S_1, S_2, S_3)$ of $\mathcal{N}$ and for any such partition $\xi_{\mathcal{S}}(y) = (\xi_{\mathcal{S},l}(\tau,\sigma, y))_{l=1}^M$ is defined by $$\xi_{\mathcal{S},l}(\tau, \sigma,y) := \left\{\begin{array}{ll}
\tau_{l} + s_{l} & \textrm{if $l \in S_2$,}\\
y_{l} & \textrm{if $l \in S_3$,} \\
\tau_{l} & \textrm{otherwise.}
\end{array} \right.$$ If $S_3 = \emptyset$, then the integral appearing in is interpreted as $J_P(\xi_{\mathcal{S}}, x)$.
The term of the sum in corresponding to the unique partition for which $S_1 = \mathcal{N}$ is simply $\Delta(\rho)J_P(\tau, x)$ where $\Delta(\rho) := \Delta_{\mathcal{N}}(\rho)$; the sum of the remaining terms is denoted by $E_{\sigma}(\rho, \tau, x)$. Thus, equals $$\label{error}
\Delta(\rho)J_P(\tau, x) + E_{\sigma}(\rho, \tau, x).$$ In conclusion, the Jacobian $J_{\sigma}$ can be expressed in terms of (an integral of) the function $J_P$ together with some error term.
Control the error. {#control-the-error. .unnumbered}
------------------
It will be shown that provided that $\delta$ is chosen sufficiently small, depending only on $d$ and $\deg P$, the right-hand summand of is subordinate to the left-hand summand. Only a bounded number of terms of are non-zero and the error is therefore a sum of $O(1)$ terms which will be estimated individually.
By the properties of the parameter tower, $s_{\nu} \lesssim \delta (\beta/\alpha)\tau_{\nu}$ for all $\nu \in \mathcal{N}$. Hence, for any $S_2 \subseteq \mathcal{N}$ one has $$\label{error estimate 1}
\prod_{\nu \in S_2}\frac{c \rho_{\nu}s_{\nu}}{\tau_{\nu}} \lesssim \delta^{\#S_2}(\beta/\alpha)^{\#S_2} \lesssim \delta^{\#S_2} \Delta_{S_2}(\rho)$$ where the final inequality is due to the definition of the blue indices. A suitable error bound would follow from a similar estimate for each of the integrals appearing in . In particular, fixing some partition $\mathcal{S} = (S_1, S_2, S_3)$ of $\mathcal{N}$, it suffices to prove $$\label{error estimate 2}
\int_{R_{\sigma, S_3}(\tau)} \left|\left(\prod_{\nu \in S_3}\frac{\partial}{\partial y_{\nu}}\right) J_P(\xi_{\mathcal{S}}(y), x)\right|\,{\mathrm{d}}y \lesssim \delta^{\#S_3} \Delta_{S_3}(\rho)|J_P(\tau, x)|.$$ Indeed, once is established, the error bound $$\begin{aligned}
\label{errorest1}
|E_{\sigma}(\rho, \tau, x)| &\lesssim& \bigg(\sum_{\substack{\mathcal{S} = (S_1, S_2, S_3) \\ S_1 \neq \mathcal{N}}} \delta^{\#S_2 + \#S_3} \prod_{j=1}^3\Delta_{S_j}(\rho)\bigg) |J_P(\tau, x)| \\
\nonumber
&\lesssim& \delta \Delta(\rho)|J_P(\tau, x)|\end{aligned}$$ immediately follows, noting the factor $\prod_{\nu \in S_3} \rho_{\nu}$ from is $O(1)$ whenever it appears in a non-zero term of the sum.
If $S_3 = \emptyset$, then is trivial. Fix a partition $\mathcal{S}$ as above with $S_3$ non-empty and some $y \in R_{\sigma, S_3}(\tau)$ and consider the ratio $$\label{ratio1}
\left| \frac{\left(\prod_{\nu \in S_3}\frac{\partial}{\partial y_{\nu}}\right)J_P(\xi_{\mathcal{S}}(y), x)}{J_P(\xi_{\mathcal{S}}(y), x)} \right|.$$ For notational convenience write $\xi = (\xi_1, \dots, \xi_n) := \xi_{\mathcal{S}}(y)$. Using the derivative estimate from Proposition \[Stovall’s observation\] one may bound by a linear combination of $O(1)$ terms (with $O(1)$ coefficients) of the form $$\label{complicated thing}
\bigg(\prod_{{\nu} \in T_1} y_{\nu}^{-1}\bigg)\bigg( \prod_{{\nu} \in T_2} y_{\nu}^{-\epsilon(\nu)}|y_{\nu} - \xi_{u({\nu})}|^{\epsilon(\nu) - 1}\bigg)\bigg( \prod_{{\nu} \in T_3}y_{\nu}^{-\epsilon(\nu)}|y_{\nu} - x_{v({\nu})}|^{\epsilon(\nu) - 1}\bigg)$$ where:
- $(T_1, T_2, T_3)$ is a partition of $S_3$;
- $u \colon T_2 \to \{1, \dots, M\}$ is a function with the property $u(j) \neq j$ for all $j \in T_2$;
- $v \colon T_3 \to \{1, \dots, n+m\}$ (with no additional conditions) and
- $\epsilon \colon T_2 \cup T_3 \to \{0,1\}$.
To prove it therefore suffices to establish a suitable bound for the integral of the product of and $|J_P(\xi, x)|$ over the set $R_{\sigma, S_3}(\tau)$. The first step is to estimate by applying the following observations.
i) Given $y \in R_{\sigma, S_3}(\tau)$, by the definition of the parameter tower the estimates $$\begin{aligned}
y_{\nu} &\geq& \tau_{\nu} = \tau_{\nu}^{1/\kappa} \tau_{\nu}^{-2K/d(d+1)} \gtrsim \alpha\tau_{\nu}^{-2K/d(d+1)};\\
|y_{\nu} - \xi_{u({\nu})}| &\geq& |\tau_{\nu} - \tau_{u({\nu})}| - \delta \alpha (\tau_{\nu}^{-2K/d(d+1)} + \tau_{u({\nu})}^{-2K/d(d+1)}); \\
|y_{\nu} - x_{v({\nu})}| &\geq& |\tau_{\nu} - x_{v({\nu})}| - \delta \alpha \tau_{\nu}^{-2K/d(d+1)}\end{aligned}$$ hold for ${\nu} \in \mathcal{N}$.
ii) Since the indices $l_{\nu}$ for $\nu \in \mathcal{N}$ are those that directly precede a blue index (and so $l_{\nu}$ is odd), Corollary \[separation\] ensures $\tau_{\nu} - \tau_u \gtrsim \alpha \tau_u^{-2K/d(d+1)}$ for all $1 \leq u < \nu$. Moreover, the ordering of the variables then guarantees $$\tau_{\nu} - \tau_u \gtrsim \alpha \tau_{\nu}^{-2K/d(d+1)} \qquad \textrm{whenever $1 \leq u < \nu$}.$$
iii) On the other hand, since the labelling $l_k$ omits the blue indices, for any $\nu \in \mathcal{N}$ and $\nu < u \leq M$ one must have $l - l_{\nu} \geq 2$ where $l$ is the index such that $\tau_u = t_l$. Consequently, by applying Corollary \[separation\] in this case one concludes that $$\tau_u-\tau_{\nu} \gtrsim \alpha \tau_{\nu}^{-2K/d(d+1)} \qquad \textrm{whenever $\nu < u \leq M$}.$$
Combining these observations one immediately deduces that $$|y_{\nu} - \xi_{u(\nu)}| \gtrsim\alpha \tau_{\nu}^{-2K/d(d+1)}$$ for all $\nu \in \mathcal{N}$, provided that $\delta$ is chosen initially to be sufficiently small in the earlier application of Lemma \[towerlem\].
It would be useful to have a similar bound for the terms $|y_{\nu} - x_l|$. At present such an estimate is not possible due to the potential lack of separation between the $\tau_{\nu}$ and $x_l$ variables. To remedy this, temporarily assume the addition separation hypothesis $$\label{sephypo1}
|\tau_{\nu} - x_l| \gtrsim \alpha\tau_{\nu}^{-2K/d(d+1)}$$ for all $\nu \in \mathcal{N}$ and all $1 \leq l \leq m+n$. Presently it is shown that this separation hypothesis leads to desirable control over the error term $E_{\sigma}(\rho, \tau, x)$; the following step is then to modify the existing set-up so that indeed holds without the need of additional assumptions.
The preceding discussion, together with the identity $|R_{\sigma, S}(\tau)| = \prod_{\nu \in S} s_{\nu}$, implies is controlled by $$\begin{aligned}
\alpha^{-\# S_3} \prod_{\nu \in S_3} \tau_{\nu}^{2K/d(d+1)} &\lesssim& \delta^{\#S_3} (\beta / \alpha)^{\#S_3/2} |R_{\sigma, S_3}(\tau)|^{-1}\\
&\lesssim& \delta^{\#S_3} |\Delta_{S_3}(\rho)||R_{\sigma, S_3}(\tau)|^{-1}\end{aligned}$$ provided that $\delta$ is chosen to be sufficiently small. Observe, both of the above inequalities are simple consequences of the definition of the blue indices. Consequently, the left-hand side of may be bounded by $$\delta^{\#S_3} \Delta_{S_3}(\rho)\frac{1}{|R_{\sigma, S_3}(\tau)|}\int_{R_{\sigma, S_3}(\tau)}|J_P(\xi_{\mathcal{S}}, x)|\,{\mathrm{d}}y$$ and so , and thence , would follow if $$|J_P(\xi(y), x)| \sim |J_P(\tau, x)| \qquad \textrm{ for all $y \in R_{\sigma, S_3}(\tau)$.}$$ This approximation is readily deduced by combining Proposition \[Stovall’s observation\] with Grönwall’s inequality (for a proof of Grönwall’s inequality see, for instance, [@Tao2006 Chapter 1]).
Hence, the estimate is established under the assumption of the separation hypothesis .
Enforce separation. {#enforce-separation. .unnumbered}
-------------------
In the previous section it was shown if were to hold for each $\nu \in \mathcal{N}$ uniformly over all $x = (x_1, \dots, x_{m+n}) \in R_{\sigma, S_3}(\tau)$, then by choosing $0< \delta \ll 1$ sufficiently small one may control the integrand by the easily-understood function $|\Delta(\rho)| |J_P(\tau, x)|$. Clearly for fixed $i$ the estimate cannot hold for at least one value of $l$, since as $x$ varies over $R(\tau)\times B_{\sigma}(\tau)$ some $x_l$ can stray close to $\tau_{\nu_i}$ in the boundary regions. To remedy this problem one simply removes a suitable small portion of $R(\tau) \times B_{\sigma}(\tau)$ from the boundary, observing that this can be done without greatly diminishing the size of the integral to be estimated. Given $0< \epsilon < 1/2$, $ 1 \leq i \leq m$ and $1 \leq j \leq n$, define the $\epsilon$-truncate of $R_i(\tau) := (\tau_{\mu_i}, \tau_{\mu_i+1})$ and $B_{\sigma, j}(\tau) := (\tau_{\nu_j} + s_{\nu_j}, \tau_{\nu_j+1})$ by $$R^{\epsilon}_i(\tau) := (\tau_{\mu_i} +\epsilon |R_i(\tau)|, \tau_{\mu_i+1}- \epsilon |R_i(\tau)|)$$ and $$B^{\epsilon}_{\sigma, j}(\tau) := (\tau_{\nu_j} + s_{\nu_j} +\epsilon |B_{\sigma, j}(\tau)|, \tau_{\nu_j+1}- \epsilon |B_{\sigma, j}(\tau)|),$$ respectively. Moreover, define the $\epsilon$-truncates of the associated rectangles to be $R^{\epsilon}(\tau) := \prod_{i=1}^m R^{\epsilon}_i(\tau)$ and $B^{\epsilon}_{\sigma}(\tau) := \prod_{j=1}^n B^{\epsilon}_{\sigma, j}(\tau)$. Lemma \[polylem\] below establishes the existence of some constant $0< c_0 < 1/2$, depending only on $d$ and $\deg P$, such that $$\label{jacoint1}
|J_{\sigma}(\rho, \tau)| \geq \bigg|\int_{D(\tau)} \wp_{\sigma}(\rho, \tau, x) \,{\mathrm{d}}x\bigg| - \frac{1}{2} \int_{D(\tau)} |\wp_{\sigma}(\rho, \tau, x)| \,{\mathrm{d}}x.$$ where $D(\tau) := R^{c_0}(\tau) \times B^{c_0}_{\sigma}(\tau)$.
It is easy to show that for all $x \in D(\tau)$ the condition holds with a uniform constant. Observe $$|B_{\sigma, j}(\tau)| = \tau_{\nu_{j+1}} - (\tau_{\nu_j} + s_{\nu_j}) = t_{l_{(\nu_j +1)}} - t_{(l_{\nu_j} +1)},$$ where the brackets in the subscript are included to aid the clarity of exposition. Since $l_{\nu_j} +1$ is, by definition, a blue index it follows that $l_{(\nu_j +1)}$ is odd and, consequently, $$|B_{\sigma, j}(\tau)| \gtrsim \alpha t_{(l_{\nu_j} +1)}^{-2K/d(d+1)} = \alpha (\tau_{\nu_j} + s_{\nu_j})^{-2K/d(d+1)}$$ by Corollary \[separation\] part i). Futhermore, recalling $s_{\nu_j} \lesssim \delta (\beta/\alpha)\tau_{\nu_j} \lesssim \tau_{\nu_j}$, it follows that $$\label{B length}
|B_{\sigma, j}(\tau)| \gtrsim \alpha \tau_{\nu_j}^{-2K/d(d+1)}.$$ Now suppose $x_l \in B^{c_0}_{\sigma, j_0}(\tau)$ for some fixed $j_0 \in \{1, \dots, n\}$. It is clear from the definition of the parameter domain that if $j \neq j_0$, then holds for $\nu = \nu_j$. Similarly, if $x_l \in R^{c_0}_{i_0}(\tau)$ for some fixed $i_0 \in \{1, \dots, m\}$, then holds for all $\nu \in \mathcal{N}$. It remains to verify when $x_l \in B^{c_0}_{\sigma, j_0}(\tau)$ and $j = j_0$, but this is immediate from the definition of the truncation and the bound .
Consequently, for $x \in D(\tau)$ and $\delta$ sufficiently small holds and thus the estimate $$\label{polynomial estimate}
|\wp_{\sigma}(\rho, \tau, x)| \gtrsim |\Delta(\rho)||J_P(\tau, x)|$$ is valid on $D(\tau)$. Furthermore, it is claimed that as $x$ varies over $D(\tau)$ the sign of $\wp_{\sigma}(\rho, \tau, x)$ is unchanged. Once this observation is established the right-hand side of can be written as $$\frac{1}{2} \int_{D(\tau)} |\wp_{\sigma}(\rho, \tau, x)| \,{\mathrm{d}}x \gtrsim |\Delta(\rho) |\int_{D(\tau)}| J_P(\tau, x)|\,{\mathrm{d}}x$$ To prove the claim, note that the ordering of the components of the $(r,t) \in \Omega$ implies the sign of $V(\tau, x)$ is fixed as $x$ varies over $D(\tau)$; the geometric inequality guaranteed by Theorem \[Dendrinos Wright theorem\] therefore ensures that the sign of $J_P(\tau, x)$ is also fixed (and is non-zero). The estimate now implies the claim.
Bound $J_P$ and apply the properties of $\Omega$. {#bound-j_p-and-apply-the-properties-of-omega. .unnumbered}
-------------------------------------------------
Combining the estimate guaranteed by Theorem \[Dendrinos Wright theorem\] and the preceding observations one deduces that $$|J_{\sigma}(\rho, \tau)| \gtrsim |\Delta(\rho) | \prod_{l=1}^{M}|L_P(\tau_l)|^{1/d} \int_{D(\tau)}\prod_{k=1}^{m+n}|L_P(x_k)|^{1/d} | V(\tau, x)| \,{\mathrm{d}}x.$$ Over the domain of integration the estimate $$| V(\tau, x)| \gtrsim \alpha^{d(d-1)/2-M(M-1)/2} |V(\tau)| \prod_{l=1}^M \tau_l^{-K(d-M)/d(d+1)} \prod_{k=1}^{m+n} x_k^{-K(d-1)/d(d+1)}$$ is valid owing to both and the additional separation enforced by truncating the set $R(\tau)$. Furthermore, the construction of the (type 1) parameter tower ensures $$\label{Vandermonde estimate}
| V(\tau)| \gtrsim \alpha^{M(M-1)/2} (\beta/\alpha)^{m/2} \prod_{l=1}^M \tau_l^{-K(M-1)/d(d+1)}.$$ Since the properties of the blue intervals imply $|\Delta(\rho) |\gtrsim (\beta/\alpha)^{n/2}$, one may combine the preceding inequalities to deduce $$\label{Jacobian bound}
|J_{\sigma}(\rho, \tau)| \gtrsim \alpha^{d(d-1)/2}(\beta/\alpha)^{(m+n)/2} \bigg(\int_{D(\tau)}\prod_{k=1}^{d-M}x_k^{2K/d(d+1)} \,{\mathrm{d}}x \bigg)\prod_{l=1}^{M}\tau_l^{2K/d(d+1)} .$$ Here the approximation $L_P(t) \sim t^{K}$ has been applied, which was a consequence of the decomposition theorem.
Finally, the integral on the right-hand side of the above expression is easily seen to satisfy $$\int_{D(\tau)}\prod_{k=1}^{d-M}x_k^{2K/d(d+1)} \,{\mathrm{d}}x \gtrsim \alpha^{d - M},$$ concluding the proof.
It remains to state and prove the lemma which justifies the estimate . In general, for $0 < \epsilon < 1/2$ the $\epsilon$-truncation $I^{\epsilon}$ of a finite open interval $I = (a, b)$ is defined as $I^{\epsilon} := (a +\epsilon(b-a), b - \epsilon(b-a))$. If $I_1, \dots, I_K$ is a family of finite open intervals, the $\epsilon$-truncation $R^{\epsilon}$ of the associated rectangle $R := \prod_{j=1}^K I_j$ is defined simply by $R^{\epsilon} := \prod_{j=1}^K I_j^{\epsilon}$.
\[polylem\] Given any $M, K \in {\mathbb{N}}$ there exists a constant $0 < c_{M,K} < 1/2$ with the following property. For all $0< \epsilon < c_{M,K}$ there exists $C_{M,K}(\epsilon) >0$ such that for any collection $I_1, \dots, I_K$ of finite open intervals with associated rectangle $R$ one has $$\int_{R \setminus R^{\epsilon}} |p(x)| \,{\mathrm{d}}x \leq C_{M,K}(\epsilon) \int_{R^{\epsilon}} |p(x)| \,{\mathrm{d}}x$$ whenever $p$ is a polynomial of degree at most $M$ in $x =(x_1, \dots, x_K)$. Moreover, $\lim_{\epsilon \rightarrow 0} C_{M,K}(\epsilon) = 0$ for any fixed $M, K$.
Once the lemma is established, taking $n,m $ and $M$ to be as defined in the previous proof and $K := m + n$, the inequality (at least in $\mathrm{Case}(1,d)$) follows by choosing $c_0$ sufficiently small so that $0 < C_{M,K}(c_0) < 1/2$.
By homogeneity it suffices to consider the case $I_1 = \dots = I_K = (0,1)$ and a simple inductive procedure further reduces the problem to the case $K = 1$. Fixing $M$ and letting $I = (0,1)$, the proof is now a simple consequence of the equivalence of norms on finite-dimensional spaces: if $C_M < \infty$ is defined to be the supremum of the ratio $\|p\|_{L^{\infty}(I)} / \|p\|_{L^1(I)}$ over all polynomials of degree at most $M$, then $$\int_{I \setminus I^{\epsilon}} |p(x)| \,{\mathrm{d}}x \leq 2\epsilon C_M \bigg(\int_{I \setminus I^{\epsilon}} |p(x)| \,{\mathrm{d}}x + \int_{I^{\epsilon}} |p(x)| \,{\mathrm{d}}x \bigg).$$ Provided that $0 < \epsilon < C_M/2$ one may take $C_{M,1}(\epsilon) := 2\epsilon C_M / (1 - 2\epsilon C_M)$, completing the proof.
The proof of the Jacobian estimates: $\mathrm{Case}(2,d+1)$ and $\mathrm{Case}(2, d)$
=====================================================================================
The argument used to prove Lemma \[jaclem1\] in $\mathrm{Case}(1,d)$ can easily be adapted to establish the result in the remaining cases. The necessary modifications are sketched below; the precise details are left to the patient reader.
Adapting the arguments to $\mathrm{Case}(2,d+1)$. {#adapting-the-arguments-to-mathrmcase2d1. .unnumbered}
-------------------------------------------------
To prove the inequality in $\mathrm{Case}(2,d+1)$ only a minor modification of the preceding argument is needed. Notice by the minimality of the parameter $N$ defined in it follows that the index $N$ is red and so $\mu_m = M$. Here $\Phi \circ \varphi^{-1}$ maps into ${\mathbb{R}}^d \times [1,2]$ and is given by $$\Phi \circ \varphi^{-1}(\rho, \tau, \sigma) = \left( \begin{array}{c}
\Psi_N(x_0, r_0; \varphi^{-1}(\rho, \tau, \sigma))\\
\rho_{\mu_m} \end{array}\right)$$ where $$\begin{aligned}
\Psi_N(x_0, r_0; \varphi^{-1}(\rho, \tau, \sigma)) &=& x_0 -r_0P(\tau_1) + \sum_{i=1}^{m-1} \rho_{\mu_i}\big( P(\tau_{\mu_i}) - P(\tau_{\mu_i +1}) \big) \\
&&+ \sum_{j=1}^n \rho_{\nu_j}\big( P(\tau_{\nu_j}+ s_{\nu_j}) - P(\tau_{\nu_j +1}) \big) + \rho_{\mu_m}P(\tau_{\mu_m}).\end{aligned}$$
The Jacobian matrix is now a $(d+1)\times (d+1)$ matrix. The columns given by differentiating $G_{\sigma}$ with respect to $\rho_{\mu_i}$ are $$\left(\begin{array}{c}
P(\tau_{\mu_i}) - P(\tau_{\mu_i+1}) \\
0
\end{array}\right) \ \ \textrm{for $j = 1, \dots, m-1$ and} \ \ \left(\begin{array}{c}
P(\tau_{\mu_m}) \\
1
\end{array}\right).$$ For remaining columns, the first $d$ components are precisely the components of the corresponding columns in the previous case and the $d+1$ component is 0. Expanding the determinant across row $(d+1)$, the methods used earlier in the proof can be applied to deduce $$J_{\sigma}(\rho, \tau) = \pm\int_{R(\tau)\times B_{\sigma}(\tau)} \wp_{\sigma}(\rho, \tau, x) \,{\mathrm{d}}x$$ where $\wp_{\sigma}(\rho, \tau, x) $ is the determinant of a $d \times d$ matrix and $$R(\tau) := \prod_{i=1}^{m-1} (\tau_{\mu_i}, \tau_{\mu_i+1}); \qquad B_{\sigma}(\tau) := \prod_{j=1}^n (\tau_{\nu_{j}} + s_{\nu_j}, \tau_{\nu_{j}+1}).$$ The key difference is now the integral is over a rectangle of dimension $m+n -1$ (rather than $m+n$). Define the truncated domain $D(\tau)$ in analogous manner to the previous case. Notice from it follows that $d-M = m+n -1$, which is precisely the dimension of the set $D(\tau)$ in the present situation. Arguing as before, the inequality also holds in this setting and from this one obtains the required estimate.
Adapting the argument to $\mathrm{Case}(2, d)$. {#adapting-the-argument-to-mathrmcase2-d. .unnumbered}
-----------------------------------------------
Here the map $\Phi \circ \varphi^{-1}$ is given by $$\begin{aligned}
\Phi\circ \varphi^{-1}(\rho, \tau, \sigma) &=& y_0 + \sum_{i=1}^m \rho_{\mu_i}\big( P(\tau_{\mu_i}) - P(\tau_{\mu_i +1}) \big) \\
&& + \sum_{j=1}^n \rho_{\nu_j}\big( P(\tau_{\nu_j}+ s_{\nu_j}(\tau,\sigma)) - P(\tau_{\nu_j +1}) \big)\end{aligned}$$ and thus the columns of the Jacobian matrix essentially agree with those of $\mathrm{Case}(1,d)$, with the exception that now there is no column corresponding to $\partial G_{\sigma} / \partial \tau_{\mu_1 - 1}$.
The above arguments now carry through almost verbatim; the only substantial difference in this situation is that the Vandermonde estimate becomes $$|V(\tau)| \gtrsim \alpha^{M(M-1)/2} (\beta/\alpha)^{(m-1)/2}$$ due to the fact that the parameter tower in this situation is of type 2, as opposed to type 1 in both of the previous cases.
A final remark
==============
\[strong type remark\] In the introduction the possibility of strengthening the restricted weak-type $(p_1, q_1)$ estimate from Proposition \[weakthm\] to a strong-type estimate was discussed. It was remarked that the strong-type estimate in dimension $d=2$ follows from a result of Gressman [@Gressman2013], but can also be established by combining the analysis contained within the present article with an extrapolation method due to Christ [@Christb] (see also [@Stovall2009]). Here some further details are sketched. The key ingredients in Christ’s extrapolation technique are certain ‘trilinear’ variants of the estimates and . Recall, to prove the weak-type bound it sufficed to show *either* *or* holds since both these estimates are equivalent. This equivalence breaks down when one passes to the trilinear setting and to establish the strong-type inequality one must prove *both* the trilinear version of *and* the trilinear version of hold. This can be achieved in the $d=2$ case by introducing an “inflation” argument (see [@Christ] and also [@Gressman2006]). One may attempt to apply the same techniques in higher dimensions but now the Jacobian arising from the inflation is rather complicated. The question of whether or not this Jacobian can be effectively estimated remains unresolved. It is possible that the inflation argument is not required when $d$ belongs to a certain congruence class modulo $3$ and potentially the strong-type bound could be established more directly from existing arguments in this situation.
Appendix: The method of Dendrinos and Stovall {#appendix-the-method-of-dendrinos-and-stovall .unnumbered}
=============================================
This final section details the construction of the sequence of sets $\{U_k\}_{k=1}^{\infty}$ featured in Lemma \[towerlem\]. The argument here is due to Dendrinos and Stovall [@Dendrinos]. At this point some preliminary definitions and remarks are pertinent. Observe $$\langle A\chi_E\,,\, \chi_F \rangle = \int_{\Sigma} \chi_F(\pi_1(x,r,t)) \chi_E(\pi_2(x,r,t)) \lambda_P(t)\,{\mathrm{d}}x{\mathrm{d}}r {\mathrm{d}}t$$ where $\Sigma := {\mathbb{R}}^d \times [1,2] \times I$ and $\pi_1 \colon \Sigma \to {\mathbb{R}}^d \times [1,2]$ and $\pi_2 \colon \Sigma \to {\mathbb{R}}^d$ are the mappings $$\pi_1(x,r,t) := (x,r), \qquad \pi_2(x,r,t) := x - rP(t).$$ Define the $\pi_j$-fibres to be the sets $\pi_j^{-1}\circ\pi_j(x,r,t)$ for $(x,r,t) \in \Sigma$ and $j = 1,2$. Thus, the $\pi_1$-fibres form a partition of $\Sigma$ into a continuum of curves (which are simply parallel lines) whilst the $\pi_2$-fibres partition $\Sigma$ into a continuum of 2-surfaces. Writing $$U := \pi_1^{-1}(F) \cap \pi_2^{-1}(E) = \left\{ (x,r,t) \in \Sigma : \pi_1(x,r,t) \in F\textrm{ and } \pi_2(x,r,t) \in E \right\}$$ it follows that $$\langle A\chi_E\,,\, \chi_F \rangle = \int_{\Sigma} \chi_U(x,r,t)\lambda_P(t)\,{\mathrm{d}}x{\mathrm{d}}r {\mathrm{d}}t.$$ The sets $\{U_k\}_{k=0}^{\infty}$ are defined recursively. To construct the initial set $U_0$, let $$B_0 := \{(x,r,t) \in U : 0 < t < (\alpha/2\kappa)^{\kappa}\}.$$ Then, recalling the definition of $\lambda_P$ and applying Fubini’s theorem, it follows that $$\begin{aligned}
\int_{\Sigma} \chi_{B_0}(x,r,t) \,\lambda_P(t){\mathrm{d}}x {\mathrm{d}}r {\mathrm{d}}t &=& \int_{F} \int_0^{(\alpha/2\kappa)^{\kappa}} \chi_E(x - rP(t))\, \lambda_P(t){\mathrm{d}}t {\mathrm{d}}x {\mathrm{d}}r \\
&\leq& \frac{1}{2}\alpha|F| = \frac{1}{2} \int_{\Sigma} \chi_U(x,r,t)\,\lambda_P(t){\mathrm{d}}x {\mathrm{d}}r {\mathrm{d}}t.\end{aligned}$$ Define $U_0 := U\setminus B_0$ so that $$\int_{\Sigma} \chi_{U_0}(x,r,t) \,\lambda_P(t){\mathrm{d}}x {\mathrm{d}}r {\mathrm{d}}t \geq \frac{1}{2}\int_{\Sigma} \chi_{U}(x,r,t) \,\lambda_P(t){\mathrm{d}}x {\mathrm{d}}r {\mathrm{d}}t.$$ Note that this definition will ensure property iii) holds for the sequence of refinements. Now suppose the set $U_{k-1}$ has been defined for some $k \geq 1$ and satisfies the conditions stipulated in the proof of Lemma \[towerlem\].
Case $k \equiv d \mod 2$. {#case-k-equiv-d-mod-2. .unnumbered}
-------------------------
In order to ensure the property holds in this case, the following refinement procedure is applied. Let $B_{k-1}$ denote the set $$\left\{ (x,r,t) \in U_{k-1} : \int_1^2\int_I \chi_{U_{k-1}}(x - rP(t) + \rho P(\tau), \rho, \tau) \,\lambda_P(\tau){\mathrm{d}}\tau{\mathrm{d}}\rho \leq 4^{-(k+1/2)} \beta \right\}.$$ The map $(\rho, \tau) \mapsto (x - rP(t) + \rho P(\tau),\rho, \tau)$ parametrises the fibre $\pi_2^{-1}(\pi_2(x,r, t))$ and so $B_{k-1}$ is precisely the set of all points belonging to $\pi_2$-fibres which have a “small” intersection with $U_{k-1}$. Removing the parts of $U_{k-1}$ lying in these fibres should not significantly diminish the measure of the set and indeed, by Fubini’s theorem and a simple change of variables, $$\begin{aligned}
\nonumber
\int_{\Sigma} \chi_{B_{k-1}}(x,r,t) \,\lambda_P(t){\mathrm{d}}t {\mathrm{d}}x{\mathrm{d}}r &=& \int_{{\mathbb{R}}^n\times[1,2]}\int_I \chi_{B_{k-1}}(x+rP(t),r,t) \,\lambda_P(t){\mathrm{d}}t {\mathrm{d}}x{\mathrm{d}}r \\
\nonumber
&\leq& \int_{\{ x \in E \,:\, T_{k-1}(x) \leq 4^{-(k+1/2)}\beta\}} T_{k-1}(x)\, {\mathrm{d}}x \\
\nonumber
&\leq& 4^{-(k+1/2)}\beta|E| \\
\label{Dendrinos and Stovall 1}
&\leq& \frac{1}{2}\int_{\Sigma} \chi_{U_{k-1}}(x,r,t) \,\lambda_P(t){\mathrm{d}}x {\mathrm{d}}r {\mathrm{d}}t\end{aligned}$$ where $$T_{k-1}(x) := \int_1^2\int_I \chi_{U_{k-1}}(x+\rho P(\tau),\rho,\tau) \,\lambda_P(\tau){\mathrm{d}}\tau{\mathrm{d}}\rho.$$ Note that the inequality is due to property i) of the sets $U_j$ for $1 \leq j \leq k-1$, stated in Lemma \[towerlem\]. Thence, letting $U_{k-1}' := U_{k-1} \setminus B_{k-1}$ it follows that $$\label{U bound}
\int_{\Sigma} \chi_{U_{k-1}'}(x,r,t) \,\lambda_P(t){\mathrm{d}}x {\mathrm{d}}r {\mathrm{d}}t \geq \frac{1}{2}\int_{\Sigma} \chi_{U_{k-1}}(x,r,t) \,\lambda_P(t){\mathrm{d}}x {\mathrm{d}}r {\mathrm{d}}t.$$ Now, recalling $I = (a,b)$, define $B_{k-1}'$ to be the set $$\left\{ (x,r,t) \in U_{k-1}' : \int_1^2\int_t^b \chi_{U_{k-1}'}(x - rP(t) + \rho P(\tau),\rho, \tau) \,\lambda_P(\tau){\mathrm{d}}\tau{\mathrm{d}}\rho \leq 4^{-(k+1)}\beta \right\}.$$ Given $x \in \pi_2(U_{k-1}')$, the fibre-wise nature of the definition of $U_{k-1}'$ implies for $(x+rP(t),r,t) \in U_{k-1}'$ if and only if $(x+rP(t),r,t) \in U_{k-1}$ and consequently $$\label{ref1}
\int_1^2\int_I \chi_{U_{k-1}'}(x+rP(t),r,t) \,\lambda_P(t){\mathrm{d}}t{\mathrm{d}}r \geq 4^{-(k+1/2)} \beta.$$ On the other hand, $$\label{ref2}
\int_1^2\int_I \chi_{B_{k-1}'}(x+rP(t),r,t) \, \lambda_P(t){\mathrm{d}}t{\mathrm{d}}r = 4^{-(k+1)}\beta.$$ Indeed, the left-hand side can be expressed as $$\nu_P\big(\big\{(r,t) \in K(x) : \nu_P\big(K(x) \cap ([1,2] \times (t,b))\big) \leq 4^{-(k+1)}\beta\big\}\big)$$ for $K(x) \subseteq [1,2] \times I$ a measurable subset. The identity is now a consequence of the fact that for any measure $\nu$ on ${\mathbb{R}}^2$ which is, say, absolutely continuous with respect to Lebesgue measure, $$\nu\big(\big\{ (r,t) \in K : \nu\big(K \cap ({\mathbb{R}}\times (t, \infty))\big) \leq u \big\}\big) = u$$ for all $0< u < \nu(K)$ and all $K \subseteq {\mathbb{R}}^2$ measurable.
Thence, combining and it follows that $$\int_1^2\int_I \chi_{B_{k-1}'}(x+rP(t),r,t) \,\lambda_P(t){\mathrm{d}}t{\mathrm{d}}r \leq \frac{1}{2} \int_1^2\int_I \chi_{U_{k-1}'}(x+rP(t),r,t) \lambda_P(t){\mathrm{d}}t{\mathrm{d}}r$$ whenever $x \in \pi_2(U_{k-1}')$. Defining $U_{k}:= U_{k-1}' \setminus B_{k-1}'$, one observes that $$\begin{aligned}
\int_{\Sigma} \chi_{U_{k}}(x,r,t) \,\lambda_P(t) {\mathrm{d}}x{\mathrm{d}}r{\mathrm{d}}t &=& \int_{\pi_2(U_{k-1}')} \int_1^2\int_I \chi_{U_k}(x+rP(t),r,t) \,\lambda_P(t){\mathrm{d}}t{\mathrm{d}}r {\mathrm{d}}x \\
&\geq& \frac{1}{2}\int_{\pi_2(U_{k-1}')} \int_1^2\int_I \chi_{U_{k-1}'}(x+rP(t),r,t) \,\lambda_P(t){\mathrm{d}}t{\mathrm{d}}r {\mathrm{d}}x \\
&=& \frac{1}{4}\int_{\Sigma} \chi_{U_{k-1}}(x,r,t) \,\lambda_P(t){\mathrm{d}}x{\mathrm{d}}r{\mathrm{d}}t.\end{aligned}$$ Moreover, the set $U_k$ is easily seen to satisfy .
Case $k \not \equiv d \mod 2$. {#case-k-not-equiv-d-mod-2. .unnumbered}
------------------------------
It remains to define the set $U_k$ under the assumption $k \not \equiv d \mod 2$, ensuring property is satisfied. Here one is concerned with the fibres of the map $\pi_1$. Define $$B_{k-1} := \left\{ (x,r,t) \in U_{k-1} : \int_I \chi_{U_{k-1}}(x,r, \tau) \,\lambda_P(\tau){\mathrm{d}}\tau \leq 4^{-(k+1/2)} \alpha \right\}.$$ Notice that the map $\tau \mapsto (x,r, \tau)$ parametrises the fibre $\pi_1^{-1}(\pi_1(x,r,t))$ and so $B_{k-1}$ is the collection of all points $(x,r,t)$ in $U_{k-1}$ which belong to $\pi_1$-fibres which have a “small” intersection with $U_{k-1}$. Reasoning analogously to the previous case, if one defines $U_{k-1}' := U_{k-1} \setminus B_{k-1}$ it follows that holds in this case. Finally, let $$B_{k-1}' := \left\{ (x,r,t) \in U_{k-1}' : \int_t^b \chi_{U_{k-1}'}(x,r, \tau) \,\lambda_P(\tau){\mathrm{d}}\tau \leq 4^{-(k+1)} \alpha \right\}$$ and $U_{k} := U_{k-1}'\setminus B_{k-1}'$. Again arguing as in the previous case, it follows that $$\int_{\Sigma} \chi_{U_{k}}(x,r,t) \,\lambda_P(t){\mathrm{d}}x {\mathrm{d}}r {\mathrm{d}}t \geq \frac{1}{4}\int_{\Sigma} \chi_{U_{k-1}}(x,r,t) \,\lambda_P(t){\mathrm{d}}x {\mathrm{d}}r {\mathrm{d}}t.$$ This recursive procedure defines a sequence of sets with all the desired properties.
[^1]: More precisely, both Bourgain and Schlag studied the circular maximal function rather than the parabolic variant discussed here. However, in this context both objects can be understood via the same techniques.
[^2]: It is remarked that this brief survey is far from complete: there are many other results and, in particular, an extensive literature investigating these problems for more general classes of curves.
[^3]: Although in the case of maximal functions one must also consider the position of the curve in the plane, see [@Iosevich1994].
[^4]: That is, for almost every $x \in {\mathbb{R}}^d$ the cardinality of the pre-image $\Phi^{-1}(\{x\})$ is no greater than some fixed (finite) constant. For further details see Lemma \[multiplicity\] below.
[^5]: For the moment curve $h(t) := (t, t^2, \dots, t^d)$, one immediately observes that $J_h(t) = cV(t)$.
[^6]: Here $\ceil{x} := \min\{ n \in {\mathbb{N}}: n \geq x\}$ and $\floor{x} := \max\{ n \in {\mathbb{N}}: n \leq x\}$ for any $x \in {\mathbb{R}}$.
[^7]: Strictly speaking, $\{\Omega_j\}_{j=1}^D$ will only satisfy a weak definition of a tower: see Remark \[measure remark\] below.
[^8]: Recall, Bezout’s theorem states that for any collection $Q_1, \dots, Q_n$ of homogeneous polynomials on ${\mathbb{C}}\mathbb{P}^n$ the number of intersection points of the associated hypersurfaces $ \{z \in {\mathbb{C}}\mathbb{P}^n : Q_j(z) = 0 \}$ (counted with multiplicity) is either uncountable or precisely $\prod_{j=1}^n \deg (Q_j)$. The real version used here follows by homogenising the polynomials and taking the domain of the resulting functions to be $\mathbb{CP}^n$. One then applies Bezout’s theorem in complex projective space, de-homogenises and restricts to real-value intersection points. See, for example, [@Hassett2007] pp 223-224.
[^9]: For notational convenience the dependence of $s_{\nu}$ on $(\tau, \sigma)$ has been suppressed.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We establish renormalizability of the full spectral action for the Yang–Mills system on a flat 4-dimensional background manifold. Interpreting the spectral action as a higher-derivative gauge theory, we find that it behaves unexpectedly well as far as renormalization is concerned. Namely, a power counting argument implies that the spectral action is superrenormalizable. From BRST-invariance of the one-loop effective action, we conclude that it is actually renormalizable as a gauge theory.'
address: 'Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands'
author:
- 'Walter D. van Suijlekom'
date: 8 February 2011
title: |
Renormalization of the spectral action\
for the Yang–Mills system
---
Introduction
============
One of the great successes of noncommutative geometry [@C94] is in its application to high-energy physics. Replacing the spacetime manifold by a noncommutative manifold, one puts the full Standard Model of elementary particles on equal geometrical footing as Einstein’s General theory of Relativity. This is worked out in full detail in [@CCM07] (see also [@CM07] and the companion [@CC10]), including the physical predictions that are a consequence of this description.
Being a geometrical description of the Standard Model that is comparable to General Relativity makes it immediate that its quantization comes with the usual problems, actually typical for the latter theory. At the moment, one works with the noncommutative manifold as setting the [*classical*]{} starting point – indeed allowing for a derivation of the full Standard Model Lagrangian at the classical level. Then, one adopts the physics textbook perturbative quantization of it, and arrive at physical predictions via the known Standard Model RG-equations. It needs no stressing that the situation around its quantization should be improved, and in the present letter we intend to take a first step in this direction.
We start with the full asymptotic expansion of the spectral action of Chamseddine and Connes [@CC96; @CC97] in the case of the Yang–Mills system on a flat background manifold. By naive power counting we show – after a suitable gauge-fixing – that the full spectral action is superrenormalizable as a higher-derivative gauge theory [@Sla71; @Sla72b] (cf. [@FS80 Section 4.4]). Then, we demonstrate that the needed counterterms are gauge invariant polynomials that can safely be added to the spectral action. This shows renormalizability of the full spectral action for the Yang–Mills action, compatibly with gauge invariance.
The Yang–Mills system
=====================
The object of study in this paper is the spectral action for the Yang–Mills (YM) system on a flat background manifold. It is given by the relatively simple formula: $$S[A] := \operatorname{Tr}f(D_A/\Lambda).$$ This [*spectral action*]{} has firm roots in the noncommutative geometrical description of the Yang–Mills system, we refer to [@CCM07] for more details. For our purposes, it suffices to know that $D_A$ is a Dirac operator with coefficients in a $SU(N)$-vector bundle equipped with a connection $A$. That is, locally we have $$D_A = i \gamma^\mu \nabla_\mu + \gamma^\mu A_\mu.$$ with $\nabla_\mu$ the spin connection on a Riemannian spin manifold $M$. For simplicity, we take $M$ to be flat ([*i.e.*]{} vanishing Riemann curvature tensor) and 4-dimensional. Furthermore, we will assume that $f$ is a Laplace transform: $$f(x) = \int_{t>0} e^{-tx^2} g(t) dt,$$ even though this assumption could be avoided by using spectral densities instead ([@EGV98] and also [@Var06 Section 8.4])
In the above notation, there is an asymptotic expansion (as $\Lambda \to \infty$): $$\label{sa-eym}
S[A] \sim \sum_{m \geq 0} \Lambda^{4-m} f_{4-m} \int_M a_m (x,D_A^2),$$ in terms of the Seeley–De Witt invariants of $D_A^2$. The coefficients are defined by $f_k := \int t^{-k/2} g(t)dt$.
Recall that the Seeley–De Witt coefficients $a_m(x,D_A^2)$ are gauge invariant polynomials in the fields $A_\mu$. Indeed, the Weitzenböck formula gives $$D_A^2 =- (\partial_\mu - i A_\mu) (\partial^\mu - i A^\mu)+ i\sum_{\mu < \nu}\gamma_\mu \gamma_\nu F_{\mu\nu}$$ in terms of the curvature $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu -i [A_\mu,A_\nu]$ of $A_\mu$. Consequently, a Theorem by Gilkey [@Gil84 Theorem 4.8.16] shows that (in this case) $a_m$ are polynomial gauge invariants in $F_{\mu\nu}$ and its covariant derivatives. The [*order*]{} $\operatorname{ord}$ of $a_m$ is $m$, where we set on generators: $$\operatorname{ord}A_{\mu_1; \mu_2\cdots \mu_k} = k.$$ Consequently, the curvature $F_{\mu\nu}$ has order $2$, and $F_{\mu_1 \mu_2; \mu_3 \cdots \mu_k}$ has order $k$. For example, $a_4(x,D_A^2)$ is proportional to $\operatorname{Tr}F_{\mu\nu}F^{\mu\nu}$ and more generally: $$a_{4+2k}(x,D_A^2) =c_k \operatorname{Tr}F_{\mu\nu} \Delta^k_A (F^{\mu\nu}) + \cO(F^3)$$ for some constants $c_k$ and the Laplacian $\Delta_A= -(\partial_\mu - i A_\mu)^2$ (see also [@Avr99] and references therein). The remainder is of third and higher order in $F$, plus its covariant derivatives, adding up to an order equal to $4+2k$.
It is the term $a_4$ that gives rise to the Yang–Mills action functional, the higher-order terms are usually ignored (being proportional to an inverse power of the ‘cut-off’ $\Lambda$). More recently, also the higher-order terms, or even the full spectral action were studied in specific cases in [@CC11; @MPT10] and from a more general point of view in [@Sui10].
The quadratic term $S_0[A]$ in $S[A]$ is given asymptotically (as $\Lambda \to \infty$) by $$S_0[A] \sim \sum_{k \geq 0} \Lambda^{-2k} f_{-2k} c_k \int \operatorname{Tr}\hat F_{\mu\nu} \Delta^{k} (\hat F^{\mu\nu})$$ where we have set $\hat F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ and $\Delta= -\partial^\mu \partial_\mu$.
We assume that the first term is the usual (free part of the) Yang–Mills action, that is, we adjust the positive function $f$ so that $f_0 c_0 =-1/4$. For the other coefficients, we have the following neat expression.[^1]
The coefficients $f_{-2k}$ are related to the $2k$’th derivatives of $f$ at zero: $$f_{-2k} = \frac{(-1)^k f^{(2k)}(0)}{(2k-1)!!}.$$
With $f(x) = \int e^{-tx^2} g(t)dt$ we derive for its derivatives: $$f^{(2k)}(x) = \int_{t>0} e^{-tx^2/2} H_{2k}(\sqrt t x) t^{k} g(t)dt$$ in terms of the Hermite polynomials $H_{n}(x) \equiv (-1)^n e^{x^2/2} (d/dx)^n e^{-x^2/2}$. Evaluating both sides at zero gives the desired result, using in addition that $H_{2k}(0)= (-1)^k (2k-1)!!$.
We end this section by introducing a formal expansion $
{\varphi}_\Lambda(\Delta) = (f_0c_0)^{-1} \sum_{k \geq0} \Lambda^{-2k} f_{-2k} c_k \Delta^k
$ (starting with $1$) so that we can write more concisely $$S_0[A] \sim -\frac{1}{4} \int \operatorname{Tr}\hat F_{\mu\nu} {\varphi}_\Lambda(\Delta) (\hat F^{\mu\nu})$$ This form motivates the interpretation of $S_0[A]$ (and of $S[A]$) as a higher-derivative gauge theory. As we will see below, this indeed regularizes the theory in such a way that $S[A]$ defines a superrenormalizable field theory.
Gauge fixing in the YM-system
=============================
We add a gauge-fixing term of the following higher-derivative form: $$\label{sa-gf}
S_\gf[A] \sim - \frac{1}{2 \xi} \int \partial_\mu A^\mu {\varphi}_\Lambda(\Delta) \left( \partial_\nu A^\nu \right)$$ We derive the [*propagator*]{} by inverting the non-degenerate quadratic form given by $S_0[A] + S_\gf[A]$: $$D_{\mu\nu}^{ab}(p; \Lambda) = \left[ g_{\mu\nu} - (1-\xi) \frac{p_\mu p_\nu}{ (p^2 +i \eta)}\right] \frac{\delta^{ab}}{(p^2 +i \eta) {\varphi}_\Lambda(p^2)}$$ which for the moment is a formal expansion in $\Lambda$. We will come back to it in more detail in the next section.
As usual, the above gauge fixing requires a Jacobian, conveniently described by a Faddeev–Popov ghost Lagrangian: $$\label{sa-gh}
S_\gh[A,\bar C,C] \sim - \int \partial_\mu \bar C {\varphi}_\Lambda(\Delta) \left( \partial^\mu C + [A^\mu,C] \right)$$ Here $C,\bar C$ are the Faddeev–Popov ghost fields and their propagator is $$\tilde D^{ab}(p; \Lambda) = \frac{\delta^{ab}}{(p^2 + i \eta) {\varphi}_\Lambda(p^2)}.$$
The sum $S[A] + S_\gf[A] + S_\gh[A,\bar C, C]$ is invariant under the BRST-transformations: $$\begin{gathered}
\label{brst}
sA_\mu = \partial_\mu C + [A_\mu,C];\qquad s C = -\half [C,C]; \qquad s \bar C = \xi^{-1} \partial_\mu A^\mu.\end{gathered}$$
First, $s(S)=0$ because of gauge invariance of $S[A]$. We compute $$s(S_\gf) = -\frac{1}{\xi} \int ( \partial_\mu A^\mu) {\varphi}_\Lambda(\Delta) \left( \partial_\nu \partial^\nu C + \partial_\nu( [A^\nu,C] \right)$$ On the other hand, $$s(S_\gh) = - \frac{1}{\xi} \int (\partial_\mu \partial^\nu A_\nu ){\varphi}_\Lambda(\Delta) \left( \partial^\mu C+ [A^\mu,C] \right)$$ which modulo vanishing boundary terms is minus the previous expression.
Note that $s^2 \neq 0$, which can be cured by standard homological methods: introduce an auxiliary (aka Nakanishi-Lautrup) field $h$ so that $\bar C$ and $h$ form a contractible pair in BRST-cohomology. In other words, we replace the above transformation in on $\bar C$ by $s \bar C = --h$ and $s h = 0$. If we replace $S_\gf + S_\gh$ by $s \Psi$ with $\Psi$ an arbitrary [*gauge fixing fermion*]{}, it follows from gauge invariance of $S$ and nilpotency of $s$ that $S + s \Psi$ is BRST-invariant. The above special form of $S_\gf+ S_\gh$ can be recovered by choosing $$\Psi =- \int {\varphi}_\Lambda(\Delta) (\bar C) \left( \half \xi h + \partial_\mu A^\mu \right).$$
One might wonder what gauge fixing condition is implemented by $S_\gf$ as in , given the presence of the term ${\varphi}_\Lambda(\Delta)$. Under suitable conditions on the function $f$, the function $x \mapsto {\varphi}_\Lambda(x)$ is positive, turning the bilinear form $$(\omega_1,\omega_2) := - \int \operatorname{Tr}\omega_1 \wedge \ast ({\varphi}_\Lambda(\Delta) \omega_2)$$ into an inner product. On the Lagrangian level, we can equally well implement the Lorenz gauge fixing condition $\partial \cdot A = 0$ using this inner product instead of the usual $L^2$-inner product. This gives rise to $S_\gf[A] = ( \partial \cdot A, \partial \cdot A)/2\xi$. Similarly, $S_\gf$ is given by the inner product $(\bar C, \partial_\mu C + [A_\mu,C])$.
Renormalization of the spectral action for the YM-system
========================================================
As said, we consider the spectral action for the Yang–Mills system as a higher-derivative field theory. This means that we will use the higher derivatives of $F_{\mu\nu}$ that appear in the asymptotic expansion as natural regulators of the theory, similar to [@Sla71; @Sla72b] (see also [@FS80 Sect. 4.4]). However, note that the regularizing terms are already present in the spectral action $S[A]$ and need not be introduced as such. Let us consider the expansion up to order $n$ (which we assume to be at least $8$), [*i.e.*]{} we set $f_{4-m} = 0$ for all $m > n$. Also, assume a gauge fixing of the form and .
Then, we easily derive from the structure of ${\varphi}_\Lambda(p^2)$ that the propagators of both the gauge field and the ghost field behave as $|p|^{-n+2}$ as $|p| \to \infty$. Indeed, in this case: $${\varphi}_\Lambda(p^2) = \sum_{k=0}^{n/2-2} \Lambda^{-2k} f_{-2k} c_k p^{2k}.$$
Moreover, the weights of the interaction in terms of powers of momenta is given by: $$\begin{array}{|c|c|c|}
\hline
\text{vertex} & \text{valence} & \max \# \text{ der}\\
\hline
\hline
\parbox{2cm}{\vspace{1mm}\includegraphics[scale=.2]{./v3.eps}} & 3 & n-3\\[2mm]
\parbox{2cm}{\includegraphics[scale=.2]{./v4.eps}} & 4 & n-4\\[2mm]
\vdots& \vdots & \vdots\\
\parbox{2cm}{\includegraphics[scale=.2]{./vn.eps}} & n & 0 \\[2mm]
\parbox{2cm}{\includegraphics[scale=.2]{./tildev.eps}} & 3 & n-3\\[2mm]
\hline
\end{array}$$ We will use $v_k$ to indicate the number of gauge interaction vertices of valence $k$, and with $\tilde v$ the number of ghost-gauge vertices.
Let us now find an expression for the [*superficial degree of divergence*]{} $\omega$ of a graph consisting of $I$ internal gauge edges, $\tilde I$ internal ghost edges, $v_k$ valence $k$ gauge vertices and $\tilde v$ ghost-gauge vertices. In 4 dimensions, we find at loop order $L$: $$\omega = 4L - I(n-2) - \tilde I (n-2) + \sum_{i=3}^n v_i (n-i) + \tilde v (n-3).$$
Let $E$ and $\tilde E$ denote the number of external gauge and ghost edges, respectively. The superficial degree of divergence of the graph equals $$\omega = (4-n)(L-1) + 4 - (E+\tilde E).$$
We use the relations $$2 I + E = \sum_i i v_i + \tilde v; \qquad
2 \tilde I + \tilde E = 2\tilde v$$ where $E$ and $\tilde E$ are the number of external gauge and ghost legs, respectively. Indeed, these formulas count the number of half (gauge/ghost) edges in a graph in two ways: from the number of edges and from the valences of the vertices. We use them to substitute for $2I$ and $2\tilde I$ in the above expression for $\omega$ so as to obtain $$\omega = 4L - In - \tilde I n + n \left(\sum_i v_i + \tilde v \right) - (E + \tilde E)$$ from which the result follows at once from Euler’s formula $L= I + \tilde I - \sum_i v_i - \tilde v -1$.
As a consequence, $\omega < 0$ if $L \geq 2$ (provided $n \geq 8$): all Feynman graphs are finite at loop order greater than 1. If $L=1$, then there are finitely many graphs which are divergent, namely those for which $E+ \tilde E \leq 4$. We conclude that the spectral action for the Yang–Mills system is superrenormalizable.
Of course, the spectral action being a gauge theory, there is more to renormalizability than just power counting: we have to establish gauge invariance of the counterterms. We already know that the counterterms needed to render the perturbative quantization of the spectral action finite are of order $4$ or less in the fields and arise only from one-loop graphs. The key property of the effective action at one loop is that it is BRST-invariant: $$s(\Gamma_{1}) = 0.$$ In particular, assuming a regularization compatible with gauge invariance, the divergent part $\Gamma_{1,\infty}$ is BRST-invariant. Results from [@Dix91; @DTV85; @DTV85b; @BDK90; @DHTV91] on BRST-cohomology for Yang–Mills type theories ascertain that the only BRST-closed functional of order 4 or less in the fields is represented by $$\delta Z \int F_{\mu\nu}F^{\mu\nu}$$ for some constant $\delta Z$. The counterterm $\Gamma_{1,\infty}$ can thus be added to $S$ and absorbed by a redefinition of the fields and coupling constant: $$\begin{gathered}
A_0 = \sqrt{1+\delta Z} A ; \qquad g_0 = \frac{g}{\sqrt{1+ \delta Z}}\end{gathered}$$ Equivalently, one could leave $A$ and $g$ invariant, and redefine $f_0 \mapsto (1+\delta Z) f_0$, leaving all other coefficients $f_{4-m}$ invariant. Intriguingly, renormalization of the spectral action for the YM-system can thus be accomplished merely by shifting the function $f$ in such a way that $f(0) \mapsto (1+\delta Z) f(0)$, whilst leaving all its higher derivatives at $0$ invariant.
The above form for $\Gamma_{1,\infty}$ can actually be established by an explicit computation in dimensional regularization following [@PS96; @PS97]. We intend to present the full details elsewhere.
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank Klaas Landsman for useful discussions and remarks. I am grateful to Dmitri Vassilevich for useful comments. NWO is acknowledged for support under VENI-project 639.031.827.
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[^1]: The coefficients $f_{2k}$ for positive $k$ were found to be the $k+1$’th moments of $f$, cf. [@CM07 Sect. 1.11] for more details.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this small note we present a Tannakian proof of the theorem of Grothendieck-Harder on the classification of torsors under a reductive group on the projective line over a field.'
author:
- Johannes Anschütz
title: A Tannakian classification of torsors on the projective line
---
Introduction
============
Let $k$ be a field, let $G/k$ be a reductive group and let ${\mathbb{P}}^1_k$ be the projective line over $k$. In this small note we present a Tannakian proof of the classification of $G$-torsors on ${\mathbb{P}}^1_k$, thereby reproving known results of A. Grothendieck [@grothendieck_sur_la_classification_des_fibres] and G. Harder [@harder_halbeinfache_gruppenschemata_ueber_vollstaendigen_kurven Satz 3.4.]. To state our main theorem we denote by $$\mathrm{Hom}^\otimes({\mathrm{Rep}}_k(G),{\mathrm{Rep}}_k({{\mathbb{G}_m}}))$$ the set of isomorphism classes of exact tensor functors $$\omega\colon {\mathrm{Rep}}_k(G)\to {\mathrm{Rep}}_k({{\mathbb{G}_m}}).$$
\[theorem: main theorem introduction\] There exists a canonical bijection $$\mathrm{Hom}^\otimes({\mathrm{Rep}}_k(G),{\mathrm{Rep}}_k({{\mathbb{G}_m}}))\cong H^1_{{\mathrm{\acute{e}t}}}({\mathbb{P}}^1_k,G).$$ In particular, there exists a canonical bijection $${\mathrm{Hom}}({{\mathbb{G}_m}},G)/{G(k)}\cong H^1_{\mathrm{Zar}}({\mathbb{P}}^1_k,G).$$
If $A\subseteq G$ denotes a maximal split torus, then $${\mathrm{Hom}}({{\mathbb{G}_m}},G)/{G(k)}\cong X_\ast(A)_+$$ is in bijection with the set of dominant cocharacters of $A\subseteq G$, which gives a very concrete description of the set $H^1_{\mathrm{Zar}}({\mathbb{P}}^1_k,G)$. Using pure inner forms of $G$ over $k$ one can describe similarly the whole set $H^1_{{\mathrm{\acute{e}t}}}({\mathbb{P}}^1_k,G)$ (cf. ).
Our proof of , which originated in questions about torsors over the Fargues-Fontaine curve (cf. [@anschuetz_reductive_group_schemes_over_the_fargues_fontaine_curve]), is based on the Tannakian description of $G$-torsors (cf. ), the Tannakian theory of filtered fiber functors (cf. [@ziegler_graded_and_filtered_fiber_functors]), the canonicity of the Harder-Narasimhan filtration (cf. ) and, most importantly, the good understanding of the category ${\mathrm{Bun}}_{{\mathbb{P}}^1_k}$ of vector bundles on ${\mathbb{P}}^1_k$ (cf. ). In particular, we use crucially the fact that $$H^1_{{\mathrm{\acute{e}t}}}({\mathbb{P}}^1_k,\mathcal{E})=0$$ for $\mathcal{E}$ a semistable vector bundle on ${\mathbb{P}}^1_k$ of slope $\geq 0$.
In a last section we mention applications of to the the computation of the Brauer group of ${\mathbb{P}}^1_k$ (avoiding Tsen’s theorem) and to the Birkhoff-Grothendieck decomposition of $G(k((t)))$.
Acknowledgment
--------------
We want to thank Jochen Heinloth for his interest and for answering several questions.
Vector bundles on ${\mathbb{P}}^1_k$
====================================
Let $k$ be an arbitrary field. We recall, in a more canonical form, the classification of vector bundles on the projective line ${\mathbb{P}}^1_k$ due to A. Grothendieck (cf. [@grothendieck_sur_la_classification_des_fibres]). Let $${\mathrm{Rep}}_k({{\mathbb{G}_m}})$$ be the category of finite dimensional representations of the multiplicative group ${{\mathbb{G}_m}}$ over $k$. More concretely, the category ${\mathrm{Rep}}_k({{\mathbb{G}_m}})$ is equivalent to the Tannakian category of finite dimensional ${\mathbb{Z}}$-graded vector spaces over $k$.
Over ${\mathbb{P}}^1_k$ there is the canonical ${{\mathbb{G}_m}}$-torsor $$\eta\colon \mathbb{A}^2_k\setminus\{0\}\to {\mathbb{P}}^1_k,\ (x_0,x_1)\mapsto [x_0:x_1],$$ also called the “Hopf bundle”. Given a representation $V\in {\mathrm{Rep}}_k({{\mathbb{G}_m}})$ the contracted product $$\mathcal{E}(V):=\mathbb{A}^2_k\setminus\{0\}\times^{{\mathbb{G}_m}}V\to {\mathbb{P}}^1_k$$ defines a (geometric) vector bundle over ${\mathbb{P}}^1_k$. The well known classification of the category $${\mathrm{Bun}}_{{\mathbb{P}}^1_k}$$ of vector bundles on ${\mathbb{P}}^1_k$ can now be phrased in the following way.
\[theorem: classification of vector bundles\] The functor $$\mathcal{E}(-)\colon {\mathrm{Rep}}_k({{\mathbb{G}_m}})\to {\mathrm{Bun}}_{{\mathbb{P}}^1_k}$$ is an exact, faithful tensor functor inducing a bijection on isomorphism classes.
However, the functor $\mathcal{E}(-)$ is not an equivalence. For example, by semi-simplicity of the category ${\mathrm{Rep}}_k({{\mathbb{G}_m}})$ every short exact sequence of ${{\mathbb{G}_m}}$-representations splits, but this is not true for short exact sequences of vector bundles on ${\mathbb{P}}^1_k$.
For $V\in {\mathrm{Rep}}_k({{\mathbb{G}_m}})$ the Harder-Narasimhan filtration of the vector bundle $$\mathcal{E}(V)$$ has a very simple description. Namely, write $$V=\bigoplus\limits_{i\in {\mathbb{Z}}} V_i$$ with ${{\mathbb{G}_m}}$ acting on $V_i$ by the character[^1] $${{\mathbb{G}_m}}\to {{\mathbb{G}_m}},\ z\mapsto z^{-i}$$ and set $$\mathrm{fil}^i(V):=\bigoplus\limits_{j\geq i} V_j$$ for $i\in {\mathbb{Z}}$. Then the Harder-Narasimhan filtration of $\mathcal{E}:=\mathcal{E}(V)$ is given by $$\ldots \subseteq \mathrm{HN}^{i+1}(\mathcal{E})\subseteq \mathrm{HN}^{i}(\mathcal{E})\subseteq\ldots \subseteq \mathcal{E}.$$ where $$\mathrm{HN}^{i}(\mathcal{E}):=\mathcal{E}(\mathrm{fil}^{i}(V)).$$
\[lemma: harder-narasimhan filtration tensor functor\] Sending a vector bundle $\mathcal{E}$ to the filtered vector bundle $\mathcal{E}$ with the Harder-Narasimhan filtration $\mathrm{HN}^\bullet(\mathcal{E})$ defines a fully faithful tensor functor $$\mathrm{HN}\colon {\mathrm{Bun}}_{{\mathbb{P}}^1_k}\to \mathrm{FilBun}_{{\mathbb{P}}^1_k}$$ into the exact tensor category of filtered vector bundles (with filtration by locally direct summands) (cf. [@ziegler_graded_and_filtered_fiber_functors Chapter 4] for a definition of $\mathrm{FilBun}_{{\mathbb{P}}^1_k}$).
This is clear from the description of the Harder-Narasimhan filtration.
We remark that the functor $\mathrm{HN}$ is *not* exact as one sees for example by looking at the Euler sequence $$0\to \mathcal{O}_{{\mathbb{P}}^1_k}(-1)\to \mathcal{O}_{{\mathbb{P}}^1_k}\oplus \mathcal{O}_{{\mathbb{P}}^1_k}\to \mathcal{O}_{{\mathbb{P}}^1_k}(1)\to 0$$ on ${\mathbb{P}}^1_k$.
Sending a filtered vector bundle $(\mathcal{E}, F^\bullet)$ to the associated graded vector bundle $$\mathrm{gr}(\mathcal{E}):=\bigoplus_{i\in {\mathbb{Z}}} F^i\mathcal{E}/F^{i+1}\mathcal{E}$$ defines an exact tensor functor $$\mathrm{gr}\colon \mathrm{FilBun}_{{\mathbb{P}}^1_k}\to \mathrm{GrBun}_{{\mathbb{P}}^1_k}$$ (cf. [@ziegler_graded_and_filtered_fiber_functors Chapter 4]).
The following lemma is immediate from , and the fact that $$H^0({\mathbb{P}}^1_k,\mathcal{O}_{{\mathbb{P}}^1_k})\cong k.$$
\[lemma: representations of gm as graded vector bundles\] The composite functor $${\mathrm{Rep}}_k({{\mathbb{G}_m}})\xrightarrow{\mathcal{E}(-)} {\mathrm{Bun}}_{{\mathbb{P}}^1_k}\xrightarrow{\mathrm{HN}} \mathrm{FilBun}_{{\mathbb{P}}^1_k}\xrightarrow{\mathrm{gr}} \mathrm{GrBun}_{{\mathbb{P}}^1_k}$$ is an equivalence of exact categories from ${\mathrm{Rep}}_k({{\mathbb{G}_m}})$ onto its essential image which consists of graded vector bundles $$\mathcal{E}=\bigoplus\limits_{i\in {\mathbb{Z}}} \mathcal{E}^i$$ such that each $\mathcal{E}^i$ is semistable of slope $i$.
Torsors over ${\mathbb{P}}^1_k$
===============================
Let $G/k$ be an arbitrary reductive group. In this section we want to classify $G$-torsors on ${\mathbb{P}}^1_k$ for the étale topology. For this we keep the notation from the last section. In particular, there is the functor $$\mathcal{E}(-)\colon {\mathrm{Rep}}_k({{\mathbb{G}_m}})\to{\mathrm{Bun}}_{{\mathbb{P}}^1_k}$$ from
In order to apply the formulations from the previous section we need a more bundle theoretic interpretation of $G$-torsors (for the étale topology). This is achieved by the Tannakian formalism (cf. [@deligne_categories_tannakiennes])
\[lemma: tannakian description of torsors\] Let $S$ be a scheme over $k$. Sending a $G$-torsor $\mathcal{P}$ over $S$ to the exact tensor functor $$\omega\colon {\mathrm{Rep}}_k(G)\to {\mathrm{Bun}}_{S},\ V\mapsto \mathcal{P}\times^{G}(V\otimes_k \mathcal{O}_S)$$ defines an equivalence from the groupoid of $G$-torsors to the groupoid of exact tensor functors from ${\mathrm{Rep}}_k(G)$ to ${\mathrm{Bun}}_S$. The inverse equivalence sends an exact tensor functor $\omega\colon {\mathrm{Rep}}_k(G)\to {\mathrm{Bun}}_{S}$ the $G$-torsor $\mathrm{Isom}^\otimes(\omega_{\mathrm{can}},\omega)$ of isomorphisms of $\omega$ to the canonical fiber functor $\omega_{\mathrm{can}}\colon {\mathrm{Rep}}_k(G)\to {\mathrm{Bun}}_S,\ V\mapsto V\otimes_k \mathcal{O}_S$.
In fact, for a general affine group scheme over $k$ one has to use the fpqc-topology in . However, as $G$ is assumed to be reductive, thus in particular smooth, a theorem of Grothendieck (cf. [@grothendieck_le_group_de_brauer_III Théorème 11.7]) allows to reduce to the étale topology.
Composing an exact tensor functor $$\omega\colon {\mathrm{Rep}}_k(G)\to {\mathrm{Bun}}_{{\mathbb{P}}^1_k}$$ with the Harder-Narasimhan functor $$\mathrm{HN}\colon {\mathrm{Bun}}_{{\mathbb{P}}^1_k}\to \mathrm{FilBun}_{{\mathbb{P}}^1_k}$$ defines a, a priori not necessarily exact, tensor functor $$\mathrm{HN}\circ \omega\colon {\mathrm{Rep}}_k(G)\to \mathrm{FilBun}_{{\mathbb{P}}^1_k}.$$
But using Haboush’s theorem reductivity of $G$ actually implies that the composition $\mathrm{HN}\circ \omega$ is still exact.
\[lemma: composition with harder-narasimhan filtration still exact\] Let $$\omega\colon {\mathrm{Rep}}_k(G)\to {\mathrm{Bun}}_{{\mathbb{P}}^1_k}$$ be an exact tensor functor. Then the composition $$\mathrm{HN}\circ \omega\colon {\mathrm{Rep}}_k(G)\to \mathrm{FilBun}_{{\mathbb{P}}^1_k}$$ is still exact.
The crucial observation is that the functors $$\omega,\ \mathrm{gr}\circ \mathrm{HN}$$ are compatible with duals, and exterior resp. symmetric products. This is clear for $\omega$ as $\omega$ is assumed to be exact and follows from for the functor $\mathrm{HN}\circ \mathrm{gr}$. In fact, for a representation $V\in {\mathrm{Rep}}_k({{\mathbb{G}_m}})$ with associated vector bundle $$\mathcal{E}:=\mathcal{E}(V)$$ we can conclude $$\Lambda^r(\mathcal{E})\cong \mathcal{E}(\Lambda^r(V)) \textrm{ resp.\ } \mathrm{Sym}^r(\mathcal{E})\cong \mathcal{E}(\mathrm{Sym}^r(V))$$ by exactness of the functor $\mathcal{E}(-)$. But by $$\mathrm{gr}\circ \mathrm{HN}\circ \mathcal{E}(-)$$ is an exact tensor equivalence of ${\mathrm{Rep}}_k({{\mathbb{G}_m}})$ with a subcategory of $\mathrm{GrBun}_{{\mathbb{P}}^1_k}$, which implies the stated compatibility with exterior and symmetric powers. Using this the proof can proceed similarly to [@dat_orlik_rapoport_period_domains Theorem 5.3.1]. We note that for a representation $V$ of $G$ there is a canonical isomorphism $$\mathrm{Sym}^r(V^\vee)\cong \mathrm{TS}_r(V)^\vee$$ from the $r$-th symmetric power $\mathrm{Sym}^r(V^\vee)$ of the dual of $V$ to the dual of the module $$\mathrm{TS}_r(V)=(V^{\otimes r})^{S_r}\subseteq V^{\otimes r}$$ of symmetric tensors. In particular, $G$-invariant homogenous polynomials on $V$ define $G$-invariant linear forms on $\mathrm{TS}_r(V)^\vee$.
Let now $0\to V\xrightarrow{f} V^\prime\xrightarrow{g} V^{\prime\prime}\to 0$ be an exact sequence in ${\mathrm{Rep}}_k(G)$. We have to check that the sequence $$0\to \tilde{\omega}(V)\xrightarrow{\tilde{\omega}(f)} \tilde{\omega}(V^\prime)\xrightarrow{\tilde{\omega}(g)} \tilde{\omega}(V^{\prime\prime})\to 0$$ with $$\tilde{\omega}:=\mathrm{gr}\circ \mathrm{HN}\circ \omega$$ is still exact. We claim that $\tilde{\omega}(f)$ is injective. This can be checked after taking the exterior power $\Lambda^{\dim V}$ of $f$ because $\tilde{\omega}$ commutes with exterior powers. In particular, to prove injectivity we can reduce the claim for general $f$ to the case $\dim V=1$. Tensoring with the dual of $V$ reduces further to the case the $V$ is moreover trivial. By Haboush’s theorem (cf. [@haboush_reductive_groups_are_geometrically_reductive]) there exists an $r>0$ and a $G$-invariant homogenous polynomial $f\in \mathrm{Sym}^r(V^\vee)$ such that $f_{|V}\neq 0$. Using the above isomorphism $\mathrm{Sym}^r(V^\vee)\cong \mathrm{TS}_r(V)^\vee$ this shows that there exists an $r> 0$ such that the morphism $$V\cong \mathrm{TS}_r(V)\xrightarrow{\mathrm{TS}_r(f)}\mathrm{TS}_r(V^\prime)$$ splits. This implies that $\tilde{\omega}(\mathrm{TS}_r(f))$ splits and thus that $\tilde{\omega}(f)$ is in particular injective because $\tilde{\omega}$ commutes with the symmetric tensors $TS_r$ as it commutes with symmetric powers and duals.
Dualizing yields that $\tilde{\omega}(g)$ is surjective at the generic point of ${\mathbb{P}}^1_k$. However, the sequence $$0\to \tilde{\omega}(V)\xrightarrow{\tilde{\omega}(f)} \tilde{\omega}(V^\prime)\xrightarrow{\tilde{\omega}(g)} \tilde{\omega}(V^{\prime\prime})\to 0$$ lies in the essential image of the functor ${\mathrm{Rep}}_k({{\mathbb{G}_m}})\to \mathrm{GrBun}_{{\mathbb{P}}^1_k}$ from . In particular, we see that the cokernel of $\tilde{\omega}(g)$ cannot have torsion, i.e., that it is zero. Finally, exactness in the middle of the sequence follows because $$\mathrm{rk}(\tilde{\omega}(V^\prime))=\mathrm{rk}(V^\prime)=\mathrm{rk}(V)+\mathrm{rk}(V^{\prime\prime})=\mathrm{rk}(\tilde{\omega}(V))+\mathrm{rk}(\tilde{\omega}(V^{\prime\prime})).$$ This finishes the proof.
We briefly recall some results about filtered fiber functors on ${\mathrm{Rep}}_kG$ (cf. [@ziegler_graded_and_filtered_fiber_functors] and [@cornut_filtrations_and_buildings]). By definition a filtered fiber functor for ${\mathrm{Rep}}_kG$ over a $k$-scheme $S$ is an exact tensor functor $$\omega\colon {\mathrm{Rep}}_k G\to \mathrm{FilBun}_S$$ into the exact tensor category of filtered vector bundles (with filtration by locally direct summands) on $S$. Associated to each filtered fiber functor $\omega$ is an exact tensor functor $$\mathrm{gr}\circ \omega\colon {\mathrm{Rep}}_k G\to \mathrm{GrBun}_S,$$ i.e., a graded fiber functor, by mapping a filtered vector bundle to its associated graded. A splitting $\gamma$ of a filtered fiber functor $\omega$ is a graded fiber functor $$\gamma\colon {\mathrm{Rep}}_kG \to \mathrm{GrBun}_S$$ such that $$\omega=\mathrm{fil}\circ \gamma$$ where the exact tensor functor $$\mathrm{fil}\colon \mathrm{GrBun}_S\to \mathrm{FilBun}_S$$ sends a graded vector bundle $$\mathcal{E}=\bigoplus\limits_{i\in {\mathbb{Z}}}\mathcal{E}^i$$ to the filtered vector bundle $(\mathcal{E}, \mathrm{fil}^\bullet \mathcal{E})$ with filtration $$\mathrm{fil}^i\mathcal{E}=\bigoplus\limits_{j\geq i}\mathcal{E}^j.$$ For a scheme $f\colon S^\prime\to S$ over $S$ let $\omega_{S^\prime}$ be the base change of the filtered fiber functor $\omega$ to $S^\prime$, i.e., $\omega_{S^\prime}$ is defined as the composition $${\mathrm{Rep}}_k G\xrightarrow{\omega} \mathrm{FilBun}_{S}\xrightarrow{f^\ast} \mathrm{FilBun}_{S^\prime},$$ which is again a filtered fiber functor. For a filtered fiber functor $\omega$ the presheaf $$\mathrm{Spl}(\omega)(S^\prime):=\{ \textrm{ set of splittings of } \omega_{S^\prime}\}$$ on the category of $S$-schemes is represented by an fpqc-torsor for the affine and faithfully flat group scheme $$U(\omega):={\textrm{Ker}}({\mathrm{Aut}}^\otimes(\omega)\to {\mathrm{Aut}}^\otimes(\mathrm{gr}\circ \omega))$$ over $S$ (cf. [@ziegler_graded_and_filtered_fiber_functors Lemma 4.20]). In particular, every filtered fiber functor $$\omega\colon {\mathrm{Rep}}_k G\to \mathrm{FilBun}_S$$ admits a splitting fpqc-locally on $S$. The group scheme $U(\omega)$ can be described more explicitely (cf. [@ziegler_graded_and_filtered_fiber_functors Theorem 4.40]). Namely there exists a decreasing filtration by normal subgroups $$U(\omega)=U_1(\omega)\supseteq\ldots \supseteq U_i(\omega)\supseteq \ldots$$ for $i\geq 1$, which has the property that for $i\geq 1$ the quotient $$\mathrm{gr}^iU(\omega):= U_i(\omega)/U_{i+1}(\omega)$$ is abelian and isomorphic to $$\mathrm{gr}^iU(\omega)\cong \mathrm{Lie}(\mathrm{gr}^iU(\omega))\cong \mathrm{gr}^i\omega(\mathrm{Lie}(G)),\ i\geq 1.$$ We can now give a proof of our main theorem about the classification of $G$-torsors on $\mathbb{P}^1_k$. We denote for a scheme $S$ over $k$ by $$\underline{{\mathrm{Hom}}}^\otimes({\mathrm{Rep}}_k(G), {\mathrm{Bun}}_S)$$ the groupoid of exact tensor functors $\omega\colon {\mathrm{Rep}}_k(G)\to {\mathrm{Bun}}_S$ and by $${\mathrm{Hom}}^\otimes({\mathrm{Rep}}_k(G), {\mathrm{Bun}}_S)$$ its set of isomorphism classes. Similarly, we use the notations $$\underline{{\mathrm{Hom}}}^\otimes({\mathrm{Rep}}_k(G), {\mathrm{Rep}}_k({{\mathbb{G}_m}}))$$ resp.$${\mathrm{Hom}}^\otimes({\mathrm{Rep}}_k(G), {\mathrm{Rep}}_k({{\mathbb{G}_m}}))$$ for the groupoid resp. the isomorphism classes of exact tensor functors $$\omega\colon {\mathrm{Rep}}_k(G)\to {\mathrm{Rep}}_k({{\mathbb{G}_m}}).$$
\[theorem: main theorem\] Let $G$ be a reductive group over $k$. Then the composition with $\mathcal{E}(-)$ defines faithful functor $$\Phi\colon \underline{{\mathrm{Hom}}}^\otimes({\mathrm{Rep}}_k(G),{\mathrm{Rep}}_k({{\mathbb{G}_m}}))\to \underline{{\mathrm{Hom}}}^\otimes({\mathrm{Rep}}_k(G),{\mathrm{Bun}}_{{\mathbb{P}}^1_k})$$ which induces a bijection $${\mathrm{Hom}}^\otimes({\mathrm{Rep}}_k(G),{\mathrm{Rep}}_k({{\mathbb{G}_m}}))\cong H^1_{{\mathrm{\acute{e}t}}}({\mathbb{P}}^1_k,G).$$ on isomorphism classes.
By the composition $${\mathrm{Rep}}_k({{\mathbb{G}_m}})\xrightarrow{\mathcal{E}(-)} {\mathrm{Bun}}_{{\mathbb{P}}^1_k}\xrightarrow{\mathrm{HN}} \mathrm{FilBun}_{{\mathbb{P}}^1_k}\xrightarrow{\mathrm{gr}} \mathrm{GrBun}_{{\mathbb{P}}^1_k}$$ is an equivalence onto its essential image. In particular, the functor $$\Phi\colon \underline{{\mathrm{Hom}}}^\otimes({\mathrm{Rep}}_k(G),{\mathrm{Rep}}_k({{\mathbb{G}_m}}))\to \underline{{\mathrm{Hom}}}^\otimes({\mathrm{Rep}}_k(G),{\mathrm{Bun}}_{{\mathbb{P}}^1_k})$$ is faithful and induces an injection on isomorphism classes. Thus we have to prove that every exact tensor functor $$\omega\colon {\mathrm{Rep}}_k(G)\to {\mathrm{Bun}}_{{\mathbb{P}}^1_k}$$ factors as $$\omega\cong \mathcal{E}(-)\circ \omega^\prime$$ for some exact tensor functor $$\omega^\prime\colon {\mathrm{Rep}}_k(G)\to {\mathrm{Rep}}_k({{\mathbb{G}_m}}).$$ Let $\tilde{\omega}:=\mathrm{HN}\circ \omega$ be the functor $$\tilde{\omega}\colon {\mathrm{Rep}}_k(G)\xrightarrow{\omega}{\mathrm{Bun}}_{{\mathbb{P}}^1_k}\xrightarrow{\mathrm{HN}} \mathrm{FilBun}_{{\mathbb{P}}^1_k}.$$ By the functor $\tilde{\omega}$ is still exact, i.e., a filtered fiber functor in the terminology of [@ziegler_graded_and_filtered_fiber_functors], and we can use the results recalled above. We get a $U(\tilde{\omega})$-torsor $$\mathrm{Spl}(\tilde{\omega})$$ of splittings of $\tilde{\omega}$. But for the filtration $$U(\tilde{\omega})\supseteq U_{2}(\tilde{\omega})\supseteq \ldots$$ the graded quotients $$\mathrm{gr}^iU(\tilde{\omega})\cong \mathrm{gr}^i\tilde{\omega}(\mathrm{Lie}(G))$$ are semistable vector bundles of slope $i\geq 1$. Hence, $$H^1_{{{\mathrm{\acute{e}t}}}}({\mathbb{P}}^1_k,\mathrm{gr}^iU(\tilde{\omega}))=0$$ because $$\mathrm{gr}^iU(\tilde{\omega})\cong \mathcal{O}_{{\mathbb{P}}^1_k}(i)^{\oplus n}$$ by . We can conclude that $$H^1_{{\mathrm{\acute{e}t}}}({\mathbb{P}}^1_k, U(\tilde{\omega}))=1,$$ hence the $U(\tilde{\omega})$-torsor $$\mathrm{Spl}(\tilde{\omega})$$ is in fact trivial, i.e., there exists a splitting $$\gamma\colon {\mathrm{Rep}}_k G\to \mathrm{GrBun}_{{\mathbb{P}}^1_k}$$ of $\tilde{\omega}$ already over ${\mathbb{P}}^1_k$. As $$\gamma\cong \mathrm{gr}\circ \tilde{\omega}$$ the functor $\gamma$ takes its image in the full subcategory $$\{\ \mathcal{E}=\bigoplus\limits_{i\in {\mathbb{Z}}} \mathcal{E}^i\in \mathrm{GrBun}_{{\mathbb{P}}^1}\ |\ \mathcal{E}^i \textrm{ semistable of slope }i\},$$ which by is equivalent to the category ${\mathrm{Rep}}_k {{\mathbb{G}_m}}$ of representations of ${{\mathbb{G}_m}}$. Thus there exists an exact tensor functor $$\omega^\prime\colon {\mathrm{Rep}}_k G\to {\mathrm{Rep}}_k {{\mathbb{G}_m}}$$ such that $$\omega\cong \mathcal{E}(-)\circ \omega^\prime,$$ by simply setting $$\omega^\prime:=\mathcal{E}_{\mathrm{gr}}(-)^{-1}\circ \mathrm{gr}\circ \tilde{\omega}$$ where $$\mathcal{E}_{\mathrm{gr}}(-)\colon {\mathrm{Rep}}_k {{\mathbb{G}_m}}\to \{\ \mathcal{E}=\bigoplus\limits_{i\in {\mathbb{Z}}} \mathcal{E}^i\in \mathrm{GrBun}_{{\mathbb{P}}^1}\ |\ \mathcal{E}^i \textrm{ semistable of slope }i\},$$ is the the equivalence of .
Let $$\omega_{\mathrm{can}}^{{\mathbb{G}_m}}\colon {\mathrm{Rep}}_k({{\mathbb{G}_m}})\to \mathrm{Vec}_k,\ V\mapsto V$$ be the canonical fiber functor of ${\mathrm{Rep}}_k({{\mathbb{G}_m}})$ over $k$. Composing with $\omega_{\mathrm{can}}^{{\mathbb{G}_m}}$ defines a morphism $$\Phi\colon \underline{{\mathrm{Hom}}}^\otimes({\mathrm{Rep}}_k(G),{\mathrm{Rep}}_k({{\mathbb{G}_m}}))\to \underline{{\mathrm{Hom}}}^\otimes({\mathrm{Rep}}_k(G),\mathrm{Vec}_k)$$ of groupoids, where the right hand side denotes the groupoid of exact tensor functors $${\mathrm{Rep}}_k(G)\to \mathrm{Vec}_k,$$ which by identifies with the groupoid of $G$-torsors on ${\mathrm{Spec}}(k)$. Geometrically, the morphism $\Phi$ can be identified on isomorphisms classes with the map $$i^\ast\colon H^1_{{{\mathrm{\acute{e}t}}}}({\mathbb{P}}^1_k, G)\to H^1_{{\mathrm{\acute{e}t}}}({\mathrm{Spec}}(k),G)$$ restricting a $G$-torsor over ${\mathbb{P}}^1_k$ to a $G$-torsor over ${\mathrm{Spec}}(k)$ along a $k$-rational point $x\in {\mathbb{P}}^1_k(k)$.
In the following lemma we analyze the fibers of this functor $\Phi$.
\[lemma: fibers are zariski torsors\] Let $\omega\colon {\mathrm{Rep}}_k(G)\to \mathrm{Vec}_k$ be an exact tensor functor and let $$H:={\mathrm{Aut}}^\otimes(\omega)$$ be the pure inner form of $G$ defined by $\omega$. Then the fiber $$\Phi^{-1}(\omega)\subseteq \underline{{\mathrm{Hom}}}^\otimes({\mathrm{Rep}}_kG,{\mathrm{Rep}}_k {{\mathbb{G}_m}})$$ is equivalent to the quotient groupoid $$[{\mathrm{Hom}}({{\mathbb{G}_m}},H)/{H(k)}]$$ of cocharacters of $H$. Moreover, passing to isomorphism classes yields a bijection $${\mathrm{Hom}}({{\mathbb{G}_m}},H)/{H(k)}\cong H^1_{\mathrm{Zar}}({\mathbb{P}}^1_k,H).$$
The first statement follows from the Tannakian formalism (cf. [@deligne_categories_tannakiennes]). Namely, $\omega$ defines an equivalence $${\mathrm{Rep}}_k(G)\cong {\mathrm{Rep}}_k(H),\ V\mapsto \omega(V)$$ and the groupoid of exact tensor functors $${\mathrm{Rep}}_k H\to {\mathrm{Rep}}_k {{\mathbb{G}_m}}$$ which commute (with a given isomorphism) with the canonical fiber functors on ${\mathrm{Rep}}_k H$ resp. ${\mathrm{Rep}}_k {{\mathbb{G}_m}}$ is equivalent to the quotient groupoid $$[{\mathrm{Hom}}({{\mathbb{G}_m}},H)/{H(k)}].$$ with $H(k)$ acting by conjugation. Clearly, for every cocharacter $$\chi\colon {{\mathbb{G}_m}}\to H$$ the push forward $$\eta\times^{{{\mathbb{G}_m}}}H$$ is an $H$-torsor, which is locally trivial in the Zariski topology, because this is true for the Hopf bundle $$\eta\colon \mathbb{A}^2_k\setminus\{0\}\to {\mathbb{P}}^1_k.$$ Let conversely $\mathcal{P}$ be an $H$-torsor over ${\mathbb{P}}^1_k$ which is trivial for the Zariski topology and let $$\omega_{\mathcal{P}}\colon {\mathrm{Rep}}_k H\to {\mathrm{Bun}}_{{\mathbb{P}}^1_k}, V\mapsto \mathcal{P}\times^{H}(V\otimes_k \mathcal{O}_{{\mathbb{P}}^1_k})$$ be the induced fiber functor (cf. ). Let $x\in {\mathbb{P}}^1_k(k)$ be a point a $k$-rational point and let $U\subseteq {\mathbb{P}}^1_k$ be open subset containing $x\in U$ such that $$\mathcal{P}_{|U}$$ is trivial. Then the exact tensor functor $${\mathrm{Rep}}_k H\xrightarrow{\omega_{\mathcal{P}}}{\mathrm{Bun}}_{{\mathbb{P}}^1_k}\xrightarrow{\mathrm{res}}{\mathrm{Bun}}_U$$ is isomorphic to the trivial fiber functor. This holds then also true after restricting to $x\in U$. Let $$\varphi\colon {\mathrm{Rep}}_k H\to {\mathrm{Rep}}_k {{\mathbb{G}_m}}$$ be an exact tensor functor such that $$\mathcal{E}(-)\circ \varphi \cong \omega_{\mathcal{P}}.$$ We can conclude that $\varphi$ preserves the canonical fiber functors on ${\mathrm{Rep}}_k H$ resp. ${\mathrm{Rep}}_k {{\mathbb{G}_m}}$ because the composition $${\mathrm{Rep}}_k {{\mathbb{G}_m}}\xrightarrow{\mathcal{E}(-)} {\mathrm{Bun}}_{{\mathbb{P}}^1_k}\xrightarrow{\mathrm{res}}{\mathrm{Bun}}_{x}\cong \mathrm{Vec}_k$$ is the canonical fiber functor. In particular, there exists a cocharacter $$\chi\colon {{\mathbb{G}_m}}\to H$$ such that $\mathcal{P}$ is obtained via pushout along $\chi$ of the Hopf bundle $$\eta\colon \mathbb{A}^2_k\setminus\{0\}\to {\mathbb{P}}^1_k.$$
Note that we have actually shown that a $G$-torsor $\mathcal{P}$ is already locally trivial for the Zariski topology if there exists some open $U\subseteq {\mathbb{P}}^1_k$ containing a $k$-rational point, such that $\mathcal{P}_{|U}$ is trivial. The classification results of Grothendieck and Harder on torsors on ${\mathbb{P}}^1_k$ (cf. [@grothendieck_sur_la_classification_des_fibres] resp. [@harder_halbeinfache_gruppenschemata_ueber_vollstaendigen_kurven]) are most concretely stated in the collowing form.
\[corollary: zariski torsors\] Let $k$ be a field and let $G/k$ be a reductive group with maximal split subtorus $A\subseteq G$. Then there exist canonical bijections $$X_\ast(A)_+\cong \mathrm{Hom}({{\mathbb{G}_m}},G)/G(k)\cong H^1_{\mathrm{Zar}}({\mathbb{P}}^1_k,G),$$ where $X_\ast(A)_+$ denotes the set of dominant cocharacters of $A\subseteq G$.
By it suffices to show $$X_\ast(A)_+\cong \mathrm{Hom}({{\mathbb{G}_m}},G)/G(k).$$ But this follows from the fact that all maximal split tori in $G$ are conjugated over $k$ and that the set of dominant cocharacters form a system of representatives for the action of the normalizer $N_G(A)$ of $A$ in $G$ on the group $X_\ast(A)$ of cocharacters for $A$.
A description of $H^1_{{\mathrm{\acute{e}t}}}({\mathbb{P}}^1_k,G)$, similar to the one of us, can be found in [@gille_torseurs_sur_la_droite_affine].
Applications
============
In this section we present some applications of the classification of torsors (following (cf. [@fargues_g_torseurs_en_theorie_de_hodge_p_adique], which discusses analogous applications to the Fargues-Fontaine curve).
The first application is the computation of the Brauer group of ${\mathbb{P}}^1_k$. For this we recall the theorem of Steinberg (cf. [@serre_galois_cohomology Chapter 3.2.3]). If $k$ is a field of cohomological dimension $\mathrm{cd}(k)\leq 1$, then Steinberg’s theorem states that $$H^1_{{{\mathrm{\acute{e}t}}}}({\mathrm{Spec}}(k),G)=1$$ for every smooth connected affine algebraic group $G/k$. In particular, the Brauer group $$\mathrm{Br}(k)=0$$ of such fields vanishes. For example, separably closed or finite fields are of cohomological dimension $\leq 1$.
\[theorem: brauer group of p1\] If $k$ is of cohomological dimension $\mathrm{cd}(k)\leq 1$, then the Brauer group $$\mathrm{Br}({\mathbb{P}}^1_k)\cong H^2_{{{\mathrm{\acute{e}t}}}}({\mathbb{P}}^1_k,{{\mathbb{G}_m}})=0$$ vanishes.
By [@grothendieck_le_group_de_brauer_II Corollaire 2.2.] there is an isomorphism $$\mathrm{Br}({\mathbb{P}}^1_k)\cong H^2_{{{\mathrm{\acute{e}t}}}}({\mathbb{P}}^1_k,{{\mathbb{G}_m}})$$ of the Brauer group $\mathrm{Br}({\mathbb{P}}^1_k)$ parametrizing equivalence classes of Azumaya algebras over $\mathcal{O}_{{\mathbb{P}}^1_k}$ with the cohomological Brauer group $H^2_{{\mathrm{\acute{e}t}}}({\mathbb{P}}^1_k,{{\mathbb{G}_m}})$. It suffices to show that for every $n\geq 0$ the canonical map $$H^1_{{\mathrm{\acute{e}t}}}({\mathbb{P}}^1_k,\mathrm{PGL}_n)\to H^2_{{{\mathrm{\acute{e}t}}}}({\mathbb{P}}^1_k,{{\mathbb{G}_m}})$$ arising as a boundary map of the short exact sequence $$1\to {{\mathbb{G}_m}}\to \mathrm{GL}_n\to \mathrm{PGL}_n\to 1$$ is trivial. Because $k$ is of cohomological dimension $\leq 1$, there exists using Steinberg’s theorem in the case $G=\mathrm{GL}_n$ or $G=\mathrm{PGL}_n$ and together with a commutative diagram $$\xymatrix{
H^1_{{{\mathrm{\acute{e}t}}}}({\mathbb{P}}^1_k, \mathrm{GL}_n)\ar[r]\ar[d]^\cong & H^1_{{\mathrm{\acute{e}t}}}({\mathbb{P}}^1_k,\mathrm{PGL}_n)\ar[d]^\cong \\
{\mathrm{Hom}}({{\mathbb{G}_m}},\mathrm{GL}_n)/{\mathrm{GL}_n(k)}\ar[r] & {\mathrm{Hom}}({{\mathbb{G}_m}},\mathrm{PGL}_n)/{\mathrm{PGL}_n(k)}.
}$$ It suffices to show that the top horizontal arrow, or equivalently the lower horizontal arrow, is surjective. But every cocharacter $$\chi\colon {{\mathbb{G}_m}}\to \mathrm{PGL}_n$$ can be lifted to $\mathrm{GL}_n$ because for the standard torus $T\cong \mathbb{G}_m^n\subseteq \mathrm{GL}_n$ there is a split exact sequence $$0\to X_\ast({{\mathbb{G}_m}})\to X_\ast(T)\to X_\ast(T/{{\mathbb{G}_m}})\to 0$$ on cocharacter groups where $T/{{\mathbb{G}_m}}$ is a maximal torus of $\mathrm{PGL}_n$.
For a general field $k$, i.e., $k$ not necessarily of cohomological dimension $\leq 1$, the Brauer group of ${\mathbb{P}}^1_k$ is given by $$\mathrm{Br}({\mathrm{Spec}}(k))\cong \mathrm{Br}({\mathbb{P}}^1_k)$$ as can be calculated from using the spectral sequence $$E^{pq}_2=H^p(\mathrm{Gal}(\bar{k}/k),H^q_{{{\mathrm{\acute{e}t}}}}({\mathbb{P}}^1_{\bar{k}},{{\mathbb{G}_m}}))\Rightarrow H^{p+q}_{{{\mathrm{\acute{e}t}}}}({\mathbb{P}}^1_k,{{\mathbb{G}_m}})$$ where $\bar{k}$ denotes a separable closure of $k$.
The next application we give is to the uniformization of $G$-torsors.
\[theorem: uniformization\] Let $k$ be a field and let $G$ be reductive group over $k$. If $x\in {\mathbb{P}}^1_k(k)$ is $k$-rational point, then every $G$-torsor $$\mathcal{P}\in H^1_{\mathrm{Zar}}({\mathbb{P}}^1_k,G)$$ which is locally trivial for the Zariski topology becomes trivial on ${\mathbb{P}}^1_k\setminus\{x\}$.
By we know that every such $G$-torsor $\mathcal{P}$ is isomorphic to the pushout $$\mathcal{P}\cong \eta\times^{{\mathbb{G}_m}}G$$ along a cocharacter $$\chi\colon {{\mathbb{G}_m}}\to G$$ of the canonical ${{\mathbb{G}_m}}$-torsor $$\eta\colon \mathbb{A}^2_k\setminus\{0\}\to {\mathbb{P}}^1_k$$ corresponding to the line bundle $\mathcal{O}_{{\mathbb{P}}^1_k}(-1)$ on ${\mathbb{P}}^1_k$. But $$\mathcal{O}_{{\mathbb{P}}^1_k}(-1)_{|{\mathbb{P}}^1_k\setminus\{x\}}$$ is trivial because ${\mathbb{P}}^1_k\setminus\{x\}\cong \mathbb{A}^1_k$. This shows the claim.
Finally, we reprove the Birkhoff-Grothendieck decomposition of $G(k((t))$ for a reductive group $G$ over $k$ (cf. [@faltings_algebraic_loop_groups_and_moduli_spaces_of_bundles Lemma 4]).
\[theorem: decomposition\] Let $A\subseteq G$ be a maximal split torus in $G$. Then there exists a canonical bijection $$X_\ast(A)_{+}\cong {G(k[t^{-1}])}\backslash G(k((t)))/{G(k[[t]])},$$ where $X_\ast(A)_+$ denotes the set of dominant cocharacters of $A\subseteq G$.
Let $x\in {\mathbb{P}}^1_k(k)$ be a $k$-rational point. By Beauville-Laszlo [@beauville_laszlo_un_lemme_de_descente] and there is an injective map $$\gamma\colon {G(k[t^{-1}])}\backslash G(k((t)))/{G(k[[t]])}\to H^1_{{{\mathrm{\acute{e}t}}}}({\mathbb{P}}^1_k,G)$$ by glueing the trivial $G$-torsor on ${\mathbb{P}}^1_k\setminus\{x\}$ with the trivial $G$-torsor on the formal completion $${\mathrm{Spec}}(\widehat{\mathcal{O}}_{{\mathbb{P}}^1_k,x})$$ along an isomorphism on ${\mathrm{Spec}}(\mathrm{Frac}(\widehat{\mathcal{O}}_{{\mathbb{P}}^1_k,x}))$. Note that $\widehat{\mathcal{O}}_{{\mathbb{P}}^1_k,x}\cong k[[t]]$. From the remark following we can conclude that the $G$-torsors obtained in this way are actually locally trivial for the Zariski topology. By we can conversely see that the image of $\gamma$ contains the set $H^1_{\mathrm{Zar}}({\mathbb{P}}^1_k,G)$. Using we can conclude that $${G(k[t^{-1}])}\backslash G(k((t)))/{G(k[[t]])}\cong H^1_{\mathrm{Zar}}({\mathbb{P}}^1_k,G)\cong X_{\ast}(A)_+.$$
[DOR10]{}
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[^1]: The sign is explained by the fact that the standard represention $z\mapsto z$ of ${{\mathbb{G}_m}}$ is sent by $\mathcal{E}(-)$ to $\mathcal{O}_{{\mathbb{P}}^1_k}(-1)$ and not to $\mathcal{O}_{{\mathbb{P}}^1_k}(1)$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study spatial noise correlations in a Si/SiGe two-qubit device with integrated micromagnets. Our method relies on the concept of decoherence-free subspaces, whereby we measure the coherence time for two different Bell states, designed to be sensitive only to either correlated or anti-correlated noise respectively. From these measurements, we find weak correlations in low-frequency noise acting on the two qubits, while no correlations could be detected in high-frequency noise. A theoretical model and numerical simulations give further insight into the additive effect of multiple independent (anti-)correlated noise sources with an asymmetric effect on the two qubits. Such a scenario is plausible given the data and our understanding of the physics of this system. This work is highly relevant for the design of optimized quantum error correction codes for spin qubits in quantum dot arrays, as well as for optimizing the design of future quantum dot arrays.'
author:
- 'Jelmer M. Boter'
- Xiao Xue
- 'Tobias S. Krähenmann'
- 'Thomas F. Watson'
- 'Vickram N. Premakumar'
- 'Daniel R. Ward'
- 'Donald E. Savage'
- 'Max G. Lagally'
- Mark Friesen
- 'Susan N. Coppersmith'
- 'Mark A. Eriksson'
- Robert Joynt
- 'Lieven M. K. Vandersypen'
title: 'Spatial Noise Correlations in a Si/SiGe Two-Qubit Device from Bell State Coherences'
---
[^1]
[^2]
[^3]
[^4]
Large-scale quantum computers will need to rely on quantum error correction (QEC) to deal with the inevitable qubit errors caused by interaction with the environment and by imperfect control signals. The noise amplitude can vary from qubit to qubit and furthermore can exhibit correlations or anti-correlations between qubits. Most QEC error thresholds, such as the 1%-threshold for the surface code [@Wang2011], are derived under the assumption of negligible correlations in qubit errors. Other approaches such as decoherence-free subspaces (DFSs) [@Lidar1998] are designed under the assumption of correlated noise, taking advantage of symmetry considerations to reduce the qubit sensitivity to external noise. Examples for quantum dot based qubits include the singlet-triplet qubit [@Levy2002; @Petta2005] and the quadrupole qubit [@Friesen2017]. In addition, QEC schemes exist that can deal with short-range correlations in the noise [@Preskill2013]. Spatial noise correlations have therefore been studied extensively, both theoretically [@Rivas2015; @Szankowski2016; @Paz-Silva2017; @Postler2018; @Kwiatkowski2018; @Premakumar2018; @Krzywda2019] and experimentally [@Monz2011; @Postler2018; @Ozaeta2019].
Semiconductor quantum dots are promising hosts for spin qubits in quantum computation [@Loss1998], because of their favorable scaling and excellent coherence properties. Silicon, in particular, has excellent properties for long-lived spin qubits: intrinsic spin-orbit coupling is weak and hyperfine interaction is small [@Zwanenburg2013]. The hyperfine interaction can even be reduced further by isotopic purification. In addition, silicon quantum dot fabrication is largely compatible with conventional CMOS industry, which allows large-scale manufacturing of silicon spin qubits and on-chip integration of classical control electronics [@Vandersypen2017]. In recent years, significant progress has been made with silicon spin qubits, showing tens of milliseconds coherence times [@Veldhorst2014], high-fidelity single- [@Veldhorst2014; @Kawakami2016; @Yoneda2018] and two-qubit gates [@Xue2019; @Huang2019], quantum algorithms [@Watson2018], strong spin-photon coupling [@Samkharadze2018; @Mi2018] and long-distance spin-spin coupling [@Borjans2019].
The most important decoherence sources in natural silicon quantum dots are the hyperfine interaction with nuclear spins and charge noise. Nuclear spin noise is typically uncorrelated between adjacent dots [@Chekhovich2013]. Charge noise is usually caused by distant fluctuating charges [@Jung2004; @Paladino2014; @Beaudoin2015], which is expected to lead to spatial correlations on the length scale of interdot distances of 100 nm or less. In the presence of a magnetic field gradient, which is commonly used for qubit selectivity and fast qubit control, qubits are sensitive to electric field fluctuations and charge noise will impact spin coherence [@Kha2015; @Kawakami2016]. However, a quantitative measurement of spatial noise correlations in an actual two-qubit device is lacking.
Here we study experimentally spatial noise correlations in a Si/SiGe two-qubit device, by preparing Bell states in either the parallel or the anti-parallel subspace, similarly to recent work with NV centers in diamond [@Bradley2019]. Via a Ramsey-style experiment, we find that Bell states in the anti-parallel subspace show a $\sim$30% longer dephasing time than those in the parallel subspace. A Hahn-echo style measurement reveals no detectable difference in the decay time for the respective Bell states. We present a simple model to describe noise correlations on two qubits, including asymmetric noise amplitudes acting on the two qubits, and study numerically the combined effect of multiple (anti-)correlated, asymmetric noise sources. We use these simulations to assess which combinations of noise sources are compatible with the observed coherence times.
![(a) Scanning electron micrograph of a similar Si/SiGe device as used in the measurements, showing the depletion gates used to define the potential landscape in the 2D electron gas accumulated by the yellow shaded gates (drawn digitally). Purple and orange circles indicate the estimated positions of the two dots, occupied by one electron each, and the ellipse indicates a sensing quantum dot. Two-qubit operations are controlled via gate voltage pulses applied to gates P1 and P2, and microwave signals for single-qubit control are applied to gates MW1 and MW2. The contours of cobalt micromagnets are indicated by the dashed black lines. (b) Energy level diagram for two qubits in an inhomogeneous magnetic field, giving rise to a difference in Zeeman energy between the two qubits.[]{data-label="fig:fig1"}](Figures_Fig1-eps-converted-to.pdf)
Figure \[fig:fig1\](a) shows a schematic of the device used in this work, which is the same as described earlier [@Watson2018; @Xue2019]. It comprises an electrostatically defined double quantum dot (DQD) in a two-dimensional electron gas (2DEG). The 2DEG is confined in a 12-nm-thick silicon quantum well, 37 nm below the surface of an undoped Si/SiGe heterostructure with natural isotope composition. On top of the heterostructure, we fabricate two gate layers with cobalt micromagnets. The device is cooled down to $T\approx30$ mK and subject to an external magnetic field of B~ext~ = 617 mT. Suitable voltages are applied to accumulation and fine gates (in the top and bottom layer, respectively) to form a DQD in the single-electron regime. Single-electron spin states are Zeeman split by the total magnetic field, and used to encode two single-spin qubits. The micromagnets ensure individual qubit addressability by a gradient in the longitudinal magnetic field, resulting in spin resonance frequencies of 18.35 GHz and 19.61 GHz for qubit 1 (Q1) and qubit 2 (Q2), respectively.
Figure \[fig:fig1\](b) shows the resulting energy level diagram for the two qubits. For perfectly correlated noise, fluctuations in the Zeeman energy for both qubits are the same: $\delta E_{Z,1} = \delta E_{Z,2} = \delta E_{Z}$. Consequently, the sum of the two qubit energies fluctuates, $\Delta(E_{Z,1} + E_{Z,2}) = 2\delta E_Z$, while their difference is not affected, $\Delta(E_{Z,1} - E_{Z,2}) = 0$. On the other hand, for perfectly anti-correlated noise $\delta E_{Z,1} = -\delta E_{Z,2}$, and the opposite holds for the sum and difference energies. Therefore, an anti-parallel Bell state, which evolves in time at a rate proportional to the difference of the single-qubit energies, will be affected by anti-correlated noise, but not by correlated noise. A parallel Bell state, which evolves in time at a rate proportional to the sum of the single-qubit energies, is sensitive to correlated noise, but not to anti-correlated noise. Such properties are exploited in DFSs and are used here as a probe for spatial correlations in the noise acting on the qubits.
Real systems are often subject to both uncorrelated and (anti-)correlated noise. Furthermore, the noise amplitudes acting on different qubits are generally different, regardless of whether the noise is uncorrelated or (anti-)\
correlated. We wish to capture all these scenarios in one unified theoretical formalism. We include pure dephasing only, which is justified by the long $T_1$ times for spin qubits compared to the experiment and coherence timescales, and assume a quasi-static Gaussian joint probability distribution for the noise acting on the two qubits. We can then express the two-qubit coherence times for an anti-parallel ($\ket{\Psi} = (\ket{\downarrow\uparrow} -i \ket{\uparrow\downarrow})/\sqrt{2}$) and a parallel ($\ket{\Phi} = (\ket{\downarrow\downarrow} -i \ket{\uparrow\uparrow})/\sqrt{2}$) Bell state quantitatively as follows (see Supplemental Material [@SuppMat]): $$\begin{split}
\left(\frac{1}{T_{2,\ket{\Psi}}^*}\right)^2 &= 2\pi^2 \left(\sigma_1^2 + \sigma_2^2 - 2\rho\sigma_1\sigma_2 \right), \\
\left(\frac{1}{T_{2,\ket{\Phi}}^*}\right)^2 &= 2\pi^2 \left(\sigma_1^2 + \sigma_2^2 + 2\rho\sigma_1\sigma_2 \right),
\end{split}
\label{eq:T2s}$$ where $\sigma_i^2$ is the variance of the noise in the resonance frequency of qubit $i$ (the single-qubit coherence time is given by $\left(\frac{1}{T_{2,i}^*}\right)^2 = 2\pi^2 \sigma_i^2$), and $\rho$ is a correlation factor ($-1 \leq \rho \leq 1$). Positive $\rho$ indicates correlations, while negative $\rho$ indicates anti-correlations.
![$T_{2,\ket{\Psi}}^*$ extracted from Eq. \[eq:T2s\] (a) as a function of correlation factor $\rho$ and noise amplitude $\sigma_1=\sigma_2$, and (b) as a function of $\sigma_1$ and $\sigma_2$ for $\rho=1$. Insets show the corresponding images for $T_{2,\ket{\Phi}}^*$. Contours correspond to (0.5, 0.75, 1.0, 1.25, 1.5, 1.75) $\mu$s. In all images an uncorrelated noise contribution corresponding to a Bell state coherence time of $2.0 \,\mu$s is added to prevent singularities.[]{data-label="fig:fig2"}](Figures_Fig2-eps-converted-to.pdf)
The effect of the noise amplitudes $\sigma_i$ and the correlation factor $\rho$ on the coherence time for the anti-parallel Bell state $T_{2,\ket{\Psi}}^*$ is visualized in Fig. \[fig:fig2\](a). Here $\sigma_1=\sigma_2$, so for $\rho=1$, $\ket{\Psi}$ forms a true DFS and the noise has no effect regardless of its amplitude. With decreasing $\rho$, $T_{2,\ket{\Psi}}^*$ decreases, as the noise becomes initially less correlated ($\rho > 0$), then uncorrelated ($\rho=0$) and eventually anti-correlated ($\rho < 0$). For $\rho=-1$, $T_{2,\ket{\Psi}}^*$ is only one fourth of the single-qubit coherence times. For $T_{2,\ket{\Phi}}^*$ the corresponding image is mirrored around $\rho=0$, see the inset of Fig. \[fig:fig2\]a, and the longest coherence time occurs for $\rho=-1$. Figure \[fig:fig2\](b) shows the effect of asymmetric noise amplitudes on the two qubits for $\rho=1$. We see that despite the maximal correlation factor, a true DFS only exists for symmetric noise ($\sigma_1=\sigma_2$) and $\ket{\Psi}$ decoheres when $\sigma_1\neq\sigma_2$. Clearly, both the asymmetry in the noise and the correlation factor impact the two-qubit coherence.
From Eq. \[eq:T2s\], we see that, as anticipated, experimental measurement of the decay times for the parallel and anti-parallel Bell states reveals whether (anti-)correlations in the noise acting on the two qubits are present. In order to quantify the correlation factor $\rho$, measurements of the single-qubit decay time are needed as well. We now summarize the experimental procedure; for more information on the measurement setup and individual qubit characteristics, see the Supplemental Material [@SuppMat] and Ref. [@Watson2018]. Q2 is initialized and read out via spin-selective tunneling to a reservoir [@Elzerman2004]. Initialization of Q1 to its ground state is done by fast spin relaxation at a hotspot [@Srinivasa2013], and read-out of Q1 is performed by mapping its spin state onto Q2 via a controlled-rotation (CROT) gate followed by spin read-out of Q2 [@Watson2018]. For single-qubit driving we exploit an artificial spin-orbit coupling, induced by cobalt micromagnets, for electric dipole spin resonance (EDSR) [@Pioro-Ladriere2007]. The two-qubit gate relies on the exchange interaction between the two qubits, controlled by gate voltage pulses. We operate in the regime where the Zeeman energy difference between the two qubits exceeds the two-qubit exchange interaction strength, hence the native two-qubit gate is the controlled-phase gate [@Meunier2011; @Veldhorst2015; @Watson2018].
![(a,c) Circuit diagrams for two-qubit experiments analogous to the measurement of Ramsey fringes. The gate sequences are designed such that single-qubit rotations are always applied simultaneously to both qubits, avoiding idle times that would lead to faster dephasing. Here $CZ_{ij}\ket{m,n} = (-1)^{\delta(i,m)\delta(j,n)}\ket{m,n}$ for $i,j,m,n \in \{0,1\}$ [@Watson2018]. (b,d) Typical $\ket{00}$ return probability as a function of delay time for (b) $\ket{\Psi}$ and (d) $\ket{\Phi}$. The data are fit with a sinusoidal function with Gaussian decay, $P_{\ket{00}} \propto e^{-\left(t/T_2^*\right)^2}$. Error bars are based on a Monte Carlo method by assuming a multinomial distribution for the measured two-spin probabilities and are $\pm 1 \sigma$ from the mean [@Watson2018]. (e) Scatter plot of decay times for $\ket{\Psi}$ and $\ket{\Phi}$ for two measurement runs separated by $\sim$50 hours (points and crosses). Every data point is averaged over $\sim$100 minutes. The average coherence times are $513 \pm 8$ ns and $387 \pm 6$ ns for $\ket{\Psi}$ and $\ket{\Phi}$, respectively. Error bars are $\pm 1 \sigma$ from the mean.[]{data-label="fig:fig3"}](Figures_Fig3-eps-converted-to.pdf)
Concretely, we perform two-qubit measurements analogous to the measurement of Ramsey fringes to measure the decay of Bell state coherences over time [@Premakumar2018]. As shown in the circuits in Figs. \[fig:fig3\](a,c), we prepare $\ket{\Psi}$ or $\ket{\Phi}$ and after a varying free evolution time we reverse the sequence to ideally return to the $\ket{00}$ state. In every run of the experiment, we measure both spins in single-shot mode and determine the two-spin probabilities from repeated experiment runs. The two-spin probabilities are normalized and a Gaussian decay is fit to the $\ket{00}$ return probability. To improve the fit of the decay, we add an evolution-time dependent phase to the first microwave pulse applied to Q2 after the delay time, so that the measured $\ket{00}$ probability oscillates. We first test the measurement procedure via artificially introduced dephasing from random rotations of each spin around its quantization axis, implemented in software via Pauli frame updates. As seen in Fig. \[fig:Supp1\] in the Supplemental Material [@SuppMat], the decay observed for the anti-parallel (parallel) Bell state is independent of the noise amplitude when the same (opposite) random rotations are applied to both spins, but increases when opposite (the same) random rotations are applied to the two spins, as expected. This validates the measurement protocol.
Figures \[fig:fig3\](b,d) show typical decay curves for $\ket{\Psi}$ and $\ket{\Phi}$, respectively, when subject to natural noise only. A scatter plot of repeated measurements, Fig. \[fig:fig3\](e), shows a systematically longer $T_2^*$ for $\ket{\Psi}$ than for $\ket{\Phi}$, indicating correlations in the noise. Using Eq. \[eq:T2s\], derived for quasi-static noise, we can extract from the decay of $\ket{\Psi}$ and $\ket{\Phi}$ a lower bound for the correlation factor, $\rho \geq 0.27\pm0.02$ (see Supplemental Material [@SuppMat]). In order to go beyond a lower bound and determine an estimate of $\rho$ from Eq. \[eq:T2s\], we also need at least one of the single-qubit dephasing times, which we measured to be $T_{2,1}^* = 0.97\pm0.02 \,\mu$s and $T_{2,2}^* = 0.59\pm0.02 \,\mu$s. Using both single-qubit $T_2^*$s in Eq. \[eq:T2s\] gives an overdetermined system of equations. We proceed by keeping $T_{2,1}^*/T_{2,2}^*$ equal to the measured ratio, and obtain a modest correlation factor, $\rho = 0.31\pm0.03$ (see Supplemental Material [@SuppMat]). In other experimental runs performed on the same sample, but separated in time by several months and with different gate voltage settings, we observed even smaller correlation factors.
We note that in keeping $T_{2,1}^*/T_{2,2}^*$ fixed, Eq. \[eq:T2s\] returns a value for $\sigma_1$ and $\sigma_2$ that is $\sim$15% larger than the measured value. The discrepancy may be in part due to the fact that the simple model that leads to Eq. \[eq:T2s\] assumes quasi-static Gaussian noise. This is a commonly made assumption in simple models of silicon spin qubits, but various experiments showed higher frequency noise to be relevant as well [@Veldhorst2014; @Yoneda2018; @Watson2018].
![Scatter plot of the two-qubit coherence times obtained in Hahn-echo style measurements for $\ket{\Psi}$ and $\ket{\Phi}$, from a fit to the data with an exponentially decaying sinusoidal function ($P_{\ket{00}} \propto e^{-t/T_2^*}$). Triangles represent data points where the Hahn echo pulses applied to both qubits are rotations around the $\hat{x}$-axis. For the circles, the rotation of Q1 is around $\hat{x}$ and the rotation of Q2 is around $\hat{y}$. Data points are averaged over $\sim$\[47, 66, 100, 148\] minutes. The average two-qubit Hahn echo coherence times are $2.03 \pm 0.09 \, \mu$s and $1.98 \pm 0.09 \, \mu$s for $\ket{\Psi}$ and $\ket{\Phi}$, respectively. Error bars are $\pm1\sigma$ from the mean.[]{data-label="fig:fig4"}](Figures_Fig4-eps-converted-to.pdf)
In order to gain insight into the frequency dependence of the spatial noise correlations, we perform measurements analogous to Hahn echo measurements. Here the delay times seen in the circuit diagrams of Fig. \[fig:fig3\](a,c) contain 180 degree rotations around the $\hat{x}$ or $\hat{y}$ axis applied to the two qubits, which reverse the time evolution resulting from static noise contributions (see the Supplemental Material [@SuppMat] for circuit diagrams and details). The results are presented in Fig. \[fig:fig4\]. The echo pulses prolong the two-qubit coherence times by a factor of $\sim 4-5$. We do not, however, observe a systematic difference in the echo decay times for the parallel versus anti-parallel Bell states, meaning there are no detectable spatial correlations in higher-frequency noise, and the correlations found in the Ramsey-style measurements of Fig. \[fig:fig3\] are mostly present in the low-frequency part of the spectrum.
To interpret the weak spatial correlations in the noise observed in the experiment, we first make a few observations. Multiple independent noise sources that each produce perfectly correlated noise ($\rho=1$) acting with the same relative amplitude on the two qubits, are equivalent to a single (stronger) source of perfectly correlated noise acting with this same relative amplitude on the two qubits. However, the effect of multiple independent asymmetric, correlated noise sources acting with randomly distributed relative amplitudes on the two qubits, rapidly becomes indistinguishable from uncorrelated noise. This is illustrated in an example simulation of the combined effect of three asymmetric, correlated noise sources, shown in the Supplemental Material [@SuppMat]. As a more extreme example, the combination of perfectly correlated and perfectly anti-correlated noise with equal amplitude, is equivalent to uncorrelated noise. All of these effects are described by (see Supplemental Material [@SuppMat]): $$\frac{T_{2,\ket{\Phi}}^*}{T_{2,\ket{\Psi}}^*} = \frac{\sigma_-}{\sigma_+} \propto \sqrt{\frac{\sum_i(\alpha_{i,1}-\alpha_{i,2})^2}{\sum_i(\alpha_{i,1}+\alpha_{i,2})^2}},
\label{eq:T2ratio}$$ where $\sigma_-$ and $\sigma_+$ are the standard deviations of the distributions of fluctuations in the difference and sum of the frequencies of the two qubits, respectively, and $\alpha_{i,j}$ is the coupling strength of noise source $i$ to qubit $j$.
We now discuss the effect of the known noise mechanisms acting on spin qubits and the expected spatial correlations for each mechanism. Fluctuating background charges in the substrate, interfaces or dielectrics directly affect the qubit splitting because of the magnetic field gradient produced by the micromagnets. When these charges are located close to the dots, they will generally couple differently to the two qubits, introducing asymmetric noise. Specifically for charge fluctuators located in between the two dots, even anti-correlated noise may result. For distant charges, the coupling becomes more symmetric, but several factors can lead to asymmetric noise amplitudes even in this case, for instance a difference in the confining potential between the two dots or a difference in the strength of the local magnetic field gradient. We have clear evidence of a pronounced difference in the confining potential of the two dots in this sample, based on the sensitivity of the respective qubit splittings to changes in gate voltages (see Supplemental Material [@SuppMat]). Similar considerations apply to the effect of gate voltage noise, which also couples to the qubit splitting through the magnetic field gradient. Another important noise source in this natural silicon substrate is hyperfine interaction with nuclear spins, for which little or no spatial correlations are expected [@Chekhovich2013].
Our expectations for the spatial noise correlations based on our understanding of the system physics are consistent with the experimental results and our theoretical observations on the combined effect of multiple noise sources. A picture emerges where noise from multiple distant charge fluctuators that affect the qubits asymmetrically due to their different confining potentials, is responsible for the (weak) spatial noise correlations at low frequency. Additional uncorrelated noise is introduced by the coupling to the nuclear spins.
In summary, we have presented an experimental study of spatial noise correlations based on the coherence of Bell states in a Si/SiGe two-qubit device. Experimentally we observe small spatial correlations in low-frequency noise, while for higher-frequency noise correlations appear to be absent. Our findings on the importance of asymmetric coupling of noise sources to two (or more) qubits can be exploited for reducing or enhancing spatial correlations in the noise in any qubit platform. For the case of spin qubits in quantum dots, this can be done for instance through a device design with engineered differences in confining potential or magnetic field gradient. In this respect, qubits encoded in two-electron spin states in dot-donor systems offer an extreme difference in confining potential [@Harvey-Collard2017b]. We anticipate that the optimization of future quantum error correction codes will go hand in hand with the design of qubits that either maximize or minimize spatial noise correlations.
Data supporting the findings of this study are available online [@Zenodo_Noise].
The authors acknowledge useful discussions with the members of the Vandersypen group, software support by F. van Riggelen, and technical assistance by M. L. I. Ammerlaan, O. W. B. Benningshof, J. H. W. Haanstra, J. D. Mensingh, R. G. Roeleveld, R. A. Schoonenboom, R. N. Schouten, M. J. Tiggelman, R. F. L. Vermeulen and S. Visser. We acknowledge financial support by Intel Corporation. Development and maintenance of the growth facilities used for fabricating samples is supported by DOE (DE-FG02-03ER46028). We acknowledge the use of facilities supported by NSF through the University of Wisconsin-Madison MRSEC (DMR-1121288). Research was sponsored by the Army Research Office (ARO), and was accomplished under Grant Number W911NF-17-1-0274. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office (ARO), or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.
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[Supplemental Material for]{}\
[**Spatial Noise Correlations in a Si/SiGe Two-Qubit Device from Bell State Coherences**]{}\
[Jelmer M. Boter,$^{1,*}$ X. Xue,$^{1,*}$ Tobias S. Krähenmann,$^1$ Thomas F. Watson,$^1$\
Vickram N. Premakumar,$^2$ Daniel R. Ward,$^2$ Donald E. Savage,$^2$ Max G. Lagally,$^2$ Mark Friesen,$^2$\
Susan N. Coppersmith,$^{2,\dagger}$ Mark A. Eriksson,$^2$ Robert Joynt,$^2$ and Lieven M. K. Vandersypen$^{1,3,\ddagger}$]{}\
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Measurement setup
=================
The measurement setup used in this work is the same as the setup used by @Watson2018 and @Xue2019. The measurements were done at a temperature of $T\approx30$ mK in an external magnetic field of B~ext~ = 617 mT. DC voltages are set via filtered lines from room-temperature digital-to-analog converters. Tektronix 5014C arbitrary waveform generators (AWGs) are connected to gates P1 and P2 via coaxial cables for gate voltage pulses. Keysight E8267D vector microwave sources are connected to gates MW1 and MW2 for EDSR. I/Q input channels of the microwave sources are connected to a master AWG to control frequency, phase and duration of the microwave bursts via I/Q modulation. The phase of the microwave drive signal determines the rotation axis in the $\hat{x}-\hat{y}$ plane of the Bloch sphere, and we update the rotating reference frame in software to perform $\hat{z}$ rotations [@Vandersypen2005]. Pulse modulation is used to increase the on/off ratio of the microwave bursts. The master AWG also controls the clock of the entire system and triggers all the other instruments. Data acquisition is done by a Spectrum M4i.44 digitizer card that is installed in the measurement computer. This card records the sensing dot current traces at a sampling rate of $\sim$60 kHz after passing through a 12-kHz Bessel low-pass filter (SIM965). Threshold detection is used to convert each trace to a single bit value (0 or 1) by the measurement computer.
Qubit characteristics
=====================
Q1 Q2
-------------------- ------------------------ ------------------------------
f 18.35 GHz 19.61 GHz
$T_1$ $>$50 ms [@Watson2018] 3.7$\pm$0.5 ms [@Watson2018]
$T^{*}_2$ 0.97$\pm$0.02 $\mu$s 0.59$\pm$0.02 $\mu$s
$T_2^{Hahn}$ 6.8$\pm$0.3 $\mu$s 2.8$\pm$0.2 $\mu$s
$F_{\ket{\Psi^+}}$
$F_{\ket{\Psi^-}}$
$F_{\ket{\Phi^+}}$
$F_{\ket{\Phi^-}}$
: Relevant single-qubit characteristics for simultaneous driving of both qubits, and Bell state fidelities F for the four Bell states. All errors are $\pm1\sigma$ from the mean.[]{data-label="tab:qubit_char"}
An upper bound on the residual exchange during single-qubit gates and free evolution of 100 kHz is determined, using the methods of @Watson2018.
Noise model
===========
We model the two-qubit system by the Hamiltonian: $$H = \frac{hf_1}{2}\sigma_1^Z + \frac{hf_2}{2}\sigma_2^Z,$$ where $h$ is the Planck constant, $f_i = \frac{g\mu_B B_i}{h}$ is the Larmor frequency for qubit $i$, $g$ is the electron g-factor, $\mu_B$ is the Bohr magneton, $B_i$ is the total magnetic field at the position of qubit $i$ and $\sigma_i^Z$ is the Pauli Z operator for qubit $i$. The two qubits are subject to dephasing noise, which we model as a fluctuating qubit frequency $f_i$. We assume Gaussian distributed noise with zero mean and covariance matrix $\Sigma$: $$\textbf{f} = (f_1,f_2) \sim \mathcal{N}((0,0),\Sigma); \,
\Sigma =
\begin{bmatrix}
\sigma_1^2 & \rho\sigma_1\sigma_2 \\
\rho\sigma_1\sigma_2 & \sigma_2^2
\end{bmatrix},$$ where $\sigma_i^2$ is the variance of the noise in $f_i$, and $\rho$ ($-1 \leq \rho \leq 1$) is a correlation factor. Positive $\rho$ indicates correlations, while negative $\rho$ indicates anti-correlations. We obtain the unitary time evolution operator by exponentiating the Hamiltonian: $$U = e^{-iHt/\hbar} =
\begin{pmatrix}
e^{-i\pi(f_1+f_2)t} & & & \\
& e^{-i\pi(f_1-f_2)t} & & \\
& & e^{i\pi(f_1-f_2)t} & \\
& & & e^{i\pi(f_1+f_2)t}
\end{pmatrix},$$ where $\hbar = \frac{h}{2\pi}$. Assuming quasi-static noise, we average over this unitary transformation by integrating over the joint probability distribution function: $$\rho(t) = \overline{U\rho(0)U^\dagger} = \frac{1}{2\pi\sqrt{\textrm{det}(\Sigma)}} \int U\rho(0)U^\dagger e^{-\textbf{f}^T\Sigma^{-1}\textbf{f}/2} d\textbf{f}.$$ The relevant expressions for anti-parallel ($\ket{\Psi}$) and parallel ($\ket{\Phi}$) Bell states are: $$\begin{split}
\bra{01}\overline{U\rho(0)U^\dagger}\ket{10} &= \frac{1}{2} \times \frac{1}{2\pi\sqrt{\textrm{det}(\Sigma)}} \int e^{-i2\pi(f_1-f_2)t} e^{-\textbf{f}^T\Sigma^{-1}\textbf{f}/2} d\textbf{f} = \frac{1}{2}\textrm{exp}\left[-2\pi^2t^2(\sigma_1^2 + \sigma_2^2 - 2\rho\sigma_1\sigma_2) \right], \\
\bra{00}\overline{U\rho(0)U^\dagger}\ket{11} &= \frac{1}{2} \times \frac{1}{2\pi\sqrt{\textrm{det}(\Sigma)}} \int e^{-i2\pi(f_1+f_2)t} e^{-\textbf{f}^T\Sigma^{-1}\textbf{f}/2} d\textbf{f} = \frac{1}{2}\textrm{exp}\left[-2\pi^2t^2(\sigma_1^2 + \sigma_2^2 + 2\rho\sigma_1\sigma_2) \right],
\end{split}$$ so the decay for anti-parallel and parallel Bell states is Gaussian with associated time scales (Eq. \[eq:T2s\] of the main text): $$\begin{split}
\left(\frac{1}{T_{2,\ket{\Psi}}^*}\right)^2 & = 2\pi^2 \left(\sigma_1^2 + \sigma_2^2 - 2\rho\sigma_1\sigma_2 \right), \\
\left(\frac{1}{T_{2,\ket{\Phi}}^*}\right)^2 & = 2\pi^2 \left(\sigma_1^2 + \sigma_2^2 + 2\rho\sigma_1\sigma_2 \right).
\end{split}
\label{eq:T2s_Supp}$$ Noting that in the case of Gaussian quasi-static noise for single-qubit decay $\left(\frac{1}{T_{2,i}^*}\right)^2 = 2\pi^2\sigma_i^2$, these expressions can be rewritten in terms of single-qubit coherence times: $$\begin{aligned}
\left(\frac{1}{T_{2,\ket{\Psi}}^*}\right)^2 & = \left(\frac{1}{T_{2,1}^*}\right)^2 + \left(\frac{1}{T_{2,2}^*}\right)^2 - 2\rho\frac{1}{T_{2,1}^*T_{2,2}^*},
\label{eq:T2Psi}\\
\left(\frac{1}{T_{2,\ket{\Phi}}^*}\right)^2 & = \left(\frac{1}{T_{2,1}^*}\right)^2 + \left(\frac{1}{T_{2,2}^*}\right)^2 + 2\rho\frac{1}{T_{2,1}^*T_{2,2}^*}.
\label{eq:T2Phi}\end{aligned}$$ Subtracting Eq. \[eq:T2Psi\] from Eq. \[eq:T2Phi\], we express the correlation factor $\rho$ in terms of the single- and two-qubit coherence times as: $$\rho = \frac{T_{2,1}^*T_{2,2}^*}{4} \left[\left(\frac{1}{T_{2,\ket{\Phi}}^*}\right)^2 - \left(\frac{1}{T_{2,\ket{\Psi}}^*}\right)^2 \right].
\label{eq:rho3}$$ In addition, Eqs. \[eq:T2Psi\] and \[eq:T2Phi\] allow one to formulate a sum rule, and to define a violation parameter $\Delta_s$ (with dimensions of a rate) that quantifies the difference between the model and experimental results: $$\left(\frac{1}{T_{2,\ket{\Psi}}^*}\right)^2 + \left(\frac{1}{T_{2,\ket{\Phi}}^*}\right)^2 - 2\left[ \left(\frac{1}{T_{2,1}^*}\right)^2 + \left(\frac{1}{T_{2,2}^*}\right)^2 \right] = \Delta_s^2,
\label{eq:Delta_s}$$ In this work, the four coherence times $T_{2,1}^*$, $T_{2,2}^*$, $T_{2,\ket{\Psi}}$ and $T_{2,\ket{\Phi}}^*$ are obtained individually and for matching model and experimental data $\Delta_s = 0$.
The model presented before assumes quasi-static noise. Without presenting the details here, a similar sum rule with corresponding violation parameter $\Delta_f$ can be obtained for non-quasi-static noise: $$\frac{1}{T_{2,\ket{\Psi}}^*} + \frac{1}{T_{2,\ket{\Phi}}^*} - 2\left(\frac{1}{T_{2,1}^*} + \frac{1}{T_{2,2}^*}\right) = \Delta_f.
\label{eq:Delta_f}$$ The violation parameter $\Delta_f$ is based on a sum rule for non-quasi-static noise, so coherence times obtained from exponential fits ($P_{\ket{00}} \propto e^{-t/T_2^*}$) have to be used in this expression.
Method verification
===================
To verify the method used in this work, we inject artificial noise in the experiments by applying random software Z rotations to the qubits and measure the coherence times for $\ket{\Phi}$ and $\ket{\Psi}$. These rotations are implemented by adding an evolution time dependent phase to the first microwave pulse after the waiting time, in addition to the phase to improve the fit of the decay. The frequency fluctuations corresponding to this extra phase are sampled from a Gaussian distribution with varying standard deviation. Adding (anti-)correlated noise is expected to have an effect on $\ket{\Phi}$ ($\ket{\Psi}$), but not on $\ket{\Psi}$ ($\ket{\Phi}$). The results of this control experiment are shown in Fig. \[fig:Supp1\]. Despite the large error bars on some of the data points, we clearly observe the expected trend, showing the method to be reliable
![Bell state coherence times for added software noise. The horizontal axis represents the amplitude of added noise in units of frequency. In the origin no noise is added, so this measurement only includes naturally existing noise. Moving to the right (left) on the horizontal axis, we increasingly add (anti-)correlated noise. Data points result from a fit to Gaussian decay, $P_{\ket{00}} \propto e^{-\left(t/T_2^*\right)^2}$. Error bars are $\pm1\sigma$ from the mean. The solid lines represent a theoretical prediction.[]{data-label="fig:Supp1"}](Fig_S1_Added_noise-eps-converted-to.pdf){width="50.00000%"}
Bell state fidelity
===================
The Bell state fidelity does affect the method used in this work. Imperfect Bell state initialization results in a finite amplitude for other off-diagonal elements of the density matrix than those of interest, which mixes in decays with different characteristic time scales. In the present experiments, the Bell states have not been characterized, but from the Bell state density matrices presented in the Supplementary Information of @Watson2018, we estimate the amplitudes of the not-intended density matrix elements to not exceed $\sim$19% of the elements of interest and most of them are much smaller. In our experiments we do not see clear deviation from a single decay.
Quantifying correlations
========================
In case the experimental data is not fully consistent with the simple quasi-static model, as quantified by the violation parameter $\Delta_s$ in Eq. \[eq:Delta\_s\], it is still possible to use this model to extract quantitative information on the correlations in the noise acting on the qubits based only on the two-qubit coherence times. From Eqs. \[eq:T2Psi\] and \[eq:T2Phi\], given the two-qubit coherence times, effective single-qubit coherence times can be calculated as: $$\left(\frac{1}{T_{2,1(2)}^*}\right)^2 = \frac{\left(\frac{1}{T_{2,\ket{\Phi}}^*}\right)^2 + \left(\frac{1}{T_{2,\ket{\Psi}}^*}\right)^2}{4} \mp \frac{1}{2}\sqrt{\left( \frac{\left(\frac{1}{T_{2,\ket{\Phi}}^*}\right)^2 + \left(\frac{1}{T_{2,\ket{\Psi}}^*}\right)^2}{2} \right)^2 - 4\left( \frac{\left(\frac{1}{T_{2,\ket{\Phi}}^*}\right)^2 - \left(\frac{1}{T_{2,\ket{\Psi}}^*}\right)^2}{4\rho} \right)^2},
\label{eq:T2_eff}$$ where the minus (plus) sign corresponds to Q1 (Q2), assuming $T_{2,1}^* \geq T_{2,2}^*$. Solutions only exist if the argument of the square root is equal to or larger than zero, so for $$|\rho| \geq \rho_{min} = \left| \frac{\left(\frac{1}{T_{2,\ket{\Phi}}^*}\right)^2 - \left(\frac{1}{T_{2,\ket{\Psi}}^*}\right)^2}{\left(\frac{1}{T_{2,\ket{\Phi}}^*}\right)^2 + \left(\frac{1}{T_{2,\ket{\Psi}}^*}\right)^2} \right|.$$ Using this simple model, we find a lower bound for the correlation factor $\rho_{min} = 0.27\pm0.02$.
Taking into account the experimental single-qubit coherence times and assuming their ratio ($\beta = \frac{T^*_{2,2}}{T^*_{2,1}}$) to be fixed, effective single-qubit coherence times can be obtained by adding Eqs. \[eq:T2Psi\] and \[eq:T2Phi\], and are given by: $$\left(\frac{1}{T_{2,1}^*}\right)^2 = \left(\frac{\beta}{T_{2,2}^*}\right)^2 = \frac{\beta^2}{2(1+\beta^2)} \left[ \left(\frac{1}{T_{2,\ket{\Phi}}^*}\right)^2 + \left(\frac{1}{T_{2,\ket{\Psi}}^*}\right)^2 \right].$$ The correlation factor $\rho$ from Eq. \[eq:rho3\] in that case is expressed as: $$\rho = \frac{\beta \left(T_{2,1}^*\right)^2}{4} \left[ \left(\frac{1}{T_{2,\ket{\Phi}}^*}\right)^2 - \left(\frac{1}{T_{2,\ket{\Psi}}^*}\right)^2 \right].
\label{eq:rho_3_fixed_ratio}$$ For the experimental value $\beta = 0.61\pm0.02$ ($T_{2,1}^* = 0.97\pm0.02 \,\mu$s and $T_{2,2}^* = 0.59\pm0.02 \,\mu$s), we find a correlation factor $\rho = 0.31\pm0.03$, and effective single-qubit coherence times $T_{2,1}^* = 0.84\pm0.03 \, \mu$s and $T_{2,2}^* = 0.51\pm 0.02 \, \mu$s.
Echo experiments
================
Dynamical decoupling sequences can be used to investigate the frequency dependence of spatial noise correlations, similar to mapping out the frequency spectrum of noise acting on a single qubit [@Muhonen2014; @Kawakami2016; @Yoneda2018].
![Circuit diagrams for two different versions (XX (a,b) and XY (c,d)) of an experiment analogous to the measurement of a Hahn echo for $\ket{\Psi}$ (a,c) and $\ket{\Phi}$ (b,d).[]{data-label="fig:Supp2"}](Figures_Supp2-eps-converted-to.pdf)
In addition to the measurements analogous to Ramsey experiments, we performed measurements analogous to a Hahn echo experiment with a single decoupling pulse on each qubit halfway the waiting time. Results are presented in Fig. \[fig:fig4\] of the main text. We performed two versions of the echo experiment to which we refer as XX and XY echo, respectively. In the XX echo experiment we apply a $\pi_X$ pulse on both qubits, which transforms $\ket{\Psi} = (\ket{\downarrow\uparrow} -i \ket{\uparrow\downarrow})/\sqrt{2}$ into $\ket{\Psi'} = (\ket{\downarrow\uparrow} +i \ket{\uparrow\downarrow})/\sqrt{2}$, and $\ket{\Phi} = (\ket{\downarrow\downarrow} -i \ket{\uparrow\uparrow})/\sqrt{2}$ into $\ket{\Phi'} = (\ket{\downarrow\downarrow} +i \ket{\uparrow\uparrow})/\sqrt{2}$, as shown in the circuits in Figs. \[fig:Supp2\](a,b). The XY echo experiment consists of a $\pi_X$ pulse on Q1 and a $\pi_Y$ pulse on Q2, which transforms $\ket{\Psi}$ and $\ket{\Phi}$ to itself, as shown in the circuits in Figs. \[fig:Supp2\](c,d). The difference between the XX and XY sequences is analogous to that between single-qubit echo pulses around $\hat{x}$ versus $\hat{y}$. We do note that for both versions of the two-qubit decoupling used in this work, the two-qubit state is taken out of the logical qubit space during the pulses.
Adding multiple independent noise sources
=========================================
To derive Eq. \[eq:T2ratio\] of the main text, consider a single noise source $i$ with coupling strength $\alpha_{i,1}$ and $\alpha_{i,2}$ (which can be expressed for instance in units of MHz/mV, if noise source $i$ is expressed in units of mV) to qubit 1 and qubit 2 respectively. The noise source fluctuates with standard deviation $\sigma$. The standard deviations of the fluctuations in the difference ($f_1-f_2$) and sum ($f_1+f_2$) of the frequencies are then given by: $$\begin{split}
\sigma_{i,-} &= \sigma|\alpha_{i,1}-\alpha_{i,2}|, \\
\sigma_{i,+} &= \sigma|\alpha_{i,1}+\alpha_{i,2}|. \\
\end{split}
\label{eq:sigma_dif_sum}$$ For $N$ independent noise sources the combined standard deviation is given by: $$\sigma^2 = \Sigma_i^N \sigma_i^2.
\label{eq:sigma_sum}$$ Combining Eqs. \[eq:sigma\_dif\_sum\] and \[eq:sigma\_sum\] gives: $$\begin{split}
\sigma_- = \sqrt{\Sigma_i^N \sigma_{i,-}^2} = \sigma\sqrt{\Sigma_i^N \left(\alpha_{i,1}-\alpha_{i,2}\right)^2}, \\
\sigma_+ = \sqrt{\Sigma_i^N \sigma_{i,-}^2} = \sigma\sqrt{\Sigma_i^N \left(\alpha_{i,1}+\alpha_{i,2}\right)^2},
\end{split}$$ where we absorb differences in standard deviations between noise sources in the coupling strengths. Since $T_2^* \propto \frac{1}{\sigma}$, this yields Eq. \[eq:T2ratio\] of the main text: $$\begin{aligned}
\frac{T_{2,\ket{\Phi}}^*}{T_{2,\ket{\Psi}}^*} &= \frac{\sigma_-}{\sigma_+} \propto \sqrt{\frac{\sum_i(\alpha_{i,1}-\alpha_{i,2})^2}{\sum_i(\alpha_{i,1}+\alpha_{i,2})^2}}.\end{aligned}$$
Simulation of multiple asymmetric noise sources
===============================================
The result of a simulation of the combined effect of three asymmetric, correlated noise sources is shown in Fig. \[fig:Supp3\]. The standard deviations of the distributions of fluctuations in difference and sum frequencies indicate that only modest correlations in the noise remain for their combined effect.
 of the main text. The coupling factors to the two qubits for these and five other gate electrodes are tabulated in Table \[tab:coupling\_factors\]. For all three gate electrodes, voltage fluctuations are sampled from a Gaussian distribution with 50 $\mu$V standard deviation. After sampling gate voltage fluctuations, the corresponding total frequency fluctuations for both qubits, and their difference and sum are calculated. The distributions of the fluctuations in (a) difference and (b) sum frequency are plotted.[]{data-label="fig:Supp3"}](Figures_Supp3-eps-converted-to.pdf)
Q1 Q2
-------- -------- -------
P1 -1 -2
P2 0.175 0.8
MW1 -0.015 0.025
MW2 0.8 8.5
B 0.43 0.36
LD -0.1 -1.44
accQD 0.9 -1.8
accRes -0.8 -3.75
: Coupling factors (in MHz/mV) of eight of the surface gate electrodes on our sample to the two qubits.[]{data-label="tab:coupling_factors"}
[^1]: These authors contributed equally to this work.
[^2]: These authors contributed equally to this work.
[^3]: Present address: School of Physics, University of New South Wales, Sydney NSW 2052, Australia.
[^4]: To whom correspondence should be addressed: [[email protected]](mailto://[email protected])
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Deep Learning (DL) algorithms have become the [*de facto*]{} Machine Learning (ML) algorithm for large scale data analysis. DL algorithms are computationally expensive – even distributed DL implementations which use MPI require days of training (model learning) time on commonly studied datasets. Long running DL applications become susceptible to faults – requiring development of a fault tolerant system infrastructure, in addition to fault tolerant DL algorithms. This raises an important question: [*What is needed from MPI for designing fault tolerant DL implementations?*]{} In this paper, we address this problem for permanent faults. We motivate the need for a fault tolerant MPI specification by an in-depth consideration of recent innovations in DL algorithms and their properties, which drive the need for specific fault tolerance features. We present an in-depth discussion on the suitability of different parallelism types (model, data and hybrid); a need (or lack thereof) for check-pointing of any critical data structures; and most importantly, consideration for several fault tolerance proposals (user-level fault mitigation (ULFM), Reinit) in MPI and their applicability to fault tolerant DL implementations. We leverage a distributed memory implementation of Caffe, currently available under the Machine Learning Toolkit for Extreme Scale (MaTEx). We implement our approaches by extending MaTEx-Caffe for using ULFM-based implementation. Our evaluation using the ImageNet dataset and AlexNet, and GoogLeNet neural network topologies demonstrates the effectiveness of the proposed fault tolerant DL implementation using OpenMPI based ULFM.'
author:
- Vinay Amatya
- Abhinav Vishnu
- Charles Siegel
- Jeff Daily
bibliography:
- 'bibTexts.bib'
title: 'What does fault tolerant Deep Learning need from MPI?'
---
Introduction {#sec:intro}
============
Deep Learning (DL) algorithms are a class of Machine Learning and Data Mining algorithms, which conduct model learning by emulating the computational structure of a mammalian brain. A deep neural network (DNN) – which stores the model of a DL algorithm – contains several [*layers*]{} of [*neurons*]{} inter-connected with [*synapses*]{}. By using deep layers, DL algorithms are able to conduct transformations on highly non-linear data, which is commonplace in many scientific datasets. DL algorithms have shown amazing results in many Computer Vision tasks [@NIPS2012_4824; @43022] and science domains such as High Energy Physics [@Baldi:2014kfa], Climate Modeling [@liu2016application] and Computational Chemistry [@goh2017deep]. DL implementations such as TensorFlow [@tensorflow2015-whitepaper], Caffe [@jia2014caffe], Theano [@bergstra+al:2010-scipy; @Bastien-Theano-2012], and Torch [@Collobert02torch:a] have become available. These implementations are primarily geared towards compute nodes that may contain a multi-core architecture (such as Intel Xeon/KNC/KNL) and/or many-core architectures (GPUs) as commonplace in Leadership Class Facilities (LCFs).
DL algorithms can be applied to a variety of input representations. The tabular input representations typically leverage Multi-layer Perceptrons (MLPs). The images, videos and speech tend to leverage the Convolutional Neural Networks (CNNs) and Recurrent Neural Networks (RNNs). The CNNs and RNNs are computationally expensive and typically require significant time for training even on relatively modest data set sizes with modest number of hidden layers. The problem is further exacerbated by the increasing number of layers (such as recently proposed Residual Networks have up to 1000 layers) and ever-increasing volume of data produced by simulations, experiments and hand-held devices. An important solution to these problems is the design and implementation of DL algorithms that are capable of execution on distributed memory systems. Table \[table:newcomparisons\] shows a table of prominent distributed DL implementations.
Name HPC Ready Fault Tolerance
----------------------------------------- ----------- -----------------
FireCaffe [@firecaffe]
S-Caffe [@scaffe]
MaTEx [@matex]
Poseidon [@poseidon]
Petuum [@petuum]
GeePS [@geeps] ?
ProjectAdam [@projectadam]
TensorFlow [@tensorflow2015-whitepaper]
MXNET [@mxnet]
CaffeonSpark [@caffeonspark]
SparkNet [@sparknet]
DogWild [@sparknet] ?
CNTK [@cntk]
Parle [@parle]
PaddlePaddle [@mxnet]
Caffe2 [@goyal:arxiv17]
Proposed FT-Caffe
: []{data-label="table:newcomparisons"}
\
An important artifact of the large scale systems is the increased frequency of faults, which are commonplace in large scale systems [@Schroeder:4775906]. Distributed DL implementations such as distributed TensorFlow, distributed memory implementations of Caffe and even recently proposed Caffe2 [@Caffe2] are primarily geared towards performance. As shown in Table \[table:newcomparisons\], we observe that state of the art HPC ready DL implementations are not fault tolerant. On the other hand, automatic fault tolerance is provided by MapReduce instantiations such as Hadoop, and Spark. However, the implementations are not HPC ready. At the same time, DL implementations are known to take days even on modest size datasets, significantly increasing the probability of observing a fault during the training phase. This raises two important questions: [*1) What are the elements of fault tolerant DL algorithms? and 2) What is needed from MPI for implementing these fault tolerant DL algorithms?*]{}
Contributions
-------------
In this paper, we address these questions and make the following contributions:
- We present the case for several types of parallelism (model, data and hybrid) as motivated from common use-cases and DNN topologies. We use this discussion to derive the suitability of fault tolerance proposals in MPI.
- We consider several design choices for implementing fault tolerant DL implementations. Specifically, we consider checkpoint-restart, Reinit (when a fault occurs, re-initialize the MPI automatically) and user-level fault mitigation.
- We consider several approaches for recovery from faults. We primarily rely on “continued execution” – where the DL implementation continues to execute by using the remaining set of compute nodes.
- We implement our design using MaTEx-Caffe and leverage the ULFM implementation available with OpenMPI. We provide an evaluation of fault tolerant MaTEx-Caffe using the ImageNet-1K dataset and widely studied neural network topologies such as AlexNet and GoogLeNet.
Our evaluation on a 16 node Intel Haswell system connected with InfiniBand indicates that the proposed fault tolerant MaTEx-Caffe is able to scale well and continues execution in the presence of actual permanent node faults. It incurs no observable overhead in the absence of faults, and provides expected performance after recovering from faults, since the overall number of compute nodes are reduced. We also observe that both Reinit [@Reinit] and ULFM [@Bland2013] proposals are suitable for addressing permanent node faults for DL algorithms. However, ULFM is simple – and versatile enough – since it obviates the need for any checkpoint/restart, re-reading of the entire dataset and allows continued execution in the presence of permanent node faults.
The rest of the paper is organized as follows: In section \[sec:background\], we present the background of the proposed research. We present elements of scalable DL algorithms in section \[sec:motivation\] and a solution space for fault tolerant MaTEx-Caffe in section \[sec:design\]. In section \[sec:existing\_proposals\] we present existing features and proposals for fault tolerance in MPI and provide implementation details in section \[sec:implDetail\]. We discuss experimental results in section \[sec:exp\], related work in section \[sec:related\] followed by conclusions in section \[sec:conclusions\].
Background {#sec:background}
==========
In this section, we provide a brief overview of the proposed research. Specifically, we focus on deep neural networks (DNNs).
Deep Neural Networks
--------------------
Symbol Meaning
---------------- ----------------------------------- -- -- -- -- -- -- -- --
$W^{(\ell)}$ weights of layer $\ell$
$b^{(\ell)}$ biases of layer $\ell$
$z^{(\ell+1)}$ $W^{(\ell)}a^{(\ell)}+b^{(\ell)}$
$a^{(\ell)}$ $\sigma(z^{(\ell)})$
$\sigma$ $\mathrm{ReLU}(x)=\max(0, x)$
$n_\ell$ number of layers
: Symbols for Backpropagation.[]{data-label="table:notation"}
\
A dataset is a collection of samples. Samples are often images, speech, text or raw vectors of numbers. ML algorithms typically split a dataset into a [*training set*]{}, used for learning the details of a model; a [*validation set*]{}, used to prevent overfitting and to tune hyper-parameters; and a [*testing set*]{}, used for the accuracy calculation after the final model is trained. Deep neural networks (DNNs) are a class of ML algorithm that learn nonlinear functions by emulating the computational structure of a mammalian brain. It consists of simple computational units called [*neurons*]{} which are connected with [*synapses*]{}.
The values of the synapses, called [*weights*]{} are learned through the [*back-propagation*]{} algorithm. It iteratively updates the weights of the DNN to find a local minimum of an objective/ cost function. With this algorithm, each sample is an input to the [*feed-forward*]{} step. The output is a [*predicted value*]{} which is compared to a [*label*]{}. The difference between the label and predicted value is used to calculate the [*gradients*]{} which are applied to update the weights. This difference is called the [*cost*]{} and the objective of training is to minimize this value on the training set while ensuring that the value on the validation set decreases as well.
Back-propagation is a special case of [*gradient descent*]{}. Any gradient descent variant uses the update rule $$\begin{aligned}
\label{graddesc}
\mathbf{w}'&=&\mathbf{w}+\lambda \nabla_{\mathbf{w}}C\\
\mathbf{b}'&=&\mathbf{b}+\lambda \nabla_{\mathbf{b}}C.\end{aligned}$$ where $\mathbf{w}$ are the weights, $\mathbf{b}$ the biases, $\lambda$ the learning rate, and $C$ is a cost function to be optimized.
We use the notation of Table \[table:notation\] and describe back-propagation in Algorithm \[alg:bp\].
\[alg:bp\] **input:** Data $X\in {\mathbb{R}}^{n\times p}$ and labels $Y\in {\mathbb{R}}^{n\times \ell}$ Compute all $z^{(\ell)}$ and $a^{(\ell)}$. $\delta^{(n_\ell)} = -(y-a^{n_{\ell}})\odot \sigma(z^{(n_\ell)})$ $\delta^{(\ell)}=W^{\ell} \delta^{(\ell+1)}\odot \sigma'(z^{(\ell)})$ $\nabla_{W^{(\ell)}}C = \delta^{(\ell+1)}{a^{(\ell)}}^T$ $\nabla_{b^{(\ell)}}C = \delta^{(\ell+1)}$
Algorithm \[alg:bp\] is most directly applicable to fully-connected neural networks. For structured data, however, convolutional neural networks (CNNs) are more useful. The fundamental unit of computation in a CNN are convolutions – which are stored as arrays of some dimension – unlike vectors in a fully-connected neural network as described above. Each neuron in a convolution layer considers input from a small window (such as `3x3,5x5`) in an image, applies a convolution and computes a value. The computation can be reduced to a matrix-vector multiplication with redundant weights, allowing the above algorithm to be applied.
Elements of Deep Learning Algorithms for Fault Tolerance Consideration {#sec:motivation}
======================================================================
In this section, we present the motivation of our work. Specifically, we consider the properties of DL algorithms, distinguishing between MLPs, CNNs and RNNs in terms of their expected execution on large scale systems. This distinction provides the necessary guidelines for requirements from MPI in terms of fault tolerance. As pointed out by Gropp and Lusk [@gropp:ijhpca02], “fault tolerance is a property of MPI programs and specification”. Hence, it is critical to consider these in conjunction. Our first element of discussion is the expected type of parallelism for scaling out DL algorithms.
Master-Slave Paradigm
---------------------
Over the last few years, several researchers have considered the possibility of scaling out DL algorithms [@NIPS2012_0598; @geeps; @projectadam; @mxnet; @caffeonspark]. The classical work in scaling out DL algorithms considered a [*master-slave*]{} paradigm, which was proposed under the DistBelief framework [@NIPS2012_0598]. It considered a hierarchical organization of [*parameter servers*]{} which would hold the latest copy of the model. The [*workers*]{} would periodically update the master with their updates and request the latest copy of the model. Several extensions to this fundamental paradigm have been proposed in the literature [@geeps; @mxnet; @firecaffe]. The limitations of the master-slave model have been well-studied in the distributed systems research [@das:arxiv16; @lian:nip15]. In addition to being a single point of failure, and a communication bottleneck, the limitation of this approach is that the convergence of master-slave paradigm worsens at scale-out. For extreme scale systems, this approach is infeasible. Hence, we disregard this approach which would be leveraged in practical deployments especially of HPC systems such as Leadership Class Facilities (LCFs).
It is also worthwhile noting that this approach is amenable to fault tolerance, especially if the reliability of the parameter server is higher than workers. A possible implementation in the master-slave paradigm is either re-spawning of new workers and splitting the original training set among these new workers (by using `MPI_Comm_spawn`) or continue executing in the presence of faults using the remaining set of compute nodes. Other researchers have made similar observations in the context of generic master-slave applications [@gropp:ijhpca02] and they are readily applicable to DL algorithms. However, due to the fundamental scaling issues of the master-slave paradigm for DL implementations, we disregard this approach from implementation.
Model Parallelism
-----------------
Another possibility which has been presented in literature for scaling out DL implementations is [*model parallelism*]{}. In this specific type, individual layers of the overall DNN model are split among different compute nodes. The training set itself is split among the compute nodes as well. Let us consider the example of the AlexNet neural network topology as shown in Figure \[fig:alexnet\_topology\]. In a sample execution of model parallelism, each of the hidden layers is resident on a single compute node.
![A pictorial representation of AlexNet [@imagenet] neural network topology[]{data-label="fig:alexnet_topology"}](./alexnet_topology.pdf){width="0.7\columnwidth"}
During the feedforward step, a batch of samples is executed on the first hidden layer. The output of the first hidden layer – which is typically referred to as activations – is forwarded to the next hidden layer, resulting in point-to-point communication between two compute nodes. This procedure is repeated untill the last layer of the DNN is reached, at which point the [*error*]{} is calculated. During the back-propagation step, the error is used to calculate the updates to the weights ([*gradients*]{}) which are communicated between compute nodes in the reverse order to the feedforward step.
Data Parallelism
----------------
A widely used option in scaling out DL implementations is [*data parallelism*]{} [@zheng:icpads16; @siegel:bigdata16; @siegel2016adaptive; @vishnu:cluster15-a; @vishnu:cluster15-b; @vishnu:ipdps16; @shohdy:icpp16; @shohdy:hipc16; @zheng:icpads16]. Under this type of parallelism, the model is replicated and the data is split among multiple compute nodes. A pictorial representation of the data parallelism is shown in the Figure \[fig:data\_parallelism\].
![A pictorial representation of data parallelism in DL algorithms using AlexNet neural network topology and four compute nodes. The model is synchronized at the end of each batch using `MPI_Allreduce` and other primitives such as NVIDIA Collective Communication Library (NCCL)[]{data-label="fig:data_parallelism"}](./data_parallelism.pdf){width="0.7\columnwidth"}
As shown in the figure, at the end of each batch each compute node (assuming that the implementation uses shared address space programming model such as OpenMP/pthread on a node) executes an `MPI_Allreduce`. By executing the all-to-all reduction primitive, the algorithm ensures that it is equivalent to the default sequential DL algorithm [@Keuper:MHLPC2015]. An important consideration for data parallelism is that strong scaling of work is essential to ensure the equivalence of the implementation to the sequential algorithm. Specifically, let us consider a batch size $b$, and let $n$ be the number of compute nodes. The overall expected complexity of the data parallelism based implementation is $\Theta\left(\frac{b}{n} + \log(n)\right)$. Naturally, the ratio of communication to computation increases with strong scaling – which is a potential downside to data parallelism. Several solutions have been proposed to handle this situation [@Krizhevsky14oneweird]. One possibility is to consider increasing the batch size and increasing the values of other hyper-parameters (such as learning rate) by Krizhevsky [@Krizhevsky14oneweird]. Recently proposed solutions such as S-Caffe [@scaffe] improve the scalability of data parallelism by leveraging the overlap of communication with computation. While a few of these approaches provide strict equivalence to the sequential algorithm, other approaches such as asynchronous variants (also referred to as asynchronous gradient descent (AGD)) are still useful, but do not provide strict equivalence to the default stochastic gradient descent (SGD) algorithm.
Scalability Analysis
--------------------
An advantage of model parallelism is its potential to scale-out the DL algorithms very well. For example, if there are 1,000 layers, then in an ideal situation each compute node may have one layer, resulting in scale-out. However, there are several reasons to not consider pure model parallelism based techniques for scaling out the DL implementations.
Method Symbol
--- ------------------------------------ ----------------- -- --
1 Features in previous layer $n_1$
2 Features in current layer $n_2$
3 Activation shape in previous layer $x_1\times x_2$
4 Window size in current layer $w_1\times w_2$
5 Strides for current layer $s_1\times s_2$
: Symbols Used For Computing Activations and Parameters[]{data-label="table:activations"}
\
For any DNN with several hidden layers, let us consider two consecutive layers: $L_1$ and $L_2$. We compute the number of parameters and activations for $L_2$ as follows: If $L_2$ is a convolution layer, then the array containing the parameters is $w_1\times w_2\times n_1\times n_2$. However, the activations are an array of size $\frac{x_1}{s_1}\times \frac{x_2}{s_2}\times n_2$. As $w_i<\frac{x_i}{s_i}$, the number of parameters is less than the number of activations. Conversely, if $L_2$ is a fully-connected layer, then there are $n_1\times n_2$ parameters and only $n_2$ activations. We note that if $L_1$ is a convolution layer, then $n_1$ must be replaced by the total number of activations of that layer, namely $\frac{x_1}{s_1}\times \frac{x_2}{s_2}\times n_1$. This implies, in general, convolutional networks have lesser communication volume of weights than activations. However, for fully-connected networks the trend is reversed. In the case of AlexNet (shown in Figure \[fig:alexnet\_topology\]), a well-established DNN architecture, this can be seen directly. The first convolutional layer contains 34,848 parameters, but has 301,056 activations, a difference of an order of magnitude. The final fully-connected layer, however, has 4,096,000 parameters and 1,000 activations.
Lessons Learned
---------------
In the previous section, we provided an in-depth discussion on the possibilities of scaling out DL implementations. While model parallelism looks attractive, in practice the ratio of communication of activations to model parameters prohibits effective scaling. This is because CNNs are increasingly becoming commonplace – including the winners of last 5 years of ImageNet classification challenge [@imagenet]. At the same time, data parallelism provides scaling out possibilities, but has limitations regarding the growth of batch size [@Keuper:MLHPC2016]. However, solutions proposed by other researchers [@Krizhevsky14oneweird; @scaffe] still make data parallelism an amenable choice for scaling out DL implementations. It is worth noting that a possibility which combines DL model and data parallelism – hybrid parallelism – has been proposed in literature as well [@das:arxiv16]. However, usually model parallelism is applied on multiple GPUs/multiple sockets on each compute node and data parallelism is applied for multiple compute nodes. In essence, we already consider hybrid parallelism, while implicitly leveraging model parallelism within a node. [**Hence, we consider data parallelism for fault tolerance considerations in rest of the paper.**]{}
Solution Space {#sec:design}
==============
In this section, we present a solution space for designing fault tolerant DL implementations using data parallelism, as discussed in the previous section. An important consideration is exploring the suitability of existing features in MPI for this purpose, with detailed considerations for the primary proposals.
Critical Data Structures in DL Algorithms
-----------------------------------------
The first step is the identification of critical data structures in DL algorithms. Specifically, there are several data structures which are used during the feed-forward and the back-propagation phase of the DL implementations. During the feed-forward step, the input dataset and the model are used – both of which are read-only during the step. However, at the back-propagation step, the model weights are updated while the input dataset remains untouched. Hence, the critical data structure for DL implementations is the model weights which are updated iteratively till convergence. It is worth noting that there are auxiliary data structures which are updated as well. As an example for DL implementations with [*momentum*]{}, data structures such as [*history*]{} are updated, which can be recalculated from the model weights. The gradients – iterative updates to the model weights – are calculated during the back-propagation step. However, they are accumulated iteratively in the model weights, and hence they are not critical.
Process Recovery Model
----------------------
An important design point is the process recovery model. Several scientific applications such as LULESH [@IPDPS13:LULESH] and NAS Parallel Benchmarks [@Bailey:SC91] typically require a fixed topology (such as a square/quadratic) in terms of number of MPI processes. However, there are no such requirements for DL implementations. Hence, it is possible to continue execution with the remaining set of compute nodes, without affecting the correctness of the DL implementation. A natural advantage of this approach is that it requires little support from process managers for practical deployments. Hence, we use this approach for designing fault tolerant DL implementation.
Suitability of Existing MPI Features and Proposals {#sec:existing_proposals}
==================================================
Suitability of Existing Error Handling in MPI
---------------------------------------------
An important design consideration is the suitability of existing error handling in MPI for designing fault tolerant DL implementations. Specifically, by initializing with `MPI_ERRORS_RETURN`, it is possible for a DL implementation to prevent aborting when a fault is detected either during MPI communication or an external system software component (such as SLURM).
By setting an explicit error handler, it is possible for the DL implementation to checkpoint their critical data structures, exit the application and re-start the application from the recent saved checkpoint. In this specific case, the critical data structure is the model parameters of the DNN, since the dataset is read-only, and it can be readily recovered from the disk/file-system. The DL implementation may be re-started using $n$ compute nodes (if spare compute nodes are available) or $n - 1$ compute nodes, since DL implementations do not have specific requirements of a topology. This approach is definitely a suitable possibility. However, it may not be necessary, since this will result in a recovery complexity of $O(n)$, a function of number of compute nodes, instead of the degree of failure. The reasoning behind this time complexity is due to the fact that the $n - 1$ compute nodes would need to read the entire dataset back from the disk (prohibitive data movement), in addition to the checkpointed model files. Hence, this approach may be considered as the baseline approach, but not necessarily as the optimal approach for handling permanent node faults in DL implementations.
Suitability of User-Level Fault Mitigation (ULFM) Proposal
----------------------------------------------------------
One of the fault tolerance proposal which has been considered for inclusion in the MPI specification for the last few years is ULFM. The salient features of ULFM are: 1) ability to provide non-collective global fault notification, 2) ability to recover from faults by fixing the [*broken*]{} communicator on the fly and 3) support for fixed/shrinking process set.
ULFM is particularly suited for applications which have small process-specific state information. Usually, resetting the global state information is non-trivial and requires writing a complex error handler. Naturally, for large-scale applications – which have been developed over decades – writing a correct error handler even for a subset of fault cases is non-trivial.
However, with data parallelism the overall state information that is required for DL implementations is minimal. Since the model is replicated across the compute nodes, the DL implementation requires no checkpointing. Hence, ULFM is potentially the right fit for implementing fault tolerant DL algorithms.
Suitability of Reinit Fault Tolerance Proposal
----------------------------------------------
Recently, Laguna [*et al.*]{} have proposed [*Reinit*]{} proposal for handling faults in MPI. The objective of Reinit is to address the limitations of ULFM, and is particularly suitable to applications where the code complexity of the recovery module is high. The salient features of the Reinit proposal are: 1) automatic re-initialization of MPI after a fault is detected, 2) semi-automatic recovery from the intermediate checkpoints, and 3) ability to handle shrinking/fixed process set.
We consider the suitability of the Reinit proposal to data parallelism based DL implementations. We observe that DL implementations would be required to check-point the model weights periodically, which would be used by the Reinit implementation during recovery. We also observe that the application would be re-started requiring the entire dataset to be read from the disk.
Lessons Learned
---------------
We observe that existing local fault notification in the MPI specification and implementations may be used for developing fault tolerant DL implementations. However, there is a significant amount of work needed within MPI and at the application level (such as intermediate checkpointing) to leverage the existing functionality. The Reinit proposal is suitable as well. However, there are two potential downsides that are readily observed: 1) Reinit would require DL implementations to consider intermediate check-pointing, when the DL algorithm does not mandate it, and 2) Reinit would require application to read the entire dataset from the disk, when reading the data could be fairly localized to the degree of failure.
ULFM has positive attributes which are definitely suitable for designing fault tolerant DL implementations. The primary functionality that is required is automatic fixing of the communicator, and reading the partial dataset from the disk, while continuing to execute with the existing set of compute nodes. The ULFM specification itself has a few implementation caveats, including the cost of fault detection (which is relatively lesser for Reinit), cost of global notification and cost of recovering the communicator. Yet, the overall cost of computation recovery is at most one batch – while in the case of Reinit, it is expected to be much higher depending upon the degree of checkpointing. Hence, we consider ULFM for implementing fault tolerant DL algorithms. In the next section, we present the implementation details for ULFM based DL algorithm.
Design and Implementation Details {#sec:implDetail}
=================================
In this section, we present design and implementation details for fault tolerant DL algorithms. We leverage the ULFM implementation provided by OpenMPI for this purpose and implement our changes in Caffe runtime. Figure \[fig:code\_snippet\] shows the overall interaction between different components.
![Overall Caffe architecture and code flow for the original code. We require changes to dataset reading and the callback for implementing fault tolerance.[]{data-label="fig:code_snippet"}](./code_snippet.pdf){width="0.8\columnwidth"}
As shown in figure, the Caffe architecture has layers such as for MLPs, CNNs and RNNs, which are defined in a prototxt file. Hence, these are already resident on disk. Our extensions of Caffe – also referred to as MaTEx-Caffe for rest of the paper – support parallel NetCDF format which requires changes for fault tolerance. Caffe runtime supports several types of solvers such as SGD, Adam and others. These solvers use a common substrate for data parallelism, where `ForwardBackward()` step computes the forward and back-propagation steps of the overall implementation. The resulting gradients are then synchronized using a callback, which is extended by MaTEx-Caffe for using `MPI_Allreduce`. The resulting synchronized gradients are then applied using the `ApplyGradients` function before the next batch of samples are ingested.
Pseudo-code Walk-through
------------------------
Figure \[fig:ulfm\_code\] shows the difference between the original non-fault-tolerant implementation of the callback `on_gradients_ready` and the fault tolerant version on the right. Figure \[fig:code\_flow\] presents the code changes for data readback in the fault tolerant version and compares it to the original code.
![Code snippet for ULFM based fault tolerant DL implementation[]{data-label="fig:ulfm_code"}](./ulfm_code.pdf){width="\columnwidth"}
### Original Callback
The original call-back receives the gradients from the `Forwardbackward` function and uses an all-to-all reduction to synchronize the gradients across all compute nodes. The resulting gradient is divided by the number of compute nodes and applied to the local model using `ApplyGradients` function.
### Callback with ULFM Changes
As shown in Figure \[fig:ulfm\_code\], the ULFM changes are handful and primarily restricted to a single callback. Specifically, when a fault is detected, we leveraget the `MPIX_Comm_shrink` function is used to shrink the communicator from original to the new communicator. Once the communicator is reset, then all to all reduction is retried till return code is `MPI_SUCCESS`.
### Data Readback for Fault Tolerant Version
Figure \[fig:code\_flow\] shows the original code flow and the code flow for fault tolerant DL. The data is read only when a fault is detected by the `on_gradients_ready`.
![Code flow for original and fault tolerant implementations for data readback[]{data-label="fig:code_flow"}](./code_flow.pdf){width="\columnwidth"}
Performance Evaluation {#sec:exp}
======================
In this section, we present a detailed performance evaluation of the proposed fault tolerant approach using ULFM. Table \[table:arch\] shows the hardware and software details of our systems. Table \[table:heur\_desc\] shows a description of proposed approaches that we have implemented. Table \[table:datasets\] provides an overview of datasets, and the associated neural networks.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Name CPU (\#cores) GPU Baseline Caffe Network MPI cuDNN CUDA Nodes \#cores
---------------------------------------------------------------------------------------------------------- --------------- ----- ---------------- --------- ----- ------- ------ ------- ---------
**[PUMA]{} & Haswell (20) & N/A & Intel-Caffe [@caffe-intel] & IB & ULFM-OpenMPI & N/A & N/A & 16 & 320\
**
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\
Name Type Implemented Description of DL Algorithm and Implementation
--------------- -------------------------------------------- ------------- -----------------------------------------------------------------------
SGD (Default) Stochastic Gradient Descent Yes Implements strong scaling by dividing batch and all-to-all reduction.
FT-SGD Fault Tolerant Stochastic Gradient Descent Yes Implements strong scaling by dividing batch and all-to-all reduction.
\
[max width=]{}
Dataset Model Description Training Samples Validation Samples Image Size Classes
------------------------------------------ ------------------------------ -------------------- ------------------ -------------------- ----------------------- --------- --
MNIST [@mnistlecun] LeNet-3 [@lecun1998gradient] Handwritten Digits 60000 10000 $28 \times28$ 10
CIFAR-10 [@Krizhevsky09learningmultiple] Caffe-default Small Images 60000 10000 $32 \times32\times3$ 10
ImageNet [@ILSVRC15] AlexNet [@NIPS2012_4824] Diverse Images 1,281,167 50,000 $256\times256\times3$ 1,000
ImageNet [@ILSVRC15] GoogLeNet Diverse Images 1,281,167 50,000 $256\times256\times3$ 1,000
Objective
---------
The objective of our performance evaluation is to understand the performance overhead of using ULFM based implementation and correctness implications (if any) of the existing ULFM implementations.
Fault Injection Methodology
---------------------------
To emulate the process faults, we insert a fault in a process by using `SIGKILL`. For controlled experiment the fault is injected after 300 batches. This effectively emulates compute node faults, since we use one process on a compute node and multiple threads for each process.
Correctness Analysis
--------------------
For understanding the correctness, we compare the [*loss curves*]{} – a measure of the error as observed during the training phase. The curves are compared for SGD and FT-SGD implementations in Figures \[fig:alexnet\_loss\_n4\], \[fig:alexnet\_loss\_n8\], and \[fig:alexnet\_loss\_n16\] using 4, 8, and 16 compute nodes respectively. The FT-SGD evaluation consists of exactly one process fault – which is usually the case in real scenarios. We observe that the loss curves for both SGD and FT-SGD implementation track each other roughly. However, the FT-SGD is behind since with reduced number of available compute nodes, the overall batch size is reduced as well. The effect is diminished on 16 compute nodes since the overall effect of losing one compute node is reduced. Similar for other datasets as shown in Figures \[fig:googlenet\_loss\_n16\], \[fig:cifar10\_loss\_n16\] and \[fig:mnist\_loss\_n16\], the convergence of FT-SGD and SGD is similar.
Performance Analysis
--------------------
Figures \[fig:alexnet\_run\_time\], \[fig:googlenet\_run\_time\], \[fig:cifar10\_run\_time\] and \[fig:mnist\_run\_time\] shows the performance comparisons of SGD and FT-SGD using AlexNet, GoogLeNet, CifarNet (CIFAR10) and LeNet-3 (MNIST). The overall evaluation uses 1024 batches, which is a relatively small number of batches in comparison to the standard number of batches such as 60K for AlexNet. For the FT-SGD evaluation, exactly one process executes `SIGKILL` after 300 batch updates, resulting in $n - 1$ remaining number of compute nodes. Since the batch on each compute node remains constant, the overall computation time as observed on each compute node is similar for FT-SGD and SGD. Also, since we have one compute node failure, the overall difference in communication time is also negligible.
Figures \[fig:alexnet\_dataload\_time\], \[fig:googlenet\_dataload\_time\], \[fig:cifar10\_dataload\_time\] and \[fig:mnist\_dataload\_time\] shows the comparison of reading the overall dataset and partial dataset when a fault occurs for AlexNet, GoogLeNet, and other datasets. These charts are specifically useful to understand the cost of reading the dataset as a function of number of compute nodes. We observe that reading the partial dataset is significantly faster than reading the entire dataset, which is not surprising. This is especially validated for ImageNet dataset which is much larger than MNIST and CIFAR10 datasets. We observe that for MNIST and CIFAR10, partial reading is actually slower, since these datasets are trivially small.
Figures \[fig:alexnet\_shrink\_time\], \[fig:googlenet\_shrink\_time\], \[fig:cifar10\_shrink\_time\] and \[fig:mnist\_shrink\_time\] shows the overhead of cumulative `MPI_Comm_shrink` as a function of number of compute nodes. These functions are executed only if a fault is detected, otherwise this code is not executed. We observe that the overhead increases with the number of compute nodes, which is expected. However, the overall time is relatively insignificant to the batch update time. Hence, the ULFM specification and the ULFM implementation are sufficient for providing functionality and performance in implementing DL algorithms.
\
\
Related Work {#sec:related}
============
\[checkpoint\_restart\]
The majority of fault tolerant solutions proposed in the literature and practice have focused on checkpoint-restart mechanisms. Under these solutions, the applications periodically save the state of the data and computation either explicitly or implicitly by using OS level approaches such as Berkeley Lab Checkpoint Restart (BLCR) or virtualization based approaches. As an example, [@Hursey:4228333; @Ma:5289172], primarily focus on checkpoint-restart method for fault tolerance, storing the checkpointed data into the filesystem, while [@Gamell:InMemory:7576485] have presented disk-less in-memory checkpointing storage-restart scheme at application level leveraging ULFM. Others [@Ouyang:CRFS:6047205] have focused on alleviating filesystem I/O bottleneck due to checkpointing, using other libraries for checkpointing; [@Kutlu:6507503] explores an algorithm based fault tolerance for data intensive algorithms with data replication techniques. Others [@Walters:4633353] have explored asynchronous decentralized replication with standard checkpoint restart techniques. Wang [*et al.*]{} [@Wang:Hybrid:5695644] discusses hybrid checkpointing, alternating between incremental checkpointing and full checkpointing, resulting in minimized checkpointed data size. All these techniques have checkpoint-restart as the fundamental method for providing fault tolerance. Guo [*et al.*]{} [@Guo:SC2015] have discussed detect-resume model for MapReduce, using MPI, in addition to Reinit model. With Detect-Resume model, the workload from the faulted process is redistributed to the remaining nodes. In their approach, the lost work is recomputed from scratch in the remaining processes, leading to a longer recovery time. However in our approach, for DL algorithms, there is no need to recompute the work from the lost process(es), hence recovery time is greatly reduced. Chakravorty [*et al.*]{} in their work [@Chakravorty:IHPC06] have explored the concept of predictive fault tolerance. While this approach introduces lesser overhead compared to the checkpoint-restart, the suitability of predictive approach is limited.
Conclusions {#sec:conclusions}
===========
In this paper, we have addressed the question of the requirements of MPI for designing fault tolerant Deep Learning (DL) algorithms. We have presented the case for several types of parallelism as motivated from common use-cases, and DNN topologies. We have used the discussion to derive the suitability of fault tolerance proposals in MPI. We have considered several design choices for implementing fault tolerant DL implementations. Specifically, we have considered checkpoint-restart, Reinit (when a fault occurs, re-initialize the MPI automatically) and user-level fault mitigation (ULFM). We have implemented our design using MaTEx-Caffe and leveraged ULFM implementation available from OpenMPI. We have provided an evaluation of fault tolerant MaTEx-Caffe using ImageNet-1K dataset and widely studied neural network topologies and datasets such as AlexNet, GoogLeNet on ImageNet dataset, and MNIST and CIFAR-10 datasets as well. Our evaluation has indicated the effectiveness of ULFM both in terms of its suitability as a specification and readiness for practical deployments.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The cloud radio access network (C-RAN) provides high spectral and energy efficiency performances, low expenditures and intelligent centralized system structures to operators, which has attracted intense interests in both academia and industry. In this paper, a hybrid coordinated multi-point transmission (H-CoMP) scheme is designed for the downlink transmission in C-RANs, which fulfills the flexible tradeoff between cooperation gain and fronthaul consumption. The queue-aware power and rate allocation with constraints of average fronthaul consumption for the delay-sensitive traffic are formulated as an infinite horizon constrained partially observed Markov decision process (POMDP), which takes both the urgent queue state information (QSI) and the imperfect channel state information at transmitters (CSIT) into account. To deal with the *curse of dimensionality* involved with the equivalent Bellman equation, the linear approximation of post-decision value functions is utilized. A stochastic gradient algorithm is presented to allocate the queue-aware power and transmission rate with H-CoMP, which is robust against unpredicted traffic arrivals and uncertainties caused by the imperfect CSIT. Furthermore, to substantially reduce the computing complexity, an online learning algorithm is proposed to estimate the per-queue post-decision value functions and update the Lagrange multipliers. The simulation results demonstrate performance gains of the proposed stochastic gradient algorithms, and confirm the asymptotical convergence of the proposed online learning algorithm.'
author:
- 'Jian Li, Mugen Peng, , Aolin Cheng, Yuling Yu, Chonggang Wang, [^1] [^2] [^3]'
title: 'Resource Allocation Optimization for Delay-Sensitive Traffic in Fronthaul Constrained Cloud Radio Access Networks'
---
Queue-aware resource allocation, hybrid coordinated multi-point transmission, fronthaul limitation, cloud radio access networks.
Introduction
============
is estimated that the demand for high-speed mobile data traffic, such as high-quality wireless video streaming, social networking and machine-to-machine communication, will get 1000 times increase by 2020[[@bib:tdscdma]]{}, which requires a revolutionary approach involving new wireless network architectures as well as advanced signal processing and networking technologies. As key components of heterogeneous networks (HetNets), low power nodes (LPNs) are deployed within the coverage of macro base stations (MBSs) and share the same frequency band to increase the capacity of cellular networks in dense areas with high traffic demands. Unfortunately, the aggressive reuse of limited radio spectrum will result in severe inter-cell interference and unacceptable degradation of system performances. Therefore, it is critical to control interference through advanced signal processing techniques to fully unleash the potential gains of HetNets. As an integral part of the LTE-Advanced (LTE-A) standards, the coordinated multi-point transmission (CoMP) technique targets the suppression of the inter-cell interference and quality of service (QoS) improvement for the cell-edge UEs. However, CoMP is faced with some disadvantages and challenges in real HetNets. The performance gain of CoMP highly depends on the perfect knowledge of channel state information (CSI) and the tight synchronization, both of which pose strict restrictions on the backhaul of LPNs. To manipulate the high density of LPNs with lowest capital expenditure (CAPEX) and operational expenditure (OPEX) effectively, the cloud radio access network (C-RAN) was proposed in[[@cran]]{} to enhance spectral efficiency and energy efficiency performances and has recently attracted intense interest in both academia and industry.
![C-RAN architecture[]{data-label="fig1"}](fig_cran.eps)
As depicted in Fig. \[fig1\], the remote radio heads (RRHs) are only configured with the front radio frequency (RF) and simple symbol processing functionalities, while the other baseband physical processing and procedures of the upper layers are executed jointly in the baseband unit (BBU) pool for UEs associating with RRHs. The LPNs are simplified as RRHs through connecting to a “signal processing cloud" with high-speed fronthaul links. To coordinate the cross-tier interference between RRHs and MBSs effectively, the BBU pool is interfaced to MBSs. Such a distributed deployment and centralized processing architecture facilitates the implementation of CoMP[[@crancluster]]{} amongst RRHs of C-RANs as well as provides ubiquitous networks coverage with MBSs. Since all the RRHs in C-RANs are connected to the BBU pool, the CoMP can be realized through virtual beamforming and the beamformers can be calculated in BBU pool. Specifically, the CoMP in downlink C-RANs can be characterized into two classes[[@comp]]{}: joint processing (JP) and coordinated beamforming (CB). For JP, the traffic payload is shared and transmitted jointly by all RRHs within the CoMP cluster[[@JP]]{}, which means multiple delivery of the same traffic payload from the centralized BBU pool to each cooperative RRH through capacity-limited fronthaul links. As for the CB, the traffic payload is only transmitted by the serving RRH, but the corresponding beamformer is jointly calculated at the centralized BBU pool to coordinate the interference to all other UEs within the CoMP cluster[[@CS]]{}. Obviously, JP achieves higher average spectrum efficiency than CB does at the expense of more fronthaul consumption, while CB requires more antennas equipped with each RRH to achieve the full intra-cluster interference coordination. However, the practical non-ideal fronthaul with limited capacity restricts the overall performances of CoMP in C-RANs.
Related Works
-------------
There exists lots of literatures aiming to alleviate the fronthaul requirement of JP without the loss of interference exploitation. The authors of[[@dynamicclustering]]{} proposed a dynamic clustered multi-cell cooperation scheme to substantially reduce the backhaul consumption by imposing restriction on the cluster size. A heuristic algorithm was proposed in[[@directional]]{} to dynamically select the directional cooperation links under a finite-capacity backhaul subject to the evaluation of benefits and costs, which cannot completely eliminate the undesired interference as in the full cooperation case. Both reweighed $l$-1 norm minimization method and heuristic iterative link removal algorithm were proposed in[[@tonyquek]]{} to reduce the user data transfer via the capacity-limited backhaul effectively by dealing with the formulated cooperative clustering and beamforming problems, which, however, are suboptimal and still suffer from a significant performance loss. A backhaul cost metric considering the number of active directional cooperation links was adopted in[[@asymmetricJP]]{}, where the design problem is minimizing this backhaul cost metric and jointly optimizing the beamforming vectors among the cooperative BSs subject to signal-to-interference-and-noise-ratio (SINR) constraints at UEs. To make a flexible tradeoff between the cooperation gain and the backhaul consumption, a rate splitting approach under the limited backhaul rate constraints was proposed in[[@streamsplitting]]{}, where some fraction of the backhaul capacity originally consumed by JP could be privately used to get more performance gains. Borrowing the idea in[[@streamsplitting]]{}, the authors of[[@r1]]{} proposed a soft switching strategy between the JP-CoMP and CB-CoMP modes under capacity-limited backhaul. Considering the high complexity and the large signaling overhead, a distributed hard switching strategy was also proposed in[[@r1]]{}. To achieve a tradeoff between diversity and multiplexing gains of multiple antennas and high spectral efficiency, the authors of[[@r2]]{} studied both the dynamic partial JP-COMP and its corresponding resource allocation in a clustered CoMP cellular networks. Generally, for the C-RANs, the potential high spectral efficiency gain of CoMP largely depends on the quality of obtained channel state information at transmitters (CSIT) as well as the fronthaul consumption. In[[@r3]]{}, channel prediction usefulness was analyzed and compared with channel estimation in downlink CoMP systems with backhaul latency in time-varying channels, considering both the centralized and decentralized JP-CoMP as well as the CB-CoMP.
However, the aforementioned works only focus on physical layer performance of spectral efficiency or energy efficiency and ignore the bursty traffic arrival as well as the delay requirement of delay-sensitive traffic. Therefore, the resulting control policy is adaptive to the channel state information (CSI) only and cannot guarantee good delay performance for delay-sensitive applications. In general, since the CSI could provide information regarding the channel opportunity while the queue state information (QSI) could indicate the urgency of the traffic flows, the queue-aware resource allocation should be adaptive to both the CSI and QSI. Furthermore, as the CSIT cannot be perfect in real systems, and systematic packet errors occur when the allocated data rate exceeds the instantaneous mutual information. Therefore, the issue of robustness against the uncertainty incurred by imperfect CSIT should also be considered in the resource allocation optimization.
There already have some research efforts on the queue-aware dynamic resource allocation in stochastic wireless networks. In paper[[@queuecluster]]{}, the authors proposed a mixed timescale delay-optimal dynamic clustering and power allocation design with downlink JP in traditional multi-cell networks. The queue-aware discontinuous transmission (DTX) and user scheduling design with downlink CB in energy-harvesting multi-cell networks was proposed in[[@dtx]]{}. A queue-weighted dynamic optimization algorithm using Lyapunov optimization approach was proposed in[[@lyapunov]]{} for the joint allocation of subframes, resource blocks, and power in the relay-based HetNets. However, all these works focus on queue-aware resource allocations in homogeneous networks or HetNets without the consideration of the imperfect CSIT. Therefore, the solutions cannot work in the C-RANs with the practical challenges of imperfect CSIT and non-ideal capacity-limited fronthaul links.
Main Contributions
------------------
To the best of our knowledge, there are lack of effective signal processing techniques and dynamic radio resource management solutions for delay-sensitive traffic in C-RANs to optimize the SE, EE and delay performances, which still remains challenging and requires more investigations. Based on the aforementioned advantages and challenges of C-RANs, the efficient CoMP scheme with tradeoff between cooperation gain and fronthaul consumption will be elaborated in this paper. Furthermore, under the average power and fronthaul consumption constraints, the dynamic radio resource management with feature of queue-awareness to maintain good delay performance for delay-sensitive traffic in stochastic C-RANs will also get studied in this paper. The major contributions of this paper are as follows.
- To allow a flexible tradeoff between cooperation gain and average fronthaul consumption, the H-CoMP scheme is proposed for the delay-sensitive traffic in C-RANs by splitting the traffic payload into shared streams and private streams. By reconstructing the shared streams and private streams and optimizing the precoders and decorrelators, the shared streams and private streams can be simultaneously transmitted to obtain the maximum achievable degree of freedom (DoF) under limited fronthaul consumption.
- Motivated by[[@survey]]{}, to minimize the transmission delay of the delay-sensitive traffic under the average power and fronthaul consumption constraints in C-RANs, the queue-aware rate and power allocation problem is formulated as an infinite horizon average cost constrained partially observed Markov process decision (POMDP). The queue-aware resource allocation policy is adaptive to both QSI and CSIT in the downlink C-RANs and can be obtained by solving a per-stage optimization for the observed system state at each scheduling frame.
- Since the optimal solution requires centralized implementation and perfect knowledge of CSIT statistics and has exponential complexity w.r.t. the number of UEs, the linear approximation of post-decision value functions involving POMDP is presented, based on which a stochastic gradient algorithm is proposed to allocate power and transmission rate dynamically with low computing complexity and high robustness against the variations and uncertainties caused by unpredictable random traffic arrivals and imperfect CSIT. Furthermore, the online learning algorithm is proposed to estimate the post-decision value functions effectively.
- The delay performances of the proposed H-CoMP and queue-aware resource allocation solution are numerically evaluated. Simulation results show that a significant delay performance gain can be achieved in the fronthaul constrained C-RANs with H-CoMP, and the queue-aware resource allocation solution is validated and effective due to the adaptiveness to both QSI and imperfect CSIT. Further, the stochastic gradient algorithms can improve the delay performances drastically, and the online learning algorithm is asymptotically converged.
The rest of this paper is organized as follows. Section II describes the system model and section III gives the design of H-CoMP scheme for the downlink C-RANs. The queue-aware resource allocation problem is formulated as POMDP in section IV and a low complexity approach is proposed in section V. The performance evaluation is conducted in section VI and section VII summarizes this paper.
$(.)^T$ and $(.)^H$ stand for the transpose and conjugate transpose, respectively. $(.)^\dagger $ stands for the pseudo-inverse. Besides, $diag(\textbf{p})$ denotes a diagonal matrix formed by the vector $\textbf{p}$.
System Models
=============
To optimize performances of downlink C-RANs, the transmission model, traffic queue dynamic model in the medium access control (MAC) layer, and the imperfect CSIT assumption in the physical layer are considered in this section.
Transmission Model
------------------
The transmission of $M$ delay-sensitive traffic payloads in downlink C-RANs with $M$ RRHs is considered. Denote $\mathcal {M} =
\{1,2,...,M\}$ as the UE set and $\mathcal {N} = \{1,2,...,M\}$ as the RRH set within the CoMP cluster. An example of C-RAN with $M =
2$ is illustrated in Fig. 2. The inter-tier interferences amongst the RRHs and MBSs are controlled by setting the maximum allowable power consumption of each RRH indicated by the MBSs through the X2 interfaces, while the intra-tier interferences in C-RANs can be eliminated by implementing the CoMP. Each RRH and UE are equipped with $N_t$ and $N_r$ antennas respectively, where $M{N_r} \geq N_t >
(M - 1){N_r}$. Within the coverage of each RRH, a served UE exists and it can also be cooperatively served by the other RRHs according to the following proposed H-CoMP scheme. In this paper, the scheduling is carried out in every frame indexed by $t$ and the frame duration is $\tau$ second.
Traffic Queue Dynamic Model
---------------------------
Let ${\bf{Q}}(t) = \{ {Q_1}(t), \ldots ,{Q_M}(t)\}$ denote the global QSI (number of bits) for $M$ queues maintained at BBU pool at the beginning of scheduling frame $t$. There will be random packet arrival ${A_i}(t)$ after ${G_i}(t)$ bits are successfully received by UE $i$ at the end of frame $t$. The random arrival process $A_i(t)$ is supposed to be independent identically distributed (i.i.d) over scheduling frame according to a general distribution with mean $\mathbb{E}\{ {A_i}(t)\} = {\lambda _i}$ and independent w.r.t $i$. Furthermore, the statistics of ${A_i}(t)$ is supposed to be unknown to the BBU. The queue dynamic of UE $i$ is then given by $${Q_i}(t + 1) = min\{ {[{Q_i}(t) - {G_i}(t)]^ + } + {A_i}(t),{N_Q}\},$$ where $[x]^+ = max\{x, 0\}$ and $N_Q$ is the maximum buffer size.
Imperfect CSIT Assumption
-------------------------
Let ${{\bf{H}}_{ji}}(t) \in {\mathbb{C}^{{N_r} \times {N_t}}}$ denote the complex channel fading coefficient between RRH $i$ and UE $j$ at frame $t$ and let $\textbf{H}(t) = \{{{\bf{H}}_{ji}}(t): j\in
\mathcal {M}, i \in \mathcal {N}\}$ denote the global CSI. Especially, every element of ${{\bf{H}}_{ji}}(t)$ is supposed to remain constant within a scheduling frame but be i.i.d over scheduling frame. The perfect knowledge of CSI is assumed to be only obtained by the UE while the imperfect CSIT ${\bf{\hat H}} =
\{{{\bf{\hat H}}_{ji}}(t) \in {\mathbb{C}^{{N_r} \times {N_t}}}:
j\in \mathcal {M}, i \in \mathcal {N}\}$ is obtained by the BBU pool. The rank of both ${\bf{\hat H}}_{ji}$ and ${\bf{H}}_{ji}$ is assumed to be $min\{N_r,N_t\}$. Furthermore, the imperfect CSIT error kernel model is given by[[@imperfectCSIT]]{}
$$\Pr [{{\bf{\hat H}}_{ji}}|{{\bf{H}}_{ji}}] = \frac{1}{{\pi {\sigma
_{ji}}}}\exp ( - \frac{{|{{{\bf{\hat H}}}_{ji}} -
{{\bf{H}}_{ji}}{|^2}}}{{{\sigma _{ji}}}}),$$
which is caused by duplexing delay in time division duplex (TDD) systems or quantization errors and feedback latency in frequency division duplex (FDD) systems. The above ${\sigma _{ji}} \in [0,1]$ indicates the CSIT quality. When ${\sigma _{ji}} = 0$, we have ${{\bf{\hat H}}_{ji}}={{\bf{H}}_{ji}}$, which corresponds to the perfect CSIT case. When ${\sigma _{ji}} = 1$, we have ${{\bf{\hat
H}}_{ji}}{{\bf{H}}_{ji}^{\dag }}$ = [**0**]{}, which corresponds to the no CSIT case.
Hybrid CoMP scheme
==================
With the limited fronthaul capacity, the maximum achievable DoF can be obtained by separating the traffic payload for UE $i$ into ${L_{(i,s)}}$ shared streams ${{\bf{s}}_{(i,s)}}$ and ${L_{(i,p)}}$ private streams ${{\bf{s}}_{(i,p)}}$ and simultaneously transmitting them with optimal precoders and decorrelators, that is the hybrid CoMP (H-CoMP) scheme. More specifically, the H-CoMP allows shared streams to be shared across the RRHs with the CoMP cluster by multiple delivery through capacity-limited fronthaul links. Meanwhile, the H-CoMP makes private streams remain private to certain RRH and the precoders are jointly calculated at the BBU pool to eliminate the intra-cluster interference. Therefore, the cooperative transmission of shared streams requires significantly more fronthaul consumption than the coordinated transmission of private streams does. In the following subsections, the traffic streams splitting model, precoder calculation and decorrelator calculation with the perfect CSIT will be thoroughly elaborated.
Traffic Streams Splitting Model
-------------------------------
To make a flexible tradeoff between the cooperation gain and average fronthaul consumption, the number of shared streams and private streams should be determined with the traffic streams splitting model. With the perfect CSIT, the zero-forcing (ZF) precoder and decorrelator designs are adopted for both shared streams and private streams. In this situation, at most ${L_{M,{N_t},{N_r}}} = {{N_t} -
(M - 1){N_r} }$ private streams can be zero-forced at RRH $i$ to eliminate interference to UE ${\rm{j}} \ne i$, i.e. $$L_{(i,p)} \leq {L_{M,{N_t},{N_r}}}.$$ Furthermore, to fully recover the ${L_{(i,p)}}$ private streams and ${L_{(i,s)}}$ shared streams at UE $i$, the constraint $$L_{(i,p)}+L_{(i,s)} \le N_r$$ should be satisfied. With the traffic streams splitting, the proposed H-CoMP allows a flexible tradeoff between the cooperation gain and the fronthaul consumption. The achievable DoFs of different schemes are compared in table I.
[|m[1.5cm]{}<|m[5cm]{}<|]{} **Scheme** & **Achievable DoF**\
CB-CoMP & $(M
-1){L_{M,{N_t},{N_r}}} + {N_r}$\
JP-CoMP & $M{N_r}$\
H-CoMP & $\{ (M - 1){L_{M,{N_t},{N_r}}} + {N_r},...,M{N_r}\}$\
Specifically, when $L_{(i,p)}+L_{(i,s)} = N_r$, the maximum DoF of $MN_rM$ can be achieved by the H-CoMP scheme, and when ${L_{(i,p)}}
= {L_{M,{N_t},{N_r}}}$, the fronthaul consumption is minimized.
Although the shared streams and private streams are superimposed in the downlink transmission of C-RANs, it is possible to eliminate the interference at RRHs and recover both of them at UEs by constructing the private streams and shared streams and designing optimal precoders and decorrelators.
Let ${{\bf{s}}_{(i,s)}} \in {\mathbb{C}^{L_{(i,s)} \times 1}}$ and ${{\bf{s}}_{(i,p)}} \in {\mathbb{C}^{L_{(i,p)} \times 1}}$ denote the shared streams and private streams respectively, where $L_{(i,s)}$ and $L_{(i,p)}$ are the number of shared streams and private streams. To facilitate the implementation of H-CoMP, the shared streams and private streams are reconstructed by inserting zero vectors as follows respectively $${{\bf{\tilde s}}_{(i,s)}} = {\{ {\bf{s}}_{(i,s)}^T,{{\textbf{0}}_{1
\times {L_{(i,p)}}}}\} ^T},$$ $${{\bf{\tilde s}}_{(i,p)}} = {\{ {\textbf{0}_{1 \times
{L_{(i,s)}}}},{\bf{s}}_{(i,p)}^T\} ^T}.$$
Let ${{\bf{\tilde W}}_{(i,s)}} \in {\mathbb{C}^{M{N_t} \times
{(L_{(i,s)} + L_{(i,p)})}}}$ and ${\tilde{\bf{ W}}_{(i,p)}} \in
{\mathbb{C}^{{N_t} \times {(L_{(i,p)} + L_{(i,s)})}}}$ denote the precoders for the reconstructed shared streams and reconstructed private streams of UE $i$ respectively. Define ${{\bf{\Lambda
}}_{(i,s)}} = diag(\sqrt {P_{(i,s)}^1} , \ldots ,\sqrt
{P_{(i,s)}^{{L_{(i,s)}}}}, {\textbf{0}_{1 \times {L_{(i,p)}}}})$ and ${{\bf{\Lambda }}_{(i,p)}} = diag({\textbf{0}_{1 \times
{L_{(i,s)}}}}, \sqrt {P_{(i,p)}^1} , \ldots ,\sqrt
{P_{(i,p)}^{{L_{(i,p)}}}} )$, where $P_{(i,s)}$ and $P_{(i,p)}$ denote the transmission power of each shared stream and private stream for UE $i$ respectively. Then the received signal vector ${{\bf{r}}_i} \in {\mathbb{C}^{{N_r} \times 1}}$ at UE $i$ is given by $$\begin{aligned}
&&{{\bf{r}}_i} = \underbrace {{{\bf{H}}_i}{{\tilde{\bf{
W}}}_{(i,s)}}{{\bf{\Lambda }}_{(i,s)}}{{\tilde{\bf{s}}}_{(i,s)}} +
{{\bf{H}}_{ii}}{{\tilde{\bf{ W}}}_{(i,p)}}{{\bf{\Lambda
}}_{(i,p)}}{{\tilde{\bf{
s}}}_{(i,p)}}}_{{\rm{the~desired~signals~for~UE~i }}} \\\nonumber &&
+ \underbrace {\sum\limits_{j \ne i} {{{\bf{H}}_i}{{\tilde{\bf{
W}}}_{(j,s)}}{{\bf{\Lambda }}_{(j,s)}}{{\tilde{\bf{s}}}_{(j,s)}}}
}_{{\rm{the~interference~from~shared~ streams}}}\\\nonumber && +
\underbrace {\sum\limits_{j \ne i} {{{\bf{H}}_{ij}}{{\tilde{\bf{
W}}}_{(j,p)}}{{\bf{\Lambda }}_{(j,p)}}{{\tilde{\bf{s}}}_{(j,p)}}}
}_{{\rm{the~interference~from~ private~streams}}} + {{\bf{n}}_i},\end{aligned}$$ where ${\textbf{H}_i} = [\begin{array}{*{20}{c}}
{{\textbf{H}_{i1}}}& \cdots &{{\textbf{H}_{iM}}}
\end{array}] \in {\mathbb{C}^{{N_r} \times M{N_t}}}$ is the aggregate complex channel fading coefficient vector from the $M$ cooperative RRHs to UE $i$, and ${{\bf{n}}_i}$ is the zero-mean unit variance complex Gaussian channel noise at UE $i$.
Precoder and Decorrelator Calculation for Shared Streams
---------------------------------------------------------
The optimal cooperative precoder at the $M$ RRHs and the decorrelator at UE $i$ should be designed to maximize the mutual information of shared streams and to eliminate the interference imposed on all the other UEs (UE $j \ne i$) as follows $$\begin{aligned}
&&\{ {\bf{\tilde W}}_{_{(i,s)}}^ * ,{\bf{\tilde U}}_{_{(i,s)}}^ * \}
= \arg \mathop {\max }\limits_{{{{\bf{\tilde W}}}_{(i,s)}},
{{{\bf{\tilde U}}}_{(i,s)}}} {\log _2}\det [\textbf{I} + \nonumber \\
&&~~~~~~~~~~~~~~~~~~~~~~~ {{{\bf{\tilde
U}}}_{(i,s)}}{{\bf{{\bf{H}}}}_i} {{{\bf{\tilde
W}}}_{(i,s)}}{\bf{\tilde W}}_{_{(i,s)}}^H{\bf{H}}_{_i}^H{\bf{\tilde
U}}_{_{(i,s)}}^H] \nonumber\\
&&s.t.~~~~~~ {{\bf{H}}_j}{{\bf{\tilde W}}_{(i,s)}} = \textbf{0}.\end{aligned}$$ Therefore, the precoder have the form of
$${{\bf{\tilde W}}_{(i,s)}^*} = {{\bf{F}}_{(i,s)}}{{{\bf{\tilde
V}}}_{(i,s)}},$$
where ${{\bf{F}}_{(i,s)}} \in {\mathbb{C}^{M{N_t} \times (M{N_t} -
(M - 1){N_r})}}$ is given by the orthonormal basis of ${\rm{nullspace([}}{\bf{H}}_1^T, \cdots ,{\bf{H}}_{i - 1}^T,
{\bf{H}}_{i + 1}^T{\rm{,}} \cdots {\rm{,}}$ ${\bf{H}}_M^T{{\rm{]}}^T})$. Let ${{\bf{H}}_i}{{\bf{F}}_{(i,s)}} =
{{\bf{U}}_{(i,s)}}{{\bf{\Sigma }}_{(i,s)}}{\bf{V}}_{(i,s)}^H$ be the singular value decomposition (SVD) of equivalent channel matrix ${{\bf{H}}_i}{{\bf{F}}_{(i,s)}}$, where the singular values in ${{\bf{\Sigma }}_{(i,s)}}$ are sorted in a decreasing order along the diagonal, ${{\bf{\tilde V}}_{(i,s)}} \in {\mathbb{C}^{(M{N_t} -
(M - 1){N_r}) \times {(L_{(i,s)}+L_{(i,p)})}}}$ is then given by the first ${L_{(i,s)}}$ columns of ${{\bf{V}}_{(i,s)}}$ as ${{\bf{\tilde
V}}_{(i,s)}} = \{ {\bf{v}}_{(i,s)}^1, \cdots
,{\bf{v}}_{(i,s)}^{{L_{(i,s)}}},{\textbf{0}}_{(M{N_t} - (M -
1){N_r}) \times L_{(i,p)}}\}$. Furthermore, the decorrelator ${\bf{\tilde U}}_{_{(i,s)}}^ *$ is given by the first ${L_{(i,s)}}$ columns of ${{\bf{U}}_{(i,s)}}$ as follows
$${\bf{\tilde U}}_{_{(i,s)}}^ * = \{ {\bf{u}}_{(i,s)}^1, \cdots
,{\bf{u}}_{(i,s)}^{{L_{(i,s)}}},{\textbf{0}}_{{N_r} \times
L_{(i,p)}}\}^H.$$
Then the recovered $L_{(i,s)}$ shared streams are given by the first $L_{(i,s)}$ rows of ${\bf{\tilde U}}_{_{(i,s)}}^ *{{\bf{r}}_i}$.
Precoder and Decorrelator Calculation for Private Streams
----------------------------------------------------------
The optimal coordinated precoder at RRH $i$ for the private streams of UE $i$ and the decorrelator at UE $i$ should be designed to maximize the mutual information of private streams and to eliminate the interference imposed on all the other UEs (UE $j \ne i$) as follows $$\begin{aligned}
&&\{ {\bf{\tilde W}}_{_{(i,p)}}^ * ,{\bf{\tilde U}}_{_{(i,p)}}^ * \}
= \arg \mathop {\max }\limits_{{{{\bf{\tilde W}}}_{(i,p)}},
{{{\bf{\tilde U}}}_{(i,p)}}} {\log _2}\det [\textbf{I} + \nonumber\\
&&~~~~~~~~~~~~~~~~~{{{\bf{\tilde U}}}_{(i,p)}}{{\bf{{\bf{H}}}}_{ii}}
{{{\bf{\tilde W}}}_{(i,p)}}{\bf{\tilde W}}_{_{(i,p)}}^H{\bf{H}}_{ii}^H{\bf{\tilde U}}_{_{(i,p)}}^H] \nonumber\\
&&s.t.~~~~~~~ {{\bf{H}}_{ji}}{{\bf{\tilde W}}_{(i,p)}} = \textbf{0}.\end{aligned}$$ Therefore the precoder has the similar form of $${{\bf{\tilde W}}_{(i,p)}^*} = {{\bf{\tilde F}}_{(i,p)}}{{{\bf{\tilde
V}}}_{(i,p)}},$$ where ${\bf{\tilde F}}_{(i,p)} = [{\textbf{0}}_{N_t \times (N_r -
L_{M,N_t,N_r})}, {\bf{F}}_{(i,p)}]$ and ${\bf{F}}_{(i,p)} \in
{\mathbb{C}^{{N_t} \times L_{M,N_t,N_r}}}$ is given by the orthonormal basis of ${\rm{nullspace([}}{\bf{H}}_{1i}^T, \cdots
,{\bf{H}}_{i-1 i}^T, {\bf{H}}_{i + 1 i}^T{\rm{,}} \cdots
{\rm{,}}{\bf{H}}_{Mi}^T{{\rm{]}}^T})$. Let ${{\bf{H}}_{ii}}{{\bf{\tilde F}}_{(i,p)}} =
{{\bf{U}}_{(i,p)}}{{\bf{\Sigma }}_{(i,p)}}{\bf{V}}_{(i,p)}^H$ be the SVD of equivalent channel matrix ${{\bf{H}}_{ii}}{{\bf{\tilde
F}}_{(i,p)}}$, where the singular values in ${{\bf{\Sigma
}}_{(i,p)}}$ are sorted in an increasing order along the diagonal, ${{\bf{\tilde V}}_{(i,p)}} \in {\mathbb{C}^{{N_r} \times
{(L_{(i,s)}+L_{(i,p)})}}}$ is then given by the last ${L_{(i,p)}}$ columns of ${{\bf{V}}_{(i,p)}}$ as ${{\bf{\tilde V}}_{(i,p)}} =
\{{\textbf{0}}_{{N_r} \times L_{(i,s)}},{\bf{v}}_{(i,p)}^{N_r -
L_{(i,p)} + 1}, \cdots ,{\bf{v}}_{(i,p)}^{{N_r}}\}$. Furthermore, the decorrelator ${\bf{\tilde U}}_{_{(i,p)}}^ *$ is given by the last ${L_{(i,p)}}$ columns of ${{\bf{U}}_{(i,p)}}$ as follows $${\bf{\tilde U}}_{_{(i,p)}}^ * = \{{\textbf{0}}_{{N_r} \times
L_{(i,s)}}, {\bf{u}}_{(i,p)}^{N_r - L_{(i,p)} + 1}, \cdots
,{\bf{u}}_{(i,p)}^{N_r}\}^H.$$ Then the recovered $L_{(i,p)}$ private streams are given by the last $L_{(i,p)}$ rows of ${\bf{\tilde U}}_{_{(i,p)}}^
*{{\bf{r}}_i}$.
[(*The Interference Nulling Between Shared Streams and Private Streams*)]{} Although the interference nulling constraints are not explicitly imposed, the interference between shared streams and private streams of UE $i$ can be still eliminated due to the fact that ${\bf{\tilde U}}_{_{(i,s)}}^
* {{\tilde{\bf{ W}}}_{(i,p)}}{{\bf{\Lambda
}}_{(i,p)}}{{\tilde{\bf{s}}}_{(i,p)}} = \textbf{0}$ and ${\bf{\tilde
U}}_{_{(i,p)}}^
* {{\tilde{\bf{ W}}}_{(i,s)}}{{\bf{\Lambda
}}_{(i,s)}}{{\tilde{\bf{s}}}_{(i,s)}} = \textbf{0}$.
The Power Consumption and Transmission Rate
-------------------------------------------
To support the cooperative transmission of $a$-th shared stream from $M$ RRHs to UE $j$, the power contributed by RRH $i$ is given by $P_{(j,s)}^a\rho _{(j,i)}^a$, where $P_{(j,s)}^a$ denote the total power to transmit the $a$-th shared stream to UE $i$ and $\rho
_{(j,i)}^a = \sum\nolimits_{x = 1}^{{N_t}} {|{{[{{\bf{\tilde
W}}^{*}_{(j,s)}}]}_{((i - 1){N_t} + x,a)}}{|^2}}$ denote the contribution by RRH $i$. To support the coordinated transmission of $a$-th private stream from RRH $i$ to UE $i$, the power of $P_{(i,s)}^a$ is needed. Therefore, with the proposed H-CoMP scheme, the total transmit power consumption at RRH $i$ is given by $$P_i = \sum\nolimits_{a = 1}^{{L_{(i,p)}}} {P_{(i,p)}^a} +
\sum\nolimits_{j = 1}^M {\sum\nolimits_{a = 1}^{{L_{(j,s)}}}
{P_{(j,s)}^a\rho _{(j,i)}^a} }\label{power}.$$
In practice, both the precoders and decorrelators are calculated at the BBU pool with the imperfect CSIT, which will cause uncertain residual interference to the recovered streams. By treating the uncertain interference as noise, the mutual information of $a$-th shared stream at ${\bf{s}}_{(i,s)}^a$ UE $i$ is given by $$C_{(i,s)}^a = {\log _2}(1 + {{\varphi}_{(i,s)}^a}P_{(i,s)}^a/(1 +
I_{(i,s)}^a)),$$ where $\varphi _{(i,s)}^a = |{\bf{\tilde
U}}_{(i,s)}^a{{\bf{H}}_i}{\bf{\tilde W}}_{(i,s)}^a{|^2} $, ${\bf{\tilde U}}_{(i,s)}^a$ and ${\bf{\tilde W}}_{(i,s)}^a$ is the $a$-th row of ${\bf{\tilde U}}_{(i,s)}^*$ and $a$-th column of ${\bf{\tilde W}}_{(i,s)}^*$ respectively, $I_{(i,s)}^a$ is the residual interference incurred by the imperfect CSIT. The mutual information of $a$-th private stream ${\bf{s}}_{(i,p)}^a$ of UE $i$ is given in a similar way. Due to the uncertainties of mutual information, the data rate successfully transmitted to UE $i$ is given by $${G_i} = ({R_{(i,s)}}\textbf{1}({R_{(i,s)}} \le {C_{(i,s)}}) +
{R_{(i,p)}}\textbf{1}({R_{(i,p)}} \le {C_{(i,p)}}))\tau,$$ where ${C_{(i,s)}} = \sum\nolimits_{a = 1}^{{L_{(i,s)}}}
{C_{(i,s)}^a} $ and ${C_{(i,p)}} = \sum\nolimits_{a =
1}^{{L_{(i,p)}}} {C_{(i,p)}^a}$ are the mutual information for shared streams and private streams respectively. ${R_{(i,s)}} $ and ${R_{(i,p)}}$ are the allocated data rate for shared streams and private streams of UE $i$ respectively.
Formulation of Queue-aware Control Problem
==========================================
To meet the urgency of the delay-sensitive traffic payloads and reduce the occurrence of packet transmission failure in the downlink C-RANs, the queue-aware resource allocation problem based on the observed system states (QSI and CSIT) will be formulated in this section.
Feasible Stationary Control Policy
----------------------------------
Considering the inter-tier interference imposed by RRHs and the energy efficient transmission of delay-sensitive traffic, the feasible resource allocation policy should satisfy the following average power consumption constraints $${P_i}{\rm{(}}\Omega {\rm{) = }}\mathop {lim}\limits_{T \to \infty }
{sup} \frac{1}{T}\sum\limits_{t = 1}^T {{{\mathbb{E}}^\Omega
}[{P_i}(t)]} \le P_i^{\max },$$ where ${\mathbb{E}^\Omega }$ indicates that the expectation is taken w.r.t the measure induced by policy $\Omega$, ${P_i}(t)$ is the total power consumption of RRH $i$ to support the H-CoMP transmission, and $P_i^{\max }$ is the maximum average power consumption indicated by MBSs. Furthermore, by varying the maximum average power consumption of each RRH, cross-tier interference could be well controlled to maintain desirable average QoS requirement for macro UEs.
It is worth noting that compared with the fronthaul consumption for traffic payload sharing, that for signaling delivery is negligible. Due to the capacity-limited fronthaul links of C-RANs, the feasible resource allocation policy also should satisfy the following average fronthaul consumption constraints $$R_i^{f}(\Omega )\!=\!\mathop {lim} \limits_{T \to \infty }\!{sup}
\frac{1}{T}\sum\limits_{t = 1}^T \!{{{\mathbb{E}}^\Omega
}[{R_{(i,p)}}(t)\! +\! \sum\limits_j\! {{R_{(j,s)}}(t)} ]} \! \le\!
R_i^{\max },$$ where ${R_{(i,p)}}(t) + \sum\nolimits_{j \in \mathcal {M}}
{{R_{(j,s)}}(t)}$ is the total data rate to be delivered to RRH $i$ through the fronthaul link connecting RRH $i$ to BBU pool, and $R_i^{\max }$ is the maximum average fronthaul consumption.
With the aforementioned resource constraints, the feasible stationary resource allocation policy for C-RANs is defined as follows.
[(*Stationary Resource Allocation Policy*)]{} A feasible stationary resource allocation policy $\Omega ({\bf{\hat
S}}) = \{ {\Omega _R}({\bf{\hat S}}),{\Omega _P}({\bf{\hat S}})\}$ is a mapping from the global observed system states ${\bf{\hat S}} =
\{ {\bf{Q}},{\bf{\hat H}}\}$ instead of the global system states ${\bf{S}} = \{ {\bf{Q}},{\bf{H}},{\bf{\hat H}}\}$ to the resource allocation actions, where ${\Omega _P}({\bf{\hat S}}) = \{
{{{P}}^a_{(i,p)}},{{{P}}^b_{(i,s)}}: 1 \leq a \leq L_{(i,p)}, 1 \leq
b \leq L_{(i,s)}, i \in M\}$ and ${\Omega _R}({\bf{\hat S}}) = \{
{{{R}}_{(i,p)}},{{{R}}_{(i,s)}}:i \in M\}$ are the power allocation policy and rate allocation policy subject to average power consumption constraints and average fronthaul consumption constraints.
Given the feasible stationary resource allocation policy $\Omega
({\bf{\hat S}})$, the induced random process ${\bf{S}} = \{
{\bf{Q}},{\bf{H}},{\bf{\hat H}}\}$ is a controlled Markov chain with the transition probability as follows $$\begin{aligned}
\Pr\!\{ {\bf{S}}(t\! +\! 1)|{\bf{S}}(t),\Omega ({\bf{\hat
S}}(t))\}\! = \Pr\{ {\bf{\hat H}}(t \!+ \!1),{\bf{H}}(t \!+\! 1)\}
\nonumber\\ \Pr\{ {\bf{Q}}(t\! + \!1)|{\bf{S}}(t),\Omega ({\bf{\hat
S}}(t))\} .\label{tansprob}\end{aligned}$$
Apparently, the queue dynamics of the $M$ UEs served by C-RAN are coupled with each other via $\Omega ({\bf{\hat S}}(t))$.
Problem Formulation
-------------------
With the positive weighting factors $\{{\beta}_i\}$ ,which indicates the relative importance of delay requirement among the $M$ users, the queue-aware resource allocation problem with average power consumption constraints and average fronthaul consumption constraints can be formulated as the following problem.
[(*Queue-aware Resource Allocation Problem*)]{} $$\begin{aligned}
&&{\min _\Omega }D({\bf{\beta }},\Omega ) = \mathop {lim} \limits_{T \to \infty } {sup}\frac{1}{T}\sum\limits_{t = 1}^T {{{\mathbb{E}}^\Omega }[\sum\limits_{i \in M} {{\beta _i}\frac{{{Q_i}}}{{{\lambda _i}}}} ]} \\
&& s.t. \rm{~~the~constraints~(17)~and~(18)~for~each~RRH} \nonumber,\end{aligned}$$ where the $\frac{{{Q_i}}}{{{\lambda _i}}}$ in objective function is the average traffic delay cost for UE $i$ by Little’s Law.
With the average power consumption constraints and average fronthaul consumption constraints in Problem 1, the occurrence of extreme instantaneous power and fronthaul consumption tends to be impossible. Furthermore, the feasible stationary resource allocation policy is defined on the observed system states ${\bf{\hat S}} = \{
{\bf{Q}},{\bf{\hat H}}\}$. Therefore, problem 1 is a constrained partially observed MDP (POMDP)[[@bertsekas]]{}, which will be solved by the following general approach.
General Approach with MDP
-------------------------
Using the Lagrange duality theory, the Lagrange dual function of problem 1 is defined as
$$J(\gamma )\! = \!\mathop {\min }\limits_\Omega L(\beta ,\gamma
,\Omega ({\bf{\hat S}}) ) \!=\! \mathop {lim}\limits_{T \to \infty
}{sup} \frac{1}{T}\!\sum\limits_{t = 1}^T {{{\mathbb{E}}^\Omega
}[g({\bf{\beta }},{\bf{\gamma }},\Omega ({\bf{\hat
S}}))]}\label{lag},$$
where $g({\bf{\beta }},{\bf{\gamma }},\Omega ({\bf{\hat S}})) =
\sum\limits_i {({\beta _i}\frac{{{Q_i}}}{{{\lambda _i}}}) + }
{\gamma _{(i,P)}}({{\rm{P}}_i} - P_i^{\max }) + {\gamma
_{(i,R)}}(R_i^{f} - R_i^{\max })$ is the per-stage system cost and ${\gamma _{(i,P)}}$ and ${\gamma _{(i,R)}}$ are the non-negative Lagrange multipliers (LMs) w.r.t the power consumption constraints and fronthaul consumption constraints. Then the dual problem of problem 1 is given by $$\mathop {\max }\nolimits_\gamma J(\gamma ).$$
Although (\[lag\]) is an unconstrained POMDP, the solution is generally nontrivial. To substantially reduce the global observed system states space, the partitioned actions are defined as follows with the i.i.d. property of the CSIT.
[(*Partitioned Actions*)]{} Given the stationary resource allocation policy $\Omega$, $\Omega
({\bf{Q}}) = \{ \Omega ({\bf{\hat S}}):\forall {\bf{\hat H}}\}$ is defined as the collections of power and rate allocation actions for all possible CSIT ${\bf{\hat H}}$ on a given QSI $\textbf{Q}$, therefore $\Omega$ is equal to the union of all partitioned actions. i.e. $\Omega = {\cup}_{\textbf{Q}}{{\Omega}(\textbf{Q})} $.
As the distribution of the traffic arrival process is unknown to the BBU, the post-decision state potential function instead of potential function will be introduced in the following theorem to derive the queue-aware resource allocation policy of eq. (\[lag\]).
[(*Equivalent Bellman Equation*)]{}\
(a)Given the LMs, the unconstrained POMDP problem can be solved by the equivalent Bellman equation as follows $$\begin{aligned}
U({\bf{\tilde Q}}) + \theta = \sum\nolimits_{\bf{A}} {\Pr
({\bf{A}})} \mathop {min}\limits_{\Omega(\textbf{Q})}g(\bf{\beta },
{\bf{\gamma }}, \textbf{Q},{\Omega(\textbf{Q})}) \nonumber\\ +
\sum\nolimits_{{\bf{\tilde Q}}'} {\Pr \{ {\bf{\tilde
Q}}'|\textbf{Q},{\Omega(\textbf{Q})}\} U({\bf{\tilde Q}}')}
\label{bellman},\end{aligned}$$ where $g(\bf{\beta }, {\bf{\gamma }},{\bf{Q}},\Omega ({\bf{Q}})) =
\mathbb{E}[g({\bf{\beta }},{\bf{\gamma }},\Omega ({\bf{\hat
S}}))|{\bf{Q}}] $ is the conditional per-stage cost and $\Pr \{
{\bf{\tilde Q}}'|{\bf{Q}},{\Omega(\textbf{Q})}\} = \mathbb{E}[\Pr
[{\bf{Q}}'|{\bf{H}},{\bf{Q}},\Omega ({\bf{\hat S}})]|{\bf{Q}}]$ is the conditional average transition kernel, $U({\bf{\tilde Q}})$ is the post-decision value function. ${\bf{\tilde Q}}$ is the post-decision state and ${\bf{\tilde Q}}' = {({\bf{Q}} - {\bf{G}})^
+ }$ is the next post-decision state, where ${\bf{Q}} = \min \{
{\bf{\tilde Q}} + {\bf{A}},{N_Q}\}$ and ${\bf{G}} = \{G_i : i \in
\mathcal {M}\}$.
(b)If there exists unique $(\theta ,\{ U({\bf{\tilde Q}})\} )$ that satisfies (\[bellman\]), then $\theta = \mathop
{min}\limits_{\Omega(\textbf{Q})} {\mathbb{E}}[g(\bf{\beta },
{\bf{\gamma }},{\bf{Q}},\Omega ({\bf{Q}}))$ is the optimal average per-stage cost for the unconstrained POMDP and the optimal resource allocation policy $\Omega $ is obtained by minimizing R.H.S of (\[bellman\]).
Please refer to Appendix A
[(*The Zero Duality Gap*)]{} Although the objective function of problem 1 is not convex w.r.t the stationary resource control policy, the duality gap between the dual problem and primal problem is zero when the condition Theorem 1 (b) is established, which implies that the primal optimal resource control policy can be obtained by solving the equivalent Bellman equation of the dual optimal problem.
[(*The Computational Complexity*)]{} Solving the equivalent Bellman equation involves ${N_Q}^M + 1$ unknowns $(\theta ,\{ U({\bf{\tilde Q}})\} )$ and ${N_Q}^M$ nonlinear fixed point equations, which means exponential state space, enormous computational complexity and full knowledge of system states transition probability in (\[tansprob\]). Therefore, a low complexity solution based on linear approximation and online learning of post-decision value functions will be further studied.
Low Complexity Approach
=======================
In this section, to substantially reduce the enormous computing complexity in centralized BBU pool, the linear approximation of post-decision value functions is utilized, upon which a stochastic gradient algorithm is proposed to obtain the QAH-CoMP policy and an online learning algorithm is proposed to estimate the post-decision value functions.
Linear Approximation of Post-decision Value Functions
-----------------------------------------------------
The linear approximation of post-decision value functions is defined by the sum of the per-queue value functions as follows[[@linearapprox]]{}
$$U({\bf{\tilde Q}}) \approx \sum\nolimits_{i \in \mathcal {M}}
{{U_i}({{\tilde Q}_i})},$$
where ${U_i}({\tilde Q_i})$ is the per-queue post-decision value functions which satisfies the following per-queue fixed point Bellman equation $$\begin{aligned}
{U_i}({{{\rm{\tilde Q}}}_i})\! + \!{\theta _i}\!&& =
\!\sum\nolimits_{{A_i}} {\Pr ({A_i})} \mathop
{min}\limits_{{\Omega}_{i}(Q_i)}[{g_i}({\beta _i},{{\bf{\gamma }}_i}
,{Q_i},{{\Omega}_{i}})\nonumber\\&& + \sum\nolimits_{{{\tilde
Q}_i}'} {\Pr \{ \tilde Q{'_i}|{Q_i},{{\Omega}_{i}}\} U(\tilde
Q{'_i})}] \label{perqueuebellman},\end{aligned}$$ where ${g_i}({\beta _i},{{\bf{\gamma }}_i} ,{Q_i},{{\Omega}_{i}}) =
\mathbb{E}[ {\beta _i}\frac{{{Q_i}}}{{{\lambda _i}}} + {\gamma
_{(i,P)}}(\sum\limits_{a = 1}^{{L_{(i,s)}}} {P_{(i,s)}^a\rho
_{(i,i)}^a} + \sum\limits_{a = 1}^{{L_{(i,p)}}} {P_{(i,p)}^a} -
P_i^{\max }) + \sum\limits_{j \in \mathcal {M}, j \ne i} {{\gamma
_{(j,P)}}}$ $ \sum\limits_{a = 1}^{{L_{(i,s)}}} {P_{(i,s)}^a\beta
_{(i,j)}^a} + {\gamma _{(i,R)}}({R_{(i,p)}}(t) + \sum\limits_{j \in
\mathcal {M}} {{R_{(j,s)}}(t)} - R_i^{\max })|Q_i]$ is the per-queue per-stage cost function. ${Q_i} = \min \{ {\tilde Q_i} +
{A_i},{N_Q}\}$ is the pre-decision state and $\tilde Q{'_i} =
{({Q_i} - {G_i})^\dag }$ is the next post-decision state. The optimality of linear approximation is established in the following lemma.
[(*The Optimality of Linear Approximation*)]{} The linear approximation is optimal only when the CSIT is perfect, which means the interference is completely eliminated with H-CoMP scheme, therefore, the queue dynamics of $M$ UEs are decoupled.
Please refer to Appendix B.
Generally, the error variance ${\sigma}_{ji}$ of the imperfect CSIT can not be large, therefore the linear approximation is asymptotically accurate with sufficiently small error variance of CSIT.
[(*The Computing Complexity*)]{} With the linear approximation, the calculation of the post-decision value functions in BBU pool is alleviated from exponential complexity $ \mathcal {O}((N_Q + 1)^M)$ to polynomial complexity $\mathcal {O}((N_Q + 1)M)$.
Low Complexity QAH-CoMP Policy
------------------------------
With the combination of the linear approximation and equivalent Bellman equation (\[bellman\]), the QAH-CoMP policy can be obtained by solving the following per-stage optimization for every observed system state, which is summarized as the following corollary.
With the observation of current system states, the per-stage optimization is given by $${\Omega ^*}({\bf{\hat S}}) \!=\! \{ \Omega _P^*({\bf{\hat
S}}),\Omega _R^*({\bf{\hat S}})\} \! =\! {\rm{arg}}\!\!\!\mathop
{min}\limits_{\Omega _P({\bf{\hat S}}),\Omega _R({\bf{\hat
S}})}\!\!\! B({\bf{\hat S}},\!{\Omega _P({\bf{\hat S}}),\Omega
_R({\bf{\hat S}})}),$$ where $B({\bf{\hat S}},{\Omega _P({\bf{\hat S}}),\Omega _R({\bf{\hat
S}})}) = \sum\limits_{i \in \mathcal {M}} {\{{\gamma _{(i,P)}}P_i
{\gamma _{(i,R)}}({R_{(i,p)}}(t) }$ ${+ \sum\limits_j
{{R_{(j,s)}}(t)} ) +
{\mathbb{E}}[{{\bf{1}}_{(i,s)}}{{\bf{1}}_{(i,p)}}]({U_i}({Q_k} -
{\tau R_{(i,s)}}}{ - \tau {R_{(i,p)}})}$ ${ - {U_i}({Q_k} - \tau
{R_{(i,p)}}) - {U_i}({Q_k} - \tau {R_{(i,s)}}) + {U_i}({Q_k}))} +
{\mathbb{E}}[{{\bf{1}}_{(i,s)}}]({U_i}({Q_k} - \tau {R_{(i,s)}}) -
{U_i}({Q_k})) + {\mathbb{E}}[{{\bf{1}}_{(i,p)}}]({U_i}({Q_k} - \tau
{R_{(i,p)}}) - {U_i}({Q_k}))\}$ is the per-stage objective, ${{\bf{1}}_{(i,p)}} = {\bf{1}}({R_{(i,p)}} \le {C_{(i,p)}})$ and ${{\bf{1}}_{(i,p)}} = {\bf{1}}({R_{(i,s)}} \le {C_{(i,s)}})$ are the indicator functions.
The per-stage optimization above is intractable due to that the expectation $\mathbb{E}$ required the explicit knowledge of CSIT errors in BBU pool. To deal with this challenge, the per-stage optimization problem can be solved by the following stochastic gradient algorithm[[@stochastic]]{}.
[(*Stochastic Gradient Algorithm*)]{}
At each frame $t > 1$, the queue-aware power and rate allocations for each UE can be obtained as the following iteration $$e_i^{t}({{\bf{\hat S}}_i}) = {[e_i^{t-1}({{\bf{\hat S}}_i}) -
{\gamma _e}(t-1)d(e_i^{t-1}({{\bf{\hat S}}_i}))]^ + },$$ where ${\gamma _e}(t)$ is the step size satisfying ${\gamma _e}(t)
> 0,\sum\nolimits_t {{\gamma _e}(t)} = \infty ,\sum\nolimits_t
{{{({\gamma _e}(t))}^2}} < \infty$ and $d(e_i^t({{\bf{\hat S}}_i}))$ is the stochastic gradient w.r.t power and rate allocation, which is summarized as follows $$\left\{ \begin{array}{l} \frac{{\partial B({\bf{\hat S}},{\Omega
_P({\bf{\hat S}}),\Omega _R({\bf{\hat S}})})}}{{\partial
P_{(i,p)}^a}} = {\gamma _{(i,P)}} + \frac{{\partial {h_i}({\bf{\hat
S}},{\Omega _P({\bf{\hat S}}),\Omega
_R({\bf{\hat S}})})}}{{\partial P_{(i,p)}^a}}\\
\frac{{\partial B({\bf{\hat S}},{\Omega _P({\bf{\hat S}}),\Omega
_R({\bf{\hat S}})})}}{{\partial P_{(i,s)}^a}} = {\gamma
_{(i,P)}}\rho _{(i,i)}^a + \frac{{\partial {h_i}({\bf{\hat
S}},{\Omega _P({\bf{\hat S}}),\Omega
_R({\bf{\hat S}})})}}{{\partial P_{(i,s)}^a}} \\~~~~~~~~~~~~~~~~~~~~~~ + \sum\limits_{j \ne i,j \in \mathcal {M}} {{\gamma _{(j,P)}}\rho _{(i,j)}^a} \\
\frac{{\partial B({\bf{\hat S}},{\Omega _P({\bf{\hat S}}),\Omega
_R({\bf{\hat S}})})}}{{\partial {R_{(i,p)}}}} = {\gamma _{(i,R)}} +
\frac{{\partial {h_i}({\bf{\hat S}},{\Omega _P({\bf{\hat S}}),\Omega
_R({\bf{\hat S}})})}}{{\partial {R_{(i,p)}}}}\\
\frac{{\partial B({\bf{\hat S}},{\Omega _P({\bf{\hat S}}),\Omega
_R({\bf{\hat S}})})}}{{\partial {R_{(i,p)}}}} = {\gamma _{(i,R)}} +
\frac{{\partial {h_i}({\bf{\hat S}},{\Omega _P({\bf{\hat S}}),\Omega
_R({\bf{\hat S}})})}}{{\partial {R_{(i,p)}}}} \\ ~~~~~~~~~~~~~~~~~
~~~~~+ \sum\limits_{j \ne i,j \in \mathcal {M}} {{\gamma _{(j,R)}}}
\end{array} \right.\label{stoch},$$ where ${h_i}({\bf{\hat S}},{\Omega _P({\bf{\hat S}}),\Omega
_R({\bf{\hat S}})}) = {{\bf{1}}_{(i,s)}}({U_i}({Q_k} - \tau
{R_{(i,s)}}) - {U_i}({Q_k})) + {{\bf{1}}_{(i,p)}}({U_i}({Q_k} - \tau
{R_{(i,p)}}) - {U_i}({Q_k})) + {{\bf{1}}_{(i,s)}}{{\bf{1}}_{(i,p)}}(
{U_i}({Q_k} - \tau {R_{(i,s)}} - \tau {R_{(i,p)}}) - {U_i}({Q_k} -
\tau {R_{(i,s)}}) - {U_i}({Q_k} - \tau {R_{(i,p)}}) +
{U_i}({Q_k}))$.
When ${\bf{\hat H}} = {\bf{ H}}$, there is no interference under the H-CoMP with perfect CSIT, and $d(e_i^t({{\bf{\hat S}}_i}))$ is deterministic instead of stochastic. Using the standard gradient update argument, the gradient search converges to a local optimum as $t \to \infty$. Therefore, the Algorithm 1 gives the asymptotically local optimal solution at small CSIT errors, which means that the explicit knowledge of imperfect CSIT is unnecessary and it is robust against the uncertainties caused by imperfect CSIT.
[(*Feedback-Assisted Realization of Algorithm 1*)]{} The calculation of stochastic gradient (\[stoch\]) in BBU pool requires some items regarding the indicator functions ${\bf{1}}_{(i,s)}$ and ${\bf{1}}_{(i,p)}$ and the differential of post-decision value functions $ U_i^{'}({\tilde Q_i})$. At each frame $t$, the indicator functions are unknown by BBU pool and have to be fed back from UEs, which is feasible due to that there are existing built-in mechanisms in wireless networks for these ACK/NACK feedback from UEs. In addition, since there is no closed-form expression of post-decision value function $U_i^{'}({\tilde Q_i})$, its differential can be estimated as follows $$U_i^{'}({\tilde Q_i}) = U_i({\tilde Q_i}) - U_i({\tilde Q_i} - 1),$$ where the online learning of $U_i({\tilde Q_i})$ will be elaborated in next subsection.
Online Learning of Per-queue Post-decision Value Functions
----------------------------------------------------------
The post-decision value functions are critical to the derivation of queue-aware resource allocation policy for C-RANs, which can be obtained by solving $N_Q$ fixed point nonlinear Bellman equations with $N_Q + 1$ variables. The offline calculation requires the explicit knowledge of conditional average transition kernel, which is infeasible. In this section, with the realtime observation of QSI and CSIT, the online learning of per-queue post-decision value functions is proposed based on the equation (\[perqueuebellman\]). Meanwhile, with the realtime resource control actions, the LMs are updated to make sure the average power consumption constraints and average fronthaul consumption constraints are satisfied[[@xrcao]]{}. The online learning of per-queue value functions and the update of LMs at centralized BBU pool are described as follows.
[(*Online Learning of Per-Queue Value Functions and Update of LMs*)]{}
Set $t = 0$, the per-queue post-decision value functions $\{U_i^{0}({{\tilde Q}_i})\}$ and LMs $\{\gamma _{(i,P)}^{0}, \gamma
_{(i,R)}^{0} \} > 0$ are initialized at the centralized BBU pool.
At the beginning of the $t$-th frame, given fixed $\{ {\eta
_j}(n)\}$, the queue-aware power and rate allocation for downlink H-CoMP transmission are determined at the BBU pool using the stochastic gradient algorithm in (\[stoch\]).
With the observation of post-decision QSI $\{{{\tilde Q}_i} \}$ and pre-decision QSI $\{{Q_i}\}$, the per-queue post-decision value function $U_i({{\tilde Q}_i})$ is online learned at BBU pool (\[valueupdate\]) for each traffic queue as follows $$\begin{aligned}
U_i^{t + 1}({{\tilde Q}_i})&& = U_i^t({{\tilde Q}_i}) + {\zeta
_u}(t)[{g_i}(\gamma _i^t,{{{\bf{\hat S}}}_i},{P_i},{R_i} +
U_i^t({Q_i} - {U_i})\nonumber\\&& - U_i^t(\tilde Q_i^0) -
U_i^t({{\tilde Q}_i})]\label{valueupdate}.\end{aligned}$$
With the observation of power and rate allocation, the $\{\gamma _{(i,P)}^{t + 1}\}$ and $\{ \gamma _{(i,R)}^{t + 1} \}$ are updated according to eq.(\[lmpower\]) and eq.(\[lmrate\]) at the BBU pool for the power consumption constraints and fronthaul consumption constraints respectively, $$\gamma _{(i,P)}^{t + 1} \!= \!{[\gamma _{(i,P)}^t + {\zeta _\gamma
}\!(t)({P_i} - P_i^{\max })]^ + }\label{lmpower},$$ $$\gamma _{(i,R)}^{t + 1}\! = \!{[\gamma _{(i,R)}^t\! + \!{\zeta
_\gamma }\!(t)({R_{(i,p)}}(t)\! + \!\sum\nolimits_{j \in \mathcal
{M}} {{R_{(j,s)}}\!(t)} \! -\! R_i^{\max })]^ + }\label{lmrate}.$$
Set $t = t + 1$ and continue to step 2 until certain termination condition is satisfied.
The ${\zeta _u}(t)$ and ${\zeta _\gamma }(t)$ in step 3 and step 4 is the iterative step size of post-decision value functions and LMs respectively. To make sure the convergence of iteration, they should satisfy the conditions as follows[[@convergence]]{}:
$${\zeta _u}(t) > 0,\sum\nolimits_t {{\zeta _u}(t)} = \infty,$$
$${\zeta _\gamma }(t) > 0,\sum\nolimits_t {{\zeta _\gamma }(t)} =
\infty,$$
$$\sum\nolimits_t{({{({\zeta _u}(t))}^2} + } {({\zeta _\gamma }(t))^2}) <
\infty,$$
$$\mathop {\lim }\limits_{t \to \infty } \frac{{{\zeta _\gamma
}(t)}}{{{\zeta _u}(t)}} = 0\label{stepsize}.$$
The condition (\[stepsize\]) implies that the LMs are relatively static during the iteration of per-queue value functions. Therefore the iteration of post-decision value functions and the iteration of LMs are done simultaneously but over two different time scales[[@onlinelearning]]{}.
It is a remarkable fact that the size of per-queue states(in bits) is still large. To accelerate the estimation of each post-state value function, the per-queue QSI space ${{\mathcal {Q}}_i}$ is partitioned into $N$ regions as (\[region\])
$${{\mathcal {Q}}_i} = \bigcup\nolimits_{n = 1}^N {{{\mathcal
{R}}_n}}\label{region}.$$
Therefore, the average value function w.r.t each region is online learned instead, then the post-decision value function of each state within the region can be estimated by interpolation method after each iteration.
Performances Evaluation
=======================
In this section, simulations are conducted to compare the performances of the proposed QAH-CoMP with various baselines in C-RANs. The delay-sensitive traffic packet arrival follows a Poisson distribution and the corresponding packet size follows an exponential distribution, which is a widely adopted traffic model[[@survey]]{}. The mean size of traffic packet is 4Mbits and the maximum buffer size is 32Mbits. The CSI ${{\bf{H}}_{ij}}$ is uniformly distributed over a state space ${\mathcal {H}}^{N_r \times
N_t}$ and the error variance of the imperfect CSIT is ${\varepsilon
_e} = 0.05$. The configuration of multi-antennas is given by $\{
{N_t} = 5,{N_r} = 2\}$ and the cluster size is $M$ = 3. Therefore, with the stream splitting of the H-CoMP scheme, there are one shared stream and one private stream to be transmitted for each UE. The total bandwidth of simulated C-RAN is 20MHz and the scheduling frame duration is 10ms. The noise power is -15dBm.
Three baselines are considered in the simulations: CB-CoMP, JP-CoMP, and channel-aware resource allocation with H-CoMP (CAH-CoMP). All these three baselines carry out rate and power allocation to maximize the average system throughput with the same fronthaul capacity and average power consumption constraint as the proposed QAH-CoMP. For the CB-CoMP baseline, the BBU pool calculates the coordinated beamformer for each RRH to eliminate the dominating intra-cluster interference. For the CAH-CoMP baseline, the proposed H-CoMP transmission is adopted, while the power allocation and rate allocation are only adaptive to CSIT.
Fig. 3 compares the delay performance of the four schemes with different packet arrival rate. The average packet delay of all the schemes increases as the average packet arrival rate increases. Compared with CB, the delay outperformance of JP weakens as the packet arrival rate increases, which is due to the fact that the fronthaul capacity becomes relatively limited with the increasing packet arrival rate. Apparently, the performance gain of QAH-CoMP compared with CAH-CoMP is contributed by power and rate allocation with the consideration of both urgent traffic flows and imperfect CSIT.
![Average packet delay vs. packet arrival rate, the maximum fronthaul consumption is $R_i^{max}$ = 20Mbits/s, the maximum transmit power is $P_i^{max}$ = 10dBm.[]{data-label="fig2"}](fig_arrival.eps)
Fig. 4 compares the delay performance of the four schemes with different maximum transmit power. The figure depicts the medium fronthaul consumption regime, in which JP-CoMP outperforms CS-CoMP due to the higher spectrum efficiency. CAH-CoMP outperforms both CS-CoMP and JP-CoMP while the outperformance of CAH-CoMP is not so obvious with relative enough fronthaul capacity. It can be observed that there is significant performance gain of the proposed QAH-CoMP compared with all the baselines across a wide range of the maximum power consumption.
![Average packet delay vs. maximum transmit power, the packet arrival rate is ${\lambda}_i$ = 2.5 packets/s, the maximum fronthaul consumption is $R_i^{max}$ = 30Mbits/s.[]{data-label="fig3"}](fig_power.eps)
Fig. 5 compares the delay performance of the four schemes with different maximum fronthaul consumption. The figure depicts the small fronthaul consumption regime, in which CAH-CoMP clearly outperforms both CS-CoMP and JP-CoMP, which is contributed by the flexible adjustment of cooperation level when the fronthaul capacity is limited. Note that the JP-CoMP has worse delay performance than CS-CoMP due to limited fronthaul capacity at first but it eventually gets performance improvement with increasing fronthaul capacity. Similarly, due to the queue-aware power and rate allocation, QAH-CoMP substantially outperforms the three baselines.
![Average packet delay vs. maximum fronthaul consumption,the packet arrival rate is ${\lambda}_i$ = 2.5 packets/s, the maximum transmit power is $P_i^{max}$ = 10dBm.[]{data-label="fig4"}](fig_fronthaul.eps)
Fig. 6 shows the convergence property of the online per-queue post-decision value functions(w.r.t ${{\mathcal {R}}_n}$, size of which is equal to mean packet size ${\bar N_i}$) learning algorithm. For viewing convenience, the post-decision value functions of the traffic queue maintained for UE 1 is plotted with the increasing of iteration step. It is significant that the learning converges extremely close to the final result after 1000 iterations.
![Per-queue Post-decision Value Functions[]{data-label="fig5"}](fig_value.eps)
Summary
=======
In this paper, an H-CoMP scheme with corresponding precoders and decorrelators are designed for the downlink fronthaul constrained C-RANs. Based on the proposed H-CoMP, a low complexity queue-aware power and rate allocation solution for the delay-sensitive traffic is then proposed using MDP and stochastic gradient algorithms. Simulation results show that the C-RANs with H-CoMP achieve more significant delay performance gains than that with CB-CoMP and JP-CoMP under the same average power and fronthaul consumption constraints, where the performance gains largely depend on the cooperation level of the proposed H-CoMP under limited fronthaul capacity. Furthermore, compared with the CAH-CoMP, the remarkable delay performance gain of QAH-CoMP is also validated by the simulation results, which is contributed by the MDP based dynamic resource allocation with the consideration of both QSI and imperfect CSIT. In the future, the theoretical analysis on the delay performance of the QAH-CoMP still remains to be an open issue, and the real experiments would be desirable to further demonstrate the effectiveness and applicability of the QAH-CoMP in fronthaul constrained C-RANs.
Proof of theorem 1
==================
According to the Proposition 4.6.1 of[[@bertsekas]]{}, the sufficient condition for optimality of problem 1 is that there exists unique $(\theta ,\{ U({\bf{S}})\} )$ that satisfies the following Bellman equation and $U({\bf{S}})$ satisfies the transversality condition $\mathop {\lim }\limits_{T \to \infty }
\frac{1}{T}{{\mathbb{E}}^\Omega }[U({\bf{S}}(T))|{\bf{S}}(0)] = 0$ for all admissible control policy $\Omega$ and initial state ${\bf{S}}(0)$. $$\begin{array}{l}
\theta + U({\bf{S}}) = \mathop {\min }\limits_{\Omega ({\bf{\hat S}})} [g({\bf{\beta }},{\bf{\gamma }},{\bf{S}},\Omega ({\bf{\hat S}})) + \sum\limits_{{\bf{S}}'} {\Pr [{\bf{S}}|{\bf{S}},\Omega ({\bf{\hat S}})]} U({\bf{S}}')] \\
= \mathop {\min }\limits_{\Omega ({\bf{\hat S}})} [g({\bf{\beta }},{\bf{\gamma }},{\bf{S}},\Omega ({\bf{\hat S}})) + \sum\limits_{{\bf{Q}}'} {\sum\limits_{{\bf{\hat H}}'} {\sum\limits_{{\bf{H}}'} {\Pr [{\bf{Q}}'|{\bf{Q}},{\bf{\hat H}},{\bf{H}},\Omega ({\bf{\hat S}})]}}} \\{{\Pr [{\bf{\hat H}}',{\bf{H}}']U({\bf{S}}')]} } \\
\end{array}$$
Then taking expectation w.r.t. ${\bf{\hat H}}',{\bf{H}}'$ on both side of the above equation, we have
$$\theta \! + \!U({\bf{Q}})\! =\! \mathop {\min }\limits_{\Omega
({\bf{Q}})} [g({\bf{\beta }},{\bf{\gamma }},{\bf{Q}},\Omega
({\bf{Q}})) + \sum\limits_{{\bf{Q}}'} {\Pr
[{\bf{Q}}'|{\bf{Q}},\Omega ({\bf{Q}})]U({\bf{Q}}')} ]$$
where $U({\bf{Q}}) = {\mathbb{E}}[U({\bf{S}})|{\bf{Q}}] =
\sum\limits_{{\bf{\hat H}}} {\sum\limits_{\bf{H}} {\Pr [{\bf{\hat
H}},{\bf{H}}]U({\bf{S}})} }$ and $\Pr [{\bf{Q}}'|{\bf{Q}},\Omega
({\bf{\hat S}})] = {\mathbb{E}}[\Pr [{\bf{Q}}'|{\bf{Q}},{\bf{\hat
H}},{\bf{H}},\Omega ({\bf{\hat S}})]|{\bf{Q}}]$. Since here we defined the post-decision State ${\bf{\tilde Q}}$, where ${\bf{Q}} =
\min \{ {\bf{\tilde Q}} + {\bf{A}},{N_Q}\}$, the equivalent Bellman equation can be transformed as the equivalent Bellman equation (\[bellman\]) in theorem 1.
Proof of lemma 1
================
With the perfect CSIT, there is no interference with the H-CoMP scheme for C-RAN, which means that the queue dynamics for every UE are completely decoupled. Detailedly speaking, ${\tilde Q_i} = {Q_i}
- {G_i}({\bf{\hat H}},{\Omega _i}({\bf{\hat S}}))$ is independent of ${Q_j}$ and ${\Omega _j}({\bf{\hat S}})$ for all $j \ne i$ due to the nonexistence of interference, therefore we have $\Pr
[{\bf{\tilde Q}}'|{\bf{Q}},\Omega ({\bf{Q}})] = \prod\nolimits_{i
\in \mathcal {M}} {\Pr [{{\tilde Q}_i}'|{\bf{Q}},\Omega ({\bf{Q}})]}
$ and $\Pr [{\tilde Q_i}'|{\bf{Q}},\Omega ({\bf{Q}})] = \Pr [{\tilde
Q_i}'|{Q_i},{\Omega _i}({\bf{Q}})] = \Pr [{\tilde
Q_i}'|{Q_i},{\Omega _i}({Q_i})]$. Suppose $U({\bf{\tilde Q}}) =
\sum\nolimits_{i \in \mathcal {M}} {{U_i}({{\tilde Q}_i})}$, by the relationship between the joint distribution and the marginal distribution, we have $$\begin{array}{l}
{~~~}\sum\nolimits_{{\bf{\tilde Q}}'} {\Pr [{\bf{\tilde Q}}'|{\bf{Q}},\Omega ({\bf{Q}})]U({\bf{\tilde Q}}')} \\
= \sum\nolimits_{{\bf{\tilde Q}}'} {\Pr [{\bf{\tilde Q}}'|{\bf{Q}},\Omega ({\bf{Q}})]\sum\nolimits_{i \in M} {{U_i}({{\tilde Q}_i}')} } \\
= \sum\nolimits_{i \in \mathcal {M}} {\sum\nolimits_{{{\tilde Q}_i}'} {\Pr [{{\tilde Q}_i}'|{\bf{Q}},\Omega ({\bf{Q}})]} } {U_i}({{\tilde Q}_i}') \\
= \sum\nolimits_{i \in \mathcal {M}} {\sum\nolimits_{{{\tilde Q}_i}'}}
{{\Pr [{{\tilde Q}_i}'|{Q_i},{\Omega _i}({Q_i})]{U_i}({{\tilde Q}_i}')} } \\
\end{array}$$
It is obvious that $g({\bf{\beta }},{\bf{\gamma }},{\bf{Q}},\Omega
(\Theta )) = \sum\nolimits_{i \in M} {{g_i}({\beta _i},{{\bf{\gamma
}}_i},{Q_i},{\Omega _i}({Q_i}))}$. Suppose $\theta =
\sum\nolimits_{i \in \mathcal {M}} {{\theta _i}}$, then the equivalent Bellman equation in (\[bellman\]) can be transformed as
$$\begin{array}{l}
{~~~~}\sum\nolimits_{i \in \mathcal {M}} {{\theta _i}} + \sum\nolimits_{i \in \mathcal {M}} {{U_i}({{\tilde Q}_i})} \\
= \sum\nolimits_{\bf{A}} {\Pr ({\bf{A}})} \mathop {\min }\limits_{\Omega (Q)} \sum\nolimits_{i \in M} {[{g_i}({\beta _i},{{\bf{\gamma }}_i},{Q_i},{\Omega _i}({Q_i}))}\\~~~{ + \sum\nolimits_{{{\tilde Q}_i}'} {\Pr [{{\tilde Q}_i}'|{Q_i},{\Omega _i}({Q_i})]{U_i}({{\tilde Q}_i}')]} } \\
\mathop = \limits^{(a)} \sum\nolimits_{i \in \mathcal {M}} {\sum\nolimits_{{A_i}} {\Pr ({A_i})} } \mathop {\min }\limits_{{\Omega _i}({Q_i})} [{g_i}({\beta _i},{{\bf{\gamma }}_i},{Q_i},{\Omega _i}({Q_i})) \\~~~+ \sum\nolimits_{{{\tilde Q}_i}'} {\Pr [{{\tilde Q}_i}'|{Q_i},{\Omega _i}({Q_i})]{U_i}({{\tilde Q}_i}')]} \\
\end{array}$$
where (a) is due to the independent assumption of the new arrival process $A_i(t)$ w.r.t $i$. Therefore, we can have the per-queue fixed point Bellman equation in (\[perqueuebellman\]) for each UE from the above equation.
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[Jian Li]{} received his B.E. degree from Nanjing University of Posts and Telecommunications, Nanjing, China, in 2010. He is currently pursuing his Ph.D. degree at the key laboratory of universal wireless communication (Ministry of Education) in Beijing University of Posts and Telecommunications (BUPT), Beijing, China. His current research interests include delay-aware cross-layer radio resource optimization for heterogeneous networks and heterogeneous cloud radio access networks.
[Mugen Peng]{} (M’05–SM’11) received the B.E. degree in Electronics Engineering from Nanjing University of Posts & Telecommunications, China in 2000 and a PhD degree in Communication and Information System from the Beijing University of Posts & Telecommunications (BUPT), China in 2005. After the PhD graduation, he joined in BUPT, and has become a full professor with the school of information and communication engineering in BUPT since Oct. 2012. During 2014, he is also an academic visiting fellow in Princeton University, USA. He is leading a research group focusing on wireless transmission and networking technologies in the Key Laboratory of Universal Wireless Communications (Ministry of Education) at BUPT, China. His main research areas include wireless communication theory, radio signal processing and convex optimizations, with particular interests in cooperative communication, radio network coding, self-organization networking, heterogeneous networking, and cloud communication. He has authored/coauthored over 40 refereed IEEE journal papers and over 200 conference proceeding papers.
Dr. Peng is currently on the Editorial/Associate Editorial Board of IEEE Communications Magazine, IEEE Access, International Journal of Antennas and Propagation (IJAP), China Communication, and International Journal of Communication Systems (IJCS). He has been the guest leading editor for the special issues in IEEE Wireless Communications, IJAP and the International Journal of Distributed Sensor Networks (IJDSN). He is serving as the track co-chair or workshop co-chair for GameNets 2014, So-HetNets in IEEE WCNC 2014, SON-HetNet 2013 in IEEE PIMRC 2013, WCSP 2013, etc. Dr. Peng was honored with the Best Paper Award in CIT 2014, ICCTA 2011, IC-BNMT 2010, and IET CCWMC 2009. He was awarded the First Grade Award of Technological Invention Award in Ministry of Education of China for his excellent research work on the hierarchical cooperative communication theory and technologies, and the Second Grade Award of Scientific & Technical Progress from China Institute of Communications for his excellent research work on the co-existence of multi-radio access networks and the 3G spectrum management in China.
[Aolin Cheng]{} received his B.E. degree in Electronic Information Science and Technology from Beijing University of Posts and Telecommunications, Beijing, China, in 2012. He is currently pursuing his M.E. degree at the laboratory of universal wireless communication (Ministry of Education) in Beijing University of Posts and Telecommunications (BUPT), Beijing, China. His current research interests include delay-aware cross-layer radio resource optimization for heterogeneous networks (HetNets), as well as stochastic approximation and Markov decision process.
[Yuling Yu]{} received the B.E. degree in Communication Engineering from Wuhan University of Technology, Wuhan, China, in 2013. She is currently pursuing her M.E. degree at the key laboratory of universal wireless communication (Ministry of Education) in Beijing University of Posts and Telecommunications (BUPT), Beijing, China. Her research focuses on delay-aware cross-layer resource optimization for heterogeneous cloud radio access networks (H-CRANs), as well as Lyapunov optimization.
[Chonggang Wang]{} (SM’09) received the Ph.D. degree from Beijing University of Posts and Telecommunications (BUPT) in 2002. He is currently a Member of Technical Staff with InterDigital Communications. His R&D focuses on: Internet of Things (IoT), Machine-to-Machine (M2M) communications, Heterogeneous Networks, and Future Internet, including technology development and standardization. He (co-)authored more than 100 journal/conference articles and book chapters. He is on the editorial board for several journals including IEEE Communications Magazine, IEEE Wireless Communications Magazine and IEEE Transactions on Network and Service Management. He is the founding Editor-in-Chief of IEEE Internet of Things Journal. He is serving and served in the organization committee for conferences/workshops including IEEE WCNC 2013, IEEE INFOCOM 2012, IEEE Globecom 2010-2012, IEEE CCNC 2012, and IEEE SmartGridComm 2012. He has also served as a TPC member for numerous conferences such as IEEE ICNP (2010-2011), IEEE INFOCOM (2008-2014), IEEE GLOBECOM (2006-2014), IEEE ICC (2007-2013), IEEE WCNC (2008-2012) and IEEE PIMRC (2012-2013). He is a co-recipient of National Award for Science and Technology Achievement in Telecommunications in 2004 on IP QoS from China Institute of Communications. He received Outstanding Leadership Award from IEEE GLOBECOM 2010 and InterDigital’s 2012 and 2013 Innovation Award. He served as an NSF panelist in wireless networks in 2012. He is a senior member of the IEEE and the vice-chair of IEEE ComSoc Multimedia Technical Committee (MMTC) (2012-2014).
[^1]: Manuscript received June 18, 2014; revised September 12, 2014; accepted October 15, 2014. The editor coordinating the review of this paper and approving it for publication was Prof. Vincenzo Piuri.
[^2]: Jian Li (e-mail: [[email protected]]{}), Mugen Peng (e-mail: [[email protected]]{}), Aolin Cheng (e-mail: [[email protected]]{}), Yuling Yu (e-mail: [[email protected]]{}) are with the Key Laboratory of Universal Wireless Communications for Ministry of Education, Beijing University of Posts and Telecommunications, China. Chonggang Wang (e-mail: [[email protected]]{}) is with the InterDigital Communications, King of Prussia, PA, USA.
[^3]: This work was supported in part by the National Natural Science Foundation of China (Grant No. 61222103, No.61361166005), the National High Technology Research and Development Program of China (Grant No. 2014AA01A701), the State Major Science and Technology Special Projects (Grant No. 2013ZX03001001), and the Beijing Natural Science Foundation (Grant No. 4131003).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We report the results of our observations of the S255IR area with the SMA at 1.3 mm in the very extended configuration and at 0.8 mm in the compact configuration as well as with the IRAM-30m at 0.8 mm. The best achieved angular resolution is about 0.4 arcsec. The dust continuum emission and several tens of molecular spectral lines are observed. The majority of the lines is detected only towards the S255IR-SMA1 clump, which represents a rotating structure (probably disk) around the young massive star. The achieved angular resolution is still insufficient for conclusions about Keplerian or non-Keplerian character of the rotation. The temperature of the molecular gas reaches 130–180 K. The size of the clump is about 500 AU. The clump is strongly fragmented as follows from the low beam filling factor. The mass of the hot gas is significantly lower than the mass of the central star. A strong DCN emission near the center of the hot core most probably indicates a presence of a relatively cold ($\lesssim 80$ K) and rather massive clump there. High velocity emission is observed in the CO line as well as in lines of high density tracers HCN, HCO$^+$, CS and other molecules. The outflow morphology obtained from combination of the SMA and IRAM-30m data is significantly different from that derived from the SMA data alone. The CO emission detected with the SMA traces only one boundary of the outflow. The outflow is most probably driven by jet bow shocks created by episodic ejections from the center. We detected a dense high velocity clump associated apparently with one of the bow shocks. The outflow strongly affects the chemical composition of the surrounding medium.'
author:
- |
I. Zinchenko, S.-Y. Liu, Y.-N. Su, S. V. Salii, A. M. Sobolev, P. Zemlyanukha, H. Beuther,\
D. K. Ojha, M. R. Samal, Y. Wang
bibliography:
- 'apj-jour.bib'
- 's255.bib'
title: 'The disk-outflow system in the S255IR area of high mass star formation'
---
Introduction
============
[The formation of high mass stars (more massive than 8–10 M$_\odot$) is still poorly understood. Studies of this process are currently a “hot topic" of astrophysical research.]{} A major unsolved problem of high mass star formation is the characterization of accretion disks around young high-mass protostars. To date, most of evidence for the existence of such disks has been indirect [e.g. @Beuther09]. If present, these disks undoubtedly play a crucial role in the star formation process. Nevertheless, the very existence of these disks, much less their physical properities, is not well-established. The importance of understanding these disks cannot be understated. For example, because of the larger masses involved, it is quite possible that high-mass disks are self-gravitating, unlike their Keplerian low-mass counterparts, implying that there may be significant dynamical differences between the high-mass and low-mass star formation processes.
S255IR is a part of the well known massive star forming complex located between the evolved Sharpless regions S255 and S257. Several authors estimated the photometric distance to the complex at about 2.5 kpc [@Russeil07; @Chavarria08; @Ojha11]. However, @Rygl10 report a distance of 1.6 kpc based on trigonometric parallax measurements of methanol masers. This seems to be the most accurate distance estimate and we adopt it here, too. [The previous studies have shown that S255IR contains a cluster of early-B-type stars [@Howard97; @Itoh01], several compact regions [@Snell86], and a number of H$_2$ emission features [@Miralles97]. The total mass of the S255IR core has been estimated from single-dish observations at $M \sim 300-400$ M$_{\odot}$ [@Zin09; @Wang11]. The estimate of the luminosity has been $L \sim 2\times 10^4$ L$_{\odot}$ [@Wang11].]{}
[The interferometric observations by @Wang11 revealed 3 continuum clumps and a high-velocity collimated outflow in the S255IR area.]{} Our Paper I [@Zin12] was devoted to the general structure and kinematics of the complex based on the SMA observations at 1.3 and 1.1 mm ( [the synthesized beam sizes were approximately $3\farcs8 \times 3\farcs0$ at 1.3 mm and $2\farcs9 \times 2\farcs6$ at 1.1 mm]{}) as well as VLA ammonia observations and GMRT low frequency continuum data. Our results as well as data by @Wang11 indicate in particular a presence of a hot rotating core and a spectacular outflow in the S255IR region. However the angular resolution was insufficient for a more detailed investigation of this core. In addition, a reliable evaluation of physical and chemical properties of this system required observations of additional molecular transitions. Therefore we performed observations of the S255IR area with the SMA at a much higher angular resolution and at higher frequencies.
Here we present observational data for S255IR obtained at 1.3 mm with the SMA in the very extended configuration and at 0.8 mm in the compact configuration. In addition we observed this area with the IRAM 30m radio telescope in order to obtain short-spacing data complementing the SMA results. We mainly discuss properties of the dense cores and high velocity outflows observed in this area.
Observations and data reduction
===============================
SMA
---
The S255 IR area was observed with the Submillimeter Array (SMA) in its compact configuration on 2010 December 14th at 350 GHz. A three field mosaic was obtained with the following phase centers: 06$^h$12$^m$53$^s$.800, 17$^\circ$59$'$22$''$.1, 06$^h$12$^m$54$^s$.8876, 17$^\circ$59$'$29$''$.3353 and 06$^h$12$^m$52$^s$.7124, 17$^\circ$59$'$14$''$.8647. The primary HPBW of the SMA antennas is 36$^{\prime\prime}$ at these frequencies. Typical system temperatures on source were between 200 K and 400 K. The resulting $uv$ coverage ranges from 10 k$\lambda$ to 90 k$\lambda$ (8 antennas in the array). 3C454.3 and Uranus were used as the bandpass calibrators and 0532+075, 0750+125 and 0530+135 were used as the complex phase and amplitude gain calibrators. A total of 8 GHz (342$-$346 GHz in the LSB and 354$-$358 GHz in the USB) was observed with the SMA bandwidth doubling correlator configuration. The spectral resolution was 0.8125 MHz.
In addition on 2011 January 07th S255IR was observed in the very extended configuration at 225 GHz. A single field was observed with the same phase center as for the central field at 350 GHz. The primary HPBW of the SMA antennas is 55$^{\prime\prime}$ at these frequencies. Typical system temperatures on source were between 90 K and 200 K. The resulting $uv$ coverage ranges from 50 k$\lambda$ to 350 k$\lambda$ (7 antennas in the array). 3C454.3 and 3C279 were used as the bandpass calibrators and 0750+125 and 0530+135 were used as the complex phase and amplitude gain calibrators. A total of 8 GHz (217.0$-$220.8 GHz in the LSB and 229.1$-$232.95 GHz in the USB) was observed with the SMA bandwidth doubling correlator configuration. The spectral resolution was 0.406 MHz and 1.625 MHz ( [given the limited correlator capability, we were not able to have the high resolution across the whole band; so, we assigned different resolutions for different tracers]{}). These data were combined with our previous data obtained at the same frequencies in the compact configuration (Paper I) which enables an investigation of the source structure in a wider range of spatial scales.
The gain calibrator flux scale, calibrated against Uranus at 350 GHz and Callisto at 225 GHz, was found to be consistent within 5% with the SMA calibrator database and estimated to be accurate within 20%.
The data calibration was carried out with the IDL superset MIR [@Scoville93], and subsequent imaging and analysis were done in MIRIAD [@Sault95]. With robust weighting for the continuum and line data, the synthesized beam sizes are approximately $2\farcs2 \times 1\farcs9$ at 350 GHz and $0\farcs5 \times 0\farcs4$ at 225 GHz. The rms noise is approximately 7 mJybeam$^{-1}$ and 1 mJybeam$^{-1}$ in the continuum images at 350 GHz and 225 GHz, respectively, and 100 mJybeam$^{-1}$ and 20 mJybeam$^{-1}$ in the spectral cubes at these frequencies at 2 kms$^{-1}$ resolution.
IRAM 30m radio telescope
------------------------
Single-dish observations of several molecular lines at the 30m IRAM radio telescope were performed in October 2012 (N$_2$H$^+$ $ J=3-2 $ at 279.5 GHz), December 2012 (SiO $J=5-4 $ at 217.1 GHz) and January 2014 (CO $ J=3-2 $ at 345.8 GHz and CS $ J=7-6 $ at 342.9 GHz). The antenna HPBW was $9\farcs3$, $11\farcs9$ and $7\farcs5$ at these frequencies, respectively. The observations were performed in the OTF mode with position switching using the HERA receiver at 217.1 GHz and EMIR receiver at higher frequencies. The central position of the maps was the same as for the primary field in the SMA observations. The reference position was selected at $-$500$''$ in right ascension from the central position. Apparently a weak CO(3–2) emission is present at the reference position resulting in a weak negative feature in the CO spectra at $ V_\mathrm{LSR}\approx 24 $ kms$^{-1}$ (the bulk of the line emission in this area is observed at $V_\mathrm{LSR}\sim 4-10$ kms$^{-1}$). This feature [does not affect significantly the observed spectra]{}. The system temperature was $ \sim 200 $ K for the N$_2$H$^+$ observations, $ \sim 300 $ K for the SiO observations and $ \sim 600 $ K for the CO and CS observations. Pointing was checked regularly on nearby strong sources and pointing errors were within a few arcseconds. The antenna temperature calibration was made by the standard chopper-wheel method.
The map size was approximately $2\farcm0 \times 2\farcm5$ for the N$_2$H$^+$ and SiO observations, covering both S255IR and S255N regions, and approximately $1\farcm5 \times 1\farcm0$ for the CO and CS observations, covering only the S255IR area. Here we discuss only the data relevant to S255IR. The data on S255N are postponed for further publications.
The data reduction was performed with the GILDAS package (http://www.iram.fr/IRAMFR/GILDAS). Then the single-dish data were combined with the SMA data using the MIRIAD procedures as described by e.g. @Wang11. The conversion to the flux density scale was made using the conversion factors presented on the IRAM 30m telescope website.
Observational results and data analysis {#sec:results}
=======================================
With the SMA we detected several tens of spectral lines in both 350 GHz and 225 GHz bands. A list of these lines, including their frequencies and energy of lower levels, is given in Tables \[table:lines-vext\], \[table:lines-comp\]. The spectral line parameters are taken from the JPL [@Pickett98] and CDMS [@Mueller01; @Mueller05] catalogs.
We present the results in the form of maps as well as spectra and line parameters at selected positions. For continuum observations we give positions, flux densities, and size estimates of the continuum sources.
[lcrrc]{} $ ^{12} $CO &2–1 &230.538000 &5.532 &\
$ ^{13} $CO &2–1 &220.398684 &5.289 &\
CH$ _{3} $OH &$6_{1} - 7_{2}$ A$^-$ &217.299202 &363.496\
&$15_{6} - 16_{5}$ A$^-$ &217.642677 &735.160 &\
&$15_{6} - 16_{5}$ A$^+$ &217.642678 &735.160 &\
&$20_{1} - 20_{0}$ E &217.886504 &497.919 &\
&$4_{2} - 3_{1}$ E &218.440063 &34.976 &\
&$8_{0} - 7_{1}$ E &220.078561 &86.051 &\
&$15_{4} - 16_{3}$ E &229.589056 &363.420&\
&$8_{-1} - 7_{0}$ E &229.758756 &78.076 &\
&$19_{5} - 20_{4}$ A$^+$ &229.864121 &567.565 &\
&$19_{5} - 20_{4}$ A$^-$ &229.939095 &567.561 &\
&$3_{-2} - 4_{-1}$ E &230.027047 &28.788 &\
&$22_{2} - 21_{-3}$ E &230.292196 &598.499 &\
&$10_{2} - 9_{3}$ A$^-$ &231.281110 &154.248&\
&$10_{2} - 9_{3}$ A$^+$ &232.418521 &154.248\
&$18_{3} - 17_{4}$ A$^+$ &232.783446 &435.360\
$^{13}$CS &5–4 &231.220686 &22.194 &\
DCN &3–2 &217.238530 &10.425 &\
HNCO &$ 10_{0,10}-9_{0,9} $ &219.798282 &47.471 &\
&$ 10_{1,9}-9_{1,8} $ &220.584762 &90.916 &\
&$ 10_{1,10}-9_{1,9} $ &218.981170 &90.569\
&$ 10_{2,9}-9_{2,8} $ &219.733850 &217.739 &\
&$ 10_{2,8}-9_{2,7} $ &219.737193 &217.739 &\
&$ 10_{3,8}-9_{3,7} $ &219.656770 &422.417 &\
&$ 10_{3,7}-9_{3,6} $ &219.656771 &422.417 &\
HC$_3$N &24–23 &218.324711 &120.504 &\
SO &$6_5 - 5_4$ &219.949433 &24.429 &\
H$_2$CO &$3_{0,3} - 2_{0,2}$ &218.222195 &10.483 &\
&$3_{2,2} - 2_{2,1}$ &218.475642 &57.608 &\
&$3_{2,1} - 2_{2,0}$ &218.760071 &57.613 &\
OCS &18–17 &218.903357 &89.304 &\
&19–18 &231.060983 &99.810 &\
HCOOH &$ 10_{0,10}-9_{0,9} $ &220.038072 &48.061 &\
&$ 10_{1,9}-9_{1,8} $ &231.505705 &53.355\
CH$_3$CN &$12_{6} - 11_{6}$ &220.594423 &315.313\
&$12_{5} - 11_{5}$ &220.641084 &236.810\
&$12_{4} - 11_{4}$ &220.679287 &172.555\
&$12_{3} - 11_{3}$ &220.709017 &122.565\
&$12_{2} - 11_{2}$ &220.730261 &86.849\
&$12_{1} - 11_{1}$ &220.743011 &65.416\
&$12_{0} - 11_{0}$ &220.747261 &58.272
[lcrrc]{}
$^{12}$CO &3–2 &345.795990 &16.596 &\
CS &7–6 &342.882850 &49.372 &\
HCO$^+$ &$4 - 3$ &356.734242 &25.682 &\
CH$ _{3} $OH &$13_{1} - 13_{0}$ A$^{-+}$ &342.729796 &211.024 &\
&$13_{-1} - 14_{-2}$ E &343.599019 &607.550&\
&$18_{2} - 17_{3}$ E &344.109039 &402.884 &\
&$19_{1} - 18_{2}$ A$^+$ &344.443433 &434.697 &\
&$16_{1} - 15_{2}$ A$^-$ &345.903916 &316.049 &\
&$18_{-3} - 17_{-4}$ E &345.919260 &442.829 &\
&$13_{0} - 12_{1}$ A$^+$ &355.602945 &193.959 &\
HCN &4–3 &354.505473 &25.251 &\
&4–3 [(0,1$^{1c}$,0)]{} &354.460435 &1049.892 &\
&4–3 [(0,1$^{1d}$,0)]{} &356.255568 &1050.021 &\
H$^{13}$CN &4–3 &345.339769 &24.861 &\
HC$^{15}$N &4–3 &344.200109 &24.779 &\
HC$_3$N &38–37 &345.609016 &306.905 &\
&39–38 &354.697456 &323.492 &\
SO &$8_8 - 7_7$ &344.310792 &70.957 &\
$^{33}$SO &$8_{9,8} - 7_{8,7}$ & 343.086102 &61.564\
&$8_{9,9} - 7_{8,8}$ & 343.087298 &61.566\
SO$_2$ &$13_{2,12} - 12_{1,11}$ &345.338538 &76.410 &\
&$12_{4,8} - 12_{3,9}$ &355.045517 &93.960 &\
&$13_{4,10} - 13_{3,11}$ &357.165390 &105.823 &\
&$15_{4,12} - 15_{3,13}$ &357.241193 &132.537 &\
&$11_{4,8} - 11_{3,9}$ &357.387580 &82.800 &\
&$8_{4,4} - 8_{3,5}$ &357.581449 &55.202 &\
&$9_{4,6} - 9_{3,7}$ &357.671821 &15.992 &\
&$7_{4,4} - 7_{3,5}$ &357.892442 &47.835 &\
&$6_{4,2} - 6_{3,3}$ &357.925848 &41.402 &\
&$17_{4,14} - 17_{3,15}$ &357.962905 &162.932 &\
H$_2$CS &$10_{0,10} - 9_{0,9}$ &342.946424 &74.132 &\
&$10_{2,9} - 9_{2,8}$ &343.322082 &126.830 &\
&$10_{3,8} - 9_{3,7}$ &343.409963 &192.613 &\
&$10_{3,7} - 9_{3,6}$ &343.414146 &192.556 &\
HCOOH &$16_{1,16} - 15_{1,15}$ &342.521225 &127.153\
&$15_{1,14} - 14_{1,13}$ &343.952413 &119.774\
&$16_{0,16} - 15_{0,15}$ &345.030596 &126.494\
&$16_{2,15} - 15_{2,14}$ &356.137250 &141.578
Data analysis
-------------
### Methanol {#sec:methanol_analysis}
For the methanol data analysis, we constructed simple radiative transfer model, which uses the large velocity gradient (LVG) approximation. Dust emission and absorption within the emission region was taken into account in the way described by @Sutton2004. We assumed that the dust particles are intermixed with gas. The same physical temperature for the gas and dust components is assumed. The molecular emission region was assumed to be spherically symmetric and uniform in H$_2$ density, gas and dust temperature, gas-to-dust ratio and methanol fractional abundance. The influence of external infrared sources was not considered. The dust opacity law was chosen as $\tau_{\mathrm{dust}} \propto \lambda^{-2}$. We adopted a gas-to-dust mass ratio of 100 and a cross section at 1 mm of $2.6 \times 10^{-25}$ cm$^2$ [@Sherwood80]. In addition to the model described in @Sutton2004 we used the collision transition rates based on the model of collisions of methanol molecules with He and para-H$_2$ molecules [@Cragg05]. Scheme of energy levels in this model includes rotational levels with the quantum numbers $J$ up to 22, $ \vert K\vert $ up to 9; the levels include the rotational levels of the ground, first and second torsionally excited states. In total, 861 levels of A-methanol and 852 levels of E-methanol were considered according to @Cragg05.
With this model we made estimates of the hydrogen number density, $n_{\mathrm{H_2}}$, specific column density of methanol, $N_{\mathrm{CH_3OH}} /\Delta V $, gas kinetic temperature, $T_\mathrm{kin}$ and fractional abundance of methanol $N_{\mathrm{CH_3OH}}/N_{\mathrm{H_2}}$, using the measured values of “quasi-thermal” (i.e. non-maser) methanol lines. The variable parameter of the source size is introduced in order to take into account beam-dilution effects.
The brightness temperatures of all detected “quasi-thermal” methanol lines as well as upper limits for the brightness temperatures of other lines were taken into account. We have searched for a set of parameters which provides the best agreement between the values of the calculated brightness temperatures ($T^{mod}_i$) and the measured brightness temperatures ($T^{obs}_i$). This corresponds to the minimum of $\chi^2 = \sum_{i}^{N} ((T^{obs}_i - T^{mod}_i )/\sigma_i)^2$, where $\sigma_i$ is observational uncertainty for a particular line.
[The sources appear inhomogeneous in some cases. However, the LVG approximation treats the problem locally and does not actually require physical homogeneity of the source. It can be well used for inhomogeneous medium [e.g. @Ossenkopf97]. We have chosen the simplest approximation for the case of the source which is smaller than the beam – introduced filling factor which is equal to the portion of the source size in the beam. The value of the filling factor is well constrained because the model is sensitive to its changes. We do not have enough observational constraints to study real clumpiness of the source. Introducing additional poorly constrained parameters will reduce validity of the modelling.]{}
### Methyl cyanide {#sec:ch3cn_analysis}
Methyl cyanide (CH$_3$CN) is a symmetric-top molecule which is an efficient “thermometer” for dense molecular gas [e.g. @Boucher80]. To derive the kinetic temperature we used the population diagram analysis which takes into account the optical depth in the CH$_3$CN lines and the beam filling factor as described by @Wang10.
Millimeter wave continuum
-------------------------
Maps of the continuum emission at 0.8 and 1.3 mm are presented in Fig. \[fig:cont\]. The 1.3 mm map was obtained by combining our new measurements in the very extended configuration with the previous measurements in the compact configuration (Paper I). It shows a rather extended emission and compact cores in the SMA1 and SMA2 clumps.
Estimates of the continuum source parameters are given in Tables \[table:cont-350\],\[table:cont-230\]. For the measurements at 1.3 mm we indicate the parameters of the compact cores seen in the very [extended]{} configuration. Parameters of the more extended components were presented in Paper I. The measured fluxes are several hundreds mJy at 350 GHz and several tens mJy at 225 GHz in the very extended configuration. The deconvolved sizes are from $ \sim 1 $ arcsecond to a few arcseconds at 345 GHz and about 0.3 arcsecond for the compact cores at 225 GHz.
[lllcccr]{} S255IR-SMA1 &6:12:54.00 &17:59:23.2 &0.50 &1.4 &0.6 &–11\
S255IR-SMA2 &6:12:53.76 &17:59:26.1 &0.56 &2.4 &2.1 &4\
S255IR-SMA3 &6:12:53.86 &17:59:23.7 &0.13 &1.3 &0.3 &44\
S255IR-SMA4 &6:12:54.01 &17:59:12.0 &0.25 &5.9 &3.8 &–82
[lllcccr]{} S255IR-SMA1 &6:12:54.010 &17:59:23.06 &0.058 &0.30 &0.27 &–19\
S255IR-SMA2 &6:12:53.779 &17:59:26.16 &0.016 &0.49 &0.31 &–27
Basic properties of the molecular emission
------------------------------------------
Here we give the general description of the observed molecular emission. A more detailed information on the relevant species and transitions is presented in the following sections.
The general morphology and kinematics of the molecular emission was described in Paper I. Our new data set contains several tens of molecular transitions. In Fig. \[fig:4x3maps\] we present representative maps obtained with the SMA at 0.8 mm.
These maps confirm that SMA1 is the brightest source of molecular emission in this area. High-excitation lines are observed exclusively towards this clump. Several lower excitation molecular lines are detected also in SMA2. Emission of HCN, SO and CH$_3$OH is observed also in SMA4. HCN and CS emission is probably present in the area to the west of SMA2, designated as S255IR-N$_2$H$^+$(1) in Paper I. [Since it is observed not in N$_2$H$^+$ only and for simplicity we shall designate it hereafter as SMA2-W]{}.
A new feature, not noticed in our Paper I and in other previous works is a molecular clump to the east of SMA1 with a rather strong emission in the HCN, HCO$^+$, CS and SO lines (Fig. \[fig:4x3maps\]). We shall designate it as SMA1-E. This clump is located near the head of the jet observed in the NIR emission (see below).
Spectra of several representative transitions towards the SMA1 and SMA2 clumps are presented in Figs. \[fig:sma1-spectra\],\[fig:sma2-spectra\].
{width="\textwidth"}
{width="\textwidth"}
As mentioned in Paper I the main emission peak toward the SMA1 clump is at $ V_\mathrm{LSR} \sim 4-5 $ kms$^{-1}$. The line width is $ \ga 5 $ kms$^{-1}$. Several lines including CO, HCN, HCO$^+$, CS, SO show broad wings indicative of high velocity outflow. We discuss this feature in the following sections. A rather strong emission is detected from vibrationally excited HCN with the excitation energy $ \sim 1000 $ K above the ground level.
The line emission from the SMA1-E clump peaks at about 8 kms$^{-1}$. The linewidth is large, $ \sim 5 $ kms$^{-1}$.
The emission from the SMA2 clump is observed at about 10 kms$^{-1}$. The lines are narrow, $ \sim 2 $ kms$^{-1}$.
With the very extended array we detected the line emission almost exclusively from the SMA1 compact core. No emission was detected in C$^{18}$O and SiO indicating an absence of compact structures in the lines of these molecules. The results of these observations are presented and discussed below.
Structure, kinematics and physical properties of dense cores
============================================================
Our data presented in Paper I and here as well as data by @Wang11 indicate the presence of four continuum clumps in the S255IR area, designated from S255IR-SMA1 to S255IR-SMA4. In addition, in Paper I we detected clumps with a rather strong molecular emission without continuum counterpart in the SMA data. One of them, S255IR-N$_2$H$^+$(1), is located close to SMA2 and was observed in the N$_2$H$^+$, NH$_3$ and several CH$_3$OH lines. Estimates of their basic physical properties were presented in Paper I. Here we investigate further these objects using the new data set.
SMA1
----
SMA1 is the brightest object in this area. As shown in Sect. \[sec:results\] most of the observed molecular lines are detected only here. The deconvolved size of the continuum source at 350 GHz measured with the SMA in the compact configuration (Table \[table:cont-350\]) is close to that found in Paper I. The flux density measured at 350 GHz is only slightly higher than the flux density obtained at 284 GHz (Paper I). It is worth noting that in Paper I we could not separate the SMA1 and SMA3 clumps. However, even if we take the integrated flux of these clumps at 350 GHz, the spectral index in the range $ 284-350 $ GHz will be only 1.6 which is inconsistent with a presumably optically thin [dust emission at these frequencies]{}. Most probably this implies a significantly larger flux loss at 350 GHz in comparison with the measurements at 284 GHz due to a smaller beam size.
The deconvolved size of the continuum source detected with the SMA in the very extended configuration is about 0$\farcs$3 (Table \[table:cont-230\]) which corresponds to about 500 AU. The observed aspect ratio for this core is close to unity.
### Kinematics
@Wang11 noticed rotation of the core around the axis roughly parallel to the outflow direction. Now, at sub-arcsecond resolution we can investigate the core kinematics on smaller scales. In Fig. \[fig:sma1\_ch3oh-vel\] we present maps of the first moment of the CH$_3$OH emission in the $4_{2} - 3_{1}$ E line and CH$_3$CN emission in the $12_{3} - 11_{3}$ line. In this figure we also indicate the axis of the jet [previously identified through]{} IR observations [@Howard97] and locations of the water masers measured with the VLBA [@Goddi07].
{width="\columnwidth"}
{width="\columnwidth"}
This figure clearly shows that the core is really rotating around the axis of the jet. The rotation velocities along the line of sight [amount to]{} a few kms$^{-1}$ [(it varies from about 3 kms$^{-1}$ to about 7 kms$^{-1}$ for CH$_3$OH $4_{2} - 3_{1}$ E and to about 8 kms$^{-1}$ for CH$_3$CN $12_{3} - 11_{3}$)]{}. In Fig. \[fig:sma1\_ch3oh-pv\] we present the position-velocity diagram for the CH$_3$OH emission in the $4_{2} - 3_{1}$ E line along the cut through the core center perpendicular to the jet axis [(PA = 157$^\circ$)]{}. It clearly shows the velocity gradient [which can be consistent with]{} Keplerian rotation but the angular resolution is still insufficient for firm conclusions about the character of this rotation.
An implicit indication of a probable further increase of the rotation velocity in the innermost part of the core comes from the line width map (Fig. \[fig:sma1\_ch3oh-width\]). It shows a significant increase of the line width in the center which can be related to rotation [(the line width increases from about 2 kms$^{-1}$ at the periphery to about 6 kms$^{-1}$ in the center)]{}.
![Map of the second moment of the CH$_3$OH emission in the $4_{2} - 3_{1}$ E line (color scale). Contours show the 1.3 mm continuum emission. [The beam for the molecular map is shown in the lower left corner.]{}[]{data-label="fig:sma1_ch3oh-width"}](sma1_ch3oh2184-width){width="\columnwidth"}
It is also worth mentioning an increased line width along the jet axis (Fig. \[fig:sma1\_ch3oh-width\]). It can be probably explained by an increased turbulence caused by the passage of the jet.
[The HCO$^+$(4–3) and probably HCN(4–3) spectra towards SMA1 (Fig. \[fig:sma1-spectra\]) show the red-shifted absorption dip which may be suggestive of infall. The CO(3–2) spectrum measured with the SMA has a similar (although broader) feature (Fig. \[fig:sma1\_abs\]). At the same time the CO(3–2) spectrum obtained from the combined SMA and IRAM-30m data shows a peak at these velocities. Most probably it means that this feature is related to an extended component resolved out by the SMA, although an infall cannot be fully excluded.]{}
![ [The SMA spectra of HCO$^+$(4–3) and CO(3–2) as well as the combined SMA+30m spectrum of CO(3–2) towards SMA1. The angular resolution is about 2 arcseconds.]{}[]{data-label="fig:sma1_abs"}](sma1_abs){width="\columnwidth"}
### Physical properties
Let us consider physical parameters of this core. We estimated gas kinetic temperature from the CH$_3$CN and CH$_3$OH observations. Modeling of the CH$_3$CN emission (Fig. \[fig:sma1\_ch3cn-fit\]) as described in Sect. \[sec:ch3cn\_analysis\] yields kinetic temperature in the range (74.9 – 197.6) K (1$\sigma$ confidence interval) towards the emission peak (Table \[table:ch3cn-sma1\]). [We have]{} detected a large number (15) of CH$_3$OH transitions towards SMA1 (Table \[table:lines-vext\]). From the CH$_3$OH analysis (Sect. \[sec:methanol\_analysis\]) we obtain $ T_\mathrm{kin} = 178 $ K with the $165-195$ K 1$\sigma$ confidence interval at the center, $ T_\mathrm{kin} = 170-200 $ K at 0$\farcs$2 to the north and $ T_\mathrm{kin} = 140-165 $ K at 0$\farcs$2 to the south (Table \[table:ch3oh-sma1\]). [Therefore there is probably a temperature gradient in the north-south direction. The temperature seems to be anti-correlated with the CH$_3$OH column density (Table \[table:ch3oh-sma1\]).]{}
![The CH$_3$CN spectrum measured towards the SMA1 emission peak and the model fit.[]{data-label="fig:sma1_ch3cn-fit"}](sma1_ch3cn-fit){width="\columnwidth"}
[lc]{} $T_\mathrm{kin}$ (K) &128.8 (74.9 – 197.6)\
$N_\mathrm{L}(\mathrm{CH_3CN})$ ($ 10^{16} $ cm$^{-2}$) & 2.12 (0.96 – 5.80)\
Beam filling factor &0.23 (0.18 – 0.36)\
$\tau(K=0)$ &2.53 (0.93 – 8.37)
![ [An example of the CH$_3$OH data fitting for the central position in SMA1. Variations of $\chi^2$ with the model parameters are shown in the 2 top and left bottom panels. The minimum of $\chi^2$ is marked by the star. Confidence intervals of 0.25, 1, 1.6 2.6 $\sigma$ are plotted on the grey scale. The confidence interval of 1 $\sigma$ is marked by the white line. The comparison of the model (stars) and observed main-beam temperatures is shown in the right bottom panel. ]{}[]{data-label="fig:sma1_ch3oh-fit"}](S255IR_SMA1_hr_mod_ch2_lines_fit){width="\columnwidth"}
[cccccc]{} 0, +0.2 & 183 (170—200)& 3.6 (2.8—5.0) & 3.2 & 14.8 (13.9—16.4) & $10^{-6} (>10^{-7})$\
0, 0 & 178 (165—195)& 5.6 (4.0—9.5) & 3.2 & 16.0 (15.2—17.2) & $10^{-6} (>10^{-7})$\
0, $-$0.2 & 153 (140—165)& 8.9 (5.0—12.6) & 0.2 & 15.2 (13.3—16.0) & $10^{-6} (>10^{-7})$
[We adopt the gas kinetic temperature of 170 K as the average value (it is close to the weighted average of the CH$_3$OH and CH$_3$CN results).]{} Assuming the same temperature (170 K) for the dust we obtain a total mass of this hot component of about 0.3 M$_\odot$. As in Paper I we assume a gas-to-dust mass ratio of 100, and adopt a dust absorption efficiency following @Ossenkopf94. The peak gas column density estimated from the continuum data is $ N(\mathrm{H_2}) \sim 3\times 10^{24} $ cm$^{-2}$. [With a]{} size of 500 AU the mean density of hot gas is about $6\times 10^8$ cm$^{-3}$. This is 2 times higher than our estimate of the SMA1 mean density in Paper I taking into account the difference in the adopted distances here and in Paper I. This increase [can be expected]{} since now we consider much smaller scales near the core center. The CH$_3$OH modeling puts no significant constraints on density. The relative abundance of CH$_3$OH is $ \sim 10^{-6} $.
Both CH$_3$CN and CH$_3$OH observations indicate beam filling factor of 0.15–0.2. This means that the source is very inhomogeneous on the 0$\farcs$4 scale and probably consists of clumps with the size $ \ll 500 $ AU and density much higher than the mean density found above. This higher density estimate does not contradict the CH$_3$OH data. A similar picture of clumpy medium was inferred from our observations of high mass star forming regions on larger scales [@Pirogov08; @Pirogov12]. Most probably it reflects turbulence in the cores.
Our data set includes several other tracers of the hot gas. One of the most important is HNCO. We detected HNCO lines in different $K_{-1}$ ladders ($ K_{-1} = 0, 1, 2, 3 $) with the excitation energies up to $ \sim 400 $ K (Table \[table:lines-vext\]). In Fig. \[fig:hnco-rd\] we present the rotational diagram for the observed HNCO transitions. This diagram was obtained from peak integrated line intensities found by 2D Gaussian fitting of the integrated intensity maps in different HNCO lines. We used the line strengths and dipole moment from the Cologne Database for Molecular Spectroscopy [@Mueller01; @Mueller05].
![The rotational diagram for HNCO towards the emission peak.[]{data-label="fig:hnco-rd"}](hnco-rd-int){width="\columnwidth"}
This diagram indicates a rotational temperature of $318\pm 70$ K (Table \[table:rd\]). [However, the $ K_{-1} = 0, 1 $ transitions can be saturated. The peak brightness temperature in these transitions is close to the brightness temperature in the apparently optically thick lines (e.g. CO and CH$_3$CN) and our modeling using RADEX [@vdTak07] shows that at the derived physical parameters and column density the optical depth in these lines is about unity. In this case the derived rotational temperature would apparently represent an upper limit to the excitation temperature.]{} The deconvolved source size is $ \sim 0\farcs3 \times 0\farcs2 $ for the $ K_{-1} = 0 $ transition and decreases for higher $K_{-1}$ ladders. The map of the integrated intensity in the HNCO $ K_{-1} = 2 $ transition is shown in Fig. \[fig:sma1\_hnco\] along with the map of the OCS emission which is another tracer of hot gas. The peaks of the HNCO and OCS emission practically coincide with the continuum peak, although distributions of these molecules seem to be somewhat different.
[lccc]{} HNCO & 318(70) &16.34(0.04) &$ \approx $0.4\
SO$_2$ & 146(16) &15.50(0.05) &$ \approx $2\
& 65(11) &15.35(0.08) &$ \approx $2
{width="\columnwidth"}
{width="\columnwidth"}
The total HNCO column density derived from the rotational diagram is $ \sim 2\times 10^{16} $ cm$^{-2}$. We used the partition function from the Cologne Database for Molecular Spectroscopy [@Mueller01; @Mueller05], too. A comparison with the total gas column density indicates HNCO abundance of $ X(\mathrm{HNCO}) \sim 10^{-8} $. This is a rather high value, close to the highest HNCO abundance derived in the survey of massive cores by @Zin00. [However, taking into account the note above about the HNCO lines saturation, this value may need a correction.]{}
In the 350 GHz band we detected many SO$_2$ lines (Table \[table:lines-comp\]). The corresponding population diagram is shown in Fig. \[fig:so2-rd\]. It indicates a range of temperatures. For low-excitation transitions the rotational temperature is $65\pm 11$ K. For the high-excitation ones it is $146\pm 16$ K. The SO$_2$ observations were performed at a much lower angular resolution (2 arcsec) than in case of HNCO and apparently include emission from both the hot core and the surrounding cloud. The derived temperatures are consistent with the other estimates for these components.
![The population diagram for SO$_2$ at the SMA1 emission peak.[]{data-label="fig:so2-rd"}](so2-rd-int){width="\columnwidth"}
Another indicator of the hot environment is the vibrationally excited HCN. We detected the emission in the $ v_2 =1 $ state, about 1000 K above the ground state. For the $ v_2 =2 $ emission the upper limit is about 5 times lower. Following the analysis presented in e.g. @Veach13 we obtain an upper limit for the excitation temperature between these states of about 500 K. This is consistent with the other estimates of the gas temperature given above. The critical density for HCN excitation is high ($ > 10^{10} $ cm$^{-3}$, @Veach13) which is consistent with our density estimates for the hot gas.
### A cold clump in the hot core?
In Fig. \[fig:sma1\_dcn\] we plot maps of the DCN $J=3-2$ and $^{13}$CS $J=5-4$ integrated line emission in the SMA1 core. The emission regions are very compact and the emission peak is shifted from the continuum peak which apparently coincides with the YSO location. The projected distance from the continuum peak is roughly 300 AU.
{width="\columnwidth"}
{width="\columnwidth"}
The velocities of the DCN and $^{13}$CS emission are practically the same as those of high excitation lines of other molecules tracing apparently the hot gas. Therefore, most probably the observed DCN emission arises within the disk and since we probably see the disk nearly face-on as discussed above, the physical distance from the center is not much larger than the projected distance. An estimate of the abundances in the LTE approximation gives $ X(\mathrm{DCN}) > 10^{-11} $ and $ X(\mathrm{^{13}CS}) > 3\times 10^{-11} $ (assuming the total mass of the clump $ <10 $ M$_\odot$ and temperature of 50–100 K). This is a “normal” value for $^{13}$CS while the derived DCN abundance implies a significant deuteriation as follows from comparison with typical HCN abundances in massive cores [e.g. @Zin09]. As pointed out in Paper I, recent modelling by @Albertsson11 shows that DCN/HCN abundance ratio sharply drops at temperatures $ \ga 80 $ K. It means that temperature of the DCN emitting clump should be rather low, much lower than the temperature of the hot gas in the disk. Another possible explanation, as also discussed in Paper I, can be a very young age of the clump, insufficient to reach the steady-state DCN/HCN abundance ratio, but this looks less probable. In principle, interiors of the accretion disk can be well shielded from an external radiation, probably providing necessary low temperatures. In any case this clump deserves further investigation in order to clarify its properties and nature. It is worth mentioning that the mass of this clump cannot be lower than $\sim 1 $ M$_\odot$, otherwise the inferred $^{13}$CS abundance would be unrealistically high. The virial mass we estimate about the same, hence the clump can be gravitationally bound.
SMA2 {#sec:sma2}
----
As in case of the SMA1 the spectral slope for the continuum emission in the frequency range 284–350 GHz is too low ($ \sim 1.3 $), which is probably caused by a higher flux loss at higher frequencies. With the SMA at sub-arcsecond resolution we detected a compact component in continuum emission of about the same size as in the SMA1. However, no high-excitation molecular lines could be detected in this area. In Paper I we derived the temperature of the SMA2 clump $ \sim 40 $ K. Now with the extended data set, from the methanol excitation analysis (Sect. \[sec:methanol\_analysis\]) we found the 1$\sigma$ confidence interval for temperature of 40–80 K. The best estimate of the methanol relative abundance is $ X(\mathrm{CH_3OH})\sim 10^{-8} $. These estimates are obtained at the 2 arcsec scale. It cannot be excluded that the temperature of the compact structure observed in the very extended configuration is somewhat higher. However it cannot be much higher since there is no sign of a higher temperature component in the molecular data. Assuming the temperature of 50 K we obtain the mass of the compact core $\sim 0.2$ M$_\odot$.
To the west from the SMA2 there is an area of molecular emission without continuum counterpart in the SMA data, observed earlier in the N$_2$H$^+$, NH$_3$ and several CH$_3$OH lines (Paper I and @Wang11). Our new methanol data analysis indicates the temperature in the range 25–65 K, density $ n(\mathrm{H_2}) $ in the range $6.3 \times10^{4}$—$2.5 \times10^{6}$, $ X(\mathrm{CH_3OH})\sim 10^{-7}$. Some of the methanol transitions are inverted in the model and can be masing. These high gas densities and rather high methanol abundances at relatively low temperatures can be explained by the influence of shock.
[cccccc]{} SMA2 & 50 (40—80)& 2.0 (1.8—3.5) & 6.3$\times 10^{6}$ ($\leq10^{8}$) & 99.3 ($\geq 84$) & $10^{-8} (\geq10^{-8})$\
SMA2-W & 40 (25—65)& 3.2 (2.2—4.5) & 1.0$\times 10^{6}$ ($6.3 \times10^{4}$—$2.5 \times10^{6}$) & 64.0 (63—70) & $10^{-7} (\geq10^{-8})$\
SMA4 & 45 (30—90)& 3.6 (0.8—7.9) & 5.6$\times 10^{4}$ ($10^{3}$—$6.3\times 10^{5}$) & 99.9 ($\geq 20$) & $10^{-7} (10^{-6}$—$10^{-9})$
SMA3 {#sec:sma3}
----
In Paper I we did not detect this component which coincides with the near-infrared source NIRS 1 [@Tamura91] and is identified as a massive disk candidate by NIR polarization observations [@Jiang08]. However, @Simpson09 argued that these measurements could be affected by instrumental effects and there is no real evidence for the “polarization disk” here. At the same time the NIR measurements indicate a possible outflow related to this object. A comparison with the measurements by @Wang11 gives a spectral index of about 3.3 in the frequency range 225–350 GHz, consistent with the optically thin dust emission. @Wang11 estimated mass of this clump at about 2 M$_\odot$. At sub-arcsecond resolution we do not see any continuum (Fig. \[fig:cont\]) or molecular emission at this position.
SMA4 {#sec:sma4}
----
This clump was first detected in Paper I. It shows a weak continuum and spectral line emission in several molecular transitions. The measured continuum flux density at 350 GHz is practically the same as at 284 GHz which also implies a much higher flux loss at the higher frequency. The temperature derived from our new methanol data analysis is about 45 K, higher than estimated in Paper I from ammonia data. However the ammonia emission here is very weak and the uncertainties are high. Some methanol transitions are inverted in the model. The influence of shock is also probable.
Morphology and properties of the outflows
=========================================
In the S255IR area high velocity emission is detected in lines of CO, SiO and several high density tracers including HCN, HCO$^+$ and CS. In Figs. \[fig:chmap-co-blue\],\[fig:chmap-co-red\] we present channel maps of the CO $ J=3-2 $ emission in the blue and red line wings, respectively. The maps of the integrated line wing emission (Fig. \[fig:co32-outflow\]) show that the CO high velocity emission observed with the SMA looks like a highly collimated bipolar outflow originating near SMA2. There is another more compact component near SMA1.
On the other hand the bipolar outflow observed in HCN and HCO$^+$ is apparently associated with SMA1 while is parallel to the CO flow (Fig. \[fig:co32-outflow\]). It is apparently associated with the jet observed in particular in the emission which is also plotted in this figure. It is worth mentioning that the extent of the jet is much smaller than the extent of the outflow seen in CO, although their orientations coincide.
The SMA data alone hint at two parallel outflows with different origins. However single-dish CO(3–2) observations with the IRAM 30m telescope show a different picture. Here we see a wider outflow clearly originating at SMA1 (Fig. \[fig:CO-30m\]). [While the emission peaks]{} coincide in the SMA and 30m maps, near the driving source the CO emission measured with the SMA traces only the northern edge of the outflow. This shows that the SMA map gives a distorted picture and in reality we have here a wider angle outflow originating at the SMA1.
Apparently this means that this northern edge of the outflow contains a relatively bright component with a characteristic scale comparable to the SMA beam. The [southern edge]{} of the outflow near the driving source [would be]{} more diffuse and resolved out by the SMA. Most probably this implies that the CO emission is formed in a compressed layer surrounding the outflow cavity. For some (unclear) reason in the northern part this layer is more pronounced. It is worth mentioning that the brightest peaks of the high velocity CO emission practically coincide in the SMA and 30m maps and also coincide with spots of molecular hydrogen emission [@Wang11] which probably indicate bow shocks. Basic physical parameters of the outflow were estimated by @Wang11 from their CO(2–1) data. Our new data are consistent with these estimates.
The position-velocity (P–V) diagram for the outflow along the cut indicated in Fig. \[fig:CO-30m\], constructed from the 30m data, is shown in the left panel of Fig. \[fig:CO-pv\]. The gap at $V_\mathrm{LSR} \sim 24$ kms$^{-1}$ is caused by the emission at the reference offset position mentioned above. [In the right panel we present the P–V diagram obtained from the SMA CO(3–2) data. The main features of these diagrams coincide. They show non-monotonic dependence of the velocity on offset, which hints at two outflow components at different distances from the driving source. These P–V diagrams are somewhat different from that presented by @Wang11. However @Wang11 plotted this diagram for a different cut, with the position angle of 75$^\circ$, which makes the direct comparison irrelevant.]{}
[In Fig. \[fig:HCO-pv\] we plot the P–V diagram for the HCO$^+$(4–3) emission as observed with the SMA, along with the part of the CO(3–2) P–V diagram for the same intervals of the offset and velocity. The diagrams are very similar, which means that both molecules trace apparently the same gas.]{}
Dense high velocity clump {#sec:hvc}
-------------------------
Our data show a [strong, compact]{} blue-shifted CS and HCN emission close to the peak of the CO blue-shifted line wing emission. The peak of this CS and HCN emission is marked by cross in Fig. \[fig:CO-30m\]. The CO(3–2), CO(2–1), HCN(4–3), CS(7–6), HCO$^+$(4–3) and N$_2$H$^+$(3–2) spectra towards this position are presented in Fig. \[fig:hvc-spectra\].
![The CO(3–2), CO(2–1), HCN(4–3), CS(7–6), HCO$^+$(4–3) and N$_2$H$^+$(3–2) spectra towards the position marked by cross in Fig. \[fig:CO-30m\].[]{data-label="fig:hvc-spectra"}](hvc-spectra){width="\columnwidth"}
One can see rather strong and broad HCN and CS lines at a central velocity of about –5 kms$^{-1}$. There is a hint of wings in these lines extending to about –20 kms$^{-1}$. The peak of the CO emission, especially in the higher $ J=3-2 $ transition is observed at more negative velocities. There is no detectable HCO$^+$(4–3), N$_2$H$^+$(3–2), CH$_3$CN and continuum emission (for N$_2$H$^+$ and CH$_3$CN we use the data from Paper I).
The spatial distribution of the CS(7–6) and CO(3–2) emission integrated in the velocity ranges –13...+1 kms$^{-1}$ and –19...+1 kms$^{-1}$, respectively, is shown in Fig. \[fig:hvc\]. The HCN distribution is very similar to that of CS. The deconvolved size of the CS emitting clump is about $ 1{\farcs}1\times 0{\farcs}3 $ which corresponds to $ 1800\,\mathrm{AU} \times 500\,\mathrm{AU} $. It is apparently located at the head of the stream observed in CO, almost exactly at the jet axis shown in Fig. \[fig:CO-30m\].
At the velocity of the CS and HCN emission peak, the brightness temperatures in the CO(3–2), CO(2–1) and HCN(4–3) lines are practically the same. The frequencies of the CO(3–2), HCN(4–3) and CS(7–6) lines are close to each other and beam parameters for them should be similar, too. However, the comparison with the CO(2–1) is [complicated by the]{} significantly different frequencies.
The CO(3–2) line is most probably saturated. Then the HCN(4–3) line should be saturated, too. The CS line is weaker implying either a relatively lower optical depth (but close to unity anyway) or a smaller size of the emission region. A simple modeling using e.g. RADEX [@vdTak07] shows that the column densities of HCN and CS required to explain such optical depths are $ \ga 10^{15} $ cm$^{-2}$. The relative abundances of these molecules are $ \le 10^{-8} $. Therefore the total gas column density in the clump is $ \ga 10^{23} $ cm$^{-2}$. Then, using the size estimated above we obtain the gas density $ n \ga 3\times 10^{6} $ cm$^{-3}$. This estimate is consistent with the observations of the HCN and CS lines which require at least such densities for excitation.
The large HCN and CS line widths indicate that the clump is gravitationally unbound. The virial mass of this clump estimated in the usual way [e.g. @Zin94] is $ M_\mathrm{vir} \sim 30 $ M$_\odot$. At the same time, [the non-detection]{} of dust emission implies an upper limit for mass orders of magnitudes lower. Therefore, the clump represents a transient entity.
It is worth mentioning [the non-detection of]{} HCO$^+$ and N$_2$H$^+$ emission. Both molecules can be destroyed by dissociative recombination [e.g. @Zin09]. Therefore their absence can indicate an enhanced ionization in the dense clump.
Ionized gas
-----------
In Fig. \[fig:co32-outflow\] we plot the map of the 15 GHz continuum emission near the SMA1/SMA2 clumps (from the VLA archival data, the angular resolution is about 4 arcsec) which shows the distribution of the ionized gas in this area. This ionized component seems to be associated with the jet traced in the emission. The GMRT map at 1280 MHz looks similar but there is a significant positional uncertainty in the GMRT data (Paper I). Properties of the continuum source were estimated in Paper I and by @Ojha11. The emission measure is $EM \sim (1-2.5)\times 10^7 $ pccm$^{-6}$. Taking into account the observed size of the continuum source, the electron density is $ n \sim 3\times 10^4 $ cm$^{-3}$.
Surroundings
============
The surroundings of the SMA1/SMA2 clumps and high velocity outflow are traced in several molecular lines. One of the most informative is the N$_2$H$^+$(3–2) transition observed with both SMA and IRAM-30m telescope. The combined map of the N$_2$H$^+$(3–2) emission is presented in Fig. \[fig:CO-30m\]. It shows an absence of N$_2$H$^+$ in the hot core, in accordance with our previous findings (Paper I). The overall morphology of the N$_2$H$^+$ emission [suggests that it originates in]{} an envelope around the central cores and the outflow lobes.
The distribution of the SiO(5–4) emission is different (Fig. \[fig:SiO+CS\]). It peaks near the SMA1 core. There is also a feature in the area of the blue outflow lobe which may be associated with the N$_2$H$^+$ emission. The CS(7–6) distribution (right panel in Fig. \[fig:SiO+CS\]) seems to be the most uniform one. It shows a rather smooth, almost spherical halo around the SMA1 and SMA2 cores. There is no sign of the outflow influence on the CS distribution. One may suspect that the optical depth in the CS line is too high and we see only the outer regions of the core. However the SMA1 and SMA2 clumps are well resolved in this map which makes such explanation less probable. Then, the total gas distribution is apparently not significantly affected, too.
Discussion {#sec:disc}
==========
The main goal of this study is the characterization of the outflow and probable accretion disk associated with the massive YSO in the S255IR clump. The data presented in the previous sections shed new light on this system.
Concerning the outflow, one of the main findings is that the SMA interferometric data alone give a rather distorted picture. They hint at two highly collimated parallel outflows with different centers of origin. However, this impression is apparently caused by a significant flux loss in the interferometric measurements. The single-dish CO observations clearly show a single less collimated outflow originating at the SMA1 core. The CO emission retrieved by the SMA near the driving source originates apparently from the northern wall of the outflow cavity. This means that this part of the wall is rather thin and bright. The absence of a noticeable CO emission at the opposite side of the wall implies a more diffuse distribution of the emission here, probably due to the density structure of the surrounding medium.
The question [arises of]{} how common is this effect in the interferometric studies of outflows. We can easily imagine a situation when an interferometric image will show multiple outflows from a single driving source, while in fact there is a single wide-angle outflow.
[The observed P–V diagrams for the outflow (Figs. \[fig:CO-pv\],\[fig:HCO-pv\]) most probably indicate periodic ejections from the driving source. Two events can be traced in the data.]{} They apparently created jet knots at different distances from the central star. The older one is responsible for the extended CO outflow. The peaks of the CO emission coincide with bright H$_2$ emission spots [@Wang11] which probably indicate bow shocks at the heads of the jets. The next ejection event created other bow shocks which are seen in particular in the emission (Fig. \[fig:co32-outflow\]). This later ejection entrains dense molecular gas observed in the wings of the HCO$^+$, HCN and CS lines. It is also traced in the CO emission as can be seen in the CO position-velocity diagram (Figs. \[fig:CO-pv\],\[fig:HCO-pv\]). Another manifestation of this activity is apparently the dense high velocity clump (Sect. \[sec:hvc\]), which most probably represents dense gas at the head of bow shock. All the jet knots lay practically on [a straight]{} line. Therefore the orientation of the jet does not change with time significantly. The age of the first, extended outflow was estimated of about 7000 years by @Wang11. Assuming the same ejection velocities for the two events and comparing distances of the bow shocks from the driving source we can conclude that the second ejection happened about 1000 years ago. Therefore the time interval between the ejection events is about 6000 years (the age estimates were done under the usual assumption of an inclination angle for the jet of 45$^\circ$).
Several entrainment mechanisms are usually discussed for molecular outflows [e.g. @Arce07; @Frank14]. In our case the most probable one is the jet bow-shock model [e.g. @Raga93]. [The main argument in favor of this model is the obvious presence of several bow shocks (traced in the H$_2$ and emission) clearly associated with the high velocity molecular gas.]{} This model is also consistent with the observed outflow morphology and kinematics. The shape of the P–V diagram differs from the frequently observed “Hubble law” and shows a range of velocities at the largest distances from the driving source. [A similar shape was observed in some other outflows which are apparently driven by bow shocks and is expected in the theoretical models [e.g. @Lee00].]{}
The jet is apparently launched from the accretion disk around the central massive young star. As mentioned above, there is a chain of water masers along the jet. For several water maser spots shown in Fig. \[fig:sma1\_ch3oh-vel\], proper motions have been measured [@Goddi07]. The velocities (from $\sim 10$ kms$^{-1}$ to $\sim 30$ kms$^{-1}$ [at the distance of 1.6 kpc]{}) are mostly perpendicular to the jet axis with outward components in some cases. [The maser velocity pattern is consistent with]{} clockwise rotation around the core center. Velocities of the masers closest to the center imply dynamical mass of the order of 20 M$_\odot$. The mass of the central star is estimated to be 27 M$_\odot$ [@Ojha11] assuming the distance of 2.5 kpc. At the distance of 1.6 kpc the mass of the star would be $23.5 \pm 3.6$ M$_\odot$. [This estimate is based on the unpublished value of the star luminosity of $(6.8\pm 2.8)\times 10^4 $ L$_\odot$ (for the distance of 1.6 kpc). Now we reconsidered the star SED by including data at millimeter wavelengths from @Zin09. This results in the luminosity of $(3.5\pm 0.3)\times 10^4 $ L$_\odot$ and mass of $20 \pm 2$ M$_\odot$.]{} It is worth mentioning that @Pashchenko03 estimated the mass of the central star to be (6–7) M$_\odot$ from their analysis of water masers variability. However, they [employed a systemic]{} velocity of 8.3 kms$^{-1}$ which contradicts our findings which clearly show [the systemic]{} velocity to be about 4.8 kms$^{-1}$. [In the following we will take the mass of 20 M$_\odot$ as the most probable value.]{} This is sufficient to explain velocities of the masers closest to the star. However maser spots located at larger distances from the center have similar velocities and imply a much larger central mass of the order of 100 M$_\odot$. This value is inconsistent with all other estimates. The corresponding mean density would be unrealistically high. This means that we probably need another explanation for the maser velocities. [Rather speculative, it can be jet rotation. Rapid rotation of protostellar jets is expected in theoretical models [e.g. @Pudritz07] and was indeed observed in some cases [@Bacciotti02].]{}
The [water masers are thought to be]{} excited by interaction of the jet with the surrounding medium. The fact that they are moving in one direction on each side of the central star implies some asymmetry in the system. There may be some misalignment between the jet and the rotation axis of the material in the outer parts of the clump [(apparently, in the axially-symmetric case the observed maser movements would be symmetric relative the axis)]{}. It can be that in the inner part the orientation of the disk plane is different. However, the difference cannot be large. We see also other indications of the jet interaction with the surrounding medium, in particular, increased line widths along the jet.
Fig. \[fig:sma1\_ch3oh-vel\] shows that the water maser condensations are shifted from the jet axis in direction of their proper motion. This projected shift is of the order of 200 AU. At the velocities mentioned above, the maser condensations would travel this distance in about 30–100 years. From this consideration it looks probable that the event created these masers happened quite recently.
The disk is apparently strongly fragmented as follows from the derived beam filling factor ($ \sim 0.2) $ for [CH$_3$CN and CH$_3$OH]{}. The mean gas density in the disk is about $6\times 10^8$ cm$^{-3}$. The density of the fragments should be much higher. Observations of vibrationally excited HCN indicate densities $ > 10^{10} $ cm$^{-3}$ which is consistent with this picture.
The derived mass of the hot gas ($\sim 0.3$ M$_\odot$) is by an order of magnitude insufficient to gravitationally bound the core. The virial mass estimated in the usual way [e.g. @Zin94] from the line width is about 7 M$_\odot$. There can be also some amount of cold material in the disk, as follows from the DCN observations. However, the total mass of the disk is apparently much lower than mass of the central star which is estimated of about 20 M$_\odot$ [(see above)]{}. Since the disk mass is significantly lower than the star mass, the disk is probably not self-gravitating and we may expect Keplerian rotation. Unfortunately the achieved angular resolution is not sufficient [to measure reliably the disk rotation curve]{}.
For a central mass of 20 M$_\odot$ the inclination angle of the disk (derived from comparison of the observed rotation line-of-sight velocity with the expected velocity for this mass) should be small, about 25$^\circ$ (Fig. \[fig:sma1\_ch3oh-pv\]), i.e. the disk is seen almost face-on. This is consistent with our maps which do not show a significant elongation of this object. [On the other hand this conclusion looks somewhat suspicious because of the well-collimated appearance of the molecular outflow. Observations at a higher angular resolution are needed to clarify the structure and kinematics of this object.]{}
[The temperature of the hot gas derived from the CH$_3$CN, CH$_3$OH and SO$_2$ observations is about 130–180 K. A similar estimate for the temperature of the hot component was obtained by us many years ago from the analysis of the IRAS data [@Zin90]. The HNCO rotational temperature is somewhat higher, about 320 K. However, as mentioned above, this value probably represents an upper limit to the excitation temperature. In addition, the HNCO excitation can be influenced by the FIR radiation.]{} The HNCO excitation in massive cores was discussed by @Zin00. That analysis shows that in the present case collisional excitation can be effective. The critical density for excitation of the $ K_{-1} = 3 $ ladder is of the order of $ 10^{10} $ cm$^{-3}$. Such density is quite possible in the SMA1 core as follows from the estimates above. [However the radiative excitation by FIR emission via the $b$-type transitions cannot be excluded, too. With the column density of $ N(\mathrm{H_2}) \sim 3\times 10^{24} $ cm$^{-2}$ the dust will be optically thick at these frequencies creating a sufficiently strong radiation field. The HNCO excitation especially in the higher $ K_{-1} $ ladders may reflect the effective temperature of this field.]{}
The highest temperature of the observed molecular material is about 300 K. [Taking the source]{} luminosity to be about $ 3.5\times 10^4 $ L$_\odot$ (see above) [this value]{} corresponds to the dust equilibrium temperature at the distance of 200–300 AU from the star. Since we do not see warmer molecular gas, [no dust should be present]{} within this radius in case of no shielding. However this [radius]{} is comparable to the observed size of the core and we do not see any central hole in the molecular distribution. [It is then possible that]{} shielded molecular clumps are [distributed]{} at smaller radii.
In this respect an interesting feature of our observations is the apparently relatively cold and rather massive clump with a strong DCN emission. It may be gravitationally bound and the mass of the clump is sufficiently high to consider it as a possible low-mass protostar.
Concerning the surroundings of the SMA1 clump, we see that the outflow strongly affects the chemical composition of the medium. The N$_2$H$^+$ molecules are destroyed along the outflow. The SiO distribution seems to be also significantly affected. At the same time there is no noticeable influence on the CS distribution. The density structure is probably not affected, too. As mentioned above (Sect. \[sec:sma2\],\[sec:sma4\]) the nearby clumps are probably influenced by shocks which are apparently associated with this outflow.
Conclusions
===========
We presented the results of our observations of the S255IR area with the SMA at 1.3 mm in the very extended configuration and at 0.8 mm in the compact configuration as well as with the IRAM-30m at 0.8 mm. The best achieved angular resolution is about 0.4 arcsec. The dust continuum emission and several tens of molecular spectral lines are observed. The majority of the lines is detected only towards the S255IR-SMA1 clump. In summary, our main findings are the following:
1\. The S255IR-SMA1 clump represents apparently a rotating structure (probably a disk) around the young massive star. The achieved angular resolution is still insufficient [to establish the character (Keplerian or non-Keplerian)]{} of the rotation. The temperature of the molecular gas reaches 130–180 K. The size of the clump is about 500 AU. It is apparently strongly fragmented as follows from the derived small ($ \sim 0.2$) beam filling factors for various molecules. The mean gas density is about $6\times 10^8$ cm$^{-3}$. The density of the fragments should be much higher which is confirmed by observations of HNCO and vibrationally excited HCN. The mass of the hot gas is $\sim 0.3$ M$_\odot$ and the total mass of the clump is significantly lower than the mass of the central star (about 20 M$_\odot$). The inclination angle of the disk (derived from comparison of the observed rotation line-of-sight velocity with the expected rotation velocity for this mass) should be small, about 25$^\circ$.
2\. We detected a strong DCN(3–2) emission near the center of the SMA1 clump. Most probably it indicates the presence of a rather large amount ($\ga 1$ M$_\odot$) of cold ($ \le 80 $ K) material. This cold clump can be gravitationally bound.
3\. High velocity emission is observed in the CO line as well as in lines of high density tracers HCN, HCO$^+$, CS and other molecules. The CO outflow is much more extended than that observed in the lines of high density tracers. Its morphology obtained from combination of the SMA and IRAM-30m data is significantly different from that derived from the SMA data alone. The CO emission detected with the SMA traces only one boundary of the outflow and leads to a rather distorted picture of the outflow structure. The velocity of the CO outflow reaches $ \sim 60 $ kms$^{-1}$.
4\. The outflow is most probably driven by the jet bow shock mechanism. The available data indicate at least two major ejection events with a time interval of several thousand years between them. The direction of the ejections does not change with time. The high velocity emission in the lines of high density tracers is associated with the peaks of the emission related to the bow shocks caused by rather recent ejections from the SMA1.
We detected a dense high velocity clump associated apparently with one of the bow shocks. It shows a strong emission in the HCN(4–3) and CS(7–6) lines. At the same time there is no detectable HCO$^+$(4–3) or N$_2$H$^+$(3–2) emission which can be probably explained by enhanced ionization.
5\. The proper motions of the water masers excited along the jet imply some misalignment of the jet with the rotation axis of the material in the outer parts of the clump. It can be that the orientation of the disk in the inner and outer parts is somewhat different. However, the difference cannot be large.
6\. The outflow strongly affects the chemical composition of the surrounding medium. The N$_2$H$^+$ molecules are destroyed along the outflow. The SiO distribution seems to be also significantly affected. At the same time there is no sign of the outflow influence on the CS distribution. The total gas distribution is apparently not significantly affected, too.
This work was supported by the Russian Academy of Sciences (Research program No. 17 of the Department of Physical Sciences), Russian Foundation for Basic Research (RFBR), National Science Council (NSC) of Taiwan and Department of Science and Technology (DST) of the Government of India in frameworks of the research grants RFBR 08-02-92001-NSC, RFBR 13-02-92697-Ind, RFBR 15-02-06098, 15-52-45057, NSC 97-2923-M-001-004-MY3, DST-RFBR INT/RUS/RFBR/P-142. A.M.S. was supported by the Russian Science Foundation (grant number 15-12-10017). S.V.S. was supported by the Ministry of Education and Science of the Russian Federation (state task No. 3.1781.2014/K). Y.W. was supported by Swiss National Science Foundation, NSFC 11303097 and 11203081, China. The research is also partly supported by the grant within the agreement No. 02.В.49.21.0003 between The Ministry of Education and Science of the Russian Federation and Lobachevsky State University of Nizhni Novgorod and by Russian Education and Science Ministry Project 3.1252.2014/k. We are grateful to Elena Trofimova for the help with the line identification and to the anonymous referee for the detailed helpful comments. The research has made use of the SIMBAD database, operated by CDS, Strasbourg, France.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We investigate the nonclassicality of a photon-subtracted Gaussian field, which was produced in a recent experiment, using negativity of the Wigner function and the non-existence of well-behaved positive $P$ function. We obtain the condition to see negativity of the Wigner function for the case including the mixed Gaussian incoming field, the threshold photodetection and the inefficient homodyne measurement. We show how similar the photon-subtracted state is to a superposition of coherent states.'
author:
- 'M. S. Kim'
- 'E. Park'
- 'P. L. Knight'
- 'H. Jeong'
title: 'Nonclassicality of a photon-subtracted Gaussian field'
---
= 10000
Introduction
============
The recent development of quantum optics has opened the possibility to generate and manipulate various non-classical light fields, which cannot be described by classical theory, in a real laboratory. It is generally accepted that the presence of a positive well-defined $P$ function (a quasiprobability function in phase space [@Sudarshan]) signals the field classical [@Mandel]; otherwise the field is categorized as nonclassical. A stronger constraint on nonclassicality is the presence of negativity in the Wigner function (another quasiprobability function) of the field [@Lee]. While a Gaussian field may not have its $P$ function, its Wigner function never becomes negative. For example, the squeezed vacuum state is represented by its Gaussian Wigner function while its $P$ function does not exist [@Barnett]. It is also known that a Gaussian field remains Gaussian by linear transformations which correspond to basic tools in a quantum optics laboratory such as a phase shifter, a beam splitter and a squeezer [@Simon; @Ekert].
Two better-known nonclassical fields are a squeezed state and a superposition of two separate coherent states (coherent-state superposition). The two kinds of states are closely related to probably the most fundamental and intriguing paradoxes in quantum theory, i.e., the Einstein-Podolsky-Rosen paradox [@EPR] for a two-mode squeezed state and the Schrödinger’s cat paradox [@Schr] for a coherent-state superposition. They are also known as useful resources for various schemes in quantum information processing. A squeezed state and a coherent-state superposition manifest different types of nonclassicality. Whereas a squeezed state is a Gaussian field, a coherent-state superposition is non-Gaussian and shows a large amount of negativity in its Wigner function. There was an early attempt to relate the two states through quantum noise of arbitrary strength [@Carmichael]. Dakna [*et al.*]{} [@Dakna] considered a connection between the two states by subtracting a precise number of photons from a squeezed field. They also showed that any quantum state can be generated from the vacuum by application of the coherent displacement operator and adding photons [@Dakna2]. On the other hand, it has been reported that by squeezing a single photon state one can generate a state which has almost unit fidelity to a coherent-state superposition of small amplitude [@Lund].
It is only very recently that a traveling non-Gaussian field was experimentally generated by subtracting a photon from a squeezed vacuum by Wenger [*et al.*]{} [@Grangier]. They used a beam splitter and a threshold detector to subtract a photon from the squeezed field, but the reconstructed Wigner function failed to show a negative value [@Grangier]. It is, thus, timely to analyze the generation of a non-Gaussian state in relation to the status of experiments. In particular, as such the state forms a starting point for distillation of a continuous-variable field for quantum information processing [@Eisert] and may improve the efficiency of quantum teleportation [@Welsch], the study will be of use. In this paper, we assess the nonclassicality of a photon-subtracted Gaussian field and study how similar this state is to a coherent-state superposition. We assess negativity of the Wigner function in conjunction with the non-existence of the positive $P$ function.
Field generated by subtracting a photon
=======================================
We would like to consider what kind of state one produces by eliminating one photon from a simple Gaussian function. A single-mode Gaussian field of its density operator $\hat{\rho}$ may be represented by the Weyl characteristic function [@note1] defined as $C(\xi)=\mbox{Tr}[\hat{D}(\xi)\hat{\rho}]$: $$C(\xi)=\exp\left(-{A\over 2}\xi_r^2-{B\over 2}\xi_i^2 \right),
\label{1}$$ where $A$ and $B$ are determined by the quadrature variances of the field. The displacement operator has been defined as $\hat{D}(\xi)=\exp(\xi\hat{a}^\dag-\xi^*\hat{a})$, where $\hat{a}$ and $\hat{a}^\dag$ are bosonic annihilation and creation operators respectively. Note also that the density operator can be obtained from the characteristic function as $$\begin{aligned}
\hat{\rho}
={1\over\pi}\int d^2\xi C(\xi)\hat{D}(-\xi),
\label{character-single}\end{aligned}$$ which can be straightforwardly obtained using identities $(1/\pi)\int
d^2\alpha|\alpha\rangle\langle\alpha|=\openone$ and [@Cahill] $$|\alpha\rangle\langle\beta|={1\over\pi}\int d^2\xi \hat{D}(-\xi)
\langle\beta|\hat{D}(\xi)|\alpha\rangle$$ where $|\alpha\rangle$ is a coherent state of amplitude $\alpha$. Even though Eq. (\[1\]) does not represent a very general Gaussian field, rotation and/or displacement operation brings any Gaussian field to this form. It is useful to start with (\[1\]) because it is extremely challenging to produce a pure squeezed state with $AB=1$ and the characteristic function (\[1\]) allows us to treat a single-mode Gaussian state of a mixed state. The uncertainty relation is given by $AB\geq 1$ and the Gaussian state is called squeezed when either $A<1$ or $B<1$.
Let us consider the experiment by Wenger [*et al.*]{} [@Grangier]. First of all, they produce a squeezed Gaussian state then this passes through a beam splitter with its transmittivity $T=t^2$, where the other input port is assumed to be served by a vacuum. At the one output of mode 2, we conditionally measure a one photon state $|1\rangle_2$. The state obtained at the other output port of mode 1 was what Wenger [*et al.*]{} produced as a non-Gaussian field in their experiment. We will evaluate the Wigner function for this field of mode 1.
By beam splitting the squeezed Gaussian field whose characteristic function is written as (\[1\]) and the vacuum of its characteristic function, $C_v(\xi)=\exp(-{1\over 2}
|\xi|^2)$, the characteristic function for the output field of modes 1 and 2 is [@KimLee] $$C_{out}(\eta,\xi)=\exp\left(-{1\over 2}{\bf x V}{\bf x}^T\right)
\label{character-out}$$ where ${\bf x} = (\eta_r, \eta_i, \xi_r, \xi_i)$ and the correlation matrix $${\bf V}=
\begin{pmatrix}
n_1 & 0 & c_1 & 0 \\
0 & n_2 & 0 & c_2 \\
c_1 & 0 & m_1 & 0 \\
0 & c_2 & 0 & m_2
\end{pmatrix}
\label{correlation}$$ with $$\begin{aligned}
n_1=TA+R,~~n_2=TB+R,~~c_1=tr(A-1),
\nonumber \\
c_2=tr(B-1),~~ m_1=RA+T,~~m_2=RB+T
\label{value}\end{aligned}$$ and $T=t^2$ and $R=r^2$.
We then use the two-mode version of (\[character-single\]) for the density operator of the output field: $$\hat{\rho}_{out}=\frac{1}{\pi^2}\int
C_{out}(\eta,\xi)\hat{D}_1(-\eta) \hat{D}_2(-\xi)d^2\eta d^2\xi.
\label{rho-out}$$ The density operator for the field of mode 1 conditioned on one-photon measurement for mode 2 is $$\hat{\rho}_1={\cal N}~_2\langle 1|\hat{\rho}_{out}|1\rangle_2.
\label{rho-1}$$ Throughout the paper, ${\cal N}$ denotes the appropriate normalization factor. For the case of Eq. (\[rho-1\]), $${\cal N}=\frac{1}{_2\langle
1|\mbox{Tr}_1[\hat{\rho}_{out}]|1\rangle_2}=
\frac{[(m_1+1)(m_2+1)]^{3/2}}{2(m_1m_2-1)}.
\label{normal}$$ With the knowledge of the one-photon Fock state expectation value of the displacement operator [@Cahill; @KimKnight] $$\langle 1|\hat{D}(-\xi)| 1\rangle=\mbox{e}^{-{|\xi|^2\over 2}}
(1-|\xi|^2),$$ the density operator is found to be $$\hat{\rho}_1={{\cal N}\over\pi^2}\int C(\eta,\xi)\hat{D}_1(-\eta)
\mbox{e}^{-{|\xi|^2\over 2}}(1-|\xi|^2)d^2\eta d^2\xi.$$ The characteristic function is then easily obtained using the identity $\mbox{Tr}[\hat{D}(\zeta)\hat{D}(-\eta)]=\pi\delta^{(2)}(\zeta-\eta)$:
$$C_1(\zeta) = \Big[1- \frac{c_1^2(m_2+1)\zeta_r^2}{(m_1+1)(m_1m_2-1)}
- \frac{c_2^2(m_1+1)\zeta_i^2}{(m_2+1)(m_1m_2-1)} \Big]
\exp\Big[-{1\over 2}\Big(n_1-{c_1^2\over m_1+1}
\Big)\zeta_r^2 - {1\over 2}
\Big(n_2-{c_2^2 \over m_2+1} \Big)\zeta_i^2\Big].%\nonumber \\
\label{character-1}$$
By Fourier transformation of the Weyl characteristic function [@Cahill2], we obtain the Wigner function. Now, the first point we are interested in is the negativity of the Wigner function. It is clear that the Fourier transform of (\[character-1\]) has the largest negativity (if any exists) at the origin of phase space and the value of the Wigner function at the point is $$W_1(0)\propto \frac{B-1}{(T+1)B+R}+\frac{A-1}{(T+1)A+R}
\label{W-0}$$ which has been obtained by substituting the parameters (\[value\]). It is obvious that if $A>1$ or $B>1$, [*i.e.*]{}, the incoming Gaussian field is not squeezed, $W(0)$ is positive everywhere. In order to find the exact condition for negativity in the Wigner function, we assume that $A<1, ~B>1$ and introduce positive parameters $x=(A+1)/(1-A)$ and $y=(B+1)/(B-1)$. Then the right-hand-side (RHS) of (\[W-0\]) becomes $$\frac{2T-x+y}{(T-x)(T+y)}$$ whose denominator is always negative. The numerator becomes positive when the transmittivity satisfies $$T>\frac{AB-1}{(1-A)(B-1)}
\label{condition-W1}$$ which always holds when the incoming Gaussian field is pure $AB=1$ (in other words, if the incoming Gaussian field is a pure squeezed state, the Wigner function always shows negativity by subtracting a photon from it).
The $P$ function of the field may be obtained using the relation between its characteristic function $C^{(p)}_1$ and the Weyl characteristic function [@Cahill2]: $$C^{(p)}(\zeta)=C(\zeta)\mbox{e}^{\frac{1}{2} |\zeta|^2}.
\label{relations}$$ With use of the characteristic function (\[character-1\]) and general Gaussian integration, we find that the $P$ characteristic function is integrable when $(n_i-1)(m_i+1)-c_i^2>0$ for $i=1,2$. By substituting the parameters (\[value\]), we find the condition equivalent to $2T(A-1)>0$ and $2T(B-1)>0$. So if the incoming field is squeezed, it is not possible to integrate $C^{(p)}$ and no $P$ function exists. Considering the positivity of the $P$ function, after a little algebra with Fourier transformation of the $P$ characteristic function, we find that the $P$ function is positive as far as it exists in this case. We conclude that the single-photon subtracted field is nonclassical (in the sense of a lack of an acceptable $P$ function) provided the original incoming field is squeezed. (However, the Wigner function does not necessarily show negativity for all those nonclassical states unless the incoming Gaussian field was pure.) Unless the incoming Gaussian field is nonclassical we cannot generate a nonclassical state by subtracting a photon from it.
This seemingly trivial result is not obvious at all as contrasted by the nonclassicality of a field by adding a photon into a Gaussian field [@Agarwal; @Lee0]. In distinction to the case of subtracting a photon, the photon-added Gaussian state always shows negativity at the origin of the phase space [@Lee0; @Mandel2; @Lee1]. By adding a photon, a highly classical state such as a high-temperature thermal state becomes non-classical, showing negativity in its Wigner function. The realization of such a photon added state is beyond the scope of the current work but we may think of a possibility within cavity quantum electrodynamics or the phonon state of a driven ion in a cavity [@Lee1].
We now introduce the coherent-superposition state [@Knight] $$\label{cat-state}
|\psi\rangle = {\cal N}(|\alpha\rangle-|
-\alpha\rangle),$$ where ${\cal N}=1/\sqrt{1-\exp[-2\alpha^2]}$, to assess its fidelity to the photon-subtracted Gaussian state. It is straightforward to calculate the characteristic function of the coherent-state superposition from Eq. (\[cat-state\]) [@Barnett]. The closeness of two states, one of which is a pure state $|\phi\rangle$ and the other (pure or mixed) is represented by its density operator $\hat{\rho}$, is measured by fidelity $\cal F$: $${\cal F} = \langle\phi|\hat{\rho}|\phi\rangle %\nonumber\\
={1\over\pi}\int d^2\zeta C_{\phi}(\zeta)C_\rho(\zeta)
\label{fidelity}$$ where the subscripts refer to the respective states.
The fidelity between $\hat{\rho}_1$ and the coherent-state superposition (\[cat-state\]) has been calculated from Eqs. (\[1\]), (\[cat-state\]), and (\[fidelity\]) and plotted in Fig. \[fig:entangle2\]. The incoming Gaussian field has been assumed a pure squeezed field. In Fig. \[fig:entangle2\], the solid line is the optimized fidelity between the photon-subtracted state and the ideal coherent-state superposition by an ideal single-photon detector. The fidelity is very high as ${\cal F}>0.99$ regardless of the transmittivity of the beam splitter when an ideal single-photon detector is used. The optimized amplitude of the ideal coherent-state superposition is $\alpha=1.16$ for the transmittivity close to unity. If the transmittivity $T$ gets smaller, the amplitude of the ideal coherent-state superposition, which maximizes the fidelity, also becomes smaller. For example, the amplitude will be $\alpha=1.02 (1.09)$ for $T=0.8$ (0.9). However, the fidelity is not sensitive to the transmittivity of the beam splitter as shown in Fig. \[fig:entangle2\] because the single photon detector successfully subtracts only one photon from the Gaussian state regardless of the transmittivity of the beam splitter. In fact, the fidelity gets slightly better as the transmittivity becomes smaller, due to the fact that both of the states are reduced to the exact single photon state as $T\rightarrow0$.
It is interesting that the fidelity between the photon subtracted field $\hat{\rho}_1$ and the coherent-state superposition is very high. This could have been guessed from their photon number distributions. The squeezed vacuum is a state with only an even number of photons [@Barnett] while the coherent-state superposition (\[cat-state\]) is a state with only an odd number of photons [@Lund]. By subtracting one photon from the squeezed state, the two states may become closer to each other. We see that the photon-subtracted squeezed field is close to the coherent-state superposition of small amplitudes. One reason can be found again in their photon number distributions. The photon number distribution of $|\psi\rangle$ peaks around $|\alpha|^2$ while that of $\hat{\rho}_1$ is a monotonous decreasing function with regard to the photon number. Thus, when $\alpha$ is small, the distributions become similar to each other. Of course, this check of the photon number distributions gives only a hint as the photon number distribution does not necessarily convey all the coherence properties of a quantum field.
![The fidelity between the photon-subtracted state and the ideal coherent-state superposition with an ideal single-photon detector (solid line) and a threshold detector (dotted line). The initial squeezing parameter is $\exp(2s)=2.36$ and the $x$ axis is the transmittivity of the beam splitter $T=t^2$. The amplitude $\alpha$ of the ideal coherent-state superposition is optimized for the maximum fidelity. The optimized amplitude $\alpha$ ranges between $1.02~({\rm when~} T=0.8)$ and $1.16~({\rm when~} T\rightarrow1)$.[]{data-label="fig:entangle2"}](fig1.eps){width="40.00000%"}
Experimental Reality
====================
As can be seen in Fig. 1, the fidelity between the ideal coherent-state superposition and the photon-subtracted state is not so sensitive to reflectivity of the beam splitter. This seemingly good result is due to an ideal single photon detector assumed for the photon-subtracted state $\hat{\rho}_1$. As mentioned, the state (\[rho-1\]) is what is wanted to achieve but the available high-efficiency photodetector is not able to discern 1 and any number of photons. Thus the state experimentally generated using such the threshold photodetector is $$\hat{\rho}_a={\cal N}\sum_{n=1}^{\infty}~_2\langle
n|\hat{\rho}_{out} |n\rangle_2 \label{any-rho}.$$ Consider the density operator for mode 1 of the output field $$\hat{\rho}_t=\mbox{Tr}_2[\hat{\rho}_{out}]=\sum_{n=0}^{\infty}
~_2\langle n|\hat{\rho}_{out}|n\rangle_2. \label{rho-t}$$ It is then clear from Eqs.(\[any-rho\]) and (\[rho-t\]) that $$\hat{\rho}_a={\cal N}(\hat{\rho}_t-~_2\langle 0|\hat{\rho}_{out}
|0\rangle_2) \label{new-rho}$$ where $$\hat{\rho}_t={1 \over \pi}\int C_{out}(\eta,0)
\hat{D}_1(-\eta)d^2\eta \label{101}$$ and $$_2\langle 0|\hat{\rho}_{out}|0\rangle_2= {1 \over \pi^2}\int
C_{out}(\eta, \xi)\mbox{e}^{-|\xi|^2/2}\hat{D}_1(-\eta)d^2\eta
d^2\xi. \label{102}$$ Using $C(\eta,\xi)$ we have already discussed, we find the characteristic function $C_a(\zeta)$ for $\hat{\rho}_a$: $$\begin{aligned}
C_a(\zeta)&=&{\cal N}\mbox{e}^{-{1\over
2}(n_1\zeta_r^2+n_2\zeta_i^2)} \Big[1 -
\frac{2}{\sqrt{(m_1+1)(m_2+1)}}\nonumber \\
&\times&\exp\Big(\frac{c_1^2}{2(m_1+1)}\zeta_r^2
+ \frac{c_2^2}{2(m_2+1)}\zeta_i^2\Big)\Big]. \label{103}\end{aligned}$$ The normalization factor is calculated as $${\cal N}=\frac{\sqrt{(m_1+1)(m_2+1)}}{\sqrt{(m_1+1)(m_2+1)}-2}.$$ The Wigner function obtained by Fourier transformation of the characteristic function (\[103\]) is what Wenger [*et al.*]{} would have reconstructed [@Grangier] if the detection efficiency of their experiment had been perfect and the modal purity unity.
Let us next consider the negativity of the Wigner function. By inspection of the characteristic function, we realize that the Wigner function has the largest negativity (if any) at the origin of the phase space and the value of the Wigner function at this point is $$\begin{aligned}
&&
W_a(0)=\frac{2{\cal N}}{\pi}\Big\{\frac{1}{\sqrt{n_1n_2}}
\nonumber \\
&&~~~
- \frac{2}{\sqrt{[n_1(m_1+1)-c_1^2][n_2(m_2+1)-c_2^2]}} \Big\}
\label{Wigner-a-origin}.\end{aligned}$$ By partly substituting the parameters (\[value\]), we find that the Wigner function becomes negative when $$\frac{2}{\sqrt{(n_1+A)(n_2+B)}}>\frac{1}{\sqrt{n_1n_2}}
\label{Wigner-a-condition}$$ which becomes a criterion for the transmittivity $$T>\frac{4-(A+1)(B+1)}{3(A-1)(B-1)}.
\label{Wigner-a-condition2}$$ For a pure squeezed Gaussian incoming field, the condition becomes $T>1/3$. It is interesting to note that regardless of the degree of squeezing (provided it is not zero), we can see the negativity in the Wigner function provided the transmittivity is larger than 1/3.
Let us assess the degree of nonclassicality by the $P$ function criterion. With use of the relation (\[relations\]) between the characteristic functions, we note that the $P$ characteristic function for $\hat{\rho}_a$ is integrable when $T(A-1)>0$ and $T(B-1)>0$. We have checked that the $P$ function is semi-positive when it exists and conclude that, for nonzero transmittivity of the beam splitter, iff the incoming Gaussian field is squeezed, the any-number photon subtracted state $\hat{\rho}_a$ is nonclassical. Again, the $P$ function criterion is weaker than the negativity criterion for the Wigner function.
We now consider how close the field obtained using the threshold detector to the coherent-state superposition (\[cat-state\]). The optimized fidelity has been calculated using Eqs. (\[cat-state\], \[fidelity\]) and (\[103\]), and plotted in Fig.\[fig:entangle2\]. It tells us that the state which is obtained by subtracting any number of photons is similar to the coherent-state superposition only when the transmittivity of the beam splitter is very high. For example, the fidelity is higher than 90% when $T>0.87$. In this case, the chance of one photon subtraction is more likely. Note that the optimized amplitude $\alpha$ ranges between $1.02~({\rm when~} T=0.8)$ and $1.16~({\rm when~} T\rightarrow1)$ in Fig. \[fig:entangle2\].
(a)
(b)
(c)
Inefficient detection and modal purity
--------------------------------------
Homodyne detection may be used to reconstruct the Wigner function for the field $\hat{\rho}_a$. Even though homodyne detectors are known for their high efficiency, the overall detection efficiency was about 75% in Wenger [*et al*]{}.’s experiment [@Grangier]. An imperfect detector is equivalent to a perfect detector with a beam splitter in front [@Yuen], where the transmittivity of the beam splitter is determined by the detection efficiency $\eta$. From ref. [@KimImoto], we note that the characteristic function for the signal field passing through a beam splitter where the other input port is served by the vacuum is $$C_{im}(\zeta)=C_a(\sqrt{\eta}\zeta)C_v(\sqrt{1-\eta}\zeta).
\label{imperfect-char}$$ Substituting Eq. (\[103\]) into Eq. (\[imperfect-char\]), we find the characteristic function for the detected field. The Fourier transform of the characteristic function shows its largest negativity at the origin of the phase space and the value there is $$W_{im}(0)\propto \frac{1}{\sqrt{vw}}-\frac{1}{\sqrt{
\Big(v-\frac{R(A-1)}{2}\Big)\Big(w-\frac{R(B-1)}{2}\Big)}}
\label{imperfect-Wigner-0}$$ where $v=T(A-1)\eta+1$ and $w=T(B-1)\eta+1$. Under the assumption $(A-1)(B-1)<0$, this becomes negative when the detection efficiency satisfies $$\eta>-\frac{1}{2T(A-1)}-\frac{1}{2T(B-1)}-\frac{R}{4T}.$$ In particular, for a pure Gaussian incoming field, the condition becomes $$\eta>\frac{1+T}{4T}.
\label{condition-imperfect}$$ The RHS is smaller than unity (the detection efficiency $\eta\leq 1$) only when $T\geq 1/3$. This is in good agreement with the perfect detection case. So, in order to see negativity in the Wigner function, the beam splitter has to have a transmittivity larger than 1/3 first and then the detection efficiency has to satisfy the condition (\[condition-imperfect\]). Wenger [*et al.*]{} employed a beam splitter with $T\approx 0.88$ in which case the detection has to be larger than a mere 53.4% to see negativity in the Wigner function.
Another important factor which degrades the quantum effect of the photon subtracted Gaussian state in a real experiment is the modal purity factor [@Grangier]. If the dark count rate of the photodetector employed to subtract a photon is non-negligible, the resulting state can be estimated in a mixture of photon subtracted squeezed state and squeezed state as $$\xi W(\alpha)+(1-\xi)W_{sq}(\alpha)$$ where $W(\alpha)$ is the Wigner function of the photon subtracted squeezed state, $W_{sq}(\alpha)$ is the Wigner function of the squeezed state, and $\xi$ corresponds to the modal purity factor, which was 0.7, in Wenger [*et al.*]{}’s experiment [@Grangier]. The Wigner functions of the photon-subtracted Gaussian state have been plotted for a number of different cases in Fig. \[fig:entangle3\]. It shows that the negativity of the Wigner function disappears when both of the homodyne efficiency $\eta$ and the modal purity $\xi$ are considered taking relevant experimental values. We suggest that either the homodyne efficiency should be improved from 0.75 to 0.9 or the modal purity factor should be improved from 0.7 to 0.9 to clearly observe the negativity of the Wigner function. In these cases, the minimum negativity will be $-0.044$ and $-0.073$, respectively.
Remarks
=======
In this paper, we are interested in the nonclassicality of a state produced by subtracting photons from a Gaussian field. Subtracting a photon does not transform a classical state into a nonclassical state whereas a nonclassical input remains nonclassical. This is in contrast to the case of adding a photon to a Gaussian field, in which case even a very chaotic field transforms into a nonclassical state [@Lee0; @Mandel2; @Lee1]. The non-Gaussian state obtained by subtracting a photon from a Gaussian field may show large negativity in its Wigner function. The condition to obtain the negativity is analyzed for a realistic case including the mixed-state input, threshold detection, inefficient homodyne detection, and modal purity. The non-Gaussian state analyzed in this paper is compared with a coherent-state superposition which may be extremely useful for fundamental and application reasons. The comparison shows the fidelity higher than 90% for the experimentally relevant situation. We compare our analysis with a recent experimental demonstration [@Grangier] of the photon-subtracted Gaussian field and suggest that either the homodyne efficiency or the modal purity factor should be improved to around $\sim 0.9$ to clearly observe the negativity of the Wigner function.
[*Note added*]{} After completion of the work, we were made aware of ref. [@nonlocality] which considers nonlocality of a photon subtracted squeezed state.
This work was supported by the UK Engineering and Physical Science Research Council, the European Union and the Korea Research Foundation (2003-070-C00024). H.J. acknowledges the Australian Research Council for financial support.
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| {
"pile_set_name": "ArXiv"
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---
abstract: |
We study the behaviour of a Hilbert geometry when going to infinity along a geodesic line. We prove that all the information is contained in the shape of the boundary at the endpoint of this geodesic line and have to introduce a regularity property of convex functions to make this link precise.\
The point of view is a dynamical one and the main interest of this article is in Lyapunov exponents of the geodesic flow.
author:
- Mickaël Crampon
bibliography:
- 'biblio.bib'
title: 'Lyapunov exponents in Hilbert geometry\'
---
Introduction
============
This article is meant to be a contribution to the understanding of Hilbert geometries, by a study of their behaviour when approaching infinity. Most of this work is part of my Ph.D. thesis, which can be found in various places on the Internet.\
Context
-------
A Hilbert geometry is a metric space $(\o,\d)$ where
- $\o$ is a *proper open convex set* of the real projective space $\R\P^n$, $n\geqslant2$; *proper* means there exists a projective hyperplane which does not intersect the closure of $\o$, or, equivalently, there is an affine chart in which $\o$ appears as a relatively compact set;
- $\d$ is the distance on $\o$ defined, for two distinct points $x,y$, by $$\d(x,y) =\frac{1}{2}|\log[a,b,x,y]|,$$ where $a$ and $b$ are the intersection points of the line $(xy)$ with the boundary $\partial \Omega$ and $[a,b,x,y]$ denotes the cross ratio of the four points : if we identify the line $(xy)$ with $\R\cup\{\infty\}$, it is defined by $[a,b,x,y]=\frac{|ax|/|bx|}{|ay|/|by|}$ .
![The Hilbert distance[]{data-label="fighilbert"}](dessins/hilbertdistance.ps)
These geometries had been introduced by Hilbert at the end of the nineteenth century as examples of spaces where lines would be geodesics, which one can see as a motivation for the fourth of his famous problems, which roughly consisted in finding all geometries satisfying this property.\
Different Hilbert geometries can have very different geometric behaviours. For example, the geometry defined by a triangle in $\R\P^2$ is isometric to the $2$-dimensional real space equipped with a norm whose ball is a regular hexagon [@delaharpe]; on the other side, the geometry defined by an ellipsoid is precisely the model that Beltrami proposed for hyperbolic geometry.\
Classifying Hilbert geometries happens to be a quite difficult task, but the global feeling is that any Hilbert geometry has an intermediate behaviour in between Euclidean and hyperbolic geometry. Most of the previous works attempted to determinate those Hilbert geometries which resembles more Euclidean or hyperbolic space:
- [@foertschkarlsson] $(\o,\d)$ is isometric to a normed vector space if and only if $\o$ is a simplex.
- [@cvv] [@vernicos] [@bernig] The following statements are equivalent:
- $\o$ is a polytope;
- $(\o,\d)$ is bi-Lipschitz equivalent to the Euclidean space;
- $(\o,\d)$ is quasi-isometric to the Euclidean space.
- [@cv] If $\o$ is strongly convex, that is, $\doo$ is of class $\C^2$ with positive definite Hessian, then $(\o,\d)$ is bi-Lipschitz equivalent to the hyperbolic space.
These results only consider polytopes or strongly convex sets and, as soon as we permit more irregularity or less symmetry, no global behaviour can be expected. Here we should recall the works of Yves Benoist who studied less regular Hilbert geometries, in particular those which admit compact quotients, called *divisible convex sets*. For the problem of classification we are concerned with here, the major achievement of Benoist is probably the characterization of Gromov-hyperbolic Hilbert geometries: they are those defined by quasi-symmetrically convex sets [@benoistqs]. About divisible convex sets, Benoist proved an hyperbolic/non hyperbolic alternative in [@benoistcv1]: if $\o$ is a divisible convex set, then the following are equivalent:
- $\o$ is strictly convex;
- $\doo$ is of class $\C^1$;
- $(\o,\d)$ is Gromov-hyperbolic.\
The goal of the present work is to get interested in all those forgotten Hilbert geometries which enjoy neither high regularity nor numerous symmetries, the strategy being the following: pick a geodesic ray (a line), follow this line to infinity and look at the geometry around it.\
Consider as an easy example the Hilbert geometry defined by a half disc, and call $a$ and $b$ the extremities of the diameter. Pick two distinct geodesic rays $c_1, c_2 : [0,+\infty) \longrightarrow \o $, ending at points $x_1$ and $x_2$ in $\doo$.
- Assume $x_1\not = x_2$. The distance between the two geodesic rays goes to infinity, except when both points $x_1$ and $x_2$ are inside the open segment $]ab[$, in which case one can parametrize the rays such that $$\lim_{t\to +\infty} \d(c_1(t),c_2(t)) = \frac{1}{2} |\log [a b x_1 x_2]|.$$
- Assume $x_1=x_2=:x$. If $x=a$ (or $x=b$), the distance between the two geodesic rays tends to some positive constant $d>0$, whose value depends on the parametrization; the smallest of which being $d = \frac{1}{2} |\log [(ab) D c_1 c_2]|$, where $D$ is the line tangent to the half-circle at $a$ (or $b$), and $[(ab) D c_1 c_2]$ denotes the cross-ratio of the four lines.\
In the other cases, one can parametrize the rays such that the distance $\d(c_1(t),c_2(t))$ decreases to $0$. Nevertheless, it does not go to $0$ at the same rate: if $x$ is in the (open) circular part, then $\d(c_1(t),c_2(t)) \sim e^{-t}$; if $x$ is in the flat part $]ab[$ then $\d(c_1(t),c_2(t)) \sim e^{-2t}$.
These simple remarks show, first, that the boundary at infinity given by asymptotic geodesic rays does not correspond to the geometric boundary $\doo$ and, second, that the geometry when going to infinity heavily depends on the point we are aiming at. This work studies this second point in details.\
For what concerns the first one, notice that geodesic and geometric boundaries will correspond if and only if the convex set is strictly convex and has $\C^1$ boundary. For polytopes or even more general non-strictly convex sets, another problem arises: there can be geodesics which are not lines. In these cases, the best thing is probably to look at the Busemann boundary, as made in [@walsh], which contains the geometric and geodesic boundaries.\
\
What we study here
------------------
In this article, we focus on those Hilbert geometries defined by a strictly convex set with $\C^1$ boundary. Since our aim is to look at the geometry around a specific geodesic line going to a point $x\in\doo$, we could equivalently assume that $x$ is an extremal point of $\o$ and that $\doo$ is $\C^1$ at $x$. This assumption is then not a very restrictive one, and we can illustrate most of the interesting behaviours; furthermore, it allows us to use and make connections with some differential and dynamical objects that I already used in a previous work [@crampon]. In section \[extension\], we explain how to get rid of this restriction and extend the main achievements.\
So we want to understand how the distance $\d(c_1(t),c_2(t))$ between two well parametrized asymptotic geodesic rays decreases to $0$ when $t$ goes to infinity. In particular, as suggested by the example of the half-disc, we would like to see when the decreasing is exponential, and in this case, to determinate the exponential rate.\
In the case of a strongly convex set, it is easy to see, as we already saw in the case of the half-disc, that $\d(c_1(t),c_2(t))\sim e^{-t}$, as in the hyperbolic space. The main result of this article about this is probably corollary \[maincorollary2\], that says that all these informations are enclosed in the shape of the boundary at the endpoint.\
I should confess that the original motivation of this work is not of a geometric nature but of a dynamical one. It is inspired by proposition 5.4 of [@crampon], which I wanted to generalize in order to understand Lyapunov exponents, decomposition and manifolds, associated to the geodesic flow of the Hilbert metric. The text is then written in this spirit, and the geodesic flow is the main object that is studied.\
The geodesic flow is the flow $\ph^t$ defined on the homogeneous tangent bundle $H\o = T\o \smallsetminus \{0\} / \R_+^*$, which consists of pairs $(x,[\xi])$, where $x$ is a point of $\o$ and $[\xi]$ a direction tangent to $\o$ at $x$. To find the image of a point $w=(x,[\xi])\in H\o$ by $\ph^t$, one follows the geodesic line $c_{w}$ leaving $x$ in the direction $[\xi]$, and one has $\ph^t(w) = (c_{w}(t), [c_w'(t)])$.\
The geodesic flow is generated by the vector field $X: H\o \longrightarrow TH\o$. If we choose an affine chart and a Euclidean norm $|\ .\ |$ on it in which $\o$ appears as a bounded convex set, then $X$ is related to the generator $X^e$ of the Euclidean geodesic flow by $X=m X^e$, where $m:H\o \longrightarrow \R$ is defined by $$m(x,[\xi]) = \frac{2}{\displaystyle\frac{1}{|xx^+|} + \frac{1}{|xx^-|}},$$ where $x^+$ and $x^-$ are the intersection points of the line $x+\R.\xi$ with the boundary $\doo$. In particular, we see that, under our hypothesis of $\C^1$ regularity of $\doo$, the function $m$ and the geodesic flow itself are of class $\C^1$.\
![The Finsler metric[]{data-label="figfinsler"}](dessins/finslermetric.ps)
This fact has to be related with the Finsler nature of the Hilbert metric. Indeed, the Hilbert metric is generated by a field of norms $F: T\o \longrightarrow \R$, with $$F(x,\xi) = \frac{|\xi|}{2} \left(\frac{1}{|xx^+|} + \frac{1}{|xx^-|}\right) = \frac{|\xi|}{m(x,[\xi])}.$$ By *generated*, we mean that the Hilbert distance between two points $x,y\in\o$ is given by $$\d(x,y) = \inf_{c : x\to y} \int_{0}^{1} F(\dot c(t))\ dt,$$ where the infimum is taken with respect to all $\C^1$ curves $c: [0,1] \longrightarrow \o$ such that $c(0)=x$, $c(1)=y$.\
\
Contents
--------
The geodesic flow of Hilbert metrics has been studied by Yves Benoist in [@benoistcv1] and by myself in [@crampon]. In the second section of this article, I recall the dynamical objects I had used in [@crampon] and the fundamental results about them; in particular, the existence of stable and unstable distributions, so that $TH\o$ admits a $\ph^t$-invariant decomposition $$TH\o = \R.X \oplus E^s \oplus E^u.$$ Stable and unstable distributions are characterized by the fact that, for a stable (resp. unstable) vector $Z\in TH\o$, the norm $\|d\ph^t(Z)\|$ decreases to $0$ when $t$ goes to $+\infty$ (resp. $-\infty$); the Finsler norm $\|\ .\ \|$ on $H\o$ that we consider here is naturally related to the Finsler metric on $\o$ (see section \[metriconHo\]).\
These two distributions are tangent to the stable and unstable foliations $W^s$ and $W^u$ of $H\o$. If one takes a point $w_0 = (x_0,[\xi_0])\in H\o$, its orbit in the future $\{\ph^t(w_0)\}_{t\geqslant 0}$ projects on the geodesic ray $c=\{x_0+\l \xi_0\}_{\l\in\R}$; the orbits, in the future, of the points $w$ in the leaf $W^s(w_0)$ passing through $w_0$ of the stable foliation, project to those geodesic rays $c$ such that $\d(c(t),c_0(t))$ tends to $0$ when $t$ goes to $+\infty$.\
The goal is then to understand how the norm $\|d\ph^t(Z)\|$ of a stable vector $Z$ goes to $0$ when $t$ goes to $+\infty$; results about distances between geodesic rays will follow by integration.\
The third and fourth parts look at the exponential growth rate of these norms $\|d\ph^t(Z)\|$, for a stable vector $Z$. This is captured by the following limit, when it exists: $$\chi(Z) = \lim_{t\to+\infty} \frac{1}{t} \log \|d\ph^t(Z)\|;$$ the quantity $\chi(Z)$ is called the Lyapunov exponent of the vector $Z$. These numbers are investigated in section 3, and section 4 shows that all the information about them is contained in the shape of the boundary at the endpoint of the geodesic ray that had been chosen. This needs the introduction of a new regularity property that we call *approximate regularity*, whose study requires some time in section 4.\
In section 5, we state the main consequences about the asymptotic behaviour of distances when following a geodesic line and explain how to extend it to the nonregular cases. We also show how Lyapunov submanifolds of the geodesic flow appear very naturally in our context.\
The sixth part is dedicated to examples, with a focus on divisible convex sets, while the last one gives connections with volume entropy, whose study might benefit from the present work.
*Unless it is explicitly stated, in particular in sections \[extension\] and \[sectionvolentropy\], the convex set $\o$ is always assumed to be strictly convex with $\C^1$ boundary.*
Foulon’s dynamical formalism and consequences
=============================================
Dynamical decomposition
-----------------------
In [@crampon], I explained why the dynamical objects introduced by Patrick Foulon in [@foulon86] to study smooth second order differential equations were still relevant and useful in the case of a Hilbert geometry defined by a strictly convex set with $\C^1$ boundary. I briefly recall them here, and refer the reader to [@crampon], [@foulon86] or the appendix of [@fouloneng].\
All the operators, functions or vector fields that we will consider are $C_X$-regular, or equivalently $C_{X^e}$-regular. That means that they are smooth in the direction $X$ of the flow. This is the essential regularity that we need because Hilbert geometries are flat geometries. Remark that this notion makes sense for those objects which are only defined along one specific orbit of the flow.\
The *vertical bundle* $VH\o$ is the smooth subbundle of vertical vectors, which are tangent to the fibers; it is defined as $VH\o=\ker d\pi$, and has dimension $n-1$. By the letter $Y$, we will always denote a $C_X$-vertical vector field. The *vertical operator* $v_X$ is well defined (this has to be checked) on $TH\o$ by $$v_X(X)=v_X(Y)=0,\ v_X([X,Y])=-Y,\ Y\in VH\o.$$ The operators $v_X$ and $v_{X^e}$ are related by $$v_X=mv_X^e.$$
The *horizontal operator* $H_{X}: VH\o \longrightarrow TH\o$ is the $C_X$-linear operator defined by $$H_{X}(Y) = -[X,Y] - \frac{1}{2} v_{X} ([X,[X,Y]]),\ Y\in VHM.$$ The *horizontal bundle* $h^XH\o$ is the $C_X$-regular subbundle defined as the image of $VHM$ by $H_X$. An important property is the one which relates the operators $H_X$ and $H_{X^e}$: $$\label{horizontal} H_{X}(Y) = mH_{X^e}(Y) + L_Y m X^e + \frac{1}{2}L_{X^e} m Y.$$
The tangent bundle of $H\o$ admits then a $C_X$-regular decomposition into $$TH\o = VH\o \oplus h^XH\o \oplus \R.X,$$ which is the counterpart of the Levi-Civita connection for Riemannian metrics.\
The $C_X$-linear operator $J^X: VH\o \oplus h^XH\o \longrightarrow VH\o \oplus h^XH\o$ is defined as $J_X=v_X$ on $h^XH\o$ and $J^X=-H_X$ on $h^XH\o$. It provides a pseudo-complex structure on $VH\o \oplus h^XH\o$: $J^X$ satisfies $J^X \circ J^X=-Id$ and exchanges $VH\o$ and $h^XH\o$.
Dynamical derivation and parallel transport {#sectionparom}
-------------------------------------------
As an analog of the covariant derivation along $X$, the *dynamical derivation* $D^{X}$ is the $C_X$-differential operator of order 1 defined by $$D^{X}(X)=0, \ D^{X}(Y)=\displaystyle -\frac{1}{2} v_{X}([X,[ X,Y]]), \ [D^{X}, H_{X}]= 0,\ Y\in VH\o.$$
Being a $C_X$-differential operator of order 1 means that for any function $f\in C_X$, $$D^{X}(fZ)=fD^{X}(Z) + (L_{X}f) Z.$$ On $VH\o$, we can write $$\label{dxhori} D^{X}(Y) = H_{X}(Y) +[X,Y].$$ The operators $D^X$ and $D^{X^e}$ are related by $$\label{formuladerivation} D^{X}=mD^{X^e} + \frac{1}{2} (L_{X^e} m)Id.$$
A vector field $Z$ is said to be *parallel* along $X$, or along any orbit of the flow if $D^{X}(Z)=0$. This allows us to consider the *parallel transport* of a $C_X$-vector field along an orbit: given $Z(w)\in T_wH\o$, the parallel transport of $Z(w)$ along $\ph.w$ is the parallel vector field $Z$ along the orbit $\ph.w$ of $w$ whose value at $w$ is $Z(w)$; the parallel transport of $Z(w)$ at $\ph^t(w)$ is the vector $Z(\ph^t(w))=T^t(Z(w)) \in T_{\ph^t(w)}H\o$. Since $D^X$ commutes with $J^X$, the parallel transport also commutes with $J^X$. If $X$ is the generator of a Riemannian geodesic flow, the projection on the base of this transport coincides with the usual parallel transport along geodesics.\
We can relate the parallel transports with respect to $X^e$ and $X$, as stated in the next lemma. This lemma is essential in this work and will be used in many different parts.
\[horver\] Let $w\in HM$ and pick a vertical vector $Y(w)\in V_wHM$. Denote by $Y$ and $Y^e$ its parallel transports with respect to $X$ and $X^e$ along the orbit $\ph.w$. Let $h=J^X(Y)$ and $h^e=J^{X^e}(Y^e)$ be the corresponding parallel transports of $h(w)=J^X(Y(w))$ and $h^e(w)=J^{X^e}(Y^e(w))$ along $\ph.w$. Then $$Y= \left(\frac{m(w)}{m}\right)^{1/2} Y^e$$ and $$h= -L_Ym\ X^e +(m(w)m)^{1/2}\ h^e - \frac{m(w)}{m}\ L_{X^e}m\ Y^e.$$
We look for the unique vector field $Y$ along $\ph.w$ such that $D^{X}(Y)=0$ and which takes the value $Y(w)$ at the point $w$. Equation (\[formuladerivation\]) gives $$D^{X}(Y)=mD^{X^e}(Y) + \frac{1}{2} L_{X}(\log m) Y.$$ Assume we can write $Y=fY^e$ along $\ph.w$. Then $f$ is the solution of the equation $$L_{X} (\log f) + \frac{1}{2} L_{X}(\log m)=0,$$ which, with $f(w)=1$, gives $$f( \ph^t(w))=\left(\frac{m(w)}{m(\ph^t(w))}\right)^{1/2}.$$ Finally, $$\label{tut}
Y(\ph^t w)= \left(\frac{m(w)}{m(\ph^t(w))}\right)^{1/2} Y^e(\ph^t w).$$
Now, using (\[dxhori\]), we have $$h=H_{X}(Y)=-[X,Y]+D^{X}(Y)=-[X,Y]$$ along $\ph.w$. Hence, from (\[tut\]), we have
$$\begin{array}{rcl}
h=-[ X,Y] & = & -L_Ym\ X^e-m\ [X^e,Y]\\\\
& = & -L_Ym\ X^e-m\ [X^e, \frac{m(w)}{m}Y^e] \\\\
& =& -L_Ym\ X^e -(m(w)m)^{1/2}\ [X^e,Y^e] + m(w) m\ L_{X^e}(m^{-1})\ Y^e\\\\
&= & -L_Ym\ X^e +(m(w)m)^{1/2}\ h^e - \frac{m(w)}{m}\ L_{X^e}m\ Y^e.
\end{array}$$
Metrics on $H\o$ {#metriconHo}
----------------
Dynamical flows are usually studied on Riemannian manifolds, and most of the definitions or theorems are stated in this context. In the case of geodesic flows on complete Riemannian manifolds $M$, $HM$ inherits a natural Riemannian metric from the base metric. In our case, we define a Finsler metric $\|\ .\ \|$ on $H\o$, using the decomposition $TH\o = \R.X \oplus h^{X}H\o \oplus VH\o$: if $Z=aX+h+Y$ is some vector of $TH\o$, we set $$\label{metriconHM} \|Z\| = \left(|a|^2+\frac{1}{2}\left((F(d\pi h))^2+(F(d\pi J^{X} (Y)))^2\right)\right)^{1/2}.$$ Since the last decomposition is only $C_X$-regular in general, $\|\ .\ \|$ is also only $C_X$-regular. It allows us to define the length of a $\C^1$ curve $c: [0,1] \rightarrow H\o$ as $$l(c) = \int_{0}^{1} \|\dot{c}(t)\|\ dt.$$ It induces a continuous metric $d_{H\o}$ on $H\o$: the distance between two points $v,w \in H\o$ is the minimal length for $\|\ .\ \|$ of a $\C^1$ curve joining $v$ and $w$.\
Remark that, if $\o\subset\R\P^2$, then $\|\ .\ \|$ is actually a $C_X$-regular Riemannian metric on $H\o$. When $\o$ is an ellipsoid, we recover the classical Riemannian metric. In any case, $\|\ .\ \|$ is obviously $J^{X}$-invariant on $h^{X}H\o \oplus VH\o$.\
Stable and unstable distributions
---------------------------------
In [@crampon], I showed why the subbundles $E^u$ and $E^s$ given by $$E^u=\{Y+J^{X}(Y),\ Y\in VH\o\},\ E^s=\{Y-J^{X}(Y),\ Y\in VH\o\},$$ naturally appeared in the study of the geodesic flow. Recall the
\[dd\] $E^u$ and $E^s$ are invariant under the flow, and if $Z^s\in E^s,\ Z^u\in E^u$, then $$d\ph^t(Z^u)= e^{t} T^t(Z^u),\ d\ph^t(Z^s)= e^{-t} T^t(Z^s).$$ The operator $J^{X}$ exhanges $E^u$ and $E^s$ and $$d\ph^t J^{X} (Z^s) = e^{2t} J^{X} (d\ph^t Z^s).$$
Remark that the second equality is just a consequence of the fact that $J^{X}$ commutes with the parallel transport: we have $$d\ph^t J^{X} (Z^s) = e^{t} T^t J^{X} (Z^s) = e^{t} J^{X} T^t(Z^s) = e^{2t} J^{X} (d\ph^t Z^s).$$ The tangent space $TH\o$ splits into $$TH\o = \R.X \oplus E^s \oplus E^u;$$ this decomposition will be called the *Anosov decomposition*. The main result about the distributions $E^u$ and $E^s$ is the following
\[stable\] Let $Z^s\in E^s,\ Z^u\in E^u$. Then $t\longmapsto \|d\ph^t Z^s\|$ is a strictly decreasing bijection from $\R$ onto $(0,+\infty)$, and $t\longmapsto \|d\ph^t Z^u\|$ is a strictly increasing bijection from $\R$ onto $(0,+\infty)$.
In what follows, the image of a point $w = (x,[\xi]) \in H\o$ under the flow is denoted by $\ph^t(w)= (x_t,[\xi_t])$, for $t\in\R$. We first need a
\[equivalents\] We have $$\frac{|x_tx^-|}{|x_tx^+|}=e^{2t}\frac{|xx^-|}{|xx^+|}.$$ In particular the following asymptotic expansion holds: $$|x_tx^+| = \frac{|xx^+|^2}{m(w)}e^{-2t} + O(e^{-4t}).$$
We have $d_{\o}(x,x_t) = t$, which implies $$e^{2t} = \frac{|xx^-|}{|xx^+|} \frac{|x_tx^-|}{|x_tx^+|},$$ and yields the result.
In order to make computations easier, we will need the following. [*A chart adapted to the point $w\in HM$ or to its orbit $\ph.w$*]{} is an affine chart where the intersection $T_{x^+}\partial\o\cap T_{x^-}\partial\o$ is contained in the hyperplane at infinity, and a Euclidean structure on it so that the line $(xx^+)$ is orthogonal to $T_{x^+}\partial\o$ and $T_{x^-}\partial\o$.
![A good chart at $w=(x,[\xi])$[]{data-label="figgoodchart"}](dessins/goodchart.ps)
\[transport\] In a good chart at $w=(x,[\xi])$ there exists a constant $C(w)$ such that, for any $Z(w) \in E^s(w)\cup E^u(w)$, $$\|T^t Z(w)\|= C(w) (|x_tx^+||x_tx^-|)^{1/2}\left(\frac{1}{|x_ty_t^+|}+\frac{1}{|x_ty_t^-|}\right),$$ where $y_t^+$ and $y_t^-$ denote the points of intersection of the line $\{x + \l d\pi(Z(w))\}_{\l\in\R}$ with $\doo$ (see figure \[parallelfigure\]).
![Parallel transport on $H\o$[]{data-label="parallelfigure"}](dessins/transport.ps)
Assume for example that $Z(w)\in E^u(w)$. Then $Z(w) = h(w) + J^X(h(w))$, for some horizontal vector $h(w)$. Let $h$ denote the parallel transport of $h(w)$, which is defined on the orbit $\ph.w$. We have $T^t Z = h + J^X(h)$ on $\ph.w$. In a good chart at $w$, lemma \[horver\] gives $$d\pi(h) = (m(w)m)^{1/2}\ d\pi(h^e);$$ in this case, since the chart is adapted, $h^e$ is just the Euclidean parallel transport of $h(w)$ along $\ph.w$. In particular, $|d\pi(h^e)| = |d\pi(h^e(w))|=|d\pi(h(w))|$. Hence $$\|T^t Z(w)\| = F(d\pi(h(\ph^t w)))= \frac{|d\pi(h(w))| m(w)}{2} m(\ph^t(w))^{1/2}\left(\frac{1}{|x_ty_t^+|}+\frac{1}{|x_ty_t^-|}\right).$$
We can now give a
Choose a stable vector $Z^s(w)\in E^s(w)$ and a chart adapted to $w=(x,[\xi])$. In that chart, the vector $d\pi(T^t Z^s(w))$ is orthogonal to $x_tx^+$ with respect to the Euclidean structure on the chart; hence so are $x_ty_t^+$ and $x_ty_t^-$. We have from lemma \[dd\], $$\|d\ph^t Z^s(w)\| = e^{-t} \|T^t Z^s(w)\|.$$ Lemma \[equivalents\] gives $$|x_tx^-| = e^{2t}|x_tx^+|\frac{|xx^-|}{|xx^+|},$$ hence from lemma \[transport\], there is a constant $C'(w)$ such that $$\|d\ph^t Z^s(w)\| = C'(w)\left(\frac{|x_tx^+|}{|x_ty_t^+|}+\frac{|x_tx^+|}{|x_ty_t^-|}\right)$$ The strict convexity of $\o$ implies that the function $h: t \mapsto \frac{|x_tx^+|}{|x_ty_t^+|} + \frac{|x_tx^+|}{|x_ty_t^-|}$ is strictly decreasing on $\R$, the $\C^1$ regularity of $\partial\o$ that $\lim_{t\to +\infty} h(t)=0$ and the strict convexity of $\o$ that $\lim_{t\to +\infty} h(t)=+\infty$.\
The same computation holds for $t \mapsto \|d\ph^{-t}(Z^u)\|$ for $Z^u\in E^u$.
Horopsheres, stable and unstable manifolds
------------------------------------------
*Horospheres* can be defined for *any* Hilbert geometry $(\o,\d)$. Pick a point $x^+\in\doo$. For any point $x\in\o$, call $(xx^+):\R\longrightarrow \o$ the geodesic line such that $(xx^+)(0)=x,\ (xx^+)(+\infty)=x^+$. Given a point $x\in\o$, there is for each point $y\in\o$ a unique time $t_y\in\R$ such that $$\lim_{t\to+\infty} \d((xx^+)(t), (yx^+)(t+t_0)) = \inf_{z\in(yx^+)} \left\{\lim_{t\to+\infty} \d((xx^+)(t), (zx^+)(t))\right\}.$$ The horosphere $\H_{x^+}(x)$ through $x$ about $x^+$ is the set of such “minimal points”: $$\H_{x^+}(x) = \{(yx^+)(t_y),\ y\in\o\}.$$ This is a continuous submanifold of $\o$.\
Come back now to a strictly convex set $\o$ with $\C^1$ boundary. In this case, as in the hyperbolic space, horospheres can also be defined as level sets of the Busemann functions $b_{x^+}(x,.)$ given by $$b_{x^+}(x,y) = \lim_{p\to x^+} \d(x,p) - \d(y,p).$$ For $w=(x,[\xi])\in H\o$, let us denote by $\H_w = \H_{x^+}(x)$ the horosphere based at $x^+=\ph^{+\infty}(w)$ and passing through $x$. The horosphere $\H_{\sigma w}$, where $\sigma: (x,[\xi])\in H\o \longmapsto (x,[-\xi])$, is the horosphere $\H_{x^-}(x)$ the horosphere based at $x^-=\ph^{-\infty}(w)$ and passing through $x$.\
The *stable and unstable manifolds* at $w_0=(x_0,[\xi_0])\in H\o$ are the $\C^1$ submanifolds of $H\o$ defined as $$W^s(w_0)=\{w=(x,[xw_0^+])\in H\o,\ x\in \H_{w}\},$$ $$W^u(w_0)=\{w=(x,[w_0^-x])\in H\o,\ x\in H_{\sigma w}\}.$$
![Stable and unstable manifolds[]{data-label="stablemanifold"}](dessins/varietestable.ps)
We can check (see [@benoistcv1]) that $$W^s(w_0)=\{w\in H\o,\ \lim_{t\to +\infty} \d(\pi\ph^t(w),\pi\ph^t(w_0))=0\} = \{w\in H\o,\ \lim_{t\to +\infty} d_{H\o}(\ph^t(w),\ph^t(w_0))=0\} ,$$ $$W^u(w_0)=\{w\in H\o,\ \lim_{t\to -\infty} \d(\pi\ph^t(w),\pi\ph^t(w_0))=0\} =\{w\in H\o,\ \lim_{t\to -\infty} d_{H\o}(\ph^t(w),\ph^t(w_0))=0\} .$$ (Recall that $\pi: H\o \longrightarrow \o$ denotes the bundle projection.) As a corollary of proposition \[stable\], we have:
The distributions $E^s$ and $E^u$ are the tangent spaces to $W^s$ and $W^u$.
\[rmkdistance\] To deduce results on $(\o,\d)$ from results on $(H\o,d_{H\o})$, it is useful to remark that the projection $\pi:H\o\longrightarrow \o$ send isometrically stable and unstable manifolds equipped with the metric induced by $\|\ .\ \|$, on horospheres, with the metric induced by $\d$.
Lyapunov exponents
==================
The goal now is to understand for a given tangent vector $Z\in TH\o$ the asymptotic behaviour of the norms $\|d\ph^t Z\|$ when $t$ goes to $\pm\infty$. In particular, we want to catch some exponential behaviour by looking at the limits, when they exist, $$\chi^{\pm}(Z)=\limsup_{t\to \pm\infty} \frac{1}{t} \log \|d\ph^t(Z)\|.$$ When $\chi^{\pm}(Z)\not= 0$, this means that $\|d\ph^t Z\|$ has exponential behaviour when $t\to\pm\infty$: for any $\epsilon>0$, there exists some $C_{\epsilon}>0$ such that, whenever $t>0$, $$C_{\epsilon}^{-1} e^{\pm(\chi^{\pm}(Z)-\epsilon) t} \leqslant \|d\ph^t(Z)\| \leqslant C_{\epsilon} e^{\pm(\chi^{\pm}(Z)+\epsilon)t}.$$ These two numbers ${\chi}^+(Z)$ and $\chi^-(Z)$ are the forward and backward Lyapunov exponents of the vector $Z$ and they are the main characters of the two next sections.
Symmetries
----------
There are lots of symmetries in our geodesic flow that we should exploit to reduce our study.\
First, thanks to the Anosov decomposition $TH\o = \R.X \oplus E^s \oplus E^u$, it is enough to study the asymptotic behaviour of the norms $\|d\ph^t Z\|$ for $Z\in E^s$ or $Z\in E^u$; of course, $d\ph^t X=X$, and we can recover the asymptotic behaviour of any vector $Z$ by decomposing it with respect to the Anosov decomposition.\
Second, thanks to the reversibility of the Hilbert metric, it suffices to study what occurs when $t$ goes to $+\infty$ by using the *flip map*: it is the $\C^{\infty}$ involutive diffeomorphism $\sigma$ defined by $$\begin{array}{lclc}
\sigma:& H\o & \longrightarrow & H\o\\
& w=(x,[\xi]) & \longmapsto & (x,[-\xi]).
\end{array}$$ The reversiblity of the Hilbert metric implies that $\sigma$ conjugates the flows $\ph^t$ and $\ph^{-t}$: $$\ph^{-t} = \sigma \circ \ph^t \circ \sigma.$$
\[flipmap\] The differential $d\sigma$ anticommutes with $J^X$, that is, $J^X\circ d\sigma = -d\sigma \circ J^X$. As a consequence, $\sigma$ preserves the decomposition $TH\o = \R.X \oplus h^XH\o \oplus VH\o$, is a $\|\ .\ \|$-isometry and exchanges stable and unstable distributions and foliations.
Clearly, $d\sigma(X)=-X$ and $d\sigma$ preserves $VH\o$. Now, just recall how $v_X$ is defined: for any $Y\in VH\o$, we have $v_X(X)=v_X(Y)=0$, and $v_X([X,Y])=-Y$, so $$d\sigma v_X(X)=v_X(d\sigma(X))=0=d\sigma v_X(Y)=v_X(d\sigma(Y)),$$ and $$v_X d\sigma([X,Y])=v_X ([d\sigma(X),d\sigma(Y)]) = v_X([-X,d\sigma(Y)] = d\sigma(Y) = -d\sigma v_X([X,Y]).$$ So $d\sigma\circ v_X = - v_X\circ d\sigma$. As for $H_X$: $$\begin{array}{rl}
d\sigma H_X(Y) = d\sigma (-[X,Y] - \displaystyle\frac{1}{2}v_X[X,[X,Y]]) & = -[d\sigma (X),d\sigma (Y)] + \displaystyle\frac{1}{2}v_X[d\sigma(X),[d\sigma(X),d\sigma(Y)]]\\\\
& = [X,d\sigma(Y)] + \displaystyle\frac{1}{2}v_X[X,[X,d\sigma(Y)]]\\\\
& = -H_X (d\sigma(Y)).
\end{array}$$ Finally, we get that $d\sigma$ and $J^X$ anticommute. This implies in particular that $\sigma$ preserves the horizontal bundle $h^XH\o$ and the metric $\|\ .\ \|$. It also gives that, if $Z=Y+J^X(Y)\in E^u$, then $d\sigma(Z) = d\sigma(Y) - J^X d\sigma (Y)\in E^s$, hence $d\sigma(E^u)=E^s$, and conversely; so $\sigma$ exchanges stable and unstable distributions and foliations.\
Now, since $\ph^{-t} = \sigma \circ \ph^t \circ \sigma$, we have $$\limsup_{t\to -\infty} \frac{1}{t} \log \|d \ph^t Z\|= \limsup_{t\to-\infty} \frac{1}{t} \log \|d\sigma d\ph^{-t} d\sigma Z\|=\limsup_{t\to+\infty} \frac{1}{-t} \log \|d\ph^{t} d\sigma Z\|$$ because $\sigma$ preserves $\|\ .\ \|$. Hence $$\label{pastfuture}
\limsup_{t\to -\infty} \frac{1}{t} \log \|d \ph^t Z\|=-\liminf_{t\to+\infty} \frac{1}{t} \log \|d\ph^{t} d\sigma Z\|.$$ This equality allows us to deduce the behaviour in the future from the one in the past: to catch the behaviour of stable vectors in the past, one can study the behaviour of unstable vectors in the future; and conversely.\
Finally the operator $J^X$ provides a symmetry between $E^u$ and $E^s$: it sends the stable vector $Z^s = Y - J^X(Y)$ to the unstable vector $Z^u = Y + J^X(Y)$. Furthermore, since $J^X$ commutes with $T^t$ and $T^t$ preserves horizontal and vertical distributions, we have $$d\ph^t Z^u = e^{t} T^t Z^u = e^t(T^t Y - J^X (T^t Y)) ,\ d\ph^t Z^s = e^{-t} T^t Z^s=e^{-t}(T^t Y - J^X (T^t Y)).$$ But, from the very definition of the metric $\|\ .\ \|$, we have $$\|T^t Z^u\| = \|T^t Z^s\| = F(d\pi J^X(T^t Y)).$$
To understand the asymptotic behaviour of the norms $\|d\ph^t Z^u\|$ and $\|d\ph^t Z^s\|$, it then suffices to understand the behaviour of the quantities $F(d\pi T^t h)$ for $h\in h^XH\o$ (recall that $J^X$ exchanges $VH\o$ and $h^XH\o$, so that $J^X(T^t Y)\in h^XH\o$). This is what we will do in the next part.
Parallel Lyapunov exponents
---------------------------
Remark that, given a point $w = (x,[\xi])\in H\o$, the projection of the horizontal space $h_w^XH\o$ at $w$ on $T\o$ is precisely the tangent space $T_x\H_w$ to the horosphere $\H_w$ at the point $x$. We now define a parallel transport along oriented geodesics on $\o$ that will contain all the information we need and become the main object of our study.\
Let fix a point $x^+\in\doo$. Denote by $W^s(x^+)=\{w\in H\o,\ \ph^{+\infty}(w)=x^+\}$ the weak stable manifold associated to $x^+$, consisting of these points $w$ that end at $x^+$. Obviously, the map $\pi$ identifies $W^s(x^+)$ with $\o$, and we will call $\pi^{-1}_{x^+}$ the inverse of $\pi_{|_{W^s(x^+)}}$; we have $\pi^{-1}_{x^+}(x) = (x,[xx^+])$.\
The radial flow $\ph_{x^+}^t$ is the flow on $\o$ defined via $$\ph_{x^+}^t = \pi\ph^t\pi^{-1}_{x^+}.$$ It is generated by the vector field $X_{x^+}$ such that $[X_{x^+}]=[xx^+]$ and $F(X_{x^+})=1$. Obviously, this flow preserves the set $\{\H_w,\ w\in W^s(x^+)\}$ of horospheres based at $x^+$, by sending $\H_w$ on $\H_{\ph^t(w)}$; also it contracts the Hilbert distance $\d$. Finally, the space $T\o$ admits a $\ph_{x^+}^t$-invariant decomposition $$T\o = \R.X_{x^+} \oplus T\H_{x^+},$$ where $T\H_{x^+}$ is the bundle over $\o$ defined as $$T\H_{x^+} = \{T_x\H_w,\ w=(x,[\xi])\in W^s(x^+)\}.$$ Furthermore, from the very definition of the radial flow, we have $d\ph_{x^+}^t = d\pi d\ph^t d\pi^{-1}_{x^+}$; so, for any vector $v\in T\H_{x^+}$, we have $$d\ph^t_{x^+}(v) = d\pi d\ph^t d\pi^{-1}_{x^+}(v),$$ where $d\pi^{-1}_{x^+}(v)$ is a stable vector. The action of $d\ph^t$ on $E^s$ can be deduced from the action of the parallel transport on $E^s$, and we now define a parallel transport on $\o$ to get the same kind of relations.
Let $x^+\in\doo$. The parallel transport $T_{x^+}^t$, $t\in\R$, in the direction of $x^+$ is defined by $$T_{x^+}^t = d\pi T^t d\pi^{-1}_{x^+}.$$
Given a vector $v\in T\o$, we deduce its parallel transport $T_{x^+}^t(v)$ by taking the unique vector $Z(v) \in E^s \oplus \R.X$ that projects on $v$, take its parallel transport $T^t Z(v)$ and project it again. Equivalently, since $E^s=\{Y-J^X(Y),\ Y\in VH\o\}$, we can also take the unique vector $h(v)$ in $\R.X \oplus h^XH\o$ that projects on $v$.\
From proposition \[stable\], we deduce that, for any $v\in T\H_{x^+}$, $$d\ph_{x^+}^t(v) = e^{-t} T_{x^+}^t(v)$$
The only thing we have to do now is to understand the behaviour of the quantities $F(T_{x^+}^t v)$ for $v\in T\H_{x^+}$.
Let $x^+\in\doo$. The *upper and lower parallel Lyapunov exponents* $\overline{\eta}(x^+,v)$ and $\overline{\eta}(x^+,v)$ of a vector $v\in T\H_{x^+}$ in the direction of $x^+$, are defined as $$\overline{\eta}(w,v)=\limsup_{t\to+\infty} \frac{1}{t} \log F( T_{x^+}^t v),\ \underline{\eta}(w,v)=\liminf_{t\to+\infty} \frac{1}{t} \log F( T_{x^+}^t v).$$
Given $w=(x,[\xi])\in W^s(x^+)$, it is not difficult to see that the numbers $\overline{\eta}(x^+,v)$ and $\underline{\eta}(x^+,v)$ can take only a finite number $\overline{p}(w)$ and $\underline{p}(w)$ of values when $v$ describes $T_x\H_w$. More precisely, there exist a $\ph^t_{x^+}$-invariant filtration $$\{0\}=\overline{H}_{0} \varsubsetneq \overline{H}_{1} \varsubsetneq \cdots \varsubsetneq \overline{H}_{\overline{p}(w)} = T_w \H_w$$ along the orbit $\ph_{x^+}.x$, and real numbers $$\overline{\eta}_{1}(w) < \cdots < \overline{\eta}_{\overline{p}(w)}(w),$$ called upper parallel Lyapunov exponents, such that for any vector $v_i\in \overline{H}_{i}\smallsetminus \overline{H}_{i-1}$, $1\leqslant i\leqslant \overline{p}(w)$, $$\limsup_{t\to\pm} \frac{1}{t} \log F(T_{x^+}^t v_i) = \overline{\eta}_{i}(w).$$ The same occurs for lower parallel Lyapunov exponents.\
As a consequence of part \[sectionlyapunovboundary\], we will have the following
Let $x^+\in\doo$. The numbers $\overline{p}$ and $\underline{p}$ are constants on $W^s(x^+)$, as well as the numbers $\overline{\eta}_i,\ 1\leqslant i\leqslant \overline{p}$ and $\underline{\eta}_i,\ 1\leqslant i\leqslant \underline{p}$.
Regular points {#sectionregular}
--------------
Recall the following general
\[regular\] Let $\ph^t$ be a $\C^1$-flow on a Finsler manifold $(W,\|\ . \ \|)$. A point $w\in W$ is said to be *regular* if there exist a $\ph^t$-invariant decomposition $$TW = E_1 \oplus \cdots \oplus E_p,$$ along the orbit $\ph.w$, called Lyapunov decomposition, and real numbers $$\chi_1(w) < \cdots < \chi_{p}(w),$$ called Lyapunov exponents, such that, for any vector $Z_i\in E_i\smallsetminus \{0\}$, $$\label{regularg}\lim_{t\to\pm\infty} \frac{1}{t} \log \|d\ph^t(Z_i)\| = \chi_i(w),$$ and $$\label{regulardet} \lim_{t\to\pm\infty} \frac{1}{t} \log |\det d\ph^t| = \sum_{i=1}^p \dim E_i\ \chi_i(w).$$ The point $w$ is said to be *forward* or *backward regular* if this behaviour occurs only when $t$ goes respectively to $+\infty$ or $-\infty$.
In this definition, we have to precise what is meant by $\det d\ph^t$, since $\|\ .\ \|$ is not a Riemannian metric. The determinant $\det d\ph^t$ just measures the effect of $\ph^t$ on volumes. But associated to the Finsler metric $\|\ .\ \|$ is the Busemann volume $vol_{W}$, which is the volume form defined by saying that, in each tangent space $T_wW$, the volume of the unit ball of $\|\ .\ \|$ is the same as the volume of the Euclidean unit ball of the same dimension. In other words, given an arbitrary Riemannian metric $g$ on $W$ with Riemannian volume $vol_g$, we have, at the point $w \in W$, $$dvol_{W}(w) = \frac{vol_g(B_g(w,1))}{vol_g(B(w,1))} dvol_g(w),$$ where $B(w,1)$ and $B_g(w,1)$ denote the unit balls in $T_wW$ for, respectively, $\|\ .\ \|$ and $g$. The determinant $\det d\ph^t$ is then defined in this way: if $A$ is some Borel subset of $T_wW$ with non-zero volume, then $$|\det d_w\ph^t | = \frac{vol_{W}(\ph^t(w)) (d\ph^t A)}{vol_{W}(w)(A)}.$$
\
Let us specify what happens in our case at a regular point $w\in H\o$. First, it has always $0$ as Lyapunov exponent since $\|d\ph^t(X)\|=1$, and we will say that $w$ has *no zero Lyapunov exponent* if the subspace $E_0$ corresponding to the exponent $0$ is $E_0=\R.X$.\
Second, proposition \[stable\] implies that $\chi(Z^s)\leqslant 0$ and $\chi(Z^u)\geqslant 0$ for any $Z^s\in E^s(w),\ Z^u\in E^u(w)$. Furthermore, if $Z^s\in E^s(w)$ and $Z^u=J^X Z^s$ is the corresponding vector of $E^u(w)$, proposition \[stable\] gives $$\chi(Z^u) = 2 + \chi(Z^s).$$ Now, choose a tangent vector $Z$ whose Lyapunov exponent is $0$. $Z$ can be written as $Z=aX+Z^u+Z^s$ for some $a\in\R,\ Z^s\in E^s,\ Z^u\in E^u$. Since $$\lim_{t\to\pm\infty} \frac{1}{t} \log \|d\ph^t(Z)\| = 0,$$ we conclude that $\chi(Z^u)=\chi(Z^s)=0$. Thus, the subspace $E_0$ corresponding to the Lyapunov exponent $0$ can be decomposed as $$E_0 = \R.X \oplus E^- \oplus E^+,$$ where $E^- \subset E^s,\ E^+ \subset E^u$.\
At a regular point, the Lyapunov decomposition can thus be written in the following way: $$\label{oseledetsdecomposition}
TH\o = E^s_0 \oplus (\oplus_{i=1}^p E^s_i) \oplus E_{p+1}^s \oplus \R.X \oplus E_0^u \oplus (\oplus_{i=1}^p E^u_i) \oplus E^u_{p+1},$$ with the relations $$E_i^s=J^X(E_i^u),\ 0\leqslant i\leqslant p+1.$$ The subspaces $E^s_0$ and $E^u_0$, or $E_{p+1}^s$ and $E_{p+1}^s$, might be $\{0\}$. The corresponding Lyapunov exponents are $$-2=\chi^-_0 < \chi_1^s < \cdots <\chi^s_p < \chi^s_{p+1} = 0 = \chi^u_0 < \chi^u_1 <\cdots<\chi^u_p < \chi^u_{p+1} = 2,$$ with $$\chi_i^u=\chi_i^s+2,\ 0\leqslant i\leqslant p.$$ If $w$ has no zero Lyapunov exponent then $E^s_0=E^u_0=E_{p+1}^s=E_{p+1}^s=\{0\}$ and all the Lyapunov exponents are [*strictly*]{} between $-2$ and $2$.\
If we now look down at the base manifold $\o$, we see that, if $w=(x,[\xi])\in H\o$ is a regular point ending at $x^+\in\doo$, the decomposition (\[oseledetsdecomposition\]) induces by projection a decomposition $$T\o = \R.X_{x^+} \oplus H_0 \oplus H_1 \oplus \cdots \oplus \cdots \oplus H_p \oplus H_{p+1}$$ along the orbit $\ph_{x^+}.x$ and there exist real numbers $-1=\eta_0<\eta_1 < \cdots < \eta_p<\eta_{p+1}=1$, that we call parallel Lyapunov exponents, such that, for any vector $v_i \in H_i\smallsetminus \{0\}$, $$\lim_{t\to+\infty} \frac{1}{t} \log F(T^t_{x^+}(v_i)) = \eta_i,$$ and $$\lim_{t\to+\infty} \frac{1}{t} \log |\det T_{x^+}^t| = \sum_{i=1}^p \dim H_i\ \eta_i.$$
We have $\R.X_{x^+}=d\pi(\R.X)$ and $$\label{dechautbas}
H_i = d\pi (E_i^s) = d\pi(E_i^u);$$ in particular, $H_0$ and $H_{p+1}$ can be $\{0\}$. Also, the parallel Lyapunov exponents $\eta_i$ are given by $$\label{lyaphautbas}
\eta_i = \chi^s_i+1 = \chi^u_i-1.$$
We then have the following characterization of regular points:
Let $x^+\in\doo$. A point $w=(x,[\xi])\in W^s(x^+)$ is regular for $\ph^t$ if and only if the point $x$ is regular for $\ph^t_{x^+}$. The decomposition and Lyapunov exponents are linked by the relations (\[dechautbas\]) and (\[lyaphautbas\]).
Obviously, all of this makes sense for forward and backward regular points.
Oseledets theorem
-----------------
The essential result about regular points is the following theorem of Oseledets, which, given an invariant probability measure of the flow, gives a condition for almost all points to be regular.
\[fullmeasure\] Let $\ph^t$ be a $\C^1$ flow on a Finsler manifold $(W,\|\ .\ \|)$ and $\mu$ a $\ph^t$-invariant probability measure. If $$\label{hyposeledets}
\frac{d}{dt}_{|_{t=0}} \log \|d\ph^{\pm t}\| \in L^1(W,\mu),$$ then the set of regular points has full measure.
Assumption $(\ref{hyposeledets})$ means that the flow does not expand or contract locally too fast. This essentially allows us to use Birkhoff’s ergodic theorem to prove the theorem.\
The next lemma proves that our geodesic flow satisfies assumption $(\ref{hyposeledets})$. Obviously, Oseledets’ theorem is not interesting on $H\o$ itself since there is no finite invariant measure. But it can be applied for any invariant measure of the geodesic flow of a given a quotient manifold $M=\o/\G$.\
Remark that our Finsler metric is $C_X$-regular so condition $(\ref{hyposeledets})$ makes sense. Furthermore, Oseledets’ theorem is usually stated on a Riemannian manifold but it is still valid for a Finsler one: using John’s ellipsoid, it is always possible to define a Riemannian metric $\|\ .\ \|_{J}$ which is bi-Lipschitz equivalent to $\|\ .\ \|$, that is, such that $$\frac{1}{\sqrt{n}} \|Z\|_{J} \leqslant \|Z\| \leqslant \sqrt{n} \|Z\|_{J},\ Z\in TW$$ where $n$ is the dimension of the manifold; of course, there is no reason for this metric $\|\ .\ \|_J$ to be even continuous but it is not important.
For any $Z^s\in E^{s},\ Z^u\in E^{u}$, we have $$-2 \leqslant \frac{d}{dt}_{|_{t=0}} \|d\ph^{t} Z^s\|\log \leqslant 0 \leqslant \frac{d}{dt}_{|_{t=0}} \|d\ph^{t} Z^u\| \leqslant 2.$$ In particular, for any $t\in\R$ and $Z\in TH\o$, $$e^{-2|t|} \|Z\| \leqslant \|d\ph^t(Z)\| \leqslant e^{2|t|} \|Z\|.$$
This lemma clearly implies the already known fact (coming from proposition \[stable\]) that Lyapunov exponents are all between $-2$ and $2$. But it is more precise: it gives that the rate of expansion/contraction is [*at any time*]{} between $-2$ and $2$, not only asymptotically, and that is what is essential to apply Oseledets’ theorem.
It is a direct corollary of proposition \[stable\]: we know that $t\mapsto \|d\ph^t Z^s\|$ is decreasing, hence $$\lim_{t\to 0}\frac{1}{t} \log \|d\ph^t Z^s\| \leqslant 0.$$ But we also know from proposition \[dd\] that $$\|d\ph^t Z^s\| = e^{-2t} \|d\ph^t J^X(Z^s)\|.$$ Since $J^X(Z^s) \in E^u$, proposition \[stable\] tells us that $t\mapsto \|d\ph^t J^X(Z^s)\|$ is increasing, hence $$\lim_{t\to 0} \frac{1}{t} \log \|d\ph^t J^X(Z^s)\| \geqslant 0$$ and $$\lim_{t\to 0} \frac{1}{t} \log \|d\ph^t Z^s\|\geqslant -2.$$ Using $J^X$, we get the second inequality, and by integrating, we get the last one.
Structure of the boundary $\doo$ {#sectionlyapunovboundary}
================================
In this part, we give a link between parallel Lyapunov exponents and the shape of the boundary at the endpoint of the orbit.\
Motivation
----------
We first give the idea in dimension $2$. Let $x^+\in\doo$, $w = (x,[\xi])\in W^s(x^+)$ and choose a vector $v$ tangent to $\H_w$ at $x$, with parallel Lyapunov exponent $\eta$. In a good chart at $w$, lemma \[transport\] gives $$F(T_{x^+}^t v) = C(w) (|x_tx^+||x_tx^-|)^{1/2}\left(\frac{1}{|x_ty_t^+|}+\frac{1}{|x_ty_t^-|}\right).$$
Assume that $|x_ty_t^-|\asymp |x_ty_t^+|$. Then $$\lim_{t\to +\infty} \frac{1}{t} \log \frac{F( T_{x^+}^t v)}{|x_tx^+|^{1/2}} = -\lim_{t\to +\infty} \frac{1}{t}\log |x_ty_t^+|,$$ hence, dividing by $\log |x_tx^+|^{1/2}$, $$\lim_{t\to +\infty}\frac{\log F( T_{x^+}^t )}{\log |x_tx^+|^{1/2}} -1 = -\lim_{t\to +\infty}\frac{\log |x_ty_t^+|}{\log |x_tx^+|^{1/2}}.$$ Since $|x_tx^+|\asymp e^{-2t}$, that yields $$\lim_{t\to +\infty}\frac{\log |x_ty_t^+|}{\log |x_tx^+|} = \frac{1+\eta}{2}.$$ Let $f: T_{x^+}\doo \longrightarrow \R^n$ be the graph of $\doo$ at $x^+$, so that $|x_tx^+|= f(|x_ty_t^+|)$. We thus obtain $$\lim_{s\to 0} \frac{\log f(s)}{\log s} = \frac{2}{1+\eta},$$ that is, for any $\epsilon>0$, there exists $C>0$ such that $$\label{eeee}
C^{-1} s^{\frac{2}{1+\eta}+\epsilon}\leqslant f(s) \leqslant C s^{\frac{2}{1+\eta}-\epsilon}.$$
This link was first established in [@crampon] for divisible convex sets, where the condition $|x_ty_t^-|\asymp |x_ty_t^+|$ is always satisfied. In order to generalize it, we must introduce new definitions. It may be a bit fastidious so you could prefer going directly to proposition \[linkboundary\], and have a look to the part in between when it is needed.
Locally convex submanifolds of $\R\P^n$
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A codimension 1 $\C^0$ submanifold $N$ of $\R^n$ is *locally (strictly) convex* if for any $x\in N$, there is a neighbourhood $V_x$ of $x$ in $\R^n$ such that $V_x\smallsetminus N$ consists of two connected components, one of them being (strictly) convex.\
A codimension 1 $\C^0$ submanifold $N$ of $\R\P^n$ is *locally* (strictly) *convex* if its trace in any affine chart is locally (strictly) convex.
Obviously, to check if $N\subset \R\P^n$ is convex around $x$, it is enough to look at the trace of $N$ in [*one*]{} affine chart at $x$. Choose a point $x\in N$ in a locally convex submanifold $N$ and an affine chart centered at $x$. We can find a tangent space $T_x$ of $N$ at $x$ such that $V_x\cap N$ is entirely contained in one of the closed half-spaces defined by $T_x$. We can then endow the chart with a suitable Euclidean structure, so that, around $x$, $N$ appears as the graph of a convex function $f: U\subset T_x \longrightarrow [0,+\infty)$ defined on an open neighbourhood $U$ of $0\in T_x$. This function is (at least) as regular as $N$, is positive, $f(0)=0$ and $f'(0)=0$ if $N$ is $\C^1$ at $x$. When $N$ is strictly locally convex, then $f$ is strictly convex, in particular $f(v)>0$ for $v\not= 0$.\
In what follows, we are interested in the shape of the boundary $\doo$ of $\o$ at some specific point, or, more generally, in the local shape of locally strictly convex $\C^1$ submanifolds of $\R\P^n$. Denote by $\mathtt{Cvx}(n)$ the set of strictly convex $\C^1$ functions $f: U\subset\R^n \longrightarrow \R$ such that $f(0)=f'(0)=0$, where $U$ is an open convex subset of $\R^n$ containing $0$. We look for properties of such functions at the origin which are invariant by projective transformations.\
Approximate $\alpha$-regularity
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We introduce here the main notion of approximate $\alpha$-regularity, describe its meaning and prove some useful lemmas.
### Definition
\[defiappreg\] A function $f\in\mathtt{Cvx}(1)$ is said to be *approximately $\alpha$-regular*, $\alpha\in [1,+\infty]$, if $$\lim_{t\to 0} \frac{\log \displaystyle\frac{f(t)+f(-t)}{2}}{\log |t|} = \alpha.$$
This property is clearly invariant by affine transformations, and in particular by change of Euclidean structure. It is in fact invariant by projective ones, but we do not need to prove it directly, since it will be a consequence of proposition \[linkboundary\].\
Obviously, the function $t\in\R \mapsto |t|^{\alpha}$, $\alpha>1$ is approximately $\alpha$-regular. To be $\alpha$-regular, with $1<\alpha<+\infty$, means that we roughly behave like $t\mapsto |t|^{\alpha}$.\
The case $\alpha=\infty$ is a particular one: $f$ is $\infty$-regular means that for any $\alpha\geqslant 1$, $f(t) \ll |t|^\alpha$ for small $|t|$. An easy example of such a function is provided by $f:t \longmapsto e^{-1/t^2}$. On the other side, $f$ is $1$-regular means that for any $\alpha>1$, $f(t) \gg |t|^\alpha$. An example of function which is $1$-regular is provided by the Legendre transform of the last one (see section \[sectionlegendre\]).\
In the case where $1<\alpha<+\infty$, we can state the following equivalent definitions. The proof is straightforward.
\[calpha\] Let $f\in\mathtt{Cvx}(1)$ and $1<\alpha<+\infty$. The following propositions are equivalent:
- $f$ is approximately $\alpha$-regular;
- for any $\epsilon>0$ and small $|t|$, $$|t|^{\alpha+\epsilon} \leqslant \frac{f(t)+f(-t)}{2} \leqslant |t|^{\alpha-\epsilon};$$
- the function $t\longmapsto \displaystyle\frac{f(t)+f(-t)}{2}$ is $\C^{\alpha-\epsilon}$ and $\alpha+\epsilon$-convex at $0$ for any $\epsilon>0$.
To understand the last proposition, we recall the following
\[deficalpha\] Let $\alpha,\beta\geqslant 1$ We say that a function $f\in\mathtt{Cvx}(n)$ is
- $\C^{\alpha}$ if for small $|t|$, there is some $C>0$ such that $$f(t) \leqslant C|t|^{\alpha};$$
- $\beta$-convex if for small $|t|$, there is some $C>0$ such that $$f(t) \geqslant C|t|^{\beta}.$$
### A useful equivalent definition
We now give another equivalent definition of approximate regularity, that shows the relation with the motivation above. Theorem \[linkboundary\] is the most important consequence of it.\
Let $f\in\mathtt{Cvx}(1)$. Denote by $f^+=f_{|_{[0,1]}}^{-1}$ and $f^-=-f_{|_{[-1,0]}}^{-1}$. These functions are both nonnegative, increasing and concave and their value at $0$ is $0$; they are $\C^1$ on $(0,1]$ and their tangent at $0$ is vertical.\
The harmonic mean of two numbers $a,b >0$ is defined as $$H(a,b) = \frac{2}{a^{-1} + b^{-1}}.$$ The harmonic mean of two functions $f,\ g: X \to (0,+\infty)$ defined on the same set $X$ is the function $H(f,g)$ defined for $x\in X$ by $$H(f,g)(x) = H(f(x),g(x)) = \frac{2}{\frac{1}{f(x)}+\frac{1}{g(x)}}.$$
\[approxeq\] A function $f\in\mathtt{Cvx}(1)$ is approximately $\alpha$-regular, $\alpha \in [1,+\infty]$ if and only if $$\lim_{t\to 0^+} \frac{\log H(f^+,f^-)(t)}{\log t} = \alpha^{-1},$$ with the convention that $\frac{1}{+\infty} = 0$.
As we will see, it is enough to take $f$ continuous, so by replacing $f^+$ and $f^-$ by $\min(f^+,f^-)$ and $\max(f^+,f^-)$, we can assume that $f^+\leqslant f^-$, that is $f(t)\geqslant f(-t)$ for $t\geqslant 0$. Now, assuming that the limit exists,
$$\lim_{t\to 0^+} \frac{\log H(f^+,f^-)(t)}{\log t} =- \lim_{t\to 0^+} \frac{\log \left(\displaystyle\frac{1}{f^+(t)} + \displaystyle\frac{1}{f^-(t)}\right)}{\log t}
= \lim_{t\to 0^+} \frac{\log f^+(t)}{\log t} - \lim_{t\to 0^+}\frac{\log \left(1 + \displaystyle\frac{f^+(t)}{f^-(t)}\right)}{\log t}.$$ Since $f^+\leqslant f^-$, the second limit is $0$, and the first one is $$\lim_{t\to 0^+} \frac{\log f^+(t)}{\log t} = \lim_{u\to 0^+} \frac{\log u}{\log f(u)}.$$
But, since $f(u)\geqslant f(-u)$ for $u\geqslant 0$, we get $$\lim_{u\to 0^+} \frac{\log u}{\log \frac{f(u)+f(-u)}{2}} = \lim_{u\to 0^+} \frac{\log u}{\log f(u) + \log \left( 1+\frac{f(-u)}{f(u)}\right)} = \lim_{u\to 0^+} \frac{\log u}{\log f(u)},$$ hence the result.
The last construction can be generalized in a way that will be useful later, for proving proposition \[linkboundary\]. Let $f\in\mathtt{Cvx}(1)$ and pick $a>0$. We define two new “inverse functions” $f_a^+(s)$ and $f^-_a(s)$ for $s\in[0,\epsilon]$, $\epsilon>0$ small enough, depending on $a$; these are positive functions defined by the equations $$f(f^+_a(s))=s-sf^+_a(s); f(-f^-_a(s))=s+sf^-_a(s).$$
\[figureapprox\]
Geometrically, for $s\in [0,\epsilon]$ on the vertical axis, the line $(as)$ cuts the graph of $f$ at two points $a^+$ and $a-$, with $s$ between $a^+$ and $a^-$; $f_a^+(s)$ and $f^-_a(s)$ are the abscissae of $a^+$ and $a^-$ (c.f. figure \[figureapprox\]). $f^+$ and $f^-$ can be considered as $f_{+\infty}^+$ and $f_{+\infty}^-$.
\[bordtordu\] Let $f\in\mathtt{Cvx}(1)$ and $a>0$. The functions $\displaystyle\frac{f_a^+}{f^+}$ and $\displaystyle\frac{f_a^-}{f^-}$ can be extended by continuity at $0$ by $$\frac{f_a^+}{f^+}(0)=\frac{f_a^-}{f^-}(0)=1.$$ In particular, for $s>0$ small enough, $$f^+(s) \asymp f_a^+(s),\ f^-(s) \asymp f_a^-(s).$$
We prove it for $f^+$ and $f_a^+$. Clearly, we have $\frac{f_a^+(s)}{f^+(s)} \leqslant 1$. Since $f$ is convex and $f(0)=0$, we get $$s-sf_a^+(s) = f(f_a^+(s)) = f\left(\frac{f_a^+(s)}{f^+(s)} f^+(s)\right) \leqslant \frac{f_a^+(s)}{f^+(s)} f(f^+(s)) = \frac{f_a^+(s)}{f^+(s)}s.$$ Hence, for $0<s\leqslant \epsilon<1$ $$\frac{f_a^+(s)}{f^+(s)} \geqslant 1-f_a^+(s) \geqslant 1 - f_a^+(\epsilon).$$ The function $\displaystyle \frac{f_a^+}{f^+}$ can even be extended at $0$ by $\displaystyle\frac{f_a^+}{f^+}(0)=1$
The result to remember is the following consequence of lemmas \[bordtordu\] and \[approxeq\]:
\[corotordu\] Pick $a>0$. A function $f\in\mathtt{Cvx}(1)$ is approximately $\alpha$-regular if and only if $$\lim_{t\to 0^+} \frac{\log H(f_a^+,f_a^-)(t)}{\log t} = \alpha^{-1}.$$
Higher dimensions
-----------------
We end this section by extending the definitions in higher dimensions:
A function $f\in\mathtt{Cvx}(n)$ is said to be *approximately regular* at $x$ if it is approximately regular in any direction, that is, for any $v\in \R^n\smallsetminus\{0\}$, there exists $\alpha(v)\in [1,\infty]$ such that $$\lim_{t\to 0} \frac{\log \displaystyle\frac{f(tv) + f(-tv)}{2}}{\log |t|} = \alpha(v).$$
Let $f\in\mathtt{Cvx}(n)$ . The upper and lower Lyapunov exponents $\overline{\alpha}(v)$ and $\underline{\alpha}(v)$ of $v\in \R^n$ are defined by
$$\overline{\alpha}(v) = \limsup_{t\to 0} \frac{\log \displaystyle\frac{f(tv) + f(-tv)}{2}}{\log |t|},$$
$$\underline{\alpha}(v) = \liminf_{t\to 0} \frac{\log \displaystyle\frac{f(tv) + f(-tv)}{2}}{\log |t|}.$$
The function is then approximately regular if and only if the preceding limits are indeed limits in $[1,+\infty]$, that is, for any $v\in \R^n$, $\overline{\alpha}(v)=\underline{\alpha}(v)$. Obviously, lemma \[approxeq\] and corollary \[corotordu\] have their counterpart in higher dimensions.
Approximate regularity of the boundary
--------------------------------------
If $\o$ is a bounded convex set in the Euclidean space $\R^n$ with $\C^1$ boundary, the graph of $\doo$ at $x$ is the function $$\begin{array}{rrcl}
f: & U\subset T_x\doo & \longrightarrow & \R^n \\
& u & \longmapsto & \{u + \l n(x)\}_{\l\in\R} \cap \doo,
\end{array}$$ where $n(x)$ denotes a normal vector to $\doo$ at $x$, and $U$ is a sufficiently small open neighbourhood of $x\in\doo$ for the function to be defined.\
We can now state our main result. Let $x^+\in\doo$. If $w=(x,[\xi])\in W^s(x^+)$ and $v\in T_x\H_w$, we denote by $p_{x^+}(v)$ the projection of $v$ on the space $T_{x^+}\doo$ in the direction $[xx^+]$. The map $p_{x^+}$ clearly induces an isomorphism $p_{x^+}(x)$ between each $T_x\H_w$ and $T_{x^+}\doo$.
\[linkboundary\] Let $\o$ be a strictly convex proper open set of $\R\P^n$ with $\C^1$ boundary. Pick $x^+\in\doo$, choose any affine chart containing $x^+$ and a Euclidean metric on it.\
Then for any $v\in T\H_{x^+}$, we have
$$\overline{\eta}(x^+,v)= \frac{2}{\underline{\alpha}(x^{+},p_{x^+}(v))}-1, \ \underline{\eta}(x^+,v)= \frac{2}{\overline{\alpha}(x^{+},p_{x^+}(v))}-1,$$ where $\underline{\alpha}(x^{+},p_{x^+}(v))$ and $\overline{\alpha}(x^{+},p_{x^+}(v))$ denote the lower and upper Lyapunov exponents of the graph of $\doo$ at $x^+$ in the direction $p_{x^+}(v)$, as defined at the very end of the last section.
\[bord\]
![For proposition \[linkboundary\]](dessins/bord.ps)
Let $w=(x,[\xi])$ be a point ending at $x^+$, $(x_t,[\xi_t]) = \ph^t(x,[\xi])$ its image by $\ph^t$, and $v\in T_x\H_w$. The vector $ T_{x^+}^t v$ is at any time contained in the plane generated by $\xi$ and $v$, thus, by working in restriction to this plane, we can assume that $n=2$.\
We cannot choose a good chart at $w$, since the chart is already fixed. But, by affine invariance, we can choose the Euclidean metric $|\ .\ |$ and $\xi_t$ so that $\xi \bot T_{x^+}\doo = \R.p_{x^+}(v)$ and $|v|=|\xi_t|=1$. Let $a$ be the point of intersection of $T_{x^+}\doo$ and $T_{x^-}\doo$. The vector $ T_{x^+} v$ always points to $a$, that is, $ T_{x^+}^t v \in \R.x_ta$. Thus,
$$F( T_{x^+}^t v) = \frac{| T_{x^+}^t v|}{2} \left(\frac{1}{|x_ty_t^+|}+\frac{1}{|x_ty_t^-|}\right),$$ where $y_t^+$ and $y_t^-$ are the intersection points of $(ax_t)$ and $\doo$. If $f:U \subset T_{x^+}\doo \longrightarrow~\R$ denotes the function whose graph is a neighbourhood of $x^+$ in $\doo$, then $$\frac{1}{2}\left(\frac{1}{|x_ty_t^+|}+\frac{1}{|x_ty_t^-|}\right) = \frac{1}{H(f_a^+,f_a^-)(|x_tx^+|)},$$ where $f_a^+$ and $f_a^-$ are defined as in corollary \[corotordu\]. This corollary tells us that $$\begin{array}{rl}
\displaystyle\limsup_{t\to+\infty} \frac{1}{t}\log\frac{1}{H(f_a^+,f_a^-)(|x_tx^+|)} & = \displaystyle\limsup_{t\to+\infty} \displaystyle-\frac{\log|x_tx^+|}{t}\displaystyle \frac{\log H(f_a^+,f_a^-)(|x_tx^+|)}{\log|x_tx^+|}\\\\
& = \displaystyle\limsup_{t\to+\infty} \displaystyle-\frac{\log|x_tx^+|}{t} \displaystyle\limsup_{s\to 0} \displaystyle\frac{\log H(f_a^+,f_a^-)(s)}{\log s} \\\\
& = \displaystyle\frac{2}{\underline{\alpha}(x^+,v)}
\end{array}$$
(recall from lemma \[equivalents\] that $|x_tx^+|= \frac{|xx^+|^2}{m(w)} e^{-2t}$). Hence $$\limsup_{t\to+\infty} \frac{1}{t} \log F( T_{x^+}^t v) = \frac{2}{\underline{\alpha}(x^+,v)} + \limsup_{t\to+\infty} \frac{1}{t} \log | T_{x^+}^t v|.$$ From our choice of Euclidean metric, we have $| T^t v(w)| \asymp \langle T^t v(w), v \rangle$. Lemma $\ref{horver}$ gives $$T_{x^+}^t v = -L_Y m(\ph^t w) \xi_t + (m(w)m(\ph^t w))^{1/2} d\pi(J^{X^e}(Y)),$$ where $Y\in VH\o$ is such that $d\pi(J^X(Y))=v(w)$; $d\pi(J^{X^e}(Y))$ is collinear to $v$ and has constant Euclidean norm, which implies that $$\lim_{t\to +\infty} \frac{1}{t}\log \langle T_{x^+}^t v, v \rangle = \lim_{t\to +\infty} \frac{1}{t}\log (m(w)m(\ph^t w))^{1/2} = -1.$$
Hence $$\overline{\eta}_{+}(w,v(w))=\limsup_{t\to+\infty} \frac{1}{t}F( T_{x^+}^t v) = \frac{2}{\underline{\alpha}(x^+,v)} -1.$$
Obviously, the same holds for lower exponents.
The last theorem tells us that the notions of Lyapunov regularity and exponents are projectively invariant, that is, it makes sense for codimension 1 submanifolds of $\R\P^n$. It then justifies the following
A locally strictly convex $\C^1$ submanifold $N$ of $\R\P^n$ is said to be *approximately regular* at $x\in N$ if its trace in some (or, equivalently, any) affine chart at $x$ is locally the graph of an approximately regular function. The numbers $\alpha_1(x)\geqslant\cdots\geqslant \alpha_p(x)$ attached to $x$ are called the *Lyapunov exponents* of $x$.
Also, remark the following properties:
\[alpha\] Let $f\in\mathtt{Cvx}(n)$. Then
- the numbers $\overline{\alpha}(v),\ v\in \R^n\smallsetminus\{0\}$, can take only a finite numbers of values. More precisely, there exist a number $\overline p$, a filtration $$\{0\} = \overline{G}_0 \varsubsetneq \overline{G}_1 \varsubsetneq \cdots \varsubsetneq \overline{G}_{\overline p}=\R^n$$ and numbers $$+\infty \geqslant \overline{\alpha}_1 > \cdots > \overline{\alpha}_{\overline p} \geqslant 1,$$ such that for any $v_i\in \overline{G}_i\smallsetminus \overline{G}_{i-1}$, $1\leqslant i \leqslant \overline p$, $$\limsup_{t\to 0} \frac{\log \displaystyle\frac{f(tv_i) + f(-tv_i)}{2}}{\log |t|} = \overline{\alpha}_i.$$ The same holds for lower Lyapunov exponents.\
- the following propositions are equivalent:
1. $f$ is approximately regular;
2. there exist a decomposition $\R^n=\oplus_{i=1}^p H_i$ and numbers $+\infty \geqslant \alpha_1 > \cdots > \alpha_p \geqslant 1$ such that the restriction $f|_{H_i\cap U}$ is approximately regular with exponent $\alpha_i$;
3. there exist a filtration $$\{0\}=G_0 \varsubsetneq G_1 \varsubsetneq \cdots \varsubsetneq G_p = \R^n$$ and numbers $+\infty \geqslant \alpha_1 > \cdots > \alpha_p \geqslant 1$ such that, for any $v_i\in G_i\smallsetminus G_{i-1}$, the restriction $f|_{\R.v_i\cap U}$ is approximately regular with exponent $\alpha_i$.
When $f$ is approximately regular, we call the numbers $\alpha_i$ the Lyapunov exponents of $f$.
The graph of $f$ can always be considered as the boundary of a strictly convex set $\o\subset \R^{n+1}$ with $\C^1$ boundary. We can then apply theorem \[linkboundary\] to this set $\o$.
Lyapunov regularity of the boundary
-----------------------------------
To characterize regular points $w\in H\o$, we need to add a property to approximate regularity because of the second point in definition \[regular\].
\[lyapunovregularity\] A function $f\in\mathtt{Cvx}(n)$ is said to be *Lyapunov regular* if
- $f$ is approximately regular with exponents $+\infty \geqslant \alpha_1 \geqslant \cdots \geqslant \alpha_p \geqslant 1$ counted with multiplicities;
- $$\lim_{t\to 0} \frac{\log \displaystyle\int_{f(u)\leqslant |t|} |u|\ du}{\log t} = \frac{1}{\alpha} ,$$ where $$\frac{1}{\alpha} = \sum_{i=1}^n \frac{1}{\alpha_i}.$$
A locally strictly convex $\C^1$ submanifold $N$ of $\R\P^n$ is said to be *Lyapunov regular* at $x\in N$ if its trace in some (or, equivalently, any) affine chart at $x$ is locally the graph of a Lyapunov regular function.
Remark that we should prove the second point in definition \[lyapunovregularity\] is projectively invariant to state the last definition. In fact, we could proceed as before in theorem \[linkboundary\] by proving the next theorem in any affine chart; but the idea is totally similar so we will not do it.
\[mainthm\] A point $w=(x,[\xi])\in H\o$ is forward regular if and only if the boundary $\doo$ is Lyapunov regular at the endpoint $x^+=\ph^{+\infty}(w)$. The Lyapunov decomposition of $T\H_{x^+}$ along $\ph_{x^+}.x$ projects under $p_{x^+}$ on the Lyapunov decomposition of $T_{x^+}\doo$, and Lyapunov exponents are related by $$\eta(v) = \frac{2}{\alpha(p_{x^+}(v))}-1,\ v\in T_x\H_w.$$
The only if part is now clear from the last theorem. Assume $\doo$ is approximately regular at $x^+$. The decomposition of $T_{x^+}\doo$ gives by projection a decomposition $$\label{rrr}
T_x\H_w = E_1 \oplus \cdots \oplus E_p,$$ such that, for any $v_i\in E_i\smallsetminus\{0\}$, $$\lim_{t\to +\infty} \frac{1}{t} \log F(T_{x^+}^t v_i) = \eta_i,$$ where $\eta_1<\cdots <\eta_p$ are the parallel Lyapunov exponents of $w$. The only thing that we have to prove is the second point in definition \[regular\], that is, $$\lim_{t\to +\infty} \frac{1}{t} \log \det T_{x^+}^t = \sum_{i=1}^p \dim E_i\ \eta_i.$$ We can assume we have chosen a good chart and the Euclidean metric so that the decomposition \[rrr\] is orthogonal. Recall that, by definition of the determinant and the Busemann volume, $$\det T_{x^+}^t = vol T_{x^+}^t(B_{x}(1)) = \frac{vol^e (T_{x^+}^t(B_{x}(1)))}{vol^e (B_{\ph^t_{x^+}(x)}(1))}.$$ Since the map $T_{x^+}^t$ is linear, the quantity $vol^e (T_{x^+}^t(B_{x}(1)))$ is just the determinant $\det^e T_{x^+}^t $ of $T_{x^+}^t$ with respect to the Euclidean metric that we have chosen; lemma \[transport\] implies that $${\det}^e T_{x^+}^t = (m(w)m(\ph^t(w))^{\frac{n-1}{2}} {\det}^e (T_{x^+}^e)^t,$$ so that $$\lim_{t\to +\infty} \frac{1}{t} \log vol^e (T_{x^+}^t(B_{x}(1))) = \frac{n-1}{2} \lim_{t\to +\infty} \frac{1}{t} \log m(\ph^t(w)) = \frac{n-1}{2} \lim_{t\to +\infty} \frac{1}{t} \log |x_tx^+| = n-1,$$ by lemma \[equivalents\].\
So we just have to study the quantity $ \frac{1}{t} \log vol^e (B_{\ph^t_{x^+}(x)}(1)).$ Call $x_t=\ph^t_{x^+}$ as usual, and for each vector $u\in T_x\H_w$, call $u_t$ the unit vector in $T_{\ph^t_{x^+}(x)}\o$ which is collinear to $u$. Since the vector $u_t$ has Finsler norm $1$, we have $$1 = \frac{|u_t|}{2}\left(\frac{1}{|x_ty_t^+|}+\frac{1}{|x_ty_t^-|}\right),$$ so $$|u_t| = \frac{2}{\frac{1}{|x_ty_t^+|}+\frac{1}{|x_ty_t^-|}} = m(x_t,[u_t]).$$ In particular, by lemmas \[transport\] and \[equivalents\], $$\lim_{t\to+\infty} \frac{1}{t} \log |u_t| = -\eta(u) + 1.$$ By convexity of the unit balls, we then get $$\lim_{t\to+\infty} \frac{1}{t} \log vol^e (B_{\ph^t_{x^+}(x)}(1)) \geqslant -\eta + (n-1) = -\frac{2}{\alpha}.$$ For the inequality from above, we just have to notice that $$vol^e (B_{\ph^t_{x^+}(x)}(1)) \leqslant vol^e \left(\o \cap T_{\ph^t_{x^+}(x)}\H_{x^+}(\ph^t_{x^+}(x))\right),$$ hence $$\begin{array}{rl}
\displaystyle\lim_{t\to+\infty} \frac{1}{t} \log vol^e (B_{\ph^t_{x^+}(x)}(1)) &\leqslant \displaystyle\lim_{t\to+\infty} \frac{1}{t} \log vol^e \left(\o \cap T_{\ph^t_{x^+}(x)}\H_{x^+}(\ph^t_{x^+}(x))\right) \\\\
&\leqslant \displaystyle\lim_{t\to+\infty} -2\displaystyle\frac{\log vol^e \left(\o \cap T_{\ph^t_{x^+}(x)}\H_{x^+}(\ph^t_{x^+}(x))\right)}{\log |x_tx^+|},
\end{array}$$ from lemma \[equivalents\]. The second property in definition \[lyapunovregularity\] implies $$\lim_{t\to+\infty} \frac{1}{t} \log vol^e (B_{\ph^t_{x^+}(x)}(1)) \leqslant -\frac{2}{\alpha}.$$ That means that $\lim_{t\to+\infty} \frac{1}{t} \log vol^e (B_{\ph^t_{x^+}(x)}(1)) = -\frac{2}{\alpha}$ and finally, $$\lim_{t\to+\infty} \frac{1}{t} \log \det T_{x^+}^t = \frac{2}{\alpha} + (n-1) = \eta.$$
In reality, I am not sure the second property in definition \[lyapunovregularity\] is necessary. I thought at the beginning it could be deduced from convexity and the other properties but I did not manage to prove it.
Lyapunov manifolds of the geodesic flow {#sectionlyapunovmanifolds}
=======================================
From the very definition of the metric $\|\ . \ \|$ (by using remark \[rmkdistance\]), we get the following corollary of theorem \[mainthm\]. Obviously, we could give an equivalent statement for non-approximately regular points by using upper and lower exponents.
\[maincorollary\] Let $\o$ be a strictly convex proper open subfset of $\R\P^n$ with $\C^1$ boundary and fix $o\in\o$. Assume $x^+\in\doo$ is approximately regular with exponents $+\infty \geqslant \alpha_1 > \cdots > \alpha_p \geqslant 1$ and filtration $$\{0\}=H_0 \varsubsetneq H_1 \varsubsetneq \cdots \varsubsetneq H_p = T_{x^+}\doo.$$ Then the horosphere $\H$ about $x^+$ passing through $o$ admits a filtration $$\{o\}=\H_0 \varsubsetneq \H_1 \varsubsetneq \cdots \varsubsetneq \H_p = \H,$$ given by $\H_i=\{x\in\H\cap (H_i\oplus \R.ox^+)\},\ 1\leqslant i \leqslant p,$ and such that $$\mathcal{H}_i\smallsetminus \mathcal{H}_{i-1} = \{x \in \H\smallsetminus\{o\},\ \lim_{t\to +\infty} \frac{1}{t} \log \d(\ph_{x^+}^t(o), \ph_{x^+}^t(x)) = \chi^s_i\},$$ with $\chi^s_i = -2+\frac{2}{\alpha_i}$.
This allows us to define Lyapunov manifolds of the geodesic flow, that is, submanifolds tangent to the subspaces appearing in the Lyapunov filtration. In the classical theory of nonuniformly hyperbolic systems, the local existence of these manifolds is a nontrivial result traditionnally achieved with the help of Hadamard-Perron theorem.\
Here these manifolds appear naturally from the decomposition of the boundary at the endpoint of the orbit we are looking at. This result can be seen as a consequence of the flatness of Hilbert geometries.
Assume $\doo$ is approximately regular at the point $x^+$. Each point $w$ of $W^s(x^+)$ is forward regular with decomposition $$TH\o = E^s_0 \oplus (\oplus_{i=1}^p E^s_i) \oplus E_{p+1}^s \oplus \R.X \oplus E_0^u \oplus (\oplus_{i=1}^p E^u_i) \oplus E^u_{p+1},$$ and Lyapunov exponents $$-2=\chi^s_0 < \chi_1^s < \cdots <\chi^s_p < \chi^s_{p+1} = 0 = \chi^u_0 < \chi^u_1 <\cdots<\chi^u_p < \chi^u_{p+1} = 2.$$ For each $w_0=(o,[ox^+])\in W^s(x^+)$, the stable manifold $W^s(w_0)$ admits a filtration by $$\{w_0\} \subset W^s_0(w_0) \varsubsetneq W^s_1(w_0) \varsubsetneq \cdots \varsubsetneq W^s_p(w_0) \subset W^s_{p+1}(w_0)=W^s(w_0),$$ with $$W^s_i(w):= \{w=(x,[xx^+])\in W^s(w_0),\ x\in \mathcal{H}_i\} = \{w \in H\o,\ \limsup_{t\to +\infty} \frac{1}{t} \log d_{H\o}(\ph^t(w_0), \ph^t(w)) \leqslant \chi^-_i\}.$$ The tangent distribution to $W^s_i(w)$ is precisely $\oplus_{k=0}^i E^s_k$. (Recall that the subspaces $E^s_0$ and $E_{p+1}^s$ can be $\{0\}$, in which case $W^s_0(w_0)=\{w\}$, and $W^s_{p}(w_0)=W^s_{p+1}(w_0)=W^s(w_0)$.)
Obviously, the last corollary can be stated also for an approximately regular point $x^-\in\doo$ and the corresponding unstable manifold $$W^u(x^-) = \{w\in H\o,\ \ph^{-\infty}(w)=x^-\}.$$
Non-strict convexity, non-$C^1$ points {#extension}
--------------------------------------
We now explain how to extend corollary \[maincorollary\] to an arbitrary convex set. Let $\o$ be *any* convex proper open subset of $\R\P^n$ and choose a point $x^+\in\doo$. The flow $\ph_{x^+}^t$ is well defined, the definition of approximate regularity given in section \[sectionlyapunovboundary\] still makes sense and the results we achieve before can be extended to this general convex set by using the following easy lemma.
Let $\o$ be any proper convex subset of $\R\P^n$ and $x\in\doo$.
- The maximal flat $$\mathcal{F}(x)=\{y\in\doo,\ [xy]\subset\doo\}$$ containing $x$ in $\doo$ is a closed convex subset of a projective subspace $\R\P^q$, for some $0\leqslant q\leqslant n-1$, whose interior is open in this $\R\P^q$ when $\mathcal{F}(x)$ is not reduced to $\{x\}$.
- The set of $C^1$ directions $$\mathcal{D}(x) = \{0\}\cup\{v\in T_x\doo\smallsetminus\{0\},\ \doo\ \text{is differentiable in the direction}\ v \}$$ is a subspace of $T_x\doo$.
The set $\mathcal{F}(x)$ is obviously closed. It is convex because of the convexity of $\o$. The projective subspace $\R\P^q$ is the one spanned by $\mathcal{F}(x)$. The second point is just a consequence of convexity.
Choose a direction $v\in T_x\doo$ in which the boundary $\doo$ is not differentiable and any vector $u\not \in T_x\doo$. We can consider the $2$-dimensional convex set $C_{v}(w)=\o \cap (\R.v \oplus \R.u)$. As we have seen in the introduction, for two distinct geodesic lines of $C_v(u)$ ending at $x$, the distance between them does not tend to $0$. Hence the negative Lyapunov exponent $\chi^s$ of such a geodesic, if it were defined, would be $\chi^s=0$; it is coherent with the fact that $\alpha(v) = 1$ and the relation $\chi^s = -2 + \frac{2}{\alpha(v)}$.\
We can now consider the subspace $\mathcal{D}(x)$ of $\C^1$ directions and the convex set $C_{x}(u) = \o \cap (D(x) \oplus \R.u)$ for an arbitrary vector $u\not \in T_x\doo$. For example, the stable manifold $\H^s_{x^+}(x)$ of $\ph_{x^+}^t$ at $x$ is the set $$\H^s_{x^+}(x) = C_x(xx^+)\cap \H_{x^+}(x).$$ The boundary $\partial C_x(u)$ is $\C^1$ at Lebesgue-almost every point $x^-$, so all we did before is relevant along Lebesgue almost-all geodesic $(x^-x^+)$. We just have to be careful for those vectors in $\span\ \mathcal{F}(x)$ which were not considered before: in such a direction $v$, the boundary is obviously $+\infty$-approximately regular, and as we have seen in the introduction, the distance between two geodesics of $C_v(u)$ with origin on the same horosphere and ending at $x^+$ goes to $0$ as $e^{-2t}$.\
As a consequence, we get that corollary \[maincorollary\] is valid for any Hilbert geometry:
\[maincorollary2\] Let $\o$ be a convex proper open subfset of $\R\P^n$ and fix $o\in\o$. Assume $x^+\in\doo$ is approximately regular with exponents $+\infty \geqslant \alpha_1 > \cdots > \alpha_p \geqslant 1$ and filtration $$\{0\} = H_0 \varsubsetneq H_1 \varsubsetneq \cdots \varsubsetneq H_p = T_{x^+}\doo.$$ Then the horosphere $\H=\H_{x^+}(o)$ about $x^+$ passing through $o$ admits a filtration $$\{o\}=\H_0 \varsubsetneq \H_1 \varsubsetneq \cdots \varsubsetneq \H_p = \H,$$ given by $\H_i=\{x\in\H\cap (H_i\oplus \R.ox^+)\},\ \leqslant i \leqslant p,$ such that $$\mathcal{H}_i\smallsetminus \mathcal{H}_{i-1} = \{x \in \H\smallsetminus\{o\},\ \lim_{t\to +\infty} \frac{1}{t} \log \d(\ph_{x^+}^t(o), \ph_{x^+}^t(x)) = \chi^s_i\},$$ with $\chi^s_i = -2+\frac{2}{\alpha_i}$.
In this last corollary, if $\mathcal{F}(x^+)$ is not reduced to $x^+$, then the subspace $H_1$ itself admits a filtration $\{0\} \varsubsetneq \span\ \mathcal{F}(x^+) \varsubsetneq H_1$; $H_1\smallsetminus \span\ \mathcal{F}(x^+)$ consists of these vectors $v$ with Lyapunov exponent $\alpha(v) = +\infty$ which are not in $\span\ \mathcal{F}(x^+)$, that is, the directions in which $\doo$ is not flat, but infinitesimally flat. Of course this also provides a filtration of $\H_1$.\
Similarly, if $\mathcal{D}(x^+)$ is not all of $T_{x^+}\doo$, we can refine the filtration into $$\cdots \varsubsetneq H_{p-1} \varsubsetneq \mathcal{D}(x^+) \varsubsetneq H_p = T_{x^+}\doo.$$ The subspace $\mathcal{D}(x^+)$ is precisely the tangent space to the stable manifold $\H^s_{x^+}(o)$ of $\ph^t_{x^+}$ at $o$, and $\H$ admits a subfiltration $$\cdots \varsubsetneq \H_{p-1} \varsubsetneq \H^s_{x^+}(o) \varsubsetneq \H_p = \H.$$
Examples
========
I do not know what can be said in general about the notion of approximate-regularity for a given strictly convex set $\o$ with $\C^1$-boundary. We can relate this with Alexandrov’s theorem which says that the boundary $\doo$ of $\o$ is $\C^2$ Lebesgue-almost everywhere. This implies that for almost every point $x\in\doo$, we have $\underline{\alpha}(v)\geqslant 2$ for all vectors $v\in T_x\doo$. It might be interesting for example to know if $\doo$ is approximately regular at almost every point.\
Here I give some more properties of approximate-regularity and study the case of divisible convex sets. In particular I show that in this case $\doo$ is approximately regular at almost every point with the same Lyapunov exponents.
Duality and approximate regularity
----------------------------------
### Legendre transform {#sectionlegendre}
Pick a function $f\in \mathtt{Cvx}(n)$. Since $f$ is $\C^1$ and strictly convex, the gradient $$\nabla: x\in U \longmapsto \nabla_x f = \left(\frac{\partial f}{\partial x_1}(x),\cdots, \frac{\partial f}{\partial x_n}(x)\right)$$ is an injective map onto a convex subset $V$ of $\R^n$. Using the gradient, a point $x$ can thus be defined by its coordinates $(x_1,\cdots,x_n)$ or by its “dual” coordinates $\left(\frac{\partial f}{\partial x_1}(x),\cdots, \frac{\partial f}{\partial x_n}(x)\right)$.\
The [**Legendre transform**]{} of $f$ is the function $f^*$ defined by $$f(x) + f^*(\nabla_x f) = \langle \nabla_x f, x \rangle.$$ It happens that the transform $f\longmapsto f^*$ is an involution of $\mathtt{Cvx}(n)$. We will see in the next section that it appears naturally when one considers the dual of a convex set. Our goal in the next section is to make a link between the shape of the boundary of the convex set and the one of its dual. For this, we study here the link between the approximate regularity of $f$ and of its Legendre transform $f^*$. I am not very familiar with Legendre transform and I did not manage to prove the next lemma in higher dimensions; but it is probably true...
\[legendre\] Assume $f\in \mathtt{Cvx}(1)$ is approximately $\alpha$-regular, $\alpha\in[1,+\infty]$. Then the Legendre transform $f^*$ of $f$ is approximately $\alpha^*$-regular with $$\frac{1}{\alpha^*} + \frac{1}{\alpha} = 1.$$
We only prove the proposition when $\alpha\in (1,+\infty)$. The Legendre transform of $f\in \mathtt{Cvx}(1)$ is given by $$f^*(f'(x))=xf'(x) - f(x).$$ By considering $f(x)+f(-x)$ instead, we can assume that $f$ is an even function, so that approximate $\alpha$-regularity gives $$\lim_{x\to 0^+} \frac{\log f(x)}{\log x} = \alpha.$$ Since $f(0)=f(x)-xf'(x) + o(x)$, we get $$\lim_{x\to 0^+} \frac{\log f'(x)}{\log x} = \alpha-1.$$ We need to understand the limit $$\lim_{x\to 0^+} \frac{\log xf'(x)-f(x)}{\log f'(x)}.$$ Fix $\epsilon>0$. There is some $x>0$ such that for $0\leqslant t\leqslant x$, we have $$\label{ggg}
t^{\alpha-1+\epsilon} \leqslant f'(t) \leqslant t^{\alpha-1-\epsilon}.$$ Remark that $$xf'(x) - f(x) = xf'(x) - \int_0^x f'(t) dt.$$ From (\[ggg\]), that means the value of $xf'(x) - f(x)$ is in between the two areas between $0$ and $x$ delimited by the line $y=f'(x)$ above and, respectively, the curves $t\mapsto t^{\alpha-1-\epsilon}$ and $t\mapsto t^{\alpha-1+\epsilon}$ below: $$f'(x)^{\frac{1}{\alpha-1-\epsilon}}f'(x) - \int_0^{f'(x)^{\frac{1}{\alpha-1-\epsilon}}} t^{\alpha-1-\epsilon}\ dt \leqslant xf'(x) - f(x) \leqslant xf'(x) - \int_0^x t^{\alpha-1+\epsilon}\ dt.$$ Hence $$(f'(x))^{\frac{\alpha-\epsilon}{\alpha-1-\epsilon}} - \frac{1}{\alpha-\epsilon} (f'(x))^{\frac{\alpha-\epsilon}{\alpha-1-\epsilon}} \leqslant xf'(x) - f(x) \leqslant xf'(x) - \frac{1}{\alpha+\epsilon} x^{\alpha+\epsilon}.$$ Using (\[ggg\]) again, we get $$\frac{\alpha-\epsilon-1}{\alpha-\epsilon}(f'(x))^{\frac{\alpha-\epsilon}{\alpha-1-\epsilon}} \leqslant xf'(x) - f(x) \leqslant x^{\alpha-\epsilon} - \frac{1}{\alpha+\epsilon} x^{\alpha+\epsilon} \leqslant x^{\alpha-\epsilon}.$$ So $$\frac{\alpha-\epsilon}{\alpha-1} = (\alpha -\epsilon)\lim_{x\to 0^+} \frac{\log x}{\log f'(x)}\leqslant \lim_{x\to 0^+} \frac{\log xf'(x)-f(x)}{\log f'(x)} \leqslant \frac{\alpha-\epsilon}{\alpha-1-\epsilon}.$$ Since $\epsilon$ is arbitrary small, we get the result.
### Dual convex set
To each convex set $\o\subset\R\P^n$ is associated its dual convex set $\o^*$. To define it, consider one of the two convex cones $C \subset \R^{n+1}$ whose trace is $\o$. The dual convex set $\o^*$ is the trace of the dual cone $$C^* = \{f\in (\R^{n+1})^*,\ \forall x\in C,\ f(x)> 0\}.$$ The cone $C^*$ is a subset of the dual of $\R^{n+1}$ but of course, it can be seen as the subset $$\{y\in \R^{n+1},\ \forall x\in C,\ \langle x,y \rangle \geqslant 0\}.$$
The set $\o^*$ can be identified with the set of projective hyperplanes which do not intersect $\overline\o$: to such a hyperplane corresponds the line of linear maps whose kernel is the given hyperplane. For example, we can see the boundary of $\o^*$ as the set of tangent spaces to $\doo$. In particular, when $\o$ is strictly convex with $\C^1$ boundary, there is a homeomorphism between the boundaries of $\o$ and $\o^*$: to the point $x\in\doo$ we associate the (projective class of the) linear map $x^*$ such that $\ker x^* = T_x\doo$.\
In the following we would like to link the shape of $\doo$ and $\doo^*$. We will work in $\R^{n+1}$ with the cones $C$ and $C^*$ where it is more usual to make computations. Choose a point $p\in\partial C$ and fix a Euclidean structure on $\R^{n+1}$ and an orthonormal basis $(u_1,\cdots,u_{n+1})$ so that $p=u_1+u_{n+1}$, $T_p\partial C = \span \{p,u_2,\cdots,u_{n+1}\}$ and $C \subset \{x=(x_1,\cdots,x_{n+1}),\ x_{n+1}>0\}$. We identify $\o$ with the intersection $C\cap\{x_{n+1}=1\}$ and the tangent space $T_p\doo$ is $p+\span \{u_2,\cdots,u_{n+1}\}$.\
Call $f : U\subset T_p\doo \longrightarrow \R$ the local graph of $\doo$ at $p$, such that, around $p$, $$\doo = \{(1-f(x_2,\cdots,x_n),x_2,\cdots,x_n,1)\}.$$
Around $p^*=(1,0,\cdots,0,-1)$, the boundary $\doo^*$ is given by $$\doo^* = \{(1,\l_2,\cdots,\l_n,-1-f^*(\l_2,\cdots,\l_n))\},$$ where $f^*$ is the Legendre transform of $f$. In other words, the local graph of $\doo^*$ at $p$ is given by the Legendre tranform $f^*$ of $f$.
Take a point $x = (1-f(x_2,\cdots,x_n),x_2,\cdots,x_n,1)\in\doo$ and call $x_{2n}=x_2u_2+\cdots+x_nu_n\in \span \{u_2,\cdots,u_{n+1}\}$ its projection on $\span \{u_2,\cdots,u_{n+1}\}$. Call $F : T_p\doo \longrightarrow \R^{n+1}$ the map given by $$F(p+x_{2n}) = p + x_{2n} - f(x_{2n})u_1 = (1-f(x_2,\cdots,x_n),x_2,\cdots,x_n,1).$$ The tangent space of $\doo$ at $x$ is then given by $$T_x\doo = x + d_xF(\span \{u_2,\cdots,u_{n+1}\}).$$ But, for $h\in\span \{u_2,\cdots,u_{n+1}\}$, we have $d_xF(h) = -d_xf(h) + h$. Hence $$T_x\doo = x + \{ h - d_xf(h),\ h\in \span \{u_2,\cdots,u_{n+1}\} \}.$$ Now the dual point of $x$ is the linear map $x^*=(x_1^*,\cdots,x_{n+1}^*)$ such that $x^*(x)=0$, $x^*(T_x\doo)=0$ and $x^*(u_1)=1$. (This last condition is just a normalization condition, since there is a line of corresponding linear maps.) The third condition gives $x_1^* = 1$. The second implies that for any $h\in\span \{u_2,\cdots,u_{n+1}\}$, $$0 = x^*(h - d_xf(h))=\langle x^* - \nabla_x f, h \rangle;$$ hence $x_{2n}^* = \nabla_x f$, that is, $x_i^* = \frac{\partial f}{\partial u_i}(x_{2n}),\ i=2,\cdots, n$. Finally, the first condition gives $$1-f(x_2,\cdots,x_n) + \langle \nabla_x f, x_{2n}\rangle + x_{n+1}^* = 0,$$ so $$x_{n+1}^* = -1-(\langle \nabla_x f, x_{2n}\rangle - f(x_{2n})).$$ By considering the set of variables $(\l_2,\cdots,\l_n)= \left(\frac{\partial f}{\partial u_2},\cdots, \frac{\partial f}{\partial u_n}\right)$, one finally gets $$x^* = (1,\l_2,\cdots,\l_n,-1-f^*(\l_2,\cdots,\l_n)).$$
From lemma \[legendre\], we get the following
Assume $\doo\subset\R\P^2$ is approximately $\alpha$-regular at the point $x$. Then $\doo^*$ is approximately $\alpha^*$-regular at the point $x^*$ with $$\frac{1}{\alpha^*} + \frac{1}{\alpha} = 1.$$
Hyperbolic isometries
---------------------
If $\o$ is strictly convex with $\C^1$ boundary, the group of isometries $Isom(\o,\d)$ of the Hilbert geometry $(\o,\d)$ consists of those projective transformations which preserve the convex set $\o$: $$Isom(\o,\d) = \{g\in PGL(n+1,\R),\ g(\o)=\o\}.$$ As in the hyperbolic space, isometries can be classified into three types, elliptic, parabolic and hyperbolic. This is proved in the forthcoming paper [@cramponmarquis].\
A hyperbolic isometry $g$ fixes exactly two points $x_g^+$ and $x_g^-$ on $\doo$. The point $x_g^+$ is the attractive point of $g$, $x_g^-$ is the repulsive point of $g$ : for any point $x\in\overline{\o}\smallsetminus\{x_g^-,x_g^+\}$, $\lim_{n\to \pm\infty} g^n(x)=x_g^{\pm}$. These two points are the eigenvectors associated to the biggest and smallest eigenvalues $\l_0$ and $\l_{p+1}$ of $g$. The isometry $g$ acts as a translation of length $\log \frac{\l_{p+1}}{\l_0}$ on the open segment $]x_g^-x_g^+[$. The following result is proved in [@crampon]:
\[lyapunovperiodic\] Let $g$ be a periodic orbit of the flow, corresponding to a hyperbolic element $g\in\Gamma$. Denote by $\l_0 > \l_1 > \cdots > \l_{p} > \l_{p+1}$ the moduli of the eigenvalues of $g$. Then
- $\g$ is regular and has no zero Lyapunov exponent;
- the Lyapunov exponents $(\eta_i(g))$ of the parallel transport along $\g$ are given by $$\eta_i(g) = -1 + 2\ \frac{\log \l_0-\log \l_i}{\log \l_{0} -\log \l_{p+1}},\ i=1\cdots p;$$
- the sum of the parallel Lyapunov exponents is given by $$\eta(g) = (n+1)\frac{\log \l_0 +\log \l_{p+1}}{\log \l_0 - \log \l_{p+1}}.$$
As a consequence of the results before, we see that, if $g$ is a hyperbolic isometry, the boundary $\doo$ is Lyapunov regular at the points $x_g^-$ and $x_g^+$, with Lyapunov exponents $$\label{exposanthyp}
\alpha_i = \frac{\log \l_{0} -\log \l_{p+1}}{\log \l_0-\log \l_i},\ i=1\cdots p.$$
The isometry $g\in Isom(\o,\d)$ acts on the dual convex set $\o^*$ by $g.y = (^tg)^{-1}(y)$. To $g\in Isom(\o,\d)$, we thus associate the isometry $g^* = (^tg)^{-1}\in Isom(\o^*,d_{\o^*})$. The dual points to $x_g^-$ and $x_g^+$ are respectively the points $x_{g^*}^+$ and $x_{g^*}^-$, at which $\doo^*$ is Lyapunov regular with Lyapunov exponents $$\alpha_i^* = \frac{\log \l_{0} -\log \l_{p+1}}{\log \l_i-\log \l_{p+1}},\ i=1\cdots p:$$ this corresponds to what gives formula (\[exposanthyp\]) for the isometry $g^{-1}$. Remark that, as expected, we have $$\frac{1}{\alpha_i^*} + \frac{1}{\alpha_i} = 1,\ i=1\cdots p.$$
Divisible convex sets
---------------------
The convex set $\o$ is said to be divisible if it admits a discrete cocompact subgroup $\G$ of projective isometries. By Selberg lemma, we can assume $\G$ has no torsion and the quotient $M=\o/\G$ is then a smooth manifold. The first example of divisible convex set is the ellipsoid, that is, the hyperbolic space. Benoist proved in [@benoistcv1] that, for a divisible convex set $\o$, the following properties were equivalent:\
- $\o$ is strictly convex;
- $\doo$ is of class $\C^1$;
- $(\o,\d)$ is Gromov-hyperbolic.\
Apart from the ellipsoid, various examples of strictly convex divisible sets have been given. Some can be constructed using Coxeter groups ([@kav], [@benoistqi]), some by deformations of hyperbolic manifolds (based on [@johnsonmillson] and [@koszul], see also [@goldman] for the $2$-dimensional case); we should also quote the exotic examples of Kapovich [@kapo] of divisible convex sets in all dimensions which are not quasi-isometric to the hyperbolic space (Benoist [@benoistqi] had already given an example in dimension 4).\
In what follows, we are given a compact manifold $M=\o/\G$, quotient of a strictly convex set $\o$ with $\C^1$ boundary.\
### Regularity of the boundary
Benoist proved that the geodesic flow on $HM$ has the Anosov property, with decomposition $$THM = \R.X \oplus E^u \oplus E^s.$$ That means there exist constants $C,\alpha>0$ such that for any $ t\geqslant 0$, $$\|d\ph^t(Z^s)\| \leqslant C e^{-\alpha t} \|Z^s\|,\ Z^s\in E^s,$$ $$\|d\ph^{-t}(Z^u)\| \leqslant C e^{-\alpha t} \|Z^u\|,\ Z^u \in E^u.$$
As a consequence, we get that the boundary $\doo$ is $\C^{\alpha}$ and $\beta$-convex for some $1<\alpha\leqslant 2 \leqslant \beta<+\infty$. This had already been remarked by Benoist in [@benoistcv1], and Guichard proved that the biggest $1<\alpha \leqslant 2$ and smallest $2 \leqslant\beta$ one can take are related to the group $\G$: $$\alpha(\o) = \sup_{g\in\G} \frac{\log \l_{0}(g) -\log \l_{p+1}(g)}{\log \l_0(g)-\log \l_1(g)}.$$ $$\beta(\o) = \inf_{g\in\G} \frac{\log \l_{0}(g) -\log \l_{p+1}(g)}{\log \l_0(g)-\log \l_p(g)}.$$ Guichard result is stated in another form: the dual group $\G^*$ also acts cocompactly on the dual convex set $\o^*$, providing another compact manifold $M^*=\o^*/\G^*$; Guichard showed that $\alpha(\o)=\alpha(\o^*)$ and $\beta(\o)=\beta(\o^*)$. In [@cramponmarquis], we will give another proof of Guichard result that we also extend to some non cocompact actions.\
The case of the ellipsoid is a particular one. Indeed, the following facts are equivalent:
- $\o$ is an ellipsoid;
- $\alpha(\o) = \beta(\o) = 2$;
- $\G$ is not Zariski-dense in $SL(n+1,\R)$;
- the parallel transport on $HM$ is an isometry;
- the Lyapunov exponents are to $-1$, $0$ and $1$, corresponding to the Anosov decomposition $THM = E^s\oplus \R.X \oplus E^u$.
### Ergodic measures
Let $\Lambda(H\o)$ be the set of regular points on $H\o$, which is obviously $\G$-invariant, and call $\Lambda$ the projection of $\Lambda(H\o)$ on $HM$. From Oseledets’ theorem, we know that for any invariant measure $m$ of the geodesic flow on $HM$, $\Lambda$ has full $m$-measure; in particular, Lyapunov exponents are defined almost everywhere. If $m$ is an ergodic measure, that is, such that invariant sets have zero or full measure, then Lyapunov exponents are constant almost everywhere: to each ergodic measure $m$ we can thus associate a number $p=p(m)$ and its parallel Lyapunov exponents $\eta_1(m) < \cdots < \eta_p(m)$.\
Kaimanovich [@kaimanovich] explained how to associate in a one-to-one way to each invariant probability measure $m$ on $HM$ a $\G$-invariant Radon measure $M=M(m)$ on the space of oriented geodesics of $\o$ given by $\partial^2\o=\doo\times\doo\smallsetminus\Delta$, where $\Delta = \{(x,x),\ x\in\doo\}$. If $m$ is ergodic, Oseledets’ theorem implies that for $M$-almost all $(x,y)\in\partial^2\o$, the geodesic from $x$ to $y$ is regular with parallel Lyapunov exponents $\eta_1(m) < \cdots < \eta_p(m)$; thus, for $M$-almost all $(x,y)\in\partial^2\o$, the boundary $\doo$ is Lyapunov regular at $x$ and $y$ with Lyapunov exponents $\alpha_i(m),\ 1\leqslant i \leqslant p$, given by $$\alpha_i(m) = \frac{2}{\eta_i(m)+1}.$$ By projecting on the first and second coordinates in $\partial^2\o$, we get for each ergodic measure $m$ two $\G$-invariant sets $\doo^-(m)$ and $\doo^+(m)$ where the boundary $\doo$ is Lyapunov regular with the same Lyapunov exponents $\alpha_i(m),\ 1\leqslant i \leqslant p$. Recall that the action of $\G$ on $\doo$ is minimal, that is, every orbit is dense; the sets $\doo^-(m)$ and $\doo^+(m)$ are then dense subsets of $\doo$.\
The diversity of invariant measures can then give an idea of the complexity of the boundary of a divisible convex set. Here are some examples.\
The easiest examples of ergodic measures are the Lebesgue measures $l_{g}$ supported by a closed orbit $g$, associated to a conjugacy class of a hyperbolic element $g\in \G$. The corresponding set of full $M(l_{g})$-measure is precisely the orbit of $(x_g^-,x_g^+)$ under $\G$ while its projections $\doo^-(m)$ and $\doo^+(m)$ are the $\G$-orbits of $x_g^-$ and $x_g^+$.\
Other examples are provided by Gibbs measure which are equilibrium states of Hölder continuous potentials $f:HM \longrightarrow \R$: the Gibbs measure of $f$ is the unique invariant probability measure $\mu_f$ such that $$h_{\mu_f} + \int f\ d\mu_f = \sup\{h_{m} + \int f\ dm,\ m\ \text{invariant probability measure}\}.$$ Two distinct potentials $f$ and $g$ have the same equilibrium states if and only if their difference is invariant under the flow. The corresponding measure $M_f$ on $\partial^2\o$ can always be written as $M_f = F M_f^s\times M_f^u$, where $F$ is a continuous function on $\partial^2\o$, and $M_f^s$ and $M_f^u$ are two finite measures on $\doo$. The three objects are determined by the potential; in particular, $M_f^u$ and $M_f^s$ are given by the Patterson-Sullivan construction, associated to the potentials $f$ and $\sigma * f$, where $\sigma$ is the flip map.\
Among them are two particular measures. The first one is the Bowen-Margulis measure $\mu_{BM}$ which is the measure of maximal entropy of the flow, that is, the equilibrium state associated to the potential $f\equiv 0$. The corresponding measure $M_{BM}$ is given by $$dM_{BM}(\xi^+,\xi^-) =e^{2\delta(\xi^+|\xi^-)_o} d\mu_{o}^2(\xi^+,\xi^-),$$ where $\mu_{o}$ is the Patterson-Sullivan measure at an arbitrary point $o\in\o$, and $(\xi^+|\xi^-)_o$ is the Gromov product $\xi^+$ and $\xi^-$ based at the point $o$. In [@crampon], I had proved that $\eta(\mu_{BM}) = \sum \eta_i(\mu_{BM}) = n-1$. Thus, we get that $\mu_o$-almost every point of $\doo$ is Lyapunov regular with exponents $\alpha_i,\ i=1,\cdots,p$, such that $\alpha = 2(n-1)$, with $\frac{1}{\alpha} = \sum_i \frac{1}{\alpha_i}$. For example, in dimension $2$, $\mu_o$-almost every point of $\doo$ is Lyapunov $2$-regular. A question I could not answer was to know if, in dimension $n\geqslant3$, there was only one parallel Lyapunov exponent if and only if $\o$ was an ellipsoid, that is, $M$ was a hyperbolic manifold.\
The second measure which is important is the Sinai-Ruelle-Bowen (SRB) measure $\mu^+$, which is the equilibrium state associated to the potential $$f^+ = \frac{d}{dt}|_{t=0} \log \det d\ph^t_{|_{E^u}}.$$ It is the only invariant measure whose conditional measures $(\mu^+)^u$ along unstable manifolds are absolutely continuous, and which satisfies the equality in the Ruelle inequality. Recall that the Ruelle inequality relates the entropy of an invariant measure $m$ to the sum of positive Lyapunov exponents $\chi^+=n-1+\eta$ of the flow: $$h_{m} \leqslant \int \chi^+\ dm.$$ Closely related to this measure is the “reverse” SRB measure $\mu^- = \sigma* \mu^+$, which is the equilibrium state of the potential $$f^- = \frac{d}{dt}|_{t=0} \log \det d\ph^t_{|_{E^s}}.$$ The measure $\mu^-$ is the only invariant measure whose conditional measures along stable manifolds are absolutely continuous.\
In the case of the ellipsoid, $\mu^+$, $\mu^-$ and $\mu_{BM}$ all coincide, since $f^+=f^-=0$, and they are all absolutely continuous; indeed, they coincide with the Liouville measure of the flow. When $\o$ is not an ellipsoid, the Zariski-density of the cocompact group $\G$ implies via Livschitz-Sinai theorem that there is no absolutely continuous measure (see [@benoistcv1]). So the three measures are distinct.\
The measures $\mu^+$ and $\mu^-$ have the same entropy $h_{SRB}$ given by $$h_{SRB} = \int \chi^+\ d\mu^+ = -\int \chi^-\ d\mu^-,$$ where $\chi^- = -(n-1)+\eta$ is the sum of negative Lyapunov exponents. In particular, since the Bowen-Margulis measure is the measure of maximal entropy and has entropy $h_{BM}\leqslant n-1$, from Ruelle inequality, we get that the almost sure value $\eta(SRB)$ (with respect to $\mu^+$ or $\mu^-$) of the sum of parallel Lyapunov exponents satisfies $\eta(SRB) < 0.$\
The measure $\mu^+$ corresponds to the measure $M^+$ on $\partial^2\o$ which can be written $M^+ = F^+ M^s \times M^u$, with $M^u$ absolutely continuous, while the measure $\mu^-$ corresponds $M^- = F^- M^u \times M^s$. In particular,
Let $\o$ be a divisible strictly convex set. Then Lebesgue-almost every point of $\doo$ is Lyapunov regular with exponents $$\alpha_i(SRB) = \frac{2}{\eta_i(SRB)+1},\ 1\leqslant i \leqslant p.$$
Since $\doo$ is also $\C^2$ Lebesgue almost-everywhere, we have that $\alpha_i(SRB)\leqslant 2$. When $\o$ is an ellipsoid, we have $p=1$ and $\alpha_1(SRB)=2$. In the other cases, the fact that $\eta(SRB)<0$ implies that $\eta_1(SRB)<0$ hence $\alpha_1(SRB)>2$. In particular, we recover the fact that the curvature of $\doo$ is concentrated on a set of Lebesgue-measure $0$ (see [@benoistcv1]).
About volume entropy {#sectionvolentropy}
====================
The volume entropy of a Riemannian metric $g$ on a manifold $M$ measures the asymptotic exponential growth of the volume of balls in the universal cover $\tilde M$; it is defined by $$\label{entvol} h_{vol}(g)=\limsup_{R\to+\infty} \frac{1}{r} \log vol_g(B(x,R)),$$
where $vol_g$ denotes the Riemannian volume corresponding to $g$. We define the volume entropy of a Hilbert geometry $(\o,\d)$ by the same formula, with respect the Busemann volume.\
Some results are already known: for instance, if $\o$ is a polytope then $h_{vol}(\o,d_{\o})=0$; at the opposite, we have the
\[volentropy\] Let $\o\subset\R\P^n$ be a convex proper open set. If the boundary $\partial\o$ of $\o$ is $\C^{1,1}$, that is, has Lipschitz derivative, then $h_{vol}(\o,d_{\o})=n-1$.
The global feeling is that any Hilbert geometry is in between the two extremal cases of the ellipsoid and the simplex. In particular, the following conjecture is still open:
\[conjectureentropy\] For any $\o\subset \R\P^n$, $$h_{vol}(\o,d_{\o})\leqslant n-1.$$
In [@bbv] the conjecture is proved in dimension $n=2$ and an example is explicitly constructed where $0<h_{vol}<1$. Following their idea for proving theorem \[volentropy\], we can get the
Let $(\o,\d)$ be any Hilbert geometry, and $\mathcal{L}$ a probability Lebesgue measure on $\doo$. Then $$h_{vol}\geqslant \int \frac{2}{\underline{\alpha}}\ d\mathcal{L},$$ where $\underline{\alpha}$ is defined by $$\frac{1}{\underline{\alpha}(x)} = \sum_{i=1}^{n} \frac{1}{\underline{\alpha}_i(x)},\ x\in\doo,$$ with $\alpha_1(x) \geqslant \cdots \geqslant \alpha_n(x)$ being the Lyapunov exponents at $x$, counted with multiplicity.
In [@bbv], the authors proved that $h_{vol}$ also measures the exponential growth rate of the volume of spheres: $$h_{vol} = \limsup_{R\to +\infty} \frac{1}{R} \log vol(S(o,R)),$$ where $S(o,R) = \{x\in\o,\ \d(o,x)=R\}$ is the sphere of radius $R$ about the arbitrary point $o$, and $vol$ denotes the Busemann volume on the sphere. This is well defined because metric balls are convex, hence $S(o,R)$ is $\C^1$ Lebesgue-almost everywhere, so we can consider the Finsler metric induced by $F$ on $S(o,R)$ and define Busemann volume.\
Fix a probability Lebesgue measure $d\xi$ on the set of directions $H_o\o$ about the point $o$, that we identify with the unit sphere $S(o,1)$. The volume of the sphere $S(o,R)$ is then given by $$vol(S(o,R)) = \int f(\xi,R)\ d\xi,$$ where $f(\xi,R) = \det dF(\xi,R)$ with $F$ being the projection about $o$ from $S(o,1)$ to $S(o,R)$. Now, using Jensen inequality and the concavity of $\log$, we get that $$h_{vol} \geqslant \limsup_{R\to +\infty} \int \frac{1}{R} \log f(\xi,R)\ d\xi;$$ then, the dominated convergence theorem gives $$h_{vol} \geqslant \int \limsup_{R\to +\infty} \frac{1}{R} \log f(\xi,R)\ d\xi.$$ But it is not difficult to see that, almost everywhere, $$\limsup_{R\to +\infty} \frac{1}{R} \log f(\xi,R) = \overline{\chi}^+(o,\xi) = \frac{2}{\underline{\alpha}(\xi^+)},$$ with $\xi^+ = \ph^{+\infty}(o,[\xi])$. Hence the result.
As a corollary, we can for example state the following result.
Let $(\o,\d)$ be any Hilbert geometry. If the boundary $\doo$ is $\beta$-convex for some $1 \leqslant \beta < +\infty$ then $h_{vol}>0$.
[**Acknowledgements.**]{} I would like to thank my two advisors Patrick Foulon and Gerhard Knieper for all the useful discussions we had in Strasbourg or in Bochum. A great thanks goes to Aurélien Bosché who helped me fighting against convex functions. Finally, I thank François Ledrappier for his interest in this work and for encouraging me to write everything down in an article.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We define a pseudo quasi-3 design as a symmetric design with the property that the derived and residual designs with respect to at least one block are quasi-symmetric. Quasi-symmetric designs can be used to construct optimal self complementary codes. In this article we give a construction of an infinite family of pseudo quasi-3 designs whose residual designs allow us to construct a family of codes with a new parameter set that meet the Grey Rankin bound.'
author:
- |
Carl Bracken\
School of Mathematical Sciences\
University College Dublin\
Ireland
title: 'Pseudo Quasi-3 Designs and their Applications to Coding Theory'
---
Introduction
============
Designs were first considered for the purpose of designing statistical experiments, but have since found applications in other areas of mathematics. The study of quasi-symmetric designs began with S.S. Shrikhande [@Sh] who considered the duals of such designs. It was shown by McGuire [@MG] that the existence of certain optimal error correcting codes was equivalent to the existence of particular quasi symmetric designs. One method of obtaining these quasi symmetric designs is by taking the derived and residual designs of a quasi-3 design (introduced by Cameron [@C]). Each quasi-3 design will give a pair of quasi symmetric designs and hence two error correcting codes. However, quasi-3 designs are quite rare so as an alternative approach we relax the conditions on the design so that it is not necessarily quasi-3 but can still give us the required quasi symmetric designs. We refer to such a design as pseudo quasi-3 and in this article we give a construction of an infinite family of these designs. This will allow us to obtain new quasi symmetric designs and their corresponding error correcting codes. We begin with some formal definitions.
A t-design with parameters $t-(v,k,\lambda)$ is a pair $D(\mathcal{X},\mathcal{B})$ where $\mathcal{X}$ is a set of points of cardinality $v$, and $\mathcal{B}$ a collection of k-element subsets of $\mathcal{X}$ called blocks, with the property that any $t$ points in $\mathcal{X} $ are contained in precisely $\lambda$ blocks.
Throughout this article we have $t=2$, that is, we are only considering different types of $2$-designs.
Let $\mathcal{B} = \{B_1, B_2, \hdots B_b\}$ be the block set of a $t$-design and let $\mathcal{X} = \{X_1, X_2, \hdots X_v\}$ be its point set. . Then the incidence matrix $(M)$ of this design is the $b \times v$ binary array, with rows indexed by the blocks and columns indexed by the points of the design and the entry $M_{ij} = 1 $ if $ X_j \in B_i $ and $ 0 $ if $ X_j \notin B_i$, i.e., the rows of $M$ are the characteristic vectors of the blocks as subsets of $\mathcal{X}$.
The dual design (denoted $D^T$) of a symmetric design $D$, is obtained by interchanging the point and block sets, while changing the relationship “contained in" to “contains".
A symmetric design is a $2$-design, where the number of points equals the number of blocks.
Symmetric designs are sometimes called square designs as the incidence matrix is a square (not neccesarily a symmetric) matrix. It is shown in [@CvL] that the dual of a symmetric design is always a symmetric design. If $D$ is a symmetric design and $D^T$ is its dual, then the incidence matrix of $D^T$ will be the transpose of the incidence matrix of $D$.
A symmetric design is said to be quasi-3 (for blocks) if it has exactly two distinct triple block intersection sizes, usually denoted $x$ and $y$ (with $x<y$).
A symmetric design can also be defined as quasi-3 for points, if there are only two possible numbers of blocks that contain any three points. It is clear that the dual of a design that is quasi-3 for points will be quasi-3 for blocks. The dual of a quasi-3 design is not necessarily quasi-3, see [@BR] for an example. Throughout this article the term “quasi-3" shall denote quasi-3 for blocks.
A quasi symmetric design is a $2$-design with only two possible block intersection sizes.
Let $D(\mathcal{X},\mathcal{B})$ be a $2-(v,k,\lambda)$ symmetric design, and $B$ a block. The derived design $(D_B)$ of $D(\mathcal{X},\mathcal{B})$ with respect to the block $B$ has block set $\mathcal{B} \setminus \{B\}$ and point set $\{x \in \mathcal{X} \ :\ x \in B \}$. $(D_B)$ is a $2-(k,\lambda,\lambda -1)$ design.
Let $D(\mathcal{X},\mathcal{B})$ be a $2-(v,k,\lambda )$ symmetric design, and $B$ a block. The residual design $(D^B)$ of $D(\mathcal{X},\mathcal{B})$ with respect to the block $B$ has block set $\mathcal{B} \setminus \{B\}$ and point set $\{x \in \mathcal{X} \ :\ x \notin B \}$. $(D^B)$ is a $2-(v-k,k- \lambda ,\lambda )$ design.
A $u \times u$ Hadamard matrix $H$ is an $u \times u$ matrix with entries $1$ and $-1$, such that $$HH^{T}=uI.$$
Hadamard matrices can only exist when $u$ is divisible by $4$ and are conjectured to exist for all such $u$. This has been verified for $u \leq 664$.
A normalised Hadamard matrix is a Hadamard matrix where both the first row and the first column consist entirely of ones.
As multiplying any row or column of a Hadamard matrix by minus one will retain the Hadamard matrix property, as will permuting rows and columns, for any Hadamard matrix there exists an equivalent normalised Hadamard matrix.
Pseudo Quasi-3 Designs
======================
The existence of a quasi-3 design with parameters $2-(4u^2, 2u^2-u, u^2-u)$ and triple block intersection sizes of $ u^2/2-u $ and $ u^2/2-u/2$ implies the existence of two quasi-symmetric designs, taken as the derived and residual designs of the quasi-3 design. One with parameters $2-( 2u^2-u, u^2-u, u^2-u-1)$ and double block intersection sizes of $ u^2/2-u $ and $ u^2/2-u/2$, the other with parameters $2-(2u^2+u, u^2, u^2-u)$ and block intersection sizes $u^2/2-u/2$ and $u^2/2$. When $u$ is a power of $2$ these quasi-3 designs can be constructed (see [@B] and [@JT2]) and hence the quasi symmetric designs are obtained. However, when $u$ is not a power of two the existence of these designs is an open problem. This family of quasi-3 designs could exist for all even $u$, no case has been ruled out or proven. It is possible that these quasi symmetric designs exist even if the quasi-3 design does not. The purpose of this article is to consider symmetric designs with a weaker property than the quasi-3 property which still give a pair of quasi symmetric designs as derived and residual designs. We are thus motivated to define the following.
A pseudo quasi-3 design is a symmetric $2$-design with the property that the block intersection sizes of all triples of blocks that contain one specified block, takes one of two distinct values.
As a convention we shall place the specified block as the first row of the incidence matrix of the design. This means that any triple of rows that contains the first row must have one of two possible intersection numbers. It should be noted that the design may have many blocks that could have this triple intersection property but in order for the design to be pseudo quasi-3, it only needs one. If all blocks are such that the triples containing them have only two possible intersection numbers then the design is quasi-3. We now obtain two quasi symmetric designs as the derived and residual designs of a pseudo quasi-3 design by projection onto the first block. In the sequel we offer a construction of pseudo quasi-3 designs. In the final section we discuss the new families of optimal error correcting codes that can be constructed from the quasi symmetric designs obtained from the pseudo quasi-3 designs.
Pseudo Quasi-3 Designs from Hadamard Matrices
=============================================
In this section we give a construction of pseudo quasi-3 designs with parameters $2-(4u^2, 2u^2-u, u^2-u)$ and triple block intersections of $ u^2/2-u $ and $ u^2/2-u/2$ provided the triple contains the first block, whenever there exist a $u \times u$ Hadamard matrix.
[**The Construction.**]{} Let $H_u$ be any normalised $u \times u$ Hadamard matrix and let $$H_{2u}= \left(\begin{array}{cc} H_u & \ \ H_u \\ H_u & -H_u
\end{array} \right).$$ $H_{2u}$ is also a normalised Hadamard matrix. Let $h_i$ denote the $i^{th}$ row of $H_{2u}$ and define $\tilde{A_i}:=h_i \otimes (h_i)^T$ and $\tilde{S_i}:=h_u \otimes (h_i)^T$, for $0 \leq i \leq 2u-1$, where $\otimes$ denotes the Kronecker product. Let $L$ be the $2u-1 \times 2u-1$ Latin square defined on the symbols $0$ to $2u-1$ with $u$ omitted by taking the cyclic shifts of
$0$ $1$ $2$ $\hdots$ $u-2$ $u-1$ $2u-1$ $2u-2$ $\hdots$ $u+3$ $u+2$ $u+1$
----- ----- ----- ---------- ------- ------- -------- -------- ---------- ------- ------- -------
\
Next we place matrix $A_i$ in position $i$ in $L$ to obtain the matrix $L(A)$. Now we use $L(A)$ and the $S_i$ matrices, as well as their transposes (denoted $S_i^T$) to construct the following matrix which we name $P_u$.
$P_u:=$
[|c|c|c|c|c|c|]{} $0$ & $S_1 $ & $ S_2 $ & $ S_3 $ & $ \hdots $ & $ S_{2u-1}$\
$S_1^T$ &\
$S_2^T$ &\
$S_3^T$ &\
$ \vdots$ &\
$S_{2u-1}^T$ &\
\
In the following theorem we demonstrate that the above matrix is a pseudo quasi-3 design.
Let $P_u$ be constructed as above. Then $P_u$ is the incidence matrix of a pseudo quasi-3 design with parameters $2-(4u^2, 2u^2-u, u^2-u)$ and triple block intersections of $ u^2/2-u $ and $ u^2/2-u/2$ when the triple contains at least one block from the first $2u$ blocks.
Proof:\
First we note that $A_0$ is the all zero matrix and the other $A_i$ matrices and the $S_i$ matrices have $2u$ rows and columns with each row having $u$ $1$’s and $u$ $0$’s. This establishes $v=4u^2$ and $k=2u^2-u$.
Any two rows of a particular $A_i$ or $S_i$ matrix, with $i \neq 0$, intersect in $u$ or $0$ places. If we compare two rows of $P_u$ from the same row of cells, the number of times they agree is determined by the agreement of the corresponding columns of $H_{2u}$.
As $H_{2u}$ is a Hadamard matrix, any two columns agree in $u$ places. However, one of these agreements corresponds to the agreement of any two rows of the all zero matrix. The remaining $u-1$ agreements yield intersections of $u$ in each cell. This gives any two such rows in $P_u$ an intersection of $u(u-1)=u^2-u$. If we compre two rows of $P_u$ from different rows of cells, we have two positions with no intersection due to the presence of exactly one all zero matrix in every row of cells. In the remaining $2u-2$ pairs of cells we have an intersection of $\frac{u}{2}$ as any two rows of $H_{2u}$ have $\frac{u}{2}$ positions in which both rows have $-1$’s. Therefore the intersection of any two such rows is $\frac{u}{2}(2u-2)=u^2-u$. This verifies that $P_u$ is a symmetric design with $\lambda = u^2-u$.
Next we consider the intersection of three blocks in which at least one of the blocks is from the first row of cells. If all three rows are from the first row of cells then the triple intersection is $u$ times the triple intersection of points in the Hadamard 3-design formed by $\left(\begin{array}{c} H_{2u}^* \\ H_{2u}^{*c}
\end{array}\right)$, where $H_{2u}^*$ is $H_{2u}$ with $1$ and $-1$ changed to $0$ and $1$ respectively and $H_{2u}^{*c}$ is the complement of $H_{2u}^*$. These designs have triple intersections of $u-1$, so the three rows of $P_u$ intersect in $\frac{u^2}{2}-u$ places.
If we take two rows from the first row of cells and one from another row, the intersection of the first two rows is in $u-1$ sections of length $u$ at the first or last $u$ positions of each cell. As the first or last $u$ positions of each $A_i$ cell, when $i$ is not $u$ or $0$, has $\frac{u}{2}$ $1$’s and $\frac{u}{2}$ $0$’s we get an intersection of $\frac{u}{2}(u-1)=\frac{u^2}{2}-\frac{u}{2}$ if the $A_0$ cell is not involved and $\frac{u}{2}(u-1)-\frac{u}{2}=\frac{u^2}{2}-u$ if $A_0$ is involved in the intersection.
A similar argument applies when we consider one row from the first row of cells and two rows from some other row of cells.
Finally we consider the case when we take one row from the first row of cells and the other two rows from two different rows of cells. Recall the intersection of any two rows from distinct matrices $A_i$ and $A_j$, with $i$ and $j$ both non zero, is $\frac{u}{2}$. This comprises of an intersection of $\frac{u}{4}$ in the first $u$ positions and an intersection of $\frac{u}{4}$ in the last $u$ positions, unless $|i-j|=u$ in which case we have an intersection of $\frac{u}{2}$ in the first $u$ positions and an intersection of $0$ in the last $u$ positions or an intersection of $0$ in the first $u$ positions and an intersection of $\frac{u}{2}$ in the last $u$ positions.
We claim that, in the position-wise differences of any two rows of $L$, precisely one of the resulting differences has magnitude $u$. To demonstate this claim we reduce all entries in $L$ modulo $u$ and observe that the resulting array is the table of Lee differences for the elements of $\mathbb{Z}_{2u-1}$. As any two elements of $\mathbb{Z}_{2u-1}$ have the same Lee distance to exactly one other element, we have zero appearing exactly once in the position-wise differences of any two rows of this array. Therefore a multiple of $u$ will appear precisely once in the differences of any two rows of the Latin square. With $\pm u$ being the only possibilities the claim is proven.
As the rows of any $S_i$ matrix consists of $u$ $0$’s followed by $u$ $1$’s or vise versa, to consider a triple intersection involving one of the first $2u$ blocks, we need only consider the intersection of the other two blocks when restricted to the first or last $u$ positions in each pair of cells. This gives a triple intersection consisting of three triples of cells with no intersection, one triple of cells with intersection of $0$ or $\frac{u}{2}$ and the remaining $2u-4$ triples intersecting in $\frac{u}{4}$ places. This yields triple intersections of $ u^2/2-u $ and $ u^2/2-u/2$ as required. [$\Box$]{}
We can now take the derived and residual designs with respect to any one of the first $2u$ blocks of $P_u$ to obtain two quasi symmetric designs.
The derived design has parameters $2-(2u^2-u, u^2-u, u^2-u-1)$ and double block intersection sizes of $ u^2/2-u $ and $ u^2/2-u/2$.
The residual design has parameters $2-(2u^2+u, u^2, u^2-u)$ and double block intersection sizes $u^2/2-u/2$ and $u^2/2$.
From this construction we now know that these designs exist whenever there exists a $u \times u$ Hadamard Matrix, which is virtually every $u$ that is a multiple of four. The parameter set for the first of these designs was already achieved in [@Bra] with a different construction. The parameters for the residual design are new.
Application to Coding Theory
============================
An error correcting code $C$ with parameters $(n,M,d)$ over an alphabet $A$ is a subset of $A^n$ with the properties that $|C|=M$ and any two elements of $C$ differ in at least $d$ coordinates. The elements of $C$ are called words and when $A=\{0,1\}$, we say that $C$ is self-complementary if for all $c \in C$, $\bar{c} \in C$ where $\bar{c}$ denotes the binary complement of $c$. In [@MG] the existence of certain quasi-symmetric designs was shown to be equivalent to the existence of a self complementary code meeting the Grey-Rankin bound with equality. The Grey-Rankin bound is an upper bound on $M$ for a fixed $n$ and $d$. It states that $$M \le \frac{8d(n-d)}{n-{(n-2d)}^2}$$ for any $(n, M, d)$ self-complementary code provided the right hand side of the inequality is positive. The following theorem is “Theorem A (part (ii))" from [@MG].
For $n$ even and $n- \sqrt{n}<2d<n $ there exists a self-complementary $(n,M,d)$ code with $M=8d(n-d)/(n-{(n-2d)}^2)$ if and only if $d$ is even and there exists a quasi-symmetric $2-(n,d,\lambda ) $ design with block intersection sizes $d/2$ and $(3d-n)/2$, where $\lambda =d(d-1)/(n-{(n-2d)}^2).$
In [@Bra] the infinite family of error correcting codes with parameters $(2u^2-u, 8u^2, u^2-u)$ were constructed using $u \times u$ Hadamard matrices. These parameters satisfy the Grey-Rankin bound with equality, therefore the above theorem implies the existence of quasi symmetric designs with parameters $2-( 2u^2-u, u^2-u, u^2-u-1)$. The $4u^2-1$ words of weight $u^2-u$ in this code form the incidence matrix of the quasi symmetric design.
The residual design of $P_u$ constructed in the last section has parameters $2-(2u^2+u, u^2, u^2-u)$. If we take its incidence matrix and its complement along with the all-zero and all-one words, we have a self complementary code with parameters $(2u^2+u, 8u^2, u^2)$. Again, these parameters meet the Grey-Rankin bound and hence the codes in this new family are optimal.
Closing Remarks and Open Proplems
=================================
In this article we have shown that there exists quasi symmetric designs with parameters $2-( 2u^2-u, u^2-u, u^2-u-1)$ and $2-(2u^2+u, u^2, u^2-u)$ whenever there exists a $u \times u$ Hadamard matrix. However, these parameters are permissable for any even $u$, so when $u$ is not divisible by $4$ another construction is needed. The only examples of quasi symmetric designs with the above parameters and $u$ not a multiple of $4$ were constructed in [@BMW] with $u=6$. This pair of designs were not taken as derived and residual designs of a pseudo quasi-3 design and the existence of such a design is open.
[**Open Problem 1.**]{} Does there exist a pseudo quasi-3 design with parameters $(144, 66, 30)$ and triple block intersections of $12$ and $15$?
It can be seen from the construction in [@BMW] that the non-existence of such a design implies the non-existence of the projective plane of order $12$.
Even if we cannot construct any more pseudo quasi-3 designs, it may be possible to obtain the remaining quasi symmetric designs by other constructions.
[**Open Problem 2.**]{} Construct quasi symmetric designs with parameters $2-( 2u^2-u, u^2-u, u^2-u-1)$ and $2-(2u^2+u, u^2, u^2-u)$ when $u>6$ and not divisible by $4$.
[00]{}
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We report the full complex dielectric function of high-purity $\textrm{Ba}_{0.68}\textrm{K}_{0.32}\textrm{Fe}_2\textrm{As}_2$ single crystals with $T_{\mathrm{c}}=38.5\ \textrm{K}$ determined by wide-band spectroscopic ellipsometry at temperatures $10\leq T\leq300\ \textrm{K}$. We discuss the microscopic origin of superconductivity-induced infrared optical anomalies in the framework of a multiband Eliashberg theory with two distinct superconducting gap energies $2\Delta_{\mathrm{A}}\approx6\ k_{\mathrm{B}}T_{\mathrm{c}}$ and $2\Delta_{\mathrm{B}}\approx2.2\ k_{\mathrm{B}}T_{\mathrm{c}}$. The observed unusual suppression of the optical conductivity in the superconducting state at energies up to $14\ k_{\mathrm{B}}T_{\mathrm{c}}$ can be ascribed to spin-fluctuation–assisted processes in the clean limit of the strong-coupling regime.'
author:
- 'A. Charnukha'
- 'O. V. Dolgov'
- 'A. A. Golubov'
- 'Y. Matiks'
- 'D. L. Sun'
- 'C. T. Lin'
- 'B. Keimer'
- 'A. V. Boris'
title: 'Eliashberg approach to superconductivity-induced infrared anomalies in $\bf\textrm{Ba}_{0.68}\textrm{K}_{0.32}\textrm{Fe}_2\textrm{As}_2$'
---
The discovery of iron-based superconductors [@kamihara] has generated significant experimental and theoretical effort to unravel the mechanism of high-temperature superconductivity in these compounds. This effort has yielded a comprehensive experimental description of the electronic structure at the Fermi level, which includes multiple Fermi surface sheets in a good agreement with density functional calculations [@Paglione_review_2010; @Johnston_Review_2010]. Partial nesting between at least two of these sheets leads to a spin-density-wave instability that renders the metallic parent compounds antiferromagnetic. In the superconducting compounds spin fluctuations become the source of strong repulsive interband interactions and might give rise to superconductivity with different signs on these sheets [@Mazin_NatureInsights_2010].
The most incisive experimental data have been obtained on high-quality single-crystals of iron pnictides with the so-called 122 structure, for instance $\textrm{BaFe}_2\textrm{As}_2$, with K substituted for Ba or Co for Fe, resulting in hole and electron doping, respectively. In all of these materials five Fermi surface sheets have been identified in calculations and confirmed by numerous independent experimental studies [@Paglione_review_2010; @Johnston_Review_2010]: in the reduced Brillouin-zone scheme these are three hole pockets at the $\Gamma$ point and two almost degenerate electron pockets at the $X$ point with nesting between hole and electron sheets. Among all of these $122$ materials, the optimally hole-doped compound $\textrm{Ba}_{0.68}\textrm{K}_{0.32}\textrm{Fe}_2\textrm{As}_2$ (BKFA) has the highest transition temperature of $38.5\ \textrm{K}$. Due to their exceptional quality, crystals of this compound are well suited as a testbed for theoretical models. A four-band Eliashberg theory with strong interband couping has already proven successful in accounting for the transition temperature, as well as the temperature dependence of the free energy and superconducting gaps of this compound [@PhysRevLett.105.027003]. This analysis has made clear that a satisfactory description of the bulk thermodynamical properties in the superconducitng state can only be obtained via strong coupling to spin fluctuations or other bosons with spectral weight below $50\ \textrm{meV}$.
![\[fig:firexperiment\] Real part of the (a) optical conductivity and (b) dielectric function in the far-infrared spectral region. Two characteristic superconductivity energy scales are present: $6\ k_{\mathrm{B}}T_{\mathrm{c}}$ and $14\ k_{\mathrm{B}}T_{\mathrm{c}}$.](Fig1.pdf){width="3.4in"}
In this Letter we extend the approach of Ref. to describe the far-infrared properties of the same BKFA single crystal. We show that major characteristic features of superconductivity can be explained within a strong-coupling Eliashberg approach with two distinct values of the superconducting energy gap $2\Delta_{\mathrm{A}}\approx6\ k_{\mathrm{B}}T_{\mathrm{c}}$ and $2\Delta_{\mathrm{B}}\approx2.2\ k_{\mathrm{B}}T_{\mathrm{c}}$, in quantitative agreement with angle-resolved photoemission [@PhysRevB.79.054517; @PhysRevLett.105.117003; @PhysRevB.83.020501], scanning-tunneling microscopy [@Shan-Wen_STM_2011] and specific-heat measurements [@PhysRevLett.105.027003]. We also demonstrate that within this approach the qualitative differences in the infrared spectra of electron- and hole-doped 122 compounds are reproduced by strong-coupling calculations in the clean and dirty limits (weak and strong impurity scattering), respectively.
The optimally-doped BKFA single crystals were grown in zirconia crucibles sealed in quartz ampoules under argon atmosphere [@Lin_BKFA_growth_2010]. From DC resistivity, magnetization and specific-heat measurements we obtained $T_{\mathrm{c}}=38.5\pm0.2\ \textrm{K}$. The sample surface was cleaved prior to every optical measurement. The full complex dielectric function $\varepsilon(\omega)$ was obtained in the range $0.01-6.5\ \textrm{eV}$ using broadband ellipsometry, as described in Ref. [@boris:027001]. In this work we focus on the itinerant charge carrier response contained within the far-infrared spectral range measured at the infrared beamline of the ANKA synchrotron light source at Karlsruhe Institute of Technology, Germany. The contribution of the interband transitions has been eliminated based on a dispersion analysis in the entire spectral range [@supplementary.material.prl].
![\[fig:scattering\] Optical scattering rate obtained from the experimental data at $41$ and $10\ \textrm{K}$ within the extended-Drude model, with the contribution of the interband transitions subtracted (blue and black lines, respectively) and at $41\ \textrm{K}$ without subtraction (green line). Dash-dotted line indicates the saturation level of the high-energy optical scattering rate.](Fig2.pdf){width="3.4in"}
The optical response of BKFA in the far-infrared spectral range is shown in Fig. \[fig:firexperiment\](a, b) respectively for the real parts of optical conductivity $\sigma(\omega)=\sigma_1(\omega)+i\sigma_2(\omega)$ and dielectric function $\varepsilon(\omega)=1+4\pi i\sigma(\omega)/\omega$. It is dominated by the contribution of the itinerant charge carriers manifested in negative values of $\varepsilon_1(\omega)$. Figure \[fig:firexperiment\](a) also reveals a peak around $15-20\ \textrm{meV}$ in $\sigma_1(\omega)$ with a concomitant upturn in $\varepsilon_1(\omega)$ at higher temperatures indicating the presence of a collective excitation. The superconducting transition is accompanied by the suppression of the optical conductivity up to $50\ \textrm{meV}$ ($14k_{\mathrm{B}}T_{\mathrm{c}}$). The corresponding missing area in $\sigma_1(\omega)$ between $41$ and $10\ \textrm{K}$, $\int_{0^+}^{6\Delta_{\mathrm{A}}}\Delta\sigma_1(\omega)d\omega=(8\lambda_{\mathrm{L}}^2)^{-1}$, manifests itself as a characteristic $-1/(\lambda_{\mathrm{L}}\omega)^2$ contribution to $\varepsilon_1(\omega)$ in Fig. \[fig:firexperiment\](b) in the superconducting state. The London penetration depth $\lambda_{\mathrm{L}}=2200$ Å extracted from these data is consistent with other measurements [@PhysRevLett.101.107004]. At energies close to the optical superconducting gap $2\Delta_{\mathrm{A}}\approx20\ \textrm{meV}$ one of the directly measured ellipsometric angles $\Psi(\omega)$ approaches its critical value of $45^\circ$ at the superconducting transition, which implies that the reflectivity of the sample approaches unity and its optical conductivity $\sigma_1(\omega)$ is close to zero. Remarkably, Fig. \[fig:firexperiment\](a) shows a quasilinear dependence of $\sigma_1(\omega)$ in the superconducting state from $2\Delta_{\mathrm{A}}$ to as high as $14k_{\mathrm{B}}T_{\mathrm{c}}$, in a stark contrast to the electron-doped 122 compounds [@PhysRevB.82.174509; @PhysRevB.81.214508; @PhysRevB.82.100506]. In the latter the optical conductivity at $2\Delta_{\mathrm{A}}$ decreases abruptly upon cooling below $T_{\mathrm{c}}$, but only a weak superconductivity-induced modification is observed at higher energies. The quasilinear behavior in BKFA cannot be reconciled with the widely used for pnictides Mattis-Bardeen theory [@Zimmermann199199], a weak-coupling extension of the BCS theory to finite impurity scattering. As all optimally-doped 122 pnictide superconductors appear to be in the strong-coupling regime, the Eliashberg theory [@PhysRev.156.470] has to be used in order to obtain an adequate description of the optical properties.
![\[fig:zerocrossing\] (a) Real part of the optical conductivity at 41K (blue line). The contribution of itinerant charge carriers (blue area) is obtained by subtracting all interband transitions $\sigma^{\mathrm{inter}}_1(\omega)$ (gray area) from the optical response. (b) Real part of the dielectric function at 41K (blue line). The free-charge-carrier response $\varepsilon^{\mathrm{it}}_1(\omega)$ (open circles) is obtained by eliminating all interband transitions $\varepsilon^{\mathrm{inter}}_1(\omega)$ (black solid line). The blue dashed line indicates the screened plasma frequency at 41K.](Fig3.pdf){width="3.4in"}
Signatures of a boson pairing mediator of the Eliashberg theory come from a qualitative analysis of the optical conductivity within the extended Drude model. It implies that the optical scattering rate is related to the far-infrared optical response as $\gamma(\omega)=\textrm{Re}[\omega_{\mathrm{pl}}^2/4\pi\sigma^{\mathrm{it}}(\omega)]$, where the superscript ’it‘ stands for ’itinerant‘ and implies that the contribution of all interband transitions [*must*]{} be subtracted from the experimentally obtained optical conductivity. The optical scattering rate shows clear evidence of an intermediate boson irrespective of complications due to the multiband character of the compound. Figure \[fig:scattering\] plots $\gamma(\omega)$ of BKFA at $41\ \textrm{K}$ (blue line) and $10\ \textrm{K}$ (black line) for $\omega_{\mathrm{pl}}=[8\textrm{SW}_{\mathrm{it}}]^{1/2}=[8\int_{0}^{\infty}\sigma_1^{\mathrm{it}}(\omega)d\omega]^{1/2}=1.6\ \textrm{eV}$ with all interband transitions subtracted in both cases ($\textrm{SW}_{\mathrm{it}}$ corresponds to the blue shaded area in Fig. \[fig:zerocrossing\](a)). In the superconducting state, no scattering is expected up to photon energies exceeding the binding energy of the Cooper pairs. Thus the onset of the optical scattering rate marks the optical energy gap $2\Delta_{\mathrm{A}}=20\ \textrm{meV}$. Saturation of $\gamma(\omega>50\ \textrm{meV})$ at $1100\ \textrm{cm}^{-1}$ indicates that the boson spectral function is contained well below $50\ \textrm{meV}$ [@Shulga1991266]. It is important to emphasize that, due to the multiband character of the iron pnictides, an analysis of the optical scattering rate in the framework of a single-band Eliashberg theory is potentially misleading. Moreover, also shown in Fig. \[fig:scattering\] is a spectrum that directly results from the experimental data, without accounting for the interband transitions. It becomes clear that an increase in the scattering rate at higher energies that might be ascribed to strong electron correlations can result from an unsubtracted contribution of the interband transitions to the complex optical conductivity. This is especially important in iron pnictides since the lowest lying interband transition at about $0.5\ \textrm{eV}$ contributes to an anomalously large value of the low-energy dielectric permittivity $\varepsilon_\infty$ [@footinbib_fir2011] due to the high polarizability of the Fe-As bonds [@2010arXiv1009.5915C]. In order to reconcile the bare plasma frequency of $1.6\ \textrm{eV}$ (see Fig. \[fig:zerocrossing\](a)) with the zero-crossing in $\varepsilon_1(\omega)$ at $0.2\ \textrm{eV}$ (blue line in Fig. \[fig:zerocrossing\](b)) $\varepsilon_\infty$ has to be as large as 60, consistent with the contribution of the interband transitions $\varepsilon^{\mathrm{inter}}_1(\omega)$ determined by means of the dispersion analysis, as shown in Fig. \[fig:zerocrossing\](b). Such $\varepsilon_\infty$ is thus an order of magnitude larger than in any other high-temperature superconductor (e. g. $\approx5$ in cuprates [@A.V.Boris04302004]). Recently, a similarly high value in a conventional superconductor was inferred from reflectivity measurements on elementary bismuth [@PhysRevLett.104.237401].
To determine the microscopic origin of the high-energy anomaly $2\Delta_{\mathrm{A}}<\hbar\omega<14k_{\mathrm{B}}T_{\mathrm{c}}$ in the real part of the optical conductivity in Fig. \[fig:firexperiment\](a) we use a four-band Eliashberg theory that proved successful in explaining thermodynamical data obtained on the same compound [@PhysRevLett.105.027003]. The $4\times4$ matrix of coupling constants and 4 densities of states characterizing this model are highly constrained by thermodynamic, transport and photoemission data [@PhysRevLett.105.027003; @PhysRevB.79.054517; @PhysRevLett.105.117003; @PhysRevB.83.020501; @Shan-Wen_STM_2011], and the same set of parameters is used here. In principle, an additional set of 4 plasma frequencies and a $4\times4$ matrix of intraband/interband impurity scattering rates has to be taken into account to describe the optical response. However, this parameter set can be strongly reduced based on the following considerations.
A substantial simplification is made possible by a projection of the four-band model onto an effective two band model motivated by the observation of two distinct groups of superconducting energy gaps in a variety of experiments [@PhysRevLett.105.027003; @PhysRevB.79.054517; @PhysRevLett.105.117003; @PhysRevB.83.020501; @Shan-Wen_STM_2011]. These gaps can be identified as a single gap $\Delta_B$ on the outer hole-like Fermi surface and a group of three gaps of magnitude $\sim\Delta_A$ on the inner hole-like and the two electron-like Fermi surfaces. Minimizing the ground-state energy subject to this grouping constraint yields an effective two-band model [@supplementary.material.prl]. Furthermore, as the superconducting transition temperature of BKFA appears to be only weakly correlated with the residual resistivity (which is a measure of the impurity scattering), off-diagonal elements of the impurity scattering matrix can be neglected (see Table S1 in Ref. ).
Given the boson spectrum centered at $13\ \textrm{meV}$ (see supplementary online material in Ref. ) consistent with the energy of the spin resonance excitation in this compound [@Osborn_INS_BKFA_2008; @footinbib_SF2011] one obtains the following two-band model coupling matrix: $\lambda_{\mathrm{AA}}=4.36,\ \lambda_{\mathrm{BB}}=0.2,\ \lambda_{\mathrm{AB}}=-0.35,\ \lambda_{\mathrm{BA}}=-0.5$, with the fractional density of states being $N_{\mathrm{A}}/(N_{\mathrm{A}}+N_{\mathrm{B}})=0.59$ [@supplementary.material.prl]. The first effective intraband coupling constant is an order of magnitude larger than predicted for the intraband electron-phonon coupling [@boeri:026403]. It does not, however, bear any physical meaning by itself but rather incorporates contributions from [*three*]{} different bands. We reiterate that the coupling matrix has been inferred from prior measurements. In this way, only two intraband impurity scattering rates enter as free parameters of the theory in addition to the plasma frequencies of the bands.
![\[fig:eliashberg\_sigma\] (a) Real part of the far-infrared conductivity obtained within the two-band Eliashberg theory (see text) at $40\ \textrm{K}$ (blue lines) and $10\ \textrm{K}$ (black lines) in the clean limit $\gamma_{\mathrm{A}}=\gamma_{\mathrm{B}}=1\ \textrm{cm}^{-1}$ (solid lines) and dirty limit $\gamma_{\mathrm{A}}=\gamma_{\mathrm{B}}=200\ \textrm{cm}^{-1}$ (dashed lines). (b) Optical scattering rate in the clean limit from the same model. The gray area shows the normalized boson spectral function $B(\omega)$ used in the calculation, displaced from zero by $2\Delta_{\mathrm{A}}$ to assist interpretation in the superconducting state [@footinbib_SF2011].](Fig4.pdf){width="3.4in"}
In our calculation we consider two clean bands with $\gamma_{\mathrm{A}}=\gamma_{\mathrm{B}}=1\ \textrm{cm}^{-1}$. As the bare plasma frequencies of all bands are similar it follows that the spectral weight of band $\mathrm{A}$ has to be much larger than that of band $\mathrm{B}$. Assigning $80\%$ of the spectral weight to the effective band we obtain the results presented as solid lines in Fig. \[fig:eliashberg\_sigma\](a). The high-energy anomaly at $14k_{\mathrm{B}}T_{\mathrm{c}}$ is naturally captured by the model without resorting to additional gaps. Its energy is given by $2\Delta_{\mathrm{A}}+\Omega$, where $\Omega$ is the characteristic frequency of the boson spectrum, as shown in Fig. \[fig:eliashberg\_sigma\](b) (gray shaded area, displaced from zero by $2\Delta_{\mathrm{A}}$). This calculation also accounts for the fact that only the biggest superconducting gap is visible in the optical responce of BKFA due to a small contribution of band B ($20\%$ of the spectral weight) to the overall optical conductivity. This leads to two possible levels of the impurity scattering rate of band B, which has to be either very small $\gamma_{\mathrm{B}}\approx1\ \textrm{cm}^{-1}$ or very large at about $1000\ \textrm{cm}^{-1}$. The latter value provides a better description of the optical scattering rate (see interactive simulation in [@supplementary.material.prl]) and DC transport [@2010arXiv1011.1900G]. However, such a large disparity between the charge carriers is hard to reconcile with the Hall and de Haas–van Alphen experiments, which imply that the impurity scattering rate of the holes is no more than one order of magnitude higher than that of the electrons [@PhysRevB.80.140508; @PhysRevLett.101.216402]. This residual uncertainty notwithstanding, our results show that the impurity scattering rate of band A must be very small, because the energy $2\Delta_{\mathrm{A}}+\Omega$ is no longer discernible in the simulated spectra when $\gamma_{\mathrm{A}}$ increases (interactive simulation in [@supplementary.material.prl]). The region of linear increase of $\sigma_1(\omega)$ is related to the linear segment in the boson spectrum and can only be observed in a very clean material.
The same reduced two-band model can be applied to the case of $\textrm{BaFe}_{1.85}\textrm{Co}_{0.15}\textrm{As}_2$ (BFCA). In this compound the spin resonance excitation occurs at a very similar energy of $10\ \textrm{meV}$ [@Inosov_BFCA_2009]. A boson spectrum centered at this energy is also consistent with Andreev-reflection measurements [@PhysRevLett.105.237002]. Recently, a comprehensive specific-heat study of this compound at different Co-doping levels has been carried out [@0295-5075-91-4-47008]. The analysis of the experimental data in the framework of the two-band $\alpha$-model indicates that the largest gap develops in the band with the largest electronic density of states, providing further evidence that several bands contribute to the strongly-coupled band in the reduced two-band model. Figure \[fig:eliashberg\_sigma\](a) (dashed lines) shows that a calculation within the same reduced two-band model qualitatively reproduces the far-infrared optical conductivity of BFCA [@PhysRevB.82.174509; @PhysRevB.81.214508; @PhysRevB.82.100506] when both bands are assumed to be dirty with $\gamma_{\mathrm{A}}=\gamma_{\mathrm{B}}=200\ \textrm{cm}^{-1}$ and a redistribution of the spectral weight between the bands is taken into account as $\omega^2_{\mathrm{pl,A}}\approx\omega^2_{\mathrm{pl,B}}$. The model captures the two prominent superconductivity-induced anomalies clearly observed in experiments: the steep onset of absorption at the value of the small gap $2\Delta_{\mathrm{B}}$ and the weaker superconductivity-induced changes of the optical conductivity extending up to $18k_{\mathrm{B}}T_{\mathrm{c}}$. The redistribution of the spectral weight between the bands in BFCA compared to BKFA implied by our analysis is justified by doping with different carriers in the two compounds, while the large difference in the their impurity scattering rates is a natural consequence of the difference in doping mechanisms by chemical substitution, which directly affects the FeAs layers in BFCA, but not in BKFA.
In summary, a qualitative description of superconductivity-induced optical anomalies in the far-infrared optical conductivity of $\textrm{Ba}_{0.68}\textrm{K}_{0.32}\textrm{Fe}_2\textrm{As}_2$ is obtained in the framework of an effective two-band Eliashberg theory with a strong coupling to spin fluctuations reduced from its four-band counterpart. The linear increase of absorption above the larger superconducting gap can only be observed when the effective band is extremely clean. The same model in the dirty limit provides a good qualitative explanation of the optical conductivity of the optimally electron-doped BFCA consistently in the strong-coupling regime.
This project was supported by the German Science Foundation under grant BO 3537/1-1 within SPP 1458. We gratefully acknowledge Y.-L. Mathis for support at the infrared beamline of the synchrotron facility ANKA at the Karlsruhe Institute of Technology and P. Popovich for taking part in some of the measurements.
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[**Supplementary online material for the article\
Eliashberg approach to superconductivity-induced infrared anomalies in $\bf\textrm{Ba}_{0.68}\textrm{K}_{0.32}\textrm{Fe}_2\textrm{As}_2$**]{} 0.17in A. Charnukha$^1$, O. V. Dolgov$^1$, A. A. Golubov$^2$, Y. Matiks$^1$, D. L. Sun$^1$, C. T. Lin$^1$, B. Keimer$^1$, and A. V. Boris$^1$ 0.05in [*$^\mathit{1}$Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany*]{}\
[*$^\mathit{2}$Faculty of Science and Technology and MESA+ Institute of Nanotechnology, 7500 AE Enschede, The Netherlands*]{}
Dispersion analysis
===================
The full complex dielectric function $\varepsilon(\omega)=\varepsilon_1(\omega)+i\varepsilon_2(\omega)$ obtained experimentally in the range from $12\ \textrm{meV}$ to $6.7\ \textrm{eV}$ was analyzed in the Drude-Lorentz model:$$\varepsilon(\omega)=1-\frac{\omega_{\mathrm{pl}}^2}{\omega^2+i\gamma\omega}+\sum_{j=1}^{n}\frac{\Delta\varepsilon_j\omega_{0j}^2}{(\omega_{0j}^2-\omega^2)-i\Gamma_j\omega}\nonumber,$$where $(\omega_{\mathrm{pl}},\ \gamma)$ are the plasma frequency renormalized by interaction with the mediating boson and optical scattering rate, and $(\Delta\varepsilon_j,\omega_{0j},\ \Gamma_j)$ are the DC permittivity contribution, center frequency and the width of the Lorentzian oscillators used to model the interband transitions, respectively. The results of this analysis are presented in Fig. \[fig:dispanalysis\] for $10\ \textrm{K}$ (blue line - experimental data, black lines - separate Lorentz contributions). To display the scale of the temperature-induced variation of the interband transitions the experimental spectrum at $300\ \textrm{K}$ is also shown (red line). The lowest interband transition in this material lies around $0.5\ \textrm{eV}$ and significantly contributes to the AC polarizability of the system, as is evident from Fig. 3(b) of the main text. The residual optical response (open circles and inset in Fig. 3(b)) was studied after the subtraction of all thus determined interband transitions down to $0.5\ \textrm{eV}$. An unscreened bare plasma frequency of $1.6\ \textrm{eV}$ at $41\ \textrm{K}$ was consistently obtained from the spectral weight of the residual response $\textrm{SW}=\int_{0}^{\infty}\sigma_1^{\mathrm{it}}(\omega)d\omega$ as $\omega_{\mathrm{pl}}=\sqrt{8\textrm{SW}}$ and a simultaneous fit of the real and imaginary parts of the dielectric function at high energies.
![\[fig:dispanalysis\]Real part of the (a) optical conductivity and (b) dielectric function (b) at 300K (red line) and 10K (blue line). Interband transitions inferred from the dispersion analysis (gray lines). The shaded region is enlarged in Fig. 3(b) of the main text.](FigS1.pdf){width="\columnwidth"}
Effective two-band model for pnictides
======================================
Multiband Eliashberg formalism
------------------------------
The Eliashberg theory of superconductivity \[S,S\] extended to the multiband case [@PhysRevB.72.024504] has already proven successful in describing the thermodynamical properties of iron pnictides [@PhysRevLett.105.027003s]. Here we employed the same formalism to account for the far-infrared optical response of these compounds. The main ingredient of the theory is the total spectral function of the electron-boson interaction $B(\omega)$ (Eliashberg function; analogous to that of the electron-phonon interaction $\alpha ^{2}F(\omega)$). In a four band system it can be decomposed into 16 functions $B(\omega) _{ij}$, where $i$ and $j$ label the four Fermi surface sheets ($i,j=1,2,3,4$). The *standard* Eliashberg functions determine the superconducting and thermodynamical properties such as the superconducting transition temperature and gaps, electronic specifc heat, de Haas-van Alphen mass renormalizations etc. and are defined as $$B(\omega)_{ij}=\frac{1}{N_{i}}\sum_{{\bf k,k}^{\prime },\nu }\left| g_{{\bf k,k}^{\prime }}^{ij,\nu }\right| ^{2}\delta
(\varepsilon _{{\bf k}}^{i})\delta (\varepsilon _{{\bf k^{\prime }}}^{j})\delta (\omega -\omega _{{\bf k-k^{\prime }}}^{\nu }),\nonumber$$ where $N_{i}$ is the partial density of states per spin on the $i$’th sheet of the Fermi surface, and $g_{{\bf k,k}^{\prime
}}^{ij}$ is the matrix element of electron-boson interactions.
Transport and electrodynamical properties are defined by 16 [*transport*]{} Eliashberg functions (which enter the Boltzmann kinetic equation) $$\begin{aligned}
B(\omega)_{tr\;ij}^{\alpha\beta}&=&\frac{1}{2N_{i}\left\langle v_{Fi}^{\alpha \text{ }2}\right\rangle }\sum_{{\bf k,k}^{\prime },\nu }\left| g_{{\bf k,k}^{\prime }}^{i,j,\nu }\right|
^{2}
\nonumber\\
&&\hskip-25pt\times(v_{Fi}^{\alpha }({\bf k})-v_{Fj}^{\beta }({\bf k}^{\prime }))^{2}\delta (\varepsilon _{{\bf k}}^{i})\delta (\varepsilon _{{\bf k}^{\prime }}^{j})\delta (\omega -\omega _{{\bf k-k}^{\prime }}^{\nu }),
\nonumber\end{aligned}$$ where $v_{Fi}^{\alpha }$ is the $\alpha $-th Cartesian component of the Fermi velocity on Fermi surface $i$. The average Fermi velocity is related to the plasma frequency by the standard expression $\omega _{pl\;i}^{2}=8\pi e^{2}N_{i}\left\langle v_{Fi}^{2}\right\rangle
=8\pi e^{2}\sum_{{\bf k}}v_{Fi}^{2}({\bf k})\delta (\varepsilon _{{\bf k}}^{i}).$ All Eliashberg functions satisfy the symmetry relations $M_{i}B _{ij}=M_{j}B _{ji},$ where $M_{i}=N_{i}$ and $M_{i}=\omega _{pl\,i}^{2}$ for the standard and transport Eliashberg functions, respectively.
Role of impurities and defects
------------------------------
Both normal and superconducting properties of a multiband superconductor significantly depend on impurity scattering. Unlike in conventional superconductivity, one has to distinguish between the intraband impurity scattering, which does not add any new physics (in the Born approximation) compared with single-band superconductivity, and *interband* scattering, which in many cases has an effect comparable to the pair-breaking effect of magnetic impurities (or of nonmagnetic impurities in superconductors with $p$- or $d$-wave pairing) [@PhysRevB.55.15146]. In this regard, the fact that no strong correlation has been observed between the residual resistivity (which indirectly characterizes the impurity scattering) and the critical temperature $T_{c}$ of the (nearly) optimally electron-doped BKFA (see Table \[table:residual\]) indicates that the level of *interband* impurities in the Born limit is very small. Thus one only needs to estimate the intraband scattering rates $\gamma_{\mathrm{A}},\ \gamma_{\mathrm{B}}$.
$T_{\textrm{c}}$, K Residual resistivity, $\textrm{m}\Omega\ \textrm{cm}$ Reference
--------------------- ------------------------------------------------------- ----------------------------
38.5 0.04 [@PhysRevLett.105.027003s]
38 0.075 [@PhysRevB.78.224512]
38 0.1 [@PhysRevLett.101.107006]
36.5 0.055 [@PhysRevB.79.174501]
: \[table:residual\]Superconducting transition temperature and residual resistivity $\rho_{40\textrm{K}}$ of (nearly) optimally hole-doped BKFA.
Theoretical model
-----------------
As a starting point we consider a $4-$band model based on the band-structure calculations with two hole bands and two electron bands crossing the Fermi level that has proven successful in accounting for the thermodynamical properties of BKFA [@PhysRevLett.105.027003s]. We use the same input parameters, namely, the densities of states $N_1=22\ \textrm{Ry-st}^{-1}$, $N_2=25\ \textrm{Ry-st}^{-1}$, and $N_3=N_4=7\ \textrm{Ry-st}^{-1}$, the first two having a hole while the other two an electron character. The main input, the spectral function of the intermediate boson, was taken following Ref. S in the form of a spin-fluctuation spectrum $\tilde{B}_{ij}(\Omega )=\lambda _{ij}f(\Omega /\Omega _{sf})$ with a linear $\omega$ dependence at low frequencies. Here $\lambda_{ij}$ is the coupling constant pairing band $i$ with band $j$ and $\Omega _{SF}$ is a characteristic spin-fluctuation frequency, the values of which correspond to those in Ref. : $\Omega_{sf}=13\ \textrm{meV}$ and$$\lambda _{ij}=\left(
\begin{array}{cccc}
0.2 & 0 & -1.7 & -1.7 \\
0 & 0.2 & -0.25 & -0.25 \\
-5.34 & -0.89 & 0.2 & 0 \\
-5.34 & -0.89 & 0 & 0.2\end{array}\right).\label{l4x4}$$Negative elements correspond to *interband* hole-electron repulsion, while the positive — to *intraband* attraction.
In order to apply this full 4-band model to description of the transport properties one has to take into account an additional set of 4 plasma frequencies and a 4x4 matrix of impurity scattering rates. The latter are difficult to determine theoretically and thus would have to be treated as free paremeters of the model. It would render the problem highly overparametrized. On the other hand, in the case of BKFA it is known that three larger gaps have approximately the same value $|\Delta_{\mathrm{A}}|\equiv|\Delta_1|\approx|\Delta_3|\approx|\Delta_4|\approx9\ \textrm{meV}$, while the smaller gap is $|\Delta_{\mathrm{B}}|\equiv|\Delta_2|\approx3\ \textrm{meV}$. One can assume this restriction exact and introduce it into the theory thus reducing the original 4-band model to a more tractable 2-band model as explained in the following section.
Reduction to a two-band model
-----------------------------
In general, the superconducting order parameters are a solution of a linear system of equations $$e_{i}=\sum_{j=1}^{4}B_{ij}(\omega)e_{j}.\label{eq:gapsystem}$$ The Eliashberg functions $B_{ij}(\omega)$ satisfy the symmetry relations $$N_iB_{ij}(\omega )=N_jB_{ji}(\omega)\label{eq:symmetryrelations}$$and, therefore, can be represented in the form $B_{ij}(\omega)=U_{ij}(\omega)N_j$, where $U_{ij}$ is a symmetrical matrix. Further, we can construct a functional $${\mathfrak{F}}\{e_i\}=\sum_{j=1}^{4}N_je_j^2-\sum_{i,j=1}^4N_ie_iU_{ij}N_je_j. \label{eq:functional}$$Equation (\[eq:gapsystem\]) then results from minimization of ${\mathfrak{F}}$ with respect to $e_i$. As mentioned above, BKFA has three gaps with very close absolute values (the first hole gap has the opposite sign with respect to the other two).
[ Real part of the optical conductivity (top left) and the optical scattering rate (top right) of $\textrm{Ba}_{0.68}\textrm{K}_{0.32}\textrm{Fe}_2\textrm{As}_2$ in the superconducting state at $10\ \textrm{K}$ (blue lines) and the normal state at $40\ \textrm{K}$ (red) as obtained experimentally (heavy) and from the effective two-band Eliashberg theory (thin). (bottom left) The spin-fluctuation spectrum used in the simulation in both the superconducting and the normal state. (bottom right) Interactive controls: adjust the sliders to select the physical parameters of the model.](FigS4.pdf "fig:"){width="\columnwidth"}](http://www.fkf.mpg.de/keimer/groups/optical/bkfafir.jar)
Minimizing functional Eq. (\[eq:functional\]) subject to the additional constraints $e_3=e_4=-e_1=\Delta
_{\mathrm{A}}$, and $e_2=-\Delta _{\mathrm{B}}$ one finds $$\left(
\begin{array}{c}
\Delta _{\mathrm{A}} \\
\Delta _{\mathrm{B}}\end{array}\right) =\left(
\begin{array}{cc}
\lambda_{AA}&\lambda_{AB}\\
\lambda_{BA}&\lambda_{BB}\end{array}\right)\left(
\begin{array}{c}
\Delta_{\mathrm{A}} \\
\Delta_{\mathrm{B}}\end{array}\right),$$where the matrix elements satisfy $$\begin{aligned}
\lambda_{AA}&=&\frac{N_1\left( \lambda_{11}-2\lambda_{13}-2\lambda_{14}\right) +N_3\lambda_{33}{+}N_4\lambda_44}{N_1+N_3+N_4}, \\
\lambda_{AB} &=&\frac{N_2\left(\lambda_{23}+\lambda_{24}\right)}{N_1+N_3+N_4}, \\
\lambda_{BA}&=&\lambda_{23}+\lambda_{24}, \\
\lambda_{BB}&=&\lambda_{22}.\end{aligned}$$Assuming the matrix elements Eq. (\[l4x4\]), the following coupling constants of the reduced 2-band model are obtained: $$\lambda_{IJ}=\left(
\begin{array}{cc}
4.36&-0.35 \\
-0.5&0.2\end{array}\right) ,\text{ }I,J=\{A,B\}.\label{l2x2}$$The partial densities of states on the Fermi level of effective band $A$ and band $B$ are$$\begin{aligned}
N_{\mathrm{A}}&=&N_1+N_3+N_4=36\ \textrm{Ry-st}^{-1},\nonumber\\ N_{\mathrm{B}}&=&N_2=25\ \textrm{Ry-st}^{-1}\label{dos2b}.\end{aligned}$$Interestingly, even though the main interactions in the 4-band model with the coupling constants Eq. (\[l4x4\]) come from the nondiagonal elements, in the reduced 2-band counterpart they are incorporated into the effective *intraband* $\lambda_{\mathrm{AA}}$ matrix element. Figure \[fig:interactive\] presents an interactive simulation of this effective two-band model (click on the figure to run the simulation and adjust the sliders to set the physical parameters of the system).
Verification of the $2\times 2$ model
-------------------------------------
The effective 2-band model closely reproduces all the predictions of the 4-band model such as the superconducting transition temperature $T_{\mathrm{c}}=38.4\ \textrm{K}$, superconducting gaps $\Delta_{\mathrm{A}}=9.7\ \textrm{meV}$ and $\Delta_{\mathrm{B}}=3.7\ \textrm{meV}$, free energy and superconducting gaps as functions of temperature, as shown in Figs. \[fig:verification\](a) and (b), respectively. The calculated densities of states $N_{\mathrm{A}}$ and $N_{\mathrm{B}}$ are very similar, in accordance with the partial Sommerfeld constants obtained in the treatment of the specific heat data in a phenomenological two-band $\alpha$-model [@PhysRevLett.105.027003s]: $\gamma_{\mathrm{A}}\simeq\gamma_{\mathrm{B}}$ (the Sommerfeld constant $\gamma$ is related to the density of states $N$ via $\gamma =\frac{2\pi}{3}N$).
![\[fig:verification\]Temperature depencence of the (a) free energy and (b) superconducting gaps the 4-band (lines) and reduced 2-band (symbol) models, with coupling matrices Eqs. (\[l4x4\],\[l2x2\]), respectively.](FigS3.pdf){width="\columnwidth"}
[1]{}
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| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'J.-M. Hameury$^1$, J.-P. Lasota$^2$ and J.-M. Huré$^3$'
title: 'A model for WZ Sge with “standard" values of $\alpha$'
---
psfig mn
\#1[to 0pt[\#1]{}]{}
Introduction
============
Dwarf novae (DN) are cataclysmic variables, which, at usually irregular intervals, undergo eruptions in which the brightness increases by 2 to 7 magnitudes. It now well established that dwarf nova eruptions have their origin in a local, thermal and viscous instability due to an abrupt change in opacities at densities and temperatures at which hydrogen is partially ionized. In the disc instability model (DIM) of dwarf nova outbursts one assumes that the $\alpha$ viscosity parameter is higher in the high state than in quiescence and on obtains in this way a global disc instability which gives a rather good description of the U Gem type dwarf nova properties. Current models of dwarf novae imply values of the $\alpha$ parameter of the order of 0.01 in the low, quiescent state, and about 4 – 10 times larger during outbursts.
The DIM in its standard version cannot however describe outbursts of Z Cam and SU UMa type of dwarf novae. In the ‘standard’ DIM one assumes that the mass transfer from the secondary star is constant, that the only instability operating in the disc is the thermal–viscous one, and one neglects illumination of the disc and of the secondary by the radiation emitted by the accretion on to the white dwarf during outbursts. It is clear that if the Z Cam and SU UMa type of dwarf nova eruptions are due to the same type of local instability as the one that operates in U Gem type dwarf nova systems at least one of the standard assumptions must be dropped.
In the case of SU UMa type dwarf novae it has been proposed by Osaki (see e.g. 1995) that their superoutbusts are due to a tidal-thermal instability (TTI). In this model, successive U Gem type outbursts (‘normal’ outbursts) lead to an accumulation of matter in the disc and an increase of its outer radius. When the disc’s outer rim enters the ‘tidal radius’ in which the 3:1 resonance operates, the tidal couple is assumed to increase the accretion rate and to trigger a superoutburst in which matter accumulated during the supercycle is dropped on to the white dwarf. The TTI model encounters several difficulties. One is that it cannot apply (see e.g Smak 1996) to the 1985 superoutburst (Mason et al. 1988) of U Gem since the tidal instability cannot operate in this system. Another one is connected with the superoutbursts of WZ Sge. The TTI model can describe this system, which shows only superoutbursts separated by very long quiescent intervals, only if one assumes a very low value of $\alpha \lta 10^{-4}$ (Smak 1993; Osaki 1995). The reason for such a low $\alpha$ in this particular system is however unexplained so that it suggests that the TTI model might not be the correct description of the WZ Sge behaviour.
Lasota et al. (1995) noticed that if the inner disc in WZ Sge were missing because of the presence of a magnetosphere (Livio & Pringle 1992) or because of evaporation (Meyer & Meyer–Hofmeister 1994) the accretion disc would be marginally stable and outbursts could be triggered by an enhanced mass transfer (EMT) from the secondary. The long recurrence time would then be the timescale of fluctuations of the mass transfer and the $\alpha$ parameter would have its ‘usual’ value. The idea that superoutbursts in general could be due to EMT has been proposed by Smak (see e.g. Smak 1996).
Warner et al. (1996) assert that in the case of a truncated inner disc one can obtain long recurrence times in the ‘standard’ DIM with a ‘standard’ value of the $\alpha$ parameter. As we shall show below, the Warner et al. (1996) model is is fact, as far as recurrence time is concerned, practically identical to the one proposed by Lasota et al. (1995). The ‘standard’ DIM with ‘standard’ values of $\alpha$ suffers however from one insurmountable difficulty: the mass contained in the quiescent accretion disc for the parameters describing WZ Sge and for $\alpha \sim 0.01$ is less than $10^{23}$ g whereas during the superoutburst of this dwarf nova more than $10^{24}$ g has been accreted by the central white dwarf (Smak 1993). In fact this difficulty is the main reason for invoking very low values of the viscosity parameter.
If one wishes to explain superoutbursts of WZ Sge without using extremely low values of $\alpha$ one cannot avoid adding mass prior to or during the eruption. As discussed by Smak (1996) there is evidence of an increased mass transfer rate during normal outbursts and superoutbursts. In the EMT scenario proposed by Smak (1996) the superoutburst begins with a major enhancement of the mass transfer caused by irradiation during the preceding normal outburst. In the case of WZ Sge which shows only superoutbursts such a trigger does not exist.
In the present article we show that if, as proposed by Lasota et al. (1995), the quiescent accretion disc in WZ Sge is marginally stable, a slight enhancement of mass transfer triggers a ‘normal’ outburst which, due to subsequent irradiation of the secondary, becomes a superoutburst since a substantial amount of matter is added to the disc during the active phase of the cycle.
WZ Sge in quiescence
====================
The orbital period of WZ Sge is 81 min, close to the minimum period of cataclysmic variables; the system is thus very compact, and the accretion disc is quite small, with an outer radius of 1.1 10$^{10}$ cm during quiescence (Smak 1993). Here and in what follows we shall use Smak’s (1993) model of WZ Sge according to which the primary and secondary masses are 0.45 and 0.06 M$_\odot$ respectively. From the luminosity of the hot spot, Smak (1993) determined a mass transfer rate in quiescence $\dot{M}_{\rm tr} \approx 2 \ 10^{15}$ g $s^{-1}$.
WZ Sge has been detected by Einstein (Eracleous et al. 1991), EXOSAT (Mukai & Shiokawa 1993) and ROSAT (van Teeseling et al. 1996); its X-ray luminosity did not significantly vary in the interval from 4 months to more than 7 years after the 1978 outburst, and stayed at the level of 3 10$^{30}$ erg s$^{-1}$. If interpreted in terms of accretion rate on to the white dwarf surface, this luminosity corresponds to an accretion rate $\dot{M}_{\rm acc}$ on to the white dwarf surface of: $$\dot{M}_{\rm acc} = 5.0 \times 10^{13} \eta_X^{-1} r_9 \left( {M_1 \over 0.45
\rm M_\odot} \right)^{-1} \left( {L_{\rm X} \over 3 \times 10^{30}} \right)
\; \rm g s^{-1} \eqno \stepeq$$ where $M_1$ is the primary mass, $r_9$ the radius in units of 10$^9$ cm, $L_{\rm X}$ the X-ray luminosity and $\eta_X \leq 1$ the efficiency conversion of gravitational energy into X-ray photons. This would imply that the disc is not very far from steady state, since the mass accretion rate did not vary significantly from soon after the outburst to now, i.e. during an interval which is comparable to the recurrence time.
Moreover, according to the DIM, in the non–equilibrium disc $\dot{M}_{\rm acc}$ has to be low enough so that the inner regions of the disc are on the cool, stable branch of the $\Sigma -
T_{\rm ef\!f}$ curve, which implies: $$\dot{M} <
\dot{M}_{\rm B} = 1.43 \times 10^{13} M_1^{-0.87} r_9^{2.60} \; \rm g
s^{-1} \eqno \stepeq$$ (Ludwig et al. 1994). Since the accretion disc has to be in this state very early after outburst one concludes, comparing Eqs. (1) and (2), that in the DIM framework the quiescent viscosity cannot be very small, i.e. that $\alpha$ cannot be very small.
Consistency between X-ray observations and Eq. (2) requires the radius of the inner edge of the disc $r_{\rm in}$ to be larger than: $$r_{\rm in} > 1.43 \times 10^{9} \eta_X^{-0.62} \left( {M_1 \over 0.45 \rm
M_\odot} \right)^{0.08} \; \rm cm \eqno \stepeq$$ This is larger than the white dwarf radius, which is easily understood if the inner parts of the disc are disrupted by either a magnetic field (see e.g. Livio & Pringle 1992), or by evaporation (Meyer & Meyer-Hofmeister 1994). At such radii however, the disc is very close to a globally stable configuration (Lasota et al. 1995; Warner et al. 1996). In the following, we assume $r_{\rm in} = 4 \times 10^9$ cm, which ensures that the total disc luminosity is less than the luminosity of the hot spot, in agreement with observations.
The outburst
============
As mentioned in the introduction two “standard $\alpha$" mechanisms have been proposed for triggering outbursts in WZ Sge. In one of them, the disc is supposed to be stable on the cool branch in quiescence, in which case outbursts have to be triggered by an increase of the mass transfer from the donor star, causing the disc to become thermally unstable (Lasota et al. 1995); in the other one, the disc is supposed to be marginally unstable and to undergo a ‘standard’ DIM outburst. (Warner et al. 1996). It must however be noted that, because the disc must remain on the cool, stable branch during quiescence, its mass cannot be more than the integral of the maximum surface density $\Sigma$, i.e. $M_{\rm d} <
M_{\rm max} = 6 \times 10^{21} \alpha^{-0.8}$ g (Smak 1993). If the mass transfer rate from the secondary remains constant, the recurrence time can be estimated as the time it takes to increase $M_{\rm d}$ to this maximum value, i.e. $$\eqalignno{
t_{\rm rec} & \sim {M_{\rm max} \over \dot{M}_{\rm tr} - \dot{M}_{\rm acc}} \cr
& \sim 4 \left( {\alpha \over 10^{-2}} \right)^{-0.8} \left( 1 -
{\dot{M}_{\rm acc} \over \dot{M}_{\rm tr}} \right)^{-1} \; \rm yr & \stepeq\cr}$$ If one requires that $\alpha$ is not extremely small, $\dot{M}_{\rm acc}$ must be very close to $\dot{M}_{\rm tr}$ (within 10%) during the whole quiescent phase in order to get very long recurrence times. It is therefore most likely that an outburst would be triggered by a small fluctuations of $\dot{M}_{\rm tr}$ that is expected to occur within the long recurrence time of WZ Sge. Since in the Lasota et al. (1995) model the disc is marginally stable, both models are in fact identical as far as the recurrence time is concerned.
In the standard DIM, the duration of the outburst is at most the time it takes to empty the disc, i.e. $$\eqalignno{ t_{\rm outb} & =
{M_{\rm max} \over \dot{M}_{\rm acc}-\dot{M}_{\rm tr}} \cr & \sim 3
\left( {\alpha \over 10^{-2}} \right)^{-0.8} \left( 1 - {\dot{M}_{\rm
tr} \over \dot{M}_{\rm acc}} \right)^{-1} \; \rm days & \stepeq\cr}$$ where we took $\dot{M}_{\rm acc} = 10^{18}$ g s$^{-1}$ (Smak 1993). The long duration of WZ Sge outbursts (about one month) implies, if $\alpha$ is not very small, that, during outburst, the mass transfer rate had increased by two orders of magnitude, and was of the same order as the accretion rate. This would very naturally result from the illumination of the secondary. Effects of irradiation of the secondary were clearly observed during the outburst of SS Cyg (Hessman et al. 1984) and an increase of $\dot{M}_{\rm tr}$ by factors $\sim$ 2 has been observed in DNs such as Z Cha and U Gem (Smak 1995). Illumination effects are expected to be even more important in WZ Sge, in which the quiescent $\dot{M}_{\rm tr}$ is particularly low, so that the secondary surface temperature is expected to be low, and which has the shortest orbital period, so that the X-ray flux heating the secondary is large. For example, Smak (1993) obtains for the effective temperature of the secondary during the outburst $T_{\rm eff, 2} \approx 17 000$ K.
In the simplest model for illumination, the mass transfer rate is proportional to $e^{-\Delta r / H}$, where $\Delta r$ is the distance between the secondary photosphere and the Lagrangian point $L_1$, and $H$ the atmospheric scale height, proportional to the secondary surface temperature. In the case of WZ Sge, $\Delta r / H$ would typically be $\gta 1$ in quiescence, and less than unity during outburst, leading to large variations of the mass transfer rate. The response of the secondary to illumination has been discussed by e.g. Osaki (1985) and Hameury et al. (1986), but the presence of screening effects, of flows from the secondary’s poles to $L_1$, and the dependence on the emitted spectrum are very complex and make it difficult to describe; the exponential dependence quoted above is certainly not a very good approximation. As a preliminary step, we assume here that $$\dot{M}_{\rm tr} = \gamma \dot{M}_{\rm acc}
\eqno \stepeq$$ with $\gamma < 1$; this is similar to the approach of Augusteijn et al. (1993) in the context of soft X–ray transients.
The expected outcome of the model is that during quiescence, the disc stays on the cool stable branch. A fluctuation of the mass transfer rate from the secondary triggers the viscous/thermal instability, most probably at the outer edge of the disc. A heat front then propagates towards the compact object. Once it reaches the inner edge of the disc, $\dot{M}_{\rm tr}$ increases up to a value $\gamma \dot{M}_{\rm acc}$, and then, since the viscous time of the disc is short as compared to the total duration of the outburst, the disc would be close to steady state, with a mass transfer rate equal to $\gamma$ times the mass accretion rate, while the disc introduces a delay equal to the viscous time $t_{\rm visc}$. This naturally produces an exponential behaviour, with a decay time equal to $t_{\rm visc} /\ln \gamma$. Eventually, $\dot{M}_{\rm tr}$ becomes less than the critical value below which the hot, stable solution in the $\Sigma - T_{\rm ef\!f}$ diagram does not exist any longer; then a cooling wave starts from the outer edge of the disc, and brings it into quiescence in a short time scale.
In order to test this, we have calculated the time-dependent evolution of a disc initially stable with a low $\dot{M}_{\rm tr}$, taken to be $1.5 \times
10^{15}$ g s$^{-1}$, which is suddenly increased to $\max (5 \times 10^{15},\
0.87 \dot{M}_{\rm acc})$ g s$^{-1}$. The enhanced mass transfer decays exponentially in 5 days. $\alpha$ is taken to be $0.01$ in the cool branch and 0.1 in the hot one. All other parameters are those quoted here for WZ Sge; the code used is described in Hameury et al. (1996). The disc is assumed to be truncated as a result of the presence of a magnetic field, so that: $$r_{\rm in} = 4 \times 10^9 \left( {\dot{M}_{\rm acc} \over 1.5 \times 10^{15}
\rm g s^{-1}} \right)^{2/7} \; \rm cm
\eqno \stepeq$$ The disc behaves exactly as described above; for illustration, the visual magnitude of the disc (i.e. that does not include contributions from the secondary or from the white dwarf) is displayed in Fig. 1. It is seen that the shape of the light curve is in good agreement with observations.
Conclusion
==========
We have shown that the unusually long recurrence time and outburst duration in WZ Sge does not require the viscosity in this system to be much lower than in all other systems; these characteristics would result from (1) a low value of the mass transfer rate, so that the system is marginally stable during quiescence; (2) a truncated disc, which is required in many other systems to account for e.g. the observed optical-UV delay; and (3) a significant illumination effect that increases the mass transfer rate from the secondary by two orders of magnitude. WZ Sge would thus be explained by a combination of the two different model proposed by Osaki (1974, 1985), in a way similar to the proposition of Duschl & Livio (1989): a fluctuation of the mass transfer rate produces a thermal/viscous disc instability that brings the disc into a hot state, leading to an sudden increase of $\dot{M}_{\rm tr}$, which then slowly decreases until a cooling wave rapidly brings the disc back into its quiescent cool state.
The possible tests of this model are not very different from those proposed by Warner et al. (1996), since the outburst is an outside-in outburst, and since the enhancement of mass transfer that triggers the outburst is not very large, so that the behaviour of the disc outer radius need not be very different from that of a pure disc instability. We do however predict an increase of the hot spot luminosity a few days before the onset of an outburst; we also predict that the occurrence of outburst should be irregular (for example of a shot noise type). Finally, the hot spot should be much brighter during outbursts; this has been observed by Patterson et al. (1975) who inferred an enhancement of mass transfer from the secondary by a factor of 60 to 1000; they deduced then that the cause of the outburst was a mass transfer instability, although they could not exclude the possibility “that a brightening of the white dwarf or disc could be the event that triggers unstable mass transfer from the secondary".
References {#references .unnumbered}
==========
Augusteijn T., Kuulkers E., Shaham J., 1993, A&A, 279, L9 Duschl W. J., Livio M., 1989, A&A, 209, 183 Eracleous M., Halpern J., Patterson J., 1991, ApJ, 290,300 Hameury J.-M., King A. R., Lasota J.-P., 1986, A&A, 162, 71 Hameury J.-M., Huré J.-M., Lasota J.-P., in preparation Hessman F.V., Robinson E.L., Nather, R.E., Zhang, E.-H., ApJ, 286, 747 Lasota J.-P., Hameury J.-M., Huré J.-M., 1995, A&A, 302, L29 Livio M., Pringle J., 1992, MNRAS, 259, 23p Ludwig K., Meyer-Hofmeister E., Ritter H., 1993, A&A, 290, 473 Mason, K.O., Cordova, F.A., Watson, M.G., King, A.R., 1988 MNRAS, 232, 779 Meyer F., Meyer-Hofmeister E., 1994, A&A, 288, 175 Mukai K., Shiokawa K., 1993, ApJ, 418, 863 Osaki Y., 1974, PASJ, 26, 429 Osaki Y., 1985, A&A, 144, 369 Osaki Y., 1995, PASJ, 47, 47 Patterson J., McGraw J.T., Coleman L., Africano J.L., 1981, ApJ, 248, 1067 Smak J., 1993, Acta astron., 43, 101 Smak J., 1995, Acta astron., 45, 355 Smak J., 1996, in [*Cataclysmic Variables and Related Objects*]{}, IAU Coll. 158, p. 45 van Teeseling A., Beuermann K., Verbunt F., 1996, A&A, in press Warner B., Livio M., Tout C. A., 1996, MNRAS, 282, 735
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We describe a novel approach for adapting an existing software model checker to perform precise runtime verification. The software under test is allowed to communicate with the wider environment (including the file system and network). The modifications to the model checker are small and self-contained, making this a viable strategy for re-using existing model checking tools in a new context.
Additionally, from the data that is gathered during a single execution in the runtime verification mode, we automatically re-construct a description of the execution environment which can then be used in the standard, full-blown model checker. This additional verification step can further improve coverage, especially in the case of parallel programs, without introducing substantial overhead into the process of runtime verification.
author:
- Katarína Kejstová
- Petr Ročkai
- Jiří Barnat
bibliography:
- 'common.bib'
title: 'From Model Checking to Runtime Verification and Back[^1]'
---
Introduction
============
While model checking is a powerful technique for software verification, it also has certain limitations and deficiencies. Many of those limitations are related to the fact that a model checker must, by design, fully isolate the program from any outside effects. Therefore, for verification purposes, the program under test is placed into an artificial environment, which gives non-deterministic (but fully reproducible) responses to the program. The existence of this model environment immediately requires trade-offs to be made. If the environment model is too coarse, errors may be missed, or spurious errors may be introduced. Creating a detailed model is, however, more costly, and the result is not guaranteed to exactly match the behaviour of the actual environment either. Moreover, a detailed model may be too rigid: programs are often executed in conditions that have not been fully anticipated, and a certain amount of coarseness in the model of the environment can highlight such unwarranted assumptions.
Many of those challenges are, however, not unique to model checking. In the context of automated testing, the test environment plays a prominent role, and a large body of work deals with related problems. Unfortunately, adapting the methods used in automated testing to the context of model checking is far from straightforward. Making existing test-based setups easier to use with model checking tools is a core contribution of this paper.
Both manual and automated testing are established, core techniques which play an important role in virtually every software development project. In a certain sense, then, testing provides an excellent opportunity to integrate rigorous tools into the software development process. A number of verification tools specifically tailored for this mode of operation have seen great success in the software development community, for instance the `memcheck` tool from the `valgrind` suite. We show that it is possible to tap into this potential also with a traditionally-designed software model checker: we hope that this will help put powerful verification technology into the hands of software developers in a natural and seamless fashion. The second important contribution of this paper, then, is an approach to build a runtime verification tool out of an existing software model checker.
Our main motivating application is extending our existing software model checker, [@barnat13:divine], with a runtime verification mode. In its latest version, has been split into a number of well-defined, reusable components [@rockai18:divm] and this presented an opportunity to explore the contexts in which the new components could be used. Based on this motivation, our primary goal is to bring traditional (software) model checking and runtime verification closer together. As outlined above, there are two sides to this coin. One is to make model checking fit better into existing software development practice, the second is to derive powerful runtime verification tools from existing model checkers. To ensure that the proposed approach is viable in practice, we have built a prototype implementation, which allowed us to execute simple C and C++ programs in the resulting runtime verifier.
The rest of the paper is organised as follows: Section \[sec:related\] describes prior art and related work, while Section \[sec:prelim\] lays out our assumptions about the model checker and its host environment. Section \[sec:passthrough\] describes adapting a model checker to also work as a runtime verifier and Section \[sec:replay\] focuses on how to make use of data gathered by the runtime verifier in the context of model checking. Section \[sec:implementation\] describes our prototype implementation based on (including evaluation) and finally, Section \[sec:conclusion\] summarises and concludes the paper.
Related Work {#sec:related}
============
There are two basic approaches to runtime verification [@havelund04:efficien]: online (real time) monitoring, where the program is annotated and, during execution, reports its actions to a monitor. In an offline mode, the trace is simply collected for later analysis. Clearly, an online-capable tool can also work in offline mode, but the reverse is not always true. An extension of the online approach allows the program to be monitored also in production, and property violations can invoke a recovery procedure in the program [@meredith12:mop]. Our work, in principle, leads to an online verifier, albeit with comparatively high execution overhead, which makes it, in most cases, unsuitable for executing code in production environments. Depending on the model checker used, it can, however, report violations to the program and invoke recovery procedures and may therefore be employed this way in certain special cases.
Since our approach leads to a runtime verification tool, this can be compared to other such existing tools. With the exception of `valgrind` [@nethercote07:valgrin], most tools in this category focus on Java programs. For instance, Java PathExplorer [@havelund04:overview.runtim] executes annotated Java byte code, along with a monitor which can check various properties, including past-time LTL. Other Java-based tools include JavaMOP [@jin12:javamop] with focus on continuous monitoring and error recovery and Java-MaC [@kim04:java.mac] with focus on high-level, formal property specification.
Our *replay mode* (described in Section \[sec:replay\]) is also related to the approach described in [@havelund00:using.runtim], where data collected at runtime is used to guide the model checker, with the aim of reducing the size of the state space. In our case, the primary motivation is to use the model checker for verifying more complex properties (including LTL) and to improve coverage of runtime verification.
Preliminaries {#sec:prelim}
=============
There are a few assumptions that we need to make about the mode of operation of the model checker. First, the model checker must be able to restrict the exploration to a single execution of the program, and it must support explicitly-valued operations. The simplest case is when the model checker in question is based on an explicit-state approach (we will deal with symbolic and/or abstract values in Section \[sec:abstract\]). If all values are represented explicitly in the model checker, exploration of a single execution is, in a sense, equivalent to simply running the program under test. Of course, since this is a model checker, the execution is subject to strict error checking.
Abstract and Symbolic Values {#sec:abstract}
----------------------------
The limitation to exploring only a single execution is, basically, a limitation on *control flow*, not on the representation of variables. The root cause for the requirement of exploring only one control flow path is that we need to insert actions into the process of model checking that will have consequences in the outside world, consequences which cannot be undone or replayed. Therefore, it is not viable to restore prior states and explore different paths through the control flow graph, which is what normally happens in a model checker. It is, however, permissible to represent data in an abstract or symbolic form, which essentially means the resulting runtime verifier will also act as a symbolic executor. In this case, an additional requirement is that the values that reach the outside world are all concrete (the abstract representation used in the model checker would not be understood by the host operating system or the wider environment). Luckily, most tools with support for symbolic values already possess this capability, since it is useful in a number of other contexts.
Environments in Model Checking {#sec:env}
------------------------------
A model checker needs a complete description of a system, that is, including any environment effects. This environment typically takes the form of code in the same language as the program itself, in our case C or C++. For small programs or program fragments, it is often sufficient to write a custom environment from scratch. This is analogous to how unit tests are written: effects from outside of the program are captured by the programmer and included as part of the test.
When dealing with larger programs or subsystems, however, the environment becomes a lot more complicated. When the program refers to an undefined function, the model checker will often provide a fallback implementation that gives completely undetermined results. This fallback, typically, does not produce any side effects. Such fallback functions constitute a form of synthetic model environment. However, this can be overly coarse: such model environment will admit many behaviours that are not actually possible in the real one, and vice versa, lasting side effects of a program action (for instance a change in file content) may not be captured at all. Those infidelities can introduce both false positives and false negatives. For this reason, it is often important to provide a more realistic environment.
A typical model checker (as opposed to a runtime verifier) cannot make use of a real operating system nor of testing-tailored, controlled environment built out of standard components (physical or virtual machines, commodity operating systems, network equipment and so on). A possible compromise is to implement an operating system which is designed to run inside a model checker, as a stand-in for the real OS. This operating system can then be re-used many times when constructing environments for model checking purposes. Moreover, this operating system is, to a certain degree, independent of the particular model checker in use. Like with standard operating systems, a substantial part of the code base can be re-used when porting the OS (that is, the host model checker is akin to a processor architecture or a hardware platform in standard operating systems).
Many programs of interest are designed to run on POSIX-like operating systems, and therefore, POSIX interfaces, along with the interfaces mandated by ISO C and C++ are a good candidate for implementation. This has the additional benefit that large parts of all these specifications are implemented in open source C and/or C++ code, and again, large parts of this code are easily ported to new kernels. Together, this means that a prefabricated environment with POSIX-like semantics is both useful for verifying many programs and relatively simple to create.
In the context of a model checker, the kernel of the operating system can be linked directly to the program, as if it were a library. In this approach, the model checker in question does not need any special support for loading kernel-like objects or even for privilege separation.
System Calls {#sec:syscall}
------------
In this section, we will consider how traditional operating systems, particularly in the POSIX family, define and implement system calls. A traditional operating system consists of many different parts, but in our context, the most important are the kernel and the user-space libraries which implement the operating system API (the most important of these libraries is, on a typical Unix system, `libc`). From the point of view of a user program, the `libc` API *is* the interface of the operating system. However, many functions which are mandated as part of this interface cannot be entirely implemented in the user space: they work with resources that the user-space code is unable to directly access. Examples of such functions would be `read` or `write`: consider a `read` from a file on a local file system. If the implementation was done in the user space, it would need direct access to the hardware, for instance the PCI bus, in order to talk to the hard drive which contains the requisite blocks of data which represent the file system. This is, quite clearly, undesirable, since granting such access to the user program would make access control and resource multiplexing impossible.
For these reasons, it is standard practice to implement parts of this functionality in separate, system-level code with a restricted interface, which makes access control and resource sharing possible. In operating system designs with monolithic kernels, this restricted interface consists of what is commonly known as system calls.[^2] A system call is, then, a mechanism which allows the user-space code to request that the system-level software (the kernel) executes certain actions on behalf of the program (subject to appropriate permission and consistency checks). The actual implementation of syscall invocation is platform-specific, but it always involves a switch from user (non-privileged) mode into kernel mode (privileged mode, *supervisor* mode or *ring 0* on x86-style processors).
On POSIX-like systems, `libc` commonly provides a generic `syscall` function (it first appeared in `3BSD`). This function allows the application to issue syscalls based on their number, passing arguments via an ellipsis (i.e. by taking advantage of variadic arguments in the C calling convention). In particular, this means that given a description of a system call (its number and the number and types of its arguments), it is possible to automatically construct an appropriate invocation of the `syscall` function.
Overview of Proposed Extensions {#sec:overview}
-------------------------------
Under the proposed extensions, we have a model checker which can operate in two modes: *run* and *verify*. In the *run* mode, a single execution of the program is explored, in the standard execution order. We expect that all behaviour checking (enforcement of memory safety, assertion checks, etc.) is still performed in this mode. The *verify* mode, on the other hand, uses the standard model checking algorithm of the given tool.
![A scheme of components involved in our proposed approach.[]{data-label="fig:scheme"}](passthrough-scheme)
The system under test (the input to this model checker), then, consists of the user program itself, along with the environment, the latter of which contains a stand-in operating system. The situation is illustrated in Figure \[fig:scheme\]. The operating system has 3 different modes:
1. a *virtual* mode, in which all interaction with the real world is simply simulated – for example, a virtual file system is maintained in-memory and is therefore part of the state of the system under test; this OS mode can be used with both *run* and *verify* modes of the model checker
2. a *passthrough* mode, which uses the `vm_syscall` model checker extension to execute system calls in the host operating system and stores a trace of all the syscalls it executed for future reference; this OS mode can only be used in the *run* mode of the model checker
3. a *replay* mode, which reads the system call trace recorded in the *passthrough* mode, but does not interact with the host operating system; this OS mode can be again used in both the *run* and *verify* mode of the model checker
Syscall Passthrough {#sec:passthrough}
===================
In order to turn a model checker into a runtime verifier, we propose a mechanism which we call *syscall passthrough*, where the virtual, stand-in operating system (see Section \[sec:env\]) gains the ability to execute syscalls in the host operating system (see also Section \[sec:syscall\]). Of course, this is generally *unsafe*, and only makes sense if the model checker can explore a single run of the program and do so *in order*.
Thanks to the architecture of system calls in POSIX-like kernels, we only need a single new primitive function to be implemented in the model checker (we will call this new primitive function `vm_syscall` from now on; first, we need to avoid confusion with the POSIX function `syscall`, second, the model checker acts as a virtual machine in this context). The sole purpose of the function is to construct and execute, in the context of the host operating system, an appropriate call to the host `syscall` function (the interface of which is explained in more detail in Section \[sec:syscall\]).
We would certainly like to avoid any system-specific knowledge in the implementation of `vm_syscall` – instead, any system-specific code should reside in the stand-in OS, which is much easier to modify than the model checker proper. To this end, the arguments to our `vm_syscall` primitive contain metadata describing the arguments `syscall` expects, in addition to the data itself. That is, `vm_syscall` needs to know whether a particular argument is an input or an output argument, its size, and if it is a buffer, the size of that buffer. The exact encoding of these metadata will be described in Section \[sec:passthrough-mc\], along with more detailed rationale for this approach.
Finally, most of the implementation work is done in the context of the (stand-in) operating system (this is described in more detail in Section \[sec:passthrough-os\]). This is good news, because most of the code in the operating system, including all of the code related to syscall passthrough, is in principle portable between model checkers.
Model Checker Extension {#sec:passthrough-mc}
-----------------------
The model checker, on the other hand, only needs to provide one additional primitive. As already mentioned, we call this primitive `vm_syscall`, and it should be available as a variadic C function to the system under test. This is similar to other built-in functions often provided by model checkers, like `malloc` or a non-deterministic choice operator. While in the program under test, invocations of such built-ins look just like ordinary C function calls, they are handled differently in the model checker and often cause special behaviour that is not otherwise available to a C program.
We would like this extension to be as platform-neutral as possible, while maintaining simplicity. Of course not all platforms provide the `syscall` primitive described in Section \[sec:syscall\], and on these platforms, the extension will be a little more complicated. Namely, when porting to a platform of this type, we need to provide our own implementation of `syscall`, which is easy to do when the system calls are available as C functions, even if tedious. In this case, we can simply assign numbers to system calls and construct a single `switch` statement which, based on a number, calls the appropriate C function.
Therefore, we can rely on the `syscall` system-level primitive without substantial loss of generality or portability. The next question to ask is whether a different extension would serve our purpose better – in particular, there is the obvious choice of exposing each syscall separately as a model checker primitive. There are two arguments against this approach.
First, it is desirable that the syscall-related machinery is all in one place and not duplicated in both the stand-in operating system and in the model checker. However, in the *virtual* and *replay* modes, this machinery must be part of the stand-in operating system, which suggests that this should be also the case in the *passthrough* mode.
Second, the number of system calls is quite large (typically a few hundred functions) and the functions are system-dependent. When the code that is specific to the host operating system resides in the stand-in operating system, it can be ported once and multiple model checkers can benefit. Of course, the stand-in operating system needs to be ported to the model checker in question, but this offers many additional advantages (particularly the virtual mode).
Now if we decide that a single universal primitive becomes part of the model checker, we still need to decide the syntax and the semantics of this extension. Since different system calls take different arguments with varying meaning, the primitive itself will clearly need to be variadic. Since one of the main reasons for choosing a single-primitive interface was platform neutrality, the primitive itself should not possess special knowledge about individual syscalls. First of all, it does not know the bit widths of individual arguments (on most systems, some arguments can be 32 bit – for instance file descriptors – and other 64 bit – object sizes, pointers, etc.). This information is crucial to correctly set up the call to `syscall` (the variadic arguments must line up). Moreover, some pointer-type arguments represent variable-sized *input* data (the buffer argument to `write`, for example) and others represent *output* data (the buffer argument to `read`). In both cases, the size of the memory allocated for the variable-sized argument must be known to `vm_syscall`, so that this memory can be correctly copied between the model checker and the system under test.
![An example invocation of `vm_syscall` performing a `read` passthrough.[]{data-label="fig:syscall"}](passthrough-syscall)
For these reasons, the arguments to `vm_syscall` also contain metadata: for each real argument that ought to be passed on to `syscall`, 2 or 3 arguments are passed to `vm_syscall`. The first one is always type information: whether the following argument is a scalar (32b or 64b integer) or a pointer, whether it is an input or an output. If the value is a scalar input, the second argument is the value itself, if it is a scalar output, the following argument is a pointer to an appropriate-sized piece of memory. If the value is a pointer, the size of the pointed-to object comes second and the pointer itself comes third. An example invocation of `vm_syscall` is shown in Figure \[fig:syscall\]. The information passed to `vm_syscall` this way is sufficient to both construct a valid call to `syscall` and to copy inputs from the system under test to the host system and pass back the outputs.
Operating System Extension {#sec:passthrough-os}
--------------------------
The `vm_syscall` interface described above is a good low-level interface to pass syscalls through to the host operating system, but it is very different from the usual POSIX way to invoke them, and it is not very intuitive or user-friendly either. It is also an unsafe interface, because wrong metadata passed to `vm_syscall` can crash the model checker, or corrupt its memory.
The proper POSIX interface is to provide a separate C function for each syscall, essentially a thin wrapper that just passes the arguments along. Calling these dedicated wrappers is more convenient, and since they are standard C functions, their use can be type-checked by the compiler. In the *virtual* mode of the operating system, those wrappers cause the execution to divert into the kernel. We can therefore re-use the entire `libc` without modifications, and implement syscall passthrough at the kernel level, where we have more control over the code.
In our OS design, the kernel implements each system call as a single C++ method of a certain class (a *component*). Which exact components are activated is decided at boot time, and it is permissible that a given system call is implemented in multiple components. Since the components are arranged in a stack, the topmost component with an implementation of a given system call “wins”. In this system, implementing a passthrough mode is simply a question of implementing a suitable passthrough component and setting it up. When `libc` invokes a system call, the control is redirected into the kernel as usual, and the passthrough component can construct an appropriate invocation of `vm_syscall`.
This construction requires the knowledge of a particular system call. Those are, luckily, more or less standardised by POSIX and the basic set is therefore reasonably portable. Moreover, we already need all of this knowledge in the implementation of the virtual mode, and hence most of the code related to the details of argument passing can be shared. As mentioned earlier, this means that the relevant `libc` code and the syscall mechanism it uses internally is identical in all the different modes of operation. The passthrough mode is, therefore, implemented entirely in the kernel of the stand-in operating system.
Tracing the Syscalls {#sec:tracing}
--------------------
The architecture of syscall passthrough makes it easy to capture argument values and results of every invoked syscall, in addition to actually passing it on to the host operating system. Namely, the implementation knows exactly which arguments are inputs and which are outputs and knows the exact size of any buffer or any other argument passed as a pointer (both input and output). This allows the implementation to store all this data in a file (appending new records as they happen). This file can then be directly loaded for use in the *replay mode* of the stand-in operating system.
Syscall Replay {#sec:replay}
==============
In a model checker, all aspects of program execution are fully repeatable. This property is carried over into the *virtual* operating mode (as described in this paper), but not into the *passthrough* mode. System calls in the host operating system are, in general, not repeatable: files appear and disappear and change content, network resources come and go and so on, often independently of the execution of the program of interest.
What the passthrough mode can do, however, is recording the answers from the host operating system (see Section \[sec:tracing\]). When we wish to repeat the same execution of the program (recall that everything apart from the values coming from `vm_syscall` is under the full control of the model checker), we do not need to actually pass on the syscalls to the host operating system: instead, we can read off the outputs from a trace. This is achieved by simply replacing all invocations of `vm_syscall` by a different mechanism, which we will call `replay_syscall`. This new function looks at the trace, ensures that the syscall invoked by the program matches the one that comes next in the trace and then simply plays back the effects observable in the program. Since the program is otherwise isolated by the model checker, those effects are limited to the changes the syscall caused in its output parameters and the value of `errno`. The appropriate return value is likewise obtained from the trace.
Motivation
----------
There are two important applications of the replay mode. First, if the model checker in question provides interactive tools to work with the state space, we can use those tools to look at real executions of the program, and in particular, we can easily step backwards in time. That is, if we have an interactive simulator (like, for example, presented in [@rockai17:simulat.llvm.bitcod]), we can derive a reversible debugger essentially for free by recording an execution in the passthrough mode and then exploring the corresponding path through the state space in the *replay* mode.
Second, if the behaviour of the program depends on circumstances other than the effects and return values of system calls, it is often the case that multiple different executions of the program will result in an identical sequence of system calls. As an example, if the program contains multiple threads, one of which issues syscalls and others only participate in computation and synchronisation, the exact thread interleaving will only have a limited effect on the order and arguments of system calls, if any. The model checker is free to explore all such interleavings, as long as they produce the same syscall trace.
That this is a practical ability is easily demonstrated. A common problem is that a given program, when executed in a controlled environment, sometimes executes correctly and other times incorrectly. In this case, by a controlled environment we mean that files and network resources did not change, and that the behaviour of the program does not depend on the value of the real-time clock. Therefore, we can reasonably expect the syscall trace to be identical (at least up to the point where the unexpected behaviour is encountered). If this is the case, the model checker will be able to reliably detect the problem based on a single syscall trace, regardless of whether the problem did or did not appear while running in the passthrough mode.
Constructing the State Space
----------------------------
As explained above, we can use the replay mode to explore behaviours of the program that result in an identical syscall trace, but are not, computation-wise, identical to the original passthrough execution. In this case, it is important that the model checker explores only executions with this property. A primitive which is commonly available in model checkers and which can serve this purpose is typically known as `assume`[^3]. The effect of this primitive is to instruct the model checker to abandon any executions where the condition of the `assume` does not hold. Therefore, our `replay_syscall`, whenever it detects a mismatch between the syscall issued by the program and the one that is next in the trace, it can simply issue `assume( false )`. The execution is abandoned and the model checker is forced to explore only those runs that match the external behaviour of the original.
Causality-Induced Partial Order
-------------------------------
The requirement that the traces exactly match up is often unnecessarily constraining. For instance, it is quite obvious that the order of two read operations (with no intervening write operations) can be flipped without affecting the outcome of either of the two reads. In this sense, such two reads are not actually ordered in the trace. This means that the trace does not need to be ordered linearly – the two reads are, instead, incomparable in the causal ordering. In general, it is impossible to find the exact causal relationships between syscalls, especially from the trace alone – a write to a file may or may not have caused certain bytes to appear on the `stdin` of the program. We can, however, construct an approximation of the correct partial order, and we can do so safely: the constructed ordering will always respect causality, but it may order certain actions unnecessarily strictly.
We say that two actions $a$ and $b$ (system call invocations) *commute* if the outcome of both is the same, regardless of their relative ordering (both $a$ and $b$ have the same individual effect, whether they are executed as $a, b$ or as $b, a$). Given a sequence of system calls that respects the causal relationships, swapping two adjacent entries which commute will lead to a new sequence with the same property. We can obtain an approximate partial order by constructing all such sequences and declaring that $a < b$ iff this is the case in all of the generated sequences.
Prototype Implementation {#sec:implementation}
========================
We have implemented the approach described in this paper, using the 4 software model checker as a base. In particular, we rely on the component in , which is a verification-focused virtual machine based on the intermediate representation (more details in Section \[sec:llvm\]). The architecture of 4, as a model checker, is illustrated in Figure \[fig:d4\]. First, we have extended with the `vm_syscall` primitive (cf. Section \[sec:passthrough\]). Taking advantage of this extension, we have implemented the requisite support code in , as described in Section \[sec:passthrough-os\]. is a pre-existing stand-in operating system component which originally supported only the *virtual* mode of operation. As part of the work presented in this paper, we implemented both a passthrough and a replay mode in .
![The architecture of 4. The shaded part is, from a model checking point of view, the system under test. However, and most of the libraries are shipped as part of .[]{data-label="fig:d4"}](passthrough-d4)
In the rest of this section, we will describe the underpinnings of 4 in more detail. The first important observation is that, since is based on interpreting bitcode, it can use a standard compiler front-end to compile C and C++ programs into the bitcode form, which can then be directly verified. We will also discuss the limitations of the current implementation and demonstrate its viability using a few examples.
Bitcode {#sec:llvm}
--------
bitcode (or intermediate representation) [@llvm16:llvm.languag] is an assembly-like language primarily aimed at optimisation and analysis. The idea is that -based analysis and optimisation code can be shared by many different compilers: a compiler front end builds simple IR corresponding to its input and delegates all further optimisation and native code generation to a common back end. This architecture is quite common in other compilers: as an example, GCC contains a number of different front ends that share infrastructure and code generation. The major innovation of is that the language on which all the common middle and back end code operates is exposed and available to 3rd-party tools. It is also quite well-documented and provides stand-alone tools to work with both bitcode and textual form of this intermediate representation.
From a language viewpoint, IR is in partial SSA form (single static assignment) with explicit basic blocks. Each basic block is made up of instructions, the last of which is a *terminator*. The terminator instruction encodes relationships between basic blocks, which form an explicit control flow graph. An example of a terminator instruction would be a conditional or an unconditional branch or a `ret`. Such instructions either transfer control to another basic block of the same function or stop execution of the function altogether.
Besides explicit control flow, also strives to make much of the data flow explicit, taking advantage of partial SSA for this reason. It is, in general, impossible to convert entire programs to a full SSA form; however, especially within a single function, it is possible to convert a significant portion of code. The SSA-form values are called *registers* in and only a few instructions can “lift” values from memory into registers and put them back again (most importantly `load` and `store`, respectively, plus a handful of atomic memory access instructions).
Runtime Verification with {#runtime-verification-with-llvm}
--------------------------
While bitcode is primarily designed to be transformed and compiled to native code, it can be, in principle, executed directly. Of course, this is less convenient than working with native code, but since the bitcode is appreciably more abstract than typical processor-level code, it is more amenable to model checking. The situation can be improved by providing tools which can work with hybrid object files, which contain both native code and the corresponding bitcode. This way, the same binary can be both executed natively and analysed by -based tools.
Extensions for Verification {#sec:extensions}
----------------------------
Unfortunately, bitcode alone is not sufficiently expressive to describe real programs: most importantly, it is not possible to encode interaction with the operating system into instructions. When is used as an intermediate step in a compiler, the lowest level of the user side of the system call mechanism is usually provided as an external, platform-specific function with a standard C calling convention. This function is usually implemented in the platform’s assembly language. The system call interface, in turn, serves as a gateway between the program and the operating system, unlocking OS-specific functionality to the program. An important point is that the gateway function itself cannot be implemented in portable .
To tackle these problems, a small set of primitives was proposed in [@rockai18:divm] (henceforth, we will refer to this enriched language as ). With these primitives, it is possible to implement a small, isolated operating system in the language alone. already provides such an operating system, called – the core OS is about 2500 lines of C++, with additional 5000 lines of code providing *virtual* POSIX-compatible file system and socket interfaces. Our implementation of the ideas outlined in Section \[sec:passthrough-os\] can, therefore, re-use a substantial part of the existing code of .
Source Code
-----------
The implementation consists of two parts. The model checker extension is about 200 lines of C++, some of which is quite straightforward. The extension is more complex: the passthrough component is about 1400 lines, while the replay component is less than 600. All the relevant source code, including the entire 4 model checker, can be obtained online[^4].
Limitations {#sec:limitations}
-----------
There are two main limitations in our current implementation. The first is caused by a simplistic implementation of the *run* mode of our model checker (see Section \[sec:overview\]). The main drawback of such a simple implementation is that syscalls that block may cause the entire model checker to deadlock. Specifically, this could happen in cases where one program thread is waiting for an action performed by another program thread. Since there is only a single model checker thread executing everything, if it becomes blocked, no program threads can make any progress. There are two possible counter-measures: one is to convert all system calls to non-blocking when corresponding `vm_syscall` invocations are constructed, another is to create multiple threads in the model checker, perhaps even a new thread for each system call. Only the latter approach requires additional modifications to the model checker, but both require modifications to the stand-in operating system.
The second limitation stems from the fact that our current `libc` implementation only covers a subset of POSIX. For instance, the `gethostbyname` interface (that is, the component of `libc` known as a resolver) is not available. This omission unfortunately prevents many interesting programs from working at the moment. However, this is not a serious limitation in principle, since the resolver component from an existing `libc` can be ported. Many of the networking-related interfaces are already present and work (in particular, TCP/IP client functionality has been tested, cf. Section \[sec:evaluation\]).
Finally, a combination of both those limitations means that the `fork` system call, which would create a new process, is not available. In addition to problems with blocking calls, there are a few attributes that are allocated to each process, and those attributes can be observed by certain system calls. For example, one such attribute is the `pid` (process identifier), obtainable with a `getpid` system call, another is the working directory of the process, available through `getcwd`. Again, there are multiple ways to resolve this problem, some of which require modifications in the model checker.
Evaluation {#sec:evaluation}
----------
Mainly due to the limitations outlined in Section \[sec:limitations\], it is not yet possible to use our prototype with many complete, real-world programs. The domain in which has been mainly used so far are either small, self-contained programs and unit tests for algorithms and data structures. Both sequential and parallel programs can be verified. The source distribution of includes about 600 test cases for the model checker, many of which also use POSIX interfaces, leveraging the existing *virtual* mode of . As a first part of our evaluation, we took all those test cases and executed them in the new *passthrough* mode, that is, in a mode when acts as a runtime verifier. A total of 595 tests passed without any problems, 3 timed out due to use of blocking system calls and 9 timed out due to presence of infinite loops. Of course, since runtime verification is not exhaustive, not all errors present in the 595 tests were uncovered in this mode.
The second part of our evaluation was to write small programs that specifically test the *passthrough* and the *replay* mode:
- `pipe`, which creates a named pipe and two threads, one writer and one reader and checks that data is transmitted through the pipe
- `rw` which simply creates, writes to and reads from files
- `rw-par` in which one thread writes data to a file and another reads and checks that data
- `network`, a very simple HTTP client which opens a TCP/IP connection to a fixed IP address, performs an HTTP request and prints the result
We tested these programs in both the *passthrough* mode and in the *replay* mode. While very simple, they clearly demonstrate that the approach works. The source code of those test programs is also available online[^5]. Clearly, our verifier incurs appreciable overhead, since it interprets the program, instead of executing it directly. Quantitative assessment of the runtime and memory overhead is subject to future work (more complex test cases are required).
Conclusions and Future Work {#sec:conclusion}
===========================
We have described an approach which allows us to take advantage of an existing software model checking tool in the context of runtime verification. On one hand, this approach makes model checking more useful by making it usable with real environments while retaining many of its advantages over testing. On the other hand, it makes existing model checking tools useful in cases when runtime verification is the favoured approach. The approach is lightweight, since the modification to the model checker is small and self-contained. The other component required in our approach, the stand-in operating system, is also reasonably portable between model checkers. The overall effort associated with our approach is small, compared to implementing two dedicated tools (a model checker and a runtime verifier).
In the future, we plan to remove the limitations described in Section \[sec:limitations\] and offer a production-ready implementation of both a passthrough and a replay mode in 4. Since the results of the preliminary evaluation are highly encouraging, we firmly believe that a runtime verification mode based on the ideas laid out in this paper will be fully integrated into a future release of .
[^1]: This work has been partially supported by the Czech Science Foundation grant No. 15-08772S and by Red Hat, Inc.
[^2]: In microkernel and other design schools, syscalls in the traditional sense only exist as an abstraction, and are implemented through some form of inter-process communication.
[^3]: The `assume` primitive is a counterpart to `assert` and has a similar interface. It is customary that a single boolean value is given as a parameter to the `assume` statement (function call), representing the assumed condition.
[^4]: <https://divine.fi.muni.cz/2017/passthrough/>
[^5]: <https://divine.fi.muni.cz/2017/passthrough/>
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'New quantum entropic inequality for states of system of $n\geq 1$ qudits is obtained. The inequality has the form of quantum subadditivity condition of bipartite qudit system and coincides with this subadditivity condition for the system of two qudits. The general statement on existence of the subadditivity condition for arbitrary probability distribution and arbitrary qudit-system tomogram is formulated. The nonlinear quantum channels creating the entangled states from separable ones are discussed.'
---
\[sh\]
****
-----------------------------------------------------------------
Subadditivity condition for spin-tomograms and density matrices
of arbitrary composite and noncomposite qudit systems
-----------------------------------------------------------------
[**V. N. Chernega, O. V. Man’ko$^*$, V. I. Man’ko** ]{}
[*P. N. Lebedev Physical Institute, Russian Academy of Sciences\
Leninskii Prospect 53, Moscow 119991, Russia*]{}
$^*$Corresponding author e-mail: [email protected]
[**Keywords:**]{} entropy, information, tomographic probability, qubits, qudit, subadditivity condition, nonlinear quantum channels.
Introduction
============
The probability distributions are characterized by Shannon entropy [@Shanon]. The state of quantum systems, identified with density matrices [@Landau; @Landau1; @vonNeuman; @vonNeuman1] are characterized by von Neumann entropy. For the pure states identified with the wave functions the von Neumann entropy is equal to zero. The entropies correspond to order in the system [@Holevo]. For complete order in the classical system the Shannon entropy equals to zero. For composite classical and quantum systems there exist some inequalities related to the entropies of the system and its subsystems. The inequalities for von Neumann entropies of bipartite quantum system mean that the sum of the entropies of the subsystems is larger or equal to the entropy of the composite system. Analogous inequality holds for Shannon entropy [@Shanon] of the bipartite system. Recently, [@Mancini96; @OlgaJRLR1997; @DodPLA; @OgaJETP; @OlgaBregence] it was shown that the quantum states can be identified with tomographic probability distributions called quantum tomograms both for discrete spin (qudit) states and for the systems with the continious variables like system of interacting oscillators. In view of this the inequalities known for classical probability distributions can be obtained also for quantum tomograms [@FoundPhysRita; @Fedele; @Vaxsha; @Turin; @ChernegaJRLRv32; @OlgaVova]. Recent review of probability vector properties both in classical and quantum domains is presented in [@JRLRN1Maromo2014]. Recently, it was clarified [@mamapapa] that the inequalities like subadditivity condition known for bipartite system can be found also for noncomposite systems. The idea of this approach is based on qubit portrait method of qudit states suggested in [@Vovf] and applied to study entanglement properties of bipartite qudit systems in [@Lupo]. There exist [@LiebSeiringer] some inequalities for von Neumann entropy of bipartite system connecting “classical” and quantum entropies. The aim of our work is to use the approach of extending the inequalities known for composite systems considered in [@Vovacompositesystems] and to obtain new inequalities for tomographic entropies both for composite and noncomposite quantum systems. The model of quantum mechanics based on classical gaussian probability distribution is elaborated in [@Khrennikov; @Khrennikov1; @Khrennikov2].
The paper is organized as follows. In second section we discuss probability vectors and entropic inequality for bipartite system. In third section we generalized subadditivity condition for arbitrary probability vector ${\bf P}$. In fourth section we review the method of the portrait of density matrices and in fifth section we discuss, as an example, system states with density $6\times6$-matrices. In sixth section we discuss nonlinear chains of maps of probability vectors. The conclusions and perspectives are given in seventh section.
Probability vectors and entropic inequality for bipartite system
================================================================
Let us consider a set of $N$ nonnegative numbers $p_1,p_2,\ldots,p_N$ such that $\sum_{k=1}^{N}p_k=1$. The set of the numbers can be identified with a probability vector ${\bf P}=(p_1,p_2,\ldots,p_N$) where the numbers $p_k$ $(k=1,2,\ldots,N)$ are related to the results of measuring a system random variable. The variable is assumed to give $N$ different values. These numbers $p_k$ provide the probability to get $k^{th}$ value of the random variable. For systems of qudits the components of probability vector ${\bf P}$ can be identified with the $n$ values of qudit state tomograms $w({\bf m},u)=\langle{\bf m}|u\rho u^+|{\bf m}\rangle$, where $\rho$ is density matrix, $u$ is unitary matrix and vector ${\bf m}=({\bf m}_1,{\bf m}_2,\ldots,{\bf m}_n)$ with ${\bf m}_k=(-j_k,-j_k+1,\ldots,j_k)$ being spin $j_k$ projection. If one considers the system which contains two subsystems (bipartite system) the measuring the values of two random variables gives the table of $n=N\cdot M$ nonnegative numbers $p_{kj}$ $(k=1,2,\ldots,N,\, j=1,2,\ldots,M)$. The numbers provide joint probability distribution associated with the results of measuring two random variables. The joint probability distribution is normalised, i.e. $$\label{eq.1}
\sum_{k=1}^N\sum_{j=1}^M p_{kj}=1.$$ If one measures only one of these two random variables the joint probability distribution determines the marginal probability distribution $$\label{eq.2}
{\cal P}_k=\sum_{j=1}^M p_{kj},\quad\sum_{k=1}^N{\cal P}_k=1.$$ Another marginal probability distribution describing the results of measuring the second random variable reads $$\label{eq.3}
\Pi_j=\sum_{k=1}^N p_{kj}, \quad\sum_{j=1}^M\Pi_j=1.$$ If the random variables are independent (there is no correlations between the subsystems of the bipartite system) the numbers $p_{kj}$ have the factorized form $$\label{eq.4}
p_{kj}={\cal P}_k\Pi_j.$$ The marginal distributions can be associated with two probability vectors $\vec{\cal P}=({\cal P}_1,{\cal P}_2,\ldots,{\cal P}_N)$ and ${\bf\Pi}=(\Pi_1,\Pi_2,\ldots,\Pi_M)$. The table of numbers $p_{kj}$ also can be described by the probability vector ${\bf P}$. In fact any column vector can be considered as rectangular matrix. Then the vector (rectangular matrix) ${\bf P}$ is expressed in terms of two rectangular matrices (vector $ \vec{\cal P}$ and ${\bf\Pi}$) as their direct product $$\label{eq.5}
{\bf P}=\vec{\cal P}\otimes{\bf \Pi},\quad {\bf P}=(P_1,P_2,\quad\ldots,\quad P_{NM}).$$ It means that we use the invertable map of natural numbers onto pairs of integers $n\Longleftrightarrow(kj)$ which explicitly reads $$\label{eq.6}
1\Longleftrightarrow 11,\quad 2\Longleftrightarrow 21,\quad\ldots,\quad N\Longleftrightarrow N1,\quad N+1\Longleftrightarrow 21,\quad \ldots,\quad n\Longleftrightarrow N\cdot M.$$ In fact we code the natural numbers $1,2,\,\ldots, \,n=N\cdot M$ by pairs of the natural numbers $(k j)$ where $k=1,2,\ldots,\,N$, $j=1,2,\ldots,\,M$. Let us for simplicity assume that $N\leq M$. Any probability distribution is characterized by Shannon entropy [@Shanon]. For example the joint probability distribution $p_{kj}$ for bipartite system has the Shannon entropy $H(1,2)$ determined as $$\label{eq.7}
H(1,2)=-\sum_{k=1}^{N}\sum_{j=1}^{M} p_{kj}\ln\,p_{kj}.$$ The marginal probability distributions have the Shannon entropies $H(1)$ and $H(2)$ of the form $$\label{eq.7}
H(1)=-\sum_{k=1}^N{\cal P}_k\,\ln{\cal P}_k,\quad H(2)=-\sum_{j=1}^M\Pi_j\,\ln\Pi_j.$$ It is worthy to note that the entropy $H(1,2)$ can be written in the form $$\label{eq.8}
H\equiv H(1,2)=-\sum_{n=1}^{N M} P_n\ln\,P_n.$$ For all these entropies we introduce the vector notations. The entropy $$\label{eq.10}
H=-{\bf P}\ln{\bf P}, \quad H(1)=-\vec{\cal P}\,\ln\,\vec{\cal P}, \quad H(2)=-{\bf \Pi}\,\ln{\bf \Pi}.$$ In the formula (\[eq.10\]) we used the following definition: $${\bf x}\ln{\bf x}=\equiv\sum_{\alpha=1}^L x_{\alpha}\ln x_\alpha.$$ It means that ${\bf x}=(x_1,x_2,\ldots,x_L)$, and $\alpha=1,2,\ldots,L.$ Using the vector notations gives the possibility to describe the Shannon entropy of bipartite system with two random variables associated with the joint probability distribution $p_{k j}$ and the system with one random variable associated with the probability distribution $p_k$ by identical formulas presented in (\[eq.10\]). The only difference between the expressions $H,\,H(1)$ and $H(2)$ is that the “scalar product” in (\[eq.10\]) is evaluated for the vectors which have different number of components. In (\[eq.10\]) the vector $\vec{\cal P}$ has $N$ components, the vector ${\bf \Pi}$ has $M$ components and the vector ${\bf P}$ has $n=N\cdot M $ components. This difference can be removed. In fact, since $\lim_{x\rightarrow 0}x\ln x=0$ we can consider vectors $\vec{\cal P}$ and ${\bf \Pi}$ as vectors with $n=N M$ components by adding the zero components to the initial vectors, i.e. $$\label{eq.11}
\vec{\cal P}=({\cal P}_1,{\cal P}_2,\ldots,{\cal P}_N,0,0,\ldots,{\cal P}_{M N}=0),$$ $$\label{eq.12}
{\bf \Pi}=( \Pi_1, \Pi_2,\ldots, \Pi_M,0,0,\ldots, \Pi_{N M}=0).$$ Using these new vectors we do not change the values of the entropies, i.e. in formulas (\[eq.10\]) we have the same expressions but all the probability–vectors ${\bf P}$, $\vec{\cal P}$ and ${\bf \Pi}$ are considered as vectors with $n=N M$ components. It is known that the marginals ${\cal P}_k$ and $\Pi_j$ of joint probability distribution $p_{k j}$ satisfy the entropic inequality called subadditivity condition which reads $$\label{eq.13}
H(1,2)\leq H(1)+H(2),$$ where the Shannon entropies are given by (\[eq.6\])-(\[eq.8\]). In explicit form this inequality reads $$\label{eq.14}
-\sum_{k=1}^N\sum_{j=1}^M p_{k j}\ln p_{k j}\leq-\sum_{k=1}^N{\cal P}_k\ln{\cal P}_k-\sum_{j=1}^M\Pi_j\ln\Pi_j.$$ For the case of independent random variables $p_{k j}={\cal P}_k\Pi_j$ one has equality $$\label{eq.15}
H(1,2)= H(1)+H(2).$$ The Shannon mutual information is defined as the difference of entropies $$\label{eq.16}
I= H(1)+H(2)-H(1,2).$$ This information satisfies the nonnegativity condition $I\geq0$.
Using vector notations we can write the subadditivity condition (\[eq.14\]) in the form $$\label{eq.17}
-{\bf P}\ln{\bf P}\leq -\vec{\cal P}\ln\vec{\cal P}-{\bf \Pi}\ln{\bf \Pi},$$ where all the probability–vectors have $N\cdot M$ components.
The Shannon information is expressed in terms of the probability $n$-vectors as $$\label{eq.18}
I= -\vec{\cal P}\ln\vec{\cal P}-{\bf \Pi}\ln{\bf \Pi}+{\bf P}\ln{\bf P}.$$ Here $n=N\cdot M$.
Generalization of subadditivity condition\
for arbitrary probability vector ${\bf P}$
==========================================
The subadditivity condition (\[eq.17\]) written as inequality for three probability $n$-vectors ${\bf P}$, $\vec{\cal P}$ and ${\bf \Pi}$ provides the possibility to generalyze the inequality and to prove that such inequality takes place for arbitrary probability $n$-vectors. To clarify this issue let us express the $n$-vectors $\vec{\cal P}$ and ${\bf \Pi}$ in terms of two stochastic $n\times n$-matrices $M_{12}$ and $M_{21}$ and vector ${\bf P}$.
In fact one can observe that the following equalities hold $$\label{eq.19}
\vec{\cal P}=M_{12}{\bf P},\quad {\bf \Pi}=M_{21}{\bf P},$$ where the stochastic matrices $M_{12}$ and $M_{21}$ read $$\label{eq.20}
M_{12}=\left(
\begin{array}{cccc}
1_M & 0_M & \ldots & 0_M \\
0_M & 1_M & \ldots & 0_M \\
\ldots & \ldots & \ldots & \ldots \\
0_M & 0_M & \ldots & 1_M \\
\cdot & 0_S & \cdot & \cdot \\
\end{array}
\right), \quad
M_{21}=\left(
\begin{array}{cccc}
1_N & 0_N & \ldots & 0_N \\
0_N & 1_N & \ldots & 0_N \\
\ldots & \ldots & \ldots & \ldots \\
0_N & 0_N & \ldots & 1_N \\
\cdot & 0_Q & \ldots & \cdot \\
\end{array}
\right).$$ Here the rectangular matrices $1_M$ and $0_M$ with one row and $M$ columns read $$\label{eq.21}
1_M=(1,1,\ldots,1), \quad 0_M=(0,0,\ldots,0).$$ The zero rectangular matrix $0_S$ has $N\cdot M-N$ rows and $N\cdot M$ columns. The $N\times N$ - blocks in the matrix $M_{21}$ are the unity $N\times N$-matrix $1_N$ and zero $N\times N$-matrix $0_N$. The zero matrix $0_Q$ contains $N\cdot M-M$ rows and $N\cdot M$ columns. Using formula (\[eq.19\]) we can rewrite subadditivity condition (\[eq.17\]) known for joint probability distribution of bipartite system in the form $$\label{eq.22}
-{\bf P}\ln{\bf P}\leq -(M_{12}{\bf P})\ln(M_{12}{\bf P})-(M_{21}{\bf P})\ln(M_{21}{\bf P}).$$ We get the inequality (\[eq.22\]) as the property of joint probability distribution of bipartite system. But it is obvious that this inequality is the inequality which is valid for arbitrary set of $n=N\cdot M$ nonnegative numbers $(p_1,p_2,\ldots,p_{N M})$. In view of this one can formulate the general statement: Given arbitrary probability vector ${\bf P}$ with $n$ components where the integer $n$ can be presented in the product form of two integers $n=N\cdot M$, $N\leq M$. Then the inequality (\[eq.22\]) holds where the matrices (\[eq.20\]) are two stochastic matrices containing only zeros and unities. The inequality (\[eq.22\]) is valid also for all $n!$ vectors ${\bf P}_{per}$ obtained from the initial vector ${\bf P}$ by means of permutations of the indices $(1,2,\ldots,n)$ labeling the vector components. It means $$\label{eq.23}
-{\bf P}\ln{\bf P}=-{\bf P}_{per}\ln{\bf P}_{per}\leq -(M_{12}{\bf P}_{per})\ln (M_{12}{\bf P}_{per})-(M_{21}{\bf P}_{per})\ln(M_{21}{\bf P}_{per})$$ It is worthy to note that the integer $n$ can have different product decomposition $n=\bar N\bar M$. The equality (\[eq.22\]) and (\[eq.23\]) take place also with new matrices $\bar M_{12}, \bar M_{21}$ obtained from (\[eq.20\]) by the substitution $N\rightarrow\bar N$ and $M\rightarrow\bar M$. It is worthy to note that the inequality (\[eq.22\]) holds for arbitrary probability vector ${\bf P}$ which corresponds to a point on simplex including the vectors which have some zero components. We use the remark to extend our inequality (\[eq.22\]) for arbitrary probability $n$-vectors including the case of prime number $n$. To write the inequality for such probability $n$-vector ${\bf P}$ we construct new vector ${\bf P}'=(p_1,p_2,\ldots,p_n,0,0,,\ldots,p_{n'}=0)$. The $n'$-vector ${\bf P}'$ has $n'$ components. We added the appropriate quantity of zero components to the initial $n$-vector ${\bf P}$ such that the new integer $n'$ has the product form $n'=N'M'$. It is clear that there are many ways to construct such vectors with different integers $n'\geq n$. All these vectors will satisfy the subadditivity condition. Another generalisation of obtained inequality can be formulated for arbitrary set of nonnegative numbers $x_1,x_2,\ldots,x_n$. These numbers correspond to a point on the cone. Using the map $$x_k\rightarrow p_k=\frac{x_k}{\sum_{j=1}^n x_j}$$ and applying the inequality (\[eq.22\]) to the vector $$\frac{{\bf x}}{\sum_{j=1}^n x_j}={\bf P}$$ we get inequality for arbitrary finite set of $n$ nonnegative numbers $x_k$, i.e. $$\label{eq.24}
-{\bf x}\ln{\bf x}\leq -(M_{12}{\bf x})\ln(M_{12}{\bf x})-(M_{21}{\bf x})\ln(M_{21}{\bf x})+(\sum_{j=1}^n x_j)\ln(\sum_{j=1}^n x_j).$$ Thus we proved that the coordinates of a point on a cone satisfy the analog of subadditivity condition with extra terms in the right-hand side of (\[eq.24\]). For arbitrary integers $n$ the stohastic matrices $M_{12}$ and $M_{21}$ can be written in fixed canonical form . We can introduce the information on the cone which is the difference of the right hand side and left-hand side of Eq.(\[eq.24\]), i.e. $$\label{eq.25}
I_{\bf x}=-(M_{12}{\bf x})\ln(M_{12}{\bf x})-(M_{21}{\bf x})\ln(M_{21}{\bf x})+{\bf x}\ln{\bf x}+(\sum_{j=1}^n x_j)\ln(\sum_{j=1}^n x_j).$$ If $\sum_{j=1}^n x_j=1$ we have the point on the simplex and the information $I_{{\bf x} }$ becomes the analog of Shannon information which we introduced for arbitrary probability distribution described by a probability vector ${\bf P}$. It reads $$\label{eq.26}
I_{\bf p}=-(M_{12}{\bf P})\ln(M_{12}{\bf P})-(M_{21}{\bf P})\ln(M_{21}{\bf P})+{\bf P}\ln{\bf P}\geq0.$$ There exist $n!$ informations $I_{{\bf p}}$ obtained from (\[eq.26\]) by replacing probability vector ${\bf P}\rightarrow {\bf P}_{per}.$ In case of bipartite systems and $n=N\cdot M$, where $N$ and $M$ correspond to outcomes of two random variables the information $I_{\bf p}$ coincides with Shannon mutual information. The meaning of introduced informations $I_{{\bf p}_{per}}$ and informations (\[eq.26\]) introduced for arbitrary probability vector ${\bf P}$ needs the additional clarification.
Portrait of density matrices
============================
We apply the analogous method to get the positive map of $n\times n$ density matrix $\rho(1,2)$ of bipartite system with $n=N\cdot M$, $N\leq M$. In fact, if the state $\rho(1,2)$ is simply separable state, i.e. $\rho(1,2)=\rho(1)\otimes\rho(2)$ and $\rho(1)$ is $N\times N$-matrix, $\rho(2)$ is $M\times M$-matrix we can observe that the $N\times N$ matrix $\rho(1)$ is given by the following procedure. Namely, the matrix elements $\rho_{k j}(1)$, $k,j,=1,2,\ldots,N$ are given as the first $N$ vector components of $N\cdot M$ vectors $\vec\rho_1(1),\vec\rho_2(1),\ldots,\vec\rho_N(1)$ where $$\label{100}
\vec\rho_1(1)=M_{12} {\bf R}_1, \quad \vec\rho_2(1)=M_{12} {\bf R}_2, \quad\dots,\quad \vec\rho_N(1)=M_{12} {\bf R}_N.$$ Here the $N\cdot M$ matrix $M_{12}$ is given by Eq.(\[eq.20\]). The $N\cdot M$ vectors ${\bf R}_j$, $j=1,2,\ldots N$ have the components $$\label{101}
({\bf R}_j)_{k\alpha}=\rho_{k j}(1)\rho_{\alpha\alpha}(2), \quad k=1,2,\ldots,N, \,\alpha=1,2,\ldots,M.$$ Thus we used the invertable map of integers $1,2,\ldots,N$, $\alpha=1,\ldots,M$ onto the pairs of integers $k,\alpha$ $1\leftrightarrow 11, \,2\leftrightarrow 21,\ldots,\,n \leftrightarrow N M$ to label the components of the vector ${\bf R}_j$.
If the matrix $\rho(1,2)$ had the generic form with matrix elements $\rho_{k\alpha j\beta}(1,2)$ we have the positive map $\rho(1,2)\rightarrow\rho(1)$ given by the same formula (\[100\]) with changed vectors ${\bf R}_j$. Namely the vectors ${\bf R}_j$ have the components $$\label{103}
({\bf R}_j)_{k \alpha}=\rho_{k\alpha\,j\alpha}(1,2).$$ There is no sum over indices $\alpha$. Thus the described construction provides the map of the $N\cdot M$ density matrix $\rho(1,2)$ onto the density matrix which also can be considered as $N\cdot M$ matrix $\bar \rho(1)$ of the form $$\label{104}
\bar\rho(1)=\left(
\begin{array}{cc}
\rho(1) & 0_1 \\
0_1^{tr} & 0_{n-N} \\
\end{array}
\right).$$ Here $0_1$ is zero rectangular matrix with $N$ rows and $n-N$ columns, the matrix $0_{n-N}$, where $n=N\cdot M$ has the zero matrix elements. Analogous construction can be applied to get the map $\rho(1,2)\rightarrow\rho(2)=\mbox{Tr}_1\rho(1,2)$. The explicit form of this map can be obtained from (\[104\]) by using the known matrix of map of vectors ${\bf a}\otimes{\bf b}\longleftrightarrow{\bf b}\otimes {\bf a}$ given by the matrix $S$ such that $$\label{105}
({\bf a}\otimes{\bf b})_k=\sum_{m=1}^n S_{km}({\bf b}\otimes{\bf a})_m.$$ Using the matrix $S$ we can reduce the problem of finding the expression for the matrix $\rho(2)$ to the problem discussed above with the replacement discussed above $1\leftrightarrow 2,\, N\leftrightarrow M$. The $N\cdot M$ matrices $\bar\rho(1)$ and $\bar\rho(2)$ satisfy the subadditivity condition $$\label{105a}
-\mbox{Tr}\bar\rho(1)\ln\bar\rho(1)-\mbox{Tr}\bar\rho(2)\ln\bar\rho(2)\geq-\mbox{Tr}\bar\rho(1,2)\ln\bar\rho(1,2).$$ Thus for arbitrary $n\times n$ matrix $\rho$, where $n=N\cdot M$ we can obtain two matrices $\rho(1)$ and $\rho(2)$ applying to the initial matrix $\rho$ the map which naturally can be applied to the bipartite matrix $\rho(1,2)$. The matrix $\rho$ can be considered as density matrix of one qudit only. Nevertheless the associated with it matrices $\bar\rho(1)$ and $\bar\rho(2)$, satisfy the subadditivity condition (\[105a\]).
Example of system states with $6\times6$ - matrices
===================================================
To demonbstrate our approach let us consider the example of $n=6$. We can consider the density matrix $\rho_{k j}$, $k,\,j\,=1,2,\ldots,6$ as the density matrix of one qudit with $j=5/2$. The matrix $\rho$ reads $$\label{106}
\left(
\begin{array}{cccccc}
\rho_{11} &\rho_{12} &\rho_{13} &\rho_{14} & \rho_{15} & \rho_{16} \\
\rho_{21} &\rho_{22} &\rho_{23} & \rho_{24} &\rho_{25} &\rho_{26} \\
\rho_{31} &\rho_{32} & \rho_{33} & \rho_{34} & \rho_{35} &\rho_{36} \\
\rho_{41} & \rho_{42} &\rho_{43} &\rho_{44} &\rho_{45} &\rho_{46} \\
\rho_{51} &\rho_{52} &\rho_{53} &\rho_{54} &\rho_{55} &\rho_{56} \\
\rho_{61} &\rho_{62} &\rho_{63} & \rho_{64} &\rho_{65} &\rho_{66} \\
\end{array}
\right)\equiv\left(
\begin{array}{cc}
\rho^{(1)}&\rho^{(2)} \\
\rho^{(3)} &\rho^{(4)} \\
\end{array}
\right).$$ Here matrices $\rho^{(k)},\, k=1,2,3,4$ are $3\times3$-matrices which constitute $\rho$. Let us take integers $N=2,\, M=3$. The $6\times6$ stochastic matrix $M_{12}$ reads $$\label{107}
M_{12}=\left(
\begin{array}{cccccc}
1 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 &0 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
\end{array}
\right).$$ The $6$-vectors ${\bf R}_1, \, {\bf R}_2$ read $$\label{108}
{\bf R}_1=\left(
\begin{array}{c}
\rho_{11} \\
\rho_{22} \\
\rho_{33} \\
\rho_{41} \\
\rho_{52} \\
\rho_{63} \\
\end{array}
\right),\quad {\bf R}_2=\left(
\begin{array}{c}
\rho_{14} \\
\rho_{25} \\
\rho_{36} \\
\rho_{44} \\
\rho_{55} \\
\rho_{66} \\
\end{array}
\right).$$ Applying the matrix $M_{12}$ to vectors (\[108\]) we get $2\times2$ - matrix $\rho(1)$ of the form $$\label{109}
\rho(1)=\left(
\begin{array}{cc}
\rho_{11}+\rho_{22}+\rho_{33} &\rho_{14}+\rho_{25}+\rho_{36} \\
\rho_{41}+\rho_{52}+\rho_{63} &\rho_{44}+\rho_{55}+\rho_{66} \\
\end{array}
\right)$$ The $6\times6$ - matrix $\bar\rho(1)$ has the form $$\label{110}
\bar\rho(1)=\left(
\begin{array}{cc}
\rho(1) &0_{24} \\
0_{24}^{tr} &0_4 \\
\end{array}
\right),\quad 0_{24}=\left(
\begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{array}
\right)$$ and $0_4$ is $4\times4$ - matrix with zero matrix elements. The matrix $\bar\rho(2)$ has the form $$\label{111}
\bar\rho(2)=\left(
\begin{array}{cc}
\rho(2) & 0 \\
0 & 0 \\
\end{array}
\right), \quad \rho(2)=\rho^{(1)}+\rho^{(4)}.$$ In general case of $n\times n$ - matrix $\rho$ one has analogous map $\rho\rightarrow\bar\rho(1)$, $\rho\rightarrow\bar\rho(2)$. The $M\times M$ - matrix $\rho(2)$ equals to the sum of $N$ blocks of the matrix $\rho$, i.e. $$\label{112}
\rho(2)=\sum_{k=1}^N\rho^{(k)}.$$ Each block $\rho^{(k)}$ is the $M\times M$ - matrix. These $N$ blocks constitute the density matrix of the state which is obtained by “decoherence” map from the initial matrix $\rho$. Namely we construct from $\rho$ the block–diagonal matrix $$\rho_d=\left(
\begin{array}{cccc}
\rho^{(1)} & 0 & 0 \\
0 & \rho^{(2)} & 0 \\
\ldots & \ldots & \ldots \\
0 & 0& \rho^{(N)} \\
\end{array}
\right)$$ keeping $N$ of the $M\times M$ - matrices and other matrix elements assume to be equal zero. Then we sum all these blocks. As result we get matrix $\rho(2)$. So, the matrix $\bar\rho(2)$ is a “portrait” of the initial matrix $\rho$. Another “portrait” is the matrix $\bar\rho(1)$ which for initial $6\times6$ - matrix $\rho$ is given by Eq.(\[110\]). There is possibility to make another map of the matrix $\rho$ onto two “portrait” matrices $\bar\rho(1)$ and $\bar\rho(2)$, namely we take $N=3$ and $M=2$. Then the $3\times3$ - matrix $\rho(1)$ reads $$\label{113}
\rho(1)=\left(
\begin{array}{ccc}
\rho_{11}+\rho_{22} &\rho_{13}+\rho_{24} &\rho_{15}+\rho_{26} \\
\rho_{31}+\rho_{42} &\rho_{33}+\rho_{44} &\rho_{35}+\rho_{46} \\
\rho_{51}+\rho_{62} &\rho_{53}+\rho_{64} &\rho_{55}+\rho_{66} \\
\end{array}
\right)$$ and the $2\times 2$ - matrix $\rho(2)$ is $$\label{114}
\rho(2)=\left(
\begin{array}{cc}
\rho_{11}+\rho_{33}+\rho_{55} &\rho_{12}+\rho_{34}+\rho_{56} \\
\rho_{21}+\rho_{43}+\rho_{65}&\rho_{22}+\rho_{44}+\rho_{66} \\
\end{array}
\right).$$ The subadditivity inequality for all the obtained pairs $\rho(1)$ and $\rho(2)$ (i.e. $\bar\rho(1)$, $\bar\rho(2)$) is given as $$\label{115}
-\mbox{Tr}(\rho(1)\ln\rho(1))-\mbox{Tr}(\rho(2)\ln\rho(2))=-\mbox{Tr}(\bar\rho(1)\ln\bar\rho(1))
-\mbox{Tr}(\bar\rho(2)\ln\bar\rho(2))
\geq-\mbox{Tr}(\rho\ln\rho).$$ The von Neumann quantum mutual information is given by difference $$\label{116}
I_{q}(\bar\rho(1),\bar\rho(2))=-\mbox{Tr}(\bar\rho(1)\ln\bar\rho(1))-\mbox{Tr}(\bar\rho(2)\ln\bar\rho(2))
+\mbox{Tr}(\rho\ln\rho).$$ The inequalities for entropies (\[115\]) are valid for $(N M)!$ matrices $\bar\rho(1)$, $\bar\rho(2)$ obtained by means of all permutations of integers $1,2,\ldots,n\rightarrow1_p,2_p,\ldots,n_p$, determining matrix elements of the $n\times n$ - matrix $\rho$. It is worthy to note that if one has any $n\times n$ - density matrix $\rho$ one cen construct the matrix $\rho_{n'}$, where $n'=n+p= N\cdot M$ which reads $$\label{117}
\left(
\begin{array}{cc}
\rho & 0 \\
0 & 0 \\
\end{array}
\right).$$ After this one obtains by described procedure all the maps $\rho_{n'}\rightarrow\bar\rho^{(n')}(1)$ and $\bar\rho^{(n')}(2)$ and the new subadditivity conditions are written for these matrices $$\label{118}
-\mbox{Tr}(\bar\rho^{(n')}(1)\ln\bar\rho^{(n')}(1))-\mbox{Tr}(\bar\rho^{(n')}(2)\ln\bar\rho^{(n')}(2))\geq
-\mbox{Tr}(\rho^{(n')}\ln\rho^{(n')})=-\mbox{Tr}(\rho\ln\rho).$$ Analogously the nonnegative mutual informations are given by the difference $$\label{119}
I_{\rho}'(\rho^{(n')}(1)),\rho^{(n')}(2))=-\mbox{Tr}(\bar\rho^{(n')}(1)\ln\bar\rho^{(n')}(1))
-\mbox{Tr}(\bar\rho^{(n')}(2)\ln\bar\rho^{(n')}(2))+\mbox{Tr}(\rho\ln\rho).$$ These informations depend on the maps of the matrix $\rho\rightarrow\rho^{(n')}(1)$, $\rho\rightarrow\rho^{(n')}(2)$. For example if one has the $5\times5$ - density matrix $\rho$ corresponding to qudit with $j=2$ (i.e. $n=5$ and one can take $n'=n+1=6$) the pairs of matrices $\rho(1)$ and $\rho(2)$ obtained by the described positive maps are, e.g. $$\label{116a}
\rho(1)=\left(
\begin{array}{ccc}
\rho_{11}+\rho_{22} &\rho_{13}+\rho_{24} &\rho_{15} \\
\rho_{31}+\rho_{42} &\rho_{33}+\rho_{44} &\rho_{35} \\
\rho_{51} &\rho_{53} &\rho_{55} \\
\end{array}
\right), \quad
\rho(2)=\left(
\begin{array}{cc}
\rho_{11}+\rho_{33}+\rho_{55} &\rho_{12}+\rho_{34} \\
\rho_{24}+\rho_{43} &\rho_{22}+\rho_{44} \\
\end{array}
\right).$$ One has $$\label{117a}
-\mbox{Tr}(\rho(1)\ln\rho(1))-\mbox{Tr}(\rho(2)\ln\rho(2))\geq -\mbox{Tr}(\rho\ln\rho).$$ Other pairs are also obtained by means of coding the pairs for generic $6\times6$ - density matrix $\rho_{k j}$ and assuming that all the matrix elements $\rho_{k6}$ and $\rho_{6 j}$ equal to zero. Then all the subadditivity conditions for qudit $j=2$ states are obtained from constructed one by permutations of the integers $1,2,3,4,5\mapsto1_p,2_p,3_p,4_p,5_p$ labeling matrix elements of the matrix $\rho_{k j},$ $(k,j=1,2,3,4,5).$
Nonlinear maps of probability vectors
=====================================
Let us discuss the possibility to make a general map of probability vector ${\bf p}=(p_1,p_2,\ldots,p_n)$ onto probability vector $\vec\Pi=(\Pi_1({\bf p}),\Pi_2({\bf p}),\ldots,\Pi_m({\bf p}))$ and the vector components of vector $\vec\Pi$, i.e. $\Pi_k({\bf p})$ are some functions of the vector ${\bf p}$. If $n=m$ and for particular case of linear functions we have the form of the map $$\label{eqq1}
\vec\Pi({\bf p})=M{\bf p},$$ where the $n\times n$ - matrix $M$ has the matrix elements with the property $\sum_{k=1}^n M_{k j}=1$. If Eq.(\[eqq1\]) provides the linear map for all the vectors ${\bf p}$ belonging to symplex the matrices $M$ are stochastic matrices with nonnegative matrix elements. If Eq.(\[eqq1\]) provides the linear map for vector belonging to some domain in the symplex the matrices $M$ can have negative matrix elements. In all these cases the matrices $M$ form a semigroup. In particular, the stochastic matrices form the semigroup. One can introduce nonlinear maps of the probability vectors, choosing the specific functions $\Pi_k({\bf p})$ which preserve the properties of nonnegativity $\Pi_k({\bf p})\geq0$ and normalization $\sum_{k=1}^n\Pi_k({\bf p})=1$. The simple example of the nonlinear map is given by the rational function of the form $$\label{eqq2}
\Pi^{(s)}_k({\bf p})=\frac{p_k^s}{\sum_{k=n_1}^{n_2}p^s_k},\quad 1\leq k=n_1, \, n_1+1,\ldots, n_2\leq n.$$ Such map for $s=1$ gives, for example, conditional probability distribution. In fact,if a joint probability distribution $P(k,j)$ is written in the form of probability vector\
${\bf p}=(P(1,1),P(1,2),\ldots,P(1,n),P(2,1),\ldots,P(n,m))$ the Bayes formula for conditional probability $${\bf p}\mapsto P(k|j)=\frac{P(k,j)}{\sum_{k=1}^n P(k,j)}$$ has the form (\[eqq2\]) with choosing corresponding indices. Particular case of this map takes place for $n_1=1, \,n_2=n$. For example, if $s=2$ one has the map $$\label{eqq3}
{\bf p}\mapsto\vec\Pi^{(2)}({\bf p})=\Large(\sum_{k=1}^n p_k^2\Large)^{-1}(p_1^2,p_2^2,\ldots,p_n^2).$$ Such maps can be considered as examples of nonlinear classical channels. In quantum case we define the nonlinear map of density $n\times n$ - matrix $\rho$ onto density $m\times m$ - matrix $R$ (i.e. $\rho_{k j}\mapsto R_{\alpha\beta}(\rho))$ preserving the properties of density matrices $R^\dag=R,\,\mbox{Tr}R=1,\,R\geq0$. The case of linear map of density matrices is the particular case of the map under discussion. For example, the positive linear map [@Stringp; @Sud61] given in the form $R_{\alpha\beta}(\rho)=\sum_{k,j=1}^n B_{\alpha\beta,k j}\rho_{kj} $ and quantum channels corresponding to completely positive maps of the density matrix play important role in studying the quantum correlations in composite systems like entanglement phenomenon. The properties of the linear map positivity or complete positivity are coded by the properties of the matrix $B_{\alpha\beta,kj}$ [@Sud61]. The nonlinear maps of the density matrices which we call “nonlinear quantum channels” are characterized by the functions $R_{\alpha\beta}(\rho_{k j})$. One of the simple examples corresponding to example of probability vector transform (\[eqq2\]) reads $$\label{eqq4}
R=\rho^s\frac{1}{\mbox{Tr}\rho^s},\quad s=2,3,\ldots,\infty.$$ The map provides the new density matrix with larger purity which in generic case for $s\rightarrow\infty$ gives the pure state. The map can create entanglement, e.g. for two-qubit $X$-states. An analog of classical Bayes formula for conditional probability distribution given by nonlinear map (\[eqq2\]) for the matrix $\rho_{k j}$ $(k,j=1,2,\ldots,n)$ has the form $$\label{eqq5}
\rho_{k j}\mapsto R^{(m)}_{k'j'}=\frac{\rho_{k' j'}}{\sum_{k=1}^m\rho_{k k}}, k',j'=1,2,\ldots, m< n.$$ The nonlinear positive map can be given in the form of map of the qudit tomogram. It is known (see, e.g. [@MarmoManko]) that density matrix of arbitrary qudit system state with density matrix $\rho(1,2,\ldots,N)$ with $N$ subsystems is described by the tomographic probability distribution (qudit tomogram) which determines the density matrix. The probability vector $\vec w(u)$ corresponding to the density matrix has the vector components depending on unitary matrix $u$ and has the form $$\label{eqq6}
\vec w(u)=|u u_o|^2\vec\rho.$$ Here $\vec\rho$ is the vector which has components equal to eigenvalues of the density matrix. The unitary matrix $u_0$ has as the columns the corresponding eigenvectors of the density matrix. Using the expression (\[eqq6\]) for tomogram of any qudit system state we formulated the general statement. Let us consider first case where the tomographic probability vector $\vec w(u)$ has $n=N\cdot M$ components. Then applying Eq.(\[eq.22\]) to the vector we have inequality $$\label{eqq7}
-|u_0|^2\vec\rho\ln|u u_o|^2\vec\rho\leq-M_{12}|u u_0|^2\vec\rho\ln(M_{12}|u u_0|^2\vec\rho)-M_{21}|u u_0|^2\vec\rho\ln(M_{21}|u u_0|^2\vec\rho).$$ This inequality is valid for tomogram of composite or noncomposite qudit system. The system is described by density matrix with eigenvalues providing $N\cdot M$ - vector $\vec\rho$ and corresponding eigenvectors combined into unitary matrix $u_0$. If the number $n\neq N\cdot M$ we introduce the vector with $n'= N\cdot M$ components where $n'=n+s$ adding $s$ extra zero components to the vector $\vec\rho$. The orthostohastic matrix $|u u_0|^2$ is extended also and replaced by the marix $$\left(
\begin{array}{cc}
|u u_0|^2 & 0 \\
0 & 1_s \\
\end{array}
\right)$$ where $1_s$ is $s\times s$-matrix.
Any map of density matrix, i.e. $\vec\rho\mapsto\vec\rho'$, $u_0\mapsto u_0'$ provides the map of the tomographic vector $\vec w(u)$. The rational map of density matrix (\[eqq4\]) is equivalent to the map of probability vector $\vec\rho$ given by (\[eqq2\]) with $n_1=1,\, n_2=n$. Thus, any linear or nonlinear map od the probability vector $\vec\rho$ which has the components equal to the density matrix eigenvalues yields the nonlinear positive map of the density matrix. Other nonlinear positive maps can be associated with change of the unitary matrix $u_0$, e.g. by means of linear map $u_0\mapsto u_0'=T u_0$, where $T$ is the unitary transform of the eigenvectors of the density matrix $\rho(1,2,\ldots,N)$. One has the entropic inequalities for the probability vector (\[eqq2\]) $(n_1=1,\,n=n_2)$ $$\label{eqq7}
-\sum_{k=1}^n\Pi_k^{(s+1)}({\bf p})\ln\Pi_k^{(s+1)}({\bf p})\leq-\sum_{k=1}^n\Pi_k^{(s)}({\bf p})\Pi_k^{(s)}({\bf p})\leq-{\bf p}\ln{\bf p}, \quad s=1,2,3,\ldots$$ and for von Neumann entropy of the density matrix $R$ (\[eqq4\]) one has $$\label{eqq8}
-\mbox{Tr}\Large[\large(\rho^s\frac{1}{\mbox{Tr}\rho^s}\large)\ln\large(\rho^s\frac{1}{\mbox{Tr}\rho^s}\large)\geq
-\mbox{Tr}\Large[\large(\rho^{s+1}\frac{1}{\mbox{Tr}\rho^{s+1}}\large)\ln\large(\rho^{s+1}\frac{1}{\mbox{Tr}\rho^{s+1}}
\large).$$ One can introduce the positive map as the convex sum of the terms $\rho^s/\mbox{Tr}\rho^s$, i.e. $$\rho\Rightarrow R=\sum_s p_s\large(\rho^s/\mbox{Tr}\rho^s\large), \quad 0\leq p_s\leq 1,\quad \sum_s p_s=1.$$ The map gives the example of nonlinear channel. For composite bipartite systems the unitary transform which makes from the entangled states, separable states can be given by matrix $A$ such that $$\label{eqq9}
A=u_{01}\otimes u_{02},$$ where the matrices $u_{ok}$ are unitary local transform matrices. For example, if the bipartite system with density matrix $\rho(1,2)$ has the eigenvectors providing unitary matrix $u_0$ such that $$\label{eqq10}
\rho(1,2)=u_0\rho_d u^+_0, \quad \rho_d=\left(
\begin{array}{cccc}
\rho_1 & 0 & 0 & 0 \\
0& \rho_2 & 0 & o \\
\ldots & \ldots & \ldots & \ldots \\
0 & 0 & 0 & \rho_n \\
\end{array}
\right)$$ any unitary matrix $A$ of the form $$\label{eqq11}
A=(u_{01}\otimes u_{02})u_0^+$$ gives the tomographic vector $$\label{eqq12}
\vec w_A(u)=|u(u_{01}\otimes u_{02})|^2\vec\rho$$ which is the tomogram of separable state. The applied transform depends on the state.
Conclusion {#conclusion .unnumbered}
==========
To conclude we formulate main results of our work. We obtain new inequalities for both probability vectors and density matrices. These inequalities are analogs of known subadditivity conditions which are valid for composite systems but the inequalities are shown to be valid for arbitrary probability vectors and arbitrary density matrix including the case of systems without subsystems. We discussed the positive nonlinear maps of probability vectors and density matrices. The nonlinear maps can be used to create entangled states from the separable states. We considered explicitly the examples of the density matrix in six-dimensional Hilbert space which can be identified either with qubit-qutrit composite system state or with the state of single qudit with $j=5/2$. It is worthy to note that one can introduce Bell inequalities for noncomposite system. Also one can study the violation of the inequalities. The Bell inequality for the qudit with $j=3/2$ has the form of the inequality for two qubit system. The entangled states for the qudit ($j=3/2$) are the states for which equality for density matrix of the state in the form of separability condition is not valid. The Bell inequality can be violated. This problem will be considered in future publication.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
In this article, it is proved that for any probability law $\mu$ over $\mathbb{R}$ and a drift field $b: \mathbb{R} \rightarrow \mathbb{R}$ and killing field $k : \mathbb{R} \rightarrow \mathbb{R}_+$ which satisfy hypotheses stated in the article and a given terminal time $t > 0$, there exists a string $m$, an $\alpha \in (0,1]$, an initial condition $x_0 \in \mathbb{R}$ and a process $X$ with infinitesimal generator $\left(\frac{1}{2}\frac{\partial^2}{\partial m \partial x} + b \frac{\partial}{\partial m} - \frac{\partial K}{\partial m}\right)$ where $k = \frac{\partial K}{\partial x}$ such that for any Borel set $B \in {\cal B}(\mathbb{R})$, $$\mathbb{P}\left ( X_t \in B | X_0 = x_0 \right ) = \alpha \mu (B).$$
Firstly, it is shown the problem with drift and without killing can be accommodated, after a simple co-ordinate change, entirely by the proof in [@N1]. The killing field presents additional problems and the proofs follow the lines of [@N1] with additional arguments.
author:
- 'John M. Noble [^1]'
title: Time Homogeneous Diffusion with Drift and Killing to Meet a Given Marginal
---
[**Key words:**]{} Time homogeneous gap diffusion, drift, killing, Kreĭn strings, marginal distribution.
Introduction
============
Results and Method of Proof
---------------------------
Let $\mu$ be a probability measure over $\mathbb{R}$, $b : \mathbb{R} \rightarrow \mathbb{R}$ and $k : \mathbb{R} \rightarrow \mathbb{R}_+$ given drift and killing functions. Set
$$\label{eqtildb} \widetilde{b}(x) = \left\{ \begin{array}{ll} b(x) & x \in \mbox{suppt}(\mu) \\ 0 & x \not \in \mbox{suppt}(\mu) \end{array}\right. , \qquad B(x) = \left\{ \begin{array}{ll} \int_{[0,x]} \widetilde{b}(y) dy & x \geq 0 \\ - \int_{[x,0)} \widetilde{b}(y) dy & x < 0 \end{array}\right.$$
where $\mbox{suppt}(\mu)$ denotes the support of the measure $\mu$. Let
$$\label{eqtildk} \widehat{k}(x) = \left\{ \begin{array}{ll} k(x) & x \in \mbox{suppt}(\mu) \\ 0 & x \not \in \mbox{suppt}(\mu) \end{array}\right. , \qquad K(x) = \left\{ \begin{array}{ll} \int_{[0,x]} \widehat{k}(y) dy & x \geq 0 \\ - \int_{[x,0)} \widehat{k}(y) dy & x < 0. \end{array}\right.$$
\[hybmu\] The target probability measure, drift and killing $(\mu, b, k)$ satisfy the following conditions.
1. $B$ from and $K$ from are absolutely continuous with respect to $\mu$.
2. \[hycond2\] Let $l_-(x) = \sup\{y \in \mbox{suppt}(\mu) \cap (-\infty, x)\}$ and let $l_+(x) = \inf\{y \in \mbox{suppt}(\mu) \cap (x,+\infty)\}$, then
$$\label{eqhypb2} \sup_{x \in \mathbb{R}}\lim_{h \downarrow 0} \int_{l_-(x) - h}^{l_+(x)+h} |\widetilde{b}(x)| dx < 1$$
where $\widetilde{b}$ is from .
3. \[hycond3\] Let $c: (0,1) \rightarrow \mathbb{R}_+$ denote the function defined by:
$$\label{eqcgamm} c(x) = \frac{\left ( \ln \frac{1}{x} \right ) - (1 - x)}{(1 - x)^2}.$$
Let $\gamma$ satisfy: $$\label{eqgamdef} \gamma = \frac{1}{2} \left( 1 - \sup_{x \in \mathbb{R}} \lim_{h \downarrow 0} \int_{l_-(x) - h}^{l_+(x)+h} |\widetilde{b}(x)| dx \right ).$$
Then $(b,\mu)$ satisfies:
$$\label{eqhypb}
\int_{-\infty}^\infty \left( \int_{0 \wedge x}^{0 \vee x} e^{F (b,y)} dy \right) \mu(dx) < +\infty$$
where
$$\label{eqefffF} F (b,y) = 2\left ( \int_{0 \wedge y}^{0 \vee y} \left | \widetilde{b}(x)\right | dx + c(\gamma ) \sup_{\underline{t} : (0\wedge y) = t_0 < \ldots < t_n = (0 \vee y)}\sum_{i=0}^{n-1} \left\{ \left(\int_{t_i}^{t_{i+1}} |\widetilde{b}(x)| dx\right)^2 \right\} \right )$$
and $\widetilde{b}$ is defined by . Here the maximum is taken over sequences of length $n$ for all $n \in \mathbb{N}$.
4. \[hypart4\] $\lim_{x \rightarrow \pm +\infty} \frac{\partial K}{\partial \mu} (x) = 0$.
Let $z_+ = \sup\{ x \in \mbox{suppt}(\mu)\}$ and $z_- = \inf\{x \in \mbox{suppt}(\mu)\}$. Then $\frac{\partial K}{\partial \mu}(x)$ is defined to be $0$ for $x > z_+$ and $x < z_-$.
This article addresses the following problem: suppose that $(\mu, b, k)$ satisfy Hypothesis \[hybmu\]. It is shown that there exists a string measure $m$, an $\alpha \in (0,1]$ and an $x_0 \in \mathbb{R}$ such that
$$\label{eqpr2} \frac{1}{2}\frac{\partial^2}{ \partial m \partial x} + \frac{\partial B}{\partial m} \nabla_m - \frac{\partial K }{ \partial m }$$
where $\nabla_m$ is defined in Section \[secdefigp\] is the infinitesimal generator of a process $X$ satisfying
$$\mathbb{P}(X_t \in B | X_0 = x_0, X_t \not \in \{D\}) = \mu(B) \qquad \forall B \in {\cal B}(\mathbb{R})$$
where $D$ is a cemetery state, $X_t \in \{D\}$ denotes that the process has been killed by time $t$ and
$$\alpha := 1 - \mathbb{P}\left( X_t \in \{D\}\right) > 0.$$
If $t$ is replaced by an exponential time, $\alpha$, $x_0$ and $m$ are uniquely determined and an explicit construction is given. If $t$ is a deterministic time, only existence is given, although the method of proof may indicate how to provide approximations.
#### Remarks on Hypothesis \[hybmu\]
1. \[rq1\] For $\gamma$ defined by , it follows from that $\gamma > 0$ (where the inequality is strict).
2. \[rq2\] For $x \in (0,1)$, the power series expansion of $1-x$ gives: $$\log \frac{1}{x} = - \log (1 - (1 - x)) = \sum_{j=1}^\infty \frac{(1 - x)^j}{j}$$
so that
$$c(x) = \sum_{j=0}^\infty \frac{(1 - x)^{j}}{j+2}.$$
It follows that $\lim_{x \uparrow 1} c(x) = \frac{1}{2}$, $c(x)$ is decreasing in the range $x \in (0,1)$, $c(x) < +\infty$ for $x > 0$ and $\lim_{x \downarrow 0} c(x) = +\infty$.
3. It is straightforward (and easier) to obtain the existence of a measure $m$ which gives an $\alpha > 0$ and a process with infinitesimal generator $$\label{eqprob2}
\left(\frac{1}{2} \frac{\partial^2}{\partial m \partial x} + \frac{\partial B}{\partial m}\nabla_m \right) - k$$ for a given drift $b$ and killing $k$, which has distribution $$\mathbb{P}(X_\tau \in \{D\}) = 1-\alpha \qquad \mathbb{P}(X_\tau \in A) = \alpha \mu (A) \qquad \forall A \in {\cal B}(\mathbb{R}),$$
where $\tau$ is the terminal time, $\mu$ is the prescribed measure, $D$ denotes the cemetery state and $\{X_\tau \in \{D\}\}$ denotes that the process has been killed by time $\tau$. As with the case discussed in this article, with similar proofs, there is uniqueness and explicit construction when stopped at an independent geometric / exponential time. When finding a process with generator given by , the hypothesis on the killing field $k$ may be relaxed; Part \[hypart4\] of Hypothesis \[hybmu\] is irrelevant for this problem, since it is only connected with ensuring that the limit of processes on atomised state spaces is not dead with probability 1 by the terminal time for a generator given by . This issue resolves itself without this hypothesis for the generator given by .
The line of proof is as follows:
1. Discrete time and finite state space are considered; conditions under which a suitable Markov chain with a given distribution when stopped at an independent geometric time are established. The solution, when it exists, is unique and the construction is explicit.
2. This is then extended to establish conditions under which there exists a Markov chain with a given distribution when stopped at an independent negative binomial time. This uses the fact that a negative binomial variable is the sum of independent identically distributed geometric variables and uses a fixed point theorem. For the problem of finding an infinitesimal generator of the form of or , substantial modifications of the arguments in [@N1] are required when killing is introduced.
3. Limits of negative binomial times by reducing the time mesh are taken to obtain a time with Gamma distribution as in [@N1]. Limits are then taken to obtain a deterministic time. The arguments are along similar lines to those of [@N1], with some crucial modifications.
4. Finally, arbitrary state space is considered. As in [@N1], the target measure is approximated by a sequence of atomised measures. The drift is dealt with by a change of co-ordinates and the sequence of atomised measures in the transformed co-ordinates is considered. The killing is dealt with by considering the process without killing, together with the conditional distribution of the killing time. Both of these converge. The problem is to ensure that the diffusion coefficient does not tend to infinity and the probability that the process has been killed does not tend to $1$ as the limit is taken. The proof requires Hypothesis \[hybmu\] Part \[hypart4\].
Background
----------
The problem of constructing a gap diffusion with a given law with compact support at an independent exponential time has been discussed fully by Cox, Hobson and Obłój in [@CHO] (2011). The problem of constructing a martingale diffusion that has law $\mu$ at a fixed time $t$ has been solved by Jiang and Tao in [@JT] (2001) under certain smoothness assumptions. Recently, Forde in [@F] (2011) extended the work of Cox, Hobson and Obłój [@CHO] to provide a process with prescribed joint law for the process at an independent exponential time $\tau$ and its supremum over the time interval $[0,\tau]$.
For any prescribed measure $\mu$, the problem of finding a martingale diffusion with given marginal $\mu$ at a fixed time $t > 0$ was solved in [@N1] (2013). Independently and simultaneously, Ekström, Hobson, Janson and Tysk [@EHJT] (2013) found a different proof; in [@EHJT], the target distribution is again approximated by atomic measures, but general results from algebraic topology to conclude existence of a limit. In [@Mon] (1972), Monroe constructs a general symmetric stable process with a prescribed marginal at a fixed time, but does not require that the resulting process satisfies a martingale property.
Motivation
----------
The subject of strong Markov processes generated by Kreĭn-Feller generalised second order differential operators and, more specifically, the inverse problem of computing a function $a$ to give a solution $f$ to the parabolic equation
$$\frac{\partial f}{\partial s} = a\left(\frac{1}{2} \frac{\partial^2}{\partial x^2} + b\frac{\partial}{\partial x} - k \right ) f$$
is of interest in its own right. Here $a$ is understood as $\frac{1}{m^\prime(x)}$ and the initial condition at $s = 0$ is a dirac mass $f(0,x) = \delta_{x_0}(x)$ at point $x_0 \in \mathbb{R}$; the end condition $f(t,x)$ for $s = t > 0$ is prescribed.
The operator $\frac{\partial^2}{\partial m \partial x}$ and its spectral theory were introduced by Kreĭn [@Kr] (1952) and, for a more developed treatment, Kac and Kreĭn [@KacKr] (1958). A lucid account of the spectral theory is given by Dym and McKean [@DMc] (1976). The operator, viewed as the generator of a strong Markov process, is discussed in Knight [@Kn] (1981) where it is referred to as a [*gap diffusion*]{} and Kotani and Watanabe [@KW] (1982) where it is referred to as a [*generalised diffusion*]{}.
In recent years, interest in gap diffusion operators and their associated processes has been strongly renewed by applications to the field of modelling financial markets. The general motivating problem within finance is that of automating the pricing and risk management of derivative securities. This is discussed by Carr and Nadtochiy in [@Carr] (2014), where the Local Variance Gamma model is developed to do this.
The addition of drift and killing have importance when the prices of both the numéraire and the asset are modelled by stochastic processes. The covariation between the price of the numéraire and the price of the asset changes the drift of the discounted asset price process, hence the requirement to incorporate a drift $b$. The inclusion of a killing field extends the class of models available.
#### Acknowledgements
I thank Peter Carr for suggesting the problem of drift and killing and indicating the importance to financial applications. I also thank an anonymous referee whose thorough reading and careful comments led to substantial improvements.
Definitions, Infinitesimal Generators and Processes {#secdefigp}
===================================================
A definition of the operator ${\cal G} = \frac{\partial^2}{\partial m \partial x}$ used in may be found in Dym and McKean [@DMc] or Kotani and Watanabe [@KW]. The Kotani Watanabe definition is more useful in this setting, because it extends to strings defined over the whole real line. The domain of the operator, denoted ${\cal D}({\cal G})$ is the space of functions $f \in {\cal B}(\mathbb{R})$ such that there exists an $m$-measurable function $g$ satisfying $\int_{x_1}^{x_2} g^2(x) m(dx) < +\infty$ for all $-\infty < x_1 < x_2 < +\infty$ such that
$$f(x) = f(x_0) + (x - x_0) f_-^\prime (x_0) + \int_{x_0}^x \int_{x_0 -}^y g(z) m(dz) dy \qquad \forall -\infty < x_0 < x < -\infty$$
where $f_-^\prime$ denotes the left derivative, $\int_a^b$ denotes integration over $(a,b]$ and $\int_{a-}^b$ denotes integration over the closed interval $[a,b]$. The quantity ${\cal G}f = \frac{\partial^2}{\partial m \partial x} f$ is defined as $g$.
The operator $\nabla_m$ is defined as follows: let $z_- = \inf\{x \in \mathbb{R} | x \in \mbox{suppt}(m)\}$ and let $z_+ = \sup\{x \in \mathbb{R} | x \in \mbox{suppt}(m)\}$. For $x \in (z_-, z_+)$, define: $$\left\{\begin{array}{l} x^*_m (x) = \lim_{\epsilon \downarrow 0} \inf\{y > x+ \epsilon | y \in \mbox{suppt}(m)\} \\ x_{*m} (x)= \lim_{\epsilon \downarrow 0} \sup\{y < x-\epsilon | y \in \mbox{suppt}(m)\}. \end{array}\right.$$
The operator $\nabla_m$ is defined on functions $f \in {\cal D}({\cal G})$ as:
$$\label{eqnabm}
\nabla_m f(x) = \left\{ \begin{array}{ll} \lim_{h \rightarrow 0} \frac{f(x^*_m(x) + h) - f(x_{*m}(x) - h)}{x^*_m - x_{*m} + 2h} & x \in (z_-,z_+) \\ 0 & \mbox{other}. \end{array}\right.$$
#### Note 1
This definition of $\nabla_m$ is the definition associated with the drifts of the Markov chains under discussion. It boils down to Equation (given later) for a discrete state space and to $\frac{\partial}{\partial x}$ for $m(dx) = dx$.
#### Note 2
The definition of the domain of the operator is not discussed further in this article, since the method of proof does not require it, but it is reasonably straightforward to show that, for any process $X$ obtained as the limit (in law) of processes with generators which converge to (as described in the article), if $f \in {\cal D}({\cal G})$, then for all $t > 0$, $F(t,.) \in {\cal D}({\cal G})$ where $F(t,x) := \mathbb{E}[f(X_t)|X_0 = x]$. It follows from the analysis given that $F$ thus defined satisfies:
$$\frac{\partial}{\partial t} F = \frac{1}{2}\frac{\partial^2}{\partial m \partial x} F + \frac{\partial B}{\partial m} \nabla_m F - \frac{\partial K}{\partial m}F.$$
When $m$ has a well defined density $m^\prime > 0$, let $a = \frac{1}{m^\prime}$ then may be written as:
$$\label{eqtwodens} a \left(\frac{1}{2}\frac{\partial^2}{\partial x^2} + b\frac{\partial}{\partial x} - k \right).$$
When a finite discrete state space ${\cal S} = \{i_1, \ldots, i_M\}$ is considered, the generator may be written as:
$$a \left(\frac{1}{2}\Delta + b\nabla - k \right).$$
where, for discrete state space, the definitions of the operators $\Delta$ and $\nabla$ are given in Definition \[deflapder\] below and, with abuse of notation, $b(i_j) : j= 2, \ldots, M-1$ here represents the sizes of the atoms of $B$ (Equation ), with $b(i_1) = b(i_M) = 0$.
\[deflapder\] Consider a state space $${\cal S} = \{i_1, \ldots, i_M\}, \qquad i_1 < \ldots < i_M.$$
For a function $f : {\cal S} \rightarrow \mathbb{R}$, the [*Laplace operator*]{} $\Delta$ is defined as:
$$\label{laplace} \left\{ \begin{array}{ll} \Delta f(i_1) = \Delta f(i_M) = 0 & \\
\Delta f(i_j) = \frac{2}{(i_{j+1} - i_j)(i_j - i_{j-1})}\left(\frac{i_j - i_{j-1}}{i_{j+1} - i_{j-1}} f(i_{j+1}) - f(i_j) + \frac{i_{j+1} - i_j}{i_{j+1} - i_{j-1}} f(i_{j-1})\right) & j = 2, \ldots, M-1 \end{array} \right.$$
The [*derivative operator*]{} $\nabla$ is defined as:
$$\label{nabla} \left\{\begin{array}{ll} \nabla f(i_1) = \nabla f(i_M) = 0 & \\
\nabla f(i_j) = \frac{f(i_{j+1}) - f(i_{j-1})}{i_{j+1} - i_{j-1}} & 2 \leq j \leq M - 1 \end{array} \right.$$
#### Remarks
1. If the function $f$ is defined on an interval $(y_0,y_1)$, $f \in C^2((y_0,y_1))$ (twice differentiable with continuous second derivative) and a sequence ${\cal S}_n$ is considered, where ${\cal S}_n = \{i_{n,1}, \ldots, i_{n,M_n}\}$, $i_{n,j} < i_{n,j+1}$, $i_{n,1} \downarrow y_0$, $i_{n,M_n} \uparrow y_1$ and $\lim_{n \rightarrow +\infty} \max_j (i_{n,j+1} - i_{n,j}) = 0$, with $\Delta_{{\cal S}_n}$ the operator defined on ${\cal S}_n$, then $\lim_{n \rightarrow +\infty} \Delta_{{\cal S}_n} f= \frac{d^2}{dx^2} f$. Note that the function $f$ has been defined on $C^2((y_0,y_1))$. The sense in which convergence is meant is: let $j_n(x) = \max\{ j : i_{n,j} \leq x\}$ then for all $x \in (y_0, y_1)$,
$$\begin{aligned}
\lefteqn{\lim_{n \rightarrow +\infty} \left | \frac{2}{(i_{n,j_n(x)+1} - i_{n,j_n(x)})(i_{n,j(x)} - i_{n,j(x)-1})}\left(\frac{i_{n,j(x)} - i_{n,j(x)-1}}{i_{n,j(x)+1} - i_{n,j(x)-1}} f(i_{n,j(x)+1})\right. \right.} \\&& \left. \left. \hspace{20mm} - f(i_{n,j(x)}) + \frac{i_{n,j(x)+1} - i_{n,j(x)}}{i_{n,j(x)+1} - i_{n,j(x)-1}}f(i_{n,j(x)-1})\right) - \frac{d^2}{dx^2}f(x) \right | = 0.\end{aligned}$$
2. If the function $f$ is defined on the whole interval $(y_0,y_1)$ and $f \in C^1(\mathbb{R})$ (differentiable, continuous first derivative) and a sequence ${\cal S}_n$ is considered where ${\cal S}_n = \{i_{1,n}, \ldots, i_{M,n}\}$, $i_{j,n} < i_{j+1,n}$ $i_{1,n} \downarrow y_0$ and $i_{M,n} \uparrow y_1$ and $$\lim_{n \rightarrow +\infty} \max_j (i_{j+1,n} - i_{j,n}) = 0,$$ with $\nabla_n$ the operator defined on ${\cal S}_n$, then $\lim_{n \rightarrow +\infty} \nabla_n f = \frac{d}{dx} f$ in the sense that for $f \in C^1((y_0,y_1))$ (differentiable with continuous derivative, for all $x \in (y_0,y_1)$ $$\lim_{n \rightarrow +\infty} \left | \frac{f(i_{j_n(x)+1}) - f(i_{j_n(x)-1})}{i_{j_n(x)+1} - i_{j_n(x)-1}} - \frac{d}{dx}f(x) \right | = 0$$ if $\lim_{n \rightarrow +\infty} \max_j(i_{n,j+1} - i_{n,j}) = 0$.
#### Notation
For finite discrete state space ${\cal S} = \{i_1, \ldots, i_M\}$, $b(i_j)$ and $k(i_j)$ are used to denote the sizes of the [*atoms*]{} of $B$ and $K$ respectively from Equations and . This is a minor abuse of notation, since in and , $b$ and $k$ are used to denote the [*derivatives*]{} of $B$ and $K$ on $\mbox{suppt}(\mu)$. This notation will be used throughout when dealing with the problem on discrete state space.
Furthermore, for finite discrete state space ${\cal S} = \{i_1, \ldots i_M\}$, let $b: {\cal S}\backslash\{i_1,i_M\} \rightarrow \mathbb{R}$ and $k :{\cal S}\backslash\{i_1,i_M\} \rightarrow \mathbb{R}_+$ denote the drift and killing respectively. The following notation will be used: let $\underline{b} = (b_2, \ldots, b_{M-1})$ where (with slight abuse of notation) $b_j = b(i_j)$ and $\underline{k} = (k_2, \ldots, k_{M-1})$ where (same notation) $k_j = k(i_j)$. The notation $\widetilde{k}_j$ will be used to denote the following:
$$\label{eqtildek} \left\{\begin{array}{ll} \widetilde{k}_j = (i_{j+1} - i_j)(i_j - i_{j-1}) k_j & j \in (2, \ldots, M-1) \\ \widetilde{k}_1 = \widetilde{k}_M = 0 \end{array}\right.$$
and $\underline{\widetilde{k}} = (\widetilde{k}_1, \ldots, \widetilde{k}_{M})$.
For all results with finite state space, the following hypothesis will be required:
\[hyb\] For a discrete, finite state space ${\cal S} = \{i_1, \ldots i_M\}$ where $i_1 < \ldots < i_M$, the vector $\underline{b}$ satisfies the condition:
$$\label{eqbhyp} -\frac{1}{i_{j+1} - i_j} < b_j < \frac{1}{i_j - i_{j-1}} \qquad j = 2, \ldots, M-1.$$
Set
$$\label{eqdefq} \left\{ \begin{array}{ll}
q_{j,j+1} = \frac{i_j - i_{j-1}}{i_{j+1} - i_{j-1}}(1 + (i_{j+1} - i_j) b_j) & j = 2, \ldots, M-1 \\
q_{j,j-1} = \frac{i_{j+1} - i_j}{i_{j+1} - i_{j-1}}(1 - (i_j - i_{j-1}) b_j) & j = 2, \ldots, M-1
\end{array} \right.$$
Condition is necessary and sufficient to ensure that $q_{j,j+1}$ and $q_{j,j-1}$ are non negative for each $j$. With these definitions of $\underline{b}$, $\underline{k}$ and $\underline{\widetilde{k}}$, the following definitions are made for the transitions (in discrete time) and the intensities (in continuous time) of the Markov processes that are of interest.
\[deftrmat\] Let $\underline{\lambda} = (\lambda_2, \ldots, \lambda_{M-1}) \in \mathbb{R}_+^{M-2}$. For $h < \frac{1}{\max_{j \in \{2,\ldots, M-1\}} \lambda_j (1 + \widetilde{k}_j)}$, let $\widetilde{P}^{(h)}(\underline{k}, \underline{\lambda})$ be the $M+1 \times M+1$ matrix defined by:
$$\label{eqptild} \left\{ \begin{array}{ll} \widetilde{P}^{(h)}_{j,M+1} (\underline{k}, \underline{\lambda}) = h \lambda_j \widetilde{k}_j & j=2,\ldots, M-1 \\
\widetilde{P}^{(h)}_{1,M+1} (\underline{k}, \underline{\lambda})= \widetilde{P}^{(h)}_{M,M+1}(\underline{k}, \underline{\lambda}) = 0 & \\ \widetilde{P}^{(h)}_{M+1,M+1}(\underline{k}, \underline{\lambda}) = 1 & \\ \widetilde{P}^{(h)}_{M+1,j}(\underline{k}, \underline{\lambda}) = 0 & j = 1, \ldots, M \\ \widetilde{P}^{(h)}_{jj}(\underline{k}, \underline{\lambda}) = 1 - \lambda_j (1 + \widetilde{k}_j)h & j = 2, \ldots, M-1 \\ \widetilde{P}^{(h)}_{11}(\underline{k},\underline{\lambda}) = \widetilde{P}^{(h)}_{MM}(\underline{k},\underline{\lambda}) = 1 & \\
\widetilde{P}^{(h)}_{12} (\underline{k}, \underline{\lambda})= \widetilde{P}^{(h)}_{M,M-1} (\underline{k}, \underline{\lambda}) = 0 & \\
\widetilde{P}^{(h)}_{j,j+1}(\underline{k}, \underline{\lambda}) = h \lambda_j q_{j,j+1} & j = 2, \ldots, M-1\\
\widetilde{P}^{(h)}_{j,j-1}(\underline{k}, \underline{\lambda}) = h \lambda_j q_{j,j-1} & j = 2, \ldots, M-1 \\ \widetilde{P}_{jk}^{(h)} (\underline{k}, \underline{\lambda}) = 0 & |j-k| \geq 2, \quad (j,k) \in \{1, \ldots, M\}^2 \end{array} \right.$$
Let $P^{(h)}(\underline{k}, \underline{\lambda})$ denote the $M \times M$ matrix defined by $P^{(h)}_{ij}(\underline{k}, \underline{\lambda}) = \widetilde{P}^{(h)}_{ij}(\underline{k}, \underline{\lambda})$ for $(i,j) \in \{1, \ldots, M\}^2$.
\[defintmat\] Let
$$\label{eqthedef} \Theta(\underline{k},\underline{\lambda}) = \lim_{h \rightarrow 0}\frac{1}{h} \left(\widetilde{P}^{(h)}(\underline{k},\underline{\lambda}) - I \right)$$
It is straightforward to see that the matrix $\Theta (\underline{k}, \underline{\lambda})$ satisfies:
$$\label{eqthetmat}
\left\{ \begin{array}{ll} \Theta_{j,M+1}(\underline{k}, \underline{\lambda}) = \lambda_j \widetilde{k}_j & j = 2, \ldots, M - 1\\
\Theta_{1,M+1} (\underline{k}, \underline{\lambda})= \Theta_{M,M+1}(\underline{k}, \underline{\lambda}) = 0 & \\
\Theta_{M+1,j}(\underline{k}, \underline{\lambda})= 0 & j = 1, \ldots, M+1\\
\Theta_{jj}(\underline{k}, \underline{\lambda}) = - \lambda_j ( 1 + \widetilde{k}_j) & j = 2, \ldots, M-1 \\
\Theta_{11}(\underline{k}, \underline{\lambda}) = \Theta_{MM}(\underline{k},\underline{\lambda}) = 0 & \\
\Theta_{12} (\underline{k}, \underline{\lambda}) = \Theta_{M,M-1}(\underline{k}, \underline{\lambda}) = 0 & \\
\Theta_{j,j+1}(\underline{k}, \underline{\lambda}) = \lambda_j q_{j,j+1} & j = 2,\ldots, M-1 \\
\Theta_{j,j-1}(\underline{k}, \underline{\lambda}) = \lambda_j q_{j,j-1} & j = 2, \ldots, M-1\\
\Theta_{j,k} (\underline{k}, \underline{\lambda}) = 0 & |j - k| \geq 2, \qquad (j,k) \in \{1, \ldots, M\}
\end{array}\right.$$
which is the [*intensity matrix*]{} of a Continuous Time Markov Chain on state space $\{1, \ldots, M+1\}$.
[**Note**]{} The dependence on $\underline{k}$ and $\underline{\lambda}$ for $\widetilde{P}^{(h)}(\underline{k}, \underline{\lambda})$, $P^{(h)}(\underline{k}, \underline{\lambda})$ and $\Theta(\underline{k}, \underline{\lambda})$ will be suppressed; these will be written as $\widetilde{P}^{(h)}$, $P^{(h)}$ and $\Theta$ respectively.
For $h < \frac{1}{\max_{j \in \{2, \ldots, M-1\}} \lambda_j ( 1 + \widetilde{k}_j)}$, $\widetilde{P}^{(h)}$ is the one-step transition matrix for a time homogeneous Markov process $X^{(h)}$, with time step length $h$, satisfying
$$\mathbb{P}(X^{(h)}_{h(t+1)} = i_k | X^{(h)}_{ht} = i_j) = \widetilde{P}^{(h)}_{jk}.$$
As discussed in [@N1], as $h \rightarrow 0$, the process $X^{(h)} \rightarrow X$ (convergence in the sense of finite dimensional marginals) to a continuous time Markov chain with intensity matrix $\Theta = \lim_{h \rightarrow 0}\frac{1}{h}(\widetilde{P}^{(h)} - I)$ from Definition \[defintmat\].
\[lmmfininfgen2\] Let ${\cal S} = \{i_1, \ldots, i_M\}$ and let $X$ be a continuous time Markov process on $S \cup \{D\}$ with transition intensity matrix from Definition \[defintmat\] Equation in the sense that
$$\left\{ \begin{array}{ll} \lim_{h \rightarrow 0}\frac{1}{h} \mathbb{P}\left(Y_h = i_k | Y_0 = i_j\right) = \Theta_{jk} & (j,k) \in \{1, \ldots, M+1\}^2 \qquad j \neq k,\\ \lim_{h \rightarrow 0}\frac{1}{h}\left(\mathbb{P}\left(Y_h = i_j | Y_0 = i_j\right) - 1 \right ) = \Theta_{jj} & j \in \{1, \ldots, M+1\} \end{array}\right.$$
Let $$\label{eqaj}\left\{\begin{array}{l} a_j = \lambda_j (i_{j+1} - i_j)(i_j - i_{j-1}) \qquad j = 2, \ldots, M-1\\ a_1 = a_M = 0\end{array}\right.$$
and (where the notation is clear) let $a : {\cal S} \rightarrow \mathbb{R}_+$ denote the function defined by $a(i_1) = a(i_M) = 0$, $a(i_j) = a_j$ for $j = 2, \ldots, M-1$. Then $Y$ has infinitesimal generator $$a \left(\frac{1}{2}\Delta + b \nabla - k\right).$$
#### Proof
Recall the definition of $\widetilde{\underline{k}}$ (Equation ). Let $f$ be a function defined on $\{i_1, \ldots, i_M\}$ and let $F(t,i_j) = \mathbb{E}_{i_j}\left [ f(Y_t)\right ]$. Then, for $j = 2, \ldots, M-1$,
$$\begin{aligned}
\frac{\partial}{\partial t} F(t, i_j) &=& \lim_{h \rightarrow 0} \frac{F(t+h, i_j) - F(t, i_j)}{h} = \lim_{h \rightarrow 0} \frac{1}{h} \mathbb{E}_{i_j} \left [f(Y_{t+h}) - f(Y_t) \right] \\ &=& \lim_{h \rightarrow 0} \frac{1}{h}\left(\mathbb{E}_{i_j}[F(t,Y_h)] - F(t,i_j)\right)\\
&=& \lambda_j\left( q_{j,j+1} F(t,i_{j+1}) - F(t,i_j) + q_{j,j-1} F(t,i_{j-1}) - \widetilde{k}_j F(t,i_j)\right) \\
&=& \lambda_j \left(\left(\frac{i_j - i_{j-1}}{i_{j+1} - i_{j-1}}F(t,i_{j+1}) - F(t,i_j) + \frac{i_{j+1} - i_j}{i_{j+1} - i_{j-1}} F(t,i_{j-1})\right)\right. \\&& + \left. \left( (i_{j+1} - i_j)(i_j - i_{j-1}) \right)b_j \left (\frac{F(t,i_{j+1}) - F(t,i_{j-1})}{i_{j+1} - i_{j-1}} \right ) - \widetilde{k}_j F(t,i_j)\right )\\
&=& a_j \left(\frac{1}{2}\Delta + b_j \nabla - k_j \right ) F(t,i_j) \end{aligned}$$
For $j \in \{1, M\}$,
$$\frac{\partial}{\partial t} F(t,i_j) = 0$$
as required.
Coordinate change to deal with the drift {#subcoc}
========================================
The addition of the drift $b$ can be dealt with through a simple change of co-ordinates, described here. The aim is to find a mapping of the process from space ${\cal S}$ (the state space of the process) to a space ${\cal R}$ such that the transformed process is drift free. For finite state space, ${\cal S} = \{i_1, \ldots, i_M\}$, $i_1 < \ldots < i_M$ the aim is to find a map $\kappa$ where (with abuse of notation) $\kappa_j = \kappa(i_j)$ for $j = 1, \ldots, M$ where $\kappa_j < \kappa_{j+1}$ for $j = 1, \ldots, M-1$ such that $q_{j,j-1}$ and $q_{j,j+1}$, for $j = 2, \ldots, M-1$, defined by , satisfy:
$$\label{eqkap1} q_{j,j+1} = \frac{\kappa_j - \kappa_{j-1}}{\kappa_{j+1} - \kappa_{j-1}} \qquad q_{j,j-1} = \frac{\kappa_{j+1} - \kappa_j}{\kappa_{j+1} - \kappa_{j-1}}.$$
Let
$$\delta_j = i_j - i_{j-1}, \qquad j = 2, \ldots, M.$$
Directly from and , it follows that $\kappa_{j} - \kappa_{j-1} = \epsilon_j$ for $j = 2, \ldots, M$ where $\epsilon_2, \ldots, \epsilon_M$ satisfy
$$\frac{\epsilon_j}{\epsilon_{j+1} + \epsilon_j} = \frac{\delta_j}{\delta_j + \delta_{j+1}} + \frac{\delta_j \delta_{j+1}}{\delta_j + \delta_{j+1}} b_j.$$
It follows that $\underline{\epsilon} = (\epsilon_2, \ldots, \epsilon_M)$ satisfies
$$\label{eqeps} \frac{\epsilon_{j+1}}{\epsilon_j} = \frac{\delta_{j+1}}{\delta_j}\frac{1 - \delta_j b_j}{1 + \delta_{j+1}b_j} \qquad j = 2, \ldots, M-1.$$
Clearly, does not determine $\underline{\kappa}$ uniquely; two additional conditions have to be specified, which represent centring and scaling. The following choice is made: let
$$\label{eqemminemplus}
e_- = \inf \left \{j : \sum_{i=1}^j p_i \geq \alpha \right \}, \qquad e_+ = \sup \left \{j : \sum_{i= j}^M p_i \geq \alpha \right \}.$$
where $0 < \alpha < 0.5$ is a number chosen such that $e_- < e_+$ (the inequality is strict). This is possible if ${\cal S}$ has $3$ or more distinct states. Let $K = (i_{e_+} - i_{e_-}) \vee 1$. Then $\underline{\epsilon} = (\epsilon_2, \ldots, \epsilon_M)$ defined by
$$\label{eqeps2} \epsilon_{j+1} = K
\frac{\delta_{j+1} \prod_{k=1}^j \left(\frac{1 - \delta_k b_k }{1 + \delta_{k+1} b_k}\right)}
{\sum_{a= e_-}^{e_+-1} \delta_{a+1} \prod_{k=1}^a \left(\frac{1 - \delta_k b_k}{1 + \delta_{k+1} b_k}\right ) }.$$
satisfies and $\kappa_{e_+} - \kappa_{e-} = K$.
Define $\underline{\kappa}$ in the following way: let $e$ satisfy $\sum_{i=1}^e p_i \geq \frac{1}{2}$ and $\sum_{i=e}^M p_i \geq \frac{1}{2}$ and set $\kappa_e = 0$. For $j \neq e$, set
$$\label{eqkappa}
\left\{ \begin{array}{ll} \kappa_{e+j + 1} = \kappa_{e + j} + \epsilon_{e+j+1} & j = 0, \ldots, M-e-1 \\ \kappa_{e - j -1} = \kappa_{e - j} - \epsilon_{e - j - 1} & j = 0, \ldots, e - 2
\end{array}\right.$$
where $\underline{\epsilon}$ is defined by . It is easy to see that $\underline{\kappa}$, thus defined, satisfies .
This choice ensures that $\underline{\kappa}$ is centred (so that in the transformed coordinates the process is a martingale with mean zero) and, furthermore, that when the discussion is extended in Section \[seccoord\] to the case of arbitrary measure $\mu$ on $\mathbb{R}$ with an appropriate sequence of atomised measures $\mu^{(N)}$, the processes with state spaces $\underline{\kappa}^{(N)}$ have suitable convergence properties.
A Function to Accommodate the Killing Field
===========================================
The method of proof adopted in this article is to try and rephrase the problem, as much as possible, in the language of [@N1] and to use as much of the technique from [@N1] as possible. The previous section introduced a co-ordinate change to deal with the drift $b$; under the co-ordinate change, the problem with drift, but without killing, reduces to that of [@N1]. The introduction of killing presents other problems: firstly, even without drift, the initial condition is no longer as clear as it was in [@N1] when the killing field is non trivial. It cannot be taken as simply the expectation of the target distribution, since the process in the time interval $[0,t]$, conditioned on being alive at time $t$, is no longer a martingale. Secondly, the process is killed at rate $\lambda_j \widetilde{k}_j$ on site $i_j$, where $\underline{\lambda}$ is the holding intensity vector which is to be computed. This feeds into the equation required to obtain the intensities and there is no longer an explicit expression, even for the process stopped at an independent exponential time, like the formula $\lambda_j = \frac{1}{p_j}{\cal F}_j(\underline{p})$ that was available in [@N1]. The function ${\cal G}$ described in this section plays the role of ${\cal F}$ in [@N1].
Let $\underline{p} = (p_1, \ldots, p_M)$ satisfy $\min_j p_j > 0$ and $\sum_{j=1}^M p_j = 1$. Let ${\cal S} = \{i_1, \ldots, i_M\}$, $i_1 < \ldots < i_M$ be a finite state space with $M$ elements; $\underline{i} = (i_1, \ldots, i_M)$ will be used to denote the elements of the space. Let $\underline{b} = (b_2, \ldots, b_{M-1})$ satisfy Hypothesis \[hyb\] and let $\underline{\kappa} = (\kappa_1, \ldots, \kappa_M)$ denote the coordinate change of $\underline{i}$ defined by Section \[subcoc\]. Let $\underline{k} = (k_2, \ldots, k_{M-1}) \in \mathbb{R}_+^{M-2}$ denote the killing field and let $\widetilde{\underline{k}}$ be defined by Equation . Let ${\cal G}_j(t,\underline{p}) : j \in \{ 1,\ldots, M\}$ be defined as follows:
$$\label{eqgeee} \left\{ \begin{array}{l} {\cal G}_1 = {\cal G}_M = 0 \\ {\cal G}_j = \frac{1}{tp_j} \frac{(\kappa_{j+1} - \kappa_{j-1})}{(\kappa_{j+1} - \kappa_j)(\kappa_j - \kappa_{j-1})}\times \left\{ \begin{array}{ll} \sum_{a = 1}^{j-1} (\kappa_j - \kappa_a)p_a (1 + t{\cal G}_a \widetilde{k}_a) & 2 \leq j \leq l-1 \\ \sum_{a = j+1}^M (\kappa_a - \kappa_j) p_a (1 + t {\cal G}_a \widetilde{k}_a) & l \leq j \leq M-1 \end{array}\right. \\ l : \kappa_{l-1} < \frac{\sum_{j=1}^{M} \kappa_j p_j (1 + t{\cal G}_j \widetilde{k}_j)}{\sum_{j=1}^M p_j (1 + t{\cal G}_j \widetilde{k}_j)} \leq \kappa_l.\end{array}\right.$$
Note that, for fixed $t$, ${\cal G} : \mathbb{S}^M \rightarrow \{0\} \times \mathbb{R}^{M-2}_+ \times \{0\}$, where $$\label{eqesem} \mathbb{S}^M = \left \{ (p_1, \ldots, p_M) : 0 < p_j < 1, \; j=1, \ldots, M,\; \sum_{j=1}^M p_j = 1 \right \}.$$
#### Notation
Throughout, a discrete target probability $\underline{p} = (p_1, \ldots, p_M)$ will be taken as a [*row*]{} vector.
The following lemma shows that such a function is well defined, which is a necessary step in accommodating the killing field.
\[lmmgeee\] For a given $\underline{p} \in \mathbb{S}^M$, there exists $({\cal G}_1,\ldots, {\cal G}_{M})$ satisfying Equation .
#### Proof
Consider $(\alpha_1, \ldots, \alpha_M)$ such that $\alpha_j > 0$ for all $j$ and $\sum_{j=1}^M \alpha_j = 1$. Now consider, for some $k \in \{2, \ldots, M-1\}$, $\beta_k \in (0,1)$ and, for $j \neq k$, $\beta_j = \frac{1 - \beta_k}{1 - \alpha_k}\alpha_j$. Then $\sum_{j=1}^M \beta_j = 1$. Let $y = \sum_{j=1}^M \kappa_j \alpha_j$ and let $z = \sum_{j=1}^M \kappa_j \beta_j$, then
$$z = y \frac{1 - \beta_k}{1 - \alpha_k} + \frac{(\beta_k - \alpha_k)}{1 - \alpha_k} \kappa_k \Rightarrow y = z + \frac{(\beta_k - \alpha_k)}{(1 - \beta_k)}(z - \kappa_k).$$
so that, for $z < \kappa_k$, it follows that $\beta_k < \alpha_k \Rightarrow y < \kappa_k$ and $\beta_k > \alpha_k \Rightarrow y < z < \kappa_k$. It therefore follows that $z < \kappa_k \Rightarrow y < \kappa_k$.
Recall that $\widetilde{k}_1 = \widetilde{k}_M = 0$ and let ${\cal G}_1^+ = {\cal G}_M^+ = {\cal G}_1^- = {\cal G}_M^- = 0$. For $j = 2, \ldots, M-1$ define ${\cal G}_j^+$ and ${\cal G}_j^-$ by:
$$\left\{ \begin{array}{ll} {\cal G}_j^- = \frac{1}{tp_j} \frac{(\kappa_{j+1} - \kappa_{j-1})}{(\kappa_{j+1} - \kappa_j)(\kappa_j - \kappa_{j-1})} \sum_{a = 1}^{j-1} (\kappa_j - \kappa_a)p_a (1 + t{\cal G}_a^- \widetilde{k}_a) & 2 \leq j \leq M-1 \\
{\cal G}_j^+ = \frac{1}{tp_j} \frac{(\kappa_{j+1} - \kappa_{j-1})}{(\kappa_{j+1} - \kappa_j)(\kappa_j - \kappa_{j-1})} \sum_{a = j+1}^M (\kappa_a - \kappa_j) p_a (1 + t {\cal G}_a^+ \widetilde{k}_a) & 2 \leq j \leq M-1 \end{array} \right.$$
Then these are well defined and positive. Define $x_{0,j}$ by: $$\left\{ \begin{array}{ll} x_{0,1} = \frac{\sum_{a=1}^M \kappa_a p_a (1 + t{\cal G}_a^+ \widetilde{k}_a)}{\sum_{a=1}^M p_a (1 + t{\cal G}_a^+ \widetilde{k}_a) } & \\
x_{0,j} = \frac{\sum_{a=1}^{j-1} \kappa_a p_a (1 + t{\cal G}_a^- \widetilde{k}_a) + \sum_{a=j}^M \kappa_a p_a (1 + t{\cal G}_a^+ \widetilde{k}_a)}{\sum_{a=1}^j p_a (1 + t{\cal G}_a^- \widetilde{k}_a) + \sum_{a=j+1}^M p_a(1 + t{\cal G}_a^+\widetilde{k}_a)} & j = 2,\ldots, M \\
x_{0,M+1} = \frac{\sum_{a=1}^M \kappa_a p_a (1 + t{\cal G}_a^- \widetilde{k}_a)}{\sum_{a=1}^M p_a (1 + t{\cal G}_a^- \widetilde{k}_a) }\end{array}\right.$$
Clearly $\kappa_1 < x_{0,j} < \kappa_M$ for each $j \in \{1, \ldots, M+1\}$. To prove the lemma, it is necessary and sufficient to show that there is a $j$ such that $\kappa_{j-1} < x_{0,j} \leq \kappa_j$. Note that $x_{0,M} < \kappa_M$. Suppose that $x_{0,j} < \kappa_{j-1}$. Then, it follows from the argument above that $x_{0,j-1} < \kappa_{j-1}$. If, furthermore, $x_{0,j-1} > \kappa_{j-2}$, then existence has been established; otherwise, proceed inductively. Since $\kappa_1 < x_{0,j} < \kappa_M$ for all $j$, the result follows.
Let
$$\label{eqx02} x_0 = \frac{\sum_{j=1}^{M} \kappa_j p_j (1 + t{\cal G}_j (t,\underline{p}) \widetilde{k}_j)}{\sum_{j=1}^M p_j (1 + t{\cal G}_j(t, \underline{p}) \widetilde{k}_j)}.$$
This will give the initial condition for the process for geometric / exponential stopping times. $x_0$ may be considered as the average of $\underline{\kappa}$ under the measure $Q(t,\underline{p}, {\cal G}(t,\underline{p}))$ where the quantity $Q$ is defined by below:
$$\label{eqqespj2} Q_j(s,\underline{p}, \underline{\lambda}) = \frac{p_j (1 + s \widetilde{k}_j \lambda_j)}{\sum_{i=1}^M p_i (1 + s \widetilde{k}_i \lambda_i)} \qquad j \in \{1, \ldots, M\}$$
(the killing field is considered fixed; this quantity will be considered as a function of time variable, the target probability and the intensities when it is used later).
Results {#secres}
=======
This section states the main results of the article, which are given as Theorems \[thgeexp2\], \[thnbt2\], \[thctlim2\] and \[thdrk\]. These theorems are stated separately, because each of them is of use in its own right. Firstly, Theorem \[thgeexp2\] concerns Exponential and Geometric times. In this setting, an explicit solution can be obtained; $\alpha$ and $\underline{\lambda}$ are determined uniquely and there are equations to produce the explicit values. Theorem \[thnbt2\] considers Negative Binomial and Gamma times. Uniqueness is not shown, but the result comes in terms of the solution to an explicit fixed point problem. Theorem \[thctlim2\] takes an appropriate limit to obtain the result for deterministic times. While the result of Theorem \[thctlim2\] is the objective, the result of Theorem \[thnbt2\] which is a step along the way has an interesting interpretation in terms of the Local Variance Gamma Model of Carr [@Carr] and is therefore stated as a theorem in its own right.
Theorems \[thgeexp2\], \[thnbt2\] and \[thctlim2\] consider discrete state spaces, while Theorem \[thdrk\] considers arbitrary probability measures where the measure and drift satisfy Hypothesis \[hybmu\].
For Theorems \[thgeexp2\], \[thnbt2\] and \[thctlim2\], let $\underline{p} = (p_1, \ldots, p_M) \in \mathbb{S}^M$, defined by . Let ${\cal S} = \{i_1, \ldots, i_M\}$, $i_1 < \ldots < i_M$ be a finite state space with $M$ elements and let $\underline{b} = (b_2, \ldots, b_{M-1})$ satisfy Hypothesis \[hyb\]. Let $\underline{k} = (k_2,\ldots, k_{M-1}) \in \mathbb{R}_+^{M-2}$ denote the killing field and let $\widetilde{\underline{k}} = (\widetilde{k}_1, \ldots, \widetilde{k}_{M})$ be defined by .
\[thgeexp2\] There exists a unique $\underline{\lambda} = (\lambda_2, \ldots, \lambda_{M-1}) \in \mathbb{R}^{M-2}_+$, $\alpha \in (0,1)$, $l \in \{2, \ldots, M\}$ and $\beta \in (0,1]$ such that for all
$$h \in \left ( 0, \min_{j \in \{2, \ldots, M-1\}} \left(\frac{1}{\lambda_j (1 + \widetilde{k}_j) }\right) \right ]$$
$\widetilde{P}^{(h)}$ given by Definition \[deftrmat\] is the one step transition matrix for a Markov chain $X^{(h)}$ with state space ${\cal S} = \{i_1, \ldots, i_M, D\}$ and time step length $h$ which satisfies
$$\alpha p_j = (1 - \beta) \mathbb{P}\left (X_{h\tau}^{(h)} = i_j | X_0^{(h)} = i_{l-1} \right ) + \beta \mathbb{P} \left (X_{h\tau}^{(h)}= i_j | X_0^{(h)} = i_l \right ) \qquad j = 1,\ldots, M,$$
where $\tau$ is independent of $X^{(h)}$ and satisfies $\tau \sim Ge(a)$ with $a = \frac{t}{t + h}$, so that $\mathbb{E}[h \tau] = \frac{a h}{1-a} = t$. The constant $\alpha$ satisfies:
$$\label{eqalph2} \alpha = \frac{1}{1 + t\sum_{j=1}^M p_j \lambda_j \widetilde{k}_j}$$
where $\lambda_1 = \lambda_M = 0$, while $\beta$ satisfies:
$$\label{eqbet2} \beta = \frac{x_0 - \kappa_{l-1}}{\kappa_l - \kappa_{l-1}}$$
where $x_0$ is defined by . The intensity vector $\underline{\lambda}$ satisfies:
$$\label{eqlamexpsol2} \lambda_j = {\cal G}_j \qquad j = 2, \ldots, M-1$$
where ${\cal G}$ is defined by , existence of such a function given by Lemma \[lmmgeee\]. Taking $h \rightarrow 0$, there exists a continuous time Markov chain $Y$ with state space ${\cal S}$, where for each $j = 2, \ldots, M - 1$, site $i_j$ has holding intensity $\lambda_j$ given by the same formula, and $Y$ satisfies
$$\alpha p_j = (1-\beta) \mathbb{P}(Y_\tau = i_j | Y_0 = i_{l-1}) + \beta \mathbb{P}(Y_\tau = i_j | Y_0 = i_l),$$
$\tau \sim Exp(1/t)$ (that is, exponential, with expected value $\mathbb{E}[\tau] = t$), $\alpha$ satisfies and $\beta$ satisfies . Let $\underline{a}$ satisfy with $\underline{\lambda}$ given by , then the infinitesimal generator of the process $Y$ is given by:
$$\label{eqinfgen2} a \left(\frac{1}{2} \Delta + b \nabla - k\right).$$
The quantity $\beta$ is interpreted in the following way: the process $X^{(h)}$ has initial condition $X^{(h)}_0 = x_0 \in (i_{l-1}, i_l]$ such that
$$\mathbb{P}(X_{0+}^{(h)} = i_l | X_0^{(h)} = x_0) = \beta \qquad \mathbb{P}(X_{0+}^{(h)} = i_{l-1}|X_0^{(h)} = x_0) = (1 - \beta).$$
Now consider negative binomial times.
\[thnbt2\] For any $r \geq 1$, there exists an $l \in \{2, \ldots, M\}$, a vector $\underline{\lambda} = (\lambda_2, \ldots, \lambda_{M-1}) \in \mathbb{R}_+^{M-2}$, a $\beta \in (0,1]$, and an $h_0 \in (0,1)$, such that for all $h \in (0,h_0)$, there is an $\alpha$ satisfying $$\label{eqalph} \alpha \in \left [ \left( 1 + \frac{(k\lambda)^* t}{r}\right)^{-r} ,1 \right ]$$
where $(k\lambda)^* = \max_{j \in \{2, \ldots, M-1\}} \widetilde{k}_j \lambda_j$ and $\widetilde{P}^{(h)}$ from Definition \[deftrmat\] is the one step transition matrix for a time homogeneous discrete time Markov chain $X^{(h)}$, time step length $h$ such that
$$(1 - \beta) \mathbb{P}(X_{h\tau}^{(h)} = i_j | X_0^{(h)} = i_{l-1}) + \beta \mathbb{P}(X_{h\tau}^{(h)} = i_j | X_0^{(h)} = i_l) = \alpha p_j \qquad j = 1,\ldots, M$$ where $\tau \sim NB(r, \frac{t}{t + hr})$, so that $\mathbb{E} \left[ h \tau \right ] = t$.
By taking the limit $h \rightarrow 0$, there is a continuous time, time homogeneous Markov chain $Y$ with transition intensity matrix $\Theta$ given by , Definition \[defintmat\], such that
$$(1 - \beta) \mathbb{P}(Y_T = i_j | Y = i_{l-1}) + \beta \mathbb{P}(Y_T = i_j |Y_0 = i_l) = \alpha p_j \qquad j = 1, \ldots, M$$
where $\alpha$ satisfies and $T \sim \mbox{Gamma}(r, \frac{r}{t})$; that is, $T$ is a Gamma time, with density function
$$\label{eqgamden} f_T(x) = \frac{r^r}{t^r}\frac{1}{\Gamma(r)}x^{r-1}e^{-xr/t} \qquad x \geq 0$$
and expected value $\mathbb{E}[T] = t$.
This is extended to deterministic time:
\[thctlim2\] For a given $t > 0$, there exists a vector $\underline{\lambda} = (\lambda_2, \ldots, \lambda_{M-1}) \in \mathbb{R}_+^{M-2}$, an
$$\label{eqalthcl2}\alpha \in \left [ \exp \left\{-t(k \lambda)^*\right\}, 1 \right ],$$
where $(k\lambda)^* = \max_{j \in \{2, \ldots, M-1\}} \lambda_j \widetilde{k}_j$, an $l \in \{2, \ldots, M\}$, a $\beta \in (0,1]$, such that $\Theta$ (Equation Definition \[defintmat\]) is the intensity matrix for a time homogeneous continuous time Markov chain $X$ such that
$$\beta \mathbb{P}(X_t = i_j | X_0 = i_l) + (1 - \beta) \mathbb{P}(X_t = i_j | X_0 = i_{l-1}) = \alpha p_j \qquad j = 1, \ldots, M.$$
Again, if $\underline{a}$ satisfies , then the infintesimal generator of $X$ is defined by .
Finally, the continuous limit in the space variable can be taken.
\[thdrk\] Let $\mu$ be a probability measure on $\mathbb{R}$, $b : \mathbb{R} \rightarrow \mathbb{R}$ a drift field and $k$ a killing field such that $(\mu, b, k)$ satisfy Hypothesis \[hybmu\], then there exists a string measure $m$, a point $x_0 \in \mathbb{R}$, an $\alpha \in (0,1]$ and a function $K$ satisfying Equation such that $\frac{1}{2}\frac{\partial^2}{\partial m \partial x} + \frac{\partial B}{\partial m}\nabla_m - \frac{\partial K}{\partial m}$ is the infinitesimal generator of a process $X$ which satisfies $$\mathbb{P}(X_t \leq x | X_0 = x_0) = \alpha \mu((-\infty, x]).$$
Here $B$ is defined by . The initial condition $X_0 = x_0$ is interpreted as follows: let ${\cal S}$ denote the support of $\mu$. Let $z_- = \sup\{y < z|y \in {\cal S}\}$ and $z_+ = \inf\{y > z | y \in {\cal S}\}$. Then there is a $\beta \in (0,1]$ such that $$\beta \mathbb{P}(X_t \leq x|X_0 = x_{0+}) + (1 - \beta) \mathbb{P}(X_t \leq x | X_0 = x_{0-}) = \alpha \mu((-\infty, x]).$$
That is, if $x_0 \not \in {\cal S}$, then the process immediately jumps into ${\cal S}$, taking values $x_{0+}$ or $x_{0-}$ with probabilities $\beta$ and $1 - \beta$ respectively:
$$\mathbb{P}(X_{0+} = x_{0+}|X_0 = x_0) = \beta \qquad \mathbb{P}(X_{0+} = x_{0-}|X_0 = x_0) = 1 - \beta.$$
Note: if $m$ has a density $m^\prime$, then the infinitesimal generator may be written as
$$\frac{1}{m^\prime}\left(\frac{1}{2}\frac{\partial^2}{\partial x^2} + b\frac{\partial}{\partial x} - k\right).$$
Proofs of the results in the absence of a killing field {#seccoord}
=======================================================
For Theorems \[thgeexp2\], \[thnbt2\] and \[thctlim2\] which consider a finite state space ${\cal S} = \{i_1, \ldots, i_M\}$, let $\underline{\kappa}$ be defined by . For $\underline{\kappa}$ so defined, the quantities $q_{j,j+1}$ and $q_{j,j-1}$ from satisfy . With $k \equiv 0$, the problem is therefore that of finding a [*martingale*]{} generalised diffusion when viewed in the changed co-ordinates described above and is therefore solved in the article [@N1].
For Theorem \[thdrk\], the proof also follows similarly to that of [@N1], with the following alterations. As in [@N1], at stage $N$, the points $i_{N,1} < \ldots < i_{N,M_N}$ are chosen in the following way: let
$$\label{eqindef}\left\{ \begin{array}{l} i_{N,1} = \inf \left \{x | \mu((-\infty,x]) \geq \frac{1}{2^N} \right \} \\
i_{N,j} = \inf \left \{x > i_{N,j-1} | \mu((i_{N,j-1},i_{N,j}]) \geq \frac{1}{2^N} \right \} \\ M_N = \inf\left \{j : 1 - \mu((-\infty, i_{N,j}]) \leq \frac{1}{2^N} \right\}. \end{array}\right.$$
As in [@N1], let $\underline{p}^{(N)}$ be defined as
$$\label{eqpenj} p_j^{(N)} = \left\{ \begin{array}{ll} \mu((-\infty, i_{N,1}]) & j = 1 \\ \mu((i_{N,j-1}, i_{N,j}]) & j = 2, \ldots, M_N - 1 \\
1 - \mu((-\infty, i_{N,M_N}]) & j = M_N.
\end{array} \right.$$
Let $\widetilde{b}$ be defined by and set:
$$\label{eqdd} b^{(N)}_j = \frac{1}{i_{N,j+1} - i_{N,j-1}}\int_{i_{N,j-1}+}^{i_{N,j+1}-} \widetilde{b}(x) dx \qquad j = 2, \ldots, M_N - 1$$
where $\int_{a+}^{b-}$ means $\int_{(a,b)}$, the integral over the open interval. Note that of Hypothesis \[hyb\] is satisfied if: $$\left\{ \begin{array}{l} -1 < \min_{j \in \{2, \ldots, M_{N}-1\}} \frac{i_{N,j+1} - i_{N,j}}{i_{N,j+1} - i_{N,j-1}}\int_{i_{N,{j-1}+}}^{i_{N,j+1}-} \widetilde{b}(x) dx,\\ \max_{j \in \{2, \ldots, M_{N}-1\}} \frac{i_{N,j} - i_{N,j-1}}{i_{N,j+1} - i_{N,j-1}}\int_{i_{N,{j-1}+}}^{i_{N,j+1}-} \widetilde{b} (x) dx < 1. \end{array}\right.$$
From and of Hypothesis \[hybmu\], it follows that there is an $N_0$ such that for all $N > N_0$
$$-1 + \gamma < \min_{j \in \{2, \ldots, M_{N}-1\}} \int_{i_{N,{j-1}+}}^{i_{N,j+1}-} \widetilde{b}(x) dx \leq \max_{j \in \{2, \ldots, M_{N}-1\}} \int_{i_{N,{j-1}+}}^{i_{N,j+1}-} \widetilde{b} (x) dx < 1 - \gamma \label{eqhypb3}$$
where $\gamma$ is from . It follows that of Hypothesis \[hyb\] is satisfied for $N > N_0$. For the remainder of the argument, only $N > N_0$ is considered. Using $\underline{b}^{(N)}$ defined by , let $\underline{\lambda}_N = (\lambda_{N,2}, \ldots, \lambda_{N,M_N-1})$ ($\lambda_{N,j}$ the holding intensity for site $i_{N,j}$, $j = 2, \ldots, M_N-1$) denote the intensity vector that provides a solution to the marginal distribution problem. Let
$$\label{eqaaa}
a^{(N)}_j = \left\{ \begin{array}{ll} \lambda_j^{(N)} (i_{N,j+1} - i_{N,j})(i_{N,j} - i_{N,j-1}) & j = 2, \ldots, M_N - 1 \\ 0 & j = 1, M_N \end{array}\right.$$
and
$$\label{eqdiscmeas} m^{(N)}(\{i_{N,j}\}) = \left\{ \begin{array}{ll} \frac{(i_{N,j+1} - i_{N,j-1})}{a_j^{(N)}} & j \in \{2, \ldots, M_N - 1\} \\ +\infty & j = 1, M_N \end{array}\right.$$
The measure $m^{(N)}$ has support $\{i_{N,1}, \ldots, i_{N,M_N}\}$. Let $z_- = \inf\{x : x \in \mbox{suppt}(\mu)$ and $z_+ = \sup\{x : x \in \mbox{suppt}(\mu)$. Then, from the arguments of [@N1], there is a limiting measure $m$ such that for any $z_- < x < y < z_+$ there exists a subsequence $(N_j)_{j \geq 1}$ satisfying
$$\label{eqlimmeas} \lim_{j \rightarrow +\infty} \sup_{ x < a < b < y} \left | m^{(N_j)}([a,b]) - m([a,b]) \right | = 0.$$
Using the notation of Section \[subcoc\], let $\delta_{N,j} = i_{N,j} - i_{N,j-1}$ for $j = 2, \ldots, M_N$, $\epsilon_{N,j} = \kappa_{N,j} - \kappa_{N,j-1}$ and
$$\label{eqemminemplusN}
e_{N-} = \inf \left \{j : \sum_{i=1}^j p_i^{(N)} \geq \alpha \right \}, \qquad e_{N+} = \sup \left \{j : \sum_{i= j}^{M_N} p_i^{(N)} \geq \alpha \right \}.$$
where $0 < \alpha < 0.5$ is a number chosen such that there exists an $a$ and a $b$ such that $a < b$ and $\alpha \geq \mu((-\infty,a])$ and $\alpha \geq \mu([b,+\infty))$ and an $N_0$ such that $$\label{eqennought} \inf_{N > N_0}|i_{N,e_{N+}} - i_{N,e_{N-}}| > 0$$
(strict inequality). Only $N > N_0$ where this condition and are satisfied will be considered. Let
$$\label{eqken} K_N = i_{N,e_{N+}} - i_{N,e_{N-}}$$
and
$$\label{eqeps2N} \epsilon_{N,j+1} = K_N
\frac{\delta_{N,j+1} \prod_{k=1}^j \left(\frac{1 - \delta_{N,k} b_k^{(N)} }{1 + \delta_{N,k+1} b_k^{(N)}}\right)}
{\sum_{a=e_{N-}}^{e_{N+}-1} \delta_{N,a+1} \prod_{k=1}^a \left(\frac{1 - \delta_{N,k} b_k^{(N)}}{1 + \delta_{N,k+1} b_k^{(N)}}\right ) }.$$
Let $e_N$ satisfy: $\sum_{i=1}^{e_N} p_j^{(N)} \geq \frac{1}{2}$ and $\sum_{i= e_N}^{M_N} p_j^{(N)} \geq \frac{1}{2}$. Set
$$\label{eqkappaN}
\left\{ \begin{array}{ll} \kappa_{N,e_N} = 0 & \\ \kappa_{N,e_N +j + 1} = \kappa_{N,e_N + j} + \epsilon_{N, e_N + j+1} & j = 0, \ldots, M_N -e_N -1 \\ \kappa_{N,e_N - j -1} = \kappa_{N,e_N - j} - \epsilon_{N,e_N - j} & j = 0, \ldots, e_N - 2.
\end{array}\right.$$
Now note that $$\left\{ \begin{array}{l} i_{N,e_{N+}} \rightarrow c_+ := \sup\left \{x \in \overline{\mbox{suppt}(\mu)} : \mu((-\infty , x)) < 1- \alpha \right \} \\ i_{N,e_{N-}} \rightarrow c_- := \inf \left \{x \in \overline{\mbox{suppt}(\mu)} : \mu((-\infty,x]) \geq \alpha \right \} \end{array}\right.$$
so that $K_N \rightarrow c_+ - c_-$, a well defined positive limit and that, by construction, $\kappa_{N,e_{N+}} - \kappa_{N,e_{N-}} = K_N$ for each $N$.
The function of of Hypothesis \[hybmu\] is to ensure that in the new co-ordinates, the process has a well defined expected value. The following lemma demonstrates that the hypothesis is sufficient for this purpose.
\[lmmkpbd\] With $b^{(N)}_j$ defined by and $(\mu, b)$ satisfying , there exists an $N_0 \in \mathbb{Z}_+$ such that
$$\label{eqkpbd} \sup_{N \geq N_0} \sum_{j=1}^{M_N} |\kappa_{N,j}|p^{(N)}_j < +\infty.$$
#### Proof of Lemma \[lmmkpbd\]
Let $\gamma$ be defined by . Let $N_0$ satisfy: for all $N > N_0$, both and hold where $e_{N+}$ and $e_{N-}$ denote the indices defined in (\[eqemminemplusN\]). Let
$$C_N = K_N \frac{\prod_{k=1}^{e_N - 1} \left(\frac{1 - \delta_{N,k} b_k^{(N)}}{1 + \delta_{N,k+1} b_k^{(N)}}\right)}{\sum_{a=e_{N-}}^{e_{N+} - 1} \delta_{N,a+1} \prod_{k=1}^{a} \left(\frac{1 - \delta_{N,k} b_k^{(N)}}{1 + \delta_{N,k+1} b_k^{(N)}}\right)}.$$
where, as above, $K_N = i_{N,e_{N+}} - i_{N,e_{N-}} = \kappa_{N,e_{N+}} - \kappa_{N,e_{N-}}$. Recall the definition of $\kappa_{N,.}$ given by , that $e_N$ is the index such that $\kappa_{N,e_N} = 0$. Also, $\delta_{N,j} = (i_{N,j} - i_{N,j-1})$. Recall the definition of $b^{(N)}$ from . Then for $j > e_N$,
$$\begin{aligned}
\kappa_{N,j} &=& \nonumber C_N \sum_{k= e_N}^{j-1} (i_{N,k+1} - i_{N,k})\prod_{l = e_N}^k \left(\frac{1 - \delta_{N,l} b_l^{(N)}}{1 + \delta_{N,l+1} b_l^{(N)}}\right)\\
& = & C_N \sum_{k= e_N}^{j-1} (i_{N,k+1} - i_{N,k})\exp \left\{ \sum_{l = e_N}^k \ln \left (1 - \delta_{N,l} b_l^{(N)} \right ) - \ln \left (1 + \delta_{N,l+1} b_l^{(N)} \right )\right\}. \label{eqkapenbd}\end{aligned}$$
Similarly, for $j < e_N$, so that $\kappa_{N,j} < 0$,
$$- \kappa_{N,j} = C_N\sum_{i = j}^{e_N - 1} (i_{N,k+1} - i_{N,k})\exp \left\{ \sum_{l = i}^{e_N-1} \ln \left (1 - \delta_{N,l} b_l^{(N)} \right ) - \ln \left (1 + \delta_{N,l+1} b_l^{(N)} \right )\right\}. \label{eqkapenbd2}$$
Note that
$$\begin{aligned}
C_N &=& K_N \left(\sum_{a = e_{N-}}^{e_N - 1} \delta_{N,a+1} \prod_{k= a+1}^{e_N-1} \left(\frac{1 + \delta_{N,k+1} b_k^{(N)}}{1 - \delta_{N,k} b_k^{(N)}}\right) + \sum_{a= e_N}^{e_{N_+ - 1}} \delta_{N,a+1} \prod_{k=e_N}^{a} \left( \frac{1 - \delta_{N,k} b_k^{(N)}}{1 + \delta_{N,k+1} b_k^{(N)}} \right)\right)^{-1}\\
&\leq & K_N \left(\sum_{a = e_{N-}}^{e_N - 1} \delta_{N,a+1} \prod_{k= a+1}^{e_N-1} \left(\frac{1 - \delta_{N,k+1} | b_k^{(N)}|}{1 + \delta_{N,k} | b_k^{(N)}|}\right) + \sum_{a= e_N}^{e_{N_+ - 1}} \delta_{N,a+1} \prod_{k=e_N}^{a} \left( \frac{1 - \delta_{N,k} |b_k^{(N)}|}{1 + \delta_{N,k+1} |b_k^{(N)}|} \right)\right)^{-1}\\
& \leq & \prod_{k=e_{N-}+1}^{e_N-1} \left(\frac{1 + \delta_{N,k} |b_k^{(N)}|}{1 - \delta_{N,k+1}|b_k^{(N)}|}\right) \prod_{k=e_N}^{e_{N+}-1} \left(\frac{1 + \delta_{N,k+1}|b_k^{(N)}|}{1 - \delta_{N,k} |b_k^{(N)}|}\right)\end{aligned}$$
where the equality
$$\sum_{a=e_{N-}}^{e_{N+}-1} \delta_{N,a+1} = \sum_{a=e_{N-}}^{e_{N+}-1} (i_{N,a+1} - i_{N,a}) = i_{N,e_{N+}} - i_{N,e_{N-}} = K_N$$
has been used. It follows from together with the definition of $c(\gamma)$ given by , the definition of $b^{(N)}_k$ given by and Remarks \[rq1\] and \[rq2\] about the function $c$ following Hypothesis \[hybmu\] that:
$$C_N \leq \exp\left\{ 2 \int_{i_{N,e_{N-}}}^{i_{N,e_{N+}}} |\widetilde{b}(x)|dx + 2 c(\gamma)\sum_{k=e_{N-}}^{e_{N+}} \left(\int_{i_{N,k}}^{i_{N,k+1}} |\widetilde{b}(x) | dx\right)^2\right\}.$$
Using $i_{N,e_{N-}} \downarrow c_-$ and $i_{N,e_{N+}} \uparrow c_+$ together with gives that $C_N$ is uniformly bounded by a constant $C < +\infty$. It follows that:
$$\begin{aligned}
\sum_j |\kappa_{N,j}| p_j^{(N)} &\leq& C \left ( \sum_{j= e_N+1}^{M_N} p_j^{(N)} \sum_{k=e_N}^{j-1} (i_{N,k+1} - i_{N,k}) \right. \\&& \times \exp\left\{ 2\int_{e_N}^{i_{N,k+1}} |\widetilde{b}(x)|dx + 2 c(\gamma) \sum_{a=e_N}^k \left(\int_{i_{N,a}}^{i_{N,a+1}} |\widetilde{b}(x)| dx \right)^2 \right\} \\&&
+ \sum_{j=1}^{e_N-1} p_j^{(N)} \sum_{k=j+1}^{e_N} (i_{N,k} - i_{N,k-1})\\&& \left. \hspace{5mm} \times \exp\left\{ 2\int_{i_{N,k-1}}^{e_N} |\widetilde{b}(x)| dx + 2 c(\gamma)\sum_{a=k-1}^{e_N} \left(\int_{i_{N,a}}^{i_{N,a+1} } |\widetilde{b}(x) | dx \right)^2 \right\} \right ) \\
&& \leq C \int_{-\infty}^\infty \left ( \int_{0 \wedge x}^{0\vee x} e^{F(b,y)} dy\right) \mu(dx)\end{aligned}$$
where $F$ is defined by . The result now follows directly from .
\[lmmkappdef\] Let $\kappa^{(N)}$ denote the function
$$\label{eqkapen}
\kappa^{(N)}(x) = \left\{ \begin{array}{lll} \kappa_{N,1} & x < i_{N,1} & \\ \kappa_{N,j} + (\kappa_{N,j+1} - \kappa_{N,j}) \frac{x - i_{N,j}}{i_{N,j+1} - i_{N,j}} & x \in [i_{N,j}, i_{N,j+1}) & 1 \leq j \leq M_{N} - 1 \\ \kappa_{N,M_N} & x \geq i_{N,M_N} & \end{array}\right.$$
There is a non-decreasing map $\kappa : \mathbb{R} \rightarrow \mathbb{R}$ such that for any $-\infty < a < b < +\infty$,
$$\label{eqkap2} \lim_{N \rightarrow +\infty} \sup_{x \in [i_{N,1} \vee a, i_{N,M_N} \wedge b]} | \kappa^{(N)}(x) - \kappa(x)| = 0$$
#### Sketch of Proof
Firstly, note that $$\label{eqderkap} \frac{d \kappa^{(N)}}{dx} = \frac{\kappa_{N,j+1} - \kappa_{N,j}}{i_{N,j+1} - i_{N,j}} = \frac{\epsilon_{N,j+1}}{\delta_{N,j+1}}\qquad x \in [i_{N,j}, i_{N,j+1}).$$
The following argument shows that $\frac{d \kappa^{(N)}}{dx}$ has a well defined limit. From the definition of $\epsilon_{N,.}$, it follows that:
$$\left. \begin{array}{l} \frac{d\kappa^{(N)}}{dx}(x) = K_N \frac{1}{\left ( \sum_{a=j+1}^{e_{N+}-1} \delta_{N,a+1} \prod_{k=j+1}^a \left(\frac{1 - \delta_{N,k} b_k^{(N)}}{1 + \delta_{N,k+1} b_k^{(N)}}\right) + \sum_{a=e_{N-}}^j \delta_{N,a+1} \prod_{k=a+1}^j \left(\frac{1 + \delta_{N,k+1} b_k^{(N)}}{1 - \delta_{N,k} b_k^{(N)}} \right) \right)} \\ x \in [i_{N,j}, i_{N,j+1}) \end{array} \right.$$
Firstly, $K_N$ has a well defined limit, which is $c_+ - c_- =: K$. Now let ${\cal D}$ denote the set of atoms of $b$. By Hypothesis \[hybmu\], this is countable. For $z \in {\cal D}$, let $z_+ = \inf\{y > z | y \in {\cal D}\}$ and let $z_- = \sup\{y < z | y \in {\cal D}\}$. Let
$$j_N(x) = \{ j : x \in [i_{N,j},i_{N,j +1})\}$$ and, for $z \in {\cal D}$, let $\Delta \widetilde{b}(z) = \lim_{h \rightarrow 0}\int_{l_-(z) - h}^{l_+(z) + h} \widetilde{b}(x) dx$, where $l_-(z)$ and $l_+(z)$ are defined in the lines above . Let $\widetilde{b}^c$ denote the continuous part of $\widetilde{b}$ (the part remaining after removing the atoms). Then, for fixed $x$,
$$\begin{aligned}
\lim_{N \rightarrow +\infty} \frac{d\kappa^{(N)}}{dx}(x) &=& K \left(\int_{c_-}^{x}e^{\int_y^x \widetilde{b}^{(c)}(z) dz}\prod_{z \in {\cal D}: y \leq z \leq x} \left ( \frac{1 + \left(\frac{z_+ - z}{z_+ - z_-} \right) \Delta \widetilde{b}(z)}{1 - \left( \frac{z - z_-}{z_+ - z_-} \right) \Delta \widetilde{b}(z)}\right)dy \right. \\&& \left. + \int_{x}^{c_+} e^{-\int_x^y \widetilde{b}^{(c)}(z)dz} \prod_{z \in {\cal D}: x \leq z \leq y} \left(\frac{1 - \left(\frac{z - z_-}{z_+ - z_-}\right)\Delta \widetilde{b}(z)}{1 + \left(\frac{z_+ - z}{z_+ - z_-}\right)\Delta \widetilde{b}(z)}\right) dy \right)^{-1}\end{aligned}$$
Let $z_- = \inf\{x \in \mbox{suppt}(\mu)\}$ and $z_+ = \sup\{x \in \mbox{suppt}(\mu)\}$. Conditions \[hycond2\] and \[hycond3\] of Hypothesis \[hybmu\] ensure that this limit is well defined on $[z_-,z_+]$. Furthermore, $i_{N,e_N}$ has a well defined limit and $\kappa^{(N)}(i_{N,e_N}) = 0$ for each $N$. The result now follows almost directly.
Let $X^{(N)}$ denote the process generated by $a^{(N)}\left(\frac{1}{2} \Delta_{{\cal S}_N} + b^{(N)}\nabla_{{\cal S}_N} \right )$ where $\Delta_{{\cal S}_N}$ and $\nabla_{{\cal S}_N}$ are the Laplacian and gradient operators respectively defined on ${\cal S}_N = \{i_{N,1}, \ldots, i_{N,M_N}\}$ (Definition \[deflapder\]) and, with $m$ satisfying , let $X$ denote the process generated by $\left(\frac{1}{2}\frac{\partial^2}{\partial m \partial x} + \frac{\partial B}{\partial m} \nabla_m\right)$. Let $Y^{(N)} = \kappa^{(N)}(X^{(N)})$ where $\kappa^{(N)}$ is defined by and the mapping $\kappa$ by and let $Y = \kappa (X)$. Then $Y^{(N)}$ is a process with state space ${\cal R}_N = \{\kappa_{N,1}, \ldots, \kappa_{N, M_N}\}$ where site $\kappa_{N,j}$ has holding intensity $\lambda^{(N)}_j$ for $j = 1, \ldots, M_N$ and, when it jumps from $\kappa_{N,j}$ for $j \in \{2, \ldots, M_N-1\}$, it jumps to $\kappa_{N,j+1}$ with probability $\frac{\kappa_{N,j} - \kappa_{N,j-1}}{\kappa_{N,j+1} - \kappa_{N,j-1}}$ and to $\kappa_{N,j-1}$ with probability $\frac{\kappa_{N,j+1} - \kappa_{N,j}}{\kappa_{N,j+1} - \kappa_{N,j-1}}$. In short, it is a process with infinitesimal generator $\frac{\widetilde{a}^{(N)}}{2}\Delta_{{\cal R}_N}$, where $\Delta_{{\cal R}_N}$ denotes the Laplace operator defined on functions on ${\cal R}_N$ (Definition \[deflapder\]) and (with reduction in the notation which is clear)
$$\widetilde{a}^{(N)}_j := \widetilde{a}^{(N)}(\kappa_{N,j}) = \left\{ \begin{array}{ll} \lambda_j^{(N)} (\kappa_{N,j+1} - \kappa_{N,j})(\kappa_{N,j} - \kappa_{N,j-1}) & j = 2, \ldots, M_N - 1 \\ 0 & j = 1, M_N.\end{array}\right.$$
Let $\widetilde{m}^{(N)}$ denote the measure supported on $\{\kappa_{N,1}, \ldots, \kappa_{N,M_N}\}$ defined by
$$\left\{ \begin{array}{l} \widetilde{m}^{(N)}(\{ \kappa_{N,j}\}) = \frac{(\kappa_{N,j+1} - \kappa_{N,j-1})}{\widetilde{a}^{(N)}_j} \\ \widetilde{m}^{(N)}(\{\kappa_{N,1}\}) = \widetilde{m}^{(N)}(\{ \kappa_{N,M_N}\}) = +\infty \end{array}\right.$$
It follows from the convergence results of and that there is a limit $\widetilde{m}$ such that for the convergent subsequence of
$$\label{eqlimmeas2}
\lim_{j \rightarrow +\infty} \sup_{ \kappa(x) < a < b < \kappa(y) } \left | \widetilde{m}^{(N_j)}([a,b]) - \widetilde{m}([a,b]) \right | = 0.$$
As in [@N1], convergence of processes is based on the following result, which is stated in Kotani-Watanabe [@KW]:
\[thchgd\] Let $W$ denote a standard Wiener process starting from $0$ and let $\phi(s,a)$ denote its local time at site $a \in \mathbb{R}$, at time $s \geq 0$. Let $m$ be a measure on $\mathbb{R}$. Let
$$T(z,s) = \int_{\mathbb{R}} \phi (s,a-z) m(da)$$
and $$T^{ -1}(z,s) = \inf\left\{r | \int_{\mathbb{R}} \phi(r,a-z) m(da) \geq s \right \}.$$
Then $Y(t,z) = z + W(T^{-1}(z,t))$ is a strong Markov process with infinitesimal generator $\frac{1}{2}\frac{\partial^2}{\partial m \partial x}$.
Let $f_0^{(N)} = \sum_{j=1}^{M_N} \kappa_{N,j} p^{(N)}_j$. It follows from Equation (\[eqkap2\]) and the definition of $\underline{p}^{(N)}$ (Equation (\[eqpenj\])) that there exists an $f_0$ such that $\lim_{N \rightarrow +\infty} |f_0^{(N)} - f_0| = 0$. It therefore follows from the convergence result together with Theorem \[thchgd\], that there is a subsequence such that for all $\epsilon > 0$
$$\lim_{j \rightarrow +\infty} \mathbb{P} \left(\sup_{0 \leq s \leq t} \left | Y^{(N_j)}_s(f_0^{(N_j)}) - Y_s(f_0) \right | > \epsilon \right) = 0,$$
where an initial condition $y$ for $Y_s^{(N)}(y)$ is interpreted as:
$$\left\{ \begin{array}{l} \mathbb{P}\left (Y_{0+}^{(N)} = \kappa_{N,l_N(y)} | Y_0^{(N)} = y \right ) = \frac{y - \kappa_{N,l_N(y)-1}}{\kappa_{N,l_N(y)} - \kappa_{N,l_N(y)-1}} =: \beta_N \\ \mathbb{P} \left (Y_{0+}^{(N)} = \kappa_{N, l_N(y)-1} | Y_0^{(N)} = y \right ) = \frac{\kappa_{N, l_N(y)} - y}{\kappa_{N, l_N(y)} - \kappa_{N, l_N(y)-1}} = 1 - \beta_N \end{array}\right.$$ and $l_N(y)$ is defined as the index such that $\kappa_{N,l_N(y)-1} < y \leq \kappa_{N,l_N(y)}$. Let $y_- = \lim_{N \rightarrow +\infty} \kappa_{N, l_N(f_0^{(N)})-1}$ and let $y_+ = \lim_{N \rightarrow +\infty} \kappa_{N, l_N(f_0^{(N)})}$. Then, when $y_- < y_+$ where the inequality is strict, the initial condition $f_0$ for process $Y(f_0)$ is interpreted as:
$$\mathbb{P}(Y_{0+} = y_+ | Y_0 = f_0) = \frac{f_0 - y_-}{y_+ - y_-} =: \beta \qquad \mathbb{P}(Y_{0+} = y_- | Y_0 = f_0) = \frac{y_+ - f_0}{y_+ - y_-} = 1 - \beta.$$
From this,
$$\lim_{j \rightarrow +\infty} \mathbb{P} \left(\sup_{0 \leq s \leq t} \left | X^{(N_j)}_s - X_s \right | > \epsilon \right) = 0$$
where $X^{(N)}$ satisfies:
$$\mathbb{P} \left (X^{(N)}_{0+} = i_{N,l_N(f_0^{(N)})} \right ) = \beta_N \qquad \mathbb{P} \left (X^{(N)}_{0+} = i_{N,l_N(f_0^{(N)})-1} \right ) = 1 - \beta_N$$
and $\left (\beta^{(N_j)}, i_{N, l_{N_j} (f_0^{(N_j)}) - 1}, i_{N, l_{N_j} (f_0^{(N_j)}) } \right ) \stackrel{j \rightarrow +\infty}{\longrightarrow} \left (\beta, x_{0-}, x_{0+} \right )$ and $X$ satisfies:
$$\mathbb{P}(X_{0+} = x_{0+}) = \beta \qquad \mathbb{P}(X_{0+} = x_{0-}) = 1 - \beta.$$
It follows that
$$\lim_{j \rightarrow +\infty} \sup_x \left |\mathbb{P}\left( X_t^{(N_j)} \leq x\right) - \mathbb{P} \left (X_t \leq x \right ) \right | = 0$$
and hence that
$$\mathbb{P} \left (X_t \leq x \right ) = \mu ((-\infty, x])$$
where $X$ is a diffusion process with infinitesimal generator $\left(\frac{1}{2}\frac{\partial^2}{\partial m \partial x} + \frac{\partial B}{\partial m} \nabla_m \right)$ as required.
Introducing the Killing Field: Preliminary Results {#seckf}
==================================================
Attention is now turned to the problem of introducing a killing field $k$. The following sections prove the theorems of the article stated in Section \[secres\]; this section presents preliminary results and notation.
The transition from finite state space to arbitrary measure on $\mathbb{R}$ follows the same proof as [@N1], together with the arguments of Section \[seccoord\], with only a few additions. Some discussion is necessary for modifying the proofs of [@N1] so that they can accommodate killing for geometric / exponential times and then to modify the fixed point theorem so that the transition can be made to negative binomial times. Once negative binomial times are accommodated, the limiting arguments to obtain the result for deterministic time are straightforward and the limiting arguments to obtain the result for arbitrary state space follow directly from the analysis of [@N1].
Recall the definitions of $\widetilde{P}$ (Equation Definition \[deftrmat\]). Let
$$\label{eqen} \widetilde{{\cal N}}_t = \frac{t+h}{h} \left (I - \frac{t}{t+h} \widetilde{P}^{(h)} \right )$$
For the problem without drift or killing, this quantity appeared crucially in establishing the result for geometric times in [@N1], with $\frac{1}{1-a} = \frac{t+h}{h}$ giving $a = \frac{t}{t+h}$.
The entries of $\widetilde{{\cal N}}$ may be computed quite easily and are given in below:
$$\label{eqtilden2} \widetilde{{\cal N}}_{t;j,k}(\underline{k}, \underline{\lambda}) = \left\{ \begin{array}{ll} 1 & k = j = M+1 \\ 0 & j = M+1, \quad k \neq M+1 \\ - t \lambda_j \widetilde{k}_j & 2 \leq j \leq M -1, \quad k = M+1 \\
0 & (j,k) = (1,M+1), \qquad (j,k) = (M,M+1) \\ 1 & (j,k) = (1,1), \qquad (j,k) = (M,M) \\ 1 + t \lambda_j ( 1 + \widetilde{k}_j ) & k = j \quad 2 \leq j \leq M-1 \\ -t\lambda_j \left(\frac{\kappa_j - \kappa_{j-1}}{\kappa_{j+1} - \kappa_{j-1}}\right) & k = j+1, \quad 2 \leq j \leq M-1 \\ -t \lambda_j \left(\frac{\kappa_{j+1} - \kappa_j}{\kappa_{j+1} - \kappa_{j-1}}\right) & k = j-1, \quad 2 \leq j \leq M-1 \\ 0 & j = 1, k \in \{2, \ldots, M\} \\ 0 & j = M, \quad k \in \{1, \ldots, M-1\} \\ 0 & (j,k) \in \{1, \ldots, M\}^2 \quad |j - k| \geq 2 \end{array}\right.$$
where $\widetilde{\underline{k}}$ is defined by . Note that this is independent of $h$. Let ${\cal N}_t$ denote the $M \times M$ matrix such that ${\cal N}_{t;j,k} = \widetilde{{\cal N}}_{t;j,k}$ for $(j,k) \in \{1, \ldots, M\}^2$.
#### Note
The notation will be suppressed; $\widetilde{{\cal N}}_t(\underline{k},\underline{\lambda})$ and ${\cal N}_t(\underline{k},\underline{\lambda})$ will be written as $\widetilde{{\cal N}}$ and ${\cal N}$ respectively. Some particular variables ($t$ or $\underline{\lambda}$) may be introduced if they are of particular concern for the point under discussion.
It is straightforward to compute that $\sum_{k = 1}^{M+1} \widetilde{{\cal N}}_{t;j,k} = 1$, but there does not seem to be a direct method to control the [*absolute*]{} values of the entries of the matrix. Control is therefore obtained by using the inverse. One result used in the sequel is that for integer $p \geq 1$, all the entries of $\widetilde{{\cal N}}^{-p}$ are non negative, bounded between $0$ and $1$ and that for each $j$, $\sum_{k = 1}^{M+1} (\widetilde{{\cal N}}^{-p}_{t})_{jk} = 1$. This follows from the following representation.
\[lmmnprep\] For integer $p \geq 1$, $\widetilde{{\cal N}}^{-p}$ has representation:
$$\label{eqtmtnm1rep} (\widetilde{{\cal N}}^{-p}_t)_{jk} = \mathbb{P}(Z_T = k | Z_0 = j)$$
where $Z$ is a continuous time Markov chain on $\{1, \ldots, M+1\}$, with intensity matrix $\Theta$ (Equation , Definition \[defintmat\]) and $T \sim \mbox{Gamma}(p,\frac{1}{t})$, using the parametrisation of a Gamma distribution from (that is the sum of $p$ independent exponential variables, each with intensity parameter $1/t$).
#### Proof of lemma \[lmmnprep\]
Let $a = \frac{t}{t + h}$. Then, for $h < \frac{1}{\max_{j \in \{2, \ldots, M-1\}} \lambda_j (1 + \widetilde{k}_j)}$, where $\widetilde{\underline{k}}$ is defined by ,
$$\begin{aligned}
(\widetilde{{\cal N}}^{-p}_t)_{i,j} &=& \left(\frac{1}{1-a} (I - a\widetilde{P}^{(h)})\right)^{-p}_{i,j} \\
&=& (1-a)^p \sum_{k=0}^\infty \binom{p + k - 1}{k} a^k (\widetilde{P}^{(h)})^k_{i,j} = (1-a)^p \sum_{k=0}^\infty \binom{p + k - 1}{k} a^k \mathbb{P}(Z^{(h)}_{kh} = j | Z^{(h)}_0 = i) \end{aligned}$$
where $Z^{(h)}$ is a Markov chain with state space $\{1, \ldots, M+1\}$ and one-step transition matrix $\widetilde{P}^{(h)}$ (where $h$ is the time step length) defined by Definition \[deftrmat\]. Since
$$\mathbb{P}(\tau = k) = \binom{p + k - 1}{k} a^k (1-a)^p \qquad k = 0,1,2, \ldots$$
is the probability mass function of an $NB(p,a)$ random variable, it follows that
$$(\widetilde{{\cal N}}^{-p})_{i,j} = \mathbb{P}\left(Z_{h\tau}^{(h)} = j|Z_0^{(h)} = i \right ) \qquad \mbox{where} \qquad \tau \sim NB \left (p, \frac{t}{t + h} \right )$$
so that $\mathbb{E}[h\tau] = \frac{hp(t/t+h)}{(h/t+h)} = pt$. The fact that $h\tau$ converges in distribution to $T \sim \mbox{Gamma} \left (p, \frac{1}{t} \right )$ as $h \rightarrow 0$, $Z^{(h)}$ converges (in the sense of finite dimensional marginals) to the required continuous time Markov chain $Z$ and $Z^{(h)}_{h\tau} \stackrel{h \rightarrow 0}{\longrightarrow_{(d)}} Z_T$ follows the proof found in [@N1].
The following precautionary lemma is introduced to deal with a problem that does not arise in [@N1]; it is necessary to establish that the Fixed Point Theorem (Theorem \[thfpt\], which is the heart of the proof) does not give a process that is dead with probability $1$ at the terminal time.
\[lmmcruclb\] For integer $p \geq 1$, there exists a constant $c > 0$, which is independent of $\underline{\lambda}$, such that for all $j \in \{1, \ldots, M\}$, $(\widetilde{\cal N}_t^{-p})_{j,M+1} < 1 - c$.
#### Proof
Recall the representation of the previous lemma: $(\widetilde{\cal N}_t^{-p})_{j,k} = \mathbb{P}(Z_T = k | Z_0 = j)$. It follows from Equation that the corresponding embedded discrete time chain has transitions $p_{j,j+1} = \frac{q_{j,j+1}}{1 + \widetilde{k}_j}$, $p_{j,j-1} = \frac{q_{j,j-1}}{1 + \widetilde{k}_j}$, $p_{j,M+1} = \frac{\widetilde{k}_j}{1 + \widetilde{k}_j}$, where $q_{j,j+1}$ and $q_{j,j-1}$ are defined by (expressed as in the drift free coordinates). These transitions do not depend on $\underline{\lambda}$. If the process reaches site $1$, it remains there; if the process reaches site $M$ it remains there. By considering a lower bound on the probability that the process reaches site $1$, it follows that, for $j \neq M$, $$1 - \widetilde{{\cal N}}_{j,M+1} \geq \prod_{i=2}^{M-1} \left(\frac{q_{i,i-1}}{1 + \widetilde{k}_i}\right) > 0$$ as required. This is a lower bound on the probability that the process never reaches the cemetery site $M+1$.
The following lemma is used in the Fixed Point Theorem, to show that as $\epsilon \rightarrow 0$, the sequence of fixed points for the approximating problems remains bounded.
\[lmmpr2\] If $\lambda_j \rightarrow +\infty$ then $({\cal N}_t^{-1})_{.j} \rightarrow \underline{0}$ and consequently $({\cal N}_t^{-p})_{.j} \rightarrow \underline{0}$ for any integer $p \geq 1$ where the notation $.j$ denotes the $j$th column of the matrix.
#### Proof
Let $\beta_{lj} = ({\cal N}_t^{-1})_{lj}$. Then $0 \leq \beta_{lj} \leq 1$. $\beta$ satisfies the following system:
$$\label{eqqlamzer2} -q_{l,l-1}\beta_{l-1,j} + \left((1 + k_l) + \frac{1}{t\lambda_l}\right)\beta_{lj} - q_{k,k+1} \beta_{k+1,j} = \left\{ \begin{array}{ll} 0 & k \neq j \\ \frac{1}{t \lambda_j} & l = j \end{array}\right.$$
where $q_{l,l-1}, q_{l,l+1}$ for $l = 2, \ldots, M-1$ is defined by and the following definition is used for $l = 1,M$:
$$\left\{ \begin{array}{ll} q_{l,l-1}\beta_{l-1,j} = 0 & l = 1 \\ q_{l,l+1}\beta_{l+1,j} = 0 & l = M. \end{array}\right.$$
From the a priori bounds on $\beta_{lj}$ (namely $0 \leq \beta_{lj} \leq 1$ and $\sum_j \beta_{lj} = 1$ for each $l$) which follow directly from Lemma \[lmmnprep\], it follows from that $\beta_{lj} \stackrel{\lambda_j \rightarrow +\infty}{\longrightarrow} 0$ for all $l = 1, \ldots, M$.
The following lemma is key to proving Theorem \[thgeexp2\], since the proof of Theorem \[thgeexp2\] boils down to solving the system of equations defined by given below.
\[lmmkey2\] Let $\underline{p}$ be a probability measure over $\{1, \ldots, M\}$. There exists a unique $\alpha \in (0,1]$, $l \in \{2, \ldots, M\}$, $x_0 \in (\kappa_{l-1}, \kappa_l]$ and $\underline{\lambda} \in \mathbb{R}_+^{M-2}$ satisfying :
$$\label{eqthg}
(\widehat{\underline{p}}\widetilde{{\cal N}}_t)_k = \left\{ \begin{array}{ll} \frac{x_0 - \kappa_{l-1}}{\kappa_l - \kappa_{l-1}} & l = k \\ \frac{\kappa_l - x_0}{\kappa_l - \kappa_{l-1}} & k = l-1 \\ 0 & \mbox{otherwise} \end{array}\right.$$
where $\widehat{p}_k = \alpha p_k$ for $k = 1,\ldots, M$ and $\widehat{p}_{M+1} = 1 - \alpha$. The solution is the following: $\alpha$ satisfies Equation (\[eqalph2\]), $x_0$ satisfies Equation (\[eqx02\]) and $\underline{\lambda} = (\lambda_2, \ldots, \lambda_{M-1})$ satisfies Equation (\[eqlamexpsol2\]), where ${\cal G}$ is defined by Equation (\[eqgeee\]) and $Q$ is defined by Equation (\[eqqespj2\]).
#### Proof
The equation given by for $k = M+1$ is: $$(1 - \alpha) \widetilde{{\cal N}}_{t;M+1,M+1} + \alpha \sum_{j=1}^M p_j \widetilde{{\cal N}}_{t;j,M+1} = 0,$$
which is:
$$1 - \alpha = \alpha t \sum_{j=1}^M p_j \lambda_j \widetilde{k}_j,$$
where $(\widetilde{k}_1, \ldots, \widetilde{k}_{M})$ is defined by . It follows that $\alpha \in (0,1]$ is required to satisfy:
$$\alpha = \frac{1}{1 + t\sum_{j=1}^M p_j \lambda_j \widetilde{k}_j}$$
so that, if there is a solution, then $\alpha$ is uniquely determined with this value. Let $x_0 \in \mathbb{R}$, $l \in \{2, \ldots, M\}$ and let $\underline{v}(l, x_0)$ satisfy:
$$v_j (x_0,l) = \left\{ \begin{array}{ll} \frac{x_0 - \kappa_{l-1}}{\kappa_l - \kappa_{l-1}} & j = l \\ \frac{\kappa_l - x_0}{\kappa_l - \kappa_{l-1}} & j = l-1 \\ 0 & j \neq l, l-1. \end{array}\right.$$
There are $M$ equations involving the $M-2$ unknowns, $\lambda_2, \ldots, \lambda_{M-1}$. These equations are:
$$\label{eqtheq2} \left\{ \begin{array}{ll} \alpha p_1 - t \lambda_2 \alpha p_2 q_{21} = v_1 & \\
-t \lambda_{j-1} \alpha p_{j-1}q_{j-1,j} + \alpha p_j (1 + t \lambda_j (1 + \widetilde{k}_j)) - t \lambda_{j+1} \alpha p_{j+1} q_{j+1,j} = v_j & j = 2, \ldots, M-1 \\
-t \lambda_{M-1} q_{M-1,M} \alpha p_{M-1} + \alpha p_M = v_M &
\end{array} \right.$$
Set $\Lambda_j = t\alpha p_j \lambda_j$ and $\widetilde{p}_j = \alpha p_j ( 1 + t \lambda_j \widetilde{k}_j)$. Since
$$\alpha = \frac{1}{1 + t \sum_{j=1}^M p_j \lambda_j \widetilde{k}_j} = \frac{1}{\sum_{j=1}^M p_j (1 + t \lambda_j \widetilde{k}_j)},$$
it follows from the definition of $Q$ (Equation ) that $\underline{\widetilde{p}} = Q (t,\underline{p})$ and $\sum_{j=1}^M \widetilde{p}_j = 1$. The system of equations may be written, with these values, as :
$$\label{eqrewrite}
\left\{\begin{array}{ll} \widetilde{p}_1 - \Lambda_2 q_{21} = v_1 & \\ - \Lambda_{j-1} q_{j-1,j} + (\widetilde{p}_j + \Lambda_j) - \Lambda_{j+1} q_{j+1,j} = v_j & j = 2, \ldots, M-1 \\
-\Lambda_{M-1}q_{M-1,M} + \widetilde{p}_M = v_M. &
\end{array}\right.$$
which is a linear system of $M$ equations with $M-2$ unknowns. To show that it is of rank at most $M-2$: summing both left hand side and right hand side give $1$ for any choice of $x_0$.
Also,
$$\sum_{j} \kappa_j v_j = \kappa_l \frac{x_0 - \kappa_{l-1}}{\kappa_l - \kappa_{l-1}} + \kappa_{l-1}\frac{\kappa_l - x_0}{\kappa_l - \kappa_{l-1}} = x_0,$$
It follows that $x_0$ is required to satisfy
$$x_0 = \alpha \sum_{j=1}^M \kappa_j p_j (1 + t \lambda_j \widetilde{k}_j) = \frac{\sum_{j=1}^M \kappa_j p_j (1 + t \lambda_j \widetilde{k}_j)}{\sum_{j=1}^M p_j (1 + t \lambda_j \widetilde{k}_j)}.$$
It follows that if there is a solution, then $\alpha$, $l$ and $x_0$ are uniquely determined with the values given in the statement of the lemma.
Since $\sum_{j=1}^M \widetilde{p}_j = 1$, it follows that the system of equations given by is that studied in [@N1]. From [@N1], it follows that $\underline{\Lambda}$ satisfies:
$$\underline{\Lambda} = {\cal L}(\widetilde{p} )$$
where
$${\cal L}_j(\underline{p} ) = \left\{ \begin{array}{ll} \frac{(\kappa_{j+1} - \kappa_{j-1})}{(\kappa_{j+1} - \kappa_j)(\kappa_j - \kappa_{j-1})} \left(\sum_{k=1}^{j-1} (\kappa_j - \kappa_k)p_k \right) & 2 \leq j \leq l-1 \\ \frac{(\kappa_{j+1} - \kappa_{j-1})}{(\kappa_{j+1} - \kappa_j)(\kappa_j - \kappa_{j-1})} \left(\sum_{k=j+1}^M (\kappa_k - \kappa_j) p _k \right) & l \leq j \leq M-1 \\ 0 & j = 1 \quad \mbox{or} \quad M. \end{array}\right.$$
Therefore any solution satisfies :
$$\label{eqlampr1}
\lambda_j \widetilde{k}_j = \frac{1}{t\alpha p_j}{\cal L}_j(Q (t,\underline{p}) ) = \frac{1 + t \lambda_j \widetilde{k}_j}{t Q (t,\underline{p},j)}{\cal L}_j(Q (t,\underline{p}) ) = \frac{1}{t}(1 + t\lambda_j \widetilde{k}_j){\cal F}_j (Q (t,\underline{p} )) \qquad j = 2, \ldots, M-1$$
where ${\cal F}$ is defined (as in [@N1]) by :
$$\label{eqeff}
{\cal F}_j(\underline{p}) = \left\{\begin{array}{ll} \frac{1}{p_j}{\cal L}_j (\underline{p}) & j = 2, \ldots, M-1 \\ 0 & j = 1, M \end{array}\right.$$
That is, $\underline{\lambda}$ is a solution if and only if $\lambda_1 = \lambda_M = 0$ and for $j = 2, \ldots, M-1$,
$$\begin{aligned}
\nonumber
\lambda_j &=& \frac{1}{t}(1 + t \lambda_j \widetilde{k}_j){\cal F}_j (Q(t,\underline{p}))\\
\nonumber &=& \frac{1}{t}(1 + t\lambda_j \widetilde{k}_j) \frac{{\cal L}_j (Q(t,\underline{p}))}{p_j (1 + t\lambda_j \widetilde{k}_j)}\frac{1}{\alpha}\\ \nonumber
&=& \frac{1}{t p_j}\frac{(\kappa_{j+1} - \kappa_{j-1})}{(\kappa_{j+1} - \kappa_j)(\kappa_j - \kappa_{j-1})} \times \left\{ \begin{array}{ll} \sum_{i = 1}^{j-1} (\kappa_j - \kappa_i)p_i (1 + t\lambda_i \widetilde{k}_i) & 2 \leq j \leq l-1 \\ \sum_{i = j+1}^M (\kappa_i - \kappa_j) p_i (1 + t \lambda_i \widetilde{k}_i) & l \leq j \leq M-1 \end{array}\right.\\
&=& {\cal G}_j \label{eqdksol}\end{aligned}$$
where ${\cal G}$ is defined by . The function ${\cal G}$ exists by Lemma \[lmmgeee\]. The proof of Lemma \[lmmkey2\] is complete.
Stopping at Independent Geometric or Exponential Time
=====================================================
The purpose of this section is to prove Theorem \[thgeexp2\].
#### Proof of Theorem \[thgeexp2\]
This is equivalent to existence and uniqueness of an $l \in \{2, \ldots, M\}$, $\beta \in (0,1]$, $\alpha \in (0,1]$ and a $\underline{\lambda} \in \mathbb{R}^{M-2}$ such that $\widetilde{P}^{(h)}$ (Definition \[deftrmat\], Equation ) is the transion matrix for a chain $X^{(h)}$ such that $\widehat{p}$ defined as:
$$\label{eqwhp}
\widehat{p}_j = \left\{ \begin{array}{ll} \alpha p_j & j = 1, \ldots, M \\ 1 - \alpha & j = M+1 \end{array}\right.$$
satisfies:
$$\widehat{p}_j = (1-a)\left((1 - \beta) (I - a\widetilde{P}^{(h)} )^{-1}_{l,j} + \beta (I - a\widetilde{P}^{(h)} )^{-1}_{l-1,j}\right)$$
where $X_0^{(h)} = x_0$ for some $x_0 \in (i_{l-1}, i_l]$ and $\beta \in (0,1]$ is a number such that $$\beta = \mathbb{P}(X_{0+}^{(h)} = i_{l-1} | X_0^{(h)} = x_0) \qquad (1-\beta ) = \mathbb{P}(X_{0+}^{(h)} = i_l | X_0^{(h)} = x_0),$$
It follows that
$$\frac{1}{1-a}\left( \underline{\widehat{p}}(I -a \widetilde{P}^{(h)} \right)_k = \frac{t+h}{h} \left( \underline{\widehat{p}}(I - \frac{t}{t+h}\widetilde{P}^{(h)} \right)_k =
\left\{ \begin{array}{ll} 1 - \beta & l = k \\ \beta & k = l-1 \\ 0 & \mbox{otherwise}, \end{array}\right.$$
which is equivalent to showing existence of an $\alpha$, $l$, $\beta$ and $\underline{\lambda}$ such that
$$( \underline{\widehat{p}} {\cal N})_k = \left\{ \begin{array}{ll} (1 - \beta) & l = k \\ \beta & k = l-1 \\ 0 & \mbox{otherwise}. \end{array}\right.$$
The result now follows directly from Lemma \[lmmkey2\] with $\underline{\kappa} = (\kappa_1, \ldots, \kappa_M)$ the change of coordinates described in Section \[subcoc\] and
$$\label{eqlamsol2}
\left\{
\begin{array}{l} \alpha = \frac{1}{1 + t\sum_{j=1}^M p_j \lambda_j \widetilde{k}_j}, \qquad \beta = \frac{\kappa_l - x_0}{\kappa_l - \kappa_{l-1}} \qquad x_0 = \frac{\sum_{j=1}^M \kappa_j p_j (1 + t \lambda_j \widetilde{k}_j)}{\sum_{j=1}^M p_j (1 + t \lambda_j \widetilde{k}_j)} \\ \lambda_j = {\cal G}_j ( \underline{p}, \underline{k} , \underline{\kappa})\end{array}\right.$$
where ${\cal G}$ is defined by and $\underline{\widetilde{k}}$ by .
The result now follows for $0 < h < \frac{1}{\max_{j \in \{2, \ldots, M-1\}} \lambda_j(1 + \widetilde{k}_j)}$. The limiting argument to obtain a continuous time process as $h \rightarrow 0$, which has the prescribed marginal when stopped at an exponential time is given in [@N1].
The case with drift and killing on a finite state space, where the process is stopped at an independent exponential time, has now been solved.
Negative Binomial, Gamma and Deterministic Time
===============================================
This section is devoted to the proofs of Theorems \[thnbt2\] and \[thctlim2\]. They follow the lines of the proofs in [@N1], with some additional ideas required to deal with the killing field.
Proof of Theorem \[thnbt2\]
---------------------------
This follows by appealing to the fixed point theorem, Theorem \[thfpt\]. As before, let $\tau \sim NB(r,a)$, with $a = \frac{t}{t+hr}$, so that $\mathbb{E}[\tau ] = \frac{ra}{1 - a} = t$. Then, with $\widetilde{P}$ defined by Equation Definition \[deftrmat\],
$$\frac{1}{1-a} (I - a\widetilde{P}^{(h)} ) = \frac{t+hr}{hr} \left( I - \frac{t}{t + hr} \widetilde{P}^{(h)} \right) = \frac{(t/r) + h}{h}\left( I - \frac{(t/r)}{(t/r) + h} \widetilde{P}^{(h)} \right ) = \widetilde{{\cal N}}_{t/r} .$$
If $\tau \sim NB(r,a)$ with $a = \frac{t}{t + hr}$, then $\underline{\lambda}$ provides a solution if and only if there is an $\alpha \in (0,1)$, an $l$ and an $x_0 \in (\kappa_{l-1}, \kappa_l]$ such that
$$\alpha p_j = \frac{\kappa_{l} - x_0}{\kappa_l - \kappa_{l-1}}\mathbb{P}(X_{h\tau}^{(h)} = i_j | X_0^{(h)} = i_{l-1}) + \frac{x_0 - \kappa_{l-1}}{\kappa_l - \kappa_{l-1}} \mathbb{P}(X_{h\tau}^{(h)} = i_j | X_0^{(h)} = i_l) \quad j = 1, \ldots, M$$
Let $\widehat{p}_j = \alpha p_j$ for $j = 1, \ldots M$ and $\widehat{p}_{M+1} = 1 - \alpha$. Then $(\alpha, x_0, \underline{\lambda})$ provide a solution if and only if
$$\begin{aligned}
\widehat{p}_j &=& \frac{\kappa_l - x_0}{\kappa_l - \kappa_{l-1}}\sum_{k=0}^\infty \mathbb{P}(X_{hk}^{(h)} = i_j | X_0^{(h)} = i_{l-1})\mathbb{P}(\tau = k) \\ && \hspace{5mm} + \frac{x_0 - \kappa_{l-1}}{\kappa_l - \kappa_{l-1}}\sum_{k=0}^\infty \mathbb{P}(X_{hk}^{(h)} = i_j | X_0^{(h)} = i_{l}) \mathbb{P}(\tau = k) \\
&=& (1-a)^r \left( \frac{\kappa_l - x_0}{\kappa_l - \kappa_{l-1}} \sum_{k=0}^\infty \binom{k+r - 1}{k} a^k (\widetilde{P}^{(h)k})_{l-1,j} + \frac{x_0 - \kappa_{l-1}}{\kappa_l - \kappa_{l-1}} \sum_{k=0}^\infty \binom{k + r - 1}{k} a^k (\widetilde{P}^{(h)k})_{l,j} \right )\\
&=& \frac{\kappa_l - x_0}{\kappa_l - \kappa_{l-1}}((I - a\widetilde{P}^{(h)})^{-r})_{l-1,j} + \frac{x_0 - \kappa_{l-l}}{\kappa_l - \kappa_{l-1}} ((I - a\widetilde{P}^{(h)})^{-r})_{l,j}\end{aligned}$$
Let $\underline{\widehat{v}}$ be the $M+1$ vector and $\underline{v}$ the $M$ vector defined by $v_l = \widehat{v}_l = \frac{x_0 - \kappa_{l-1}}{\kappa_l - \kappa_{l-1}}$, $v_{l-1} = \widehat{v}_{l-1} = \frac{\kappa_l - x_0}{\kappa_{l} - \kappa_{l-1}}$ and $v_j = 0$ for $j \neq l-1, l$. It follows that a solution is provided by any $\alpha \in (0,1)$, $\underline{\lambda}$ and $x_0$ such that
$$\underline{\widehat{p}} \widetilde{{\cal N}}_{t/r}^r = \underline{\widehat{v}}$$
holds. Let
$$\label{eqqdef2} \underline{q} = \frac{1}{\sum_{j,k} p_j {\cal N}_{t/r;j,k}^{(r-1)} }\underline{p} {\cal N}_{t/r}^{(r-1)},$$
then $\underline{\lambda}$ provides a solution for all $h \in \left (0, \frac{1}{\max_{j \in \{2, \ldots, M-1\}} \lambda_j (1 + \widetilde{k}_j)}\right)$ where $\widetilde{\underline{k}}$ is defined by , if and only if there is an $\alpha \in (0,1]$ such that $ \alpha \underline{q} {\cal N}_{t/r}(\underline{\lambda}) = \underline{v}$. It follows from Lemma \[lmmkey2\] that $\underline{\lambda}$ is a solution if and only if
$$\label{eqsollmm2} \lambda_j = {\cal G}_j \left (Q\left(\frac{t}{r},\underline{q}\right ) \right ) \qquad j = 1, \ldots, M$$
where ${\cal G}$ is defined by and $Q$ by . Here
$$\alpha = \frac{1}{1 + \frac{t}{r}\sum_{j=1}^M q_j \lambda_j \widetilde{k}_j} \qquad x_0 = \frac{\sum_{j=1}^M \kappa_j q_j (1 + \frac{t}{r} \lambda_j \widetilde{k}_j)}{\sum_{j=1}^M q_j (1 + \frac{t}{r} \lambda_j \widetilde{k}_j)}.$$
The existence of a $\underline{\lambda}$ satisfying follows from Theorem \[thfpt\], which gives existence of a fixed point. For the bounds on $\alpha$, let $\sigma = \inf\{t : X_t \in \{ D\} \}$, then $$\mathbb{P}(\sigma \geq (n+1)h) \geq (1 - h (\lambda k)^*)^n$$ so that, for $\tau \sim NB\left(r,\frac{t}{t+hr} \right )$ independent of $\sigma$ and using $a = \frac{t}{t+hr}$, for $h < \frac{1}{(\lambda k)^*}$,
$$\begin{aligned}
\alpha &=& \mathbb{P}(h \tau < \sigma) = \sum_{n=0}^\infty \mathbb{P}(\sigma > nh|\tau = n)\mathbb{P}(\tau = n)\\
&=& \sum_{n=0}^\infty \mathbb{P}(\sigma \geq (n+1)h)\mathbb{P}(\tau = n) \geq \sum_{n=0}^\infty (1 - h (\lambda k)^*)^{n} \binom{n+r-1}{n}a^n(1-a)^{r}\\
&=& \frac{ (1 - a)^{r}}{(1 - a(1- h (\lambda k)^*))^{r}}= \left(1 + \frac{t(\lambda k)^*}{r}\right)^{-r},\end{aligned}$$
as required. These results hold for all $h \in \left ( 0, \frac{1}{(\lambda k)^*}\right )$ and hence in the continuous time limit as $h \rightarrow 0$. Details of the convergence of finite dimensional marginals are given in [@N1].
Proof of Theorem \[thctlim2\] {#mod}
-----------------------------
This follows almost directly from the proof of Theorem \[thnbt2\]; the problem is to show that when the limit is taken, the result is non-trivial. Let $\underline{\lambda}^{(r)}$ denote a solution for the process stopped at an independent $\mbox{Gamma}(r, \frac{r}{t})$ time (parametrisation: the second parameter is an intensity parameter, as with ). Let $\mathbb{P}(X_{T_{r}}^{(r)} \in \{D\}) = 1 - \alpha_{r}$ (where $\{D\}$ denotes the ‘cemetery’; $1 - \alpha_r$ is the probability that the process has been killed by time $T_r$). Note that $\lambda_1 = \lambda_M = 0$ (and hence there is no killing of the process once it has reached sites $i_1$ or $i_M$). By the proof of Lemma \[lmmcruclb\], this implies that $\mathbb{P}(X_{T_{r}}^{(r)} \in \{D\}) > \prod_{j=1}^{M-1} \left(\frac{q_{M-j,M-j-1}}{1 + \widetilde{k}_{M-j}}\right)$, where $q$ is defined by . (This lower bound comes from considering the embedded discrete time process; when it jumps from site $j$, it jumps to $j+1$ or $j-1$ or $D$ with probabilities $\frac{q_{j,j+1}}{1 + \widetilde{k}_j}$, $\frac{q_{j,j-1}}{1 + \widetilde{k}_j}$ and $\frac{\widetilde{k}_j}{1 + \widetilde{k}_j}$ respectively. Once it reaches site $1$, it remains there for all time). This lower bound does not depend on $r$. It follows that $\inf_r \alpha_r > 0$.
Now suppose that there is a subsequence $r_k$ and a $j \in \{2, \ldots, M-1\}$ such that $\lambda_j^{(r_k)} \rightarrow 0$ then, in the limit, if the process reaches site $i_j$, it remains there for all time, so that either $p_{j+1} = \ldots = p_M = 0$ (if $x_0 \leq i_j$) which is a contradiction, or $p_1 = \ldots = p_{j-1} = 0$ (if $x_0 \geq i_j$), again a contradiction.
It follows that there are two constants $0 < c < C < +\infty$ such that $c < \inf_j \inf_r \lambda_j^{(r)} \leq \sup_j \sup_r \lambda_j^{(r)} < C$ and hence it follows that there is a limit point $\underline{\lambda}$ of $\underline{\lambda}^{(r)}$ which provides a solution. The lower bound follows by taking the limit as $r\rightarrow +\infty$ in .
#### Note
At this point there is a (minor) divergence when one tries to establish existence of $m$ such that the generator defined by has the required properties. When considering this problem, there are killing rates $k_1$ and $k_M$ on sites $i_1$ and $i_M$ respectively, which are not necessarily $0$. But after the process reaches either of these sites, the killing rate is exponential and therefore the process survives with positive probability for any finite time and it is straightforward to obtain an upper bound on the killing probability which is strictly less than $1$ when the process is stopped at a Gamma$\left(r, \frac{r}{t}\right)$ time for fixed $t > 0$; the upper bound is independent of $r \geq 1$.
Fixed Point Theorem
-------------------
For fixed $\underline{k}$ let $\underline{h} : \mathbb{R}^M \times \mathbb{R}^{M-2}_+ \rightarrow \mathbb{R}^M$ denote the function defined by:
$$\label{eqhdef} \underline{h}(\underline{p}, \underline{\lambda}) = \frac{1}{\sum_{j,k} p_j {\cal N}_{jk}(\underline{\lambda})}\underline{p} {\cal N}(\underline{\lambda}) = \frac{1}{\sum_j p_j (1 + \frac{t}{r} \lambda_j \widetilde{k}_j)} \underline{p} {\cal N}(\underline{\lambda}).$$
where $\widetilde{\underline{k}}$ is defined by . Directly from the definition, for any $\underline{p} \in \mathbb{R}^M_+$ and $\underline{\lambda} \in \mathbb{R}^{M-2}_+$,
$$\label{eqsumh} \sum_j h_j(\underline{p}, \underline{\lambda}) = 1.$$
From the definition, it is also clear that $\underline{h}(\alpha\underline{p}, \underline{\lambda}) = \underline{h}(\underline{p}, \underline{\lambda})$ for any $\underline{p} \in \mathbb{R}^M_+$, $\alpha \in \mathbb{R} \backslash \{0\}$ and $\underline{\lambda} \in \mathbb{R}_+^{M-2}$.
For $r \geq 2$, set
$$\label{eqhrdef} \underline{h}^{(r)}(\underline{p}, \underline{\lambda}) = \underline{h}(\underline{h}^{(r-1)}(\underline{p}, \underline{\lambda}), \underline{\lambda}).$$
\[thfpt\] Set
$$\label{eqadef2} {\cal A} (\underline{\lambda}, \underline{p})(j):= {\cal G}_j \left(\frac{t}{r}, Q \left (\frac{t}{r}, h^{(r-1)}(\underline{p} , \underline{\lambda}), \underline{\lambda}\right ) \right ) \qquad j = 2, \ldots, M-1$$
where $Q$ defined by and ${\cal G}$ by . There exists a solution $\underline{\lambda}$ to the equation
$$\label{eqfpeq} \underline{\lambda} = {\cal A}(\underline{\lambda}, \underline{p}).$$
which satisfies $\underline{\lambda} \in \mathbb{R}^{M-2}_+$.
#### Proof of Theorem \[thfpt\]
The proof follows the lines of [@N1]. As in [@N1], for $\underline{p} \in \mathbb{R}^M$, set
$$\label{eqcee} C(\underline{p}, \epsilon) = \sum_{j=1}^M \left(\frac{p_j}{\sum_{k=1}^M (p_k \vee \epsilon)} \vee \epsilon \right).$$
For any $\underline{p} \in \mathbb{R}^M$ and $\epsilon \in (0,1)$, $C(\underline{p}, \epsilon) \leq M$. For $\underline{p} \in \mathbb{R}^M$ such that $\sum_{k=1}^M (p_k \vee 0) \geq 1$, it follows that for any $\epsilon \in [0,1)$, $\sum_{k=1}^M (p_k \vee \epsilon) \geq 1$ and hence that
$$M \geq C(\underline{p},\epsilon) \geq \sum_{j=1}^M \left(\frac{p_j}{\sum_{k=1}^M (p_k \vee \epsilon)} \vee \frac{\epsilon}{\sum_{k=1} (p_k \vee \epsilon) } \right) = 1.$$
Let ${\cal P}^{(\epsilon)}: \mathbb{R}^M \rightarrow \mathbb{R}_+^M$ denote the function
$$\label{eqp} {\cal P}^{(\epsilon)}_j(\underline{p}) = \frac{1}{C(\underline{p}, \epsilon)}\left(\frac{p_j}{\sum_{k=1}^M (p_k \vee \epsilon)} \vee \epsilon \right)$$
where $C$ is defined by so that $\sum_{j=1}^M {\cal P}_j^{(\epsilon)}(\underline{p}) = 1$. It follows that for any $\underline{p} \in \mathbb{R}^M$,
$$\min_j {\cal P}_j^{(\epsilon)}(\underline{p}) \geq \frac{\epsilon}{M}.$$
Set
$$\label{eqaeps2} {\cal A}^{(\epsilon)}(\underline{\lambda}, \underline{p})(j) = {\cal G}_j\left ( \frac{t}{r}, Q \left (\frac{t}{r}, {\cal P}^{(\epsilon)}(h^{(r-1)}(\underline{p}, \underline{\lambda})), \underline{\lambda} \right ) \right ).$$
For each $\epsilon > 0$, there exists a $K(\epsilon) < +\infty$ such that $$\sup_{\underline{\lambda} \in \mathbb{R}^{M-2}_+} \max_{j \in \{2, \ldots, M-1\}} {\cal A}^{(\epsilon)}(\underline{\lambda}, \underline{p})(j) \leq K (\epsilon).$$
#### Proof
Consider Equation . If $p_j \geq \epsilon$ for all $j \in \{1, \ldots, M\}$, then ${\cal A}^{(\epsilon)}(\underline{\lambda}, \underline{p})(j) \leq f_j$ where $f_j$ satisfies
$$\left\{ \begin{array}{ll} f_j = a + b\sum_{i=1}^{j-1} f_i & j = 2, \ldots, M \\ f_1 = 0 & \\
a = \frac{1}{\epsilon} \max_j\frac{r(\kappa_{j+1} - \kappa_{j-1})(\kappa_M - \kappa_1)}{t(\kappa_{j+1} - \kappa_j)(\kappa_j - \kappa_{j-1})}, & b = a t \max_j \widetilde{k}_j
\end{array} \right.$$
where $\widetilde{k}_j : j = 1, \ldots, M$ is defined by . The solution to this equation is
$$f_1 = 0 \qquad f_j = a(1 + b)^{j-2} \qquad j \geq 2.$$
and hence
$$0 \leq \min_j {\cal G}_j \leq \max_j {\cal G}_j \leq a (1 + b)^{M-2}.$$
This depends on $\epsilon$, but it does not depend on $\underline{\lambda}$.
It is clear from the construction that for fixed $\epsilon > 0$, ${\cal A}^{(\epsilon)}(., \underline{p})$ is continuous in $\underline{\lambda}$. Therefore, by the Schauder Fixed Point Theorem, there is a solution $\underline{\lambda}^{(\epsilon)}$ to the equation
$$\underline{\lambda} = {\cal A}^{(\epsilon)}(\underline{\lambda}, \underline{p}).$$
Let $\underline{\lambda}^{(\epsilon)}$ denote a fixed point (solution) and let
$$\label{eqheps} \underline{h}_\epsilon = {\cal P}^{(\epsilon)}(h^{(r-1)}(\underline{p}, \underline{\lambda}^{(\epsilon)})),$$
where ${\cal P}^{(\epsilon)}$ is defined by , so that
$$\label{eqlamep} \lambda^{(\epsilon)}_j = {\cal G}_j \left(\frac{t}{r}, Q \left(\frac{t}{r},\underline{h}_\epsilon \right ) \right ) \qquad j = 2, \ldots, M-1$$
where ${\cal G}$ is defined by and $Q$ is defined by . It is required to show:
- $\limsup_{\epsilon \rightarrow 0}\max_{j \in \{2, \ldots, M-1\}} \lambda_j^{(\epsilon)} < +\infty$
- $\liminf_{\epsilon \rightarrow 0} \min_{j \in \{2, \ldots, M-1\}} \lambda_j^{(\epsilon)} > 0$
#### Showing $\limsup_{\epsilon \rightarrow 0} \max_{j \in \{2, \ldots, M-1\}} \lambda_j^{(\epsilon)} < +\infty$
It follows from , using the definition of ${\cal G}$ (Equation ) and the definition of $Q$ (Equation ) that:
$$\begin{aligned}
\lefteqn{ \nonumber\frac{t}{r} h_{\epsilon,j} \lambda^{(\epsilon)}_j \left(1 + \frac{t}{r} \widetilde{k}_j \lambda^{(\epsilon)}_j\right)}\\&& \label{eqhepp2} = \frac{(\kappa_{j+1} - \kappa_{j-1})}{(\kappa_{j+1} - \kappa_j)(\kappa_j - \kappa_{j-1})}\times \left\{ \begin{array}{ll} \sum_{i=1}^{j-1} (\kappa_j - \kappa_i) h_{\epsilon,i} \left( 1 + \frac{t}{r} \widetilde{k}_i \lambda_i^{(\epsilon)}\right)^2 & 2 \leq j \leq l-1 \\
\sum_{i=j+1}^{M} (\kappa_i - \kappa_j) h_{\epsilon,i} \left( 1 + \frac{t}{r}\widetilde{k}_i \lambda_i^{(\epsilon)}\right)^2 & l \leq j \leq M -1
\end{array}\right. \end{aligned}$$
It follows that
$$\label{eqhepconv} h_{\epsilon,j} \stackrel{\lambda_j^{(\epsilon)} \rightarrow +\infty}{\longrightarrow} 0.$$
This can be seen inductively from : recall that $h_{\epsilon,j} > 0$ for each $j$ and $\sum_{j=1}^M h_{\epsilon,j} = 1$. Since $\lambda_1^{(\epsilon)} = \lambda_M^{(\epsilon)} = 0$, the result is clearly true for $j = 2$ and $M-1$ and, furthermore, there are uniform bounds on $\frac{t}{r} h_{\epsilon,2}\lambda_2^{(\epsilon)}\left(1 + \frac{t}{r}\widetilde{k}_2 \lambda_2^{(\epsilon)} \right )$ and $\frac{t}{r} h_{\epsilon,M-1}\lambda_{M-1}^{(\epsilon)}\left(1 + \frac{t}{r}\widetilde{k}_{M-1} \lambda_{M-1}^{(\epsilon)} \right )$. From this, it follows that there are uniform bounds on $\frac{t}{r} h_{\epsilon,2} \left(1 + \frac{t}{r}\widetilde{k}_2 \lambda_2^{(\epsilon)} \right )^2$ and $\frac{t}{r} h_{\epsilon,M-1} \left(1 + \frac{t}{r}\widetilde{k}_{M-1} \lambda_{M-1}^{(\epsilon)} \right )^2$. Inductively, it follows that there are uniform bounds on $\frac{t}{r} h_{\epsilon,j} \left(1 + \frac{t}{r}\widetilde{k}_j \lambda_j^{(\epsilon)} \right )^2$ which hold for all $j$ and hence follows from .
Set
$$K(\underline{\lambda}, \epsilon) := \sum_{j=1}^M \left ( h_j^{(r-1)}(\underline{p}, \underline{\lambda}) \vee \epsilon \right)\qquad \mbox{and} \qquad K_\epsilon := K(\underline{\lambda}^{(\epsilon)},\epsilon).$$
From , it follows that $K_\epsilon \geq 1$. Set
$$C_\epsilon = C \left (\underline{h}^{(r-1)}(\underline{p}, \underline{\lambda}^{(\epsilon)}) \right )$$
where $C$ is the function defined by .
Let
$$\widetilde{{\cal N}}^{(\epsilon)} = \widetilde{\cal N}(\underline{\lambda}^{(\epsilon)}), \qquad {\cal N}^{(\epsilon)} = {\cal N}(\underline{\lambda}^{(\epsilon)}).$$
The first equality below follows from the definition of $\underline{h}_\epsilon$ by and . The second equality follows from the definition of $\underline{h}$ by and $\underline{h}^{(r)}$ given by together with the identity: $\sum_{j=1}^M h_j^{(r-1)} = 1$, which follows from and . Recall that $\underline{p}$ and $\underline{h}$ are taken as [*row*]{} vectors.
$$h_{\epsilon,j} = \frac{1}{C_\epsilon}\left(\frac{1}{K_\epsilon}h^{(r-1)}_j(\underline{p}, \underline{\lambda}^{(\epsilon)}) \vee \epsilon \right ) = \frac{1}{C_\epsilon K_\epsilon} \frac{1}{\sum_{k} (\underline{p} ({\cal N}^{(\epsilon)})^{r-1})_{k}} (\underline{p} ({\cal N}^{(\epsilon)})^{r-1})_{j} \vee \frac{\epsilon}{C_\epsilon}$$
where $(\underline{p} ({\cal N}^{(\epsilon)})^{r-1})_{j} = \sum_{k} p_k (({\cal N}^{(\epsilon)})^{r-1})_{k,j}$, $(({\cal N}^{(\epsilon)})^{r-1})_{k,j}$ being the $(k,j)$ component of the matrix $({\cal N}^{(\epsilon)})^{r-1}$.
Set
$$\label{eqhhat} \widehat{{\cal H}}_{\epsilon;.,j} =
\left\{
\begin{array}{lll}
\frac{1}{C_\epsilon K_\epsilon \sum_{k} (\underline{p} ({\cal N}^{(\epsilon)})^{r-1})_{k}} (({\cal N}^{(\epsilon)})^{r-1})_{.j} & h_{\epsilon,j} > \frac{\epsilon}{ C_\epsilon} \\ \frac{\epsilon}{C_\epsilon} & h_{\epsilon,j} = \frac{\epsilon}{ C_\epsilon} & \end{array} \right.$$
then, since $\sum_{j=1}^M p_j = 1$,
$$\label{eqhep} \underline{h}_{\epsilon} = \underline{p} \widehat{{\cal H}}_{\epsilon}.$$
Define $\underline{p}^{(\epsilon)}$ as:
$$\label{eqpdef} \underline{p}^{(\epsilon)} := \frac{1}{\sum_{l} ( \underline{p} \widehat{{\cal H}}_{\epsilon} ({\cal N}^{(\epsilon)})^{-(r-1)})_{l}} \underline{p} \widehat{\cal H}_{\epsilon} ({\cal N}^{(\epsilon)})^{-(r-1)}.$$
By construction, $\sum_{j=1}^M p_j^{(\epsilon)} = 1$. Furthermore, it follows from the definition that $\underline{p}^{(\epsilon)}$ satisfies:
$$\label{eqpdef2} \underline{p}^{(\epsilon)} = \frac{1}{\sum_{l} ( \underline{h}_{\epsilon} ({\cal N}^{(\epsilon)})^{-(r-1)})_{l}} \underline{h}_{\epsilon}({\cal N}^{(\epsilon)})^{-(r-1)}.$$
From the characterisation given by Lemma \[lmmnprep\], it follows that $0 \leq ((\widetilde{\cal N}^{(\epsilon)})^{-(r-1)})_{j,k} \leq 1$ for each $(j,k) \in \{1, \ldots, M+1\}^2$. Furthermore, $h_{\epsilon;j} \geq 0$ for all $\epsilon > 0$ and all $j \in \{1, \ldots, M\}$. From this, it follows that $p^{(\epsilon)}_j \geq 0$ for each $j \in \{1, \ldots, M\}$.
From and , it follows that
$$\underline{h}_{\epsilon} = \left( \sum_{l} \left (\underline{p} \widehat{{\cal H}}_{\epsilon} ({\cal N}^{(\epsilon)})^{-(r-1)}\right )_l \right) \underline{p}^{(\epsilon)}({\cal N}^{(\epsilon)})^{r-1}.$$
Set
$$\label{eqsdef} S_\epsilon = \left \{ \beta|h_{\epsilon, \beta} = \frac{\epsilon}{C_\epsilon} \right \}$$
Let $Y$ be a continuous time Markov chain with state space $\{1, \ldots, M+1\}$ with transition intensity matrix given by Equation (\[eqthetmat\]), Definition \[defintmat\]. Let $T$ denote an independent time with distribution $T \sim \mbox{Gamma} \left (r-1,\frac{r}{t}\right )$ (using the notation of ). Let $$c(m) = 1 - \mathbb{P}(Y_T = M+1 | Y_0 = m).$$
It follows from that $((\widetilde{{\cal N}}^{(\epsilon)})^{-(r-1)})_{M+1,j} = 0$ for $j = 1, \ldots, M$. From this it follows that
$$(\widetilde{{\cal N}}^{(\epsilon)})^{-(r-1)} = \left(\begin{array}{c|c} ({\cal N}^{(\epsilon)})^{-(r-1)} & - ({\cal N}^{(\epsilon)})^{-(r-1)} \underline{v} \\ \hline \underline{0} & 1 \end{array}\right )$$
where $\underline{0}$ is an $M$-row vector of $0$s, and $\underline{v}$ is the $M$-column vector with $v_j = (\widetilde{{\cal N}}^{(\epsilon) r-1 })_{j,M+1}$, $j = 1, \ldots, M$. It follows that for $m_2 \in S_\epsilon$,
$$\sum_k (({\cal N}^{(\epsilon)})^{-(r-1)})_{m_1,k}\widehat{{\cal H}}_{k,m_2} = \frac{\epsilon}{C_\epsilon} \sum_{k=1}^M (({\cal N}^{(\epsilon)})^{ -(r-1)})_{m_1,k} = \frac{\epsilon}{C_\epsilon}(1 - ((\widetilde{\cal N}^{(\epsilon)})^{-(r-1)})_{m_1,M+1}) = \frac{\epsilon}{C_\epsilon} c(m_1).$$
It follows that:
$$\label{eqnmn} \left(({\cal N}^{(\epsilon)})^{-(r-1)} \widehat{{\cal H}}_\epsilon \right)_{m_1,m_2} = \left\{ \begin{array}{ll}\frac{1}{C_\epsilon K_\epsilon\sum_{jk} p_j (({\cal N}^{(\epsilon)})^{ r-1})_{j,k}} I(m_1,m_2) & m_2 \not \in S_\epsilon \\ \frac{\epsilon}{C_\epsilon} c(m_1) & m_2 \in S_\epsilon \end{array}\right.$$
where $I(m_1,m_2) = \left\{ \begin{array}{ll} 1 & m_1 = m_2 \\ 0 & m_1 \neq m_2 \end{array}\right.$. Set
$$\label{eqefff} F_\epsilon := ({\cal N}^{(\epsilon)})^{-(r-1)} \widehat{{\cal H}}_\epsilon \qquad \mbox{so that} \qquad ({\cal N}^{(\epsilon)})^{r-1} F_\epsilon = \widehat{{\cal H}}_\epsilon.$$
It follows (from , using in the denominator) that:
$$\label{eqpprerewrite} \underline{p}^{(\epsilon)} = \frac{1}{\sum_{j,k } h_{\epsilon,j} (({\cal N}^{(\epsilon)})^{ -(r-1)})_{j,k}} \underline{p} {\cal N}^{(\epsilon) r-1} F_\epsilon ({\cal N}^{(\epsilon)})^{ -(r-1)}.$$
Let $\Lambda_\epsilon$ be the matrix such that
$$\Lambda_{\epsilon; m_1,m_2} = \left\{ \begin{array}{ll} 1 & m_1 = m_2 \qquad m_2 \not \in S_\epsilon \\ 0 & \mbox{otherwise}. \end{array}\right.$$
Let ${\cal I}_\epsilon$ denote the matrix with entries:
$${\cal I}_{\epsilon; m_1,m_2} = \left\{\begin{array}{ll} c(m_1) & m_2 \in S_\epsilon \\ 0 & \mbox{otherwise} \end{array}\right.$$
Then ${\cal I}_\epsilon$ has column $(c(1), \ldots, c(M))^t$ for each $m_2 \in S_\epsilon$ and the remaining columns are columns of $0$s. Then may be written, using $F_\epsilon$ from as:
$$F_\epsilon = \frac{1}{C_\epsilon K_\epsilon \sum_{k} (\underline{p} ({\cal N}^{(\epsilon)})^{r-1})_{k}}\Lambda_\epsilon + \frac{\epsilon}{C_\epsilon } {\cal I}_\epsilon,$$
so that
$$\label{eqppostrewrite}
\underline{p}^{(\epsilon)} = \frac{1}{\sum_k \left(\underline{h}_\epsilon ({\cal N}^{(\epsilon)})^{-(r-1)} \right )_k }\left( \frac{\underline{p} ({\cal N}^{(\epsilon)})^{r-1} \Lambda_\epsilon ({\cal N}^{(\epsilon)})^{-(r-1)}}{C_\epsilon K_\epsilon \sum_k (\underline{p} ({\cal N}^{(\epsilon)})^{r-1})_k} + \frac{\epsilon}{C_\epsilon} ({\cal N}^{(\epsilon)})^{r-1} {\cal I}_\epsilon ({\cal N}^{(\epsilon)})^{-(r-1)}\right).$$
Note that, since $((\widetilde{{\cal N}}^{(\epsilon)})^{ -(r-1)})_{jk} = (({\cal N}^{(\epsilon)})^{-(r-1)})_{jk}$ for $(j,k) \in \{1, \ldots, M\}^2$, and $$\sum_{k=1}^{M+1} ((\widetilde{{\cal N}}^{(\epsilon)})^{-(r-1)})_{jk} = 1 \qquad \forall j \in \{1, \ldots, M\},$$ it follows that
$$\label{eqhnk} \sum_k \left(\underline{h}_\epsilon ({\cal N}^{(\epsilon)})^{-(r-1)} \right )_k = 1 - \sum_k h_{\epsilon;k} ((\widetilde{{\cal N}}^{(\epsilon)})^{-(r-1)})_{k,M+1} > c_1 > 0$$
for a $c_1 > 0$ which does not depend on $\epsilon$, by Lemma \[lmmcruclb\].
#### Note
Similarly to the note at the end of Subsection \[mod\], this is the other point where an additional argument is required when an infinitesimal generator given by , since there is killing at sites $i_1$ and $i_M$ at rates $k_1$ and $k_M$ respectively, which are not necessarily $0$. The modification is similar. Consider the proof of Lemma \[lmmcruclb\]; after the process eventually reaches state $1$ or state $M$, which it does with positive probability, the killing rate after it hits these sites is bounded; it has rate $k_1$ on site $1$ and $k_M$ on site $M$ and with Generator the kill rate does not depend on $\underline{\lambda}$. Hence a $c_1 > 0$ may be obtained independent of $\epsilon$ such that holds.
For any invertible matrix ${\cal S}$, the eigenvalues of ${\cal S}^{-1}{\cal A} {\cal S}$ are the same as the eigenvalues of ${\cal A}$. It follows that the eigenvalues of $({\cal N}^{(\epsilon)})^{r-1} \Lambda_\epsilon ({\cal N}^{(\epsilon)})^{-(r-1)}$ are the eigenvalues of $\Lambda_\epsilon$; $0$ with multiplicity equal to the number of elements of $S_\epsilon$ and the remaining eigenvalues all $1$. Similarly, the eigenvalues of ${\cal I}_\epsilon$ are bounded independently of $\epsilon$, since each entry of the $M\times M$ matrix lies in $[0,1]$. It follows that
$$\lim_{\epsilon \rightarrow 0}\frac{\epsilon}{C_\epsilon} \underline{p} ({\cal N}^{(\epsilon)})^{r-1} {\cal I}_\epsilon ({\cal N}^{(\epsilon)})^{ -(r-1)} = 0.$$
It now follows directly that if $K_\epsilon \sum_{k} (\underline{p} ({\cal N}^{(\epsilon)})^{r-1})_{k} \stackrel{\epsilon \rightarrow 0}{\longrightarrow} + \infty$, then $\underline{p}^{(\epsilon)}\stackrel{\epsilon \rightarrow 0}{\longrightarrow} \underline{0}$, contradicting the fact that $p_j^{(\epsilon)} \geq 0$ for each $j$ and $\sum_j p_j^{(\epsilon)} = 1$ for each $\epsilon \in (0,1)$.
Therefore:
$$0 \leq \inf_\epsilon K_\epsilon \left(\sum_{k} (\underline{p} ({\cal N}^{(\epsilon)})^{r-1})_{k} \right) \leq \sup_\epsilon K_\epsilon \left(\sum_{k} (\underline{p} ({\cal N}^{(\epsilon)})^{r-1})_{k} \right) < +\infty.$$
From the definition of $K_\epsilon$,
$$K_\epsilon \left(\sum_{k} (\underline{p} ({\cal N}^{(\epsilon)})^{ r-1})_{k} \right) = \sum_{k=1}^M \left( (\underline{p} ({\cal N}^{(\epsilon)})^{r-1})_{k} \vee \left(\sum_{k} (\underline{p} ({\cal N}^{(\epsilon)})^{r-1})_{k}\right)\epsilon \right).$$
From the above, $$0 \leq \inf_\epsilon \sum_{k} (\underline{p} ({\cal N}^{(\epsilon)})^{ r-1})_{k} \leq \sup_\epsilon \sum_{k} (\underline{p} ({\cal N}^{(\epsilon)})^{r-1})_{k} < +\infty$$
and
$$\sup_\epsilon \max_k \left((\underline{p} ({\cal N}^{(\epsilon)})^{ r-1})_{k} \vee 0 \right) < + \infty,$$
from which
$$\sup_\epsilon \max_k \left | (\underline{p} ({\cal N}^{(\epsilon)})^{ r-1})_{k} \right | < + \infty.$$
Set $\lambda^{*(\epsilon)} = \max_j \lambda_j^{(\epsilon)}$ and let
$${\cal N}^{*(\epsilon)} = \frac{1}{\lambda^{*(\epsilon)}}{\cal N}^{(\epsilon)}$$
(that is, divide every element by $\lambda^{*(\epsilon)}$). Then if there is a sequence $\epsilon_n \rightarrow 0$ such that $\lambda^{*(\epsilon_n)} \stackrel{n \rightarrow +\infty}{\longrightarrow} + \infty$, any limit point ${\cal N}^*$ of ${\cal N}^{*(\epsilon_n)}$ satisfies
$$0 = \underline{p}{\cal N}^{*(r-1)}.$$
It follows from the construction of ${\cal N}^*$ that the rank $\rho$ of ${\cal N}^*$ is the number of components of $\underline{\lambda}$ such that $\lim_{n \rightarrow +\infty}\frac{\lambda_j^{(\epsilon_n)}}{\lambda^{*(\epsilon_n)}} > 0$, where where $\underline{\lambda}^{(\epsilon_n)}$ is a sequence that gives the limit point. This is seen as follows: consider the lowest index $k_1$ such that $\lim_{n \rightarrow +\infty} \frac{\lambda_{k_1}^{(\epsilon_n)}}{\lambda^{*(\epsilon_n)}} > 0$, then ${\cal N}^*$ in the limit, column $k_1 - 1$ will have exactly one entry; element ${\cal N}^*_{k_1, k_1 - 1}$ will be the only non-zero element of column $k_1$. Suppose $k_1 < \ldots < k_\rho$ are the relevant indices, then the columns $({\cal N}^*_{.,k_1-1}, \ldots, {\cal N}^*_{.,k_\rho-1})$ provide an upper triangular matrix, with elements ${\cal N}^*_{k_j, k_j-1} \neq 0$ and ${\cal N}_{p,k_j - 1} = 0$ for all $p \geq k_j + 1$, proving that ${\cal N}^*$ is of rank $\rho$.
Therefore ${\cal N}^{*(r-1)}$ is of rank $\rho$ and the non-zero rows of ${\cal N}^{*(r-1)}$ are those corresponding to the indices $k : \lim_{n \rightarrow +\infty} \frac{\lambda_k^{(\epsilon_n)}}{\lambda^{*(\epsilon_n)}} > 0$. Since the space spanned by the $\rho$ rows is of rank $\rho$, it follows that $p_k = 0$ for each of these $p_k$, which is a contradiction (since, by hypothesis, $p_k > 0$ for each $k$). Hence
$$\sup_\epsilon \lambda^{*(\epsilon)} < +\infty.$$
#### Showing that $\inf_\epsilon \min_{j \in \{2, \ldots, M-1\}} \lambda_j^{(\epsilon)} > 0$.
Now suppose that there is a subsequence $\lambda_j^{(\epsilon_n)} \stackrel{n \rightarrow +\infty}{\longrightarrow} 0$ for some $j \in \{2, \ldots, M-1\}$. As before, $\lambda^{*(\epsilon)} = \max_{j \in \{2, \ldots, M-1\}} \lambda^{(\epsilon)}_j$. Recall the representation from Lemma \[lmmnprep\], that
$$(\widetilde{{\cal N}}^{-(r-1)})_{j,k} (\underline{\lambda}) = \mathbb{P}(Y_T = k | Y_0 = j)$$
where $Y$ is a continuous time Markov chain with state space $\{1, \ldots, M+1\}$, with intensity matrix given by Equation , Definition \[defintmat\] and $T$ is an independent random variable with distribution $T \sim \mbox{Gamma}(r-1,\frac{r}{t})$ (using parametrisation found in ). Let $\pi_T$ denote the density function of the random variable $T$. Recall that $\sup_\epsilon \max_k \lambda^{(\epsilon)}_k < +\infty$ and suppose that $\lambda_j^{(\epsilon_n)} \rightarrow 0$
If $\lambda^{(\epsilon_n)}_j \rightarrow 0$ for some $m_1 \leq j < m_2$ where $m_1 < m_2$, then, letting $\tau_j = \inf\{r | X_r = j\}$ and $\pi_j (dr) $ the probability measure such that $\mathbb{P}(\tau_j \in A) = \int_A \pi_j (dr)$, then
$$\begin{aligned}
\lefteqn{ \mathbb{P}(Y_T^{(\epsilon_n)} = m_2 |Y_0^{(\epsilon_n)} = m_1 ) = \int_0^\infty \mathbb{P}(Y_s^{(\epsilon_n)} = m_2|Y_0^{(\epsilon_n)} = m_1, T = s)\pi_T(s) ds}\\&&
\int_0^\infty \int_0^s \mathbb{P}(Y_{r}^{(\epsilon_n)} = j |Y_0^{(\epsilon_n)} = m_1)\mathbb{P}(Y_{s - r}^{( \epsilon_n)} = m_2 | Y_0^{(\epsilon_n)} = j) \pi_j(dr) \pi_T(s)ds \end{aligned}$$
so that if $\lambda_{j}^{(\epsilon_n)} \rightarrow 0$, then $\mathbb{P}(Y_s^{(\epsilon_n)} = k|Y_0^{(\epsilon_n)} = j) \rightarrow 0$ for all $k$. It follows that
$$\label{eqytepn} \mathbb{P}(Y_T^{(\epsilon_n)} = m_2 |Y_0^{(\epsilon_n)} = m_1) = ({\cal N}^{-(r-1)})_{m_1,m_2}(\underline{\lambda}^{(\epsilon_n)}) \rightarrow 0$$
for all $(m_1, m_2)$ such that $m_1 \leq j < m_2$. Similarly, if $\lambda^{(\epsilon_n)}_j \stackrel{n \rightarrow +\infty}{\longrightarrow} 0$ for some $m_1 \geq j > m_2$ where $m_1 > m_2$, then holds.
It follows that $ {\cal N}^{-(r-1)}_{kp} (\underline{\lambda}^{(\epsilon_n)}) \rightarrow 0$ for all $(k,p)$ such that $k \leq j < p$ or $k \geq j > p$.
Furthermore, it follows from that for any sequence with limit point $\underline{\lambda}^{(0)}$ such that $\lambda^{(\epsilon_n)}_j\rightarrow 0$ for some $j \in \{2, \ldots, M-1\}$, there is an $l \in \{1, \ldots, M\}$ such that if $j \leq l-1$, then $h_k^{(r-1)}(\underline{p}, \underline{\lambda}^{(0)}) \leq 0$ for all $1 \leq k \leq j - 1$ and if $j \geq l$ then $h_k^{(r-1)} (\underline{p}, \underline{\lambda}^{(0)}) \leq 0$ for all $j+1 \leq k \leq M$.
Now recall the definition of $h^{(r-1)}$ ( and ) from which it follows that
$$h^{(r-1)}(\underline{p},\underline{\lambda}) = \left(\sum_{jk} p_j \left({\cal N}^{r-1}(\underline{\lambda})\right)_{jk} \right)^{-1} \underline{p} {\cal N}^{r-1}(\underline{\lambda}).$$
From this it follows that:
$$\underline{p} = \underline{p}{\cal N}^{(r-1)}(\underline{\lambda}) {\cal N}^{-(r-1)} (\underline{\lambda}) = \left ( \sum_{j,k} p_j ({\cal N}^{r-1})_{j,k}(\underline{\lambda})\right) \underline{h}^{(r-1)}(\underline{p}, \underline{\lambda}){\cal N}^{-(r-1)}(\underline{\lambda}).$$
Recall the definition of $\lambda^{(\epsilon)}$ given by and let $l$ denote the index from the definition of ${\cal G}$ in . With $\underline{\lambda} = \underline{\lambda}^{(0)}$ and considering the zeroes of ${\cal N}^{-(r-1)}(\underline{\lambda}^{(0)})$, it follows that if $j \leq l-1$, then $p_1 \leq 0, \ldots, p_{j-1} \leq 0$, which is a contradiction. If $j \geq l$, then $p_{j+1} \leq 0, \ldots, p_M \leq 0$, which is a contradiction.
It follows that any limit point $\underline{\lambda}$ satisfies $0 < \min_{j \in \{2, \ldots, M-1\}} \lambda_j \leq \max_{j \in \{2, \ldots, M-1\}} \lambda_j < +\infty$, consequently that $h_{0,1} > 0, \ldots, h_{0,M} > 0$, therefore $\underline{h}(\underline{p}, \underline{\lambda}) = \underline{h}_0$ and therefore $\underline{\lambda}$ satisfies Equation . Theorem \[thfpt\] is proved.
Proof of Theorem \[thdrk\]
==========================
Following the proof of Theorem \[thctlim2\], the theorem is already proved for a finite state space ${\cal S} = \{i_1, \ldots, i_M\}$; let $\underline{a}$ satisfy $a_j = \lambda_j (i_{j+1} - i_j)(i_j - i_{j-1})$ then, following Lemma \[lmmfininfgen2\], the continuous time, time homogeneous Markov process $X$ that satisfies Theorem \[thctlim2\] has infinitesimal generator
$${\cal L} = a\left(\frac{1}{2}\Delta + b\nabla - k\right )$$
where the operators $\Delta$ and $\nabla$ are defined by Equations and respectively, where ${\cal L}$ means:
$${\cal L}f(i_j) = a_j \left(\frac{1}{2}\Delta + b_j \nabla - k_j \right) f(i_j) \qquad j = 1, \ldots, M \qquad a_1 = a_M = 0.$$
For a probability distribution $\mu$ over $\mathbb{R}$, the proof follows the same lines as the proof already given for $k \equiv 0$. Set
$$\label{eqesen} {\cal S}_N = \{i_{N,1}, \ldots, i_{N,M_N}\}$$
and let $\underline{p}^{(N)}$ be defined by Equation . Let $\underline{\lambda}_N$ denote a solution to the terminal distribution problem for distribution $\underline{p}^{(N)}$ over space ${\cal S}_N$. Let
$$a^{(N)}_j = a^{(N)}(i_{N,j}) = \lambda^{(N)}_j (i_{N,j+1} - i_{N,j})(i_{N,j} - i_{N,j-1}) \qquad j = 2, \ldots, M_N - 1$$
and let ${\cal L}^{(N)}$ be the infinitesimal generator defined by
$${\cal L}^{(N)} f(i_{N,j}) = a^{(N)}(i_{N,j}) \left(\frac{1}{2}\Delta_N + b^{(N)}_j \nabla_N - k^{(N)}_j \right) f(i_{N,j}) \qquad j = 2, \ldots, M_N - 1$$
where $\Delta_N$ and $\nabla_N$ are the Laplacian and gradient operators defined on ${\cal S}_N$ (Definition \[deflapder\]), the approximate drift field $(b^{(N)}_2, \ldots, b^{(N)}_{M_N-1})$ defined by and
$$\label{eqkconv} k^{(N)}_j =
\frac{1}{i_{N,j+1} - i_{N,j}}\int_{i_{N,j}-}^{i_{N,j+1}} \widehat{k} (x) dx \qquad j = 2, \ldots, M_N-1$$
where $\widehat{k}$ is from , $\int_{a-}^b$ means integration over the interval $[a,b)$. Then ${\cal L}^{(N)}$ is the infinitesimal generator of the process $X^{(N)}$ with state space ${\cal S}_N \cup \{D\}$, where $D$ denotes a cemetery, such that there is an $l_N$, an $\alpha_N \in (0,1)$ and a $\beta_N \in (0,1)$ such that
$$\label{eqtildyn} \left\{\begin{array}{l} \beta_N \mathbb{P} \left ( X_t^{(N)} = i_{N,j} | X_0 = i_{N,l_N} \right ) + (1-\beta_N )\mathbb{P} \left ( X_t^{(N)} = i_{N,j} | X_0 = i_{N,l_N -1} \right ) = \alpha_N p^{(N)}_j \\ j = 1, \ldots, M_N \\ \beta_N \mathbb{P} \left ( X_t^{(N)} \in \{ D \} | X_0 = i_{N,l_N } \right ) + (1-\beta_N )\mathbb{P} \left ( X_t^{(N)} \in \{ D \} | X_0 = i_{N,l_N -1} \right ) = 1 - \alpha_N \end{array}\right.$$
The quantity $\beta_N$ may be interpreted in the following way: there is a point $x_{N,0} \in (i_{N,l_N-1}, i_{N,l_N}]$, denoting the initial condition, such that
$$\mathbb{P} \left (X_{0+}^{(N)} = i_{N,l_N} | X_0^{(N)} = x_{N,0} \right ) = \beta_N, \qquad \mathbb{P} \left (X_{0+}^{(N)} = i_{N,l_N-1} | X_0^{(N)} = x_{N,0} \right ) = 1 - \beta_N.$$
Let $\widetilde{X}^{(N)}$ denote the process with infinitesimal generator $a^{(N)}\left(\frac{1}{2}\Delta_N + b^{(N)} \nabla_N\right)$, then $$X^{(N)}_t = \left\{\begin{array}{ll} \widetilde{X}^{(N)}_t & t \leq \tau_N \\ D & t > \tau_N \end{array}\right.$$
where $\tau_N$ is a random time satisfying $$\label{eqtauen} \mathbb{P}(\tau_N \geq s |( \widetilde{X}^{(N)}_.)) = \exp\left\{ -\int_0^s a^{(N)}(\widetilde{X}_r^{(N)})k( \widetilde{X}^{(N)}_r) dr\right \} \qquad s \geq 0.$$
It follows from Theorem \[thchgd\] that for a sequence $z_j \rightarrow z$, there exists a $\widetilde{X}$ such that
$$\label{eqxlim}\lim_{j \rightarrow +\infty} \mathbb{P} \left( \sup_{0 \leq s \leq t} \left | \widetilde{X}_s^{(N_j)}(z_{N_j}) - \widetilde{X}_s(z) \right | > \epsilon \right ) = 0.$$
Let $\tau$ denote a random time satisfying
$$\label{eqrttau} \mathbb{P}(\tau \geq s |(\widetilde{X}_.)) = \exp\left\{ -\int_0^s \frac{dK}{dm} (\widetilde{X}_r) dr\right \} \qquad s \geq 0.$$
It follows from Equations (\[eqtauen\]) and (\[eqrttau\]) that for a sequence $z_{N_j} \rightarrow z$ such that $z_{N_j} \in {\cal S}_{N_j}$ for each $j$ (${\cal S}_N$ defined by ),
$$\lim_{j \rightarrow +\infty} \left | \mathbb{P}\left ( \left \{ \tau^{(N_j)} \leq t \right \} \right ) - \mathbb{P} \left ( \left \{ \tau \leq t \right \} \right )\right | = 0$$
and
$$\lim_{j \rightarrow +\infty} \sup_{x \in \mathbb{R}} \left | \mathbb{P}(\{\widetilde{X}^{(N_j)}_{t}(z_{N_j}) \leq x\} \cap \{\tau^{(N_j)} > t \}) - \mathbb{P}(\{\widetilde{X}_t(z) \leq x \} \cap \{ \tau > t\})\right | = 0$$
from which it follows that $X$ is a process with infinitesimal generator $\frac{1}{2} \frac{d^2}{dm dx} + \frac{\partial B}{\partial m}\nabla_m - \frac{dK}{dm}$ with the required distribution at the prescribed time $t > 0$, provided $\inf_N \alpha_N > 0$.
Finally, it has to be shown that for the sequence of measures $m^{(N)}$ there does not exist a subsequence such that $m^{(N_j)} \rightarrow 0$, which would correspond to $\mathbb{P}(X_t^{(N_j)} \in \{D\}) \stackrel{j \rightarrow +\infty}{\longrightarrow} 1$.
Let $m^{(N)}$ denote the sequence of measures corresponding to the atomised state spaces. Let $e_{N-}$ and $e_{N+}$ be the indices defined by . Let $K_N = m^{(N)}([i_{e_{N-}}, i_{e_{N+}}])$ and $\widehat{m}^{(N)} = \frac{m^{(N)}}{K_N}$. If $K_{N_j} \rightarrow 0$, then there exists a limit point $\widehat{m}$ of $\widehat{m}^{(N_j)}$ such that ${\cal L} := \frac{1}{2}\frac{\partial^2}{\partial \widehat{m} \partial x} + \frac{\partial B}{\partial \widehat{m}} \nabla_{\widehat{m}} - \frac{\partial K}{\partial \widehat{m}}$ is the infinitesimal generator of a process $X$ which, conditioned on being alive, has stationary distribution $\mu$.
Let $Y$ denote the process with infinitesimal generator $\frac{1}{2}\frac{\partial^2}{\partial \widehat{m} \partial x} + \frac{\partial B}{\partial \widehat{m}} \nabla_{\widehat{m}}$ and let $P(t;x,dy)$ denote its transition kernel. Let $p(t;x,y) = \frac{P(t;x,dy)}{\widehat{m}(dy)}$. Let $\phi(y) = \frac{\mu(dy)}{\widehat{m}(dy)}$. Then $\phi(y)$ has representation
$$\label{eqphirep} \phi(y) = \int \mu (dx) \int p(t;x,y) \frac{\mathbb{E}_x \left [ e^{-\int_0^t \frac{\partial K}{\partial \mu}(Y_s)\phi(Y_s) ds} | Y_t = y \right ]}{\mathbb{E}_x \left [ e^{-\int_0^t \frac{\partial K}{\partial \mu}(Y_s)\phi (Y_s) ds} \right ]}.$$
This can be seen as follows: the transition kernel $Q(t;x,dy)$ for the process $X$ satisfies
$$\left\{ \begin{array}{l} \frac{\partial}{\partial t} Q(t;x,A) = \frac{1}{2}\frac{\partial^2}{\partial \widehat{m} \partial x} Q(t;x,A) + \frac{\partial B}{\partial \widehat{m}} \nabla_{\widehat{m}} Q(t;x,A) - \frac{\partial K}{\partial \widehat{m}} Q(t;x,A) \\ Q(t;x,A) = {\bf 1}_A(x). \end{array}\right.$$
Using $\frac{\partial K}{\partial \widehat{m}} = \frac{\partial K}{\partial \mu}\frac{\partial \mu}{\partial \widehat{m}} = \frac{\partial K}{\partial \mu} \phi$, this has representation:
$$Q(t;x,A) = \mathbb{E}_x \left [ {\bf 1}_A(Y_t)e^{-\int_0^t \frac{\partial K}{\partial \mu}(Y_s)\phi(Y_s) ds}\right ] \qquad A \in {\cal B}(\mathbb{R}).$$
Conditioning on being alive, $\mathbb{P}(X_t \in A | X_0 = x, X_t \not \in \{D\}) = \frac{Q(t;x,A)}{Q(t;x, \mathbb{R})}$, so that
$$\mu (A) = \int \mathbb{P}(X_t \in A | X_0 = x, X_t \not \in \{D\}) \mu (dx)$$
from which Equation follows. This holds for all $t > 0$. Firstly, it follows from this that $\phi(y) \stackrel{y \rightarrow \pm \infty}{\longrightarrow} 0$. Secondly, by the hypothesis on $b$ and $\mu$, it follows from Lemma \[lmmkappdef\] that $Y$ may be put into ‘martingale’ coordinates (described in Section \[subcoc\]). Let $z_+ = \sup\{x | x \in \mbox{suppt}(\mu)\}$ and $z_- = \inf\{x | x \in \mbox{suppt}(\mu)\}$. It follows from basic properties of martingales that if either $z_- > -\infty$ or $z_+ < +\infty$, then $Y_t$ has a well defined limit almost surely, otherwise $|Y_s| \stackrel{s \rightarrow +\infty}{\longrightarrow} +\infty$. In all cases, it follows from Hypothesis \[hybmu\] Part \[hypart4\] that $\frac{\partial K}{\partial \mu} (Y_s) \stackrel{s \rightarrow +\infty}{\longrightarrow} 0$. From this, it follows that for fixed $(x,y) \in (z_-,z_+)$,
$$\limsup_{t \rightarrow +\infty} \frac{\mathbb{E}_x \left [ e^{-\int_0^t \frac{\partial K}{\partial \mu}(Y_s)\phi(Y_s) ds} | Y_t = y \right ]}{\mathbb{E}_x \left [ e^{-\int_0^t \frac{\partial K}{\partial \mu}(Y_s)\phi (Y_s) ds} \right ]} < 1.$$
It follows from the existence of a transformation to martingale coordinates that $p(t;x,y) \rightarrow 0$ for all $(x,y) \in (z_-,z_+)$, from which it follows that $\phi(y) \equiv 0$, hence $\mu \equiv 0$ and a contradiction has been obtained.
Conclusion and Further Study
============================
The article [@N1] established existence of generalised diffusion to meet a given marginal for any probability measure over $\mathbb{R}$. This article deals with the introduction of drift and killing and establishes conditions on given drift and killing under which there exists a ‘clock’ such that the process, conditioned on being alive at a fixed time $t$, has the prescribed marginal.
The open problem of interest is to determine the extent to which the conditions on the drift and killing are merely technicalities to make the proofs work, or whether counter examples can be obtained. In particular, can one find a solution to the problem if there exists a string $m$ such that $\frac{1}{2}\frac{\partial^2}{\partial m \partial x} + \frac{\partial B}{\partial m}\nabla_m - \frac{\partial K}{\partial m}$ is the generator of a process which, conditioned on being alive, has [*invariant measure*]{} $\mu$? This situation (of course) does not arise in the absence of drift and killing.
Another problem of great interest is to explore the connections between the method given here and the Local Variance Gamma Model by Peter Carr, discussed in [@Carr]. This model considers a process composed with a Gamma process. This boils down to a generalised diffusion stopped at an exponential time. There are further developments in [@Carr] and it is of interest to explore the connections between the process stopped at a Gamma time described here, with the problem of introducing more uniform maturity spacings for the problem of calibrating to meet multiple smiles discussed in [@Carr].
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Noble, J.M. \[2013\][*Time Homogeneous Diffusions with a Given Marginal at a Deterministic Time*]{} Stochastic Processes and Applications vol. 123 no. 3 pp 675 - 718
[^1]: email address: [[email protected]]{}\
accepted for publication: Stochastic Processes and their Applications
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In graph theory a partition of the vertex set of a graph is called equitable if for all pairs of cells all vertices in one cell have an equal number of neighbours in the other cell. Considering the implications for the adjacency matrix one may generalize that concept as a block partition of a complex square matrix s.t. each block has constant row sum. It is well known that replacing each block by its row sum yields a smaller matrix whose multiset of eigenvalues is contained in the initial spectrum. We generalize this approach to weighted row sums and rectangular matrices and derive an efficient unitary transformation which approximately block triangularizes a matrix w.r.t. an arbitrary partition. Singular values and Hermiticity (if present) are preserved. The approximation is exact in the equitable case and the error can be bounded in terms of unitarily invariant matrix norms.'
author:
- Mario Thüne
bibliography:
- 'MTh\_EEPfEBT\_ref.bib'
title: Exploiting Equitable Partitions for Efficient Block Triangularization
---
Introduction
============
Equitable Partitions
--------------------
Let $\Gamma$ be a (multi-)graph and let $\mathbf{A}$ be its adjacency matrix, whose entries $a_{vw}$ are the number of edges connecting vertices $v$ and $w$. Let $\Pi=\left(c_1,\ldots,c_k\right)$ be a partition of the vertex set of $\Gamma$ into $k$ cells, inducing a block partition of $\mathbf{A}$, i.e. a simultaneous (disjoint and exhaustive) partition of its rows and columns. It is convenient to define an *indicator matrix* of a partition as
\[defB\] $$\mathbf{B}=\left(b_{vi}\right)\in\left\{0,1\right\}^{N\times k}\text{ with }b_{vi}=\left\{\begin{array}{ll}1&\text{, if item $v$ is in cell $i$}\\0&\text{, else.}\end{array}\right.$$
The partition $\Pi$ is called equitable if all vertices of $\Gamma$ in the same cell have the same number of neighbours in any cell. Equivalently, we may call it equitable if each induced submatrix of $\mathbf{A}$ has constant row sum. The equitable partitions of $\mathbf{A}$ ordered by refinement form a lattice which contains the trivial equitable partition, in which every cell has size exactly one, as the minimum. From the definition it follows that a partition of $\mathbf{A}$ is equitable if and only if there exits a matrix $\mathbf{\Theta}=\left(\theta_{ij}\right)$ s.t. $$\label{equiCentral}
\mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{\Theta}
\quad\text{i.e.}\quad
\forall\ i,j\in\left\{1,\ldots,k\right\}\ \forall\ u\in\left\{1,\ldots,n_i\right\}\ \sum\limits_v^{n_j}\mathbf{A}_{ij,{uv}}=\theta_{ij}.$$ The matrix $\mathbf{\Theta}$ is called the quotient of the partition. Its entries $\theta_{ij}$ are the constant row sums of the matrix blocks $A_{ij}$ induced by the cells $c_i$ and $c_j$, which are the number of edges connecting a fixed vertex in $c_i$ to vertices in $c_j$.
Applications in Graph and Matrix Theory
---------------------------------------
The notion of equitable partitions was developed in graph theory. In network analysis the same concept is also known as *exact coloration* [@EB94RegEqu] or *exact role assignment* [@lerner2005assignments]. It is closely related to *graph fibration* [@BV02FibGra] and arises naturally in the context of graph automorphisms problems since every non trivial automorphism induces a non trival equitable partition. As graph invariants which can be searched for quickly using quite efficient algorithms, equitable partitions are useful in attacking graph isomorphism problems. In that context they are also known as *1-dimensional Weisfeiler-Lehman stabilizers* [@CFI92OptLow].Block partitioned matrices s.t. each block has constant row sum are called *block-stochastic matrices*. In the context of markov chains the technique of *lumping* exploits equitable partitions in order to reduce the number of states [@Buchholz94]. The quotient $\mathbf{\Theta}$ is also known as the *front divisor* [@CDS80SpeGra]. Famously, the spectrum of $\mathbf{\Theta}$, called the main spectrum, is a subset of the spectrum of $\mathbf{A}$ since the columns of $\mathbf{B}$ span an invariant subspace if holds. Therefore, there is a similarity transformation which $2\times2$ block triangularizes $\mathbf{A}$ s.t. one diagonal block is the quotient. Such a transformation can be constructed and applied efficiently in a way provided in [@H59AppThe], [@Chang2011559]. The block triangularization method given below differs from that approach in order to fit in a generalized framework of equitability and provides efficient unitary transformations.
Aim and Outline
---------------
We will generalize the notion of ordinary equitable partitions to arbitrary weighted partitions of the rows and columns of complex matrices. According to a given partition we derive an efficient and stable unitary similarity transformation in order to $2\times2$ block triangularize the matrix up to an error term, which is minimized w.r.t. to several matrix norms and vanishes if and only if exact equitability holds. The transformation can be computed in $O\left(N\right)$ and applied in $O\left(N^2\right)$. It can be further generalized enabling the application to rectangular matrices while maintaining the unitarity property. However the further generalized transformation does only preserve the singular values, but (in general) not the spectrum.Despite offering insides into the structure of objects represented by a graph or a matrix our notion of equitability and its corresponding transformation may be used for compression and for preprocessing eigen and singular value problems. Although describing our transformation as an efficient compression method may seem to suggest that the exploited structure is, in a sense, rare, the concept of equitable partitions, as indicated above, is rather common in various applications, where it is found directly in the studied problem or as an interesting exceptional or ideal case. The usefulness might be increased in particular by the fact that deviations from an exact equitability may be allowed within our framework. In order to get used to the concept and some notation, we briefly discuss in section the special case of an ordinary unweighted equitable partition of a complex square matrix including the derivation of the associated efficient unitary block triangularization and we give an example. In the main part, section , we consider weighted not necessarily equitable partitions introducing the deviation matrix and give our main theorem. In section we consider non exact equitability as an eigenvalue perturbation, give a short overview of several other known generalizations of equation , and briefly consider the problem of finding an equitable partition. Our further generalized version of the concept applicable to rectangular matrices can be found in the appendix.Throughout the article we use the apostrophe to denote the complex conjugated transpose without distinguishing between real and complex operands and we utilize the following notation
Let $n\in\mathbb{N}$. $\mathbf{j}_n=(\hspace{0.1em}\underbrace{1,\ldots,1}_{\text{$n$ times}}\hspace{0.1em})'$ and $\mathbf{f}_n=(1,\hspace{-0.5em}\underbrace{0,\ldots,0}_{\text{$(n-1)$ times}}\hspace{-0.5em})'$.
Unweighted Equitable Partitions {#secUEP}
===============================
Indicator Matrix and Quotient
-----------------------------
Let $\mathbf{A}\in\mathbb{C}^{N\times N}$ and let $\Pi=\left(c_1,\ldots,c_k\right)$ be a simultaneous (disjoint and exhaustive) partition of its rows and columns with *indicator matrix* $\mathbf{B}$ as in definition . Let $\mathbf{A}_{ij}\in\mathbf{C}^{n_i\times n_j}$ be the matrix block in $\mathbf{A}$ induced by row cell $c_i$ and column cell $c_j$. Let $n_i$ be the size of the cell $c_i$ and let $$\mathbf{N}=\left(\mathbf{B}'\mathbf{B}\right)^{\frac{1}{2}}=\operatorname{diag}\left(\sqrt{n_1},\ldots,\sqrt{n_k}\right)$$ We introduce the front quotient, the rear quotient and the Rayleigh quotient respectively as $$\mathbf{E}^{-}=\mathbf{N}^{-2}\mathbf{B}'\mathbf{A}\mathbf{B},\quad
\mathbf{E}^{+}=\mathbf{B}'\mathbf{A}\mathbf{B}\mathbf{N}^{-2},\quad
\mathbf{E}^{\mathrm{0}}=\mathbf{N}^{-1}\mathbf{B}'\mathbf{A}\mathbf{B}\mathbf{N}^{-1}$$ We call $\mathbf{A}$ *front equitable* (i) and respectively *rear equitable* (ii) w.r.t. $\mathbf{B}$ if $$\label{FReq_part}
\left(i\right)\ \mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{E}^{-}\quad,\quad
\left(ii\right)\ \mathbf{B}'\mathbf{A}=\mathbf{E}^{+}\mathbf{B}'.$$ It is easy to see that for Hermitian matrices row equitability and column equitability imply each other. For the rest of this section we assume front equitability, i.e. $$\label{fronteq}
\mathbf{A}_{ij}\mathbf{j}_{n_j}=e^{-}_{ij}\mathbf{j}_{n_i}\quad \forall\ i,j\in\left\{1,\ldots,k\right\}.$$
Block Triangularization
-----------------------
In order to block triangularize $\mathbf{A}$ according to $\mathbf{B}$ we utilize the Householder matrices $$\label{EPhouseholder}
\mathbf{H}_i=\mathbf{I}_{n_i}-2\frac{\mathbf{y}_i\mathbf{y}_i'}{\mathbf{y}_i'\mathbf{y}_i}\ ,\quad
\mathbf{y}_i=\mathbf{j}_{i}+\sqrt{n_i}\mathbf{f}_{n_i}.$$ The following useful relations are easily verified $$\label{EPrelations}
\mathbf{H}_i\mathbf{f}_{n_i}=-\frac{1}{\sqrt{n_i}}\mathbf{j}_{n_i}\quad,\quad
\mathbf{H}_i'\mathbf{j}_{n_i}=-\sqrt{n_i}\mathbf{f}_{n_i}.$$ In order to simplify notations but w.l.o.g. we assume *suitable indexing* which means that $\mathbf{A}$ and $\mathbf{B}$ are indexed in such a way that for $u$ in cell $c_i$ and $v$ in cell $c_j$ it holds that $i<j$ implies $u<v$. Then our proposed transformation of $\mathbf{A}$ can be written conveniently in matrix form using the matrix $$\mathbf{\tilde{H}}=\operatorname{diag}\left(\mathbf{H}_1,\ldots,\mathbf{H}_k\right),$$ which is explicitly block diagonal and, according to , unitary. $$\label{transformA}
\mathbf{\tilde{A}}=\mathbf{\tilde{H}}'\mathbf{A}\mathbf{\tilde{H}}
=\left(\begin{array}{ccc}\mathbf{\tilde{A}}_{11}&\cdots&\mathbf{\tilde{A}}_{1k}\\
\vdots&\ddots&\vdots\\
\mathbf{\tilde{A}}_{k1}&\cdots&\mathbf{\tilde{A}}_{kk}
\end{array}\right)
\quad\text{with}\quad
\mathbf{\tilde{A}}_{ij}=\mathbf{H}_{i}'\mathbf{A}_{ij}\mathbf{H}_{j}.$$ By and there exists a matrix $\mathbf{E}=\left(e_{ij}\right)$ s.t. $$\mathbf{\tilde{A}}_{ij}\mathbf{f}_{j}=e_{ij}\mathbf{f}_{n_i},$$ which immediately shows that each $\mathbf{\tilde{A}}_{ij}$ is block triangular with the left upper block being the scalar $e_{ij}$. Therefore, there is a readily available, in general not unique permutation matrix $\mathbf{\Omega}$ such that $$\mathbf{\hat{A}}=\mathbf{\Omega}'\mathbf{\tilde{A}}\mathbf{\Omega}=
\left(\begin{array}{cc}\mathbf{E}&\mathbf{D}\\\mathbf{0}&\mathbf{F}\end{array}\right)$$ is explicitly block triangular. Since the applied transformations are unitary, the spectrum and the singular values of $\mathbf{A}$ are preserved. We will refer to $\mathbf{F}$, which in general depends on the indexing of $\mathbf{A}$ and on $\mathbf{\Omega}$, as a *factor*. One shows that all factors are unitarily equivalent and that by similarity $$\sigma\left(\mathbf{A}\right)=\sigma\left(\mathbf{E}\right)+\sigma\left(\mathbf{F}\right).$$ Additionally, if $\mathbf{v}$ is an eigenvector of $\mathbf{\hat{A}}$ then $\mathbf{\tilde{H}}\mathbf{\Omega}\mathbf{v}$ is an eigenvector of $\mathbf{A}$ to the same eigenvalue. One also shows that $\mathbf{D}$ vanishes if and only if rear equitability holds. The computational costs for the transformation $\mathbf{\tilde{H}}$ are of order $O\left(n_in_j\right)$ on each subblock for we apply only matrix vector multiplication and matrix addition since $\mathbf{H}_i$ is a rank one update of the identity. Therefore, the total costs are of order $O\left(N^2\right)$. Since $\mathbf{\tilde{H}}\mathbf{\Omega}$ is unitary, Hermiticity (if present) of $\mathbf{A}$ is preserved. Numeric stability is supported by using Householder matrices. Note that in this section we constructed $\mathbf{\tilde{H}}$ s.t. $\mathbf{E}=\mathbf{E}^{\mathrm{0}}$. In the general case those two matrices are unitarily equivalent but not necessarily identical.
Example
-------
Let $$\mathbf{A}_0=\left(\begin{array}{cccccc}
1&2&3&3&3&2\\
2&4&3&1&2&1\\
3&3&1&4&1&1\\
3&1&4&0&2&3\\
3&2&1&2&3&2\\
2&1&1&3&2&4
\end{array}\right)\quad\text{and}\quad
\mathbf{P}_0=\left(\begin{array}{cccccc}
1&0&0&0&0&0\\
0&1&0&0&0&0\\
0&0&0&0&0&1\\
0&0&0&1&0&0\\
0&0&0&0&1&0\\
0&0&1&0&0&0
\end{array}\right)$$ One verifies that $\mathbf{A}_0$ is (unweighted) front equitable w.r.t. $\Pi_0=\left(1\vert2,6\vert3,4,5\right)$. Using the permutation $\mathbf{P}_0$ we can transform it into the suitably indexed form $$\mathbf{A}=\mathbf{P}_0'\mathbf{A}_0\mathbf{P}_0=\left(\begin{array}{cccccc}
1&2&2&3&3&3\\
2&4&1&1&2&3\\
2&1&4&3&2&1\\
3&1&3&0&2&4\\
3&2&2&2&3&1\\
3&3&1&4&1&1
\end{array}\right),$$ which is (unweighted) front equitable w.r.t. $\Pi=\left(1\vert2,3\vert4,5,6\right)$ with front quotient $$\mathbf{E}^{-}=\left(\begin{array}{ccc}
1&4&9\\
2&5&6\\
3&4&6
\end{array}\right).$$ One may employ $$\mathbf{H}_1=\mathbf{H}\left(\mathbf{j}_1\right)=-1,$$ $$\mathbf{H}_2=\mathbf{H}\left(\mathbf{j}_2\right)=-\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1&1\\1&-1\end{array}\right),$$ $$\mathbf{H}_3=\mathbf{H}\left(\mathbf{j}_3\right)=-\frac{1}{\sqrt{3}}\left(\begin{array}{ccc}1&1&1\\1&\frac{1+\sqrt{3}}{-2}&\frac{1-\sqrt{3}}{-2}\\1&\frac{1-\sqrt{3}}{-2}&\frac{1+\sqrt{3}}{-2}\end{array}\right)$$ and $\mathbf{\tilde{H}}=\operatorname{diag}\left(\mathbf{H}_1,\mathbf{H}_2,\mathbf{H}_3\right)$ to transform $\mathbf{A}$ s.t. $$\mathbf{\tilde{A}}=\mathbf{\tilde{H}}'\mathbf{A}\mathbf{\tilde{H}}=\left(\begin{array}{cccccc}
1&\frac{4}{\sqrt{2}}&0&\frac{9}{\sqrt{3}}&0&0\\
\frac{4}{\sqrt{2}}&5&0&6\frac{\sqrt{2}}{\sqrt{3}}&0&0\\
0&0&3&0&\mbox{-}3\scalebox{0.8}{\mbox{+}}\sqrt{3}&\mbox{-}3\mbox{-}\sqrt{3}\\
\frac{9}{\sqrt{3}}&6\frac{\sqrt{2}}{\sqrt{3}}&0&6&0&0\\
0&0&\mbox{-}3\scalebox{0.8}{\mbox{+}}\sqrt{3}&0&\sqrt{3}\mbox{-}1&\mbox{-}6\\
0&0&\mbox{-}3\mbox{-}\sqrt{3}&0&\mbox{-}6&\mbox{-}\sqrt{3}\mbox{-}1
\end{array}\right).$$ Using the permutation $\mathbf{\Omega}=\left(\begin{array}{cccccc}
1&0&0&0&0&0\\
0&1&0&0&0&0\\
0&0&0&1&0&0\\
0&0&1&0&0&0\\
0&0&0&0&1&0\\
0&0&0&0&0&1
\end{array}\right)$ we obtain the matrix $$\mathbf{\hat{A}}=\mathbf{\Omega}'\mathbf{\tilde{A}}\mathbf{\Omega}=\left(\begin{array}{cccccc}
1&\frac{4}{\sqrt{2}}&\frac{9}{\sqrt{3}}&0&0&0\\
\frac{4}{\sqrt{2}}&5&6\frac{\sqrt{2}}{\sqrt{3}}&0&0&0\\
\frac{9}{\sqrt{3}}&6\frac{\sqrt{2}}{\sqrt{3}}&6&0&0&0\\
0&0&0&3&\mbox{-}3\scalebox{0.8}{\mbox{+}}\sqrt{3}&\mbox{-}3\mbox{-}\sqrt{3}\\
0&0&0&\mbox{-}3\scalebox{0.8}{\mbox{+}}\sqrt{3}&\sqrt{3}\mbox{-}1&\mbox{-}6\\
0&0&0&\mbox{-}3\mbox{-}\sqrt{3}&\mbox{-}6&\mbox{-}\sqrt{3}\mbox{-}1
\end{array}\right),$$ which is explicitly reducible. Since $\mathbf{A}$ is Hermitian, the (unweighted) partition $\Pi$ induces front and rear equitability and we actually obtain a block diagonal form. Note that both blocks are Hermitian but the front quotient $\mathbf{E}^{-}$ is not. One verifies that $$\mathbf{E}=\left(\begin{array}{ccc}
1&\frac{4}{\sqrt{2}}&\frac{9}{\sqrt{3}}\\
\frac{4}{\sqrt{2}}&5&6\frac{\sqrt{2}}{\sqrt{3}}\\
\frac{9}{\sqrt{3}}&6\frac{\sqrt{2}}{\sqrt{3}}&6
\end{array}\right)=\operatorname{diag}\left(1,2,3\right)^{\frac{1}{2}}\mathbf{E}^{-}\operatorname{diag}\left(1,2,3\right)^{-\frac{1}{2}}.$$ Let $\mathbf{F}$ denote the lower diagonal block of $\mathbf{\hat{A}}$. Let $\mathbf{V}_{\mathbf{E}}$ and $\mathbf{V}_{\mathbf{F}}$ be the eigenvector matrices of $\mathbf{E}$ and $\mathbf{F}$, respectively. Then one shows that $$\mathbf{V}=\mathbf{P}_0\mathbf{\tilde{H}}\mathbf{\Omega}\left(\begin{array}{cc}\mathbf{V}_{\mathbf{E}}&\mathbf{0}\\\mathbf{0}&\mathbf{V}_{\mathbf{F}}\end{array}\right)$$ is an eigenvector matrix of $\mathbf{A}$. Note that $\mathbf{A}$ and $\mathbf{V}$ need more storage than $\mathbf{\hat{A}}$, $\mathbf{V}_{\mathbf{E}}$, and $\mathbf{V}_{\mathbf{F}}$. The transformations $\mathbf{P}_0$, $\mathbf{\tilde{H}}$ and $\mathbf{\Omega}$ follow from $\Pi_0$ which can be stored as a vector. Due to the small size the blocks of $\mathbf{\tilde{H}}$ were given explicitly as dense matrices. For larger problems one would prefer the usual sparse form as a rank one update of the identity given in .
Weighted Equitable Partitions {#secWEP}
=============================
Preliminaries
-------------
In this section we generalize equitable partitions and accordingly the proposed block triangularization method for square matrices. We introduce the generalized quotient defined for arbitrary partitions of a matrix as a generalization of front and rear quotient. We also introduce the deviation vectors and the deviation matrix and utilize the norm of the latter in order to quantify deviations of a given partition from our generalized notion of equitability. The generalization of the efficient unitary similarity transformation introduced above yields a block triangularization up to an error term due to the deviation from equitability. A further generalization applicable to rectangular matrices preserving only singular values but in general not the spectrum is discussed in the appendix.Note that whenever we invert a matrix explicitly (i.e. not by complex conjugated transposition) this matrix is diagonal. The occasional uses of the pseudo inverse with the property $$c^{\dagger}=\left\{\begin{array}{cc}
0&,c=0\\ \frac{1}{c}&,\text{else}
\end{array}\right. ,\quad c\in\mathbb{C}$$ may be regarded as merely technical.
Complex Householder Transformations
-----------------------------------
This subsubsection aims at the transformation in definition and its properties given in . We consider elementary unitary matrices (EUMs) which are rank (at most) one updates of the identity and necessarily (in order to be unitary) [@S95EleUni] of the form $$\mathbf{U}\left(\gamma,\mathbf{y}\right)=\mathbf{I}-\frac{2}{1+i\gamma}\left(\mathbf{y}'\mathbf{y}\right)^{\dagger}\mathbf{y}\mathbf{y}',\quad\mathbf{y}\in\mathbb{C}^{n},\gamma\in\mathbb{R}.$$ EUMs are a complex generalization of real Householder matrices [@H58UniTri], [@L96ComEle]. We observe that for $c\in\mathbb{C}\setminus\left\{0\right\}$ and $\mathbf{P}$ being a permutation matrix $$\label{eqEscale}
\mathbf{U}\left(\gamma,c\mathbf{y}\right)=\mathbf{U}\left(\gamma,\mathbf{y}\right)\ ,\quad
\mathbf{U}\left(\gamma,\mathbf{P}\mathbf{y}\right)=\mathbf{P}\mathbf{U}\left(\gamma,\mathbf{y}\right)\mathbf{P}'.$$ Let $\mathbf{x}$ and $\mathbf{z}$ be non vanishing complex vectors. We seek an EUM mapping $\mathbf{z}$ into the direction of $\mathbf{x}$, i.e. a complex vector $\mathbf{y}$ and a real number $\gamma$ s.t. $$\label{eqEytox}
\mathbf{U}\left(\gamma,\mathbf{y}\right)\mathbf{z}=\alpha\mathbf{x}\quad\text{with}\quad\alpha\in\mathbb{C}\setminus\left\{0\right\},$$ which implies that $\lVert\mathbf{x}\rVert$ and $\lVert\mathbf{z}\rVert$ determine $\alpha$ up to a phase factor $$\sqrt{\left(\mathbf{U}\left(\gamma,\mathbf{y}\right)\mathbf{z}\right)
'\left(\mathbf{U}\left(\gamma,\mathbf{y}\right)\mathbf{z}\right)}=
\lVert\mathbf{z}\rVert=\left|\alpha\right|\lVert\mathbf{x}\rVert.$$ Again by $\mathbf{y}$ is a linear combination of $\mathbf{x}$ and $\mathbf{z}$, namely $$\label{u=y+ax=cu}
\mathbf{z}-\alpha\mathbf{x}=\frac{2}{1+i\gamma}\left(\mathbf{y}'\mathbf{y}\right)^{\dagger}\left(\mathbf{y}'\mathbf{z}\right)\mathbf{y}.$$ Since according to scaling of $\mathbf{y}$ does not change $\mathbf{U}\left(\gamma,\mathbf{y}\right)$ we may choose $\mathbf{y}=\mathbf{x}-\frac{1}{\alpha}\mathbf{z}$. We are particular interested in the case $\mathbf{z}=\mathbf{f}_n$. Setting $\alpha=\beta\left|\alpha\right|$ and using , we reach in the non trivial case, $\mathbf{y}\neq0$, $$\label{eqGamma}
\gamma\left(\mathbf{x},\beta\right)=
\left(\lVert\mathbf{x}\rVert-\operatorname{Re}\left(\beta x^1\right)\right)^{\dagger}\operatorname{Im}\left(\beta x^1\right).$$ Thus, the required EUM of $\mathbf{f}_n$ into the direction of $\mathbf{x}$ is determined up to a complex parameter $\beta$ lying on the unit circle. We introduce $$\label{H=U}
\mathbf{H}\left(\mathbf{x},\beta\right)=
\mathbf{U}\left({\gamma\left(\mathbf{x},\beta\right)},
\mathbf{x}-\lVert\mathbf{x}\rVert\overline{\beta}\mathbf{f}_n\right)\text{ with }
\left|\beta\right|=1$$ and give an explicit definition.
\[def\_H(x)\] Let $\mathbf{x}\in\mathbb{C}^n$ with $\lVert\mathbf{x}\rVert>0$, let $x^1=\mathbf{f}_n'\mathbf{x}$ denote its first entry and let $\beta$ be a complex number with $\left|\beta\right|=1$, then $$\mathbf{H}\left(\mathbf{x},\beta\right)=\left\{\begin{array}{cc}
\mathbf{I}_n & ,\frac{1}{\lVert\mathbf{x}\rVert}\mathbf{x}=\overline{\beta}\mathbf{f}_n\\
\mathbf{I}_n+\frac{
\left(\mathbf{x}-\lVert\mathbf{x}\rVert\overline{\beta}\mathbf{f}_n\right)
\left(\mathbf{x}-\lVert\mathbf{x}\rVert\overline{\beta}\mathbf{f}_n\right)'}
{\lVert\mathbf{x}\rVert\overline{\beta}\left(
\overline{x^1}-\lVert\mathbf{x}\rVert\beta\right)} & ,\text{ else. }
\end{array}\right.$$
Using $\mathbf{y}=\mathbf{x}-\lVert\mathbf{x}\rVert\overline{\beta}\mathbf{f}_n$ we may rewrite $$\mathbf{H}\left(\mathbf{x},\beta\right)=\mathbf{I}_n+
\frac{\beta}{\lVert\mathbf{x}\rVert}\left(\mathbf{y}'\mathbf{f}_n\right)^{\dagger}
\mathbf{y}\mathbf{y}'=\mathbf{I}_n-\left(\mathbf{x}'\mathbf{y}\right)^{\dagger}\mathbf{y}\mathbf{y}'.$$ And we summarize the following properties $$\label{eq_relations_H}
\mathbf{H}\left(\mathbf{x},\beta\right)\mathbf{f}_n
=\frac{\beta}{\lVert\mathbf{x}\rVert}\mathbf{x}\quad
\text{and}\quad\mathbf{H}\left(\mathbf{x},\beta\right)'\mathbf{x}
=\frac{\lVert\mathbf{x}\rVert}{\beta}\mathbf{f}_n.$$ Since $\mathbf{H}\left(\mathbf{x},\beta\right)$ is a rank one update of $\mathbf{I}_n$, it can be stored with $O\left(n\right)$ and multiplied with a square matrix of size $n$ in $O\left(n^2\right)$. Note that $\mathbf{H}\left(\mathbf{x},\beta\right)$ crucially depends on the ordering of the entries of $\mathbf{x}$, $$\label{eq_permdependent_H}
\mathbf{H}\left(\mathbf{P}'\mathbf{x},\beta\right)\neq\mathbf{P}'\mathbf{H}\left(\mathbf{x},\beta\right)\mathbf{P}\quad\text{for general $\mathbf{x}$ and permutation matrix $\mathbf{P}$}.$$ Although its norm is determined to be $1$, the actual choice of $\beta$ is arbitrary. We may exploit that freedom in order to enhance the numerical properties of $\mathbf{H}\left(\beta,\mathbf{x}\right)$. Particular useful is a choice s.t. $\beta x^1\in\mathbb{R}$, implying $\gamma=0$ by and leading to a Hermitian matrix. Furthermore, for real $\mathbf{x}$, $\beta\in\left\{-1,1\right\}$ ensures a real matrix. A practical recommendation might be
\[beta0\] $\beta_0\left(\mathbf{x}\right)=\left\{\begin{array}{ll}
-\frac{\overline{x^1}}{\left|x^1\right|}&, x^1\neq0\\
1&, x^1=0\end{array}\right.\quad,\ \mathbf{x}\in\mathbb{C}^n$,
which supports numerical stability and coincides with the usual recommendation for the numerical construction of a real Householder matrix. In the previous section we applied $\beta_0$ tacitly.
Weighted Partition, Quotient and Deviation Matrix
-------------------------------------------------
Let $\Pi=\left(c_1,\ldots,c_k\right)$ be a partition of $\{1,\ldots,N\}$ into $k$ cells with indicator matrix $\mathbf{B}$. Let $\mathbf{w}\in\mathbb{C}^{N}$ and $\mathbf{A}\in\mathbb{C}^{N\times N}$. We introduce the *weighted indicator matrix* .
\[defWIM\] Let $\Pi=\left(c_1,\ldots,c_k\right)$ be a partition of $\left\{1,\ldots,N\right\}$. Let $\mathbf{w}\in\mathbb{C}^N$ and let $w^v$ denote its $v$-th entry. $$\mathbf{W}=\left(w_{vi}\right)\in\mathbb{C}^{N\times k}\text{ with }w_{vi}=\left\{\begin{array}{ll}w^v&\ v\in c_i\\0&\text{, else}\end{array}\right.$$
A weighted indicator matrix $\mathbf{W}$ is called *admissible* if $\lVert\mathbf{w}_i\rVert$ for all vector blocks $\mathbf{w}_i$ induced by $c_i$. This implies that $\mathbf{W}'\mathbf{W}$ is invertible and ultimately ensures that the complete spectrum of the quotient, to be defined below, is contained in the spectrum of $\mathbf{A}$. For the rest of this section we assume admissibility.We call $\mathbf{W}$ *suitably indexed* if for $u\in c_i,v\in c_j$ it holds that $i<j$ implies $u<v$. In that case the index set is ordered block wise and $\mathbf{W}$ is explicitly block diagonal. In order to simplify the exposition, we may w.l.o.g. assume a suitable indexing.
\[defFM\]Let $\mathbf{A}\in\mathbb{C}^{N\times N}$ and let $\mathbf{W}$ be an admissible weighted indicator matrix and let $\alpha\in\mathbb{R}$. The *generalized quotient* $\mathbf{E}^{\alpha}$ is given by $$\mathbf{E}^{\alpha}=\left(\mathbf{W}'\mathbf{W}\right)^{-\frac{1-\alpha}{2}}\mathbf{W}'
\mathbf{A}\mathbf{W}\left(\mathbf{W}'\mathbf{W}\right)^{-\frac{1+\alpha}{2}}.$$
We call $\mathbf{E}^{0}$ the *Rayleigh quotient*. The matrix entries of $\mathbf{E}^{\alpha}$ are $$e^{\alpha}_{ij}=\left(\frac{1}{\lVert\mathbf{w}_i\rVert}\right)^{\left(1-\alpha\right)}
\mathbf{w}_i'\mathbf{A}_{ij}\mathbf{w}_j
\left(\frac{1}{\lVert\mathbf{w}_j\rVert}\right)^{\left(1+\alpha\right)}.$$ Since $\mathbf{E}^{\alpha}=\left(\mathbf{W}\mathbf{W}\right)^{\frac{\alpha}{2}}\mathbf{E}^0\left(\mathbf{W}\mathbf{W}\right)^{-\frac{\alpha}{2}}$, all generalized quotients are similar. We distinguish the *front quotient* $\mathbf{E}^{-}=\mathbf{E}^{-1}$ and the *rear quotient* $\mathbf{E}^{+}=\mathbf{E}^{1}$. The matrix $\mathbf{A}$ is called *front equitable* w.r.t. $\mathbf{W}$ if and only if $$\mathbf{A}\mathbf{W}=\mathbf{W}\mathbf{E}^{-}
\quad\text{, i.e.}\quad
\forall\ i,j\in\left\{1,\ldots,k\right\}\ \mathbf{A}_{ij}\mathbf{w}_{j}=e^{-}_{ij}\mathbf{w}_{i}$$ and we call $\mathbf{A}$ *rear equitable* w.r.t. $\mathbf{W}$ if and only if $$\mathbf{W}'\mathbf{A}={\mathbf{E}^{+}}\mathbf{W}'
\quad\text{, i.e.}\quad
\forall\ i,j\in\left\{1,\ldots,k\right\}\ \mathbf{w}_{i}'\mathbf{A}_{ij}={e^{+}_{ij}}\mathbf{w}_{j}'.$$
Maintaining the notation above the *front* and *rear deviation vectors* are defined respectively as $$\mathbf{t}^{-}_{ij}=\frac{1}{\lVert\mathbf{w}_j\rVert}\left(\mathbf{A}_{ij}\mathbf{w}_{j}-e^{-}_{ij}\mathbf{w}_{i}\right)
\quad\text{ and }\quad
{\mathbf{t}^{+}_{ij}}=\frac{1}{\lVert\mathbf{w}_i\rVert}\left({\mathbf{w}_{i}'\mathbf{A}_{ij}}-{e^{+}_{ij}}\mathbf{w}_{j}'\right)',$$ and the *front* and *rear deviation matrices* are $$\mathbf{T}^{\pm}=\left(\begin{array}{ccc}
\mathbf{t}^{\pm}_{11}&\cdots&\mathbf{t}^{\pm}_{1k}\\
\vdots&\ddots&\vdots\\
\mathbf{t}^{\pm}_{k1}&\cdots&\mathbf{t}^{\pm}_{kk}
\end{array}\right)\in\mathbb{C}^{N\times k}\ \ ,\textrm{i.e.}\ \
\begin{array}{lll}
\mathbf{T}^{-}=\left(\mathbf{A}\mathbf{W}-\mathbf{W}\mathbf{E}^{-}\right)\left(\mathbf{W}'\mathbf{W}\right)^{-\frac{1}{2}}\\
\\
\mathbf{T}^{+}=
\left(\mathbf{A}'\mathbf{W}-\mathbf{W}{\mathbf{E}^{+}}'\right)\left(\mathbf{W}'\mathbf{W}\right)^{-\frac{1}{2}}.
\end{array}$$
The entries of $\mathbf{E}^{\pm}$ and the deviation vectors have an intuitive interpretation in the framework of ordinary equitability arising for $\mathbf{w}_i=\mathbf{j}_{n_i}$. Then $e^{-}_{ij}$ and $\lVert\mathbf{t}^{-}_{ij}\rVert$ ($e^{+}_{ij}$ and $\lVert\mathbf{t}^{+}_{ij}\rVert$) are the mean and the standard deviation of the row (column) sums of $\mathbf{A}_{ij}$.Scaling the vector blocks $\mathbf{w}_{i}$ by $\mu_i\in\mathbb{C}\setminus\left\{0\right\}$ changes the entries of the generalized quotient to $\mu_i^{\alpha}e^{\alpha}_{ij}\mu_j^{-\alpha}$ although such a transformation sustains equitability (if present). Note that $e^{0}_{ij}$ and $\lVert{\mathbf{t}^{\pm}_{ij}}\rVert$, and therefore the singular values of $\mathbf{T}^{\pm}$, are independent of such a scaling. By definition, $\mathbf{T}^{\pm}$ is an all zero matrix if and only if its respective equitability holds. At the end of this section, we will consider suitable norms of $\mathbf{T}^{\pm}$ as measures for deviation from equitability.
(Approximate) Block Triangularization
-------------------------------------
Let $\mathbf{W}$ be an admissible weighted indicator matrix of a partition $\Pi=\left(c_1,\ldots,c_k\right)$ with weight vector $\mathbf{w}\in\mathbb{C}^{N}$ and indicator matrix $\mathbf{B}$. Let $\mathbf{w}_i$ be induced by $c_i$. Replacing $\mathbf{w}_i$ by $\mathbf{f}_{n_i}$ for all $i$ yields the new vector $\mathbf{f}$. Let $\mathbf{N}=\left(\mathbf{W}'\mathbf{W}\right)^{\frac{1}{2}}$ and let $\mathbf{V}=\operatorname{diag}\left(\beta_1,\ldots,\beta_k\right)$ be a unitary diagonal matrix of size $k$. We introduce $$\mathbf{Y}\left(\mathbf{W},\mathbf{V}\right)=\mathbf{Y}\left(\mathbf{w},\Pi,\mathbf{V}\right)=\operatorname{diag}\left(\mathbf{w}\right)\mathbf{B}
-\operatorname{diag}\left(\mathbf{f}\right)\mathbf{B}\mathbf{N}\mathbf{V}',$$ which has the form of a weighted indicator matrix. The actual choice of $\mathbf{V}$ is a priori arbitrary. This freedom may be exploited in order to enhance the numerical properties of the transformation matrix given in the next definition.
\[defH\] Let $\mathbf{Y}$ be derived from an admissible weighted indicator matrix $\mathbf{W}$ and a unitary diagonal matrix $\mathbf{V}$ as above, then $$\mathbf{H}\left(\mathbf{W},\mathbf{V}\right)=\mathbf{I}_N-\mathbf{Y}\left(\mathbf{W}'\mathbf{Y}\right)^{\dagger}\mathbf{Y}'.$$
Since $\mathbf{Y}$ and $\mathbf{W}$ have the same block diagonal form, $\mathbf{Y}'\mathbf{W}$ is a diagonal matrix and $\mathbf{H}\left(\mathbf{V},\mathbf{W}\right)$ is block diagonal, hence its numerical properties are comparable to those of a single Householder matrix. In particular, the costs for computing and storing are of order $O\left(N\right)$, and it can be applied to a square matrix in $O\left(N^2\right)$. For suitably indexed $\mathbf{W}$ the block diagonal form of $\mathbf{H}\left(\mathbf{W},\mathbf{V}\right)$ is explicit, $$\mathbf{H}\left(\mathbf{W},\mathbf{V}\right)=\operatorname{diag}\left(\mathbf{H}\left(\mathbf{w}_1,\beta_1\right),\ldots,\mathbf{H}\left(\mathbf{w}_k,\beta_k\right)\right).$$ The diagonal blocks are given in definition . For $\mathbf{A}\in\mathbb{C}^{N\times N}$ we consider $$\mathbf{\tilde{A}}=\mathbf{H}\left(\mathbf{W},\mathbf{V}\right)'\mathbf{A}\mathbf{H}\left(\mathbf{W},\mathbf{V}\right)\quad\text{with}\quad\mathbf{\tilde{A}}_{ij}=\mathbf{H}\left(\mathbf{w}_i,\beta_i\right)'\mathbf{A}_{ij}\mathbf{H}\left(\mathbf{w}_j,\beta_j\right).$$ By the properties of the $\mathbf{H}\left(\mathbf{w}_i,\beta_i\right)$ it follows that $$\begin{aligned}
\mathbf{\tilde{A}}_{ij}\mathbf{f}_{n_j}&\sim \mathbf{f}_{n_i}\ \forall i,j
\text{ if and only if }\mathbf{A}\text{ is front equitable w.r.t. }\mathbf{W},\\
\mathbf{f}_{n_i}'\mathbf{\tilde{A}}_{ij}&\sim \mathbf{f}_{n_j}'\ \forall i,j
\text{ if and only if }\mathbf{A}\text{ is rear equitable w.r.t. }\mathbf{W}.\end{aligned}$$ If we consider for a moment front (row) equitability, the first column (row) of each block $\mathbf{\tilde{A}}_{ij}$ would be all zero from its second to last entry. This implies an implicit block triangular form of $\mathbf{\tilde{A}}$, which can be made explicit by the following permutation mapping the first index of each cell accordingly into $\left\{1,\ldots,k\right\}$.
\[defOmega\] Let $\mathbf{n}=\left(n_1,\ldots,n_k\right)$ be a sequence of $k$ positive integers with $\sum_{i=1}^{k}n_i=N$. The permutation $\Omega_{\mathbf{n}}: \left\{1,\ldots,N\right\}\to\left\{1,\ldots,N\right\}$ is defined by $$\Omega_{\mathbf{n}}\left(m_i+\sum\limits_{j=i}^{i-1}n_j\right)
=\left\{
\begin{array}{ll}
i & ,m_i=1\\
k-i+m_i+\sum\limits_{j=i}^{i-1}n_j & ,m_i\in\left\{2,\ldots,n_i\right\}
\end{array}
\right.$$ with $i\in\left\{1,\ldots,k\right\}$.
We proceed with the general case and give the following theorem, which may be seen as a corollary of theorem . In order to keep this section self contained, it is proven independently.
\[theo1\] Let $\Pi=\left(c_1,\ldots,c_k\right)$ be an admissible partition for $\mathbf{A}\in\mathbb{C}^{N\times N}$ and $\mathbf{w}\in\mathbb{C}^{N}$ with weighted indicator matrix $\mathbf{W}\in\mathbb{C}^{N\times k}$, generalized quotient $\mathbf{E}^{\alpha}$ and deviation matrices $\mathbf{T}^{\pm}$. Let $\mathbf{V}=\operatorname{diag}\left(\beta_1,\ldots,\beta_k\right)$ be a unitary diagonal matrix and let $\mathbf{\tilde{H}}=\mathbf{H}\left(\mathbf{W},\mathbf{V}\right)$ as in definition and let $\mathbf{\Omega}$ be the permutation matrix corresponding to $\Omega_{\left(\left|c_1\right|,\ldots,\left|c_k\right|\right)}$. Let $$\mathbf{\tilde{A}}=\mathbf{\tilde{H}}'\mathbf{A}\mathbf{\tilde{H}}=
\left(\begin{array}{ccc}
\mathbf{\tilde{A}}_{11}&\cdots&\mathbf{\tilde{A}}_{1k}\\
\vdots&\ddots&\vdots\\
\mathbf{\tilde{A}}_{k1}&\cdots&\mathbf{\tilde{A}}_{kk}\\
\end{array}\right)
,\quad
\mathbf{\tilde{A}}_{ij}=\mathbf{H}\left(\mathbf{w}_{i},\beta_i\right)'\mathbf{A}_{ij}\mathbf{H}\left(\mathbf{w}_{j},\beta_j\right)$$ and $$\mathbf{\hat{A}}=\mathbf{\Omega}'\mathbf{\tilde{A}}\mathbf{\Omega}=
\left(\begin{array}{cc}
\mathbf{E}^{\phantom{-}}&{\mathbf{D}^{+}}'\\
\mathbf{D}^{-}&\mathbf{F}^{\phantom{+}}
\end{array}\right)\quad\text{with}\quad
\mathbf{E}\in\mathbb{C}^{k\times k}.$$ Then $\mathbf{\hat{A}}$ is unitarily similar to $\mathbf{A}$, the upper left block $\mathbf{E}$ is unitarily similar to the Rayleigh quotient $\mathbf{E}^0$ and the off-diagonal blocks $\mathbf{D}^{\pm}$ have the same singular values as $\mathbf{T}^{\pm}$, respectively. Additionally, any eigenvector $\mathbf{\hat{z}}$ of $\mathbf{\hat{A}}$ yields an eigenvector $\mathbf{z}=\mathbf{\tilde{H}}\mathbf{\Omega}\mathbf{\hat{z}}$ of $\mathbf{A}$ to the same eigenvalue.
Unitary similarity to $\mathbf{A}$ follows from the unitarity of $\mathbf{\tilde{H}}$ and $\mathbf{\Omega}$.Considering the matrix blocks $\mathbf{\tilde{A}}_{ij}$ of $\mathbf{\tilde{A}}$ induced by cells $c_i$ and $c_j$ we have $$e_{ij}=\mathbf{f}_{n_i}'\mathbf{\tilde{A}}_{ij}\mathbf{f}_{n_j}=
\frac{\overline{\beta_i}}{\lVert\mathbf{w}_i\rVert}
\frac{\beta_j}{\lVert\mathbf{w}_j\rVert}
\mathbf{w}_{i}'\mathbf{A}_{ij}\mathbf{w}_{j}=
\frac{\beta_j}{\beta_i}e^{0}_{ij}.$$ By $\mathbf{\Omega}$ those $e_{ij}$ are mapped accordingly into the upper left block $\mathbf{E}$. Therefore, we may rewrite $\mathbf{E}=\mathbf{V}'\mathbf{E}^0\mathbf{V}$, which proofs unitary similarity of $\mathbf{E}$ and $\mathbf{E}^{0}$.In order to show that $\mathbf{D}^{\pm}$ is unitarily equivalent to $\mathbf{T}^{\pm}$, we observe that by the properties of $\mathbf{\Omega}$ we can write $\mathbf{D}^{\pm}$ as $$\mathbf{D}^{\pm}=\left(\begin{array}{ccc}
\mathbf{d}^{\pm}_{11}&\cdots&\mathbf{d}^{\pm}_{1k}\\
\vdots&\ddots&\vdots\\
\mathbf{d}^{\pm}_{k1}&\cdots&\mathbf{d}^{\pm}_{kk}\\
\end{array}\right)\in\mathbb{C}^{\left(N-k\right)\times k},$$ wherein $\mathbf{d}^{-}_{ij}$ is the first column and ${\mathbf{d}^{+}_{ij}}'$ is the first row of the matrix block $\mathbf{\tilde{A}}_{ij}$ starting from the second entry. We have $$\left(\begin{array}{c}\hspace{-0.5em}0\\\mathbf{d}^{-}_{ij}\end{array}\right)=
\mathbf{\tilde{A}}_{ij}\mathbf{f}_{n_j}-e^{-}_{ij}\mathbf{f}_{n_i}=
\mathbf{H}\left(\beta_i,\mathbf{w}_i\right)'\mathbf{t}^{-}_{ij},$$ $$\left(0,{\mathbf{d}^{+}_{ij}}'\ \right)=
\mathbf{f}_{n_i}'\mathbf{\tilde{A}}_{ij}-e^{+}_{ij}\mathbf{f}_{n_j}'=
{\mathbf{t}^{+}_{ij}}'\mathbf{H}\left(\beta_j,\mathbf{w}_j\right),$$ which shows that $$\label{OHT=0D}
\mathbf{\Omega}'\mathbf{\tilde{H}}'\mathbf{T}^{\pm}=
\left(\begin{array}{c}
\mathbf{0}^{\phantom{\pm}}\\\mathbf{D}^{\pm}
\end{array}\right).$$ The eigenvector relation can be shown by applying $\mathbf{\tilde{H}}\mathbf{\Omega}$ from the left to $$\lambda\mathbf{\hat{z}}=\mathbf{\hat{A}}\mathbf{\hat{z}}=
\mathbf{\Omega}'\mathbf{\tilde{H}}'\mathbf{A}\mathbf{z}.$$
Deviation from Equitability
---------------------------
Let $\lVert\cdot\rVert_{U}$ denote a unitarily invariant norm.
\[cor1\] $$\lVert\mathbf{D}^{\pm}\rVert_{U}=\lVert\mathbf{T}^{\pm}\rVert_{U}.$$
\[cor2\] Let $\mathbf{T}^{-}_{\mathbf{\Theta}}=
\left(\mathbf{A}\mathbf{W}-
\mathbf{W}\mathbf{\Theta}\right)\mathbf{N}^{-1}$ and $\mathbf{T}^{+}_{\mathbf{\Theta}}=\mathbf{N}^{-1}\left(\mathbf{W}'\mathbf{A}-
\mathbf{\Theta}\mathbf{W}'\right)$ with $\mathbf{N}=\left(\mathbf{W}'\mathbf{W}\right)^{\frac{1}{2}}$. Then $$\lVert\mathbf{T}^{\pm}\rVert_{U}=
\min_{\mathbf{\Theta}}\lVert\mathbf{T}^{\pm}_{\mathbf{\Theta}}\rVert_{U}.$$ The minimum is unique if $\lVert\cdot\rVert_{U}$ is a Schatten norm.
Applying $\mathbf{\Omega}'\mathbf{\tilde{H}}'$ from the left and $\mathbf{V}'$ from the right to $\mathbf{T}^{-}_{\mathbf{\Theta}}$ yields $$\begin{aligned}
\lVert\mathbf{\Omega}'\mathbf{\tilde{H}}'\mathbf{T}^{-}_{\mathbf{\Theta}}\mathbf{V}'\rVert_U&=
\lVert\mathbf{\bar{A}}
\mathbf{\Omega}'\mathbf{\tilde{H}}'
\mathbf{W}\mathbf{N}^{-1}\mathbf{V}-
\mathbf{\Omega}'\mathbf{\tilde{H}}'\mathbf{W}\mathbf{\Theta}\mathbf{N}^{-1}\mathbf{V}\rVert_U\nonumber\\
&=\lVert\left(\begin{array}{c}\mathbf{E}^{\phantom{-}}\\\mathbf{D}^{-}\end{array}\right)
-\left(\begin{array}{c}
\mathbf{V}'\mathbf{N}
\mathbf{\Theta}\mathbf{N}^{-1}\mathbf{V}\\
\mathbf{0}\end{array}\right)\rVert_U\end{aligned}$$ using $\mathbf{\Omega}'\mathbf{\tilde{H}}'\mathbf{W}=
\left(\begin{array}{c}
\mathbf{V}'\mathbf{N}\\
\mathbf{0}\end{array}\right)
$. The last term is readily minimized for $$\Theta=\mathbf{N}^{-1}\mathbf{V}\mathbf{E}\mathbf{V}'\mathbf{N}=\left(\mathbf{W}'\mathbf{W}\right)^{-\frac{1}{2}}\mathbf{E}^{0}\left(\mathbf{W}'\mathbf{W}\right)^{\frac{1}{2}}=\mathbf{E}^{-}.$$ Obviously, the minimization is unique for several $\lVert\cdot\rVert_{U}$ including the Schatten norms. A similar proof applies for $\mathbf{T}^{+}$.
The idea underlying the last proof is essentially the same as in [@Dax20101234 proof of theorem 11]. A particular useful choice for ${\lVert\cdot\rVert_{U}}$ might be the Frobenius norm, which upper bounds the spectral norm. Its square is simply the sum of the squared norms of the deviation vectors. One may also think of other characterizations for approximate equitable partitions which have moderate computational costs, for instance the number of nonzero columns of $\mathbf{T}^{\pm}$, which upper bounds the rank.
Discussion and Remarks {#secDMR}
======================
Relating Equitability Deviation and Spectral Deviation
------------------------------------------------------
Since $\mathbf{\hat{A}}$ and $\mathbf{A}$ are unitarily similar and by corollaries and of theorem , we may in a sense ’measure’ the deviation of a partition from being equitable by using a suitable unitarily invariant norm of $\mathbf{T}^{\pm}$, yielding a norm of $\mathbf{D}^{\pm}$, which in turn may serve as a measure for the deviation of the joint eigenvalue sets or the joint singular value sets of $\mathbf{E}$ and $\mathbf{F}$ from the respective values of $\mathbf{A}$.As an example we consider the spectral norm and the eigenvalue bound of Weyl for Hermitian matrices. Assuming Hermiticity we may set $\mathbf{D}^{\pm}=\mathbf{D}$ and $$\mathbf{\hat{A}}=\left(\begin{array}{cc}\mathbf{E}&\mathbf{0}\\
\mathbf{0}&\mathbf{F}\end{array}\right)+\left(\begin{array}{cc}\mathbf{0}&\mathbf{D}'\\
\mathbf{D}&\mathbf{0}\end{array}\right).$$ Let $\mu_1\leq\ldots\leq\mu_N$ be the joint spectrum of Hermitian $\mathbf{E}$ and $\mathbf{F}$, $\lambda_1\leq\ldots\leq\lambda_N$ the eigenvalues of $\mathbf{A}$ and let $\tau_{\text{spec}}$ be the largest singular value of $\mathbf{D}$. We have $$\left|\mu_i-\lambda_i\right|\leq\tau_{\text{spec}}\ ,\ 1\leq i\leq N$$ by the Weyl inequalities. Many more pertubation bounds on eigenvalues and singular values and thier corresponding vectors are feasible, e.g. [@EI98AbsPer],[@Deif1995403],[@E85OptBou].
Cognate Concepts
----------------
The notion of quasi-block-stochastic matrices of Kuich [@K68QuaBlo] as a generalization of quasi-stochastic matrices [@H55QuaSto] bears a close resemblance to our notion of equitability. A minor difference is that for quasi-block-stochastic matrices it is required that the first entry of each $\mathbf{v}_i$ has to be $1$. Kuich also describes how to exploit this structure to triangularize a (real) matrix by a (real, in general not unitary) similarity transformation using a theorem of Haynsworth [@H59AppThe]. Another similar but less general concept is used by Fiol and Carriga and is called *pseudo-regular* partitions. It considers a positive eigenvector $\mathbf{v}$ of binary matrices [@F99EigInt pp. 278/9]. The partition $\Pi$ of the matrix is pseudo-regular if $\mathbf{v}$ and $\Pi$ induce a (weighted) equitable partition. Since $\mathbf{v}$ is fixed up to a positive scale factor, the pseudo-quotient (i.e. front divisor) is unique.There are some more techniques in network analysis which can be described as variations of and which are used to partition the node set of a graph (=assigning roles) according to structural properties and to derive a smaller graph (the quotient or image graph) which gives a condensed representation of essential relations between the cells (=roles) of that partition. Some of those are without apparent regard to the spectrum. For instance, Kate and Ravindran introduced *epsilon equitable* partitions for (an adjacency matrix $\mathbf{A}$ of) a simple graph [@KR09EpsEqu]. Let $\Pi=\left(c_1,\ldots,c_k\right)$ be a partition of the node set of $\mathbf{A}$. Let $\mathbf{A}_{ij}$ be induced by the $i$-th row cell and the $j$-th column cell. Let $\mathbf{r}_{ij}=\mathbf{A}_{ij}\mathbf{j}_{n_j}$ be a column vector of length $n_i=\left|c_i\right|$. If $$\forall\ i,j\in\left\{1,\ldots,k\right\}\quad\max\limits_{1\leq v,w\leq n_i} \left|\mathbf{r}_{ij,v}-\mathbf{r}_{ij,w}\right|\leq\epsilon$$ then $\Pi$ is called $\epsilon$-equitable. The ordinary equitable partition arises for $\epsilon=0$. Another variation of can be employed to describe the concept of *regular equivalence* [@EB94RegEqu], which is defined by the restriction that for a partition $\Pi$ any vector $\mathbf{r}_{ij}=\mathbf{A}_{ij}\mathbf{j}_{n_j}$ must have either no zero entry or all entries zero i.e. $$\forall\ i,j\in\left\{1,\ldots,k\right\}\quad\prod_v\mathbf{r}_{ij,v}=0\Rightarrow\sum_v\left|\mathbf{r}_{ij,v}\right|=0.$$
Finding Equitable Partitions
----------------------------
There are several algorithms for finding ordinary equitable partitions of graphs and matrices, for instance [@B99CompEP], [@GKMS14]. We sketch the most often employed top-down approach made suitable to the case of finding an ordinary front equitable partition of a complex matrix $\mathbf{A}$. At each step one considers a temporary partition (initially often the single cell partition) and (sequentially) subdivides any cell $c_i$ for any $j$ according to the entries of $\mathbf{A}_{ij}\mathbf{j}_{n_j}$, called colors, s.t. each subcell is induced by a unique color, until this subdivision is non trivial, resulting in a refined partition. One iterates until any feasible subdivision is trivial, i.e. the final partition is the unique coarsest front equitable refinement (w.r.t. to the initial partition). Of course, this can be adapted for the weighted case. However, if the weight vector $\mathbf{w}$ has no zero entries one may employ the sketched procedure for the unweighted case readily by considering the matrix $\operatorname{diag}\left(\mathbf{w}\right)^{-1}\mathbf{A}\operatorname{diag}\left(\mathbf{w}\right)$. This follows by left multiplication of $\operatorname{diag}\left(\mathbf{w}\right)^{-1}$ to the equitability condition $$\mathbf{A}\operatorname{diag}\left(\mathbf{w}\right)\mathbf{B}=\operatorname{diag}\left(\mathbf{w}\right)\mathbf{B}\mathbf{E}^{-}.$$ In general, the choice of a weight vector $\mathbf{w}$ may be guided by insights into the problem underlying the considered matrix $\mathbf{A}$. In search for $\mathbf{w}$, one may also exploit that the columns of the weighted indicator matrix $\mathbf{W}$ are a basis for the linear span of all eigenvectors of $\mathbf{A}$ corresponding to eigensolutions of $\mathbf{E}^{-}$. As an example, let $\mathbf{x}$ and $\mathbf{y}$ be two such eigenvectors for different eigenvalues. Considering them separately using the top down approach above one finds the single cell partition since $\mathbf{x}$ and $\mathbf{y}$ are eigenvectors. This may be avoided by using a non trivial linear combination, which lies in the linear span of the columns of $\mathbf{W}$ but is not an eigenvector.How to find partitions with suitably small but non zero deviation from equitability is out of the scope of this article.
Generalization as a Singular Value Decomposition
================================================
Our proposed method for block triangularization can be described as an employment of a one-step singular value decomposition (SVD) of the weighted indicator matrix $\mathbf{W}$ as $$\mathbf{W}=\left[\mathbf{\tilde{H}}\mathbf{\Omega}\right]\left(\begin{array}{c}\mathbf{N}\\\mathbf{0}\end{array}\right)\mathbf{V}'$$ wherein the square diagonal matrix $\mathbf{N}$ contains the singular values of $\mathbf{W}$ and $\mathbf{V}=\operatorname{diag}\left(\beta_1,\ldots,\beta_k\right)$ is unitary diagonal. In deed, if we interpret $\mathbf{f}_{n_i}$ and the vector blocks $\mathbf{w}_i$ as matrices in $\mathbb{C}^{n_i\times1}$, then $\mathbf{w}_i$ has the SVD $$\mathbf{w}_i=
\mathbf{H}\left(\mathbf{w}_i,\beta_i\right)
\frac{\lVert\mathbf{w}_i\rVert}{\beta_i}\mathbf{f}_{n_i}=
\mathbf{H}\left(\mathbf{w}_i,\beta_i\right)
\left(\begin{array}{c}\lVert\mathbf{w}_i\rVert\\\mathbf{0}\end{array}\right)
\overline{\beta_i}.$$ In that view, one may obtain a generalization by replacing the non vanishing vector blocks $\mathbf{w}_i$ by rectangular matrix blocks $\mathbf{W}_i$ with maximal column rank. In the remainder of this section we build on this idea and derive an approximate block triangularization of a rectangular matrix $\mathbf{A}\in\mathbb{C}^{m\times n}$, using given SVDs of a pair of block diagonal matrices with maximal column rank, acting on the rows and columns of $\mathbf{A}$ respectively and separately.For notational convenience we define a $2\times 1$ block matrix with empty lower block and the identity matrix in the square upper block.
\[defInr\] Let $r$ and $n$ be positive integers with $r\leq n$. $$\mathbf{I}_n^r=
\left(\begin{array}{c}\mathbf{I}_r\\\mathbf{0}\end{array}\right)\in\left\{0,1\right\}^{n\times r}.$$
We may identify $\mathbf{I}_{n}^1=\mathbf{f}_{n}$. As a block diagonal generalization we define
\[defInrBD\] Let $\mathbf{r}=\left(r_1,\ldots,r_k\right)$ and $\mathbf{n}=\left(n_1,\dots,n_k\right)$ be ordered sequences of positive integers, s.t. $r_i\leq n_i$. $$\mathbf{I}_{\mathbf{n}}^{\mathbf{r}}=\operatorname{diag}\left(\mathbf{I}_{n_1}^{r_1},\ldots,\mathbf{I}_{n_k}^{r_k}\right).$$
We will also utilize the following permutation.
\[defOmGen\] Let $\mathbf{r}=\left(r_1,\dots,r_k\right)$ and $\mathbf{n}=\left(n_1,\dots,n_k\right)$ be ordered sequences of positive integers s.t. $\forall\ i\in\left\{1,\ldots,k\right\}\ r_i\leq n_i$. Let $r=\sum_ir_i$ and $n=\sum_in_i$. Then $\Omega_{\mathbf{n}}^{\mathbf{r}}: \left\{1,\ldots,n\right\}\to\left\{1,\ldots,n\right\}$ is defined by $$\Omega_{\mathbf{n}}^{\mathbf{r}}\left(s_i+\sum\limits_{j=1}^{i-1}n_j\right)=\left\{\begin{array}{cc}
s_i+\sum\limits_{j=1}^{i-1}r_j&,0<s_i\leq r_i\\
s_i+r+\sum\limits_{j=1}^{i-1}\left(n_j-r_j\right)&,r_i<s_i\leq n_i\end{array}\right.$$ with $i\in\left\{1,\ldots,k\right\}$.
$\Omega_{\mathbf{n}}^{\mathbf{r}}$ maps the first $r_i$ elements of cell $i$ into the first $r$ elements.
\[PropOmega\] In the notation of definitions and above, let $\mathbf{\Omega}$ be the permutation matrix corresponding to $\Omega_{\mathbf{n}}^{\mathbf{r}}$. Then $\mathbf{\Omega}'\mathbf{I}_{\mathbf{n}}^{\mathbf{r}}=\mathbf{I}_n^r.$
\[defAllGen\] Let $\mathbf{W}^{-}\in\mathbb{C}^{m\times q}$ be a block diagonal matrix with $l$ diagonal blocks $\mathbf{W}^{-}_i\in\mathbb{C}^{m_i\times q_i}$ of rank $q_i$ and let $\mathbf{W}^{+}\in\mathbb{C}^{n\times r}$ be a block diagonal matrix with $k$ diagonal blocks $\mathbf{W}^{+}_i\in\mathbb{C}^{n_i\times r_i}$ each of rank $r_i$. Let singular value decompositions for the $\mathbf{W}^{-}_i$ be given by $$\mathbf{W}^{-}_i=\mathbf{U}^{-}_i\mathbf{S}^{-}_i{\mathbf{V}^{-}_i}'=
\mathbf{U}^{-}_i\left(\mathbf{I}_{m_i}^{q_i}\mathbf{N}^{-}_i\right){\mathbf{V}^{-}_i}'$$ with square unitary $\mathbf{U}^{-}_i\in\mathbb{C}^{m_i\times m_i}$ and $\mathbf{V}^{-}_i\in\mathbb{C}^{q_i\times q_i}$, and with $\mathbf{S}^{-}_i=\mathbf{I}_{m_i}^{q_i}\mathbf{N}^{-}_i$ wherein $\mathbf{N}^{-}_i\in\mathbb{R}^{q_i\times q_i}$ is a square diagonal matrix with positive diagonal elements. Let $\mathbf{U}^{-}\in\mathbb{C}^{m\times m}$, $\mathbf{S}^{-}\in\mathbb{C}^{m\times q}$, $\mathbf{N}^{-}\in\mathbb{R}^{q\times q}$, and $\mathbf{V}^{-}\in\mathbb{C}^{q\times q}$ be block diagonal with $l$ diagonal blocks given by $\mathbf{U}^{-}_i$, $\mathbf{S}^{-}_i$, $\mathbf{N}^{-}_i$, and $\mathbf{V}^{-}_i$, respectively. Let $\mathbf{\Omega
}^{-}$ be the permutation matrix corresponding to the permutation $\Omega^{\left(q_1,\dots,q_l\right)}_{\left(m_1,\dots,m_l\right)}$ s.t. ${\mathbf{\Omega}^{-}}'\mathbf{S}^{-}=\mathbf{I}_m^q\mathbf{N}^{-}$. This induces a singular value decomposition of $\mathbf{W}^{-}$ as $$\mathbf{W}^{-}=\mathbf{U}^{-}\left({\mathbf{\Omega}^{-}}'\mathbf{S}^{-}\right){\mathbf{V}^{-}}'=\mathbf{U}^{-}\left(\mathbf{I}_m^q\mathbf{N}^{-}\right){\mathbf{V}^{-}}'.$$ Let the corresponding relations hold for $\mathbf{W}^{+}$ and let $\mathbf{A}\in\mathbb{C}^{m\times n}$. Define the *Rayleigh quotient* $\mathbf{E}^{\mathrm{0}}\in\mathbb{C}^{q\times r}$ as a block matrix with $$\mathbf{E}^{0}=\left(\begin{array}{ccc}
\mathbf{E}^{0}_{11}&\cdots&\mathbf{E}^{0}_{1n}\\
\vdots&\ddots&\vdots\\
\mathbf{E}^{0}_{1m}&\cdots&\mathbf{E}^{0}_{mn}
\end{array}\right)
,\quad
\mathbf{E}^{0}_{ij}=
\mathbf{V}^{-}_{i}{\mathbf{I}^{q_i}_{m_i}}'{\mathbf{U}^{-}_{i}}'
\mathbf{A}_{ij}
\mathbf{U}^{+}_{j}\mathbf{I}^{r_j}_{n_j}{\mathbf{V}^{+}_{j}}'\in\mathbb{C}^{q_i\times r_j}.\nonumber$$ The *front* and [rear deviation matrices]{}, $\mathbf{T}^{-}\in\mathbb{C}^{m\times r}$ and $\mathbf{T}^{+}\in\mathbb{C}^{n\times q}$ respectively, are block matrices with $$\begin{aligned}
\mathbf{T}^{-}_{ij}=&
\phantom{(}\mathbf{A}_{ij}\phantom{)}\mathbf{U}^{+}_{j}\mathbf{I}^{r_j}_{n_j}{\mathbf{V}^{+}_{j}}'-
\mathbf{U}^{-}_{i}\mathbf{I}^{q_i}_{m_i}{\mathbf{V}^{-}_{i}}'\phantom{(}\mathbf{E}^{0}_{ij}\phantom{)'}\in\mathbb{C}^{m_i\times r_i},\nonumber\\
\mathbf{T}^{+}_{ij}=&
\left(\mathbf{A}_{ij}\right)'\mathbf{U}^{-}_{i}\mathbf{I}^{q_i}_{m_i}{\mathbf{V}^{-}_{i}}'-
\mathbf{U}^{+}_{j}\mathbf{I}^{r_j}_{n_j}{\mathbf{V}^{+}_{j}}'\left(\mathbf{E}^{0}_{ij}\right)'\in\mathbb{C}^{n_i\times q_i}.\nonumber\end{aligned}$$
In the notation of definition , $\mathbf{E}^{0}$, $\mathbf{T}^{-}$, and $\mathbf{T}^{+}$ are identical for all singular value decompositions of $\mathbf{W}^{-}$ and $\mathbf{W}^{+}$ which obey the block diagonal form.
$\mathbf{E}^{0}$ and $\mathbf{T}^{\pm}$ can be entirely expressed in terms of $\mathbf{W}^{\pm}$ since $$\mathbf{U}^{-}\mathbf{I}^{\mathbf{q}}_{\mathbf{m}}{\mathbf{V}^{-}}'=\mathbf{W}^{-}\left({\mathbf{W}^{-}}'\mathbf{W}^{-}\right)^{-\frac{1}{2}}
\text{and}\ \
\mathbf{U}^{+}\mathbf{I}^{\mathbf{r}}_{\mathbf{n}}{\mathbf{V}^{+}}'=\mathbf{W}^{+}\left({\mathbf{W}^{+}}'\mathbf{W}^{+}\right)^{-\frac{1}{2}}.$$ $$\begin{aligned}
\mathbf{E}^{0}&=\left({\mathbf{W}^{-}}'\mathbf{W}^{-}\right)^{-\frac{1}{2}}{\mathbf{W}^{-}}'\mathbf{A}\mathbf{W}^{+}\left({\mathbf{W}^{+}}'\mathbf{W}^{+}\right)^{-\frac{1}{2}},
\\
\mathbf{T}^{-}&=
\mathbf{A}\mathbf{W}^{+}
\left({\mathbf{W}^{+}}'\mathbf{W}^{+}\right)^{-\frac{1}{2}}-
{\mathbf{W}^{-}}'\left({\mathbf{W}^{-}}'\mathbf{W}^{-}\right)^{-\frac{1}{2}}
\mathbf{E}^{0},
\\
\mathbf{T}^{+}&=
\mathbf{A}'\mathbf{W}^{-}
\left({\mathbf{W}^{-}}'\mathbf{W}^{-}\right)^{-\frac{1}{2}}-
{\mathbf{W}^{+}}'
\left({\mathbf{W}^{+}}'\mathbf{W}^{+}\right)^{-\frac{1}{2}}
{\left(\mathbf{E}^{0}\right)}'.\end{aligned}$$
\[theoGen\] In the notation of definition above, let $$\mathbf{\hat{A}}={\mathbf{\Omega}^{-}}'{\mathbf{U}^{-}}'\mathbf{A}\mathbf{U}^{+}\mathbf{\Omega}^{+}=
\left(\begin{array}{cc}
\mathbf{E}^{\mathrm{\phantom{f}}}&{\mathbf{D}^{+}}'\\
\mathbf{D}^{-}&\mathbf{F}^{\mathrm{\phantom{r}}}
\end{array}\right)\quad\text{with}\quad\mathbf{E}\in\mathbb{C}^{q\times r}.$$ and let $\lVert\cdot\rVert_{U}$ be a unitarily invariant matrix norm. Then $\mathbf{E}$ and $\mathbf{E}^{0}$ are unitarily equivalent, $\mathbf{D}^{\pm}$ and $\mathbf{T}^{\pm}$ have the same singular values, respectively, and $$\begin{aligned}
\lVert\mathbf{T}^{-}\rVert_{U}=&\min_{\mathbf{\Theta}}\lVert
\mathbf{A}\mathbf{U}^{+}\mathbf{I}^{\mathbf{r}}_{\mathbf{n}}{\mathbf{V}^{+}}'-
\mathbf{U}^{-}\mathbf{I}^{\mathbf{q}}_{\mathbf{m}}{\mathbf{V}^{-}}'\mathbf{\Theta}\rVert_{U}\nonumber\\
\lVert\mathbf{T}^{+}\rVert_{U}=&\min_{\mathbf{\Theta}}\lVert
\mathbf{A}'\mathbf{U}^{-}\mathbf{I}^{\mathbf{q}}_{\mathbf{m}}{\mathbf{V}^{-}}'-
\mathbf{U}^{+}\mathbf{I}^{\mathbf{r}}_{\mathbf{n}}{\mathbf{V}^{+}}'\mathbf{\Theta}'
\rVert_{U}.\nonumber\end{aligned}$$
Unitary equivalence of $\mathbf{E}$ and $\mathbf{E}^{0}$ follows from $$\mathbf{E}={\mathbf{I}^{q}_{m}}'\mathbf{\hat{A}}{\mathbf{I}^{r}_{n}}=
\operatorname{diag}\left(
\mathbf{U}^{-}_{1}\mathbf{I}^{q_1}_{m_1},\ldots,
\mathbf{U}^{-}_{l}\mathbf{I}^{q_l}_{m_l}\right)'
\mathbf{A}
\operatorname{diag}\left(
\mathbf{U}^{+}_{1}\mathbf{I}^{r_1}_{n_1},\ldots,
\mathbf{U}^{+}_{k}\mathbf{I}^{r_k}_{n_k}\right),$$ which uses proposition , yielding $\mathbf{E}^{0}=\mathbf{V}^{-}\mathbf{E}{\mathbf{V}^{+}}'.$That $\mathbf{D}^{-}$ and $\mathbf{T}^{-}$ share the same multiset of singular values follows from $$\begin{aligned}
\left(\begin{array}{c}\mathbf{0}^{\phantom{-}}\\\mathbf{D}^{-}\end{array}\right)&=
\mathbf{\hat{A}}\mathbf{I}^{r}_{n}-\mathbf{I}^{q}_{m}\mathbf{E}=
{\mathbf{\Omega}^{-}}'{\mathbf{U}^{-}}'\mathbf{A}
\mathbf{U}^{+}\mathbf{\Omega}^{+}\mathbf{I}^{r}_{n}-
\mathbf{I}^{q}_{m}{\mathbf{V}^{-}}'\mathbf{E}^{0}{\mathbf{V}^{+}}\nonumber\\
&={\mathbf{\Omega}^{-}}'{\mathbf{U}^{-}}'\mathbf{T}^{-}{\mathbf{V}^{+}}.\end{aligned}$$ The proof for $\mathbf{D}^{+}$ and $\mathbf{T}^{+}$ is analogous.Applying the unitary matrices ${\mathbf{\Omega}^{-}}'{\mathbf{U}^{-}}'$ from the left and ${\mathbf{V}^{+}}$ from the right to the second term in the penultimate equation of theorem yields $$\begin{aligned}
\min_{\mathbf{\Theta}}
\lVert\mathbf{\tilde{A}}
{\mathbf{\Omega}^{+}}'\mathbf{I}^{\mathbf{r}}_{\mathbf{n}}-
{\mathbf{\Omega}^{-}}'\mathbf{I}^{\mathbf{q}}_{\mathbf{m}}
{\mathbf{V}^{-}}'\mathbf{\Theta}\mathbf{V}^{+}\rVert_U
=\lVert\left(\begin{array}{c}\mathbf{E}^{\phantom{-}}\\\mathbf{D}^{-}\end{array}\right)
-\left(\begin{array}{c}
{\mathbf{V}^{-}}'
\mathbf{\Theta}\mathbf{V}^{+}\\
\mathbf{0}\end{array}\right)\rVert_U.\end{aligned}$$ The last term is readily minimized for $\mathbf{\Theta}={\mathbf{V}^{-}}\mathbf{E}{\mathbf{V}^{+}}'=\mathbf{E}^{0}.$ The minimum is obviously unique if $\lVert\cdot\rVert_U$ is a Schatten norm. A similar proof applies for the minimum property of $\mathbf{T}^{+}$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The dynamics of a 2[*D*]{} site percolation model on a square lattice is studied using the hierarchical approach introduced by Gabrielov et al., [*Phys. Rev. E*]{}, [**60**]{}, 5293-5300, 1999. The key elements of the approach are the tree representation of clusters and their coalescence, and the Horton-Strahler scheme for cluster ranking. Accordingly, the evolution of percolation model is considered as a hierarchical inverse cascade of cluster aggregation. A three-exponent time-dependent scaling for the cluster rank distribution is derived using the Tokunaga branching constraint and classical results on percolation in terms of cluster masses. Deviations from the pure scaling are described. An empirical constraint on the dynamics of a rank population is reported based on numerical simulations.'
author:
- 'Ilya Zaliapin, Henry Wong, Andrei Gabrielov'
title: 'Inverse cascade in percolation model: hierarchical description of time-dependent scaling'
---
Introduction
============
Percolation model is probably the simplest and best studied system that experiences (geometrical) phase transition of the second kind [@SA]. It is widely used as a toy model for spatially distributed stochastic processes, such as diffusion in disordered media, forest fires, gelation, semiconduction, etc. [@Sornette04; @SA]. Importantly for our study, percolation model presents a transparent mechanism of the process of hierarchical aggregation (coagulation). This process has been actively employed for describing the essential properties of material fracture and earthquake nucleation [@BSLA97; @BSL97; @GKZN00; @NS90; @NG91; @NTG95; @SB01; @ZKG03], starting from the pioneering works of Allegre et al. [@ALP82] and Newman and Knopoff [@NK82; @NK83; @KN83]. In this paper we describe the evolution of percolation model in terms of an inverse cascade of hierarchical cluster aggregation.
An early idea of hierarchical aggregation was introduced by Newman and Knopoff in the “crack-fusion” model for repetitive cycles of large earthquakes [@NK82; @NK83; @KN83; @NK90]. Their model focused on processes of small cracks fusions into successively larger ones, accommodating the influence of mainshocks and aftershocks, juvenile crack genesis from tectonic stresses, crack healing, and anelastic-creep induced time delays, plus other effects. Turcotte et al. [@Tur+99] have reinstated this line of research considering a log-binned description of hierarchical aggregation and performing numerical tests to study its scaling properties. Gabrielov et al. [@GNT99] first have employed the Horton-Strahler hierarchical ranking [@Horton45; @NTG97] to construct an exactly solvable model of a general inverse cascade process. The Horton-Strahler ranks (see Sect. \[HS\]) that came from hydrology and have been not well known in physical applications happened to be more natural than cluster masses (sizes, areas) in describing the aggregation process. Moreover, the ranks are shown essential for formulating the analytical models [@GNT99]. Recent efforts deal with studying the aggregation dynamics and its various scalings via exactly solvable hierarchical models and extensive simulations [@MNTG04].
Below we focus on the evolution of the first spanning cluster in the the classical site-percolation model, and decribe it as a consecutive hierarchical fusion of smaller clusters into larger ones. Noteworthy, we are interested not in a final solution of a percolation state, but in an evolutionary path leading from the juvenile single-particle clusters to a self-similar population of clusters of arbitrary large size (limited by the finiteness of the lattice), the percolation cluster included. Thus we depart from the steady-state assumption of [@GNT99; @MNTG04; @Tur+99] as well as from the asymptotic focus on the percolation onset typical for the classical percolation studies [@SA].
Specifically, we follow [@GNT99] and represent each cluster by a time-oriented tree that reflects the history of cluster formation. The model dynamics is then described in terms of the corresponding trees using the well-developed theory of hierarchical scaling complexities [@BP97; @NTG97]. An important role is played by the the Horton-Strahler scheme that provides a natural ranking for the tree-based structures. Another important element is the Tokunaga classification that defines a special subclass of trees with self-similar branching. A large number of hierarchies observed in nature are shown to be Tokunaga trees [@NTG97]; this is also the case for the clusters in percolation model [@GNT99; @MNTG04]. We use the Tokunaga constraint together with classical results on percolation dynamics (in terms of cluster masses) to derive time-dependent scaling laws for rank distribution of clusters. Importantly, we report a three-exponent scaling for the dynamics of a population of clusters of a given rank, in deviation from the two-exponent scaling well-known for the population of a given mass [@SA; @Margolina+84]. We also analyze deviations from the pure scaling and confirm our results by numerical simulations.
The inverse cascades and aggregation (coagulation) processes are important for evolution of many natural hazardous processes: earthquakes, landslides, and forest fires are argued to follow the hierarchical aggregation dynamics [@TMGR02; @MNTG04]. A general review of the theory and models of kinetics of irreversible aggregation is given by Leyvraz [@Leyvraz03]. An alternative approach to analytical modeling, based on ideas from [@GNT99], but using equations that are consistent with the mass action law of chemical kinetics, can be found in da Costa et al. [@CGW02].
The paper is organized as follows. The percolation model is described in Sect. \[model\]; this section also introduces tree representation of clusters and the Horton-Strahler ranking. In Sect. \[MRD\] we derive the average mass of clusters of a given rank using the Tokunaga constraint on cluster branching. This result will be actively used in consecutive sections. Section \[RD\] is devoted to the time-dependent rank distribution of clusters. First (Sect. \[rankdist\]), we establish the exponential rank distribution at percolation using the result of Sect. \[MRD\]. We then proceed with time-dependent rank distribution; Sect. \[dynrankdist\] introduces the three-exponent scaling for ranks and compares it to the well-known Stauffer’s two-exponent scaling for cluster masses. Scaling for ranks averaged over the entire evolution of the percolation cluster is derived in Sect. \[AS\]; this result is motivated by the heuristic studies that typically use averaged observations on a system. Time-dependent finite-size corrections to the established scalings are described in Sect. \[corrections\]. Our study of rank distributions is concluded in Sect. \[MDGR\] by describig the time-dependent behavior of the total mass of clusters of a given rank. Sect. \[CFS\] analyzes fractal properties of clusters and reports sharp increase of cluster fractal dimension in the vicinity of percolation. Sect. \[DC\] uses simulations to establish a notable constraint on the dynamics of a rank populations. The results are discussed in Sect. \[discussion\].
Model
=====
Dynamics
--------
We consider the classical 2[*D*]{} site-percolation model [@SA]. The model dynamics starts with an empty $L\times L$ square lattice. At each step a particle is dropped into a randomly chosen unoccupied site; thus each site can be either occupied by one and only one particle or empty. Two sites are considered [*neighbors*]{} if they share one side; each site on a square lattice has four neighbors. Cluster is defined as a group of occupied neighbor sites [@SA]. Time refers to the steps at which particles drop onto the lattice. Since we do not have annihilation of particles, time is formally equivalent to the number of particles on the lattice. It is convenient to normalize time by the lattice size $L^2$ so it varies from $\rho=0$ at the start to $\rho=1$ when all sites are occupied. During the system evolution, occupied sites start to aggregate and clusters begin to form. Once a sufficient number of particles is accumulated, a percolation cluster is formed connecting the opposite sides of the lattice vertically and/or horizontally.
The density $\rho$ increases monotonically from zero to its critical value $\rho_c$ at percolation. For an infinite lattice $\rho_c \approx 0.59274606$ [@NZ01], while for a finite lattice it is smaller [@SA]: $$\label{rhofin}
\rho_c(L) = \rho_c - cL^{-3}.$$
Many phenomena encountered in the percolation model mimic what we see when the phase transitions of the second kind occur. Note however that these phenomena are of purely geometrical and statistical rather than physical nature. Indeed, the physical percolation theory is largerly predicated in this geometrical model and there are many empirical links between them; this is why the percolation model is said to be an example of the [*geometrical*]{} phase transition of the second kind, and why its nomenclature emerges from that of the physical critical phenomena.
The theoretical description of the percolation dynamics is conventionally given in terms of the cluster masses [@SA]; and most of the universal scalings – a benchmark of phase transitions of second kind – deal with parameters expressed via the mass distribution of clusters. However, if one is interested in analytical description of the aggregation process, the mass description happens to be inferior to the hierarchical rank approach [@GNT99; @MNTG04]. Properly defined ranks not only allow one to construct exactly solvable models of aggregation, but also they are more feasible for observations in practice. In addition, they reflect the individual history of cluster formation. Below we follow the hierarchical approach of Gabrielov et al. [@GNT99] to study the percolation dynamics.
Tree representation of clusters
-------------------------------
Each cluster in our model is represented by a tree that reflects the time-dependent formation of a cluster (its history), and is a subject for quantitative analysis. Specifically, each one-particle cluster is represented by a trivial tree consisting of a single node. When two clusters are merged together their trees are also merged by adding a new node (parent) for which they become children (and siblings to each other.) In our model, the coalescence of two or more clusters can only be materialized by adding to the lattice a new particle which will be a neighbor to one or more existing clusters. Figure \[fig\_model1\]a illustrates the four possible types of coalescence. We call $k$-coalescence in a situation when a newly dropped particle (marked [**N**]{} in the figure) is a neighbor to $k$ existing clusters (gray numbered sites). Numerical simulations on a square lattice with $L=2,000$ suggest the following relative frequencies $Q_k$ of $k$-coalescences: $Q_1\approx0.628$, $Q_2\approx0.318$, $Q_3\approx0.052$, $Q_4\approx0.002$. Figures \[fig\_model1\]b,c illustrate how a tree is formed for different coalescence types. There are two basic situations: When a new particle is a neighbor to only one existing cluster, it is considered as an individual one-particle cluster that is connected to the existing one. The connecting node of the tree in this case does not correspond to a particle on the lattice (panel b). When a new particle is droped in a neighbor to two, three, or four existing clusters, it is not condidered as an individual cluster. Instead, it corresponds to the connecting node in the tree (panel c). Thus, the connecting node in a tree may or may not correspond to a lattice particle depending on the coalescence type. The branching parameter (number of children for a given parent) of a tree for any cluster varies between 2 and 4. Note that both 1- and 2-coalescences result in merging only two clusters; accordingly, most of the observed coalescences (about 95%) involve only two clusters while coalescence of three or four clusters is extremely rare.
The consecutive process of tree formation for a simple four-particle cluster is illustrated in Fig. \[fig\_model2\]. Importantly, the individual evolution of a cluster is crucial in constructing the corresponding hierarchical tree. To construct the tree one needs to consider all consecutive coalescences that have formed the cluster, not only its final shape. Therefore, it is clear that the same tree may correspond to clusters of different shape: Figure \[fig\_model3\]a shows two 11-particle clusters that both correspond to the same tree shown in panel b. Therefore, working with trees, we unavoidably narrow the information about the cluster population. Notice however both trees capture an excessively larger amount of information than mere cluster masses. Summing up, the time evolution of a cluster is neccesary and sufficient to uniquely determine the corresponding tree, while the inverse is not true. The problems of describing the set of trees that might correspond to a given cluster, and the set of clusters that correspond to a given tree is beyond the scope of this paper. Next, we describe the ranking of clusters, presenting a conventional alternative to the logarithmic binning of cluster masses.
Horton-Strahler ranking {#HS}
-----------------------
The appropriate ordering of trees (clusters) is very important for meaningful description and analysis of the model dynamics. The problem of such an ordering is not trivial since the clusters may grow and coalesce in a variety of peculiar ways. An advantageous way to solve this problem is given by the Horton-Strahler topological classification of ramified patterns [@Horton45; @Strahler; @BP97] illustrated in Fig. \[fig\_model3\]b: One assigns ranks to the nodes of a tree, starting from $r=1$ at leaves (clusters consisting of one particle.) When two or more clusters with ranks $r_i$, $i=1,\dots,n$ merge together, a new cluster is formed with the rank [@BP97]: $$r=\left\{\begin{array}{ll}
r_1+1,&\mbox{ if }r_i=r_1~\forall~i=1,\dots,n\\
\max{(r_i)},&\mbox{ otherwise.}
\end{array}\right.$$ The rank of a cluster is that of the root of the corresponding tree. It is possible to consider an alternative definition of ranks: When at least two clusters with rank $r$ coalesce, and other coalescing clusters have a lower rank, the rank of a new cluster becomes $r+1$. Clearly, the two definition coincide when only two clusters coalesce. The results reported in this paper are independent of the particular definition, since coalescence of more than two clusters (especially of high ranks) is a rare event.
Originally introduced in geomorphology by Horton [@Horton45] and later refined by Strahler [@Strahler], this classification is shown to be inherent in various geophysical, biological, and computational applications [@BP97; @GNT99; @MNTG04; @NTG97; @Toro01; @TPN98].
Mass-rank distribution {#MRD}
======================
In this section we derive the distribution of the average mass $m_r$ of rank $r$ clusters. It will be used consequently to connect various mass and rank scaling laws. First, we define the branching ratio $T_{ij}$ for a given cluster (tree) as the number $N_{ij}$ of subclusters (nodes) of rank $i$ that joined subcluster (node) of rank $j$, averaged over subclusters (nodes) of rank $j$ [@NTG97; @Tokunaga]: $$T_{ij}=\frac{N_{ij}}{N_j}.$$
Next we note that the mass of a rank $r$ cluster is the sum of two $r-1$ cluster masses that formed the cluster (we ignore the possibility for three or more clusters to coalesce at the same step), plus a unit mass of a joining particle, plus the mass of all the lower-rank clusters that joined the considered cluster, hence: $$\begin{aligned}
\label{mr}
m_1&=&1\nonumber\\
m_2&=&(2\,m_1+P)+T_{12}(m_1+P)\nonumber\\
m_3&=&(2\,m_2+1)+T_{23}(m_2+1)+T_{13}(m_1+P)\nonumber\\
&\dots&\nonumber\\
m_k&=&(2\,m_{k-1}+1)+\sum_{i=1}^{k-1}T_{k-i\,k}(m_{k-i}+1)
-(1-P)T_{1\,k},~k\ge 3.\end{aligned}$$ Here the coefficient $P$ addresses the possibility for a one-particle cluster to join another cluster in two ways: via a one-particle connector (with probability $P$) or directly (with probability $1-P$); the clusters with $r>2$ can only join other clusters using a one-particle connector.
It was predicted by Gabrielov et al. [@GNT99] and later confirmed by simulations [@MNTG04] that clusters in percolation model obey the Tokunaga scaling [@Tokunaga] asymptotically in $k$: $$\label{tokunaga}
T_{i\,i+k}=T_k=s_0s^{k-1}.$$ This rewrites Eq. (\[mr\]) for $k\ge 3$ as $$m_k=(2\,m_{k-1}+1)+\sum_{i=1}^{k-1}T_{i}(m_{k-i}+1)
-(1-P)T_{k-1}.$$ Assuming the mass-rank relation in the form $m_r=c^{r-1}$, $c>1$ we obtain $$\begin{aligned}
\label{mr1}
c\,^{k-1}&=&2c\,^{k-1}+1+\sum_{i=1}^{k-1}s_0s\,^{i-1}
\left(c\,^{k-i-1}+1\right)-
(1-P)s_0\,s\,^{k-1}\nonumber\\
&=&c\,^{k-2}\left[
2+\frac{1}{c\,^{k-2}}
+s_0\frac{1-\left(s/c\right)^{k-1}}
{1-s/c}
+\frac{s_0}{c\,^{k-2}}\,\frac{s\,^{k-1}-1}{s-1}
-(1-P)s_0\left(s/c\right)^{k-2}\right].\nonumber\end{aligned}$$ It is easily checked that this equation has a solution only if $c>s$; thus $s/c<1$ and for large $k$ then follows $$c^{k-1} = c^{k-2}\left[2+\frac{s_0}{1-s/c}\right]$$ leading to the final equation $$c^2-c(2+s+s_0)+2s=0$$ with solution: $$\label{c}
c=\frac{2+s+s_0\pm\sqrt{(2+s+s_0)^2-8s}}{2}.$$
Remarkably, the model of Gabrielov et al. [@GNT99] predicts in a Euclidean (assuming clusters of regular, non-fractal, shape) limit of an inverse cascade model $$s_0\approx 0.55495813,~~ s=1/s_0\approx1.80193774,
~~{\rm and~~} c=1/s_0^2\approx3.24697602.$$ The Eq. (\[c\]) in this case gives $c(s_0,s)=3.24697960$ (this is the only solution such that $c>s$), which is remarkably close (6 digits) to the result of [@GNT99]. Furthermore, the non-Eucledian (assuming fractal shape of clusters) steady-state simulations of Morein et al. [@MNTG04] suggest $$s\approx3.0253, \qquad s_0\approx0.6993,
\qquad c\approx4.325,$$ which exactly solves Eq. (\[c\]). We found it quite amazing that our complimentary set of assumptions used to derive (\[c\]) lead to the same numerical results as analytical study [@GNT99] and simulations of [@MNTG04]. This suggests an underlying connection between our approaches to describe the hierarchical aggregation.
The observed mass-rank distribution of clusters at percolation is shown in Fig. \[fig\_MR\]; it obeys the exponential relation $$\label{MR}
m_r= 10^{\,\gamma (r-1)} = c^{\,r-1},$$ with $\gamma\approx 0.625$, $c=10^{\gamma}\approx 4.2$. Our simulation suggest that the mass distribution within a given rank is approximately lognormal (not shown) with the mean given by Eq. (\[MR\]) and a rank-independent standard deviation.
The relation (\[MR\]) is a key element in our further analysis. As we will show, the distribution of cluster ranks at percolation (Sect. \[rankdist\]) and its finite-size corrections (Sect. \[corrections\]) are obtained from the corresponding classical laws for masses by simple substituting the relation (\[MR\]). At the same time, one of the most important results: the time dependent rank distribution can not be obtained this way and requires an additional treatment (Sect. \[dynrankdist\]).
The exponential relation of Eq. (\[MR\]) happens to be valid over the entire time interval $0<\rho\le\rho_c$. The corresponding dynamics of $c(\rho)$ is shown in Fig. \[fig\_c\]: it grows with time from about 2.0 at the earliest stages to 4.2 at percolation. This growth reflects the fact that clusters become more weighty with time due to coupling with the clusters of lower ranks (which does not change the rank but increases the mass). The growth is not monotonous; it is accompanied by pronounced log-periodic oscillations which are associated with creation of new ranks. The log-periodic oscillations that accompany general power-law increase of observed parameters have been found in many systems including hierarchical models of defect development [@NTG95], biased diffusion on random lattices [@SS98], diffusion-limited aggregation (DLA) [@Sor+96], and others. Log-periodic oscillations can be naturally explained by the Discrete Scale Invariance (DSI) [@Sornette04], which occurs in a system whose observables scale only for a discrete set of values. A famous example of DSI is given by the Cantor set that pocesses a discrete scale symmetry: In order to superimpose its scaled image onto the original, one has to stretch it by the discrete factors $3^n$, $n=1,2,\dots$, not a continuous set of values. The Cantor set and percolation belong to systems with built-in geometrical hierarchy, leading to DSI. In our particular system, ranks take only a countable set of values. Creation of new ranks necessarily disrupt the system in a discontinuous way resulting in the log-periodicity.
Now we return to the numerical value of parameter $c$. In steady-state simulations of [@MNTG04] $c = 4.325$, which is reasonably close to what we observe at percolation. Recall that the models of [@GNT99; @MNTG04] use the “fractal correction” $\epsilon$ to the cluster shape; this correction affects the rate $r_{ij}$ of clusters coalescence: $$r_{ij}\approx \epsilon^{-|j-i|}L_iL_j,$$ where $L_i$ is the total boundary size of the clusters of rank $i$. The correction $\epsilon$ can be expressed as $$\epsilon=\frac{1}{\sqrt{c}}\,\frac{c-1}{c-2},$$ which, together with results of Fig. \[fig\_c\], shows that in the percolation model $\epsilon$ decreases in time passing the Euclidean limit $\epsilon=1$ [@GNT99] at $(\rho_c-\rho)\approx 0.14$ and approaching the steady-state “fractal” $\epsilon\approx 0.68$ [@MNTG04] at $\rho=\rho_c$. The interval $2< c \le 4.2$ observed during $0<\rho\le \rho_c$ corresponds to $0.68\le\epsilon<\infty$.
Rank distribution {#RD}
=================
This section is devoted to establishing various time-dependent scaling laws for clusters of a given rank. We will see that it is tipically impossible to derive such laws by applying the mass-rank relation (\[MR\]) to the coresponding well-known laws for cluster masses. This illustrates an original character and richness of the rank description and prompts for developing new methods of analysis. We start with the simplest problem: rank distribution at percolation.
Distribution at percolation {#rankdist}
---------------------------
We start recalling the well-known cluster mass distribution at percolation [@SA]: $$\label{GRm}
n_m(\rho_c)\sim q_0\,m^{-\tau},$$ where $n_m(\rho_c)$ is the number of clusters of mass $m$ per lattice site, and the Fisher exponent $\tau=187/91\approx 2.05$ is universal for 2[*D*]{} systems [@Fisher67; @SA]. Figure \[fig\_GRm\] illustrates the mass distribution at percolation for a system with $L=2000$; to smooth out statistical fluctuations it shows the number of clusters with mass equal to or larger than $m$: $\sum_{m'\ge m} n_m(\rho_c)$. Equation (\[GRm\]) suggests the slope $\tau-1\approx\,1.05$, while the observed slope $0.96$ is somewhat less than that. This is due to the impact of two concurrent phenomena: so-called “deviation from scaling” at small $m$ [@Hoshen+79] and finite-size effects at large $m$ [@Margolina+84; @Hoshen+79]; they are discussed below in Sect. \[corrections\].
Now, we use Eq. (\[GRm\]) to derive the distribution of the number $n_r(\rho_c)$ of the clusters of rank $r$ at percolation. Taking summation over all clusters of rank $r$ and mass $m$ we obtain: $$\begin{aligned}
\label{nrm0}
n_r(\rho_c)&=&\sum n_{r,m}(\rho_c)
=q_0\,\sum_{m_{\rm lo}}^{m_{\rm up}}\,m^{-\tau}\nonumber\\
&\sim&\frac{q_0}{\tau-1}
\left[(m_{\rm lo})^{-\tau+1}-(m_{\rm up})^{-\tau+1}\right]\nonumber\\
&=&\frac{q_0}{\tau-1}
\left[\left(\frac{m_{\rm lo}}{ m_r }\right)^{-\tau+1}
-\left(\frac{m_{\rm up}}{ m_r }\right)^{-\tau+1}\right]
m_r^{-\tau+1}.\end{aligned}$$ Our simulations suggest (not shown) that the mass distribution within a given rank is lognormal with a rank-independent standard deviation. Thus, for arbitrary upper and lower quantiles $m_{\rm up}$, $m_{\rm lo}$ of this distribution the values $$\frac{m_{\rm lo(up)}}{m_r}$$ are rank independent. Using this, we finally express $n_r(\rho_c)$ via $m_r$: $$\label{nrm}
n_r(\rho_c)=p_0\, m_r ^{-\tau+1}
\propto m_r ^{-1.05}.$$
The power law (\[nrm\]) is observed in a steady-state aggregation model of [@MNTG04] with index $1.147$. This index increase comparing to our $1.05$ is due to the fact that in [@MNTG04] intermediate clusters are removed from the lattice providing extra space for a larger number of smaller clusters.
Combining the mass-rank relation (\[MR\]) with (\[nrm\]) we obtain the following exponential rank distribution at percolation: $$\begin{aligned}
\label{GRr}
n_r(\rho_c)&\sim& p_0\, m_r ^{-\tau+1}
=p_0\,\left(c^{-\tau+1}\right)^{r-1}
= p_1\,10^{-b\,r}\end{aligned}$$ with $$p_1=p_0\,c^{\tau-1},~b=(\tau-1)\,\log_{10}c\approx 0.62.$$
This is indeed what we observe in Fig. \[fig\_GRr\] where the rank distribution $n_r$ at percolation is shown by the dash-dotted line. The study [@MNTG04] suggests $c^{1-\tau}=0.186$ while our predictions and observations lead to $c^{1-\tau}\approx 4.2^{-1.05} = 0.22$. The two values are in good agreement, the slight difference is explained, as in Eq. (\[nrm\]), by removal of intermediate clusters in [@MNTG04]. Next we consider the rank distribution for $\rho\ne\rho_c$.
Dynamical rank distribution: three-exponent scaling {#dynrankdist}
---------------------------------------------------
Here we expand results of the previous section by establishing the time-dependent rank dustribution. First, we consider the dynamics of rank population.
### Temporal dynamics of rank population
The dynamics of the total number $(n_r\cdot L^2)$ of the clusters of a given rank $r$ is illustrated in Fig. \[fig\_rank\] for $r=5,6,7$. The population follows a characteristic bell-shaped trajectory, with percolation at its rightward limb. As in the case of mass description, one does not observe steady-state behavior in the cluster dynamics: The population of each rank steadily develops to its peak as a result of merging of the clusters of lower ranks; then it starts decreasing, giving birth to the clusters of higher ranks. As naturally follows from the model definition, the peak of the population of a higher rank comes after the peak of a lower rank. Figure \[fig\_rank\_all\] shows the population dynamics for the ranks $1\le r \le 11$ in semilogarithmic scale. Here one clearly sees the similarity in the dynamics of different ranks. Note that this figure is remarkably similar to Fig. 7 from [@Tur+99] that shows the dynamics of clusters with logarithmically binned masses. We now proceed by establishing the appropriate time-dependent scaling laws.
### Time-dependent mass distribution
Recall that the temporal dynamics of the cluster mass distribution is given by the two-exponent scaling law [@Stauffer75; @SA; @Margolina+84]: $$\label{2exp}
n_m(\rho)\sim m^{-\tau}\,f_0(z),\qquad
z=(\rho_c-\rho)m^{\sigma}+z_0,$$ with $\sigma=1/2$. The function $f_0$ has a bell-shaped form with maximum to the left of percolation; it can be roughly approximated by a Gaussian function [@Hoshen+79; @Margolina+84]: $$\label{f0}
f_0(z)\propto \exp\left(-a\,z^2\right).$$ Note that the shift $z_0$ is independent of $m$.
Considered as a function of $m$, the two-exponent scaling explains the power law mass distribution (\[GRm\]) at percolation (with $q_0=f_0(z_0)$) as well as the downward bend for $\rho<\rho_c$, clearly observed in Fig. \[fig\_GRm\] (dashed line); while as a function of $\rho$ it describes the bell-shaped dynamics of clusters with given mass $m$.
### Time-dependent rank distribution {#rscale}
Combining the scaling laws (\[MR\]) and (\[2exp\]) one formally obtains the two-exponent scaling for rank dynamics. However, the two exponent scaling does not work for ranks; to show this we assume more generally $$\label{r2exp}
n_r(\rho)\sim g_0(z)10^{-br},~
z=(\rho_c-\rho)h(r)+z'_0,$$ which is consistent with the exponential rank distribution of Eq.(\[GRr\]) at percolation (with $p_0=g_0(z'_0)$). Possible deviations from the pure exponential law at $\rho<\rho_c$ (clearly observed in Fig. \[fig\_GRr\]) and dynamics of a given rank (see Figs. \[fig\_rank\],\[fig\_rank\_all\]) are described by specific form of the functions $g_0(\cdot)$ and $h(\cdot)$. Following [@Hoshen+79] we define $$\label{nu}
\nu_r(z):=\frac{n_r(z)}{n_r(z'_0)}
=\frac{g_0(z)}{g_0(z'_0)}.$$ and choose $h(\cdot)$ in such a way that positions of the peaks of $\nu_r(z)$ coincide for different $r$; it is always possible by choosing the appropriate time change $h(r)$. Figure \[fig\_nu\]a shows the ratio $\nu_r(z)/\nu_1(z)$ for $r=2,3,6,8$. One can see that the two-exponent scaling does not work in our case: the curves do not coincide.
Nevertheless, the simple scaling picture is restored by introducing the additional, third, shift exponent: $$\label{r3exp}
h(r)= a_1\,10^{\sigma_1\,r},\qquad
z'_0(r)=a_2\, 10^{-\sigma_2\,r}.$$ Function $g_0$ still can be approximated by a Gaussian function $$\label{g0}
g_0(z)\propto \exp\left(-\frac{z^2}{2}\right).$$
Once the correct scaling form is established, the use of (\[MR\]) is again legitimate, and the exponent $\sigma_1$ in Eq. (\[r3exp\]) can be evaluated as: $$\sigma_1=\sigma\log_{10} \hat{c}
\approx 0.24,$$ where $\hat{c}\approx 3$ is the median of $c$ values observed during $\rho<\rho_c$. The observed exponent $\sigma_1\approx 0.23$ (not shown) is fairly close to its predicted value. The shift exponent is estimated as $\sigma_2\approx 0.03$; while scale coefficients are $a_1\approx 1.54$, $a_2\approx 1.43$. The function $g_0(z)$ that uses these estimates is shown in Fig. \[fig\_g0\] where different symbols depict clusters of different ranks. The collapse is obvious, confirming the validity of the three-exponent scaling (\[r2exp\]), (\[r3exp\]), (\[g0\]).
In the scaling for cluster masses, the time renormalization $(\rho_c-\rho)m^{\sigma}$ collapses the dynamics of mass $m$ clusters onto the master curve $f_0(z-z_0)$ with its only peak shifted by $z_0$ leftward from percolation; the shift $z_0$ is mass independent. Similarly, in the scaling for ranks the time renormalization $(\rho_c-\rho)10^{\sigma_1\,r}$ collapses the dynamics of rank $r$ clusters onto the master curve $g_0(z-z'_0)$, although the shift now is rank dependent and is given by $10^{\sigma_2\,r}$. To illustrate this, we show the position of percolation on the righthand limb of the Gaussian $g_0(z-0.51)$ in Fig. \[fig\_nu\]b. The higher the rank, the closer the position of percolation to the peak of $g_0$.
Averaged scaling {#AS}
----------------
In applications, it is often impossible to measure the size distribution of system elements at a given time instant. Moreover, sometimes the instantaneous size distribution does not exist at all: This is indeed the case for the systems described by marked point processes widely used to model seismicity, volcano activity, starquakes, etc. [@DVJ]. In such situations one uses the averaged measurements. For instance, the famed Gutenberg-Richter law [@GR54; @Tur97; @BZ03] that gives exponential approximation to the size distribution of earthquakes (via their magnitudes) is valid only after appropriate averaging over a wide spatio-temporal domain. This explains the importance of the question: How do the distributions of Eq. (\[2exp\]), (\[r2exp\]) change after temporal averaging?
We answer this question for averaging over $0\le\rho\le\rho_c$. For the mass distribution this leads to: $$\begin{aligned}
\label{avem}
\widehat{n_m}&:=&\int_0^{\rho_c} n_m(\rho)d\rho
=\int_0^{\rho_c} f_0(z)\,m^{-\tau}d\rho\nonumber\\
&\propto&\int_0^{\rho_c}\exp\left\{-a\left[(\rho_c-\rho)m^{\sigma}
-z_0\right]^2\right\}m^{-\tau}d\rho\nonumber\\
&\propto& m^{-\tau-\sigma}\int_{u_1}^{u_2}
\exp\left\{u^2/2\right\}du\nonumber\\
&\propto& m^{-\tau-\sigma}~~(\approx m^{-5/2}).\end{aligned}$$ Here the last step neglects the weak dependence of the integral on $m$ (and uses the values $\tau\approx 2.0$, $\sigma=1/2$). The validity of (\[avem\]) is confirmed by the observed averaged mass distribution shown by the solid line in Fig. \[fig\_GRm\]. The averaged mass distribution is similar to that at percolation: it retains the power-law form while the slope is increased by $1/2$ due to averaging.
Similarly, we obtain for ranks: $$\begin{aligned}
\label{aver}
\widehat{n_r}&:=&\int_0^{\rho_c} n_r(\rho)d\rho
=\int_0^{\rho_c} g_0(z)\,10^{-br}d\rho\nonumber\\
&\propto&\int_0^{\rho_c}\exp\left\{-a'\left[
(\rho_c-\rho)10^{\sigma_1\,r}-a\,10^{-\sigma_2\,r}\right]^2
\right\}10^{-b\,r}d\rho\nonumber\\
&\propto& 10^{-(\sigma_1+b)\,r}\int_{u_1}^{u_2}
\exp\left\{u^2/2\right\}du\nonumber\\
&\propto& 10^{-(\sigma_1+b)\,r} =
10^{r\,(1-\sigma-\tau)\log_{10}\tilde{c}}=
10^{-r\,\alpha_r}.\end{aligned}$$ The exponent $\alpha_r$ may vary from 0.71 to 0.93 depending on $3.0\le\tilde{c}\le 4.2$ (the range of $c$ values for the time when at least three ranks have been formed so the estimation of the distribution slope is meaningful). Simulations suggest (solid line in Fig. \[fig\_GRr\]) $\alpha_r=0.87$, which is in good agreement with our prediction. Again, the averaged rank distribution retains the exponential form of the distribution at percolation; while its index has increased due to averaging.
Correction to simple scaling {#corrections}
----------------------------
Due to finiteness of the lattice, the results of previous sections require some corrections to match exactly the simulated rank distributions. The appropriate corrections are described below.
### Corrected scaling at percolation
The pure power and exponential laws in Figs. \[fig\_GRm\], \[fig\_GRr\] are just first-order approximations to the observed cluster distributions at percolation. In both cases one sees the downward bending for small clusters and upward bending for large clusters. These are not due to statistical fluctuations. The downward bending for small clusters is explained by “deviations from scaling” [@Hoshen+79]: it can be shown analytically that the small clusters do not yet obey the general scaling law of Eqs. (\[GRm\]), (\[GRr\]) which holds only for large enough masses (ranks). The upward bend at large clusters is due to finite-size effects [@Hoshen+79; @Margolina+84]: each large cluster that reaches outside the lattice boundary is “seen” as a number of smaller clusters, thus creating the upward deviation from the pure power (exponential) law. This phenomenon is especially important when the system is close to percolation and clusters of arbitrary large sizes have already been formed. The appropriate scale corrections for the mass distribution were studied by Hoshen et al. [@Hoshen+79] and Margolina et al. [@Margolina+84].
To study the above phenomena it is convenient to consider the normalized functions $$N_m:=m^{\tau-1}\sum_{m'\ge m} n_{m'},\qquad
N_r:=10^{br} n_r,$$ which, in the absence of scale corrections, would become constants: $$N_m=\frac{q_0}{\tau-1},\qquad
N_r=p_1\,c^{\tau-1}.$$ The function $N_r$ is shown in Fig. \[fig\_Nr\]a; it clearly deviates from the horizontal plateau at both sides.
In case of the mass distribution, the corrections to scaling are given by [@Margolina+84]: $$\label{GRmc}
n_m(\rho_c)\simeq m^{-\tau}
\left(q_0+q_1\,m^{-\Omega}+q_L\,m^{1/D}L^{-1}\right),$$ where $\Omega \approx 0.75$, $1/D=48/91$ is the universal mean cluster radius exponent, and $q_0,q_1,q_L$ are independent of $s$ and $L$. The first additional term describes the deviation from scaling for small clusters, while the second one is responsible for finite-size effects.
For rank distribution, the “deviations from scaling” at lower clusters are only observed for $r=1$; while the finite-size effects at large clusters are clearly present for many ranks. Accordingly, we propose the following correction to scaling for the rank distribution: $$\label{GRrc}
n_r(\rho_c)\simeq 10^{-br}
\left(p_0+p_L\,10^{d\,r}L^{-1}\right),~r>1.$$ with $$d=\frac{1}{D}\,\log_{10}c \approx 0.33.$$ The observed value of $d$ can be estimated by plotting $(n_r\,10^{br}-p_0)$ as a function of $r$ as shown in Fig. \[fig\_Nr\]b. The observed ranks $4\le r \le 9$ follow the predicted scaling (\[GRrc\]) nicely.
Importantly, the corrections to scaling (\[GRrc\]) act at all cluster sizes, so they can not be neglected even for the intermediate clusters, not only for the largest ones. Indeed, their effect decreases with $L$, but this decrease is very slow. Notably, as shown by Morein et al. [@MNTG04] (their Fig. 5) even for lattices as large as $L=30,000$ during the process when clusters as large as 2% of the lattice size are removed, the cluster size distribution clearly exhibits the upward deviations at large ranks ($r=11,12,13$.) For smaller systems these deviations become dominant and may lead to an artificial decrease of the observed slope of cluster size distribution; this is demonstrated in Fig. \[fig\_GRm\],\[fig\_GRr\] and is also seen in the analysis of Turcotte et al. [@Tur+99] (their Fig. 9).
### Dynamics of scaling corrections {#scalecorr}
Since the finite size effects play an important role in shaping the observed cluster size distribution, it is worth studying their dynamics. Specifically, we will be interested in transition of the cluster size distribution from the convex shape (in semi- or bilogarithmic scale) at $\rho\ll\rho_c$ to formation of the upward bend at percolation.
For this we introduce a measure of convexity for the rank distribution, defined as an area between $\log n_r(\rho)$ and a chord connecting its first and last points as shown in Fig. \[fig\_mu\] (the point $r=1$ is not considered being affected by the deviations from scaling): $$\label{mconv}
\mu(\rho):=\int_{2}^{r_{\rm max}}
\left[\log_{10} n_r(\rho)-(A\,r+B)\right]
\,d\,r,$$ with $$A=\frac{\log_{10}\left(n_{r_{\rm max}}/n_2\right)}
{r_{\rm max}-2},
~B=\log_{10}\,n_2-2A.$$ The values of $\mu$ are positive when $n_r(\rho)$ is convex in semilogarithmic scale, negative when it is concave, and vanish when it is linear. The measure $\mu(\rho)$ averaged over 1,000 runs on the lattice $L=2000$ is shown in Fig. \[fig\_mu\]; the bell-shaped form of $\mu$ is decorated by the logperiodic oscillations for $(\rho_c-\rho)>10^{-2}$ explained by creation of new ranks, which temporarily increases convexity. Zero level is crossed at about $(\rho_c-\rho)=2\cdot 10^{-3}$, after that the rank distribution is concave. A detailed analysis (not shown) demonstrates that the distribution is never exactly linear; the transition from convex to concave shape is realized through the wave-shaped form when the distribution is still convex for the lower $r$, but is already concave for the higher ones. Qualitatively the same picture is observed for the mass distribution $n_m(\rho)$ (in bilogarithmic scale).
The transformation of the cluster size distribution prior to percolation is not unlike a well-known pattern “upward bend” first described by Narkunskaya and Shnirman [@NS90; @NS94] in an early static model of defect development. Later it was found in steel samples and seismicity of California [@RKB97], and confirmed by the dynamical modeling of failure in a hierarchical system (so-called colliding cascade models) [@GKZN00; @ZKG03].
Mass dynamics of a given rank {#MDGR}
-----------------------------
Here we consider the dynamics of total and average mass of rank $r$ clusters: $$\label{mass}
M_r=\sum m\,n_{rm},\qquad
m_r =\frac{\sum m\,n_{rm}}{\sum n_r}
=\frac{M_r}{n_r}.$$ Here $n_{rm}$ denotes the number of clusters of rank $r$ and mass $m$. Figure \[fig\_avem\] shows $n_r$, $M_r$, and $ m_r $ for rank 5; the similar picture is observed for other ranks. It is tempting to use Gaussian approximation for $M_r$ and predict the Gaussian dynamics of $ m_r $ (as a ratio of two Gaussians) and relate their parameters. Detailed analysis however demonstrates that under this approach the peak of $ m_r $ for ranks $r\ge 9$ should be observed after percolation; while in simulations this peak is always prior to percolation (not shown). Note that one still might approximate $M_r$ and $ m_r $ by Gaussians with properly scaled parameters; such approximations will be good for rough curve fitting, but will fail to reproduce deeper properties of cluster dynamics. This demonstrates the general limitations of Gaussian approximations in the percolation problem.
Cluster fractal structure {#CFS}
=========================
In this section we evaluate the fractal structure of clusters considering the mass-circumference relation $$\label{circum}
m \propto C^{D_r},$$ where $C$ is the number of empty neighbors of a cluster of mass $m$. For percolation cluster at infinite grid we have $D_r=1$, which shows that the percolation cluster is a “linear” rather than a space-filling object [@SA]. Figure \[fig\_D\]a shows the cluster masses as a function of their circumference for different ranks. It is easily seen how the linear scaling $D_r=1$ gradually develops as rank increases. Figure \[fig\_D\]b shows the index $D_r$ estimated for $1\le r\le 9$.
Figure \[fig\_fin\] shows the dynamics of $D_5$ prior to percolation; noteworthy, its steady state behavior is altered by a gradual increase as $\rho\to\rho_c$. Similar increase is observed for clusters of other ranks.
To explain the increase of $D_r$ we recall that the rate of cluster coalescence is proportional to their circumference (see e.g. [@GNT99]). Thus, for a given mass, clusters with a lower $D_r$ have larger circumference, and a higher chance to coalesce. When a sufficient number of rank $r$ clusters have been formed, the clusters with low $D_r$ start to coalesce leaving the high-$D_r$ clusters on the grid.
Another reason for the increase of $D_r$ is the finite-size effects. Specifically, this is an effect of having clusters that on an infinite grid have already gained higher ranks, but on our finite lattice are still small.
Dynamical constraint {#DC}
====================
Here we report an interesting regularity in rank dynamics that put a notable constraint on analytical modeling of percolation process. Specificaly, we consider the slope between two consecutive points of the rank distribution: $$\theta_r(\rho):=\log\,\frac{n_r(\rho)}{n_{r+1}(\rho)}.$$ Dynamics of $\theta_4$ is shown in Fig. \[fig\_peaks\]a together with that of $n_6$. Noteworthy, the peaks of two curves (minimum of $\theta_4$ and maximum of $n_6$) coincide. This happens to be true for all ranks: positions of corresponding peaks are shown as a function of rank in Fig. \[fig\_peaks\]b. Such perfect matching is very unlikely to be accidental. Thus we conjecture that in order for $n_r(\rho)$ to properly describe the time-dependent behavior of rank population, the following system of differential equations must have a solution: $$\label{constraint}
\left\{\begin{array}{l}
\dot{\theta}_r=0\\
\dot{n}_{r+2}=0
\end{array}
\right.
\Rightarrow
\left\{\begin{array}{l}
\dot{n_r}\,n_{r+1}-n_r\dot{n}_{r+1}=0\\
\dot{n}_{r+2}=0
\end{array}
\right.$$
Applying this constraint to the three-exponent scaling of Eqs. (\[r2exp\]),(\[r3exp\]),(\[g0\]) we find $$\label{sigmas}
\sigma_2=\sigma_1+\log_{10}\left(1-10^{-2\sigma_1}\right).$$ According to (\[sigmas\]), the observed value $\sigma_1=0.23$ gives $\sigma_2=0.04$, which is 33% larger than the observed value $\sigma_2=0.03$. The discrepancy is due to the approximate character of the Gaussian approximation (\[g0\]) for $g_0$.
Discussion
==========
The goal of this study was to describe the evolution of percolation model in terms of consecutive aggregation of smaller clusters into larger ones using the Horton-Strahler hierarchical scheme. First, this contributes to a novel understanding of the percolation phenomenon as a time-dependent hierarchical inverse cascade process. Second, this allows one to test the validity of the approach introduced by Gabrielov et al. [@GNT99] and further developed by Morein et al. [@MNTG04] for a steady-state approximation to a general aggregation process.
We considered dynamical and scaling properties of site-percolation on a 2[*D*]{} square lattice. Following [@GNT99] we described clusters by hierarchical trees that reflect the history of cluster formation; the Horton-Strahler scheme was used to rank the trees and thus the corresponding clusters. We concentrated on the development of the first percolation cluster, thus working with a system that does not exhibit the steady-state dynamics, contrary to the studies [@GNT99; @MNTG04] that have developed mean-field steady-state approximations to the system.
Combining the results obtained in the classical percolation studies with the Tokunaga constraint on the cluster branching structure we derived various rank-dependent scaling laws connecting the number $n_r$ of clusters of rank $r$, their average mass $m_r$, and the rank $r$. We have compared the parameters of these laws with those predicted and observed by [@GNT99; @MNTG04] in steady-state aggregation models. The values of parameters are shown to be in a perfect agreement, confirming the validity of the approach used in [@GNT99; @MNTG04]. In absence of the steady-state behavior, we derived the time-dependent versions of the scaling laws. We reported the three-index scaling (\[r2exp\]), (\[r3exp\]) for the number $n_r(\rho)$ of clusters of a given rank, which deviates from the classical two-exponent scaling for masses.
We studied in detail the transition of the system from earlier stages to the vicinity of percolation and reported several characteristic phenomena observed as $\rho\to\rho_c$. They include transformation of the cluster size distribution not unlike that observed in seismicity, steel samples, and previous models of hierarchical fractures [@NS90; @RKB97; @GKZN00; @ZKG03]; and increase of the cluster fractal dimension. In our simple model these phenomena are partly explained (qualitatively as well as quantitatively) by finite-size effects; nevertheless we believe that they should not be neglected as irrelevant side-effects of numerical simulation. In fact, in practice we often work with systems that are described by intermediate depth hierarchies (in other words they have intermediate number of degrees of freedom). The percolation results related to small and intermediate lattices might be of high relevance in describing such systems. In addition, simulations on large lattices ($L=30,000$) performed by Morein et al. [@MNTG04] show that finite-size effects are still present even for large $L$.
We have formulated an empirical constraint of Eq. (\[constraint\]) for the time-dependent behavior of rank population size $n_r(\rho)$; the constraint follows very clearly from the observed values of $n_r(\rho)$. It would be interesting to check this condition in real systems traditionally described by the percolation model.
Our closing remark is on the index $\tau$ of cluster mass distribution at percolation (Eq. (\[GRm\])). The studies of Gabrielov et al. [@GNT99] and Morein et al. [@MNTG04] predict $\tau=2$; which slightly deviates from the well established theoretical value of the Fisher exponent $\tau=187/91\approx 2.05$. The index of the mass distribution is an essential characteristic of a system, thus even this slight difference of 2.5% might seem disappointing implying the intrinsically approximate character of the modeling of [@GNT99; @MNTG04]. In fact, this implication is not true. To validate the approach of [@GNT99; @MNTG04] we notice that the Fisher exponent is tightly connected to the precise count of cluster particles on a site-level, hardly feasible in practice. At the same time, the studies [@ZLK99; @CZ03] have demonstrated that when we “characterize the size distribution of clusters in a way that circumvents the site-level description” considering any “macroscopic measure of the length scale of the cluster”, the exponent of the corresponding scaling law becomes $2$, universally for all 2[*D*]{} systems. An example of a “macroscopic measure” is the linear size in arbitrary direction, the radius of gyration, the diameter of the covering disk, etc. Clearly, the modeling of [@GNT99; @MNTG04] deals with such a macroscopic measure of cluster size, and hence predicts the correct scaling exponent.
We are sincerely grateful to Bill Newman for numerous focused discussions; his critics and advice have helped significantly in organizing and presenting the results. We thank Vladimir Keilis-Borok, Gleb Morein, and Donald Turcotte for their continuous interest to the work and David Shatto for help in preparing the manuscript. This study was partly supported by NSF, grant ATM 0327558.
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Zaliapin, I., Keilis-Borok, V., Ghil, M. (2003) A Boolean delay equation model of colliding cascades. Part II: Prediction of critical transitions. [*J. Stat. Phys.*]{} [**111**]{}, (3-4): 839-861.
![Multiple coalescence of clusters. a) Coalescence of clusters is materialized by adding to the lattice a new particle [**N**]{} (black) that is a neighbor to one, two, three, or four existing clusters (numbered gray sites). The relative frequencies $Q_k$ , $k=1,2,3,4$ of $k$-coalescences based on similations with $L=2,000$ are shown in the figure. The corresponding tree is constructed as shown in panel b) (for $k=1$) and c) (for $k=2$). The cases $k=3,4$ are analogous to $k=2$. Note that about 95% of coalescences result in merging two clusters. See text for details.[]{data-label="fig_model1"}](figure1.eps)
![Tree representation of clusters: scheme. The dynamics is from left to right. At first step particle [**A**]{} is dropped onto the lattice and a one-particle cluster is formed; it is represented by a one-node tree. At second step another one-particle cluster [**B**]{} is formed; it is represented by another one-node tree. At third step new particle [**C**]{} coalesces with cluster [**A**]{} to form two-particle cluster [**AC**]{}. This cluster is represented by a three-node tree; note that the connecting node of the tree does not correspond to any particle. At fourth step new particle [**D**]{} connects existing clusters [**AC**]{} and [**B**]{} to form four-particle cluster [**ABCD**]{}. This cluster correspond to a five-node tree.[]{data-label="fig_model2"}](figure2.eps)
![a) Non-uniqueness of tree representation. Two different 11-particle clusters that correspond to the same tree shown in panel b). Particles have been dropped according to their alphabet marks; so first was the particle [**A**]{}, then [**B**]{}, etc. b) Horton-Strahler ranking: illustration. The ranks are shown next to the tree nodes.[]{data-label="fig_model3"}](figure3.eps)
![Mass-rank distribution observed on a 2,000$\times$2,000 lattice at percolation. Dots – individual clusters, balls – average mass $m_r$ within a given rank. Line shows the relation $m_r=\left[10^{\,0.625}\right]^{\,r-1}=4.2^{\,r-1}$.[]{data-label="fig_MR"}](figure4.eps)
![Parameter $c$ of the mass-rank relation $m_r=c^{r-1}$ as a function of time. At percolation $c(\rho_c)\approx 4.2$; the Euclidean limit of [@GNT99] corresponds to $c=3.25$, it is reached at $\rho_c-\rho\approx 0.14$.[]{data-label="fig_c"}](figure5.eps)
![Mass distribution of clusters observed on a $2,000\times 2,000$ lattice at percolation $\rho=\rho_c$ (dash-dotted line), $\rho=0.48$ (dashed line), and averaged over $0<\rho<\rho_c$ (solid line). To smooth out statistical fluctuations we show the cumulative distribution: $\propto\sum_{m'>m}n_m$. For comparison, all curves are normalized to unity at $m=1$.[]{data-label="fig_GRm"}](figure6.eps)
![Rank distribution of clusters observed for $2,000\times 2,000$ lattice at percolation $\rho=\rho_c$ (dash-dotted line), $\rho=0.29$ (dashed line), and averaged over the percolation cycle $0<\rho<\rho_c$ (solid line). For comparison, all curves are normalized to unity at $r=1$.[]{data-label="fig_GRr"}](figure7.eps)
![Dynamics of population $n_r\cdot L^2$ of a given rank, $r=5,6,7$ for $L=2,000$. Moment of percolation is depicted by a vertical dashed line.[]{data-label="fig_rank"}](figure8.eps)
![Dynamics of population $n_r$ of a given rank, $1\le r\le11$ in semilogarithmic scale. Moment of percolation is depicted by a vertical dashed line. (Cf. Fig. 7 from [@Tur+99]).[]{data-label="fig_rank_all"}](figure9.eps)
![Scaling for rank dynamics. a) Ratios $\nu_r(z)/\nu_1(z)$ do not collapse thus rejecting the two-exponent scaling hypothesis; see details in Sect. \[rscale\]. b) Position of percolation on the normalized Gaussian $g_0(z-0.51)$; see details in Sect. \[rscale\].[]{data-label="fig_nu"}](figure10.eps)
![Three-exponent scaling for rank dynamics. The master Gaussians $g_0(z)$ for different ranks collapse when using the renormalization given by Eqs. (\[r2exp\]),(\[r3exp\]).[]{data-label="fig_g0"}](figure11.eps)
![Corrections to scaling. The pure exponential rank distribution of Eq. (\[GRr\]) suggests a horizontal plateau for the normalized function $N_r=10^{br}n_r$, while the observed values clearly deviate from the plateau at small and large clusters (panel a). The large cluster deviation is due to finite size efffects and is described by an exponential correction of Eq. (\[GRrc\]) with $d\approx 0.33$ (panel b).[]{data-label="fig_Nr"}](figure12.eps)
![Dynamics of scale corrections. A convexity measure $\mu(\rho)$ is defined by Eq. (\[mconv\]) and illustrated in the insert. It is positive for convex, and negative for concave rank distribution. The downward bend of the rank distribution observed at early stages ($\mu>0$) is changed to the upward one ($\mu<0$) for $(\rho_c-\rho)<2\cdot10^{-3}$. See details in Sect. \[scalecorr\].[]{data-label="fig_mu"}](figure13.eps)
![Dynamics of number of clusters $n_r$ (solid line), total mass $M_r$ (dashed line), and average mass $m_r$ (dash-dotted line) for clusters of rank $r=5$.[]{data-label="fig_avem"}](figure14.eps)
![Fractal structure of clusters. a) Mass-circumference relation for clusters of different ranks. The asymptotic power relation with slope 1.0 is gradually develops as rank increases. b) Values of fractal dimension $D_r$ (Eq. (\[circum\])) for different ranks. Both panels correspond to a $2,000\times2,000$ lattice at percolation.[]{data-label="fig_D"}](figure15.eps)
![Premonitory increase of cluster fractal dimension. The steady-state dynamics of fractal dimension $D_5$ (Eq. \[circum\]) changes, and $D_5$ starts to increase, as system approaches percolation. Similar phenomenon is observed for other ranks. []{data-label="fig_fin"}](figure16.eps)
![Dynamical constraint for $n_r(\rho)$. Dynamics of $\theta_4=\log(n_4/n_5)$ and $n_6$ is shown in panel a: peaks of two curves coincide. The similar phenomenon is observed for other ranks: panel b shows the times of maxima of $n_r$ (circles) and minima of $\theta_{r-2}$ (triangles) for $3\le r\le 9$.[]{data-label="fig_peaks"}](figure17.eps)
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We study one-sided Markov shifts, corresponding to positively recurrent Markov chains with countable (finite or infinite) state spaces. The following classification problem is considered: when two one-sided Markov shifts are isomorphic up to a measure preserving isomorphism ? In this paper we solve the problem for the class of $\rho$-uniform (or finitely $\rho$-Bernoulli) one-sided Markov shifts considered in .
We show that every ergodic $\rho$-uniform Markov shift $T$ can be represented in a canonical form $T = T_G $ by means of a canonical (uniquely determined by $T$) stochastic graph $G$. In the canonical form, two such shifts $T_{G_1}$ and $T_{G_2}$ are isomorphic if and only if their canonical stochastic graphs $G_1$ and $G_2$ are isomorphic.
address:
- 'Address: [*Ben Zion Rubshtein, Dept. of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel.*]{}'
- ' E-mail: [*[email protected]*]{}'
author:
- 'Ben-Zion Rubshtein'
title: ' On a class of one-sided Markov shifts'
---
Introduction {#Intro}
============
In this paper we consider the classification problem for one-sided Markov shifts with respect to measure preserving isomorphism. Let $G$ be a finite or countable stochastic graph, i.e. a directed graph, whose edges $g \in G$ are equipped with positive weights $p(g)$. The weights $p(g)$ determine transition probabilities of a Markov chain on the discrete state space $G$. The corresponding one-sided Markov shift $T_G$ acts on the space $(X_G,m_G)$, where $ X_G = G^{{\mathbb{N}}}$ and $m_G$ is a stationary (probability) Markov measure on $X_G$. We deal only with irreducible positively recurrent Markov chains, so that such a Markov measure exists and the shift $T_G$ is an ergodic endomorphism of the Lebesgue space $(X_G,m_G)$. The problem under consideration is : When for given two stochastic graphs $G_1$ and $G_2$, does there exist an isomorphism $\; \Phi { : }X_{G_1} {\rightarrow}X_{G_2} \;$ such that $\; m_{G_2} = m_{G_1} \circ \Phi^{-1} \;$ and $\; \Phi \circ T_{G_1} \;=\; T_{G_2} \circ \Phi \;$.
It is obvious, that any (weight preserving) graph isomorphism $\phi { : }G_1 {\rightarrow}G_2$ generates such an isomorphism $\Phi = \Phi_\phi$, but nonisomorphic graphs can generate the same shift $T_G$.
Recently J. Ashley, B. Marcus and S. Tuncel [@AsMaTu] solved the classification problem for one-sided Markov shifts corresponding to [**finite**]{} Markov chains. They used an approach which is based on the following important fact: Two one-sided Markov shifts $T_{G_1}$ and $T_{G_2}$ (on finite state spaces) are isomorphic iff there exists a common extension $G$ of $G_1$ and $G_2$ by right resolving graph homomorphisms of degree $1$. The result was proved implicitly in [@BoTu], where regular isomorphisms and right closing maps for two-sided Markov shifts were studied (See also [@As], [@KiMaTr], [@Tr], [@Ki] and references cited there)
It should be noted that the classification problem for two-sided shifts is quite different from the one-sided case. Namely, any mixing two-sided Markov shift is isomorphic to the Bernoulli shift with the same entropy [@FrOr] and two-sided Bernoulli shifts are isomorphic iff they have the same entropy by the Sinai-Ornstein theorem [@Si]-[@Or].
On the other hand, let $T_\rho$ be the one-sided Bernoulli shift with a discrete state space $(I,\rho)$, where $I$ is a finite or countable set, $1 < |I| \leq \infty$, and $\rho = \{\rho_i\}_{i
\in I}$, $\sum_{i} \rho_i = 1$, $\rho_i > 0$. The endomorphism $T_\rho$ acts as the one-sided shift on the product space $(X_\rho , m_\rho) = \prod_{n=1}^{\infty} (I, \rho)$. Consider the measurable partition $T_\rho^{-1} {\varepsilon}= \{ T_\rho^{-1} x \;,\; x \in X_\rho \}$ generated by $T_\rho$ on $X_\rho$. The partition admits an independent complement ${\delta}$, which is not unique in general, but necessarily has the distribution $\rho$. This implies that one-sided Bernoulli shifts $T_{\rho_1}$ and $T_{\rho_2}$ are isomorphic iff the distributions $\rho_1$ and $\rho_2$ coincide.
This simple observation motivates the following definition. An endomorphism $T$ of a Lebesgue space $(X, m)$ is called ${\boldsymbol{\rho}}$[**-uniform**]{} (or [**finitely**]{} ${\boldsymbol{\rho}}$[**-Bernoulli**]{} according to ) if the measurable partition $T^{-1} {\varepsilon}= \{ T^{-1} x \;,\;\; x \in X \}$ admits an independent complement ${\delta}$ with $distr \; {\delta}= \rho$. We denote by ${\mathcal {UE}(\rho)}$ the class of all $\rho$-uniform endomorphisms.
Recall that the cofiltration $\xi (T)$ generated by an endomorphism $T$ is the decreasing sequence $\{\xi_n\}_{n=1}^{\infty}$ of the measurable partitions $\xi_n = T^{-n} {\varepsilon}$ of the space $X$ onto inverse images $T^{-n}x$. If two endomorphisms $T_1$ and $T_2$ are isomorphic, i.e. there exists an isomorphism $\Phi$ such that $\; \Phi \circ T_1 = T_2 \circ \Phi \;$, then $\; \Phi (T_1^{-n}x) = T_2^{-n} (\Phi x) \;$ for almost all $x \in X$, i.e. $\Phi (T_1^{-n} {\varepsilon}) = T_2^{-n} {\varepsilon}$ for all $n$. This means that the cofiltrations $\xi(T_1)$ and $\xi(T_2)$ are isomorphic.
If $T \in {\mathcal {UE}(\rho)}$, the cofiltration $\xi(T)$ is not necessarily isomorphic to the [**standard**]{} cofiltration $\xi(T_\rho)$, generated by the Bernoulli shift $T_\rho$. However, it is [**finitely isomorphic**]{} to $\xi(T_\rho)$, i.e. for every $n \in {{\mathbb{N}}}$ there exists an isomorphism $\Phi_n$ such that $\Phi_n (T^{-k} {\varepsilon}) =T_\rho^{-k} {\varepsilon}$ for all $1 \leq k
\leq n$.
The isomorphism problem for $\rho$-uniform endomorphisms is decomposed into the following two parts: When are the cofiltrations $\xi(T_1)$ and $\xi(T_2)$ isomorphic? When are $T_1$ and $T_2$ isomorphic provided that $\xi(T_1) = \xi(T_2)$?
In particular, for given $ T \in {\mathcal {UE}(\rho)}$: When is the cofiltration $\xi(T)$ standard, i.e. isomorphic to $\xi(T_\rho)$ ? When are $T_1$ and $T_2$ isomorphic provided that $\xi(T_1) = \xi(T_2)$ ?
All these problems are quite nontrivial even in the dyadic case $\rho = (\frac{1}{2},\frac{1}{2})$. Various classes of decreasing sequences of measurable partitions were considered by A.M. Vershik -, V.G. Vinokurov [@Vi], A.M. Stepin
[@St] and by author ,,-. A new remarkable progress in the theory is due to J. Feldman, D.J. Rudolph, D. Heicklen and Ch. Hoffman (See [@FeR], [@HeHo], [@HeHoR], [@Ho], [@HoR]). Note also that, as it was shown in , a $\rho$-uniform one-sided Markov shift $T_G$ is isomorphic to the Bernoulli shift $T_\rho$ iff the cofiltration $\xi(T_G)$ is isomorphic to standard cofiltration $\xi(T_\rho)$.
The purpose of this paper is to classify the $\rho$-uniform one-sided Markov shifts. We show that every ergodic $\rho$-uniform Markov shift $T$ can be represented in a [**canonical form**]{} $T = T_G $ by means of a [**canonical**]{} (uniquely determined by $T$) stochastic graph $G$. In the canonical form, two such shifts $T_{G_1}$ and $T_{G_2}$ are isomorphic if and only if their canonical stochastic graphs $G_1$ and $G_2$ are isomorphic.
First we consider (Section \[s2\]) general $\rho$-uniform endomorphisms and use the following results from . Any ergodic $T \in {\mathcal {UE}(\rho)}$ can be represented as a skew product over $T_\rho$ on the space $X_\rho \times Y_d$, $d \in {{\mathbb{N}}}\cup \{\infty\} $, where $Y_d$ consists of $d$ atoms of equal measure $\frac{1}{d}$ for $d < \infty$ and $Y_\infty$ is a Lebesgue space with no atoms, (see Section \[ss2.2\] below). According to we introduce the [**minimal index**]{} $d(T)$ of $T \in {\mathcal {UE}(\rho)}$ as the minimal possible $d$ in the above skew product representation of $T$. The index $d(T)$ is an invariant of the endomorphism $T$ and $d(T) = 1$ iff $T$ is isomorphic to the Bernoulli shift $T_\rho$.
Other important invariants of $T \in {\mathcal {UE}(\rho)}$ (introduced also in ) are the [**partitions**]{} ${\boldsymbol{\gamma}}{{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$, ${\boldsymbol{\beta}}{{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$ and the [**index**]{} ${\mathbf{d}}_{{\boldsymbol{\gamma}}{ : }{\boldsymbol{\beta}}}
{{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$. The partition ${\gamma}(T)$ is the smallest (i.e. having the most coarse elements) measurable partition of $X$ such that almost all elements of the partition $\; {\beta}_n := {\gamma}(T) \vee
T^{-n} {\varepsilon}\;$ have homogeneous conditional measures for all $n$. The corresponding tail partition is defined by $\; {\beta}(T) =
\bigwedge_{n=1}^{\infty} {\beta}_n \geq {\gamma}(T) \;$. and the index $d_{{\gamma}{ : }{\beta}}(T) $ is the number of elements of ${\beta}(T)$ in typical elements of the partition ${\gamma}(T)$ (Proposition \[gga:gb\]).
It was proved in that $\; d(T) = d_{{\gamma}{ : }{\beta}}
(T) < \infty \;$ for any $\rho$-uniform one-sided Markov shift $T
= T_G$. This result implies, in particular, that $T_G$ is [**simple**]{} in the sence of Definition \[def SUE\]. The classification of general simple $\rho$-uniform endomorphisms is reduced to a description of equivalent $d$-extensions of the Bernoulli shift $T_\rho$ (Theorem \[simple\]).
Next we turn to $\rho$-uniform one-sided Markov shifts.
It is easy to see that a Markov shift $T_G$ is $\rho$-uniform iff the graph $G$ satisfies the following condition: For any vertex $u$ the set $G_u$ of all edges starting in $u$, equipped with the corresponding weights $\; p(g) \;,\; g \in G_u \;$, is isomorphic to $(I,\rho)$. This means that the transition probabilities of the Markov chain (starting from any fixed state) coincide with $\;
\rho(i) \;,\; i \in I \;$, up to a permutation. We call these graphs and Markov chains ${\boldsymbol{\rho}}$[**-uniform.**]{} In particular, $(I,\rho)$ itself is considered as a $\rho$-uniform graph having a single vertex. The corresponding Markov shift is the Bernoulli shift $T_\rho$.
Following [@AsMaTu] we use in the sequel graph homomorphisms of the form $\; \phi { : }G_1 {\rightarrow}G_2 \;$, which are assumed to be [**weight preserving**]{} and [**deterministic**]{}, i.e. right resolving in the terminology of [@AsMaTu], (see Definition \[hom\] for details). Thus a stochastic graph $G$ is $\rho$-uniform iff there exists a homomorphisms $\; \phi { : }G {\rightarrow}I \;$.
Two particular kinds of homomorphisms are of special interest in our explanation, they are homomorphisms of [**degree 1**]{} and [**d-extensions**]{}. A homomorphism $\; \phi { : }G_1 {\rightarrow}G_2 \;$ has degree $1$, $\; d(\phi) = 1\;$, if the corresponding factor map $\; \Phi_\phi { : }X_{G_1} {\rightarrow}X_{G_2} \;$ is an isomorphism. So that $\; \Phi_\phi \circ T_{G_1} \;=\; T_{G_2}
\circ \Phi_\phi \;$, i.e. $T_{G_1}$ and $T_{G_2}$ are isomorphic.
The d-extensions homomorphism are defined in Section \[ss3.2\] by the condition: $\; |\phi^{-1}g| = d \;,\; g \in G \;$. They can be described (up to equivalence) by the [**graph skew products**]{}, (see Example \[GSP\] and Definition \[def GSP\] in Section \[ss3.2\]).
As the first step to the construction of the canonical graph we show (Theorem \[phi bar\]) that any homomorphism $\; \phi { : }G {\rightarrow}I \;$ can be extended to a $d$-extension ${{\bar \phi}}$ by homomorphisms of degree $1$ (See Diagram \[diag phi bar\]). To this end we consider a ${\mathbf{d}}$[**-contractive**]{} semigroup ${{\mathcal S}}(\phi)$, associated with the homomorphism $\phi$, and the corresponding [**persistent**]{} sets (Section \[ss4.4\]). Thus we reduce the classification problem to the study of diagrams of the form $$\label{pipsi}
(\pi,\psi) \;{ : }\;
\xymatrix{ {{\bar H}}\ar[r]^{\pi} & H \ar[r]^{\psi} & I}$$ where ${{\bar H}}$ is a $d$-extension, $\psi$ is a degree $1$ homomorphism and the shift $T_{{\bar H}}$ is isomorphic to the shift $T_G$.
The second step is to minimize $d$ in the above Diagram \[pipsi\]. We show (Theorem \[phi bar d(T)\]) that, passing possibally to a “$n$-stringing” graph $G{^{(n)}}$, one can choose the minimal $ d = d(T)$. Note that the result is based on .
The third final step is to reduce the homomorphism $\psi$ in Diagram \[pipsi\] as much as possible. Let ${{\mathcal Ext}^d(I,\rho)}$ denotes the set of all $d$-extensions of the Bernoulli graph $(I,\rho)$ of the form (\[pipsi\]). We show that ${{\mathcal Ext}^d(I,\rho)}$ can be equipped with a natural [**partial order**]{} “$\preceq$” and [**equivalence relation**]{} “$\sim$” (Definition \[partial order\]). The minimal elements of ${{\mathcal Ext}^d(I,\rho)}$ with respect to the order are called [**irreducible**]{} (Definition \[irreduc\]). We describe these irreducible $(\pi,\psi)$-extensions by means of the persistent $d$-partitions, associated with elements of ${{\mathcal Ext}^d(I,\rho)}$ (Theorem \[reduc part\]).
Now we can formulate the main result of the paper (Theorems \[canon form\] and \[classification\]).
- [*Let $T_G$ be a $\rho$-uniform ergodic one-sided Markov shift. A stochastic graph ${{\bar H}}= {{\bar H}}(G)$ is said to be a canonical graph for the shift $T$ if there exists an irreducible $(\pi,\psi)$-extension (\[pipsi\]) from ${{\mathcal Ext}^d(I,\rho)}$ with $\; d = d(T) \;$ such that the shift $T_{{\bar H}}$ is isomorphic to $T_G$.*]{}
- [*Any $\rho$-uniform ergodic one-sided Markov shift can be represented in the canonical form $T = T_{{\bar H}}$ by a canonic graph ${{\bar H}}= {{\bar H}}(G)$.*]{}
- [*In this canonical form, two shifts $T_{{{\bar H}}_!}$ and $T_{{{\bar H}}_2}$ are isomorphic iff the canonical graphs ${{\bar H}}_1$ and ${{\bar H}}_1$ are isomorphic, and iff the corresponding irreducible $(\pi,\psi)$-extensions are equivalent.*]{}
The paper is organized as follows.
In Section \[s2\] we study general $\rho$-uniform endomorphisms (class ${\mathcal {UE}(\rho)}$) and [**simple**]{} $\rho$-uniform endomorphisms (subclass ${\mathcal {SUE}(\rho)}$). Following , we introduce the [**partitions**]{} ${\boldsymbol{\gamma}}{{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$, ${\boldsymbol{\beta}}{{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$ and the [**index**]{} ${\mathbf{d}}_{{\boldsymbol{\gamma}}{ : }{\boldsymbol{\beta}}} {{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$. Two main conclusions of the section are Theorem \[simple\] (classification of simple $\rho$-uniform endomorphisms) and Theorem \[simp mar\], which states that every ergodic $\rho$-uniform one-sided Markov shift $T_G$ is simple and $\;
d(T_G) \;=\; d_{{\gamma}{ : }{\beta}}(T_G) \;<\; \infty \;$.
In Section \[s3\] we consider general properties of stochastic graphs and their homomorphisms. In particular, we define ${\boldsymbol{\rho}}$[**-uniform** ]{} graphs corresponding to $\rho$-uniform Markov shifts. We prove that the index $d(T_G)$ of any ergodic $\rho$-uniform Markov shift $T_G$ is finite (Theorem \[zet del\]). This follows from the finiteness of the degree $d(\phi)$ of any homomorphism $\phi { : }G {\rightarrow}I$ from any $\rho$-uniform graph $G$ onto the standard Bernoulli graph $(I,\rho)$. The degree $d(\phi)$, in turn, can be computed by means a special [**d-contractive**]{} semigroup ${{\mathcal S}}(\phi)$, induced by $\phi$ (Theorem \[d(phi)\]).
Section \[s4\] contains some essential stages of the proof of Main Theorems \[canon form\] and \[classification\]. Homomorphisms of degree $1$ and extensions of the Bernoulli graph are considered in Sections \[ss4.1\] and \[ss4.2\]. Theorem \[Equ ext\] (Section \[ss4.3\]) reduces the classification of skew product over Markov shifts $T_H$ to the classification of the corresponding graph skew product over $H$. In Sections \[ss4.4\] and \[ss4.5\], we study the set ${{\mathcal Ext}^d(I,\rho)}$ of all $(\pi,\psi)$-pairs of the form (\[pipsi\]). The main result of Section \[s4\] is Theorem \[reduc ext\], which claims the existence and uniqueness of the irreducible $(\pi,\psi)$-pair $(\pi_*,\psi_*)$, majorized by a given $(\pi,\psi) \in {{\mathcal Ext}^d(I,\rho)}$.
In Section \[s5\] we prove Main Theorems \[canon form\] and \[classification\] and give some consequences and examples. As a consequence we prove also (Theorem \[common exten 1\]) that two shifts $T_{G_1}$ and $T_{G_2}$ are isomorphic iff the graphs $G_1$ and $G_2$ have a common extension of degree $1$.
We do not study here the classification problem for general, not necessarily $\rho$-uniform, one-sided Markov shifts as well as the classification problem of the cofiltrations, generated by the shifts. Our approach seems to be a good tool to this end and we hope to deal with these two problems in another paper.
We do not also consider the classification problem of one-sided Markov shifts with infinite invariant measure, in particular, of null-recurrent one-sided Markov shifts. One can find a good introduction to the topic and more references in [@Aar Chapters 4 and 5].
Class of $\rho$-uniform endomorphisms {#s2}
=======================================
Lebesgue spaces and their measurable partitions {#ss2.1}
-------------------------------------------------
We use terminology and results of the Rokhlin’s theory of Lebesgue spaces and their measurable partitions (See , ). An improved and more detailed explanation can be found in [@ViRuFed]. We fix the terms “homomorphism , isomorphism, endomorphism” only for [**measure preserving**]{} maps of Lebesgue spaces.
Let $(X, {{\mathcal F}}, m)$ be a Lebesgue space with $ mX=1$. The space $X$ is called [**homogeneous** ]{} if it is non-atomic or if it consists of $d$ points of measure $\; \frac{1}{d} \;,\; d
\in {{\mathbb{N}}}\;$.
Let $ {\zeta}$ be a partition of $X$ onto mutually disjoint sets $ C
\in {\zeta}$. The element of $ {\zeta}$ containing a point $ x $ is denoted by $ C_{\zeta}(x) $. The partition $ {\zeta}$ is measurable iff there exists a measurable function $ f { : }X {\rightarrow}{{\mathbb{R}}}$ such that $$x \stackrel{{\zeta}}{\sim} y \Longleftrightarrow
C_{\zeta}(x) = C_{\zeta}(y) \Longleftrightarrow f(x)=f(y) \;,\; x,y \in X$$ Elements of $ {\zeta}$ are considered as Lebesgue spaces $\; (C, {{\mathcal F}}^C , m^C) \;,\; C \in {\zeta}\;$, with canonical system of conditional measures $\; m^C \;,\;C \in {\zeta}\;$. We shall denote also by $ m(A|C)$ the conditional measures $\; m^C(A \cap C) \;$ of a measurable set $ A \in {{\mathcal F}}$ in the element $ C$ of $ {\zeta}$.
Two measurable partitions ${\zeta}_1$ and ${\zeta}_2$ are said to be [**independent** ]{} (${\zeta}_1 \perp {\zeta}_2 $) if the corresponding ${\sigma}$-algebras $ {{\mathcal F}}({\zeta}_1) $ and $ {{\mathcal F}}({\zeta}_2)$ are independent , where ${{\mathcal F}}({\zeta})$ denotes the $m$-completion of the ${\sigma}$-algebra of all measurable ${\zeta}$-sets. We shall write also $\; {\zeta}_1 \perp {\zeta}_2 \pmod {\zeta}\;$ if the partitions ${\zeta}_1$ and $ {\zeta}_2 $ are [**conditionally independent**]{} with respect to the third measurable partition ${\zeta}$. This means that $$m(A \cap B \;|\; C_{\zeta}(x)) =
m(A|C_{\zeta}(x)) \cdot m(B \;|\; C_{\zeta}(x))$$ for all $ A \in {{\mathcal F}}({\zeta}_1), B \in {{\mathcal F}}({\zeta}_2) $ and a.a. $ x \in X $.
We denote by $ {\varepsilon}= {\varepsilon}_X $ the partition of $X$ onto separate points and by $\nu = \nu_X$ the trivial partition of $X$.
An [**independent complement**]{} of ${\delta}$ is a measurable partition $ \eta $ such that $\; {\zeta}\perp \eta \;$ and $\; {\zeta}\vee \eta =
{\varepsilon}\;$. The partition ${\zeta}$ admits an independent complement iff almost all elements $ (C,m^C) $ of ${\zeta}$ are mutually isomorphic. The collection of all independent complements of ${\zeta}$ is denoted by $IC({\zeta})$.
We shall use induced endomorphisms, which are defined as follows. Let $ A \in {{\mathcal F}}$ , $ mA > 0$ and $T$ be an endomorphism of $(X,m)$. Then the return function $$\label{ret fun}
{\varphi}_A(x) := min \{ n \geq 1 { : }T^{n}x \in A \}
\;\;,\;\; x \in A$$ is finite a.e. on $A$. The [**induced endomorphism**]{} $T_A$ on $A$ is defined now by $\; T_Ax = T^{{\varphi}_A(x)}x \;$. It is an endomorphism of $ (A,{{\mathcal F}}\cap A, m{|}_A) $ and it is ergodic if $T$ is ergodic .
$|E|$ denotes the cardinality of the set $E$
Classes ${\mathcal UE} {\boldsymbol{(}}{\boldsymbol{\rho}}{\boldsymbol{)}}$ and index ${\mathbf{d}}{{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$. {#ss2.2}
---------------------------------------------------------------------------------------------------------------------------------------------------
Let $(I,\rho)$ be a finite or countable state space $$\rho = \{ \rho(i) \;,\; i \in I \} \;,\;\; \rho (i) > 0
\;\;,\;\; \sum_{i \in I}{\rho(i)} = 1 .$$
\[def rho Bern\] An endomorphism $T$ of a Lebesgue space $(X,m)$ is said to be ${\boldsymbol{\rho}}$[**-unform**]{} or [**finitely**]{} ${\boldsymbol{\rho}}$[**-Bernoulli**]{} endomorphism $\; (T \in {\mathcal {UE}(\rho)}\;$, if there exists a discrete measurable partition ${\delta}$ of $X$, which satisfies the following condition:
1. $\; distr \; {\delta}= \rho \;$, i.e. $\; {\delta}= \{ B(i) \}_{i \in I}$ with $\;\;m(B(i)) = \rho (i), \;\; i \in I $,
2. $\; {\delta}\in IC (T^{-1} {\varepsilon}) $ , i.e. ${\delta}\perp T^{-1} {\varepsilon}\;$ and $\; {\delta}\vee T^{-1}{\varepsilon}= {\varepsilon}$.
So $ {\mathcal {UE}(\rho)}$ denotes the class of all $\rho$-unform endomorphisms. Denote by ${\Delta}_{\rho}(T)$ the set of all partitions ${\delta}$ satisfying the condition $(i)$ and $(ii)$. Then $T \in {\mathcal {UE}(\rho)}$ means ${\Delta}_\rho(T) \neq\emptyset $.
For $T \in {\mathcal {UE}(\rho)}$ and ${\delta}\in {\Delta}_{\rho}(T)$ define $$\label{delta n}
{\delta}{^{(n)}}= T^{-n+1} {\delta}\;\;,\;\;\;
{\delta}{^{(n)}}= \{ T^{-n+1}B(i) \}_{i \in I } \;\;,\;\; n \in {{\mathbb{N}}}$$ Then $\; distr \;{\delta}_n = \rho \;$ and the partitions $ \; {\delta}_1 \;,\; {\delta}_2 \;,\;{\delta}_3 \;,\; \ldots \; $ are independent.
The partitions $$\label{delta (n)}
{\delta}{^{(n)}}= \bigvee_{k=1}^{n} {\delta}_{k} \;\;,\;\;\;
{\delta}^{(\infty)} = \bigvee_{k=1}^{\infty} {\delta}_k$$ satisfy for all $n$ the conditions $${\delta}{^{(n)}}\in IC(T^{-n} {\varepsilon}) \;\;,\;\;
{\delta}{^{(\infty)}}\perp T^{-n} {\varepsilon}\pmod { {\delta}{^{(\infty)}}\wedge T^{-n} {\varepsilon}}$$ and $${\delta}{^{(\infty)}}\vee T^{-n} {\varepsilon}= {\varepsilon}\;\;,\;\;
{\delta}{^{(\infty)}}\wedge T^{-n} {\varepsilon}\;=\; T^{-n} {\delta}{^{(\infty)}}\;.$$ In particular, let $ T=T_\rho$ be a Bernoulli endomorphism, which acts on the space $$(X_\rho, m_\rho) = \prod_{n=1}^{\infty} (I,\rho)$$ as the one-sided shift $$T_{\rho} x = \{ x_{n+1}\}_{n=1}^{\infty} \;\; , \;\;\;
x = \{ x_n \}_{n=1}^{\infty} \in X_\rho \;\;.$$ We can set $$\label{delta rho}
{\delta}_\rho = \{ B_\rho(i) \}_{i \in I} \;\;,\;\; B_\rho(i) =
\{ x = \{ x_{n}\}_{n=1}^{\infty} \in X_{\rho}
\;\; { : }\;\; x_1=i \} \;.$$ Then $\; {\delta}_\rho \in {\Delta}_\rho (T_\rho )$ and ${\delta}_{\rho}$ is an one-sided Bernoulli generator of $T_\rho$, that is $${\delta}_{\rho}^{(\infty)} = \bigvee_{n=1}^{\infty} T^{-n+1}{\delta}_{\rho}
= {\varepsilon}_{X_{\rho}} \;.$$
In general case, for $T \in {\mathcal {UE}(\rho)}$ and ${\delta}\in {\Delta}_\rho(T)$, the partition ${\delta}^{(\infty)}$ does not equal ${\varepsilon}$, but we can define the canonical factor map $$\Phi_{{\delta}} { : }X \ni x {\rightarrow}\Phi_{{\delta}} (x) = \{i_{n} (x) \}_{n=1}^{\infty} \in X_{\rho} \;,$$ where $i_{n}(x) \in I$ is uniquely defined by the inclusion $ T^{n}x \in B(i_{n}(x)) \in {\delta}$.
The homomorphism $\Phi_{{\delta}}$ satisfies $\; \Phi_{{\delta}} \circ T = T_{\rho} \circ \Phi_{{\delta}} \;$ and it determines the following representation of $T$ by a skew product over $ T_{\rho} $ (See ).
\[T decompos\] Let $T \in {\mathcal {UE}(\rho)}$ be an endomorphism of $(X,m)$ and ${\delta}\in {\Delta}_\rho(T)$. Then
1. There exists an independent complement ${\sigma}$ of the partition ${\delta}{^{(\infty)}}$.
2. The pair $\; ({\delta}{^{(\infty)}}, {\sigma}) \;$ induces decomposition of the space $(X,m)$ into the direct product $\; (X_\rho \times Y \;,\; m_{X_\rho} \times m_{Y})$ such that the factor map $\Phi_{\delta}$ coincides under the decomposition with the canonical projection $$\pi \; { : }\; X_\rho \times Y \ni (x,y) {\rightarrow}x \in X_\rho$$ and $${\delta}= \pi^{-1} {\delta}_{\rho}
\;\;, \;\; {\delta}{^{(\infty)}}= \pi^{-1} {\varepsilon}_{X_\rho}
= {\varepsilon}_{X_\rho} \times \nu_Y
\;\;,\;\; {\sigma}= \nu_{X_\rho} \times {\varepsilon}_Y$$
3. The endomorphism $T$ is identified with the following skew product over $ T_\rho $ $$\label{skew1}
{{\bar T}}(x,y) = ( T_\rho x , A(x) y)
\;\;,\;\; (x,y) \in X_{\rho} \times Y$$ where $\; \{ A(x), x \in X_\rho \} \;$ is a measurable family of automorphisms of $Y$.
4. If $T$ is ergodic, $Y$ is a homogeneous Lebesgue space.
Every homogeneous Lebesgue space $Y$ is isomorphic to $\; Y_d
\;.\;d \in {{\mathbb{N}}}\cup \{\infty\} \;$, where $\; Y_{\infty} \;$ is the Lebesgue space with a continuous measure and $\; Y_d \;$,$\;d \in
{{\mathbb{N}}}\;$, consists of $d$ points of measure $\frac{1}{d}$. Thus for any ergodic $T$ endomorphism $\; T \in {\mathcal {UE}(\rho)}\;$ and $\; {\delta}\in
{\Delta}_{\rho} (T) \;$ there exists $\; d = d(T,{\delta}) \in {{\mathbb{N}}}\cup
\{\infty\} \;$ such that $$u_{{\delta}^{(\infty)}}(x)
:= m^{C_{{\delta}^{(\infty)}} (x)}(\{x\}) = \frac{1}{d}$$ for a.a. $x \in X $.
\[def d(T)\]
1. The number $\; d(T,{\delta}) \;$ will be called the [**index**]{} of $T \in {\mathcal {UE}(\rho)}$ with respect to $\; {\delta}\in {\Delta}_{\rho}(T) \;$.
2. The [**minimal index**]{} $d(T)$ of $T$ is defined as $$\label{d(T)}
d(T) \;=\; min \; \{\; d(T,{\delta}) \;,\; {\delta}\in {\Delta}_{\rho} (T)\}$$
Note that an ergodic endomorphism $T$ is isomorphic to the Bernoulli shift $T_\rho$ iff $T \in {\mathcal {UE}(\rho)}$, and $d(T) = 1$, that is, there exists ${\delta}\in {\Delta}_\rho(T)$ such that $d(T,{\delta}) = 1$, i.e. ${\delta}{^{(\infty)}}= {\varepsilon}$.
Partitions ${\boldsymbol{\alpha}}{{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$, ${\boldsymbol{\beta}}{{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$, ${\boldsymbol{\gamma}}{{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$ and indices ${\mathbf{d}}_{{\boldsymbol{\alpha}}} {{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$, ${\mathbf{d}}_{{\boldsymbol{\gamma}}{ : }{\boldsymbol{\beta}}} {{\boldsymbol{(}}{\mathbf{T}}{\boldsymbol{)}}}$ {#ss2.3}
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Let $T$ be an endomorphism of $(X,m)$ and let $ \{ \xi_n
\}_{n=1}^{\infty} $ be the decreasing sequence of measurable partitions $\; \xi_n := T^{-n}{\varepsilon}\;$, generated by $T$. The element of $\xi_n$ , containing a point $x \in X$, has the form $\; C_{\xi_n}(x) = T^{-n}(T^n x) \;$,
In order to introduce the partitions ${\gamma}(T)$ and ${\beta}(T)$, consider the measurable functions $$u_n(x) = m^{C_{\xi_n}(x)} (C_{\xi_{n-1}}(x)) \;\;\;,
\; n \in {{\mathbb{N}}}\;\;\; , \; x \in X \;,$$ where $\xi_0 := {\varepsilon}$. With these $\; u_n { : }X {\rightarrow}[0,1]
\;$ we can consider the measurable partitions $${\gamma}_n = \bigvee_{k=1}^{n}u_{k}^{-1} {\varepsilon}_{[0,1]}
\;\; ,\;\; n \in {{\mathbb{N}}}\;,$$ generated by $\;u_k\;,\; k \leq n \;$, and also $$\label{gga gbn gb}
{\gamma}= \bigvee_{n=1}^{\infty} {\gamma}_n \;\;,\;\;
{\beta}_n = {\gamma}\vee T^{-n}{\varepsilon}\;\;,\;\;
{\beta}= \bigwedge_{n=1}^{\infty} {\beta}_n$$ We shall write $\; {\gamma}_n(T) \;,\; {\gamma}(T)\;,\; {\beta}_n(T)\;,\;
{\beta}(T) \;$ to indicate $T$, if it will be necessary.
\[gga:gb\] Suppose that $T\in {\mathcal {UE}(\rho)}$ and T is ergodic. Then there exists $d \in {{\mathbb{N}}}\cup \{\infty\}$ such that $$m^{C_{{\gamma}}(x)}(C_{{\beta}}(x))=\frac{1}{d}$$ for a.a. $x\in X$.
We may define now the index $\; d_{{\gamma}{ : }{\beta}}(T) \;$ of an ergodic endomorphism $T \in {\mathcal {UE}(\rho)}$ as the number $d$ constructed in Proposition \[gga:gb\], i.e. $$d_{{\gamma}{ : }{\beta}}(T) := (m^{C_{\gamma}(x)}(C_{\beta}(x)))^{-1}$$ for a.e. $x \in X$.
We shall use the following properties of the partitions (\[gga gbn gb\])
\[gga gd d(T)\] Suppose that $\; T \in {\mathcal {UE}(\rho)}\;$, let $\; {\delta}\in {\Delta}_\rho(T) \;$ and the partitions $\; {\delta}_n \;,\; {\delta}{^{(n)}}\;,\; {\delta}{^{(\infty)}}\;$ defined by (\[delta n\]) and (\[delta (n)\]). Then
1. $\; {\gamma}{^{(n)}}\le {\delta}_n
\;\;,\;\; {\beta}_n \perp {\delta}{^{(n)}}\pmod {{\gamma}_n}
\;\;,\;\; n \in {{\mathbb{N}}}\;.$
2. ${\gamma}\le {\delta}{^{(\infty)}}\;\;,\;\; {\beta}\perp {\delta}{^{(\infty)}}\pmod {{\gamma}} \;.$
3. $\; d_{{\gamma}{ : }{\beta}}(T) \; \leq \; d(T) \;$
We shall also use the [**tail**]{} measurable partition $\; {\alpha}(T) := \bigwedge_{n=1}^{\infty} T^{-n} {\varepsilon}\;$. An endomorphism $T$ is called [**exact**]{} if ${\alpha}(T) = \nu$. The [**tail index**]{} $d_{\alpha}(T)$ (which is, in fact, the [**period** ]{} of $T$) is defined as follows: $d_{\alpha}(T)= \infty$ if $
X{/}_{{\alpha}(T)}$ is a continuous Lebesgue space and $d_{{\alpha}}(T) = d$ if $ X {/}_{{\alpha}((T)}$ consists of $d$ atoms of measure $\frac{1}{d}$. So that $d_{\alpha}(T) \in {{\mathbb{N}}}\cup \{\infty\}$.
It is easily to see, that $$\label{ga gb gga}
T^{-1} {\alpha}= {\alpha}\;\;,\;\; {\alpha}\vee {\gamma}\le {\beta}\;\;,\;\;
T^{-1} {\gamma}\le {\gamma}\;\;,\;\; T^{-1} {\beta}\le {\beta}$$ and $\; {\alpha}\perp {\delta}{^{(\infty)}}\;$ for any $ {\delta}\in {\Delta}_{\rho}(T) $.
Turning to the canonical projection $\Phi_{\delta}$ we have
\[ga(T rho)\]
1. $ {\alpha}(T_\rho) = \nu
\;\;,\;\; {\beta}(T_\rho) = {\gamma}(T_{\rho}) \;$.
2. $ {\gamma}_n(T) = \Phi_{{\delta}}^{-1}{\gamma}_n(T_\rho)
\;\;,\;\; {\gamma}(T) = \Phi_{{\delta}}^{-1}{\gamma}(T_\rho) \;$.
The stated above propositions were proved in .
Simple ${\boldsymbol{\rho}}$-uniform endomorphisms {#ss2.4}
--------------------------------------------------
We use now the partitions ${\gamma}(T)$ and $ {\beta}(T)$ to introduce an important subclass of the class ${\mathcal {UE}(\rho)}$
\[def SUE\] An endomorphism $T \in {\mathcal {UE}(\rho)}$ of a Lebesgue space $(X,m)$ is said to be a [**simple**]{} $\rho$-uniform endomorphism $(T \in {\mathcal {SUE}(\rho)})$, if there exists partition ${\delta}\in {\Delta}_\rho(T)= IC (T^{-1} {\varepsilon})$ such that $$\label{SUE1}
{\delta}^{(\infty)} \vee {\beta}(T)= {\varepsilon}$$
We denote by ${\mathcal {SUE}(\rho)}$ the class of all simple $\rho$-uniform endomorphisms.
The Bernoulli endomorphism (one-sided Bernoulli shift) $T=T_{\rho}$ belongs to ${\mathcal {SUE}(\rho)}$. In this case there exists a partition $ {\delta}= {\delta}_{\rho} \in {\Delta}_{\rho}(T)$ such that $
{\delta}{^{(\infty)}}= {\varepsilon}$ and hence $ {\beta}(T) \vee {\delta}{^{(\infty)}}= {\varepsilon}$
\[rem SUE\] It is easily to show that the condition (\[SUE1\]) holds iff there exists an independent complement ${\sigma}\in IC({\delta}^{(\infty)}) $ of ${\delta}^{(\infty)}$ that satisfies $$\label{SUE2}
{\sigma}\vee {\gamma}(T) = {\beta}(T) \;\;,\;\;
{\sigma}\in IC ({\delta}^{(\infty)}) \;\;,\;\;
{\delta}\in {\Delta}_\rho(T)= IC (T^{-1} {\varepsilon})$$
\[simple1\] Suppose $\;T \in {\mathcal {UE}(\rho)}\;$ is ergodic and $\; d(T) < \infty \;$. Then $T$ is simple iff $\; d(T) = d_{{\gamma}{ : }{\beta}}(T) \;$.
Since $\; d(T) < \infty \;$ we have, by Proposition \[gga gd d(T)\] (iii), that $\; d_{{\gamma}{ : }{\beta}}(T) \leq d(T)
< \infty \;$. Definition of the index $d_{{\gamma}{ : }{\beta}}(T)$ (Proposition \[gga:gb\]) means that $\; m^{C_{\gamma}(x)}(C_{\beta}(x)) = d^{-1} \;$ for a.a. $x \in X$ and $\; d = d_{{\gamma}{ : }{\beta}}(T) \in {{\mathbb{N}}}\;$. Almost every element of ${\gamma}(T)$ consists precisely of $d$ elements of the partition ${\beta}(T)$. On the other hand there exists ${\delta}\in {\Delta}_\rho(T)$ such that almost every element of the corresponding partition ${\delta}{^{(\infty)}}$ consists precisely of $d(T)$ points, $d \leq d(T)$. By Proposition \[gga gd d(T)\] (ii) we have $${\beta}(T) \perp {\delta}{^{(\infty)}}\pmod {{\gamma}(T)} \;\;,\;\;
{\beta}(T) \wedge {\delta}{^{(\infty)}}= {\gamma}(T) \;.$$ Whence, the condition \[SUE1\] holds iff $\; d(T) = d \;$.
Let $\; T \in {\mathcal {SUE}(\rho)}\;$. By Proposition \[T decompos\] any choice of the partition ${\sigma}$ in the equality (\[SUE2\]) determines a skew product representation (\[skew1\]) of $T={{\bar T}}$ over $T_\rho$. Herewith, all statements of Proposition \[T decompos\] hold and we have also by (\[SUE2\]) and Proposition \[ga(T rho)\]) $$\label{SUE3}
{\beta}(T_\rho)={\gamma}(T_\rho)
\;\;,\;\; {\gamma}({{\bar T}}) = {\gamma}(T_\rho) \times \nu_Y
\;\;,\;\; {\beta}({{\bar T}}) = {\gamma}(T_\rho) \times {\varepsilon}_Y$$ These arguments imply the following
\[simple\] Let $T$ be a $\rho$-uniform simple endomorphism, $\; T \in {\mathcal {SUE}(\rho)}\;$.
1. $T$ can be represented in the skew product form (\[skew1\]) $T={{\bar T}}$ over $T_\rho$ $$\label{Tbar}
{{\bar T}}(x,y) = ( T_\rho x , A(x) y)
\;\;,\;\; (x,y) \in X_{\rho} \times Y$$ with a measurable family $\; \{ A(x) \;,\; x \in X_\rho \} \;$ of automorphisms of $Y$ such that $\; {\beta}({{\bar T}}) = {\gamma}(T_\rho) \times {\varepsilon}_Y \;$.
2. Two such skew product endomorphisms ${{\bar T}}_k$, $k=1,2,$ $$\label{Tbar 1,2}
{{\bar T}}_k (x,y) = ( T_\rho x , A_k(x) y)
\;\;,\;\; (x,y) \in X_{\rho} \times Y$$ are isomorphic iff the corresponding families $\; A_1(x) \;$ and $\; A_2(x) \;$ are cohomologous, i.e. $$\label{A2W=WA1}
A_2(x) W(x) = W(T_\rho x) A_1(x) \;\;,\;\; x \in X_\rho$$ for a measurable family of $\; \{ W(x) \;,\; x \in X_\rho \} \;$ of automorphisms of $Y$.
Part (i) follows from Proposition \[T decompos\] with (\[SUE3\]).
Let ${{\bar T}}_1$ and ${{\bar T}}_2$ be two skew product endomorphisms of the form (\[Tbar 1,2\]). Denote $\; {{\tilde{W}}}(x,y):=(x,W(x)y) \;$. Then (\[A2W=WA1\]) implies $ {{\bar T}}_2 \circ S = S \circ {{\bar T}}_1 $ if we use the automorphism $S = {{\tilde{W}}}$.
Conversely, suppose there exists an automorphism $S$ such that $ {{\bar T}}_2 \circ S = S \circ {{\bar T}}_1 $. Then the partitrions $${{\bar {\gamma}}}:= {\gamma}({{\bar T}}_1) = {\gamma}({{\bar T}}_2) = {\gamma}(T_\rho) \times \nu_Y$$ and $${{\bar {\beta}}}:= {\beta}({{\bar T}}_1) = {\beta}({{\bar T}}_2) = {\gamma}(T_\rho) \times {\varepsilon}_Y$$ are invariant with respect to $S$. Moreover, ${{\bar {\gamma}}}$ is element-wise invariant with respect to $S$. Hence, $S {|}_C
({{\bar {\beta}}}{|}_C) = {{\bar {\beta}}}{|}_C $ for almost every element $C \in {{\bar {\gamma}}}$. The restriction $S{|}_C$ induces a factor automorphism $W_C$ on the factor space $ C{/}_{{{\bar {\beta}}}{|}_C} \cong Y $. We obtain a measurable family $\; W(x) := W_{C(x)} \;,\; x \in
X_\rho \;$, of automorphisms of $Y$. Since the partition $\;
{{\bar {\gamma}}}= {\gamma}(T_\rho) \times \nu_Y \;$ is ${{\bar T}}_1)$- and ${{\bar T}}_2$-invariant, the functions $ A_1(x) $ and $ A_2(x) $ (as well as $W(x)$) are constant on elements of ${\gamma}(T_\rho)$. Therefore the equality $ {{\bar T}}_2 \circ S = S \circ {{\bar T}}_1 $ implies $
{{\bar T}}_2 \circ {{\tilde{W}}} = {{\tilde{W}}} \circ {{\bar T}}_1 $ and (\[A2W=WA1\]) holds.
Consider two special cases.
[**Absolutely non-homogeneous ${\boldsymbol{\rho}}$.**]{} The distribution $\; \rho = \{\rho(i) \}_{i \in I} \;$ is called absolutely non-homogeneous if $\; \rho(i) \neq \rho(j) \;$ for all $\;i \neq j\;$.
In this case we have $\; {\gamma}_1(T) \vee T^{-1}{\varepsilon}= {\varepsilon}\;$. On the other hand $\; {\gamma}_1(T) \perp T^{-1}{\varepsilon}\;$. Thus $\; {\Delta}_\rho(T) \;$ consists of the only partition, which is $\; {\delta}= {\gamma}_1(T) \;$. Hence $${\delta}{^{(\infty)}}= {\gamma}(T) \;\;,\;\;
{\beta}_n(T) = {\gamma}_n(T) \vee T^{-n}{\varepsilon}= {\varepsilon}\;,$$ $${\beta}(T) = \bigwedge_{n=1}^{\infty} {\beta}_n(T) = {\varepsilon}\;\;,\;\;
{\beta}(T) \vee {\delta}{^{(\infty)}}= {\varepsilon}$$ Thus we have
Every $\rho$-uniform endomorphisms with absolutely non-homogeneous $\;\rho \;$ is simple.
[**Homogeneous ${\boldsymbol{\rho}}$.**]{} We have another extremal case if $\;\rho \;$ is homogeneous, i.e. if for some $\; l \in {{\mathbb{N}}}\;$,$\; I = \{1,2, \ldots ,l\;\} \;$ and $\; \rho(i) = l^{-1} \;$ for all $\; i \in I \;.$
All the functions $ \;u_n \;$, which generate the partitions $\; {\gamma}_n(T) \;$, are constant, $$u_n(x) \;=\; m^{C_{\xi_n }(x)} (C_{\xi_{n-1}}(x))
\;=\; l^{-1} \;,\; n \in {{\mathbb{N}}}\;,\; x \in X$$ We have $\; {\gamma}(T) = {\gamma}_n(T) = \nu \;,$ and $\; {\beta}_n(T) =
T^{-n}{\varepsilon}\;$, whence, $\; {\beta}(T) = \bigwedge_{n=1}^{\infty}
T^{-n}{\varepsilon}= {\alpha}(T) \;$. Therefore, for any $\;{\delta}\in {\Delta}_{\rho}(T)\;$ the equality (\[SUE1\]) is equivalent to $\; {\delta}{^{(\infty)}}\vee {\alpha}(T) = {\varepsilon}\;$. On the other hand $\; {\delta}{^{(\infty)}}\perp {\alpha}(T) \;$ for every $\;{\delta}\in {\Delta}_{\rho}(T)\;$.
Thus we have for $ T \in {\mathcal {UE}(\rho)}$ with homogeneous $\rho$
Let $ T \in {\mathcal {UE}(\rho)}$ with homogeneous $\rho$. Then
1. $T$ is simple iff there exists $\;{\delta}\in {\Delta}_{\rho}(T)\;$ such that $ {\delta}{^{(\infty)}}\in IC({\alpha}(T)) $.
2. The skew product decomposition in Theorem \[simple\] is a direct product $\; T_\rho \times S \;$ with $ S = T {/}_{{\alpha}(T)} $.
3. Two such direct products $\; T_\rho \times S_1 \;$ and $\; T_\rho \times S_2 \;$ are isomorphic iff the automorphisms $ S_1 $ and $ S_2 $ are isomorphic.
4. If, in addition, $T$ is exact, i.e. $\;{\alpha}(T) = \nu \;$, then $T$ is simple iff $T$ is isomorphic to $\; T_\rho \;$.
It is easily to construct a skew product $T$ over $T_\rho$, which is exact and has entropy $\; h(T) > h(T_\rho) = \log l \;$. Every such endomorphism is $\rho$-uniform, $\; T \in {\mathcal {UE}(\rho)}\;$, but it is not isomorphic to $T_\rho$, whence, it is not simple. See also [@FeR], [@HeHo], [@HeHoR], [@Ho], [@HoR]), for more interesting examples of such kind of endomorphisms.
It can be shown that there exist non-simple exact endomorphisms in each class $\; {\mathcal {UE}(\rho)}\;$ in the case, when $\rho$ is not absolutely non-homogeneous, i.e. $\; \rho(i) = \rho(j)\;$ for some $\;i,j \in I\;$.
The following result plays an important role in present paper.
\[simp mar\] Every ergodic $\rho$-uniform one-sided Markov shift $T_G$, corresponding to a positively recurrent Markov chain on a finite or countable state space, is simple and $$\label{d=dg:b}
d(T_G) \;=\; d_{{\gamma}{ : }{\beta}}(T_G) \;<\; \infty \;.$$
The last statement \[d=dg:b\] was proved in . It implies that $T_G$ is simple by Proposition \[simple1\].
Stochastic graphs and their homomorphisms. {#s3}
============================================
Stochastic graphs and Markov shifts {#ss3.1}
-----------------------------------
We need some terminology concerning stochastic graphs and their homomorphisms.
Consider a directed graph with countable (finite or infinite) set $G$ of edges. Denote by $G{^{(0)}}$ the set of all vertices of the graph. We also denote by $s(g)$ the starting vertex and by $t(g)$ the terminal vertex of an edge $g \in G$ $$\xymatrix{ t(g) & s(g) \ar[l]_{g} }$$ The maps $$s \; { : }\; G \ni g \; {\rightarrow}\; s(g) \in G{^{(0)}} \;\; \;,\; \;\;
t \; { : }\; G \ni g \; {\rightarrow}\; t(g) \in G{^{(0)}}$$ completely determine the structure of the graph $G$,
In the sequel we assume that both the sets $$_vG = \{ g \in G \;{ : }\; t(g) = v \} \;\;\;,\;\;\;
G_u = \{ g \in G \;{ : }\; s(g)= u \}$$ are not empty for all vertices $ \;u\;,v\;\in G{^{(0)}} $.
Denote by $\; G{^{(n)}}\;$ the set of all paths of length $n$ in $G$ , i.e. $$\label{G(n)}
G{^{(n)}}= \{ g_1 g_2 \ldots g_n \in G^n \;{ : }\;
s(g_1)=t(g_2) , \ldots , s(g_{n-1})=t(g_n) \}$$ A graph $G$ is said to be [**irreducible**]{} if for every pair of vertices $ \;u\;,v\;\in G{^{(0)}} $ there exists a finite $G$-path $ \;g_1g_2 \ldots g_n \in G{^{(n)}}\;$ such that $\; u=s(g_n) \;$ and $\; v=t(g_1) \;$.
Take into account that we use here and in the sequel the notation $\; g_1 \; g_2\; \ldots \; g_n \;$ for [**backward**]{} paths $$\xymatrix{
t(g_1) & s(g_1) = t(g_2) \ar[l]_-{g_1}
& s(g_2) = t(g_3) \ar[l]_-{g_2}
& \ar[l]_-{g_3} }
\;\;\dots\;\;
\xymatrix{ & s(g_n) \ar[l]_-{g_n} }$$ A graph $G$ is called [**stochastic**]{} if its edges $g$ are equipped with positive numbers $ p(g) $ such that $\; \sum_{g \in
G_u}\;p(g) \;=\; 1 \;$ for all $\; u \; \in G{^{(0)}} \; $. The weights $\; p(g) \;,\; g \in G \;$, determine the backward transition probabilities of the Markov chain induced by $G$.
We shall assume in the sequel that there exist stationary probabilities $ p{^{(0)}}(u) > 0 $ on $ G{^{(0)}}$ such that $$\label{PR}
\sum_{u \in G{^{(0)}}}\;p{^{(0)}}(u) \;=\; 1 \;\;\;,\;\;\;
\sum_{g \in _vG} p(g) p{^{(0)}} (s(g)) \;=\; p{^{(0)}} (v)$$ for all vertices $ \;u\;,v\;\in G{^{(0)}} $.
It is known that the stationary probabilities on $G{^{(0)}}$ exist iff the corresponding to $G$ Markov chain is positively recurrent. If, in addition, the irreducibility condition hold, the stationary probabilities $\;p{^{(0)}}(u) \;,\; u \in G{^{(0)}} \;$ on the vertices are uniquely determined by the transition probabilities $\;
p(g)\;,\;g \in G\;$ on the edges.
Thus any stochastic graph $(G,p)$ induces a Markov chain on the state space $G$ with the transition probabilities matrix $\; P
\;=\;{(P(g,h))}_{g\in G , h \in G} \;$, where $$P(g,h) \;=\;
\left\{
\begin{array}{ll}
p(h) \;, & if \;\; t(g)=s(h) \; \\
0 \;\;, & otherwise.
\end{array}
\right.$$ In the sequel we mainly deal with stochastic graphs, which induce irreducible positively recurrent Markov chains.
The one-sided Markov shift $ T_G $, induced by the stochastic graph $ G $ , is defined as follows. Let $$X_G = \{x={\{g_{n}\}}_{n=1}^{\infty} \in G^{{{\mathbb{N}}}} \;{ : }\;
s(g_1)=t(g_2)\;,\; s(g_2)=t(g_3) \;,\; \dots \; \}$$ and the Markov measure $m_G$ on $ X_G $ is given by $$p(g_{1}) \; p(g_{2}) \; \ldots \; p(g_n) \; p{^{(0)}}(s(g_n))$$ on the cylindric sets of the form $$A(g_1\;g_2\; \ldots \;g_n)
\;:=\; \{x={\{x_k\}}_{k=1}^{\infty} \in X_G
\;{ : }\; x_1=g_1 \;,\; \ldots \;,\; x_n=g_n \;\}$$ where $\; g_1\;g_2\; \ldots \;g_n) \; \in \; G{^{(n)}}\;$ is a $G$-path of length $n$ in $G$.
The one-sided shift $T_G$ acts on the probability space $\;( X_G , m_G) \; $ by $$T_G({\{x_n\}}_{n=1}^{\infty}\;) \;=\; {\{x_{n+1}\}}_{n=1}^{\infty}$$ and $T_G$ preserves the Markov measure $m_G$. The shift $T_G$ is ergodic iff the graph $G$ is irreducible. Under the irreducibility condition, the stationary probabilities $p{^{(0)}}$ on $G{^{(0)}}$ and, hence, the $T_G$-invariant Markov measure $m_G$ are uniquely determined by the stochastic graph $\; (G,p) \;$.
The coordinate functions $$Z_n \;{ : }\; X_G \ni x={\{x_k\}}_{k=1}^{\infty}\; {\rightarrow}\; x_n
\in G \;\;,\;\;n \in {{\mathbb{N}}}$$ form a stationary Markov chain on $ ( X_G , m_G) $ with the backward transition probabilities $$P(g,h) \;=\;m_G\{\; Z_n = h \;|\; Z_{n+1} = g \;\} \;=\; p(h)
\;\;,\;\; n \in {{\mathbb{N}}}$$ for all $\; (h,g) \in G^{(2)} \;$.
Consider now the partitions $${\zeta}_n = {Z_n}^{-1}{\varepsilon}_G =
T_G^{-n+1}{\zeta}_1 = \{ T_G^{-n+1}A(g) \}_{g \in G}
\;\;,\;\; n \in {{\mathbb{N}}}\;,$$ generated by $Z_n$ on $ X_G$, where $$A(g) \;=\;
\{\; x={\{x_k\}}_{k=1}^{\infty} \in X_G \;{ : }\; x_1=g \;\}$$ Setting $ {\zeta}= {\zeta}_1 $ and $ T = T_G $, we have $$\label{gen par}
\bigvee^{\infty}_{n=1}T^{-n+1}{\zeta}= {\varepsilon}$$ $$\label{Mar par}
{\zeta}\perp \bigvee^{\infty}_{n=1}T^{-n}{\zeta}\pmod { T^{-1}{\zeta}}$$
Recall that a measurable partition ${\zeta}$ of $(X,m)$ is said to be a [**one-sided Markov generator**]{} or [**one-sided Markov generating partition**]{} for an endomorphism T of $(X,m)$, if the above conditions (\[gen par\]) and (\[Mar par\]) hold.
The partition $ {\zeta}_G $ will be called the [**standard**]{} one-sided Markov generator of the one-sided Markov shift $T_G$ on $ X_G $.
\[Bern graph\] Let $(I,\rho)$ be a finite or countable alphabet and $$\rho = \{ \rho(i) \;,\; i \in I \} \;,\;\; \rho (i) > 0
\;\;,\;\; \sum_{i \in I}{\rho(i)} = 1 .$$ be a probability on $I$. We shall consider $(I,\rho)$ as a stochastic graph, which has the set of edges $i \in I$ with weights $\; \rho(i) \;$ and a single vertex, denoted by$\;"o"\;$. So $G{^{(0)}} = \{o\}$ is a singleton and $ s(i) = t(i) = o $ for all $i \in I$. We shall say that $(I,\rho)$ is the [**standard Bernoulli graph.**]{}
For instance, $\; \xymatrix{ o \ar@(ul,dl) []_{p} \ar@(dr,ur) []_{q} } \;$ if $\; |I| = 2 \;$ and $\; \rho = (p,q) \;$.
The corresponding to $(I,\rho)$ one-sided Markov shift $T_I$ coincides with the Bernoulli shift $ T_I = T_\rho $. The generating partition $ {\zeta}_I$ coincides with the standard Bernoulli generator $\; {\delta}_\rho = \{ B_\rho(i) \}_{i \in I} \;$, defined by (\[delta rho\]).
[**Induced shift**]{} $\; {\mathbf{T}}_{\mathbf{u}}\;$. For any $\; u \in G{^{(0)}} \;$, denote $$D(u) \;:=\; \{\; x={\{x_k\}}_{k=1}^{\infty} \in
X_G \;{ : }\; t(x_1)=u \;\} \;\;,\;\; u \in G{^{(0)}} \;.$$ and consider the partition $\; {\zeta}{^{(0)}} := {\{ D(u) \}}_{u \in G{^{(0)}}} \;$ on the space $X_G$. The partition ${\zeta}{^{(0)}}$ is a Markov partition with respect to shift $T_G$ , i.e. $$\label{Mar par 0}
{\zeta}{^{(0)}} \perp T_G^{-1} {\varepsilon}_{X_G} \pmod { T_G^{-1}{\zeta}{^{(0)}} } \;,$$ but it is not a one-sided generator for $T_G$, in general.
We shall use in the sequel the endomorphisms $ T_u := (T_G)_{D(u)} $, induced by the shift $T_G$ on elements $ D(u) $ of ${\zeta}{^{(0)}}$, $\; u \in G{^{(0)}} \;$. The Markov property (\[Mar par 0\]) provides that for every $u \in G{^{(0)}} $ the induced endomorphism $ T_u $ is a Bernoulli shift. More exactly, in accordance with the general definition of return functions (\[ret fun\]) we have $${\varphi}_u(x)={\varphi}_{D(u)}(x) := min \{ n \geq 1 { : }T_G^{n}x \in D(u) \}
\;\;,\;\; x \in D(u)$$ and $$T_u x \;=\; T_G^{{\varphi}_u(x)} x \;\;,\;\; x \in X_G \;.$$ Take $\; I_u = \bigcup_{n=1}^\infty I_{u,n} \;$, where $I_{u,n}$ be the set of all $ g_1g_2 \ldots g_n \in G{^{(n)}}$ such that $$\label{I u n}
t(g_1) = s(g_n) = u \;\;,\;\;
s(g_k) = t(g_{k+1}) \neq u \;,\; k = 1,2, \ldots ,n-1 \;.$$ Define also $\; \rho_u = \{\rho_u(i)\}_{i \in I_u} \;$ by $$\label{rho u}
\rho_u(i) = p(g_1) p(g_2) \ldots p(g_n) \;\;,\;\;
i = g_1g_2 \ldots g_n \in I_{u,n} \;.\; n \in {{\mathbb{N}}}\;.$$ For any $\; i = g_1g_2 \ldots g_n \in I_{u,n} \;$ we set $ B_u(i) := A(g_1g_2 \dots g_n)$ and consider the partition $\; {\zeta}_u = \{ B_u(i)\}_{i \in I_u} \;$, whose elements are enumerated by the alphabet $I_u$. The Markov property (\[Mar par 0\]) implies that the partitions $\; T_u^{-n} {\zeta}_u \;,\; n \in {{\mathbb{N}}}\;$ are independent. Thus
\[Tu\] The induced endomorphism $T_u$ is isomorphic to the Bernoulli shift $T_{\rho_u}$ and $ {\zeta}_u $ is a one-sided Bernoulli generator of$T_u$.
Graph homomorphisms and skew products {#ss3.2}
-------------------------------------
Now we want to establish the class of graph homomorphisms that we shall use.
\[hom\] Let $\; G \;$ and $\; H \;$ be two stochastic graphs.
1. A map $\; \phi { : }G {\rightarrow}H \;$ is a [**graph homomorphism**]{} if there exists a map $\; \phi{^{(0)}} : G{^{(0)}} {\rightarrow}H{^{(0)}} \;$ such that $$s(\phi (g)) = \phi{^{(0)}} (s(g)) \;\;\;,\;\;\;
t(\phi (g)) = \phi{^{(0)}} (t(g))$$ for all $\; g \in G \;$. (Note that, if the map $\; \phi{^{(0)}} \;$ exists it is unique.)
2. A graph homomorphism $\;\phi { : }G {\rightarrow}H \;$ is [**deterministic**]{} if $\; \phi{^{(0)}} (G{^{(0)}}) = H{^{(0)}} \;$ and for every $\; u \in G{^{(0)}} \;$ the restriction of $\phi $ on $ G_u$ $$\phi {|}_{G_u} \;{ : }\; G_{u} \; \rightarrow \;H_{\phi{^{(0)}} (u)}$$ is a bijection of this set onto $\; H_{\phi{^{(0)}} (u)} \;$.
3. A graph homomorphism is [**weight preserving**]{} or ${\mathbf{p}}$[**-preserving**]{} if $\; p(\phi (g)) \;=\; p(g) \;$ for all $\; g \in G \;$.
Two edges $g_1$ and $g_2$ are said to be [**congruent**]{} if $$s(g_1)=s(g_2) \;\;,\;\; t(g_1)=t(g_2) \;\;,\;\; p(g_1)=p(g_2) \;.$$ The map $ \phi{^{(0)}} $ in the above definition is uniquely determined by $ \phi $, but $ \phi{^{(0)}} $ does not determines $
\phi$ if $G$ has congruent edges.
Anyway one can use a more explicit notation $$( \phi , \phi{^{(0)}} ) \;{ : }\; (G , G{^{(0)}}) \; {\rightarrow}\; (H , H{^{(0)}} )$$ for the homomorphism $\;\phi { : }G {\rightarrow}H \;$.
We shall denote by $\; {{\mathcal Hom}}(G,H) \;$ the set of all weight preserving deterministic graph) homomorphisms $\;\phi { : }G
{\rightarrow}H \;$. In the sequel the term “homomorphism” always means just [**weight preserving deterministic graph homomorphism**]{}.
\[factor map\] Let $\; \phi { : }G {\rightarrow}H\;$ be a map.
1. If $ \phi $ is a graph homomorphism, it induces a factor map $$\Phi_\phi \;{ : }\; X_G {\rightarrow}X_H \;\;,\;\;
\Phi_\phi ( {\{x_n\}}_{n=1}^{\infty} ) \;=\;
{\{ \phi ( x_n ) \}}_{n=1}^{\infty}$$ such that $\; \Phi_\phi \circ T_G \;=\; T_H \circ {\Phi}_{\phi}\;.$
2. If, in addition, $\phi $ is weight preserving, the factor map $ \Phi_\phi $ is measure preserving, $\; ( m_H = m_G \circ {\Phi_\phi}^{-1} ) $.
3. If $\;\phi\;$ is also deterministic, the shift $T_G$ can be represented as a skew product $$\label{skew prod}
{{\bar T}}(x,y) = ( T_H x \;,\; A(x) y) \;\;,\;\; (x,y) \in X_H \times Y$$ where $\; \{ A(x) \;,\; x \in X_H \} \;$ is a measurable family of automorphisms of $Y$.
4. If $T_G$ is ergodic, $Y$ is a homogeneous Lebesgue space.
Parts (i) and (ii) follow directly from Definition \[hom\]. Part (iii) and (iv) can be proved by analogy with Proposition \[T decompos\]
Moreover
\[fin ind\] Let $\phi \in {{\mathcal Hom}}(G,H) $ and suppose that the shift $ T_G $ is ergodic. Then there exists $\;d \in {{\mathbb{N}}}\;$ such that $\; | \Phi_\phi^{-1}(x) | \;=\; d \;$ for almost all $\; x \in X_H \;$. That is, in the skew product (\[skew prod\]) the space $Y$ is finite, $\; |Y| = d \;$.
Note that Theorem \[fin ind\] claims the finiteness of $d$ even in the case, when the graph $G$ is not finite, i.e. $ |G| = \infty
$. This is a consequence of positive recurrence of the corresponding to $G$ Markov chain. The skew product decomposition (\[skew prod\]) of $T_G$ over $T_H$ is a $d$-extension.
Theorem \[fin ind\] was proved earlier in a particular case, when $H$ is a Bernoulli graph, i.e. when $ H{^{(0)}} = \{ o \} $ is a singleton (See Theorem 3.3 and Corollary 3.4 from , and also Theorem \[zet del\] below).
We omit the proof of Theorem \[fin ind\] in general case , since only the pointed out particular case is considered in this paper.
\[def d(phi)\]. The integer $d$ in Theorem \[fin ind\], i.e. the degree of the factor map $\Phi_\phi$ will be called the [**degree**]{} of the homomorphism $\phi$.
Denoting the degree by $\; d(\phi) \;$, we have $\; d(\phi) = |\Phi_\phi^{-1}(x)| \;$ for a.a. $ x \in X_H $.
The following construction plays a central role in our explanation.
\[GSP\]
Let $\;d \in {{\mathbb{N}}}\;$ and let $\; Y_d = \{ 1,2, \ldots , d \} \;$ consists of $d$ points of measure $\frac{1}{d}$. Denote by $\; {{\mathcal A}_d}= {\mathcal A}(Y_d) \;$ the full group of all permutations of $Y_d$.
Given a stochastic graph $ H $, equipped with a function $\; a { : }H \ni h {\rightarrow}a(h) \in {{\mathcal A}_d}\;$, we construct a stochastic graph $ {{\bar H}}_a $ and a homomorphism $\; \pi_H { : }{{\bar H}}_a {\rightarrow}H \;$ by $${{\bar H}}_a \;=\; H \;\times \; Y_d \;\; , \;\;
{{{\bar H}}_a}{^{(0)}} \;=\; H{^{(0)}} \; \times \; Y_d \; ,$$ with $$s({{\bar h}}) = (s(h),y) \;\;,\;\;
t({{\bar h}}) = (t(h),a(h)y) \;\;,\;\; p({{\bar h}}) = p(h)$$ for $\; {{\bar h}}= (h,y) \in {{\bar H}}_a = H \times Y_d \;$ and also $$p{^{(0)}}({{\bar u}}) = p{^{(0)}}(u) \;\;,\;\;
{{\bar u}}= (u,y) \in {{{\bar H}}_a}{^{(0)}} = H{^{(0)}} \times Y_d$$. The natural projection $$\pi_H { : }{{\bar H}}_a = H \times Y_d {\rightarrow}H \;\;,\;\;
{\pi_H}{^{(0)}} { : }{{\bar H}}_a^{(0)} = H{^{(0)}} \times Y_d {\rightarrow}H{^{(0)}}$$ is a homomorphism.
\[def GSP\] We shall say that the graph $ {{\bar H}}_a $ is a [**skew product**]{} over $H$ and the homomorphism $\;\pi_H { : }{{\bar H}}_a {\rightarrow}H \;$ is a [**graph skew product**]{} (or [**GSP**]{}) $d$-extension of $H$ .
In the above construction we have $\; | {\pi_H}^{-1} (h) | = d \;$ for all $\; h \in H \;$ and this is, in fact, a characteristic property of the graph skew product $d$-extension in the following sense
\[hom equi\] Two homomorphisms $\; \phi_k { : }G_k {\rightarrow}H \;,\; k = 1,2 \;$ are said to be [**equivalent**]{} if $\; \phi_2 = {\kappa}\circ \phi_1 \;$ for an appropriate isomorphism $\; {\kappa}{ : }G_1 {\rightarrow}G_2 \;$.
\[d-uniform\] Let $\; d \in {{\mathbb{N}}}\;$. A homomorphism $\; \phi \in {{\mathcal Hom}}(G,H) \;$ is called a [**$d$-extension**]{} if $$\label{phi-1=d}
|{\phi}^{-1} (h)| = d \;\;,\;\; h \in H \;.$$
\[d-unif=GSP\] Any $d$-extension $ \phi { : }G {\rightarrow}H $ is equivalent to a GSP $d$-extension $\; \pi_H { : }{{\bar H}}_a {\rightarrow}H \;$.
Let $\; \phi \in {{\mathcal Hom}}(G,H) \;$ is a $d$-extension. Since $\phi$ is deterministic the restrictions $\; \phi{|}_{G_u}
\;$ are bijections between $G_u$ and $ H_{\phi{^{(0)}}(u)} $ for all $
u \in H{^{(0)}} $. Hence the condition (\[phi-1=d\]) is equivalent to $$|{\phi{^{(0)}}}^{-1}(u)| = d \;,\; u \in H{^{(0)}} \;.$$ For each $u \in H{^{(0)}}$ we can choose a bijection $ w_u $ of $ {\phi{^{(0)}}}^{-1} (u) $ onto $ Y_d $. With any fixed choice of these bijections we set $$H \ni h {\rightarrow}a(h) = w_{t(h)} \circ {w_{s(h)}}^{-1} \in {{\mathcal A}_d}\;,$$ and consider the corresponding skew product graph $ {{\bar H}}_a $. The bijections $ w_u $ uniquely determine an isomorphism $\; {\kappa}{ : }G {\rightarrow}{{\bar H}}_a \;$ such that $\; \phi =\pi_H \circ {\kappa}\;$.
\[re GSP shift\] The Markov shift $T_{{{\bar H}}_a}$ corresponding to a graph skew product $ {{\bar H}}_a $ can be identified with the skew product endomorphism $ {{\bar T}}_{H,a} $, defined by $${{\bar T}}_{H,a} (x,y) = (T_H x , {a(x_1)}^{-1}y) \;\;,\;\;
x ={\{ x_n \}}_{n=1}^{\infty} \in X_H \;,\; y \in Y_d \;.$$ and thus any $d$-extension is a homomorphism of degree $d$.
Indeed, the shift $T_{{{\bar H}}_a}$ acts on the space $$X_{{{\bar H}}_a} = \{ {\{ (x_n,y_n) \}}_{n=1}^{\infty} \;{ : }\;
x = {\{ x_n \}}_{n=1}^{\infty} \in X_H \;,\;
y_n = a(x_n)y_{n+1} \in Y_d \}$$ and the map $$\Psi \;{ : }\; X_{{{\bar H}}_a} \ni {\{ x_n \}}_{n=1}^{\infty} {\rightarrow}({\{ x_n \}}_{n=1}^{\infty},y_1) \in X_H \times Y_d$$ realizes the identification, that is, $\; m_H \otimes m_{Y_d} = m_{{{\bar H}}_a} \circ \Psi^{-1} \;$ and $\; {{\bar T}}_{H,a} \circ \Psi = \Psi \circ T_{{{\bar H}}_a} \;$. Note also that $$\label{Psi gz}
\Psi \; {\zeta}_{{{\bar H}}_a} \;=\; {\zeta}_H \times {\varepsilon}_{Y_d} \;\;,\;\;
\Psi \; {\zeta}_{{{\bar H}}_a}{^{(0)}} \;=\; {\zeta}_H
{^{(0)}} \times {\varepsilon}_{Y_d} \;.$$
Consider now two skew product endomorphisms $\; {{\bar T}}_{H,a_k} \;$, corresponding to graph skew products $ {{\bar H}}_{a_k} $ with two functions $\; a_k { : }H {\rightarrow}{{\mathcal A}_d}\;,\; k=1,2 \;$.
\[cohom\]
1. Two functions $\; a_k { : }H {\rightarrow}{{\mathcal A}_d}\;,\; k=1,2 \;$, are said to be [**cohomologous**]{} with respect to $H$ if there exists a map $\; w { : }H{^{(0)}} {\rightarrow}{{\mathcal A}_d}\;$ such that $$\label{a cohom}
a_2(h) w(s(h)) = w(t(h)) a_1(h) \;,\; h \in H$$
2. Two measurable functions $\; A_k { : }X_H {\rightarrow}{{\mathcal A}_d}\;,\; k=1,2 \;$ are said to be [ **cohomologous**]{} with respect to $T_H$ if there exists a measurable map $\; W { : }X_H {\rightarrow}{{\mathcal A}_d}\;$ such that $$\label{A cohom}
A_2(x) W(x) = W(T_H x) A_1(x) \;,\; x \in X_H$$
In accordance with Definitions \[hom equi\] and \[cohom\] we can say now that the homomorphisms $\;\pi_H { : }{{\bar H}}_{a_k} {\rightarrow}H $ are equivalent iff the functions $\; a_k : H {\rightarrow}{{\mathcal A}_d}\;,\; k=1,2 \;$ are cohomologous with respect to $H$.
The equality (\[a cohom\]) is equivalent to (\[A cohom\]) if we take $$A_k(x) = {a_k(x_1)}^{-1} \;,\; k=1,2 \;\;,\;\; W(x) = w(t(x_1))$$ for $\; x = {\{ x_n \}}_{n=1}^{\infty} \in X_H \;$ and given $a_k$ and $w$. Hence if $a_1$ and $a_2$ are cohomologous with respect to $H$, then $A_1$ and $A_2$ cohomologous with respect to $T_H$.
We shall show in Section \[ss4.3\] that the inverse is also true.
\[triv exten\] Let $\chi { : }H {\rightarrow}H_1 $ be a homomorphism and $ \pi_1 : {{\bar H}}_1 {\rightarrow}H_1 $ be a $d$-extension of $H_1$ generated by a function $a_1 : H_1 {\rightarrow}{{\mathcal A}_d}$. Setting $ a(h) :=
a_1(\chi(h)) $ we obtain a $d$-extension $ \pi { : }{{\bar H}}:= {{\bar H}}_a
{\rightarrow}H $ of $H$. The map $ {{\bar \chi}}(h,y) := ({\kappa}(h),y) \;,\; (h,y)
\in {{\bar H}}$ is a homomorphism and the diagram $$\label{diag ext}
\xymatrix{
{{\bar H}}\ar[d]^{\pi} \ar[r]^{{{\bar \chi}}} & {{\bar H}}_1 \ar[d]^{\pi_1} \\
H \ar[r]^{\chi} & H_1 }$$ commutes. The homomorphism $ {{\bar \chi}}$ is called a [**trivial**]{} extension of ${{\bar \chi}}$. If, in addition, $ d(\chi) = 1 $, then $ d({{\bar \chi}}) = 1 $ and hence the corresponding endomorphisms ${{\bar T}}_{H,a}$ and ${{\bar T}}_{H_1,a_1}$ are isomorphic.
Stochastic ${\boldsymbol{\rho}}$-unform graphs {#ss3.3}
----------------------------------------------
We continue to consider $\;(I,\rho)\;$ as the standard Bernoulli stochastic graph, (Example \[Bern graph\])
\[rho uni\] A stochastic graph $ (G,p) $ is called ${\boldsymbol{\rho}}$[**-uniform**]{} if there exists a homomorphism $\; \phi \in {{\mathcal Hom}}(G,I) \;$.
For any such homomorphism $\phi$ and for every $ u \in G{^{(0)}} $ $$\phi \;{|}_{G_u} \;{ : }\;
( G_u \;,\; p \;{|}_{G_u} )\; {\rightarrow}\; (I,\rho)$$ is a weight preserving bijection. Thus the atomic probability spaces $\; ( G_u , p {|}_{G_u} ) \;$ are isomorphic to $(I,\rho)$ for every $ u \in G{^{(0)}} $.
\[p3.4\] $\; T_G \in {\mathcal {UE}(\rho)}\;$ iff $G$ is $\rho$-uniform.
Consider the partition $\; \xi_1 := T_{G}^{-1} {{\varepsilon}}_{X_G} \;$ generated by the shift $T_G$. The Markov property of the measure $m_G$ on $X_G$ implies $$m^{C_{\xi_1}(x)}(\{x\}) \;=\; m_G \{\; Z_1=x_1 \;|\; Z_2=x_2 \;\}
\;=\; p(x_1)$$ for a.a. $\; x={\{x_{n}\}}_{n=1}^{\infty} \in X_G \;$, Here $\; m^{C_{\xi_1}(x)}(\{x\}) \;$ is the conditional measure of the point $x$ in the element $C_{\xi_1}(x) = T_G^{-1}T_Gx $ of the partition $ \xi_1 $. Hence for every $ u \in G{^{(0)}} $ almost all elements $(C,m_C)$ of the partition $ \xi_1 $ are isomorphic to $\; ( G_u , p {|}_{G_u})
\;$ on the set $\; \{ x={\{x_n\}}_{n=1}^{\infty} \in X_G
\;{ : }\; s(x_1) = u \} \;$. But $\;T_G \in {\mathcal {UE}(\rho)}\;$ iff a.a. elements $(C,m_C)$ are isomorphic to $(I,\rho)$. Hence $\;T_G \in
{\mathcal {UE}(\rho)}\;$ iff $\;( G_u , p{|}_{G_u})\;$ are isomorphic to $(I,\rho)$ for every $ u \in G{^{(0)}} $.
Let $G$ be a $\rho$-uniform graph and $\; \phi \in {{\mathcal Hom}}(G,I) \;$. Consider the partition $\; \phi^{-1} {\varepsilon}_I = \{ \phi^{-1} (i) \;,\; i \in I \;\}$ of $G$. The first coordinate function $$Z_1 \;{ : }\; X_G \ni x={\{x_k\}}_{k=1}^{\infty}
\; {\rightarrow}\; x_1 \in G$$ generates the following partition $${\delta}_\phi \;=\; Z_1^{-1} ( \phi^{-1} {\varepsilon}_I )$$ of the space $ X_G $. Elements of $\; {{\delta}}_{\phi} \;$ have the form $$B(i) = Z_{1}^{-1} ( \phi^{-1} (i) ) =
\{ x={\{x_{k}\}}_{k=1}^{\infty} \in X_G \;{ : }\;
{\phi}(g) = i \} \;,\; i \in I$$ Using the standard Markov generator $${\zeta}_G = Z_{1}^{-1}{\varepsilon}_G = {\{A(g)\}}_{g \in G} \;\;,\;\;
A(g) = Z_{1}^{-1}(g)$$ of $T_G$, we have $$B(i) = {\bigcup}_{g \in {\phi}^{-1}(i)} A(g)$$ and $$m_G(B(i)) = {\sum}_{g \in {\phi}^{-1}(i)} p(g) p{^{(0)}}(s(g))
= \rho(i) {\sum}_{u \in G{^{(0)}}} p{^{(0)}}(u) = \rho(i)$$ for $\; i \in I \;$. Hence for $\; {\delta}\;=\; {\delta}_\phi \;$ we have $$\label{del phi}
{\delta}\in IC (\; T_{G}^{-1} {\varepsilon}_{X_G} \;) =
{\Delta}_\rho( T_G ) \;\;,\;\; {\delta}\leq {\zeta}_G$$ Denoting by $\; {\Delta}_\rho(T_G , {\zeta}_G) \;$ the set of all $ {\delta}$ that satisfy (\[del phi\]), we have also
\[d(T,zeta)\] $\; {\Delta}_\rho(T_G , {\zeta}_G) \;$ is precisely the set of all $ {\delta}$ of the form $\; {\delta}\;=\; {\delta}_\phi \;$.
Now we introduce a semigroup $\; {{\mathcal S}}(\phi) \;$ of maps $\; f { : }G{^{(0)}} {\rightarrow}G{^{(0)}} \;$ induced by the homomorphism $\phi$.
Let $\; i \in I \;$. Since $\; \phi \;$ is deterministic the restriction $\; \phi {|}_{G_u} \;{ : }\; G_u \; {\rightarrow}I \;$ is a bijection of $G_u$ onto $I$ for every $u \in G{^{(0)}}$. Hence for any pair $(i,u)$ there exists an unique $\; g_{i,u} \;$ such that $\; \phi (
g_{i,u} ) = i \;$ and $\; s( g_{u,i} ) = u \;$. Putting $\; f_iu =
g_{i,u} \;$, we get a map $\; f_i { : }G{^{(0)}} {\rightarrow}G{^{(0)}} \;$. Let $\; {{\mathcal S}}(\phi) \;$ be the semigroup generated by the maps $\;
\{ f_i \;,\; i \in I \} \;$.
Let $ {{\mathcal F}}{{\mathcal S}}(I) $ be the set of all finite words $\; i_1i_2
\dots i_n \;$ in the alphabet $I$. We shall consider $ {{\mathcal F}}{{\mathcal S}}(I) $ as a free semigroup with the generating set $I$ and with juxtaposition multiplication: $$i_1i_2 \dots i_m \cdot j_1j_2 \dots j_n
\;=\; i_1i_2 \dots i_mj_1j_2 \dots j_n$$ and set $$f_{i_1i_2 \dots i_n} \;=\; f_{i_1} \circ f_{i_2} \circ \; \dots \;
\circ f_{i_{n}} \;\;,\;\; i_1i_2 \dots i_n \; \in \; I^n$$ Then $\; i_1i_2 \dots i_n {\rightarrow}f_{i_1i_2 \dots i_n} \;$ is a homomorphism from the semigroup $ {{\mathcal F}}{{\mathcal S}}(I) $ onto the semigroup $${{\mathcal S}}(\phi) = \{ f_{i_1i_2 \dots i_n} \;,\;i_1i_2 \dots i_n
\in {{\mathcal F}}{{\mathcal S}}(I) \} \;,$$ generated by $\; \{ f_i \;,\; i \in I \} \;$.
Now we can describe the partitions $${\delta}_\phi = \{ B(i) \}_{i \in I} \;\;,\;\;
{\delta}_\phi{^{(n)}}\;=\; \bigvee_{k=1}^{n} T_{G}^{-k+1} {\delta}_\phi
\;,\; n \in {{\mathbb{N}}}$$ as follows.
First recall that the partition $\; {\zeta}{^{(0)}}$ consists of the atoms $D(u) = Z^{-1}(_uG) \;,\;u \in G{^{(0)}} \;$ and rename the elements $\; A(g) \;,\; g \in G \;$ of the partition ${\zeta}_G $ by $$D(i,u) := A(g_{i,u}) \;\;,\;\; u \in G{^{(0)}} \;,\; i \in I \;.$$ Then for all $i \in I$ and $u \in G{^{(0)}}$ we have $\; D(i,u) = B(i) \cap T_{G}^{-1}D(u) \;$, $$D(i,u) = \{ x={ \{ x_n \} }_{n=1}^{\infty} \in X_G
\;{ : }\; t(x_1) = u \;,\; \phi(x_1) = i \}$$ and $$\label{Bi Diu}
B(i) = \bigcup_{u \in G{^{(0)}}} D(i,u) \;\;\;,\;\;\;
D(u) = \bigcup_{v { : }f_i(v)=u} D(i,v) \;.$$ Further for any $\; g_1g_2 \dots g_n \in G{^{(n)}}\;$ there exists a unique pair $\; (i_1i_2 \dots i_n,u) \in I^n \times G{^{(0)}} \;$ such that $$\label{u i t}
u = s(g_n) \;,\; i_k = \phi (g_k) \;,\; t(g_k) = f_{i_k} (s(g_k))
\;,\; k = 1,2, \dots ,n$$ Hence any atom $\; A(g_1g_2\; \dots \,g_n) \;$ of the partition $\; {\zeta}_G{^{(n)}}= \bigvee_{k=1}^{n} T^{-k+1} {\zeta}_G \;$ can be renamed by $\;D(i_1i_2 \dots i_n,u) = A(g_1g_2\; \dots \,g_{n}) \;$, where the pair $\; (i_1i_2 \dots i_n,u) \;$ satisfies (\[u i t\]). By (\[Bi Diu\]) any atom $\; B(i_1i_2\; \dots \;i_n) \;$ of the partition $\; {\delta}_\phi{^{(n)}}= \bigvee_{k=1}^{n} T_{G}^{-k+1}
{\delta}_\phi \;$ has the form $$B(i_1i_2\; \dots \;i_n)
\;=\; \bigcup_{u \in G{^{(0)}}} D(i_1i_2 \dots i_n,u)$$ and since $$D(i_1i_2...i_n,u) \;=\; B(i_1i_2\; \dots \;i_n) \cap T^{-n} D(u)$$ we have $$m_G(B(i_1i_2\; \dots \;i_n)) \;=\;
\rho (i_1) \rho (i_2) \; \dots \; \rho (i_n),$$ $$m_G( D(i_1i_2 \dots i_n,u) ) \;=\;
\rho (i_1) \rho (i_2) \; \dots \; \rho (i_{n})p{^{(0)}}(u).$$ Any ${\zeta}_G{^{(0)}}$-set has the form $$D(E) = \{ x={ \{ x_n \} }_{n=1}^{\infty} \in X_G
\;{ : }\; t( x_1 ) \in E \}.$$ for a subset $\; E \subseteq G{^{(0)}} \;$. Then for any $ i_1i_2 \; \dots \; i_n \in I^n $ $$D(E) \cap B(i_1i_2\; \dots \;i_n) \;=\;
\bigcup_{u { : }f_{i_1i_2 \;\dots\; i_n} (u) \in E }
D(i_1i_2 \dots i_n,u).$$ Hence $$\label{D(E)}
m_G ( D(E) \; | \; B(i_1i_2 \; \dots \; i_n) ) \;=\;
p{^{(0)}} ( f^{-1}_{i_1i_2\; \dots \; i_n} (E) ).$$ Next theorem is basic for our explanation. Let $${\delta}_{\phi}^{(\infty)} \;=\; \bigvee_{n=1}^{\infty} {\delta}_\phi{^{(n)}}\;=\; \bigvee_{n=1}^{\infty} T_{G}^{-n+1} {\delta}_\phi$$
\[zet del\] Let $\; \phi \in {{\mathcal Hom}}(G,I) \;$. Then
1. $\; {\zeta}_G \; \vee \; {\delta}_{\phi}^{(\infty)}
\;=\; {\varepsilon}_{X_G} \;.$
2. $\; d(T_G) \;\leq\; d(T_G,{\delta}_{\phi}) \;=\;
d(\phi) \;<\; \infty \;$
It was proved in that if ${\zeta}$ is a one-sided Markov generator of $T$ and $$T \in {\mathcal {UE}(\rho)}\;,\; {\delta}\in {\Delta}_\rho(T) \;,\; {\delta}\; \leq \; {\zeta}\; ,$$ then $\; {\zeta}\vee {\delta}^{(\infty)} = {\varepsilon}\;$. Hence (i) follows by putting $ {\delta}= {\delta}_\phi $ and $ {\zeta}= {\zeta}_G $ .
Since the partition $ {\zeta}_G $ is finite or countable the equality (i) implies that almost all elements $(C,m_C)$ of the partition ${\delta}_{\phi}^{(\infty)}$ are atomic. Taking in to account the ergodicity of $T_G$, we see that almost all elements of ${\delta}_{\phi}^{(\infty)}$ consist of $d$ atoms of measure $\frac{1}{d}$ for an natural $d$. Herewith by Definitions \[def d(T)\] and \[def d(phi)\] we have $\; d =
d(T_G,{\delta}_{\phi}) = d(\phi)$ and, whence, (ii) follows.
We need the following sharp version of Part (i) of Theorem \[zet del\]
\[zet0 del\] $\; {\zeta}{^{(0)}}_G \; \vee \; {\delta}_{\phi}^{(\infty)} \;=\; {\varepsilon}_{X_G} \;$
Choose an increasing sequence of positive numbers $c_n > 0$ and an increasing sequence of finite subsets $E_n$ of $G{^{(0)}}$ such that $$\label{zet0 del1}
\bigcup_{n=1}^\infty E_n = G{^{(0)}} \;\;\;.\;\;\;
\sum_{n=1}^\infty(1-c_n) < \infty \;\;\;.\;\;\; p{^{(0)}}(E_n)>c_n \;.$$ Since $|E_n|<\infty$ there exist $\; i{^{(n)}}_1i{^{(n)}}_2 \dots i{^{(n)}}_{k_n} \in I^{k_n} \;$ and $\; f_n := f_{i{^{(n)}}_1i{^{(n)}}_2 \dots i{^{(n)}}_{k_n}} \in {{\mathcal S}}(\phi) \;$ such that $$|f_n(E_n)| \;=\; \min \{ |f(E_n)| \;{ : }\; f \in {{\mathcal S}}(\phi) \}.$$ The choice of $f_n$ provides that all restrictions $\; f{|}_{f_n(E_n)} \;,\; f \in {{\mathcal S}}(\phi) \;$ are bijections.
Consider the sets $$\begin{split}
& B_n := B(i{^{(n)}}_1i{^{(n)}}_2 \dots i{^{(n)}}_{k_n}) \;\;,\;\; \\
& B'_n := B_n \bigcap T_G^{-k_n}D(E_n) \;=\;
\bigcup_{u \in E_n}D(i{^{(n)}}_1i{^{(n)}}_2 \dots i{^{(n)}}_{k_n},u)
\end{split}$$ and also $$F_n \;:=\; \{ x \in X_G \;{ : }\;
T^{{\omega}_n(x)+n}x \in B^\prime _n \} \;,$$ where $${\omega}_n(x) \;:=\; \min \{k \geq 0 \;{ : }\; T^{n+k}x \in B_n \}$$ Then it is not hard to see that $$m_G(F_n) \;=\; m_G(B'_n \;|\; B_n) \;=\; p{^{(0)}}(E_n) \;>\; c_n \;.$$ Set $\; F \;:=\; \liminf_{n {\rightarrow}\infty}F_n \;$. Then we have $\; m_G(F)=1 \;$, since $\;\sum(1-c_n) < \infty \;$. By constructing, the set $F$ has the following property. Suppose $\; x={\{x_k\}}_{k=1}^{\infty} \;$ and $\;y={\{y_k\}}_{k=1}^{\infty} \;$ belong to $F$ and $$\; \Phi_{{\delta}_\phi}(x) \;=\; \Phi_{{\delta}_\phi}(y)
\;=\; {\{i_k\}}_{k=1}^{\infty}
\;\in \; X_\rho \;.$$ If $\; s(x_m) \neq s(y_m) \;$ for some $m \geq 1 $, then $$t(x_k) = f_{i_k}s(x_k) \neq t(y_k) = f_{i_k}s(y_k) \;\;,\;\;
k \;=\; 1,\,2,\;\dots \;m \;.$$ In other words, if $t(x_1) = t(y_1)$ and $\; \Phi_{{\delta}_\phi}(x) \;=\; \Phi_{{\delta}_\phi}(y) \;$ then $x=y$. Thus $\; {\zeta}{^{(0)}}_G \; \vee \; {\delta}_{\phi}^{(\infty)} \;=\;
{\varepsilon}_{X_G} \;$ on the set $F$ of measure $1$.
Semigroup $ {{\mathcal S}}({\boldsymbol{\phi}})$ and persistent ${\mathbf{d}}$-sets. {#ss3.4}
------------------------------------------------------------------------------------
Let $U$ be a finite or countable set.
\[L set\] Let $ {{\mathcal S}}$ be a semigroup of maps $\;f \;{ : }\; U \;{\rightarrow}\; U\;$ on $U$ and let $\; d \in {{\mathbb{N}}}\;$. Call the semigroup ${{\mathcal S}}$ ${\mathbf{d}}$[**-contractive**]{} if there exists a subset $\; L\; \subseteq U \;$ such that
1. $ |f(L)| \;=\; |L| \;=\; d \;$ for all $\; f \in {{\mathcal S}}\;.$
2. For every finite subset $\; E \subset U \;$ there exists $\; f \in {{\mathcal S}}\;$ with $\; f(E) \subseteq L \;$.
The sets $L$, satisfying (i) and (ii), will be called [**persistent ${\mathbf{d}}$-sets**]{} with respect to $ {{\mathcal S}}$.
Denote by $\; {{\mathcal L}}({{\mathcal S}}) \; $ the set of all such $L$. We have directly from the definition:
- For $\; L \in {{\mathcal L}}({{\mathcal S}}) \;$ and $\; f \in {{\mathcal S}}\;$ the restriction $\; f {|}_{L} \;{ : }\; L \;{\rightarrow}\; f(L) \;$ is a bijection and $\; f(L) \; \in \; {{\mathcal L}}( {{\mathcal S}}) \;$.
- The semigroup $ {{\mathcal S}}$ acts transitively on $ {{\mathcal L}}( {{\mathcal S}}) $, i.e. for every pair $\; L_1 \;,\;L_2\; \in \; {{\mathcal L}}({{\mathcal S}}) \;$ there exists $\; f \in {{\mathcal S}}\;$ such that $\; f(L_1) \;=\; L_2 \;$.
- The integer $d$ is equal to $$\label{d(G)}
d( {{\mathcal S}}) \; := \; \sup_{E \subseteq U \;{ : }\; |E| < \infty}
\;\; \min_{ f \in {{\mathcal S}}} \;\; |f(E)| \; .$$ and $\; d({{\mathcal S}}) \;=\; \min_{ f \in {{\mathcal S}}} \;\; |f(U)| \;$ if $\; |U| \;<\; \infty \;$.
Let $G$ be a $\rho$-uniform stochastic graph and $ \phi \in {{\mathcal Hom}}(G,I) $ be a homomorphism $ \phi { : }G {\rightarrow}I $.
Return to the semigroup $\; {{\mathcal S}}(\phi) \;$ which acts on $\; U = G{^{(0)}} \;$ .
\[d(phi)\] Let $T_G$ be an ergodic one-sided Markov shift corresponding to a $\rho$-uniform stochastic graph $G$ and let $ \phi \in {{\mathcal Hom}}(G,I)$. Then the semigroup $ {{\mathcal S}}(\phi)$ is $d$-contractive on $G{^{(0)}} $ and $$\label{d=d(G)}
d \;=\; d({{\mathcal S}}(\phi)) \;=\; d(T_G , {\delta}_\phi) \;=\; d(\phi)$$
To prove the theorem we shall use the partition ${\zeta}{^{(0)}}_G$ on $G{^{(0)}}$. Recall that ${\zeta}{^{(0)}}_G$ consists of all atoms of the form $\; D(u) = Z_1^{-1} (_uG) \;,\; u \in G{^{(0)}}. \;$. For any subset $E$ of $G{^{(0)}} $ we denote $$D(E) = \{ x={\{x_{n}\}}_{n=1}^{\infty} \in X_G \;{ : }\;
t( x_1 ) \in E \} = \bigcup_{u \in E} D(u) \;,$$ i.e. $ D(E) $ is a $ {\zeta}{^{(0)}}_G$-set corresponding to $E$ in the space $X_G$.
It follows from Theorem \[zet del\] Part (ii) that almost all elements $(C,m_C)$ of the partition ${\delta}_{\phi}^{(\infty)}$ are isomorphic to $Y_d$, where $\; d = d(T_G,{\delta}_{\phi}) \in {{\mathbb{N}}}\;$. Hence $$m( \{x\} \;|\; C_{{\delta}_{\phi}^{(\infty)}}(x)) \;=\; \frac{1}{d}$$ for a.e. $x \in X_G$. Then Lemma \[zet0 del\] implies that there exists a measurable family $\; \{ l(x) \;,\; x \in X \} \;$ of subsets $\; l(x) \subseteq G{^{(0)}} \;$ such that $$\label{l(x)=d}
m ( D(l(x)) \;|\; C_{{\delta}_{\phi}^{(\infty)}} (x) ) \;=\; 1
\;\;,\;\; |l(x)| \;=\; d$$ almost everywhere on $ X_G $.
For any $ L \subseteq G{^{(0)}} $ denote $$\check{L} \;:=\; \{ x \in X_G \;{ : }\; l(x) = L \} \;\;,\;\;
{{\mathcal L}}\;:=\; \{ L \subseteq G{^{(0)}} \;:\; m_G ( \check{L} ) > 0 \} \;,$$ i.e. ${{\mathcal L}}$ is the (finite or countable) set of all essential values of the function $\;x \;{\rightarrow}\; l(x) \;$. We show that $\; {{\mathcal L}}\subseteq {{\mathcal L}}({{\mathcal S}}(\phi)) \;$, i.e. that every $\; L \in {{\mathcal L}}\;$ satisfies the conditions (i) and (ii) of Definition \[L set\].
Take any finite subset $E \subseteq G{^{(0)}}$ and choose $\; c > 0
\;$ such that $ \; c \;<\; \min_{u \in E}p{^{(0)}}(u) \;$. For $\; L \in {{\mathcal L}}\;$ and almost all $\; x={\{x_{n}\}}_{n=1}^{\infty} \in \check{L} \;$ we have by (\[D(E)\]) $$lim_{n {\rightarrow}\infty} m(D(L) \;|\; C_{{\delta}_\phi{^{(n)}}}(x) \}
\;=\; m_G (D(l(x)) \;|\; C_{{\delta}_\phi{^{(\infty)}}} (x)\} \;=\; 1$$ and by (\[D(E)\]) $$m(D(L) \;|\; C_{{\delta}_\phi{^{(n)}}} (x)\} \;=\;
p{^{(0)}}( f^{-1}_{x_1x_2\; \ldots \;x_n} (L)).$$ Hence we can choose $n$ and $\; (x_1x_2\; \ldots \;x_{n}) \in G{^{(n)}}\;$ such that $$m(B(x_1x_2\; \dots \;x_n) \cap \check{L} ) > 0$$ and then $$p{^{(0)}} (f^{-1}_{x_1x_2\; \dots \;x_{n}} (L)) \;>\; 1-c$$ The choice of $c$ provides $\; f^{-1}_{x_1x_2\; \ldots \;x_n}(E) \supseteq L \;$ and thus Part (ii) of Definition \[L set\] holds. Part (i) follows from the equalities $$\label{f l(x)}
f_{x_1x_2\; \dots \;x_{n}} l(T_G^nx) \;=\; l(x) \;\;,\;\; |l(x)| = d
\;\;,\;\; x \in X_G$$ by the definition of $l(x)$ . We have proved the inclusion $\; {{\mathcal L}}\subseteq {{\mathcal L}}({{\mathcal S}}(\phi)) \;$ , which implies that the semigroup ${{\mathcal S}}$ is $d$-contractive with $\; d = d(T_G , {\delta}_\phi) $
\[L=L(phi)\] $\; {{\mathcal L}}= {{\mathcal L}}({{\mathcal S}}(\phi)) \;$,
It was proved above that $\; {{\mathcal L}}\subseteq {{\mathcal L}}({{\mathcal S}}(\phi)) \;$. Take $ M \in {{\mathcal L}}({{\mathcal S}}(\phi)) $ and $ L \in {{\mathcal L}}$. Since also $ L \in {{\mathcal L}}({{\mathcal S}}(\phi))$, there exists $i_1i_2 \dots
i_{n} \in I^n $ such that $$\; f_{x_1x_2\;...\;x_{n}} (M) \;=\; L
\;=\; l(x) \;,\; x \in \check{L} \;$$ Then the equality (\[f l(x)\]) implies $\; M = l(T^{n}x) \;$ on a set of positive measure in $X$ and hence $\; M \in {{\mathcal L}}\;$. Thus $\; {{\mathcal L}}\supseteq {{\mathcal L}}({{\mathcal S}}(\phi)) \;$.
Note that the notion of $d$-contractive semigroup was introduced in , where an analog of Theorem \[d(phi)\] was also proved. Definition \[L set\] is a generalization of what is called “point collapsing” by M. Rosenblatt , in the case $ |U| < \infty $ and $ d=1 $ . The case , when $ |U| = \infty $ and $ d=1 $, was considered in [@KuMuTo].
Graph skew product representation. {#ss3.5}
-----------------------------------
From now on let $G$ be a $\rho$-uniform stochastic graph, which is irreducible and satisfies the positive recurrence condition.
\[phi bar\] Let $\; \phi \in {{\mathcal Hom}}(G,I) \;$ be a homomorphism of degree $\; d = d(\phi) \;$. Then there exists a commutative diagram $$\label{diag phi bar}
\xymatrix{ {{\bar H}}\ar[d]^{{{\bar \phi}}} \ar[r]^{{{\bar \psi}}} & G \ar[d]^{\phi} \\
H \ar[r]^{\psi} & I }$$ where the graph $ {{\bar H}}= {{\bar H}}_a $ is a graph skew product over $H$, generated by a function $\; a { : }H \ni h {\rightarrow}a(h) \in {{\mathcal A}_d}\;$, the homomorphism $ {{\bar \phi}}$ coincides with the natural projection $\pi_H$, and both the homomorphisms $\; {{\bar \psi}}\in {{\mathcal Hom}}({{\bar H}},G) \;$ and $\; \psi \in {{\mathcal Hom}}(H,I)\;$ are of degree 1. In particular, $\; ({{\bar \phi}},\psi) \in {{\mathcal Ext}^d(I,\rho)}\;$
We construct a commutative diagram $$\label{diag phi hat}
\xymatrix{
{{\hat H}}\ar[d]^{{{\hat \phi}}} \ar[r]^{{{\hat \psi}}} & G \ar[d]^{\phi} \\
H \ar[r]^{\psi} & I }$$ such that the homomorphism $\; {{\hat \phi}}\in {{\mathcal Hom}}({{\hat H}},H) \;$ is a $d$-extension (See Definition \[d-uniform\]) and both homomorphisms $\; {{\hat \psi}}\in {{\mathcal Hom}}({{\hat H}},G) \;$ and $\; \psi \in {{\mathcal Hom}}(H,I) \;$ are of degree 1 .
We shall use the persistent $d$-sets $ {{\mathcal L}}= {{\mathcal L}}({{\mathcal S}}(\phi)) $ of the semigroup $ {{\mathcal S}}(\phi) $, described in Theorem \[d(phi)\] (Section\[ss3.4\]). Since $ {{\mathcal L}}$ is finite or countable we can enumerate the set by an alphabet $J$, setting $ {{\mathcal L}}= \{ L_j \;,\; j \in J \} $.
Recall that the semigroup $ {{\mathcal S}}(\phi) $ is $d$-contractive with $ d = d({{\mathcal S}}(\phi)) $ (Theorem \[d(phi)\]). For any pair $\; i \in I \;,\; j \in J \;$ we have $\; |f_i(L_j)|
= |L_j| =d \;$ and the restrictions $\; f_i {|}_{L_j} \;$ is a bijection of $\; L_j \;$ onto $\;f_i(L_j) \;$, whence, $\;f_i(L_j)
\in {{\mathcal L}}\;$ for all $i$ and $j$. For any $\;i \in I \;$ denote by $f_i^J$ the map $\; J {\rightarrow}J \;$, which is defined by $\; f_i^J j
= j' \;$, where $\; f_i(L_j) = L_{j'} \;$.
To construct Diagram \[diag phi hat\] define first $\; \psi { : }H {\rightarrow}I \;$ with $ \; H := I \times J \;$,$\; H{^{(0)}} := J \;$, where $$s(i,j) = j \;\;,\;\; t(i,j) = f_i^J j \;\;,\;\; p(i,j) = \rho(i)
\;\;,\;\; \psi (i,j) = i \;.$$ Next set $${{\hat H}}{^{(0)}} \;:=\;
\{ (j,u) \in J \times G{^{(0)}} \;{ : }\; j \in J \;,\; u \in L_j \}
\;\;,\;\; {{\hat H}}\;:=\; I \times {{\hat H}}{^{(0)}}$$ with $$s(i,j,u) = (j,u) \;\;,\;\; t(i,j,u) = (f_i^J j , f_i u) \;\;,\;\;
p(i,j,u) = \rho(i) \;.$$ Finally, we define the maps ${{\hat \psi}}$ and ${{\hat \phi}}$ by $${{\hat \phi}}{ : }{{\hat H}}\ni (i,j,u) {\rightarrow}(i,j) \in H \;\;,\;\;
{{\hat \psi}}{ : }{{\hat H}}\ni (i,j,u) {\rightarrow}g_{i,u} \in G \;.$$ where $g_{i,u}$ is uniquely determined by the conditions $s(g) =
u$ and $\phi(g) = i$.
It follows directly from this constructing that $H$ and ${{\hat H}}$ are stochastic graphs, and that ${{\hat \phi}}$ , ${{\hat \psi}}$ and $\psi$ are homomorphisms, and that Diagram \[diag phi hat\] commutes. Point out only that ${{\hat \phi}}$ is a $d$-extension, since $ |L_j| = d $ for all $j$ and hence ${{\hat \phi}}$ is of degree $d$. This implies that $\hat{\psi}$ and $\psi$ are of degree 1 , since $\phi$ is of degree $d$.
It remains to apply Proposition \[d-unif=GSP\] to the homomorphism $\; {{\hat \phi}}{ : }{{\hat H}}{\rightarrow}H \;$.
Next we construct d-extensions with a minimal possible $d$. Let $G$ as above be a $\rho$-uniform stochastic graph. Recall that $G{^{(n)}}$ denotes the set of all $n$-paths in $G$, see (\[G(n)\]). We shall consider $G{^{(n)}}$ as a stochastic graph with the set of vertices $G^{(n-1)}$, where for any $\; g{^{(n)}}=
g_1\;g_2\; \ldots \;g_n \;$ $$s(g{^{(n)}}) \;=\; g_2\;g_3\; \ldots \;g_n \;\;,\;\;
t(g{^{(n)}}) \;=\; g_1\;g_2\; \ldots \;g_{n-1}$$ and $\; p(g{^{(n)}}) \;=\; p(g_1) p(g_2) \ldots p(g_n) \;$. If $G$ is $\rho$-uniform, the “$n$-stringing” graph $G{^{(n)}}$ is also $\rho$-uniform. The natural projection $$\pi{^{(n)}}{ : }G{^{(n)}}\ni g{^{(n)}}=(g_1\;g_2\;
\ldots \;g_n) {\rightarrow}g_1 \in G$$ is a homomorphism and $\; \phi \circ \pi{^{(n)}}\in {{\mathcal Hom}}(G{^{(n)}},I) \;$ for any $\; \phi \in {{\mathcal Hom}}(G,I) \;$. However, if $(I,\rho)$ has congruent edges there exist $\; \phi_1 \in {{\mathcal Hom}}(G{^{(n)}},I) \;$, which are not of the above form $\; \phi_1 = \phi \circ \pi{^{(n)}}\;$. It is an obvious fact, that $d(\pi{^{(n)}}) = 1$, i.e. $\;
\Phi_{\pi{^{(n)}}} : X_{G{^{(n)}}} {\rightarrow}X_G$ is an isomorphism. We use the index $d(T,{\delta})$ and the minimal index $d(T)$, which were defined by Definition \[def d(T)\].
\[phi bar d(T)\] Let $G$ be a $\rho$-uniform stochastic graph, which is irreducible and satisfies the positive recurrence condition. Then there exist an integer $n \in {{\mathbb{N}}}$, a homomorphism $\; \phi
\in {{\mathcal Hom}}(G{^{(n)}},I) \;$ and a commutative diagram $$\label{diag bar H Gn}
\xymatrix{
{{\bar H}}\ar[d]^{{{\bar \phi}}} \ar[r]^{{{\bar \psi}}} & G{^{(n)}}\ar[d]^{\phi} \\
H \ar[r]^{\psi} & I }$$ such that
1. The graph ${{\bar H}}= {{\bar H}}_a $ is a skew product over a graph $H$, generated by a function $\; a { : }H \ni h {\rightarrow}a(h) \in {{\mathcal A}_d}\;\;,\;\;d \in {{\mathbb{N}}}\;$, and the homomorphism $ {{\bar \phi}}$ coincides with the natural projection $\pi_H$ of ${{\bar H}}$ onto $H$,
2. $\; d = d(\phi) = d(T_G) \;$,
3. The homomorphisms $\; {{\bar \psi}}\in {{\mathcal Hom}}({{\bar H}},G) \;$ and $\; \psi \in {{\mathcal Hom}}(H,I) \;$ are of degree 1 .
Let ${\zeta}= {\zeta}_G $ be the standard Markov generator of the shift $T_G$. It was proved in that there exist $ n \in {{\mathbb{N}}}$ and $ {\delta}\in {\Delta}_\rho(T_G) $ such that $${\delta}\leq {\zeta}{^{(n)}}:= \bigvee_{k=1}^{n} T^{-k+1} {\zeta}\;\;,\;\; d(T,{\delta}) = d(T) \;.$$ Take ${\zeta}{^{(n)}}$ and $G{^{(n)}}$ instead ${\zeta}$ and $G$ in Proposition \[d(T,zeta)\]. Then we obtain $\; \phi \in {{\mathcal Hom}}(G{^{(n)}},I) \;$ with $\; {\delta}= {\delta}_\phi \;$ and thus, by using Corollary \[phi bar\], we complete the proof.
Homomorphisms and finite extensions. {#s4}
====================================
Homomorphisms of degree 1 {#ss4.1}
-------------------------
Let $H$ be a $\rho$-uniform graph and consider a homomorphism $\; \psi { : }H {\rightarrow}I \;$. Suppose that $\psi$ is of degree 1. By Theorem \[d(phi)\] the semigroup ${{\mathcal S}}(\psi)$, generated by $\; f_i = f_i^\psi \;,\; i \in I \;$, is $1$-contractive and all its persistent sets are singletons. Using $\psi$ we can identify the graph $H$ with $I \times J$, where $\; J = H{^{(0)}} \;$ and for any $ h = (i,j) \in H $ $$\psi (h)=i \;,\; s(h)=j \;\;,\;\; t(h)= f_i j
\;\;,\;\; p(h)=\rho (i)$$ Since $d(\psi)=1$ the partition ${\delta}_\psi$ is a one-sided Bernoulli generator for the Markov shift $T_H$. The factor map $\; \Phi_{{\delta}_\psi} \;{ : }\; X_H {\rightarrow}X_\rho \;$ is an isomorphism, $\; \Phi_{{\delta}_\psi} \;\circ\; T_H = T_{\rho}
\;\circ\; \Phi_{{\delta}_\psi}$ and we can consider the Markov partitions $\; {\zeta}_\rho := \Phi_{{\delta}_\psi}({\zeta}_H) \;$ and $\;
{\zeta}{^{(0)}}_\rho := \Phi_{{\delta}_\psi}({\zeta}_H{^{(0)}}) \;$ for $T_\rho$ on $X_\rho$, which correspond to the Markov partitions ${\zeta}_H$ and ${\zeta}_H{^{(0)}}$ for $T_H$ on $X_H$. The partition ${\delta}_\rho = \Phi_{{\delta}_\psi}({\delta}_H) $ coincides with the standard Bernoulli generator of the Bernoulli shift $T_\rho$.
Thus we have, with the notations from Section \[ss3.3\], $${\delta}_\rho = \{ B_\rho (i) \}_{i \in I} \;\;,\;\;
{\zeta}_\rho = \{ D_\rho (i,j) \}_{ (i,j) \in I\times J} \;\;,\;\;
{\zeta}{^{(0)}}_\rho = \{ D_\rho (j) \}_{ j \in J}$$ where $$D_\rho (i,f_i(j)) = B_\rho (i) \cap T_\rho^{-1} D_\rho (j)
\;\;,\;\; (i,j) \in I\times J$$ Hence the homomorphism $\psi$ is determined by the partitions $\;
{\delta}_\rho \;,\; {\zeta}_\rho \;,\; {\zeta}{^{(0)}}_\rho \;$ uniquely up to equivalence (see Definition \[hom equi\]).
Our aim now is to construct a [**common extension of degree 1**]{} for two homomorphisms of degree 1.
\[degree 1\] Let $\; \psi_1 { : }H_1 {\rightarrow}I \;,\; \psi_2 { : }H_2 {\rightarrow}I
\;$ be two homomorphisms of $\rho$-uniform graphs $H_1$ and $H'_2$ onto the Bernoulli graph $ (I,\rho) $ and suppose that $ \psi_1 $ and $ \psi_2 $ are of degree 1. Then there exist a $\rho$-uniform graph $H$ and homomorphisms $\psi$, $\chi_1$ and $\chi_2 $ of degree 1, for which the following diagram commutes: $$\label{diag degree 1}
\xymatrix{
& H_2 \ar[rr]^{\psi_2} & & I \\
H \ar[ur]^{\chi_2} \ar[urrr]^{\psi} \ar[rr]_{\chi_1} & &
H_1 \ar[ur]_{\psi_1} & }$$
The homomorphism $\psi$ will be called a [**common extension**]{} of $\psi_1$ and $\psi_2$ of degree $1$. Denote by $ ({\zeta}_1 ,
{\zeta}{^{(0)}}_1) $ and $ ({\zeta}_2 , {\zeta}{^{(0)}}_2) $ the pairs of Markov partitions of the space $X_\rho$, which correspond to the homomorphisms $\psi_1$ and $\psi_2$. Here we omit the subscript “$\rho$” and mark the partitions and their elements by subscripts “$1$” and “$2$”, respectively.
We have to construct the desired $H$ and $\; \psi { : }H {\rightarrow}I
\;$ by means of the partitions $${\zeta}:= {\zeta}_1 \vee {\zeta}_2 \;\;,\;\;
{\zeta}{^{(0)}} := {\zeta}_1^{(0)} \vee {\zeta}_2^{(0)} \;.$$ By the identification $H_1= I \times J_1$ and $H_2= I \times J_2$, we have $${\zeta}_1 = \{ D_1 (i,j_1) \}_{ (i,j_1) \in I \times J_1} \;\;,\;\;
{\zeta}_1^{(0)} = \{ D_1 (j_1) \}_{ j_1 \in J_1} \;\;,$$ $${\zeta}_2 = \{ D_2 (i,j_2) \}_{ (i,j_2) \in I \times J_2} \;\;,\;\;
{\zeta}_2^{(0)} = \{ D_2 (j_2) \}_{ j_2 \in J_2} \;\;,$$ and then the partition $ {\zeta}{^{(0)}} $ consists of all elements $$D(j) = D_1 (j_1) \cap D_2 (j_2) \;\;,\;\; j=(j_1,j_2) \in J \;.$$ where the set $J$ is defined by $$\label{eq J}
J := \{ j=(j_1,j_2) \;{ : }\; p{^{(0)}} (j) :=
m_\rho ( D_1 (j_1) \cap D_2(j_2)) >0 \} \subset J_1 \times J_2 \;.$$ For any $i \in I$ and $ j=(j_1,j_2) \in J $ we set $ f_i j := (f_{1,i} j_1 ,f_{1,i} j_2 )$. Then $$\begin{split}
D(f_i j) :=
& D_1 (f_{1,i} j_1 ) \cap D _2(f_{2,i} j_2)
\supseteq D_1(i,j_1) \cap D_2(i,j_2) = \\
& B(i) \cap T_\rho^{-1} (D_1 ( j_1) \cap D_2( j_2)) =
B(i) \cap T_\rho^{-1} D(j)
\end{split}$$ Since $ {\delta}$ and $ T_\rho^{-1} {\varepsilon}$ are independent, this implies $$p{^{(0)}} (f_i j) = m_\rho ( D(f_i j)) \geq
m_\rho(B(i) \cap T_\rho^{-1} D(j)) = \rho(i) p{^{(0)}} (j) \;.$$ Hence $f_i j \in J$ for all $j \in J$ and $ i \in I $ .
Thus we are able to define a stochastic graph $H := I \times J $ with $H{^{(0)}} := J$ such that for any $j \in H{^{(0)}} $ and $h=(i,j) \in H $ $$s(h) := j \;,\; t(h)
:= f_i (j) \;,\; p(h) := \rho (i) \;,\; \psi (h) := i \;.$$ The construction provides that $H$ is a $\rho$-uniform graph , $p{^{(0)}}$ is a stationary probability on $H{^{(0)}} $ and $\psi { : }H
{\rightarrow}I $ is a homomorphism of index $1$. Moreover, if we set $$\chi_1(h) := (i,j_1) \;,\; \chi_2(h) := (i,j_2) \;\;,\;\;
h=(i,j_1,j_2) \in H= I \times J_1 \times J_2 \;,$$ then $\chi_1 { : }H {\rightarrow}H_1$ and $\chi_2 : H {\rightarrow}H_2$ are homomorphisms and Diagram \[diag degree 1\] commutes.
We shall use also the following sharpening of the previous theorem, which can be proved in a similar way.
\[degree 1 sharp\] Let $${\kappa}_1 { : }H_1 {\rightarrow}H_0 \;\;,\;\; {\kappa}_2 { : }H_2 {\rightarrow}H_0
\;\;,\;\; \psi_0 { : }H_0 {\rightarrow}I$$ be homomorphisms of $\rho$-uniform graphs $H_1 \;,\; H_2$ and $H_0$ and suppose they are of degree 1. Then there exist a $\rho$-uniform graph $H$ and homomorphisms $\chi$, $\chi_1$ and $\chi_2 $ of degree 1, for which the following diagram commutes $$\label{diag degree 1 sharp}
\xymatrix{
& H_2 \ar[rr]^{{\kappa}_2} & & H_0 \ar[rr]^{\psi_0} & & I \\
H \ar[ur]^{\chi_2} \ar[urrr]^{\chi} \ar[rr]_{\chi_1} & &
H_1 \ar[ur]_{{\kappa}_1} & & & }$$
Note that this theorem holds without adding of homomorphism $\psi_0$ i.e. for graphs, which are not necessary $\rho$-uniform, but we do not use the fact in this paper.
Extensions of Bernoulli graphs. {#ss4.2}
--------------------------------
Consider a very special case of the graph skew product construction ${{\bar H}}_a$ (see Example \[GSP\]), when the graph $H$ is the standard Bernoulli graph $(I,\rho)$. Let $ d \in {{\mathbb{N}}}$ and let $\; a { : }I {\rightarrow}{{\mathcal A}_d}\;$ be a function on $I$ with the values $\; a(i) , i \in I , \;$ in the group ${{\mathcal A}_d}$ of all permutations of $\; Y_d = \{1,2, \ldots ,d \} \;$. According to the general GSP-construction we have $\; {{\bar I}}_a = I
\times Y_d \;$ , $\; {{{\bar I}}_a}{^{(0)}} = Y_d \;$ and $\; \pi { : }{{\bar I}}_a {\rightarrow}I \;$, where for any $ {{\bar h}}= (i,y) \in {{\bar I}}_a $ $$s({{\bar h}}) = y \;,\; t({{\bar h}}) = a(i)y \;,\; \pi ({{\bar h}}) = i \;,\;
p({{\bar h}}) = \rho(i) \;,\, p{^{(0)}} (y) = d^{-1} \;.$$ The stochastic graph ${{\bar I}}_a$ is $\rho$-uniform and it is irreducible iff the group ${\Gamma}(a) $, generated by $\; \{ a(i) , i
\in I \} \subseteq {{\mathcal A}_d}\;$, is transitive on $Y_d$.
As it was noted in Section \[ss3.2\] (see Remark \[re GSP shift\]) the Markov shift $T_{{{\bar I}}_a} $ is isomorphic to the skew product ${{\bar T}}_{\rho,a}$ , which acts on $X_\rho \times Y_d$ by $$\label{bT rho a}
{{\bar T}}_{\rho,a} (x,y) = (T_\rho x , {a(x_1)}^{-1} y) \;\;,\;\;
x ={\{ x_n \}}_{n=1}^{\infty} \in X_\rho \;,\; y \in Y_d \;.$$
\[th rho cohom\] Let $\; \pi_k \;{ : }\; {{\bar I}}_{a_k} {\rightarrow}I \;,\; k = 1,2, \;$ be two $d$-extensions of the Bernoulli graph $(I,\rho)$, generated by functions $ a_k { : }I {\rightarrow}{{\mathcal A}_d}$, respectively. Let the functions $\; A_k { : }X_\rho {\rightarrow}{{\mathcal A}_d}\;,\; k=1,2, \;$ are defined by $$\label{Am(x)}
A_k(x) := {a_k(x_1)}^{-1} \;\;,\;\; x
={\{x_n\}}_{n=1}^{\infty} \in X_\rho \;.$$ If there exists a measurable function $ W { : }X_\rho {\rightarrow}{{\mathcal A}_d}$ such that $$\label{eq rho cohom}
A_2(x) \cdot W(x) = W(T_\rho x) \cdot A_1(x) \;\;,\;\; x \in X_\rho$$ then $W(x)$ does not depend on $x$ , i.e. $W(x) = w_0 \in {{\mathcal A}_d}$ a.e. on $X_\rho$. Thus $A_1$ and $A_2$ are cohomologous with respect to $T_\rho$ iff $a_1$ and $a_2$ are conjugate in ${{\mathcal A}_d}$, i.e. $\; a_2(i) \cdot w_0 = w_0 \cdot a_1(i) \;,\; i \in I \;$.
Note that the last equality means the equivalence of $a_1$ and $a_2$ in the sense of Definition \[cohom\] , since $I{^{(0)}} = \{ o
\}$.
To proof the theorem we need the following simple lemma.
\[le ext Ber\] Let ${\Gamma}$ be a finite group with the identity element $e$. For any $ b { : }I {\rightarrow}{\Gamma}$ denote $$\label{be(x)}
B{^{(n)}}(x) \;:=\; b(x_1) \cdot b(x_2) \cdot \ldots \cdot b(x_n)
\;\;,\;\; x ={\{x_n\}}_{n=1}^{\infty} \in X_\rho$$ and $$\label{gwb(x)}
{\omega}_b(x) \;:=\; min \{ n \in {{\mathbb{N}}}\;{ : }\; B{^{(n)}}(x) = e \}
\;\;,\;\; x \in X_\rho \;.$$ Then the transformation $T_\rho^{{\omega}_b}$, defined by $$X_\rho \ni x {\rightarrow}T_\rho^{{\omega}_b} x := {T_\rho}^{{\omega}_b(x)}x \in
X_\rho \;,$$ is an ergodic endomorphism of $X_\rho$, which is in fact a one-sided Bernoulli shift.
Consider the ${\Gamma}$-extension of the graph $(I,\rho)$ generated by $b$.
Namely, set $\; {{\tilde{I}}}_b := \; I \times {\Gamma}\;$ and $\; {{{\tilde{I}}}_b}{^{(0)}} := {\Gamma}\;$ with $$s({\tilde{i}}) = (s(i) , {\gamma}) \;,\; t({\tilde{i}}) = (t(i) , b(i) \cdot {\gamma})
\;,\; p({\tilde{i}}) = \rho(i) \;,\; p{^{(0)}} ({\gamma}) = {|{\Gamma}|}^{-1}$$ The skew product endomorphism $ {{\tilde{T}}}_{\rho,b} $ corresponding to the stochastic graph $ {{\tilde{I}}}_b$, acts on the space $X_\rho \times {\Gamma}$ by $${{\tilde{T}}}_{\rho,b} (x , {\gamma}) = (T_\rho x \;,\; B(x) \cdot {\gamma})
\;\;,\;\; x ={\{ x_n \}}_{n=1}^{\infty} \in X_\rho
\;,\; {\gamma}\in {\Gamma}\;.$$ where $\; B(x) := {b(x_1)}^{-1} \;$. The skew product $ {{\tilde{T}}}_{\rho,b} $ can be identified with the Markov shift $T_{{{\tilde{I}}}_b}$ (see Remark \[re GSP shift\] ). Under this identification the partition ${\zeta}^{(0)}_{{{\tilde{I}}}_b}$ coincides with the partition $${\zeta}{^{(0)}} = \nu_{X_\rho} \times {\varepsilon}_{{\Gamma}} =
\{{\tilde{E}}({\gamma}) \}_{{\gamma}\in {\Gamma}} \;,$$ where $${\tilde{E}}({\gamma}) :=
X_\rho \times \{{\gamma}\} \subseteq X_\rho \times {\Gamma}\;\;.\;\; {\gamma}\in {\Gamma}\;.$$ For any ${\gamma}\in {\Gamma}$ consider the endomorphism $ ({{\tilde{T}}}_{\rho,b})_{{\tilde{E}}({\gamma})} $ induced by $ {{\tilde{T}}}_{\rho,b} $ on the set $ {\tilde{E}}({\gamma}) $. Let $${\varphi}_{{\tilde{E}}({\gamma})} \;{ : }\; {\tilde{E}}({\gamma}) \ni (x,{\gamma}) {\rightarrow}{\varphi}_{{\tilde{E}}({\gamma})}(x,{\gamma}) \in {{\mathbb{N}}}$$ be the corresponding return functions (\[ret fun\]).
Since we use the left shifts on ${\Gamma}$ in the definition of the skew product $ {\tilde{T}}_b $ and they commute with the right shifts, we have $${\varphi}_{{\tilde{E}}({\gamma})}(x,{\gamma}) =
{\varphi}_{{\tilde{E}}({\gamma}\cdot {\beta})}(x,{\gamma}\cdot {\beta})
\;,\; {\gamma},{\beta}\in {\Gamma}\;,\; x \in X_\rho \;.$$ Hence with (\[gwb(x)\]) and (\[be(x)\]) we have $${\omega}^b(x) = {\varphi}_{{\tilde{E}}({\gamma})}(x,{\gamma}) \;\;,\;\;$$ and $${{{\tilde{T}}}_b}^{{\omega}^b(x)}(x,{\gamma}) = (T^{{\omega}^b(x)}x,{\gamma})
\;.\; {\gamma}\in {\Gamma}\;,\; x \in X_\rho \;,$$ Thus $T^{{\omega}^b}$ is isomorphic to the endomorphisms $(T_{{{\tilde{I}}}_b})_{D({\gamma})}$ induced by the Markov shift $T_{{{\tilde{I}}}_b}$ on elements $D({\gamma})$ of the partition $ {{\zeta}^{(0)}_{{{\tilde{I}}}_b}}$. So that $T^{{\omega}^b}$ is a Bernoulli shift by Proposition \[Tu\].
[**Proof of Theorem \[th rho cohom\]** ]{} For given two functions $a_1$ an $a_2$ put $$b { : }I \ni i {\rightarrow}b(i) :=
(a_1(i),a_2(i)) \in {\Gamma}:= {{\mathcal A}_d}\times {{\mathcal A}_d}\;.$$ and denote for $k=1,2$ $$A_k^{{\omega}_b}(x) :=
A_k(T_\rho^{{\omega}_b(x)-1}x) \cdot \ldots \cdot A_k(T_\rho x)
\cdot A_k(x) \;\;,\;\; k=1,2$$ with $A_1$ and $A_2$ defined by (\[Am(x)\]). Then by definition of $b$ and ${\omega}_b$ we have $$A_2^{{\omega}_b}(x) := A_1^{{\omega}_b}(x) = e \;\;,\;\; x \in X_\rho \;,$$ where $e$ is the identity of ${{\mathcal A}_d}$. The equality (\[eq rho cohom\]) implies $$A_2^{{\omega}_b}(x) \cdot W(x) = W(T_\rho^{{\omega}_b} x) \cdot A_1^{{\omega}_b}(x)$$ and then $\; W(T_\rho^{{\omega}_b} (x) = W(x) \;$ a.e. on $X_\rho$. By Lemma \[le ext Ber\] $\; T_\rho^{{\omega}_b} $ is ergodic and hence $W(x)$ is constant a.e. on $X_\rho$.
Equivalent extensions. {#ss4.3}
----------------------
Let $d \in {{\mathbb{N}}}$, and $H$ be an irreducible positively recurrent stochastic graph. Given a function $\; a { : }H \ni h {\rightarrow}a(h)
\in {{\mathcal A}_d}\;$ consider the graph skew product $d$-extension ${{\bar H}}_a$ of $H$ generated by the function $a$ (See Example \[GSP\]). Recall that the skew product endomorphism $ {{\bar T}}_{H,a} $, corresponding to $ {{\bar H}}_a $, acts on the space $X_H \times Y_d $ by $${{\bar T}}_{H,a} (x,y) = (T_H x , A(x) y)
\;\;,\;\; x ={\{ x_n \}}_{n=1}^{\infty} \in X_H \;,\; y \in Y_d \;.$$ where $\; A(x) := {a(x_1)}^{-1} \;$. We shall use Definition \[cohom\]
\[Equ ext\] Let $\; \pi_k { : }{{\bar H}}_{a_k} {\rightarrow}H \;,\; k = 1,2 \;$, be two $d$-extensions of $H$ generated by functions $a_1$ and $a_2$, respectively, and let the functions $\; A_k { : }X_H {\rightarrow}{{\mathcal A}_d}\;,\; k=1,2 \;$ are defined by $$\label{Ak = ak -1}
A_k(x) := a_k(x_1)^{-1}
\;\;,\;\; x ={\{ x_n \}}_{n=1}^{\infty} \in X_H \;.$$ Then the following two conditions are equivalent
1. $A_1$ and $A_2$ cohomologous with respect to $T_H$, i.e. there exists a measurable map $\; W { : }X_H {\rightarrow}{{\mathcal A}_d}\;$ such that $$\label{A coho}
A_2(x) \cdot W(x) = W(Tx) \cdot A_1(x) \;\;,\;\; x \in X_H \;,$$
2. $a_1$ and $a_2$ cohomologous with respect to $H$, i.e. there exists a map $\; w { : }H{^{(0)}} {\rightarrow}{{\mathcal A}_d}\;$ such that $$\label{a coho}
a_2(h) \cdot w(s(h)) = w(t(h)) \cdot a_1(h) \;,\; h \in H$$
It is obvious that (\[a coho\]) implies (\[A coho\]) with $$\label{W(x)}
W(x) = w(t(x_1)) \;\;,\;\; x ={\{ x_n \}}_{n=1}^{\infty} \in X_H$$ That is (ii) implies (i).
To prove the converse, suppose that (\[A coho\]) holds with a suitable measurable function $\; W { : }X_H {\rightarrow}{{\mathcal A}_d}\;$.
We have to show that the function $W(x)$ necessarily has the form (\[W(x)\]), i.e. $W(x)$ is constant on each element $D(u) =
Z_1^{-1}(_uH)$ of the partition $\; {\zeta}_H{^{(0)}} = \{ D(u) \;,\; u
\in H{^{(0)}} \} \;$.
To this purpose we shall use induced endomorphisms, which are defined as follows.
Fix an atom $D(u)$ of the partition $ {\zeta}_H{^{(0)}} $ and consider the endomorphism $ T_u := (T_H)_{D(u)} $, induced by the shift $T_H$ on $D(u)$, see Section \[ss3.1\]. In accordance with the general definition (\[ret fun\]), the return function $${\varphi}_u(x)={\varphi}_{D(u)}(x) := min \{ n \geq 1 { : }T_H^{n}x \in D(u) \}
\;\;,\;\; x \in D(u)$$ induces $T_u$ by $\; T_ux = T_H^{{\varphi}_u(x)}x \;$. By Proposition \[Tu\] the induced endomorphism $T_u$ is isomorphic to the Bernoulli shift $T_{\rho_u}$, where $\; I_u= \bigcup_{n=1}^\infty
I_{u,n} \;$, and $\; \rho_u = \{\rho_u(i)\}_{i \in I_u} \;$ are defined by (\[I u n\]) and (\[rho u\]). That is, $I_{u,n}$ consists of all $\; h_1h_2 \ldots h_n \in H{^{(n)}}\;$ such that $$t(h_1) = s(h_n) = u \;\;,\;\; s(h_m) = t(h_{m+1}) \neq u \;,\;
m = 1,2, \ldots ,n-1$$ and $$\rho_u(i) = p(h_1) p(h_2) \ldots p(h_n) \;\;,\;\;
i = h_1h_2 \ldots h_n \in I_{u,n} \;.\; n \in {{\mathbb{N}}}\;.$$ For any $u \in H{^{(0)}}$ and $k=1,2$ set $$A_k^{{\varphi}_u}(x) := A_k(T^{{\varphi}_u(x)-1}x)
\cdot \ldots \cdot A_k(Tx) \cdot A_k(x)
\;\;.\;\; x \in D(u)$$ and $$a_k^u (i) := a_k(h_1) \cdot a_k(h_2) \cdot \ldots \cdot a_k(h_n)
\;\;,\;\; i = h_1h_2 \ldots h_n \in I_{u,n} \;.$$ Then $$A_k^{{\varphi}_u}(x) = {a_k^u(i)}^{-1} \;\;,\;\;
x \in B_u(i) \subseteq D(u) \;.\; i \in I_u \;,$$ where $$B_u(i) := \{ x ={\{ x_n \}}_{n=1}^{\infty} \in D(u) \;{ : }\;
(x_1x_2 \ldots x_n) = i \in I_{u,n} \;.$$ Then the equality (\[A coho\]) implies $$\label{Au cohom}
A_2^{{\varphi}_u}(x) \cdot W(x) =W(T_u x) \cdot A_1^{{\varphi}_u}(x)
\;\;.\;\; x \in D(u) \;,$$ i.e. $A_1^{{\varphi}_u}$ and $A_2^{{\varphi}_u}$ are cohomologous on $D(u)$ with respect to $ T_u = T_H^{{\varphi}_u} $.
Since for any fix $u \in D(u)$ the partition $\; {\zeta}_u = \{ B_u(i)
\}_{i \in I_u} \;$ is a one-sided Bernoulli generator for $T_u$, we may apply Theorem \[th rho cohom\] with the Bernoulli shift $
T_u = T_{\rho_u}$ and with the functions $a_k^u \;,\; k=1,2$. Therefore, it follows from (\[Au cohom\]) that there exists $\;
w(u) \in {{\mathcal A}_d}\;$ such that $\; W(x)=w(u) \;$ a.e. on $ D(u) $.
For every $u$ we have now an element $w(u)$ such that $ W(x) =
w(u) = w(t(x_1))$ for a.e. $ x \in D(u) $. Hence $ W(x) $ is of the form (\[W(x)\]), $\; W(T_H x) =
w(s(x_1)) \;$. Thus (\[A coho\]) implies (\[a coho\]).
As a consequence we obtain
\[Equ ext 1\] Let $\; \pi_k { : }{{\bar H}}_{a_k} {\rightarrow}H \;,\; k = 1,2 \;$, be two $d$-extensions of $H$ generated by functions $a_1$ and $a_2$, respectively. Let also $\psi { : }H {\rightarrow}I $ be an homomorphism of degree $1$. Suppose $ d = d({{\bar T}}_{H,a_1}) =
d({{\bar T}}_{H,a_2}) $. Then the endomorphisms ${{\bar T}}_{H,a_1}$ and ${{\bar T}}_{H,a_2}$ are isomorphic iff $a_1$ and $a_2$ cohomologous with respect to $H$.
Since $d(\psi) = 1$ the factor map $\Psi := \Phi_\psi
{ : }X_H {\rightarrow}X_I$ is an isomorphism. Consider two skew products over the Bernoulli shift $T_\rho$ $${{\bar T}}_k (x,y) = ( T_\rho x , B_k(x) y)
\;\;,\;\; (x,y) \in X_\rho \times Y_d
\;\;,\;\; k = 1,2$$ where $ B_k(x) := A_k(\Psi^{-1}x) $ and $A_k$ induced by $a_k$ as above (\[Ak = ak -1\]). Each of the shifts $T_{{{\bar H}}_{a_k}}$ is a simple $\rho$-uniform endomorphism by Theorem \[simp mar\]. The skew products ${{\bar T}}_{H,a_k}$ as well as the shifts $T_{{{\bar H}}_{a_k}}$, are isomorphic to ${{\bar T}}_k$. They are $\rho$-uniform endomorphisms and $d = d({{\bar T}}_1) = d({{\bar T}}_2).$ By Theorem \[simple\] ${{\bar T}}_1$ and ${{\bar T}}_2$ are isomorphic iff the functions $B_1$ and $B_2$ are cohomologous with respect to $T_\rho$. This means that the functions $A_1$ and $A_2$ are cohomologous with respect to $T_H$. Finally, by Theorem \[Equ ext\] the last condition holds iff $a_1$ and $a_2$ cohomologous with respect to $H$.
${{\mathbf{G}}{\mathbf{S}}{\mathbf{P}}}$-extensions and persistent ${\mathbf{d}}$-partitions. {#ss4.4}
----------------------------------------------------------------------------------------------
Let $H$ be a stochastic graph and $(I,\rho)$ a standard Bernoulli graph. In this section we study extensions of the form $$\label{pi psi}
(\pi,\psi) \;{ : }\; \xymatrix{ {{\bar H}}\ar[r]^{\pi} & H \ar[r]^{\psi} & I}$$ where the graph $H$ be an extension of the Bernoulli graph $(I,\rho)$ by a homomorphism $\psi$ of degree $ d(\psi) = 1 $ and $ {{\bar H}}= {{\bar H}}_a $ be a graph skew product $d$-extension of $H$ generated by a function $ a { : }H {\rightarrow}{{\mathcal A}_d}$ (See Example \[GSP\]). The diagrams of the above form (\[pi psi\]) will be referred to ${\boldsymbol{(}}{\boldsymbol{\pi}}, {\boldsymbol{\psi}}{\boldsymbol{)}}$[**-extensions**]{}. We assume that the graph $ {{\bar H}}$ is irreducible, i.e. the corresponding Markov shift $T_{{\bar H}}$ and skew product ${{\bar T}}_{H,a} $ are ergodic.
Fixing an extension (\[pi psi\]) and setting $\; J = H{^{(0)}} \;$, we identify $H$ with $ I \times J $ such that $$\psi (h) = i \;,\; s(h) = j \;,\; t(h) = f_i (j)
\;,\; p(h) = \rho(i)$$ for any $ h = (i,j) \in H = I \times J $. Here the maps $ f_i { : }J {\rightarrow}J $ are uniquely determined by $$f_ij = t(i,j) \;\;,\;\; (i,j) \in I \times J$$ and the semigroup $ {{\mathcal S}}(\psi) $, generated by $ \{ f_i \;,\; i
\in I \} $ is $1$-contractive, since $ d(\psi) = 1 \;$ (Theorem \[d(phi)\]).
The $d$-extension $ {{\bar H}}= {{\bar H}}_a $ is described now as follows: $$\label{bar H}
{{\bar H}}= I \times J \times Y_d \;\; , \;\;
{{\bar H}}{^{(0)}} = H{^{(0)}} \times Y_d = J \times Y_d \;,$$ where for any $ {{\bar h}}= (i,j,y) $ $$\label{stp bar H}
s({{\bar h}}) = (j,y) \;,\; t({{\bar h}}) = (f_ij , a(i,j)y) \;,\;
p({{\bar h}}) = \rho(i) \;,\; a(h) = a(i,j)$$ The homomorphisms $\psi$, $\pi$ and $\phi = \psi \circ \pi $ are defined by $$\pi ({{\bar h}}) = h = (i,j) \;\;,\;\; \pi{^{(0)}} (j,y) = j \;\;,\;\;
\phi ({{\bar h}}) = \psi (h) = i \;,$$ where $ d(\phi) = d(\pi) = d $ and the diagram $$\label{diag I J Yd}
\xymatrix{
I \times J \times Y_d \ar[d]^{\pi} \ar[dr]^{\phi} & \\
I \times J \ar[r]^{\psi} & I }$$ commutes. The semigroup ${{\mathcal S}}(\phi)$ can be described now as a $d$-extension ${{\bar {{\mathcal S}}}}= {{\bar {{\mathcal S}}}}(\pi,\psi)$ of the semigroup ${{\mathcal S}}(\psi)$.
Set $$\label{bfi (j,y)}
{{\bar f}}_i(j,y) := t(i,j,y) = (f_i j \;,\; a(i,j) y)
\;,\; (j,y) \in J \times Y_d \;,\; i \in I \;.$$ The maps $ {{\bar f}}_i $ act on $J \times Y_d $.
The semigroup ${{\bar {{\mathcal S}}}}$, generated by $\; {{\bar f}}_i \;,\; i \in I \;$, consists of all maps of the form: $$\{ {{\bar f}}_{i_1i_2 \ldots i_n} \;,\; i_1i_2 \ldots i_n \in I^n \;,\;
n \in {{\mathbb{N}}}\} \;,$$ where $${{\bar f}}_{i_1i_2 \ldots i_n}(j,y) = (f_{i_1i_2 \ldots i_n} j ,
a(i_1i_2 \ldots i_n,j) y)$$ and $$a(i_1i_2 \ldots i_n,j) := a(i_1,f_{i_2i_3 \ldots i_n} j )
\ldots a(i_{n-1},f_{i_n} j ) a(i_n,j) \;.$$ Note that $\; {{\mathcal S}}(\psi) \ni f {\rightarrow}{{\bar f}}\in {{\bar {{\mathcal S}}}}\;$ is an isomorphism between the semigroups.
\[S(phi) bar H\] The semigroup ${{\bar {{\mathcal S}}}}$ is $d$-contractive and its persistent $d$-sets are of the form $${{\mathcal L}}({{\bar {{\mathcal S}}}}) = \{ L_j , j \in J \} \;\;,\;\; L_j := \{ j \} \times Y_d \;.$$
By Theorem \[d(phi)\] the semigroup $ {{\bar {{\mathcal S}}}}$ is $d$-contractive and ${{\mathcal S}}(\psi)$ is $1$-contractive, since $ d(\phi) = d $ and $ d(\psi) = 1 $.
For any finite set $ F \subseteq J \times Y_d $ the set $ E :=
\pi{^{(0)}} (F) \subseteq J $ is also finite. Since the semigroup ${{\mathcal S}}(\psi)$ is $1$-contractive there exist $ i_1i_2 \ldots i_n \in
I^n $ and $j \in J$ such that $\; f_{i_1i_2 \ldots i_n}(E) = \{j\}
\;$ and hence $\; {{\bar f}}_{i_1i_2 \ldots i_n}(F) \subseteq L_j \;$. On the other hand $ d = |L_j| = |{{\bar f}}_i(L_j)| $ for all $i \in I ,
j \in J$. Thus the sets $L_j$ and only they are persistent sets for the semigroup ${{\bar {{\mathcal S}}}}$.
For every $E \subseteq J$ set $ {{\bar E}}:= {\pi{^{(0)}}}^{-1}E = E \times
Y_d $.
\[pers part\] Let the semigroup ${{\bar {{\mathcal S}}}}$ be as above.
1. A subset $ R $ of $ {{\bar H}}{^{(0)}} = J \times Y_d $ will be called [**transversal**]{} with respect to $\pi{^{(0)}}$ if $ \pi{^{(0)}} (R) = H{^{(0)}} = J $ and the restriction $ \pi{^{(0)}} {|}_R { : }R {\rightarrow}J $ is a bijection. A partition $\; r = \{ R_1,R_2, \ldots ,R_d \} \;$ will be called [**transversal**]{} with respect to $\pi{^{(0)}}$ if all the set $R_1,R_2, \ldots ,R_d$ are transversal.
2. A transversal partition $r$ will be called [ **persistent**]{} with respect to semigroup $ {{\bar {{\mathcal S}}}}$, if for every transversal partition $r_1$ and every finite subset $E \subseteq J$ there exists $ {{\bar f}}\in {{\bar {{\mathcal S}}}}$ such that $\; {{\bar f}}^{-1}r_1 \;{|}_{{\bar E}}= r \;{|}_{{\bar E}}\;.$
Denote by ${{\mathcal R}}$ the set of all transversal partitions and by ${{\mathcal R}}({{\bar {{\mathcal S}}}})$ the set of all persistent partitions for the semigroup ${{\bar {{\mathcal S}}}}$.
If a set $R$ is transversal then for any $i \in I$ the set ${{\bar f}}^{-1}R$ is transversal. Hence for any $r = \{ R_1,R_2, \ldots
,R_d \} \in {{\mathcal R}}$ and $ {{\bar f}}\in {{\bar {{\mathcal S}}}}$ we have $${{\bar f}}^{-1} r := \{ \; {{\bar f}}^{-1}R_1 , {{\bar f}}^{-1}R_2, \ldots ,
{{\bar f}}^{-1}R_d \; \} \in {{\mathcal R}}\;.$$ Further we shall use this action $${{\mathcal R}}\ni r \; {\rightarrow}\; {{\bar f}}^{-1} r \in {{\mathcal R}}\;\;,\;\; {{\bar f}}\in {{\bar {{\mathcal S}}}}$$ of the semigroup ${{\bar {{\mathcal S}}}}$ on ${{\mathcal R}}$. The following lemma shows that $ {{\mathcal R}}({{\bar {{\mathcal S}}}})$ is an attracting set for ${{\mathcal R}}$ with respect to the action in a natural sense.
\[gL(phi)\]
1. The set ${{\mathcal R}}({{\bar {{\mathcal S}}}})$ is not empty.
2. ${{\bar f}}^{-1} {{\mathcal R}}({{\bar {{\mathcal S}}}}) \subseteq {{\mathcal R}}({{\bar {{\mathcal S}}}})$ for all ${{\bar f}}\in {{\bar {{\mathcal S}}}}$
3. ${{\mathcal R}}({{\bar {{\mathcal S}}}})$ is the least subset of ${{\mathcal R}}$ with the property (ii).
Consider a subset ${{\mathcal R}}_0 ({{\bar {{\mathcal S}}}})$ of ${{\mathcal R}}$ consisting of all $r \in {{\mathcal R}}$ having the following property:
1. For any finite subset $E \subseteq J $ there exists $\; f \in {{\mathcal S}}(\psi) $ such that $f(E)$ is a singleton, i.e. $|f(E)|=1$, and $ r \;{|}_{{\bar E}}= {{\bar f}}^{-1}{\varepsilon}\;{|}_{{\bar E}}\;$, where $ {{\bar E}}:= E \times Y_d $ and $ {\varepsilon}= {\varepsilon}_{J \times Y_d} $.
We show that ${{\mathcal R}}({{\bar {{\mathcal S}}}}) = {{\mathcal R}}_0({{\bar {{\mathcal S}}}}) \neq \emptyset$.
Take a sequence $\; E_n \nearrow J \;,\; |E_n| < \infty \;$. Since $d({{\mathcal S}}(\psi)) = 1 \;$ we can find a sequence $g_n \in {{\mathcal S}}(\psi)$ such that for all $n \in {{\mathbb{N}}}$ and $ f_n := g_n \cdot \ldots \cdot g_2 \cdot g_1$, the set $f_n (E_n)$ is single-point, i.e. $|f_n (E_n)|=1$. Using the decreasing sequence of partitions $${\varepsilon}\geq {{\bar f}}_1^{-1} {\varepsilon}\geq {{\bar f}}_2^{-1} {\varepsilon}\geq \; \ldots \;
\geq {{\bar f}}_n^{-1} {\varepsilon}\geq \ldots$$ set $\; r_0 := \bigwedge_{n=1}^{\infty} {{{\bar f}}_n}^{-1}{\varepsilon}\;$. Since $|f_n (E_n)|=1$ the restriction $r_0 \;{|}_{{{\bar E}}_n} $ consists of $d$ sets, whose projections on $J$ are $E_n$. Hence $
r_0 \in {{\mathcal R}}$ and $r_0 \;{|}_{{{\bar E}}_n} = {{\bar f}}_n^{-1}{\varepsilon}\;{|}_{{{\bar E}}_n} $. We see that $r_0 \in {{\mathcal R}}_0({{\bar {{\mathcal S}}}})$, i.e. ${{\mathcal R}}_0$ is not empty,
Let $r \in {{\mathcal R}}_0({{\bar {{\mathcal S}}}})$ and $r_1 \in {{\mathcal R}}$. For any finite subset $E \subseteq J$ there exists ${{\bar f}}\in {{\mathcal R}}({{\bar {{\mathcal S}}}})$ such that $ r
\;{|}_{{\bar E}}= {{\bar f}}^{-1}{\varepsilon}\;{|}_{{\bar E}}\;$. Then $${{\bar f}}^{-1}r_1 \;{|}_{{\bar E}}\leq {{\bar f}}^{-1}{\varepsilon}\;{|}_{{\bar E}}= r \;{|}_{{\bar E}}\;,$$ Since each of the partitions consists of $d$ elements, we have also ${{\bar f}}^{-1}r_1 \;{|}_{{\bar E}}= r \;{|}_{{\bar E}}$. Hence $r \in {{\mathcal R}}({{\bar {{\mathcal S}}}})$.
Conversely, let $r \in {{\mathcal R}}({{\bar {{\mathcal S}}}})$ and $E$ be a finite subset of $J$. There are exist ${{\bar f}}\in {{\bar {{\mathcal S}}}}$ and $r_1 \in {{\mathcal R}}$ such that $
r_1 \;{|}_{{\bar E}}= {{\bar f}}^{-1}{\varepsilon}\;{|}_{{\bar E}}\;$. On the other hand, since $r \in {{\mathcal R}}({{\bar {{\mathcal S}}}})$, we can choose ${{\bar f}}_1
\in {{\bar {{\mathcal S}}}}$ for which $ {{\bar f}}_1^{-1}r_1 \;{|}_{{\bar E}}= r \;{|}_{{\bar E}}$. Hence $$r \;{|}_{{\bar E}}= {{\bar f}}_1^{-1}r_1 = {{\bar f}}_1^{-1}{{\bar f}}^{-1}{\varepsilon}\;{|}_{{\bar E}}= ({{\bar f}}^{-1}{{\bar f}}_1)^{-1}{\varepsilon}\;{|}_{{\bar E}}.$$ We see that $r \in {{\mathcal R}}_0({{\bar {{\mathcal S}}}})$ and thus ${{\mathcal R}}({{\bar {{\mathcal S}}}}) = {{\mathcal R}}_0$ and Part $(i)$ follows.
Parts $(ii)$ and $(iii)$ follow in the same manner by the definition of ${{\mathcal R}}({{\bar {{\mathcal S}}}})$ and by the equality ${{\mathcal R}}({{\bar {{\mathcal S}}}}) =
{{\mathcal R}}_0({{\bar {{\mathcal S}}}})$.
Irreducible ${\mathbf{d}}$-extensions. {#ss4.5}
---------------------------------------
In this section we continue to study $(\pi,\psi)$-extensions of the form (\[pi psi\]) $$(\pi,\psi) \;{ : }\; \xymatrix{
{{\bar H}}\ar[r]^{\pi} & H \ar[r]^{\psi} & I } \;,$$ where the graph $H$ is an extension of the Bernoulli graph $(I,\rho)$ by a homomorphism $\psi$ of degree $ 1 $ and $ {{\bar H}}=
{{\bar H}}_a $ be a GSP $d$-extension of $H$, generated by a function $ a
{ : }H {\rightarrow}{{\mathcal A}_d}$.
Fix $d$ and $(I,\rho)$ and consider the set $\; {{\mathcal Ext}^d(I,\rho)}\;$ of all $(\pi,\psi)$-extensions of the form (\[pi psi\]). This set is equipped with a natural partial order and with an equivalence relation as follows
\[partial order\] Let $\; (\pi,\psi) { : }\xymatrix{ {{\bar H}}\ar[r]^{\pi} & H \ar[r]^{\psi} & I } \;$ and $\; (\pi_1,\psi_1) { : }\xymatrix{{{\bar H}}_1 \ar[r]^{\pi_1} & H_1 \ar[r]^{\psi_1} & I } \;$ be two $(\pi,\psi)$-extensions from ${{\mathcal Ext}^d(I,\rho)}$. Let $a$ and $a_1$ be the functions, which generate the extensions ${{\bar H}}$ and ${{\bar H}}_1$, respectively.
1. A homomorphism $\; {{\bar {\kappa}}}{ : }{{\bar H}}{\rightarrow}{{\bar H}}_1 \;$ is said to be a [**trivializable**]{} $d$-extension of a homomorphism $\; {\kappa}{ : }H {\rightarrow}H_1 \;$, if the square part of Diagram \[diag order\] ( below ) commutes and the functions
$a_1 \circ \chi$ and $a$ are cohomologous with respect to $H$.
2. We shall say that $$(\pi_1,\psi_1) \; \preceq \; (\pi,\psi)$$ if there is a commutative diagram $$\label{diag order}
\xymatrix{
{{\bar H}}\ar[d]^{{{\bar {\kappa}}}} \ar[r]^{\pi}
& H \ar[d]^{{\kappa}} \ar[dr]^{\psi} & \\
{{\bar H}}_1 \ar[r]^{\pi_1}
& H_1 \ar[r]^{\psi_1} & I }\;,$$ where $\; {{\bar {\kappa}}}\in {{\mathcal Hom}}({{\bar H}},{{\bar H}}_1) \;$ is a trivializable $d$-extension of $\; {\kappa}\in {{\mathcal Hom}}(H,H_1) \;$.
3. We shall say that $$(\pi_1,\psi_1) \; \sim \; (\pi,\psi) \;$$ if there is commutative Diagram \[diag order\], where both $\; {\kappa}\;{ : }\; H {\rightarrow}H_1 \;$ and its $d$-extension $\; {{\bar {\kappa}}}\;{ : }\; {{\bar H}}{\rightarrow}{{\bar H}}_1 \;$ are isomorphisms.
In connection with Part (i) of the definition, note that an extension $\; {{\bar {\kappa}}}\;{ : }\; {{\bar H}}{\rightarrow}{{\bar H}}_1 \;$ is trivializable iff it is equivalent to a trivial extension of $\; {\kappa}\;:\; H
{\rightarrow}H_1 \;$ (see Remark \[triv exten\]).
It can be checked also that $\; (\pi_1,\psi_1) \preceq (\pi,\psi)
\;$ and $\; (\pi,\psi) \preceq (\pi_1,\psi_1) \;$ imply $\;
(\pi_1,\psi_1) \sim (\pi,\psi) \;$, but we do not use the fact in this paper.
Our aim now is to describe “minimal” elements of $\; (\; {{\mathcal Ext}^d(I,\rho)}\;,\; \preceq \;) \;$.
\[irreduc\] An extension $(\pi,\psi) \in {{\mathcal Ext}^d(I,\rho)}$ is called [**irreducible**]{} if $(\pi_1,\psi_1) \sim (\pi,\psi)$ as soon as $ (\pi_1,\psi_1) \in
{{\mathcal Ext}^d(I,\rho)}$ and $(\pi_1,\psi_1) \preceq (\pi,\psi)$.
\[reduc ext\] For any $\; (\pi,\psi) \in {{\mathcal Ext}^d(I,\rho)}\;$ there exists a unique up to equivalence irreducible $(\pi,\psi)$-extension $\; (\pi_*,\psi_*) \in {{\mathcal Ext}^d(I,\rho)}\;$ such that $\; (\pi_*,\psi_*) \preceq (\pi,\psi) \;$.
To prove the theorem we fix a pair $(\pi,\psi) \in {{\mathcal Ext}^d(I,\rho)}$ and again use the identification (\[bar H\]). Namely, $$\label{diagr I J Yd}
(\pi, \psi) \;\;:\;\;
\xymatrix{
{{\bar H}}= I \times J \times Y_d \ar[r]^-\pi
& H = I \times J \ar[r]^-\psi & I }$$ where $H{^{(0)}} = J$ and ${{\bar H}}{^{(0)}} = H{^{(0)}} \times Y_d = J \times Y_d $ as in Section \[ss4.4\].
We construct the desired irreducible $(\pi_*,\psi_*)$-extension and a corresponding commutative diagram $$\label{diag reduc ext}
\xymatrix{
{{\bar H}}\ar[d]^{{{\bar {\kappa}}}_*} \ar[r]^{\pi}
& H \ar[d]^{{\kappa}_*} \ar[dr]^{\psi} & \\
{{\bar H}}_* \ar[r]^{\pi_*}
& H_* \ar[r]^{\psi_*} & I }$$ by means of the semigroup $ {{\bar {{\mathcal S}}}}= {{\bar {{\mathcal S}}}}(\pi,\psi) $ and its persistent partitions ${{\mathcal R}}({{\bar {{\mathcal S}}}})$.
\[reduc part\] A partition $\xi$ of $J=H{^{(0)}}$ is called [**reducing**]{} partition if the following two conditions hold
1. $f^{-1}\xi \leq \xi$ for all $f \in {{\mathcal S}}(\psi)$, i.e. $\xi$ is ${{\mathcal S}}(\psi)$-invariant
2. For any element $C \in \xi$ denote ${{\bar C}}:= \pi^{-1}C$ and let $r {|}_{{\bar C}}$ be the restriction of the partition $r \in {{\mathcal R}}({{\bar {{\mathcal S}}}})$ on the set ${{\bar C}}$. Then all the partitions $\; r {|}_{{\bar C}}\;,\; r \in {{\mathcal R}}({{\bar {{\mathcal S}}}}) \;$ coincide with each other.
Consider the set $\Xi$ of all reducing partitions $\xi$ on $H{^{(0)}}$.
For any $\xi \in \Xi$ we have $\; {\pi{^{(0)}}}^{-1}\xi = \xi \times \nu_{Y_d} \;$ and the partition $\; {\pi{^{(0)}}}^{-1}\xi \vee r \;$ does not depend on the choice of $r \in {{\mathcal R}}({{\bar {{\mathcal S}}}})$. So that we may set $$\label{bxi}
{{\bar \xi}}:= {\pi{^{(0)}}}^{-1}\xi \vee r \;\;,\;\; \xi \in \Xi$$ and ${{\bar \Xi}}:= \{{{\bar \xi}}\;{ : }\; \xi \in \Xi \}$ on ${{\bar H}}{^{(0)}}$.
Since $\xi$ is ${{\mathcal S}}(\psi)$-invariant and ${{\mathcal R}}({{\bar {{\mathcal S}}}})$ is ${{\bar {{\mathcal S}}}}$-invariant by Lemma \[gL(phi)\], the partition ${{\bar \xi}}$ is also ${{\bar {{\mathcal S}}}}$-invariant.
Therefore we may introduce the [**factor pair**]{} $$\label{factor pair}
\xymatrix{
{{\bar H}}{/}_{{\bar \xi}}\ar[r]^{\pi{/}_\xi}
& H{/}_\xi \ar[r]^{\psi{/}_\xi} & I }$$ Namely, we set $$H{/}_\xi := I \times J{/}_\xi \;\;,\;\;
{{\bar H}}{/}_{{\bar \xi}}:= I \times J{/}_\xi \times Y_d$$ Any element of ${{\bar \xi}}$ consists of $d$ elements of the form $\;
R_y^C \;,\; y \in Y_d \;$, where $C \in \xi$ and $\pi{^{(0)}}(R_y^C) =
C$. Hence. by possibly passing to an equivalent extension, we may assume that $R_y^C = C \times \{y\}$, i.e. ${{\bar \xi}}= \xi \times
{\varepsilon}_{Y_d}$. This means that the function $a = a(i,j)$, generating the extension ${{\bar H}}= {{\bar H}}_a$, does not depend on $j$ on the elements of $\xi$. Hence the equalities (\[stp bar H\]) and (\[bfi (j,y)\]) well define $a{/}_\xi$ and $ {{\bar H}}{/}_{{\bar \xi}}:=
(H{/}_\xi)_{a{/}_\xi}$. Thus we have shown
\[natur proj\] For any $\xi \in \Xi$ the natural projections $${\pi_{{\bar \xi}}}{^{(0)}} { : }{{\bar H}}{^{(0)}} {\rightarrow}{{\bar H}}{^{(0)}}{/}_{{\bar \xi}}\;\;,\;\;
{\pi_\xi}{^{(0)}} { : }H{^{(0)}} {\rightarrow}H{^{(0)}}{/}_\xi$$ uniquely determine $(\pi{/}_\xi,\psi{/}_\xi) \in {{\mathcal Ext}^d(I,\rho)}$ such that $ (\pi{/}_\xi,\psi{/}_\xi) \preceq (\pi,\psi) $ with the coresponding commutative diagram $$\label{diag /xi}
\xymatrix{
{{\bar H}}\ar[d]^{\pi_{{\bar \xi}}} \ar[r]^{\pi}
& H \ar[d]^{\pi_\xi} \ar[dr]^{\psi} & \\
{{\bar H}}{/}_{{\bar \xi}}\ar[r]^{\pi{/}_\xi}
& H{/}_\xi \ar[r]^{\psi{/}_\xi} & I }$$
Conversely
\[convers\] For any $\; (\pi_1,\psi_1) \; \preceq \; (\pi,\psi) \;$ there exists $\xi \in \Xi$ such that $(\pi{/}_\xi,\psi{/}_\xi) \sim (\pi_1,\psi_1)$
Take the map $ {\kappa}{^{(0)}} { : }H{^{(0)}} {\rightarrow}{H_1}{^{(0)}} $ induced by homomorphism $ {\kappa}{ : }H {\rightarrow}H_1 $ from Diagram \[diag ext\] and set $\xi := {{\kappa}{^{(0)}}}^{-1}{\varepsilon}_{{H_1}{^{(0)}}}$. Then $\xi \in \Xi$ and it is desired
[**Proof of Theorem \[reduc ext\].**]{} It is easily to see that $\Xi$ is a lattice, i.e. $ \xi_1 \vee \xi_2 \in \Xi $ and $ \xi_1
\wedge \xi_2 \in \Xi $ for all $\xi_1 , \xi_2 \;\in\;\Xi $. Herewith, $\Xi$ has the least element. Denote the least element by $\xi_* $ and let $ {{\bar \xi}}_* := \overline{(\xi_*)}$ be the corresponding partition of ${{\bar H}}{^{(0)}}$. Note that $ {{\bar \xi}}_*$ is the least element of ${{\bar \Xi}}$. Herewith $ {{\bar \xi}}_*$ is the least partition of ${{\bar H}}{^{(0)}}$ such that for all $r \in {{\mathcal R}}({{\bar {{\mathcal S}}}})$ and every $C \in \xi$ the restriction $r \;{|}_{{\bar C}}$ consists precisely of $d$ elements.
Putting $\xi = \xi_*$ in Diagram \[diag /xi\] (Proposition \[natur proj\]) we obtain Diagram \[diag reduc ext\] with $$H_* = H{/}_{\xi_*} \;\;,\;\; {{\bar H}}_* = {{\bar H}}{/}_{{{\bar \xi}}_*}
\;\;,\;\; \pi_* = \pi{/}_{\xi_*} \;\;,\;\; \psi_* =
\psi{/}_{\xi_*} \;.$$ and $ (\pi_*,\psi_*) \preceq (\pi,\psi) $.
Using by the above propositions and Lemma \[gL(phi)\], we see that the pair $(\pi_*,\psi_*)$ is irreducible and that it is the only (up to equivalence) irreducible pair majorized by $
(\pi,\psi) $.
\[irred ext\] The above arguments show that a pair $ (\pi,\psi) \in {{\mathcal Ext}^d(I,\rho)}$ is irreducible iff $ (\pi_*,\psi_*) = (\pi,\psi) $, i.e. iff $ \xi_*
= {\varepsilon}_{H{^{(0)}}}$. The last equality means that the persistent partitions $ {{\mathcal R}}({{\bar {{\mathcal S}}}}) $ separate the points of $H{^{(0)}}$ in the following sense: for every pair $ u_1,u_2 \in H{^{(0)}} $ there exist $R_1 \in r_1 \in {{\mathcal R}}({{\bar {{\mathcal S}}}})$ and $R_2 \in r_2 \in {{\mathcal R}}({{\bar {{\mathcal S}}}})$ such that $${\pi{^{(0)}}}^{-1}(u_1) \cap R_1 \cap R_2 \neq \emptyset \;\;,\;\;
{\pi{^{(0)}}}^{-1}(u_2) \cap R_1 \cap R_2 = \emptyset \;.$$
Canonical form and classification. {#s5}
==================================
Main Theorems. {#ss5.1}
---------------
The following two theorems claim the existence and uniqueness of the canonical form of $\rho$-uniform one-sided Markov shifts.
\[canon form\] Let $G$ be a $\rho$-uniform stochastic graph, which is irreducible and positively recurrent. Then there exists a $(\pi,\psi)$-extension $$\label{pi,psi}
(\pi,\psi) \;{ : }\;
\xymatrix{ {{\bar H}}\ar[r]^{\pi} & H \ar[r]^{\psi} & I }$$ such that
1. The shifts $T_G$ and $T_{{\bar H}}$ are isomorphic,
2. $(\pi,\psi) \in {{\mathcal Ext}^d(I,\rho)}$, where $d = d(T_G)$ is the minimal index of the shift $T_G$,
3. The extension $(\pi,\psi) $ is irreducible.
Combining the results of Theorems \[reduc ext\] and \[phi bar d(T)\] we obtain from Diagrams \[diag reduc ext\] and \[diag bar H Gn\] the following commuting diagram $$\label{diag main}
\xymatrix{
{{\bar H}}\ar[dd]^{{{\bar {\kappa}}}} \ar[rd]^{\pi} \ar[rr]^{{{\bar \psi}}}
&& G{^{(n)}}\ar[dd]^{\phi} \ar[rr]^{\pi{^{(n)}}}
&& G \\
& H \ar[d]^{{\kappa}} \ar[dr]^{\psi} & && \\
{{\bar H}}_* \ar[r]^{\pi_*}
& H_* \ar[r]^{\psi_*} & I && }$$ Here, $\pi$, $\pi_*$ and $\phi$ are homomorphisms of degree $d = d(T_G)$, all other homomorphisms are of degree $1$, and the extension $(\pi_*,\psi_*) \in {{\mathcal Ext}^d(I,\rho)}$ is irreducible.
Since $G$ and $ {{\bar H}}_* $ have a common extension ${{\bar H}}$ of degree $1$, the shifts $T_G$ and $T_{{{\bar H}}_*}$ are isomorphic. Thus the the extension $(\pi_*,\psi_*) $ is desired.
\[def canon form\] We shall say that $T_{{\bar H}}$ is a [**canonical form**]{} of the shift $T_G$, if there exists an extension (\[pi,psi\]) satisfying the conditions of Theorem \[canon form\]. Herewith the graph ${{\bar H}}$ is said to be the [**canonical graph**]{} for $T_G$.
Theorem \[canon form\] states the existence of the canonical form. Turn to the uniqueness.
\[classification\] Let $G_1$ and $G_2$ be two $\rho$-uniform stochastic graphs, which are irreducible and satisfy the positive recurrence condition. Suppose the shifts $T_{G_1}$ and $T_{G_1}$ are represented in the canonical form $T_{{{\bar H}}_1}$ and $T_{{{\bar H}}_2}$, respectively, and let $$\label{pi12,psi12}
(\pi_k,\psi_k) \;{ : }\;
\xymatrix{
{{\bar H}}_k \ar[r]^{\pi_k} & H \ar[r]^{\psi_k} & I } \;\;,\;\; k=1,2$$ be corresponding canonical $(\pi,\psi) $-extensions.
Then the following conditions are equivalent
1. The shifts $T_{G_1}$ and $T_{G_2}$ are isomorphic, $(T_{G_1} \sim T_{G_2})$.
2. The graphs ${{\bar H}}_1$ and ${{\bar H}}_2$ are isomorphic, $({{\bar H}}_1 \sim {{\bar H}}_2)$.
3. The extensions $(\pi_1,\psi_1) $ and $(\pi_1,\psi_1) $
are equivalent,$ ( (\pi_1,\psi_1) \sim (\pi_2,\psi_2) ) $.
By the definition we have $T_{G_1} \sim T_{{{\bar H}}_1}$ , $T_{G_2} \sim T_{{{\bar H}}_2}$ and $$(\pi_1,\psi_1) \sim (\pi_2,\psi_2) \;\; {\Longrightarrow}\;\; {{\bar H}}_1 \sim {{\bar H}}_2
\;\; {\Longrightarrow}\;\; T_{{{\bar H}}_1} \sim T_{{{\bar H}}_2}$$ Thus we need to prove only $$\label{RRaro}
T_{{{\bar H}}_1} \sim T_{{{\bar H}}_2} \;\; {\Longrightarrow}\;\; (\pi_1,\psi_1) \sim (\pi_2,\psi_2)$$ Suppose $T_{{{\bar H}}_1} \sim T_{{{\bar H}}_2} $ and let $\; a_k { : }H_k
{\rightarrow}{{\mathcal A}_d}\;,\; k=1,2 \;$, be the functions generating ${{\bar H}}_k$, where $\; d = d(T_{{{\bar H}}_1}) = d(T_{{{\bar H}}_2}) \;$.
Since both of $ \psi_1 { : }H_1 {\rightarrow}I $ and $ \psi_2 : H_2
{\rightarrow}I $ are of degree $1$, we can apply Theorem \[degree 1\] and to construct a common extension $H$ of $H_1$ and $H_2$. Herewith, the corresponding Diagram \[diag degree 1\] commutes and the homomorphisms $\; \psi { : }H {\rightarrow}I \;$ , $\; \chi_1
{ : }H {\rightarrow}H_1 \;$ and $\; \chi_2 { : }H {\rightarrow}H_2 \;$ are of degree $1$.
By Remark \[triv exten\] each of homomorphisms $\; \chi_k { : }H_{b_k} {\rightarrow}H_k \;,\; k = 1,2 \;$ admits the trivial extension $\; {{\bar \chi}}_k { : }{{\bar H}}_{b_k} {\rightarrow}{{\bar H}}_k \;$ with the commuting diagram $$\label{diag ext k=1,2}
\xymatrix{
{{\bar H}}_{b_k} \ar[d]^{\pi_{b_k}} \ar[r]^{{{\bar \chi}}_k}
& {{\bar H}}_k \ar[d]^{\pi_k} \\
H \ar[r]^{\chi_k} & H_k }$$ Here ${{\bar \chi}}_k$ is of degree $1$ and $b_k := a_k \circ \chi_k $ for $k = 1,2$. Since $ d({{\bar \chi}}_1) = d({{\bar \chi}}_2) = 1 $ we have $T_{{{\bar H}}_1} \sim T_{{{\bar H}}_{b_1}} $ and $T_{{{\bar H}}_2} \sim T_{{{\bar H}}_{b_2}}
$. Therefore $T_{{{\bar H}}_1} \sim T_{{{\bar H}}_2} $ implies that the skew products ${{\bar T}}_{H,b_1}$ and ${{\bar T}}_{H,b_2}$ are isomorphic.
Thus we have two GSP $d$-extensions $\; \pi_{b_k} { : }{{\bar H}}_{b_k} {\rightarrow}H \;,\; k = 1,2 \;,\;$ of $H$ and a homomorphism $\psi : H {\rightarrow}I $ of degree $1$. Herewith, the number $d$ is the minimal index of ${{\bar T}}_{H,b_1}$ and ${{\bar T}}_{H,b_2}$. By Theorem \[Equ ext 1\] the functions $b_1$ and $b_2$ are cohomologous with respect to $H$. Hence two constructed $(\pi,\psi)$-extensions $$(\pi_{b_k},\psi) \;{ : }\;
\xymatrix{ {{\bar H}}_{b_k} \ar[r]^{\pi}
& H \ar[r]^{\psi} & I } \;\;,\;\; k=1,2$$ are equivalent, $\; (\pi_{b_1},\psi) \sim (\pi_{b_2},\psi) \;$.
On the other hand by constructing both two diagrams $$\xymatrix{
{{\bar H}}_{b_k} \ar[d]^{{{\bar {\kappa}}}_k} \ar[r]^{\pi_{b_k}}
& H \ar[d]^{{\kappa}} \ar[dr]^{\psi} & \\
{{\bar H}}_k \ar[r]^{\pi_k}
& H_k \ar[r]^{\psi_k} & I } \;\;,\;\; k=1,2$$ commute. This means that $\; (\pi_1,\psi_1) \; \preceq \;
(\pi_{b_1},\psi) \;$ and $\; (\pi_2,\psi_2) \; \preceq \;
(\pi_{b_2},\psi) \;$.
The pairs $(\pi_1,\psi_1)$ and $(\pi_2,\psi_2)$ are irreducible and they are majorized by equivalent pairs. Hence they are equivalent.
We have shown (\[RRaro\]).
As a consequence we have also
\[common exten 1\] Under conditions of Theorem \[classification\] the shifts $T_{G_1}$ and $T_{G_2}$ are isomorphic iff the graphs $G_1$ and $G_2$ have a common extension of degree $1$, i.e. there exists a diagram $$\label{diag com ext 1}
\xymatrix{
G_! & G \ar[l]_{\phi_1} \ar[r]^{\phi_2} & G_2 }$$ where homomorphisms $\phi_1$ and $\phi_2$ are of degree $1$.
By Theorem \[phi bar d(T)\] we have two diagram of homomorphisms $$\label{diag com ext 2}
\xymatrix{
G_k & G_k{^{(n)}}\ar[l]_{\pi{^{(n)}}}
& {{\bar H}}_k \ar[l]_{{{\bar \psi}}_k} \ar[r]^{\pi_k}
& H_k \ar[r]^{\psi_k} & I } \;\;\;\; k=1,2$$ where $d(\pi{^{(n)}}) = d({{\bar \psi}}_k) = d(\psi_k) = 1 $ and $\pi_k$ is a $d$-extension. So that $(\pi_k,\psi_k) \in {{\mathcal Ext}^d(I,\rho)}$.
By Theorem \[reduc ext\] each pair $(\pi_k,\psi_k) \;,\; k=1,2
\;$ majorizes an irreducible pair from $ {{\mathcal Ext}^d(I,\rho)}$. If the the shifts $T_{G_1}$ and $T_{G_2}$ are isomorphic the irreducible pairs are equivalent (Theorem \[classification\]) and we may assume without loss of generality that they coincide with each other.
Thus there exists $(\pi_0,\psi_0) \in {{\mathcal Ext}^d(I,\rho)}$ with two commuting diagrams $$\label{diag reduc ext k=1,2}
\xymatrix{
{{\bar H}}_k \ar[d]^{{{\bar {\kappa}}}_k} \ar[r]^{\pi_k}
& H_k \ar[d]^{{\kappa}_k} \ar[dr]^{\psi_k} & \\
{{\bar H}}_0 \ar[r]^{\pi_0}
& H_0 \ar[r]^{\psi_0} & I } \;\;\;\; k=1,2$$ Passing possibly to equivalent extensions we may also assume that ${{\bar {\kappa}}}_1$ and ${{\bar {\kappa}}}_2$ are trivial extensions of ${\kappa}_1$ and ${\kappa}_2$.
By Theorem \[degree 1 sharp\] and Remark \[triv exten\] we find a common extension of degree 1 $$\label{diag com ext 3}
\xymatrix{ H_1 & H \ar[l]_{\chi_1} \ar[r]^{\chi_2} & H_2 }$$ of $H_1$ and $H_2$ with the trivial extensions $$\label{diag com ext 4}
\xymatrix{
{{\bar H}}_! & {{\bar H}}\ar[l]_{{{\bar \chi}}_1} \ar[r]^{{{\bar \chi}}_2} & {{\bar H}}_2 }$$ of $\chi_1$ and $\chi_2$ such that the corresponding diagram $$\label{diag com ext 5}
\xymatrix{
& {{\bar H}}_2 \ar[dd]_(.75){\pi_2} \ar[rr]^{{{\bar {\kappa}}}_2}&
& {{\bar H}}_0 \ar[dd]^{\pi_0} \ar[drrrr]^{{{\bar \psi}}_0} & &&& \\
{{\bar H}}\ar[dd]_{\pi} \ar[ur]^{{{\bar \chi}}_2} \ar[rr]^(.75){{{\bar \chi}}_1}
\ar[rrru]^{{{\bar \chi}}}
&& {{\bar H}}_1 \ar[ur]_{{{\bar {\kappa}}}_1} \ar[dd]^(.25){\pi_1} &&&&& I \\
& H_2 \ar[rr]^(.75){{\kappa}_2}&
& H_0 \ar[rrrru]^{\psi_0} &&&& \\
H \ar[ur]^{\chi_2} \ar[rr]_{\chi_1} \ar[rrru]^{\chi}
& & H_1 \ar[ur]_{{\kappa}_1} & && && \\ }$$ commutes. Therefore we have $$\label{diag com ext 6}
\xymatrix{
G_1 & G_1{^{(n)}}\ar[l]_{\pi{^{(n)}}} & {{\bar H}}_1 \ar[l]_{{{\bar \psi}}_1}
& {{\bar H}}\ar[l]_{{{\bar {\kappa}}}_1} \ar[r]^{{{\bar {\kappa}}}_2} & {{\bar H}}_2 \ar[r]^{\psi_2}
& G_2{^{(n)}}\ar[r]^{\pi{^{(n)}}} & G_2 }$$ Putting $\; G := {{\bar H}}\;$ and $\; \phi_k := {{\bar {\kappa}}}_k \circ {{\bar \psi}}_k
\circ \pi{^{(n)}}\;$ for $\; k = 1,2 \;$, we obtain the desired common extension of degree $1$ (\[diag com ext 1\]).
Consequences and examples. {#ss5.2}
--------------------------
Consider some particular cases.
[**Extensions of Bernoulli graphs.**]{} Let $(I,\rho)$ be a standard Bernoulli graph and let $ d \in {{\mathbb{N}}}$. Let $\; a { : }I {\rightarrow}{{\mathcal A}_d}\;$ be a function $\; a { : }I {\rightarrow}{{\mathcal A}_d}\;$ on $I$ with the values $\; a(i) \;,\; i \in I \;,$ in the group ${{\mathcal A}_d}$ of all permutations of $\; Y_d = \{1,2, \ldots ,d \} \;$. Consider a $d$-extension $\; {{\bar I}}_a \;$ generated by the function $a$ (See Section \[ss4.2\]). We assume that the group $ {\Gamma}(a) $, generated by $\; a(i) , i \in
I, \;$ acts transitively on $Y_d$. This provides that the shift $T_{{{\bar I}}_a} $ and the skew product ${{\bar T}}_{I,a}$ are ergodic.
We want to clarify: when is $\; {{\bar I}}_a \;$ the canonical graph for the corresponding Markov shift $T_{{{\bar I}}_a}$ (Definition \[def canon form\]). Let $\pi { : }{{\bar I}}_a {\rightarrow}I$ be the projection and $ (\pi,\psi) \in {{\mathcal Ext}^d(I,\rho)}$. Since every homomorphism $ \psi { : }I
{\rightarrow}I $ is an automorphism, the pair $(\pi,\psi)$ is irreducible. Therefore $\; {{\bar I}}_a \;$ is a the canonical graph iff $\; d(T_{{{\bar I}}_a}) = d \;$.
\[d = d(T bI a)\] If the function $a$ satisfies the following condition $$\label{modular a}
\rho(i) = \rho(i') \;\; {\Longrightarrow}\;\; a(i) = a(i')
\;\;,\;\; i,i' \in I$$ then $\; d(T_{{{\bar I}}_a}) = d \;$.
Suppose the condition (\[modular a\]) holds. The Markov shift $T_{{{\bar I}}_a} $ is isomorphic to the skew product ${{\bar T}}= {{\bar T}}_{\rho,a}$, which acts on $X_\rho \times Y_d$ by (\[bT rho a\]). So that we have $d(T_{{{\bar I}}_a}) = d({{\bar T}}) $ and by Theorem \[simp mar\] $\; d({{\bar T}}) \;=\; d_{{\gamma}{ : }{\beta}}({{\bar T}})
\;$.
A direct computation, using (\[modular a\]), the definition of ${\gamma}(T)$ and ${\beta}({{\bar T}})$ and Proposition \[ga(T rho)\], shows that $${\beta}({{\bar T}}) = {\gamma}(T_\rho) \times {\varepsilon}_{Y_d} \;\;\;,\;\;\;
{\gamma}({{\bar T}}) = {\gamma}(T_\rho) \times \nu_{Y_d} \;.$$ This means that any element of ${\gamma}({{\bar T}})$ consists precisely of $d$ elements of the partition ${\beta}({{\bar T}})$. By the definition of the index $ d_{{\gamma}{ : }{\beta}}({{\bar T}}) $ we have $\; d_{{\gamma}{ : }{\beta}}({{\bar T}}) = d \;$. Thus $\; d(T_{{{\bar I}}_a}) = d \;$.
Taking into account Theorem \[th rho cohom\] we have
\[cor modular a\] Let $\; \pi_k { : }{{\bar I}}_{a_k} {\rightarrow}I \;,\; k = 1,2, \;$ be two $d$-extensions of the Bernoulli graph $(I,\rho)$, generated by functions $ a_k { : }I {\rightarrow}{{\mathcal A}_d}$, respectively, and suppose both the functions $\; a_k \;,\;k=1,2 \;$ satisfy the condition \[modular a\]. Then the Markov shifts $T_{{{\bar I}}_{a_1}}$ and $T_{{{\bar I}}_{a_2}}$ are isomorphic iff $ a_1 $ and $ a_2 $ are conjugate in ${{\mathcal A}_d}$, i.e. there exists $w_0 \in {{\mathcal A}_d}$ such that $\; a_2(i) \cdot w_0 = w_0
\cdot a_1(i) \;,\; i \in I \;$.
\[rem modular a\] It can be proved that for $d$-extension ${{\bar I}}_a$, the condition \[modular a\] is equivalent to $\; d(T_{{{\bar I}}_a}) = d \;$.
[**Absolutely non-homogeneous ${\boldsymbol{\rho}}$.**]{} Consider the case , when $\rho$ is absolutely non-homogeneous (see Section \[ss2.4\]). this means that $\; \rho(i) \neq \rho(i') \;$ for all $\;i \neq i' \;$ from $I$, i.e. the Bernoulli graph $(I,\rho)$ has no congruent edges.
In this case for any $\rho$-uniform graph $G$ there exists a [**unique**]{} homomorphism $\; \phi { : }G {\rightarrow}I \;$. Therefore Theorem \[phi bar d(T)\] can be sharpened as follows
\[phi bar d(T) sharp\] Let $G$ be a $\rho$-uniform stochastic graph, which is irreducible and satisfies the positive recurrence condition. Suppose that $\rho$ is absolutely non-homogeneous. Then there exist a unique homomorphism $\; \phi \in {{\mathcal Hom}}(G,I) \;$ and a commutative diagram $$\label{diag bar H G}
\xymatrix{ {{\bar H}}\ar[d]^{\pi} \ar[r]^{{{\bar \psi}}} & G \ar[d]^{\phi} \\
H \ar[r]^{\psi} & I }$$ such that
1. The pair $\; (\pi,\psi) \in {{\mathcal Ext}^d(I,\rho)}$ is a $(\pi,\psi)$-extension.
2. $\; d = d(\phi) = d(T_G) \;$,
A natural question, which is arisen in connection with the previous theorem is:
\[gen road prob\] Does Theorem \[phi bar d(T)\] hold with $n=1$ in general case, when $\rho$ is not necessarily absolutely non-homogeneous, i.e. when $(I,\rho)$ has congruent edges ?
As we know, the problem is open even in the case, when the graph $G$ is finite (See [@AsMaTu] and references therein.)
[**Homogeneous ${\boldsymbol{\rho}}$ and Road Problem** ]{} Consider a special case, when $\rho$ is homogeneous, i.e. $\; \rho(i) =
l^{-1} \;,\; i \in I \;$ with an integer $l = |I| \in {{\mathbb{N}}}$. Theorem \[simp mar\] and arguments adduced in Section \[ss2.4\] imply
\[\] Suppose $\rho$ is homogeneous. Then every ergodic $\rho$-uniform Markov shift $\; T_G \;$ is isomorphic to a direct product $\; T_\rho \times {\sigma}_d \;$ of the Bernoulli shift $T_\rho$ and a cyclic permutation ${\sigma}_d$ of $Y_d$, where $d$ is the period of $ T_G $. If, in addition, $ T_G
$ is exact, then it is isomorphic to the Bernoulli shift $T_\rho$, herewith, there exists $n \in {{\mathbb{N}}}$ and a homomorphism $\; \phi
{ : }G{^{(n)}}{\rightarrow}I \;$ of degree $1$.
The result was proved earlier in for finite $G$ and in for general case.
If $G$ is finite and $\rho$ is homogeneous Question \[gen road prob\] is a reformulation of well-known Road Coloring Problem (See [@Fr], [@O'B], [@AdGoWe], [@Ki]). As we know, the problem is still open.
Some (p,q)-uniform graphs. {#ss5.4}
--------------------------
We construct some simple examples to illustrate the case, when the $\psi$-part in the canonical pair $(\pi,\psi)$ is not trivial.
Let $\; I = \{0,1\}\;$ and $\; \rho = (p,q) \;$, where $\; 0 < p < 1 \;$ and $\; q=1-p \;$. Given $\; n \in {{\mathbb{N}}}\;$ consider the following random walk on $\; J_n := \{ 1,2, \ldots ,n \} \;$ $$\label{FDR}
\xymatrix@C=3pc{
1 \ar@(ul,dl) []_{q} \ar@/_/ [r]_{p}
& 2 \ar@/_/ [l]_{q} \ar@/_/ [r]_{p}
& \;\;\;\;\hdots\;\;\;\;
\ar@/_/ [l]_{q} \ar@/_/ [r]_{p}
& n \ar@/_/ [l]_{q} \ar@(dr,ur) []_{p} } \;,$$ which is known as a [**Finite Drunkard Ruin**]{}. We set here: $\; H:=I \times J_n \;$ , $\; H{^{(0)}} := J_n \;$ and $$s(h) = j \;,\; t(h) = f_ij \;,\; \psi (h) = i
\;\;,\;\; h = (i,j) \in H \;,$$ where the maps $\; f_i { : }J_n/ {\rightarrow}J_n \;,\; i = 0,1 \;,$ are defined by $$f_1 j = \min{(j+1,n)} \;\;,\;\; f_0 j = \max{(j-1,1)}
\;\;\;,\;\;\; j \in J_n$$ and the weights of edges $\; p(h) \;,\; h \in H \;$ are given according to (\[FDR\]) by $\; p(1,j) = p \;$, $\; p(0,j) = q
\;$.
Then the finite stochastic graph $H$ is irreducible and $\rho$-uniform, $\; \psi \in {{\mathcal Hom}}(H,I) \;$. The semigroup ${{\mathcal S}}(\psi)$, generated by $\{f_0,f_1\}$, is $1$-contractive, since $\;
(f_0)^n (J_n) = \{1\} \;$. Whence, $\; d(\psi) = 1 \;$ and the Markov shift $T_H$ is isomorphic to the Bernoulli shift $T_\psi$.
Given $p$ and $n$ we construct a ${{\mathbb{Z}}}_2$-extension ${{\bar H}}_a$ of the graph $H$, where $a : H {\rightarrow}{{\mathbb{Z}}}_2$ and ${{\mathbb{Z}}}_2 := \{0,1\}$ be the cyclic group of order $2$.
Define $\; a : H=I \times J_n \ni h=(i,j) {\rightarrow}a(h) \in {{\mathbb{Z}}}_2 \;$ by $$\label{a(h)}
a(i,j) \;=\; \left\{
\begin{array}{ll}
1 \;\;,\;\; & if \;\; (i,j) \;=\; (1,1) \; \\
0 \;\;,\;\; & if \;\; (i,j) \;\neq\; (1,1) \;.
\end{array}
\right.$$ Then the corresponding graph ${{\bar H}}_a$ has the form $$\label{FDR2}
\xymatrix@C=3pc{
**[r] 11 \ar@(ul,dl) []_{q} \ar [rd]_(.75){p}
& 21 \ar@/_/ [l]_{q} \ar@/_/ [r]_{p}
& \;\;\;\hdots\;\;\;
\ar@/_/ [l]_{q} \ar@/_/ [r]_{p}
& **[l] n1 \ar@/_/ [l]_{q} \ar@(dr,ur) []_{p}
& z=1 \\
**[r] 10 \ar@(dl,ul) []^{q} \ar [ru]^(.75){p}
& 20 \ar@/^/ [l]^{q} \ar@/^/ [r]^{p}
& \;\;\;\hdots\;\;\;
\ar@/^/ [l]^{q} \ar@/^/ [r]^{p}
& **[l] n0 \ar@/^/ [l]^{q} \ar@(ur,dr) []^{p}
& z=0 }$$ for $\; n > 2 \;.$ and $$\label{FDR3}
\xymatrix@C=3pc{
**[r] 11 \ar@(ul,dl) []_{q} \ar@/^3pc/ [d]^{p}
&& **[r] 11 \ar@(ul,dl) []_{q} \ar [rd]_(.75){p}
& **[l] 21 \ar@/_/ [l]_{q} \ar@(dr,ur) []_{p}
& z=1 \\
**[r] 10 \ar@(ul,dl) []_{q} \ar@/_2pc/ [u]^{p}
&& **[r] 10 \ar@(dl,ul) []^{q} \ar [ru]^(.75){p}
& **[l] 20 \ar@/^/ [l]^{q} \ar@(ur,dr) []^{p}
& z=0 }$$ for two special cases $\; n = 1,2 \;$
Suppose $\; p \neq q \;$. We claim in this case that for all $n
\in {{\mathbb{N}}}$ the graphs (\[FDR3\]) and (\[FDR2\]) are canonical. Indeed, $\; d(\pi_H) = d(T_{{{\bar H}}_a}) = 2 \;$, since $ \rho = (p,q)
$ is absolutely non-homogeneous. In order to check the irreducibility of the $2$-extension $\; (\pi_H,\psi) { : }{{\bar H}}_a
{\rightarrow}H {\rightarrow}I \;$ consider the semigroup ${{\bar {{\mathcal S}}}}= {{\bar {{\mathcal S}}}}(\pi,\psi)$ and its persistent partitions ${{\mathcal R}}({{\bar {{\mathcal S}}}})$.
The semigroup ${{\bar {{\mathcal S}}}}$ is generated by $\; \{{{\bar f}}_i , i\in I \}$, where $${{\bar f}}_i \; (j\;,\;z) \;=\; (f_i \; j \;,\; z + a(i,j) \pmod 2 ) \;\;,\;\;
(j,z) \in J_n \times {{\mathbb{Z}}}_2 \;.$$ A direct computation shows that for $\; n = 1,2 \;$ any transversal partition of $J_n \times {{\mathbb{Z}}}_2 $ is persistent in the sense of Definition \[pers part\] , and for $\; n > 2 \;$ there exists a non-persistent transversal partition. Naimly, the partition, consisting of two sets of the following “alternating” form $$\{ (1,0) , (2,1) , (3,0) , (4,1) , \ldots \}
\;\;,\;\; \{ (1,1) , (2,0) , (3,1) , (4,0) , \ldots \} \;,$$ is so. Moreover, this is the only transversal partition, which is not persistent. This implies that for every $ n \in {{\mathbb{N}}}$ the persistent partitions ${{\mathcal R}}({{\bar {{\mathcal S}}}})$ separate points of $ J_n \times
{{\mathbb{Z}}}_2 $ in the sense of Remark \[irred ext\] and the $2$-extension $(\pi_H, \psi)$ is irreducible. Thus
- For all $n \in {{\mathbb{N}}}$ and $p \neq q$ the graphs ${{\bar H}}_a$ are canonical graphs for the corresponding shifts $T_{{{\bar H}}_a}$.
Just in the same way we can consider the following [**Infinite Drunkard Ruin**]{} $$\label{FDR4}
\xymatrix@C=3pc{
**[r] 1 \ar@(ul,dl) []_{q} \ar@/_/ [r]_{p}
& 2 \ar@/_/ [l]_{q} \ar@/_/ [r]_{p}
& \;\;\;\hdots\;\;\;
\ar@/_/ [l]_{q} \ar@/_/ [r]_{p}
& n \ar@/_/ [l]_{q} \ar@/_/ [r]_{p}
& \ar@/_/ [l]_{q} \;\;\;\hdots }$$ where $\; H:=I \times {{\mathbb{N}}}\;$ , $\; H{^{(0)}} := {{\mathbb{N}}}\;$.
Suppose $\; p < q \;$. Then the corresponding Markov chain is positively recurrent and the Markov shift $T_H$ is isomorphic to the Bernoulli shift $T\rho$.
Again define the functions $\; a : H=I \times {{\mathbb{N}}}\ni h=(i,j) {\rightarrow}a(h) \in {{\mathbb{Z}}}_2 \;$ by (\[a(h)\]). Then ${{\mathbb{Z}}}_2$-extension ${{\bar H}}_a$ of the graph $H$ (\[FDR5\]) has the form $$\label{FDR5}
\xymatrix@C=3pc{
**[r] 11 \ar@(ul,dl) []_{q} \ar [rd]_(.75){p}
& 21 \ar@/_/ [l]_{q} \ar@/_/ [r]_{p}
& \;\;\;\hdots\;\;\; \ar@/_/ [l]_{q} \ar@/_/ [r]_{p}
& n1 \ar@/_/ [l]_{q} \ar@/_/ [r]_{p}
& \ar@/_/ [l]_{q} \;\;\;\hdots \\
**[r] 10 \ar@(dl,ul) []^{q} \ar [ru]^(.75){p}
& 20 \ar@/^/ [l]^{q} \ar@/^/ [r]^{p}
& \;\;\;\hdots\;\;\; \ar@/^/ [l]^{q} \ar@/^/ [r]^{p}
& n0 \ar@/^/ [l]^{q} \ar@/^/ [r]^{p}
& \ar@/^/ [l]^{q} \;\;\;\hdots }$$ It can be shown in this case that any transversal set is persistent. Thus
- If $p < q$ the graph ${{\bar H}}_a$ (\[FDR5\]) is the canonical graph for the shift $T_{{{\bar H}}_a}$.
Note that the shift $T_{{{\bar H}}_a}$ is a ${{\mathbb{Z}}}_2$-extension of the Bernoulli shift $T_{p,q}$, therefore, $T_{{{\bar H}}_a}$ has a $4$-element one-sided generator. On the other hand the shift is not isomorphic to Markov shifts on finite state spaces. Thus
- If $p < q$ the one-sided Markov shift $T_{{{\bar H}}_a}$ has no finite one-sided Markov generator.
[9]{}
J. Aaronson. An introduction to infinite ergodic theory. Math. Surveys and Monographs, vol. 50, 1997.
A.L. Adler, L.W. Goodwyn, B. Weiss. Equivalence of topological Markov shifts. Isr. J. Math. 27(1977), 49-63.
J. Ashley. Resolving factor maps for shifts of finite type with equal entropy. Ergod. Th. Dyn. Syst. 11(1991), 219-240.
J. Ashley, B. Marcus, S. Tuncel. The classification of one-sided Markov chains. Ergod. Th. Dyn. Syst. 17(1997), 269-295.
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J. Friedman. On the Road Coloring Problem. Proc. AMS 110(1990), 1133-1135.
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D. Heicklen, C. Hoffman. $(T,T^{-1})$ is not standard. Ergod. Th. Dyn. Syst. 18(1998), 875-878.
D. Heicklen, C. Hoffman, D.J. Rudolph. Entropy and dyadic equivalence of random walks on a random scenery. Adv. Math. 156(2000), 157-179.
C. Hoffman. A zero entropy $T$ such that the $(T,Id)$-endomorphism is non standard. Proc. AMS 128(2000), 183-188.
C. Hoffman, D. Rudolph. Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann, of Math. 156(2002), 79-101.
M. Keane, M. Smorodinsky. Finitary isomorphism of irreducible Markov shifts. Israel J. Math. 34(1979), 281-286.
B.P. Kitchens. Symbolic dynamic. Springer-Verlag, Berlin, 1998.
B.P. Kitchens, B. Marcus, P. Trow. Eventual factor maps and compositions of closing maps. Ergod. Th. Dyn. Syst. 11(1991), 85-113.
W. Krieger. On finitary isomorphism of Markov shifts that have finite expected coding time. Z. Wahr. 65(1983), 323-328.
J. Kubo, H. Murata, H, Totoki. On the isomorphism problem for endomorphisms of Lebesgue space. Publ. RIMS. Kyoto Univ. 9(1974), 285-296.
D. Ornstein. Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4(1970), 337-352.
O’Brien. The Road Coloring Problem. Isr. J. Math. 39(1981), 145-154.
V. Rokhlin. On the fundamental ideas of the measure theory. Mat. Sbor. 25(1949), 107-150 (Russian), Trans. AMS. 71(1952). 1-54.
V. Rokhlin. Lectures on the entropy theory of transformation with invariant measure. Usp. Mat. Nauk. 22(1967), 3-56 (Russian). Russ. Math. Surveys 22(1967), 1-52.
M. Rosenblatt. Stationary processes as shifts of functions of independent random variables. J. Math. Mech. 8(1959), 665-681.
M. Rosenblatt. Markov processes. Structure and asymptotic behavior. Springer Grundl. Math. 184, Berlin, 1971.
B.Z. Rubshtein. On decreasing sequences of measurable partitions. Sov. Math. Dok. 13(1972), 962-965.
B.Z. Rubshtein. Decreasing sequences of measurable partitions generated by endomorphisms. Ups. Math. Nauk 28(1973), 247-248.
B.Z. Rubshtein. Generating partitions of Markov endomorphism. Func. Anal. Appl. 8(1974), 84-85.
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B.Z. Rubshtein. Lacunary isomorphism of decreasing sequences of measurable partition. Israel J. Math. 97(1997), 317-345.
B.Z. Rubshtein, On finitely Bernoulli one-sided Markov shifts and their cofiltrations, Ergod. Th. Dyn. Syst, 19(1999), 1527-1524.
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P. Trow. Degrees of finite to one factor maps. Israel J. Math. 71(1990), 229-238.
A.M. Vershik. A lacunary isomorphism theorem for monotone sequences of measurable partitions. Func. Anal. Appl. 2(1968). 17-21.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
High resolution radio observations remain the most direct way to study the formation and evolution of radio jets associated with the accretion onto massive black holes. We report preliminary results of our seven year VLBA observational program to understand the nature of relativistic beaming in blazars and the surrounding environment of massive black holes.
Most blazars show an apparent outward flow away from an active core. However, in a few sources the motion appears inward, most likely the result of projection of a curved trajectory which bends back toward along the line of sight. The apparent motion of jet features is not always oriented along the direction separating the feature from the core, and in a few cases we have observed a clear change in the direction and velocity of a feature as it flows along the jet. In other sources, the motion appears to follow a simple ballistic trajectory. We find no simple relation between the time scales of flux density changes and apparent component velocities.
author:
- 'K. I. Kellermann, M. L. Lister, and D. C. Homan'
- 'E. Ros and J. A. Zensus'
- 'M. H. Cohen and M. Russo'
- 'R. C. Vermeulen'
title: Superluminal Motion and Relativistic Beaming in Blazar Jets
---
\#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} =
\#1 1.25in .125in .25in
= 100000
Background
==========
It is generally accepted that the blazar phenomenon is due to the anisotropic boosting of the radiation along the direction of motion which gives rise to an apparent enhanced luminosity at all wavelengths if the observer is located close to the direction of motion. There are many observations which support this interpretation including the one sided appearance of blazar jets and the rapid flux density variability observed at many wavelengths. However, the only direct observations of relativistic motion are at radio wavelengths when motion close to the line of sight produces a compression of the time frame resulting in apparent superluminal motion. High resolution interferometric radio images are able to measure such motions which are typically less than one milliarcsecond per year.
Since 1994, we have been using the NRAO Very Long Baseline Array (VLBA) at 15 GHz (2 cm) to study this relativistic outflow in a sample of quasars and active galactic nuclei (AGN). Our goal is to understand the nature of the relativistic flow and the origin and propagation of relativistic jets. In particular, we want to know how blazars differ from other quasars and active galactic nuclei. The high resolution radio images often show pronounced bends and twists. We want to know whether or not the flow appears ballistic, that is if individual features have straight trajectories as would occur from a precessing nozzle, or, whether features follow the curvature of the jet characteristic of plasma instabilities. Are there changes in the speed or direction of features as they propagate down the jet? Does the moving pattern actually reflect the bulk flow velocity, or is there a separate pattern velocity, for example reflecting the propagation of shocks along the jet? Is there a characteristic Lorentz factor for different classes of AGN? If not, what is the distribution of Lorentz factors and what determines their value? Blazars typically show pronounced flux density variations on time scales ranging from minutes to years. Do these flux density outbursts reflect the origin of new superluminal components, and how do the time scales of intensity variations relate to apparent velocity?
Our full sample consists of 173 galaxies, quasars, and BL Lac Objects. In order to relate our observations to relativistic beaming models, we wished to define a complete un-biased flux density limited sample. However, as there are no sky surveys at 15 GHz, and AGN are generally flux density variable, there is no simple objective way of obtaining a precisely defined flux density limited sample.
Assuming a constant intrinsic value of the Lorentz factor, $\gamma$, then if the bulk velocity is equal to the pattern velocity, it is easy to calculate the distribution of observed apparent velocity (e.g., Vermeulen & Cohen 1994). However, if there is a distribution of $\gamma$’s, then an analytic solution is more difficult. One of our goals was to compile a sample whose properties can be compared with Monte Carlo simulations of relativistic beaming, so we selected sources on the basis of the parsec scale flux density only, ignoring any contribution from extended (kiloparsec-scale) structure that is not necessarily beamed. Our sample includes all known sources which meet the following criteria.
- Declination $> -20^\circ$
- Galactic latitude $|b| > 2.5^\circ$
- Total 2 cm VLBA flux density $> 1.5$ Jy, ($>2$ Jy if below the celestial equator) at any epoch since 1995.
We refer to this sample as an [*unbiased representative sample*]{}. We have, so far, good multi epoch observations of 96 sources. Observations in progress are expected to increase the number of sources to what will be a complete unbiased sample of about 120 sources. We have constructed the sample by reference to the Kühr 1 Jy catalog (Kühr 1981), the VLA calibrator manual, the JVAS survey, the VLBA Calibrator Survey (Beasley et al. 2002), the 22 GHz VLBI survey of Moellenbrock et al. (1996) the high-frequency peaked samples of Ter[ä]{}sranta et al. (2001) and Dallacasa et al. (2000), and the UMRAO (http://www.astro.lsa.umich.edu/obs/radiotel/umrao.html) and RATAN (Kovalev et al. 1999) monitoring programs. Although our selection method is somewhat complex, it is based on the directly-measured compact flux density, and does not use a single-epoch spectral index criterion to estimate compact flux density. Also the fact that survey membership is not determined from a single “snapshot” epoch means that we are not excluding potentially interesting sources simply because they happened to be in a low state at the time of the original investigation.
In this paper we present data on the 157 individual features found in the 96 sources in our full sample which have core-jet structure for which we have obtained good multi-epoch data on their motion and which have measured redshifts. We have generally observed each source about once per year, but more often for those sources with rapid changes and less often for those with slow changes. Each image typically has an angular resolution better than 1 milliarcsec, rms noise about 250 $\mu$Jy, and dynamic range better than 1000:1. Images of all of our observations are available at http://www.nrao.edu/2cmsurvey which will soon be supplemented by material describing the motions observed in each source. Throughout this paper we use a cosmology with $H_{0}=65$ km/sec/Mpc, $\Omega_{\Lambda}=0.7$, and $\Omega_{m}=0.3$.
Statistics of Superluminal; Motion
==================================
In spite of decades of studying superluminal source motions, the details of the kinematics have remained elusive. One of the problems is, that contrary to indications of early observations (e.g., Cohen et al. 1977), the radio jets often do not contain simple well defined moving components. Instead, the jets may show a complex brightness distribution with regions of enhanced intensity that may brighten and fade with time. Some features appear to move; others are stationary, or may break up into two or more separate features, and it is often unclear how these moving features are related to the actual underlying relativistic flow.
With these sensitive, high-resolution, high-dynamic range images from the VLBA, we are generally able to define one or more components in each source which have lasted for the duration of our observing program. Figure 1 shows the distribution of apparent linear velocity for the 157 components contained in our full sample that have well-determined motions. This includes 104 quasar components, 31 BL Lac components, and 22 components associated with the nucleus of an active galaxy.
Figure 1 is in marked contrast to early discussions of superluminal motion, which indicated typical values of $\gamma$ in the range 5 to 10 (Cohen et al. 1977, Porcas 1987). We believe that these earlier studies were biased in favor of faster apparent velocities since they were not based on unbiased samples, but rather used velocities which had been reported in the literature and which contained a disproportionate fraction of high velocities, as the slow velocities being “uninteresting” and more difficult to measure, were usually not followed up with further observations nor were they reported in the literature.
Most of the sources in our sample are quasars and their velocity distribution is peaked near low values of [*v/c*]{} between zero and ten, but there is a tail extending out to [*v/c*]{} $\sim$ 34. Features associated with the active nuclei of galaxies all appear to have motions in the range $0 < v/c < 8$, while the BL Lac objects appear more uniformly distributed over the entire range from 0 to 35.
Several of the observed velocities are negative, that is the jet component appears to be approaching rather than receding from the core. However, most of these reported negative velocities are consistent, within the errors, with no significant motion. In other cases, there is evidence of a newly emerging component ejected from the core, and the combination is not resolved by our beam. This causes an apparent shift in the position of the core and a corresponding decrease in the apparent separation of the core and jet component. It is also possible that the true core is not seen, possibly due to absorption, and that both of the components we are observing are part of the moving jet. In a few cases, such as 0454+844, 0735+178, and 1128+385, the apparent decrease in component separation from the core may be due to component motion away from the core along a highly curved jet which bends back toward the line of sight so that the apparent projected separation from the core appears to decrease with time.
We interpret our observations based on relativistic beaming models which assume all sources are relativistic with an intrinsic velocity close to the speed of light described by a Lorentz factor, $\gamma$. In a simple ballistic model, in which all jets have the same Lorentz factor, the effect of Doppler boosting increases the probability of observing sources close to the line of sight. In the case of a flux limited sample, angles close to $1/\gamma$ are commonly observed where $\beta_{\mathrm app}$ $\sim \gamma$ (e.g., Vermeulen & Cohen 1994, Vermeulen 1995). If there is no Doppler boosting, then most sources are expected to lie near the plane of the sky with an apparent velocity near [*c*]{}. The observed velocities do not show the expected concentration near the upper end of the distribution corresponding to the simple single-gamma ballistic model. Ekers & Laing (1990) have commented that light echo models, in particular, which do not invoke any Doppler boosting, are consistent with this kind of observed velocity distribution. Our observed distributions actually peak at lower values than expected from simple light echo models. Moreover, we find a strong correlation between apparent velocity and apparent radio luminosity as expected if the apparent radio luminosity is enhanced by Doppler boosting (Lister et al. 2003a).
Lister and Marscher (1997) have shown that an observed velocity distributions similar to that shown in Figure 1, may be reproduced with a power law distribution of intrinsic Lorentz factors. Our data are consistent with such a distribution having a large excess of small Lorentz factors contained in a volume limited sample (Lister et al. in preparation). Alternatively, the bulk velocity flow which determines the amount of Doppler boosting may be less than the pattern flow which may reflect the propagation of shock fronts rather than the relativistic flow (e.g., Vermeulen $\&$ Cohen 1994). The interpretation of jet kinematics is further complicated if there is a distribution of velocities within a single jet. For example, there may be a fast inner jet surrounded by a more slowly moving outer sheath. In such cases the appearance and apparent velocity will be a complex function of the orientation with respect to the line of sight
Kinematics of Curved jets
=========================
Many of the jets we have observed show pronounced curvature sometimes with multiple oscillations characteristic of plasma instabilities. Individual components may follow a wide range of trajectories. In some cases, such as 3C 273 (1226+023), the location of the bend appears to propagate away from the center of activity. In such cases the motion may be described as ballistic, that is components appear to move along a straight trajectory, but one which may not be pointed toward the most compact feature assumed to be the core. In other cases, such as 3C 279 (1253$-$055), the component motion is more complex and appears to flow along a non radial or even curved trajectory (Homan et al. in preparation).
Figure 2 illustrates an example of non radial motion in the jet of the quasar 3C390.3 (1845+797). In this case the two prominent components appear to be moving with similar velocities of $\sim 2.5c$, but along slightly different trajectories of 27 and 30 degrees. Neither are aligned with the direction of the core or with the extended jet feature which points toward the distant hot spot along position angle = 35.
The Ultra Luminous Infra Red Galaxy 1345+125 (4C 12.50) which has a very curved jet along with a weak counterjet, is consistent with a conical helix of wavelength 280 pc that is the result of Kelvin-Helmholtz instabilities driven by a slow precession of the jet nozzle with a half-angle of 23 degrees oriented along an angle of 82 degrees to the line of sight (Lister et al. 2003b).
Time Variability and Motions
============================
It is generally assumed that the large rapid changes in flux density which are frequently observed in blazars are the result of Doppler boosting combined with time compression of the relativistic plasma moving nearly along the line of sight. Small changes in the direction of the flow then can lead to large changes in the apparent luminosity and velocity. Provided that the bulk flow velocity and the pattern velocity are the same, we would expect to see a relation between variability time scale and the apparent transverse linear velocity. L[ä]{}hteenm[ä]{}ki $\&$ Valtaoja (1999) have calculated Doppler boosting factors from Mets[ä]{}hovi 1.3 cm and 8 mm variability data assuming an intrinsic brightness temperature of $5\times 10^{10}$K characteristic of a self absorbed synchrotron source where the particle and magnetic energy is in equilibrium.
We have compared our VLBA observations with both the Mets[ä]{}hovi and 2 cm University of Michigan (UMRAO) variability data. The UMRAO data covers a longer time period and is more densely sampled; however events which appear to be well defined at 8 mm and 1.3 cm are often blended at the longer wavelength. Figure 3 shows that the observations essentially all fit inside the $\gamma = 30$ curve, as they should for a flux density limited sample (Lister $\&$ Marsher 1997), but the detailed distribution does not match the expected one. Further, there are a number of highly variable sources sources, such as 3C 454.3, CTA 102, and 2134+00 which show little or no significant motions. Each of these sources has a highly bent or oscillating jet. Possibly, the bright features which we have observed in these sources reflect stationary positions along the jet where the relativistic flow is oriented close to the line of sight thus giving rise to enhanced radiation due to Doppler boosting. There does appear to be a well defined upper limit to the measured apparent velocity which is close to the variability Doppler factor. Calculations of the variability Doppler factor using intrinsic brightness temperatures closer to the inverse Compton limit lead to even larger scatter and the lack of any clear envelope to the distribution. Also, we note that there is a large dispersion between between Doppler factors deduced from the UMRAO 2 cm data and the shorter wavelength Mets[ä]{}hovi data, so the robustness of the Doppler factors calculated in this way appears to be very uncertain. The relation between $\beta_{\mathrm app}$ and $D_{\mathrm var}$ will be discussed in more detail by Cohen et al. (in preparation).
We acknowledge valuable discussions with Hugh and Margo Aller, Yuri Kovalev jr. and Matthias Kadler.
Beasley, A. J., Gordon, D., Peck, A. B., Petrov, L., MacMillan, D. S., Fomalont, E. B., & Ma, C. 2002, , 141, 13
Cohen, M. H. et al. 1977, Nature, 268, 405
Dallacasa, D., Stanghellini, C., Centonza, M., & Fanti, R. 2000, , 363, 887
Ekers, R., & Laing, R. A. 1990, in Parsec Scale Radio Jets, eds. J.A. Zensus and T. Pearson (Cambridge: University Press), 333
Kovalev, Y. Y., Nizhelsky, N. A., Kovalev, Y. A., Berlin, A. B., Zhekanis, G. V., Mingaliev, M. G., & Bogdantsov, A. V. 1999, , 139, 545
Kühr, H., Witzel, A., Pauliny-Toth, I. I. K., & Nauber, U. 1981, , 45, 367
L[ä]{}hteenm[ä]{}ki, A. & Valtaoja, E. 1999, , 521, 493
Lister, M. Ł, 2003a in [*Active Galactic Nuclei: from Central Engine to Host Galaxy*]{}, eds.: S. Colin, F. Combes, & Shlosman, to be published in ASP Conference Series.
Lister, M., L. 2003b, , in press
Lister, M. L. & Marscher, A. P. 1997, , 476, 572
Moellenbrock, G. A. et al. 1996, , 111, 2174
Porcas, R. W. 1987, [*Superluminal Radio Sources*]{}, eds. J. A. Zensus & T. J. Pearson, Cambridge University Press, pg. 12
Ter[" a]{}sranta, H., Urpo, S., Wiren, S., & Valtonen, M. 2001, , 368, 431
Vermeulen, R. C. & Cohen, M. H. 1994, , 430, 467
Vermeulen, R. C. 1995, PNAS, 92, 11385
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A multi-cell mobile edge computing network is studied, in which each user wishes to compute the product of a user-generated data matrix with a network-stored matrix through data uploading, distributed edge computing, and output downloading. Assuming randomly straggling edge servers, this paper investigates the interplay among upload, compute, and download times in high signal-to-noise ratio regimes. A policy based on cascaded coded computing and on coordinated and cooperative interference management in uplink and downlink is proposed and proved to be approximately optimal for sufficiently large upload times. By investing more time in uplink transmission, the policy creates data redundancy at the edge nodes to reduce both computation times by coded computing, and download times via transmitter cooperation. Moreover, it allows computing times to be traded for download times.'
author:
- '[^1]'
bibliography:
- 'refer.bib'
title: 'Multi-Cell Mobile Edge Coded Computing: Trading Communication and Computing for Distributed Matrix Multiplication'
---
=2.1pt =2.1pt
[1.0]{}
Introduction
============
Mobile edge computing (MEC) is an emerging network architecture that enables cloud-computing capabilities at the edge nodes (ENs) of mobile networks[@MEC]. MEC makes it possible to offer mobile users applications, such as recommendation or gaming services, that require significant storage and computing resources, by task offloading. A key problem, which is the subject of this paper, is to understand the interplay and performance trade-offs between communication (in both uplink and downlink) and computing during the offloading process in multi-cell MEC networks (see Fig. \[sysModel\]).
To this end, this study focuses on the baseline problem of computing the product between user-generated data vectors $\mathbf{u}$’s and a network-stored matrix $\mathbf{A}$. Examples of applications include recommendation systems based on collaborative filtering, in which the user-generated data $\mathbf{u}$ corresponds to user profile vectors, while the network-side matrix $\mathbf{A}$ collects the profile vectors of a certain class of items, e.g., movies. Note that matrix $\mathbf{A}$ may be very large in practice, preventing a simple solution whereby users download and store the matrix for local computation. Matrix multiplication, as many other more complex computations[@bekkerman2011scaling], can be decomposed into subtasks and distributed across multiple servers. In MEC networks, the servers are embedded in distinct ENs, and hence distributed computation at the edge requires input data uploading via the uplink, computation at the ENs, and output result downloading via the downlink.
In the process discussed above, the overall latency is the sum of three components, namely, the times needed for uploading, for computing at the ENs, and for downloading. This paper is devoted to studying the interplay and trade-offs among these three components from an information-theoretic standpoint. The key idea is that investing more time in any one of the three steps may be instrumental in reducing the time needed for subsequent steps thanks to coded computing [@fundamental_tradeoff; @Speeding_Up; @songzestraggler; @jingjing; @stagglerexploit; @8006960; @improvedJingjing] and cooperative transmission [@7857805; @sengupta2017fog; @FanXu; @TaoCache]. As explained next, both coded computing and cooperative transmissions leverage forms of spatial redundancy.
{width="4.4in" height="2.4in"}
\[sysModel\]
Coded computing was introduced in [@Speeding_Up] for a master-slave system with ideal communication links and linear computations. It aims at reducing the average latency caused by distributed servers with random computing times, i.e., the problem of so-called *straggling servers*[@dean2013tail], through linear coding of the rows of matrix $\mathbf{A}$. Linear coding assigns each server a flexible number of encoded rows of matrix $\mathbf{A}$. Thanks to coding, specifically to Maximum Distance Separable (MDS) codes, assigning more coded rows at the servers reduces the number of servers that need to complete their computations in order to recover the desired outputs.
A simple way to ensure spatial redundancy is to assign repeatedly the same rows of matrix $\mathbf{A}$ across multiple ENs. While this does not provide the same robustness against stragglers as MDS coding, it allows ENs to compute common outputs, i.e., computation replication. This in turn makes it possible for the ENs to cooperate for transmission to the users in the downlink, which can reduce the download latency in an interference-limited system such as that in Fig. \[sysModel\]. This idea has been explored by [@8007057; @jingjing; @KKLMEC] for task offloading in multi-cell MEC systems and by [@li2019wireless] for data shuffling in wireless MapReduce systems. Note that the same idea has also been explored in the context of multi-cell caching systems for content delivery in [@7857805; @sengupta2017fog; @FanXu; @TaoCache].
In fact, in the MEC system of Fig. \[sysModel\], investing more time for uplink communication allows the same user-generated input vectors to be received by more ENs, which enhances spatial redundancy. The spatial redundancy generally introduces a heavier computation load, which in turn can increase the robustness against straggling servers via coded computing and reduce download latency by enhancing transmission cooperation opportunities at the ENs.
Based on these observations, this paper tackles the question: *Given an upload latency, what is the optimal trade-off between computing and download latencies?* We focus on the high signal-to-noise ratio (SNR) regime in order to highlight the role of interference management as enabled by spatial redundancy. The main prior works in this regard are [@jingjing] and [@KKLMEC]. The work [@jingjing] proposed a computing and downloading strategy by making the simplified assumption that the upload time is unconstrained so that the input vectors from all users are available at all ENs. The work [@KKLMEC], on the other hand, characterized the trade-off between upload and download latencies by assuming that the computing time at each EN is deterministic (in contrast to random) so that coded computing is not needed. Moreover, [@KKLMEC] adopts a general task model, rather than matrix multiplication as studied in this work.
In contrast to [@jingjing] and [@KKLMEC], in this paper, we parameterize the trade-off region between computing and download latencies in terms of the upload latency. We propose a joint task assignment, input upload, compute, and output download policy that generalizes that of [@jingjing] and [@KKLMEC], which is based on a cascade of MDS and repetition codes[@improvedJingjing], by accounting also for the more sophisticated coordination strategies proposed in [@FanXu]. Furthermore, we provide a converse result, which demonstrates that the achievable upload time is optimal, and the compute time and download time are within constant multiplicative gaps to their respective lower bounds.
Paper organization: Sec. \[problemformu\] presents the problem formulation and definitions. Main results are presented in Sec. \[mainresults\]. The proposed achievable schemes are summarized in Sec. \[AchievableScheme\].
Notations: $[a\!:\!b]$ denotes the set $\{a\!+\!1,a\!+\!2,\dots,b\}$, $[K]$ denotes the set $[1\!:\!K]$, and $(x)^+$ denotes $\max\{x,0\}$.
Problem Formulation {#problemformu}
===================
MEC Network Model
-----------------
As shown in Fig. \[sysModel\], we consider a multi-cell MEC network consisting of $K$ single-antenna ENs communicating with $M$ single-antenna users via a shared wireless channel. Denote by $\mathcal{K}\!=\!\{1,2,\ldots,K\}$ the set of ENs and by $\mathcal{M}\!=\!\{1,2,\ldots,M\}$ the set of users. Each EN is equipped with an edge server. Let $h^u_{ki}$ denote the uplink channel fading from user $i\!\in\!\mathcal{M}$ to EN $k\!\in\!\mathcal{K}$, and $h^d_{ik}$ denote the downlink channel fading from EN $k\!\in\! \mathcal{K}$ to user $i\!\in\!\mathcal{M}$, both of which are independent and identically distributed (i.i.d.) for all pairs $(i,k)$ according to some continuous distribution. All users and ENs can acquire the full channel state information about uplink channel $\mathbf{H}^u\!\triangleq\!\{h^u_{ki}\!:k\!\in\!\mathcal{K},i\!\in\!\mathcal{M}\}$ and downlink channel $\mathbf{H}^d\!\triangleq\!\{h^d_{ik}\!:i\!\in\!\mathcal{M},k\!\in\!\mathcal{K}\}$.
We consider the matrix-vector product computation task $\mathbf{v}\!=\!\mathbf{A}\mathbf{u}$, where $\mathbf{u}\!\in\!\mathbb{F}^{n\times1}_{2^B}$ is the user-generated input vector, $\mathbf{v}\!\in\!\mathbb{F}^{m\times 1}_{2^B}$ is the output vector, $\mathbf{A}\! \in\!\mathbb{F}^{m\times n}_{2^B}$ is a data matrix available at the network end, and $B$ is the size (in bits) of each element. Each user $i$ has $N$ input vectors $\mathbf{u}_{i,j}$, $j\!\in\![N]$, and wishes to compute the $N$ output vectors $$\label{output}
\mathbf{v}_{i,j}\!=\! \mathbf{A}{\mathbf{u}_{i,j}},~\text{for}~j\!\in\![N].$$ The matrix $\mathbf{A}$ is partially stored across the ENs, which conduct the product operations in a distributed manner. To this end, each EN $k$ has a storage capacity of $\mu m nB$ bits, and it can hence store a fraction $\mu\!\in\![\frac{1}{K},1]$ of the rows of matrix $\mathbf{A}$. Specifically, during an offline storage phase, an encoding matrix $\mathbf{E}_k\!\in\!\mathbb{F}^{\mu m\times m}_{2^B}$ is used to generate a coded matrix $\mathbf{A}_k\!=\!\mathbf{E}_k\mathbf{A}$, which is then stored at EN $k$, as in [@songzestraggler; @improvedJingjing].
Task Offloading Procedure
-------------------------
The task offloading procedure proceeds via task assignment, input uploading, edge computing, and output downloading.
### Task Assignment
A task assignment scheme is defined through sets $\{\mathcal{U}_{i,\mathcal{K}^{'}}\!:i\!\in\!\mathcal{M},\mathcal{K}^{'}\!\subseteq\! \mathcal{K}\}$, where $\mathcal{U}_{i,\mathcal{K}^{'}}\!\subseteq\!\{\mathbf{u}_{i,j}\}^N_{j=1}$ denotes the subset of input vectors from user $i$ that are assigned only to the subset of ENs $\mathcal{K}^{'}$ for computation. Since each input vector must be computed, we have the condition $\bigcup_{\mathcal{K}^{'}\!\subseteq \mathcal{K}} \mathcal{U}_{i,\mathcal{K}^{'}}\!=\!\{\mathbf{u}_{i,j}\}^N_{j=1}$ for $i\!\in\!\mathcal{M}$. Furthermore, by definition, we have the relation $\mathcal{U}_{i,\mathcal{K}^{'}}\!\bigcap \mathcal{U}_{i,\mathcal{K}^{''}}\!=\!\varnothing$ for $\mathcal{K}^{'}\!\neq\!\mathcal{K}^{''}$. The subset of input vectors from all users assigned to each EN $k$ is denoted as $\mathcal{U}_k\!=\!\bigcup_{{i}\in\mathcal{M},\,\mathcal{K}^{'}\!\subseteq \mathcal{K}:\,k\in\mathcal{K}^{'}} \mathcal{U}_{i,\mathcal{K}^{'}}$.
\[defenition1\] (Repetition Order) For a given task assignment scheme $\{\mathcal{U}_{i,\mathcal{K}^{'}}\}_{i\in\mathcal{M},\mathcal{K}^{'}\subseteq\mathcal{K}}$, the repetition order $r$, with $1\!\le\!r \!\le \!K$, is defined as average input data redundancy, i.e., the total number of input vectors assigned to the $K$ ENs (counting repetitions) divided by the total number of input vectors of the $M$ users, i.e., $r\!\triangleq\!\frac{\sum_{k\in\mathcal{K}}|\mathcal{U}_{k}|}{MN}$.
### Input Uploading
At run time, each user $i$ maps its input vectors $\{\mathbf{u}_{i,j}\}^N_{j=1}$ into a codeword $\mathbf{X}^u_i\!\triangleq\!\left(X^u_i(t)\right)^{T^{u}}_{t=1}$ of length $T^u$ symbols under the power constraint $(T^{u})^{-1}\mathbb{E}\big[||\mathbf{X}^u_i||^2\big]\!\le\! P^u$. Note that $X^u_i(t)\!\in\! \mathbb{C}$ is the symbol transmitted at time $t\!\in\![T^{u}]$. At each EN $k\!\in\!\mathcal{K}$, the received signal $Y^u_k(t) \!\in\! \mathbb{C}$ at time $t\!\in\![T^{u}]$ can be expressed as $Y^u_k(t)\!=\!\sum_{i \in \mathcal{M}}h^u_{ki}(t)X^u_i(t)\!+\!Z^u_{k}(t)$, where $Z^u_{k}(t)\!\sim\!\mathcal{CN}(0,1)$ denotes the noise at EN $k$. Each EN $k$ decodes the sequence $\left(Y^u_k(t)\right)^{T^u}_{t=1}$ into an estimate $\{\widehat{\mathbf{u}}_{i,j}\}$ of the assigned input vectors $\{\mathbf{u}_{i,j}\!:\mathbf{u}_{i,j}\!\in\!\mathcal{U}_k\}$.
### Edge Computing
After the uploading phase is completed, each EN $k$ computes the products of the assigned estimated input vectors in set $\mathcal{U}_{k}$ with its stored coded model $\mathbf{A}_k$. The computing time for EN $k$ to complete the computation of the corresponding $\mu m |\mathcal{U}_{k}|$ row-vector products is modeled as $$\label{computingtime}
T^c_k = \mu m |\mathcal{U}_{k}| \omega_k, ~\text{for}~k\in\mathcal{K}.$$ In (\[computingtime\]), random variable $\omega_k$ represents the time needed by EN $k$ to compute a row-vector product, and is modelled as an exponential distribution with mean $1/\eta$ (see, e.g., [@Speeding_Up; @songzestraggler; @8006960]). The MEC network waits until the fastest $q$ ENs, denoted as subset $\mathcal{K}_q\!\subseteq\!\mathcal{K}$, have finished their tasks before returning the results back to users in the downlink. The cardinality $|\mathcal{K}_q|\!=\!q$ is referred to as *the recovery order*. The rest of $K\!-\!q$ ENs are known as *stragglers*. The resulting (random) duration of the edge computing phase is hence given by $T^c=\max_{k\in\mathcal{K}_q} T^c_k$.
### Output Downloading
At the end of the edge computing phase, each EN $k\!\in\!\mathcal{K}_q$ obtains the coded outputs $\mathcal{V}_k\!\triangleq\!\left\{\mathbf{v}_{i,j,k}\!=\!\mathbf{A}_k\widehat{\mathbf{u}}_{i,j}\!:\mathbf{u}_{i,j}\!\in\!\mathcal{U}_{k}\right\}$. Every EN $k$ in $\mathcal{K}_q$ then maps $\mathcal{V}_k$ into a length-$T^d$ codeword $\mathbf{X}^{d}_k\!\triangleq\!\left(X^{d}_{k}(t)\right)^{T^{d}}_{t=1}$ with an average power constraint $(T^{d})^{-1}\mathbb{E}\big[||\mathbf{X}^d_k||^2\big]\!\le\! P^d$. For each user $i\!\in\!\mathcal{M}$, its received signal $Y^{d}_i(t) \!\in\! \mathbb{C}$ at time $t\!\in\![T^{d}]$ is given by $Y^{d}_i(t)\!=\!\sum_{k \in \mathcal{K}_q}h^d_{ik}(t)X^{d}_{k}(t)\!+\!Z^{d}_{i}(t)$, where $Z^{d}_{i}(t)\!\sim\!\mathcal{CN}(0,1)$ is the noise at user $i$. Each user $i$ decodes the sequence $(Y^{d}_i(t))^{T^d}_{t=1}$ to obtain an estimate $\{\widehat{\mathbf{v}}_{i,j,k}\}_{j\in[N],k\in\mathcal{K}_q}$ of the coded outputs, from which it obtains an estimate $\{\widehat{\mathbf{v}}_{i,j}\}_{j\in[N]}$ of its desired outputs. This is possible if the estimated coded outputs $\{\widehat{\mathbf{v}}_{i,j,k}\}_{j\in[N],k\in\mathcal{K}_q}$ contain enough information to guarantee the condition $H(\{\mathbf{v}_{i,j}\}_{j\in[N]}|\{\widehat{\mathbf{v}}_{i,j,k}\}_{j\in[N],k\in\mathcal{K}_q})\!=\!0$. The overall error probability is given as $\mathrm{P}_e\!\triangleq\!\mathbb{P}\big(\bigcup^{M~~N}_{i=1,j=1} \left\{\widehat{\mathbf{v}}_{i,j}\!\neq\!\mathbf{v}_{i,j}\right\}\!\big)$. A task offloading policy is said to be feasible when the error probability $\mathrm{P}_e\!\to\!0$ as $B\!\to\!\infty$.
Performance Metric
------------------
The performance of the considered MEC network is characterized by the latency triplet due to task uploading, computing, and downloading, which we measure in the high-SNR regime by following [@sengupta2017fog].
\[defenition2\] The normalized uploading time (NULT), normalized computing time (NCT), and normalized downloading time (NDLT) achieved by a feasible policy with repetition order $r$ and recovery order $q$ are defined, respectively, as
$$\begin{aligned}
\tau^{u}(r) &\triangleq \lim_{P_u\to\infty} \lim_{B\to\infty} \frac{\mathbb{E}_{\mathbf{H}^u}[T^u]}{ NnB/\log P^u}, \label{NULTtau}\\
\tau^c (r,q) &\triangleq \lim_{m\to\infty}\frac{\mathbb{E}\left[T^c\right]}{ Nm/\eta},\label{NCTtau}\\
\tau^{d}(r,q) &\triangleq \lim_{P_d\to\infty}\lim_{m\to\infty}\lim_{B\to\infty} \frac{\mathbb{E}_{\mathbf{H}^d}[T^d]}{NmB/\log P^d}. \label{NDLTtau}\end{aligned}$$
The definitions (\[NULTtau\]) and (\[NDLTtau\]) have been also adopted in [@KKLMEC], by normalizing the delivery times to those of reference interference-free systems (with high-SNR rates $\log P^u$ and $\log P^d$, respectively). Similarly, the computing time in definition (\[NCTtau\]) is normalized by the average time needed to compute over all the input vectors of a user. To avoid rounding complications, in definition (\[NCTtau\]) and (\[NDLTtau\]), we let the output dimension $m$ grow to infinity.
For a given NULT $\tau^u$, the compute-download latency region is defined as the union of the set for all NCT-NDLT pairs $(\tau^c, \tau^d)$, i.e., $$\begin{aligned}
\label{region}
\!\!\!\mathscr{T}^{*}(\tau^u)\!\triangleq\!\big\{\!(\tau^c, \tau^{d})\!: (\tau^u(r),\tau^c(r,q),\tau^{d}(r,q))~&\text{is achievable for}~\text{some $(r,q)$ and}\,\,\tau^u\!\le\!\tau^u(r),\nonumber\\
\!&~~~~~~~~~~~~~~\tau^{c}\!\ge\!\tau^{c}(r,q),~\text{and}~\tau^{d}\!\ge\!\tau^{d}(r,q)\big\}\!.\!\!\end{aligned}$$
\[remarkconvex\] (Convexity of compute-download latency region.) For an input data assignment policy $\{\mathcal{U}_{i,\mathcal{K}^{'}}\}_{i\in\mathcal{M},\mathcal{K}^{'}\subseteq\mathcal{K}}$ with a repetition order $r$, fix an input uploading strategy achieving NULT $\tau^u$. Consider two policies $\pi_1$ and $\pi_2$ that differ may in their computation and download pahses, and achieve NCT-NDLT pairs $\left(\tau^c_{1}, \tau^d_{1}\right)$ and $\left(\tau^c_{2}, \tau^d_{2}\right)$, respectively. For any ratio $\lambda\!\in\![0,1]$, there exists a policy that achieves the NCT-NDLT pair $\lambda\left(\tau^c_{1}, \tau^d_{1}\right) \!+\!(1\!-\!\lambda)\left(\tau^c_{2}, \tau^d_{2}\right)$ for the same NULT $\tau^u$. To this end, assuming $m$ is sufficiently large, matrix $\mathbf{A}$, correspondingly, all output vectors (\[output\]) are split horizontally so that $N\lambda m$ and $N(1\!-\!\lambda)m$ outputs are processed by using policies $\pi_1$ and $\pi_2$, respectively. By the linearity of the NCT (\[NCTtau\]) and NDLT (\[NDLTtau\]) with respect to the output size, the claimed pair of NCT and NDLT is achieved. This implies the region (\[region\]) is convex.
The region $\mathscr{T}^{*}(\tau^u)$ captures the trade-offs between computation and download communication latencies for a fixed upload communication latency. The region in (\[region\]) is convex thanks to time- and memory-sharing arguments in a manner similar to[@sengupta2017fog Lemma 1] (the same is not true for the region of achievable triplets $(\tau^u, \tau^c, \tau^d)$).
{width="3.8in" height="2.3in"}
\[mds\_repetition\]
Main Results {#mainresults}
============
In this section, we first present the inner and outer bounds on the compute-download latency region, and then discuss some consequences of the main results in terms of the tradeoffs among upload, compute, and download latencies. Then, we specialize the main results to a number of simple set-ups to illustrate the connections with existing works.
Key Ideas {#keyidea}
---------
We start by outlining the main ideas that underpin the proposed policy. For a repetition-recovery order pair $(r,q)$, as shown in Fig. \[mds\_repetition\], during task assignment, matrix $\mathbf{A}$ is encoded by a cascade of an MDS code of rate $1/\rho_1$ and a repetition code of rate $1/\rho_2$. As in [@Speeding_Up; @songzestraggler; @jingjing], MDS codes can alleviate the impact of stragglers on the computation latency by decreasing the admissible values for the number $q$ of non-straggling ENs. Repetition coding can instead reduce the download latency by enabling cooperative transmission among multiple ENs computing the same outputs[@jingjing; @KKLMEC], as discussed below.
In the input upload phase, each user divides its $N$ input vectors into $\binom{K}{r}$ subsets $\{\mathcal{U}_{i,\mathcal{K}^{'}}\}$, with each subset uploaded to a distinct subset $\mathcal{K}^{'}$ of $r$ ENs. Thus, as shown in Fig. \[mds\_repetition\], in the computing phase, each input vector of any user is computed by a subset of $p_1$ non-straggling ENs with $p_1$ being at least $r\!-(\!K\!-q)$ and at most $\min\{r,q\}$. Therefore, since each encoded row of $\mathbf{A}$ is replicated at a subset of $\rho_2$ ENs, after computation, each MDS-encoded row-vector product result for a user will be replicated at a subset of $p_2$ non-straggling ENs, with $p_2$ being at least $\max\{\rho_2\!-\! K\!+\!p_1,1\}$ and at most $\min\{p_1,\rho_2\}$.
In the output download phase, as proposed in [@FanXu], each subset of $p_2$ ENs computing the same coded outputs can first use zero-forcing (ZF) precoding to null the interfering signal caused by common outputs at a subset of $p_2-\!1$ undesired users. When the number of undesired users does not exceed $p_2-\!1$, i.e., when $M\!-\!1\!\le\!p_2-\!1$, by ZF precoding, each user only receives its desired outputs with all undesired outputs being cancelled out. When this condition is violated, i.e., when $M\!>\!p_2$, after ZF precoding, each output still causes interferences to $M\!-\!p_2$ undesired users. As detailed in [@FanXu], interference alignment can be applied in cascade to the ZF precoders in order to mitigate the impact of these interfering signals.
Bounds
------
The scheme summarized above and detailed in Sec. \[AchievableScheme\] achieves the following region.
\[achievableresults\] (Inner bound). For the described MEC network with $M$ users and $K$ ENs, each with storage capacity $\mu\!\in\![\frac{1}{K},1]$, the following communication-computation latency triplet $\big(\tau^{u}_a(r),$ $\tau^c_a(r,q),\tau^{d}_a(r,q)\big)$ is achievable
$$\begin{aligned}
\!\tau^{u}_a(r)&\!=\! \frac{(M\!-\!1)r \!+\! K}{K}, \label{achieveUpoad} \\
\!\tau^c_a(r,q)&\!=\!\frac{Mr\mu(H_k\!-\!H_{K-q})}{K}\!, \label{achievecompute}\\
\!\tau^d_a(r,q)&\!=\!\sum\limits^{\min\{r,q\}}_{p_1\!=r-\!K\!+q}\!\!\!\!\!B_{p_1}\!\!\left(\sum\limits^{l_{max}}_{p_2=l_{p_1}}\!\!\frac{B_{p_2}}{d^d_{p_1,M,p_2}}\!+\!
\!\frac{B_{l_{p_1}\!-1}}{d^d_{p_1,M,l_{p_1}\!-1}}\!\right)\!, \label{achievedownload}\end{aligned}$$
for any repetition order $r$ and recovery order $q$ in the set
$$\label{regionR}
\mathcal{R}\!\triangleq\!\big\{(r,q)\!:r\!\in\![K],q\!\in\![K],~\text{and}~ (r\!-\!K\!+\!q)\mu\!\ge\!1\big\},$$
where
$H_K\!=\!\sum^K_{k=1}\frac{1}{k}$, $B_{p_1}\!=\!\binom{q}{p_1}\binom{K-q}{r-p_1}\frac{1}{\binom{K}{r}}$, $B_{p_2}\!=\!\binom{p_1}{p_2}\binom{K-p_1}{\rho_2-p_2}\frac{\rho_1}{\binom{K}{\rho_2}}$, $B_{l_{p_1}\!-1}\!=\!1\!-\!\!\sum^{l_{max}}_{p_2=l_{p_1}}\!\!B_{p_2}$
;
$$\label{dofdd}
d^d_{p_1,M,p_2}=\left\{
\begin{aligned}
&1, &p_2\!\ge\!M~~~~\\
&\frac{\binom{p_1}{M-1}(M\!-\!1)}{\binom{p_1}{M-1}(M\!-\!1)+1}, &~~p_2\!=\!M\!-\!1\\
&\max\left\{d', \frac{p_2}{M}\right\}, &~~p_2\!\le\!M\!-\!2
\end{aligned}
~,\right.$$
with $d'\!\triangleq\!\max_{1\le t\le p_2}\frac{p_1-t+1}{M+p_1-2t+1}$;
$$\begin{aligned}
\label{theoremcoderate}
\!\!\rho_2\!=\!\inf\!\bigg\{\rho\!:\!\!\binom{K}{\rho}\!-\!\binom{2K\!-\!r\!-\!q}{\rho}\!\ge\! \frac{1}{\rho_1}\binom{K}{\rho},\rho_1\rho\!=\!K\mu,\rho_1\!\in\!\big\{1,K\mu/(K\mu\!-\!1),K\mu/(K\mu\!-\!2)\cdots,K\mu\big\}\!\bigg\};\!\!\end{aligned}$$
and
$$\begin{aligned}
\label{replicationmin}
\!\!\!l_{p_1} \!\!=\! \inf\!\bigg\{l\!:\!\!\sum\nolimits^{l_{max}}_{p_2=l}\!\!B_{p_2}m\!\le\!m, l\!\in\![l_{min}\!:\!l_{max}], l_{max}\!=\!\min\{p_1,\rho_2\}, l_{min}\!=\!\max\{\rho_2\!-\! K\!+\!p_1,1\} \bigg\}.\!\end{aligned}$$
For an NULT $\tau^u\!=\!\tau^u_a(r)$ given in (\[achieveUpoad\]) for some $r$, an inner bound $\mathscr{T}_{in}(\tau^{u})$ on the compute-download latency region is given as the convex hull of the set $\big\{\!\left(\tau^c_a(r,q),\tau^d_a(r,q)\right)\!:\!q\!\in\!\big[\lceil\!\frac{1}{\mu}\!\rceil\!+\!K\!-\!r\!:\!K\big]\!\big\}$.
We also have the following converse.
\[lower\_bound\] (Converse). For the same MEC network, the set of all admissible pairs $(r,q)$ is included in the set $\mathcal{R}$ in (\[regionR\]). Furthermore, any feasible communication-computation latency triplet $\left(\tau^{u}(r),\tau^{c}(r,q),\tau^{d}(r,q)\right)$ for pairs $(r,q)$ in $\mathcal{R}$ is lower bounded as
$$\begin{aligned}
\!\tau^{u}(r)&\!\ge\! \tau^u_a(r),\!\label{lowerupload}\\
\!\tau^{c}(r,q)&\!\ge\!\tau^{c^*}\!(r,q)\!=\!\max\limits_{t\in[q-1]}\!\!\frac{(H_K\!-\! H_{K-q+t})(r\!-\!K\!+\!t)^{+}M\mu }{t}\!,\!\label{lowercompute} \\
\!\tau^{d}(r,q)&\!\ge\!\tau^{d^*}\!(r,q)\!=\!\max_{t\in\{1,\cdots,\min\{q,M\}\}}\!\!\!\frac{M\!-\!(M\!-\!t)(q\!-\!t)\frac{r}{K}\mu}{t}\!.\! \label{lowerdownload}\end{aligned}$$
For an NULT $\tau^u\!=\!\tau^u_a(r)$ in (\[achieveUpoad\]) for some $r$, an outer bound $\mathscr{T}_{out}(\tau^{u})$ of the compute-download latency region is given as the convex hull of set $\big\{\!\!\left(\tau^{c^*}\!\!(r,q),\tau^{d^*}\!\!(r,q)\right)\!\!:\!q\!\in\!\!\big[\lceil\!\frac{1}{\mu}\!\rceil\!+\!K\!-r\!:\!\!K\big]\!\big\}$.
*Proof:* By considering all possible task assignments and the effect of random stragglers, bound (\[lowerupload\]) is derived via genie-aided arguments; (\[lowercompute\]) is derived by some basic inequalities; The proof of (\[lowerdownload\]) follows and generalizes steps in [@sengupta2017fog Eq. (63)-(65)]. The detailed proof is available in the Appendix. Fig. \[latencytriples\] plots the derived inner and outer bounds on the compute-download latency region $\mathscr{T}^{*}(\tau^{u})$ for $M\!\!=\!\!K\!\!=\!\!10$ and for two different values of $\tau^u$. First, we observe that, *as $q$ increases, the NDLT is reduced at the expense of an increasing NCT*: A larger $q$ enables more opportunities for transmission cooperation at the ENs during output downloading, while increasing, on average, the time required for $q$ ENs to complete their tasks. Furthermore, comparing Fig. \[bR\] with Fig. \[sR\], we also see that *allowing for a longer upload time increases the compute-download latency region*. This is because more information is uploaded to ENs over a larger latency $\tau^u$. Thus, on the one hand, users can wait for fewer ENs to finish their computing tasks, reducing the NCT; and, on the other hand, the increased duplication of outputs also increases opportunities for transmission cooperation to reduce the NDLT.
Optimality
----------
\[gapLemma\] (Optimality). For a given $r\!\in\![1,K]$, it is not possible to reduce the achievable NULT $\tau^u_a(r)$ in (\[achieveUpoad\]) while still guaranteeing the feasibility of a triplet $\left(\tau^{u}_a(r),\tau^{c},\tau^{d}\right)$. Furthermore, for a sufficiently large NULT $\tau^u\!\ge\!\tau^u_a(K\!-\!n_1)$ and small recovery order $q\!\le\!K(1\!-\!1/n_2)\!+\!1$, with integers $0\!\le\!n_1\!\!<\!q/2$ and $n_2\!\ge\!1$, the multiplicative gap between the achievable NCT in (\[achievecompute\]) and its lower bound $\tau^{c^*}$ in (\[lowercompute\]) satisfies the inequality $\tau^{c}/\tau^{c^*}\!\le\!(1\!+\!n_1) (1\!+\!n_2)$. Finally, for a sufficiently large NULT $\tau^u\!\ge\!\tau^u_a(K\!-\!n)$, with integer $n\!\ge\! 0$, the multiplicative gap between the achievable NDLT in (\[achievedownload\]) and its lower bound $\tau^{d^*}$ in (\[lowerdownload\]) satisfies the inequality $\tau^d/\tau^{d^*}\!\le\!2(1\!+\!n\mu)$, and hence, if $r\!=\!K$, we have $\tau^d/\tau^{d^*}\!\le\!2$.
The multiplicative gaps in Fig. \[latencytriples\] are consistent with Lemma \[gapLemma\], since $\tau^c/\tau^{c^*}\!=\!3.61\!<\!22$ at $(r,q)\!=\!(9,10)$ (i.e., $n_1\!=\!1$ and $n_2\!=\!10$) and $\tau^d/\tau^{d^*}\!=\!1.32\!<\!3.2$ at $(r,q)\!=\!(9,3)$ (i.e., $n\!=\!1$).
Special Cases
-------------
In the special case when $r\!=\!K$, hence ignoring limitations on the uplink transmission, the achievable NDLT (\[achievedownload\]) reduces to the normalized communication delay in [@jingjing Eq. (13)], when using only ZF precoding in downlink. Furthermore, when setting $q\!=\!K$, hence ignoring stragglers[’]{} effects, and $\mu\!=\!1$, i.e., ignoring ENs[’]{} storage constraint, the achievable NDLT (\[achievedownload\]) reduces to $\tau^d\!=\!M/\min\{K,M\}$, which recovers the communication load in [@8007057 Remark 5], and the NDT with cache-aided EN cooperation in [@sengupta2017fog Eq. (25)].
Details on The Proposed Policy {#AchievableScheme}
==============================
Consider $\mu\!\in\!\!\{1/K,2/K,\cdots\!,1\}$[^2], a repetition order $r\!\in\![K]$, and a recovery order $q\!\in\![K]$ in the feasible set $\mathcal{R}$ in (\[regionR\]). Note that each input vector is replicated on at least $r\!-\!(K\!-\!q)$ non-stragglers, so set $\mathcal{R}$ ensures that any subset of $r\!-\!K\!+\!q$ ENs can store at least $m$ coded rows of $\mathbf{A}$ to multiply each input.
### Task Assignment {#achieveuploadscheme}
Following the discussion in Sec. \[keyidea\], each EN $k$ is assigned $M\binom{K\!-\!1}{r-\!1}N/\binom{K}{r}\!=\!MNr/K$ inputs corresponding to subsets $\{\mathcal{U}_{i,\mathcal{K}^{'}}\!: i\!\in\!\mathcal{M}, \mathcal{K}^{'}\!\subseteq\!\mathcal{K}, |\mathcal{K}^{'}|\!=\!r, k\!\in\!\mathcal{K}^{'}\}$.
### Input Uploading {#uploadtime}
For uplink transmissions, each user needs to communicate with all $\binom{K}{r}$ subsets of ENs of cardinality $r$. Therefore, the uplink channel is an X-multicast channel with $M$ transmitters, $K$ receivers, and size-$r$ multicast group. As proved in [@DOfNiesen Theorem 2], its optimal per-receiver DoF is given by $d^{u}_{r}\!=\!Mr\!/(Mr\!+\!K\!\!-\!r)$. The per-receiver rate of this channel in the high SNR regime can be approximated as $d^{u}_{r}\!\times\!\log\!P_u\!+\!o(\log\!P_u)$, so the uploading time can be approximately expressed as $T^u\!=\!\frac{MNr}{K}nB/(d^{u}_{r}\log\!P_u\!+o(\log\!P_u))$. Let $P_u\!\to\!\infty$ and $B\!\to\!\infty$, by Definition \[defenition2\], the NULT at repetition order $r$ is obtained as in (\[achieveUpoad\]).
### Edge Computing
Following Sec. \[keyidea\], the cascaded MDS-repetition code rate pair satisfies $\rho_1\!\!\in\!\!\left\{1,K\mu/(K\mu\!-\!1),K\mu/(K\mu\!-\!2),\cdots,K\mu\right\}$ and $\rho_2\!\!=\!\!K\mu/\rho_1$ under the constraint of the total storage size $k\mu$. Then, we split the coded matrix $\mathbf{A}_c$ into $\binom{K}{\rho_2}$ submatrices $\{\mathbf{A}_{c,\mathcal{K}^{''}}\!\}$, each stored at a distinct subset $\mathcal{K}^{''}\!$ of $\rho_2$ ENs. Then, as shown in Fig. \[mds\_repetition\], any subset of $r\!-\!K\!+\!q$ non-straggling ENs must store at least $m$ encoded rows to compute all outputs, which is ensured by condition $\rho_1m\!-\!\binom{K\!-(r\!-\!K\!+q)}{\!\rho_2\!}\rho_1m/\binom{\!K\!}{\!\rho_2\!}\!\ge\!m$ given in (\[theoremcoderate\]). Further, in order to create more spatial redundancy, the parameter $\rho_2$ is maximize as in (\[theoremcoderate\]) under the recovery condition. As an example, in Fig. \[downlinkchannel\], for $K\!\!=\!\!M\!\!=\!\!5$, $m\!=\!40$, $\mu\!\!=\!\!3/5$, $q\!\!=\!\!3$, and $r\!\!=\!\!4$, by (\[theoremcoderate\]), we have $(\rho_1,\rho_2)\!=\!(3/2,2)$ such that $\mathbf{A}$ is encoded into $60$ rows and then split into $\binom{5}{2}\!\!=\!\!10$ submatrices, each with $6$ rows replicated at $2$ ENs. By the given task input assignment $\{\mathcal{U}_{i,\mathcal{K}^{'}}\}$, each EN $k$ computes $MNr\mu m/K$ row-vector products. Thus, by Definition \[defenition2\] and [@arnold2008first Eq. (4.6.6)], the NCT is given by $\tau^c(r,q)\!=\lim\limits_{m\to\infty}\!\frac{\mathbb{E}\left[\left\{\frac{MNr\mu m}{K}\omega_k\right\}_{q:K}\right]}{Nm/\eta}
\!=\!\frac{Mr\mu\mathbb{E}\left[\{\omega_k\}_{q:K}\right]\!\eta}{K}
\!=\!\frac{Mr\mu(H_k\!-\!H_{K-q})}{K}$.
{width="4.1in" height="2.5in"}
\[downlinkchannel\]
### Output Downloading
Following Sec. \[keyidea\], for each user, the number of input vectors computed by $p_1$ non-straggling ENs equals $\binom{K\!-q}{r-p_1}N/\binom{K}{r}\!=\!\big(B_{p_1}/\binom{q}{p_1}\big)N$. Furthermore, the number of encoded rows of $\mathbf{A}$ replicated at $p_2$ non-straggling ENs is $\binom{K-p_1}{\rho_2-p_2}\rho_1m/\binom{K}{\rho_2}\!=\!\big(B_{p_2}/\binom{p_1}{p_2}\big)m$. Thus, as discussed in Sec. \[keyidea\], when $p_2\!\ge\!M$, each subset of $p_2$ ENs computing the same $M\big(B_{p_1}/\binom{q}{p_1}\big)N\big(B_{p_2}/\binom{p_1}{p_2}\big)m$ outputs can cooperatively transmit these outputs to $M$ users via ZF precoding. In contrast, when $p_2\!<\!M$, each subset of $p_2$ ENs partitions each common output into $\binom{M-1}{p_2-1}$ submessages, and first use ZF precoding to null the interference caused by each submessage at a distinct subset of $p_2\!-\!1$ undesired users. Then, by cascading ZF precoding with asymptotic interference alignment, the rest of the interfering signals from each subset of $t-\!1$ ENs can be aligned into a distinct subspace at each user. For example, for $p_2\!=\!M\!-\!1$, each submessage only causes interference to one user, so all interfering signals at each user can be aligned into one common subspace. The resulting downlink is a cooperative-X channel with $p_1$ transmitters, $M$ receivers, and size-$p_2$ cooperation group. By [@FanXu Lemma 1], an achievable per-receiver DoF of this channel is given as (\[dofdd\]). Similar to the calculation of NULT in Sec. \[uploadtime\], by Definition \[defenition2\], the NDLT for each user to download the outputs replicated at $p_2$ non-stragglers is given by $$\tau^d_{p_1,p_2}\!=\!\frac{B_{p_1}B_{p_2}/\binom{q}{p_1}}{d^d_{p_1,M,p_2}}. \label{p_1,p_2}$$
Due to the MDS coding, the total number of coded outputs available on the $p_1$ ENs may exceed the number $m$ needed to recover the outputs of each input vector. Denote by $l_{p_1-1}$, $\max\{\rho_2\!-\!K\!+\!p_1,1\}\!\le\!l_{p_1-1}\!\le\!\min\{p_1,\rho_2\}$, the minimum degrees of replication of needed coded outputs on the $p_1$ ENs, it is determined by (\[replicationmin\]), so the number of needed coded outputs replicated at $l_{p_1}\!\!-\!1$ ENs equals $M(B_{p_1}/\binom{q}{p_1})NB_{l_{p_1}\!-1}m$, where $B_{l_{p_1}-1}\!=\!1\!-\!\!\sum^{l_{max}}_{p_2=l_{p_1}}\!\!B_{p_2}$. Note that $B_{l_{p_1}\!-\!1}m/\binom{p_1}{l_{p_1}\!-1}$ can be seen as an integer for infinitely large $m$ since $\big(\!B_{l_{p_1}\!-1}m\!\!\!\mod{\!\binom{p_1}{l_{p_1}\!-1}}\big)/m\!\le\!\binom{p_1}{l_{p_1}\!-1}/m\!\to\! 0$ as $m\!\to\!\infty$. Hence, any exclusive subset of $l_{p_1}\!-\!1$ ENs can cooperatively transmit $\big(B_{p_1}/\binom{q}{p_1}\big)N\big(B_{l_{p_1}\!-1}/\binom{p_1}{l_{p_1}\!-1}\big)m$ common outputs to each of $M$ users, so the downlink channel is also a cooperative-X channel with $p_1$ transmitters, $M$ receivers, and cooperation group size $l_{p_1}\!-\!1$. Similar to (\[p\_1,p\_2\]), the NDLT for each user to download the outputs replicated at $l_{p_1}\!\!-\!1$ non-stragglers is given by $$\tau^d_{p_1,l_{p_1}-1}\!=\!\frac{B_{p_1}B_{l_{p_1}-1}/\binom{q}{p_1}}{d^d_{p_1,M,l_{p_1}-1}}.$$ Therefore, by considering all the inputs computed by $p_1$ ENs, with $p_1$ from $r\!-\!(K\!-\!q)$ to $\min\{r,q\}$, and all the outputs replicated at $p_2$ ENs, with $p_2$ from $l_{p_1-1}$ to $\min\{p_1,\rho_2\}$, and by summing all download times, the NDLT is obtained in (\[achievedownload\]).
For example, in Fig. \[downlinkchannel\], for inputs $\{\mathbf{u}_{i,1}\}^5_{i=1}$ computed by $p_1\!\!=\!\!2$ ENs, there are $30$ outputs $\{\mathbf{a}_{25}\mathbf{u}_{i,1},\ldots,\mathbf{a}_{30}\mathbf{u}_{i,1}\}^5_{i=1}$ replicated at $p_2\!=\!2$ ENs. These $30$ outputs can be cooperatively transmitted back to the users via ZF precoding, resulting in a 2-transmitter 5-receiver MISO broadcast channel that is a special case of cooperative-X channels under full transmitter cooperation. As a result, an NDLT of $3/40$ is achieved. After this round of transmission, users still need $34\!\times\!5\!=\!170$ outputs inside the blue dashed rectangle in Fig. \[downlinkchannel\], which can be transmitted by the 2 ENs via interference alignment. The downlink is a 2-transmitter 5-receiver X channel that is a special case of cooperative-X channels with size-$1$ cooperation group, yielding the NDLT of $51/100$. Thus, the NDLT for outputs of $\{\mathbf{u}_{i,1}\}^5_{i=1}$ is $3/40\!+\!51/100\!=\!117/200$. Then, the input vectors $\{\mathbf{u}_{i,2}\}^5_{i=1},\{\mathbf{u}_{i,3}\}^5_{i=1}$ are also computed by $p_1\!\!=\!\!2$ ENs, their outputs can be transmitted in a similar way, which achieves an NDLT of $(117/200)\!\times\!2\!=\!117/100$. Likewise, for the inputs $\{\mathbf{u}_{i,4}\}^5_{i=1},\{\mathbf{u}_{i,5}\}^5_{i=1}$ computed by $p_1\!\!=\!\!3$ ENs, the 3-transmitter 5-receiver cooperative X-channel with size-$2$ cooperation group, and 3-transmitter 5-receiver X-channel are formed to transmit the total $400$ outputs, yielding an NDLT of $(21/100\!+\!77/300)\!\times\! 2\!=\!14/15$. Thus, in this example, the total NDLT at $(r,q)\!=\!(4,3)$ is $14/15\!+\!(117/200)\!\times\!3\!=\!1613/600$.
### Inner Bound of Compute-Download Latency Region
For an NULT $\tau^u\!=\!\tau^u_a(r)$ given in (\[achieveUpoad\]) for some $r\!\in\!\mathcal{R}$, where $\mathcal{R}$ is given by (\[regionR\]), the feasible values of recovery order $q$’s should satisfy $\lceil\frac{1}{\mu}\rceil\!+\!K\!-r\!\le\!q\!\le\!K$. The non-integer $q$ can be rewritten as $q\!=\!\lambda\lceil q\rceil\!+\! (1\!-\!\lambda)\lfloor q\rfloor$ for some $\lambda\!\in\![0,1]$. Based on Remark \[remarkconvex\], we can combine our proposed policies at $(r,\lceil q\rceil)$ and $(r,\lfloor q\rfloor)$ via time- and memory-sharing methods to achieve the NCT-NDLT pair $\left(\tau^{c}_a(r,q),\tau^{d}_a(r,q)\right)\! = \!\lambda \left(\tau^{c}_a(r,\lceil q\rceil),\tau^{d}_a(r,\lceil q\rceil)\right) \!+ (1\!-\!\lambda)\!\left(\tau^{c}_a(r,\lfloor q\rfloor),\tau^{d}_a(r,\lfloor q\rfloor)\right)$. In fact, for any two integer-valued $q_1$ and $q_2$, any convex combination of achievable pairs $(\tau^{c}_a(r,q_1),\tau^{d}_a(r,q_1)$ and $(\tau^{c}_a(r,q_2),\tau^{d}_a(r,q_2))$ can also be achieved. So an inner bound $\mathscr{T}_{in}(\tau^{u})$ of the compute-download latency region is given as the convex hull of set $\big\{\!\left(\tau^{c}_a(r),\tau^d_a(r,q)\right)\!:\!q\!\in\!\big[\lceil\frac{1}{\mu}\rceil\!+\!K\!-\!r\!:\! K\big]\big\}$.
Appendix: Converse Proofs {#appendix-converse-proofs .unnumbered}
=========================
In this appendix, we prove the lower bounds in Theorem \[lower\_bound\] and the multiplicative gaps in Lemma \[gapLemma\]. For a repetition-recovery order pair $(r,q)$, as discussed, each input will be replicated on at least $r\!-\!(K\!-\!q)$ non-stragglers. The condition $(r-\!K\!+q)\mu\!\ge\!1$ must be satisfied such that any subset of $r\!-\!K\!+\!q$ non-stragglers are able to provide sufficient information to compute the outputs of all users. This proves that no pair $(r,q)$ is feasible outside the feasible set $\mathcal{R}$ in (\[regionR\]). Consider an arbitrary user input assignment policy $\big\{\mathcal{U}_{i,\mathcal{K}^{'}}\!: i\!\in\!\mathcal{M},\mathcal{K}^{'}\!\subseteq\!\mathcal{K}, |\mathcal{K}^{'}|\!=\!r\big\}$. The input vectors from user $i$ assigned to EN $k$ are denoted as set $\mathcal{I}_{i,k}\!\triangleq\!\big\{\mathcal{U}_{i, \mathcal{K}^{'}}\!\big\}_{\mathcal{K}^{'}\subset\mathcal{K}:k\in\mathcal{K}^{'}}$ for $i\!\in\!\mathcal{M}$ and $k\!\in\!\mathcal{K}$, and the size of $\mathcal{I}_{i,k}$ is denoted as $\gamma_{i,k}NnB$ bits, where the ratio $\gamma_{i,k}$ satisfies $$\begin{aligned}
&\sum\limits_{k\in\mathcal{K}}\gamma_{i,k} = r ,~ i \in\mathcal{M} \label{cons111}\\
&~0\le \gamma_{i,k} \le 1,~i \in\mathcal{M},~\text{and}~k\in\mathcal{K}. \label{cons222}\end{aligned}$$ In the rest of this appendix, we first derive the lower bounds on the NULT, NCT, and NDLT for a particular task assignment policy $\big\{\mathcal{U}_{i,\mathcal{K}^{'}}\!\big\}$ with repetition-recovery order $(r,q)\!\in\!\mathcal{R}$. Then, we minimize these lower bounds over all feasible task assignment policies to obtain the minimum NULT $\tau^{u^*}$, NCT $\tau^{c^*}$, and NDLT $\tau^{d^*}$. Then, for a fixed lower bound of NULT at $r\!\in\!\big[\lceil\frac{1}{\mu}\rceil,K\big]$, by convexity of the compute-download latency region, an outer bound of this region is given as described in Theorem \[lower\_bound\].
Lower Bound and Optimality of NULT
----------------------------------
For a particular task assignment policy $\big\{\mathcal{U}_{i,\mathcal{K}^{'}}\!\big\}$, we use genie-aided arguments to derive a lower bound on the NULT. Specifically, for any EN $k$ and user $i_o$, consider the following three disjoint subsets of task input vectors (or messages): $$\begin{aligned}
\mathcal{W}_{r} &= \{\mathcal{U}_{i, \mathcal{K}^{'} }: i\in\mathcal{M}, k\in\mathcal{K}^{'}\},\\
\mathcal{W}_{t} &= \{\mathcal{U}_{i, \mathcal{K}^{'} }:i=i_o,k\notin \mathcal{K}^{'} \},\\
\overline{\mathcal{W}} &= \{\mathcal{U}_{i,\mathcal{K}^{'}}: i\ne i_o~\text{and}~k\notin \mathcal{K}^{'}\}.\end{aligned}$$ The set $\mathcal{W}_{r}$ indicates the input messages from all users assigned to EN $k$ or all input messages that EN $k$ needs to decode, which satisfies $|\mathcal{W}_{r}|\!=\!\sum_{i\in\mathcal{M}}\gamma_{i,k}NnB$. The set $\mathcal{W}_{t}$ indicates the input messages from user $i_o$ assigned to all ENs in $\mathcal{K}$ excluding EN $k$, which satisfies $|\mathcal{W}_{t}|\!=\!(1\!-\!\gamma_{i_o,k})NnB$. The last set $\overline{\mathcal{W}}$ indicates all input messages from users in $\mathcal{M}$ excluding user $i$ assigned to ENs in $\mathcal{K}$ excluding EN $k$.
Let a genie provide the messages $\overline{\mathcal{W}}$ to all ENs, and additionally provide messages $\mathcal{W}_{r}$ to ENs in $\mathcal{M}/\{k\}$. The received signal of EN $j$ can be represented as $$\begin{aligned}
\mathbf{y}_j &= \sum\limits^{M}_{i=1,i\ne i_o}\mathbf{H}^u_{ji}\mathbf{X}_{i} + \mathbf{H}^u_{ji_o}\mathbf{X}_{i_o}+\mathbf{Z}^u_j,
$$ where the diagonal matrices $\mathbf{H}^u_{ji}$, $\mathbf{X}_{i}$, and $\mathbf{Z}^u_j$ denotes the channel coefficients from user $i$ to EN $j$, the signal transmitted by user $i$, and the noise received at EN $j$, respectively, over the block length $T^u$. The ENs in $\mathcal{M}/\{k\}$ have messages $\overline{\mathcal{W}}\!\cup\!\mathcal{W}_{r}$, which include the input messages that EN $k$ should decode and input messages transmitted by all users excluding user $i_o$. By this genie-aided information, each EN $j\!\in\!\mathcal{M}/\{k\}$ can construct the transmitted symbols $\{\mathbf{X}_i\!:i\!\ne\!i_o\}$ and subtract them from the received signal. So we can rewrite the signal received at EN $j\!\ne\!k$ as $$\label{equation1122}
\bar{\mathbf{y}}_j = \mathbf{y}_j - \sum\limits_{i\in\mathcal{M}/\{i_o\}}\mathbf{H}^u_{ji}\mathbf{X}_{i} = \mathbf{H}^u_{ji_o}\mathbf{X}_{i_o} + \mathbf{Z}^u_j.$$ Each EN $j\!\in\!\mathcal{M}/\{k\}$ needs to decode the input messages in subset of $\mathcal{W}_{t}$ assigned to it, denoted as $\mathcal{W}^j_{t}$. By Fano[’]{}s inequality and (\[equation1122\]), we have $$\label{fano1}
H(\mathcal{W}^j_{t}|\mathbf{y}_j,\overline{\mathcal{W}},\mathcal{W}_{r}) \le T^u \epsilon, ~~j\in\mathcal{M}/\{i\}.$$ Since EN $k$ can decode the input messages $\mathcal{W}_{r}$ assigned to it, by Fano[’]{}s inequality, we also obtain $$\label{Fano11}
H(\mathcal{W}_{r}|\hat{\mathbf{y}}_k,\overline{\mathcal{W}}) \le T^u\epsilon .$$ Then, EN $k$ can construct the transmitted symbols $\{\mathbf{X}_i\!:i\!\ne\!i_o\}$ based on genie-aided messages $\overline{\mathcal{W}}$ and its decoded messages $\mathcal{W}_{r}$, and subtract them from its received signal, obtaining
$$\bar{\mathbf{y}}_k = \mathbf{y}_k - \sum\limits_{i\in\mathcal{M}/\{i_o\}}\mathbf{H}^u_{ki}\mathbf{X}_{i} = \mathbf{H}^u_{ki_o}\mathbf{X}_{i_o} + \mathbf{Z}^u_k.$$
Reducing the noise in the constructed signal $\bar{\mathbf{y}}_k$ and multiplying it by $\mathbf{H}^u_{ji_o}\left(\mathbf{H}^u_{ki_o}\right)^{-1}$, we obtain $$\bar{\mathbf{y}}^{j}_k = \mathbf{H}^u_{ji_o}\left(\mathbf{H}^u_{ki_o}\right)^{-1} \bar{\mathbf{y}}_k = \mathbf{H}^u_{ji_o}\mathbf{X}_{i_o} + \hat{\mathbf{Z}}^u_j,$$ where $\hat{\mathbf{Z}}^u_j$ is the reduced noise. By (\[equation1122\]), we see that $\bar{\mathbf{y}}^{j}_k $ is a degraded version of $\bar{\mathbf{y}}_j $ for EN $j\!\in\!\mathcal{M}/\{i\}$. Hence, for the messages that ENs in $\mathcal{M}/\{i\}$ can decode, EN $k$ must also be able to decode them, and we have $$\label{fano2}
\!\!\!H(\mathcal{W}^j_{t}|\hat{\mathbf{y}}_k,\!\overline{\mathcal{W}},\!\mathcal{W}_{r})\!\le\! H(\mathcal{W}^j_{t}|\mathbf{y}_j,\!\overline{\mathcal{W}},\!\mathcal{W}_{r})\! \le\! T^u \epsilon, j\!\!\in\!\!\mathcal{M}\!/\!\{i\}\!.\!\!$$ Using genie-aided information, receiver cooperation, and noise reducing as discussed above can only improve channel capacity. Thus, we obtain the following chain of inequalities,
$$\begin{aligned}
&|\mathcal{W}_{r}|+|\mathcal{W}_{t}|\nonumber\\
&= H(\mathcal{W}_{r},\mathcal{W}_{t}) \nonumber\\
&\stackrel{(a)}{=}\!H(\mathcal{W}_{r},\mathcal{W}_{t}|\overline{\mathcal{W}})\nonumber\\
&\stackrel{(b)}{=}\!I(\mathcal{W}_{r},\mathcal{W}_{t};\hat{\mathbf{y}}_k|\overline{\mathcal{W}}) + H(\mathcal{W}_{r},\mathcal{W}_{t}|\hat{\mathbf{y}}_k,\overline{\mathcal{W}})\nonumber\\
&\stackrel{(c)}{=}\!I(\mathcal{W}_{r},\mathcal{W}_{t};\hat{\mathbf{y}}_k|\overline{\mathcal{W}}) + H({\mathcal{W}_{r}|\hat{\mathbf{y}}_k,\overline{\mathcal{W}}})+H({\mathcal{W}_{t}|\hat{\mathbf{y}}_k,\mathcal{W}_{r},\overline{\mathcal{W}}})\nonumber\\
& \le I(\mathcal{W}_{r},\mathcal{W}_{t};\hat{\mathbf{y}}_k|\overline{\mathcal{W}})+ H({\mathcal{W}_{r}|\hat{\mathbf{y}}_k,\overline{\mathcal{W}}})\!+\! \!\!\!\sum\limits_{j\in\mathcal{M}/\{k\}}\!\!\!\!\!H({\mathcal{W}^j_{t}|\hat{\mathbf{y}}_k,\mathcal{W}_{r},\overline{\mathcal{W}}})\nonumber\\
&\stackrel{(d)}{\le} I(\mathcal{W}_{r},\mathcal{W}_{t};\hat{\mathbf{y}}_k|\overline{\mathcal{W}}) + T^u \epsilon+ \sum\limits_{j\in\mathcal{M}/\{k\}} T^u\epsilon\nonumber\\
&\stackrel{(e)}{\le} I(\mathbf{x}_1,\mathbf{x}_2,\cdots,\mathbf{x}_{a_i},\mathbf{x}_{i_o}:\hat{\mathbf{y}}_i|\overline{\mathcal{W}}) + M T^u\epsilon \nonumber\\
&\stackrel{(f)}{\le}T^u \log P_u + M T^u\epsilon, \label{inequality}\end{aligned}$$
where (a) is due to the independence of messages, (b) and (c) are based on the chain rule, (d) follows Fano[’]{}s inequalities (\[Fano11\]) and (\[fano2\]), (e) uses the data processing inequality, and (f) follows the DoF bound of MAC channels. Dividing (\[inequality\]) by $NnB\!/\!\log\!P_u$, and let $P_u\!\to\!\infty$ and $\epsilon\!\to\!0$ as $B\to\infty$, we have $$\begin{aligned}
\!\!\!\tau^u\!&\ge\!\frac{|\mathcal{W}_{r}|\!+\!|\mathcal{W}_{t}|}{NnB}\!=\!\sum\limits_{i\in\mathcal{M}}\!\!\gamma_{i,k} \!+\! 1\!-\gamma_{i_o,k}\! = \!\sum_{i\in\mathcal{M}/\{i_o\}}\!\!\!\gamma_{i,k}\! +\!1.\!\!\end{aligned}$$ Hence, the NULT for a particular task assignment $\boldsymbol{\gamma}\!\triangleq\![\gamma_{i,k}]_{i\in\mathcal{M},k\in\mathcal{K}}$ satisfies $\tau^u \!\ge\! \sum_{i\in\mathcal{M}/\{i_o\}}\!\gamma_{i,k} \!+\! 1$ for $k\!\in\!\mathcal{K},i_o\!\in\!\mathcal{M}$, i.e., the minimum NULT for task assignment policy $\boldsymbol{\gamma}$ is lower bounded by
$$\tau^{u^*}(r,\boldsymbol{\gamma}) \ge \max\limits_{ k\in\mathcal{K},\forall i_o\in\mathcal{M}}~ \sum_{i\in\mathcal{M}/\{i_o\}}\gamma_{i,k} + 1.$$
Furthermore, the minimum NULT over all feasible task assignment is given as $\tau^{u^*}\!(r)\!=\!\min\limits_{\boldsymbol{\gamma}}\tau^{u^*}\!(r,\boldsymbol{\gamma})$, i.e., it can be lower bounded by the optimal solution of the optimization problem
$$\begin{aligned}
\mathcal{P}_1:\quad&\min\limits_{\boldsymbol{\gamma}}~\max\limits_{ k\in\mathcal{K},\forall i_o\in\mathcal{M}}~ \sum_{i\in\mathcal{M}/\{i_o\}}\gamma_{i,k} + 1 \nonumber\\
\mathnormal{s.t.}&\quad (\ref{cons111}), (\ref{cons222}).\nonumber\end{aligned}$$
By defining a new variable $\lambda_{k,\bar{i}_o}\!=\!\sum\limits_{i\in\mathcal{M}/\{i_o\}}\gamma_{i,k}$, Problem $\mathcal{P}_1$ can be transformed into
$$\begin{aligned}
\mathcal{P}_2:\quad&\min\limits_{\boldsymbol{\lambda}}~\max\limits_{ k\in\mathcal{K},\forall i_o\in\mathcal{M}} \lambda_{k,\bar{i}_o} + 1 \nonumber\\
\mathnormal{s.t.}&\quad \sum_{k \in\mathcal{K}}\lambda_{k,\bar{i}_o} = r(M-1), ~ i_o\in\mathcal{M} \\
&\quad ~0\le \lambda_{k,\bar{i}_o} \le M-1,~ k \in\mathcal{K}.\end{aligned}$$
We can easily prove that the optimal solution to $\mathcal{P}_2$ is given by $\lambda^*_{k,\bar{i}_o}\!=\!r(M\!-\!1)/K$ for $k\!\in\!\mathcal{K}$ and $i_o\!\in\!\mathcal{M}$, which is unique and can be proved by contradiction. In turn, we use $\{\lambda^*_{k,\bar{i}_o}\}$ to construct a feasible solution to $\mathcal{P}_1$ by letting $\gamma^*_{i,k}\!=\!\lambda^*_{k,\bar{i}_o}/(M\!-\!1)$ for $i\!\in\!\mathcal{M}$ and $k\!\in\!\mathcal{K}$, and hence obtain the optimal solution to $\mathcal{P}_1$ as $\gamma^*_{i,k}\!=\!r/K$. Therefore, at repetition order $r$, the minimum NULT $\tau^{u^*}(r)$ is lower bounded by
$$\label{lowerbound_tauu}
\tau^{u^*}(r)\ge \frac{r(M-1)+K}{K}.$$
The lower bound of NULT in Theorem \[lower\_bound\] is thus proved. It is seen that (\[lowerbound\_tauu\]) is the same as (\[achieveUpoad\]) in Theorem \[achievableresults\], so the achievable NULT in (\[achieveUpoad\]) is optimal, which proves the optimality of upload times stated in Lemma \[gapLemma\].
Lower Bound and Multiplicative Gap Analysis of NCT {#binaryconverseup}
--------------------------------------------------
### Lower bound
Let $\{X_k\}_{q:K}$ denote the $q$-th smallest value of $K$ variables $\{X_k\}^{K}_{k=1}$ and $q\!:\!K$ denote the index of $q$-th smallest variable. For a particular task assignment policy $\big\{\mathcal{U}_{i,\mathcal{K}^{'}}\!\big\}$ satisfying (\[cons111\]) and (\[cons222\]) and a recovery order $q$, the computing time when the $q$-th fastest EN finishes its assigned tasks is lower bounded by $$\begin{aligned}
T^c_{q:K} &= \bigg\{\!\mu m\sum\limits_{i\in\mathcal{M}}\gamma_{i,k}N \omega_k\!\bigg\}_{\!\!q:K}\nonumber\\
&\stackrel{(g)}{\ge} \max\limits_{t\in[q-1]}\bigg\{\!\mu m \bigg\{\sum\limits_{i\in\mathcal{M}}\gamma_{i,k}N \bigg\}_{\!\!t:K}\!\!\!\cdot\left\{\omega_k\right\}_{q-t:K}\!\bigg\},\end{aligned}$$ where $(g)$ follows the fact that for $K$ product values like the form $\{x_ky_k\}^K_{k=1}$, there must exist $q$ values whose product term (either $x_k$ or $y_k$) is not larger than that of $\{x_ky_k\}_{q:K}$. We thus have $\{x_ky_k\}_{q:K}\!\ge\!\{x_k\}_{t:K}\{y_k\}_{q-t:K}$, $t\!\in\![q\!-\!1]$. Taking the expectation on $T^c_{q:K}$, we have
$$\begin{aligned}
\mathbb{E}\left[ T^c_{q:K} \right] & \ge \mathbb{E}\left[\max\limits_{t\in[q-1]}\bigg\{\!\mu m \bigg\{\sum\limits_{i\in\mathcal{M}}\gamma_{i,k}N \bigg\}_{\!\!t:K}\!\!\!\cdot\left\{\omega_k\right\}_{q-t:K}\!\bigg\}\right]\nonumber \\
& \stackrel{(h)}{\ge} \max\limits_{t\in[q-1]}\bigg\{\!\mu m \bigg\{\sum\limits_{i\in\mathcal{M}}\gamma_{i,k}N \bigg\}_{\!\!t:K}\!\!\!\cdot\mathbb{E}\left[\left\{\omega_k\right\}_{q-t:K}\right]\!\bigg\} \nonumber\\
& \stackrel{(i)}{=} \max\limits_{t\in[q-1]}\frac{(H_K - H_{K-q+t})\mu m}{\eta} \left\{\sum\limits_{i\in\mathcal{M}}\gamma_{i,k}N \right\}_{\!\!t:K},\end{aligned}$$
where (h) follows $\mathbb{E}\big[\max\limits_{t}x_t\big]\!\ge\!\max\limits_{t}\mathbb{E}\left[x_t\right]$, (i) uses the $(q\!-\!t)$-th order statistic of $K$ i.i.d exponential random variables. The second term denotes the $t$-th smallest value among $K$ EN workload sizes. By (\[cons111\]) and (\[cons222\]), for $\forall i\!\in\!\mathcal{M}$, we let $\gamma_{i,k}\!=\!1$, $k\!=\!t\!+\!1\!:\!K,t\!+\!2\!:\!K,\cdots,K\!:\!K$, so the sum of the $t$ smallest values (i.e., $1\!:\!K$, $\cdots$,$t\!:\!K$) is lower bounded by $(r\!-\!K\!+\!t)^{+}NM$. Since the second term also represents the largest value among those $t$ smallest EN workload sizes, so this term can be further lower bounded by the average value $(r\!-\!K\!+\!t)^{+}NM/t$. So the average time for the $q$ fastest ENs to finish their tasks is lower bounded by $T^c(r,q)\!\ge\!\max\limits_{t\in[q-1]}\frac{(H_K - H_{K-q+t})\mu m}{\eta}\frac{(r-\!K\!+t)^{+}NM}{t}$. Normalizing it by $Nm/\eta$, the lower bound of the NCT is given by $$\tau^{c^*}(r,q)\ge \max\limits_{t\in[q-1]}\frac{(H_K - H_{K-q+t})(r-\!K\!+t)^{+}M\mu }{t}. \label{low_compu}$$
### Multiplicative gap
The multiplicative gap between the achievable NCT in Theorem \[achievableresults\] and the lower bound (\[low\_compu\]) satisfies $$\begin{aligned}
\frac{\tau^c(r,q)}{\tau^{c*}(r,q)} &\le \min\limits_{t\in[q-1]}\frac{Mr\mu(H_k\!-\!H_{K-q})t}{K(H_K - H_{K-q+t})(r-\!K\!+t)^{+}M\mu}\nonumber \\
&\le \min\limits_{t\in[q-1]} \frac{t}{(r-\!K\!+t)^{+}} \cdot\left(1+\frac{H_{K-q+t}\!-\!H_{K-q}}{H_K - H_{K-q+t}}\right)\nonumber\\
&\le \frac{q/2}{(r-\!K\!+q/2)^{+}} \cdot \left(1+\frac{\sum^{K-q/2}_{K-q+1}1/i}{\sum^{K}_{K-q/2+1}1/i}\right)\nonumber\\
&\le \frac{q/2}{(r-\!K\!+q/2)^{+}} \cdot \left(1+\frac{\frac{q}{2}\frac{1}{K-q+1}}{\frac{q}{2}\frac{1}{K}}\right) \nonumber\\
&= \frac{q/2}{(r-\!K\!+q/2)^{+}} \cdot \left(1+\frac{K}{K-q+1}\right).\end{aligned}$$ When $r\!\ge\!K\!-\!n_1$ and $q\!\le\!K(1\!-\!1/n_2)\!+\!1$ with integers $0\!\le\!n_1\!<\!q/2$ and $n_2\!\ge\!1$, we have $\frac{q/2}{(r-\!K\!+q/2)^{+}}\!\le\!\frac{q/2}{q/2-n_1}\!\le\!n_1\!+\!1$ and $K/(K\!-q+\!1)\!\le\!n_2$, respectively, and consequently, we have $\tau^c/\tau^{c*}\!\le\!(1\!+n_1 )(1\!+n_2)$. Since the upload time is optimal and increases strictly with $r$, the repetition order satisfies $r\!\ge\!\!K\!-n_1$ when the upload time $\tau^u\!\ge\!\tau^u_a(K\!-\!n_1)$.
Lower Bound and Multiplicative Gap Analysis of NDLT {#binaryconverseup}
---------------------------------------------------
### Lower bound
For a particular task assignment policy $\big\{\mathcal{U}_{i,\mathcal{K}^{'}}\!\big\}$ satisfying (\[cons111\]) and (\[cons222\]), and a particular subset of $q$ ENs denoted as $\mathcal{K}_q\!\subseteq\!\mathcal{K}$ whose outputs are available, each EN $k\!\in\!\mathcal{K}_q$ is assigned $r_{i,k}N$ input vectors from each user $i\!\in\!\mathcal{M}$ and can store $\mu m$ rows of $\mathbf{A}$. Since each user $i$ wants $mN$ row-vector product results $\{\mathbf{v}_{i,j}\!=\!\mathbf{A}_{m\times n}\mathbf{u}_{i,j}\}_{j\in[N]}$, it is equivalent to state that each EN $k$ can store $r_{i,k}\mu$ fractional outputs desired by each user $i$, denoted as $\mathcal{S}_{i,k}\!\triangleq\!\{\mathbf{A}_k\mathbf{u}_{i,j}\!:\mathbf{u}_{i,j}\!\in\!\mathcal{U}_{i,\mathcal{K}^{'}},k\!\in\!\mathcal{K}^{'}\}$ and with size $|\mathcal{S}_{i,k}|\!=\!\gamma_{i,k}\mu NmB$ bits, where $\gamma_{i,k}$ satisfies (\[cons111\]) and (\[cons222\]). Thus, the policy $\big\{\mathcal{U}_{i,\mathcal{K}^{'}}\!\big\}_{i\in\mathcal{M},k\in\mathcal{K}}$ with an available EN set $\mathcal{K}_q$ is equivalent to a particular computation results distribution $\{\mathcal{S}_{i,k}\}_{i\in\mathcal{M},k\in\mathcal{K}_q}$. Let $\mathcal{M}_t\!\subseteq\!\mathcal{M}$ denote an arbitrary subset of $t$ users and $\mathcal{Q}_{q-t}\!\subseteq\!\mathcal{K}_q$ denote an arbitrary subset of $q\!-\!t$ ENs. Also, we have $\mathcal{M}_{M-t}\!=\!\mathcal{M}/\mathcal{M}_t$ and $\mathcal{Q}_t\!=\!\mathcal{K}_q/\mathcal{Q}_{q-t}$. For a particular computation results distribution $\{\mathcal{S}_{i,k}\}_{i\in\mathcal{M},k\in\mathcal{K}_q}$, we adopt the arguments proved in [@sengupta2017fog Lemma 6] to derive the lower bound of the NDLT, i.e., intuitively, *given any subset of $t$ signals received at $t\!\le\!\min\{q,M\}$ users, denoted as $\{Y_i\}_{i\in\mathcal{M}_t}$, and the stored computation results information of $q-t$ ENs, denoted as $\{\mathcal{S}_{i,k}\}_{i\in\mathcal{M},k\in\mathcal{Q}_{q-t}}$, all transmitted signals $\{X_{k}\}_{k\in\mathcal{K}}$ and all the desired outputs $\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M},j\in[N]}$ can be resolved in the high-SNR regime.* First, we have the following equality,
$$\begin{aligned}
&MNmB \!= \! H\left(\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M},j\in[N]}\right) \nonumber\\
\!&= \! I\left(\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M},j\in[N]};\{Y_i\}_{i\in\mathcal{M}_t},\{\mathcal{S}_{i,k}\}_{i\in\mathcal{M},k\in\mathcal{Q}_{q\!-\!t}}\right) \!+ \! H\left(\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M},j\in[N]}|\{Y_i\}_{i\in\mathcal{M}_t},\{\mathcal{S}_{i,k}\}_{i\in\mathcal{M},k\in\mathcal{Q}_{q\!-\!t}}\right).\label{ineq1}\end{aligned}$$
For the first term, following steps in [@sengupta2017fog Eq. (64)], we have
$$\begin{aligned}
&I\left(\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M},j\in[N]};\{Y_i\}_{i\in\mathcal{M}_t},\{\mathcal{S}_{i,k}\}_{i\in\mathcal{M},k\in\mathcal{Q}_{q\!-\!t}}\right) \nonumber\\
&=I\left(\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M},j\in[N]};\{Y_i\}_{i\in\mathcal{M}_t}\right) + I\left(\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M},j\in[N]};\{\mathcal{S}_{i,k}\}_{i\in\mathcal{M},k\in\mathcal{Q}_{q\!-\!t}}|\{Y_i\}_{i\in\mathcal{M}_t}\right)\nonumber\\
&\le I\left(\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M},j\in[N]};\{Y_i\}_{i\in\mathcal{M}_t}\right) +
I\left(\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M},j\in[N]};\{\mathcal{S}_{i,k}\}_{i\in\mathcal{M},k\in\mathcal{Q}_{q\!-\!t}},\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M}_t,j\in[N]}|\{Y_i\}_{i\in\mathcal{M}_t}\right)\nonumber\\
&=I\left(\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M},j\in[N]};\{Y_i\}_{i\in\mathcal{M}_t}\right) +
I\left(\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M},j\in[N]};\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M}_t,j\in[N]}|\{Y_i\}_{i\in\mathcal{M}_t}\right)
\nonumber\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\!+\!I\left(\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M},j\in[N]};\{\mathcal{S}_{i,k}\}_{i\in\mathcal{M},k\in\mathcal{Q}_{q\!-\!t}},|\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M}_t,j\in[N]},\{Y_i\}_{i\in\mathcal{M}_t}\right)\nonumber\\
&\le I\left(\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M},j\in[N]};\{Y_i\}_{i\in\mathcal{M}_t}\right) +
H\left(\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M}_t,j\in[N]}|\{Y_i\}_{i\in\mathcal{M}_t}\right)
\nonumber\\
&~~~+H\left(\{\mathcal{S}_{i,k}\}_{i\in\mathcal{M},k\in\mathcal{Q}_{q\!-\!t}}|\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M}_t,j\in[N]},\{Y_i\}_{i\in\mathcal{M}_t}\right)
-H\left(\{\mathcal{S}_{i,k}\}_{i\in\mathcal{M},k\in\mathcal{Q}_{q\!-\!t}}|\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M},j\in[N]},\{Y_i\}_{i\in\mathcal{M}_t}\right)\nonumber\\
&\!\stackrel{(j)}{\le}\!h\left(\{Y_i\}_{i\in\mathcal{M}_t}\right)-h\left(\{Y_i\}_{i\in\mathcal{M}_t}|\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M},j\in[N]}\right) +
tNmB\epsilon + H\left(\{\mathcal{S}_{i,k}\}_{i\in\mathcal{M},k\in\mathcal{Q}_{q\!-\!t}}|\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M}_t,j\in[N]}\right)\nonumber\\
& \stackrel{(k)}{\le} tT\log\left(2\pi e(\Lambda P^d+1)\right) - h\left(\{n_i\}_{i\in\mathcal{M}_t}|\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M},j\in[N]}\right) +
tNmB\epsilon + \sum\limits_{k\in\mathcal{Q}_{q\!-\!t}}H\left(\{\mathcal{S}_{i,k}\}_{i\in\mathcal{M}}|\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M}_t,j\in[N]}\right)\nonumber\\
& \stackrel{(l)}{\le}\!tT\log\left(2\pi e(\Lambda P^d+1)\right) - tT\log\left(2\pi e \right) +
tNmB\epsilon + \sum\limits_{k\in\mathcal{Q}_{q\!-\!t}}\sum\limits_{i\in\mathcal{M}_{M\!-\!t}}H\left(\mathcal{S}_{i,k}\right)\nonumber\\
&\le tT\log\left(\Lambda P^d+1\right) + tNmB\epsilon + \sum\limits_{k\in\mathcal{Q}_{q\!-\!t}}\sum\limits_{i\in\mathcal{M}_{M\!-\!t}}\gamma_{i,k}\mu NmB, \label{ineq2}
\end{aligned}$$
where, in step $(j)$, $\{Y_i\}$ are continuous random variables, the third term uses Fano[’]{}s inequality, the fourth term is because dropping the condition increases the entropy, the last term in last step is 0 since the storage information $\{\mathcal{S}_{i,k}\}$ are the functions of $\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M}_t,j\in[N]}$; In step $(k)$, the first term uses [@sengupta2017fog Lemma 5], and note that $\Lambda$ defined in [@sengupta2017fog Lemma 5] is a constant only depending on downlink channel coefficients in $\mathbf{H}^d$. For the second term, by [@sengupta2017fog Lemma 6] that proves the adopted argument, we have $$\begin{aligned}
\!\!\!H\left(\{\mathbf{v}_{i,j}\}_{i\in\mathcal{M},j\in[N]}|\{Y_i\}_{i\in\mathcal{M}_t},\{\mathcal{S}_{i,k}\}_{i\in\mathcal{M},k\in\mathcal{Q}_{q\!-\!t}}\right)
\!\le\!tNmB\epsilon\!+\!T\log\det\left( \mathbf{I}_{M-t} +\tilde{\mathbf{H}}^d(\tilde{\mathbf{H}}^d)^{H}\right)\!,\!\! \label{ineq3}\end{aligned}$$ where the $(M\!-\!t)\!\times\!(M\!-\!t)$ matrix $\tilde{\mathbf{H}}^d$ defined in [@sengupta2017fog Lemma 6] only depends on the channel matrix $\mathbf{H}^d$, and $\mathbf{I}_{M\!-\!t}$ is a $(M\!-\!t)\!\times\!(M\!-\!t)$ identity matrix. The expressions of $\tilde{\mathbf{G}}$ and $\Lambda$ are omitted here since they can be treated as constants.
Substituting (\[ineq2\]) and (\[ineq3\]) into (\[ineq1\]), we have
$$\begin{aligned}
\!\!\!M NmB\!\le\!tT\log\left(\Lambda P^d+1\right) + 2tNmB\epsilon \! +\!\!\sum\limits_{k\in\mathcal{Q}_{q\!-\!t}}\sum\limits_{i\in\mathcal{M}_{M\!-\!t}}\!\!\! \! \gamma_{i,k}\mu NmB\!+\! T\log\det\! \left( \mathbf{I}_{M-t}\!+\!\tilde{\mathbf{H}}^d(\tilde{\mathbf{H}}^d)^{H}\right)\!,\!\!\end{aligned}$$
Moving $T$ to the left side and dividing by $\frac{NmB}{\log P^d}$, we have
$$\begin{aligned}
\frac{T}{NmB/\log P^d}\ge \frac{M\!-\!\sum\limits_{k\in\mathcal{Q}_{q\!-\!t}}\sum\limits_{i\in\mathcal{M}_{M\!-\!t}}\!\!\gamma_{i,k}\mu\!-\!2t\epsilon}{t}
\cdot\!\frac{t\log P^d}{t\log\left(\Lambda P^d\!+\!1\right)\!+\!\log\det\left( \mathbf{I}_{M-t} \!+\!\tilde{\mathbf{H}}^d(\tilde{\mathbf{H}}^d)^{H}\right)}.\end{aligned}$$
Taking $P^d\to\infty$ and $\epsilon\to 0$ as $B\to\infty$, the NDLT under the output distribution $\{\mathcal{S}_{i,k}\}_{i\in\mathcal{M},k\in\mathcal{Q}_{q\!-\!t}}$ is lower bounded by $$\tau^{d^*}\!(\mathcal{K}_{q},\mathcal{Q}_{q-t},r) \ge \frac{M\!-\!\!\sum\limits_{k\in\mathcal{Q}_{q\!-\!t}}\sum\limits_{i\in\mathcal{M}_{M\!-\!t}}\!\!\!\gamma_{i,k}\mu}{t},~ \forall \mathcal{Q}_{q-t}\subseteq\mathcal{K}_q.\!\!$$ Note that the adopted argument holds for any subset of $q\!-\!t$ ENs. Thus, by tasking the sum over all possible subset $\mathcal{Q}_{q-t}\subseteq\mathcal{K}_q$, we have $$\begin{aligned}
\binom{q}{q\!-\!t}\tau^{d^*}(\mathcal{K}_{q},q-t,r)& \ge \sum\limits_{\mathcal{Q}_{q-t}\subseteq\mathcal{K}_q} \!\!\!\frac{M\!-\!\!\sum\limits_{k\in\mathcal{Q}_{q\!-\!t}}\sum\limits_{i\in\mathcal{M}_{M\!-\!t}}\!\!\!\gamma_{i,k}\mu}{t}\nonumber\\
& = \frac{\binom{q}{q-t}M\!-\!\!\sum\limits_{i\in\mathcal{M}_{M\!-\!t}}\sum\limits_{\mathcal{Q}_{q-t}\subseteq\mathcal{K}_q}\sum\limits_{k\in\mathcal{Q}_{q\!-\!t}}\!\!\!\gamma_{i,k}\mu}{t}\nonumber\\
& = \frac{\binom{q}{q-t}M\!-\!\!\sum\limits_{i\in\mathcal{M}_{M\!-\!t}}\binom{q-1}{q-t-1}\sum\limits_{k\in\mathcal{K}_q}\!\!\gamma_{i,k}\mu}{t}.\!\!\end{aligned}$$ For the particular policy $\big\{\mathcal{U}_{i,\mathcal{K}^{'}}\!\big\}$ with repetition order $r$ and satisfying (\[cons111\]) and (\[cons222\]), this lower bound also holds for any subset $\mathcal{K}_{q}$ since $K\!-\!q$ stragglers occur randomly, by taking the sum over all possible subsets $\mathcal{K}_{q}\subseteq\mathcal{K}$, we have
$$\begin{aligned}
\!\!\!\!\binom{\!K\!}{\!q\!}\!\binom{\!q\!}{\!q\!-\!t\!}\tau^{d^*}\!(q,q\!-\!t,r)
& \ge \sum\limits_{\mathcal{K}_{q}\subseteq\mathcal{K}}\!\!\!\frac{\binom{q}{q-t}M\!-\!\!\sum\limits_{i\in\mathcal{M}_{\!M\!-\!t}}\!\!\!\!\binom{q-1}{q-t-1}\!\!\sum\limits_{k\in\mathcal{K}_q}\!\!\gamma_{i,k}\mu}{t}\nonumber\\
& = \frac{\binom{K}{q}\binom{q}{q-t}M\!-\!\!\!\sum\limits_{i\in\mathcal{M}_{\!M\!-\!t}}\!\!\!\binom{q-1}{q-t-1}\!\sum\limits_{\mathcal{K}_{q}\subseteq\mathcal{K}}\sum\limits_{k\in\mathcal{K}_q}\!\!\gamma_{i,k}\mu}{t}\nonumber\\
& = \frac{\binom{K}{q}\binom{q}{q-t}M\!-\!\!\!\sum\limits_{i\in\mathcal{M}_{M\!-\!t}}\!\!\!\binom{q-1}{q-t-1}\binom{K-1}{q-1}\!\!\sum\limits_{k\in\mathcal{K}}\!\!\gamma_{i,k}\mu}{t}\nonumber\\
& \stackrel{(m)}{=}\frac{\binom{K}{q}\binom{q}{q-t}M\!\!-\!(\!M\!\!-\!t)\binom{q-1}{q-t-1}\binom{K-1}{q-1}r\mu}{t},\!\!\label{eq46}\end{aligned}$$
where $(m)$ is due to (\[cons111\]). Remanaging (\[eq46\]), the lower bound of NDLT at the pair $(r,q)$ is given by
$$\begin{aligned}
\tau^{d^*}(q,q-t,r)&\ge \frac{M-(M\!-\!t)(q-t)\frac{r}{K}\mu}{t}, \label{tau_t}\end{aligned}$$
Since the argument we adopt to derive (\[tau\_t\]) holds for $1\!\le\!t\!\le\!\min\{q,M\}$, the lower bound of NDLT at $(r,q)$ can be optimized as $$\label{lowerbound_taud}
\tau^{d^*}(r,q)\ge \max_{t\in\{1,\cdots,\min\{q,M\}\}}\frac{M-(M\!-\!t)(q-t)\frac{r}{K}\mu}{t}.$$
### Multiplicative gap
By (\[achievedownload\]), the achievable NDLT is upper bounded by
$$\begin{aligned}
\tau^d&=\sum\limits^{\min\{r,q\}}_{p_1=r-K+q}B_{p_1}\left(\sum\limits^{l_{max}}_{p_2=l_{min}}\frac{B_{p_2}}{d^d_{p_1,M,p_2}}\!+\!\frac{B_{l_{p_1}\!-\!1}}{d^d_{p_1,M,l_{p_1}\!-\!1}}\right)\nonumber\\
&\stackrel{(m)}{\le}\sum\limits^{\min\{r,q\}}_{p_1=r-K+q}B_{p_1}\frac{\sum\limits^{l_{max}}_{p_2=l_{min}}\!B_{p_2}\!+\!B_{l_{p_1}\!-\!1}}{d^d_{p_1,M,1}}\nonumber\\
&\stackrel{(n)}{\le}\sum\limits^{\min\{r,q\}}_{p_1=r-K+q}\binom{q}{p_1}\binom{K-q}{r-p_1}\frac{1}{\binom{K}{r}}\cdot\frac{1}{d^d_{r-K+q,M,1}}\nonumber\\
&=\frac{1}{d^d_{r-K+q,M,1}},\end{aligned}$$
where $(m)$ is because $d^d_{p_1,M,p_2}$ increases with $p_2$ [@gckkl Lemma 1] and $(n)$ is because $d^d_{p_1,M,1}\!=\!p_1/(p_1\!+\!M\!-\!1)$ increases with $p_1$. By (\[lowerbound\_taud\]), we have $\tau^{d^*}(r,q)\!\ge\!M/\min\{q,M\}$, so the multiplicative gap satisfies
$$\begin{aligned}
\frac{\tau^d}{\tau^{d^*}}&\le\frac{\min\{q,M\}}{d^d_{r-K+q,M,1}} =\frac{\min\{q,M\}(r\!-\!K\!+\!q\!+\!M\!-\!1)}{(r\!-\!K\!+\!q)M}.\end{aligned}$$
If $q\!\le\!M$, we have $\frac{\tau^d}{\tau^{d^*}}\!\le\!\frac{q}{r-K+q}(\frac{q}{M}\!+\!\frac{M-1}{M}\!-\!\frac{K-r}{M})\!\le\!\frac{2q}{r-K+q}\!\le\!\frac{2q}{q-n}\!\le\!2(n\mu\!+\!1)$ for $r\!\ge\!K\!-\!n$; Otherwise, we have $\frac{\tau^d}{\tau^{d^*}}\!\le\!1\!+\!\frac{M-1}{r-K+q}\!\le\!1\!+\!\frac{q-1}{q-n}\!\le\!2\!+\!(n\!-\!1)\mu$ for $r\!\ge\!K\!-\!n$. Here, integers $n$ satisfies $n\!\le\!q\!-\!\frac{1}{\mu}$ due to $(r\!-\!K\!+\!q)\mu\!\ge\!1$. In summary, since $2(n\mu\!+\!1)\!>\!2\!+\!(n\!-\!1)\mu$, we have $\frac{\tau^d}{\tau^{d^*}}\!\le\!2(n\mu\!+\!1)$ for $r\!\ge\!K\!-\!n$. Furthermore, when the upload time $\tau^u\!\ge\!\tau^u_a(K\!-n)$, the repetition order satisfies $r\!\ge\!K\!-\!n$. Thus, when $r\!=\!K$, or equivalently, $\tau^u\!\ge\!\tau^u_a(K)$, we have $\frac{\tau^d}{\tau^{d^*}}\!\le\!2$.
### Outer Bound of Compute-Download Latency Region
By the feasible set $\mathcal{R}$ in (\[regionR\]) and the convexity of $\mathscr{T}^{*}(\tau^{u})$ in Remark \[remarkconvex\], for an NULT $\tau^u\!=\!\tau^u_a(r)$ in (\[achieveUpoad\]) for some $r$, an outer bound $\mathscr{T}_{out}(\tau^{u})$ of the compute-download latency region is given as the convex hull of set $\big\{\!\!\left(\tau^{c^*}\!\!(r,q),\tau^{d^*}\!\!(r,q)\right)\!\!:\!q\!\in\!\!\big[\lceil\!\frac{1}{\mu}\!\rceil\!+\!K\!-r\!:\!\!K\big]\!\big\}$.
[^1]: A short version is to appear in IEEE ISIT 2020. The work by K. Li and M. Tao is supported by the National Natural Science Foundation of China under Grant 61941106 and the National Key R$\&$D Project of China under Grant 2019YFB1802702. The work by J. Zhang and O. Simeone is funded by the European Research Council under the European Union’s Horizon 2020 Research and Innovation Programme (Grant Agreement No. 725731).
[^2]: For general $u\!\in\![\frac{1}{K},1]$ satisfying $K\mu\!=\!\beta \lceil K\mu\rceil\!+\!(1\!-\!\beta)\lfloor K\mu\rfloor$, we can use memory- and time-sharing methods to achieve the linear combinations of the latency triplets achieved at integers $\lceil K\mu\rceil$ and $\lfloor K\mu\rfloor$ for a fixed $r$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we establish a uniqueness theorem for algebraically nondegenerate meromorphic maps of $\C^m$ into $\C P^n$ and slowly moving hypersurfaces $Q_j \subset\C P^n,$ $j=1,\dots,q$ in (weakly) general position, where $q$ depends effectively on $n$ and on the degrees $d_j$ of the hypersurfaces $Q_j$.'
author:
- '$\quad$Gerd Dethloff and Tran Van Tan'
date: '$\quad$'
title: |
A UNIQUENESS THEOREM\
FOR MEROMORPHIC MAPS\
WITH MOVING HYPERSURFACES
---
Introduction
============
One of the most striking consequences of Nevanlinna’s theory was his “five values” theorem, which says that if $f$ and $g$ are non-constant meromorphic functions on $\C$ such that $f^{-1}(a_i)=g^{-1}(a_i)$ for five distinct points $a_i$ in the extended complex plane, then $f=g$. This theorem is an example of what is now known as “uniqueness theorem". In 1975, Fujimoto generalized this result of Nevanlinna to the case of meromorphic maps of $\C^m$ into $\C P^n.$ In the last years, many uniqueness theorems for meromorphic maps with hyperplanes (both for fixed and for moving ones) have been established.
to 5cm
[Mathematics Subject Classification 2000: Primary 32H30; Secondary 32H04, 32H25, 14J70.]{}
[Key words: Nevanlinna theory, Second Main Theorem, Uniqueness Theorem.]{}
[The first named author was partially supported by the Fields Institute Toronto. The second named author was partially supported by the post-doctoral research program of the Abdus Salam International Centre for Theoretical Physics.]{}
For the case of hypersurfaces, however, there are so far only the uniqueness theorem of Thai and Tan [@ThT] for the case of Fermat moving hypersurfaces and the one of Dulock and Ru [@DR2] for the case of (general) fixed hypersurfaces. More precisely, in [@DR2], Dulock and Ru prove that one has a uniqueness theorem for algebraically non-degenate holomorphic maps $f,g:\C \rightarrow \C P^n$ satisfying $f=g$ on $\cup_{i=1}^q(f^{-1}(Q_i)\cup g^{-1}(Q_i)),$ with respect to $q > (n+1)+\frac{2Mn}{\tilde d} + \frac{1}{2}$ fixed hypersurfaces $Q_i \subset \C P^n$ in general position, where $\tilde d$ is the minimum of the degrees of these hypersurfaces and $M$ is the truncation level in the Second Main Theorem for fixed hypersurface targets obtained by An-Phuong [@AP] with $\epsilon = \frac{1}{2}$. Their method of proof comes from their paper [@DR1], where they prove a uniqueness theorem for holomorphic curves into abelian varieties.
In this paper, by a method different to the one used by Dulock and Ru, we prove a uniqueness theorem for the case of slowly moving hypersurfaces (Corollary \[3.2\] below). More precisely, we prove that one has a uniqueness theorem for algebraically non-degenate meromorphic maps $f,g:\C^m \rightarrow \C P^n$ satisfying $f=g$ on $\cup_{i=1}^q(f^{-1}(Q_i)\cup g^{-1}(Q_i))$ with respect to $q > (n+1) + \frac{2nL}{\tilde d} + \frac{1}{2}$ moving hypersurfaces $Q_i \subset \C P^n$ in (weakly) general position, where $\tilde d$ is the minimum of the degrees of these hypersurfaces and $L$ is the truncation level in the Second Main Theorem for moving hypersurface targets obtained by the authors in [@DT1] with $\epsilon = \frac{1}{2}$. Moreover, under the additional assumption that the $f^{-1}(Q_i)$, $i=1,...,q$ intersect properly, $q > (n+1) + \frac{2L}{\tilde d} + \frac{1}{2}$ moving hypersurfaces are sufficient. We remark that in the special case of fixed hypersurfaces, our result gives back the uniqueness theorem of Dulock and Ru (remark that $L \leq M$ in this case). Moreover, we give our uniqueness theorem in a slightly more general form (Theorem \[3.1\] below), requiring assumptions on the $(p-1)$ first derivatives of the maps, which gives in return a better bounds on the number of moving hypersurfaces in $\C P^n$, namely $q > (n+1) + \frac{2nL}{p\tilde d} + \frac{1}{2}$ respectively $q > (n+1) + \frac{2L}{p\tilde d} + \frac{1}{2}$.
Preliminaries
=============
For $z = (z_1,\dots,z_m) \in \C^m$, we set $\Vert z \Vert = \Big(\sum\limits_{j=1}^m |z_j|^2\Big)^{1/2}$ and define $$\begin{aligned}
B(r) &= \{ z \in \C^m : \Vert z \Vert < r\},\quad
S(r) = \{ z \in \C^m : \Vert z \Vert = r\},\\
d^c &= \dfrac{\sqrt{-1}}{4\pi}(\overline \partial - \partial),\quad
{\mathcal}V = \big(dd^c \Vert z\Vert^2\big)^{m-1},\; \sigma = d^c
\text{log}\Vert z\Vert^2 \land \big(dd^c\text{log}\Vert z
\Vert\big)^{m-1}.\end{aligned}$$ Let $L$ be a positive integer or $+\infty$ and $\nu$ be a divisor on $\C^m.$ Set $ |\nu| = \overline {\{z : \nu(z) \neq 0\}}.$ We define the counting function of $\nu$ by $$\begin{aligned}
N^{(L)}_\nu(r) := \int\limits_1^r \frac{n^{(L)}(t)}{t^{2m-1}}dt\quad
(1 < r < +\infty),\end{aligned}$$ where $$\begin{aligned}
n^{(L)}(t) &= \int\limits_{|\nu | \cap B(t)} \text{min}\{\nu
,L\}\cdot {\mathcal}V\ \quad
\text{for}\quad m \geq 2 \ \text{and}\\
n^{(L)}(t) &= \sum_{|z| \leq t}\text{min}\{ \nu(z),L\} \qquad\quad
\text{for}\quad m = 1.\end{aligned}$$
Let $F$ be a nonzero holomorphic function on $\C^m$. For a set $\alpha = (\alpha_1,\dots,\alpha_m)$ of nonnegative integers, we set $|\alpha| := \alpha_1 + \dots + \alpha_m$ and $\mathcal D^\alpha F
:= \dfrac{\partial^{|\alpha|}} {\partial^{\alpha_1}z_1 \cdots
\partial^{\alpha_m}z_m}\,\cdotp$ We define the zero divisor $\nu_F$ of $F$ by $$\begin{aligned}
\nu_F(z) = \max \big\{ p : \mathcal D^\alpha F(z) = 0 \ \text{for
all $\alpha$ with}\ |\alpha| < p \big\}.\end{aligned}$$
Let $\varphi$ be a nonzero meromorphic function on $\C^m$. The zero divisor $\nu_\varphi$ of $\varphi$ is defined as follows: For each $a \in \C^m$, we choose nonzero holomorphic functions $F$ and $G$ on a neighborhood $U$ of $a$ such that $\varphi = \dfrac{F}{G}$ on $U$ and $\text{dim}\big(F^{-1}(0) \cap G^{-1}(0)\big) \leq m-2$, then we put $\nu_\varphi(a) := \nu_F(a)$.
Set $N_\varphi^{(L)}(r):=N_{\nu_\varphi}^{(L)}(r).$ For brevity we will omit the character ${}^{(L)}$ in the counting function if $L=+\infty.$
Let $f$ be a meromorphic map of $\C^m$ into $\C P^n$. For arbitrary fixed homogeneous coordinates $(w_0: \cdots : w_n)$ of $\C P^n$, we take a reduced representation $f = (f_0 : \cdots : f_n)$, which means that each $f_i$ is a holomorphic function on $\C^m$ and $f(z)
= (f_0(z) : \cdots : f_n(z))$ outside the analytic set $\{ z :
f_0(z) = \cdots = f_n(z) = 0\}$ of codimension $\geq 2$. Set $\Vert
f \Vert = \max \{ |f_0|, \dots , |f_n| \}$.
The characteristic function of $f$ is defined by $$\begin{aligned}
T_f(r) := \int\limits_{S(r)}\text{log}\Vert f \Vert \sigma -
\int\limits_{S(1)} \text{log}\Vert f \Vert \sigma ,\quad 1 < r < +\infty.\end{aligned}$$
For a meromorphic function $\varphi$ on $\C^m$, the characteristic function $T_\varphi(r)$ of $\varphi$ is defined by considering $\varphi$ as a meromorphic map of $\C^m$ into $\C P^1$.
Let $f$ be a nonconstant meromorphic map of $\C^m$ into $\C P^n$. We say that a meromorphic function $\varphi$ on $\C^m$ is “small" with respect to $f$ if $T_\varphi(r) = o(T_f(r))$ as $r \to \infty$ (outside a set of finite Lebesgue measure).
Denote by $\mathcal M$ the field of all meromorphic functions on $\C^m$ and by ${\mathcal}K_f$ the subfield of $\mathcal M$ which consists of all “small" (with respect to $f$) meromorphic functions on $\C^m$.
For a homogeneous polynomial $Q \in {\mathcal}M[x_0,\dots,x_n]$ of degree $d \geq 1$ we write $Q = \sum\limits_{I \in {\mathcal}T_{d}} a_{I}x^I,$ where $ {\mathcal}T_d := \big\{ (i_0,\dots,i_n) \in \N_0^{n+1} : i_0 + \dots + i_n = d
\big\}$ and $x^I = x_0^{i_0} \cdots x_n^{i_n}$ for $x =
(x_0,\dots,x_n)$ and $I = (i_0, \dots,i_n) \in {\mathcal}T_d.$ Denote by $Q(z)= Q(z)(x_0, \dots, x_n)=\sum\limits_{I \in {\mathcal}T_{d}} a_{I}(z)x^I$ the homogeneous polynomial over $\C$ obtained by evaluating the coefficients of $Q$ at a specific point $z \in \C^m$ in which all coefficient functions of $Q$ are holomorphic.
Let $Q \in {\mathcal}M[x_0,\dots,x_n]$ of degree $d \geq 1$ with $Q(f):=Q(f_0, \dots , f_n) \not\equiv 0$. We define $$N^{(L)}_f(r,Q) := N^{(L)}_{Q(f)}(r)\;\;\text{and}\;\;
f^{-1}(Q):=\{z: \nu_{Q(f)}>0\}.$$
The *First Main Theorem* of Nevanlinna theory gives, for $Q = \sum\limits_{I \in {\mathcal}T_{d}} a_{I}x^I$ with $Q(f):=Q(f_0, \dots , f_n) \not\equiv 0$ : $$N(r, Q)\leq d\cdot T_f(r)+ O\big(\sum_{I \in {\mathcal}T_d}T_{{a_{I}}}(r)\big).$$
Let $$\begin{aligned}
Q_j = \sum\limits_{I \in {\mathcal}T_{d_j}} a_{jI}x^I \quad (j = 1,\dots,q)\end{aligned}$$ be homogeneous polynomials in ${\mathcal}K_f[x_0,\dots,x_n]$ with $\text{deg}\,Q_j = d_j \geq 1.$ Denote by ${\mathcal}K_{\{Q_j\}_{j=1}^q}$ the field over $\C$ of all meromorphic functions on $\C^m$ generated by all quotients $\big\{\frac{a_{jI_{1}}}{a_{jI_{2}}} :a_{jI_{2}}\not\equiv 0,
I_{1}, I_{2}
\in {\mathcal}T_{d_j}; j \in \{1,\dots,q\} \big\}$. We say that $f$ is algebraically nondegenerate over ${\mathcal}K_{\{Q_j\}_{j=1}^q}$ if there is no nonzero homogeneous polynomial $Q \in {\mathcal}K_{\{Q_j\}_{j=1}^q}[x_0,\dots,x_n]$ such that $Q(f_0,\dots,$ $f_n) \equiv 0$.
We say that a set $\{Q_j\}_{j=1}^q$ $(q \geq n+1)$ of homogeneous polynomials in ${\mathcal}K_f [x_0,\dots,$ $x_n]$ is admissible (or in (weakly) general position) if there exists $z \in \C^m$ in which all coefficient functions of all $Q_j$, $j=1,...,q$ are holomorphic and such that for any $1 \leq j_0 < \dots < j_n \leq q$ the system of equations $$\begin{aligned}
\label{zz}
\left\{ \begin{matrix}
Q_{j_i}(z)(x_0,\dots,x_n) = 0\cr
0 \leq i \leq n\end{matrix}\right.\end{aligned}$$ has only the trivial solution $(x_0, \dots , x_n) = (0,\dots,0)$ in $\C^{n+1}$. We remark that in this case this is true for the generic $z \in \C^m$.
In order to prove our result for (weakly) general position (under the stronger assumption of pointwise general position this can be avoided), we finally will need some classical results on resultants, see Lang [@b10], section IX.3, for the precise definition, the existence and for the principal properties of resultants, as well as Eremenko-Sodin [@b4], page 127: Let $\big\{Q_j\big\}_{j=0}^n$ be a set of homogeneous polynomials of common degree $d \geq 1$ in ${\mathcal}K_f[x_0,\dots,x_n]$ $$\begin{aligned}
Q_j = \sum_{I \in {\mathcal}T_d} a_{jI}x^I,\quad a_{jI} \in {\mathcal}K_f \quad
(j = 0,\dots,n).\end{aligned}$$ Let $T = (\dots,t_{kI},\dots)$ ($k \in \{0,\dots,n\}$, $I \in {\mathcal}T_d$) be a family of variables. Set $$\begin{aligned}
\widetilde Q_j = \sum_{I \in {\mathcal}T_d} t_{jI}x^I \in \Z[T,x],\quad
j = 0,\dots, n.\end{aligned}$$ Let $\widetilde R \in \Z[T]$ be the resultant of $\widetilde Q_0, \dots,
\widetilde Q_n$. This is a polynomial in the variables $T = (\dots,t_{kI},\dots)$ ($k \in \{0,\dots,n\}$, $I \in {\mathcal}T_d$) with integer coefficients, such that the condition $\widetilde R (T) =0$ is necessary and sufficient for the existence of a nontrivial solution $(x_0, \dots , x_n) \not= (0,\dots,0)$ in $\C^{n+1}$ of the system of equations $$\begin{aligned}
\label{z}
\left\{ \begin{matrix}
\widetilde Q_{j}(T)(x_0,\dots,x_n) = 0\cr
0 \leq i \leq n\end{matrix}\right. \:.\end{aligned}$$ From equations (\[z\]) and (\[zz\]) is follows immediately that if $$\big\{Q_j= \widetilde Q_j(a_{jI})(x_0, \dots, x_n)\,, \:j=0, \dots , n\big\}$$ is an admissible set, $$R := \widetilde R(\dots, a_{kI}, \dots) \not\equiv 0\,.\label{zzz}$$ Furthermore, since $a_{kI} \in {\mathcal}K_f$, we have $R \in {\mathcal}K_f$. We finally will use the following result on resultants, which is contained in Theorem 3.4 in [@b10] (see also Eremenko-Sodin [@b4], page 127, for a similar result):
\[lang\] There exists a positive integer $s$ and polynomials $\big\{\widetilde
b_{ij}\big\}_{0 \leq i, j \leq n}$ in $\Z[T,x]$, which are (without loss of generality) zero or homogenous in $x$ of degree $s-d$, such that $$\begin{aligned}
x_i^s \cdot \widetilde R = \sum_{j=0}^n \widetilde b_{ij} \widetilde Q_j\quad
\text{for all}\ i \in \{0,\dots,n\}.\end{aligned}$$
If we still set $$\begin{aligned}
b_{ij} = \widetilde b_{ij}\big((\dots,a_{kI},\dots), (f_0,\dots,f_n)\big),\quad
0 \leq i, j \leq n,\end{aligned}$$ we get
$$\begin{aligned}
\label{res}
f_i^s \cdot R = \sum_{j=0}^n b_{ij} \cdot Q_j(f_0,\dots,f_n)\quad
\text{for all}\ i \in \{0,\dots,n\}.\end{aligned}$$
In particular, if $D \subset \C^m$ is a divisor contained in all divisors $f^{-1}(Q_j)$, $j=0,...,n$, then $R$ vanishes on $D$: This follows from (\[res\]) since $f=(f_0:...:f_n)$ is a reduced representation (and it follows in principle already directly from the definition of the resultant).
Main result
===========
Let $f,g$ be nonconstant meromorphic maps of $\C^m$ into $\C P^n$. Let $\big\{Q_j\big\}_{j=1}^q$ be an admissible set of homogeneous polynomials in ${\mathcal}K_f [x_0,\dots,x_n]$ with $\deg Q_j = d_j \geq
1$. Denote by $d, d^*,\tilde{d}$ respectively the least common multiple, the maximum number and the minimum number of the $d_j$’s. Put $N=d\cdot(4(n+1)(2^n-1)(nd+1)+n+1)$. Set $t_{\{Q_j\}_{j=1}^q}=1$ if the field ${\mathcal}K_{\{Q_j\}_{j=1}^q}$ coincides with the complex number field $\C$ (ie. all $Q_j$ are fixed hypersurface targets) and $$t_{\{Q_j\}_{j=1}^q}
=\Bigg(\binom{n+N}{n}^2.\binom{q}{n}+\big[\frac{\big(
\binom{n+N}{n}^2.\binom{q}{n}-1 \big).\log\big(
\binom{n+N}{n}^2.\binom{q}{n}\big)}{\log(1+\frac{1}{4\binom{n+N}{n}N})}+1\big]^2\Bigg)^{
\binom{n+N}{n}^2.\binom{q}{n}-1}$$ if ${\mathcal}K_{\{Q_j\}_{j=1}^q}\ne\C,$ where we denote $[x]:=\max\{k\in \Z: k\leq x\}$ for a real number $x.$ Let $L=[\frac{d^*\cdot\binom{n+N}{n}t_{\{Q_j\}_{j=1}^q}-d^*}{d}+1].$
With these notations, we state our main result:
\[3.1\] a) Assume that $f,g$ are algebraically nondegenerate over ${\mathcal}K_{\{Q_j\}_{j=1}^q}$ and satisfy\
i) $\mathcal D^\alpha\big(\frac{f_k}{f_s}\big)=\mathcal
D^\alpha\big(\frac{g_k}{g_s}\big)$ on $\big(\cup_{i=1}^q(f^{-1}(Q_i)\cup g^{-1}(Q_i))\big)\backslash \big(Zero
(f_s.g_s)\big),$ for all $|\alpha | < p,\ 0\leq k\ne s
\leq n,$ where $p$ is a positive integer and $(f_0:\cdots:f_n),$ $(g_0:\cdots:g_n)$ are reduced representations of $f,g$ respectively.\
Then for $q>n+\frac{2nL}{p\tilde{d}}+\frac{3}{2}$ , we have $f\equiv g.$\
b) Assume that $f,g$ as in a) satisfy i) and\
ii) $\dim \big(f^{-1}(Q_i)\cap f^{-1}(Q_j)\big)\leq m-2$ for all $1\leq i<j\leq q$.\
Then for $q>n+\frac{2L}{p\tilde{d}}+\frac{3}{2}$ , we have $f\equiv g.$
We note that if $p=1$ the condition $i)$ becomes the following usual condition: $f=g$ on $\cup_{i=1}^q(f^{-1}(Q_i)\cup g^{-1}(Q_i)),$ and we state this case again explicitly because of its importance:
\[3.2\] a) Assume that $f,g$ are algebraically nondegenerate over ${\mathcal}K_{\{Q_j\}_{j=1}^q}$ and satisfy\
i) $f=g$ on $\cup_{i=1}^q(f^{-1}(Q_i)\cup g^{-1}(Q_i))$.\
Then for $q>n+\frac{2nL}{\tilde{d}}+\frac{3}{2}$ , we have $f\equiv g.$\
b) Assume that $f,g$ as in a) satisfy i) and\
ii) $\dim \big(f^{-1}(Q_i)\cap f^{-1}(Q_j)\big)\leq m-2$ for all $1\leq i<j\leq q$.\
Then for $q>n+\frac{2L}{\tilde{d}}+\frac{3}{2}$ , we have $f\equiv g.$
In order to prove Theorem 3.1, we need the following two results. The first one is similar to Lemma 5.1 in Ji [@J], the second one is a special case of our main result in [@DT1].
\[3.3\]Let $A_1,\dots,A_k$ be pure $(m-1)$- dimensional analytic subsets of $\C^m.$ Let $f_1, f_2$ be meromorphic maps of $\C^m$ into $\C P^n$. Then there exists a dense subset $\mathcal
C\subset \C^{n+1}\backslash \{0\}$ such that for any $c=(c_0,\dots,c_n)\in \mathcal C$ the hyperplane $H_c$ defined by $c_0w_0+\cdots+c_nw_n=0$ satisfies: $\dim\big(\cup_{j=1}^kA_j\cap
f_i^{-1}(H_c)\big)\leq m-2, i\in\{1,2\}.$
[**Proof of Proposition \[3.3\]:**]{} For any irreducible pure $(m-1)-$dimensional component $\sigma$ of $\cup_{j=1}^k A_j$ we set $$K_\sigma^i =\big\{(t_0,\dots,t_n) \in \C^{n+1} :
\sum\limits_{s=0}^nt_sf_{is} =0 \text{\ \ on\ }\sigma \big\}\ ,\quad
i\in\{1, 2\},$$ where $(f_{i0}:\cdots:f_{in})$ are reduced representations of $f_i$. Then $K_\sigma^i$ is a complex vector subspace of $\C^{n+1}$. Since $\text{dim}\{ f_{i0} = \cdots = f_{in} = 0\}\leq m-2,$ we get that $\sigma\backslash\bigcup\limits_{i\in\{1, 2\}}\{ f_{i0} =
\cdots = f_{in} = 0\}\ne\varnothing.$ This implies that $\dim K_\sigma^i \leqslant n$. Let $K = \bigcup\limits_{i\in\{1, 2\}}\bigcup\limits_\sigma K_\sigma^i,$ then $K$ is a union of at most a countable number of at most $n-$dimensional complex vector subspaces in $\C^{n+1}$. Let $\mathcal{C} = \C^{n+1}\backslash K$. Then $\mathcal{C}$ meets the requirement of the Proposition.
\[3.4\] Under the same assumption as in Theorem \[3.1\], we have $$\begin{aligned}
(q-n-\frac{3}{2}) T_f(r) \leq \sum_{j=1}^q \frac{1}{d_j}
N^{(L)}_f(r,Q_j),\end{aligned}$$ for all $r \in [1, +\infty)$ excluding a Borel subset $E$ of $[1,
+\infty)$ with $\displaystyle{\int\limits_E} dr < + \infty$.
[**Proof of Theorem \[3.4\]:**]{} This is the special case of the Main Theorem and Proposition 1.2. in [@DT1] for $\epsilon = \frac{1}{2}$ and where we estimate the different $d_j$’s in the numerators of the expressions entering into the truncation level $L$ by $d^*$.\
[**Proof of Theorem \[3.1\]:**]{} Assume that $f\not\equiv g.$ We first prove the following\
[**Claim:**]{} There exist (fixed) hyperplanes $H_i: a_{i0}w_0+\dots a_{in}w_n=0\;(i=1,2)$ in $\C
P^n$ such that $S=S_{H_1, H_2}(f,g):=
\frac{H_1(f)}{H_2(f)}-\frac{H_1(g)}{H_2(g)}\not\equiv 0$ and $$\begin{aligned}
\dim(f^{-1}(Q_j)\cap f^{-1}\big(H_i)\big)\leq m-2,\; \dim(g^{-1}(Q_j)\cap g^{-1}\big(H_i)\big)\leq
m-2\label{1}\end{aligned}$$ for all $j\in\{1,\dots,q\}$, $i\in\{1,2\}.$\
[**Proof of the Claim**]{}: By assumption i) of Theorem \[3.1\] we have pure $(m-1)-$dimensional analytic sets $$\begin{aligned}
A_j:=f^{-1}(Q_j)=g^{-1}(Q_j) \subset
\C^m, \:
j=1,\dots, q\,.\label{2}\end{aligned}$$ By Proposition \[3.3\] there exists a dense subset $\mathcal
C\subset \C^{n+1}\backslash \{0\}$ such that for any $c=(c_0,\dots,c_n)\in \mathcal C$ the hyperplane $H_c$ defined by $c_0w_0+\cdots+c_nw_n=0$ satisfies (\[1\]), that is $$\dim(A_j\cap f^{-1}\big(H_c)\big)\leq m-2,\; \dim(A_j\cap g^{-1}\big(H_c)\big)\leq
m-2$$ for all $j\in\{1,\dots,q\}$. Since $f,g$ are algebraically nondegenerate over ${\mathcal}K_{\{Q_j\}_{j=1}^q}$, so in particular algebraically nondegenerate over $\C$, we have that $L_c(f)\not\equiv 0$ and $L_c(g)\not\equiv 0$ are holomorphic functions for all $c=(c_0,\dots,c_n)\in \mathcal C$, where $L_c(f):= \sum_{i=0}^nc_if_i$ with a reduced representation $f=(f_0:\dots :f_n)$ and $L_c(g):= \sum_{i=0}^nc_ig_i$ with a reduced representation $g=(g_0:\dots :g_n)$. Finally for $c^{(1)}, c^{(2)}\in \mathcal C$, we put $S_{c^{(1)},c^{(2)}}(f,g):=
\frac{L_{c^{(1)}}(f)}{L_{c^{(2)}}(f)}-\frac{L_{c^{(1)}}(g)}{L_{c^{(2)}}(g)}$. In order to prove the Claim it suffices to show that for some $c^{(1)}, c^{(2)}\in \mathcal C$, $S_{c^{(1)},c^{(2)}}(f,g) \not\equiv 0$. Assume the contrary. Then for all $0 \leq i < j \leq n$ there exist sequences $(c^{(1)})_{\nu}$, $(c^{(2)})_{\nu}$, $\nu \in \N$, of elements in $\mathcal C$ such that $L_{(c^{(1)})_{\nu}}(f) \rightarrow f_i$ and $L_{(c^{(2)})_{\nu}}(f) \rightarrow f_j$. From this we get $$0\equiv
S_{(c^{(1)})_{\nu},(c^{(2)})_{\nu}}(f,g) \rightarrow \frac{f_i}{f_j} -
\frac{g_i}{g_j}\,,$$ what implies $0 \equiv \frac{f_i}{f_j} -
\frac{g_i}{g_j}$ for all $0\leq i<j\leq n$, contradicting the assumption $f\not\equiv g$. This proves the claim.
Since $f=g$ on $\cup_{j=1}^q f^{-1}(Q_j),$ for any generic point $$z_0\in\cup_{j=1}^q
f^{-1}(Q_j)\backslash \big( f^{-1}(H_2)\cup g^{-1}(H_2)\big)$$ (outside an analytic subset of codimension at least 2), there exists $s\in\{0,\dots,n\}$ such that both of $f_s(z_0)$ and $g_s(z_0)$ are different from zero. Then by assumption i) we have $$\begin{aligned}
\mathcal D^\alpha S(z_0)&=\mathcal D^\alpha\bigl (
\frac{H_1(f)}{H_2(f)}-\frac{H_1(g)}{H_2(g)}\bigl )(z_0)\\&
=\mathcal D^\alpha\bigl (
\frac{\sum_{v=0}^{n}\frac{f_v}{f_s}a_{1v}}{\sum_{v=0}^{n}\frac{f_v}{f_s}a_{2v}}-\frac{\sum_{v=0}^{n}\frac{g_v}{g_s}a_{1v}}{\sum_{v=0}^{n}\frac{g_v}{g_s}a_{2v}}\bigl
)(z_0)=0\end{aligned}$$ for all $ |\alpha |< p.$
This implies that
$$\begin{aligned}
\nu_S\ge p \;\text{on}\;\cup_{j=1}^q f^{-1}(Q_j)\backslash
\big(A\cup f^{-1}(H_2)\cup g^{-1}(H_2)\big).\label{3}\end{aligned}$$
where $A$ is an analytic subset of codimension at least 2.
Now we will estimate the divisors $\nu_{Q_j \circ f}$ by making use of the resultants: In fact, for any $J=\{j_0,...,j_n\} \subset \{1,2,...,q\}$, let $R_J$ be the resultant of of $Q_{j_0},...,Q_{j_n}$. Then if $D \subset \C^m$ is a divisor contained in all divisors $f^{-1}(Q_{j_k})$, $k=0,...,n$, then $R_J$ vanishes on $D$. Thus, we get $$\begin{aligned}
\label{n}
\sum_{j=1}^q\text{min}\{1, \nu_{Q_j \circ f}\} \leq n \cdot \text{min}\{1, \sum_{j=1}^q \nu_{Q_j \circ f}\} + (q-n) \cdot \text{min}\{1, \sum_{|J|=n+1}\nu_{R_J}\}\end{aligned}$$
By (\[1\]), (\[2\]),(\[3\]), (\[n\]), by the First Main Theorem and since $R_J \in {\cal K}_f$, we have $$\begin{aligned}
\sum_{j=1}^qN_g^{(1)}(r,Q_j)=\sum_{j=1}^qN_f^{(1)}(r,Q_j) &\leq
\frac{n}{p}N_S(r) + o(T_f(r))\label{4a}\end{aligned}$$ Furthermore, by the First Main Theorem $$\begin{aligned}
N_S(r)
&\leq
T_{\frac{H_1(f)}{H_2(f)}-\frac{H_1(g)}{H_2(g)}}(r)+O(1)\notag\\
&\leq
T_{\frac{H_1(f)}{H_2(f)}}(r)+T_{\frac{H_1(g)}{H_2(g)}}(r)+O(1)\notag\\
&\leq T_f(r)+T_g(r)+O(1).\label{4}\end{aligned}$$ Thus, $$\begin{aligned}
\sum_{j=1}^q\big(N_f^{(1)}(r,Q_j)+N_g^{(1)}(r,Q_j)\big)\leq
\frac{2n}{p}\big(T_f(r)+T_g(r)\big)+o(T_f(r)).\label{5}\end{aligned}$$ By Theorem \[3.4\] and by the First Main Theorem, we have $$\begin{aligned}
(q-n-\frac{3}{2})T_f(r)\leq\sum_{j=1}^q\frac{1}{d_j}N_f^{(L)}(r,Q_j)\notag\\
\leq\sum_{j=1}^q\frac{L}{d_j}N_f^{(1)}(r,Q_j)=\sum_{j=1}^q\frac{L}{d_j}N_g^{(1)}(r,Q_j)\leq
qLT_g(r)+o(T_f(r))\label{6}\end{aligned}$$ for all $r \in [1, +\infty)$ excluding a Borel subset $E$ of $(1,
+\infty)$ with $\displaystyle{\int\limits_E} dr < + \infty$ (note that $Q_j\in\mathcal K_f[x_0,\dots,x_n]).$
This implies that $\mathcal K_f\subset\mathcal K_g.$ Then $\{Q_j\}_{j=1}^q\subset\mathcal K_g[x_0,\dots,x_n].$ So we can apply Theorem \[3.4\] for both meromorphic maps $f$ and $g$ with moving hypersurfaces $\{Q_j\}_{j=1}^q.$ By Theorem \[3.4\] and by the First Main Theorem, we have $$\begin{aligned}
(q-n-\frac{3}{2})\big(T_f(r)+T_g(r)\big)\leq\sum_{j=1}^q\frac{1}{d_j}\big(N_f^{(L)}(r,Q_j)+N_g^{(L)}(r,Q_j)\big)\notag\\
\leq\frac{L}{\tilde{d}}\sum_{j=1}^q\big(N_f^{(1)}(r,Q_j)+N_g^{(1)}(r,Q_j)\big)\label{7}
\end{aligned}$$ for all $r \in [1, +\infty)$ excluding a Borel subset $E$ of $(1,
+\infty)$ with $\displaystyle{\int\limits_E} dr < + \infty$.
Combining with (\[5\]), we get $$\begin{aligned}
(q-n-\frac{3}{2})\big(T_f(r)+T_g(r)\big)\leq\frac{2nL}{p\tilde{d}}\big(T_f(r)+T_g(r)\big)+o(T_f(r))
\label{8}\end{aligned}$$ for all $r \in [1, +\infty)$ excluding a Borel subset $E$ of $(1,
+\infty)$ with $\displaystyle{\int\limits_E} dr < + \infty$. This is a contradiction, since $q>n+\frac{2nL}{p\tilde{d}}+\frac{3}{2}$, thus finishing the proof of part a).\
In order to prove b), we observe that under the additional assumption ii), we can improve (\[4a\]), namely we get, by using (\[1\]), (\[2\]), (\[3\]) and assumption ii) $$\begin{aligned}
\sum_{j=1}^qN_g^{(1)}(r,Q_j)=\sum_{j=1}^qN_f^{(1)}(r,Q_j) &\leq
\frac{1}{p}N_S(r)\label{4ab}\end{aligned}$$ This improves (\[5\]), namely we get from (\[4\]) and (\[4ab\]): $$\begin{aligned}
\sum_{j=1}^q\big(N_f^{(1)}(r,Q_j)+N_g^{(1)}(r,Q_j)\big)\leq
\frac{2}{p}\big(T_f(r)+T_g(r)\big)+O(1).\label{5b}\end{aligned}$$ Using this (\[8\]) becomes, by using now (\[7\]) and (\[5b\]): $$\begin{aligned}
(q-n-\frac{3}{2})\big(T_f(r)+T_g(r)\big)\leq\frac{2L}{p\tilde{d}}\big(T_f(r)+T_g(r)\big)+O(1)
\label{8b}\end{aligned}$$ for all $r \in [1, +\infty)$ excluding a Borel subset $E$ of $(1,
+\infty)$ with $\displaystyle{\int\limits_E} dr < + \infty$. This is a contradiction, since $q>n+\frac{2L}{p\tilde{d}}+\frac{3}{2}$, thus finishing the proof of part b).
[99]{}
T. T. H. An and H. T. Phuong, *An explicit estimate on multiplicity truncation in the Second Main Theorem for holomorphic curves encountering hypersurfaces in general position in projective space*, Preprint (2007), to appear in Houston J. Math..
G. Dethloff and T. V. Tan, *A second main theorem for moving hypersurface targets*, Preprint (2007), to appear in Houston J. Math..
G. Dethloff and T. V. Tan, *Uniqueness theorems for meromorphic maps with few hyperplanes*, Preprint (2007), to appear in Bull. Sci. Math..
M. Dulock and M. Ru *Uniqueness of holomorphic curves into abelian varieties*, Preprint (2008), to appear in Trans. Amer. Math. Soc..
M. Dulock and M. Ru *A uniqueness theorem for holomorphic curves encountering hypersurfaces in projective space*, Preprint (2008), to appear in Complex Variables and Elliptic Equations.
A. E. Eremenko and M. L. Sodin, *The value distribution of meromorphic functions and meromorphic curves from the point of view of potential theory*, St. Petersburg Math. J. **3** (1992), 109-136.
S. Ji, *Uniqueness problem without multiplicities in value distribution theory*, Pacific J. Math. **135** (1988), 323-348.
S. Lang, Algebra (third edition), Addision - Wesley, 1993.
L. Smiley, *Geometric conditions for unicity of holomorphic curves*, Contemp. Math. **25** (1983), 149-154.
D. D. Thai and T. V. Tan, *Uniqueness problem of meromorphic mappings for moving hypersurfaces,* Preprint (2007).
Gerd Dethloff$^{1-2} $\
$^1$ Université Européenne de Bretagne, France\
$^2$ Université de Brest\
Laboratoire de mathématiques\
UMR CNRS 6205\
6, avenue Le Gorgeu, BP 452\
29275 Brest Cedex, France\
e-mail: [email protected]\
Tran Van Tan\
Department of Mathematics\
Hanoi National University of Education\
136-Xuan Thuy street, Cau Giay, Hanoi, Vietnam\
e-mail: [email protected]; [email protected]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: '[We provide algorithms to compute a complete irredundant set of extremely strong Shoda pairs of a finite group $G$ and the set of primitive central idempotents of the rational group algebra $\mathbb{Q}[G]$ realized by them. These algorithms are also extended to write new algorithms for computing a complete irredundant set of strong Shoda pairs of $G$ and the set of primitive central idempotents of $\mathbb{Q}[G]$ realized by them. Another algorithm to check whether a finite group $G$ is normally monomial or not is also described.]{}'
author:
- |
Gurmeet K. Bakshi and Sugandha Maheshwary[[^1] [^2]]{}\
[*Centre for Advanced Study in Mathematics,*]{}\
[*Panjab University, Chandigarh 160014, India*]{}\
[*email: [email protected] and [email protected]* ]{}
bibliography:
- 'ReferencesBM3.bib'
title: '**Extremely strong Shoda pairs with `GAP`**'
---
rational group algebra, primitive central idempotents, strong Shoda pairs, extremely strong Shoda pairs, normally monomial groups.\
[**MSC2000:** ]{}Primary: 20C05, 16S34; Secondary: 68W30.
Introduction
============
Let $G$ be a finite group and let $\mathbb{Q}[G]$ be the rational group algebra of $G$. A *strong Shoda pair* of $G$, introduced by Olivieri, del R[í]{}o and Sim[ó]{}n [@Oli], is a pair $(H,K)$ of subgroups of $G$ with the subgroups $H$ and $K$ satisfying some technical conditions. In [@BM], a strong Shoda pair $(H,K)$ with $H$ normal in $G$ is called as an *extremely strong Shoda pair* of $G$. An important property ([@Oli], Proposition 3.3) of the strong Shoda pairs of $G$ is that each such pair $(H,K)$ determines a *primitive central idempotent* of $\mathbb{Q}[G]$, called the *primitive central idempotent of* $\mathbb{Q}[G]$ *realized by* $(H,K)$, which is denoted by $e(G,H,K)$. Let $E$ be the set of all primitive central idempotents of $\mathbb{Q}[G]$ and $E_{SSP}$ (resp. $E_{ESSP}$) be the set of primitive central idempotents of $\mathbb{Q}[G]$ realized by the strong Shoda pairs (resp. extremely strong Shoda pairs) of $G$. The groups $G$ for which $E=E_{SSP}$ are called *strongly monomial* groups and are known to constitute a large class of monomial groups, including abelian-by-supersolvable groups [@Oli]. Also, in [@BM], it has been proved that $E=E_{SSP}=E_{ESSP}$ if, and only if, $G$ is a *normally monomial* group i.e., every complex irreducible character of $G$ is induced from a linear character of a normal subgroup of $G$. The `GAP` [@GAP] package `Wedderga` [@Wedd] features the function `PrimitiveCentralIdempotentsByStrongSP(QG);` that computes the set $E_{SSP}$ for the rational group algebra $\mathbb{Q}[G]$ and the function `StrongShodaPairs(G);` that determines a subset $X$ of strong Shoda pairs of $G$ such that $(H,K)\mapsto e(G,H,K)$ defines a bijection from $X$ to $E_{SSP}$. Such a set $X$ is called a *complete irredundant set of strong Shoda pairs* of $G$. These functions are based on the search algorithms provided by Olivieri and del R[í]{}o [@OliA]. Another relevant feature of `Wedderga` is the function `IsStronglyMonomial(G);` which checks whether the group $G$ is strongly monomial or not. Using this function, it has been revealed in [@Olte] that all the monomial groups of order less than $1000$ are strongly monomial.\
In this paper, we provide an algorithm to compute a complete irredundant set of extremely strong Shoda pairs of $G$. This algorithm is based on the work in [@BM]. We further extend this algorithm by combining it with the search algorithm provided by Olivieri and del R[í]{}o [@OliA] to obtain a new algorithm that computes a complete irredundant set of strong Shoda pairs of $G$. As a consequence, we obtain algorithms to write the sets $E_{ESSP}$ and $E_{SSP}$ of primitive central idempotents of $\mathbb{Q}[G]$ realized by extremely strong Shoda pairs of $G$ and those re alized by strong Shoda pairs of $G$ respectively. Another algorithm to check whether a finite group $G$ is normally monomial or not also follows as a consequence. These algorithms are given in Section 3 and enable us to write the following functions in `GAP` language:
- `ExtStrongShodaPairs(G);` which computes a complete irredundant set of extremely strong Shoda pairs of $G$ i.e., a subset $X$ of extremely strong Shoda pairs of $G$, such that such that $(H,K)\mapsto e(G,H,K)$ gives a bijection from $X$ to $E_{ESSP}$.
- `StShodaPairs(G);` which computes a complete irredundant set of strong Shoda pairs of $G$.
- `PrimitiveCentralIdempotentsByExtSSP(QG);` which computes the set of primitive central idempotents of $\mathbb{Q}[G]$ realized by extremely strong Shoda pairs of $G$.
- `PrimitiveCentralIdempotentsByStSP(QG);` which computes the set of primitive central idempotents of $\mathbb{Q}[G]$ realized by strong Shoda pairs of $G$.
- `IsNormallyMonomial(G);` which checks whether the group $G$ is normally monomial or not.
Using the function `IsNormallyMonomial(G);` we have searched for normally monomial groups among the groups in `GAP` library of small groups. The search indicates that the class of normally monomial groups is a substantial class of monomial groups. It may also be mentioned that if $G$ is a normally monomial group, then the output obtained by the functions `StShodaPairs(G);` and `PrimitiveCentralIdempotentsByStSP(QG);` is same as that obtained by `ExtStrongShodaPairs(G);` and `PrimitiveCentralIdempotentsByExtSSP(QG);` respectively. Furthermore, for a finite group $G$, the functions `StShodaPairs(G);` and `PrimitiveCentralIdempotentsByStSP(QG);` are alternative to the functions `StrongShodaPairs(G);` and `PrimitiveCentralIdempotentsByStrongSP(QG);` respectively, which are currently available in `Wedderga`.
In Section 4, we compare the runtimes of the function `StShodaPairs(G);` with `StrongShodaPairs(G);` for a large and evenly spread sample of groups of order up to $2000$. For this sample, the functions `PrimitiveCentralIdempotentsByStSP(QG);` and `PrimitiveCentralIdempotentsByStrongSP(QG)`; are also compared for runtimes. It is observed that these new functions show significant improvement in the time taken to compute the same outputs. Further, in order to observe the performance separately for solvable and non solvable groups, we also compared the runtimes of `StShodaPairs(G);` with `StrongShodaPairs(G);` for another two samples. The sample of solvable groups consists of all the groups of odd order up to 2000, and the other sample consists of all non solvable groups of order up to 2000. It is observed that the performance of `StShodaPairs(G);` is exceptionally better in comparison with that of `StrongShodaPairs(G);` for solvable groups. However, in the case of non solvable groups, the performance of the two functions is almost identical. Finally, we describe the reasons for the difference in the performance of these functions.
Notation and Preliminaries
==========================
Throughout this paper, $G$ denotes a finite group. By $H \leq G$ (resp. $H\unlhd G$), we mean that $H$ is a subgroup (resp. normal subgroup) of $G$. For $H \leq G$, $[G:H]$ denotes the index of $H$ in $G$, $N_{G}(H)$ denotes the normalizer of $H$ in $G$, $\operatorname{core}_{G}(H)=\bigcap_{x\in G}xHx^{-1}$ and $\hat{H}=\frac{1}{|H|}\sum_{h \in H}h$, where $|H|$ is the order of $H$. For $K\unlhd H \leq G$, write $$\varepsilon (H,K) := \begin{cases} \hat{H}, & {\rm if}~ H = K; \\ \prod (\hat{K}-\hat{L}), & {\rm otherwise,} \end{cases}$$ where $L$ runs over the minimal normal subgroups of $H$ containing $K$ properly. Set
$e(G,H,K)$ := the sum of all the distinct $G$-conjugates of $\varepsilon(H,K)$.
Let $\varphi$ denote the Euler phi function. Denote by $\operatorname{Irr}(G)$, the set of all complex irreducible characters of $G$. For $\chi \in \operatorname{Irr}(G)$, $\mathbb{Q}(\chi)$ denotes the field obtained by adjoining to $\mathbb{Q}$, all the character values $\chi(g),~g \in G$, and $\operatorname{Gal}(\mathbb{Q}(\chi)/\mathbb{Q})$ is the Galois group of the extension $\mathbb{Q}(\chi)$ over $\mathbb{Q}$.
It is well known that $\chi \mapsto e_{\mathbb{Q}}(\chi):=\frac{\chi(1)}{|G|}\sum_{\sigma \in \operatorname{Gal}(\mathbb{Q}(\chi)/\mathbb{Q}) }\sum_{g \in G}\sigma(\chi(g^{-1}))g$ defines a surjective map from $\operatorname{Irr}(G)$ to the set of primitive central idempotents of the rational group algebra $\mathbb{Q}[G]$. If $\chi$ is the trivial character of $G$, then it is easy to see that $e_{\mathbb{Q}}(\chi)=\hat{G}$.\
Olivieri et al [@Oli] proved the following:
\[t1\][([[@Oli], Lemma 1.2, Theorem 2.1]{})]{}
1. If $\chi$ is a non trivial linear character of $G$ with kernel $N$ then $$\label{e1}
e_{\mathbb{Q}}(\chi)=\varepsilon(G,N).$$
2. If $\chi$ is monomial, i.e. $\chi$ is induced from a linear character $\psi$ of a subgroup $H$ of $G$, then there exists $\alpha \in \mathbb{Q}$ such that $$ e_{\mathbb{Q}}(\chi)= \alpha e(G, H, K),$$ where $K$ is the kernel of the character $\psi$. Furthermore $\alpha=1$, if the distinct $G$-conjugates of $\varepsilon(H,K)$ are mutually orthogonal.
Shoda (see [@curt], Theorem 45.2) gave a criteria to decide if the character of $G$ induced by a linear character $\psi$ of a subgroup $H$ of $G$ is irreducible, in terms of $H$ and the kernel $K$ of $\psi$. A pair $(H,K)$ of $G$ satisfying this criteria is called a *Shoda pair* ([@Oli], Definition 1.4) of $G$. A *strong Shoda pair* ([@Oli], Definition 3.1) of $G$ is a pair $(H,K)$ of subgroups satisfying the following conditions:
> (i)
>
> : $K \unlhd H \unlhd N_{G}(K)$;
>
> (ii)
>
> : $H/K$ is cyclic and a maximal abelian subgroup of $N_{G}(K)/K$;
>
> (iii)
>
> : the distinct $G$-conjugates of $\varepsilon(H,K)$ are mutually orthogonal.
>
As the name suggests, each strong Shoda pair of $G$ is also a Shoda pair of $G$ ([@Oli], Proposition 3.3). A strong Shoda pair $(H,K)$ of $G$ is called an *extremely strong Shoda pair* of $G$, if $H\unlhd G$. Observe that $(G,G)$ is always an extremely strong Shoda pair of $G$.
From Theorem \[t1\], it follows that if $(H, K)$ is a strong Shoda pair of $G$, then $e(G,H,K)$ is a primitive central idempotent of $\mathbb{Q}[G]$, called [*the primitive central idempotent of $\mathbb{Q}[G]$ *realized by $(H,K)$**]{}. For a strong Shoda pair $(H,K)$ of $G$, we denote by $\operatorname{dim}(H,K)$, the $\mathbb{Q}$-dimension of the simple component $\mathbb{Q}[G]e(G,H,K)$ of $\mathbb{Q}[G]$. In view of ([@Oli], Proposition 3.4), $\operatorname{dim}(H,K)$ equals $\varphi([H:K])[N_{G}(K):H][G:N_{G}(K)]^{2}.$ Two strong (resp. extremely strong) Shoda pairs $(H_{1}, K_{1})$ and $(H_{2}, K_{2})$ of $G$ are said to be [*equivalent*]{} if $e(G, H_{1}, K_{1})=e(G, H_{2}, K_{2})$. A complete set of representatives of distinct equivalence classes of strong (resp. extremely strong) Shoda pairs of $G$ is called a [*complete irredundant set of strong [(]{}resp. extremely strong[)]{} Shoda pairs*]{} of $G$.
We now recall the method given in [@BM] to compute a complete irredundant set of extremely strong Shoda pairs of a finite group $G$.
Let $\mathcal{N}$ be the set of all the distinct normal subgroups of $G$. For $N \in \mathcal{N}$, let $A_{N}$ be a normal subgroup of $G$ containing $N$ such that $A_{N}/N$ is an abelian normal subgroup of maximal order in $G/N$. Note that the choice of $A_{N}$ is not unique. However, we need to fix one such $A_{N}$. For a fixed $A_{N}$, set\
$\begin{array}{lll}
\mathcal{D}_{N}: & {\rm the ~set ~of ~all ~subgroups~} D ~{\rm of} ~A_{N}~ {\rm containing ~} N~{\rm such ~that~}
\operatorname{core}_{G}(D)=N, \\& A_{N}/D ~{\rm is ~ cyclic ~ and ~ is ~a ~ maximal~ abelian ~ subgroup~ of~} N_{G}(D)/D. \vspace{.2cm}\\ \mathcal{T}_{N}: & {\rm a ~set~ of ~representatives~ of~ } \mathcal{D}_{N} {\rm ~ under ~the~ equivalence~ relation~ defined~ by} \\ & {\rm
conjugacy~ of~ subgroups~in~} G. \vspace{.2cm} \\ \mathcal{S}_{N}: & \{( A_{N},D)~|~ D \in \mathcal{T}_{N}\}.\end{array}$
\[t2\][([@BM], Theorem 1)]{} Let $G$ be a finite group. Then,\
[(i)]{} $\cup_{N \in \mathcal{N} }\mathcal{S}_{N} $ is a complete irredundant set of extremely strong Shoda pairs of $G$.\
[(ii)]{} $\{ e(G, A_{N}, D)\, |\, ( A_{N},D) \in \mathcal{S}_{N},~ N \in \mathcal{N} \}$ is a complete set of primitive central idempotents of $\mathbb{Q}[G]$ if, and only if, $G$ is normally monomial.
It may be noted that in Theorem \[t2\], the choice of $A_{N}$ is irrelevant. For $N \in \mathcal{N}$, let $A_{N}'$ be another normal subgroup of $G$ containing $N$ such that $A_{N}'/N$ is an abelian normal subgroup of maximal order in $G/N$ and let $\mathcal{D}_{N}'$, $\mathcal{T}_{N}'$ and $\mathcal{S}_{N}'$ be defined corresponding to $A_{N}'$. Then any pair in $\mathcal{S}_{N}'$ is equivalent to a pair in $\mathcal{S}_{N}$ and vice versa. This is because, if $(A_{N}',D')\in \mathcal{S}_{N}'$ and $\psi$ is a linear character of $A_{N}'$ with kernel $D'$, then $\psi^{G}$ is irreducible and hence by ([@BM], Lemma 1), there exists $(A_{N},D)\in \mathcal{S}_{N}$ such that $e_{\mathbb{Q}}(\psi^{G})=e(G,A_{N},D).$ However, in view of Theorem \[t1\], $e_{\mathbb{Q}}(\psi^{G})=e(G,A_{N}',D').$ This gives that $(A_{N},D)$ is equivalent to $(A_{N}',D')$. The reverse conclusion holds similarly.
[([@BM], Corollary 1)]{}\[c1\] If $G$ is a normally monomial group, then $\bigcup_{N \in \mathcal{N}}\mathcal{S}_{N}$ is a complete irredundant set of strong Shoda pairs of $G$.
\[c2\][([@BM], Corollary 2)]{} A finite group $G$ is normally monomial if, and only if, $$\sum_{N\in \mathcal{N}}\sum_{(A_{N},D) \in \mathcal{S}_{N}}dim(A_{N},D) = |G|.$$
Algorithms
==========
We shall use the notation developed in the previous section.
Extremely Strong Shoda Pairs
----------------------------
We provide Algorithm 1, which computes the set $ESSP$, which is a complete irredundant set of extremely strong Shoda pairs of a given finite group $G$. This algorithm is based on Theorem \[t2\]. It mainly requires the set $\mathcal{N}$ of normal subgroups of $G$ and the computation of $\mathcal{S}_{N}$ for each $N\in \mathcal{N}$. The set $\mathcal{S}_{N}$ is computed as explained in Section 2 and by using Lemmas \[l1\]-\[l2\] to avoid unnecessary computations.
\[l1\] For a normal subgroup $N$ of $G$, the following hold:
(i)
: If $G/N$ is abelian, then $$\mathcal{S}_{N} = \begin{cases} \{(G, N)\}, ~& {\rm if}~G/N~{\rm is~ cyclic;}\\ \emptyset, & {\rm otherwise.}
\end{cases}$$
(ii)
: If $G/N$ is non abelian and $A_{N}/N$ is cyclic, then $$\mathcal{S}_{N} = \begin{cases} \{(A_{N}, N)\}, & ~{\rm if}~ A_{N}/N ~{\rm is~ a~ maximal~ abelian~ subgroup ~of} ~G/N;\\ \emptyset, & {\rm otherwise.}
\end{cases}$$
[**Proof.**]{} Follows immediately from the definition of $\mathcal{S}_{N}$. $\Box$
\[l3\] If $\mathcal{M} \subseteq \mathcal{N}$ is such that $$\sum_{N\in \mathcal{M}}\sum_{(A_{N},D) \in \mathcal{S}_{N}}dim(A_{N},D)=|G|,$$ then $\mathcal{S}_{N}= \emptyset~ $ for all $~N \in \mathcal{N}\setminus \mathcal{M}$.
[**Proof.**]{} The primitive central idempotents $e(G,A_{N},D)$ for $(A_{N},D)\in \mathcal{S}_{N} $, $N \in \mathcal{N}$, are distinct, as $\bigcup_{N \in \mathcal{N}}\mathcal{S}_{N}$ is a complete irredundant set of extremely strong Shoda pairs of $G$. Therefore, $\bigoplus_{N \in \mathcal{N}}\bigoplus_{(A_{N},D)\in \mathcal{S}_{N}}\mathbb{Q}[G]e(G,A_{N},D)$ is a direct summand of $\mathbb{Q}[G]$, and hence its $\mathbb{Q}$-dimension is at most $|G|$. Consequently, $$\begin{aligned}
|G| &\geq & \sum_{N\in \mathcal{N}}\sum_{(A_{N},D)\in \mathcal{S}_{N}} dim(A_{N},D) \nonumber \\
&\geq & \sum_{N\in \mathcal{M}}\sum_{(A_{N},D)\in \mathcal{S}_{N}} dim(A_{N},D) ~~~~(\because\mathcal{M} \subseteq \mathcal{N}) \nonumber\\
&=& |G|.\end{aligned}$$ This yields that $\mathcal{S}_{N} =\emptyset$ for all $N\not \in \mathcal{M}$ and completes the proof.$~\Box$
\[l2\] If $(H,K)$ is a strong Shoda pair of $G$ with $N=\operatorname{core}_{G}(K)$, then the centre of $G/N$ must be cyclic.\
In particular, if $N \in \mathcal{N}$ is such that the centre of $G/N$ is not cyclic, then $\mathcal{S}_{N}= \emptyset$.
[**Proof.**]{} Let $aK$ be a generator of $H/K$ and let $\zeta$ be a primitive $m^{th}$ root of unity, where $m=[H:K]$. Consider the linear representation $\rho: H \rightarrow \mathbb{C}$ given by $x\mapsto \zeta^{i}$, if $xK=a^{i}K$, for $x \in H$. Since $(H,K)$ is a strong Shoda pair, $\rho^{G}$ is an irreducible representation of $G$. Now, as $\operatorname{ker}(\rho^{G})=\bigcap_{x\in G}x(\operatorname{ker}\rho)x^{-1}=\bigcap_{x\in G}xKx^{-1}=\operatorname{core}_{G}(K)=N$, the result follows from ([@IM], Lemma 2.27).$~\Box$\
\[A1\]
$\mathcal{N}:=$Normal subgroups of $G$ (in decreasing order) $ESSP:=[[G,G]]$ $SumDim:=1$
We now describe Algorithm 1. The first step of the algorithm is to compute the list $\mathcal{N}$ of all the normal subgroups of $G$ in decreasing order. If $N=G$, we have $\mathcal{S}_{N}= \{(G,G)\}$. Therefore, we initially set the list $ESSP$, which is the list of extremely strong Shoda pairs of $G$ found at any stage of computation, to be `[[G,G]]`. As $\operatorname{dim}(G,G)=1$, we set $SumDim$, which denotes the sum of $\mathbb{Q}$-dimensions of the simple components of $\mathbb{Q}[G]$ corresponding to the elements in $ESSP$, to be 1. For $N\in \mathcal{N}$, $N \neq G$, if $N$ contains the commutator subgroup $G'$ of $G$, then the corresponding set $\mathcal{S}_{N}$ is computed using Lemma \[l1\](i). Otherwise, $\mathcal{S}_{N}$ is computed using Theorem \[t2\] along with Lemmas \[l1\] and \[l2\]. In either of the two cases, if $\mathcal{S}_{N}\neq \emptyset$, then the elements of $\mathcal{S}_{N}$ are added to the list $ESSP$. Also, the sum of $\mathbb{Q}$-dimensions of simple components of $\mathbb{Q}[G]$ corresponding to the extremely strong Shoda pairs of $G$ in $\mathcal{S}_{N}$ is added to $SumDim$. In view of Lemma \[l3\], the process stops when either $SumDim=|G|$ or when all the normal subgroups of $G$ are exhausted. The normal subgroups $N$ of $G$ are selected in decreasing order i.e., if the normal subgroup $N_{1}$ is chosen before the normal subgroup $N_{2}$, then $|N_{1}|\geq|N_{2}|$. This has been done keeping in view the ease of computation of $\mathcal{S}_{N}$, if $G/N$ has small order. This algorithm enables us to write the function `ExtStrongShodaPairs(G);` in `GAP` language.
Strong Shoda Pairs
------------------
We next describe Algorithm \[A2\] to compute the set $StSP$, which is a complete irredundant set of strong Shoda pairs of a given finite group $G$.\
\[A2\]
$StSP$:= A complete irredundant set of extremely strong Shoda pairs of $G$;\
$SumDim$:=the sum of $\mathbb{Q}$-dimensions of simple components of $\mathbb{Q}[G]$ corresponding to the primitive central idempotents realized by the extremely strong Shoda pairs of $G$;
Initially, $StSP$ is the list $ESSP$ of extremely strong Shoda pairs of $G$ obtained using Algorithm \[A1\]. Also, $SumDim$ is set to be the the sum of $\mathbb{Q}$-dimensions of simple components of $\mathbb{Q}[G]$ corresponding to the primitive central idempotents realized by extremely strong Shoda pairs of $G$. In case $SumDim=|G|$, by Corollaries \[c1\] and \[c2\], $StSP$ is a complete irredundant set of strong Shoda pairs of $G$ and the algorithm terminates. Otherwise, to find the remaining strong Shoda pairs of $G$, we make use of the algorithm provided by Olivieri and del R[í]{}o [@OliA] with desired modifications. For a strong Shoda pair $(H,K)$ of $G$, we use the fact that $G/\operatorname{core}_{G}(K)$ must be cyclic (Lemma \[l2\]). Moreover, if $(H,K)$ realizes a primitive central idempotent of $\mathbb{Q}[G]$ different from the one realized by an extremely strong Shoda pair of $G$, then none of $H$ or $K$ is normal in $G$. This algorithm allows us to write the function `StShodaPairs(G);` in `GAP` language.
Primitive Central Idempotents
-----------------------------
The algorithm to compute the primitive central idempotents of $\mathbb{Q}[G]$ realized by extremely strong Shoda pairs of $G$ is similar to Algorithm 1. The only difference is that at any stage of the computation, instead of collecting the elements of $\mathcal{S}_{N}$, one collects the primitive central idempotents realized by them. Using this algorithm, we write the function `PrimitiveCentralIdempotentsByExtSSP(QG);` in `GAP` language which computes the set of primitive central idempotents realized by extremely strong Shoda pairs of $G$. To compute the primitive central idempotent $e(G,H,K)$ of $\mathbb{Q}[G]$ realized by the strong Shoda pair $(H,K)$ of $G$, we use the function `Idempotent_eGsum(QG,H,K);` currently available in `Wedderga`.
Similarly, the algorithm to compute the primitive central idempotents of $\mathbb{Q}[G]$ realized by strong Shoda pairs of $G$ is obtained by a slight modification of Algorithm \[A2\] and the corresponding function `PrimitiveCentralIdempotentsByStSP(QG);` is also obtained.
Normally Monomial Groups
------------------------
The algorithm to check whether a finite group $G$ is normally monomial or not is obtained by replacing the result $ESSP$ of Algorithm \[A1\] with $SumDim$. In view of Corollary \[c2\], $G$ is normally monomial if, and only if, $SumDim$=$|G|$. This algorithm enables us to write the function `IsNormallyMonomial(G);` in `GAP` language.
Using the function `IsNormallyMonomial(G);` we have found by a computer search that $98.84\%$ of the monomial groups of order up to $500$ are normally monomial. Also, $97.88\%$ of all the finite groups of order up to $500$ are normally monomial. An exhaustive computer search also yields that among the groups of odd order up to $2000$, the only groups which are not normally monomial are:
> `SmallGroup(375,2); SmallGroup(1029,12); SmallGroup(1053,51); SmallGroup(1125,3); SmallGroup(1125,7); SmallGroup(1215,68); SmallGroup(1875,18); SmallGroup(1875,19);`
It may be pointed out that all the groups in the above list, except the second and the third, are non monomial.\
Runtime Comparison
==================
We now present an experimental runtime comparison between the following two sets of functions for different samples of groups:
1. `StrongShodaPairs(G);` with `StShodaPairs(G);`
2. `PrimitiveCentralIdempotentsByStrongSP(QG);` with\
`PrimitiveCentralIdempotentsByStSP(QG);`
For a given sample of groups, let $t(n)$ be the average of the runtimes, taken in milliseconds, for the groups in $S$ of order $n$ for $n\geq 1$. If the sample contains no group of order $n$, then set $t(n)=0$. Define $T(n)= \sum_{i=1}^{n}t(i),~n\geq 1$.
We now describe the first sample $S$ which consists of $31272$ groups of order up to $2000$. For $1 \leq n \leq 2000$, $n\neq 1024$, if the number of non isomorphic groups of order $n$ is less than $200$, then $S$ contains all the groups of order $n$. Otherwise, we include in the sample $S$, at least $100$ groups of order $n$, which are evenly spread in the `GAP` library of small groups. The groups of order 1024 are excluded because of their non availability in `GAP` library. For this sample, the graph of $n$ versus $T(n)$ for the comparison of the functions `StrongShodaPairs(G);` and `StShodaPairs(G);` is presented in Fig.1. Also, Fig.2 presents the runtime comparison of the function `PrimitiveCentralIdempotentsByStrongSP(QG);` with the function `PrimitiveCentralIdempotentsByStSP(QG);`
\[f1\]
\[f2\] 
We next compare the runtimes of `StrongShodaPairs(G);` and `StShodaPairs(G);` for a sample of solvable and that of non solvable groups. The sample $S_{1}$ of solvable groups consists of all the groups of odd order up to 2000 and the sample $S_{2}$ consists of all the non solvable groups of order up to 2000. The graph of $n$ versus $T(n)$ for these samples are presented in Figs.3 and 4 respectively.
\[f3\]
\[f4\] 
In Figs.1-4, `SSP` and `StSP` are the curves for the functions `StrongShodaPairs(G);` and `StShodaPairs(G);` respectively and `PCIsBySSP` and `PCIsByStSP` represent the curves for the functions `PrimitiveCentralIdempotentsByStrongSP(QG);` and `PrimitiveCentralIdempotentsByStSP(QG);` respectively. These experiments have been performed on the computer with Intel Core i7-4770 CPU @ 3.40GHz Dual Core, 4GB RAM.\
The overall improvement in the performance of `StShodaPairs(G);` in comparison to `StrongShodaPairs(G);` is mainly due to following differences in their respective algorithms:
[2]{}
`StShodaPairs(G);`
`StrongShodaPairs(G);`
[2]{}
1.
: Begins by computing all the normal subgroups of $G$. The conjugacy classes of subgroups of $G$ are computed only if $G$ is not normally monomial.
<!-- -->
1.
: Always begins by computing all conjugacy classes of subgroups of $G$. It may be pointed out that generating the full subgroup lattice of $G$ restricts the efficiency when $G$ has large order.
[2]{}
2.
: Firstly, the extremely strong Shoda pairs of $G$ are computed. If $G$ is not normally monomial, then the remaining strong Shoda pairs of are found by the search algorithm of `StrongShodaPairs(G);`, with slight modifications.
<!-- -->
2.
: There is no distinction between the computation of extremely strong Shoda pairs and that of strong Shoda pairs of $G$.
[2]{}
3.
: Extremely strong Shoda pairs of $G$ are computed using Theorem\[t2\], which ensures that each time a new extremely strong Shoda pair is constructed, it is necessarily inequivalent to any of the extremely strong Shoda pair already obtained.
<!-- -->
3.
: When a new strong Shoda pair of $G$ is discovered, it is not necessarily inequivalent to the ones already discovered. Each time a new strong Shoda pair is found, the algorithm computes the corresponding primitive central idempotent of $\mathbb{Q}[G]$ and checks its equivalence.
The above differences also result in the improved performance of the function `PrimitiveCentralIdempotentsByStSP(QG);` in comparison to that of the function `PrimitiveCentralIdempotentsByStrongSP(QG);` which is currently available in `Wedderga`.
**Acknowledgements**\
The authors are grateful to the anonymous referees for their valuable comments and suggestions which have helped to write the paper in the present form.
[^1]: Research supported by CSIR, India (File No: SPM-09/135(0107)/2011-EMR-I) is gratefully acknowledged
[^2]: Corresponding author
| {
"pile_set_name": "ArXiv"
} |
---
author:
- Andrew Hubery
title: 'From Triangulated Categories to Lie Algebras: A Theorem of Peng and Xiao'
---
In his seminal article [@Ringel], Ringel showed how to associate to any finitary ring $\Lambda$ an associative unital algebra $\mathcal H(\Lambda)$, with structure constants encoding information about extensions between finite modules. This generalised the Hall algebra [@Hall; @Steinitz], which deals with the ring of $p$-adic integers ${\mathbb Z}_p$ and finite $p$-groups.
In the subsequent article [@Ringel2] it is shown that if $\Lambda$ is a representation-directed algebra over a finite field $k$, then the structure constants are given by evaluating integer polynomials. Using these Hall polynomials as structure constants, one may therefore form the generic Ringel-Hall algebra over ${\mathbb Z}[T]$. Let $\mathfrak n(\Lambda)$ be the subgroup of $\mathcal H(\Lambda)$ generated by the indecomposable modules. If we specialise $T\mapsto1$, then ${\mathbb Z}\otimes_{{\mathbb Z}[T]}\mathfrak n(\Lambda)$ becomes a Lie subalgebra of ${\mathbb Z}\otimes_{{\mathbb Z}[T]}\mathcal H(\Lambda)$. In fact, over the rational numbers, ${\mathbb Q}\otimes_{{\mathbb Z}[T]}\mathcal H(\Lambda)$ is isomorphic to the universal enveloping algebra of ${\mathbb Q}\otimes_{{\mathbb Z}[T]}\mathfrak n(\Lambda)$.
In particular, let $\Lambda$ be a representation-finite hereditary $k$-algebra and let $\mathfrak g=\mathfrak n_-\oplus\mathfrak h\oplus\mathfrak n_+$ be the semisimple complex Lie algebra of the same type as $\Lambda$. Then ${\mathbb Z}\otimes_{{\mathbb Z}[T]}\mathfrak n(\Lambda)$ can be identified with the Chevalley ${\mathbb Z}$-form of $\mathfrak n_+$, and ${\mathbb Z}\otimes_{{\mathbb Z}[T]}\mathcal H(\Lambda)$ becomes the Kostant ${\mathbb Z}$-form of the universal enveloping algebra $U(\mathfrak n_+)$ [@Ringel3].
For a general finite dimensional hereditary $k$-algebra $\Lambda$ one considers the composition algebra, the subalgebra generated by the simple modules. This also has a generic version, constructed as a subalgebra of a direct product over infinitely many finite fields of composition algebras [@Ringel4]. Green showed in [@Green] that the generic composition algebra (after twisting the multiplication via the Euler form of the category $\bmod\Lambda$) is isomorphic to the quantum group of the same type as $\Lambda$.
Therefore, we can realise the quantum group of any symmetrisable Kac-Moody Lie algebra via the module categories of finite dimensional hereditary $k$-algebras.
A natural question is whether it is possible to obtain the full (quantised) enveloping algebra, or at least the full Lie algebra. The latter question was answered by Peng and Xiao in [@PX1] for the affine Lie algebras of type $\widetilde{\mathbb A}$, and in [@PX2] for the simple complex Lie algebras, using the root category $\mathcal D^b(\bmod\Lambda)/T^2$. Finally, these methods were generalised to all 2-periodic triangulated $k$-categories (satisfying some finiteness conditions), and in particular to the root category of a finite dimensional hereditary $k$-algebra [@PX3].
In particular, given any symmetrisable generalised Cartan matrix, the (derived) Kac-Moody Lie algebra can be realised via the root categories (together with the Grothendieck groups) of finite dimensional hereditary $k$-algebras.
Unfortunately, it is unlikely that the approach taken by Peng and Xiao yields an associative algebra.
In this article, we offer a simplified and more intuitive proof of the theorem of Peng and Xiao [@PX3]. We begin by providing a more categorical proof of the associativity of multiplication for the standard Ringel-Hall algebras. Until now the proof has relied on counting filtrations, and so cannot easily be adapted to triangulated categories. The new proof exhibited in Section 2 works in the setting of an exact category, with the key result being Proposition \[P1\]. Replacing the pull-back/push-out diagram by the Octahedral Axiom immediately yields the analogous Proposition \[P2\] for triangulated categories. This clarifies and improves the results in Section 6 of [@PX3].
In Section 4 we offer a unified approach to the calculations in Section 7 of [@PX3], and then use these results to prove the Jacobi identity. We note that in Case (I), an extra argument is required to prove that coefficient of $\tilde h_X$ vanishes. This argument is missing in [@PX3].
In the final section we consider when we can endow the Lie algebra with a symmetric bilinear form. Contrary to [@PX3], this is not always possible, even for the root category of a finite dimensional hereditary $k$-algebra, since it is not in general defined on the Cartan subalgebra. The best we can hope for is to define the form on $\mathfrak h\times\mathfrak h_1$, where $\mathfrak h_1$ is a subgroup of the Cartan subalgebra. The proof that this is invariant then follows from our previous considerations on the Jacobi identity.
We remark that Toën has recently shown how to construct an associative algebra from a dg-category [@Toen] (under some finiteness assumptions), and it will clearly be of interest to investigate the connections between his derived Hall algebra and the quantised enveloping algebras.
[**Acknowledgements**]{} These notes were prepared for a series of lectures at Universität Paderborn. I would like to thank all involved for helpful comments and suggestions, and in particular H. Asashiba, W. Crawley-Boevey, B. Deng, H. Krause and J. Xiao.
The Main Theorem
================
Let $k$ be a finite field and let $\mathcal T$ be an idempotent complete triangulated $k$-category. We shall also assume that $\mathcal T$ is skeletally small, so the isomorphism classes of objects form a set, and 2-periodic, i.e. $T^2\cong1$, where $T$ is the shift of $\mathcal T$. Note that for $X$ indecomposable, $\operatorname{End}X$ is a finite dimensional local $k$-algebra. Set $d(X):=\dim(\operatorname{End}X/\mathrm{rad}\operatorname{End}X)$.
The Grothendieck group $\mathcal G$ of $\mathcal T$ is the quotient of the free abelian group with generators the isomorphism classes of objects in $\mathcal T$ by the relations $[X]+[Y]-[L]$ whenever there exists an exact triangle $Y\to L\to X\to TY$. Denote by $h_X$ the image of $[X]$ in $\mathcal G$. Note that $h_{TM}=-h_M$ for all objects $M$, and that the group is generated by all $h_X$ for $X$ indecomposable. The Grothendieck group is called proper if $h_X\neq0$ in $\mathfrak h$ for all indecomposable objects $X$.
We call $\mathcal T$ finitary if all homomorphism spaces are finite sets and, given a fixed element $h\in\mathcal G$, there are only finitely many isomorphism classes of indecomposable objects $X$ (up to shift) with $h_X=h$.
Let $k$ be a finite field with $q_k$ elements and let $\mathcal T$ be a skeletally small and idempotent complete triangulated $k$-category. Assume also that $\mathcal T$ is finitary and 2-periodic with proper Grothendieck group. Then we can associate to $\mathcal T$ a Lie algebra $\mathfrak g(\mathcal T)$ over the ring ${\mathbb Z}/(q_k-1){\mathbb Z}$.
The rest of the article is devoted to proving this theorem.
The Ringel-Hall Algebra of an Exact Category
============================================
We recall the definition of an exact category. Let $\mathcal A$ be an additive category and let $\mathcal E$ be a class of kernel-cokernel pairs $(f,g)$, closed under isomorphism. We call $f$ an inflation, $g$ a deflation and $(f,g)$ a conflation.
The pair $(\mathcal A,\mathcal E)$ is an exact category in the sense of Quillen [@Quillen] (see also the appendix to [@Keller]) if the following axioms hold .
1. $0\xrightarrow{1} 0$ is a deflation.
2. The composition of two deflations is again a deflation.
3. For $f:Y\to L$ and a deflation $m:M\to L$ there exists a pull-back $$\xymatrix{L'\ar@{.>}[r]^{f'}\ar@{.>}[d]^{m'} &M\ar[d]^m\\Y\ar[r]^f &L}$$ with $m'$ a deflation.
4. For $m':L'\to Y$ and an inflation $f':L'\to M$ there exists a push-out $$\xymatrix{L'\ar[r]^{f'}\ar[d]^{m'} & M\ar@{.>}[d]^m\\Y\ar@{.>}[r]^f &L}$$ with $f$ an inflation.
We recall the following result from [@Keller].
The product of two inflations is again an inflation. Moreover, the pair $((f'\, -m')^t,(m\,f))$ obtained from either (Ex2) or (Ex3) is a conflation.
The Grothendieck group $\mathcal G$ of $\mathcal A$ is the quotient of the free abelian group with generators the isomorphism classes of objects of $\mathcal A$ by the relations $[X]+[Y]-[L]$ whenever there exists a conflation $Y\to L\to X$. Denote by $h_X$ the image of $[X]$ in $\mathcal G$.
We call $\mathcal A$ finitary if each homomorphism group is finite and, given any $h\in\mathcal G$, there exist only finitely many isomorphism classes $[X]$ with $h_X=h$. N.B. The category of finite modules over a finitary ring, in the sense of Ringel [@Ringel2], is then a finitary abelian category.
Let $\mathcal A$ be a finitary and skeletally small exact category and let $W_{XY}^L$ denote the set of all conflations $Y\to L\to X$. The group $\operatorname{Aut}(X,Y):=\operatorname{Aut}X\times\operatorname{Aut}Y$ acts on $W_{XY}^L$ via $$\xymatrix{Y\ar[r]^f\ar[d]^\eta &L\ar[r]^g\ar@{=}[d] &X\ar[d]^\xi\\
Y\ar[r]^{\overline f} &L\ar[r]^{\overline g} &X}$$ and we denote the quotient set by $V_{XY}^L$. Since $f$ is monic and $g$ epic this action is free, so $$F_{XY}^L:=\left|V_{XY}^L\right|=\frac{\left|W_{XY}^L\right|}{\left|\operatorname{Aut}(X,Y)\right|}.$$
The Ringel-Hall algebra $\mathcal H(\mathcal A)$ is defined as follows. Form the free ${\mathbb Z}$-module on generators indexed by the set of isomorphism classes of objects, writing $u_X$ for $u_{[X]}$, and use the numbers $F_{XY}^L$ as structure constants. That is, $$u_Xu_Y:=\sum_{[L]}F_{XY}^Lu_L.$$ We note that this sum is finite and that $u_0$ is a unit for the multiplication. We now prove that the multiplication is associative.
Let us define an action of $\operatorname{Aut}(X,Y,Z,L):=\operatorname{Aut}X\times\operatorname{Aut}Y\times\operatorname{Aut}Z\times\operatorname{Aut}L$ on the pairs of conflations $W_{XY}^L\times W_{LZ}^M$ via $$\xymatrix{Y\ar[r]^f\ar[d]^\eta &L\ar[r]^g\ar[d]^\lambda &X\ar[d]^\xi\\
Y\ar[r]^{\overline f} &L\ar[r]^{\overline g} &X} \qquad
\xymatrix{Z\ar[r]^l\ar[d]^\zeta &M\ar[r]^m\ar@{=}[d] &L\ar[d]^\lambda\\
Z\ar[r]^{\overline l} &M\ar[r]^{\overline m} &L}$$ Dually we have an action of $\operatorname{Aut}(X,Y,Z,L')$ on the pairs $W_{XL'}^M\times W_{YZ}^{L'}$.
We fix a complete set of representatives $\mathcal M$ for all isomorphism classes of objects of $\mathcal A$.
\[P1\] For $X,Y,Z,M\in\mathcal M$ there is a bijection between the sets $$\bigcup_{L\in\mathcal M}\frac{W_{XY}^L\times W_{LZ}^M}{\operatorname{Aut}(X,Y,Z,L)}\longleftrightarrow\bigcup_{L'\in\mathcal M}\frac{W_{XL'}^M\times W_{YZ}^{L'}}{\operatorname{Aut}(X,Y,Z,L')}.$$
Consider the pair $((f,g),(l,m))\in W_{XY}^L\times W_{LZ}^M$. Let $L'\in\mathcal M$ be the representative of the pull-back $$\xymatrix{L'\ar[r]^{f'}\ar[d]^{m'}&M\ar[d]^m\\Y\ar[r]^f&L}$$ Take $l':Z\to L'$ such that $f'l'=l$ and $m'l'=0$. Then $l'$ is a kernel for the deflation $m'$, so $(l',m')$ is a conflation. Similarly, setting $g':=gm$, then $g'$ is a deflation and $f'$ is a kernel for $g'$, so $(f',g')$ is a conflation. Thus we have a commutative diagram $$\xymatrix{Z\ar[d]^{l'}\ar@{=}[r] &Z\ar[d]^l\\L'\ar[d]^{m'}\ar[r]^{f'} &M\ar[d]^m\ar[r]^{g'} &X\ar@{=}[d]\\Y\ar[r]^f &L\ar[r]^g &X}$$ with all rows and columns being conflations. In particular, we obtain the pair $((f',g'),(l',m'))$ in $W_{XL'}^M\times W_{YZ}^{L'}$.
Now consider the pair $((\overline f,\overline g),(\overline l,\overline m))=(\xi,\eta,\zeta,\lambda)\cdot((f,g),(l,m))$. Using the same construction as above, we obtain $((\overline f',\overline g'),(\overline l',\overline m'))$ in $W_{XL''}^M\times W_{YZ}^{L''}$, where $L''\in\mathcal M$ is the representative of the pull-back of $\overline m$ along $\overline f$.
There exists $\lambda'$ giving a morphism of conflations $$\xymatrix@C=15pt{L'\ar[rr]^-{(f'\,-m')^t}\ar@{.>}[d]^{\lambda'} &&M\oplus Y\ar[rr]^-{(m\,f)}\ar[d]^-{\left(\begin{smallmatrix}1\\&\eta\end{smallmatrix}\right)} &&L\ar[d]^\lambda\\
L''\ar[rr]^-{(\overline f'\,-\overline m')^t} &&M\oplus Y\ar[rr]^-{(\overline m\,\overline f)} &&L}$$ It follows that $\lambda'$ is an isomorphism, hence $L''=L'$ and $\lambda'\in\operatorname{Aut}L'$.
Similarly we obtain $\xi'$ and $\zeta'$ giving isomorphisms of conflations $$\xymatrix{L'\ar[r]^{f'}\ar[d]^{\lambda'} &M\ar[r]^{g'}\ar@{=}[d] &X\ar@{.>}[d]^{\xi'}\\
L'\ar[r]^{\overline f'} &M\ar[r]^{\overline g'} &X} \qquad
\xymatrix{Z\ar[r]^{l'}\ar@{.>}[d]^{\zeta'} &L'\ar[r]^{m'}\ar[d]^{\lambda'} &Y\ar[d]^\eta\\
Z\ar[r]^{\overline l'} &L'\ar[r]^{\overline m'} &Y}$$ Thus $((\overline f',\overline g'),(\overline l',\overline m'))=(\xi',\eta,\zeta',\lambda')\cdot((f',g'),(l',m'))$ and so the two pairs of conflations lie in the same $\operatorname{Aut}(X,Y,Z,L')$-orbit.
In fact, since $l'$ is monic and $g'$ epic, we deduce that $\zeta'=\zeta$ and $\xi'=\xi$.
This proves that the map from left to right is well-defined. Using the symmetry of the situation, we obtain the required bijection.
Let $N_{XYZ}^{LM}$ denote the size of the quotient set $\frac{W_{XY}^L\times W_{LZ}^M}{\operatorname{Aut}(X,Y,Z,L)}$. Since the action is again free, we have $$N_{XYZ}^{LM}=\frac{\left|W_{XY}^L\times W_{LZ}^M\right|}{\left|\operatorname{Aut}(X,Y,Z,L)\right|}=F_{XY}^L F_{LZ}^M.$$ Hence we have the associativity of the multiplication $$\sum_{[L]}F_{XY}^LF_{LZ}^M=\sum_{[L']}F_{XL'}^MF_{YZ}^{L'}.$$
The Ringel-Hall algebra $\mathcal H(\mathcal A)$ of a finitary and skeletally small exact category $\mathcal A$ is an associative, unital algebra.
Suppose further that $\mathcal A$ is a $k$-category for some finite field $k$. Set $q_k:=\left|k\right|$ and write $\mathfrak n$ for the subgroup of $\mathcal H(\mathcal A)$ generated by the $u_X$ for $X$ indecomposable.
Over the ring ${\mathbb Z}/(q_k-1)$, $\mathfrak n$ becomes a Lie subalgebra of $\mathcal H(\mathcal A)$.
It is enough to show that $F_{XY}^L-F_{YX}^L\equiv 0\mod (q_k-1)$ whenever $X$ and $Y$ are indecomposable and $L$ is decomposable (c.f. [@Ringel4]). We may assume that $X\not\cong Y$, since otherwise the result is trivial. So suppose that $L=L'\oplus L''$ and consider a conflation $Y\xrightarrow{(f'\,f'')^t} L'\oplus L''\xrightarrow{(g'\,g'')} X$.
If $L\not\cong X\oplus Y$ then $k^*$ acts freely on the set $V_{XY}^L$, hence $F_{XY}^L\equiv0$. For, let $\big(\eta,\big(\begin{smallmatrix}1\\&\kappa\end{smallmatrix}\big),\xi\big)\in\operatorname{Aut}(Y,L,X)$ with $\kappa\in k^*$ be an automorphism of this conflation. Since $\mathcal A$ is idempotent complete, $\operatorname{End}Y$ is a finite dimensional local $k$-algebra. As neither $f'$ nor $f''$ is 0, both $\eta-1$ and $\eta-\kappa$ are nilpotent. Therefore $\kappa-1$ is also nilpotent, so $\kappa=1$.
Now suppose that $L=X\oplus Y$, so that every conflation is split. We may assume that $f''=1$, $g'=1$ and $f''=-g'=\theta:Y\to X$. Thus $V_{XY}^L\cong\operatorname{Hom}(Y,X)$ and hence $F_{XY}^{X\oplus Y}\equiv 1\mod (q_k-1)$. Similarly $F_{YX}^{X\oplus Y}\equiv1$ and we are done.
Triangulated Categories
=======================
We would like to imitate this result for a triangulated category $\mathcal T$, and in particular for the root category of a finite dimensional hereditary algebra.
We first recall the definition of a triangulated category. Let $\mathcal T$ be an additive category endowed with an equivalence $T$ and let $\mathcal E$ be a class of triangles $Y\xrightarrow{f} L\xrightarrow{g} X\xrightarrow{h} TL$ closed under isomorphism. We call the triangles in $\mathcal E$ exact.
The pair $(\mathcal T,\mathcal E)$ is a triangulated category if the following axioms hold [@Puppe; @Verdier] (see also [@Krause; @Neeman]).
1. The triangle $0\to X\xrightarrow{1} X\to 0$ is exact for all $X$.
2. Each map $f$ fits into an exact triangle $(f,g,h)$.
3. The triangle $(f,g,h)$ is exact if and only if $(g,h,-Tf)$ is exact.
4. Given two exact triangles $(f,g,h)$ and $(f',g',h')$ and two maps $\eta$ and $\lambda$ such that $\lambda f=f'\eta$, there exists a morphism $(\eta,\lambda,\xi)$ of exact triangles $$\xymatrix{Y\ar[r]^f\ar[d]^\eta &L\ar[r]^g\ar[d]^\lambda &X\ar[r]^h\ar@{.>}[d]^\xi &TY\ar[d]^{T\eta}\\
Y'\ar[r]^{f'} &L'\ar[r]^{g'} &X'\ar[r]^{h'} &TY'}$$
5. For $f:Y\to L$ and $m:M\to L$ there exists a morphism of triangles $$\xymatrix{L'\ar[r]^{f'}\ar[d]^{m'} &M\ar[r]^{g'}\ar[d]^m &X\ar[r]^{h'}\ar@{=}[d] &TL'\ar[d]^{Tm'}\\
Y\ar[r]^f &L\ar[r]^g &X\ar[r]^h &TY}$$ such that the left hand square is homotopy cartesian with differential $\delta=h'g$. That is, the triangle $((f'\,-m')^t,(m\,f),h'g)$ is exact.
We shall need the following results about triangulated categories. (In fact, none of the following needs the axiom (Tr$4'$)). A triangle split if it is isomorphic to a triangle of the form $Y\xrightarrow{(0\,1)^t} X\oplus Y\xrightarrow{(1\,0)} X\xrightarrow{0} TY$.
\[LemA\] Let $Y\xrightarrow{f} L\xrightarrow{g} X\xrightarrow{h} TY$ be a triangle.
1. If in (Tr3) $\eta$ and $\lambda$ are isomorphisms, then so is $\xi$;
2. This triangle is split if and only if $h=0$;
3. Suppose $L=M\oplus P$ and that $f=(f'\,0)^t$. Then $(f,g,h)$ is isomorphic to $Y\xrightarrow{(f'\,0)^t} M\oplus P\xrightarrow{\big(\begin{smallmatrix}g'\\&1\end{smallmatrix}\big)} N\oplus P\xrightarrow{(h'\,0)} TY$ and $(f',g',h')$ is exact. A similar result also holds whenever $g=(g'\,0)$.
The first part is Proposition 1.1.20 in [@Neeman], and the second part is Corollary 1.2.7.
The third occurs as Lemma 2.5 in [@PX2], but the proof can be simplified as follows. Form an exact triangle $Y\xrightarrow{f'} M\xrightarrow{g'} N\xrightarrow{h'} TY$. The direct sum of triangles $Y\xrightarrow{(f'\,0)^t} M\oplus P\xrightarrow{\big(\begin{smallmatrix}g'\\&1\end{smallmatrix}\big)} N\oplus P\xrightarrow{(h'\,0)} TY$ is exact by Proposition 1.2.1 of [@Neeman], and there exists a morphism $(1,1,\xi)$ from $(f,g,h)$ to this triangle, which must be an isomorphism.
We recall that (Tr4’) is equivalent to the following axiom (Tr4), or the Octahedral Axiom (see [@Neeman] or the appendix in [@Krause]).
1. Given exact triangles $(f,g,h),(l,m,n)$ and $(l',m',n')$ with $n'=nf$, there exists an exact triangle $(f',g',h')$ such that the following diagram commutes $$\xymatrix{Z\ar@{=}[r]\ar[d]^{l'} &Z\ar[d]^l\\
L'\ar@{.>}[r]^{f'}\ar[d]^{m'} &M\ar@{.>}[r]^{g'}\ar[d]^m &X\ar@{.>}[r]^-{h'}\ar@{=}[d] &TL'\ar[d]^{Tm'}\\
Y\ar[r]^f\ar[d]^{n'} &L\ar[r]^g\ar[d]^n &X\ar[r]^-{h} &TY\\
TZ\ar@{=}[r] &TZ}$$ Moreover, the middle left square is homotopy cartesian with differential $\delta=h'g=(Tl')n$.
We call $\mathcal T$ finitary if each homomorphism group is finite and, given any element $h$ of the Grothendieck group $\mathcal G$ of $\mathcal T$, there exist only finitely many isomorphism classes of indecomposables $[X]$ (up to shift) with $h_X=h$. In this case we have another characterisation of split triangles: the triangle $(f,g,h)$ is split if and only if $L\cong X\oplus Y$. For, we can apply the cohomological functor $\operatorname{Hom}(-,TY)$. By counting arguments, $\operatorname{Hom}(h,TY)=0$, so $h=0$.
Now let $\mathcal T$ be a finitary and skeletally small triangulated category. We prove that the analogue of Proposition \[P1\] still holds for $\mathcal T$.
Denote by $W_{XY}^L$ the set of all exact triangles $Y\to L\to X\to TY$. The group $\operatorname{Aut}(X,Y)=\operatorname{Aut}X\times \operatorname{Aut}Y$ acts via $$\xymatrix{Y\ar[d]^\eta\ar[r]^f &L\ar@{=}[d]\ar[r]^g &X\ar[d]^\xi\ar[r]^-h &TY\ar[d]^{T\eta}\\
Y\ar[r]^{\overline f} &L\ar[r]^{\overline g} &X\ar[r]^-{\overline h} &TY}$$ and we write $V_{XY}^L$ for the quotient. Note that this action is not free.
Let $\operatorname{Aut}(X,Y,Z,L)$ act on $W_{XY}^L\times W_{LZ}^M$ via $$\xymatrix{Y\ar[r]^f\ar[d]^\eta &L\ar[r]^g\ar[d]^\lambda &X\ar[r]^-h\ar[d]^\xi &TY\ar[d]^{T\eta}\\
Y\ar[r]^{\overline f} &L\ar[r]^{\overline g} &X\ar[r]^-{\overline h} &TY} \qquad
\xymatrix{Z\ar[r]^l\ar[d]^\zeta &M\ar[r]^m\ar@{=}[d] &L\ar[r]^-n\ar[d]^\lambda &TZ\ar[d]^{T\zeta}\\
Z\ar[r]^{\overline l} &M\ar[r]^{\overline m} &L\ar[r]^-{\overline n} &TZ}$$ Dually, we have an action of $\operatorname{Aut}(X,Y,Z,L')$ on the set $W_{XL'}^M\times W_{YZ}^{L'}$.
We fix a complete set of representatives $\mathcal M$ for all isomorphism classes of objects of $\mathcal T$.
\[P2\] For $X,Y,Z,M\in\mathcal M$ there is a bijection between the sets $$\bigcup_{L\in\mathcal M}\frac{W_{XY}^L\times W_{LZ}^M}{\operatorname{Aut}(X,Y,Z,L)}\longleftrightarrow\bigcup_{L'\in\mathcal M}\frac{W_{XL'}^M\times W_{YZ}^{L'}}{\operatorname{Aut}(X,Y,Z,L')}.$$
Consider the pair $((f,g,h),(l,m,n))\in W_{XY}^L\times W_{LZ}^M$. Applying (Tr4), we obtain for some $L'\in\mathcal M$ a pair $((f',g',h'),(l',m',n'))\in W_{XL'}^M\times W_{YZ}^{L'}$ fitting into a commutative diagram such that the middle left square is homotopy cartesian with differential $\delta=h'g=(Tl')n$.
Now consider the pair $((\overline f,\overline g,\overline h),(\overline l,\overline m,\overline n))=(\xi,\eta,\zeta,\lambda)\cdot((f,g,h),(l,m,n))$. Using (TR4) we again find for some $L''\in\mathcal M$ a pair $((\overline f',\overline g',\overline h'),(\overline l',\overline m',\overline n'))\in W_{XL''}^M\times W_{YZ}^{L''}$ fitting into a commutative diagram such that the middle left square is homotopy cartesian with differential $\overline\delta=\overline h'\overline g=(T\overline l')\overline n$.
Using (Tr3) we can find $\lambda'$ giving a morphism of exact triangles $$\xymatrix@C=15pt{L'\ar[rr]^-{(f'\,-m')^t}\ar@{.>}[d]^{\lambda'} &&M\oplus Y\ar[rr]^-{(m\,f)}\ar[d]^-{\left(\begin{smallmatrix}1\\&\eta\end{smallmatrix}\right)} &&L\ar[rr]^\delta\ar[d]^\lambda &&TL'\ar@{.>}[d]^{T\lambda'}\\
L''\ar[rr]^-{(\overline f'\,-\overline m')^t} &&M\oplus Y\ar[rr]^-{(\overline m\,\overline f)} &&L\ar[rr]^{\overline\delta} &&TL''}$$ It follows that $\lambda'$ is an isomorphism, hence $L''=L'$ and $\lambda'\in\operatorname{Aut}L'$.
Similarly we obtain $\xi'$ and $\zeta'$ giving isomorphisms of exact triangles $$\xymatrix{L'\ar[d]^{\lambda'}\ar[r]^{f'} &M\ar@{=}[d]\ar[r]^{g'} &X\ar@{.>}[d]^{\xi'}\ar[r]^{h'} &TL'\ar[d]^{T\lambda'}\\L'\ar[r]^{\overline f'} &M\ar[r]^{\overline g'} &X\ar[r]^{\overline h'} &TL'} \qquad
\xymatrix{Z\ar@{.>}[d]^{\zeta'}\ar[r]^{l'} &L'\ar[d]^{\lambda'}\ar[r]^{m'} &Y\ar[d]^\eta\ar[r]^{n'} &TZ\ar@{.>}[d]^{T\zeta'}\\Z\ar[r]^{\overline l'} &L'\ar[r]^{\overline m'} &Y\ar[r]^{\overline n'} &TZ}$$
We deduce that $((\overline f',\overline g',\overline h'),(\overline l',\overline m',\overline n'))=(\xi',\eta,\zeta',\lambda')\cdot((f',g',h'),(l',m',n'))$ and so the two pairs of exact triangles lie in the same $\operatorname{Aut}(X,Y,Z,L')$-orbit.
This shows that the map from left to right is well-defined. Using the symmetry of the situation, we obtain the required bijection.
**Remark.** In fact, it follows from the proof that $\bar h'(\xi'-\xi)g=0$ and $\bar l'(\zeta'-\zeta)T^{-1}n=0$. However, unlike for an exact category, we cannot deduce that $\xi'=\xi$ and $\zeta'=\zeta$. Similarly, none of the actions described above is free. In particular, it is not clear whether we can define an associative algebra using the numbers $F_{XY}^L:=\left|V_{XY}^L\right|$ as structure constants.
Reduction modulo $q_k-1$
========================
From now on, let $k$ be a finite field with $q_k$ elements, and $\mathcal T$ a finitary and skeletally small triangulated $k$-category. Furthermore, we assume that $\mathcal T$ has split idempotents. Thus the endomorphism ring of an indecomposable object is a finite dimensional local $k$-algebra. For an indecomposable $X$, denote by $d(X)$ the dimension $\dim_k(\operatorname{End}X/\mathrm{rad}\operatorname{End}X)$.
**Remark.** This is not a serious restriction on our triangulated category, since we can always form the idempotent completion [@BS].
For fixed indecomposable objects $X,Y$ and $Z$ we will calculate the numbers $F_{XY}^L, F_{LZ}^M$ and $N_{XYZ}^{LM}:=\left|\frac{W_{XY}^L\times W_{LZ}^M}{\operatorname{Aut}(X,Y,Z,L)}\right|$. It will be convenient to work over the subring $R\subset{\mathbb Q}$ obtained by localising ${\mathbb Z}$ at the set of numbers coprime to $(q_k-1)$.
The method of proof here and later is based on the following observation. If a finite group $G$ acts on a finite set $X$, then $$\frac{\left|X\right|}{\left|G\right|}=\sum_{x\in X/G}\frac{1}{\left|\operatorname{Stab}x\right|}.$$
The numbers $F_{XY}^L$
----------------------
\[L1\]
1. If $L\not\cong M\oplus TZ$, then $F_{LZ}^M\equiv \frac{\left|W_{LZ}^M\right|}{\left|\operatorname{Aut}(Z,L)\right|} \mod (q_k-1)$.
2. If $L\cong M\oplus TZ$, then $F_{LZ}^M=1$ and $\left|W_{LZ}^M\right|=\frac{\left|\operatorname{Aut}L\right|}{\left|\operatorname{Hom}(TZ,M)\right|}$.
Suppose first that $L\not\cong M\oplus TZ$. Using the observation, we shall prove that the stabilisers are all bijective to vector spaces and hence have size 1 modulo $(q_k-1)$.
For $t=(l,m,n)\in W_{LZ}^M$ define $$S(t):=\{(\zeta-1,0,\lambda-1)\in\operatorname{End}t \mid (\zeta,\lambda)\in\operatorname{Stab}(t)\}.$$ Here we have written $\operatorname{End}t$ for the subalgebra of $\operatorname{End}Z\times\operatorname{End}M\times\operatorname{End}L$ consisting of all triples $(\zeta,\mu,\lambda)$ giving a morphism of the triangle $t$.
Clearly $S(t)$ is in bijection with $\operatorname{Stab}(t)$ and contains 0. For $(\bar\zeta,0,\bar\lambda)\in S(t)$ we know that $\bar\zeta$ is either nilpotent or an isomorphism, and that $l\bar\zeta=0$. Since $l\neq 0$, $\bar\zeta$ is nilpotent.
Let $(\bar\zeta_i,0,\bar\lambda_i)$ for $i=1,2$ be two elements of $S(t)$ and let $\alpha\in k$. Then $\bar\zeta:=\bar\zeta_1+\alpha\bar\zeta_2$ is nilpotent, and hence $\zeta:=1+\bar\zeta$ is an isomorphism. Set $\lambda:=1+\bar\lambda_1+\alpha\bar\lambda_2$, so that $(\zeta,1,\lambda)\in\operatorname{End}t$. Thus $\lambda$ is an isomorphism and so $(\zeta,\lambda)\in\operatorname{Stab}(t)$. Therefore $S(t)$ is a vector subspace of $\operatorname{End}t$. This proves Part 1.
Now suppose that $L=M\oplus TZ$. There is a unique orbit, represented by the triangle $Z\xrightarrow{0} M\xrightarrow{(1\,0)^t} M\oplus TZ\xrightarrow{(0\,1)} TZ$. Clearly $(\zeta,\lambda)$ stablises this triangle if and only if $\lambda=\left(\begin{smallmatrix}1&\theta\\0&T\zeta\end{smallmatrix}\right)$ for some $\theta:TZ\to M$.
\[L2\]
1. If $L\not\not\cong 0$, then $F_{XY}^L\equiv \frac{\left|W_{XY}^L\right|}{\left|\operatorname{Aut}(X,Y)\right|} \mod (q_k-1)$.
2. If $L\cong X\oplus Y$ and $X\not\cong Y$, then $F_{XY}^L=\frac{\left|W_{XY}^L\right|}{\left|\operatorname{Aut}(X,Y)\right|}=\left|\operatorname{Hom}(Y,X)\right|$.
3. We have $\left|W_{XY}^{X\oplus Y}\right|=\frac{\left|\operatorname{Aut}(X\oplus Y)\right|}{\left|\operatorname{Hom}(X,Y)\right|}$.
This first part follows from the previous lemma, since $X\cong L\oplus TY$ and $X$ and $Y$ indecomposable implies $L\cong0$.
Suppose $L=X\oplus Y$ and $X\not\cong Y$. Since every triangle $(f,g,h)$ is split, each orbit is represented by a triangle of the form $((\theta\,1)^t,(1\,-\theta),0)$ for some $\theta:Y\to X$. These orbits are all distinct, and have trivial stabilisers.
Finally consider $W_{XY}^{X\oplus Y}$. By rotation, this set is in bijection with $W_{X\oplus YT^{-1}X}^Y$, whose size we know by the previous lemma.
The numbers $N_{XYZ}^{LM}$
--------------------------
\[L3\] If $L\not\cong M\oplus TZ$, then $N_{XYZ}^{LM} \equiv F_{XY}^LF_{LZ}^M$.
If $L=0$, then it is clear that $N_{XYZ}^{LM}=F_{XY}^LF_{LZ}^M$, and this equals 0 unless both $X\cong TY$ and $Z\cong M$, in which case it equals 1.
Suppose $L\not\cong 0$ and let $(t,u):=((f,g,h),(l,m,n))\in W_{XY}^L\times W_{LZ}^M$. For $\beta$ an automorphism write $\bar\beta:=\beta-1$ and define $$S(t,u):=\{((\bar\eta,\bar\lambda,\bar\xi),(\bar\zeta,0,\bar\lambda))\in\operatorname{End}t\times\operatorname{End}u \mid (\xi,\eta,\zeta,\lambda)\in\operatorname{Stab}(t,u)\}.$$ As before, $S(t,u)$ is in bijection with $\operatorname{Stab}(t,u)$ and contains the zero morphism.
For $((\bar\eta,\bar\lambda,\bar\xi),(\bar\zeta,0,\bar\lambda))\in S(t,u)$ we know that $\bar\zeta$ is nilpotent, say $\bar\zeta^{r-1}=0$. Then $n\bar\lambda^{r-1}=0$, so $\bar\lambda^{r-1}=m\theta$ for some $\theta:L\to M$. Hence $\bar\lambda^r=0$, since $\bar\lambda m=0$.
We now have $(\bar\eta^r,0,\bar\xi^r)\in\operatorname{End}t$. If $\bar\xi^r$ is an automorphism, then $g=0$ and the rotation $(-T^{-1}h,f,g)$ of $t$ is split. Since $X$ and $Y$ are indecomposable, $L\cong0$, a contradiction. Hence $\bar\xi$ is nilpotent. Similarly $\bar\eta$ is nilpotent.
Let $((\bar\eta_i,\bar\lambda_i,\bar\xi_i),(\bar\zeta_i,0,\bar\lambda_i))$ for $i=1,2$ be two elements of $S(t,u)$ and let $\alpha\in k$. Set $\bar\xi:=\bar\xi_1+\alpha\bar\xi_2$ and $\xi:=1+\bar\xi$, and similarly for $\eta,\zeta$ and $\lambda$. Then $\bar\xi,\bar\eta$ and $\bar\zeta$ are nilpotent, so $\xi,\eta$ and $\zeta$ are isomorphisms. Since $((\eta,\lambda,\xi),(\zeta,1,\lambda))\in\operatorname{End}t\times\operatorname{End}u$, $\lambda$ must also be an isomorphism and so $(\xi,\eta,\zeta,\lambda)\in\operatorname{Stab}(t,u)$. Therefore $S(t,u)$ is a vector subspace of $\operatorname{End}t\times\operatorname{End}u$.
It follows that $N_{XYZ}^{LM}\equiv\frac{\left|W_{XY}^L\times W_{LZ}^M\right|}{\left|\operatorname{Aut}(X,Y,Z,L)\right|}$, which we know by the previous subsection is congruent to the product $F_{XY}^LF_{LZ}^M$.
\[L4\] Let $L=M\oplus TZ$ with $M\not\cong 0$, so that $F_{LZ}^M=1$ and $\left|W_{LZ}^M\right|=\frac{\left|\operatorname{Aut}L\right|}{\left|\operatorname{Hom}(TZ,M)\right|}$.
1. If $L\not\cong X\oplus Y$, then $N_{XYZ}^{LM} \equiv \frac{\left|W_{XY}^L\right|}{\left|\operatorname{Aut}(X,Y,Z)\right|}$.
2. If $L\cong X\oplus Y$, then there are three subcases.
- If $M\not\cong Y$, then $N_{XYZ}^{LM}=1$.
- If $M\not\cong X$, then $N_{XYZ}^{LM}-1\equiv\frac{\dim\operatorname{Hom}(Y,X)}{d(X)}$.
- If $M\cong X\cong Y$, then $N_{XYZ}^{LM}-2 \equiv \frac{\dim\mathrm{rad}\operatorname{End}X}{d(X)}$.
Every orbit is represented by a pair $(t,u)=\big((f,g,h),(0,(1\,0)^t,(0\,1))\big)$. We shall use the terminology of the previous lemma. Then $\lambda=\left(\begin{smallmatrix}1&\theta\\0&T\zeta\end{smallmatrix}\right)$ for some $\theta:TZ\to M$.
Write $f=(f_1\,f_2)^t$ and $g=(g_1\,g_2)$. Then $$f_1\bar\eta=\theta f_2, \quad f_2\bar\eta=(T\bar\zeta)f_2, \quad \bar\xi g_1=0, \quad \bar\xi g_2=g_1\theta+g_2(T\bar\zeta), \quad \bar\eta h=h\bar\xi.$$
1\. If $L\not\cong X\oplus Y$ then $\bar\xi,\bar\eta$ and $\bar\zeta$ are all nilpotent. For, if $\bar\xi$ is an automorphism, then $g_1=0$ and $Y\cong M\oplus Y'$ by Lemma \[LemA\]. Since $Y$ is indecomposable and $M\not\cong 0$, we must have that $Y\cong M$ and $X\cong TZ$, hence $L\cong X\oplus Y$, a contradiction. Thus $\bar\xi$ is nilpotent. Now, if $\bar\eta$ is an automorphism, then $h=0$. Thus $t$ is split and $L\cong X\oplus Y$, a contradiction. Finally, if $\bar\zeta$ is an automorphism, then $f_2=0$ and Lemma \[LemA\] applies to give $X\cong M$ and $Y\cong TZ$, hence $L\cong X\oplus Y$, a contradiction.
Therefore $S(t,u)$ is a vector subspace of $\operatorname{End}t\times\operatorname{End}u$ and the result follows, using the formula for $\left|W_{LZ}^M\right|$.
2\. Now suppose that $L\cong X\oplus Y$. Then $t$ is split and $h=0$. Also, either $f_1$ or $f_2$ is an isomorphism.
Suppose that $f_2$ is an isomorphism, so we may identify $Y=TZ$ and $X=M$. Then there is a unique orbit for $(t,u)$, given by the pair of split triangles. It follows that $\xi=1$, $\theta=0$ and $\eta=T\zeta$, so the stabiliser is isomorphic to $\operatorname{Aut}Y$.
Conversely, suppose that $f_2$ is not an isomorphism. Then we may identify $Y=M$ and $X=TZ$, and $t$ is represented by $((1\,\phi)^t,(-\phi\,1),0)$ with $\phi:Y\to X$ not an isomorphism. We note the conditions $$\xi\phi=\phi, \quad T\zeta=\xi+\phi\theta, \quad \eta=1+\theta\phi, \quad \lambda=\left(\begin{smallmatrix}1&\theta\\0&T\zeta\end{smallmatrix}\right).$$ Since $\phi$ is not an isomorphism, $\eta$ is necessarily invertible, and $\zeta$ and $\lambda$ are invertible if and only if $\xi$ is. Therefore the stabiliser is determined by $\theta$ and $\xi$ such that $\bar\xi\phi=0$.
If $\phi=0$, then the stabiliser is isomorphic to $\operatorname{Aut}X\times\operatorname{Hom}(X,Y)$. Otherwise, if $\phi\neq0$, then $\bar\xi$ must be nilpotent and the stabiliser is in bijection with some subspace of $\mathrm{rad}\operatorname{End}X\times\operatorname{Hom}(X,Y)$.
We conclude that if $M\not\cong Y$, then $N_{XYZ}^{LM}=1$.
If $M\not\cong X$ then $N_{XYZ}^{LM}-1$ is congruent to $$\frac{\left|W_{XY}^L\times W_{LZ}^M\right|}{\left|\operatorname{Aut}(X,Y,Z,L)\right|}-\frac{1}{\left|\operatorname{Aut}X\right|\left|\operatorname{Hom}(X,Y)\right|}=\frac{\left|\operatorname{Hom}(Y,X)\right|-1}{\left|\operatorname{Hom}(X,Y)\right|\left|\operatorname{Aut}X\right|}$$ using Lemma \[L2\]. Since $\operatorname{End}X$ is local, we can write $\left|\operatorname{Aut}X\right|=q^r(q^{d(X)}-1)$, where $r=\dim\mathrm{rad}\operatorname{End}X$. Thus $$N_{XYZ}^{LM}-1 \equiv \frac{1}{q^{r+\dim\operatorname{Hom}(X,Y)}}\cdot\frac{q^{\dim\operatorname{Hom}(Y,X)}-1}{q^{d(X)}-1} \equiv \frac{\dim\operatorname{Hom}(Y,X)}{d(X)}.$$
Finally, if $M=X=Y=TZ$, then $N_{XYZ}^{LM}-2$ is congruent to $$\frac{\left|W_{XX}^{X^2}\right|}{\left|\operatorname{Aut}X\right|^3\left|\operatorname{End}X\right|}-\frac{1}{\left|\operatorname{Aut}X\right|\left|\operatorname{End}X\right|}-\frac{1}{\left|\operatorname{Aut}X\right|}.$$ From Lemma \[L2\], $\left|W_{XX}^{X^2}\right|=\left|\operatorname{Aut}X^2\right|/\left|\operatorname{End}X\right|$. Using the notation above, $\left|\operatorname{Aut}X^2\right|=q^{4r+d(X)}(q^{d(X)}+1)(q^{d(X)}-1)^2$, so that $$N_{XYZ}^{LM}-2 \equiv \frac{1}{q^{2r+d(X)}}\cdot\frac{q^r-1}{q^{d(X)}-1}\equiv \frac{r}{d(X)}.$$
We state the dual result for the numbers $\hat N_{XYZ}^{ML'}:=\left|\smash[t]{\frac{W_{XL'}^M\times W_{YZ}^{L'}}{\operatorname{Aut}(X,Y,Z,L')}}\right|$, with the special case being when $L'\cong M\oplus T^{-1}X$.
\[L5\] If $L'\not\cong M\oplus T^{-1}X$, then $\hat N_{XYZ}^{ML'}\equiv F_{XL'}^MF_{YZ}^{L'}$.
On the other hand, let $L'=M\oplus T^{-1}X$ with $M\not\cong0$.
1. If $L'\not\cong Y\oplus Z$, then $\hat N_{XYZ}^{ML'}\equiv\frac{\big|W_{YZ}^{L'}\big|}{\left|\operatorname{Aut}(X,Y,Z)\right|}$.
2. If $L'\cong Y\oplus Z$, then there are three subcases.
- If $M\not\cong Y$, then $\hat N_{XYZ}^{ML'}=1$.
- If $M\not\cong Z$, then $\hat N_{XYZ}^{ML'}-1\equiv\frac{\dim\operatorname{Hom}(Z,Y)}{d(Z)}$.
- If $M\cong Y\cong Z$, Then $\hat N_{XYZ}^{ML'}-2\equiv\frac{\dim\mathrm{rad}\operatorname{End}Z}{d(Z)}$.
The Jacobi Identity
===================
In order to prove the Jacobi identity, we shall need two further assumptions on our triangulated category $\mathcal T$. Firstly, we need that $\mathcal T$ is 2-periodic, i.e. $T^2\cong1$. We note that this is satisfied for the root category $\mathcal D^b(\bmod\Lambda)/T^2$ for $\Lambda$ a finite dimensional hereditary $k$-algebra [@PX2] (see also [@Keller2]). We shall also need that for $X$ indecomposable, $h_X\neq0$ in $\mathcal G$. The latter condition is referred to as proper in [@PX3].
We continue to work over the ring $R/(q_k-1)$.
For $X$ indecomposable, define $\tilde h_X:=\frac{h_X}{d(X)}$ in ${\mathbb Q}\otimes_{\mathbb Z}\mathcal G$ and set $\mathfrak h$ to be the integer lattice generated by the $\tilde h_X$. Define a bilinear form on $\mathfrak h$ by setting $(h_X\,|\,h_Y)$ to be the alternating sum $$\dim\operatorname{Hom}(X,Y)-\dim\operatorname{Hom}(X,TY)+\dim\operatorname{Hom}(Y,X)-\dim\operatorname{Hom}(Y,TX).$$ Note that $(\tilde h_X\,|\,h_Y)\in{\mathbb Z}$ for all indecomposable objects $X$ and $Y$.
Define $\mathfrak n$ to be the free abelian group on generators indexed by the isomorphism classes of indecomposable objects, writing $u_X$ for $u_{[X]}$.
In this section, $X,Y$ and $Z$ will again denote indecomposable objects.
We define a bracket product on $\mathfrak g:=\mathfrak h\oplus\mathfrak n$ via the formulae $$\begin{aligned}
{}
[u_X,u_Y] &:= \begin{cases}
\sum_{L\,\mathrm{ind}}(F_{XY}^L-F_{YX}^L)u_L &\textrm{if }X\not\cong TY;\\
\tilde h_X &\textrm{if }X\cong TY,
\end{cases}\\{}
[\tilde h_X,u_Y] &:= -(\tilde h_X\mid h_Y)u_Y, \quad [u_Y,\tilde h_X]=(\tilde h_X\mid h_Y)u_Y\\{}
[\tilde h_X,\tilde h_Y] &:= 0.\end{aligned}$$ Clearly this is antisymmetric. It is well defined from the finiteness condition on the Grothedieck group.
We wish to prove the Jacobi identity $$[[u_X,u_Y],u_Z]-[[u_X,u_Z],u_Y]-[[u_Z,u_Y],u_X]\equiv 0$$ in the following three cases.
1. $X,Y\not\cong TZ$ and $X\not\cong TY$. In this case, $$[[u_X,u_Y],u_Z]=\sum_{\substack{L,M\,\mathrm{ind}\\L\not\cong TZ}}(F_{XY}^L-F_{YX}^L)(F_{LZ}^M-F_{ZL}^M)u_M - (F_{XY}^{TZ}-F_{YX}^{TZ})\tilde h_Z.$$
2. $X,Y\not\cong TZ$ and $X\cong TY$. In this case, $[[u_X,u_Y],u_Z]=-(\tilde h_X\,|\,h_Z)u_Z$.
3. $X\cong Y\cong TZ$. In this case, the Jacobi identity is trivial.
N.B. It is enough to consider these cases. For, if $X\cong TZ$ and $Y\not\cong TZ$, then either $X\not\cong TY$ and we can use the triple $(u_Z,u_X,u_Y)$, or $X\cong TY$ and we can use the triple $(u_Y,u_Z,u_X)$. Similarly if $X\not\cong TZ$ and $Y\cong TZ$.
\[L6\] If $L$ is decomposable or 0, then $F_{XY}^L-F_{YX}^L\equiv 0$.
For $L=0$ this is clear, so suppose that $L$ is decomposable, say $L=M\oplus TZ$ with $Z$ indecomposable and $M\not\cong 0$. If $L\not\cong X\oplus Y$, then by Lemmas \[L2\] and \[L4\], $$F_{XY}^L\equiv\frac{\left|W_{XY}^L\right|}{\left|\operatorname{Aut}(X,Y)\right|}\equiv\left|\operatorname{Aut}Z\right|N_{XYZ}^{LM}\equiv 0.$$ If $L=X\oplus Y$ but $X\not\cong Y$, then by Lemma \[L2\], $F_{XY}^L=\left|\operatorname{Hom}(Y,X)\right|\equiv 1$. Finally, if $X\cong Y$, then the left hand side is clearly 0.
This implies that in the expression for $[[u_X,u_Y],u_Z]$ in Case (I), we can remove the restriction that $L$ is indecomposable. Moreover, since the Grothedieck group is proper, we can remove the restriction that $L\not\cong TZ$. For, $F_{TZZ}^M\neq0$ implies $h_M=0$, so that $M$ is decomposable.
**Case (I)** The coefficient $c_M$ of $u_M$ in the Jacobi identity equals the alternating sum $$c_M=\Delta_{XYZ}^M+\Delta_{YZX}^M+\Delta_{ZXY}^M-\Delta_{YXZ}^M-\Delta_{ZYX}^M-\Delta_{XZY}^M,$$ where $$\Delta_{XYZ}^M:=\sum_LF_{XY}^LF_{LZ}^M-\sum_LF_{XL}^MF_{YZ}^L.$$ N.B. This summation makes sense in $R/(q_k-1)$, since if $L$ is decomposable and not isomorphic to $X\oplus Y$, then $F_{XY}^L\equiv 0$.
Using Lemmas \[L3\] and \[L5\], Proposition \[P2\] and that $T^2=1$ we see that $$\Delta_{XYZ}^M \equiv F_{XY}^{M\oplus TZ}F_{M\oplus TZZ}^M-F_{XM\oplus TX}^MF_{YZ}^{M\oplus TX}-N_{XYZ}^{M\oplus TZM}+\hat N_{XYZ}^{MM\oplus TX}.$$ Substituting this into $c_M$, the part involving the $F$s can be expressed as $$(F_{XY}^{M\oplus TZ}-F_{YX}^{M\oplus TZ})F_{M\oplus TZ Z}^M - F_{XM\oplus TX}^M(F_{YZ}^{M\oplus TX}-F_{ZY}^{M\oplus TX})$$ together with two other terms given by cyclically permuting $(X,Y,Z)$. These all vanish by Lemma \[L6\].
It remains to consider the terms involving $N$ and $\hat N$. By our conditions on $(X,Y,Z)$, we must always be in Case 1 of Lemmas \[L4\] and \[L5\]. We see that $\frac{\left|W_{XY}^{M\oplus TZ}\right|}{\left|\operatorname{Aut}(X,Y,Z)\right|}$ occurs once from $N_{XYZ}^{M\oplus TZ M}$ and once from $\hat N_{ZXY}^{MM\oplus TZ}$, and these cancel in $c_M$. Similarly for each of the other permutations.
We deduce that the coefficient of $u_M$ in the Jacobi identity is 0.
It remains to consider the terms lying in $\mathfrak h$. That is, we need to check that $$-(F_{XY}^{TZ}-F_{YX}^{TZ})\tilde h_Z + (F_{XZ}^{TY}-F_{ZX}^{TY})\tilde h_Y + (F_{ZY}^{TX}-F_{YZ}^{TX})\tilde h_X \equiv 0.$$
\[L7\] Let $t:Y\to Z\to X\to TY$ be a triangle. Then $\operatorname{End}t$ is local.
We will show that if one of $\xi,\eta$ or $\zeta$ is nilpotent, then they all are.
Suppose that $\xi$ is nilpotent, say $\xi^m=0$. Then $(\eta^m,\zeta^m,0)\in\operatorname{End}t$, and since $t$ is not split, $\eta^m$ must be nilpotent. Similarly, after rotating the triangle, we see that $\zeta$ is nilpotent.
Set $d(t):=\dim(\operatorname{End}t/\mathrm{rad}\operatorname{End}t)$. The above proof shows that we have a natural monomorphism $$\operatorname{End}t/\mathrm{rad}\operatorname{End}t\to\operatorname{End}X/\mathrm{rad}\operatorname{End}X,$$ and hence $d(t)$ divides $d(X)$, and similarly $d(Y)$ and $d(Z)$.
The group $\operatorname{Aut}(X,Y,Z)$ acts on the set $W_{XY}^{TZ}$, so $$\frac{\left|W_{XY}^{TZ}\right|}{\left|\operatorname{Aut}(X,Y,Z)\right|}=\sum_t\frac{1}{\left|\operatorname{Aut}t\right|},$$ where the sum is taken over all orbits. Applying Lemma \[L1\], we get $$F_{XY}^{TZ}\equiv \frac{\left|W_{XY}^{TZ}\right|}{\left|\operatorname{Aut}(X,Y)\right|}=\frac{\left|W_{XY}^{TZ}\right|\left|\operatorname{Aut}(Z)\right|}{\left|\operatorname{Aut}(X,Y,Z)\right|} = \sum_t\frac{\left|\operatorname{Aut}Z\right|}{\left|\operatorname{Aut}t\right|}\equiv \sum_t\frac{d(Z)}{d(t)}.$$
If $t\in W_{XY}^{TZ}$, then $h_X+h_Y+h_Z=0$ and so $$\frac{d(X)}{d(t)}\tilde h_X+\frac{d(Y)}{d(t)}\tilde h_Y+\frac{d(Z)}{d(t)}\tilde h_Z=0$$ By rotating triangles, we deduce that $$-F_{XY}^{TZ}\tilde h_Z \equiv -\sum_t\frac{d(Z)}{d(t)}\tilde h_Z \equiv \sum_t\left(\frac{d(X)}{d(t)}\tilde h_X+\frac{d(Y)}{d(t)}\tilde h_Y\right) \equiv F_{YZ}^{TX}\tilde h_X+F_{ZX}^{TY}\tilde h_Y.$$
This completes the proof of the Jacobi identity in Case (I).
**Case (II)** Since the Grothendieck group is proper, $u_X$ does not occur in $[u_X,u_Z]$. Furthermore, all numbers of the form $F_{XTX}^M$ vanish. Therefore $$-[[u_X,u_Z],u_{TX}]-[[u_Z,u_{TX}],u_X]=\sum_Mc_Mu_M,$$ where $$c_M=\Delta_{XTXZ}^M+\Delta_{TXZX}^M+\Delta_{ZXTX}^M-\Delta_{TXXZ}^M-\Delta_{ZTXX}^M-\Delta_{XZTX}^M.$$ In particular, this lies entirely in $\mathfrak n$.
As before, the part of $c_M$ involving the $F$s vanishes. For the terms involving $N$ and $\hat N$, we have two cases, depending on whether $M\cong Z$ or not. If $M\not\cong Z$, then we are always in Case 1 of Lemmas \[L4\] and \[L5\], and $c_M\equiv 0$ as in Case (I). So suppose that $M=Z$. Then, for example, $$N_{XTXZ}^{M\oplus TZ M}\equiv \tfrac{\left|W_{XTX}^{M\oplus TZ}\right|}{\left|\operatorname{Aut}(X,Y,Z)\right|}, \quad N_{TXZX}^{M\oplus TXM}\equiv 1+\tfrac{\dim\operatorname{Hom}(Z,TX)}{d(X)}, \quad N_{ZXTX}^{M\oplus XM}\equiv 1.$$ A short calculation (noting $\operatorname{Hom}(TX,Z)\cong\operatorname{Hom}(X,TZ)$) reveals that $c_Z=(\tilde h_X\,|\,h_Z)$. Since $[[u_X,u_{TX}],u_Z]=-(\tilde h_X \mid h_Z)u_Z$, we are done.
We have therefore shown that the Jacobi identity is satisfied for all triples $(u_X,u_Y,u_Z)$.
Now consider the triple $(u_X,u_Y,\tilde h_Z)$. Then $[[u_X,u_Y],\tilde h_Z]$ equals 0 if $X\cong TY$; otherwise we have $$[[u_X,u_Y],\tilde h_Z] = \sum_{L\,\mathrm{ind}}(F_{XY}^L-F_{YX}^L)[u_L,\tilde h_Z] = (\tilde h_Z\mid h_X+h_Y)[u_X,u_Y].$$ The Jacobi identity follows immediately. The remaining case $(u_X,\tilde h_Y,\tilde h_Z)$ is easily checked.
This completes the proof of the Main Theorem.
The Symmetric Bilinear Form
===========================
It would clearly be of interest to extend the symmetric bilinear form to an invariant form on the whole of the Lie algebra $\mathfrak g/(q_k-1)\mathfrak g$. Unfortunately, this is not always possible since although $(\tilde h_X\,|\,h_Y)\in{\mathbb Z}$, it is not true that $(\tilde h_X\,|\,\tilde h_Y)\in{\mathbb Z}$. In fact, this even fails for the root category $\mathcal D^b(\bmod \Lambda)/T^2$ of a finite dimensional hereditary $k$-algebra $\Lambda$ (for example, when $\Lambda$ is the $\mathbb F_4$-species of type $\mathbb G_2$).
One important special case when we can define the bilinear form is when $\Lambda=kQ$ is the path algebra of a quiver. For, the Grothendieck group is in this case freely generated by the simple modules, and these all have trivial endomorphism rings.
In general, define $\mathfrak h_1$ to be the sublattice of $\mathfrak h$ generated by $h_X$ for $X$ indecomposable. Set $\mathfrak g_1:=\mathfrak h_1\oplus\mathfrak n$ and extend the bilinear form on $\mathfrak h\times\mathfrak h_1$ to the whole of $\mathfrak g\times\mathfrak g_1$ via $$(\tilde h_X\mid u_Y):=0, \quad (u_X\mid u_Y):=\begin{cases}1 &\textrm{if }X\cong TY;\\0 &\textrm{otherwise}.\end{cases}$$ This form is then invariant over $R/(q_k-1)$ in the sense that $$\begin{aligned}
(\tilde h_X\mid[u_Y,u_Z])d(Z) &\equiv -([\tilde h_X,u_Y]\mid u_Z)\\
\textrm{and}\quad d(X)([u_X,u_Y]\mid u_Z) &\equiv (u_X\mid[u_Y,u_Z])d(Z).\end{aligned}$$ For the first condition, both sides are 0 unless $Y\cong TZ$, in which case they are both congruent to $(\tilde h_X|h_Y)$. For the second, we need to show that $$d(X)\big(F_{XY}^{TZ}-F_{YX}^{TZ}\big)\equiv d(Z)\big(F_{YZ}^{TX}-F_{ZY}^{TX}\big).$$ As in Case (I) of the Jacobi identity, we know that $$d(X)F_{XY}^{TZ}\equiv\sum_t\frac{d(X)d(Z)}{d(t)}\equiv d(Z)F_{YZ}^{TX},$$ and we are done.
[99]{} P. Balmer and M. Schlichting, ‘Idempotent completion of triangulated categories’, *J. Algebra* 236 (2001) 819–834. J.A. Green, ‘Hall algebras, hereditary algebras and quantum groups’, *Invent. Math.* 120 (1995) 361–377. P. Hall, ‘The algebra of partitions’, in *Proc. 4th Canad. Math. Congress* (Banff, 1957) (Univ. of Toronto Press, Toronto, 1959) 147–159. B. Keller, ‘Chain complexes and stable categories’, *Manus. Math.* 67 (1990) 379–417. B. Keller, ‘On triangulated orbit categories’, preprint <http://www.math.jussieu.fr/~keller/publ/triaorbit.ps> . H. Krause, ‘Derived categories, resolutions, and Brown representability’, <http://wwwmath.upb.de/~hkrause/publ/chicago.ps> . D. Quillen, *Higher algebraic K-theory, I*, Springer Lecture Notes in Mathematics 341 (1973) 85–147. A. Neeman, *Triangulated Categories*, Annals of Mathematical Studies 148 (Princeton University Press, Princeton, 2001). L. Peng and J. Xiao, ‘A realisation of affine Lie algebras of type $\tilde A_{n-1}$ via the derived categories of cyclic quivers’, in *Representation theory of algebras* (Cocoyoc, 1994) CMS Conf. Proc. 18 (Amer. Math. Soc., Providence, 1996) 539–554. L. Peng and J. Xiao, ‘Root categories and simple Lie algebras’, *J. Algebra* 198 (1997) 19–56. L. Peng and J. Xiao, ‘Triangulated categories and Kac-Moody Lie algebras’, *Invent. Math.* 140 (2000) 563–603. D. Puppe, ‘On the formal structure of stable homotopy theory’, in *Colloquium on algebraic topology*, Matematisk Institut, Aarhus Universitet (Aarhus, 1962) 65–71. C.M. Ringel, ‘Hall algebras and quantum groups’, *Invent. Math.* 101 (1990) 583–591. C.M. Ringel, ‘Hall algebras’, in *Topics in Algebra, Part 1*, Banach Center Publ. 26, Part 1 (PWN, Warsaw, 1990) 433–447. C.M. Ringel, ‘Hall polynomials for the representation-finite hereditary algebras’, *Adv. Math.* 84 (1990) 137–178. C.M. Ringel, ‘From representations of quivers via Hall and Loewy algebras to quantum groups’, *Contemp. Math.* 131 (1992) 381–401. E. Steinitz, ‘Zur Theorie der Abel’schen Gruppen’, Jahresber. Deutsch. Math.-Verein 9 (1901) 80–85. B. Toën, ‘Derived Hall algebras’, preprint <http://front.math.ucdavis.edu/math.QA/0501343> . J.L. Verdier, ‘Des catégories dérivées des catégories abéliennes’, *Astérique* 239 (1996)
[Andrew Hubery\
Universität Paderborn, Germany\
[email protected]]{}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We introduce a method to estimate the size of the domain of definition of the solutions of a meromorphic vector field on a neighborhood of its pole divisor. The corresponding techniques are, in a certain sense, quantitative versions of some well-known phenomena related to the presence of metrics with positive curvature. Several applications of these ideas are provided including a type of “confinement theorem” for solutions of complete polynomial vector fields on ${{\mathbb C}}^n$ and obstructions for certain (germs of) vector fields to be realized by a global holomorphic vector field on a compact Kähler manifold. As a complement a new approach to certain classical equations is proposed and detailed in the case of Halphen equations.'
author:
- 'Julio C. Rebelo & Helena Reis'
title: Uniformizing complex ODEs and applications
---
Introduction
============
The object of this paper is a method to investigate the domain of definition of solutions for holomorphic (meromorphic) vector fields. This method is quite general and applicable to arbitrarily high dimensions whereas it provides new results already in dimension $3$. As a consequence this work is essentially constituted by two parts, the first one corresponding to the general setting and central results which will then be exploited in the second part to yield the mentioned applications. To greater or lesser extent, these applications arise from following the solution of a complex (polynomial or rational) vector field over “special real paths going off to infinity”.
This Introduction is aimed only at stating the main applications considered in this work. These results were chosen not with a purpose of being the sharpest possible but rather of indicating ways of exploiting our method which is considered to be the main contribution of this work. In Section 2 we shall provide a more detailed discussion explaining our point of view and underlining the common structures lying behind the theorems below. It is also to be noted that our applications concern very special types of vector fields (or of differential equations) such as complete vector fields and Halphen equations. Nonetheless the setting is also well-adapted to investigating differential equations having meromorphic solutions including several classical equations appearing in Mathematics and Physics. For example equations concerned with the works of Painlevé and Chazy are among those having a very immediate connection with our work. Whereas these possibilities are not developed here, they seem to provide a clear indication that further applications of our techniques will be found in a near future.
Let us begin by considering a complete polynomial vector field $X$ on ${{\mathbb C}}^n$ of degree at least $2$. This means that its complex solutions are defined for all $T \in {{\mathbb C}}$. In the special case in which $X$ is also completely integrable, its orbits can be compactified into rational curves by adding to them some “singular points of $X$ at infinity”. This fact can be interpreted as a type of confinement phenomenon for the corresponding solutions. Our first results can be regarded as a weaker, whereas sharp, type of confinement phenomenon holding for every solution of a complete polynomial vector field. To state it, we proceed as follows. Since $X$ is polynomial it defines a singular holomorphic foliation ${\mathcal{D}}$ on ${{\mathbb C}}P(n) = {{\mathbb C}}^n \cup \Delta_{\infty}$ viewed as a compactification of ${{\mathbb C}}^n$. Consider a leaf $L$ of ${\mathcal{D}}$ (details on the definition of “leaf” in the singular context can be found in Section 3). On $L$ two (singular) oriented real one-dimensional foliations ${\mathcal{H}}, \, {\mathcal{H}}^{\perp}$ are going to be defined. They will depend on the leaf $L$ of ${\mathcal{D}}$ is a regular way as it will be apparent from their definitions (cf. Sections 3 and 6). More importantly, ${\mathcal{H}}, \, {\mathcal{H}}^{\perp}$ are mutually orthogonal with respect to the conformal structure of $L$, in fact, they agree respectively with the real foliation and the purely imaginary foliation induced on $L$ by a certain Abelian form. Since $L$ is endowed with a conformal structure, it makes sense to define also foliations ${\mathcal{H}}^{\theta}$ whose (oriented) trajectories makes an angle $\theta$ with the oriented trajectories of ${\mathcal{H}}$ ($\theta \in [-\pi /2, \pi /2]$). The trajectories of these foliations define the “directions of confinement” for $L$ as it is seen from Theorem A below. In the sequel $\Phi : {{\mathbb C}}\times {{\mathbb C}}^n \rightarrow {{\mathbb C}}^n$ stands for the holomorphic flow generated by $X$ whereas ${\rm Sing}\, ({\mathcal{D}})
\subset {{\mathbb C}}P(n)$ denotes the singular set of ${\mathcal{D}}$.
[**Theorem A**]{}. [*Suppose that $X$ is a complete polynomial vector field as above. Fix an arbitrarily small neighborhood $V$ of $({\rm Sing}\, ({\mathcal{D}}) \cap \Delta_{\infty}) \cup
{\rm Sing}\, (X)$ in ${{\mathbb C}}P(n)$ and suppose we are given a point $p \in {{\mathbb C}}^n$, $X (p) \neq 0$ and an angle $\theta \in
(-\pi/2 ,\pi/2)$. Denote by $L_p$ (resp. $l_p^{+,\theta}$) the leaf of ${{\mathcal F}}$ through $p$ $($resp. the semi-trajectory of ${\mathcal{H}}^{\theta}$ initiated at $p)$ and consider the lift $c:[0, \infty) \rightarrow {{\mathbb C}}$ of $l_p^{+,\theta}$ by $\Phi$, i.e. $t \in [0, \infty) \mapsto
\Phi (c(t), p)$ is a one-to-one parametrization of $l_p^{+,\theta}$ $(c(0)=0)$. Then there is a constant $C$ such that $${\rm meas} \, \left( \{ t \in [0, \infty) \; ; \; \Phi (c(t), p) \not\in V \} \right) < C \, ,$$ where ${\rm meas}$ stands for the usual Lebesgue measure on ${{\mathbb R}}$.* ]{}
The preceding theorem states that the trajectory $l_p^{+,\theta}$ spend most of its “life” in the neighborhood $V$ and hence arbitrarily close to the singular points of ${\mathcal{D}}$. Furthermore the constant $C$ varies continuously with $\theta$. In particular if we consider a compact interval $[-\pi/2 + \delta ,
\pi /2 - \delta] \subset (-\pi/2 ,\pi/2)$ then $C$ can be chosen so that the above estimate holds for every $\theta \in [-\pi/2 + \delta , \pi /2 - \delta]$. The existence of $C$ uniform for $\theta \in [-\pi/2 + \delta , \pi /2 - \delta]$ allows us to generalize the statement to paths $c \subset L_p$ more general than the trajectories of ${\mathcal{H}}^{\theta}$, for example we may consider paths $c$ as before such that the angle made at the point $c(t)$ by the speed vector $c'(t)$ and the foliation ${\mathcal{H}}$ lies in $[-\pi/2 + \delta , \pi /2 - \delta]$ for all $t$. Details on these statements are left to the reader.
Confinement phenomena contrast with ergodicity so that it is natural to search for a variant of this theorem focusing on the “area” defined in ${{\mathbb C}}$ by those values of $T$ for which $\Phi (T,p) \in V$. This variant might be viewed, in particular, as a “super non-ergodic” phenomenon for complete vector field. To state it, let $B_r \subset {{\mathbb C}}$ denote the disc of radius $r$ about $0 \in {{\mathbb C}}$. A continuous path properly embedded $c: (-\infty, \infty) \rightarrow {{\mathbb C}}$ is a [*separating curve*]{} if it is of class $C^{\infty}$ with possible exception of a discrete set and it is either periodic or it satisfies the condition $\lim_{t \rightarrow -\infty} c(t) = \infty$, $\lim_{t \rightarrow \infty} c(t) = \infty$ (where the last conditions means that the curve eventually leaves every compact subset of ${{\mathbb C}}$. A separating curve divides ${{\mathbb C}}$ in at least two connected component and at least one of these components is unbounded. Then we have:
[**Theorem A’**]{}. [*Let $X$, $V$, $L_p$ and $p \in {{\mathbb C}}^n$ be as above. Consider the parametrization of $L_p$ by ${{\mathbb C}}$ (possibly as a covering map) which is given by $\Phi_p (T) = \Phi (T, p)$. Then there exists a separating curve $c: (-\infty, \infty) \rightarrow {{\mathbb C}}$, $\Phi_p(c(0)) =p$, and an unbounded component $\mathcal{U}^+$ of ${{\mathbb C}}\setminus c(t)$ such that the following holds: the set $\mathcal{T}_V \subset \mathcal{U}^+ \subset {{\mathbb C}}$ defined by $$\mathcal{T}_V = \{ T \in \mathcal{U}^+ \subset {{\mathbb C}}\; ; \; \Phi (T,p) \in V \} \,$$ satisfies $$\lim_{r \rightarrow \infty} \frac{ {\rm Meas}\, (\mathcal{T}_V \cap B_r)}
{{\rm Meas}\, (\mathcal{U}^+ \cap B_r)} =1 \, ,$$ where ${\rm Meas}$ stands for the usual Lebesgue measure of ${{\mathbb C}}$ ($\simeq {{\mathbb R}}^2$).*]{}
Unlike most standard averaging theorems, the statement above holds for every point $p \in {{\mathbb C}}^n$ and not only for almost all points. Besides it is easy to conclude from the proof given in Section 6.1 that for almost all points $p$ the corresponding separating curve is smooth. In fact, this separating curve is nothing but a geodesic (“straight line”) for a suitable flat structure on ${{\mathbb C}}$ which is naturally comparable with the standard one (for further details see Section 6.1). In particular it follows that ${\rm Meas}\, (\mathcal{U}^+ \cap B_r)$ is, in fact, comparable to the measure of the large discs $B_r$. Since discs of this nature are used in the construction of Ahlfors currents, the previous statement may look unnatural since the latter cannot charge singular points. Explanation for this difference is however easy since the construction of Ahlfors currents is based on the “global volume” and this may have little relation with the asymptotic behavior of an actual solution. To be more precise fix a diffeomorphism between ${{\mathbb C}}$ and a leaf $L$, for example a time-$t$ diffeomorphism $\Phi_t$ induced by the corresponding vector field. To construct Ahlfors current the ambient manifold is equipped with a Hermitian metric which is then pulled back by $\Phi_t$ to yield a metric $d_{{{\mathbb C}}}$ on ${{\mathbb C}}$. Ahlfors current is then constructed by choosing a suitable sequence of discs $B_{r_i}$ whose radii $r_i$ are measured with respect to $d_{{{\mathbb C}}}$ and satisfy $r_i \rightarrow \infty$. Clearly a “small” neighborhood $V$ of a singular point in $M$ has small diameter for the fixed Hermitian metric and so does a connected component $(L \cap V)_0$ of $L \cap V$. The diameter (resp. the “area”), of $\phi_t^{-1} ((L \cap V)_0)$ w.r.t. $d_{{{\mathbb C}}}$ is therefore small as well. Now we should note that $d_{{{\mathbb C}}}$ may differ markedly from the Euclidean metric on ${{\mathbb C}}$ thus it might happen, for example, that the euclidean area of $\phi_t^{-1} ((L \cap V)_0)$ is “large”. The proofs of the preceding theorems will make it clear that this phenomenon is precisely what happens in these cases. As a conclusion, whereas Ahlfors currents are among the most efficient tools for studying (singular) holomorphic foliations possessing leaves covered by ${{\mathbb C}}$, they might be less so when considering specific differential equations.
The statement of Theorems A and A’ indicate that the structure of the singularities of ${\mathcal{D}}$ lying in $\Delta_{\infty}$ must bear significant information on the global dynamics of corresponding vector fields. According to this principle, it is natural to wonder that complete vector fields whose associated foliations ${\mathcal{D}}$ have only “simple singularities” in $\Delta_{\infty}$ must be amenable to a detailed global analysis. For “simple” singularities we shall mean the following types of singular points $q \in \Delta_{\infty}$ for ${\mathcal{D}}$:
1. Non-degenerate singularities: this means that ${\mathcal{D}}$ can locally be represented by a vector field having non-degenerate linear part at $q$ (i.e. the Jacobian matrix of $X$ at $q$ is invertible, equivalently, it possesses $n$ eigenvalues different from zero). Besides, since resonances may arise, we assume that $q$ is [*not of Poincaré-Dulac type*]{}, i.e. if all the eigenvalues of ${\mathcal{D}}$ at $q$ belong to ${{\mathbb R}}_+^{\ast}$ then ${\mathcal{D}}$ must be locally linearizable about $q$.
2. Codimension $1$ saddle-nodes: these are singularities of ${\mathcal{D}}$ lying in $\Delta_{\infty}$ whose eigenvalue associated to the direction transverse to $\Delta_{\infty}$ is equal to [*zero*]{} whereas it has $n-1$ eigenvalues different from zero and corresponding to directions contained in $\Delta_{\infty}$. Again we require that the $(n-1)$-dimensional singularity induced on the plane $\Delta_{\infty}$ should not be a singularity of Poincaré-Dulac type.
Note that singular points of ${\mathcal{D}}$ as in item (1) above are necessarily isolated though this is no longer valid for Codimension $1$ saddle-nodes. Next we have:
[**Theorem B**]{}. [*Let $X$ be a complete polynomial vector field on ${{\mathbb C}}^n$ whose singular set has codimension at least $2$. Suppose that all singularities of ${\mathcal{D}}$ lying in $\Delta_{\infty}$ are as in items (1) or (2) above. Then the foliation ${\mathcal{D}}$ associated to $X$ defines a rational pencil on ${{\mathbb C}}P(n)$.*]{}
Theorem B will be proved in Section 6.2. The statement of this theorem may be compared to results of [@bruno] for complete polynomial vector fields on ${{\mathbb C}}^2$. It is to be noted that the results of [@bruno] chronologically preceded the classification obtained in [@marco3]. Also the recent paper [@guillotreb] contains a very general classification theorem for meromorphic vector fields admitting “maximal solutions” on complex surfaces and these include complete vector fields as in [@marco3]. All these questions are however wide open for $n \geq 3$ and our Theorem B appears as a contribution to them.
To have a better appreciation of the difficulties involved in these problems, let us consider the case of semi-complete vector fields (as considered for example in [@guillotreb]). Recall that a vector field is said to be semi-complete on a domain $U$ if its solution $\phi$ verifying $\phi (0) =p \in U$ is defined on a [*maximal domain*]{} of ${{\mathbb C}}$ for all $p \in U$. Here a domain $V \subseteq {{\mathbb C}}$ where the solution $\phi$ is defined is said to be [*maximal*]{} if for every point $\hat{T}$ in the boundary $\partial V$ of $V$ and every sequence $\{ T_i \} \subset V$ such that $T_i \rightarrow \hat{T}$, the sequence $\phi (T_i)$ leaves every compact set in $U$, cf. Section 2 for further details. Clearly a complete vector field is automatically semi-complete since we can take $V = {{\mathbb C}}$ so that $\partial V =\emptyset$. If we consider polynomial semi-complete vector fields on ${{\mathbb C}}^n$ then even the quadratic homogeneous case is already hard to understand once $n \geq 3$. In fact, A. Guillot has conducted detailed research about semi-complete quadratic homogeneous vector fields (with simple singularities at infinity) in [@guillotFourier], [@guillotIHES]. In [@guillotFourier] he introduced certain lattices (of coefficients) where all these vector fields are to be found whereas in [@guillotIHES] he studied the special case of Halphen’s vector fields and the related problem about actions of ${\rm PSL}\, (2, {{\mathbb C}})$ on compact $3$-manifolds. The beauty and depth of these results motivated us to try to apply our techniques to vector fields satisfying the conditions stated in [@guillotFourier]. To abridge notations, we shall refer to these vector fields by saying that they belong to Guillot lattice (the reader interested in the case $n = 3$ will find the document [@guillotThesis] particularly useful).
Note that a semi-complete vector field of ${{\mathbb C}}^3$ belonging to Guillot lattice may not be complete and its orbits (thought of as leaves of the associated foliation) may be hyperbolic Riemann surfaces. This is actually what occurs with Halphen vector fields except for a few special cases cf. [@guillotIHES], [@guillotThesis] or yet Section 7. In this Introduction by a Halphen vector field it is always meant a “hyperbolic” Halphen vector field. These vector fields are semi-complete and the maximal domains of definitions for their solutions may be either a bounded region of ${{\mathbb C}}$ or a (hyperbolic) unbounded region (for example the complement of a suitable disc).
More generally [@guillotFourier] contains a classification of semi-complete quadratic vector fields up to a finite number of vector fields. A full understanding of the dynamics of these possible “exceptional” cases seems to be a very hard problem. In any event, among vector fields belonging to Guillot lattice we shall also consider those satisfying the following condition: no singularity of the associated foliation lying in the hyperplane at infinity has all its eigenvalues lying in ${{\mathbb R}}_+$. They will be referred to as vector fields having no dicritical singularity at infinity. We can now state Theorem C.
[**Theorem C**]{}.
*Suppose that $X$ is a holomorphic vector field defined compact manifold $M$. Consider a singularity $p \in M$ of $X$ and denote by $X_k$ the first non-zero homogeneous component of the Taylor series of $X$ at $p$. Suppose that one of the following condition holds:*
$\bullet$ $X_k$ is a multiple of a vector field in Guillot lattice having no dicritical singularity at infinity.
$\bullet$ $X_k$ is a multiple of a hyperbolic Halphen vector field (in case $M$ has dimension $3$).
Then $M$ does not a carry a Kähler structure.
Note that Theorem C is somehow sharp in the sense that [@guillotIHES] contains examples of compact $3$-manifolds equipped with a global holomorphic vector field exhibiting the singularity of a hyperbolic Halphen vector field. Naturally the corresponding manifolds are not Kähler.
Theorem C will be proved in Section 7. The proof is indeed very short though based on the material developed in the preceding sections. The rest of the section will be taken up by a discussion of the main dynamical issues associated to Halphen vector fields. The results presented there are definitely not new as they can all be found in [@guillotIHES] together with a large amount of additional information. Yet the discussion conducted in Section 7 makes the article self-contained in the sense that proofs for the properties of these vector fields needed for Theorem C are provided. Additional motivation to conduct this discussion lies in the fact that the arguments provided are in line with the spirit of our method and differ from the original arguments of [@guillotIHES] and can be extended to encompass more general situations, see [@guillotOsaka]. Finally they lend themselves to a further new application: for $n=3$ they allow us to classify the first non-zero homogeneous component of a singularity of a complex vector field on a Kähler threefold, a result to be compared with Theorem C of [@ghreb]. Whereas this classification will not be carried out here, at the end of the paper we shall describe a step-by-step procedure to arrive to it.
[**Acknowledgements**]{}: The first author wants to express his gratitude to A. Guillot for many discussions concerning several objects present in this work and, in particular, for having explained to him several points of [@guillotIHES]. The second author is partially supported by Fundação para a Ciência e Tecnologia (FCT) through CMUP, through the Post-Doc grant SFRH/BPD/34596/2007 and through the project PTDC/MAT/103319/2008.
Overview of methods and results
===============================
This section contains a general description of the structure of the paper as well as some “qualitative” explanation of our techniques. “Quantitative” information required by the corresponding proofs will be supplied in the course of this paper. Some complements to the theorems stated in the Introduction will also be provided along with further comments on their contents.
First a point should be made about the vector fields and/or differential equations considered in this work. This is due to the fact that they are far from being “generic”. For example complete vector fields on ${{\mathbb C}}^n$ are very “non-generic” among polynomial/rational vector fields or yet among singular holomorphic foliations on projective spaces. Indeed the leaves of a foliation on ${{\mathbb C}}P(n)$ induced by a complete vector field are quotients of ${{\mathbb C}}$ as Riemann surfaces and this, by itself, is already very “non-generic”. Whereas they are “non-generic” their interest can hardly be questioned since, for example, they constitute a natural Lie algebra for the group of algebraic automorphisms of ${{\mathbb C}}^n$ and they remain an object intensively discussed as shown by the recent works of A. Bustinduy, L. Giraldo and others (cf. [@giraldo1], [@giraldo2] and their references). Actually in the study of differential equations we often encounter very special (i.e. “non-generic”) examples that turn out to play crucial roles in the theory. Apart from complete vector fields, our techniques also apply to semi-complete ones, i.e. to those vector fields admitting “maximal solutions” defined on subsets of ${{\mathbb C}}$ (cf. below). Halphen vector fields as studied in [@guillotIHES] satisfy this condition and they are discussed in Section 7. Their importance is clear since they appear in Mathematical Physics, in the theory of Ricci flow (namely the case of Ricci flow on homogeneous spaces) as well as in number theory through the celebrated functions $P, \, Q , \, R$ of Ramanujan. Yet another class of special equations/vector fields that fits in the pattern of our theory consists of those equations having meromorphic solutions defined on all of ${{\mathbb C}}$, in fact, they can be thought of as suitable semi-complete vector fields having meromorphic solutions defined on all of ${{\mathbb C}}$. These equations constitute a classical object of complex analysis and they include several well-known equations due to Painlevé and Chazy whose importance can hardly be overstated. This last direction of research however will not be exploited in this work though we hope it will be developed in the future. Another promising direction that is left for future investigation concerns the connections of our work with the point of view developed by X. Gomez-Mont and his collaborators, see [@xavier1], [@xavier2]. In fact, the study of some phenomena singled out in [@xavier2] by exploiting a simple “real” variant of our method seems to be very promising.
Let us now begin to outline the structure of this paper. Consider a polynomial vector field $X$ on ${{\mathbb C}}^n$ and denote by ${\mathcal{D}}$ the associated foliation induced on ${{\mathbb C}}P(n)$. Let $X_d$ stand for the top-degree homogeneous component of $X$ (having degree $d \geq 2$) and suppose that $X_d$ is not a multiple of the radial vector field. Under this assumption, the foliation ${\mathcal{D}}$ leaves the hyperplane at infinity $\Delta_{\infty}
= {{\mathbb C}}P(n) \setminus {{\mathbb C}}^n$ invariant. In addition, and modulo a minor remark discussed in Section 3, this foliation coincides with the foliation induced on $\Delta_{\infty}$ by $X_d$. The latter foliation can alternatively be seen as follows. The homogeneous character of $X_d$ implies that the direction of $X_d$ is invariant by homotheties of ${{\mathbb C}}^n$. Thus these directions have a well-defined projection on the quotient space of ${{\mathbb C}}^n$ by radial lines, i.e. on ${{\mathbb C}}P(n-1)$. Naturally we have $\Delta_{\infty} \simeq {{\mathbb C}}P(n-1)$ and the two foliations in question turn out to coincide under this identification. Furthermore they also coincide with the foliation induced on the exceptional divisor $\Delta_0 \simeq {{\mathbb C}}P(n-1)$ by the punctual blow-up of $X_d$ at the origin. The foliation associated to $X_d$ on ${{\mathbb C}}P (n)$ is going to be denoted by ${{\mathcal F}}$ and its restriction to $\Delta_{\infty}$ by ${{\mathcal F}}_{\infty}$. If $L_{\infty}$ is a leaf of ${{\mathcal F}}_{\infty}$ then the “cone over $L_{\infty}$” is invariant by ${{\mathcal F}}$.
Fundamentally our method consists of estimating the “speed” of the vector field $X$ near $\Delta_{\infty}$. This is done in two steps. First whereas $X$ has poles over $\Delta_{\infty}$, we can locally eliminate from $X$ the unbounded factor so as to obtain a “local regular vector field” about every regular point $p \in \Delta_{\infty}$ of ${{\mathcal F}}_{\infty}$. These “local vector fields” however do not patch together in a “foliated” global vector field since two representatives obtained through overlapping coordinates differ in general by a multiplicative constant. This means that they define, nonetheless, an affine structure (induced by $X_d$ or by $X$) on every leaf of ${{\mathcal F}}_{\infty}$. Versions of this affine structure already appeared in [@guillotFourier] and in some previous work of the first author under the name of “renormalized time-form”, it also plays an important role in [@guillotreb]. For the present work, the interest of this affine structure arises from the fact that it lends itself well to provide precise estimates of the flow of $X$ as long as precise estimates for the “distance” from the orbit in question to $\Delta_{\infty}$ are available.
Here it comes the second ingredient of our construction, namely a quantitative measure of “the rate of approximation” of a leaf of ${{\mathcal F}}$ to $\Delta_{\infty}$. Because $\Delta_{\infty} \subset {{\mathbb C}}P(n)$ and the Fubini-Study metric on ${{\mathbb C}}P(n)$ has positive curvature, it is well-known that complex submanifolds always bend themselves towards $\Delta_{\infty}$. In our case, this implies that the distance (relative to the Fubini-Study metric) of a leaf $L$ of ${{\mathcal F}}$ to $\Delta_{\infty}$ can never reach a local minimum unless this minimum is [*zero*]{}. The second ingredient of our method is reminiscent from this remark. Indeed we shall use the euclidean metric on suitably chosen affine coordinates, which is especially well-adapted to work with the above mentioned affine structure. Besides by exploiting the fact that we have leaves of an actual foliation (rather than general complex submanifolds) a quantitative version of the rate of approximation of a leaf to $\Delta_{\infty}$ is obtained. The phenomenon goes essentially as follows. At each regular point $p$ of a leaf $L$ of ${{\mathcal F}}$ we can single out the steepest descent direction of $L$ towards $\Delta_{\infty}$, namely the negative of the gradient of the distance function restricted to $L$. This yields a singular real one-dimensional oriented foliation ${\mathcal{H}}$ on $L$. Furthermore the conformal structure of $L$ is such that the foliation ${\mathcal{H}}^{\perp}$ orthogonal to ${\mathcal{H}}$ is constituted by level curves for the mentioned distance function. Roughly speaking an exponential rate of approximation for $L$ to $\Delta_{\infty}$ over the trajectories of ${\mathcal{H}}$ can then be derived. This estimate combines to the “uniform” estimates involved in the mentioned foliated affine structure to produce accurate estimates for the time taken by the flow of $X$ over trajectories of ${\mathcal{H}}$. In particular it is shown that the time taken by $X$ to cover an entire (infinite) trajectory is [*finite*]{} provided that the trajectory remains away from the singularities of ${{\mathcal F}}$ lying in $\Delta_{\infty}$. This results is then sharpened in Section 5 by allowing the trajectory to accumulate on (simple) singular points and still obtaining an analogous estimate. In particular there is only one special type of “simple” singularity that may yield an “endpoint” for the trajectories of ${\mathcal{H}}$ and, in this case, this will be an intersection point between the leaf $L$ and the hyperplane $\Delta_{\infty}$: the corresponding trajectory of ${\mathcal{H}}$ should then be thought of as being “finite”. Switching back and forward between estimates involving $X_d$ and estimates involving $X$ is not very complicated since $X$ is close to $X_d$ near $\Delta_{\infty}$.
The material mentioned above is covered in Sections 3, 4 and 5. Armed with these results we begin in Section 6 to prove the theorems stated in the Introduction. Theorems A and A’ are very natural. Since $X$ is complete its integral over a trajectory of ${\mathcal{H}}$ cannot converge. Besides this trajectory can never “reach” $\Delta_{\infty}$ since $X$ is complete on ${{\mathbb C}}^n$. This contrasts with our previous estimate asserting convergence of the integral in question provided that the corresponding trajectory of ${\mathcal{H}}$ remains away from the singularities of ${{\mathcal F}}$ (or ${\mathcal{D}}$) lying in $\Delta_{\infty}$. The only possible explanation for this apparent contradiction is that the flow of $X$ spend all but a finite amount of its existence on arbitrarily small neighborhoods of these singularities. The proof of Theorem A’ goes along similar ideas. In fact, the estimates carried over trajectories of ${\mathcal{H}}$ remain valid for every oriented foliation ${\mathcal{H}}^{\theta}$ forming an angle $\theta \in
(-\pi/2 , \pi/2)$ with ${\mathcal{H}}$. Once again the foliations ${\mathcal{H}}^{\theta}$ are well-defined thanks to the existence of a preferred conformal structure on the leaves of ${{\mathcal F}}, \, {\mathcal{D}}$. Modulo fixing a base point and using the obvious identifications, the union of all these trajectories span an unbounded region of ${{\mathbb C}}$ viewed as the domain of definition of the solution in question. The area of this region is “large” and, heuristically, can be imagined as “half of the area of large discs”.
These two theorems are clearly sharp in the sense that the leaves of ${\mathcal{D}}$, in general, can only confine themselves at singularities over certain special directions. In other words, we cannot have the whole complement of a compact part of a leaf being confined at a singularity. For example a simply connected leaf having a “punctual” end becomes a rational curve modulo adding the end to the leaf as it promptly follows from the classical Remmert-Stein theorem. Also simple complete polynomial vector fields such as $$y \frac{\partial}{\partial y} + xy \left[ x \frac{\partial}{\partial x} - y \frac{\partial}{\partial y} \right]$$ on ${{\mathbb C}}^2$ already exhibit non-properly embedded (simply connected) orbits accumulating on all of the “line at infinity” in the projective plane. Leaves that are “cylinders” are easier to handle as it is known since [@suzuki]. Yet they may have transcendent ends.
In view of Theorems A and A’, it is natural to imagine that the singular set of ${\mathcal{D}}$ contains significative information about the global geometry of complete polynomial vector fields. Theorem B is a contribution to the study of these vector fields as well as a test for the extent to which their global dynamics can be determined by the structure of their singularities. As mentioned the idea is to consider only “simple singularities” and to check what can be said about the vector field. From this point of view Theorem B is totally satisfactory since its statement is clearly sharp.
The proof of Theorem B is arguably the best application of our techniques and its main ingredients are going to be described in the sequel. The central difficulty is to guarantee the existence of a “dicritical singularity” for ${\mathcal{D}}$ in $\Delta_{\infty}$, i.e. a linearizable singularity all of whose eigenvalues belong to ${{\mathbb R}}_+$. The existence of this type of singularity implies in particular that the generic orbit of the vector field $X$ is of type ${{\mathbb C}}^{\ast}$ in the sense of [@suzuki] and several additional properties follow at once. To ensure the existence of this singularity is however a subtle question that can be approached as follows. First we replace $X$ by its top-degree homogeneous component $X_d$ along with its associated foliation denoted by ${{\mathcal F}}$. The property of having a dicritical singularity at $\Delta_{\infty}$ is common to ${\mathcal{D}}$ and ${{\mathcal F}}$ so that it is more convenient to work with a homogeneous vector field. Nonetheless we should take into account that $X_d$ is no longer complete but only [*semi-complete*]{}. Recall that a vector field $Z$ defined on an open set $U$ is said to be semi-complete (in $U$) if for every $p \in U$ there exists a solution $\phi : V \subset {{\mathbb C}}\rightarrow U$ of $X$, with $\phi (0) =p$, such that whenever a sequence $\{ T_i \}
\subset V$ converges to a point $\hat{T}$ in the boundary of $V$ the corresponding sequence $\phi (T_i)$ leaves every compact set in $U$. In this sense $V$ is a maximal domain of definition for the solution $\phi$ of $Z$. Being only semi-complete, it may happen that $X_d$ “reaches the infinity in finite time” and this gives rise to some additional difficulties.
An important difficulty arising from the difference between complete and semi-complete vector fields lies in the fact that the leaves of the foliation associated to a semi-complete vector field may be hyperbolic Riemann surfaces. An example of this phenomenon is, indeed, provided by Halphen vector fields, cf. Section 7. However, in the case of the foliation ${{\mathcal F}}$ associated to $X_d$ it can be proved that the corresponding leaves are still quotients of ${{\mathbb C}}$. This is done by resorting to a result due to Brunella concerning the pluri-subharmonic variation of the foliated Poincaré metric, cf. [@marco2]. The solutions of $X_d$ will therefore be meromorphic functions defined on ${{\mathbb C}}$ or in ${{\mathbb C}}$ minus a point. Next we bring in our results involving the time taken by $X_d$ to cover trajectories of ${\mathcal{H}}$ in the singular context (here it is used the main result of Section 5, namely Theorem \[maintheo\]). Theorem \[maintheo\] immediately implies that the solutions cannot be meromorphic on all of ${{\mathbb C}}$ and, by exploiting some additional properties of the foliations ${\mathcal{H}}, \, {\mathcal{H}}^{\perp}$, a contradiction ensuring the existence of the desired dicritical singularity is finally obtained.
Let us now make some comments about the assumption that the singularities of ${\mathcal{D}}$ lying in $\Delta_{\infty}$ are simple. First a reader more familiar with complete polynomial vector fields might be tempted to think that the condition imposed on the singularities of ${\mathcal{D}}$ immediately implies that the vector field $X$ must have degree $2$. This is however not the case. For example some of these singularities may be a codimension 1 saddle-node singularity (as in item 2 of the definition of simple singularities given in the Introduction). Also the statement of Theorem B may be extended to encompass fair more general singularities among the very general class of “absolutely isolated singularities”, cf. [@canoetc]. While we shall not seek to accurately establish any of these extensions, at the very end of Section 5 the reader will find some information on the structure of more degenerate singularities for which our methods will still work. It is also interesting to observe that our techniques apply equally well to rational vector fields and not only to polynomial ones. In practice this change the “expected” order of $\Delta_{\infty}$ as component of the “pole divisor” of the vector field in question and leads to numerous additional possibilities whose analysis may partially be facilitated by our ideas. In particular several Painlevé equations fall in this class of problems.
As mentioned A. Bustinduy, L. Giraldo and their collaborators have been investigating properties of the solutions of complete vector fields through various methods such as the theory of Nevanlinna and Andersen-Lempert theories, see [@giraldo1], [@giraldo1] and their references. Similarly, if take into account a classical result due to Forstneric [@forst], our method is likely to find some applications in the theory of holomorphic differential equations blowing-up in finite real time. This should lead to some progress in questions similar to those treated by Fornaess and Grellier in [@fornaess] which itself connects with previous works by a number of authors including possible applications in the spirit of [@fornaessbuzzard].
Finally in view of the beautiful results obtained by A. Guillot in [@guillotFourier], [@guillotIHES], it is natural to try to apply our methods to quadratic semi-complete vector fields as those considered in the mentioned works. The material prepared for the proofs of Theorems A, A’ and B allows us to show that one such vector field having no dicritical singularity at infinity has leaves that are hyperbolic Riemann surfaces. This quickly leads to Theorem C by resorting again to Brunella’s result on the variation of the Poincaré metric, see [@marco2]. Halphen vector fields are however very special in the sense that they [*do have*]{} dicritical singularities at infinity and still the leaves of their associated foliation may be hyperbolic Riemann surfaces. Whereas these results, and many others, are due to A. Guillot and appear in [@guillotIHES], we found it was worth re-obtaining them by following our general point of view. This discussion takes up most of the last section of this paper. In particular it involves some considerations about convergence of Poincaré series that differ from their classical theory.
Another motivation for us to revisit Guillot’s work on Halphen vector fields is to pave the way for one further application of our method. As briefly stated in the Introduction, the arguments developed in our discussion about Halphen vector fields can be adapted to provide a classification of the first homogeneous components at a singular point of a globally defined holomorphic vector field on a compact Kähler threefold. In addition the same classification applies equally well to the top degree homogeneous component of a complete polynomial vector field on ${{\mathbb C}}^3$. However, unlike the other results of this paper, our techniques does not allow to derive a similar classification for dimensions higher than $3$. In fact, dimension $3$ turns out to be special for two reasons: on one hand the groups of automorphisms of compact Riemann surfaces are easy to describe and, on the other hand, the existence of invariant surfaces for the corresponding foliations enables us to apply a powerful theorem due to McQuillan which is available only in dimension $2$. Details on this procedure are to be found in Section 7.3.
To close this discussion we would like to point out a curious remark involving Theorem C and, more generally, semi-complete homogeneous vector fields. In fact singularities of homogeneous vector field on ${{\mathbb C}}^3$ (or ${{\mathbb C}}^n$) possess a natural meromorphic “dual” represented by a neighborhood of the (hyper-) plane at infinity (even though this neighborhood cannot be collapsed to a singular point). More precisely, as detailed in Section 3, the blow-up at the origin of a homogeneous vector field leads to an exceptional divisor sharing a natural “duality” with the divisor obtained at infinity of the corresponding projective space. We shall refer to a neighborhood of the hyperplane at infinity as the [*dual singularity*]{} (assuming that the singularity of a homogeneous polynomial vector field is implicitly fixed). By virtue of Theorem C and of the global realization of Halphen vector fields constructed in [@guillotIHES], it is natural to ask whether the dual singularity of a hyperbolic Halphen vector field can be realized by a complete meromorphic vector field on a complex $3$-manifold not necessarily compact (where by complete meromorphic vector field it is meant a meromorphic vector field that is complete in the complement of its pole locus). The answer to this question turns out to be negative as it follows from the discussion in Section 7.
Homogeneous vector fields and their foliations {#sechom}
==============================================
Unless otherwise stated, throughout this paper all homogeneous vector fields have degree $d \geq 2$ and are supposed not to be a multiple of the radial vector field. In this section we shall work in dimension $3$ rather than in ${{\mathbb C}}^n$ just to abridge notations since all arguments presented in the sequel can be carried over word-by-word to higher dimensions.
Consider a homogeneous polynomial vector field $X$ of degree $d \geq 2$ defined on ${{\mathbb C}}^3$. Since $X$ is homogeneous, its associated foliation ${{\mathcal F}}$ is invariant by homotheties of the form $(x, y, z) \mapsto (\lambda x,\lambda y ,\lambda z)$, $\lambda \in {{\mathbb C}}^{\ast}$ and, therefore, also induces a foliation on ${{\mathbb C \mathbb P}}(2)$. An alternative way to look at this situation consists of punctually blowing-up $X$ at the origin of ${{\mathbb C}}^3$. We denote by ${{\widetilde{\mathbb C}}}^3$ the corresponding blow-up of ${{\mathbb C}}^3$ and by $\Delta_0 =\pi^{-1} (0)$ the resulting exceptional divisor, where $\pi: {{\widetilde{\mathbb C}}}^3 \mapsto {{\mathbb C}}^3$ represents the punctual blow-up of ${{\mathbb C}}^3$ at the origin. The transform (blow-up) ${{\widetilde{X}}}$ (resp. ${{\widetilde{\mathcal F}}}$) of $X$ (resp. ${{\mathcal F}}$) vanishes identically over $\Delta_0$ (resp. leaves $\Delta_0$ invariant), as it follows from the fact that the degree of $X$ is strictly greater than $1$ (resp. that $X$ is not a multiple of the radial vector field).
Recalling also that ${{\widetilde{\mathbb C}}}^3$ can be viewed as a line bundle over $\Delta_0 =\pi^{-1}
(0)$, let ${{\mathcal P}}_0$ denote the bundle projection ${{\mathcal P}}_0: {{\widetilde{\mathbb C}}}^3
\rightarrow \Delta_0$. This line bundle can be compactified into a projective line bundle by adding the “section at infinity” $\Delta_{\infty}$. Denoting by $M$ the total space of the resulting projective line bundle, it follows that $M$ comes with two bundle projections ${{\mathcal P}}_0, \, {{\mathcal P}}_{\infty}$ realizing it as a projective bundle respectively over $\Delta_0, \, \Delta_{\infty}$. The manifold $M$ is also isomorphic to the blow-up of ${{\mathbb C \mathbb P}}(3)$ at the origin. The vector field ${{\widetilde{X}}}$ can meromorphically be extended to $M$ so that it induces a holomorphic foliation, still denoted by ${{\widetilde{\mathcal F}}}$, on all of $M$. Besides ${{\widetilde{\mathcal F}}}$ leaves both $\Delta_0, \, \Delta_{\infty}$ invariant since $X$ is homogeneous and it is not a multiple of the radial vector field. The foliation induced on $\Delta_0$ (resp. $\Delta_{\infty}$) by restriction of ${{\widetilde{\mathcal F}}}$ is going to be denoted by ${{\widetilde{\mathcal F}}}_0$ (resp. ${{\widetilde{\mathcal F}}}_{\infty}$). Because ${{\widetilde{\mathcal F}}}$ comes from a homogeneous vector field, these foliations coincide with the restrictions of ${{\widetilde{\mathcal F}}}$ to $\Delta_0, \, \Delta_{\infty}$. As to the vector field ${{\widetilde{X}}}$, its pole divisor coincides with $\Delta_{\infty}$ and it has order $d-1>0$. The zero divisor of ${{\widetilde{X}}}$ is the union of $\Delta_0$ (a component of order $d-1 >0$) with the transform of the zero divisor of $X$.
Because our foliations are singular, we shall consider a definition of “regular leaf” allowing, in some cases, a leaf to “go through” a singularity. For some applications, it may be of interest to use the definition given in [@marco2] which we briefly recall for the convenience of the reader. Consider the $n$-dimensional polydisc ${\bf D}^n$ and the trivial fibration ${\bf D}^n = {\bf D}^{n-1} \times {\bf D} \rightarrow {\bf
D}^{n-1}$. A meromorphic map $f : {\bf D}^n \rightarrow M$ is said to be a [*foliated meromorphic immersion*]{} if the following conditions are verified.
1\. The indeterminacy set $I (f)$ of $f$ intersects each vertical fiber of ${\bf D}^n$ over a discrete set.
2\. $f$ is an immersion on the complement of $I (f)$ and, in this complement, takes vertical fibers to leaves of ${{\widetilde{\mathcal F}}}$ (more generally of the foliation under consideration).
Consider a regular point $p$ in $M \setminus {\rm Sing}\, ({{\widetilde{\mathcal F}}})$. Restricted to $M \setminus {\rm Sing}\, ({{\widetilde{\mathcal F}}})$ the foliation ${{\widetilde{\mathcal F}}}$ is regular so that the leaf $L_p$ through $p$ has an obvious sense. A closed subset $K \subset L_p$ is called a [*vanishing end*]{} of $L_p$ if all the conditions below are satisfied:
1\. $K$ is isomorphic to the punctured disc and the holonomy of the restriction of ${{\widetilde{\mathcal F}}}$ to $M \setminus {\rm Sing}\,
({{\widetilde{\mathcal F}}})$ corresponding to the cycle $\partial K$ has finite order $k$.
2\. There is a foliated meromorphic immersion $f : {\bf D}^n \rightarrow M$ such that
2.a
: $I (f) \cap (\{ 0\} \times {\bf D}) = 0 \in {\bf D} \subset {{\mathbb C}}$, where $\{ 0 \}$ stands for the origin of ${\bf D}^{n-1} \subset {{\mathbb C}}^{n-1}$.
2.b
: The image of $f$ restricted to $(\{ 0\} \times {\bf D})$ is the interior of $K$. Furthermore $f: (\{ 0\} \times {\bf D})
\rightarrow {\rm Int} \, (K)$ is a regular covering of degree $k$, where ${\rm Int} \, (K)$ stands for the interior of $K$.
Given a regular point $p \in M \setminus {\rm Sing}\, ({{\widetilde{\mathcal F}}})$, consider the regular leaf $L_p$ through $p$ relative to the restriction of ${{\widetilde{\mathcal F}}}$ to $M \setminus {\rm Sing}\, ({{\widetilde{\mathcal F}}})$. If $L_p$ possesses no vanishing ends, then the regular leaf of ${{\widetilde{\mathcal F}}}$ containing $p$ is exactly $L_p$. Otherwise this leaf will consist of $L_p$ with the ends of the vanishing ends added to it. Naturally when no misunderstanding is possible, this leaf will still be denoted by $L_p$. Here we observe that the operation of adding an end to $L_p$ should be understood in the sense of orbifolds: the multiplicity of the added point will precisely be the order $k$ of the holonomy relative to $\partial K$. Naturally such orbifolds can [*a posteriori*]{} be made into Riemann surfaces by standard normalization.
Let us now continue with the discussion of the structure of the foliation ${{\widetilde{\mathcal F}}}$ over $M$. Naturally the singular set of ${{\widetilde{\mathcal F}}}$ has codimension at least $2$. Besides this singular set is saturated (i.e. invariant) by the fibers of ${{\mathcal P}}_0$ (resp. ${{\mathcal P}}_{\infty}$) due to the invariance of ${{\mathcal F}}$ by homotheties of the form $(x, y, z) \mapsto (\lambda x,
\lambda y, \lambda z)$, $\lambda \in {{\mathbb C}}^{\ast}$. In particular, the foliations ${{\widetilde{\mathcal F}}}_0, \, {{\widetilde{\mathcal F}}}_{\infty}$ automatically have singular sets of codimension at least $2$ [*inside $\Delta_0, \, \Delta_{\infty}$*]{} (in other words the intersection of the singular set of ${{\widetilde{\mathcal F}}}$ with ${{\widetilde{\mathcal F}}}_0, \, {{\widetilde{\mathcal F}}}_{\infty}$ yields a set of codimension at least $2$ inside $\Delta_0, \, \Delta_{\infty}$).
Consider a non-algebraic leaf $L$ of ${{\widetilde{\mathcal F}}}$ not contained in $\Delta_0 \cup \Delta_{\infty}$. The projection of $L$ onto $\Delta_0$ (resp. $\Delta_{\infty}$), ${{\mathcal P}}_0 (L) =L_0$ (resp. ${{\mathcal P}}_{\infty} (L) =L_{\infty}$), is clearly a leaf of ${{\widetilde{\mathcal F}}}_0$ (resp. ${{\widetilde{\mathcal F}}}_{\infty}$) since the initial vector field $X$ is homogeneous. Furthermore one immediately checks that the restriction of ${{\mathcal P}}_0$ (resp. ${{\mathcal P}}_{\infty}$) to $L$ realizes $L$ as an Abelian covering of $L_0$ (resp. $L_{\infty}$). Therefore we conclude:
1. The non-compact leaves $L, \, L_0, \, L_{\infty}$ have all the same nature: either they are covered by ${{\mathbb C}}$ or they are covered by the unit disc $D$.
2. $L_0, \, L_{\infty}$ are isomorphic as Riemann surfaces. $L$ is an Abelian covering of $L_0, \, L_{\infty}$.
In this way, we may focus on the behavior of ${{\widetilde{X}}}$ near its pole divisor $\Delta_{\infty}$ or near $\Delta_0$ according to our convenience. Next consider a leaf $L_{\infty}$ of ${{\widetilde{\mathcal F}}}_{\infty}$. By [*the cone over $L_{\infty}$*]{} it is meant the $2$-dimensional immersed singular surface ${{\mathcal P}}_{\infty}^{-1} (L_{\infty})$ which is invariant by ${{\widetilde{\mathcal F}}}$. In other words, if $\psi(T) = (x(T), y(T),0)$, $T \in \Omega \subseteq {{\mathbb C}}$, is a local parametrization of $L_{\infty}$, then the cone is parameterized by $\Phi(T,z) = (x(T),
y(T),z)$, $z \in {{\mathbb C}}$. The singular points of ${{\mathcal P}}_{\infty}^{-1}
(L_{\infty})$ belong to fibers sitting over the singular set of ${{\widetilde{\mathcal F}}}_{\infty}$ which, we recall, may intersect $L_{\infty}$ non-trivially due to the above given definition of “regular leaf”. Away from its singularities, ${{\mathcal P}}_{\infty}^{-1} (L_{\infty})$ can be viewed as a complex surface equipped with a singular holomorphic foliation. Let us then denote by $S$ this surface and by ${{\widetilde{\mathcal F}}}_S$ the foliation on $S$ obtained by restriction of ${{\widetilde{\mathcal F}}}$ to $S$. Note that $S$ is invariant under the automorphism $(x,y,z)\mapsto
(x,y,{\lambda}z)$, ${\lambda}\in {{\mathbb C}}^*$ and so is the foliation ${{\widetilde{\mathcal F}}}_S$.
Since $S$ is a 2-dimensional variety, ${{\widetilde{\mathcal F}}}_S$ is a codimension $1$ singular foliation on $S$ and hence it is transversely conformal. This allows us to keep good control of the directions over which the leaves of ${{\widetilde{\mathcal F}}}_S$ “become closer one to the others” modulo choosing an auxiliary Hermitian metric. The idea is well-known and can be found, for instance, in [@Ghys-bourb]. In our case however, we shall use an explicit parametrization. For this, let $M$ be equipped with affine coordinates $(x , y, z)$ on $M$ such that
1. $\{ z=0\} \subset \Delta_{\infty}$, $(x , y) \in
{{\mathbb C}}^2$, $z \in {{\mathbb C}}$.
2. the vector field ${{\widetilde{X}}}$ is given by $${{\widetilde{X}}}= \frac{1}{z^{d-1}} \left[ F (x , y) \frac{\partial}{\partial
x} + G (x , y) \frac{\partial}{\partial y} + zH (x , y)
\frac{\partial}{\partial z} \right] \label{tXX}$$ where $F ,G$ are polynomials of degree either $d$ or $d+1$ and $H$ is a polynomial of degree $d$ (the independence of $F ,G$ and $H$ on $z$ is a consequence of the homogeneity of $X$).
3. The projection ${{\mathcal P}}_{\infty} : M \rightarrow \Delta_{\infty}$ in the above coordinates becomes $(x , y, z) \mapsto (x , y)$.
Note that $\Delta_{\infty}$ is itself isomorphic to ${{\mathbb C \mathbb P}}(2)$. In particular the affine coordinates $(x ,y)$ for $\Delta_{\infty}$ defines a notion of “line at infinity” for $\Delta_{\infty}$ itself. We shall denote this “line” by $\Delta_{\infty}^{(x,y)}$. In particular we note that the domain of definition of the coordinates $(x , y, z)$ coincides with $M \setminus (\Delta_0 \cup {{\mathcal P}}_{\infty}^{-1} (\Delta_{\infty}^{(x,y)}))$. Naturally the choice of the affine coordinates $(x,y)$ and of the line $\Delta_{\infty}^{(x,y)}$ are not canonical. For a generic choice of these coordinates, $\Delta_{\infty}^{(x,y)}$ does not contain singular points of the corresponding foliation on $\Delta_{\infty}$ and, besides, $\Delta_{\infty}^{(x,y)}$ is not invariant by this foliation. Now we have:
\[lineatinfinitygeneric\] Suppose that the affine coordinates $(x,y)$ are chosen so that the resulting “line at infinity” $\Delta_{\infty}^{(x,y)}$ is not invariant by the corresponding foliation on $\Delta_{\infty}$. Then the top-degree component of the vector field ${{\widetilde{X}}}$ has the form $$\label{topdegree}
f(x , y) [x \partial /\partial x + y \partial /\partial y + z {\partial /\partial z}]$$ for a degree $d$ homogeneous polynomial $f$.
. Suppose that the initial homogeneous vector field $X$ is given by $X = A {\partial /\partial x}+ B {\partial /\partial y}+ C{\partial /\partial z}$. Then, with the above notations, ${{\widetilde{X}}}= z^{1-d} [F \partial /\partial x
+ G \partial /\partial y + zH \partial /\partial z]$ where $F(x,y) = A(x,y,1) - x C(x,y,1)$, $G(x,y) = B(x,y,1) - y C(x,y,1)$ and $H(x,y) = - C(x,y,1)$. Thus the assumption on the degree of $F$ (resp. $G, \, H$) follows from the fact that we can assume $A$ (resp. $B, \, C$) not divisible by $z$. This also implies that the top degree of $F$ (resp. $G, \, H$) is given by $x$ (resp. $y, \, z$) times the top degree of $C$, i.e. the top degree component of ${{\widetilde{X}}}$ has the form (\[topdegree\]).
A further comment concerning the difference between the foliation ${{\widetilde{\mathcal F}}}_{\infty}$ induced by ${{\widetilde{X}}}$ on $\Delta_{\infty}$ and the corresponding foliation ${{\widetilde{\mathcal F}}}$ in the $3$-dimensional space is also needed. To be more precise, consider the vector field ${{\widetilde{X}}}$ given by Formula (\[tXX\]) in the coordinates $(x,y,z)$. If $F,G$ have only trivial common factors, then the foliation induced by ${{\widetilde{X}}}$ on $\Delta_{\infty}$ is given in $(x,y, \{z=0 \})$ coordinates by $F \partial /\partial x + G \partial /\partial y$. Suppose now that $F$ and $G$ possess nontrivial common factors. Set $\textsc{P} =
{\rm g.c.d.}\, (F,G)$ so that $F = \textsc{P}. a(x,y)$ and $G =
\textsc{P} . b(x,y)$ with $a,b$ having only trivial common factors. In this case the foliation ${{\widetilde{\mathcal F}}}_{\infty}$ is actually represented by the vector field $a(x,y) \partial /\partial x + b(x,y) \partial
/\partial y$. If $\textsc{P}$ also divides $H$, then this foliation still coincides with the restriction of ${{\widetilde{\mathcal F}}}$ to $\Delta_{\infty}$. Thus the only relevant case occurs when $\textsc{P}$ does not divide $H$. After eliminating all common factors, we can suppose that ${\rm g.c.d.}\, (\textsc{P}, H)$ is invertible. Summarizing this discussion, we see that the foliation associated to ${{\widetilde{X}}}$ is given by a polynomial vector field of the form $$Y = \textsc{P} \left[ a (x,y) \frac{\partial}{\partial
x} + b (x , y) \frac{\partial}{\partial y} \right] + z H (x , y)
\frac{\partial}{\partial z} \, ,$$ where ${\rm g.c.d.}\, (\textsc{P}, H)$ is constant. It follows in particular that the projective curve $\{ \textsc{P}
=0 \} \subset \Delta_{\infty}$ (if not empty) is constituted by singularities of ${{\widetilde{\mathcal F}}}$ whereas its “generic” point is regular for ${{\widetilde{\mathcal F}}}_{\infty}$. Besides there are two different possibilities that need to be considered:
(a)
: $\{ \textsc{P} = 0 \} \subset \Delta_{\infty}$ is invariant by ${{\widetilde{\mathcal F}}}_{\infty}$.
(b)
: $\{ \textsc{P} =0 \} \subset \Delta_{\infty}$ is not invariant by ${{\widetilde{\mathcal F}}}_{\infty}$.
\[willitbeuseful?\] [In much of what follows the possibility of having $\{ \textsc{P} = 0 \}$ invariant by ${{\widetilde{\mathcal F}}}_{\infty}$ will be excluded from the discussion since it does not take place in the case of semi-complete (and hence of complete) vector fields, cf. Proposition \[aproposition\].]{}
Our purpose is now to equip the leaves of ${{\widetilde{\mathcal F}}}$ in $\Delta_{\infty}$ with an abelian form $\omega_1$ naturally related to the holonomy of the leaf in question. This will be done in the affine ${{\mathbb C}}^3$ where the coordinates $(x,y,z)$ are defined. With the preceding notations, let us fix a regular leaf $L_{\infty} \subset \Delta_{\infty}$ and a point $p \in L_{\infty}$. Assume that $p$ is a regular point for ${{\widetilde{\mathcal F}}}$ or that $p$ is a singular point for the same foliation being $L_{\infty}$ a connected separatrix for ${{\widetilde{\mathcal F}}}$ at $p$. Under this assumption, the leaf $L_{\infty}$ can locally be parametrized in the form $(x,y(x))$, or $(x(y),y)$, and $z=0$. It suffices to consider a local parametrization of the form $(x,y(x))$ since the other possibility is analogous. The vector field ${{\widetilde{X}}}$ then yields $$\frac{dz}{dx} = \frac{H(x,y(x))}{F(x,y(x))} z \; .$$ Therefore $$z = z_0 \exp \left[ \int_{x_0}^x \frac{H(x,y(x))}{F(x,y(x))} dx \right] \; . \label{omega1}$$ Thus we define an abelian form $\omega_1$ on $L_{\infty}$ by declaring that the coefficient of $\omega_1$ at $(x,y(x))$ is nothing but $-H(x,y(x)) / F(x,y(x))$ (the minus sign is only a matter of convention). In particular we note that possible non-trivial common factors between $F,H$ are automatically canceled out in the definition of $\omega_1$. If the leaf were parameterized in the form $(x(y),y)$, the analogous result would yield for coefficient $-H(x,y(x)) / G(x,y(x))$. The form $\omega_1$ is the “logarithmic derivative of the holonomy” for the foliation ${{\widetilde{\mathcal F}}}_S$ induced on the cone $S$ over $L_{\infty}$. This means the following: let $L$ be a leaf of ${{\widetilde{\mathcal F}}}_S$ and consider a path $c:[0,1] \mapsto
L$, on $L$. Denoting by ${\rm Hol}(c)$ the holonomy associated to $c$, we have $$({\rm Hol} (c))^{\prime}(c(0)) = e^{-\int_c \omega_1} \, ,$$ where ${\rm Hol} (c)$ is identified with a map between open sets of ${{\mathbb C}}$ equipped with the coordinate $z$.
Fixed a regular leaf $L_{\infty} \subseteq \Delta_{\infty}$ of ${{\widetilde{\mathcal F}}}$ there are real trajectories, or paths, contained in $L_{\infty}$ and possessing a contractive holonomy. To construct these trajectories we proceed as follows. The Abelian form $\omega_1$ induces on $L_{\infty}$ a pair of real 1-dimensional oriented singular foliations: the foliations given by $\{ {\rm
Im}(\omega_1) = 0\}$ and by $\{ {\rm Re}(\omega_1) = 0\}$. Denote by $\mathcal{H}$ the oriented foliation defined by $\{ {\rm
Im}(\omega_1) = 0\}$, being the orientation determined by the positivity of ${\rm Re}(\omega_1)$, i.e. if $\phi(t)$ is a parametrization of a leaf of $\mathcal{H}$ then ${\rm Re}(\omega_1.\phi^{\prime}(t))
= \omega_1.\phi^{\prime}(t) > 0$. Each oriented trajectory of the foliation $\mathcal{H}$ will be called a real trajectory.
It is clearly necessary to have information about the singular set of $\mathcal{H}$. Since $\mathcal{H}$ depends only on the foliation associated to ${{\widetilde{X}}}$ (rather than of ${{\widetilde{X}}}$ itself), we identify four “critical regions” (possibly) giving rise to singularities for $\mathcal{H}$, namely:
1. Singular points of ${{\widetilde{\mathcal F}}}_{\infty}$.
2. Points in the curve $\{ H =0 \}$ (assuming, as already done, that ${\rm g.c.d.}\, (\textsc{P}, H)$ is a constant).
3. Points in the curve $\{ \textsc{P} =0 \}$.
4. The line at infinity $\Delta_{\infty}^{(x,y)} \subset \Delta_{\infty}$ (defined by means of the affine coordinates $(x,y)$).
In the sequel we shall determine the structure of the foliation ${\mathcal{H}}$ in cases 2, 3 and 4 above. The discussion of singular points of ${{\widetilde{\mathcal F}}}_{\infty}$ will mostly be carried out in Section 4. Note that points belonging simultaneously to $\{ H =0 \}$ and to $\{ \textsc{P} =0 \}$ are somehow “more degenerate” and they might require a more detailed analysis. Yet these points are in finite number so that for global dynamics considerations as those carried out in the subsequent sections they can be ignored. Actually in most of our applications we shall deal with the case where $\textsc{P}$ is constant. As to points belonging to the line at infinity, they will all be treated with the same argument whether or not they belong to the (closures of) $\{ H =0 \}$ or to $\{ \textsc{P} =0 \}$. Finally, concerning the case (4) above, it naturally depends on the choice of the affine coordinates $(x,y)$. A generic choice of these leads to a line at infinity $\Delta_{\infty}^{(x,y)}$ that neither is invariant by ${{\widetilde{\mathcal F}}}_{\infty}$ nor it contains singularities of it.
Let us begin with the curve $\{ H =0 \}$ corresponding to zeros of $\omega_1$.
\[tiposing\] Let $p \in \Delta_{\infty}$ be a regular point of ${{\widetilde{\mathcal F}}}$. Assume that $p$ lies in the curve $\{ H =0 \} \cap \Delta_{\infty}$ (but not in $\{ \textsc{P} =0 \}$). Then $p$ is a singular point for $\mathcal{H}$. Besides the local structure of $\mathcal{H}$ restricted to the leaf of ${{\widetilde{\mathcal F}}}$ through $p$ is a saddle with $2m$ (real) separatrices (for $m \geq 2$).
Consider local coordinates $(u,v,w)$ about $p$ where the leaf $L_p$ of ${{\widetilde{\mathcal F}}}$ through $p$ locally coincides with $\{ v=w=0\}$. Also we can suppose that $F (p) \neq 0$ modulo substituting $F$ by $G$. In fact, otherwise either $p$ would be a singular point of ${{\widetilde{\mathcal F}}}$ or a regular point of ${{\widetilde{\mathcal F}}}$ lying in the curve $\{ \textsc{P} =0 \}$. As already mentioned, these cases will be treated later.
We then conclude that the restriction of $\omega_1$ to $L_p$ is a holomorphic $1$-form about $p$ with a zero at $p$. The structure of the real foliation induced near a zero of a holomorphic $1$-form on a Riemann surface is always a saddle as in the statement. Here $m$ is precisely the order of $p$ as zero of $\omega_1$.
Let us now work out the behavior of ${\mathcal{H}}$ at points of $\{ \textsc{P} =0 \}$. Clearly it is sufficient to consider the domain of definition of the coordinates $(x,y,z)$. Let $\textsc{P} = \textsc{P}_1^{k_1} \cdots \textsc{P}_l^{k_l}$ be the decomposition of $\textsc{P}$ into irreducible components along with their corresponding multiplicity. We just need to consider the curve $\{ \textsc{P}_1^{k_1} =0 \}$.
\[curvaP=0\] Suppose that $\{ \textsc{P}_1 =0 \}$ is not invariant by ${{\widetilde{\mathcal F}}}_{\infty}$. If $k_1 \geq 2$ then $\omega_1$ has a pole of order $k_1 \geq 2$ at a generic point $p$ of this curve so that ${\mathcal{H}}$ has a saddle-singularity at $p$. On the other hand, if $k_1=1$, then $p$ is a simple pole for $\omega_1$ whose residue (not necessarily real) equals $H(p)/ F^{\ast} (p)$ where $F^{\ast} = F/ \textsc{P}_1$.
Again it is enough to consider generic points where $\{ \textsc{P}_1 =0 \}$ is transverse to ${{\widetilde{\mathcal F}}}_{\infty}$. At a generic point $p$ of $\{ \textsc{P}_1 =0 \}$ we have that $F^{\ast}(p) \ne 0$. Moreover, since we are assuming ${\rm g.c.d.}\, (\textsc{P}, H)$ to be invertible, we can also assume that $H(p) \ne 0$. The $1$-form $\omega_1$ has therefore a pole of order $k_1$ whose coefficient is equal to $H(p)/F^{\ast}$.
Let us now consider points belonging to the line at infinity $\Delta_{\infty}^{(x,y)} \subset \Delta_{\infty}$.
\[lineatinfinity\] Points belonging to $\Delta_{\infty}^{(x,y)}$ yield source singularities for ${\mathcal{H}}$ provided that the coordinates $(x,y)$ are generically chosen.
As already seen, $\Delta_{\infty}^{(x,y)}$ does not contain singular points of ${{\widetilde{\mathcal F}}}_{\infty}$. Besides $\Delta_{\infty}^{(x,y)}$ is not invariant by ${{\widetilde{\mathcal F}}}_{\infty}$ either. It then follows that each point $p$ in $\Delta_{\infty}^{(x,y)}$ locally belongs to a unique leaf $L_p$ of ${{\widetilde{\mathcal F}}}_{\infty}$ (by virtue of the definition of leaf given earlier in this section, this is not necessarily true if $\Delta_{\infty}^{(x,y)}$ is invariant by, or contains singular points of, ${{\widetilde{\mathcal F}}}_{\infty}$). Thus the map that assigning to $p$ the residue at $p$ of the $1$-form $\omega_1$ is globally defined on $\Delta_{\infty}^{(x,y)}$. It is lack of continuity is associated to points of tangency between $\Delta_{\infty}^{(x,y)}$ and ${{\widetilde{\mathcal F}}}_{\infty}$ since, on a neighborhood of these, a same leaf of ${{\widetilde{\mathcal F}}}$ intersects $\Delta_{\infty}^{(x,y)}$ more than once. Yet to establish the statement it suffices to check this residue equals $1$ at a generic point of $\Delta_{\infty}^{(x,y)}$. Indeed, this will imply that the residue must be real strictly positive at every point $p$ in $\Delta_{\infty}^{(x,y)}$ so that all these points constitute source singularities for ${\mathcal{H}}$.
Let us then consider those points where $\Delta_{\infty}^{(x,y)}$ is transverse to ${{\widetilde{\mathcal F}}}_{\infty}$. Let $(u,v,w)$ be new local affine coordinates for $M$ where $w$ is the coordinate transverse to $\Delta_{\infty}$ and such that the line at infinity $\Delta_{\infty}^{(x,y)}$ is given by $\{u = 0\}$. The standard change of coordinates associated is then given by $(u,v,w) \longmapsto (1/u, v/u, w) = (x,y,z)$. In these new coordinates, the vector field ${{\widetilde{X}}}$ becomes (up to multiplication by $w^{1-d}$) $$-u^2 F(1/u, v/u) \frac{\partial}{\partial u} + u(-vF(1/u, v/u) + G(1/u, v/u)) \frac{\partial}{\partial v}
+ w H(1/u, v/u) \frac{\partial}{\partial w} \; .$$ Recall that the polynomial vector field $F(x,y) \partial /\partial x + G(x,y) \partial /\partial y$ has degree $d+1$. Furthermore its component of degree $d+1$ has the form $f(x , y) [x \partial
/\partial x + y \partial /\partial y]$ where $f$ is homogeneous of degree $d$. In particular $u^2 F(1/u, v/u)$ has a pole of order $d-1$ over $\{ u=0\}$. Similarly the top-degree homogeneous component of $-vF(1/u, v/u) + G(1/u, v/u)$ vanishes identically so that $\Delta_{\infty}^{(x,y)}$ represents a polar component of degree $d-1$ for the component of ${{\widetilde{X}}}$ in the $v$-direction as well. Finally the order of poles of $H(1/u,v/u)$ over $\Delta_{\infty}^{(x,y)}$ equals $d$. Formula (\[omega1\]) then shows that $\omega_1$ has poles of order $1$ over $\Delta_{\infty}^{(x,y)}$. Indeed the principal part of $\omega_1$ is simply $1/u$, since the top degree component is given by Equation (\[topdegree\]). The statement follows.
To close this section, let us introduce the global notion of trajectory for the foliation $\mathcal{H}$ under the condition that the trajectory in question remains away from the singular set of ${{\widetilde{\mathcal F}}}$. It also convenient to consider the standard flat metric of ${{\mathbb C}}^3$ where the initial affine coordinates $(x,y,z)$ are defined. Roughly speaking a real trajectory is said to be [*finite*]{} if its length, with respect to the above metric, is finite. A precise definition of this length requires to define what are the [*endpoints of a trajectory*]{} which will be done in the sequel, cf. Remark \[rem\] below. To motivate this definition let $l$ be a real trajectory of $\mathcal{H}$ and let $c:[0,1]
\rightarrow l$ be a parametrization of the segment of this trajectory joining $p_0 = c(0)$ to $p_1 = c(1)$. Then the holonomy map ${\rm
Hol}(c): \Sigma_0 \rightarrow \Sigma_1$, where $\Sigma_0, \, \Sigma_1$ are vertical complex lines equipped with the coordinates $z$, satisfies $$\label{contr}
|({\rm Hol}(c))^{\prime}| = e^{- Re(\int_c \omega_1)} < 1 \, .$$ Clearly this formula means that the holonomy is contractive. Roughly speaking, the role played by these trajectories in our discussion can be summarized as follows. Nearby a sink $p$ all real trajectories converge to $p$. Estimate (\[contr\]) guarantees that the distance of the leaves of ${{\widetilde{\mathcal F}}}_S$ to $L_{\infty}$ has a local minimum at $p$ (which may well be zero). On the other hand, nearby a source $p$, all real trajectories go away from $p$. This means that the distance of the leaves of ${{\widetilde{\mathcal F}}}_S$ to $L_{\infty}$ reaches a local maximum at $p$. It remains to consider the case of saddle singularities of $\mathcal{H}$. Indeed it can happen that one of these trajectories meets, as a separatrix, a saddle singularity of $\mathcal{H}$. In this case the trajectory in question can be continued as follows: if our trajectory defines a separatrix for a saddle singularity of $\mathcal{H}$, this saddle singularity also possesses repelling proper directions (in number correspondent to the number of attractive ones). Thus the mentioned trajectory can naturally be continued by following a separatrix associated to a repelling direction. The continued trajectory still yields contractive holonomy. Alternately we may consider a slight perturbation of the trajectory in question on a neighborhood of the saddle singularity so as to avoid it to reach the singularity itself.
\[rem\] [More generally, it follows from the preceding that only sinks or sources singularities of $\mathcal{H}$ (corresponding to maxima or minima for the distance function between leaves) can provide genuine [*endpoints*]{} for a trajectory of $\mathcal{H}$. Otherwise we shall allow the trajectory to be continued regardless of whether or not it passes several times over the same point. In particular a regular [*periodic trajectory of $\mathcal{H}$ necessarily has infinite length.*]{}]{}
Going back to our specific case in which $\omega_1$ is characterized by Formula (\[omega1\]), it follows that the local trajectories of ${\mathcal{H}}$ are determined as the lift in $T_{(x,y(x))}L_{\infty}$ of the vector $v$ where $v$ is such that $v.H(x,y(x)) / F(x,y(x))$ belongs to ${{\mathbb R}}_-$. Also it is to be noted that the abelian form $\omega_1$ is independent of the leaf in the same cone $S$. In fact, Equation (\[omega1\]) shows that it depends solely on $L_{\infty}$. These remarks can be summarized as follows.
1. The trajectory of ${\mathcal{H}}$ through a point $(p_1, p_2,p_3)$ projects on the trajectory of ${\mathcal{H}}$ through the point $(p_1,p_2,0)$ which, in addition, is globally contained in the plane $\{z=0\}$.
2. Since the absolute value of the coordinate “$z$” is always decreasing over a trajectory of ${\mathcal{H}}$, it follows that the trajectory of ${\mathcal{H}}$ through $(p_1, p_2,p_3)$ has infinite length if and only if the the trajectory of ${\mathcal{H}}$ through $(p_1, p_2,0)$ has infinite length.
The following simple lemma will also be important later on. For this lemma we should take into account that, whereas $\Delta_{\infty}^{(x,y)}$ can be chosen “generic”, it always possesses points of tangency with the foliation ${{\widetilde{\mathcal F}}}_{\infty}$.
\[remainingcompact\] For a generic choice of the affine coordinates $(x,y)$ an oriented trajectory of ${\mathcal{H}}$ never intersect $\Delta_{\infty}^{(x,y)}$. Besides there is a compact part $K \subset {{\mathbb C}}^3$ and a constant $C_K$ so that the following holds: every segment of trajectory $l$ of ${\mathcal{H}}$ whose total length is greater than $C_K$ verifies the condition that the part of $l$ lying in ${{\mathbb C}}^3 \setminus K$ is less than, say, $1/10$ of the total length of the segment in question.
Let $q_1, \ldots , q_r$ be the points where $\Delta_{\infty}^{(x,y)}$ is tangent to ${{\widetilde{\mathcal F}}}_{\infty}$ and fix a small neighborhood $W_i$ of $q_i$, $i=1, \ldots ,r$. Then there is a “tubular neighborhood” $V$ of $\Delta_{\infty}^{(x,y)} \setminus \bigcup_{i=1}^r W_i$ so that the following holds: for every point $p \in \partial V \setminus \bigcup_{i=1}^r W_i$ the trajectory of ${\mathcal{H}}$ through $p$ is transverse to $\partial V$ and oriented outwards $V$. In other words, no trajectory of ${\mathcal{H}}$ may enter $V$ without entering first some $W_i$.
On the other hand the structure of ${\mathcal{H}}$ trajectories on $W_i$ is easy to describe. If $\Delta_{\infty}^{(x,y)}$ is “sufficiently generic”, then the tangency of $\Delta_{\infty}^{(x,y)}$ at ${{\widetilde{\mathcal F}}}_{\infty}$ at $q_i$ is quadratic (for all $i \in \{1, \ldots, r\}$). In particular if $L_i$ is the local leaf of ${{\widetilde{\mathcal F}}}_{\infty}$ through $q_i$, the point $q_i$ is itself a source-type singularity for the trajectories of ${\mathcal{H}}$. Thus no trajectory of ${\mathcal{H}}$ actually intersects $\Delta_{\infty}^{(x,y)}$. Finally if $V$ and the neighborhoods $W_i$ are sufficiently small, then the length of a segment of trajectory lying in $V\cup \bigcup_{i=1}^r W_i$ is less than, say, $1/30$ the length of the segment of same trajectory in $K={{\mathbb C}}^3 \setminus V\cup \bigcup_{i=1}^r W_i$ which is determined by two “successive” passages of the trajectory in question through $V\cup \bigcup_{i=1}^r W_i$. The statement then follows.
\[lastsection3\] [In certain cases it may be useful to make a “non-generic” choice of the affine coordinates $(x,y)$ so as to have a line at infinity $\Delta_{\infty}^{(x,y)}$ passing through certain singular points of ${{\widetilde{\mathcal F}}}_{\infty}$. We shall briefly mention one situation of this type later on, cf. Remark (\[tobeincluded\]).]{}
Renormalization in the exceptional divisor
==========================================
Let us continue our discussion of homogeneous polynomial vector fields. We shall keep the notations of Section 3 emphasizing the $3$-dimensional case though all the results presented below are valid in arbitrary dimensions.
Again $X$ will stand for a homogeneous polynomial vector field of degree $d \geq 2$ whose associated foliation is denoted by ${{\mathcal F}}$. The presence of a vector field enables us to endow every regular leaf $L$ of ${{\mathcal F}}$ with an Abelian $1$-form, denoted by $dT$ and defined by the equation $dT.X =1$. The $1$-form $dT$ is going to be called the time-form induced by $X$. When $X$ is complete, given a curve $c:[0,1] \rightarrow L$ joining two points in $L$, $c(0)$ and $c(1)$, the integral $\int_c dT$ measures the time needed to cover $c$ from $c(0)$ to $c(1)$ following the flow of $X$. Again ${{\widetilde{X}}}$ stands for the vector field induced by $X$ on $M$. Throughout this section generic affine coordinates $(x,y)$ are chosen (as in the context of the previous section).
Since the vector field ${{\widetilde{X}}}$ has poles over $\Delta_{\infty}$, the time-form is not defined for the regular leaves of ${{\widetilde{\mathcal F}}}_{\infty}$. However, it is possible to define a “renormalized time-form” on a neighborhood of each regular point $p$ of a leaf $L_{\infty} \subseteq
\Delta_{\infty}$. This goes as follows. Let $L_{\infty} \subseteq \Delta_{\infty}$ be a regular leaf of ${{\widetilde{\mathcal F}}}$ and let $p \in L_{\infty}$ be a regular point of $L_{\infty}$ which is not singular for ${{\widetilde{\mathcal F}}}$. Choose local coordinates $(x',y',z')$ around $p$ where the foliation becomes locally given by the vector field $\partial / \partial x'$. Besides we also impose $\{ z'=0 \} \subset
\Delta_{\infty}$. In these coordinates the vector field ${{\widetilde{X}}}$ is given by ${z'}^{1-d} f(x',y',z')
\partial/\partial x'$. The “renormalized time-form” over $L_{\infty}$ is then defined as $dx'/f(x',0,0)$. In other words, it is obtained from ${{\widetilde{X}}}$ by eliminating its pole component. Naturally there is no canonical choice for the coordinate $z'$ and this prevents us from having a global definition for the “renormalized time-form”. More precisely the local form $dx'/f(x',0,0)$ is not globally defined on $L_{\infty}$ because, when a change of coordinates is performed, two local definitions of this “renormalized time-form” will agree only up to a multiplicative constant. Therefore, whereas the previous construction does not define an Abelian form on $L_{\infty}$, it endows $L_{\infty}$ with an affine structure (for further details we refer to [@guillotreb]). The purpose of this section is to exploit this affine structure to estimate the domain of definition of the solutions of ${{\widetilde{X}}}$. As it will be seen, precise estimates can be obtained in this way as long as the “evolution” of the coordinate “$z$” is well-controlled.
Although we have defined the “renormalized time-form” only at regular points of ${{\widetilde{\mathcal F}}}$, this form admits a natural asymptotic extension to the singularities of ${{\widetilde{\mathcal F}}}$ lying in $\Delta_{\infty}$. Details on these extensions will be given as they become necessary.
Fix a point $p_0$ contained in the singular set of ${{\widetilde{\mathcal F}}}_{\infty}$. Suppose that the restriction of ${{\widetilde{\mathcal F}}}$ to a neighborhood of $p_0$ is given by Equation (\[tXX\]) so that $${{\widetilde{X}}}= \frac{1}{z^{d-1}} \left[ F(x,y) \frac{\partial }{\partial x}
+ G(x,y) \frac{\partial }{\partial y} + zH(x,y) \frac{\partial
}{\partial z} \right].$$ With the notations of Section \[sechom\], let $\textsc{P} = {\rm
g.c.d.}\, (F,G)$ so that $F = \textsc{P}. a (x,y)$ and $G =
\textsc{P} . b (x,y)$. Denoting by $\overline{\textsc{P}}$ the greatest common divisor between $\textsc{P}$ and $H$. Thus we can set $\textsc{P} = \overline{\textsc{P}} \textsc{P}^{\ast}$ and $H = \overline{\textsc{P}} H^{\ast}$. It follows that $$\label{campo}
{{\widetilde{X}}}= \frac{\overline{\textsc{P}}}{z^{d-1}} \left[ \textsc{P}^{\ast}(x,y) \left( a (x,y)
\frac{\partial }{\partial x} + b (x,y) \frac{\partial }{\partial y}
\right) + zH^{\ast} (x , y) \frac{\partial}{\partial z} \right]$$ being $p_0 \simeq (0,0,0)$. If $\textsc{P}^{\ast}$ is not constant, the curve in $\Delta_{\infty}$ induced by $\{ \textsc{P}^{\ast} =0\}$ is singular for ${{\widetilde{\mathcal F}}}$, though its generic points are regular for ${{\widetilde{\mathcal F}}}_{\infty}$. From this point of view $\{ \textsc{P}^{\ast} =0\}$ may or may not be invariant by ${{\widetilde{\mathcal F}}}_{\infty}$. Again to abridge the discussion we shall make the following assumption valid for the rest of this paper unless otherwise stated:
[**General assumption**]{}: No irreducible component of the curve $\{ \textsc{P}^{\ast} =0\}$ is invariant by ${{\widetilde{\mathcal F}}}_{\infty}$.
This condition does not affect the general character of the results stated in the Introduction. For example, concerning the statements of Theorems A, A’ and B, we have the following:
\[aproposition\] Assume that $X$ is a homogeneous semi-complete vector field with degree greater than or equal to $3$. Suppose that ${{\widetilde{X}}}$ is as in Equation (\[campo\]). Then $\textsc{P}^{\ast} = 0$ is not invariant by ${{\widetilde{\mathcal F}}}_{\infty}$. In fact, a regular leaf of ${{\widetilde{\mathcal F}}}_{\infty}$ cannot be contained in the singular set of ${{\widetilde{\mathcal F}}}$.
Let $L$ be a regular leaf of ${{\widetilde{\mathcal F}}}_{\infty}$ and assume for a contradiction that $L$ is contained in the singular set of ${{\widetilde{\mathcal F}}}$. Since ${\rm g.c.d.}\, (a, b)$ is locally invertible (cf. Equation (\[campo\])), $\textsc{P}^{\ast}$ must vanish identically over $L$. Denote by $\textsc{P}_1$ the irreducible component of $\textsc{P}^{\ast}$ vanishing over $L$.
Let $m \geq 1$ be the order of $\textsc{P}^{\ast}_1$ over $L$. In other words, at a generic smooth point of $L$ we can find local coordinates $(x,y,z)$, $\{ z=0\} \subset \Delta_{\infty}$, where $L$ is identified with $\{y=0, z=0\}$. In these coordinates we have that $\textsc{P}^{\ast}_1(x,y) = y^m$ and $H(0,0) \neq 0$. Besides if the point is sufficiently generic we also have $a(0,0) \neq 0$.
Because $L$ is invariant for the foliation, it follows that $b$ is divisible by $y$. Recalling also that $m \geq 1$, it follows that, in the above local coordinates, the first non-trivial homogeneous component of ${{\widetilde{X}}}$ at the origin is given by $${{\widetilde{X}}}^H = z^{1-d} y^k \left[ {\alpha}y \frac{\partial }{\partial x} + {\lambda}z \frac{\partial }{\partial z} \right]$$ for some constants ${\lambda}= H(0,0) \in {{\mathbb C}}^{\ast}$, $k \geq 0$ and ${\alpha}\in {{\mathbb C}}$ (${\alpha}\ne 0$ if and only if $m =1$). The hyperplanes $\{y
= {\rm cte}\}$ are invariant for the associated foliation. For each nonvanishing constant ${\rm cte}$ sufficiently close to zero, the leaves are such that $$\dot{z} = {\rm cte}^k {\lambda}z^{2-d}$$ However, since $d \geq 3$, the corresponding vector field is clearly not semi-complete what contradicts the fact that $L$ is invariant by ${{\widetilde{\mathcal F}}}_{\infty}$. The proposition follows.
The above proof also yields:
\[trivializingP\] Assume that $d \geq 3$. Then $\textsc{P}^{\ast}$ must be constant provided that ${{\widetilde{X}}}$ is semi-complete. If $d=2$ and $\textsc{P}^{\ast}$ is not constant, then the foliation ${{\widetilde{\mathcal F}}}_{\infty}$ is induced by a vector field of degree $0$ or $1$.
By construction, the renormalized time-form is defined only for regular leaves of the foliation (no matter whether or not they are contained in the zero/pole divisor). The interest of the above lemma lies precisely in the statement that, under the conditions above, [*no leaf of ${{\widetilde{\mathcal F}}}_{\infty}$ is contained in the singular set of ${{\widetilde{\mathcal F}}}$*]{} so that the “renormalized time-form” can be defined over every leaf $L_{\infty} \subseteq \Delta_{\infty}$. Furthermore if $X$ is semi-complete but has degree $2$, the fact that $\textsc{P}$ is not invertible implies that $X$ is a linear function times a linear vector field. Clearly this case can directly be treated so that it is going to be left to the reader. Summarizing for the rest of this section and for all of Section 5, the “general assumption” stated above will be assumed without further comments.
Consider the foliation ${{\widetilde{\mathcal F}}}_{\infty}$ on $\Delta_{\infty}$. Fix a regular leaf $L_{\infty} \subseteq \Delta_{\infty}$ of ${{\widetilde{\mathcal F}}}$ and let $S$ be the cone over $L_{\infty}$, $\mathcal{P}^{-1}_{\infty}(L_{\infty})$. Denote by $\mathcal{H}$ the oriented $1$-dimensional real foliation induced by the Abelian form $\omega_1$ (cf. Section \[sechom\]). It is also useful to consider other foliations similar to $\mathcal{H}$. For this let us consider an angle $\theta \in (-\pi/2, \pi/2)$. Denote by $\mathcal{H}^{\theta}$ the oriented foliation whose (oriented) trajectories make an angle of $\theta$ with the (oriented) trajectories of $\mathcal{H}$. It is clear that these foliations are well-defined under the same conditions that $\mathcal{H}$. It is also clear that the holonomy maps of ${{\widetilde{\mathcal F}}}_S$ obtained over the trajectories of $\mathcal{H}^{\theta}$ are still contractions as in Formula (\[contr\]) (up to a constant). In the sequel we denote by $l^{\theta}$ an oriented trajectory of $\mathcal{H}^{\theta}$.
Given (a segment of) a trajectory $l_p$ of $\mathcal{H}$ (resp. $l_p^{\theta}$ of $\mathcal{H}^{\theta}$), we are interested in the value of the integral $\int_{l_p} dT$ (resp. $\int_{l_p^{\theta}} dT$). To investigate the behavior of this integral, it is clear that the singularities of ${{\widetilde{\mathcal F}}}_{\infty}$ will pose further difficulties. Thus it is natural to begin with (segments of) trajectories of $\mathcal{H}$ (resp. $\mathcal{H}^{\theta}$) that remain away from the corresponding singular set. For this let $W$ be a sufficiently small open neighborhood of the singular set of ${{\widetilde{\mathcal F}}}$ on $\Delta_{\infty}$. Let $l_p$ (resp. $l_p^{\theta}$) be (a segment of) a trajectory of ${\mathcal{H}}$ (resp. ${\mathcal{H}}^{\theta}$). We can now state one of our main results. Despite our $3$-dimensional setting, the reader will immediately check that this result holds in arbitrary dimensions (as it is always the case in the present section).
\[introducelabel4\] Suppose that $l_p$ (resp. $l_p^{\theta}$) is contained in $\Delta_{\infty} \setminus W$. Then $\int_{l_q} dT$ (resp. $\int_{l_q^{\theta}} dT$) converges for all $q = (p,z_0) \in \mathcal{P}^{-1}_{\infty}(p,0)$, where $l_q$ (resp. $l_q^{\theta}$) denotes the lift of $l_p$ (resp. $l_p^{\theta}$) to the leaf of ${{\widetilde{\mathcal F}}}$ through $q$ and where $dT$ stands for the time form associated to ${{\widetilde{X}}}$. More precisely, assuming $W$ fixed, there exists a constant $C$ (varying continuously with $\theta$) such that for every path $c :[0,1] \rightarrow L$, $c(0) =q$, with image contained in a trajectory $l_q^{\theta}$ of ${\mathcal{H}}^{\theta}$ and not intersecting $W$, we have $$\int_c dT \leq \int_{l_q^{\theta}} dT < C \vert z_0 \vert^{d-1} \, .$$
It suffices to prove the statement for the case of a trajectory $l_p$ of ${\mathcal{H}}$ since the adaptations needed for trajectories of ${\mathcal{H}}^{\theta}$ are clear. Also we can suppose without loss of generality that the length of $l_p$ is infinite. Finally we recall that the affine coordinates $(x,y)$ were generically chosen in the sense of Section 3. Also let $W$ be the previously chosen open neighborhood of the singular set of ${{\widetilde{\mathcal F}}}$ on $\Delta_{\infty}$. Assume that $l_p$ is connected and totally contained in $\Delta_{\infty} \setminus W$. Since the intersection of $\Delta_{\infty} \setminus W$ with the singular set of ${{\widetilde{\mathcal F}}}_{\infty}$ is empty, the only singularities of $\mathcal{H}$ that can appear on $l_p$ are saddles. However, as noted in Remark \[rem\], the corresponding trajectories of $\mathcal{H}$ can locally be deformed to avoid these singularities. Moreover the uniform contractive character of the corresponding holonomy maps is still kept by these deformed trajectories. As to the polar set of $\omega_1$, we recall that this set is contained in the line at infinity $\Delta_{\infty}^{(x,y)}$. Thanks to Lemma \[remainingcompact\], every sufficiently long segment of $l_p$ remains most “of its length” on a fixed compact part of the affine ${{\mathbb C}}^2$ associated to the coordinates $(x,y)$. Since $F$ is clearly bounded on this compact part, the estimates of Lemma \[remainingcompact\] allow us to conclude the following: every sufficiently long segment $c_p$ of $l_p$ can be split into a concatenation $c_1 \ast c_2 \ast \cdots \ast c_k$ such that:
1. The image of $c_i$, for $i$ odd, is contained in the compact set $K$. Besides at points belonging to these segments the absolute value of $\omega_1$ is bounded from below, i.e. $|\omega_1| \geq {\alpha}> 0$.
2. If $i_0$ is odd, then the sum of the lengths of all even $i$’s, $i <i_0$, is less than, say, $2/3$ the sum of the lengths of $c_1, \ldots ,c_{i_0}$.
3. The absolute value of the coordinate “$z$” decreases monotonically over the segment $c_p$.
Fix $q \in \mathcal{P}^{-1}_{\infty}(p)$ and let $L$ be the leaf through $q$. Consider the lift of $l_p$ to $L$ and denote it by $l_q$. Note that $l_q$ is an oriented trajectory of $\mathcal{H}$ over $L$. We want to express $l_q$ in the coordinates $(x,y,z)$ considered in the preceding section. In fact, the goal will be to compute the value of its last coordinate $z$. For this consider a connected oriented path $c$ contained in $l_p$, joining $p$ to another point of $l_p$. Consider also a lift of $c$ contained in $l_q$. The $z$-coordinate of the mentioned lift is given by $$z = z_0 e^{-\int_c \omega_1}$$ where $z_0$ is the $z$-coordinate of $q$. In other words, $z_0$ is the “height" of $q$ relatively to $L_{\infty}$. In particular $$\begin{aligned}
|z| &=& \left|z_0 e^{-\int_c \omega_1}\right| = |z_0| e^{-{\rm Re}
\int_c \omega_1} = |z_0| e^{-\int_0^1 {\rm Re} (\omega_1(c(t)).
c^{\prime}(t)) dt}\\
&=& |z_0| e^{-\int_0^1 |(\omega_1(c(t)). c^{\prime}(t)| dt} \leq
|z_0| e^{-\int_0^1 {\alpha}|c^{\prime}(t)| dt /3} = |z_0| e^{-{\alpha}{\rm
length}(c) /3}\end{aligned}$$ This estimate shows that, whenever a segment of $l_p$ having length equal to $3\ln(2)/ 2{\alpha}$ is lifted in a regular leaf of ${{\widetilde{\mathcal F}}}$ projecting over $L_{\infty}$, the height of the final point of the lift in question is at most $1/2$ of the height of its initial point.
Now the integral $\int_{l_q} dT$ can be estimated as follows. The time-form on $L$ is given, in local coordinates, by $$dT = \frac{z^{d-1}}{F(x,y)}dx.$$ Since $l_p$, the projection of $l_q$ by $\mathcal{P}_{\infty}$, is contained on a compact set not intersecting the singular set of ${{\widetilde{\mathcal F}}}_{\infty}$, the absolute value of $F(x,y)$ is bounded from below, i.e. $$|F(x,y)| \geq {\beta}> 0, \qquad {\rm for \;\; \; all } \; \; \; (x, y)
\in \Delta_{\infty} \setminus W.$$ Otherwise we replace $F$ by $G$ (recall that we are dealing only with regular points of ${{\widetilde{\mathcal F}}}_{\infty}$). Hence, considering $l_q$ as the concatenation of segments having length equal to $3\ln(2)
/2 {\alpha}$, $l_q = \sum_{i=0}^{\infty}l_{i,q}$, it follows that $$\begin{aligned}
\left| \int_{l_q} dT \right| &=& \left| \sum_{i=0}^{\infty} \int_{l_{i,q}} \frac{z^{k-1}}{F(x, y)} dx \right|
\leq \sum_{i=0}^{\infty} \left| \int_0^1 \frac{z_{i,q}^{d-1}(t)}{F(x_{i,q}(t),y_{i,q}(t))} x^{\prime}_{i,q}(t) dt \right|\\
&\leq& \sum_{i=0}^{\infty} \int_0^1 \frac{|z_{i,q}(t)|^{d-1}}{|F(x_{i,q}(t),y_{i,q}(t))|} |x^{\prime}_{i,q}(t)| dt
\leq \sum_{i=0}^{\infty} \int_0^1 \frac{|z_0|^{d-1}(\frac{1}{2})^{i(d-1)}}{{\beta}} |l^{\prime}_{i,p}(t)| dt\\
&\leq& \frac{|z_0|^{d-1}}{{\beta}} {\rm length }(l_{i,p}) \sum_{i=0}^{\infty}\left( \frac{1}{2^{d-1}} \right)^{i}
= \frac{3|z_0|^{d-1} {\rm ln}(2)}{2{\alpha}{\beta}} \frac{1}{1 - (\frac{1}{2})^{d-1}} < \infty\end{aligned}$$ where $l_{i,q}(t) = (x_{i,q}(t), y_{i,q}(t), z_{i,q}(t))$, $t \in
[0,1]$, is such that $l_q = \sum_{i=0}^{\infty} l_{i,q}$ and $\mathcal{P}_{\infty}(l_{i,q}) = l_{i,p}$. The proposition follows.
What precedes shows that the integral mentioned above is indeed bounded on $\Delta_{\infty} \setminus W$. Our next goal will be to get rid of the condition on $W$, i.e. we want to allow the trajectory $l_p$ (resp. $l_p^{\theta}$) to accumulate on the singular set of ${{\widetilde{\mathcal F}}}$ in $\Delta_{\infty}$. This will lead us to study the behavior of this integral over segments of trajectories of ${\mathcal{H}}$ (resp. ${\mathcal{H}}^{\theta}$) that are close to the singularities of ${{\widetilde{\mathcal F}}}$. This local analysis will be the object of the Section 4. Nonetheless, to finish the current section, let us give an elementary general result concerning trajectories ${\mathcal{H}}, \, {\mathcal{H}}^{\theta}$ that are contained in a local separatrix for a singularity of ${{\widetilde{\mathcal F}}}, \, {{\widetilde{\mathcal F}}}_{\infty}$. This goes as follows. Consider again a vector field ${{\widetilde{X}}}$ as in Equation (\[campo\]). Let $p \in \Delta_{\infty}$ be a singular point of ${{\widetilde{\mathcal F}}}$ and consider a (germ of) analytic curve $Sep \subset \Delta_{\infty}$ passing through $p$ and not entirely contained in the singular set of ${{\widetilde{\mathcal F}}}$. Besides let $\gamma (t)$ denote an irreducible Puiseaux parametrization for $Sep$ defined on a neighborhood of $0 \in {{\mathbb C}}$. Finally denote by $k$ the order at $0 \in {{\mathbb C}}$ of the vector field obtained by pulling-back the restriction of the vector field $\textsc{P}(x,y) ( a (x,y) \frac{\partial }{\partial x} + b (x,y) \frac{\partial }{\partial y} )$ to $Sep$ by $\gamma$. Our assumption concerning $Sep$ ensures that $k$ is a strictly positive integer. Now we have:
\[separatrixndimensions\] Consider the restriction of $\omega_1$ to $Sep$. Then the nature of $\omega_1$ (restricted to $Sep$) at $p$ is determined by the order $l$ at $0 \in {{\mathbb C}}$ of the function $t \mapsto H \circ \gamma (t)$ as follows:
- If $l =k$ then $\omega_1$ is holomorphic and regular at $p$. If $l > k$ then $\omega_1$ is holomorphic and the restriction of ${\mathcal{H}}$ to $Sep$ has a saddle singularity at $p$.
- If $l=k-1$ then $\omega_1$ has a simple pole at $p$. The residue of the pole is $-H(0,0) /F(0,0)$. In particular the restriction of ${\mathcal{H}}$ to $Sep$ has a sink (source) at $p$ provided that $H(0,0)/F(0,0) \in {{\mathbb R}}_+$ (resp. $H(0,0)/F(0,0) \in {{\mathbb R}}_+$).
- If, in addition, $X$ is semi-complete and $l < k-1$ then $d=2$. Furthermore $\omega_1$ has a pole of order 2 or greater at $p$. The foliation ${\mathcal{H}}$ has a saddle singularity at $p$.
The cone over $Sep$ can locally be parameterized by $(t, z) \mapsto (\gamma (t) ,z)$. Since this cone is left invariant by ${{\widetilde{\mathcal F}}}$, we can pull-back (the restriction of) ${{\widetilde{\mathcal F}}}, \, {{\widetilde{X}}}$ to the coordinates $(t,z)$. We then obtain a vector field $Y$ given by $$Y = z^{1-d} [ f(t) \partial /\partial t + z(H \circ \gamma) (t) \partial /\partial z]$$ where the order at $0 \in {{\mathbb C}}$ of the function $f(t)$ is $k$. In these coordinates the form $\omega_1$ restricted to the axis $\{ z=0\}$ is nothing but $H \circ \gamma (t) dt/f(t)$. Therefore to conclude the statement it suffices to check that $d=2$ for $l < k-1$. To do this, note that the condition $k > l+1$ implies that the first non-trivial homogeneous component of ${{\widetilde{X}}}$ at $p$ has the form $z^{2-d} H \circ \gamma (t) \partial /\partial z$. However the latter vector field is not semi-complete for $d \geq 3$.
To close this section let us point out again that the preceding statements hold in arbitrary dimensions despite our choice of emphasizing the case of dimension $3$. The reader will have no difficulty to realize that our proofs work equally well in the $n$-dimensional case.
Structure of $\mathcal{H}$ near singular points of ${{\widetilde{\mathcal F}}}_{\infty}$
========================================================================================
Next we are going to establish an extension of Theorem \[introducelabel4\] by allowing the trajectories of ${\mathcal{H}}, \, {\mathcal{H}}^{\theta}$ to accumulate on singularities of ${{\widetilde{\mathcal F}}}, \, {{\widetilde{\mathcal F}}}_{\infty}$. These singularities however will be supposed to be “simple enough”. These assumptions are, in fact, necessary since reasonably simple singularities, such as “saddle-nodes” slightly more degenerated than those considered in the statement of Theorem B already give rise to new complications preventing us from generalizing Theorem \[introducelabel4\] without further information.
As in Sections 3 and 4, we still keep notations and provide arguments emphasizing the $3$-dimensional case. The extensions of the arguments to higher dimensions however does not present any additional difficulty. Again we consider the foliation ${{\widetilde{\mathcal F}}}$ associated to a homogeneous semi-complete vector field on ${{\mathbb C}}^n$ and assume that the singularities of ${{\widetilde{\mathcal F}}}$ lying in $\Delta_{\infty}$ are simple in the sense stated in the Introduction. In particular, if $n=3$, then this condition is equivalent to saying that the singularities of ${{\widetilde{\mathcal F}}}_{\infty}$ possess exactly two eigenvalues different from [*zero*]{}. Moreover if these eigenvalues have the form $1,N$ with $N \in \mathbb{Z}_+$, then we also ask the singularity to be linearizable (i.e. not to be conjugate to its Poincaré-Dulac normal form, cf. for example [@arnold]). With these assumptions Theorem \[introducelabel4\] admits the following extension:
\[maintheo\] Suppose that $X$ (homogenous) is semi-complete and assume that all the singularities of ${{\widetilde{\mathcal F}}}_{\infty}$ are simple (in the sense indicated in the Introduction). Suppose that there is $\theta \in (-\pi/2, \pi/2)$ and a point $p\in \Delta_{\infty}$ such that the trajectory $l_p^{\theta}$ of $\mathcal{H}^{\theta}$ through $p$ has infinite length. Then $\int_{l_q} dT$ converges for all $q \in \mathcal{P}^{-1}_{\infty}(p)$, where $l_q$ denotes the lift of $l_p$ to the leaf through $q$ and $dT$ is the time form associated to ${{\widetilde{X}}}$.
Applications of Theorems \[introducelabel4\] and \[maintheo\] will be provided in the forthcoming sections. The present section will entirely be devoted to the proof of Theorem \[maintheo\].
In view of Proposition \[aproposition\] (cf. also Corollary \[trivializingP\]), we can assume that $\textsc{P}^{\ast}$ is constant (and thus equal to $1$) so that $\textsc{P} = \overline{\textsc{P}}$. Hence $X$ is a homogeneous polynomial vector field multiplied by a homogeneous polynomial. Since multiplying a homogeneous vector field by a homogeneous polynomial does not affect our arguments, which are essentially determined by the underlying foliation, we are going to assume throughout the sequel that $\textsc{P} = \overline{\textsc{P}}\textsc{P}^{\ast} \equiv 1$ to abridge notations.
To begin with fix a point $p_0 \in \Delta_{\infty}$ contained in the singular set of ${{\widetilde{\mathcal F}}}$. Recall that the two eigenvalues of ${{\widetilde{\mathcal F}}}_{\infty}$ at $p_0$ are supposed to be different from zero (and, in case they are of the form $1,N$ with $N \in \mathbb{Z}_+$, ${{\widetilde{\mathcal F}}}_{\infty}$ is supposed not to be conjugate to its Poincaré-Dulac normal form). To be more precise, a vector field representing ${{\widetilde{\mathcal F}}}$ about $p_0$ and having a singular set of codimension $\geq 2$ can be written as $$\bar{X} = F (x,y) \frac{\partial }{\partial x} + G (x,y)
\frac{\partial }{\partial y} + zH(x,y) \frac{\partial }{\partial z}$$ with ${\rm g.c.d.}\, (F,G) = 1$. Note that $\bar{X}$ differs from our vector field ${{\widetilde{X}}}$ by a multiplicative function having the form $z^{1-d} f(x,y)$ where $f(0,0) \neq 0$. The condition about the singularities of ${{\widetilde{\mathcal F}}}_{\infty}$ simply means that the vector field $Z = F (x,y) \partial /\partial x + G (x,y) \partial
/\partial y$ has eigenvalues ${\lambda}_1, \, {\lambda}_2$ at $p_0$ with ${\lambda}_1 {\lambda}_2 \neq 0$. Furthermore if ${\lambda}_1, \, {\lambda}_2$ is of the form $1,N$ with $N \in \mathbb{Z}_+$, then $Z$ is linearizable (recalling that a non-linearizable vector field verifying the preceding conditions must be conjugate to $(Nx + y^N) \partial /\partial x
+ y \partial /\partial y$). This summarizes the assumption of Theorem \[maintheo\].
Fixed a separatrix $Sep$ for ${{\widetilde{\mathcal F}}}_{\infty}$ at $p_0 \in \Delta_{\infty}$, we have:
\[thislabelwasmissing\] Assume that $H(0,0) = 0$. Then the Abelian form $\omega_1$ on the cone over $Sep$ is holomorphic.
Since $p_0$ has non-degenerate linear part for ${{\widetilde{\mathcal F}}}_{\infty}$, it follows that whether or not $Sep$ is smooth, the order $k$ at $0 \in {{\mathbb C}}$ of the vector field obtained by pulling-back the restriction of the vector field $Z$ to $Sep$ by an irreducible Puiseaux parameterization $\gamma$ equals $1$. The statement then results from Lemma \[separatrixndimensions\].
It follows from this lemma that $p_0$ is either a regular point or a saddle singularity for ${\mathcal{H}}$. In both cases the singular point in question can be avoided without disrupting the contractive character of the corresponding holonomy map (Remark \[rem\]). In particular this type of singularity can be ruled out from the subsequent discussion. Let us now consider the case where $H(0,0) \ne 0$.
\[hreal\] For $X$ semi-complete assume that $H(0,0) \neq 0$. Then $d=2$ and $H(0,0)/{\lambda}_1 \in {{\mathbb Q}}$.
Let us consider the restriction of ${{\widetilde{X}}}$ to the invariant manifold $\{x=0, \, y=0\}$. This restriction is naturally a semi-complete vector field that does not vanish identically, since we are assuming $H(0,0) \ne 0$. The restriction is, in fact, given by $H(0,0) z^{2-d} {\partial /\partial z}$. The semi-complete assumption implies that $d=2$.
Let us now denote by ${{\widetilde{\mathcal F}}}_S$ the foliation induced on the $2$-plane sitting over the invariant manifold $\{y=0\}$. Obviously $Sep$ is a separatrix for ${{\widetilde{\mathcal F}}}_S$. The restriction of ${{\widetilde{X}}}$ to this plane is clearly semi-complete. Besides this restriction is not identically zero since ${{\widetilde{X}}}= z^{1-d} \bar{X}$ up to an invertible factor. In particular the first homogeneous component at the origin of this restriction is therefore semi-complete as well. Nonetheless, the homogeneous component in question is given by $z^{-1} [ {\lambda}_1 x {\partial /\partial x}+ H(0,0) z {\partial /\partial z}]$. Now note that its asymptotic order, which is equal to $1 - H(0,0)/{\lambda}_1$, must belong to $[0,2] \cap {{\mathbb Q}}$ since the vector field is semi-complete. The result follows.
In particular the quotient of the eigenvalues of ${{\widetilde{\mathcal F}}}_{\infty}$ at the singular point $p_0$ is a rational number (and thus real). In this case the $1$-form $\omega_1$ has a simple pole at the origin ($\simeq p_0$). Therefore there are two cases to be considered according to whether ${\lambda}_1/{\lambda}_2 \in {{\mathbb Q}}_+$ or ${\lambda}_1/{\lambda}_2 \in {{\mathbb Q}}_-$. The first case is easy to be treated.
\[quotientinR+\] With the preceding notations suppose that ${\lambda}_1/{\lambda}_2 \in {{\mathbb Q}}^+$. Then $p_0$ is a sink $($resp. source$)$ singularity for ${\mathcal{H}}$ provided that $H(0,0)/{\lambda}_1 > 0$ $($resp. $H(0,0)/{\lambda}_1 < 0)$. In either case, $p_0$ yields a terminal point for the trajectories of ${\mathcal{H}}$.
It suffices to consider the case $H(0,0)/{\lambda}_1 > 0$. Clearly the structure of ${\mathcal{H}}$ over the two (smooth) separatrices of ${{\widetilde{\mathcal F}}}_{\infty}$ at $p_0$ corresponds to sinks. As to the remaining leaves, recall that they all accumulate on the origin. Furthermore the structure of ${\mathcal{H}}$ over regular points of these leaves has to be of the same nature as the corresponding structure over the smooth separatrizes. That is to say that all these trajectories point inward the singularity $p_0 \simeq (0,0)$. The lemma is proved.
So let us assume from now on that ${\lambda}_1/{\lambda}_2 \in {{\mathbb Q}}^-$. The restriction of ${{\widetilde{\mathcal F}}}_{\infty}$ to a neighborhood of $p_0$ admits $2$ separatrices. These separatrices are the unique leaves (of the restriction of ${{\widetilde{\mathcal F}}}_{\infty}$ to a neighborhood of $p_0$) “radially” accumulating on the singular point $p_0$. In vague terms, the separatrices are the only leaves of ${{\widetilde{\mathcal F}}}_{\infty}$ accumulating on $p_0$ if we ignore the effect of the local holonomy of this foliation. Denote by $Sep$ one of them. The restriction of $\mathcal{H}$ to $Sep$ may have a singular point at $p_0 \in Sep$. The nature of this singular point depends also on the sign of the quotient $H(0,0)/{\lambda}_1$. If $H(0,0)/{\lambda}_1 > 0$ then $p_0$ corresponds to a sink of $\mathcal{H}$ (or of $\omega_1$ by a small abuse of notation) over $Sep$. Conversely, in the case where $H(0,0)/{\lambda}_1 < 0$, the singular point corresponds to a source. We note however that $H(0,0)/{\lambda}_1$ and $H(0,0)/{\lambda}_2$ have opposite signs. This implies that if $p_0$ is a sink of $\omega_1$ for one of the separatrices then $p_0$ is a source for the other.
The above indicated issue about source and sinks singularities appearing on the two separatrices of a singularity $p_0$ as before deserves some comments. First, if we consider real trajectories of $\mathcal{H}$ in the separatrix admitting $p_0$ as a sink, then these trajectories will reach a future endpoint at $p_0$. Somehow compensating the existence of this future endpoint, in the other separatrix new ${\mathcal{H}}$-trajectories are issued. These phenomena can however occur for only finitely many leaves of our foliation since each separatrix of a singularity as above can give rise to only one global leaf of ${{\widetilde{\mathcal F}}}, \, {{\widetilde{\mathcal F}}}_{\infty}$. In particular it will play no significant role in the proof of any of the theorems stated in the Introduction. In this concern, a far more important observation concern those ${\mathcal{H}}$-trajectories whose projection on $\Delta_{\infty}$ enters a small neighborhood of $p_0$ but are not contained in the corresponding separatrix of $p_0$. In fact, these trajectories can naturally be continued through the “saddle” associated to the singularity so as to eventually leave a fixed neighborhood of $p_0$. Indeed the foliation $\mathcal{H}$ is regular over all leaves of ${{\widetilde{\mathcal F}}}_{\infty}$ different from the two separatrices on a neighborhood of $p_0$. Besides, as we are going to see next, the “continued” trajectory keeps the contractive character of its holonomy.
Suppose then that the eigenvalues ${\lambda}_1, {\lambda}_2$ at $p_0$ satisfies ${\lambda}_1/{\lambda}_2 \in {{\mathbb Q}}^-$. Let us still assume that $H(0,0) \ne 0$ so that it can be normalized to be $1$. Let $U_{{\varepsilon}} = \{(x,y,z): |x|, |y| \leq {\varepsilon}\}$ be a small neighborhood of the origin $\simeq p_0$, not containing other singular points of ${{\widetilde{\mathcal F}}}_{\infty}$. Fix a regular leaf $L_{\infty} \subseteq \Delta_{\infty}$ intersecting $U$ and consider a real trajectory $l \subseteq L_{\infty}$. For these singularities we have:
\[siegeldomain\] The integral $\int_{l_q \cap U} dT$ is uniformly bounded for every $p \in l$ and $q \in {{\mathcal P}}^{-1}_{\infty}(p)$.
\[explainingstatement\] [It should be emphasized that the trajectory $l_q$ in the statement is viewed as a global trajectory of $\mathcal{H}$. In other words, the intersection $l_q \cap U$ possesses, in general, infinitely many connected components. The proposition indeed claims that the sum of the integrals of $dT$ over all these connected components is uniformly bounded.]{}
[*Proof of Proposition \[siegeldomain\]*]{}. Without loss of generality we set ${\lambda}_1 \in {{\mathbb R}}^+$. Since, locally, the only codimension $1$ component of the divisor of zeros/poles of ${{\widetilde{X}}}$ is constituted by the hyperplane at infinity (in coordinates given by $\{z=0\}$), it follows that ${{\widetilde{X}}}$ can be written as $$z^{1-d} \left[ F(x,y) \frac{\partial}{\partial x} + G(x,y)
\frac{\partial }{\partial y} + zH(x,y) \frac{\partial }{\partial z}
\right]$$ in suitable coordinates. In the above formula, we have $F(x,y) = x({\lambda}_1 + {\rm h.o.t.})$ and $G(x,y) = y({\lambda}_2 + {\rm h.o.t.})$.
Recall that ${\lambda}_1 > 0$ (resp. ${\lambda}_2 < 0$). Consider now the restriction of $\omega_1$ to the $x$-axis (resp. $y$-axis). The residue of $\omega_1$ at $0 \simeq p_0$ with respect to the mentioned axis is equal to $-H(0,0)/{\lambda}_1$ (resp. $-H(0,0)/{\lambda}_2$). Then it follows that the restriction of ${\mathcal{H}}$ to the $x$-axis (resp. $y$-axis) possesses a sink singularity (resp. source singularity) at $p_0 \simeq 0$. Hence the real trajectories contained in the $x$-axis approaches $p_0$. Similarly, those trajectories contained in the $y$-axis move away from $p_0$. It is easy to describe the behaviour of ${\mathcal{H}}$ on the regular leaves of $U$ not accumulating at $p_0$: over a real trajectory of ${\mathcal{H}}|_U$ the absolute value of $x$ decreases while the absolute value of $y$ increases. In other words, a real trajectory moves away from the invariant plane $\{y=0\}$ while approaches the plane $\{x=0\}$. In particular, whenever a (global) real trajectory $l$ enters the open set $U_{{\varepsilon}}$ it necessarily leaves $U_{{\varepsilon}}$ as well.
In this way the only possibility for a real trajectory (not contained in the global leaves containing the axes $\{ y=z=0\}$ and $\{x=z=0\}$) to accumulate on the singular point $p_0$ happens when this trajectory enters infinitely many times the open set $U_{{\varepsilon}}$. The sequence of points defined by the moment in which the mentioned trajectory enters $U_{{\varepsilon}}$ must also contain a subsequence that converges for a point in the $x$-axis. Also, in this case, it is immediate to check that the [*length of each connected component of $l \cap U_{{\varepsilon}}$*]{} is bounded above by some uniform constant.
For each leaf of ${{\widetilde{\mathcal F}}}\cap U$ not contained in the invariant plane $\{x=0\}$ the time-form is given by $$\label{timeform}
dT = \frac{z^{d-1}}{F(x,y)}dx.$$ The leaf can locally be parametrized by the $x$-variable under the form $(x, y(x), z(x))$ where $z$ is given by Equation (\[omega1\]). The form of $F$ and $G$ allows us to see that $y(x) = y_0 (x/x_0)^{{\lambda}_2/{\lambda}_1}g(x)$ for some bounded holomorphic function $g$ on ${{\mathbb C}}\setminus {{\mathbb R}}^-_0$ verifying $\lim_{x \rightarrow x_0}g(x) = 1$. In turn, the coordinate $z$ is given by $z = z_0 e^{ -\int_{x_0}^x \omega_1 }$, where $\omega_1$ coincides with $-H(x,y(x))/F(x,y(x))dx$. Therefore, substituting $y$ and $z$ on Equation (\[timeform\]), we obtain $$dT = z_0^{d-1} \frac{1}{F(x,y(x))} e^{ -(d-1) \int_{x_0}^x \omega_1
} dx. \label{introducelabel1}$$
Since we need to estimate the integral of the time-form over oriented real trajectories of ${\mathcal{H}}$, let us start by controlling the exponential term. Since $H(0,0) = 1$, it follows that $$-\frac{H(x,y)}{F(x,y)} = -\frac{1}{{\lambda}_1 x} \left( 1 + p(x,y)
\right)$$ for some holomorphic function $p(x,y)$ on a neighborhood of the origin verifying $p(0,0) = 0$. In particular, if $\varepsilon$ is sufficiently small, the absolute value of $p(x,y)$ is bounded above by a small constant $0 < \delta << 1$ on $U_{{\varepsilon}}$. If $l$ is a trajectory of ${\mathcal{H}}$ then $\int_l \omega_1$ is a positive real number. Therefore $$\left| e^{-(d-1)\int_l \omega_1} \right| = e^{-(d-1) {\rm Re}\int_l
\omega_1} = e^{-(d-1)\int_l \omega_1}.$$
Consider a (connected) segment of the real trajectory $l$ joining $x_0$ to $x$ where both points are contained in the neighborhood in question. Denote by $\phi : [0, 1] \rightarrow L$ a parametrization of this segment satisfying $\phi(0) = x_0$ and $\phi(1) = x$. Up to a change of coordinates, “close" to a rotation, we can assume that the (connected) segment $\phi ([0,1])$ is totally contained in the real axis. In fact, we can assume that it is contained on its positive component. In particular, we can take $\phi(t) = x_0 + t(x - x_0)$. It then follows $$\begin{aligned}
\int_l \omega_1 &= \int_0^1 \omega_1. \phi = \int_0^1 -\frac{\phi^{\prime}(t)}{{\lambda}_1 \phi(t)} (1 + p(\phi(t),
y(\phi(t))) dt\\
&= \frac{1}{{\lambda}_1} \int_0^1 -\frac{x - x_0}{x_0 + t(x - x_0)} (1 +
p(\phi(t), y(\phi(t))) dt\\
&\geq \frac{1 - \delta}{{\lambda}_1} \int_0^1 -\frac{x - x_0}{x_0 + t(x -
x_0)}dt = \frac{1 - \delta}{{\lambda}_1} \ln \left( \frac{x_0}{x} \right) \, .\end{aligned}$$ Therefore we obtain $$\left| e^{ -(d-1) \int_l \omega_1 } \right| \leq C x^{\frac{(d-1)(1
- \delta)}{{\lambda}_1}} \label{introducelabel2}$$ where $C$ is a constant depending on $d$, ${\lambda}_1$, $\delta$ and $x_0 (= {\varepsilon})$. In more accurate terms, $C = {\varepsilon}^{\frac{(1-d)(1-\delta)}{{\lambda}_1}}$. In fact, this estimation should be multiplied by a constant representing the supremum of the absolute value of the determinant of the change of coordinates. However we can, basically, include this quantity on $C$ since the absolute value of the determinant is bounded above on $U_{{\varepsilon}}$. Indeed its value is very close to $1$ since the change of coordinates is “close" to a rotation. In this sense, the constant $C$ does not depend on the segment of the real trajectory.
Now recall that $F(x,y) = {\lambda}_1 x(1 + f(x,y))$, for some holomorphic function $f$ on $U_{{\varepsilon}}$ satisfying $f(0,0)=0$. Modulo reducing ${\varepsilon}$, we can assume that $|f(x,y)|$ is bounded above by a small constant $0 < \tau << 1$. Therefore, the coefficient of the time-form satisfies $$|dT| \leq |z_0|^{d-1} \frac{C}{{\lambda}_1 (1 - \tau)} x^{\frac{(d-1)(1 -
\delta)}{{\lambda}_1} - 1} \, .$$ Since the exponent of $x$ is greater than $-1$, the primitive of the coefficient of the time-form, up to the term $z_0^{d-1}$, is a bounded holomorphic function. Consequently, the integral of the time-form, up the same term, over each connected component $l_i$ of $l
\cap U_{{\varepsilon}}$ is bounded above. In fact, there is a positive constant $K$, not depending on the trajectory of ${\mathcal{H}}$, such that $$\left| \int_{l_i} z_0^{1-d} dT \right|< K.$$
Finally the integral of the time form along $l_i$ is now bounded by $K$ times the absolute value of a positive power of the variable $z$ in the moment that the trajectory $l$ enters the open set $U_{{\varepsilon}}$ or, equivalently, on the starting point of $l_i$. We denote by $z_i$ the value $z$ at the starting point of $l_i$. As already mentioned, the holonomy of ${{\widetilde{\mathcal F}}}$ is contractive. Therefore, since the length of the real trajectory between two consecutive starting points of $l \cap U_{{\varepsilon}}$ is bounded from below, the sequence $z_i$ is such that $|z_{i+1}|/|z_i|
\leq k$, for some constant $0 < k < 1$, since the trajectories of $\mathcal{H}$ have contractive holonomy. Thus $$\left| \int_{l \cap U} dT \right| \leq \sum \left| \int_{l_i} dT
\right| \leq \sum K|z_i|^{d-1} \leq \sum K|z_0|^{d-1}k^{i(d-1)} =
\frac{K|z_0|^{d-1}}{1-k^{d-1}}$$ which means that the integral is uniformly bounded as desired.
Let us now provide the proof of Theorem \[maintheo\].
The proof follows immediately from the combination of Theorem \[introducelabel4\] with Proposition \[siegeldomain\].
[**$\bullet$ A comment about a natural relaxation of the condition on the singularities of ${{\widetilde{\mathcal F}}}$**]{}.
To close this section, and before beginning the applications of the material so far developed, we would like to indicate a far more general setting to which the previous results, especially Theorem \[maintheo\] and Proposition \[siegeldomain\], can be extended. The point of this generalization is that, to large extent, the results valid for “simple singularities” will remain valid in the context of “absolutely isolated singularities”, cf. [@canoetc].
To explain how to work out the mentioned generalizations, first consider the case where the vector field ${{\widetilde{X}}}$ has the form $$\label{tXXxy}
x^n y^m z^{1-d} \left[ F(x,y) \frac{\partial}{\partial x} + G(x,y)
\frac{\partial }{\partial y} + zH(x,y) \frac{\partial }{\partial z}
\right]$$ for some strictly positive integers $m,n$ and for $F,G$ as above. We would like to point out that, unfortunately, in this case the statement of Proposition \[siegeldomain\] is no longer valid even for (locally) semi-complete vector fields. However, the statement still holds if $m, n$ are negative integers. This is a key to adapt Theorem \[maintheo\] and Theorem B to the large class of vector fields leading to foliations whose singularities are “absolutely isolated”. In fact, consider (non-homogeneous) polynomial vector fields having only “absolutely isolated singularities” in the “hyperplane at infinity” (cf. Section 6.2), then these singularities “at infinity” can be reduced through a sequence of punctual blow-ups. In turn this will (almost always) yield a divisor over which the transform of the initial vector field will have poles. In other words, locally around the new (reduced) singular points we are going to find the situation represented by Formula (\[tXXxy\]) with non-positive corresponding integers $m,n$. Therefore the preceding strategy of establish the convergence of the natural improper integrals will still be applicable.
Applications to complete vector fields
======================================
Ends of solutions of complete polynomial vector fields on ${{\mathbb C}}^n$
---------------------------------------------------------------------------
This first application concerns Theorem \[introducelabel4\]. Consider a complete polynomial vector field $X$ defined on ${{\mathbb C}}^n$. Set $X=\sum_{i=0}^d X_i$ where $X_i$ stands for the homogeneous component of degree $i$ of $X$. To keep as much as possible the notations used in the previous sections, the foliation associated to $X$ will be denoted by ${\mathcal{D}}$ whereas ${{\mathcal F}}$ will stand for the foliation associated to the top-degree homogeneous component $X_d$. We shall always assume that $d \geq 2$.
Recall that both foliations ${\mathcal{D}}$ and ${{\mathcal F}}$ admit holomorphic extensions to ${{\mathbb C}}P(n)$ and these extensions are also denoted by ${\mathcal{D}}$ and ${{\mathcal F}}$.
\[suzuki1.1\] The homogeneous vector field $X_d$ is not a multiple of the radial vector field $${\mathcal{E}}= x_1 \partial /\partial x_1 + \cdots + x_n \partial /\partial x_n \; .$$
First note that the vector field $X_d$ is semi-complete on all of ${{\mathbb C}}^n$ since it is the top-degree homogeneous component of a complete vector field. Next suppose we have $X_d = P {\mathcal{E}}$ where $P$ is a polynomial of degree $d-1$. The semi-complete character of $X_d$ then implies that $d=2$. Therefore it only remains to exclude this last possibility. For this, note that the generic leaf $L$ of ${\mathcal{D}}$ intersects the hyperplane at infinity of ${{\mathbb C}}P(n)$ transversely at a regular point $p$ for ${\mathcal{D}}$. Besides the point $p$ is regular for the restriction of $X$ to $L$. In other words, the flow of $X$ reaches the hyperplane at infinity in finite time. This is impossible since $X$ is complete on ${{\mathbb C}}^n$. The proof of the lemma is over.
Again let $\Delta_{\infty}$ denote the hyperplane at infinity in ${{\mathbb C}}P(n)$. It follows from the preceding that $\Delta_{\infty}$ is invariant by both ${\mathcal{D}}$ and ${{\mathcal F}}$. Besides the foliations [*induced*]{} on $\Delta_{\infty}$ by ${\mathcal{D}}, \, {{\mathcal F}}$ turn out to coincide. The foliation induced by ${{\mathcal F}}$ on $\Delta_{\infty}$ will be denoted by ${{\mathcal F}}_{\infty}$. Also $\Delta_{\infty}$ corresponds to the divisor of poles for both $X, \, X_d$.
Since ${{\mathcal F}}$ is given by a homogeneous vector field, the ideas developed in Sections 2 and 3 can directly be applied to it. In turn, near $\Delta_{\infty}$, the foliation ${\mathcal{D}}$ becomes very close to ${{\mathcal F}}$. In the sequel we are going to combine these two issues in order to establish Theorem A.
Let us begin by choosing affine coordinates $(x_1, \ldots, x_{n-1}, z)$ analogous to those used in Sections 3, 4. Namely the hyperplane $\{ z=0\}$ is contained in $\Delta_{\infty}$ and the plane at infinity $\Delta_{\infty}^{1,\ldots ,n-1}$ defined by the affine coordinates $x_1, \ldots, x_{n-1}$, where $z=0$ is fixed, is not invariant by the restrictions of either ${\mathcal{D}}$ or ${{\mathcal F}}$ to $\Delta_{\infty}$. We are then able to apply the results of Section 4 to the foliation ${{\mathcal F}}$. In particular, the leaves of ${{\mathcal F}}$ are equipped with the (singular) real foliations ${\mathcal{H}}^{\theta}$ where $\theta$ is chosen in the interval $(-\pi/2 , \pi/2)$. For the rest of this section, these foliations will be denoted by ${\mathcal{H}}_{{{\mathcal F}}}$ (resp. ${\mathcal{H}}^{\theta}_{{{\mathcal F}}}$). To define a suitable version of these real trajectories in the leaves of ${\mathcal{D}}$ we proceed as follows. Given a point $p=
(x_1^0, \ldots, x_{n-1}^0, z^0)$ with $z^0 \neq 0$, let $L_p$ denote the leaf of ${\mathcal{D}}$ through $p$. To define the foliation ${\mathcal{H}}_{{\mathcal{D}}}$ at $p$, we consider the function $(x_1, \ldots, x_{n-1}, z) \mapsto \vert z \vert$ restricted to $L_p$. The tangent vector to ${\mathcal{H}}_{{\mathcal{D}}}$ at $p$ is simply the negative of the gradient of the function in question. Once ${\mathcal{H}}_{{\mathcal{D}}}$ is defined the foliations ${\mathcal{H}}^{\theta}_{{\mathcal{D}}}$ have an obvious definition.
The next step in our construction consists of investigating the basic properties of ${\mathcal{H}}_{{\mathcal{D}}}$ in analogy with the properties of ${\mathcal{H}}_{{{\mathcal F}}}$ considered in Sections 3 and 4. Recalling that ${\mathcal{D}}, \, {{\mathcal F}}$ induce the same foliation ${{\mathcal F}}_{\infty}$ on $\Delta_{\infty}$, consider a point $(x_1^0, \ldots ,x_{n-1}^0, 0) \in \Delta_{\infty}$ that is regular for the restrictions of both ${\mathcal{D}}, \, {{\mathcal F}}$ to $\Delta_{\infty}$. Then we have:
\[suzuki2.2\] The direction of ${\mathcal{H}}_{{\mathcal{D}}}$ at the point $(x_1^0, \ldots ,x_{n-1}^0, z)$ converges uniformly to the direction of ${\mathcal{H}}_{{{\mathcal F}}}$ at $(x_1^0, \ldots ,x_{n-1}^0, 0)$. In particular the foliation ${\mathcal{H}}_{{\mathcal{D}}}$ can be extended to the regular part of ${\mathcal{D}}$ in $\Delta_{\infty}$ and this extended foliation coincides with ${\mathcal{H}}_{{{\mathcal F}}}$ on $\Delta_{\infty}$.
Since the behavior of ${\mathcal{D}}$ near $(x_1^0, \ldots ,x_{n-1}^0, 0)$ is dominated by the component $X_d$ of $X$, it suffices to check that the trajectories of ${\mathcal{H}}_{{{\mathcal F}}}$ admit a definition analogous to the one given above for the trajectories of ${\mathcal{H}}_{{\mathcal{D}}}$. In other words, it suffices to prove that the direction of ${\mathcal{H}}_{{{\mathcal F}}}$ at $(x_1^0, \ldots ,x_{n-1}^0, z)$ coincides with the gradient of the function $(x_1, \ldots, x_{n-1}, z) \mapsto \vert z \vert$ restricted to the leaf of ${{\mathcal F}}$ through $(x_1^0, \ldots ,x_{n-1}^0, z)$. This is however an immediate consequence of Formula (\[omega1\]). The lemma is proved.
To help us to explain how to derive properties of $X, \, {\mathcal{D}}$ from properties of $X_d, \, {{\mathcal F}}$, it is convenient to consider a small neighborhood (in the $n$-dimensional ambient space) $V$ of $({\rm Sing}\, ({\mathcal{D}}) \cap \Delta_{\infty}) \cup {\rm Sing}\, (X)$, where ${\rm Sing}\, ({\mathcal{D}})$ (resp. ${\rm Sing}\, (X)$) stands for the singular set of ${\mathcal{D}}$ (resp. $X$). Next denote by $U$ a neighborhood of $\Delta_{\infty} \setminus V$. Also, in order to keep a “uniform contractive character” over trajectories of ${\mathcal{H}}_{{{\mathcal F}}}^{\theta}$, we fix some (small) $\epsilon >0$ and consider only those values of $\theta$ belonging to the interval $(-\pi/2 + \epsilon, \pi/2 - \epsilon)$.
Note that all the endpoints belonging to $U$ for trajectories of ${\mathcal{H}}_{{{\mathcal F}}}^{\theta}$ are situated over $\Delta_{\infty}^{1,\ldots ,n-1}$. In particular a trajectory of ${\mathcal{H}}_{{{\mathcal F}}}^{\theta}$ through an affine point $(x_1^0, \ldots ,x_{n-1}^0, 0) \in U \cap \Delta_{\infty}$ will never approach $\Delta_{\infty}^{1,\ldots ,n-1}$ unless it first enters $V$. Modulo choosing the neighborhood $U$ sufficiently narrow, the restriction to $U$ of the foliation ${\mathcal{D}}$ is very close to the (restriction to $U$ of the) foliation ${{\mathcal F}}$. A similar statement holds for the foliations ${\mathcal{H}}_{{\mathcal{D}}}$ and ${\mathcal{H}}_{{{\mathcal F}}}$ thanks to Lemma \[suzuki2.2\]. In particular we obtain the following:
\[suzuki3.3\] Let $\epsilon >0$ be fixed. Consider a point $p=(x_1^0, \ldots ,x_{n-1}^0, z^0) \in U$ and denote by $l_p^{+,\theta}$ the semi-trajectory of ${\mathcal{H}}_{{\mathcal{D}}}^{\theta}$ initiated at $p$ for $\theta \in [-\frac{\pi}{2} + \epsilon, \frac{\pi}{2} - \epsilon]$. Consider a path $c :[0,1] \rightarrow l_p^{+,\theta} \cap U$, with $c(0) =p$, and set $c(1) = (x_1^1, \ldots ,x_{n-1}^1, z^1)$. Then there is a constant ${\rm Cte}$ depending solely on $\epsilon$ such that the following condition is always verified: whenever the length of $c$ exceeds ${\rm Cte}$, we have the estimate $\vert z^1 \vert < \vert z^0 \vert /2$.
It follows immediately from the proof of Theorem \[introducelabel4\] concerning the foliation ${{\mathcal F}}$. More precisely, it was shown that for the analogue statement where the foliation ${\mathcal{H}}_{{\mathcal{D}}}^{\theta}$ is replaced by the foliation ${\mathcal{H}}_{{{\mathcal F}}}$ it suffices to choose ${\rm Cte}$ as $3\ln(2)/2{\alpha}$. The present statement follows from the fact that inside $U$ the foliation ${\mathcal{H}}_{{{\mathcal F}}}$ is “very close” to ${\mathcal{H}}_{{\mathcal{D}}}$.
One last ingredient is still needed for the proof of Theorem A. Let $l_p^{+,\theta}$ be a trajectory as in Lemma \[suzuki3.3\] and denote by $L_p$ the leaf of ${\mathcal{D}}$ containing $l_p^{+,\theta}$. The idea behind the statement of Theorem A consists of estimating the integral of $dT_L$ over $l_p^{+,\theta}$ where $dT_L$ stands for the time-form induced by $X$ on $L_p$. In Section 4 suitable estimates for this type of integral were obtained in the case of homogeneous vector fields. The estimate is based on the “renormalized time-form” induced on $\Delta_{\infty}$ by the vector field and on the evolution of the distance of the points to $\Delta_{\infty}$ (the “height” of the points). As to the height of points, the preceding lemma provides a suitable control of their evolution over trajectories of ${\mathcal{H}}_{{\mathcal{D}}}^{\theta}$ in the case of non-homogeneous polynomial vector fields. Finally we recall that the foliations induced by $X$ and by $X_d$ on $\Delta_{\infty}$ turn out to coincide and the same holds for the “renormalized time-forms” induced on $\Delta_{\infty}$ by $X$ and by $X_d$.
We are now ready to prove Theorem A.
Consider the foliation ${\mathcal{D}}$ induced by $X$ on ${{\mathbb C}}P(n)$ and let $\Delta_{\infty}$ be as above. Let $V$ denote the given neighborhood of $({\rm Sing}\, ({\mathcal{D}}) \cap \Delta_{\infty}) \cup {\rm Sing}\, (X)$ and fix $\epsilon > 0$. Next choose a neighborhood $U$ of $\Delta_{\infty} \setminus V$ so that the statement of Lemma \[suzuki3.3\] is valid. It is sufficient to prove the theorem for the foliation ${\mathcal{H}}_{{\mathcal{D}}}$ since the adaptations needed to the general case of the foliations ${\mathcal{H}}_{{\mathcal{D}}}^{\theta}$, $\theta \in (-\pi/2 + \epsilon , \pi/2 -\epsilon)$, are clear.
Consider a point $p=(x_1^0, \ldots ,x_{n-1}^0, z^0) \in U \setminus V$. Denote by $l_p^+$ (resp. $L_p$) the semi-trajectory of ${\mathcal{H}}_{{\mathcal{D}}}$ initiated at $p$ (resp. the leaf of ${\mathcal{D}}$ through $p$). Suppose first that $l_p^+$ is entirely contained in $U$. To explain the structure of the proof of our theorem, we shall first prove that the preceding assumption contradicts the fact that the vector field $X$ is complete. For this we are going to show that the integral of the time-form $dT_L$ induced by $X$ on $L_p$ over $l_p^+$ is convergent. Since it clearly accumulates on $\Delta_{\infty}$ (in particular $l_p^+$ leaves every compact set contained in $L_p$) the convergence of the mentioned integral contradicts the completeness of $X$. Let us also point out that our claim reduces to Theorem \[introducelabel4\] in the case of homogeneous vector fields.
To adapt the proof of Theorem \[introducelabel4\] to the present case where $X$ is not homogeneous we proceed as follows. The choice of the coordinates $(x_1, \ldots, x_{n-1}, z) = (\underline{x}, z)$ allows us to write $X_d$ in the form $$X_d = z^{1-d} [F_1 (\underline{x}) \partial /\partial x_1 + \cdots + F_{n-1} (\underline{x}) \partial /\partial x_{n-1} + z H (\underline{x}) \partial /\partial z]$$ whereas the vector field $X$ becomes $$X = z^{1-d} [ F_1^{\ast} (\underline{x}, z) \partial /\partial x_1 + \cdots + F_{n-1}^{\ast} (\underline{x}, z) \partial /\partial x_{n-1} +
z H^{\ast} (\underline{x}, z) \partial /\partial z \!] .$$ Besides the coefficients $F_i, \, F_i^{\ast}$, $i=1, \cdots , n-1$, (resp. $H_i, \, H_i^{\ast}$) are related by the formulas $$F_i^{\ast} (x_1, \ldots , x_{n-1}, z) - F_i (x_1, \ldots , x_{n-1}) = z P_i (x_1, \ldots , x_{n-1}, z)$$ (resp. $H^{\ast} (x_1, \ldots , x_{n-1}, z) - H (x_1, \ldots , x_{n-1}) = z Q (x_1, \ldots , x_{n-1}, z)$) where $Q, \, P_i$ are polynomials in the variables in question. Next note that the time-form $dT_L$ is given by $$dT_L = \frac{z^{d-1} }{F_1^{\ast} (x_1, \ldots , x_{n-1}, z)} \, dx_1 = \cdots = \frac{z^{d-1} }{F_{n-1}^{\ast} (x_1, \ldots , x_{n-1}, z)} \, dx_{n-1} \, .$$ Now since $U$ does not intersect the singular set of ${\mathcal{D}}$, we can suppose without loss of generality that $F_1^{\ast} (x_1, \ldots , x_{n-1}, z)$ is bounded from below by a positive constant $\beta$ in $U$, otherwise we replace $F_1^{\ast}$ by a suitable $F_i^{\ast}$. This last estimate combined to Lemma \[suzuki3.3\] then shows that the integral of $dT_L$ over $l_p^+$ is bounded by simply repeating the calculations performed in the proof of Theorem \[introducelabel4\].
We are then led to conclude that the semi-trajectory $l_p^+$ must intersect the neighborhood $V$ of ${\rm Sing}\, ({\mathcal{D}}) \cap \Delta_{\infty}$ regardless of how small is $V$. In particular it may happen that $l_p^+$ accumulates on singular points of ${\mathcal{D}}$ lying in $\Delta_{\infty}$. In this case the integral of $dT_L$ over $l_p^+$ cannot be bounded without additional conditions. Fortunately to establish Theorem A it is not needed to keep track of the amount of “time” that $l_p^+$ spend [*inside*]{} $V$ but rather of the amount of time that $l_p^+$ spend away from $V$. To be more precise let us prove the following:
[*Claim*]{}. The distance between the trajectory $l_p^+$ and the hyperplane $\Delta_{\infty}$ cannot have a minimum unless this minimum is [*zero*]{}. In this latter case the intersection point $l_p^+ \cap \Delta_{\infty}$ is never reached by the flow of $X$.
Before starting the proof of the claim, it is convenient to make some general comments regarding the possibility of having a point $q$ in $l_p^+ \cap \Delta_{\infty}$. A first case where this may happen arises from the definition of “leaf” given in Section 2, borrowed from [@marco2]. According to this definition the leaf $L_p$ of ${\mathcal{D}}$ may contain a singular point of ${\mathcal{D}}$ lying in $\Delta_{\infty}$. In fact, in this case, a local branch of $L_p$ about $q$ defines an irreducible separatrix for ${\mathcal{D}}$ at $q$. It is then natural to think of $q$ as belonging to $l_p^+$. More generally, it may happen that $l_p^+$ converges to a point $q$ lying in $\Delta_{\infty}$ whether or not $q$ belongs to $L_p$. With a small abuse of notation, the point $q$ may be thought of as belonging to $l_p^+$. In all these cases the statement of Theorem A is obvious: the completeness of $X$ implies that the integral of $dT_L$ over a local branch of $l_p^+$ converging to $q$ is infinite. So $l_p^+$ enters every given neighborhood of $q$ and remains inside “forever”. The statement then follows from observing that $q$ must be a singular point of ${\mathcal{D}}$ since $\Delta_{\infty}$ is invariant by ${\mathcal{D}}$.
A further reduction in the proof of Theorem A is possible even though not strictly needed. Namely, with the above notations, we can suppose that (a local branch of) $l_p^+$ never converges to a point $q$ that is singular for $X$ (and in particular for ${\mathcal{D}}$). In fact, if this point belongs $\Delta_{\infty}$ then the theorem results immediately as previously seen. Similarly, if $q \in {\rm Sing}\, (X) \setminus \Delta_{\infty}$, then the theorem follows from the standard results on existence and uniqueness of solutions for regular ordinary differential equations.
[*Proof of the Claim*]{}. Given what precedes, let us suppose for a contradiction that $q$ is a point of minimum for the mentioned distance and that $q$ lies in ${{\mathbb C}}P(n) \setminus \Delta_{\infty}$. First we are going to prove that the point $q$ must belong to the domain of definition of the coordinates $(x_1, \ldots , x_{n-1}, z)$. Since $X$ is not homogeneous and $q$ is not in $\Delta_{\infty}$, this assertion is not an immediate consequence of Lemma \[lineatinfinity\]. Thus, in order to prove it, suppose that $c: [0,1) \rightarrow l_p^+$ is a local parametrization of $l_p^+$ with $\lim_{t\rightarrow 1^-} c(t) =q$. Setting $c(t) = (x_1 (t), \ldots ,x_{n-1} (t), z(t))$, it follows that $z(t)$ is locally bounded at $q$. If, in addition, $(x_1 (t), \ldots ,x_{n-1} (t))$ leaves the domain of definition of coordinates $(x_1, \ldots , x_{n-1}, z)$, then by using standard coordinates of ${{\mathbb C}}P(n)$ whose domain contains $\Delta_{\infty}$, it immediately follows from the bounded character of $z(t)$ that $\lim_{t\rightarrow 1^-} c(t) =q$ actually belongs to $\Delta_{\infty}$. In this case the theorem follows as already shown.
Summarizing the above discussion, we can suppose that $q = (q_1, \ldots, q_n)$ is a regular point for $X, \, {\mathcal{D}}$ belonging to the domain of definition of the coordinates $(x_1, \ldots , x_{n-1}, z)$ and verifying $q_n \neq 0$. A final contradiction can now be obtained as follows. Let $\Phi (T) = (\Phi_1 (T), \ldots , \Phi_n (T))$ be a local parametrization of $L_p$ about $q$ ($\Phi (0) =q$). Since $q$ is a regular point for $X$, the holomorphic map $T \mapsto \Phi_n (T) \in {{\mathbb C}}$ is not constant and hence it has isolated critical points. From the local inversion theorem and the fact that an open set remains open when an [*isolated point of its boundary*]{} is added to it, it follows that $T \mapsto \Phi_n (T)$ is an open map what in turn contradicts the assumption that $\vert \Phi_n \vert$ has a (positive) local minimum at $T=0$. The claim is proved.
To finish the proof of Theorem A consider now the semi-trajectory $l_p^+$. The above discussion shows that $l_p^+$ accumulates on $\Delta_{\infty}$, in particular $l_p^+$ leaves compact sets of $L_p$. The completeness of $X$ then implies $$\int_{l_p^+} dT_L = \infty \; .$$ Consider a decomposition $l_p^+ = c_1 \ast c_2 \ast \cdots$ of $l_p^+$ in finitely or infinitely many paths such that $c_k$ is contained in $U$ for $k$ odd and $c_k$ is contained in $V$ for $k$ even. The statement is now reduced to prove that $$\sum_{k=0}^{\infty} \left[ \int_{c_{2k+1}} dT_L \right] < \infty \, .$$ The last estimate however follows from the same argument employed above in the case where $l_p^+$ was entirely contained in $U$. It suffices to observe that the claim guarantees that $\vert c_{2(k+1) +1} (0) \vert < \vert c_{2k+1} (1) \vert$. The theorem is proved.
Let us now establish Theorem A’.
Consider again a fixed point $p$ and let $\Phi_p : {{\mathbb C}}\rightarrow L_p$ be given by $\Phi_p (T) = \Phi (T, p)$ where $L_p$ stands for the leaf of ${\mathcal{D}}$ through $p$. In the affine coordinates $(x_1, \ldots ,x_{n-1}, z)$ the map $\Phi_p$ becomes $(\Phi_1 (T), \ldots , \Phi_n (T))$. In particular this allows us to define the Abelian form $\eta$ on ${{\mathbb C}}$ by letting $\eta = - \Phi_n^{'} dT / \Phi_n$. Next, if the oriented foliation ${\mathcal{H}}$ is restricted to $L_p$, then we can consider the corresponding pulled-back oriented foliation $\Phi_p^{\ast} {\mathcal{H}}$ on ${{\mathbb C}}$. In this direction we have:
[*Claim 1*]{}: The oriented foliation $\Phi_p^{\ast} {\mathcal{H}}$ coincides with the real (positive) foliation induced by $\eta$.
[*Proof of Claim 1*]{}. Consider a point $q = \Phi_p (T_0) \in L_p$ that is regular for ${\mathcal{H}}$. The direction of ${\mathcal{H}}$ at $q$ is determined by the negative of the gradient of the “height” function $(x_1, \ldots ,x_{n-1}, z) \mapsto \Vert z \Vert$ restricted to $L_p$. In the coordinate $T$ this function is simply $T \mapsto \Vert \Phi_n (T) \Vert$. The gradient direction of this latter function is determined by the condition that $\Phi_n^{'} (T_0) (T-T_0)$ must be aligned with $\Phi_n (T_0)$. This amounts to saying that the direction of $\Phi_p^{\ast} {\mathcal{H}}$ at $T_0$ is nothing but the positive real direction of $\eta$.
To abridge notations the foliation $\Phi_p^{\ast} {\mathcal{H}}$ will be denoted by $\{ {\rm Arg}\, \eta =0\}$. More generally the pull-back by $\Phi_p$ of the foliations ${\mathcal{H}}^{\theta}$ coincide with $\{ {\rm Arg}\, \eta =\theta\}$, in particular $\{ {\rm Arg}\, \eta =\pi/2\}$ is the foliation orthogonal to $\{ {\rm Arg}\, \eta =0\} = \Phi_p^{\ast} {\mathcal{H}}$.
The separating curve $c_0$ to be chosen is given by the trajectory of $\{ {\rm Arg}\, \eta =\pi/2\}$ through $T_0$ i.e. a “leaf” of ${\mathcal{H}}^{\theta}$ for $\theta = \pi/2$. Geometrically $\Phi_p (c_0)$ is the curve determined in $L_p$ by the intersection of $L_p$ itself with the real hypersurface $\{ \Vert z \Vert =
\vert \Phi_n (T_0) \vert \}$. This curve may be closed. Next we choose the component $\mathcal{U}^+$ of ${{\mathbb C}}\setminus c_0$ that corresponds to the saturated of $T_0$ by the spray of trajectories of $\{ {\rm Arg}\, \eta =\theta\}$ issued from $T_0$ with $\theta \in (-\pi/2 ,\pi/2)$. To check that $\mathcal{U}^+$ is unbounded just notice that a trajectory $l_p^{+, \theta} \subset L_p$, $\theta \in (-\pi/2 ,\pi/2)$, issued from $p$ will, by construction, accumulate on $\Delta_{\infty}$ unless it accumulates on a singularity of $X$ (in ${{\mathbb C}}^n$). The statement being clear in the latter case, let us consider that $l_p$ accumulates on $\Delta_{\infty}$. In particular it leaves every compact set contained in $L_p$. Since $X$ is complete, it results that the integral of the corresponding time-form over $l_p^{+, \theta}$ is unbounded. Next note that the preimage of $l_p^{+, \theta}$ by $\Phi_p$ is the trajectory of $\{ {\rm Arg}\, \eta =\theta\}$ issued from $T_0$. Furthermore the preimage by $\Phi_p$ of the time-form induced on $L_p$ is nothing but the canonical form $dT$ on ${{\mathbb C}}$. Thus the integral of the time-form over segments of $l_p^{+, \theta}$ is equal to the integral of the form $dT$ over corresponding segments of the mentioned trajectory. It then follows that the trajectory in question must leave every compact set contained in ${{\mathbb C}}$ what shows that $\mathcal{U}^+$ is unbounded.
Summarizing to show that $\mathcal{U}^+$ satisfies all the conditions in the statement it only remains to check that $$\lim_{r \rightarrow \infty} \frac{ {\rm Meas}\, (\mathcal{T}_V \cap B_r)}
{{\rm Meas}\, (\mathcal{U}^+ \cap B_r)} =1 \, . \label{thelastone}$$
To begin with, note that $\eta$ is holomorphic in $\mathcal{U}^+$ since $\Phi_n (T)$ never reaches $0 \in {{\mathbb C}}$. Furthermore the trajectories of ${\mathcal{H}}^{\theta}$, $\theta \in (-\pi/2 ,\pi/2)$, approach $\{ z=0\}$. These trajectories, in fact, remain in a compact part of the domain of definition of the coordinates $(x_1, \ldots ,x_{n-1}, z)$ since the “infinity” of $\{ z=0\}$ consists of poles with residue equal to $1$ for the abelian form $\omega_1$ in Section 3, cf. Lemma \[lineatinfinity\]. The existence of poles with residue equal to $1$ implies that on a neighborhood of this “plane at infinity” the trajectories of $\{ {\rm Arg}\, \eta =\pi/2\}$ are closed curves and the trajectories of ${\mathcal{H}}^{\theta}$ for $\theta \in (-\pi/2 ,\pi/2)$ point outward these closed curves. The preceding claim then becomes clear. As a consequence of this, we conclude that the absolute value of the coefficient of $\eta$ is uniformly bounded in $\mathcal{U}^+ \setminus \mathcal{T}_V$ since away from $\mathcal{T}_V$ we can choose the denominator of $\omega_1 = -H/F$ to be bounded from below by a positive constant. Next observe that $\eta$ defines a singular flat structure on $\mathcal{U}^+$ and for this flat structure the trajectories of $\{ {\rm Arg}\, \eta =\theta\}$ are geodesics (“straight lines”). This leads us to
[*Claim 2*]{}: Given $\varepsilon >0$, there is $\delta >0$ such that the saturated $\mathcal{U}_{\delta}^+$ of $T_0$ by trajectories of $\{ {\rm Arg}\, \eta =\theta\}$ for $\theta \in (-\pi/2 +\delta ,\pi/2 - \delta)$ verifies $$\liminf_{r \rightarrow \infty} \frac{ {\rm Meas}\, [(\mathcal{U}_{\delta}^+ \cap B_r) \cup (\mathcal{T}_V \cap (\mathcal{U}^+ \setminus \mathcal{U}^+_{\delta}))]}
{{\rm Meas}\, (\mathcal{U}^+ \cap B_r)} > 1- \varepsilon \, .$$
[*Proof of Claim 2*]{}. The statement would be clear if the flat structure induced by $\eta$ were the standard flat structure of ${{\mathbb C}}$. Consider an arc of circle $S_{r_0}$ (about $T_0$) of radius $r_0$ whose interior contains no singular points of $\eta$. Consider also two trajectories $l_{\theta_1}, \, l_{\theta_2}$ issued from $T_0$. Since $X$ is complete these trajectories intersect $S_{r_0}$ at points $T_1, T_2$. Because $\eta$ is closed, the integral of $\eta$ over the boundary of the triangle whose sides are the segments of $l_{\theta_1}, \, l_{\theta_2}$ delimited by $T_0$ and $T_1, T_2$ and the corresponding arc of $S_{r_0}$ determined by $T_1, T_2$ equals zero. Finally since the coefficient of $\eta$ is uniformly bounded (and bounded from below if we stay away from its singular points), we conclude that the length of the arc of $S_{r_0}$ determined by $T_1, T_2$ is bounded by ${\rm Const} r_0 \vert \theta_1 - \theta_2 \vert$ for every pair $\theta_1, \theta_2$. From this we obtain the desired estimate provided that $S_{r_0}$ contains no singularities of $\eta$ in its interior.
To finish the proof of the claim, we need to consider the effect of the singularities of $\eta$. These singularities are of saddle type since $\eta$ is holomorphic on $\mathcal{U}^+$. For every singular point of $\eta$ we consider a disc of uniform radius about the corresponding point in $L_p$. In the complement of the union of these discs $\omega_1$, and therefore $\eta$ in the coordinate $T$, is bounded from below by a positive constant. The claim will be proved if the union of these discs in the coordinate $T$ has area less than $\varepsilon r/2$ for $r$ large. In fact, in the complement of this union $\eta$ is bounded from below by a positive constant and from above by the previous constant so that the preceding argument can be applied in finitely many regions of a ball $B_r$. Finally to justify the previous claim note that, in order to prove the desired estimate, we only need to consider those discs about points in $L_p$ that lie in the complement of $V$. It then follows that in these discs the norm of $X$ is bounded from below by a positive constant so that their size in the coordinate $T$ is uniformly bounded. Besides the distance in the leaf $L_p$ between every two discs as before is bounded from below by a positive constant. Though this property is not directly reflected in the coordinate $T$ since the norm of $X$ increases (i.e. $X$ becomes “faster”), the size of the corresponding neighborhoods reduces proportionally to the increase of the norm of $X$. This quickly leads to the desired conclusion and establishes the claim.
In view of Claim 2 to end the proof of Theorem A’ it suffices to show that $$\lim_{r \rightarrow \infty} \frac{ {\rm Meas}\, (\mathcal{T}_V \cap B_r \cap \mathcal{U}_{\delta}^+)}
{{\rm Meas}\, (\mathcal{U}_{\delta}^+ \cap B_r)} =1 \, ,$$ for fixed positive $\delta$. To do this consider $r$ given. Next observe that every for $\theta \in (-\pi/2 +\delta ,\pi/2 - \delta)$ the corresponding trajectory $l_{\theta}$ of $\{ {\rm Arg}\, \eta =\theta\}$ issued from $T_0$ intersects the boundary $\partial B_r$ of $B_r$ since $X$ is complete. Let $T_{\theta, r}$ be this intersection point and denote by $l_{\theta, r}$ the segment of $l_{\theta}$ delimited by $T_0$ and $T_{\theta, r}$. According to Theorem A, there is uniform constant ${\rm Cte}$ (depending neither on $\theta$ nor on $r$) such that the length of the segments of $l_{\theta, r}$ corresponding to those instants where $\Phi (T)$ remains away from $V$ is bounded by ${\rm Cte}$ whereas the length of $l_{\theta, r}$ goes to infinity as $r \rightarrow \infty$. The statement of Theorem A’ now results from a standard application of Fubini’s theorem.
Complete polynomial vector fields on ${{\mathbb C}}^n$ with simple singularities at infinity
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In this section we shall give an application of Theorem \[maintheo\] that cannot be obtained from Theorem \[introducelabel4\] alone. Let $X$ be a complete polynomial vector fields on ${{\mathbb C}}^n$ and denote by ${\mathcal{D}}$ its associated foliation. Recall that we make no distinction between ${\mathcal{D}}$ viewed as a foliation on ${{\mathbb C}}^n$ and ${\mathcal{D}}$ viewed as a foliation on ${{\mathbb C}}P(n)$. Again $X_d$, $d \geq 2$, will denote the homogeneous component of highest degree of $X$. The foliation associated to $X_d$ is denoted by ${{\mathcal F}}$ and can also be viewed as a foliation on both ${{\mathbb C}}^n$ or ${{\mathbb C}}P(n)$. Recall that the singularities of ${\mathcal{D}}$ in the hyperplane at infinity $\Delta_{\infty}$ are “simple” in the sense of Conditions 1 and 2 given in the Introduction (just before the statement of Theorem B). It then follows that they are isolated inside $\Delta_{\infty}$ (but maybe not inside ${{\mathbb C}}P(n)$). Furthermore note that if a non-constant polynomial $P$ divides $X_d$ it must also divide $X$. Otherwise the curve induced on $\Delta_{\infty}$ by $\{ P=0\}$ would contain singularities of ${\mathcal{D}}$ whose linear part is degenerate what is impossible. In turn this implies that the singular sets of ${\mathcal{D}}$ and of ${{\mathcal F}}$ coincide on $\Delta_{\infty}$. Furthermore $X_d$ must have a singular set of codimension at least $2$ since the singular set of $X$ has codimension two or greater.
Let us also observe that the restriction of ${\mathcal{D}}$ to $\Delta_{\infty}$ coincides with the foliation ${{\mathcal F}}_{\infty}$ induced by ${{\mathcal F}}$. Also if $q\in \Delta_{\infty}$ is a (necessarily common) singular point of ${\mathcal{D}}, \, {{\mathcal F}}$, then the corresponding linear parts of these foliations at $q$ turn out to coincide.
\[Cstar1.1\] With the definition of leaf given in Section 3 (borrowed from [@marco2]), the leaves of ${{\mathcal F}}_{\infty}$ are either rational curves or Riemann surfaces uniformized by ${{\mathbb C}}$.
We need to show that the leaves of ${{\mathcal F}}_{\infty}$ cannot be hyperbolic Riemann surfaces. Since ${{\mathbb C}}P(n)$ has a Kähler structure, it follows from the main result of [@marco2] that the set of parabolic leaves of ${\mathcal{D}}$ is a pluri-polar set unless it coincides with the whole space. Since the leaves of ${\mathcal{D}}$ contained in ${{\mathbb C}}^n$ are clearly parabolic, we conclude that the leaves of ${\mathcal{D}}$ contained in $\Delta_{\infty}$ must also be parabolic (or rational). The latter leaves however are precisely the leaves of ${{\mathcal F}}_{\infty}$.
The next step is to consider the foliation ${{\widetilde{\mathcal F}}}$ induced on the manifold $M$ by ${{\mathcal F}}$ as in Sections 3, 4 and 5. In particular the foliation ${{\widetilde{\mathcal F}}}_{\infty}$ is naturally identified with ${{\mathcal F}}_{\infty}$. In view of the existence of the projections ${{\mathcal P}}_0, \, {{\mathcal P}}_{\infty}$ introduced in Section 3, Lemma \[Cstar1.1\] implies that no leaf of ${{\widetilde{\mathcal F}}}$ is hyperbolic. Note that this conclusion cannot directly be derived from the vector field $X_d$ since $X_d$ need not be complete (though it is semi-complete).
The next proposition is crucial for the proof of Theorem B. In turn its proof relies heavily on Theorem \[maintheo\].
\[hyperbolicleaves\] There exists a singularity of ${{\widetilde{\mathcal F}}}_{\infty}$ providing a sink singularity for ${\mathcal{H}}$ (resp. ${\mathcal{H}}^{\theta}$) restricted to a generic leaf of ${{\widetilde{\mathcal F}}}_{\infty}$.
Let us suppose for a contradiction that the statement is false. In view of the material in Section 3, it follows that the trajectories of ${\mathcal{H}}$ have no future endpoint. Therefore these trajectories have all infinite length and this will be exploited in the proof. To begin with note that $L_q$ is not contained in a rational curve otherwise ${{\widetilde{\mathcal F}}}_{\infty}$ would be a rational pencil on $\Delta_{\infty}$. In this case the foliation ${\mathcal{H}}$ would have sink singularities when restricted to a generic leaf of ${{\widetilde{\mathcal F}}}_{\infty}$.
Our strategy to obtain a contradiction will consist of showing that a “generic” leaf of ${{\widetilde{\mathcal F}}}_{\infty}$ cannot be a “parabolic” (i.e. non-hyperbolic) Riemann surface. For this consider $q \in \mathcal{P}^{-1}_{\infty}(p)$ as in the statement of Theorem \[maintheo\] and denote by $L_q$ the leaf of ${{\widetilde{\mathcal F}}}$ through $q$. In particular ${{\mathcal P}}_{\infty} (L_q) =L_{\infty}$. Now note that ${{\widetilde{X}}}_d$ is regular over $L_q$. Since ${{\widetilde{X}}}_d$ is semi-complete, this leaf is recovered by an open set ${\bf U} \subseteq {{\mathbb C}}$ which is the domain of definition of the semi-complete flow on $L_q$. In fact, the assumption of having a semi-complete vector field is equivalent to saying that ${\bf U}$ is a maximal domain of definition for the solution $\phi$ of the equation associated to this vector field with initial condition at $q$. Now the lift of $l_q$ in ${\bf U}$ has finite length for the natural euclidean metric of ${{\mathbb C}}$. Indeed, this length is nothing but the integral $\int_{l_q} dT$. Hence the mentioned lift converges to a point $t_0$ in the boundary of ${\bf U}$.
Recalling that $L_q$ must be a non-hyperbolic Riemann surface, and noting that it cannot be contained in a rational curve, it follows that ${\bf U} = {{\mathbb C}}\setminus \{ t_0\}$. In particular, $L_q$ cannot be compactified in an elliptic curve. Thus $L_q$ is actually not contained in any compact curve. Consider now the plane ${{\mathbb C}}$ equipped with the coordinate $t$. In particular the canonical form $dt$ coincides with the pull-back by $\phi$ of the time-form induced on $L_q$ by $X$. Also denote by ${\mathcal{H}}_t$ the lift of the foliation ${\mathcal{H}}$ (restricted to $L_q$) to the ${{\mathbb C}}$-plane. ${\mathcal{H}}$ is naturally a foliation defined on ${\bf U} = {{\mathbb C}}\setminus \{ t_0\}$. Yet we have the following claim whose meaning is fully explained below.
[*Claim*]{}. The point $t_0$ represents a “sink singularity” for ${\mathcal{H}}_t$.
[*Proof of the Claim*]{}. Clearly $L_q$ possesses a cylindrical end. In other words, the map $\phi$ allows us to realize a punctured neighborhood of $t_0 \in {{\mathbb C}}$ as an end of $L_q$. Next consider a point $q' = \phi (t')$ for $t'$ near $t_0$. The trajectory of ${\mathcal{H}}$ through $q'$ is infinite and thus the integral of the corresponding time-form over this trajectory must converge again to a point lying in the boundary of ${\bf U} = {{\mathbb C}}\setminus \{ t_0\}$. We then conclude that the mentioned integral converges to $t_0$. In the ${{\mathbb C}}$-plane equipped with the coordinate $t$, the preceding translates into the fact that the integral of the canonical form $dt$ over the lift (by $\phi$) of the ${\mathcal{H}}$-trajectory through $q'$ converges to $t_0$. This lift however is nothing but the trajectory of ${\mathcal{H}}_t$ through $t'$, it then follows that the trajectory in question converges to $t_0$ and this is the contents of the claim.
The above proof needs further comments. As mentioned ${\mathcal{H}}_t$ is the foliation induced on the plane ${{\mathbb C}}$ by the $1$-form $\phi^{\ast} \omega_1$ which is meromorphic on ${\bf U} = {{\mathbb C}}\setminus \{ t_0\}$ but not necessarily at $t_0$. Indeed, $t_0$ may be an essential singularity for $\phi^{\ast} \omega_1$. Previously in our discussion of singularities for this type of foliation, the “classification” as saddle, source or sink applied only for meromorphic forms (often quoted as abelian forms) and therefore they need not apply to essential singular points of the corresponding forms. This is why the expression sink singularity was written between quotes in the statement of the claim. Whereas no general discussion of singularities obtained at essential singular points of $1$-forms was carried out, the preceding proof shows that $t_0$ topologically looks like a sink for ${\mathcal{H}}_t$ in the sense that ${\mathcal{H}}_t$-trajectories through nearby points converge to $t_0$. To avoid confusion, this type of situation will be referred to from now on as constituting an [*improper sink*]{} for ${\mathcal{H}}$. Thus in the “time coordinate” $t$, the points $t_0$ is viewed as a “topological sink” for ${\mathcal{H}}_t$ (or for ${\mathcal{H}}$) in the sense discussed above. Nonetheless if we look at the actual ${\mathcal{H}}$-trajectories in the leaf $L_q$, those trajectories previously viewed as “converging to $t_0$” become trajectories of ${\mathcal{H}}$ having infinite length.
Summarizing what precedes, the foliation ${\mathcal{H}}_t$ on the plane ${{\mathbb C}}$ has a unique improper sink singularity, corresponding to $t=t_0$, and no (ordinary) sink singularity. Furthermore all trajectories of ${\mathcal{H}}_t$ converge to $t_0$. Indeed, the integral of the time-form over every trajectory of ${\mathcal{H}}$ converges to a point in the boundary of ${\bf U} = {{\mathbb C}}\setminus \{ t_0\}$ and hence to $t_0$ itself.
To finish the proof of the lemma, we are going to show that the situation described above cannot happen. In fact, let $t^{\ast}_1$ be a [*source singularity of ${\mathcal{H}}_t$*]{}. Note that $t^{\ast}_1$ exists since this type singularity is produced by the intersections of ${{\mathcal P}}_{\infty} (L_q)$ with the hyperplane at infinity $\Delta_{\infty}^{(x,y)}$ in the coordinates $(x,y)$ (i.e. the hyperplane at infinity corresponding to affine coordinates for $\Delta_{\infty}$). Since $\Delta_{\infty}^{(x,y)}$ neither is invariant by ${{\widetilde{\mathcal F}}}_{\infty}$ nor there are singularities of ${{\widetilde{\mathcal F}}}_{\infty}$ lying in $\Delta_{\infty}^{(x,y)}$, it follows that every leaf of ${{\widetilde{\mathcal F}}}_{\infty}$ intersects non-trivially $\Delta_{\infty}^{(x,y)}$. These intersections produce then source singularities for ${\mathcal{H}}_t$. These source singularities also have residue equal to $1$.
Next note that away from saddle-singularities of ${\mathcal{H}}_t$, this foliation can be given a structure of transverse riemannian foliation: just parametrize the trajectories of the orthogonal foliation by the integral of $\phi^{\ast} \omega_1$. Thus there is a region $I_1^{\ast}$ on a small loop $c$ about $t_0$ such that the integral of $\phi^{\ast} \omega_1$ over $I_1^{\ast}$ equals $-1$ (the negative of the residue of $\phi^{\ast} \omega_1$ at $t^{\ast}_1$). In fact, $I_1^{\ast}$ is obtained by the intersections with $c$ of the trajectories emanated from $t^{\ast}_1$ (recalling that they all converge to $t_0$). However, $c$ can be chosen so that $\phi^{\ast} \omega_1$ is bounded on a neighborhood of $c$ and hence the integral of $\phi^{\ast} \omega_1$ over $c$ is well-defined. To derive a final contradiction, it suffices to observe that in the complement of the small disc bounded by $c$, there are infinitely many source singularities $t^{\ast}_1, t^{\ast}_2, \ldots$ for ${\mathcal{H}}_t$. In fact, under this assumption, each of these singularities determine as above a region $I_k^{\ast}$ over which the integral of $\phi^{\ast} \omega_1$ equals $-1$. Finally all these regions $I_1^{\ast}, I_2^{\ast}, \cdots$ are clearly pairwise disjoint what immediately leads to a contradiction with the existence of a bound for $\phi^{\ast} \omega_1$ on a neighborhood of $c$.
Finally it only remains to check the existence of infinitely many source singularities $t^{\ast}_1, t^{\ast}_2, \ldots$ in the complement of the disc (containing $t_0$) and bounded by $c$. Let us then suppose that the number of the mentioned source singularities is finite. It then follows that the solution $\phi$ is bounded in the complement of a compact region of the plane ${{\mathbb C}}$. Hence this solution can be extended to a neighborhood of the infinity in ${{\mathbb C}}$. Now the image of $\infty$ in ${{\mathbb C}}\cup \{ \infty \} \setminus \{ t_0 \}$ under the extension of $\phi$ can be nothing but a singular point of the complete vector field $X$ on the affine ${{\mathbb C}}^3$. Modulo using the standard Remmert-Stein theorem, it follows that $L_q$ contains a local separatrix for an affine singularity $p$ of $X$. In other words, a generic leaf of ${{\widetilde{\mathcal F}}}$ defines a local separatrix of ${{\widetilde{\mathcal F}}}$ at one of its singular points. This possibility can be ruled out since the statement of Theorem B would follow immediately from considering the restriction of $X$ to these separatrices on a neighborhood of the corresponding singular points (where $X$ is actually holomorphic). We are then left with a contradiction that finishes our proof.
For the purpose of proving Theorem B, Proposition \[hyperbolicleaves\] can be summarized by saying that there is a singularity $q \in \Delta_{\infty}$ of ${{\widetilde{\mathcal F}}}$ all of whose eigenvalues $\lambda^q_1, \ldots, \lambda^q_n$ belong to ${{\mathbb R}}_+^{\ast}$. This assertion can, in turn, slightly be improved. In fact, recall that $Q . Y_1$ is a semi-complete vector field where $Y_1$ is a linear vector field with real positive eigenvalues $\lambda^q_1, \ldots , \lambda^q_n$. Suppose first that $Q$ is a rational function whose both pole divisor and zero divisor contain the origin. A simple analysis of the possibilities shows that under these conditions we must have $\{ \lambda^q_1, \ldots , \lambda^q_3 \} \subset {{\mathbb Z}}_+$. The other possibility for $Q$ is that it has a (unique) polar component of degree $d-1$ passing through $q$ (actually coinciding with $\Delta_{\infty}$) and empty zero divisor. In this case the foliation ${{\widetilde{\mathcal F}}}$ has a smooth separatrix transverse to $\Delta_{\infty}$. By restricting $X_d$ to this separatrix the semi-complete condition implies that $d$ must be equal to $2$. Besides the local holonomy of the separatrix in question must be trivial (cf. for example [@guillotThesis]). It then follows that each of the eigenvalues $\lambda^q_1 ,\ldots , \lambda^q_{n-1}$ is a multiple of the eigenvalue $\lambda^q_n$. Hence we can set $\lambda^q_n =1$ and $\lambda^q_1 , \ldots , \lambda^q_{n-1} \in {{\mathbb Z}}_+$. This refined statement will lead us to
\[Cstar3.3\] The set formed by the separatrices of ${\mathcal{D}}$ at $q$ contains non-trivial open sets of ${{\mathbb C}}^n$.
Consider the vector field $X$ (resp. foliation ${\mathcal{D}}$) on a neighborhood of the singularity $q \in \Delta_{\infty}$. What precedes ensures us that the $n$ eigenvalues $\lambda^q_1, \ldots , \lambda^q_3$ of ${\mathcal{D}}$ at $q$ are strictly positive integers. Therefore they belong to the Poincaré domain. In particular the corresponding local vector field is either linearizable or it admits a Poincaré-Dulac normal form. Since the latter possibility was ruled out by assumption, our local vector field must be linearizable at $q$. The statement follows now from the fact that $\{ \lambda^q_1, \ldots , \lambda^q_3 \} \subset {{\mathbb Z}}_+$.
Consider a local leaf of ${\mathcal{D}}$ defining a separatrix $\mathcal{S}$ for ${\mathcal{D}}$ at $q$ as above. Since $\mathcal{S}$ may be singular at $q$, we consider also a local irreducible Puiseux parametrization $\gamma (t)$ for $\mathcal{S}$ where $t$ is defined on a neighborhood of $0 \in {{\mathbb C}}$ and $\gamma (0) =q$. Since $\mathcal{S}$ is invariant by $X$, we consider the pull-back $\gamma^{\ast} X$ by $\gamma$ of the restriction of $X$ to $\mathcal{S}$. The irreducible character of $\gamma$ ensures that $\gamma^{\ast} X$ is holomorphic and semi-complete on a neighborhood of $0 \in {{\mathbb C}}$. Therefore the order of $\gamma^{\ast} X$ at $0 \in {{\mathbb C}}$ belongs to the finite set $\{ 0, 1, 2\}$. Furthermore, we can actually exclude the case in which this order equals zero since otherwise points of $L$ would reach $q \in \Delta_{\infty}$ in finite time and this contradicts the fact that $X$ is complete on ${{\mathbb C}}^n$.
Suppose now that the order of $\gamma^{\ast} X$ is $2$. Modulo adding the point $q$ to the global leaf $L$, it follows that $L$ is a Riemann surface equipped with a complete holomorphic vector field having a quadratic singularity. Thus $L$ must be the Riemann sphere and hence (of closure) compact. This being valid for set of leaves of ${\mathcal{D}}$ containing open sets of ${{\mathbb C}}^n$ (cf. Lemma \[Cstar3.3\]) it follows that ${\mathcal{D}}$ defines a rational pencil on ${{\mathbb C}}^n$ (and in particular it is completely integrable).
Finally consider the case where the order of $\gamma^{\ast} X$ at $0 \in {{\mathbb C}}$ equals $1$. In this case the time-form induced on $L$ by $X$ has a non-trivial period. Besides the restriction of $X$ to the (normalization of) the closure of $L$ contains a singularity so that the (closure of) $L$ cannot be an elliptic curve. It then follows that $L$ cannot be an orbit of type ${{\mathbb C}}$. Now it follows from Lemma \[Cstar3.3\] and from Suzuki’s theorem [@suzuki] that the generic orbit of $X$ has type ${{\mathbb C}}^{\ast}$. A theorem due to Brunella then guarantees the existence of a non-constant first integral $R$ for ${\mathcal{D}}$, cf. [@marco1].
To conclude that $X$ is completely integrable, i.e. that ${\mathcal{D}}$ is a rational pencil we shall proceed by induction on the dimension $n$. Indeed it will suffice to check the theorem for $n=3$ since the induction step will then become clear. Therefore let us fix a point $p \in {{\mathbb C}}^n$ that is regular for $X$. Denote by $L_p$ the leaf of ${\mathcal{D}}$ through $p$ and let $S$ stand for the surface level of $R$ containing $p$. The preceding argument also holds for the restriction of $X$ to $S$. In fact, consider the foliation ${\mathcal{D}}_S$ obtained on $S$ by restricting ${\mathcal{D}}$. The assumptions made about the structure of the singularities of ${\mathcal{D}}$ lying in $\Delta_{\infty}$ automatically implies similar assumptions for the singularities of ${\mathcal{D}}_S$ lying in (the desingularization of) $S \cap \Delta_{\infty}$. In other words, we can assume without loss of generality that the “generic” orbit of $X$ [*restricted to the affine part of $S$*]{} still is of type ${{\mathbb C}}^{\ast}$. In fact the results of Suzuki are valid for Stein manifolds. Similarly the above mentioned theorem of Brunella also holds for the restriction of $X$ to $S$. It means that ${\mathcal{D}}_S$ also possesses a non-constant rational first integral $R_2$. However, for $n=3$, $S$ is an algebraic surface of dimension $2$. The existence of $R_2$ then implies that the restriction of ${\mathcal{D}}$ to $S$ is a pencil in a natural sense. Thus the closure of $L_p$ is, in fact, an algebraic curve (possibly singular). Because the restriction of $X$ to this compact curve contains a singular point it must be a rational curve, modulo normalization. Summarizing the closure of $L_p$ is a (possibly singular) rational curve. The theorem is proved when $n=3$. The general case follows easily by induction.
\[tobeincluded\]
Throughout this paper, the affine coordinates $(x,y,z)$ (as well as their higher dimensional versions) used in the construction of $M$ where generically chosen in the sense that $\Delta_{\infty}^{(x,y)}$ remained away from the singularities of ${{\widetilde{\mathcal F}}}_{\infty}$. In particular, in the context of Theorem B, this was useful to establish Proposition \[hyperbolicleaves\] guaranteing the existence of a singular point all of whose eigenvalues are strictly positive. This argument will be repeated in the context of Theorem C to be proved in Section 7.1. The point we want to make here is that further information, such as the existence of a second singularity with similar property, may be obtained by choosing the coordinates $(x,y,z)$ so as to make $\Delta_{\infty}^{(x,y)}$ to pass through the first singularity having only strictly positive eigenvalues.
The above mentioned choice of coordinates $(x,y,z)$ is particularly useful because, under very general conditions, the “source character” of points lying in $\Delta_{\infty}^{(x,y)}$ may cancel out the “sink character” of the singularity in question. In other words, we can avoid the singularity fixed in the beginning of producing “future endpoints” for the trajectories of ${\mathcal{H}}$. If we know that the leaves of ${{\widetilde{\mathcal F}}}_{\infty}$ are parabolic Riemann surfaces, it then follows the existence of some other point in $\Delta_{\infty}$ yielding a sink singularity for ${\mathcal{H}}$
Theorem C and Halphen vector fields
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This section contains the proof of Theorem C along with a discussion of some of the results obtained in [@guillotIHES] with proofs obtained through the methods developed in the course of this work. The results presented here about Halphen vector fields are not new and often less precise than the original theorems of A. Guillot so that they can be regarded simply as further illustrations of our techniques. Nonetheless the setting introduced here makes sense for much more general vector fields transverse to singular fibrations. In particular we plan to apply similar ideas to some Painlevé equations as well as to other classical equations including several Chazy’s equations. This discussion will also motivate a conjecture that would greatly enhance the applicability of all these ideas.
Guillot’s lattices of quadratic vector fields : examples and results
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This paragraph concerns an application of our techniques to the work of A. Guillot in [@guillotFourier], [@guillotIHES] (cf. also [@guillotThesis] for an introduction to both papers). Let us place ourselves in the context of [@guillotFourier]. We are therefore dealing with semi-complete (homogeneous) quadratic vector fields $X^2$ with isolated singularities defined on ${{\mathbb C}}^{n}$. The foliation ${{\widetilde{\mathcal F}}}$ associated to $X^2$ on $M$ leaves $\Delta_{\infty}$ (resp. $\Delta_0$) invariant. The pole divisor (resp. zero divisor) of the lift to $M$ of $X^2$ consists of $\Delta_{\infty}$ (resp. $\Delta_0$) with multiplicity exactly $1$. Finally ${{\widetilde{\mathcal F}}}$ has exactly $2^n -1$ singularities on $\Delta_{\infty}$ (resp. $\Delta_0$) and all of them possess $n$ eigenvalues different from [*zero*]{}.
Let us denote by $p_1, \ldots , p_{2^n -1}$ (resp. $q_1, \ldots , q_{2^n -1}$) the singularities of ${{\widetilde{\mathcal F}}}$ in $\Delta_0$ (resp. their dual singularities in $\Delta_{\infty}$). Following [@guillotFourier], [@guillotThesis] convention, the eigenvalues of ${{\widetilde{\mathcal F}}}$ at $p_i$ are $1, u_1^i , \ldots , u_{n-1}^i$ where $1$ is the eigenvalue corresponding to the radial direction. According to Guillot, for a semi-complete vector field $X^2$ as above, $u_1^i, \ldots , u_n^i$ are all integers. Besides setting $\xi_i = u_1^i \times \cdots \times u_{n-1}^i$, $X^2$ yields a solution for the equation $$\sum_{i=1}^{2^n-1} \frac{1}{\xi_i} = (-1)^{n+1} \, . \label{egyptianfraction}$$ In particular the problem of precisely classifying vector fields as above is naturally related to these egyptian fractions. In general this problem is quite intricate as attested by the multitude of interesting examples presented in the above mentioned works. Our contribution to this problem begins with the following lemma:
\[Guillotlattice1.1\] The leaves of a vector field in Guillot lattice having no dicritical singularity at infinity are hyperbolic Riemann surfaces (where by dicritical it is meant that all the $n$ eigenvalues of the singularity are positive real).
If there is no dicritical singularity in $\Delta_{\infty}$, there cannot exist a future endpoint for the trajectories of ${\mathcal{H}}, \, {\mathcal{H}}^{\theta}$, $\theta \in (-\pi/2, \pi/2)$. In other words, the trajectories of the foliations $\mathcal{H}^{\theta}$ must be infinite. Thus we can apply Theorem \[maintheo\] and the argument of Proposition \[hyperbolicleaves\] to conclude the statement.
The converse of Lemma \[Guillotlattice1.1\] does not hold. A particularly interesting example being provided by Halphen vector fields (defined on ${{\mathbb C}}^3$). The dynamics and geometry of these vector fields are beautifully described in [@guillotIHES], the reader is referred to this paper for background information. From our point of view, these vector fields provide an interesting example illustrating some aspects of our discussion.
First we note that Halphen vector fields possess dicritical singularities in $\Delta_{\infty}$ (as well as in $\Delta_0$). Yet, under natural additional conditions, they are semi-complete and possess leaves that are isomorphic (as Riemann surface) to the unit disc. This deserves some further comments. It was seen that the existence of dicritical singularities is a necessary condition for the leaves of ${{\mathcal F}}, \, {{\widetilde{\mathcal F}}}$ to be non-hyperbolic Riemann surfaces. Indeed a more precise statement holds: in order to have non-hyperbolic leaves, “most” ${\mathcal{H}}$ trajectories induced on the leaves of ${{\mathcal F}}_{\infty}$ must be of finite length. To have finite length implies that the leaf must have both “future and past” ends and, in turn, he existence of “future ends” leads to the existence of the dicritical singularity in our context. In the case of Halphen vector fields, it can be shown by using the ideas of [@guillotIHES] that the trajectories of ${\mathcal{H}}$ are, indeed, finite and that they tile the corresponding leaves of ${{\mathcal F}}$. This might suggest that the leaves of ${{\mathcal F}}$ ought to be “parabolic Riemann surfaces” what is not the case. Explanation for their hyperbolic character is provided [@guillotIHES]) by carefully exploiting the intimate connection between these equations and the Lie algebra of ${\rm PSL}\, (2, {{\mathbb C}})$. Whereas we can hardly improve on [@guillotIHES]), we shall provide in Section 7.2 an explanation of this fact based on the ideas introduced in this paper.
We are now ready to prove Theorem C.
The fact that the singularity produced by a vector field of special type having no dicritical singularity at infinity cannot be realized in a Kähler manifold follows from the combination of Lemma \[Guillotlattice1.1\] with Brunella’s theorem as used in the proof of Lemma \[Cstar1.1\]. In fact, if there were a Kähler manifold equipped with a holomorphic vector field $X$ exhibiting this type of singularity at a point $p$, the blow-up of $X$ at $p$ would endow the exceptional divisor with a foliation whose leaves are hyperbolic Riemann surfaces. Since the remaining leaves associated to the orbits of $X$ must be parabolic (being globally defined $X$ is automatically complete), a contradiction results.
The same argument applies to the case of (hyperbolic) Halphen vector fields since these vector fields are known to have leaves that are isomorphic to the unit disc, cf. [@guillotIHES] or the discussion carried out below.
Consider now the dual singularity of vector field of special type having no dicritical singularity at infinity. This means that we have a neighborhood of $\Delta_{\infty}$ embedded in a manifold $N$. The argument of Theorem \[maintheo\] that an orbit of the vector field entering this neighborhood will contain trajectories converging to the pole locus and such that the integral of the corresponding time-form is finite. This contradicts the completeness of the vector field in the complement of its pole locus. Again the argument for Halphen vector fields is totally analogous.
Poincaré-type series and Halphen vector fields
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As a matter of fact, Halphen vector fields constitute a particularly remarkable example of semi-complete vector field belonging to Guillot’s lattice whose geometry and dynamics is nicely described in [@guillotIHES]. As mentioned in this section we are going to discuss some aspects of these vector fields from an alternate point of view. Needless to say that the reader is referred to [@guillotIHES] for a fuller discussion.
In the sequel, we shall work on ${{\mathbb C}}^3$. Let ${\mathcal{E}}$ be the radial vector field ${\mathcal{E}}= x\partial /\partial x + y \partial /\partial y + z
\partial /\partial z$ and set $Y = \partial /\partial x + \partial /\partial y + \partial /\partial z$. A quadratic homogeneous vector field $X$ is said to be a Halphen vector field if it satisfies $[Y,X] = 2{\mathcal{E}}$. It then follows that the triplet $Y, {\mathcal{E}}, X$ form a Lie algebra isomorphic to the Lie algebra of ${\rm PSL}\, (2, {{\mathbb C}})$. If $Y,X$ and ${\mathcal{E}}$ are identified to the vector fields they induce on $M$ as in Sections 3 and 4 and expressed in the corresponding coordinates $(x,y,z)$, then ${\mathcal{E}}$ becomes $z^{-1} \partial /\partial z$. In other words $X,Y$ commute up to a vertical vector field. Thus $X$ preserves the projection on $\Delta_{\infty}$ of the foliation induced by $Y$. In other words, the foliation ${{\mathcal F}}_{\infty}^X$ induced by $X$ on $\Delta_{\infty}$ is transverse to the foliation induced by $Y$ on $\Delta_{\infty}$. Since $Y$ was a constant vector field, the foliation it induced on $\Delta_{\infty}$ is simply a linear pencil of rational curves. Summarizing ${{\mathcal F}}_{\infty}^X$ is transverse to a linear pencil of rational curves.
Once the above observation is made, it is easy to work out the structure of ${{\mathcal F}}_{\infty}^X$. It leaves exactly $3$ projective lines $C_1, C_2, C_3$ invariant and these $3$ lines intersect mutually at a radial singularity $P \in \Delta_{\infty}$. Indeed the eigenvalues of ${{\mathcal F}}_{\infty}^X$ at $P$ are $1,1$ whereas the eigenvalues of ${{\mathcal F}}^X$ (the foliation induced by $X$ on all of $M$) at $P$ are $1,1,-1$ (the $-1$ eigenvalue being associated to the direction “$z$”). Also, if $P' \in \Delta_0$ is the singularity of ${{\mathcal F}}^X$ “dual” to $p$, then the eigenvalues of ${{\mathcal F}}^X$ at $p'$ are $1,1,1$. For $i=1,2,3$, let $p_i, q_i$ denote the remaining two singularities of ${{\mathcal F}}^X$ over $C_i$. We assume that $X$ is semi-complete though this is not really indispensable in what follows. According to Halphen’s results revisited by Guillot, it easily follows that the eigenvalues of ${{\mathcal F}}_X$ at $p_i$ (resp. $q_i$) and have the form $-1,-1, m_i$ (resp. $-1, -1, -m_i$), with $m_i \in {{\mathbb N}}^{\ast}$. The converse also holds though it is harder to prove, the argument being based on relating the semi-complete character of the vector field to the “uniform” character of its transversely projective structure (for a proof of the converse see [@guillotIHES]). In any event it is also easy to check that ${{\mathcal F}}_X$ is locally linearizable about $p_i$ (resp. $q_i$). Finally note that the convention used above concerning the order of the eigenvalues of ${{\mathcal F}}_X$ at $p_i$ (resp. $q_i$) is such that the first eigenvalue corresponds to the vertical direction “$z$”, the second to the curve $C_i$ and the third to a direction contained in $\Delta_{\infty}$ and transverse to $C_i$. This convention is slightly different from [@guillotIHES] since the eigenvalue associated to the direction “$z$” is denoted by $-1$ rather than $1$. This change of sign is due to the fact that we consider singularities in $\Delta_{\infty}$ whereas Guillot considers singularities in $\Delta_0$.
The dynamics of ${{\mathcal F}}_{\infty}^X$ is fully encoded in its global holonomy group with respect to a fixed line $\overline{C}$ in the above mentioned linear pencil that is transverse to ${{\mathcal F}}_{\infty}^X$. The preceding also shows that this holonomy group coincides with the subgroup of ${\rm PSL}\, (2, {{\mathbb C}})$ generated by three elements $\xi_1, \xi_2, \xi_3$ which are associated to the local holonomy of each of the three invariant lines. In particular we have $$\xi_1 \xi_2 \xi_3 = \xi_1^{m_1} = \xi_2^{m_2}= \xi_3^{m_3} = {\rm id} \, .$$ In other words, it is a triangle group whose dynamics on $S^2$ is well known: provided that $$\frac{1}{m_1} + \frac{1}{m_2} + \frac{1}{m_3} < 1 \, , \label{stillacrossreference}$$ this group is conjugate to a subgroup of ${\rm PSL}\, (2, {{\mathbb C}})$ and thus it leaves a circle $\Lambda_{\infty} \subset S^2
\simeq \overline{C}$ invariant. Besides each connected component of $\overline{C} \setminus \Lambda_{\infty}$ is invariant by the action. In fact, on these components the action is properly discontinuous whereas it is minimal when restricted to the circle $\Lambda_{\infty}$ itself. Similarly is the case where $1/m_1 + 1/m_2 +1/m_3 =1$, the resulting groups are well-known groups of affine diffeomorphisms associated to special tiles of the plane. When $1/m_1 + 1/m_2 +1/m_3 >1$ the resulting group is indeed finite and thus it is easy to see that all leaves are compact.
[A note of caution in what precedes concerns the fact the quasi-isometric type of the holonomy group does not [*a priori*]{} determine the quasi-isometric type of the leaves of ${{\mathcal F}}_X$ since the latter are not everywhere transverse to the associated fibration: besides the existence of singularities, there are $3$ fibers of this fibration that are invariant under ${{\mathcal F}}_X$. In particular it is not clear that the leaves of ${{\mathcal F}}_X$ must be hyperbolic once Estimate (\[stillacrossreference\]) is verified.]{}
A similar picture is valid for the foliation ${{\mathcal F}}^X$ associated to $X$ on $M$. Clearly ${{\mathcal F}}^X$ is transverse to the codimension $1$ foliation defined by the “cone” over the leaves of ${{\mathcal F}}_{\infty}^Y$ and its dynamics is also encoded in its corresponding holonomy group. This is still generated by the local holonomy [*with respect to $M$*]{} of each of the mentioned three invariant lines. Each of the three generators is now realized as an automorphism $\Xi_i$ of ${\mathbb{F}}_1$, the line bundle over ${{\mathbb C}}P(1)$ with Chern class equal to $1$. In our context this line bundle is the cylinder over $\overline{C} \subset \Delta_{\infty}$. Besides ${\mathbb{F}}_1$ can be obtained by gluing together two copies of ${{\mathbb C}}\times {{\mathbb C}}$ with coordinates $(w,z)$ and $(w', z')$ according to the equation $(w', z') = (1/w , wz)$. The coordinate “$z$” of the first copy can be identified with the previous affine coordinate “$z$” for ${{\mathbb C}}^3$. The automorphism $\Xi_i$ fixes the null-section and thus it can be expressed in the mentioned coordinates as $$\Xi_i (w,z) = (\xi_i (w) , B_i(w) z)$$ where $\xi_i (w)$ is a homography. Furthermore it is also know that $\Xi_i^{m_i} = {\rm id}$ on ${\mathbb{F}}_1$ and $\xi_i^{m_i} = {\rm id}$ on $\overline{C}$. Thus $$B_i(w) \times B_i( \xi_i (w)) \times \cdots \times B_i (\xi_i^{m_i-1} (w)) = 1 \, . \label{product=1}$$
Next recall that the Möbius group has a natural extended action to ${\mathbb{F}}_1$ consisting of multiplying vectors in the fibers by the square root of its derivative. In other words, if $\xi$ is a homography, then its extended action on a pair $(w,z) \in {\mathbb{F}}_1$ is simply $$\xi . (w,z) = (\xi (w) , \sqrt{\xi' (w)} \, z ) \, .$$ It is to be noted that the square root of the derivative of a homography is well-defined so that the claim follows from observing that ${\mathbb{F}}_1 \otimes {\mathbb{F}}_1$ is isomorphic to the tangent bundle of $\overline{C}$. In particular a more explicitly expression for $\Xi_i$ can be derived as follows. Let $q_i, q_{i+1}$ ($\neq 0, \infty$) denote the two fixed points of $\xi_i$ in $\overline{C}$. The transformation $$\sigma_i (w) = \frac{w-q_i}{w-q_{i+1}} \; \; \; \; ; \; \; \; \; \sigma_i^{-1} (w) =\frac{q_{i+1} w -q_i}{w-1}$$ conjugates $\xi_i$ to a homography fixing $0, \infty$. In this coordinate $\overline{w}$, $\xi_i$ must take on the form $\overline{w} \mapsto k_i^2 \overline{w}$ where $k_i^2 = e^{2\pi \sqrt{-1}/m_i}$ since $\xi_i^{m_i} = {\rm id}$ on $\overline{C}$. Furthermore, in the coordinate $\overline{w}$ it is clear that $B_i (\overline{w})$ must be constant so as to allow $\Xi_i$ to have a holomorphic extension to a fibered neighborhood of $\infty$. Setting $B_i (\overline{w}) =B_i$ for this particular choice of coordinates, it follows that the expression of $\Xi_i$ in the initial coordinate $w$ is given by $$\begin{aligned}
\Xi_i (w,z) &=& (\xi_i (w) , B_i (w) z) = \left( \sigma_i^{-1} (k_i^2 \sigma_i (w)) , \sqrt{(\sigma_i^{-1})'\vert_{k_i^2 \sigma_i (w)}} B_i \sqrt{\sigma_i' (w)} z \right) \\
& = & \left( \frac{(k_i^{-1} q_i -k_i q_{i+1}) w + q_i q_{i+1} (k_i -k_i^{-1})}{(k_i^{-1} -k_i) w + (k_i q_i -k_i^{-1} q_{i+1})} ,
\frac{k_i^{-1} B_i (q_i -q_{i+1}) z}{(k_i^{-1} -k_i) w + (k_i q_i -k_i^{-1} q_{i+1})} \right) \\
& = & (\xi_i (w) , k_i^{-1} B_i \sqrt{\xi_i' (w)} z ) \, .\end{aligned}$$ The above formulas are going to enable us to understand the solutions of Halphen vector fields from the point of view worked out in this work. Let us first consider the special case where $1/m_1 + 1/m_2 +1/m_3 =1$. In this case the three homographies $\xi_1 ,\xi_2 , \xi_3$ share a common fixed point. Choosing coordinates $(w,z)$ where this point is $\infty$ it follows that $\xi'(w)$ is constant and thus $\Xi_i (w,z) = (\xi_i (w) , A_i z)$ for certain constants $A_i$, $i=1,2, 3$. As a sort of converse to Theorem \[maintheo\], we obtain the following:
\[entireHalphen\] Under the above conditions all the leaves of ${{\mathcal F}}^X$ are uniformized by ${{\mathbb C}}$ as Riemann surfaces.
The proof is rather simple. Let $L$ be a generic leaf of ${{\mathcal F}}_X$ and let ${\mathbb{F}}_1$ be as above. Because all the $A_i$’s are constant, the intersection points $p_1, p_2 \ldots$ between $L$ and ${\mathbb{F}}_1$ have their distance to $\Delta_{\infty}$ bounded from below by a positive constant. Thus the time-form $dT_L$ induced by $X$ on $L$ is uniformly bounded on a neighborhood $W$ of $\{ p_1, p_2 \ldots \}$. Consider now the maximal domain ${\bf U} \subseteq {{\mathbb C}}$ of a solution $\phi$ of $X$. Suppose for a contradiction that ${{\mathbb C}}\setminus {\bf U} \neq \emptyset$ and choose a point $T \in {{\mathbb C}}$ lying in the boundary $\partial {\bf U}$ of ${\bf U}$. Finally let $t_1, t_2, \ldots $ be a sequence of points in ${\bf U}$ converging to $T$. Given the above mentioned structure of ${{\mathcal F}}_X$ as a foliation transverse to a fibration, we can assume without loss of generality that $\phi (t_j)$ lies in $W$. Since $dT_L$ is uniformly bounded on $W$, it follows that $\phi$ is defined on a disc of uniform (positive) radius about each $t_j$. This is however impossible since $t_j \rightarrow T \in \partial {\bf U}$. The proposition is proved.
From now on let us focus on the more interesting case where $\xi_1, \xi_2, \xi_3$ generate a hyperbolic triangle group. The fixed points of $\xi_1, \xi_2, \xi_3$ are three (mutually different) points $q_1 ,q_2, q_3 \in \overline{C}$. By this we mean that for each $i \in \{ 1,2,3 \}$, the homography $\xi_i$ fixes the points $q_i, q_{i+1}$ (where $q_{3+1} =q_1$). In the present case there is no coordinate $w$ where all the $\xi_i$ become affine maps. Thus we shall need to work with the full information provided by the action of $\Xi_1, \Xi_2, \Xi_3$ on ${\mathbb{F}}_1$. We are going to show that the geometry of the leaves is related to the Poincaré series with exponent $1/2$. Next let us denote by $\Gamma$ the group generated by $\xi_1, \xi_2, \xi_3$ and consider its Cayley graph with respect to the generating set given by $\xi_1, \xi_2, \xi_3$ and their inverses. Choose a “geodesic ray” $\gamma_0 ={\rm id}, \gamma_1, \gamma_2 , \ldots$ in the Cayley graph going from the identity to an “end” of the graph. We have:
\[hyperbolic-vs-entire\] Let $L$ denote a leaf of ${{\mathcal F}}_X$ passing through a point $(w_0, z_0)$. As a Riemann surface $L$ is hyperbolic provided that the series $$S (w_0) = \sum_{j=0}^{\infty} \Vert \gamma_j (w_0) \Vert^{1/2} \label{georay}$$ is convergent for more than one geodesic ray. If this series diverges for all geodesic rays as above then $L$ is a quotient of ${{\mathbb C}}$.
The proof consists of elaborating further on the argument used in Proposition \[entireHalphen\]. Again denote by ${\bf U}$ the maximal domain of definition of the solution $\phi$ of $X$ satisfying $\phi (0) =(w_0, z_0)$, $z_0 \neq 0$. Denote by $L_{\infty}$ the projection of $L$ on $\Delta_{\infty}$. By virtue of the structure of the foliation ${{\mathcal F}}_{\infty}^X$ on $\Delta_{\infty}$, we know that $L_{\infty}$ is a ramified covering of ${{\mathbb C}}P(1)$ where the ramified points sit over three points of ${{\mathbb C}}P(1)$ (identified to the three invariant fibers of ${{\mathcal F}}_{\infty}^X$). We can then think of $L_{\infty}$ as being the universal covering of ${{\mathbb C \mathbb P}}(1)$ minus $3$ points modulo adding to it the ramification points. In particular there is a natural sense in considering [*fundamental domains*]{} in $L_{\infty}$. The leaf $L_{\infty}$ can then be considered as the union of the corresponding fundamental domains $L_{\infty}^{(0)}, L_{\infty}^{(1)}, \ldots$ such that $L_{\infty}^{(j)} = \gamma_j (L_{\infty}^{(0)})$. These domains have natural lifts to the leaf $L \subset M\setminus (\Delta_0 \cup \Delta_{\infty})$ which, modulo re-numeration, will be denoted by $L^{(0)}, L^{(1)}, \ldots$ in such way that $L^{(j)} = \Gamma_j (L^{(0)})$ where $\Gamma_j$ is the automorphism of ${\mathbb{F}}_1$ corresponding to the action of $\gamma_j$ on $\overline{C}$.
On $L$ (or on its universal covering if necessary), we define the map $$\mathcal{D}_L (p) = \int_{(w_0, z_0)}^p dT_L$$ where $dT_L$ stands for the time-form induced on $L$. Since $X$ is semi-complete, $\mathcal{D}_L$ provides a diffeomorphism from the (universal covering of) $L$ to ${\bf U}$. Let then $U_{(j)}$ be the image of $L^{(j)}$ by $\mathcal{D}_L$. The above assertion implies that the set $U_{(j)}$ “tile” ${\bf U}$ without overlapping and modulo adding the image of ramification points (involved in the preceding definition of the fundamental domains $L_{\infty}^{(0)}, L_{\infty}^{(1)}, \ldots$). Next recall that the affine structure on $L_{\infty}$ is uniformly bounded (from below and by above). Combining this fact to the expression for $dT_L$ arising from Formula \[tXX\], with $d=2$, it follows the existence $0<r_j < R_j$, $j =1, 2 ,\ldots$, satisfying the conditions below.
1. There are constants $0<c<C$, independent of $j$, such that $$c \Vert \pi_2 (\Xi_j (w_0, z_0)) \Vert r_j < R_j < C \Vert \pi_2 (\Xi_j (w_0, z_0)) \Vert r_j \, ,$$ where $\pi_2$ stands for the projection on the second coordinate (i.e. the fiber of ${\mathbb{F}}_1$).
2. The image of $U_{(j)}$ under $\mathcal{D}_L$ contains a ball of radius $r_j$ about $\mathcal{D}_L (\Xi_j (w_0, z_0))$. Similarly the same image is contained in ball of radius $R_j$ about $\mathcal{D}_L (\Xi_j (w_0, z_0))$.
It then becomes clear that ${\bf U}$ must be the whole ${{\mathbb C}}$ provided that the series $\sum_{j=0}^{\infty} \Vert \pi_2 (\Xi_j (w_0, z_0) \Vert$ diverges for every geodesic ray. Conversely, if this series is convergent, then we can easily construct a “small” piece of continuum contained in the boundary of ${\bf U} \subset {{\mathbb C}}$. Thus ${\bf U}$ must be a hyperbolic domain so that $L$ itself must be a Riemann surface covered by the unity disk. To conclude the proof of the proposition it is therefore sufficient to check that the series $\sum_{j=0}^{\infty} \Vert \pi_2 (\Xi_j (w_0, z_0) \Vert$ converges (resp. diverges) if and only if so does the “reduced Poincaré series” in the statement. This is however clear since $\vert k_i^{-1} B_i \vert =1$ as an immediate consequence of Formula (\[product=1\]). The proof of the proposition is over.
Next we are going to show that the series (\[georay\]) always converges provided that $w_0 \in \overline{C} \setminus
\Lambda_{\infty}$. Some indications concerning the behavior of this series in the case $w_0 \in \Lambda_{\infty}$ will also be provided. As mentioned the series in question differs from the usual Poincaré series since the sum is not carried over the entire group but only over those elements belonging to a chosen “geodesic ray”. Indeed the “full” Poincaré series of $\Gamma$ with exponent $1/2$ diverges as it follows from well-known results due mainly to Sullivan (see [@bulletin] for an overview of the standard theory).
\[alwaysconverging\] The series (\[georay\]) converges provided that $w_0 \in \overline{C} \setminus \Lambda_{\infty}$.
The argument is rather simple. Consider the action of $\Gamma$ in the $w$-plane. This action preserves a circle identified to $\Lambda_{\infty}$. In particular $\Gamma$ is realized as a Fuchsian group, i.e. a discrete group of automorphisms of the hyperbolic disc. Next, since $\Gamma$ acts on the hyperbolic disc, it is easy to see that the convergence of this series does not depend on $w_0$. In other words, the series (\[georay\]) converges for $w_0$ if and only if it converges at $0$. It is then sufficient to check that the series converges for $w_0 =0$ (identified to the origin of the disc). For this consider again the geodesic ray $\gamma_0 ={\rm id}, \gamma_1, \gamma_2 , \ldots$ in the Cayley graph of $\Gamma$ and set $a_j = \gamma_j (0)$. In the mentioned Cayley graph the distance between the identity and $\gamma_j$ is obviously $j$. The existence of a quasi-isometry between this graph and the hyperbolic disc implies that the hyperbolic distance $d_H (0, \gamma_j (0))$ between $0$ and $\gamma_j (0)$ satisfies $$c j < d_H (0, \gamma_j (0)) < C j$$ for appropriate uniform constants $C > c >0$. The standard formula for the length of a minimizing geodesic in the hyperbolic unit disc joining $0$ to a point $a$ of this disc (naturally satisfying $\Vert a \Vert <1$) yields $$\frac{e^{cj} -1}{e^{cj} + 1} \leq \Vert a_j \Vert \leq \frac{e^{Cj} -1}{e^{Cj} + 1} \, .$$ On the other hand the coefficients of the hyperbolic metric at $0$ and at $a_j$ allow us to obtain a formula for the derivative of $\gamma_j$ at $0$. Combined with the above estimates this formula gives $$\Vert \gamma_j' (0) \Vert \leq \frac{4e^{Cj}}{(1+ e^{Cj})^2}$$ and thus $\Vert \gamma_j' (0) \Vert^{1/2} \leq 2 e^{Cj/2} / (e^{Cj} +1)$. The convergence of the mentioned series follows immediately.
To close this discussion let us briefly indicate the behavior of the series (\[georay\]) for points $w_0$ lying in $\Lambda_{\infty}$. Since $\gamma_j$ takes $0$ to $a_j$, it follows that $\gamma_j (w) = e^{2\pi i \theta} (w+a_j)/
(1 + \overline{a}_j w)$, for some $\theta \in [0,1)$. In particular $$\Vert \gamma_j' (w) \Vert = \frac{1 - \Vert a_j \Vert^2}{(1 + \overline{a}_j w)^2} \, .$$ Because $\Gamma$ is discrete, we know that $\Vert a_j \Vert \rightarrow 1$ when $j \rightarrow + \infty$. Next set $a_j = \Vert a_j \Vert e^{i\theta_j}$ and $w_0 = e^{-i\theta_j -\pi + \alpha_j}$ so that $$\Vert \gamma_j' (w_0) \Vert = \frac{1 - \Vert a_j \Vert^2}{1 + \Vert a_j \Vert^2 -2\Vert a_j \Vert \cos (\alpha_j)} \, . \label{justtobequoted}$$ Next note that $\Vert \gamma_j' (w_0) \Vert > 1/2$ as long as $\cos (\alpha_j) > \Vert a_j \Vert$. In particular if there are infinitely many indices $j$ satisfying this condition that the corresponding series will diverge. Clearly the existence of infinitely many indices as indicated means that some subsequence of the $a_j$ converge “almost radially” for $-w_0$. Conversely if the denominator in (\[justtobequoted\]) is bounded from below by some positive constant, then the argument used in Lemma \[alwaysconverging\] ensures again the convergence of series (\[georay\]). In general we are led to a finer analysis taking into account the “conic approximation” of $-z_0$ by the sequence $a_j =\gamma_j (0)$. Not surprisingly the behavior of series (\[georay\]) on $\Lambda_{\infty}$ depends on the initial point $w_0$: for some values of $w_0$ it diverges whereas for others it is convergent.
Most properties of Halphen vector fields become encoded in the extended dynamics of the group generated by $\Xi_1, \Xi_2, \Xi_3$ on ${\mathbb{F}}_1$. For example the study of first integrals for Halphen vector fields amounts to searching functions that are invariant by this action. In particular on each connected component of $\overline{C} \setminus \Lambda_{\infty}$ it is not hard to construct “automorphic functions” for this group so that on the corresponding open sets on $M$ the Halphen vector field possesses a holomorphic first integral. It is also not very hard to check that the Halphen vector field does not admit a holomorphic (or meromorphic) first integral on the set corresponding to $\Lambda_{\infty}$ (which has real dimension equal to $5$). Yet on the latter set, there is a real-valued first integral for the equation that is actually globally defined on $M$. We shall not pursue this type of discussion here not only because Halphen vector fields were detailed studied in [@guillotIHES] but also because the corresponding issues will be no longer in line with the main ideas of this paper.
However, inasmuch none of the above results about Halphen vector fields is new, our method is applicable in other situations since, in most of our discussion, we have only needed the fact that Halphen vector fields are transverse to a singular rational fibration. Since many remarkable equations, such as Painlevé equations, admit a formulation in terms of rational vector fields transverse to suitable singular foliations, it is very reasonable to expect that similar arguments can provide new insight in other situations of interest an example of it being provided by the material in Section 7.3.
It also interesting to consider differential equations whose solutions are meromorphic functions (defined on ${{\mathbb C}}$). This is a very classical topic not only in differential equations and integrable systems but also in Nevanlinna’s theory. The methods presented in the course of this work seem well-designed to investigate some of these issues. In fact, for example, Painlevé equations equations may be expressed as a rational vector field on ${{\mathbb C}}^3$. More generally every non-autonomous second-order differential equations is equivalent to a rational vector field on ${{\mathbb C}}^3$ which, actually, has a very special nature. Whereas these vector fields may be rational rather than “polynomial”, this is of relatively little importance for our methods. Essentially the pole divisor may be displaced from the “plane at infinity” to some other algebraic surface and our methods will still be effective in this case.
In the case where we have vector fields as above having solutions that are [*meromorphic functions defined on ${{\mathbb C}}$*]{}. Consider also the foliation ${{\mathcal F}}_{\infty}$ induced on the “plane at infinity” by this vector field. Modulo very mild assumptions concerning the singularities of the corresponding vector fields at infinity, the methods of Theorem \[maintheo\] and of Proposition \[hyperbolicleaves\] will imply that all the ${\mathcal{H}}$-trajectories defined on ${{\mathcal F}}$ must be of finite length (in particular they must have future-end singularities). This seems to be a very restrictive condition on the foliation ${{\mathcal F}}_{\infty}$ and it is therefore conceivable to expect this issue can further be exploited to shed some new light in these very classical problems.
In general foliations for which all ${\mathcal{H}}$-trajectories are of finite length seem very special. In this direction we would like to propose:
[**Conjecture**]{}: Foliations all of whose ${\mathcal{H}}$-trajectories are of finite length are always transverse to a singular fibration (possibly modulo passing to some abelian covering).
If this conjecture holds, then the ideas developed here can be applied to every rational vector field since either we will have ${\mathcal{H}}$-trajectories with infinite length to “produce” hyperbolic domains or we shall have a type of Poincaré series for the corresponding global holonomy group to be studied.
Singular points of holomorphic vector fields on compact Kähler threefolds
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As promised this last section is devoted to describing a procedure leading to the solution of the following problem:
[*Problem*]{}: Let $X$ be a globally defined holomorphic vector field on a compact $3$-dimensional Kähler manifold $N$. Suppose that $p \in N$ is such that $X(p) = 0$. Find all possible normal forms for the first non-zero homogeneous component $X^h$ of $X$ at $p$.
Whereas we shall not work out the desired list of normal forms here, we are going to give a detailed description of a procedure eventually leading to it. Naturally to fully implement this procedure is a reasonably elaborated task that would take quite a few pages so that there is no point in trying to carry it out in this paper. On the other hand, let us point out that the same “list” of normal forms also contains all the possibilities for the top degree homogeneous component of a complete polynomial vector field on ${{\mathbb C}}^3$. Indeed all the arguments presented in the subsequent discussion can immediately be transferred to the case of complete polynomial vector fields by using the methods already explained in the previous sections.
To begin with by means of a local coordinate we identify $X$ to a holomorphic vector field defined on a neighborhood of $(0,0,0) \in {{\mathbb C}}^3$. In particular $X^h$ is simply the first non-zero homogeneous component of the Taylor series of $X$ at the origin so that it is defined on all of ${{\mathbb C}}^3$. Besides, we can assume that $X^h$ is not a multiple of the radial vector field since otherwise there is nothing to be proved. In particular $X^h$ induces a vector field. This allows us to consider $X^h$ as being defined in the manifold $M$ obtained by blowing-up ${{\mathbb C}}P(3)$ at the “origin” exactly as previously done in this work. In particular $X^h$ induces a foliation ${{\mathcal F}}^h$ on $M$ and ${{\mathcal F}}^h$ leaves invariant the planes $\Delta_0$ and $\Delta_{\infty}$. As already seen, the fact that $X$ is a globally defined vector field on a compact Kähler manifold implies the following:
$\bullet$ $X^h$ is semi-complete.
$\bullet$ The leaves of ${{\mathcal F}}^h$ are not hyperbolic Riemann surfaces. In particular the foliations ${{\mathcal F}}^h_0, \, {{\mathcal F}}^h_{\infty}$, obtained by restriction of ${{\mathcal F}}^h$ to $\Delta_0, \, \Delta_{\infty}$, have no hyperbolic leaf.
We can also assume that the foliation ${{\mathcal F}}^h$ does not coincide with a pencil of rational or of elliptic curves for otherwise normal forms for $X^h$ can easily be obtained. Indeed recall that rational pencils on $\Delta_0$ were classified long ago by Suzuki cf. [@suzuki]. Although the analogous question about elliptic pencils is a lot more complicated, the fact that $X^h$ is semi-complete makes the question much more treatable. So we shall look for the cases where the foliations ${{\mathcal F}}^h_0, \, {{\mathcal F}}^h_{\infty}$ do not coincide with a pencil. These foliations are clearly identified through the construction of Section $3$ and, as usual, it suffices to work with ${{\mathcal F}}^h_{\infty}$. Since ${{\mathcal F}}^h_{\infty}$ is defined on the $2$-dimensional manifold $\Delta_{\infty} \simeq {{\mathbb C}}P(2)$, a deep result by McQuillan implies that they are transverse to a rational or to an elliptic pencil defined on $\Delta_{\infty}$, cf. [@book]. The result of McQuillan itself, whereas very powerful, falls short of solving our question. in fact, our problem is in principle harder than the problem of classifying (non-homogeneous) polynomial vector fields on ${{\mathbb C}}^2$ which has itself required non-trivial elaboration on McQuillan techniques cf. [@marco3], [@bruno] (see also [@guillotreb] for a more general result dispensing with McQuillan’s theorem).
In any event we shall combine McQuillan’s theorem with the ideas developed in Section 7.2 to work out normal forms for $X^h$. This goes as follows. As in the previous paragraph, the global dynamics of ${{\mathcal F}}^h_{\infty}$ is encoded in its (global) holonomy group which is itself a subgroup of the autormorphism group of a curve $\overline{C}$. The curve $\overline{C}$ is either rational or elliptic. To abridge the discussion let us consider only the case in which $\overline{C}$ is a rational curve. The global holonomy group $\Gamma$ is then a subgroup of ${\rm PSL}\, (2, {{\mathbb C}})$. Next we have:
\[finishing1\] The holonomy group $\Gamma$ is conjugate to a subgroup of ${\rm Aff}\, ({{\mathbb C}})$.
Suppose for a contradiction the statement is false. It follows that $\Gamma$ is not solvable. Therefore either $\Gamma$ is discrete or it is dense in ${\rm PSL}\, (2, {{\mathbb C}})$. If $\Gamma$ is discrete then the estimates carried out in the proof Lemma \[alwaysconverging\] can be repeated to obtain a contradiction with the fact the leaves of ${{\mathcal F}}^h$ are not hyperbolic Riemann surfaces. Similarly, if $\Gamma$ is dense in ${\rm PSL}\, (2, {{\mathbb C}})$, then $\Gamma$ contains an infinite discrete subgroup $\Gamma_0$. The ends of the Cayley graph of $\Gamma_0$ corresponds to ends of the leaves of ${{\mathcal F}}^h$. Thus once again the estimates of Lemma \[alwaysconverging\] applied to $\Gamma_0$ leads to a contradiction. The lemma is proved.
Lemma \[finishing1\] allows us to suppose that $\Gamma \subseteq {\rm Aff}\, ({{\mathbb C}})$. Resuming the notations of the preceding paragraph, we choose generators $\Xi_1, \ldots , \Xi_r$ for $\Gamma$. Again we set $\Xi_i (w,z) = (\xi_i (z) , A_i z)$ where the coordinates $(w,z)$ are chosen so that the common fixed point of the homographies $\xi_i$ is identified to $\infty$. In particular each $A_i$ is constant in the sense that it does not depend on the variable $z$. Now the argument used in Proposition \[entireHalphen\] combined to the fact the leaves of ${{\mathcal F}}^h$ are parabolic Riemann surfaces immediately yields:
\[finishing2\] For every $i \in \{ 1, \ldots ,r\}$, we have $\Vert A_i \Vert =1$.
It follows from Lemma \[finishing1\] that ${{\mathcal F}}^h$ possesses an algebraic curve $\overline{B}$ that is not a fiber of the rational fibration in question. Indeed $\overline{B}$ corresponds to the fixed point of $\Gamma$ on $\overline{C}$. We shall now use some of the techniques introduced in [@guillotreb]. The semi-complete vector field $X^h$ induces a [*uniformizable*]{} affine structure on $\overline{B}$. In particular $\overline{B}$ must be either rational or elliptic. Furthermore the corresponding uniformizable affine structures were also classified in the same paper. When $\overline{B}$ is an elliptic curve the affine structure is either the standard “flat” structure or the so-called Hopfian torus. The case of Hopfian torus can however be ruled out since it would imply that some $A_i$ has modulos different from $1$, thus contradicting Lemma \[finishing2\]. The other possibility corresponds to having $\overline{B}$ as a rational curve. The number of possibilities for the affine structure in question is slightly larger. With the notations of [@guillotreb], we have:
1. Rational orbifold of type $n >0$.
2. The parabolic cylinder.
3. The orbifold $(2,2, \infty)$. [@guillotreb]
4. One of the elliptic orbifolds $(2,3,6)$, $(2,4,4)$, $(3,3,3)$, $(2,2,2,2)$.
It also clear that the singularities of the affine structure on $\overline{B}$ corresponds to invariant fibers of the rational pencil. The converse is less clear. In fact, the crucial step to be able to carry out the classification of the possible homogeneous vector field $X^h$ is as follows.
\[finishing3\] Suppose that $p \in \overline{B}$ is an intersection point of $\overline{B}$ with an invariant fiber of the fibration in question. Then the affine structure induced by $X^h$ on $\overline{B}$ cannot be regular at $p$.
If we take for grant the statement above, the classification of $X^h$ can be concluded as follows:
- $\overline{B}$ must be a rational curve. Indeed, over an elliptic curve the uniformizable affine structure have no singular point. Thus the fibration transverse to ${{\mathcal F}}^h$ has no invariant fiber. Therefore ${{\mathcal F}}^h$ must be transverse to the mentioned fibration everywhere and this is impossible since ${{\mathcal F}}^h$ has singular points.
- From now on we consider that $\overline{B}$ is a rational curve. Let us first suppose that the affine structure induced on $\overline{B}$ by $X^h$ is the parabolic cylinder. In this case it becomes clear that ${{\mathcal F}}^h$ is induced by a linear vector field so that nothing else need to be said about this case.
- Next we recall a general statement of [@guillotreb], namely the so-called “Fundamental Lemma” asserting in particular the following: suppose that $p \in \overline{B}$ is a singular point for the affine structure on $\overline{B}$. Then the local holonomy of $\overline{B}$ about $p$ has order determined by the ramification index of the affine structure at $p$. This order is finite unless there is no ramification.
- Complementing the previous comment, let us point out that the singular points of the above described uniformizable affine structures where ramification vanishes are precisely the two singular points of the “parabolic cylinder” structure and the point indicated as $\infty$ in the orbifold $(2,2,\infty)$.
- We now return to the list of possible affine structures. The case of the “rational orbifold of type $n >0$” can immediately be ruled out. The local holonomy maps are finite and they commute. Thus the holonomy group $\Gamma$ is finite and therefore this affine structure can only be produced if ${{\mathcal F}}^h$ is itself a pencil.
- The elliptic orbifolds $(2,3,6)$, $(2,4,4)$, $(3,3,3)$. Here we immediately conclude that $\Gamma$ must be a discrete subgroup of ${\rm Aff}\, ({{\mathbb C}})$. These cases therefore correspond to the three “special” Halphen vector fields singled out in Proposition \[entireHalphen\].
- Finally it remains to consider the cases of the orbifolds $(2,2, \infty)$ and $(2,2,2,2)$. These two groups have similar representations (with infinite image) on ${\rm Aff}\, ({{\mathbb C}})$. The resulting “Riccati equation” is therefore uniquely determined. It would only remains to check whether or not is comes from a semi-complete homogeneous polynomial vector field. This is probably not the case.
To finish this discussion let us indicate how we can try to establish the statement \[finishing3\] or, if this statement is false, the understand the structure of invariant fibers giving rise to regular points for the affine structure of $\overline{B}$. The idea to approach this problem relies on adapting the “combinatorics” developed in Section 6 of [@guillotreb] to the current case. For this adaptation to make sense the key remark is the “reduction theorem” proved in the Appendix of [@julioindiana]. In fact, according to these results the singularities of the vector field $X^h$ (defined on a $3$-dimensional ambient) can be reduced through usual “cylindric” blow-ups. With this reduction performed, let us consider a singularity $q$ of the corresponding reduced foliation $\widetilde{{{\mathcal F}}}^h$. The singularity $q$ belongs to an invariant plane for $\widetilde{{{\mathcal F}}}^h$ and, locally, we can find coordinates $(u,v,w)$ about $q$ where $\widetilde{{{\mathcal F}}}^h$ coincides with the local orbits of a vector field has the form $$a(u,v) \partial /\partial u + b(u,v) \partial /\partial v + w c(u,v) \partial /\partial w \, ,$$ where $q \simeq (0,0,0)$. Besides the singularity of the $2$-dimensional vector field $a(u,v) \partial /\partial u + b(u,v) \partial /\partial v$ at $(0,0) \in {{\mathbb C}}^2$ is redued in the usual Seidenberg sense. All this follows from the reduction procedure presented in [@julioindiana].
Next observe that, if $a(u,v) \partial /\partial u + b(u,v) \partial /\partial v$ has two eigenvalues different from zero at $(0,0) \in {{\mathbb C}}^2$ then it possesses two separatrices $S_1, S_2$ (which in the $3$-dimensional ambient are contained in the plane $\{ w=0\}$). Furthemore the ramification indices of the affine structures induced in these separatrices can be put in natural correspondence. This correspondence can be viewed as a version of the [*reciprocity relation*]{} in Section 6 of [@guillotreb] (Equation (15)) and it is enough to start up the combinatoric study of the corresponding arrangement of invariant curves.
A new difficulty appearing in our case corresponds to the case in which $a(u,v) \partial /\partial u + b(u,v) \partial /\partial v$ possesses exactly one eigenvalue equal to zero at $(0,0) \in {{\mathbb C}}^2$, i.e. the corresponding $2$-dimensional singularity is a saddle-node. In this case the reciprocity relation does not have an immediate generalization and this might create some new difficulties. However the detailed understanding of these singularities obtained in Section 3 of [@julioindiana] is likely to help us to fill in all the gaps.
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[Julio Rebelo]{}\
Institut de Mathématiques de Toulouse\
118 Route de Narbonne\
F-31062 Toulouse, FRANCE.\
[email protected]
[Helena Reis]{}\
Centro de Matemática da Universidade do Porto,\
Faculdade de Economia da Universidade do Porto,\
Portugal\
[email protected]\
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Adoption of innovations, products or online services is commonly interpreted as a spreading process driven to large extent by social influence and conditioned by the needs and capacities of individuals. To model this process one usually introduces behavioural threshold mechanisms, which can give rise to the evolution of global cascades if the system satisfies a set of conditions. However, these models do not address temporal aspects of the emerging cascades, which in real systems may evolve through various pathways ranging from slow to rapid patterns. Here we fill this gap through the analysis and modelling of product adoption in the world’s largest voice over internet service, the social network of Skype. We provide empirical evidence about the heterogeneous distribution of fractional behavioural thresholds, which appears to be independent of the degree of adopting egos. We show that the structure of real-world adoption clusters is radically different from previous theoretical expectations, since vulnerable adoptions—induced by a single adopting neighbour—appear to be important only locally, while spontaneous adopters arriving at a constant rate and the involvement of unconcerned individuals govern the global emergence of social spreading.'
author:
- 'Márton Karsai [^1]'
- Gerardo Iñiguez
- Riivo Kikas
- Kimmo Kaski
- János Kertész
title: |
**Local cascades induced global contagion:\
How heterogeneous thresholds, exogenous effects, and unconcerned behaviour govern online adoption spreading**
---
**keywords:** cascading behaviour, social spreading phenomena, complex contagion, adoption thresholds
Introduction {#introduction .unnumbered}
============
Spreading of opinions, frauds, behavioural patterns, and product adoptions are all examples of social contagion phenomena where collective patterns emerge due to correlated decisions of a large number of individuals. Although these choices are personal, they are not independent but potentially driven by several processes such as social influence [@Centola2010Spread], homophily [@McPherson2001], and information arriving from external sources like news or mass media [@Toole2011Modeling]. Social contagion evolves over networks of interconnected individuals, where links associated with social ties transfer influence between peers [@RevModPhys.81.591]. Several earlier studies aimed to identify the dominant mechanisms at play in social contagion processes [@Rogers2003Diffusion; @Granovetter1978Threshold; @schelling1969models; @Axelrod1997Dissemination]. One key element, termed behavioural threshold by Granovetter [@Granovetter1978Threshold], is defined as *“the number or proportion of others who must make one decision before a given actor does so”*. Following this idea various network models have been introduced [@Watts2002Simple; @Handjani1997Survival; @valente-thresholds-1996; @Watts2007Influentials; @Melnik2013Multistage; @Gomez2010Modeling] to understand the threshold-driven spreading, commonly known as *complex contagion* [@Centola2007Complex]. Although these models are related to a larger set of collective dynamics, they are particularly different from *simple contagion* where the exposure of nodes is driven by independent contagion stimuli [@Barrat2008Dynamical; @Bass1969]. In addition, collective adoption patterns may appear as a consequence of homophilic structural correlations, where connected individuals adopt due to their similar interests and not due to direct social influence. Distinguishing between the effects of social influence and homophily at the individual level remains as a challenge [@Aral2009; @Shalizi2011]. Furthermore, in real social spreading phenomena all these mechanisms are arguably present. However, while in the case of homophily the adoption behaviour is only seemingly correlated, and for simple contagion only the number of exposures matters, in complex contagion the fraction of adopting neighbours relative to the total number of partners determines whether a node adopts or not, capturing the natural mechanisms involved in individuals’ decision makings [@Holt06; @bikhchandani-hirshleifer-welch-92; @Karsai2014Complex]. Due to this additional complexity, threshold models are able to emulate system-wide adoption patterns known as global cascades.
Behavioural cascades are rare but potentially stupendous social spreading phenomena, where collective patterns of exposure emerge as a consequence of small initial perturbations. Some examples are the rapid emergence of political and grass-root movements [@GonzalezBailon2011Dynamics; @BorgeHolthoefer2011Structural; @EllisInformation], fast spreading of information [@Dow2013Anatomy; @Gruhl2004Information; @Banos2013Role; @Watts2007Influentials; @Hale2013Regime; @Leskovec2005Patterns; @Leskovec2007Dynamics; @Goel2012Structure] or behavioural patterns [@Fowler2009Cooperative], etc. The characterisation [@Goel2012Structure; @BorgeHolthoefer2013Cascading; @Hackett2013Cascades; @Gleeson2008Cascades; @Brummitt2011; @GhoshCascadesArxiv2010] and modelling [@Watts2002Simple; @Hurd2013Watts; @Singh2013Thresholdlimited; @Gleeson2007Seed] of such processes have received plenty of attention and provide some basic understanding of the conditions and structure of empirical and synthetic cascades. However, these studies commonly fail in addressing the temporal dynamics of the emerging cascades, which may vary considerably between different cases of social contagion. Moreover, they have not answered why real-world cascades can evolve through various dynamic pathways ranging from slow to rapid patterning, especially in systems where the threshold mechanisms play a role and social phenomena spread globally. Besides the case of rapid cascading mentioned above, an example of the other extreme is the propagation of products in social networks [@Bass1969], where adoption evolves gradually even if it is driven by threshold mechanisms and may cover a large fraction of the total population [@Karsai2014Complex]. This behaviour characterises the adoption of online services such as Facebook, Twitter, LinkedIn and Skype (Fig.\[fig:1\]a), since their yearly maximum relative growth of cumulative adoption [@SocialMedia] (for definition see Material and Methods (MM)) is lower than in the case of rapid cascades as suggested e.g. by the Watts threshold (WT) model.
To fill this gap in the modelling of social diffusion, here we will analyse and model real-world examples of social contagion phenomena. Our aim is to identify the crucial mechanisms necessary to consider in models of complex contagion to match them better with reality, and define a model that incorporates these mechanisms and captures the possible dynamics leading to the emergence of real-world global cascades. We follow the adoption dynamics of the Skype paid service “buy credit” for $89$ months since 2004, which evolves over the social network of one of the largest voice over internet providers in the world. Data includes the time of first payment of each user, an individual and conscious action that tracks adoption behaviour. In addition we follow the “subscription" service over $42$ months since 2008 (for results see Supplementary Information \[SI\]). In contrast to other empirical studies where incomplete knowledge about the underlying social network leads to unavoidable bias [@Karsai2014Complex], we use here the largest connected component of the aggregated free Skype service as the underlying structure, where nodes are Skype users and links confirmed contacts between them. This is a good approximation since it maps all connections in the Skype social network without sampling, and the paid service is only available for individuals already enrolled in the Skype network. The underlying structure is an aggregate from September 2003 to November 2011 (i.e. over $99$ months) and contains roughly 4.4 billion links and 510 million registered users worldwide [@SkypeIPO]. The data is fully anonymised and considers only confirmed connections between users (for more data details see SI).
In what follows we first provide empirical evidence of the distribution of individual adoption thresholds and other structural and dynamical features of a worldwide adoption cluster. We incorporate the observed structural and threshold heterogeneities into a dynamical threshold model where multiple nodes adopting spontaneously (i.e. firstly among their neighbours) are allowed [@Ruan2015]. We find that if the fraction of users who reject to adopt the product is large, the system enters a quenched state where the evolution and structure of the global adoption cluster is very similar to our empirical observations. Model calculations and the analysis of the real social contagion process suggest that the evolving structure of an adoption cluster differs radically from what has been proposed earlier [@Watts2002Simple], since it is triggered by several spontaneous adoptions arriving at a constant rate, while stable adopters who are initially resisting exposure, are actually responsible for the emergence of global social adoption (Fig. \[fig:1\]b and c).
Results {#results .unnumbered}
=======
Social contagion phenomena can be modelled as binary-state processes evolving on networks and driven by threshold mechanisms. In these systems individuals are represented by nodes, each being either in a susceptible (0) or adopter (1) state and influencing each other by transferring information via social ties [@Granovetter1978Threshold]. Nodes are connected in a network with degree distribution $P(k)$ and average degree $z = \langle k \rangle$. In addition, each node has an individual threshold $\phi \in [0, 1]$ drawn from a distribution $P(\phi)$ with average $w = \langle \phi \rangle$. This threshold determines the minimum fraction of exposed neighbours that triggers adoption and captures the resistance of an individual against engaging in spreading behaviour. Once a node reaches its threshold, it switches state from $0$ to $1$ and keeps it until the end of the dynamics. In his seminal paper about threshold dynamics, Watts [@Watts2002Simple] classified nodes into three categories based on their threshold and degree. He identified [*innovator*]{} nodes that spontaneously change state to $1$, thus starting the process. Such nodes have a trivial threshold $\phi=0$. Then there are nodes with threshold $0 < \phi \leq 1/k$, called [*vulnerable*]{}, which need one adopting neighbour before their own adoption. Finally, there are more resilient nodes with threshold $\phi>1/k$, denoted as [*stable*]{}, referring to individuals in need of strong social influence to follow the actions of their acquaintances.
In the WT model [@Watts2002Simple], small perturbations (like the spontaneous adoption of a single seed node) can trigger global cascading patterns. However, their emergence is subject to the so-called [*cascade condition*]{}: the innovator seed has to be linked to a percolating vulnerable cluster, which adopts immediately afterwards and further triggers a global cascade (i.e. a set of adopters larger than a fixed fraction of the finite network). The cascade condition is satisfied if the network is inside a bounded regime in $(w, z)$-space [@Watts2002Simple]. This regime depends on degree and threshold heterogeneities [@Watts2002Simple] and may change its shape if several innovators start the process [@Singh2013Thresholdlimited].
Empirical observations {#empirical-observations .unnumbered}
----------------------
Degree and threshold heterogeneities are indeed present in the social network of Skype. The degree distribution $P(k)$ is well approximated by a lognormal function $P(k) \propto k^{-1} e^{-(\ln k - \mu_D)^2/(2\sigma_D^2)}$ ($k \geq \kmin$) with parameters $\mu_D=1.2$, $\sigma_D=1.39$ and $\kmin=1$ (Fig. \[fig:1\]d), giving an average degree $z = 8.56$ (for goodness of fit see SI). Moreover, at the time of adoption we can measure the threshold $\phi=\Phi_k/k$ of a user by counting the number $\Phi_k$ of its neighbours who have adopted the service earlier. We then group users by degree and calculate the distribution $P(\Phi_k)$ of the integer threshold $\Phi_k$ [@Gleeson2008Cascades] (Fig. \[fig:1\]e). By using the scaling relation $P(\Phi_k, k) = k P(\Phi_k/k)$ all distributions collapse to a master curve well approximated by a lognormal function $P(\phi) \propto \phi^{-1} e^{-(\ln\phi - \mu_T)^2/(2\sigma_T^2)}$, with parameters $\mu_T=-2$ and $\sigma_T=1$ as constrained by the average threshold $w = 0.19$ (see MM and SI). Note that we observe qualitatively the same scaling and lognormal shape of the threshold distribution for another service (see SI). These empirical observations, in addition to the broad degree distribution, provide quantitative evidence about the heterogeneous nature of adoption thresholds.
{width="90.00000%"}
Since we know the complete structure of the online social network, as well as the first time of service usage for all adopters, we can follow the temporal evolution of the adoption dynamics. By counting the number of adopting neighbours of an ego, we identify innovators ($\Phi_k=0$), and vulnerable ($\Phi_k=1$) or stable ($\Phi_k>1$) nodes. The adoption rates for these categories behave rather differently from previous suggestions [@Watts2002Simple] (Fig. \[fig:1\]f). First, there is not only one seed but an increasing fraction of innovators in the system who, after an initial period, adopt approximately at a constant rate. Second, vulnerable nodes adopt approximately with the same rate as innovators suggesting a strong correlation between these types of adoption. Third, the overall adoption process accelerates due to the increasing rate of stable adoptions induced by social influence. At the same time a giant adoption cluster grows and percolates through the whole network (Fig. \[fig:3\]a, main panel). Despite of this expansion dynamics and connected structure of the service adoption cluster, the service reaches less than $6\%$ of the total number of active Skype users over a period of $7$ years [@SkypeIPO]. Therefore we ask whether one can refer to these adoption clusters as cascades. They are not triggered by a small perturbation but induced by several innovators; their evolution is not instantaneous but ranges through several years; and although they involve millions of individuals, they reach only a reduced fraction of the whole network. To answer we incorporate the above mentioned features into a dynamical threshold model [@Ruan2015] with a growing group of innovators and investigate their effect on the evolution of global social adoption. Note that we also perform a null model study to demonstrate, on the system level, that social influence dominates the contagion process, but not homophily (see section S3 of the SI, together with another empirical spreading scenario in S7.1).
Model {#model .unnumbered}
-----
Our modelling framework is an extension to conventional threshold dynamics on networks studied by Watts, Gleeson, Singh, and others, where all nodes are initially susceptible and innovators are only introduced as an initial seed of arbitrary size [@Watts2002Simple; @Gleeson2007Seed; @Singh2013Thresholdlimited]. Apart from the threshold rule discussed above, our model considers two additional features: (i) a fraction $r$ of ‘immune’ nodes that never adopt, indicating lack of interest in the service; (ii) due to external influence, susceptible nodes adopt the innovation spontaneously (i.e. become innovators) throughout time with constant rate $p_n$, rather than only at the beginning of the dynamics. In this way, the dynamical evolution of the system is completely defined by the online social network, the distribution $P(\phi)$ and the parameters $r$, $p_n$. For the sake of simplicity we consider a configuration-model network, i.e., we ignore correlations in the social network and characterise it solely by its degree distribution $P(k)$. Furthermore, node degrees and thresholds are considered to be independent [@Gleeson2008Cascades; @gleeson2013binary; @gleeson2011high].
Our threshold model, which has also been introduced in [@Ruan2015], can be studied analytically by extending the framework of approximate master equations (AMEs) for monotone binary-state dynamics recently developed by Gleeson [@Gleeson2008Cascades; @gleeson2013binary; @gleeson2011high], where the transition rate between susceptible and adoption states only depends on the number $m$ of network neighbours that have already adopted. We describe a node by the property vector $\kvec = (k, c)$, where $k = k_0, k_1, \ldots k_{M-1}$ is its degree and $c = 0, 1, \ldots, M$ its type, i.e. $c = 0$ is the type of the fraction $r$ of immune nodes, while $c \neq 0$ is the type of all non-immune nodes that have threshold $\phi_c$. In this way $P(\phi)$ is substituted by the discrete distribution of types $P(c)$ (for $c > 0$). The integer $M$ is the maximum number of degrees (or non-zero types) considered in the AME framework, which can be increased to improve the accuracy of the analytical approximation at the expense of speed in its numerical computation (see S4.2). Under these conditions, the AME system describing the dynamics of the threshold model is reduced to the pair of ordinary differential equations (see SI),
\[eq:reducedAMEs\] $$\begin{aligned}
\dot{\rho} &= h(\nu, t) - \rho, \\
\dot{\nu} &= g(\nu, t) - \nu,\end{aligned}$$
where $\rho(t)$ is the fraction of adopters in the network, $\nu(t)$ is the probability that a randomly chosen neighbour of a susceptible node is an adopter, and the initial conditions are $\rho(0) = \nu(0) = 0$. Here, $$\label{eq:hTerm}
h = (1 - r) \Big[ \ft + (1 - \ft) \sum_{\kvec | c \neq 0} P(k) P(c) \sum_{m \geq k\phi_c} \Bkm(\nu) \Big],$$ and, $$\label{eq:gTerm}
g = (1 - r) \Big[ \ft + (1 - \ft) \sum_{\kvec | c \neq 0} \frac{k}{z} P(k) P(c) \sum_{m \geq k\phi_c} \Bkom(\nu) \Big],$$ where $\ft = 1 - (1 - \pr) e^{-\pr t}$, $\pr = p_n / (1 - r)$, and $\Bkm(\nu) = \binom{k}{m} \nu^m (1 - \nu)^{k - m}$ is the binomial distribution. The fraction of adopters $\rho$ is then obtained by solving Eq. (\[eq:reducedAMEs\]) numerically. Since susceptible nodes adopt spontaneously with rate $p_n$, the fraction of innovators $\rho_0(t)$ in the network is given by (see S4.3), $$\label{eq:innovFrac}
\rho_0 = \pr \int_0^t (1 - r - \rho) dt.$$
![[**Threshold model for the adoption of online services.**]{} [**(a-b)**]{} Surface plot of the normalised fraction of adopters $\rho / (1 - r)$ in $(w, z)$-space, for $r = 0.73$ and $t = 89$. Contour lines signal parameter values for which $20\%$ of non-immune nodes have adopted, for fixed $r$ and varying time (a), and for fixed time and varying $r$ (b). The continuous contour line and dot indicate parameter values in the last observation of Skype s3. A regime of maximal adoption ($\rho \approx 1 - r$) grows as time goes by, and shrinks for larger $r$. [**(c)**]{} Time series of the fraction of adopters $\rho$ for fixed $p_n = 0.00019$ and varying $r$ (main), and for fixed $r = 0$ and varying $p_n$ (inset). These curves are well approximated by the solution of Eq. (\[eq:reducedAMEs\]) for $k_0 = 3$, $k_{M-1} = 150$ and $M = 25$ (dashed lines). The dynamics is clearly faster for larger $p_n$. As $r$ increases, the system enters a regime where the dynamics is slowed down and adopters are mostly innovators. [**(d)**]{} Final fraction of innovators $\rho_0(\infty)$ and time $t_c$ when $50\%$ of non-immune nodes have adopted as a function of $r$, both simulated and theoretical. The crossover to a regime of slow adoption is characterised by a maximal fraction of innovators and time $t_c$. Unless otherwise stated, $p_n=0.00019$ and we use $N=10^4$, $\mu_D=1.09$, $\sigma_D=1.39$, $\kmin=1$, $\mu_T=-2$, and $\sigma_T=1$ to obtain $z = 8.56$ and $w = 0.19$ as in Skype s3. The difference in $\mu_D$ between data and model is due to finite-size effects (see Materials and Methods). Numerical results are averages over $10^2$ (a-b) and $10^3$ (c-d) realisations. \[fig:2\]](Fig2.pdf){width="60.00000%"}
We also implement the threshold model numerically via a Monte Carlo simulation in a network of size $N$, with a lognormal degree distribution and a lognormal threshold distribution as observed empirically. Thus, we can explore the behaviour of $\rho$ and $\rho_0$ as a function of $z$, $w$, $p_n$ and $r$, both in the numerical simulation and in the theoretical approximation given by Eqs. (\[eq:reducedAMEs\]) and (\[eq:innovFrac\]). For $p_n > 0$ some nodes adopt spontaneously as time passes by, leading to a frozen state characterised by a final fraction $\rho(\infty) = 1 - r$ of adopters. However, the time needed to reach such state depends heavily on the distribution of degrees and thresholds, as signalled by a region of large adoption ($\rho \approx 1 - r$) that grows in $(w, z)$-space with time (contour lines in Fig. \[fig:2\]a). If we fix a time in the dynamics and vary the fraction of immune nodes instead, this region shrinks as $r$ increases (contour lines in Fig. \[fig:2\]b). In other words, the set of networks (defined by their average degree and threshold) that allow the spread of adoption is larger at later times in the dynamics, or when the fraction of immune nodes is small. When both $t$ and $r$ are fixed, the normalised fraction of adopters $\rho / (1 - r)$ gradually decreases for less connected networks with larger thresholds (surface plot in Fig. \[fig:2\]a and b).
For $r \approx 0$ the critical fraction of innovators necessary to trigger a cascade of fast adoption throughout all susceptible nodes may be identified as the inflection point in the time series of $\rho$ (Fig. \[fig:2\]c, inset). The adoption cascade appears sooner for larger $p_n$, since this parameter regulates how quickly the critical fraction of innovators is reached. Yet as we increase $r$ above a threshold $r_c$, the system enters a regime where rapid cascades disappear and adoption is slowed down. The crossover between these regimes is gradual, as seen in the shape of $\rho$ for increasing $r$ (Fig. \[fig:2\]c, main panel). We may identify $r_c$ in various ways: by the maximum in both the final fraction of innovators $\rho_0(\infty)$ and the critical time $t_c$ when $\rho = (1-r)/2$ (Fig. \[fig:2\]d), or as the $r$ value where the inflection point in $\rho$ disappears. These measures indicate $r_c \approx 0.8$ for the chosen parameters. All global properties of the dynamics (like the functional dependence of $\rho$ and $\rho_0$) are very well approximated by the solution of Eqs. (\[eq:reducedAMEs\]) and (\[eq:innovFrac\]) (dashed lines in Fig. \[fig:2\]c and d). Indeed, the AME framework is able to capture the shape of the $\rho$ time series, the crossover between regimes of fast and slow adoption, as well as the maximum in $\rho_0(\infty)$ and $t_c$.
Validation {#validation .unnumbered}
----------
To better understand how innovation spreads throughout real social networks, we take a closer look at the internal structure of the service adoption process. By taking into account individual adoption times we construct an evolving adoption network with links between users who have adopted the service before time $t$ and are connected in the social structure. In order to avoid the effect of instantaneous group adoptions (evidently not driven by social influence), we only consider links between nodes who are neighbours in the underlying social network and whose adoption did not happen at the same time. This way links in the adoption graph indicate ties where social influence among individuals could have existed. The size distribution $P(s_a)$ of connected components in the adoption network shows the emergence of a giant percolating component over time (Fig. \[fig:3\]a), along with several other small clusters. Moreover, after decomposition we observe that the giant cluster does not consist of a single innovator seed and percolating vulnerable tree [@Watts2002Simple], but builds up from several innovator seeds that induce small vulnerable trees locally (Fig. \[fig:3\]b), each with small depth (Fig. \[fig:3\]d) [@Bakshy11; @Goel2012Structure]. At the same time the stable adoption network (considering connections between all stable adopters at the time) has a giant connected component, indicating the emergence of a percolating stable cluster with size comparable to the largest adoption cluster (Fig. \[fig:3\]a, inset). These observations suggest a scenario for the evolution of the global adoption component different from earlier threshold models [@Watts2002Simple]. It appears that here multiple innovators adopt at different times and trigger local vulnerable trees (Fig.\[fig:1\]b), which in turn induce a percolating component of connected stable nodes that holds the global adoption cluster together (Fig.\[fig:1\]c). Consequently, in the structure of the adoption network primary triggering effects are important only locally, while external and secondary triggering mechanisms seem to be responsible for the emergence of global-scale adoption.
![**Empirical cluster statistics and simulation results.** **(a)** Empirical connected-component size distribution at different times for the adoption \[$P(s_a)$, main panel\] and stable adoption \[$P(s_s)$, inset\] networks, with $s_a$ and $s_s$ relative to system size. **(b)** Empirical connected-component size distribution $P(s_v)$ for the relative size of innovator-induced vulnerable trees at different times. **(c)** Average size of the largest ($LC$) and 2nd largest ($LC^{2nd}$) components of the model network (‘Net’), adoption network (‘Casc’), stable network (‘Stab’), and induced vulnerable trees (‘Vuln’) as a function of $r$. Dashed lines show the observed relative size of the real $LC$ of the adopter network in $2011$ \[see main panel in (a)\] and the predicted $r$ value. **(d)** Distribution $P(d)$ of depths of induced vulnerable trees in both data and model for several $r$ values, showing a good fit with the data for $r=0.73$. The difference in the tail is due to finite-size effects. **(e)** Correlation $\langle s_v \rangle (k)$ between innovator degree and average size of vulnerable trees in both data and model with the same $r$ values as in (e). Model calculations for (d) and (e) correspond to networks of size $N=10^6$ and are averaged over $10^2$ realisations. \[fig:3\]](Fig3.pdf){width="60.00000%"}
To model the observed dynamics and explore the effect of immune nodes, we perform extensive numerical simulations of the threshold model with parameters determined directly from the data (see MM and SI). We use a network structure with empirical degree and threshold distributions and fix $p_n=0.00019$ as the constant rate of innovators, implying that the time scale of a Monte Carlo iteration in the model is 1 month. We measure the average size of the largest ($LC$) and second largest ($LC^{2nd}$) connected components of the background social network, and of the stable, vulnerable and global adoption networks, as a function of the fraction of immune nodes $r$. After $T=89$ iterations (matching the length of the real observation period) we identify three regimes of the dynamics (Fig. \[fig:3\]c): if $0<r<0.6$ (dark-shaded area) the spreading process is very rapid and evolves in a global cascade, which reaches most of the nodes of the shrinking susceptible network in a few iteration steps. About $10\%$ of adopters are connected in a percolating stable cluster, while vulnerable components remain very small in accordance with empirical observations. In the crossover regime $0.6<r<0.8$ (light-shaded area), the adoption process slows down considerably (Fig. \[fig:2\]d, lower panel), as stable adoptions become less likely due to the quenching effect of immune nodes. The adoption process becomes the slowest at $r_c=0.8$ (Fig. \[fig:2\]d, lower panel) when the percolating stable cluster falls apart, as demonstrated by a peak in the corresponding $LC^{2nd}$ curve (Fig. \[fig:3\]c, lower panel). Finally, around $r=0.9$ the adoption network becomes fragmented and no global diffusion takes place. We repeat the same calculations for another service and find qualitatively the same picture, but with the crossover regime shifted towards larger $r$ values due to the different parametrisation of the model process. Note that another possible reason for the slow adoption could be the time users wait between their threshold has been reached and actual adoption. We test for the effect of this potential scenario on the empirical curves but find no qualitative change in the dynamics (see SI).
We can use these calculations to estimate the only unknown parameter (the fraction $r$ of immune nodes in Skype) by matching the size of the largest component ($LC_{Net}$) between real and model adoption networks at time $T$. Empirically, this value is the relative size corresponding to the last point on the right-hand side of the distribution for $2011$ (Fig. \[fig:3\]a, main panel). The corresponding value in the model is $r = 0.73$ (dashed lines in Fig. \[fig:3\]c; also Fig. \[fig:2\]a and b), suggesting that the real adoption process lies in the crossover regime. The other analysed service turns out to lie right of the crossover regime, which explains its large innovator adoption rate and reduced size of stable and vulnerable adoption clusters (see SI).
To test the validity of the prediction of $r$ we perform three different calculations. First we measure the maximum relative growth rate of cumulative adoption and find a good match between model and data (Skype s3 and Model Skype s3 in Fig. \[fig:1\]a). In other words, the model correctly estimates the speed of the adoption process. Second, we measure the distribution $P(d)$ of depths of induced vulnerable trees (Fig. \[fig:3\]d). Finally, in order to verify earlier theoretical suggestions [@Singh2013Thresholdlimited], we look at the correlation $\langle s_v \rangle (k)$ between the degree of innovators and the average size of vulnerable trees induced by them (Fig. \[fig:3\]e). We perform the last two measurements on the real data and in the model process for $r=0.6$ and $0.9$, as well as for the predicted value $r=0.73$. In the case of $\langle s_v \rangle (k)$, we find a strong positive correlation in the data, explained partially by degree heterogeneities in the underlying social network, but surprisingly well emulated by the model. More importantly, although both quantities appear to scale with $r$, measures for the estimated $r$ value fit the empirical data remarkably well, confirming our earlier validation based on the matching of relative component sizes (for further discussion see SI).
Discussion {#discussion .unnumbered}
==========
Although some products and innovations diffusing in society may cover a large fraction of the population, their spreading tends to follow slow cascading patterns, the dynamics of which have been modelled before by simple diffusion models like that of Bass [@Bass1969]. However, this approach neglects threshold mechanisms that arguably drive the decision making of single individuals. On the other hand, threshold models study the conditions for cascades in global diffusion but do not address their temporal evolution, which is clearly a relevant factor in real-world adoption processes. These models are commonly used to predict rapid cascading patterns of adoption, which is a more realistic scenario for the spreading of information, opinions, or behavioural patterns but are not observed in the case of product or innovation diffusion where adoption requires additional efforts, e.g., free or paid registration. Here we provide a solution for this conundrum by analysing and modelling the worldwide spread of an online service in the techno-social communication network of Skype. Beyond the novel empirical evidence about heterogeneous adoption thresholds and non-linear dynamics of the adoption process, we identify two additional components necessary to introduce in the modelling of product adoption, namely: (a) a constant flow of innovators, which may induce rapid adoption cascades even if the system is initially out of the cascading regime; and (b) a fraction of immune nodes that forces the system into a quenched state where adoption slows down. These features are responsible for a critical structure of empirical adoption components that radically differs from previous theoretical expectations. We incorporate these mechanisms into a threshold model controlled by the rate of innovators and the fraction of immune nodes. The model is able to reproduce several pathways ranging from cascading behaviour to more realistic dynamics of innovation adoption. By constraining the model with empirically determined parameters, we provide an estimate for the real fraction of susceptible agents in the social network of Skype, and validate this prediction through correlated structural features matching empirical observations.
Our aim in this study was to provide empirical observations as well as methods and tools to model the dynamics of social contagion phenomena with the hope it will foster thoughts for future research. One possible direction would be the observation of the reported structure and evolution of the global adoption cluster in other systems similar to the ones studied in [@BorgeHolthoefer2011Structural; @Dow2013Anatomy; @Gruhl2004Information; @Goel2012Structure; @BorgeHolthoefer2013Cascading; @Bakshy11]. Other promising directions could be the consideration of homophilic or assortative structural correlations, the evolving nature of the underpinning social structure as studied in [@Karsai2014Complex], interpersonal influence, or the effects of leader-follower mechanisms on the social contagion process. Finally, we hope that the reported results may improve efficiency in the strategies of enhancing the diffusion of products and innovations, by shifting attention from the creation of short-lived perturbations to the sustenance of external input.
### Competing interest statement {#competing-interest-statement .unnumbered}
The authors have no competing interests.
### Authors’ contributions statement {#authors-contributions-statement .unnumbered}
M.K., G.I., R.K., K.K, and J.K designed the research and participated in writing the manuscript. R.K. and M.K. analysed the empirical data, G.I. made the analytical calculations, and M.K. and G.I. performed the numerical simulations.
### Acknowledgements {#acknowledgements .unnumbered}
The authors gratefully acknowledge the support of M. Dumas, A. Saabas, and A. Dumitras from STACC and Microsoft/Skype Labs as well as constructive comments by J. Saramäki and T. N[ä]{}si.
### Funding statement {#funding-statement .unnumbered}
G.I. acknowledges the Academy of Finland, and J.K. the CIMPLEX FET Open H2020 EU project for support. This research was partly funded by Microsoft/Skype Labs.
Material and Methods {#material-and-methods .unnumbered}
====================
Data description {#data-description .unnumbered}
================
We use a static representation of the Skype social network aggregated over 99 months between September 2003 and November 2011. We follow the adoption of the “buy credit” paid service for $89$ months starting from 2004, and the paid service “subscription” for $42$ months starting from 2008 (for further details about the network and service see SI). By considering the online social structure and adoption times, we identify users as innovator, vulnerable, or stable nodes based on the number $\Phi_k$ of adopting neighbours at the time of exposure. Thresholds are calculated as $\phi=\Phi_k/k$ for users with $k$ contacts. The adoption network is constructed by considering confirmed social links between users who adopted the service earlier than $t$. In order to avoid the effect of instantaneous group adoptions (evidently not driven by social influence), we only consider links between nodes who are neighbours in the underlying social network and whose adoption did not happen at the same time. Note that for the categorisation of nodes we use only the adoption time and the state of their peers, and thus real categories may differ slightly. For example, an innovator may appear as a vulnerable or stable node, even if its decision was not driven by social influence but some of its peers adopted earlier. To consider this bias we measure effective rates of adoption for the model process as well, just like for the empirical case (Fig.\[fig:1\]) and section S3.
Maximum relative growth rate {#maximum-relative-growth-rate .unnumbered}
============================
This measure is obtained by taking the maximum of the yearly adoption rate (yearly count of adoptions) normalised by the final observed adoption number of a given service. It characterises the maximum speed of adoption a service experienced during its history and takes values between 0 (no cascade) and 1 (instantaneous cascade). We repeat this measurement for the estimated number of registered users of Facebook, Twitter, and LinkedIn [@SocialMedia], as well as for the number of active users of Skype and three paid Skype services. Adoption rates for Facebook, Twitter, and LinkedIn correspond to the period between 2006 and 2012, and for Skype and its services to the interval from release date until 2011.
Empirical parameter estimation {#empirical-parameter-estimation .unnumbered}
==============================
We use the Skype data to directly determine all model parameters, apart from the fraction $r$ of immune nodes. To best approximate the degree distribution of the real network, after testing different candidate functions (see SI) we select a lognormal function $P(k) = e^{ -(\ln k-\mu_D)^2 / (2\sigma_D^2) } / (k\sigma_D\sqrt{2\pi})$ with parameters $\mu_D=1.2$ and $\sigma_D=1.39$ and minimum degree $\kmin = 1$, leading to the average degree $z = 8.56$. To account for finite-size effects in the model results for low $N$ (Fig. \[fig:2\]), we decrease $\mu_D$ slightly to obtain the same value of $z$ as in the real network.
The threshold distribution of each degree group collapses to a master curve after normalisation by using the scaling relation $P(\Phi_k,k)=k P (\Phi_k/k)$. This master curve can be well approximated by the lognormal distribution $P(\phi) = e^{ -(\ln \phi-\mu_T)^2 / (2\sigma_T^2) } / (\phi\sigma_T\sqrt{2\pi})$, with parameters $\mu_T=-2$ and $\sigma_T=1$ as determined by the empirical average threshold $w = 0.19$ and standard deviation $0.233$ (for further details see SI). We estimate a rate of innovators $p_n = 0.00019$ by fitting a constant function to $R_i(t)$ for $t > 2\tau$ (Fig. \[fig:1\]f). The fit to $\pn$ also matches the time scale of a Monte Carlo iteration in the model to 1 month. Model results (Fig. \[fig:3\]d and e) are calculated with $r = 0.73$ and $p_n = 0.00019$. Simulation results in Fig. \[fig:3\]c (d and e) are averaged over $100$ configuration-model networks of size $N=10^5$ ($10^6$) after $T=89$ iterations, matching the length of the observation period in Skype.
Model description {#model-description .unnumbered}
=================
We characterise the static social network by the extended distribution $\Pk$, where $\Pk = r P(k)$ for $c = 0$ and $\Pk = (1 - r) P(k) P(c)$ for $c > 0$. Non-immune, susceptible nodes with property vector $\kvec$ adopt spontaneously at a constant rate $\pn$, else they adopt only if a fraction $\phi_c$ of their $k$ neighbours have adopted before. These rules are condensed in the probability $\Fkm dt$ that a node will adopt in a small time interval $dt$, given that $m$ of its neighbours are already adopters, $$\label{eq:thresRule}
\Fkm =
\begin{cases}
\pr & \text{if} \quad m < k \phi_c \\
1 & \text{if} \quad m \geq k \phi_c
\end{cases}, \quad \forall m \; \text{and} \; k, c \neq 0,$$ with $F_{(k,0),m} = 0$ $\forall k, m$ and $F_{(0,c),0} = \pr$ $\forall c \neq 0$ (for immune and isolated nodes, respectively). The dynamics of adoption is well described by an AME for the fraction $\skm(t)$ of $\kvec$-nodes that are susceptible at time $t$ and have $m=0,\ldots,k$ adopting neighbours [@porter2014; @gleeson2013binary; @gleeson2011high], $$\label{eq:AMEsThres}
\dskm = -\Fkm \skm -\bs (k - m) \skm + \bs (k - m + 1) \skmo,$$ where $\bs(t) = \frac{\sumk \Pk \summ (k - m) \Fkm \skm(t)}{\sumk \Pk \summ (k - m) \skm(t)}$. To reduce the dimensionality of Eq. (\[eq:AMEsThres\]) we consider the ansatz $\skm = \Bkm (\nu) e^{-\pr t}$ for $m < k\phi_c$, leading to the condition $\dot{\nu} = \bs (1 - \nu)$. With $\rho = 1 - \sumk \Pk \summ \skm$ and some algebra, this condition is reduced to Eq. (\[eq:reducedAMEs\]) (see SI).
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{#section .unnumbered}
Detailed data description {#sec:ddescr}
=========================
This study has been conducted on a dataset of the social network of Skype. The centrepiece of the dataset is the *contact network*, where nodes represent users and edges between pairs of users exist if they are in each other’s contact lists. A user’s contact list is composed of *friends*. If user $u$ wants to add another user $v$ to his/her contact list, $u$ sends $v$ a contact request, and the edge is established at the moment $v$ approves the request (or not, if the contact request is rejected). Each edge is labelled with a time stamp indicating the moment the contact request was approved. As the underpinning social structure we consider the static representation of the Skype social network, aggregated for $99$ months between September 2003 and November 2011. The largest connected component of this structure includes roughly 510 million users and 4.4 billion edges.
As the chosen service evolving on the Skype network, we follow how users purchase “credits” for calling phones. For each user, the dataset includes the date when he/she first adopted the paid product “buy credit” (first credit purchase, for all purposes). We select this service since its lifetime of $89$ months is considerably long (it was introduced in 2004), and it can be adopted by registered Skype users only. This way the aggregated Skype network provides a complete description of the mediating social structure, which allows us to calculate the correct degree and adoption threshold for all individuals. To make additional observations and to further test our model, we perform calculations on a second paid service called “subscription”, which was introduced in April 2008, lasts for over $42$ months, and can also be adopted by registered Skype users only. Results regarding this service are presented in Section \[sec:addserv\].
By considering the online social structure and the adoption times we identify users as innovator, vulnerable, or stable nodes based on the number $\Phi_k$ of adopting neighbours at the time of exposure. Thresholds are calculated as $\phi=\Phi_k/k$ for users with $k$ contacts. The adoption network is constructed by considering confirmed social links between users who adopted the service earlier than the time of observation $t$. In order to avoid the effect of instantaneous group adoptions (evidently not driven by social influence), we only consider links between nodes who are neighbours in the underlying social network and whose adoption did not happen at the same time. Note that for the categorization of nodes we use only the adoption time and the state of their peers, and thus ‘real’ categories may differ slightly. For example, an innovator may appear as a vulnerable or stable node, even if its decision was not driven by social influence but some of its peers adopted earlier. To consider this bias we also measure ‘effective’ rates of adoption for the model process, just like for the empirical case (Fig.1, main text) and section S3.
The dataset does not include identity information. All usernames are anonymized and there is no way of inferring a user’s identity solely from the profile. The dataset does not contain any information about interpersonal interactions, apart from the contact list.
Empirical determination of model parameters {#sec:pars}
===========================================
Parameters in the model are the rate of innovators $p_{n}$, the degree distribution $P(k)$, the threshold distribution $P(\phi)$, and the fraction of immune nodes $r$. Other than $r$, all of them can be estimated from the data as follows.
Rate of innovators
------------------
As discussed in the main text, the rate of spontaneous adoption saturates approximately to a constant value after an initial transition period, which allows us to determine the rate of innovators by fitting a constant function on the curve after time $2\tau$. We estimate this rate to be $p_{n} = 0.00019$, as demonstrated in Fig. S\[fig:NullModelRate\]a where the dashed line assigns the fitted constant function.
Degree distribution {#sec:degDistrFit}
-------------------
Degrees in the aggregated Skype network are broadly distributed with a fat tail corresponding to strong degree heterogeneities. To characterize this distribution analytically we select two candidate distribution functions. The first is a shifted power-law distribution function of the form, $$P(k)=\frac{\gamma-1}{C+k_{min}}\left( \frac{C+k}{C+k_{min}} \right)^{-\gamma} \hspace{.2in} \mbox{for} \hspace{.2in} k_{min}\leq k,
\label{eq:shSF}$$ where $k$ denotes the degree, $\gamma$ is the power-law exponent scaling the tail of the distribution, and $k_{min}$ is the minimum degree (in our case 1). $C$ is a constant scaling the shift of the distribution, which can be determined as $C=z(\gamma-2)-k_{min}(\gamma-1)$ since we know the average degree $z = 8.56$ of the empirical network. This way our only free parameter during the fit is the degree exponent $\gamma$. After fitting this function by using the non-linear least-square method, we obtain a relatively good match with the empirical distribution (Fig. S\[fig:degdistr\]a) for exponent $\gamma=3.61$.
\
Our second candidate function is a lognormal distribution function of the form, $$P(k)=\frac{1}{k\sigma_D\sqrt{2\pi}}e^{-\frac{(\ln k-\mu_D)^2}{2\sigma_D^2}} \hspace{.2in} \mbox{for} \hspace{.2in} k_{min}\leq k,
\label{eq:logn}$$ where $\mu_D$ and $\sigma_D$ are the scaling parameters. After fitting this function by using the non-linear least square method with two free parameters ($\mu_D$ and $\sigma_D$), we obtain an excellent fit with the empirical distribution for parameters $\mu_D=1.2$ and $\sigma_D=1.39$.
To select the best candidate function, we calculate the corresponding Jensen-Shannon ($JS$) divergence values [@Lin1991] between the empirical and fitted distributions. As a result we find that for the shifted power-law function the best fit provides $JS=0.0257$, while for the lognormal distribution we get $JS=0.0051$. Thus we select the lognormal distribution as the best analytical function describing the degree distribution of the empirical network.
Threshold distribution {#sec:thrDistr}
----------------------
The adoption threshold $\phi$ of a node is defined as $\phi = \Phi_k / k$, i.e. the fraction of adopting neighbours that trigger the adoption of the central node. Therefore it can only take certain fractional values determined by the degree $k$. Although thresholds are defined as a fraction, by considering nodes of the same degree we can focus on the integer threshold $\Phi_k$, defined as the number of a node’s neighbours who have adopted the service earlier.
In our method we first group nodes of the same degree, record their integer thresholds, and then calculate the threshold distribution for each degree group, as shown in the main text (Fig. 1e, inset). These distributions collapse to a master curve after normalization by using the scaling relation $P(\Phi_k,k)=k P (\Phi_k/k)$ (Fig. 1e, main panel). Moreover, this master curve can be well approximated by a lognormal distribution of the form, $$P(\phi)=\frac{1}{\phi \sigma_T\sqrt{2\pi}}e^{-\frac{(\ln\phi-\mu_T)^2}{2\sigma_T^2}},
\label{eq:thrLogn}$$ where $\mu_T=-2$ and $\sigma_T=1$, as determined by the empirical average threshold $w = 0.19$ and standard deviation (STD) $0.233$.
These findings indicate that although individual thresholds are strongly determined by degree, their distribution is degree-invariant, suggesting that the fraction of adopting friends rather than its absolute number is relevant during the service adoption process. The estimated empirical values of parameters are summarized in Table \[table:pars\].
$p_{n}$ $\langle k \rangle$ $\mu_D$ $\sigma_D$ $w$ $STD(\phi)$ $\mu_T$ $\sigma_T$
----------- --------------------- --------- ------------ -------- ------------- --------- ------------
$0.00019$ $8.56$ $1.2$ $1.39$ $0.19$ $0.233$ $-2$ $1$
: Estimated empirical parameters for service “buy credit”.[]{data-label="table:pars"}
Social influence - null model study {#sec:sinf}
===================================
Studies of social contagion phenomena assume that social influence is responsible for the correlated adoption of connected people. However, an alternative explanation for the observed correlated adoption patterns is homophily: a link creation mechanism by which similar egos get connected in a social structure. In the latter case, the correlated adoption of a connected group of people would be explained by their similarity and not necessarily due to social influence. Homophily and influence are two processes that may simultaneously play a role during the adoption process. Nevertheless, distinguishing between them on the individual level is very difficult using our or any similar datasets [@Shalizi2011]. Fortunately, at the system level one may decide which process is dominant in the empirical data. To do that first we need to elaborate on the differences between these two processes.
Influence-driven adoption of an ego can happen once one or more of its neighbours have adopted, since then their actions may influence the decision of the central ego. Consequently, the time ordering of adoptions of the ego and its neighbours matters in this case. Homophily-driven adoption is, however, different. Homophily drives social tie formation such that similar people tend to be connected in the social structure. In this case connected people may adopt because they have similar interests, but the time ordering of their adoptions would not matter. Therefore, it is valid to assume that adoption could evolve in clusters due to homophily, but these adoptions would appear in a random order.
To test our hypothesis we define a null model where we take the adoption times of each adopter and shuffle them randomly among all adopting egos. This way a randomly selected time is assigned to each adopter, while the adoption rate and the final set of adopters remain the same. Moreover, this procedure only destroys correlations between adoption events induced by social influence, but keeps the social network structure and node degrees unchanged. In this way, during the null model process the same egos appear as adopters, but the time series of adoption may in principle change (or not), corresponding to social influence (or homophily) as a dominant factor during the adoption process. If adoption is mostly driven by homophily, the rates of adoption would not change considerably beyond statistical fluctuations. On the other hand, if social influence plays a role in the process, rates of adoption in the null model should be very different from the empirical curves, implying that the time ordering of events matters in the adoption process. In this case, the rate of innovators should be higher than in the empirical data, since nodes that are in the adoption cluster originally but not directly connected would have a greater chance to appear as innovators, due to a random adoption time that is not conditional to the time ordering of the adopting neighbours.
After calculating the adoption rates of different user groups in the shuffled null model sequence we observe the latter situation: the rate of innovators becomes dominant, while the rate of stable and vulnerable adoptions drops considerably as they appear only by chance. This suggests that the temporal ordering of adoption events matters a lot in the evolution of the observed adoption patterns, and thus social influence may play a strong role here. Of course one cannot decide whether influence is solely driving the process or homophily has some impact on it; in reality it probably does to some extent. However, we can use this null model measure to demonstrate the presence and importance of the mechanism of influence during the adoption process.
Threshold model {#sec:thresModel}
===============
Model description {#ssec:modDesc}
-----------------
This model emulates the rise and temporal evolution of system-wide adoption cascades in complex social networks [@watts2002; @singh2013; @Ruan2015]. Note that this model has been introduced in [@Ruan2015], where its general scaling behaviour has been explored. In a system of fixed size, a node has social interactions with $k$ other agents and is characterized by a continuous adoption threshold $\phi$. When faced with the prospect of adopting a given innovation, product, or fad, susceptible individuals adopt spontaneously with rate $\pn$. Otherwise, the node adopts if at least a fraction $\phi$ of its $k$ neighbours have adopted before (the so-called ‘threshold rule’). Moreover, a fraction $r$ of the system is ‘immune’ to the innovation, in the sense that these agents never adopt regardless of their values of $k$ and $\phi$. The distributions of degrees and thresholds, $P(k)$ and $P(\phi)$ (as well as the values of $\pn$ and $r$), thus determine the average topological state and dynamical evolution of the system.
The model may be implemented numerically via a Monte Carlo simulation of the rules described above in a system of size $N$. Here, the dynamical state of the system is determined by the adoption state (0 or 1) of all nodes, which change in asynchronous random order in a series of time steps. Once an agent adopts and its state changes from 0 to 1, it remains so for the rest of the dynamics, thus ensuring a frozen final state for the finite system where no more adoptions arise. Each time step consists of $N$ node updates: In each node update, a randomly selected node (non-immune and in state 0) adopts spontaneously with probability $\pr = \pn / (1 - r)$ [^2]; else it adopts only if the threshold rule is satisfied. The rescaled rate $\pr$ is necessary if we wish to obtain a rate $\pn$ of innovators in early times of the dynamics, regardless of the value of $r$. Finally, we assume that agents with $k=0$ receive no social influence (for any value of $\phi$), and therefore can only adopt spontaneously. We will now explore this dynamics with numerical simulations and a rate equation formalism.
Stochastic binary-state dynamics {#ssec:stocDynam}
--------------------------------
Here we extend an approximate master equation (AME) formalism for stochastic binary-state dynamics as developed recently by Gleeson [@porter2014; @gleeson2013; @gleeson2008; @gleeson2011]. In a stochastic binary-state dynamics, each node in the network can take one of two possible states (susceptible or adopter in the language of innovation adoption) at any point in time, and state-switching happens randomly with probabilities that only depend on the current state of the updating agent and on the states of its neighbours. This general definition includes the threshold model described above as a special case. Such formalism considers configuration-model networks, that is, an ensemble of networks specified by the degree distribution $P(k)$ but otherwise maximally random [@newman2010].
All relevant properties used to describe a node are included in the vector $\kvec = (k, c)$, where $k = k_0, k_1, \ldots k_{M-1}$ is the degree of the node and $c = 0, 1, \ldots, M$ a dummy variable that labels its ‘type’, i.e. any other property that characterizes the node apart from its degree. In the case of our threshold model, $c = 0$ is the type of the fraction $r$ of immune nodes, while $c \neq 0$ labels the type of all non-immune nodes with given threshold $\phi_c$. The various values of $c \neq 0$ correspond then to different adoption thresholds $\phi_c$. The integer $M$ is the maximum number of degrees/types considered in the AME framework, which can be increased to improve the accuracy of the analytical approximation at the expense of speed in its numerical computation[^3]. Any pair of nodes with identical values of $\kvec$ are considered equivalent in this level of description, forming a node class with the same average dynamics. Moreover, $P(k)$ and $P(\phi)$ can be generalized to the joint distribution $\Pk$ giving the probability that a randomly selected node has property vector $\kvec$ (i.e. degree $k$ and type $c$). Here it is useful to define $P(c)$ as the distribution of all non-zero types, $c = 1, \ldots, M$. If degrees and thresholds are chosen independently, like in our model, then $\Pk = r P(k)$ for $c = 0$ and $\Pk = (1 - r) P(k) P(c)$ for $c > 0$. The distribution $P(c)$ is, in other words, a discrete, rescaled version of the continuous threshold distribution $P(\phi)$.
In the language of innovation adoption, the dynamics of a node is determined by the number $m = 0, 1, \ldots k$ of its neighbours that have already adopted when the node is deciding whether to adopt or not. During a small time interval $dt$, a susceptible node (in state 0) adopts with probability $\Fkm dt$, while an adopter (in state 1) becomes susceptible with probability $R_{\kvec, m} dt$. The functions $\Fkm$ and $R_{\kvec, m}$, known as infection and recovery rates, respectively, determine the temporal evolution of the node class $\kvec$. In the particular case of threshold models, a so-called monotone dynamics, $R_{\kvec, m} = 0$ $\forall\, \kvec, m$ (since no adopters become susceptible again). As for $\Fkm$, the rules of spontaneous and threshold adoption imply, $$\label{eq:thresRule}
\Fkm =
\begin{cases}
\pr & \text{if} \quad m < k \phi_c \\
1 & \text{if} \quad m \geq k \phi_c
\end{cases}, \quad \forall m \; \text{and} \; k, c \neq 0,$$ that is, a node adopts the innovation either spontaneously with rate $\pr$, or with probability 1 if its number of adopting neighbours equals or exceeds the integer threshold $\Phi_k = \lceil k \phi_c \rceil$. Immune nodes ($c = 0$) have an infection rate of $F_{(k,0),m} = 0$ $\forall k, m$, while for isolated nodes ($k = 0$) $F_{(0,c),0} = \pr$ $\forall c \neq 0$. In other words, immune nodes never adopt, and isolated nodes can only adopt spontaneously. We note that $\Fkm$ is written in terms of $\pr = \pn / (1 - r)$, not $\pn$, in order to counter the trivial decrease in the rate of spontaneous adoption for non-zero $r$.
Let us now turn to the rate equations for our threshold model, called AMEs in the formalism by Gleeson. We denote by $\skm(t)$ the fraction of $\kvec$-class nodes that are susceptible at time $t$ and have $m$ adopting neighbours. Therefore, the fraction of agents with property vector $\kvec$ that are adopters at time $t$ is $\pk(t) = 1 - \sum_{m=0}^k \skm (t)$, and the fraction of adopters in the system is $\rho (t) = \sumk \Pk \pk(t)$. Here, the sum over classes means a sum over all degrees and types, i.e. $\sumk \bullet = \sum_k \sum_c \bullet$. Assuming a monotone dynamics ($R_{\kvec, m} = 0$), the AMEs for $\skm$ can be written as [@porter2014; @gleeson2013; @gleeson2011], $$\label{eq:AMEsThres}
\frac{d \skm}{dt} = -\Fkm \skm -\bs (k - m) \skm + \bs (k - m + 1) \skmo,$$ where $m = 0, \ldots, k$, $s_{\kvec, -1} \equiv 0$, $\Fkm$ follows Eq. (\[eq:thresRule\]), and $\bs(t)$ (the rate at which edges between pairs of susceptible nodes transform to edges between a susceptible agent and an adopter) is given by, $$\label{eq:rateBs}
\bs(t) = \frac{\sumk \Pk \summ (k - m) \Fkm \skm(t)}{\sumk \Pk \summ (k - m) \skm(t)}.$$ If at time $t = 0$ we randomly choose a fraction $\rho (0) = \sumk \Pk \pk(0)$ of nodes as seed for the adoption process, the initial conditions for Eq. (\[eq:AMEsThres\]) are $\skm (0) = [1 - \pk(0)] \Bkm [\rho(0)]$, with $\pk(0)$ the initial fraction of adopters in class $\kvec$ and $\Bkm$ a binomial factor, $$\label{eq:BinomFac}
\Bkm(\rho) = \binom{k}{m} \rho^m (1 - \rho)^{k - m}.$$
The solution $\skm(t)$ of the AME system in Eq. (\[eq:AMEsThres\]) provides a very accurate description of the dynamics of our model, yet its dimension (i.e. number of equations to solve) grows quadratically with the number of degrees and linearly with the number of threshold values considered. Fortunately, the AMEs for our model can be mapped to a reduced-dimension system with a derivation similar to the one used by Gleeson in the case of the Watts threshold model [@watts2002; @singh2013].
Reduced-dimension AMEs {#ssec:redAMEs}
----------------------
To reduce the dimension of Eq. (\[eq:AMEsThres\]), we need to consider system-wide quantities that are more aggregated than $\skm$. One of them is the probability that a randomly chosen node is an adopter, $\rho(t) = 1 - \sumk \Pk \summ \skm (t)$, i.e. the fraction of adopters in the network. The other one is the probability that a randomly chosen neighbour of a susceptible node is an adopter, $\nu(t) = \sumk \Pk \summ m \skm(t) / \summ k \skm(t)$.
We start by proposing an exact solution for the AME system in terms of the following ansatz, $$\label{eq:AMEansatz}
\skm(t) = [1 - \pk(0)] \Bkm [\nu(t)] e^{-\pr t}
\quad \text{for} \; m < k\phi_c \; \text{and} \; c \neq 0,$$ and $s_{(k,0),m} = \Bkm(\nu)$ for $c = 0$, where $\Bkm$ follows Eq. (\[eq:BinomFac\]). The meaning of the ansatz in Eq. (\[eq:AMEansatz\]) is quite intuitive and considers two processes. First, a susceptible agent with degree $k$ and $m$ adopting neighbours is not selected as part of the initial adoption seed with probability $1 - \pk(0)$ and is connected to $m$ adopters with the binomially distributed probability $\Bkm(\nu)$. Second, for $m < k\phi_c$ a susceptible node does not fulfill the threshold rule and can only adopt spontaneously with probability $e^{-\pr t}$, since the system is progressively been filled by adopters. Considering these two processes as independent we end up with the product in Eq. (\[eq:AMEansatz\]). Finally, since immune nodes do not adopt and are distributed randomly over the network, $s_{(k,0),m}$ is determined only by a binomial factor.
The next step is to insert the ansatz (\[eq:AMEansatz\]) into the AME system (\[eq:AMEsThres\]) and derive a set of differential equations for the aggregated quantities $\rho$ and $\nu$. Taking the time derivative $\dskm$ of Eq. (\[eq:AMEansatz\]) (i.e. the left-hand side of Eq. (\[eq:AMEsThres\])) we get, $$\label{eq:ansatzINames1}
\dskm = \left( \left[ \frac{m}{\nu} - \frac{k-m}{1-\nu} \right] \dot{\nu} - \pr \right) \skm.$$ Then, we use the threshold rule (\[eq:thresRule\]) for $m < k\phi_c$, the ansatz (\[eq:AMEansatz\]) and the binomial identity, $$\label{eq:binomIdent}
\Bkmo(\nu) = \frac{1-\nu}{\nu} \frac{m}{k-m+1} \Bkm(\nu),$$ in the right-hand side of Eq. (\[eq:AMEsThres\]) to obtain, $$\label{eq:ansatzINames2}
-\Fkm \skm -\bs (k - m) \skm + \bs (k - m + 1) \skmo = \left[ -\pr + \bs \left( m - k + \frac{1-\nu}{\nu}m \right) \right] \skm.$$ Equating Eqs. (\[eq:ansatzINames1\]) and (\[eq:ansatzINames2\]) as in the AME system (\[eq:AMEsThres\]) leads to, $$\label{eq:condNu}
\dot{\nu} = \bs (1 - \nu),$$ a condition on $\nu$ so that the ansatz (\[eq:AMEansatz\]) is a solution of Eq. (\[eq:AMEsThres\]). This differential equation has the initial condition $\nu(0) = \rho(0)$, obtained by evaluating Eq. (\[eq:AMEansatz\]) at $t = 0$ and comparing with the expression $[1 - \pk(0)] \Bkm [\rho(0)]$, which corresponds to a random distribution of initial adopters among $\kvec$ classes. Furthermore, by assuming a (yet to be determined) function $g(\nu, t)$ such that $\dot{\nu} = g(\nu, t) - \nu$, Eq. (\[eq:condNu\]) reduces to, $$\label{eq:condBeta}
\bs = \frac{g(\nu, t) - \nu}{1 - \nu}.$$
Now, we consider the following general result derived by Gleeson in [@gleeson2013] (Eqs. (F6)–(F10) therein), $$\label{eq:GleesonEq}
\sumk \Pk \summ (k - m) \skm = z (1 - \nu)^2,$$ with $z = \sum_k k P(k)$ the average degree in the network. Eq. (\[eq:GleesonEq\]) is valid for functions $\skm$ and $\nu$ that satisfy Eqs. (\[eq:AMEsThres\]) and (\[eq:condNu\]) respectively, for any $\Fkm$ and random initial conditions on $\skm$ and $\nu$, and is thus applicable in our case. Our goal here is to use Eq. (\[eq:GleesonEq\]) to find an expression for $g(\nu)$ and therefore write the differential equation (\[eq:condNu\]) explicitly. Noting that the left-hand side of Eq. (\[eq:GleesonEq\]) is the denominator in the definition (\[eq:rateBs\]) of $\bs$ and that $F_{(k,0),m} = 0$ (i.e. immune nodes do not adopt), Eq. (\[eq:rateBs\]) gives, $$\begin{aligned}
\label{eq:betaExpl1}
\bs &= \frac{1 - r}{z (1 - \nu)^2} \left[ \pr \sumkc P(k) P(c) \summLess (k - m) \skm + \sumkc P(k) P(c) \summMore (k - m) \skm \right] \nonumber\\
&= \frac{1}{z (1 - \nu)^2} \left[ \sumk \Pk \summ (k - m) \skm - r \sum_k P(k) \summ (k - m) s_{(k,0),m} \right. \nonumber\\
&\quad \left. - (1 - r) (1 - \pr) \sumkc P(k) P(c) \summLess (k - m) \skm \right],\end{aligned}$$ where we have written $\Pk$ explicitly as $\Pk = r P(k)$ for $c = 0$ and $\Pk = (1 - r) P(k) P(c)$ for $c > 0$, in order to notice the dependence on $r$. Then, we insert the ansatz (\[eq:AMEansatz\]) (with its special case $s_{(k,0),m} = \Bkm(\nu)$ for immune nodes), as well as the identities $(k - m) \Bkm(\nu) = k (1 - \nu) \Bkom(\nu)$ and $\summLess \Bkom(\nu) = 1 - \summMore \Bkom(\nu)$ to obtain, $$\begin{aligned}
\label{eq:betaExpl2}
\bs &= \frac{1}{1 - \nu} \Bigg( (1 - r) \Bigg[ 1 - (1 - \pr) e^{-\pr t} \Bigg. \Bigg. \nonumber\\
&\quad \left. \left. + (1 - \pr) e^{-\pr t} \sumkc \frac{k}{z} P(k) P(c) \left( \pk(0) + [1 - \pk(0)] \summMore \Bkom(\nu) \right) \right] - \nu \right).\end{aligned}$$
A comparison of Eqs. (\[eq:condBeta\]) and (\[eq:betaExpl2\]) gives us the following expression for $g(\nu, t)$, $$\label{eq:gFactor}
g(\nu, t) = (1 - r) \left( \ft + (1 - \ft) \sumkc \frac{k}{z} P(k) P(c) \left[ \pk(0) + [1 - \pk(0)] \sum_{m \geq k\phi_c} \Bkom(\nu) \right] \right),$$ where we define $\ft$ as $\ft = 1 - (1 - \pr) e^{-\pr t}$. Thus, the AME system (\[eq:AMEsThres\]) gets reduced to the differential equation $\dot{\nu} = g(\nu, t) - \nu$, with $g(\nu, t)$ given explicitly by Eq. (\[eq:gFactor\]).
Even though the equation $\dot{\nu} = g(\nu, t) - \nu$ is closed and in this sense equivalent to Eq. (\[eq:AMEsThres\]), we can also derive the corresponding equation for $\rho$, since we are mainly interested in the temporal evolution of the fraction of adopters in the network. From the definition of $\rho$ and Eq. (\[eq:AMEsThres\]) we have, $$\begin{aligned}
\label{eq:rhoDeriv1}
\dot{\rho} = - \sumk \Pk \summ \dskm &= \sumk \Pk \summ \Fkm \skm \nonumber\\
&\quad + \bs \sumk \Pk \summ \big[ (k - m) \skm - (k - m + 1) \skmo \big],\end{aligned}$$ where the second term in the right-hand side telescopes to zero. Then, we use an algebraic manipulation similar to that of Eqs. (\[eq:betaExpl1\]) and (\[eq:betaExpl2\]) to obtain, $$\begin{aligned}
\label{eq:rhoDeriv2}
& \sumk \Pk \summ \Fkm \skm = (1 - r) \left( \pr \sumkc P(k) P(c) \summLess \skm + \sumkc P(k) P(c) \summMore \skm \right) \nonumber\\
&= (1 - r) \left( 1 - (1 - r) (1 - \pr) \sumkc P(k) P(c) \summLess \skm \right) - \rho \nonumber\\
&= (1 - r) \left( \ft + (1 - \ft) \sumkc P(k) P(c) \left[ \pk(0) + [1 - \pk(0)] \sum_{m \geq k\phi_c} \Bkm(\nu) \right] \right) - \rho.\end{aligned}$$ In this way, Eqs. (\[eq:rhoDeriv1\]) and (\[eq:rhoDeriv2\]) can be rewritten as $\dot{\rho} = h(\nu, t) - \rho$, where, $$\label{eq:hFactor}
h(\nu, t) = (1 - r) \left( \ft + (1 - \ft) \sumkc P(k) P(c) \left[ \pk(0) + [1 - \pk(0)] \sum_{m \geq k\phi_c} \Bkm(\nu) \right] \right).$$
Joining all of these results, the AME system (\[eq:AMEsThres\]) gets reduced to the system of two ordinary differential equations,
\[eq:reducedAMEs\] $$\begin{aligned}
\dot{\rho} &= h(\nu, t) - \rho, \\
\dot{\nu} &= g(\nu, t) - \nu,\end{aligned}$$
with the quantities $g(\nu)$ and $h(\nu)$ given explicitly by Eqs. (\[eq:gFactor\]) and (\[eq:hFactor\]).
The system (\[eq:reducedAMEs\]) can be solved numerically to obtain $\rho(t)$ and thus characterize the temporal evolution of the adoption process. Let us further separate the fraction of adopters as $\rho(t) = \rho_0(t) + \rho_1(t)$, where $\rho_0$ and $\rho_1$ are the fractions of innovators and induced adopters (i.e. vulnerable and stable nodes), respectively. Now consider the identity, $$\label{eq:suscepIdent}
1 - \rho = \sumk \Pk \summ \skm = r + (1 - r) \sumkc P(k) P(c) \summ \skm = r + \rho_s,$$ where $\rho_s(t)$ is the fraction of non-immune, susceptible nodes that can eventually adopt, either spontaneously or not. Since such susceptible nodes adopt spontaneously at a rate $\pr$, the rate equation for innovators is $\dot{\rho}_0 = \pr \rho_s$. Then, with Eq. (\[eq:suscepIdent\]) we obtain, $$\label{eq:innovRateEq}
\rho_0(t) = \pr \int_0^t [1 - r - \rho(t)] dt,$$ which can be calculated explicitly with the numerical solution of Eq. (\[eq:reducedAMEs\]).
Waiting time of adoption {#sec:tw}
========================
\
Another reason behind the non-rapid evolution of the adoption process could be the time users wait after their personal adoption threshold is reached and before adopting the service. This lag in adoption can be due to individual characteristics, or come from the fact that social influence does not spread instantaneously (as commonly assumed in threshold models, including ours). However, the waiting time $\tau_w$ can be estimated by measuring the time difference between the last adoption in a user’s egocentric network and the time of adoption. We define $\tau_w=0$ for innovators, but $\tau_w$ can take any positive value for vulnerable and stable adopters up to the length of the observation period.
Waiting times are broadly distributed for adopters (Fig. S\[fig:WaitingT\]a), meaning that many users adopt the service shortly after their personal threshold is reached, but a considerable fraction waits long before adopting the service. The heterogeneous nature of waiting times may be a key element behind the observed adoption dynamics. One way to figure out the effect of waiting times on the speed of cascade evolution is by removing them. We can extract waiting times from adoption times and thus calculate rescaled adoption times. The rescaled adoption time of a user is the last time when his/her fraction of adopting neighbours changed and the adoption threshold was (hypothetically) reached. After this procedure we can calculate a new adoption rate function by using rescaled adoption times and compare it to the original. From Fig. S\[fig:WaitingT\]b we can conclude that although adoption becomes faster, the rescaled adoption dynamics is still not rapid. On the contrary, it suggests that the rescaled adoption dynamics is still very slow and quite similar to the original. Consequently, waiting times cannot explain the observed dynamics of adoption.
Note that long waiting times can have a further effect on the measured dynamics. After the ‘real’ threshold of a user is reached and he/she waits to adopt, some neighbours may adopt the product. Hence all observed measures are in this sense ‘effective’: observed thresholds are larger or equal than real thresholds; the innovator rate is smaller or equal; the vulnerable and stable rates will be larger or equal; and waiting times will be shorter or equal than the real values. Consequently the process may be actually faster than that we observe in Fig. S\[fig:WaitingT\]b after removing effective waiting times. However, this bias becomes important only after the majority of individuals in the social network has adopted the service and the spontaneous emergence of adopting neighbours becomes more frequent. As the fraction of adopters in our dataset is always less than $6\%$ [@SkypeIPO], we expect minor effects of this observational bias on measurements.
Empirical and model cluster statistics {#sec:clustStats}
======================================
\
As described in the main text, we perform extensive model calculations using empirically determined parameters to estimate the only unknown parameter, the fraction of immune nodes $r$. We match the relative size of the largest connected component of the real adoption network with its corresponding measure in the model at the end of the observation period, and estimate the fraction of immune nodes in the real system as $r=0.73$. To support our estimation we also measure the distribution $P(d)$ of the depth of induced vulnerable trees and the correlation $\langle s_v \rangle (k)$ between the degree of innovator nodes and the average size of induced vulnerable trees in the model, and match them with the equivalent empirical measures. To provide further support for the estimated $r$ value we show the dependence of these quantities of different $r$ values.
We measure $P(d)$ and $\langle s_v \rangle (k)$ for $r=0.6$ and $0.9$, as well as for the predicted value $r=0.73$ (Fig. S\[fig:ClusStat\]). It is clear that both quantities scale with $r$. For smaller $r$ more nodes are susceptible for adoption, allowing deeper and larger vulnerable trees, while for larger $r$ no large induced cluster can emerge as the system is forced into a quenched state. Moreover, measures for the estimated $r$ value fit the empirical data considerably well. This collapse is remarkable, since we neglect any higher-order structural and temporal correlations in the model (like assortative mixing, community structure, bursty adoption patterns, periodic activity fluctuations, etc.), which are present in the empirical system. Differences in the tails of the measures are due to finite-size effects since the modelled network is two orders of magnitude smaller than the empirical social structure. Note that although we can look for an $r$ fraction that produces a better fit between model and data in terms of $P(d)$ and $\langle s_v \rangle (k)$, the collapse in Fig. S\[fig:ClusStat\] demonstrates the quality of an independent procedure of estimating $r$ (i.e. by matching the relative size of components). Therefore, these results are intended for validation only and not as a method to estimate the correct value of $r$.
Calculations for additional service {#sec:addserv}
===================================
\
Empirical observations
----------------------
In order to support our empirical observations and modelling of the social spreading of Skype, we examine the adoption dynamics of an additional paid service called “subscription”, introduced in April 2008 and with adoption data for over $42$ months until the end of the observation period. This service is only available for registered Skype users, and we can therefore use the accumulated static Skype network as background social structure. In order to investigate the adoption of this service we repeat all calculations described previously. First we measure the decoupled rate of innovator, vulnerable, and stable adopters (Fig. S\[fig:srv4RateTh\]a). We see that after a short initial period innovators adopt approximately with a constant rate, setting the model parameter to $p_n=0.00012$. Moreover, here innovators dominate social spreading since the rate of vulnerable and stable adoptions is relatively low.
We also measure the integer threshold distribution for different degree groups (Fig. S\[fig:srv4RateTh\]b, inset) just as described in Section \[sec:thrDistr\]. These distributions scale together after normalization with the scaling relation $P(\Phi_k,k)=k P (\Phi_k/k)$ (Fig. S\[fig:srv4RateTh\]b, main panel) and are well approximated by a lognormal distribution \[Eq. (\[eq:thrLogn\])\] with parameters $\mu_T=-3.73$ and $\sigma_T=1.39$, as determined by the average threshold $w=0.063$ and STD $0.153$. Note that since the adoption dynamics of this service is dominated by innovators, the average threshold $w$ is smaller than in the case of the “buy credit” service. All parameters are summarized in Table \[table:parssrv4\]. Since the background network is the same for both services, network parameters are those of Table.\[table:pars\].
\
Although the adoption process is dominated by innovators, a giant connected component evolves in the adoption network (Fig. S\[fig:srv4RateTh\]a, main panel). On the other hand, its relative size is considerable smaller than for the “buy credit” service. The stable adoption network is also dominated by a giant component, but its relative size is even smaller when compared to the adoption network (Fig. S\[fig:srv4RateTh\]a, inset). Moreover, the largest vulnerable trees are only two orders of magnitude smaller than the stable giant cluster (Fig. S\[fig:srv4RateTh\]b). For comparison, this difference is five order of magnitude for the “buy credit” service.
$p_{n}$ $w$ $STD(\phi)$ $\mu_T$ $\sigma_T$
----------- --------- ------------- --------- ------------
$0.00012$ $0.063$ $0.153$ $-3.73$ $1.39$
: Estimated empirical parameters for service “subscription”.[]{data-label="table:parssrv4"}
Model and validation
--------------------
![ Average size of the largest ($LC$) and 2nd largest ($LC^{2nd}$) components of the model network (‘Net’), model adoption network (‘Casc’), model stable network (‘Stab’), and induced vulnerable trees (‘Vuln’) as a function of $r$. Dashed lines show the observed relative size of the real $LC$ of the adopter network in 2011 (Fig. S\[fig:srv4adClust\], main panel) and the predicted $r$ value. The lower panel depicts the time $t_{50\%}$ when the adoption process has reached $50\%$ of the susceptible network as a function of $r$. We use $100$ realizations of configuration-model networks with size $N=10^5$ and lognormal degree distribution parametrized as described in Section \[sec:degDistrFit\]. Model calculations correspond to the parameters of Table \[table:parssrv4\] for $42$ iteration steps (matching the length of the observation period).[]{data-label="fig:srv4model"}](LC12_R_types_T100_fig_lgn_srv4.pdf){width="110mm"}
We repeat all model calculations with the parameters of the “subscription” service to see whether we can recover its adoption dynamics by using the dynamical threshold model introduced in the main text and in Section \[sec:thresModel\]. We check the dependence on $r$ of the average size of the largest connected component of the network ($LC$) of susceptible nodes available for the adoption process, the adoption network, the stable adoption network, and of vulnerable trees (Fig. S\[fig:srv4model\], upper panel). In addition we record the average size $LC^{2nd}$ of the second largest connected component (Fig. S\[fig:srv4model\], middle panel). Finally we show the time when the adoption process has reached the $50\%$ of available susceptible nodes in the adoption network (Fig. S\[fig:srv4model\], lower panel).
The $r$ dependence of the adoption process appears to be qualitatively similar to our earlier calculations on the “buy credit” service, but there are remarkable differences. Firstly, the crossover regime (depicted by the light grey area in Fig. S\[fig:srv4model\]) is shifted towards larger $r$ values due to the different threshold distribution and innovator adoption rate. Secondly, after matching the relative size of the largest connected component of the empirical adoption network (last point on the right-hand side of Fig. S\[fig:srv4adClust\], main panel), the predicted $r=0.928$ is out of the crossover regime. At this point the background social network is still not fragmented (as evidenced by the black line in Fig. S\[fig:srv4model\], which has not reached its maximum yet) and it allows for the emergence of large connected adoption clusters. It is very sparse, however, which explains: (a) the dominating innovator adoption rate observed empirically; (b) the reduced size of the giant component of the adoption and stable adoption networks; and (c) the relatively large innovator trees as compared to the stable adoption network components. We observe that the largest vulnerable trees are smaller than the largest stable clusters in the empirical data, while the opposite is true for the model. A possible explanation of this difference is the assumption in the model that the network is degree-uncorrelated. This is a necessary approximation in order to treat the model analytically, but it might not hold for the empirical network. All in all, this picture suggests that the “subscription” service is out of the rapid and even the crossover cascading regimes, and that its dynamics is mostly driven by independent innovators rather than social influence, on a network of which a large majority is not susceptible to innovation.
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[^1]: Corresponding author: [email protected]
[^2]: We define $\pr = 1$ for $\pn > 1 - r$.
[^3]: Explicitly, rather than using $k_0 = k_{\text{min}}$, $k_{M-1} = N - 1$ and $M = N - k_{\text{min}}$ (i.e, considering all possible degrees in the empirical/simulated network), we take a small $k_0 > k_{\text{min}}$ and large $k_{M-1} < N - 1$, $M < N - k_{\text{min}}$, with the other $M - 2$ degree values equidistantly distributed between $k_0$ and $k_{M-1}$, thus disregarding some degrees and gaining speed in the computation of the AMEs. Similarly, the $M$ threshold values corresponding to nonzero types are uniformly distributed in the open interval $(0, 1)$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper we study an integral invariant which obstructs the existence on a compact complex manifold of a volume form with the determinant of its Ricci form proportional to itself, in particular obstructs the existence of a Kähler-Einstein metric, and has been studied since 1980’s. We study this invariant from the view point of locally conformally Kähler geometry. We first see that we can define an integral invariant for coverings of compact complex manifolds with automorphic volume forms. This situation typically occurs for locally conformally Kähler manifolds. Secondly, we see that this invariant coincides with the former one. We also show that the invariant vanishes for any compact Vaisman manifolds.'
address:
- 'Department of Mathematics, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro, Tokyo 152–8551, Japan'
- 'Department of Mathematics, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro, Tokyo 152–8551, Japan'
- 'Univ. of Bucharest, Faculty of Mathematics, 14 Academiei str, 70109 Bucharest, Romania, and Institute of Mathematics “Simion Stoilow" of the Romanian Academy, 21, Calea Grivitei str., 010702-Bucharest, Romania.'
author:
- Akito Futaki
- Kota Hattori
- Liviu Ornea
date: 'May 24, 2011'
title: An integral invariant from the view point of locally conformally Kähler geometry
---
Introduction
============
In [@futaki83.1], the first author introduced an integral invariant defined on Fano manifolds and showed that it obstructs the existence of Kähler-Einstein metrics. More precisely, if $M$ is a Fano manifold of dimension $n$ and $$\omega = i\, g_{i{{\overline j}}}\, dz^i \wedge d{{\overline z}}^j$$ is a Kähler form representing $2\pi c_1(M)$, there exists a real smooth function $F \in C^\infty(M)$ such that the Ricci form $$\rho(\omega) = - i \partial {{\overline \partial}}\log \det g$$ is written as $$\rho(\omega) - \omega = i \partial {{\overline \partial}}F$$ since both $\rho(\omega)$ and $\omega$ represent $2\pi c_1(M)$. Then the invariant is defined as a character $f$ of the Lie algebra $\mathfrak h(M)$ of all holomorphic vector fields on $M$ into $\mathbb C$ and is expressed for $X \in \mathfrak h(M)$ by $$\label{invariant1} f(X) = \int_M\, XF\, \omega^n.$$ This invariant was later extended in various ways. We first briefly review the various extension.
The first line of extension is as invariants for compact Kähler manifolds with fixed Kähler class. Let $(M, [\omega])$ be a compact Kähler manifold $M$ with a Kähler class $[\omega]$. Then we can extend $f$ as an obstruction to the existence of a Kähler form in $[\omega]$ of constant scalar curvature ([@futaki83.2], [@calabi85]). This is defined by the same formula (\[invariant1\]) if we replace the condition of $F$ by $$\sigma - \int_M \sigma \omega^n/\vol(M) = \Delta F$$ where $\sigma$ denotes the scalar curvature of $\omega$. When $[\omega]$ is an integral class this was further reformulated by Donaldson [@donaldson02] as an invariant for polarized schemes, and was used to define the notion of K-stability. In a guise the reformulated invariant was expressed as slopes for subschemes by Ross and Thomas [@rossthomas06]. For Fano manifolds with the anticanonical class, the invariant $f$ has recently been extended to an obstruction to the existence of Kähler-Einstein metrics with cone singularities along a divisor (Donaldson [@donaldson1102], Li [@ChiLi1104]), and it is used to define logarithmic K-stability. Around the same time as the work [@futaki83.2] and [@calabi85], the invariant $f$ obstructing the constant scalar curvature Kähler metric was extended further by Bando [@bando83] to a family of invariants $f_k$, $k=1, \cdots, n$, where $f_k$ obstructs the existence of a Kähler form in $[\omega]$ such that the $k$-th Chern form $c_k(\omega)$ is harmonic. Notice that the scalar curvature is constant if and only if the first Chern form is harmonic by the second Bianchi identity. Thus $f_1$ coincides with $f$. Bando’s idea can be further extended to transverse Kähler geometry of compact Sasaki manifolds [@FOW].
The second line of extension was obtained in [@futakimorita85], but this extension is obtained by relating the invariant $f$ for Fano manifolds to invariants classically known in the theory of the equivariant cohomology. Again, let $M$ be a Fano manifold and $\omega = i\, g_{i{{\overline j}}}\, dz^i \wedge d{{\overline z}}^j$ is a Kähler form representing $2\pi c_1(M)$. By the solution by Yau [@yau77] to the Calabi conjecture, there exists a Kähler form $\eta$ representing $2\pi c_1(M)$ such that $\rho(\eta) = \omega$. Then we can rewrite $f$ as in (\[invariant1\]) in terms of $\eta$ and obtain \[invariant2\] f(X) = \_M X ()\^n where \[invariant3\] X\^n = i(X) \^n and $i(X)$ denotes the interior product by $X$, see [@futakimorita85] or [@futaki88] for the detail. Note that, instead of the Kähler form $\eta$, we may use any volume form $\Omega$ and its Ricci form $$\rho(\Omega) = - i \p\bp \log \O.$$ Then we may write (\[invariant2\]) as \[invariant4\] f(X) = \_M X (Ø)\^n where \[invariant5\] XØ= i(X) Ø. We can prove that $f$ is then independent of the choice of $\Omega$, and thus we do not need to assume that $M$ is Fano or Kähler. Thus we obtained an invariant for (possibly non-Kähler) complex manifolds. This last invariant is the one we wish to study in this paper. Note also that we can rewrite (\[invariant4\]) as \[invariant6\] f(X) = - \_M X () . Therefore the invariant $f$ is an obstruction to the existence of a volume form $\O$ such that $\rho(\O)^n/\O$ is constant.
We remark in passing that there is a larger family of invariants including these two lines of extension ([@futaki04-1]). Among them we have a family of invariants which obstructs asymptotic Chow semistability of polarized manifolds ([@futaki04-1], [@FOS08]). By computing them for a 7-dimensional toric Fano manifold suggested by Nill and Paffenholz [@NillPaffen], Ono, Sano and Yotsutani [@OSY09] showed that there is a Kähler-Einstein Fano manifold which is asymptotically unstable.
Now let us turn to the study of the invariant defined by (\[invariant4\]) or (\[invariant6\]). Let $M$ be a compact connected complex manifold of dimension $n$. Consider a covering space $\pi : \wt M \to M$ with the group $\Gamma$ of the deck transformations, and let $\chi : \Gamma \to \mathbb R^+$ be a homomorphism. A volume form $\Omega$ on $\wt M$ is said to be automorphic with respect to $\chi$ if, for any $\gamma \in \Gamma$, $$\label{Intro1}
\gamma^\ast \Omega = \chi(\gamma)\Omega.$$ Such a covering with automorphic volume form naturally occurs for locally conformally Kähler manifolds as we shall see in the next section.
Given such a covering $\wt M$ with automorphic volume form with respect to $\chi$ we have a Ricci form $\rho_\O$ of $\O$ defined on $\wt M$ by \[Intro2\] \_Ø= - i Ø. Since $\O$ is automorphic, $\rho_\O$ is invariant under the action of $\Gamma$, and thus descends to a $2$-form on $M$ which is denoted by the same notation $\rho_\O$. This represents the first Chern class $2\pi c_1(M)$, and also $2\pi c_1(\wt M)$ upstairs if $\wt M$ is compact.
Denote by $\mathfrak h(M)$ and $\mathfrak h(\wt M)$ the Lie algebras of all holomorphic vector fields on $M$ and $\wt M$ respectively. Denote also by $\mathfrak h_\Gamma (\wt M)$ the Lie subalgebra of $\mathfrak h(\wt M)$ consisting of all holomorphic vector fields on $\wt M$ which are invariant under the action of $\Gamma$. Then a holomorphic vector field in $\mathfrak h_\Gamma(\wt M)$ descends to a holomorphic vector field on $M$, and thus $\mathfrak h_\Gamma(\wt M)$ can be naturally regarded as a Lie subalgebra of $\mathfrak h(M)$. For an $X$ in $\mathfrak h_\Gamma(\wt M)$, its divergence $\div X$ is defined by \[Intro3\] XØ= i(X) Øwhere $i(X)$ denotes the interior product by $X$. Since $\O$ is automorphic and $X$ is invariant under $\Gamma$ it follows that $\div X$ is invariant under $\Gamma$ and that $\div X$ descends to a smooth function on $M$.
We define a linear function $f : \mathfrak h_\Gamma(\wt M) \to {{\mathbb C}}$ by \[Intro4\] f(X) = \_M X \_Ø\^n.
The main theorem of this paper is the following.
\[Intro5\] $\mathrm{(a)}$ Let $M$ be a compact connected complex manifold and $\wt M$ its covering space with the group $\Gamma$ of deck transformations. Suppose that we are given a character $\chi : \Gamma \to \mathbb R^+$. Then $f$ is independent of the choice of the volume form automorphic with respect to $\chi$.\
$\mathrm{(b)}$ The invariants defined by (\[invariant4\]) and (\[Intro4\]) coincide.\
A locally conformally Kähler manifold $(M,J,g)$ is said to be a Vaisman manifold if there is a metric in the conformal class of $g$ for which the Lee form is parallel, see section 2 for more detail.
\[Intro6\] The invariant in the previous theorem vanishes for any compact Vaisman manifold.
This paper is organized as follows. In section 2 we summarize the basics of locally conformally Kähler geometry, and give a proof of Theorem \[Intro6\]. In section 3 we give a proof of Theorem \[Intro5\]. In section 4 we compute the invariant for the one point blow-up of the Hopf surface, and see that this surface gives an example of nontrivial invariant.
Locally conformally Kähler manifolds
====================================
Let $(M,J)$ be a connected complex manifold of complex dimension $n \ge 2$ with $J$ a complex structure. A locally conformally Kähler structure (LCK structure for short) on $(M,J)$ is a covering $$\Gamma \to (\widetilde M,\wt J,\wt \o) \to (M,J)$$ where $\widetilde M$ is a covering space of $M$, $\wt \o$ a Kähler form on $\widetilde M$, and $\Gamma$ the group of deck transformations acting on $\widetilde M$ as holomorphic homotheties. Thus there is a homomorphism $\chi : \Gamma \to \mathbb R^+$ satisfying $$\gamma^\ast \wt \o = \chi(\gamma) \wt \o.$$ A $p$-form $\alpha$ on $\widetilde M$ ia said to be automorphic if $\gamma^\ast \alpha = \lambda(\gamma) \alpha$ for some character $\lambda : \Gamma \to \mathbb R^+$. The above Kähler form $\wt \o$ is an example of an automorphic $2$-form.
There is an equivalent definition of an LCK structure described as follows. An LCK structure is a collection of an open covering $M = \cup_{\alpha \in \Lambda} U_\alpha$ and Kähler metrics $g_\alpha$ on $U_\alpha$ satisfying $$g_\alpha = c_{\alpha\beta} g_\beta$$ on $U_\alpha \cap U_\beta$ with $c_{\alpha\beta} \in \mathbb R_+$. Then $\{c_{\a\b}\}$ gives a cocycle. Let $\theta$ be a representative as a closed one form defining the same cohomology class as $\{\log c_{\a\b}\}$. Thus we have $d\theta = 0$, and locally $\theta|_{U_\a} = df_\a$ for a smooth function $f_\a$ on $U_\a$ with $f_\b - f_\a = \log c_{\a\b}$ and $e^{f_\a}g_\a = e^{f_\b} g_\b$ on $U_\a \cap U_\b$. Therefore $g := e^{f_\a}g_\a$ defines a global Hermitian metric locally conformal to a Kähler metric. The $1$-form $\theta$ is called the Lee form. Let $\omega$ be the fundamental $2$-form defined by $$\omega(X,Y) = g(JX,Y).$$ Then one easily shows that $$\begin{aligned}
d\omega &=& \theta \wedge\omega \label{lck1}\\
d\theta &=& 0. \label{lck2}
\end{aligned}$$ As an equivalent third definition we may say that an Hermitian manifold $(M,J,g)$ is a locally conformally Kähler manifold if the fundamental $2$-form $\omega$ of $g$ satisfies (\[lck1\]) and (\[lck2\]).
When we say $(M,J,g)$ is an LCK manifold we assume that $\theta \ne 0$, that is, $(M,J,g)$ is not globally Kähler.
The equivalence between the second and the third definitions is obvious. To see that first implies the third, suppose that we are in a situation of the first definition. Let $L$ be the $\mathbb R$-bundle given by $\wt M \times_\chi \mathbb R$. Since $\chi$ is $\mathbb R^+$-valued, $L$ is oriented and thus is a trivial bundle. It follows that $L$ has a nowhere zero section which defines a positive $\chi$-equivariant function $\phi$ on $\wt M$. Then $\omega := \phi^{-1}\wt\omega$ is a $\Gamma$-invariant positive $2$-form. This $\omega$ satisfies the third definition with $\theta = - \log \phi$.
We need only to show that the second implies the first. Suppose that we have Kähler forms $\omega_\a$ on $U_\a$ such that $\omega_\a = c_{\a\b} \omega_\b$ with $c_{\a\b} \in \mathbb R^+$. Then $\{c_{\a\b}\} \in H^1(M, \mathbb R^{+\delta})$ defines a flat principal $\mathbb R^+$-bundle. The holonomy gives a character $\chi : \Gamma = \pi_1(M) \to \mathbb R^+$. Let $L = \wt M \times_\chi \mathbb R$ be the associated $\mathbb R$-bundle. We may regard $\{\omega_\a\}$ as a section of $$L\otimes \Lambda^2T^{\ast} M = (\wt M \times_\chi \mathbb R)\otimes \Lambda^2T^\ast M = \wt M \times_\chi (\mathbb R \otimes p^\ast\Lambda^2T^{\ast} M)$$ where $p : \wt M \to M$ is the projection. Thus $\{\omega_\a\}$ defines a $\chi$-equivariant closed $2$-form $\wt \o$ on $\wt M$. This completes the equivalence of the three definitions of LCK structures.
Recall that a locally conformally Kähler manifold $(M,J,g)$ is said to be a Vaisman manifold if there is a metric in the conformal class of $g$ for which the Lee form is parallel. It is shown in [@KamishimaOrnea05] that a Vaisman manifold is obtained as a quotient of the Kähler cone $C(S)$ of a Sasakian manifold $S$ by a subgroup $\Gamma$ of the homotheties acting freely and properly discontinuously. Then the proof follows from Lemma \[Ricci\] below since for the Reeb vector field $\xi = Jr\frac\p{\p r}$ on $C(S)$, $\xi - iJ\xi$ is a holomorphic flow and the Ricci tensor degenerates on this orbit. See the arguments below for the notations.
Recall that a Riemannian manifold $(S, g)$ of dimension $2m + 1$ is a Sasakian manifold if the cone $(C(S), \bar g) = (\mathbb R^+ \times S,
dr^2 + r^2 g)$ is a Kähler manifold. Here $r$ is the standard coordinate on $\mathbb R^+$. The metric $\bar g$ is a warped product metric, and the Riemannian geometry of $C(S)$ is easily studied from that of $S$. Let $\bar\nabla$ and $\nabla$ be the Levi-Civita connections on $C(S)$ and $S$ respectively. Let $X, Y$ be tangent vector fields on $S$, which are naturally regarded as vector fields on $C(S)$ by the product structure $C(S) = \mathbb R^+ \times S$. Consider a vector field $\Psi := r\frac \p{\p r}$ on $C(S)$. It is generally true for cone manifolds that \[Ricci1\] |\_X = |\_ X = X and that \[Ricci2\] |\_X Y = \_X Y - g(X,Y) . Let $\bar R$ be the curvature tensors on $C(S)$. Then using (\[Ricci1\]) and (\[Ricci2\]) we obtain $$\begin{aligned}
\bar R(X,\Psi, Y, \Psi) &=& \bar g (- \bar\nabla_X \bar\nabla_{\Psi} Y + \bar\nabla_{\Psi}\bar\nabla_X Y + \bar\nabla_{[X, \Psi]}Y, \Psi\\
&=& \bar g (- \bar\nabla_X Y + \bar\nabla_{\Psi}(\nabla_X Y - g(X,Y)\Psi), \Psi)\\
&=& - \bar g(\nabla_X Y - g(X,Y)\Psi, \Psi) + \bar g(\nabla_X Y - g(X,Y)\Psi, \Psi)\\
&=& 0.\end{aligned}$$ This implies that \[Ricci3’\] |Ric(, ) = 0 where $\bar Ric$ denotes the Ricci tensor on $C(S)$. Let $J$ be the complex structure on $C(S)$. The vector field $\xi = Jr\frac\p{\p r}$ on $C(S)$ is called the Reeb vector field, and it is a standard fact in Sasakian geometry that $\xi - iJ\xi$ is a holomorphic vector field. Since the Ricci tensor on a Kähler manifold is $J$-invariant, (\[Ricci3’\]) implies \[Ricci4’\] |Ric(, ) = 0. From (\[Ricci3’\]) and (\[Ricci4’\]) we have proved the following.
\[Ricci\] The Ricci tensor on $C(S)$ vanishes on the plane spanned by $\xi$ and $J\xi = - r \frac \p{\p r}$.
For a Vaisman manifold $M$ we can also find an LCK metric $g$ for which the Ricci form $\rho(g)$ satisfies $\rho(g)^n = 0$, showing that the integrand of (\[invariant4\]) and (\[invariant6\]) vanishes. Recall that the Kähler form $\wt \o$ on $C(S)$ is written as $\wt \o = dd^c r^2$, see for example [@FOW]. As we have seen in Lemma \[Ricci\], we have $\rho(\wt \o)^{m+1} = 0$. Note that $n= m+1$ here. Then $\wt \o/r^2 = \frac 1 {r^2} dd^c r^2$ defines a $\Gamma$-invariant 2-form and descends to $M$. Since $dd^c \log r$ is the transverse Kähler form $\omega^T$ on the Sasaki manifold $S$ (but regarded as lifted to $C(S)$), the Ricci form of $\wt \o/r^2$ is equal to $\rho(\wt \o) + 2(m+1)\omega^T$. This also degenerates on the orbit of the flow generated by $\xi - iJ\xi$. Hence we have $\rho(\wt \o/r^2)^{m+1} = 0$ on the Vaisman manifold $M$.
Proof of Theorem \[Intro5\]
===========================
In this section we prove the following result.
\[result1\] Let $M$ be a compact connected complex manifold and $\wt M$ its covering space with the group $\Gamma$ of deck transformations. Suppose that we are given a character $\chi : \Gamma \to \mathbb R^+$. Then $f$ defined by (\[Intro4\]) is independent of the choice of the automorphic volume form $\Omega$ and its character $\chi$.
Theorem \[Intro5\] follows from Theorem \[result1\] because (a) in Theorem \[Intro5\] is obtained by comparing $(\O_1, \chi)$ and $(\O_2, \chi)$, and (b) in Theorem \[Intro5\] is obtained by comparing $(\O_1, \chi)$ and $(\O_2, 1)$.
Let $M$ be a compact connected complex manifold and $\wt M$ be a covering space of $M$ with the group $\Gamma$ of deck transformations. Let $\O$ be a smooth volume form on $\wt M$ automorphic with respect to $\chi : \Gamma
\to \mathbb R^+$. If $z^1, \cdots, z^n$ are local holomorphic coordinates on $\wt M$, the volume form $\O$ can be expressed as \[proof1\] Ø= aidz\^1 d[[z]{}]{}\^1 i dz\^n d[[z]{}]{}\^n where $a$ is a local positive smooth function. The Ricci form $\rho_\O$ and the divergence $\div X$ can be expressed using $a$ as \[proof2\] \_Ø= - ia, and \[proof3\] X = \_[i=1]{}\^n + X a. From (\[proof3\]) we obtain \[proof4\] X = i(X) a.
Let $\O_0, \O_1$ be volume forms automorphic with respect to $\chi_0,\chi_1 : \Gamma \to \mathbb R^+$, respectively. Then $\O_i$ can be expressed as Ø\_0 &=& a idz\^1 d[[z]{}]{}\^1 i dz\^n d[[z]{}]{}\^n,\
Ø\_1 &=& a idz\^1 d[[z]{}]{}\^1 i dz\^n d[[z]{}]{}\^n, where the positive real valued function $\varphi$ on $\wt M$ is given by $\O_1 = \varphi \O_0$. Then we have \[proof6\] \^\*= for all $\gamma\in \Gamma$.
Let $\O_t$ be \[proof7\] Ø\_t = \^t a idz\^1 d[[z]{}]{}\^1 i dz\^n d[[z]{}]{}\^n, for each $0 \le t \le 1$. Then each $\O_t$ is automorphic with respect to a character $\chi_t:=\chi_0^{1-t}\chi_1^t$. Thus a smooth family of linear maps $f_t:\mathfrak h_\Gamma (\wt M) \to \mathbb{C}$ is defined by \[proof8\] f\_t(X) = \_M \_t X \_[Ø\_t]{}\^n, where $\div_t X$ is the divergence determined by $\O_t$. Then it suffices to show that \[proof9\] f\_t(X) = 0 for all $X \in \mathfrak h_\Gamma (\wt M)$.
It is easy to see \[proof10\] d[dt]{} (\_t X) = X() and \[proof11\] d[dt]{} \_[Ø\_t]{} = - i. Then we have && d[dt]{} \_M \_t X \_[Ø\_t]{}\^n\
&=& \_M X( ) \_[Ø\_t]{}\^n - \_M \_t X i( ) n\_[Ø\_t]{}\^[n-1]{}\
&=& \_M X( ) \_[Ø\_t]{}\^n + \_M (\_tX (i ) n\_[Ø\_t]{}\^[n-1]{})\
&& - \_M (\_tX) (i ) n\_[Ø\_t]{}\^[n-1]{}. Although $\varphi$ is not $\Gamma$-invariant, $\p \log \varphi$ is $\Gamma$-invariant from (\[proof6\]) and descends to a $1$-form on $M$. Since $\div_tX$ and $\rho_{\O_t}$ are also defined globally on $M$, we can deduce \[proof12\] \_M (\_tX (i ) n\_[Ø\_t]{}\^[n-1]{}) = 0 from Stokes’ Theorem. Therefore we have && d[dt]{} \_M \_t X \_[Ø\_t]{}\^n\
&=& \_M X( ) \_[Ø\_t]{}\^n - \_M (\_tX) (i ) n\_[Ø\_t]{}\^[n-1]{}\
&=& \_M X( ) \_[Ø\_t]{}\^n\
&& - \_M (i(X)(\^t a)) (i) n\_[Ø\_t]{}\^[n-1]{}\
&=& \_M X( ) \_[Ø\_t]{}\^n + \_M (i(X)\_[Ø\_t]{}\^n) \
&=& \_M i(X) (\_[Ø\_t]{}\^n ) = 0 since $\rho_{\O_t}^n \wedge \p \log \varphi \equiv 0$ because of dimension reasons. This completes the proof of Theorem \[result1\].
In general the vanishing of $f$ is the obstruction to the existence of an automorphic volume form on $\wt M$ with $\rho_\O = 0$. But if $\wt M = M$ or $\chi$ is trivial, the vanishing of $f$ obstructs the existence of a volume form with $\rho_\O^n = k\O$ for some constant $k$.
The localization formula and an example
=======================================
Now that the invariant is independent of the choice of $(\O, \chi)$ we may use an old result to compute the case when $\chi$ is trivial. This is a residue formula for holomorphic vector fields.
Let $X$ be a holomorphic vector field in $\frak h (M)$. Define a section $L(X)$ of the endomorphism bundle $\mathrm{End}(TM)$ of the holomorphic tangent bundle $TM$ by \[proof13\] L(X)Y = \_XY - \[X,Y\]. Suppose that the zero set $\mathrm{zero}(X)$ of $X$ consists of smooth submanifolds $\{Z_\lambda\}_{\lambda \in \Lambda}$. Then $L(X)$ induces a section $L^\nu(X)$ of the endomorphism bundle of the normal bundle $\nu(Z_\lambda) = (TM|_{Z_\lambda})/TZ_{\lambda}$ of $Z_\lambda$.
\[zero1\] If $L^\nu(X)$ is nonsingular at every $q \in \mathrm{zero}(X)$, we have the following localization formula (1[2]{})\^n (n+1)f(X) = \_\_[Z\_]{}((X + c\_1(M))\^[n+1]{}|\_[Z\_]{})/(L\^(X) + i[2]{}K) where $K$ is the curvature form of $\nu(Z_\lambda)$ with respect to the induced Hermitian connection.
We provide an explicit computation of non-zero invariant on the blow-up at a point of a Hopf surface which by [@tr; @vu] is an LCK manifold, using the localization formula.
Let $H^2$ be a Hopf surface that we regard as $\CM^2\setminus \{0\}/\ZM$, where $\ZM$ is generated by the transformation $(z_1,z_2)\mapsto (2z_1,2z_2)$. We choose the fundamental domain on $\CM^2\setminus \{0\}$ to be $\{(z_1,z_2)\ |\ 1 \leq |z_1|^2+|z_2|^2\leq 2\}$.
Let $M$ be the blow-up of $H^2$ at the point $(0, \frac 32)$. It will be convenient to change the coordinates $(z_1,z_2)$ into $(w_1,w_2)$ by: $$w_1=z_1,\quad w_2=z_2-\frac 32.$$ Then the blow-up takes place at the origin $(w_1,w_2)=(0,0)$ and the exceptional divisor $E$ is $\{(w_1:w_2)\}\cong \CM P^1\subset M$.
Let $X=z_1\frac \p{\p z_1}$ be the radial (global) vector field on $\CM^2$. Its zero set contains $(0,\frac 32)$. We shall equally denote by $X$ its lift to $M$. Its zero set on $M$ will certainly contain $\{z_1=0\}$, but also some other point that we now determine.
Take first local coordinates on $M$ around $(1:0)\in E$ to be $$\z_1=w_1, \quad \z_2=\frac{w_2}{w_1}.$$ This change of coordinates is consistent with the coordinates on the exceptional divisor. In the new coordinates, $X$ is written as $$X=\z_1\frac \p{\p \z_1}-\z_2\frac \p{\p \z_2}.$$ In these coordinates $(\z_1,\z_2)$, $\mathrm{zero}(X)=\left\{ (0,0)\right\}.$ Hence, the zero is on $E$ (as $\z_1=0$). On the other hand, $\z_2=0$ implies $w_2=0$. Thus, the isolated zero of $X$ is $(w_1:w_2)=(1:0)$.
Now recall that, in general, if a holomorphic vector field $Y=a\frac \p{\p\z_1}+b\frac \p{\p\z_2}$, then $$\label{zero}
L(Y)(\frac \p{\p\z_j})|_{\mathrm{zero}(Y)}=-\Ll_Y \frac \p{\p\z_j}+\nabla_X\frac \p{\p\z_j}=\frac{\p a}{\p\z_j}\frac \p{\p\z_1}+\frac{\p b}{\p\z_j}\frac \p{\p\z_2},$$ as the $\nabla_Y=0$ on the zero set of $Y$.
Hence, in our case, for the point $(1:0)$, the localization formula reduces to: $$\frac{\tr(L(X))^3}{\det(L^\nu(X))}=\frac{\left(\tr\begin{pmatrix}1&0\\0&-1\end{pmatrix}\right)^3}{\det \begin{pmatrix}1&0\\0&-1\end{pmatrix}}=0,$$ as the normal bundle of the point equals the tangent bundle at the point, and this is trivial, hence $\Theta=0$.
So, the isolated zero does not contribute to the value of the invariant.
For the dimension 1 component of $\mathrm{zero}(X)$, take on $M$, around $(0:1)$, the coordinates: $$\z_1=w_2,\quad \z_2=\frac{w_1}{w_2}.$$ In these coordinates $X$ takes the form $$X=\z_2\frac \p {\p\z_2},$$ and $\mathrm{zero}(X)=\left\{(\z_1,0)\right\}$, a line represented by $(0:1)$. It is the proper transform of $\{z_1=0\}$. Using , we find now $$L(X)=\begin{pmatrix}0&0\\0&1\end{pmatrix}.$$ The localization formula gives: $$\int_Z\frac{\left(\tr \left( \begin{pmatrix}0&0\\0&1\end{pmatrix} +\frac{\sqrt{-1}}{2\pi}\Theta\right)\right)^3}{1+\frac{\sqrt{-1}}{2\pi}\Theta^\nu}=
\int_Z\frac{\left(1+c_1(Z)+c_1(\nu(Z))\right)^3}{1+c_1(\nu(Z))},$$ where $\nu(Z)$ is the normal bundle of the zero set $Z=\mathrm{zero}(X)$.
Observe that $\nu(Z)=-[E]$. Indeed, if $\pi:M \to H^2$ denotes the natural projection, then: $$[\pi(Z)]=[\pi(Z+E)],$$ and hence (as they are trivial line bundles), $$0=\pi^*[\pi(Z)]=[Z+E]=[Z]+[E]=[\nu(Z)]+[E].$$ On the other hand, $c_1(Z)=0$, as $Z$ is an elliptic curve. We obtain: $$\begin{aligned}
\int_Z\frac{\left(1+c_1(Z)+c_1(\nu(Z))\right)^3}{1+c_1(\nu(Z))}
&=& \int_Z(1-c_1([E]))^3 (1+c_1([E])))\\
&=& \int_Z(-3c_1([E]))+c_1([E])))\\
&=& Z\cdot (-2[E]))=-2.\end{aligned}$$ In conclusion, $3(\frac 1{2\pi})^2f(X)=-2\neq 0$.
[100]{}
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We give an analytical derivation of the mass gap of the $O(N)$ sigma models and investigate a large-order behavior of the weak coupling asymptotic expansion for the energy. For sufficiently large $N$ the series is sign-oscillating, which is expected from the large $N$ solution of the sigma model. However, for $N=3$ and $N=4$ the series are sign positive.'
author:
- Dmytro Volin
title: 'From the mass gap in $O(N)$ to the non-Borel-summability in $O(3)$ and $O(4)$ sigma-models'
---
Introduction
============
The $O(N)$ sigma-model is often considered as a toy model for the quantum cromodynamics. It is asymptotically free and dynamically generates a mass scale $\Lambda$ (analog of $\Lambda_{QCD}$), although its classical formulation does not contain any dimensionful parameters. It is widely accepted that the asymptotic states of the sigma model are the massive particles in the vector representation of the $O(N)$ group. The mass of the particles should be proportional to the only mass scale of the theory: $\mass=c\ \Lambda$.
The mass $\mass$ is a physical quantity, while $c$ and $\Lambda$ depend on the regularization scheme. The coefficient $c$ cannot be determined from the perturbation theory. Luckily, the $O(N)$ sigma-model can be studied nonperturbatively due to its integrability. The explicit expression for the coefficient $c$ in the $\overline{MS}$ scheme was found in [@Hasenfratz:1990ab; @Hasenfratz:1990zz]: \[c\] c=(8e)\^[1[N-2]{}]{}1. We explain the idea of this calculation in the next section. In order to obtain $c$, one has to solve the integral equation (\[iequation\]) in the weak-coupling regime at the leading and subleading order. At the leading order the solution was found by the application of the generalized Wiener-Hopf method [@Forgacs:1991rs]. The subleading order, up to our knowledge, was found only numerically, although with a high precision, which allowed to conjecture the expression (\[c\]). The goal of this paper is to solve the integral equation (\[iequation\]) analytically at subleading and higher orders in a recursive manner.
Another motivation for this work comes from the AdS/CFT correspondence where the appearance of the O(6) sigma-model raised the necessity of the solution of the equation (\[iequation\]) and its generalization at next-to the subleading order [@Bajnok:2008it].
Armed with the recursive procedure, we made an estimation for the large-order behavior of the coefficients of expansion and found unexpected properties for $N=3$ and $N=4$ (see the discussion session).
Integral equation
=================
The idea of [@Hasenfratz:1990ab; @Hasenfratz:1990zz] was to consider the sigma-model in the presence of a magnetic field $h$ coupled to a $U(1)$ charge. When the value of $h$ exceeds the mass gap, a finite density $\rho$ of equally polarized particles is created. At large values of $h$ the free energy of the system can be computed perturbatively due to the asymptotic freedom. Knowing the free energy density $f[h]$, we can find the energy density $\varepsilon[\rho]$ through the Legendre transform: =\_[h]{}(f\[h\]+h).
It is convenient to introduce a running coupling constant $\a[\mu]$ via the relation \[al\] 1+(-1)=$$\frac{2\pi\mu\Delta}{\Lambda_{\overline{MS}}}$$, =1[N-2]{}. The perturbative QFT predicts the following expansion for the energy density: \[energy\] =$\a+\frac 12\a^2+\Delta\sum_{n=3}^\infty \chi_n \a^n$+$\frac{\Lambda_{\overline{MS}}^2}{\rho^2}$, where $\a$ is evaluated at the scale $\mu=\rho$.
The energy of the system in the large volume can be calculated also from the asymptotic Bethe Ansatz, which explicitly contains the mass $\mass$. The considered energy density $\varepsilon$ is recovered in the thermodynamical limit, in which the number of particles $K$ and the length of the system $L$ both go to infinity with fixed $\rho=K/L$. In this limit the Bethe Ansatz reduces to the integral equation for the density of the rapidity distribution $\chi[\theta]$: \[iequation\] &-&\_[-B]{}\^[B]{}K\[-’\]d’=,\^2<B\^2;\
K\[\]&=&1[2i]{}d[d]{}S\_0\[\],\
S\_0\[\]&=&-/. .The energy density and the density of particles are given by =\_[-B]{}\^B, =\_[-B]{}\^B. We see that $\varepsilon$ depends on $\rho$ through the parameter $B$. To compare with the expansion (\[energy\]) we have to consider the large $\rho$, or equivalently large $B$, asymptotics of the integral equation (\[iequation\]).
The integral equation (\[iequation\]) can be rewritten in a nice form in terms of the resolvent for the function $\chi[\theta]$. For this we first notice that the kernel $K[\theta]$ can be represented as \[K\] K\[\]= 1[2i]{} (-) 1, where $D=e^{i\pi\partial_\theta}$ is a shift operator and $(1+D^{\pm 1})^{-1}=1-D^{\pm 1}+D^{\pm 2}-\ldots$. This representation for the kernel can be easily derived if to notice the formal equality $\G[i\theta/2\pi]\simeq (1/\theta)^{\frac {D^2}{1-D^2}}$.
The resolvent of $\chi[\theta]$ defined by \[resolvent\] R\[\]=\_[-B]{}\^B d’ is analytic everywhere except on the support $[-B,B]$ of $\chi[\theta]$. The residue of $R[\theta]$ at infinity equals to $2\pi\rho/\mass$. The density distribution $\chi[\theta]$ can be read from the discontinuity of the resolvent on the interval $[-B,B]$: \[discontinuity\] =-1[2i]{}$R[\theta+i0]-R[\theta-i0]$. Using (\[K\]), (\[resolvent\]), and (\[discontinuity\]) we can rewrite the integral equation (\[iequation\]) as \[requation\] R\[+i0\]-R\[-i0\]=\
=-2i , \^2<B\^2.
Leading order solution for the energy density
=============================================
In the following we will neglect the terms that give exponentially suppressed contribution to the value of $\varepsilon$. In this approximation we have \[e1\] \_[0]{}\^B e\^e\^B\_[-]{}\^0 e\^, =B+z2. In other words, $\varepsilon$ receives the main contribution from the vicinity of the branch points $\pm B$. Therefore we will consider the double scaling limit \[doublescaling\] B,, z=2(-B) .
From (\[discontinuity\]) and (\[e1\]) we can express the energy density in terms of the inverse Laplace transform of the resolvent: \[energylaplace\] =\[1/2\], \[s\]\_[-i+0]{}\^[i+0]{}e\^[sz]{}R\[z\]. In the double scaling limit the equation (\[requation\]) is simplified: \[zequation\] R\[z+i0\]-R\[z-i0\]=\
=-i e\^[B]{}e\^[z2]{}, z<0.
The inverse Laplace transform of (\[zequation\]) is straightforward: \[laplaceequation\] $
\frac{e^{i (1 - 2\Delta)\pi s}\hat{R}[s\! -\! i 0]}{\cos[\pi (s-i0)]} +
\frac{e^{-i (1 - 2\Delta)\pi s}\hat{R}[
s\! +\! i 0]}{\cos[\pi (s+i0)]} $ =\
&&= e\^B $\frac 1 {s + \frac 1 2 - i 0} - \frac 1 {s + \frac 1 2 +
i 0}$,s<0.To find the correct solution to (\[laplaceequation\]) we demand the following analytical properties for $\hat R[s]$ at each order of $1/B$ expansion:
$\hat R[s]$ is analytic everywhere except on the negative real axis,
$\hat R[s]$ has simple poles at $s=-n/(2\Delta)$ and simple zeroes at $s=-1/2-n$, where $n$ is a positive integer,
$\hat R[s]$ is expanded in negative powers of $s$ at infinity. The presence of zeroes in the resolvent can be directly seen from the equation (\[laplaceequation\]). The solution of the correspondent homogeneous equation should have in addition a zero at $s=-1/2$. The origin of the other stated analytical properties are explained in the appendix. The most general solution of the equation (\[laplaceequation\]) which satisfies stated analyticity properties is the following: \[sol1\] R\[s\]&=&$\frac{A}{s+\frac 12}+Q[s]$,A=e\^[-12+B+]{},\
&=&1e\^[(1-2)s-2s]{},\
Q\[s\]&=&1[Bs]{} \_[n,m=0]{}\^. The dependence of $Q[s]$ on $B$ is not a consequence of (\[laplaceequation\]) and can be deduced from the considerations of the next section. We see that $Q[s]$ does not contribute to the leading order of the $1/B$ expansion. This solution should be compared with the expression for $G_\pm[i\xi]$ of [@Forgacs:1991rs].
The leading order of $\varepsilon$ is given by $\mass\, A\Phi[1/2]e^B/(4\pi)$.
The particle density and subleading corrections
===============================================
First note that if we apply the operator $\frac 1{D^{-\Delta}\!-D^{\Delta}}(D^{-1/2}+D^{1/2})$ to the equation (\[requation\]) we will get \[nondoublescaling\] $D^{\Delta-1/2}R[\theta+i0]+D^{-\Delta+1/2}R[\theta-i0]$=0. In the previous section we found the most general solution in the double scaling limit. Still we have to fix unknown coefficients $Q_{n,m}$. For this we consider a different regime. We take again $B\to\infty$ but now we will be interested in the values of the resolvent $R[\theta]$ at the distances of the order $B$ or larger from the branch points $\theta=\pm B$. In this case the shift operator can be expanded in the Taylor series $D=1+i\pi\partial_\theta-\frac 12\pi^2(\partial_\theta)^2+\ldots$. Solving the equation (\[nondoublescaling\]) perturbatively we can expand the resolvent in the inverse powers of $B$: \[sol2\] R\[\]=\_[n,m=0]{}\^\_[k=0]{}\^[m+n]{}$$\frac{\theta\!-\!B}{\theta\!+\!B}$$\^k, where $\e[k]=k\ \textrm{mod}\ 2$. The perturbative meaning of the expansion (\[sol2\]) is most easily seen in terms of the variable $u=\theta/B$. The solution (\[sol2\]) gives us the value for the particle density from the residue of the resolvent at infinity: =$c_{0,0,0}+\sum_{m=1}^\infty \frac{c_{0,m,0}-2c_{0,m,1}}{B^m}$.
If we reexpand (\[sol2\]) in the double scaling limit (\[doublescaling\]), we should recover the solution obtained in the previous section. This condition uniquely fixes all the coefficients $c_{n,m,k}$ and $Q_{n,m}$.
The expansion (\[sol2\]) in the double scaling limit organizes at each order of $1/B$ as a $1/z$ expansion. Therefore we should compare it with the Laplace transform of the small $s$ expansion of $\hat R[s]$. As an illustration, we give here the terms of these expansions which are relevant for calculation of the leading and the subleading orders of $\rho$ and $\varepsilon$: &R&=+- c\_[0,0,0]{}+\
&+&,\
&&\_0\^ds e\^[-sz]{}R\[s\]=-A\
&+&$-2+\frac{2\Delta\log\frac{2e}{\Delta}-1-(1-2\Delta)\log z}{z}$.
Results and discussions
=======================
From the results of the previous sections we find the expressions for $\rho$ and $\varepsilon$ at the leading and subleading orders: &=&\^2$1+\frac 1{4B}$,\
&=&$\sqrt{B}-\frac {\frac 32+(1\!-\!2\Delta)\log 4B}{4\sqrt{B}}$. Resolving the parametric dependence we obtain exactly $\a$ and $\a^2$ terms in the expansion (\[energy\]) if $\a$ is defined as \[am\] 1+(-1)=+$$\(\frac{8}{e}\)^\Delta\frac{2\pi}{\Gamma[\Delta]}$$. Comparing (\[am\]) with (\[al\]) we confirm the result (\[c\]).
Comparing the solutions (\[sol1\]) and (\[sol2\]) one can find the higher order corrections to the energy density. At first four loops they are given by: &\_3&=12, \_4=- (24 (3) \^2 +.\
&&.+8 \^2-42 (3) -28 +21 (3)-8)\
&\_5&=-(456 (3) \^3+24 \^3-918 (3) \^2.-\
&&.-60 \^2+609 (3) -140 -105 (3)-24).The two-loop result $\chi_3$ coincides with the field theory calculations in [@Bajnok:2008it]. The fact that the energy can be expanded in power series over the running coupling constant (\[am\]) is a nontrivial property of the integral equation (\[iequation\]). This is a strong check of the validity of the bootstrap approach. This hidden renorm-group dynamics of the integral equation was explained in [@Forgacs:1991rs].
We have found analytical expressions for $\chi_n$ up to $n=26$ [^1]. This allowed us to estimate the large $n$ behavior of $\chi_n$: \[largeorder\] \_na\_n\[\]. For $\Delta=0$ we have $a_n[0]=(-1)^{n-1}$. This result is consistent with the fact that in the large $N$ limit the leading IR renormalon pole in the Borel plane is absent [@David:1982qv]. The large-order behavior of the coefficients $\chi_n$ is therefore governed by the leading UV renormalon pole leading to the Borel-summable oscillating behavior (\[largeorder\]). Surprisingly, $a_n\simeq 1.09$ for $\Delta=1$ and $a_{n+1}\simeq n^{-1}(2.09 - 0.43(n\ \rm{mod}\ 2))$, [*i.e.*]{} the series is non-Borel-summable. The Borel ambiguity is of the order ${\Lambda^2}/{\rho^2}$ as it should be from the field theory point of view (see (\[energy\])).
For arbitrary $\Delta\geq 1$ the behavior of the coefficients $a_n$ interpolates between those for $\Delta=0$, $1/2$ and $1$. We estimate the sign oscillation of the coefficients for sufficiently large $n$ and $\Delta<\Delta_c\simeq 0.4$. For $1\leq \Delta\leq \Delta_c$ all the $a_n$ are positive. For $\Delta >1$ the asymptotic behavior of $\a_n$ is given by $a_n\simeq -\a_1 n^{\a_2} \Delta^{n-2}$, where $\a_1$ is positive.
The non-Borel-summability in the $O(N)$ sigma models for $N=3$ and $N=4$ is a quite unexpected property and should be understood better. For $N=3$ one might try to explain it by the presence of the instantons. For $N=4$ our observation is supported by the non-Borel-summability in the SU(N) principal chiral field model (PCF) at large $N$ [@Fateev:1994ai] (since the $O(4)$ sigma-model can be viewed as an $SU(2)$ PCF model).
The method used in this paper was developed in the papers [@Kostov:2008ax; @Volin:2008kd] and with no doubts can be applied to similar systems such as $\mathbb{CP}^n$, PCF or Gross-Neveu sigma-models.
Appendix: Analytical properties of the resolvents
=================================================
There is yet another representation of the integral equation (\[iequation\]). To derive it we have to extend this equation to the whole real axis. For this we introduce a new function $\chi_h[\theta]$ (density of holes) which has the support complementary to $[-B,B]$. The function $\chi_h[\theta]$ is defined to be such that the equation \[iequation2\] \_h\[’\]+-\_[-B]{}\^[B]{}d” K\[’-”\]=\
=is valid for any real $\theta'$.
Integrating the equation (\[iequation2\]) with the Cauchy kernel $\frac 1{\theta-\theta'}$, we get \[requation2\] R\[\]+R\_h\[\]&=&T\[\], [Im]{}\[\]0, where $R_h$ is the resolvent for $\chi_h$ and $T$ is the resolvent for the r.h.s of (\[iequation2\]).
The function $\hat R[s]$ is defined via the inverse Laplace transform (see (\[energylaplace\])) and is analytic for $\rm{Re}[s]>0$. We define the analytical continuation of $\hat R[s]$ to $\rm{Re}[s]<0$ via the path that does not cross the ray $s<0$. To study the analytical properties of $\hat R[s]$ in the region $\rm{Re}[s],\rm{Im}[s]<0$ we consider the equation (\[requation2\]) for $\rm{Im}[\theta]$ in the double scaling limit and apply the integral $\int_{i\infty-0}^{-i\infty-0} \frac{dz}{2\pi i}e^{sz}$. The result can be written as R\[s\]&=&$\ \hat T[s]+\frac {\mass}{2}e^B\frac 1{s+\frac 12}\ $, [Im]{}\[s\]<0,\
T\[s\]&=&\_[i-0]{}\^[-i-0]{} e\^[sz]{} T\[z\]. Since $T[z]$ is analytic everywhere except on the ray $z>0$ the function $\hat T[s]$ is analytic for $\rm{Re}[s]<0$. We conclude that $\hat R[s]$ is analytic everywhere in the considered region and has poles on the ray $s<0$.
Similarly we consider the region $\rm{Re}[s]<0,\rm{Im}[s]>0$. This explains the pole structure of the resolvent.
Since $K[\theta-\theta']$ is analytic at $\theta=B$, $\chi[\theta]$ is analytic as well. This means that expansion of $R[z]$ at $z=0$ is given by $R[z]=\log[z](\b_0+z\b_1+z^2\b_2+\ldots)$. The inverse Laplace transform of this expansion gives us the stated behavior of $\hat R[s]$ at infinity.
Note that the stated analytical properties of $\hat R[s]$ are valid only in the double scaling limit. The inverse Laplace transform of $R[\theta]$ at finite $B$ is an entire function in the $s$-plane. This phenomena is illustrated by the function $f[s]=\frac 1{s}-\frac 1{s}e^{-\frac sB}$. This is an entire function. But since we are interested in $f[1/2]$ in the large $B$ limit the exponentially suppressed term can be neglected and effectively we get a pole.\
[**Acknowledgments**]{}
The author thanks to B. Basso, G. Korchemsky, I.Kostov and D.Serban for many useful discussions. The author is grateful to B.Basso and G.Korchemsky for providing a comprehensive introduction into the subject. This work has been supported by the European Union through ENRAGE network (contract MRTN-CT-2004-005616).
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[^1]: the *Mathematica* code for the calculations will be published in the PhD thesis
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present an *ab initio* quantum theory of the metal-insulator transition in Ti$_2$O$_3$. The recently developed cluster LDA+DMFT scheme is applied to describe the many-body features of this compound. The conventional single site DMFT cannot reproduce a low temperature insulating phase for any reasonable values of the Coulomb interaction. We show that the non-local Coulomb interactions and the strong chemical bonding within Ti-Ti pair is the origin of the small gap insulating ground state of Ti$_2$O$_3$.'
author:
- 'A.I. Poteryaev'
- 'A.I. Lichtenstein'
- 'G. Kotliar'
title: |
Non-local Coulomb interactions and metal-insulator transition in Ti$_2$O$_3$:\
a cluster LDA+DMFT approach.
---
The complicated electronic structure and the nature of the metal-insulator transition in Ti$_{2}$O$_{3}$ and V$_{2}$O$_{3}$ has been the object of intensive experimental and theoretical investigation over the past half century [@Imada]. Recent progress in high-energy photo-emission spectroscopy [@Spring8] and correlated electrons dynamical-mean field theory (DMFT) [@review] has shed new light on the first-order metal-insulator transition in V$_{2}$O$_{3}$. It has been shown that an realistic description of the metallic and insulating phases of V$_{2}$O$_{3}$ can be obtained from the combination of a band structure scheme with the local electron-electron interaction given from DMFT [@Held]. The correlation effects in Ti$_{2}$O$_{3}$ are less clear but angle resolved photo-emission experiment [@Smith] shows a strong reduction of the Ti 3$d$-bandwidth compared to band structure calculations. The important question is related to the mechanism of the small, about 0.1 eV, semiconductor band-gap formation. The generally accepted view is that the metal-insulator transition is related to the decrease of the $c/a$ ratio in rhombohedral Ti$_{2}$O$_{3}$ and the formation of a Ti-Ti pair along $z$-axis [@Goodenough]. Below the broad (almost 250 K in width) metal-insulator transition at around 470 K the Ti-Ti pair distance is seen to decrease without any change of the rhombohedral structure or the formation of long-range antiferromagnetic order [@mag]. This is in contrast to the case of V$_{2}$O$_{3}$ where the V-V pair distance increases within a monoclinic distortion in the antiferromagnetic phase [@Imada].
Ti$_{2}$O$_{3}$ has an $\alpha $-Al$_{2}$O$_{3}$ corundum structure (Fig. \[structure\]) in the metallic and insulating phases with two formula units per rhombohedral cell [@Abrahams_Rice]. Each Ti atom is surrounded by the octahedron of oxygens leading to the large $t_{2g}$-$e_{g}^{\sigma }$ splitting. The trigonal distortion gives an additional splitting of $t_{2g}$ bands into $e_{g}^{\pi }$-$a_{1g}$ states and $a_{1g}$ subbands of Ti-Ti pair form strong bonding-antibonding counterparts (Fig. \[structure\]). In principle, the large decrease of the Ti-Ti distance could split further an occupied single-degenerate $a_{1g}$ states from a double-degenerate $e_{g}^{\pi }$ states of $t_{2g}$ subband and form the insulating $d^{1}$ configuration of this Ti compound. Nevertheless, state of the art LDA calculations have shown that for reasonable Ti-Ti pair distances Ti$_{2}$O$_{3}$ will stay metallic [@Mattheiss].
In order to investigate the role of electron-electron interactions in the formation of this insulating low-temperature phase one needs an accurate estimation of the $a_{1g}$ and $e_{g}$ bandwidths in this complex structure [@Zeiger]. For example a simple free $[$Ti$_{2}$O$_{9}]^{12-}$ cluster mean-field investigation can easily produce a semiconducting gap due to drastic underestimation of the $a_{1g}$ and $e_{g}$ bandwidths [@Nakatsugawa]. On the other hand a more accurate band structure calculation within the unrestricted Hartree-Fock approximation results in large gap antiferromagnetic state [@Catti]. Thus it is crucial to use both the correct Green-function embedding of the Ti-Ti pairs as well as a more accurate treatment of the electron-electron interaction.
The role of metal-metal pair formation and the “molecular” versus band pictures of the electronic structure have attracted much attention in these compounds [@pair]. The combination of a strong on-site Coulomb interaction and the large anisotropy between the hopping parameters in and perpendicular to the pair direction can favor a localized molecular-orbital picture of the insulating phase. However, realistic tight-binding calculations for V$_{2}$O$_{3}$ have shown the importance of long-range hopping parameters [@Elfimov]. It is also unclear how good an on-site approximation is for the electron-electron interaction. Since the pair forms a natural “molecular like” element in the corundum-type Ti$_{2}$O$_{3}$ structure it might be expected that non-local electron correlations are important in this system. Thus an approach which combines pair and beyond pair hopping with non-local electron interactions would be seem to be ideal for this problem.
In this letter we apply for the first time a method, the cluster DMFT scheme [@cluster; @cdmft], which contains all the physics of correlated pairs in crystals to determine the origin of the insulating phase and the metal-insulator transition in Ti$_{2}$O$_{3}$. A numerically exact multi-orbital Quantum Monte-Carlo (QMC) scheme is deployed for the solution of the cluster DMFT problem and an accurate first principles tight-binding parametrization used for the one electronic structure. Our strategy here is to investigate the gap formation using single site [@anisimov] and cluster LDA+DMFT with only local correlations included. We then deploy the full non-local CDMFT and in this way are able to directly elucidate the impact non-local Coulomb interactions have on the physics. We show that the competition between strong bonding within the Ti-Ti pair and localization from correlation effects leads to the unique situation of the small semiconducting gap structure in Ti$_{2}$O$_{3}$ oxide and that non-local Coulomb correlations are of crucial importance for the physics of this small gap insulators.
![Left - rhombohedral unit cell of Ti$_{2}$O$_{3}$ corundum structure. Titanium ions are indicated by the red color, oxygens by green, and the pair of Ti atoms in $z$ direction by blue. Right - schematic representation of the $t_{2g}$ splitting in Ti$_{2}$O$_{3}$ (top part) and the intersite Coulomb interaction matrix (bottom part). $w$=0 in all our calculations.[]{data-label="structure"}](str1.ps){width=".5\textwidth"}
![Left - rhombohedral unit cell of Ti$_{2}$O$_{3}$ corundum structure. Titanium ions are indicated by the red color, oxygens by green, and the pair of Ti atoms in $z$ direction by blue. Right - schematic representation of the $t_{2g}$ splitting in Ti$_{2}$O$_{3}$ (top part) and the intersite Coulomb interaction matrix (bottom part). $w$=0 in all our calculations.[]{data-label="structure"}](scheme.ps){width=".7\textwidth"}
[ $$\! \! \! \! \! \! \! \! \! \! \! \! \! \! \! U_{mm^{\prime}}^{ij} = \left(
\begin{array}{ccc}
\textcolor{blue}{V_{e_g}} & w & \textcolor{red}{W} \\
w & \textcolor{blue}{V_{e_g}} & \textcolor{red}{W} \\
\textcolor{red}{W} & \textcolor{red}{W} & \textcolor{green}{V_{a_{1g}}}
\end{array}
\right)$$ ]{}
We start with the orthogonal LDA Hamiltonian $H_{mm^{\prime }}^{LDA}(\mathbf{k})$ in the massively downfolded N-th order muffin-tin orbital representation [@OKA] ($m$ corresponds to the 12 $t_{2g}$ orbitals of two Ti-Ti pairs in rhombohedral unit cell) and include different Coulomb interactions (see Fig. \[structure\]). DMFT results for the local and non-local Coulomb interactions are presented in Figs. \[dos\_ss\],\[dos\_2x2\].
The bare LDA density of states (DOS) is shown in Fig. \[dos\_2x2\] by the dashed lines for the low temperature structure (LTS, $\sim$300 K [@Abrahams_Rice]) and high temperature structural (HTS, $\sim$870 K [@Abrahams_Rice]) data on the upper and lower panels respectively. Both LTS and HTS electronic structures are metallic within the LDA scheme. The $a_{1g}$ subband (green dashed line in the Fig. \[dos\_2x2\]) has a strong bonding-antibonding splitting in contrast to the $e_{g}^{\pi}$ subbands (red dashed line). The bandwidth of the HTS is approximately 2.8 eV and smaller than the bandwidth of the LTS (3.2 eV) due to the reduction of the $t_{a_{1g},a_{1g}}$ hopping from -0.85 to -0.63 eV.
The cluster DMFT maps the many-body crystal system onto an effective self-consistent multi-orbital quantum impurity-cluster problem [@cluster; @cdmft]. The corresponding Green-function matrix for the Ti-Ti cluster in the LDA+DMFT scheme is calculated via the BZ-integration $$\mathbf{G}(i\omega _{n})=\sum_{\mathbf{k}}[(i\omega _{n}+\mu )\mathbf{1}-%
\mathbf{H}^{LDA}(\mathbf{k})-\mathbf{\Sigma }(\omega _{n})]^{-1}, \label{BZI}$$ where $\mu$ is the chemical potential defined self-consistently through the total number of electrons, $\omega _{n}=(2n+1)\pi T$ are the Matsubara frequencies for temperature $T\equiv \beta ^{-1}$ ($n=0,\pm 1,...$) and $\sigma$ is the spin index. The Hamiltonian and the self-energy matrix have the following super-matrix form corresponding to the symmetry of two Ti-Ti pairs in the unit cell $${\scriptstyle
\left(
\begin{array}{cccc}
\mathbf{H}_{11}+\mathbf{\Sigma}_{11} & \mathbf{H}_{12}+\mathbf{\Sigma}_{12}
& \mathbf{H}_{13} & \mathbf{H}_{14} \\
\mathbf{H}_{21}+\mathbf{\Sigma}_{21} & \mathbf{H}_{22}+\mathbf{\Sigma}_{11}
& \mathbf{H}_{23} & \mathbf{H}_{24} \\
\mathbf{H}_{31} & \mathbf{H}_{32} & \mathbf{H}_{33}+\mathbf{\Sigma}_{11}
& \mathbf{H}_{34}+\mathbf{\Sigma}_{12} \\
\mathbf{H}_{41} & \mathbf{H}_{42} & \mathbf{H}_{43}+\mathbf{\Sigma}_{21}
& \mathbf{H}_{44}+\mathbf{\Sigma}_{11}
\end{array}
\right), }$$ where $\mathbf{H}_{ij}(\mathbf{k})$ and $\mathbf{\Sigma }_{ij}(\omega _{n})$ are 3$\times $3 matrices for the $t_{2g}$ states and $\mathbf{\Sigma }_{11}$ and $\mathbf{\Sigma }_{12}$ correspond to the intrasite and intersite contributions to the self-energy respectively.
In the self-consistent cluster DMFT scheme the local Green-function (\[BZI\]) should coincide with the corresponding solution of the effective two-site quantum impurity problem [@review] $$\mathbf{G}_{\sigma }(\tau -\tau ^{\prime })=-\frac{1}{Z}\int D[\mathbf{c},%
\mathbf{c}^{+}]e^{-S_{eff}}\mathbf{c}(\tau )\mathbf{c}^{+}(\tau ^{\prime })
\label{path}$$ here $\mathbf{c}(\tau )$=$[c_{im\sigma }(\tau )]$ is the super-vector of the Grassman variables, $Z$ is the partition function, $i$ runs over Ti-Ti pair and $m$ runs over $e_{g}^{\pi }$ or $a_{1g}$ orbitals. The effective cluster action $S_{eff}$ is defined in terms of so-called “bath” Green function matrix [@review] $\mathcal{G}_{\sigma}^{-1}(\omega _{n})=\mathbf{G}_{\sigma }^{-1}(\omega _{n})+
\mathbf{\Sigma }_{\sigma }(\omega _{n})$ which describes the energy, orbitals, spin and temperature dependent interactions of particular cluster with the rest of the crystal [$$\begin{aligned}
S_{eff} & = & -\int_{0}^{\beta} d\tau \int_{0}^{\beta} d\tau^{\prime}
Tr[\mathbf{c}^{+}(\tau)\mathcal{G}^{-1}(\tau,\tau^{\prime})
\mathbf{c}(\tau^{\prime})] + \\
& \frac{1}{2} & \sum_{im,jm^{\prime},\sigma} [ U_{mm^{\prime}}^{ij}
n_{im}^{\sigma}n_{jm^{\prime}}^{-\sigma}+
(U_{mm^{\prime}}^{ij}-J_{mm^{\prime}}^{ij})n_{im}^{\sigma}n_{jm^{\prime}}^{\sigma}]
\notag \end{aligned}$$ ]{} here $n_{im\sigma }$=$c_{im\sigma }^{+}c_{im\sigma }$. We have parameterized the screened local Coulomb and exchange matrices ($U_{mm^{\prime }}^{ii}$ and $J_{mm^{\prime }}^{ii}$) for the $t_{2g}$ electrons in terms of average Coulomb and exchange integrals [@Kotliar] and used a simple approximation to the intersite $U_{mm^{\prime }}^{ij}$ interactions as shown in the Fig. \[structure\].
The multi-band impurity QMC scheme [@Hirsch; @rozenberg] has been used for the numerically exact calculation of the cluster Green function (eq. \[path\]). The number of auxiliary Ising fields in the discrete Hirsh-Fye transformation were 15 and 58 for the local and non-local Coulomb interaction schemes respectively. For accurate QMC integration we used of the order of 10$^{6}$ sweeps, with 8000 $\mathbf{k}$-points for the BZ-integration. Within 15-20 DMFT iterations convergence in the self-energy was reached. The maximum entropy method [@MEM] has been used for analytical continuation of the diagonal part of the Green function matrix to the real energy axis.
![Density of states for the single site (upper panel) and cluster (lower panel) DMFT calculations with different values of the Coulomb repulsion $U$ and $J$=0.5 eV. Inset: $N(0)$ $versus$ Coulomb parameter. Filled dark green circles - DMFT results with $J$=0.5 eV, filled orange squares - DMFT with $J$=0 eV. Open blue circles and violet squares are CDMFT with $J$=0.5 and 0 eV respectively.[]{data-label="dos_ss"}](dos_ss.ps){width=".45\textwidth" height=".4\textwidth"}
Firstly, in Fig.\[dos\_ss\] we show the total density of states for both conventional single site (DMFT) and the cluster (CDMFT) dynamical mean field theory where only local electron correlations have been included. The QMC simulation has been carried out for $\beta$=20 eV$^{-1}$ which corresponds to a temperature of $T$$\simeq$580 K which is on the border of the metal-insulator transition. In the upper panel of Fig. \[dos\_ss\] are shown the results of DMFT calculations with $U$=2,3,4 eV and exchange parameter $J$=0.5 eV. For all values of Coulomb interactions there is a peak below the Fermi level at around -0.5 eV, predominantly of $%
a_{1g}$ character with in all cases the same intensity. Above the Fermi level there are two peaks. The first is at 0.5 eV and has $e_{g}$ character while the other peak is strongly dependent on the Coulomb parameter and can be associated with an upper Hubbard band. A lower Hubbard band can be seen at around -2 eV. We see that for all values of $U$ the the shape of pseudogap is unchanged and the system remains metallic. On the lower part of Fig. \[dos\_ss\] the results of the CDMFT calculation are shown for the same values of the Coulomb and exchange parameters. The general structure of the DOS is seen to be similar to the single site calculation, however one may note interesting differences. The lower $a_{1g}$ quasiparticle band is decreased in intensity and shifted towards the Fermi level from -0.6 eV to -0.3 eV on increasing $U$ from 2 to 4 eV. This has the result that for $U$=4 eV the pseudogap is now located directly at Fermi level, whereas for other $U$-values and for all DMFT results it lies on the slope of quasiparticle peak.
Using the temperature DOS at the Fermi level, defined as $N(0) \equiv -ImG(\omega_0)/\pi$ with $\omega_0=\pi T$ we are able to estimate at what values of $U$ the system will become insulating. This is indicated in the inset in the upper panel of the Fig. \[dos\_ss\]. We see that for the single site calculations $N(0)$ depends weakly on $U$ and the system will remain metallic up to very large values, about 8 eV, of the Coulomb parameter. On the other hand for the cluster calculation $N(0)$ is seen to decrease strongly as a function of $U$ for both values of exchange parameter, and the critical value for an insulating solution is now lower at $U$$\sim$5-6 eV. As expected for the $d^1$ configuration the finite value of the exchange parameter effectively decreases the Coulomb interaction matrix. We see the single site results are in greater contradiction to the experiment as compared to LDA (see Fig. \[dos\_2x2\]): the local Coulomb interaction leads to the reduction of the bonding-antibonding splitting of the $a_{1g}$ subband and this acts to suppress gap formation. On the other hand in the cluster case a small semiconducting gap is developed for large $U$ due to dynamical antiferromagnetic correlation in the Ti-Ti pair.
Nevertheless, using either the DMFT or CDMFT schemes with only local correlations there remains a dramatic absence of gap formation in Ti$_2$O$_3$. We now deploy the full non-local correlation in CDMFT to the effect of non-local correlations on low and high temperature electronic structure. We have used different values of the non-local Coulomb parameters and found that the most important correspond to non-diagonal interactions. For both structures we have chosen values of $U$=2 eV and $J$=0.5 eV which are close to those from constrained LDA estimations [@uvalue], while the off diagonal Coulomb parameter $W$ has been chosen at $W$=0.5 eV [@wvalue]. On the upper panel of Fig. \[dos\_2x2\] is shown the total and partial DOS for $\beta$=20 eV$^{-1}$. Shown also is the LDA result. One can see that for the reasonable parameters chosen we can reproduce the correct value of the semiconducting gap $\sim$0.1 eV while keeping the bonding-antibonding splitting on the LDA level. In the lower panel the high temperature metallic solution corresponding to $\beta$=10 eV$^{-1}$ is shown. Here we emphasize that the proper inclusion of the structural effect on the LDA level is important as evinced by the fact that for $\beta$=20 eV$^{-1}$ and high temperature hamiltonian we again obtain a metallic solution. The $e_g$ states are similar for both LTS and HTS calculations with a small shift of occupied part in LTS case. However, the difference between the LTS and HTS phases is more pronounced for the $a_{1g}$ states. The bonding-antibonding splitting in the LTS is about 2 eV whereas in the HTS case it is only 1.5 eV. The occupied $a_{1g}$ states in the LTS phase are shifted down opening the insulating gap. The important difference between the large $U$ and small $U$ plus non-local $W$ is the absence of well defined Hubbard bands. This absence makes possible a critical test of the theory proposed here, and thus it would be very interesting for photo-emission experiments to check the existence or not of a lower Hubbard band at around -2 eV.
![Partial and total CDMFT (solid line) compared to the LDA (dashed) DOS with $W$=0.5 eV and $V_{a_{1g}}$=$V_{e_{g}}$=0. Total DOS are shown by black, the $e_{g}$ states by red, and $a_{1g}$ states by green. On the upper panel the low temperature structure and $\beta$=20 eV$^{-1}$ are used. For the lower panel the high temperature structure and $\beta$=10 eV$^{-1}$ are used. The diagonal and the biggest $a_{1g}$ off-diagonal Green functions $G(\tau)$ are shown in the upper inset. In the lower inset the $Re \Sigma_{a_{1g},a_{1g}}$ with $W$=0.5 eV are shown by blue, $Re \Sigma^{\prime}_{a_{1g},a_{1g}}$ with $W$=0 eV by orange and $Im \Sigma_{a_{1g}}$ are shown by green[]{data-label="dos_2x2"}](dos_2x22.ps){width=".45\textwidth"}
We have shown that the cluster LDA+DMFT calculation with a moderate Coulomb repulsion among the $a_{1g}$ orbitals is essential to produce the high temperature semimetallic state and the low temperature insulating state. To understand the role play of the intersite Coulomb interaction we focus on the the quantity $t_{a_{1g},a_{1g}} + Re \Sigma_{a_{1g},a_{1g}}(i \omega)$ which we can interpret as a frequency dependent “effective $a_{1g}-a_{1g}$ hopping” which describes the hopping matrix element in the titanium pair. We find that this quantity is surprisingly frequency dependent (see lower inset of Fig. \[dos\_2x2\]).
We conclude that the main role of the intersite Coulomb interaction is dynamic (the Hartree contribution to this quantity is small) and results in the effective $a_{1g}-a_{1g}$ hopping that [*increases*]{} as the frequency decreases. This enhancement produces a strong level repulsion of the bonding antibonding $a_{1g}$ levels, lowering the $a_{1g}$ level relative to the $e_g$ level at the low frequency. This effect combined with a small narrowing of the $a_{1g}$ band opens the $e_g-a_{1g}$ band gap which results in the insulating state. We checked that this enhancement of the effective hopping as frequency is decreased is absent if we turned off the intersite Coulomb repulsion.
This effect is the cluster DMFT analog of a mechanism first discussed in the context of the single impurity model by Haldane [@haldane]. He observed, that a Coulomb repulsion between the impurity level and additional conduction electron states or screening channels, [*enhances*]{} the hybridization (single impurity analog of the hopping matrix element) as we renormalize to low frequency.
We would like to acknowledge O.K. Andersen, V.I. Anisimov, A. Georges and M.I. Katsnelson for useful discussions. This work was supported by the Netherlands Foundation for the Fundamental Study of Matter (FOM). GK was supported by the ONR, grant N000140210766. The authors are grateful to the Kavli Institute of Theoretical Physics, Santa Barbara, for hospitality during the initial stages of this work. This research was supported in part by the National Science Foundation under Grant No. PHY99-07949.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present an algorithm *CRE*, which either finds a Hamilton cycle in a graph $G$ or determines that there is no such cycle in the graph. The algorithm’s expected running time over input distribution $G\sim G(n,p)$ is $(1+o(1))n/p$, the optimal possible expected time, for $p=p(n) \geq 70n^{-\frac{1}{2}}$. This improves upon previous results on this problem due to Gurevich and Shelah, and to Thomason.'
author:
- 'Yahav Alon [^1]'
- 'Michael Krivelevich [^2]'
title: Finding a Hamilton cycle fast on average using rotations and extensions
---
\[section\] \[section\] \[thmtool\][Corollary]{} \[thmtool\][Lemma]{} \[thmtool\][Definition]{} \[thmtool\][Proposition]{} \[thmtool\][Claim]{}
Introduction {#sec-intro}
============
Hamilton cycles are a central topic in modern graph theory, a fact that extends to the field of random graphs as well, with numerous and diverse results regarding the appearance of Hamilton cycles in random graphs obtained over many years.\
Consider the random graph model $G(n,p)$, in which every one of the edges of $K_n$ is added to $G$ with probability $p$ independently of the other edges. A classical result by Koml[ó]{}s and Szemer[é]{}di [@KS83], and independently by Bollob[á]{}s [@B84], states that a random graph $G\sim G(n,p)$, with $np-\ln n - \ln \ln n \rightarrow \infty$, is with high probability Hamiltonian. It should also be noted that if $np -\ln n - \ln \ln n \rightarrow -\infty$ then with high probability $\delta (G) \leq 1$, and thus $G$ is not Hamiltonian.\
In fact, a stronger result was proved by Bollob[á]{}s in [@B84] and by Ajtai, Koml[ó]{}s and Szemer[é]{}di in [@AKS85]. It states that the *hitting time* of graph Hamiltonicity is with high probability equal to the hitting time of the property $\delta (G) \geq 2$. In other words: if one adds edges to an empty graph on $n$ vertices in a random order, then with high probability the exact edge whose addition to the graph has increased its minimal degree to $2$, has also made the graph Hamiltonian.\
In light of this, one can ask whether there exists a computationally efficient way to find a Hamilton cycle in a graph $G$, or to determine that it contains none, provided that $G$ is sampled from the probability space $G(n,p)$ with $np-\ln n - \ln \ln n \rightarrow \infty$.\
The answer to this question differs greatly depending on how one defines the term “computationally efficient".\
For example, if our interest lies in finding an algorithm with a fast worst case time complexity, that is, its running time on any input is bounded by some “small" function of the number of vertices $n$, we might get disappointed. This is due to the fact that the graph Hamiltonicity problem is a well known NP-complete problem (see e.g. [@GJL]), and as such no polynomial time algorithm solving it is known. In fact, the best known worst case complexity algorithm is achieved by dynamic programming algorithms (see Bellman [@BELL] and Held, Karp [@HK]), with asymptotic time $O\left( 2^n\cdot n^2 \right)$.\
That said, different models of complexity may yield very different results. Consider for example a model in which an algorithm is allowed to return the result *“failure"*, admitting that it has failed to find a Hamilton cycle in the input graph (without providing a proof that there is none), under the condition that if $p \geq f(n)$ and $G\sim G(n,p)$ then the probability that the algorithm fails on input $G$ is of order $o(1)$.\
In this model, much faster algorithms are available. A notable example is given in a 1987 paper by Bollob[á]{}s, Fenner and Frieze [@BFF], who present an algorithm *HAM1* with time complexity $O\left( n^{4+\varepsilon} \right)$ with $\varepsilon > 0$ arbitrarily small, that either finds a Hamilton cycle or returns *“failure"*. They further show that if the input graph $G$ is distributed $G\sim G(n,p)$, for any $p=p(n)$, then $$\lim _{n\rightarrow \infty} Pr[\mbox{\emph{HAM1} finds a Hamilton cycle in }G] = \lim _{n\rightarrow \infty} Pr[G\ \mbox{is Hamiltonian}].$$ Combined with the above stated fact that if $np-\ln n - \ln \ln n \rightarrow \infty$ then $G$ is with high probability Hamiltonian, this means that for $p \geq \frac{\ln n + \ln \ln n + \omega (1)}{n}$ the probability that *HAM1* returns *“failure"* is indeed $o(1)$.\
Another example of a fast algorithm that is not likely to return *“failure"* is given in [@FKSV], where the authors choose to measure the complexity by the number of positive edge query results the algorithm requires. They show an algorithm that requires $(1+o(1))n$ successful queries, and fails with probability $o(1)$ on graphs distributed according to $G(n,p)$, with $p \geq \frac{\ln n + \ln \ln n + \omega (1)}{n}$.\
An intuitive measure of complexity which seems interesting to consider is the *expected* running time. Denote by $T_A(G)$ the running time of some algorithm *A* on an input graph $G$. Say $G\sim G(n,p)$, how small can $\mathbb{E}\left[ T_A(G) \right]$ be?\
If we assume that there is no polynomial time algorithm that finds a Hamilton cycle in a graph, then finding an algorithm with polynomial expected running time is in some sense a more difficult problem than that of finding a polynomial time algorithm that fails with probability $o(1)$: if the expected time is polynomial, it means that those cases on which the running time is super-polynomial take up at most $n^{-\omega (1)}$ of the probability space. So such an algorithm can be used to construct a polynomial time algorithm that returns *“failure"* with probability $n^{-\omega (1)}$.\
Bollob[á]{}s, Fenner and Frieze [@BFF] used their algorithm *HAM1* to construct a slightly modified algorithm *HAM*, which applies an exponential running time algorithm on inputs on which *HAM1* returned *“failure"*, and prove that the expected running time of *HAM* on $G\sim G\left( n,\frac{1}{2} \right)$ is polynomial in $n$.\
Gurevich and Shelah [@GS87] improved upon this result, by presenting an algorithm *HPA*, which finds a Hamiltonian $s-t$ path in a graph $G$, with a linear expected running time, where this time the input is assumed to be distributed according to distribution $G(n,p)$, with $p\in [0,1]$ being a constant (not necessarily $\frac{1}{2}$). This can easily be altered into an algorithm that finds a Hamilton cycle rather than a Hamilton $s-t$ path. They did this by presenting three consecutive algorithms *HPA1, HPA2, HPA3*, such that failure of one algorithm to find a Hamilton $s-t$ path results in the next one being called, and such that *HPA1* takes linear time and $$Pr[\mbox{\emph{HPAi} fails on }G]\cdot \mathbb{E}\left[ T_{HPA(i+1)}(G)\right] = O(n).$$ They further show that their result is optimal for this range of $p$, by proving a stronger claim: If $A$ is an algorithm for finding a Hamilton cycle and $p \geq \frac{3 \ln n}{n}$, $G\sim G(n,p)$, then $\mathbb{E}\left[ T_A(G)\right] \geq n / p$. This result can be obtained by observing that in order to find a Hamilton cycle in a graph $G$, the algorithm must sample at least $n$ existing edges of $G$, which means that the expected number of queried pairs of vertices in $A$ must be at least the expected number of queries required for finding $n$ edges, which is exactly $n/p$.\
Further improvement was later given by Thomason [@THOM], who presented an algorithm *A*, similarly constructed of three consecutive algorithms *A1,A2,A3*. The expected running time of *A* is asymptotically optimal up to multiplication by a constant (that is $\mathbb{E}\left[ T_A(G) \right] = O(n/p)$), for a wider class of random graphs: whenever $p\geq 12n^{-\frac{1}{3}}$.\
For further reading on the algorithmic aspects of random graphs, including Hamiltonicity, we refer to [@FM].\
In this paper we present a new algorithm *CRE* (Cycle rotation extension) for finding a Hamilton cycle, and prove that if $p\geq 70 n^{-\frac{1}{2}}$ and $G\sim G(n,p)$ then $\mathbb{E}\left[ T_{CRE}(G) \right] = (1+o(1))n/p$. This constitutes a substantial progress in a long-standing open problem on Hamiltonicity of random graphs (see e.g., **Problem 16** in [@FRI]).\
Formally, we prove the following main result:
Let $p\geq 70 n^{-\frac{1}{2}}$ and let $G\sim G(n,p)$. There is an algorithm for finding a Hamilton cycle in a graph, with expected running time $(1+o(1))n/p$ on $G$.
As the algorithm’s name suggests, we will try and employ techniques inspired by P[ó]{}sa’s *rotation-extension*, which were introduced by P[ó]{}sa in 1976 [@POS] in his research of Hamiltonicity in random graphs. Informally put, *rotation-extension* is a technique which under certain conditions allows one to gradually extend paths or cycles in a graph, by finding (through a process usually referred to as a rotation) a large number of pairs of vertices, such that the existence of an edge between any of these pairs enables one to get a longer path or cycle (an extension) using this edge.\
Similarly to the previous results, we will define *CRE* by aligning three algorithms, each calling the next one in case of failure. In essence, the three algorithms will be:
- *CRE1* – A simple greedy algorithm, tasked with optimizing the expected time complexity.
- *CRE2* – The main algorithm, tasked with finding a Hamilton cycle in polynomial time in all but an exponentially small fraction of the probability space.
- *CRE3* – An exponential running time algorithm tasked with finding a Hamilton cycle in the graph when the previous two algorithms failed. This algorithm is identical to *HPA3*.
In Section \[sec-per\] we present some preliminaries. In Section \[sec-alg\] we present the *CRE* algorithm, and prove its correctness. In Section \[sec-time\] we prove that the expected running time of *CRE* is $(1+o(1))n/p$. In Section \[remarks\] we add some concluding remarks.
Preliminaries {#sec-per}
=============
In this section we provide several definitions and results to be used in the following sections.\
Throughout the paper, it is assumed that all logarithmic functions are in the natural base, unless explicitly stated otherwise.\
We suppress the rounding notation occasionally to simplify the presentation.\
The following standard graph theoretic notations will be used:
- $N_G(U)$ : the external neighbourhood of a vertex subset $U$ in the graph $G$, i.e. $$N_G(U) = \lbrace v \in V(G)\setminus U:\ v\ \mbox{has\ a\ neighbour\ in}\ U \rbrace.$$
- $e_G(U)$: the number of edges spanned by a vertex subset $U$ in a graph $G$. This will sometimes be abbreviated as $e(U)$, when the identity of $G$ is clear from the context.
- $e_G(U,W)$: the number of edges of $G$ between the two disjoint vertex sets $U,W$. This will sometimes be abbreviated as $e(U,W)$ when $G$ is clear from the context.
Furthermore, given a cycle or a path $S$ in a graph, with some orientation, we denote:
- $S^{-1}$: the cycle composed of the vertices and edges of $S$, but with the opposite orientation.
- $s_S(v)$: the successor of a vertex $v \in S$ on $S$, according to the given orientation. When the identity of the cycle is clear, we will write $s(v)$.
- $s_S(U)$: the set of successors $\{s_S(u):\ u\in U\}$. When the identity of the cycle is clear, we will write $s(U)$.
- $p_S(v)$: the predecessor of a vertex $v \in S$ on $S$, according to the given orientation. When the identity of the cycle is clear, we will write $p(v)$.
- $p_S(U)$: the set of predecessors $\{p_S(u):\ u\in U\}$. When the identity of the cycle is clear, we will write $p(U)$.
- $S(v \rightarrow u)$: the path $\left( v,s_S(v),s^2_S(v),...,p_S(u),u \right) \subseteq S$.
Gearing towards our concrete setting of a graph $G$ distributed according to $G(n,p)$ with $p\geq 70n^{-\frac{1}{2}}$, given a graph $G$, we will define the set of vertices with small degree (with regards to the expected degree) in $G$:
\[small\] Let $G$ be a graph on $n$ vertices. The set $\mathit{SMALL}(G)$ is defined as $$\mathit{SMALL}(G):= \lbrace v\in V(G) \mid d(v) < 40\sqrt{n} \rbrace .$$
We shall also make use of the following definition:\
Let $\Gamma = \left( X \cup Y,E \right)$ be a bipartite graph. An edge subset $M \subseteq E(\Gamma )$ is called a *$\leq 2$-matching* from $X$ to $Y$ if each vertex of $X$ is incident to at most $2$ edges in $M$, and each vertex of $Y$ is incident to at most one edge in $M$. A *maximum $\leq 2$-matching* in $\Gamma$ is a $\leq 2$*-matching* with the maximum possible number of edges.
We note that given a bipartite graph $\Gamma = \left( X \cup Y ,E\right)$, a *maximum $\leq 2$-matching* from $X$ to $Y$ can be found in time $(|X|+|Y|)^{O(1)}$ by using the *MaxFlow* algorithm.\
For some of our probabilistic bounds, we will use the following standard result throughout the paper:
\[chernoff\] [*(Chernoff bound for binomial tails, see e.g. [@CHER])*]{} Let $X\sim Bin(n,p)$. Then for every $\delta > 0$, $Pr[X < np - \delta ] \leq \exp \left( -\frac{\delta ^2}{2np} \right) .$
The *CRE* algorithm {#sec-alg}
===================
We now present the three components of the *CRE* algorithm, and prove that they are sound. Recall that each component can either fail or return a result, which is either a Hamilton cycle in the input graph or a declaration that there is none. The *CRE* algorithm itself will be:\
$CRE(G)$:\
*If* $CRE1(G)$ *did not fail, return the result of* $CRE1(G)$. *Otherwise:*\
*If* $CRE2(G)$ *did not fail, return the result of* $CRE2(G)$. *Otherwise:*\
*Return the result of* $CRE3(G)$.
*CRE1* {#subalg1}
------
We present the algorithm *CRE1*. This algorithm will be a greedy algorithm, tasked with optimizing the expected running time. As such, we aim for it to have the following properties, whenever $p\geq 70 n^{-\frac{1}{2}}$:
- $\mathbb{E}\left[ T_{CRE1}(G) \right] =(1+o(1))n / p$;
- $Pr[CRE1\mbox{ returns }``failure"] \cdot \mathbb{E}\left[ T_{CRE2}(G) \right] = o(n/p)$.
In the algorithm description we will assume that $V(G)=[n]$.\
**The *CRE1* algorithm description:**\
- Attempt to construct a path $P_1$ in $G\left( [n/2] \right)$ by greedily querying for a neighbour of the current last vertex in the path from outside the path, until the path’s end vertex does not have any neighbours among the remaining vertices. If $\frac{n}{2}-|P_1| > \sqrt{n} \log n$, return *“Failure"*. Denote this path by $P_1 = (v_1,...,v_{n/2-n_1})$, with $n_1 = |[n/2]\setminus P_1|$.\
Attempt to construct a path $P_2$ in $G\left( [n/2+1,n] \right)$ in the same manner, and return *“Failure"* if $\frac{n}{2}-|P_2| > \sqrt{n} \log n$. Denote $P_2 = (u_1,...,u_{n/2-n_2})$.
- Find indices $i,j,k,l$ with minimal $i+j+k+l$, such that $(v_i,u_{n/2-n_2-j}),(v_{n/2-n_1-k},u_l) \in E(G)$. If $i+j+k+l > \sqrt{n} \log n$, return *“Failure"*. Otherwise, denote by $S_0$ the cycle: $$S_0 := P_1(v_i\rightarrow v_{n/2-n_1-k})\cup \{ (v_{n/2-n_1-k},u_l)\} \cup P_2(u_l\rightarrow u_{n/2-n_2-j}) \cup \{(v_i,u_{n/2-n_2-j}) \}.$$
- Initialize $i=0$, and repeat the following loop until no vertices are left outside the cycle $S_i$. Choose some vertex $v\notin S_i$. For ease of description we will assume that $v\in [n/2]$. In the complementing case, the description is completely symmetrical, replacing $P_2$ with $P_1$, $n_2$ with $n_1$ and so on.\
Create a set $X = \{x_1,...,x_{\sqrt[3]{n}}\}$ of neighbours of $v$ on $(P_2 \cap S_i)\setminus \{ u_{n/2-n_2-j} \}$ that have not been used in this step, with $z:=x_{\sqrt[3]{n}}$ being the maximal one with respect to $P_2$. Return *“failure"* if no such $\sqrt[3]{n}$ vertices exist. Otherwise, create a set $Y=\{y_1,...,y_{\sqrt[3]{n}}\}$ of neighbours of $s_{S_i}( z )$ on $(P_1 \cap S_i) \setminus \{v_i\}$. Return *“failure"* if no such $\sqrt[3]{n}$ vertices exist. Finally, find a pair $x\in X\setminus \{ z \}, y\in Y$ such that $\left( s_{S_i}(x), p_{S_i}(y) \right) \in E(G)$. If no such pair exists, return *“failure"*. Otherwise, set $$\begin{array}{rl}
S_{i+1}:= & \{(v,x)\} \cup S_i^{-1}(x\rightarrow y) \cup \{ (y,s(z)) \} \cup S_i(s(z) \rightarrow p(y)) \cup \{ (p(y),s(x)) \}\\
& \cup S_i(s(x)\rightarrow z ) \cup \{ (z,v) \};
\end{array}$$ $i:=i+1$.
*CRE2* {#subalg2}
------
We present a description of *CRE2*, followed by a proof that the algorithm is sound, that is, if *CRE2* does not fail on a graph $G$ then it returns a Hamilton cycle that is a subgraph of $G$ if and only if $G$ is Hamiltonian.\
**The *CRE2* algorithm description:**\
- Determine $\mathit{SMALL}(G)$ (see Def. \[small\]) by going over all vertices and checking their degrees in $G$. If the resulting set is larger than $ 2\sqrt{n}$, return *“Failure"*.
- Find a *maximum $\leq 2$-matching* $M$ in $G$ from $\mathit{SMALL}(G)$ to $V(G)\setminus \mathit{SMALL}(G)$. Denote by $U$ the subset of vertices in $V(G)\setminus \mathit{SMALL}(G)$ that have degree $1$ in $M$. If $|U| \leq |\mathit{SMALL}(G)|$, add arbitrary vertices to $U$ until it is of size $|\mathit{SMALL}(G)|+1$.
- Using the dynamic programming algorithm (*HPA3*, see description in Section \[subalg3\]), find a Hamilton cycle in the graph with vertex set $U\cup \mathit{SMALL}(G)$ and edge set $E_G\left( U\cup \mathit{SMALL}(G)\right) \cup \left( U \times U \right)$. If no such cycle exists, determine that $G$ is not Hamiltonian. Otherwise, denote this cycle by $C$.\
Let $\mathit{NE} = (U\times U)\cap C \setminus E(G)$, let $|\mathit{NE}|=r$, and denote the members of $\mathit{NE}$ by $\{e_1,...,e_r\}$.
- For each $1 \leq j \leq r$ find a path $P_j$ of length at most $4$ connecting the two vertices of $e_j$, with all of its internal vertices in $G\setminus \left( \bigcup\limits _{k =1}^{j-1} P_k \cup \mathit{SMALL}(G) \cup U \right)$, using *BFS*. If for some $j$ no such path exists, return *“Failure"*. Otherwise, set $i=0$ and denote the resulting cycle by $S_0 = \left( C\cup \bigcup\limits _{j=1}^r P_j \right) \setminus \mathit{NE}$.
- Attempt to add at least one vertex of $V(G)\setminus V(S_i)$ to $S_i$ by doing the following:\
Using BFS, determine all connected components of $G\setminus S_i$, and denote by $V_i$ a largest connected component. If $|S_i|\geq 0.99n$ and $|V_i| \leq 15 \sqrt{n}$, go to *Step 6*. Otherwise, choose an arbitrary orientation to $S_i$ and let $U_i := s(N_G(V_i)\cap S_i)$. If $U_i$ is an independent set, return *“Failure"*. Otherwise, let $(u,w)$ be an edge in $U_i$, let $u^{\prime} = p(u),w^{\prime} = p(w)$ and let $P$ be a path, with all its internal vertices in $V_i$, connecting $u^{\prime}$ to $w^{\prime}$ (this path was uncovered in the BFS stage). Without loss of generality, $u$ precedes $w$ on $S_i$. Set $S_{i+1}$ to be:\
$$S_{i+1}= S_i(w\rightarrow u^{\prime}) \cup P \cup S_i^{-1}( w^{\prime} \rightarrow u) \cup \lbrace (u,w) \rbrace .$$ Set $i=i+1$, and return to *Step 5*.
- While there is some vertex $v\in V(G)\setminus V(S_i)$, attempt to add it to $S_i$ by exhaustively searching for two vertices $u,w \in N_G(v)\cap S_i$, a set $E_1 \subseteq E(S_i)$ of size at most 4, and a set $E_2 \subseteq E_G(V(S_i))\setminus E(S_i)$ of size $|E_1|-1$, such that $S_{i+1} :=(S_i\setminus E_1) \cup E_2 \cup \{ (u,v) , (v,w) \}$ is a cycle of size $|S_i|+1$. If no such $u,w,E_1,E_2$ exist, return *“failure"*.\
\[sound\] If $G$ is a graph such that *CRE2* does not result in failure when applied to $G$, then *CRE2* returns a Hamilton cycle if and only if $G$ is Hamiltonian. Furthermore, if *CRE2* returns a Hamilton cycle then it is a subgraph of $G$.
In each step $E(S_i) \subseteq E(G)$ and $S_i\subsetneq S_{i+1}$. So it is clear that if the algorithm returns a Hamilton cycle then it is indeed a Hamilton cycle contained in $G$.\
The complementing case is *CRE2* declaring that $G$ is not Hamiltonian. This can only occur in Step 3, if the algorithm failed to find a Hamilton cycle in the graph consisting of vertices $\mathit{SMALL}(G)\cup U$ and edges $E_G(\mathit{SMALL}(G) \cup U)\cup (U\times U)$, which we will denote by $H$. Since the dynamic programming algorithm was used to find such a cycle, failure to find one means that it does not exist in $H$, so it remains to be shown that if $G$ is Hamiltonian then $H$ must also be Hamiltonian. We provide a proof of this due to Thomason [@THOM].\
Let $G^*$ denote the graph obtained by adding to $G$ all the non-edges with both vertices in $G\setminus \mathit{SMALL}(G)$. Assume that $G$ is Hamiltonian. Then $G^*$ must also be Hamiltonian.\
For some Hamilton cycle $C$, define its *kernel set* to be the edge subset $C\setminus E_{G^*}(V(G)\setminus \mathit{SMALL}(G))$. The kernel set of a Hamilton cycle consists of a set of disjoint paths in $G\setminus E_G(V(G)\setminus \mathit{SMALL}(G))$, containing between them all of $\mathit{SMALL}(G)$, whose endvertices lie in $V(G)\setminus \mathit{SMALL}(G)$.\
Let $C$ be a Hamilton cycle in $G^*$ such that the number of edges from $M$ contained in its kernel set is maximised. Denote $\mathit{SMALL}(G) = W_0 {\mathbin{\mathaccent\cdot\cup}}W_1 {\mathbin{\mathaccent\cdot\cup}}W_2$, where $W_i$ is the subset of $\mathit{SMALL}(G)$ joined by $i$ edges of the kernel set to $V(G)\setminus \mathit{SMALL}(G)$. Let $K\subseteq V(G)\setminus \mathit{SMALL}(G)$ be the set of vertices joined by the kernel set to $\mathit{SMALL}(G)$. Then any vertex in $W_i$ matches to at most $2-i$ vertices in $U\setminus K$, for otherwise if $x\in W_i$ and $(x,y) \in M$, where $y\notin K$, we can remove a kernel set edge from $x$, replace it with $(x,y)$, and create a new kernel set (of another Hamilton cycle $C^{\prime}$) with more edges from $M$ in it. Now, for each vertex in $K$, choose an edge of the kernel set incident to it arbitrarily. Then a vertex of $W_i$ is incident with at most $i$ of these edges. So these edges, along with the edge set $M\cap (\mathit{SMALL}(G) \times (U\setminus K))$, together form a $\leq 2$-matching of order $|U\cup K|$. Since the largest $\leq 2$-matching has order exactly $|U|$, we see that $K\subseteq U$. It now follows from the definition of a kernel set that we can construct a Hamilton cycle in $H$, as claimed.\
*CRE3* {#subalg3}
------
The final part of *CRE* is *CRE3*, an algorithm with the following desired properties:
- The time complexity of *CRE3* is $2^{2n}\cdot n^{O(1)}$;
- The space complexity of *CRE3* is linear in $n$;
- The result of *CRE3* is either a Hamilton cycle contained in the input graph, or a declaration that the graph is not Hamiltonian if the input graph contains none.
Luckily, such an algorithm already exists — the algorithm *HPA3* presented by Gurevich and Shelah in [@GS87]. For completeness we give a brief description of the algorithm. For proof of the properties, see the original paper. We note that, as mentioned in Section \[sec-intro\], an algorithm with time complexity $O\left( 2^n\cdot n^2 \right)$ is known. The downside of this algorithm is that it also has exponential space complexity. This is not a very big issue for us, since our interests in this paper lie exclusively in time complexity, but since we can get a similar algorithm, but with linear space, with its time complexity still sufficiently small for our purposes, this is the one we chose.\
The algorithm *HPA3*, given a graph $G$ and two vertices $s,t\in V(G)$, finds a Hamilton path in $G$ from $s$ to $t$. First we note that converting this algorithm into an algorithm for finding a Hamilton cycle is very simple: choose an arbitrary vertex in $G$, say $s$, and iterate $HPA3(G\setminus (s,t),s,t)$ over all $t\in N_G(s)$. If for some $t$ a Hamilton $s-t$ path $P$ is found then $P\cup (s,t)$ is a Hamilton cycle in $G$. If all iterations fail, then surely $G$ cannot be Hamiltonian.\
*HPA3* is defined recursively, as follows:\
$HPA3(G,s,t):$\
*If $V(G)=\{s,t\}$, return $(s,t)$ if it is an edge, and* **“No such path"** if it is not an edge. *Otherwise:*\
*For all $c\in V(G)\setminus \{s,t\}$ and for all $A\subseteq V(G)\setminus \{s,t,c\}$ of size $\lfloor \frac{n-3}{2} \rfloor$:*\
*If $HPA3(A,s,c)$ and $HPA3(G\setminus A,c,t)$ are successful, return $HPA3(A,s,c) \cup HPA3(G\setminus A,c,t)$*;\
*otherwise, continue.*\
*If loop failed, return* **“No such path".**
Expected time complexity of *CRE* {#sec-time}
=================================
In this section we aim to prove that the algorithm described in Section \[sec-alg\] meets the time complexity goals we had set, that is: if $p \geq 70n^{-\frac{1}{2}}$, then the expected running time over $G(n,p)$ is $(1+o(1))n/p$. Since $$\begin{array}{rcl}
\mathbb{E}\left[ T_{CRE}(G) \right] & \leq & \mathbb{E}\left[ T_{CRE1}(G) \right] + Pr[CRE1\mbox{ fails}]\cdot \mathbb{E}\left[ T_{CRE2}(G)\, |\, CRE1\mbox{ fails} \right] \\
& & + Pr[CRE2\mbox{ fails}]\cdot \mathbb{E}\left[ T_{CRE3}(G) \right] ,
\end{array}$$ it is sufficient to prove that the following hold:
- $\mathbb{E}\left[ T_{CRE1}(G) \right] = (1+o(1))n/p$;
- $Pr[CRE1\mbox{ fails}]\cdot \mathbb{E}\left[ T_{CRE2}(G) \, |\, CRE1\mbox{ fails} \right] = o(n/p)$;
- The probability that *CRE2* returns *“failure"* is $2^{-2n} \cdot n^{-\omega (1)}$;
- The running time of *CRE3* is $ 2^{2n} \cdot n^{O(1)}$.
A proof of the last point is provided in [@GS87]. We now provide proofs for the other three points.\
Expected running time of *CRE1* {#subsec-time1}
-------------------------------
\[runtime1\] If $p\geq 70n^{-\frac{1}{2}}$, $G\sim G(n,p)$, then $\mathbb{E}\left[ T_{CRE1}(G) \right] = (1+o(1))n/p$.
The expected running time of *CRE1* is the sum of the expected running times of its three steps.
- In Step one *CRE1* samples edges, until it reaches at most $n-2$ successes, which means that the expected time of this step is at most $(n-2)/p$;
- In Step 2 *CRE1* samples edges until it finds two existing edges. So the expected running time of this step is $2/p$;
- In Step 3 *CRE1* repeats a loop at most $\sqrt{n}\log n$ times. In each time, it samples edges until it finds $2\sqrt[3]{n}+1$ existing ones. So the expected running time of this step is at most $\sqrt{n}\log n \cdot \left( 2\sqrt[3]{n} + 1 \right) /p = o(n/p)$.
Overall, we get the desired sum of $(1+o(1))n/p$.
Probability of failure of *CRE1* {#subsec-prob1}
--------------------------------
\[prob1\] Let $p\geq 70n^{-\frac{1}{2}}$ and let $G\sim G(n,p)$. Then the probability that $CRE1(G)$ returns the result *“failure"* is $o(n^{-60})$.
We note that since no edge is sampled twice during the run of *CRE1*, all the possible events that lead to failure are independent. We bound from above the probability of each of these events occurring.
1. *CRE1* fails if at some point in Step 1 the last vertex in $P_1$ has no neighbours in the set $[n/2]\setminus P_1$, and if at that point this set is larger than $\sqrt{n}\log n$. The probability of this occurring is at most the probability that among $\frac{n}{2}$ independent random variables distributed $Bin\left( \sqrt{n}\log n,p \right)$ at least one is equal to zero. We bound this probability by applying the union bound: $$\begin{array}{rcl}
Pr[n_1 \geq \sqrt{n}\log n] & \leq & 0.5n\cdot (1-p)^{\sqrt{n}\log n} \\
& \leq & 0.5n\exp (-70\log n) \\
& = & o(n^{-60}).
\end{array}$$
2. $$Pr[n_2 \geq \sqrt{n}\log n] = Pr[n_1 \geq \sqrt{n}\log n] = o(n^{-60}).$$
3. Step 2 results in failure if the minimal indices $i,j,k,l$ for which $\left(v_i,u_{n/2-n_2-j} \right) ,\left(v_{n/2-n_1-k},u_l \right)$ are in $E(G)$ satisfy $i+j+k+l > \sqrt{n}\log n$, and in particular $i+j> 0.5 \sqrt{n}\log n$ or $k+l> 0.5 \sqrt{n}\log n$. There are $\binom{0.5 \sqrt{n}\log n}{2} \geq 0.1n\log ^2n$ pairs $i,j$ (or $k,l$) with $i+j \leq 0.5 \sqrt{n}\log n$, for which an edge query resulted in failure. Applying the union bound we get $$\begin{array}{rcl}
Pr[i+j+k+l > \sqrt{n}\log n] & \leq & 2Pr[i+j > 0.5\sqrt{n}\log n] \\
& \leq & (1-p)^{0.1n\log ^2 n} = n^{-\omega (1)}.
\end{array}$$
4. If Step 3 resulted in failure, say in the $m$’th iteration, then there was some vertex $v$ outside of $S_m$ such that one of the following happened:
1. $v$ did not have $\sqrt[3]{n}$ neighbours in (wlog) $(P_2 \cap S_m)\setminus \{ u_{n/2-n_2-j} \}$ that have not been used in iterations 0 to $i-1$;
2. $s_{S_m}(z)$ did not have $\sqrt[3]{n}$ neighbours in $(P_1 \cap S_m)\setminus \{ v_i \}$;
3. $s(X)$ and $p(Y)$ did not have any edge between them.
Since up to the $m$’th iteration, at most $\sqrt{n}\log n \cdot \sqrt[3]{n} = o(n)$ vertices of $S_m$ have been used, the probability of *(a)* and *(b)* is at most the probability that $Bin(n/6,p)<\sqrt[3]{n}$. So: $$\begin{array}{rcl}
Pr[\mbox{Step 3 failed}] & \leq & n\cdot \left( 2\cdot Pr\left[ Bin(n/6,p)<\sqrt[3]{n} \right] + (1-p)^{\sqrt[3]{n}(\sqrt[3]{n}-1)} \right) \\
& \leq & n\cdot \left( \exp \left( -\Omega (\sqrt{n}) \right) + \exp \left( -\Omega (\sqrt[6]{n}) \right) \right) = n^{-\omega (1)}.
\end{array}$$
So all of the events that lead to failure have probability $o(n^{-60})$, and therefore the probability of failure is also $o(n^{-60})$, as we have set out to prove.
Expected running time of *CRE2* {#subsec-time2}
-------------------------------
\[runtime2\] Let $p=p(n)\geq 70n^{-\frac{1}{2}}$. Then $Pr[CRE1\mbox{ fails}]\cdot \mathbb{E}\left[ T_{CRE2}(G)\, |\, CRE1\mbox{ fails} \right] = O(1)$, where the input to both algorithms is distributed according to $G\sim G(n,p)$.
Denote $Pr[CRE1\mbox{ fails}] := p_1$. Except for Step 3, all steps of *CRE2* have time complexity at most $O(n^5)$, regardless of the input graph. As for Step 3, since $|U\cup \mathit{SMALL}(G)|\leq 3|\mathit{SMALL}(G)|$, the expected runtime of this step (assuming we reach it) is $$\begin{array}{rcl}
\mathbb{E}\left[ T_{Step\ 3}(G)\, |\, CRE1\mbox{ fails} \right] & = &\sum _{k=1}^{2\sqrt{n}} k^{O(1)}2^{6k}\cdot Pr\left[ |\mathit{SMALL}(G)| = k\, | \, CRE1\mbox{ fails}\right]\\
& \leq & {p_1}^{-1} \cdot \sum _{k=1}^{2\sqrt{n}} k^{O(1)}2^{6k}\cdot Pr\left[ |\mathit{SMALL}(G)| = k\right].
\end{array}$$ We bound each term from above, using the Chernoff bound (Lemma \[chernoff\])
$$\begin{array}{rcl}
k^{O(1)}2^{6k}\cdot Pr\left[ |\mathit{SMALL}(G)| = k\right] & \leq & k^{O(1)}2^{6k} \cdot \binom{n}{k} \cdot Pr\left[ Bin(k(n-k),p) \leq \frac{3}{4} knp \right] \\
& \leq & \exp \left( O(\log k) + 6k + k\log n - \Omega (knp) \right) = o\left( n^{-1} \right),
\end{array}$$
hence the value of the entire sum above is at most $o(1)$.
So overall $$Pr[CRE1\mbox{ fails}]\cdot \mathbb{E}\left[ T_{CRE2}(G)\, |\, CRE1\mbox{ fails} \right] = p_1 \cdot O\left( n^5 + {p_1}^{-1} \right) = O(1).$$
Probability of failure of *CRE2* {#subsec-prob2}
--------------------------------
Let $G\sim G(n,p)$, where $p = p(n) \geq 70n^{-\frac{1}{2}}$.\
We will call an event $A$ *rare* if $Pr[A] = 2^{-2n} \cdot n^{-\omega (1)}$. Our goal is to prove that $CRE2(G)$ resulting in *failure* is a rare event. We aim to do this by presenting a graph property $(P)$ such that:
- $G \notin (P)$ is rare;
- If $G \in (P)$ then *CRE2* deterministically either finds a Hamilton cycle or determines that the graph is not Hamiltonian.
Define the graph property $(P)$ as follows: $$\forall U,W \subseteq V(G)\ disjoint\ subsets \,: e(U,W)> |U|\cdot |W| \cdot p\left( 1-\sqrt{\frac{n^{1.5}}{10|U|\cdot |W|}} \right) .$$ (In particular, if $|U|\cdot|W| \geq \frac{n^{1.5}}{10}$ then $e(U,W) \geq 1.)$\
\[prob2\] If $p = p(n) \geq 70n^{-\frac{1}{2}}$ and $G\sim G(n,p)$, then $G \notin (P)$ is rare.
We bound from above the probability that $G \notin (P)$.\
Let $U,W \subseteq V(G)$ be two disjoint sets, and assume that $|U|\cdot|W|\geq \frac{n^{1.5}}{10}$. By the Chernoff bound (Lemma \[chernoff\]), the probability of $e(U,W) \leq |U|\cdot |W| \cdot p\left( 1-\sqrt{\frac{n^{1.5}}{10|U|\cdot |W|}} \right)$ is at most
$$Pr\left[ Bin\left( |U|\cdot |W|,p\right) \leq |U|\cdot |W| \cdot p \left( 1-\sqrt{\frac{n^{1.5}}{10|U|\cdot |W|}} \right) \right] \leq \exp \left(- \frac{1}{20} \cdot n^{1.5}p \right) \leq e^{-3.5n}.\\$$ Finally, by the union bound we get that the probability that exist such $U,W$ is at most $3^n\cdot e^{-3.5n} = 2^{-2n} \cdot n^{-\omega (1)}$, as desired.
In order to prove that *CRE2* does not result in *“failure"* on an input graph $G$ satisfying $(P)$ for $p=p(n)\geq 70n^{-\frac{1}{2}}$, we will show that none of the four stages that may result in *“failure"* does so on such an input.\
In the following lemmas it is assumed, without stating explicitly, that $p(n)\geq 70n^{-\frac{1}{2}}$.
\[failure1\] Let $G$ be a graph on $n$ vertices satisfying $(P)$. Then *Step 1* does not return *“Failure"* on input $G$.
*CRE2* fails this step if and only if $|\mathit{SMALL}(G)| \geq 2\sqrt{n}$. Let $A \subseteq \mathit{SMALL}(G)$ be some subset of size $2\sqrt{n}$. So $A$ and $V(G)\setminus A$ are two disjoint subsets with $|A|\cdot |V(G)\setminus A|\geq 1.9n^{1.5}$, but $$e\left( A,V(G)\setminus A \right) \leq 40\sqrt{n}|A| \leq \left( 1-\frac{1}{\sqrt{19}} \right) \cdot |A|\cdot |V(G)\setminus A|\cdot p,$$ a contradiction to $G$ satisfying $(P)$.
\[failure4\] Let $G$ be a graph on $n$ vertices satisfying $(P)$. Then *Step 4* does not return *“Failure"* on input $G$.
Say we failed to find a path of length at most $4$ between the vertices of some non-edge $e_i := (u_1,u_2)\in U\times U$ in the graph $H_i := G\setminus \left( \bigcup\limits _{j =1}^{i-1} P_j \cup \mathit{SMALL}(G) \cup U \right)$. Since $u_1, u_2\notin \mathit{SMALL}(G)$, it holds that $$|N_{H_i}(u_1)|,|N_{H_i}(u_2)| \geq 40\sqrt{n}-6\cdot |\mathit{SMALL}(G)| \geq 25\sqrt{n}.$$ Let $D_2(G,v)$ denote the set of vertices in a graph $G$ of distance at most 2 from a vertex $v$. Because there is no path of length at most 4, the sets $D_2(H_i,u_1), D_2(H_i, u_2)$ do not intersect each other, which means that one of them, WLOG $D_2(H_i,u_1)$, is of size at most $\frac{1}{2}n$. But then we have $$|N_{H_i}(u_1)|\cdot |H_i\setminus \left( D_2(H_i,u_1) \cup \{u_1\} \right) | \geq 25\sqrt{n} \cdot \left( n-12\sqrt{n}-\frac{1}{2}n -1 \right) \geq 10n^{1.5},$$ $$e\left( N_{H_i}(u_1),H_i\setminus \left(N_{H_i}(u_1) \cup D_2(H_i,u_1) \cup \{u_1\} \right) \right) =0,$$ which means $G \notin (P)$, a contradiction.
\[failure5\] Let $G$ be a graph on $n$ vertices satisfying *(P)*. Then *Step 5* does not return *“Failure"* on input $G$.
Say we failed at some time $i$, that is: the constructed vertex set $U_i$ is an independent set. Recall that $U_i$ is the set of successors along $S_i$ of vertices in $N_G(V_i)\cap S_i$, where $V_i$ is a maximum sized connected component of $G\setminus S_i $. Let $W_i=N_G(V_i)\cap S_i$. Consider the following cases:
1. $|U_i| \geq n^{\frac{3}{4}}$. Let $A_1,A_2 \subseteq U_i$ be two disjoint subsets of size $\frac{1}{2}n^{\frac{3}{4}}$. So $|A_1|\cdot |A_2| = \frac{1}{4}n^{1.5}$, but $e(A_1,A_2)=0$, a contradiction.
2. $|V_i| > n - 30\sqrt{n}$. Observe two facts:
- By Def. \[small\], since $|V(G)\setminus V_i|< 30\sqrt{n} <40\sqrt{n}$, we get that $\forall v\in S_i\setminus \mathit{SMALL}(G):\ N_G(v)\cap V_i \neq \emptyset$;
- Since $|\mathit{SMALL}(G)| < \frac{1}{2}|S_0| \leq \frac{1}{2}|S_i|$, there are two vertices $w_1,w_2 \in S_i\setminus \mathit{SMALL}(G)$ such that $w_1=s_{S_i}(w_2)$.
So $w_1,w_2$ belong to $W_i$, and their successors are connected by an edge, which means that the algorithm could not have failed.
3. $15\sqrt{n} \leq |V_i| \leq n-30\sqrt{n}$. Observe that if the algorithm failed then $|W_i|=|U_i| \leq \min \lbrace \frac{1}{2}|S_i| , n^{3/4} \rbrace$, and therefore we have
- $|V_i| + |V(G)\setminus (V_i \cup W_i)| \geq n-n^{\frac{3}{4}}$;
- $|V_i| \geq 15\sqrt{n}$;
- $|V(G)\setminus (V_i \cup W_i)| = |V(G)\setminus V_i | - | W_i| \geq |V(G)\setminus V_i | - \frac{1}{2} | S_i| \geq \frac{1}{2} |V(G) \setminus V_i| \geq 15\sqrt{n}$.
So $V_i$ and $V(G) \setminus (V_i \cup W_i)$ are two sets, with $|V_i|\cdot |V(G) \setminus (V_i \cup W_i)| \geq 10n^{1.5}$, but $e(V_i,V(G)\setminus (V_i \cup W_i))=0$, a contradiction to our assumption that $G\in (P)$.
4. $|V_i|\leq 15\sqrt{n},\ |S_i| < 0.99n$. Then all connected components of $G\setminus S_i$ are of size at most $15\sqrt{n}$, and the sum of their sizes is at least $0.01n$. So the vertices of $V(G)\setminus S_i$ can be partitioned into two sets $A_1,A_2$ such that each one of them is a union of connected components, and $|A_1|,|A_2| \geq n^{\frac{3}{4}}$. But then $|A_1|\cdot |A_2| \geq n^{1.5}$ and $e(A_1,A_2)=0$, a contradiction.
The complementing case to those already covered is when $|V_i|\leq 15\sqrt{n},\ |S_i| \geq 0.99n$, which can only occur in Stage 6.
\[failure6\] Let $G$ be a graph on $n$ vertices satisfying *(P)*. Then *Step 6* does not return *“Failure"* on input $G$.
We show that under the assumption that $G\in (P)$, the cycle $S_i$ contains two vertices $u,w$ and two edge subsets $E_1,E_2$ as described in *Step 6*. Since the algorithm searches for such $u,w,E_1,E_2$ exhaustively, and only returns *“Failure"* upon failing the search, this means that if $G\in (P)$ the algorithm does not fail.\
Recall that in this stage we can assume that $|S_i| \geq 0.99n$ and that all connected components of $G\setminus S_i$ are of size at most $15\sqrt{n}$. It follows that for every $v\in V(G)\setminus S_i$ we have $|N_G(v)\cap S_i| \geq d_G(v)-|V_i| \geq 20\sqrt{n}$. Observe that if $ |N_G(v)\cap S_i| > \frac{1}{2}n$ then $v$ has two neighbours adjacent on $S_i$, say $u,w$, so setting $E_1 = (u,w),\ E_2=\emptyset$ results in a cycle as desired, so we can assume that $|N_G(v)\cap S_i| \leq \frac{1}{2}n$.\
Let $U_i := p(N_G(v)\cap S_i)$. Since $|U_i|,|S_i\setminus U_i|\geq 20\sqrt{n}$ and $|U_i|+|S_i\setminus U_i|\geq 0.99n$, we get that $|U_i|\cdot |S_i\setminus U_i| \geq 10n^{1.5}$, and therefore $e(U_i,S_i\setminus U_i) \geq 0.9p|U_i|\cdot |S_i\setminus U_i| \geq 0.4|U_i|np$. It follows that there is some $u \in N_G(v)\cap S_i$ such that $d_{S_i}(p(u)) \geq 0.4np \geq 20\sqrt{n}$. Denote $t_0:=p(u)$, and let $Q$ be the path $\{v\} \cup S_i(u\rightarrow t_0)$.\
Define the following three special vertices on $Q$:
- $c_v$ : a vertex on $Q$ such that $|N_{Q(v\rightarrow c_v)}(v)| = \lfloor \frac{1}{2}|N_{Q}(v)| \rfloor$;
- $c_{t_0}$ : a vertex on $Q$ such that $|N_{Q(c_{t_0}\rightarrow t_0)}(t_0)| = \lfloor \frac{1}{2}|N_{Q}(t_0)| \rfloor$;
- $c$ : a vertex on $Q$ such that $|Q(v\rightarrow c)| = \lfloor \frac{1}{2}|Q| \rfloor$.
We will assume that $c_v$ and $c$ precede $c_{t_0}$ on $Q$, and remark that the proof is quite similar for the complementing cases, in which $c_{t_0}$ precedes one or both of $c_v,c$, with some minor changes required to some of the definitions down the line.\
Denote: $Q_1 := Q(v\rightarrow c_v),\ Q_2:= Q(c_{t_0}\rightarrow t_0),\ Q_3:= Q(v\rightarrow c)$.\
We now aim to show that $E_1,E_2,w$ as required exist in the graph, with respect to the already chosen $u$, by using rotations and extensions.\
Let $W_i$ be the set $N_{Q_1}(v)$ and $T_i$ the set $s_Q(N_{Q_2}(t_0))$. By our choices of $v,t_0,c_v,c_{t_0}$ we know that $|W_i|,|T_i| \geq 10\sqrt{n}$. Now, construct the set $O_i$ as follows:\
For each vertex $x\in W_i$ and for each $y\in N_{Q_3}(p_Q(x))\setminus \{x\}$ add $s_Q(y)$ to $O_i$ if $y \in Q(v\rightarrow x)$ and add $p_Q(y)$ to $O_i$ if $y \in Q(x\rightarrow c)$.
The size $|O_i|$ is at least $0.2n$.
By our construction, $|O_i| \geq |N_{Q_3}(p_Q(W_i))| - |W_i|$. If $|O_i| < 0.2 n$ then $| Q_3 \setminus N_{Q_3}(p_Q(W_i))| \geq 0.25 n - |W_i|$, and $p_Q(W_i),\ Q_3 \setminus N_{Q_3}(p_Q(W_i))$ are two sets that have no edges between them, but the product of their sizes is at least $2n^{1.5} $, a contradiction.
There is an edge between $O_i$ and $T_i$.
The two sets are disjoint, and $|O_i|\cdot |T_i| \geq 2n^{1.5}$.
\(v) at (-0.2\*, 1.5\*) ; (u) at (-0.5\*, ) ; (pw) at (0, ) ; (w) at (, ) ; (cv) at (, ) ; (s) at (, 0) ; (ss) at (, -) ; (c) at (, -) ; (ct0) at (0, -) ; (pt) at (-, -) ; (t) at (-, 0) ; (t0) at (-, ) ;
(,0) arc \[start angle=0, end angle=90-, radius=\]; (0,) arc \[start angle=90, end angle=135-, radius=\]; (0,) arc \[start angle=90, end angle=90-, radius=\]; (,0) arc \[start angle=0, end angle=-, radius=\]; (-,) arc \[start angle=135, end angle=135-, radius=\]; (-,0) arc \[start angle=180, end angle=135, radius=\]; (-,0) arc \[start angle=180, end angle=180+, radius=\]; (0,-) arc \[start angle=270, end angle=180+, radius=\]; (0,-) arc \[start angle=-90, end angle=-, radius=\];
(ss) to (pw); (t) to (s); (pt) to (t0); (v) to (w); (v) to (u);
Let $s\in O_i,\ t\in T_i$ be such that $(s,t)\in E(G)$, and let $w \in W_i$ be a vertex that caused $s$ to be added to $O_i$. Finally, define:
- $E_1 := \{(u,t_0),\ (p_Q(w),w),\ (s,s_Q(s)),\ (p_Q(t),t) \}$;
- $E_2 := \{(p_Q(w),s_Q(s)),\ (p_Q(t)),t_0),\ (s,t)\}$.
Then $E_1,E_2,u,w$ are as required by the algorithm (see Fig. 1 for illustration).
Concluding remarks {#remarks}
==================
To summarise, we have presented an algorithm *CRE* which is comprised of three aligned algorithms, in the spirit of previous results, and utilises rotations and extensions in order to find a Hamilton cycle in a graph, and proved that its expected running time on a random graph $G\sim G(n,p)$ is optimal, for $p\geq 70n^{-\frac{1}{2}}$.
We note that even if we make changes to some parameters in our algorithm, $p=\Omega \left( n^{-\frac{1}{2}} \right)$ seems to be the lowest range of probability for which our expected running time bound works, at least with our current argument. The reason for this is the existence of some bottlenecks along the proof, where smaller orders of magnitude of the edge probability no longer work. Such a bottleneck can be observed, for example, in Step 4 of *CRE2*, where the algorithm tries to connect some set of paths into a cycle that contains them, by finding paths between pairs of endpoints of paths one by one. In our proof we use the fact that the total length of the paths is highly likely to be much smaller than the minimum degree of the vertices at the endpoints of the paths (that is to say that the complement event is rare, i.e., has probability $2^{2n}\cdot n^{-\omega (1)}$). This is due to the fact that, on the one hand, all of the paths’ endpoints have degrees at least comparable to the expected average degree of the graph, since by our construction none of the endpoints are members of $\mathit{SMALL}(G)$ – the set of vertices with very small degrees. On the other hand, the total number of vertices in the union of all the paths is not likely to be very big, since this vertex set contains at most $6\cdot |\mathit{SMALL}(G)|$ vertices, a size likely to be much smaller than the average degree of the graph for our parameters, as we observed that $\mathit{SMALL}(G)$ is highly likely to be of size much smaller than $np$. If $p =o\left( n^{-\frac{1}{2}} \right)$, however, then the event “$|\mathit{SMALL}(G)|>np$" has probability $2^{-o(n)}$, and in particular it is no longer rare. In other words, the probability that one of the paths’ endpoints has all its neighbours residing in the union of $\mathit{SMALL}(G)$ and previously constructed paths is $2^{-o(n)}$, and the expected runtime of *CRE* might no longer even be polynomial.\
And so, we leave it as an open question whether a polynomial expected running time Hamiltonicity algorithm exists for edge probability $p =o\left( n^{-\frac{1}{2}} \right)$.\
**Acknowledgements.** The authors would like to express their thanks to the referees of the paper, and to Samotij Wojtek, for their valuable input towards improving the presentation of our result.\
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M. Garey, D. Johnson and L. Stockmeyer, *Some simplified NP-complete graph problems*, Theoretical Computer Science 1.3 (1976), 237–267.
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[^1]: ‡School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, 6997801, Israel. Email: [email protected].
[^2]: School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, 6997801, Israel. Email: [email protected]. Partially supported by USA-Israel BSF grant 2014361, and by ISF grant 1261/17.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we develop necessary and sufficient conditions for the validity of a martingale approximation for the partial sums of a stationary process in terms of the maximum of consecutive errors. Such an approximation is useful for transferring the conditional functional central limit theorem from the martingale to the original process. The condition found is simple and well adapted to a variety of examples, leading to a better understanding of the structure of several stochastic processes and their asymptotic behaviors. The approximation brings together many disparate examples in probability theory. It is valid for classes of variables defined by familiar projection conditions such as the Maxwell–Woodroofe condition, various classes of mixing processes, including the large class of strongly mixing processes, and for additive functionals of Markov chains with normal or symmetric Markov operators.'
address:
- 'POMI (Saint Petersburg Department of the Steklov Institute of Mathematics), 27 Fontanka emb., Saint Petersburg 191023, Russia. '
- 'Department of Mathematical Sciences, University of Cincinnati, P.O. Box 210025, Cincinnati, OH 45221-0025, USA. '
author:
-
-
title: On the functional central limit theorem via martingale approximation
---
Introduction and results
========================
The objective of this paper is to find a characterization of stationary stochastic processes that can be studied via a martingale approximation in order to derive the functional central limit theorem for processes associated with partial sums.
There are several ways to present the results since stationary processes can be introduced in several equivalent ways. We assume that $(\xi
_{n})_{n\in\mathbb{Z}}$ denotes a stationary Markov chain defined on a probability space $(\Omega,\mathcal{F},P)$ with values in a measurable space $(S,\mathcal{A})$. The marginal distribution and the transition kernel are denoted by $\pi(A)=P(\xi_{0}\in A)$ and $Q(\xi_{0},A)=P(\xi_{1}\in A|
\xi
_{0})$, respectively. In addition, $Q$ denotes the operator [acting via $(Qf)(\xi)=\int
_{S}f(s)Q(\xi,\mathrm{d}s).$ Next, let $\mathbb{L}_{0}^{2}(\pi)$ be the set of functions on $S$ such that $\int f^{2}\,\mathrm{d}\pi<\infty$ and $\int f\,\mathrm{d}\pi=0.$ Denote by $\mathcal{F}_{k}$ the $\sigma$-field generated by $\xi_{i}$ with $i\leq k,$ $X_{i}=f(\xi_{i})$, $S_{n}=\sum _{i=0}^{n-1}X_{i}$ (i.e., $S_{0}=0,S_{1}=X_{0}$, $S_{2}=X_{0}+X_{1},\dots$). For any integrable variable $X$, we define $\mathbb{E}_{k}(X)=\mathbb{E}(X|\mathcal{F}_{k}).$ In our notation, $\mathbb{E}_{0}(X_{1})=Qf(\xi_{0})=\mathbb{E}(X_{1}|\xi
_{0}).$ ]{} We also set $\mathcal F_{-\infty}=\bigcap_{k \in\mathbb Z}
\mathcal F_{k} $ .
Throughout the paper, we assume $f\in\mathbb{L}_{0}^{2}(\pi)$; in other words, we assume that $\Vert X\Vert _{2}=(\mathbb{E}[X_{1}^{2}])^{1/2}<\infty$ and $\mathbb{E}[X_{1}]=0.$
Note that any stationary sequence $(Y_{k})_{k\in\mathbb{Z}}$ can be viewed as a function of a Markov process $\xi_{k}=(Y_{i};i\leq k)$ for the function $g(\xi_{k})=Y_{k}$.
The stationary stochastic processes may be also introduced in the following, alternative, way. Let $T\dvtx \Omega\mapsto\Omega$ be a bijective bimeasurable transformation preserving the probability. Let $\mathcal{F}_{0}$ be a sub-$\sigma$-algebra of $\mathcal{F}$ satisfying $\mathcal{F}_{0}\subseteq
T^{-1}(\mathcal{F}_{0})$. We then define the non-decreasing filtration $(\mathcal{F}_{i})_{i\in\mathbb{Z}}$ by $\mathcal{F}_{i}=T^{-i}(\mathcal
{F}_{0})$. Let $X_{0}$ be a random variable which is ${}\mathcal{F}_{0}$-measurable. We define the stationary sequence $(X_{i})_{i\in\mathbb
{Z}}$ by $X_{i}=X_{0}\circ T^{i}$.
In this paper, we shall use both frameworks.
In order to analyze the asymptotic behavior of the partial sums $S_{n}=\sum_{i=0}^{n-1}X_{i},$ Gordin, in [@g], proposed to decompose the sums related to the original stationary sequence into the sum $$S_{n}=M_{n}+R_{n} \label{martaprox}$$ of a square-integrable martingale $M_{n}=\sum_{i=0}^{n-1}D_{i}$ adapted to $\mathcal{F}_{n}$, whose martingale differences $(D_{i})$ are stationary, and a so-called coboundary $R_{n}$, that is, a telescoping sum of random variables with the basic property that $\sup_{n}\mathbb{E}(R_{n}^{2})<\infty$. More precisely, $X_{n}=D_{n}+Z_{n}-Z_{n-1},$ where $Z_{n}$ is another stationary sequence in $\mathbb{L}_{2}$. The limiting properties of the martingales can then be transported from the martingale to the general sequence. In the context of Markov chains, the existence of such a decomposition is equivalent to the solvability of the Poisson equation in $\mathbb{L}_{2}$.
For proving a central limit theorem for stationary sequences, a weaker form of martingale approximation has been pointed out by many authors (see, e.g., [@mpu2] for a survey). Recently, two interesting papers, one by Dedecker, Merlevède and Volný [@dmv] and the other by Zhao and Woodroofe [@Zw2], provided necessary and sufficient conditions for martingale approximation with an error term in (\[martaprox\]) satisfying $$\mathbb{E}\bigl((S_{n}-M_{n})^{2}\bigr)/n\rightarrow0. \label{MA}$$ This decomposition is strong enough for transporting the conditional central limit theorem from sums of stationary martingale differences in $\mathbb{L}_{2}$ to $S_{n}/\sqrt{n}.$ By conditional CLT, as discussed in [@DM02], we understand, in this context, that for any continuous function $f$ such that $|f(x)|/(1+x^{2})$ is bounded and for any $k \ge0,$ $$\biggl\Vert \mathbb{E}_{k}\bigl(f\bigl(S_{n}/\sqrt{n}\bigr)\bigr)- \int_{- \infty}^{\infty} f\bigl(x
\sqrt{\eta}\bigr) g(x)\,\mathrm{d}x\biggr\Vert _1 \mathop{\longrightarrow}_{n \to\infty} 0,
\label{CLT}$$ where $g$ is the standard normal density and $\eta\ge0$ is an invariant function satisfying $$\mathop{\lim}_{n \to\infty}\biggl\Vert\frac{\mathbb
{E}_{0}(S_n^2)}{n}-\eta
\biggr\Vert_1=0.$$ Here, and throughout the paper, we denote by $\Vert \cdot\Vert _{p}$ the norm in $\mathbb{L}_{p}$.
An important extension of this theory is to consider the conditional central limit theorem in its functional form. For $t\in\lbrack0,1]$, define $$S_n(t)=S_{[nt]}+(nt-[nt])X_{[nt]},$$ where $[x]$ denotes the integer part of $x$. Note that $S_n(\cdot)/\sqrt{n}$ is a random element of the space $C([0,1])$ endowed with the supremum norm $\Vert \cdot\Vert _{\infty}.$ Then, by the conditional CLT in the functional form (FCLT), we understand that for any continuous function $f\dvtx C([0,1]) \to\mathbb R$ such that $x \mapsto|f(x)|/(1+\Vert x\Vert _{\infty}^2)$ is bounded and for any $k\ge
0$, we have $$\biggl\Vert \mathbb{E}_{k}\bigl(f\bigl(S_n/\sqrt{n}\bigr)\bigr)- \int_{C([0,1])}\bigl(f\bigl(x
\sqrt{\eta}\bigr)\bigr)\,\mathrm{d}W(x)\biggr\Vert _1\mathop{\longrightarrow}_{n \to\infty} 0. \label{FCLT}$$ Here, $W$ is the standard Wiener measure on $C([0,1])$.
It is well known that a martingale with stationary differences in $\mathbb{L}_{2}$ satisfies this type of behavior with $\eta=
\lim_{n \to\infty} \sum_{l=0}^{n-1}D_{l}^2/n$ in $\mathbb
{L}_{1}$ – this is at the heart of many statistical procedures. This conditional form of the invariance principle is a stable type of convergence that makes possible the change of measure with another absolutely continuous measure, as discussed in [@b; @ru; @hh].
With such a result in mind, the question is now to find necessary and sufficient conditions for a martingale decomposition with the error term satisfying $$\mathbb{E}\Bigl(\max_{1\leq j\leq n}(S_{j}-M_{j})^{2}\Bigr)\big/n\rightarrow0 .
\label{maxcond}$$
In order to state our martingale approximation result, for fixed $m$, we consider the stationary sequence $$Y_{0}^{m}=\frac{1}{m}\mathbb{E}_{0}(X_{1}+\cdots+X_{m}),\qquad
Y_{k}^{m}=Y_{0}^{m}\circ T^{k}. \label{defY}$$ In the language of Markov operators, we then have $$Y_{0}^{m}=\frac{1}{m}(Qf+\cdots+Q^{m}f)(\xi_{0}) .$$ It is convenient to introduce a seminorm notation, namely,$$\Vert Z\Vert _{M^{+}}=\lim\sup_{n\rightarrow\infty}\frac{1}{\sqrt{n}}\Biggl\Vert \max_{1\leq k\leq n}\Biggl| \sum_{j=1}^{k} Z \circ T^j
\Biggr| \Biggr\Vert_{2}$$ on the space of all $Z \in L^2_0$ with $\Vert Z\Vert _{M^{+}} < \infty.$
\[T\]\[maxgencopy(2)\]Assume that $(X_{k})_{k\in Z}$ is a stationary sequence of centered square-integrable random variables. Then$$\Vert Y_{0}^{m}\Vert _{M^{+}}\rightarrow0\qquad\mbox{as }m\rightarrow\infty
\label{MAX}$$ if and only if there exists a martingale with stationary increments satisfying [(\[maxcond\])]{}. Such a martingale is unique if it exists. In particular, [(\[MAX\])]{} implies [(\[FCLT\])]{}.
As a consequence of the proof of Theorem \[T\], we also obtain the following result that adds a new equivalent condition to the characterizations by Dedecker, Merlevède and Volný [@dmv] and Zhao and Woodroofe [@Zw2]. With $(Y_{k}^{m})_{k\in Z}$ defined by [(\[defY\])]{} and the seminorm notation $$\Vert Y_{0}^{m}\Vert _{+}=\lim\sup_{n\rightarrow\infty}\frac{1}{\sqrt{n}}\Biggl\Vert \sum_{j=1}^{n}Y_{j}^{m}\Biggr\Vert _{2}$$ we have the following characterization.
\[T2\]\[maxgencopy(1)\]\[Tcopy(1)\]\[maxgen\]Assume that $(X_{k})_{k\in Z}$ is as in Theorem \[T\]. Then $$\Vert Y_{0}^{m}\Vert _{+}\rightarrow0\qquad\mbox{as }m\rightarrow\infty
\label{MA1}$$ if and only if there exists a stationary martingale satisfying [(\[MA\])]{}. Such a martingale is unique if it exists. In particular, [(\[MA1\])]{} implies [(\[CLT\])]{}.
Our approach is constructive. If the stationary sequence is supposed to be ergodic, then the constructed martingale differences are also ergodic and therefore the conditional theorems (\[CLT\]) and (\[FCLT\]) can be easily transported to the original processes satisfying (\[MA1\]) and (\[MAX\]), respectively, with $\eta=\Vert D_{0}\Vert _{2}$.
A natural and useful question is to provide classes of stochastic processes that have a martingale decomposition with an error term satisfying [(\[maxcond\])]{}, in other words, to provide sharp sufficient conditions for such a decomposition. Obviously, a maximal inequality is needed in order to verify this condition. We shall combine our approach with several maximal inequalities. One is due to Rio [@rio], formula (3.9), page 53; for related inequalities, see [@Pel] and [@DR].
- For any stationary process with centered variables in $\mathbb{L }_{2}
$, $$\mathbb{E}\Bigl(\max_{1\leq i\leq n}S_{i}^{2}\Bigr)\leq8n\mathbb{E}(X_{0}^{2})+16
\sum_{k=2}^{n}\mathbb{E}|X_{0}\mathbb{E}_{0}(S_{k}-S_1)|.
\label{Rio}$$
Another inequality comes from [@PU1], Proposition (2.3); see also [@puw], Theorem 1, for the inequality in $\mathbb{L}_{p}$.
- For any stationary process with centered variables in $\mathbb{L}_{2}$, $$\begin{aligned}
\label{PU}
\mathbb{E}\Bigl(\max_{1\leq i\leq n}S_{i}^{2}\Bigr) & \leq& n\Biggl(2\Vert
X_{0}\Vert_{2}+3\sum_{j=0}^{r-1}\frac{\Vert \mathbb
{E}_{0}(S_{2^{j}})\Vert _{2}}{2^{j/2}}\Biggr)^{2}
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
& \leq& n\Biggl(2\Vert X_{0}\Vert_{2}+80\sum_{j=1}^{n}\frac{\Vert \mathbb{E}_{0}(S_{j})\Vert _{2}}{j^{3/2}}\Biggr)^{2},\end{aligned}$$ where $2^{r-1} < n\leq2^{r}.$
The following maximal inequality is a particular case of Dedecker and Merlevède [@DM02], Proposition 6; see [@Wu], Theorem 1, for the inequality in $\mathbb{L}_{p}$.
- For any stationary process with centered variables in $\mathbb{L}_{2}$ such that $\mathbb E(X_{0}|\mathcal{F}_{-\infty})=0$ almost surely, we have$$\mathbb{E}\Bigl(\max_{1\leq i\leq n}S_{i}^{2}\Bigr)\leq4n\Biggl(\sum
_{i=0}^{\infty}\Vert
\mathbb{E}_{-i}(X_{0})-\mathbb{E}_{-i-1}(X_{0})\Vert_{2}\Biggr)^{2}
.\label{DM}$$
Another inequality we use for additive functionals of stationary reversible Markov chains is a consequence of Wu [@wu], Corollary 2.7 and relation (2.5) in the same paper (note that there is a typographical error in this relation, namely, a square should be added to the norm); see also [@svy]:
- Assume $(\xi_{n})_{n\in\mathbb{Z}}$ is a stationary, reversible Markov chain and $X_n=f(\xi_n)$ with $f\in\mathbb
{L}_{0}^{2}(\pi)$. Then, for every $n\geq1$,$$\mathbb{E}\Bigl(\max_{1\leq i\leq n}S_{i}^{2}\Bigr)\leq
(24n+3)\sum _{n=0}^{\infty}\mathbb E(X_{0}X_{n}),
\label{LW}$$ provided the series on the right-hand side is convergent.
This inequality, originally stated for the ergodic case, extends without changes to the general case.
By combining the martingale decomposition in Theorem \[T\] with these maximal inequalities, we point out various classes of stochastic processes for which a conditional functional limit theorem holds. These include mixing processes and classes of Markov chains.
Proof of Theorem \[T\]
======================
The proof of this theorem has several steps.
*Step Construction of the approximating martingale*.
The construction of the martingale decomposition is based on averages. It was introduced by Wu and Woodroofe [@W] (see their definition (6) on page 1677) and further developed in [@Zw2], extending the construction in [@Heyde] and [@gl]; see also [@1994], Theorem 8.1, and [@KV]. We give the martingale construction here for completeness.
We introduce a parameter $m\geq1$ (kept fixed for the moment) and define the following stationary sequence of random variables:$$\theta_{0}^{m}=\frac{1}{m}\sum_{i=1}^{m}\mathbb{E}_{0}(S_{i}),\theta_{k}^{m}=\theta_{0}^{m}\circ T^{k}.$$ Set $$D_{k}^{m}=\theta_{k+1}^{m}-\mathbb{E}_{k}(\theta_{k+1}^{m}),\qquad
M_{n}^{m}=\sum_{k=0}^{n-1}D_{k}^{m}. \label{martd}$$ Then $(D_{k}^{m})_{k\in\mathbb{Z}}$ is a stationary martingale difference sequence and $(M_{n}^{m})_{n\geq0}$ is a martingale. Thus, we have$$X_{k}=D_{k}^{m}+\theta_{k}^{m}-\theta_{k+1}^{m}+\frac{1}{m}\mathbb{E}_{k}(S_{k+m+1}-S_{k+1})$$ and therefore$$\begin{aligned}
\label{martdec}
S_{k}& =&M_{k}^{m}+\theta_{0}^{m}-\theta_{k}^{m}+\sum
_{j=1}^{k}\frac{1}{m}\mathbb{E}_{j-1}(S_{j+m}-S_{j})
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
& =&M_{k}^{m}+\theta_{0}^{m}-\theta_{k}^{m}+\overline{R}_{k}^{ m},\end{aligned}$$ where we have made use of the notation$$\overline{R}_{k}^{ m}=\sum _{j=1}^{k}\frac{1}{m}\mathbb{E}_{j-1}(S_{j+m}-S_{j}).$$ Observe that $$\overline{R}_{k}^{ m}=\sum_{j=0}^{k-1}Y_{j}^{m}.\label{rests}$$ With the notation $$R_{k}^{m}=\theta_{0}^{m}-\theta_{k}^{m}+\overline{R}_{k}^{ m},
\label{rest}$$ we have $$S_{k}=M_{k}^{m}+R_{k}^{m}. \label{deco}$$
*Step Sufficiency*.
We show that $\Vert Y_{0}^{m}\Vert _{M^{+}}\rightarrow0$ as $m\rightarrow\infty
$ is sufficient for (\[maxcond\]).
The starting point is the construction of the martingale differences, as in (\[martd\]). By the martingale property and (\[deco\]), for all positive integers $m^{\prime}
$ and $m^{\prime\prime}$, we have $$\Vert D_{0}^{m^{\prime}}-D_{0}^{m^{\prime\prime}}\Vert _{2}=\frac{1}{\sqrt
{n}}\Vert M_{n}^{m^{\prime}}-M_{n}^{m^{\prime\prime}}\Vert _{2}=\frac{1}{\sqrt
{n}}\Vert R_{n}^{m^{\prime}}-R_{n}^{m^{\prime\prime}}\Vert _{2} .$$ We now let $n\rightarrow\infty.$ By relation (\[rest\]) and stationarity, it follows that $$\begin{aligned}
\lim\sup_{n\rightarrow\infty}\frac{1}{\sqrt{n}}\Vert R_{n}^{m^{\prime
}}-R_{n}^{m^{\prime\prime}}\Vert _{2}&=&\lim\sup_{n\rightarrow\infty
}\frac{1}{\sqrt{n}}\Vert \overline{R}_{n}^{ m^{\prime}}-\overline{R}_{n}^{
m^{\prime
\prime}}\Vert _{2} \\
&\leq&\lim\sup_{n\rightarrow\infty}\frac{1}{\sqrt{n}}(\Vert \overline{R}_{n}^{ m^{\prime}}\Vert _{2}+\Vert \overline{R}_{n}^{ m^{\prime\prime
}}\Vert _{2}).\end{aligned}$$ By (\[MAX\])$,$ the limit when $m^{\prime}$ and $m^{\prime\prime}$ both tend to $\infty$ is then $0$, giving that $(D_{0}^{m})$ is Cauchy in $\mathbb{L}_{2}$ and therefore convergent. Denote its limit by $D_{0}$. Then $M_{n}=\sum _{k=0}^{n-1}D_{k}$ is a martingale with the desired properties. To see this, we start from the decomposition in relation ([martdec]{}) and obtain $$|S_{k}-M_{k}|\leq|M_{k}^{m}-M_{k}|+|\theta_{k}^{m}-\theta_{0}^{m}|+|\overline{R}_{k}^{m}|.$$ Then$$\begin{aligned}
\frac{{1}}{\sqrt{n}}\Bigl\Vert \max_{1\leq k\leq n}|S_{k}-M_{k}|
\Bigr\Vert_{2}&\leq&
\frac{{1}}{\sqrt{n}}\Bigl\Vert \max_{1\leq k\leq n}|M_{k}^{m}-M_{k}|
\Bigr\Vert _{2} \\
&&{}+\frac{{1}}{\sqrt{n}}\Vert \theta_{0}^{m}\Vert _{2}+\frac{{1}}{\sqrt{n}}\Bigl\Vert \max_{1\leq k\leq n}|\theta_{k}^{m}| \Bigr\Vert_{2}+\frac{{1}}{\sqrt{n}}\Bigl\Vert
\max_{1\leq k\leq n}|\overline{R}_{k}^{ m}| \Bigr\Vert_{2}.\end{aligned}$$ By Doob’s maximal inequality for martingales and by stationarity, we conclude that $$\frac{{1}}{\sqrt{n}}\Bigl\Vert \max_{1\leq k\leq n}|M_{k}^{m}-M_{k}|
\Bigr\Vert _{2}\leq
\Vert D_{0}^{m}-D_{0}\Vert _{2}.$$ For $m$ fixed, since $(\theta_{k}^{m})_{k\in Z}$ is a stationary sequence of square-integrable random variables, for any $A>0$, we have$$\begin{aligned}
\frac{{1}}{n}\mathbb{E}\Bigl[\max_{1\leq k\leq n}|\theta_{k}^{m}|^{2}\Bigr]
&\leq&\frac{A^{2}}{n}+\frac{1}{n}\sum_{k=1}^{n}\mathbb{E[}|\theta
_{k}^{m}|^{2}I(|\theta_{k}^{m}|>A)] \\
&=&\frac{A^{2}}{n}+\mathbb{E[}|\theta_{0}^{m}|^{2}I(|\theta_{0}^{m}|>A)]\end{aligned}$$ and then, clearly,$$\lim_{n\rightarrow\infty}\frac{{1}}{n}\mathbb{E}\Bigl[\max_{1\leq k\leq
n}|\theta_{k}^{m}|^{2}\Bigr]=0. \label{maxnegl}$$ Then, taking into account [(\[rests\])]{}, we easily obtain$$\lim\sup_{n\rightarrow\infty}\frac{{1}}{\sqrt{n}}\Bigl\Vert \max_{1\leq
k\leq
n}|S_{k}-M_{k}| \Bigr\Vert_{2}\leq\Vert D_{0}^{m}-D_{0}\Vert _{2}+\Vert Y_{0}^{m}\Vert _{M^{+}}$$ and the result follows by letting $m\rightarrow\infty$, from the fact that $D_{0}^{m}\rightarrow D_{0}$ in $\mathbb{L}_{2}$. It is easy to see that the martingale is unique.
*Step Necessity*.
Assume that the martingale approximation (\[maxcond\]) holds. With the notation $R_{n}=S_{n}-M_{n},$ we then have $$\lim_{n\rightarrow\infty}\frac{{1}}{\sqrt{n}}\Bigl\Vert \max_{1\leq k\leq
n}|R_{k}| \Bigr\Vert _{2}=0.$$ In particular, this approximation implies that $$\lim_{n\rightarrow\infty}\frac{1}{\sqrt{n}}\max_{1\leq k\leq
n}\Vert \mathbb E(S_{k}|\mathcal{F}_{0})\Vert _{2}=0. \label{natural}$$ From $$\Vert \overline{R}_{n}^{ n}\Vert _{2}\leq\Vert \mathbb
E(S_{n}|\mathcal{F}_{0})\Vert _{2},$$ we deduce that $$\Vert R_{n}^{n}\Vert _{2}=\Vert \theta_{0}^{n}-\theta_{n}^{n}+\overline{R}_{n}^{ n}\Vert _{2}\leq2\Vert \theta_{0}^{n}\Vert _{2}+\Vert \overline{R}_{n}^{ n}\Vert _{2}\leq3\max_{1\leq k\leq n}\Vert \mathbb E(S_{k}|\mathcal
{F}_{0})\Vert _{2},$$ whence, by (\[natural\]), it follows that $$\lim_{n\rightarrow\infty}\frac{\Vert R_{n}^{n}\Vert _{2}}{\sqrt{n}}=0.$$
As a consequence, we obtain$$\mathbb{E}(D_{0}^{n}-D_{0})^{2}=\frac{\mathbb
{E}(M_{n}^{n}-M_{n})^{2}}{n}=\frac{\mathbb{E}(R_{n}^{n}-R_{n})^{2}}{n}\rightarrow0\qquad\mbox{as }
n\rightarrow\infty.$$ This shows that $D_{0}^{n}\rightarrow D_{0}$ in $\mathbb{L}_{2}.$ By the triangle inequality, followed by Doob’s inequality, for any positive integer $m$, we have $$\begin{aligned}
\frac{{1}}{\sqrt{n}}\Bigl\Vert \max_{1\leq k\leq n}|R_{k}^{m}| \Bigr\Vert _{2}
&\leq&
\frac{{1}}{\sqrt{n}}\Bigl\Vert \max_{1\leq k\leq n}|R_{k}| \Bigr\Vert _{2}+\frac
{{1}}{\sqrt{n}}\Bigl\Vert \max_{1\leq k\leq n}|M_{k}^{m}-M_{k}| \Bigr\Vert_{2} \\
&\leq&\frac{{1}}{\sqrt{n}}\Bigl\Vert \max_{1\leq k\leq n}|R_{k}| \Bigr\Vert _{2}+\Vert D_{0}^{m}-D_{0}\Vert .\end{aligned}$$ Now, letting $n\rightarrow\infty$ followed by $m\rightarrow\infty,$ we obtain $$\lim_{m\rightarrow\infty}\lim\sup_{n\rightarrow\infty}\frac
{{1}}{\sqrt{n}}\Bigl\Vert \max_{1\leq k\leq n}|R_{k}^{m}| \Bigr\Vert _{2}=0. \label{rest1}$$ Now, observe that by [(\[rest\])]{}, $R_{n}^{m}-\overline{R}_{n}^{
m}=\theta
_{0}^{m}-\theta_{n}^{m}$. Then, for every fixed $m,$ by (\[maxnegl\]), we have $$\frac{{1}}{\sqrt{n}}\Bigl\Vert \max_{1\leq k\leq n}|\theta_{0}^{m}-\theta
_{k}^{m}| \Bigr\Vert _{2}\mathop{\rightarrow}_{n\rightarrow\infty}0.$$ Thus, we conclude from [(\[rest1\])]{} that $$\lim_{m\rightarrow\infty}{\Vert }Y_{0}^{ m}{\Vert }_{M^{+}}=0$$ and the necessity follows.
Applications
============
Applications using projective criteria
--------------------------------------
The first application involves the class of variables satisfying the Maxwell–Woodroofe condition [@mw].
\[underMW\] Assume that $$\Delta(X_{0})=\sum_{k=1}^{\infty}\frac{\Vert \mathbb
{E}_{0}(S_{k})\Vert _{2}}{k^{3/2}}<\infty. \label{MW}$$ The martingale approximation [(\[maxcond\])]{} then holds.
In order to verify condition (\[MAX\]) of Theorem \[T\], we apply inequality (\[PU\]) to the stationary sequence $(Y_{k}^{m})_{k\in
\mathbb{Z}}$ defined by (\[defY\]). Then $$\Biggl\Vert \max_{1\leq j\leq n}\Biggl|\sum_{k=0}^{j-1}Y_{k}^{m}\Biggr|
\Biggr\Vert_{2}\leq n^{1/2}\bigl(2\Vert Y_{0}^{m}\Vert _{2}+80\Delta(Y_{0}^{m})\bigr).$$ First, note that by Peligrad and Utev [@PU1], Proposition 2.5, we know that condition (\[MW\]) implies that $\Vert Y_{0}^{m}\Vert _{2}\rightarrow
0.$ We complete the proof by showing that $$\Delta(Y_{0}^{m})\mathop{\longrightarrow}_{m\rightarrow\infty}0.$$ Since $\Vert Y_{0}^{m}\Vert _{2}\rightarrow0,$ by the triangle inequality and stationarity, every term of the series on the right-hand side of the equality $$\Delta(Y_{0}^{m})=\sum_{k=1}^{\infty}\frac{1}{k^{3/2}}\Vert \mathbb{E}_{0}(Y_{0}^{m}+\cdots+Y_{k-1}^{m})\Vert _{2}$$ tends to $0$ as $m\rightarrow\infty.$ Furthermore, because $$\begin{aligned}
\Vert \mathbb{E}_{0}(Y_{0}^{m}+\cdots +Y_{k-1}^{m})\Vert _{2}&=&\Biggl\Vert\mathbb{E}_{0}\Biggl(\frac{1}{m}\sum_{l=1}^{m}\sum_{i=0}^{k-1}\mathbb
{E}_{i}(X_{i+l})\Biggr)\Biggr\Vert _{2} \\
&\leq&\Vert \mathbb{E}_{0}(X_{0}+\cdots+X_{k-1})\Vert _{2},\end{aligned}$$ each term in $\Delta(Y_{0}^{m})$ is dominated by the corresponding term in $\Delta(X_{0})$, the latter being independent of $m$. The result follows from the above considerations, along with the Lebesgue dominated convergence theorem for the counting measure.
For the sake of applications, we give the following corollary.
Assume that$$\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}\Vert\mathbb{E}_{0}(X_{n})\Vert
_{2}<\infty. \label{mixg1}$$ The martingale representation [(\[maxcond\])]{} then holds.
The fact that (\[mixg1\]) implies (\[MW\]) was observed in Maxwell and Woodroofe [@mw].
We shall now combine Theorem \[T\] with Rio’s maximal inequality (\[Rio\]) to obtain the following proposition.
\[underRio\]Assume that for any $j\geq0$,$$\Gamma_{j}=\sum_{k\geq j}\Vert X_{j}\mathbb{E}_{0}(X_{k})\Vert _{1}<\infty\quad \mbox{and}\quad \frac{1}{m}\sum_{j=0}^{m-1}\Gamma_{j}\rightarrow0
\qquad\mbox{as } m \rightarrow\infty. \label{condRio}$$ The martingale representation [(\[maxcond\])]{} then holds.
In order to verify condition (\[MAX\]), we now apply the maximal inequality (\[Rio\]) to $(Y_{k}^{m})_{k\geq1}$ defined by (\[defY\]). We conclude that for $n\geq m$, $$\begin{aligned}
\Biggl\Vert \max_{1\leq j\leq n}\Biggl|\sum_{k=0}^{j-1}Y_{k}^{m}\Biggr|\Biggr \Vert
_{2}^{2}&\leq&
8n\Vert Y_{0}^{m}\Vert _{2}^{2}+16\sum_{j=1}^{n-1}\Vert Y_{0}^{m}\mathbb{E}_{0}(Y_{1}^{m}+\cdots+Y_{j}^{m})\Vert _{1} \\
&\leq&8n(12m+1)\Vert Y_{0}^{m}\Vert _{2}^{2}+16\sum
_{j=m+1}^{n-1}\Vert Y_{0}^{m}\mathbb{E}_{0}(Y_{m+1}^{m}+\cdots+Y_{j}^{m})\Vert _{1},\end{aligned}$$ where, in the last sum, we have implemented a decomposition into two terms to deal with overlapping blocks. So, for an absolute constant $C$, $$\frac{1}{n}\Biggl\Vert \max_{1\leq j\leq n}\Biggl|\sum_{k=0}^{j-1}Y_{k}^{m}\Biggr| \Biggr\Vert _{2}^{2}\leq C\Biggl(\frac{\Vert \mathbb{E}_{0}(S_{m})\Vert _{2}^{2}}{m}+\frac
{1}{n}\sum_{l=m+1}^{n-1}\Vert Y_{0}^{m}\mathbb{E}_{0}(Y_{m+1}^{m}+\cdots
+Y_{l}^{m})\Vert _{1}\Biggr).$$ Since, for any $l>m$, $$\begin{aligned}
\Vert Y_{0}^{m}\mathbb{E}_{0}(Y_{m+1}^{m}+\cdots+Y_{l}^{m})\Vert _{1}& \leq&\frac
{1}{m}\sum_{j=1}^{m}\sup_{i>m}\Vert (\mathbb{E}_{0}(X_{j}))\mathbb{E}_{0}(X_{i}+\cdots+X_{i+l})\Vert _{1} \\
&\leq&\frac{1}{m}\sum_{j=1}^{m}\sum_{k\geq m}\Vert \mathbb
{E}_{0}(X_{j})\mathbb{E}_{0}(X_{k})\Vert _{1}\end{aligned}$$ and also $$\Vert \mathbb{E}_{0}(S_{m})\Vert _{2}^{2}\leq
2\sum _{j=0}^{m-1}\sum _{k=j}^{m-1}\Vert \mathbb{E}_{0}(X_{j})\mathbb{E}_{0}(X_{k})\Vert _{1},$$ we then obtain, by the properties of conditional expectations, that for a certain absolute constant $C^{\prime}$, $$\frac{1}{n}\Biggl\Vert \max_{1\leq j\leq n}\Biggl|\sum
_{k=0}^{j-1}Y_{k}^{m}\Biggr| \Biggr\Vert _{2}^{2} \leq \frac{C^{\prime}}{m}\sum
_{j=0}^{m}\sum_{k\geq j}\Vert X_{j}\mathbb{E}_{0}(X_{k})\Vert _{1}$$ and the result follows from condition (\[condRio\]), by first letting $n\rightarrow\infty$, followed by $m\rightarrow\infty$.
The projective criteria in the next proposition were studied in [@hh; @HA; @GG], among others.
Assume $$\mathbb E(X_{0}|\mathcal{F}_{-\infty})=0\quad \mbox{almost surely
and}\quad
\sum_{i=1}^{\infty}\Vert\mathbb{E}_{-i}(X_{0})-\mathbb{E}_{-i-1}(X_{0})\Vert_{2}<\infty. \label{P}$$ The martingale approximation [(\[maxcond\])]{} then holds.
The validity of this proposition easily follows by verifying condition (\[MAX\]) via maximal inequality (\[DM\]) applied to $(Y_{k}^{m})_{k\geq1}$ defined by (\[defY\]). Indeed, by (\[DM\]), the triangle inequality and stationarity, we have $$\begin{aligned}
\frac{1}{\sqrt{n}}\Biggl\Vert \max_{1\leq j\leq n}\Biggl|\sum_{k=0}^{j-1}Y_{k}^{m}\Biggr|\Biggr\Vert_{2}&\leq&2\sum_{i=0}^{\infty}\Vert\mathbb{E}_{-i}(Y_{0}^{m})-\mathbb
{E}_{-i-1}(Y_{0}^{m})\Vert_{2} \\
&\leq&\frac{2}{m}\sum_{i=0}^{\infty}\sum_{k=1}^{m}\Vert\mathbb
{E}_{-i}(X_{k})-\mathbb{E}_{-i-1}(X_{k})\Vert_{2}.\end{aligned}$$ Now, by stationarity, change of order of summation and change of variable, $$\frac{1}{\sqrt{n}}\Biggl\Vert\max_{1\leq j\leq n}\Biggl|\sum_{k=0}^{j-1}Y_{k}^{m}\Biggr|
\Biggr\Vert_{2}\leq\frac{2}{m}\sum_{k=1}^{m}\sum_{j=k}^{\infty}\Vert\mathbb{E}_{-j}(X_{0})-\mathbb{E}_{-j-1}(X_{0})\Vert_{2}.$$ To verify condition (\[MAX\]), we let $n\rightarrow\infty$ followed by $m\rightarrow\infty.$ Note that the term on the right-hand side of the previous inequality tends to $0$ as $m\rightarrow\infty$, by (\[P\]).
Application to mixing sequences
-------------------------------
The results in the previous section can be immediately applied to mixing sequences, leading to the sharpest possible results and providing additional information about the structures of these processes. Examples include various classes of Markov chains and Gaussian processes.
We shall introduce the following mixing coefficients: for any two $\sigma$-algebras $\mathcal{A}$ and $\mathcal{B}$, define the strong mixing coefficient $\alpha(\mathcal{A},\mathcal{B)}$, $$\alpha(\mathcal{A},\mathcal{B)=}\sup\{|\mathbb{P}(A\cap B)-\mathbb{P}(A)
\mathbb{P}(B)|;A\in\mathcal{A},B\in\mathcal{B\}},$$ and the $\rho$-mixing coefficient, known also as the maximal coefficient of correlation $\rho(\mathcal{A},\mathcal{B})$, $$\rho(\mathcal{A},\mathcal{B})=\sup\{\operatorname{Cov}(X,Y)/\Vert X\Vert
_{2}\Vert
Y\Vert_{2}\dvtx X\in\mathbb{L}_{2}(\mathcal{A}),Y\in\mathbb{L}_{2}(
\mathcal{B})\}.$$
For the stationary sequence of random variables $(X_{k})_{k\in
\mathbb{Z}},$ we also define $\mathcal{F}_{m}^{n}$, the $\sigma$-field generated by $X_{i}$ with indices $m\leq i\leq n$. $\mathcal{F}^{n}$ denotes the $
\sigma$-field generated by $X_{i}$ with indices $i\geq n$ and $\mathcal
{F}_{m}$ denotes the $\sigma$-field generated by $X_{i}$ with indices $i\leq m.
$ The sequences of coefficients $\alpha(n)$ and $\rho(n)$ are then defined by $$\alpha(n)=\alpha(\mathcal{F}_{0},\mathcal{F}_{n}^{n})\quad \mbox{and}\quad \rho(n)=\rho(\mathcal{F}_{0},\mathcal{F}^{n}).$$ Equivalently (see [@rick], Chapter 4), $$\rho(n)=\sup\{\Vert\mathbb{E}(Y|\mathcal{F}_{0})\Vert_{2}/\Vert Y\Vert
_{2}\dvtx Y\in\mathbb{L}_{2}(\mathcal{F}^{n}),\mbox{ }\mathbb{E}(Y)=0\}.$$ Finally, we say that the stationary sequence is strongly mixing if $\alpha(n)\rightarrow0$ as $n\rightarrow\infty$ and $\rho$-mixing if $\rho(n)\rightarrow0$ as $n\rightarrow\infty$.
An interesting application of Proposition \[underMW\] is to $\rho$-mixing sequences. It is well known that the central limit theorem and its invariance principle hold for stationary centered sequences with finite second moments under the assumption $$\sum_{k=1}^{\infty}\rho(2^{k})<\infty, \label{condrho}$$ where $\rho(n)=\rho(\mathcal{F}_{0},\mathcal{F}^{n}\mathcal{)}$. Let us recall that the central limit theorem is due to [@II], while the invariance principle is found in [@pm; @shao; @u89; @u91]. The fact that condition (\[condrho\]) is sharp in this context is due to [@rick], Volume 1, page 367, and Volume 3, Theorem 34.13. Bradley’s example shows that if (\[condrho\]) fails, then $S_{n}/ \Vert S_{n}\Vert _2$ might have non-degenerate non-normal distributions as weak limit points.
As a corollary of Proposition \[underMW\], we obtain the conditional invariance principle for $\rho$-mixing sequences.
Assume $\sum_{k=1}^{\infty}\rho(2^{k})<\infty.$ The martingale representation [(\[maxcond\])]{} then holds.
As in [@mpu2], for a positive constant $C$, we have $$\sum_{r=0}^{\infty}\frac{\Vert
\mathbb{E}(S_{2^{r}}|\mathcal{F}_{0})\Vert_{2}}{2^{r/2}}\leq C
\Vert X_0\Vert _2\sum_{j=0}^{\infty}\rho(2^{j}).$$
To obtain sharp results for strongly mixing sequences, we shall use Proposition \[underRio\].
According to Doukhan, Massart and Rio [@DMR], a condition that is optimal for CLT or the invariance principle for strongly mixing sequences is $$\label{condalha}
\sum _{k\geq1}{\mathbb{E}}X_{0}^{2}I\bigl(|X_{0}|\geq
Q_{|X_{0}|}(2\alpha_{k})\bigr)<\infty,$$ where $Q_{|X_{0}|}$ denotes the cadlag inverse of the function $t\rightarrow P(|X_{0}|>t).$ Also under this condition, we add the additional information given by Theorem \[T\].
Assume that condition [(\[condalha\])]{} is satisfied. The martingale representation [(\[maxcond\])]{} then holds.
We shall just verify the condition of Proposition \[underRio\]. Note that on the set $[0,{\ P}(|Y|>0)]$, the function $H_{Y}\dvtx x\rightarrow
\int_{0}^{x}Q_{Y}(u)\,\mathrm{d}u$ is an absolutely continuous and increasing function with values in $[0,{\ E}|Y|]$. Denote by $G_{Y}$ the inverse of $H_{Y}$. With this notation, by Merlevède and Peligrad [@mp], relation (4.84), we have $$\Vert X_{j}\mathbb{E}(X_{k}|\mathcal{F}_{0})\Vert _{1}\leq3\int_{0}^{\Vert
{\mathbb{E}}(X_{k}|\mathcal{F}_{0})\Vert_{1}}Q_{|X_{0}|}\circ G(u)\,\mathrm{d}u$$ and we then majorize the right-hand side in the previous inequality by Dedecker and Doukhan [@dd], Proposition 1, to obtain $$\Vert X_{j}\mathbb{E}(X_{k}|\mathcal{F}_{0})\Vert _{1}\leq6\int_{0}^{2\alpha
(k)}Q_{|X_{0}|}^{2}\,\mathrm{d}u .$$ Therefore,$$\begin{aligned}
\sum_{k\geq j}\Vert X_{j}\mathbb{E}_{0}(X_{k})\Vert _{1}&\leq&6\sum_{k\geq
j}\int_{0}^{2\alpha(k)}Q_{|X_{0}|}^{2}\,\mathrm{d}u \\
&\leq&6\sum_{k\geq j}{\mathbb{E}}X_{0}^{2}I\bigl(|X_{0}|\geq Q_{|X_{0}|}(2\alpha
_{k})\bigr) \rightarrow0\qquad \mbox{as }j\rightarrow\infty
.\end{aligned}$$
Note that the coefficient $\alpha(k)$ is defined by using only one variable in the future. Moreover, by the Cauchy–Schwarz inequality, condition (\[condalha\]) is satisfied if the variables have finite moments of order $2+\delta$ for a $\delta>0$ and $$\sum_{k\geq1}\alpha(k)^{\delta/(2+\delta)}<\infty.$$ An excellent source of information for classes of mixing sequences and classes of Markov chains satisfying mixing conditions is the book by Bradley [@rick]. Further applications can be obtained by using the coupling coefficients in [@dp].
Application to additive functionals of reversible Markov chains
---------------------------------------------------------------
For reversible Markov processes (i.e., $Q=Q^{\ast}$), the invariance principle under an optimal condition is known since Kipnis and Varadhan [@KV]. The following is a formulation in terms of martingale approximation.
Let $(\xi_{i})_{i\in\mathbb{Z}}$ be a stationary reversible Markov chain and $f\in\mathbb{L}_{0}^{2}(\pi)$ with the property $$\lim_{n\rightarrow\infty}\frac{\operatorname{var}(S_{n})}{n}\rightarrow\sigma
_{f}^{2}<\infty.
\label{revcondCLT}$$ The martingale approximation satisfying [(\[maxcond\])]{} then holds.
We have to verify condition [(\[MAX\])]{}. Denote by $\rho_{f}$ the spectral measure of $f$ corresponding to the self-adjoint operator $Q$ on $\mathbb{L}_{2}(\pi)$. It is well known that the assumption [(\[revcondCLT\])]{} for $f \in\mathbb L^2_0$ implies that $ \int _{-1}^{1}(1-t)^{-1}\rho_{f}(\mathrm{d}t)<\infty$ (see [@KV]). Define $Y_{0}^{m}$ by [(\[defY\])]{}. By the maximal inequality [(\[LW\])]{}, we have$$\frac{1}{n}\mathbb{E}\Biggl(\max_{1\leq j\leq n}\Biggl|\sum
_{k=0}^{j-1}Y_{k}^{m}\Biggr|\Biggr)^{2}\leq27\sum_{k\geq0}\mathbb{E(}Y_{0}^{m}Y_{k}^{m}),$$ provided that the sum on the right-hand side is finite. To prove it, by using spectral calculus for the self-adjoint operator $Q$, we obtain $$\sum_{k\geq0}\mathbb{E(}Y_{0}^{m}Y_{k}^{m})\leq\frac{1}{m^{2}}\int _{-1}^{1}\frac{(1+t+\cdots+t^{m-1})^{2}}{(1-t)}\rho_{f}(\mathrm{d}t)$$ and therefore, for every positive integer $m>0,$$$\Vert Y_{0}^{m}\Vert _{M^{+}}^{2}\leq27\int _{-1}^{1}\frac{(1+t+\cdots+t^{m-1})^{2}}{m^{2}(1-t)}\rho_{f}(\mathrm{d}t).$$ Since $\int _{-1}^{1}(1-t)^{-1}\rho_{f}(\mathrm{d}t)<\infty$, the right-hand side is finite and, by the dominated convergence theorem,$$\lim_{m\rightarrow\infty}\Vert Y_{0}^{m}\Vert _{M^{+}}^{2}=0.$$
Similar results are expected to hold for other classes of stationary and ergodic Markov chains when $Q~$is not necessarily self-adjoint, but instead satisfies a quasi-symmetry or strong sector condition, or is symmetrized. See [@wu] and [@svy] for these related processes.
Application to additive functionals of normal Markov chains
-----------------------------------------------------------
For additive functionals of normal Markov chains $(QQ^{\ast}=Q^{\ast
}Q),$ the central limit theorem below is a result of Gordin and Lifshitz [@gl]. As an application of Theorem \[T2\], we give an alternative proof.
Let $\rho_{f}$ be the spectral measure on the closed unit disk $D
\subset\mathbb C$ corresponding to the function $f\in\mathbb{L}_{0}^{2}(\pi)$.
\[normal\] Let $(\xi_{i})_{i\in\mathbb{Z}}$ be a stationary normal Markov chain and a function $f\in\mathbb{L}_{0}^{2}(\pi),$ satisfying the condition $$\int _{D}\frac{1}{|1-z|}\rho_{f}(\mathrm{d}z)<\infty.
\label{normcond}$$ The martingale approximation [(\[MA\])]{} then holds.
According to Theorem \[T2\], we have to verify condition (\[MA1\]). By using spectral calculus as in [@1994], Chapter 4, after some computations, we get$$\lim\sup_{n\rightarrow\infty}\frac{1}{n}\Biggl\Vert \sum_{k=0}^{n-1}Y_{k}^{m}\Biggr\Vert _{2}^{2}\leq4\int _{D}\frac{|1+z+\cdots+z^{m-1}|^{2}}{m^{2}|1-z|}\rho_{f}(\mathrm{d}z)$$ and condition (\[MA1\]) is therefore satisfied by condition (\[normcond\]) and the dominated convergence theorem.
Condition [(\[normcond\])]{} has an interesting equivalent formulation in terms of conditional moments that is in the spirit of (and which implies) the Mawxell–Woodroofe condition [(\[MW\])]{}.
Condition [(\[normcond\])]{} is equivalent to $$\sum_{k=1}^{\infty}\frac{\Vert \mathbb
{E}_{0}(S_{k})\Vert _{2}^{2}}{k^{2}}<\infty. \label{Gap}$$
Condition [(\[Gap\])]{} is further implied by $$\sum_{k=1}^{\infty}\Vert \mathbb{E}_{0}(X_{k})\Vert _{2}^{2}<\infty.
\label{cor2}$$
The equivalence in the above remark can be found in [@Cuny], Lemma 2.1. The fact that [(\[cor2\])]{} implies [(\[Gap\])]{} is easily established, much like the proof that [(\[mixg1\])]{} implies [(\[MW\])]{}.
Acknowledgements {#acknowledgements .unnumbered}
================
Mikhail Gordin was supported in part by a Charles Phelps Taft Memorial Fund grant and RFBR Grant 10-01-00242\_a. Magda Peligrad was supported in part by a Charles Phelps Taft Memorial Fund grant and NSA Grant H98230-09-1-0005. The authors are grateful to the referees for carefully reading the paper and for numerous suggestions that improved the presentation of the paper.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A logarithmic oscillator (in short, log-oscillator) behaves like an ideal thermostat because of its infinite heat capacity: when it weakly couples to another system, time averages of the system observables agree with ensemble averages from a Gibbs distribution with a temperature $T$ that is given by the strength of the logarithmic potential. The resulting equations of motion are Hamiltonian and may be implemented not only in a computer but also with real-world experiments, e.g., with cold atoms.'
author:
- 'Michele Campisi, Fei Zhan, Peter Talkner, and Peter Hänggi'
title: 'Logarithmic oscillators: ideal Hamiltonian thermostats'
---
Thermostats play an important role in computational physics [@Klages]. They provide effective and useful methods to simulate the action of a thermal environment on systems of physical and chemical interest. Mathematically speaking, their salient feature is to produce “thermostated dynamics” of the system of interest: that is, they are meant to impose long-time averages of system observables that coincide with Gibbs-ensemble averages at a given temperature $T$. Widely used thermostats are: the Langevin thermostat [@Ermak80JCmptP169], Andersen’s stochastic collision thermostat [@Andersen80JCP72] and the Nosé-Hoover deterministic thermostat [@Nose84JCP81; @Hoover85PRA31; @Martyna92HCP97].
Here we present a thermostat differing in various respects from the previously reported ones. Our main result is that a *logarithmic oscillator* (or a “log-oscillator” as we shall call it below), weakly coupled to the system of interest (in short “the system” in what follows) leads to thermostated system dynamics. In its simplest 1D version the system+log-oscillator Hamiltonian reads: $$H= \sum_i \frac{p_i^2}{2m_i} + V(\mathbf q) +
\frac{P^2}{2M}+T\ln\frac{|X|}{b}+ h(\mathbf q,X)
\label{eq:H}$$ where $\mathbf{p}=(p_1, \dots, p_N),\mathbf{q}=(q_1, \dots, p_N),m_i$ are the momenta, positions, and masses of the particles composing the system; $X,P,M$ are the log-oscillator position, momentum, and mass, respectively; $b>0$ sets the length scale of the log-oscillator and $T$ is the thermostat temperature; $V(\mathbf{q})$ is the system inter-particle potential and $h(\mathbf q,X)$ denotes a [*weak*]{} interaction energy that couples the log-oscillator to the system. When the total Hamiltonian $H$ is ergodic, the system+log-oscillator trajectory samples the microcanonical ensemble, and the system trajectory samples the canonical ensemble at temperature $T$. This continues to hold if the 1D log-oscillator is replaced by higher dimensional log-oscillators, for example for a charged particle in the attractive logarithmic 2D Coulomb field generated by a long charged wire.
Compared to the previously reported thermostats the present thermostat exhibits an evident advantage. The Hamiltonian (\[eq:H\]) or its higher dimensional versions can be readily implemented in a physical experiment. In Fig. \[fig:sketch\] we show a possible implementation. The system is composed of a gas of neutral atoms confined into a box. The thermostat is an ion subject to the attractive 2D coulomb potential generated by a thin oppositely charged wire, $|Q \lambda|/2 \pi \varepsilon_0 \ln \rho$. Here $Q>0$ is the charge of the ion, $\lambda<0$ the linear charge density of the wire, $\rho$ the distance between wire and particle, and $\varepsilon_0$ the electric permittivity of vacuum. Through short-range repulsive interactions the ion thermalizes the neutral gas to the temperature $T=|Q \lambda|/\pi \varepsilon_0$. Another possibility for the realization of a log-oscillator is by means of a laser beam with an intensity profile of logarithmic form coupled non-resonantly to an atom [@Schleich10PRA82]. This could be realized to thermostat cold atomic gases [@Bloch08RMP80].
Atomic systems in isolation from the environment naturally sample the microcanonical ensemble. For small systems this sampling may considerably differ from the canonical one and can result in distinctive thermodynamic features such as negative specific heats. These were experimentally investigated with small atomic clusters [@Schmidt01PRL86; @Gobet02PRL89]. Typically it is difficult to have a small isolated system sample the canonical Gibbs distribution. Our method opens this possibility. More generally, by using a single log-oscillator as an environment simulator, our method allows to experimentally study thermostated small systems in isolation from the real environment. One advantage of this paradigm is that our method would allow to control a thermal parameter, the temperature $T$, by means of mechanical parameters, e.g., with reference to Fig. \[fig:sketch\], the charge density $\lambda$ on the wire.
Just like the Nosé-Hoover thermostat, our thermostat is deterministic and time-reversible, but at variance with Nosé-Hoover dynamics which are not Hamiltonian [@Klages; @Kusnezov], our thermostated dynamics are manifestly Hamiltonian. There exist “generalized Hamiltonian formalisms” [@Klages] for the Nosé-Hoover dynamics in the literature. The most prominent examples use Nosé’s Hamiltonian [@Nose84JCP81]: $H_\text{Nos\'e}=\sum {{p}_i^2}/{2m_i {X}^2} + V({\mathbf{q}}) +
{{P}^2}/{2M}+f T\ln {{X}}$ or Dettmann’s Hamiltonian [@Dettmann; @Hoover12arXiv]: $H_D= X H_\text{Nos\'e} $. At variance with our Hamiltonian in Eq. (\[eq:H\]), these involve the non standard kinetic terms, ${{p}_i^2}/{2m_i {X}^2}$ and ${{p}_i^2}/{2m_i {X}}$, respectively, which, due to the dependence on the log-oscillator position, cannot readily be realized in an experiment. Further, while Dettmann’s Hamiltonian produces thermostated trajectories only for a specific value of the energy (i.e., $H_D=0$) our method thermalizes the system irrespective of the energy value. We elucidated these issues further in Ref. [@Campisi12arXiv]. The usefulness and importance of the Nosé-Hoover equations as a computational thermostat are beyond question [@HooverBook].
*Theory.–* Before we shall provide the formal argument we present a physical explanation indicating why it is plausible that the Hamiltonian in Eq. (\[eq:H\]) leads to thermostated system dynamics. Consider the isolated 1D log-oscillator: $$H_{\text{log}} = \frac{P^2}{2M}+T\ln\frac{|X|}{b}\, .
\label{eq:Hlog}$$ Applying the virial theorem, $
\left\langle
P \partial H_\text{log}/\partial P \right\rangle=
\left\langle
X \partial H_\text{log}/\partial X \right\rangle
$, to the 1D log-oscillator, we obtain $
\left\langle {P^2}/{M} \right\rangle = T
$ where $\langle \cdot \rangle$ denotes the time average. That means that all trajectories of a log-oscillator have the same average kinetic energy [@Schleich10PRA82], i.e., the same kinetic temperature $\langle P^2/M \rangle=T$, regardless of their energy $E$. This implies $\partial T/\partial E=0$. Recalling the definition of heat capacity, $C=
\partial E/\partial T$, one finds that the log-oscillator exhibits a spectacular property: its heat capacity is infinite, which is the defining feature of an ideal thermostat. Since the log-oscillator may only exist in the state of temperature $T$, we expect that a system will reach this same temperature $T$ when it is weakly coupled to the log-oscillator.
To formally prove that the log-oscillator induces thermostated dynamics of the system at the temperature $T$, we recall the general expression for the probability density function $p(\mathbf{q},\mathbf{p})$ to find a system at the point $(\mathbf{q},\mathbf{p})$ of its phase space when it is weakly coupled to a second system \[the log-oscillator in the present case\], provided that the compound system probability distribution is microcanonical. It reads [@Khinchin49Book] $$\begin{aligned}
p(\mathbf{q},\mathbf{p})=\frac{\Omega_ {\text{log}}[E_{\text{tot}}-H_S(\mathbf{q},
\mathbf{p})]}{
\Omega(E_{\text{tot}})}\, ,
\label{eq:khinchin-formula}\end{aligned}$$ where $E_{\text{tot}}$ is the total (conserved) energy of the compound system. With $E$ denoting the log-oscillator energy, $$\Omega_{\text{log}}(E) = \int \mathrm{d}{X}
\mathrm{d}{P}\delta[E-H_{\text{log}}(X, P)]
\label{eq:Omega-B}$$ is the density of states of the log-oscillator, and $$\Omega(E_{\text{tot}}) = \int \mathrm{d}{X}
\mathrm{d}{P}\mathrm{d}\mathbf{q}\mathrm{d}\mathbf{p}\,
\delta[E_{\text{tot}}-H(\mathbf{q},\mathbf{p},X,P)]$$ is the density of states of the compound system. Here $\delta(...)$ denotes Dirac’s delta function and $H_S$ is the system Hamiltonian.
According to Eq. (\[eq:khinchin-formula\]) the density of states of the log-oscillator defines the shape of the distribution of the system. Performing the integration in Eq. (\[eq:Omega-B\]) with the log- oscillator Hamiltonian, Eq. (\[eq:Hlog\]), one obtains for the density of states of the log-oscillator the expression $$\Omega_{\text{log}}(E)= 2 b\sqrt{2 \pi M/T}\, e^{E/T}\, .
\label{eq:Omega-log}$$ Inserting Eq. (\[eq:Omega-log\]) into Eq. (\[eq:khinchin-formula\]) yields the Gibbs distribution for the system, $$\begin{aligned}
p(\mathbf{q},\mathbf{p})={e^{-H_S(\mathbf{q},\mathbf{p})/T}
}/{Z(T)} \, ,
\label{eq:canonical}\end{aligned}$$ regardless of the energy $E_{\text{tot}}$ assigned to the compound system. Here $Z(T)=\int \mathrm{d}\mathbf{q}\mathrm{d}\mathbf{p}e^{-
H_S(\mathbf{q},\mathbf{p})/T}$ is the system canonical partition function.
Also a $f$-dimensional log-oscillator $
H(\mathbf{X},\mathbf{P})=\mathbf{P}^2/(2M) +f T/2 \ln
(\mathbf{X}^2/b^2)$ \[where $\mathbf{X}$ and $\mathbf{P}$ are vectors of size $f$\] results in the exponential density of states $\Omega_{\text{log}} \propto e^{E/T}$. Therefore, $f$-dimensional log-oscillators induce thermostated dynamics as well.
So far we have left the system-thermostat interaction $h(\mathbf{q},X)$ unspecified. As in standard statistical mechanics where a heat bath with many degrees of freedom replaces the single log-oscillator [@Khinchin49Book], $h(\mathbf{q},X)$ must comply with two requirements: (i) it must be sufficiently weak that it can completely be neglected in the calculation of the probability density $p(\mathbf{q},\mathbf{p})$. This assumption guarantees the applicability of Eq. (\[eq:khinchin-formula\]) provided that the total system stays in microcanonical equilibrium. In order that this equilibrium state actually is reached from arbitrary initial conditions it is necessary (ii) that the total dynamics is ergodic. To meet these two requirements, short-range repulsive interactions typically suffice, see the numerical examples below. Note that with a short-range repulsive interaction, the fraction of time during which the log-oscillator interacts with any other particle is much smaller than one. This assures that the average interaction energy represents only a small part of the total energy, and hence the weak coupling assumption implied by Eq. (\[eq:khinchin-formula\]) is met.
*Numerics.–* In order to corroborate our statement we performed 1D and 3D molecular dynamics simulations using symplectic integrators [@Hairer06Book].
In our first numerical experiment we used two point particles of mass $m$ in a 1D box of length $L$ and placed a log-oscillator of mass $M$ and strength $T$ between them, see the inset in Fig. \[fig:plot\]. The three particles interact with each other and with the fixed walls via the truncated Lennard-Jones potential, reading
$$V_{LJ}(q)=
\left\{
\begin{array}{ll}
0\, , & |q| > 2^{1/6}\sigma \\
4\varepsilon \left[\left(\frac{\sigma}{q}\right)^{12}-\left(\frac{\sigma}{q}
\right)^ { 6 } \right ] +\varepsilon\, , & |q| < 2^{1/6}\sigma
\end{array}
\right. \, ,
\label{eq:V(x)}$$
that is $h(q_1,q_2,X)=\sum_i V_{LJ}(|q_i-X|)$, and $V(q_1,q_2)=V_{LJ}(|q_1-q_2|)+\sum_i [V_{LJ}(|q_i+L/2|)+V_{LJ}(|q_i-L/2|)]$ where $L$ is the box length. In the simulations we adopted $m,\sigma$ and $\varepsilon$, as the units of mass, length, and energy, respectively. In order to avoid the singularity of the logarithmic potential at the origin we replaced it with the following potential: $$\varphi_{b}(X)= \frac{T}{2}\ln\frac{X^2+b^2}{b^2} \, .
\label{eq:phi(x)}$$ For all simulations we used the value $b=\sigma$. This truncation results in a correction of the density of states (\[eq:Omega-log\]), which vanishes as the energy $E_{\text{tot}}$ increases. Fig. \[fig:plot\] depicts the probability density function, $\varrho(E_S)$ of finding the system consisting of the two orange particles depicted in the inset at the kinetic energy $E_S$ in a molecular dynamics simulation at total energy $E_{\text{tot}}$. According to Eq. (\[eq:canonical\]) this should be of the form $\varrho(E_S)\propto e^{-E_S/T}\Omega_S(E_S)\propto e^{-E_S/T}$, where $\Omega_S(E_S)$ is the system density of states. Note that $\Omega_S(E_S)$ is constant in the case of a system Hamiltonian composed of two quadratic degrees of freedom. The numerically computed curve excellently fits the desired canonical distribution with the expected temperature $T$. The simulation energy $E_{\text{tot}}$ was chosen large enough, so that the error introduced by the replacement of the purely logarithmic potential with the truncated one, was negligible. The box length was taken such that it exceeded the maximal excursion of the log-oscillator $x_\text{max}=2 \sigma \sqrt{e^{2E_\text{tot}/T}-1}$. Otherwise the log-potential would be effectively cut-off by the box-potential and consequently the exponential shape of the density of states would be destroyed.
Our second numerical experiment considers as thermal bath a charged particle in the electric field generated by a long and oppositely charged wire: the so-called 2D Coulomb potential which is of logarithmic form, Fig. \[fig:sketch\]. The charged particle Hamiltonian reads: $$H(P_x,P_y,P_z,X,Y,Z)=\frac{P_x^2+P_y^2+P_z^2}{2M} +T \ln
\frac{X^2+Y^2}{b^2}
\label{eq:H-coulomb2D}$$ where $T=|Q \lambda|/\pi \varepsilon_0 $. Assuming that the motion is confined in the $Z$ direction by two rigid walls parallel to the $XY$ plane and separated by a distance $L_z$, one obtains for the density of states the expression $
\Omega_{\text{log}}(E)= \pi^{5/2} (2 M)^{3/2} L_z b^2 T^{1/2} e^{E/T}
$. Thus we expect the system to behave as a thermostat. In our simulation we let this thermostat weakly interact with a neutral gas of 3 particles confined in a box, and recorded the probability $p(v)$ to find the absolute value of any of the $3\times 3$ velocity components of the neutral gas at value $v$ during the simulation. As with the 1D simulation, the 3+1 particles were interacting with each other and with the fixed box walls via the truncated Lennard-Jones potential, Eq. (\[eq:V(x)\]). The logarithmic potential is truncated in the same way as in the 1D case, Eq. (\[eq:phi(x)\]), that is we used the potential $$\varphi(X,Y)=T\ln[(X^2+Y^2+b^2)/b^2]\, .
\label{eq:H-coulomb2D-trunc}$$ The results are displayed in Fig. \[fig:plot2\]. The truncation of the logarithmic potential entails a deviation of the density of states from the exponential form: $\Omega_{\text{log}}(E)\propto e^{E/T}[\sqrt{\pi} - 2\Gamma(3/2,E/T)]$ where $\Gamma(a,x)$ is the upper incomplete gamma function. Note that with $E/T \gg 1$ this deviation vanishes exponentially as $\Omega_\text{log}(E) \propto e^{E/T}(\sqrt{\pi}-2\sqrt{E/T}e^{-E/T}$), where we have used the asymptotic expansion of the upper incomplete Gamma function [@AbramowitzBook]. This leads to a deviation of the distribution $p(v)$ from the Maxwellian form. For a fixed simulation energy $E_{\text{tot}}$, this deviation in $p(v)$ becomes more pronounced as the number of degrees of freedom composing the system increases, cf. the inset in Fig. \[fig:plot2\]. This can be compensated by increasing the simulation energy $E_{\text{tot}}$. We estimate that this scales as $E_\text{tot} \gtrsim c 3NT/2 \simeq c \langle H_S \rangle$, with some constant $c$ depending on the required degree of approximation.
*Remarks.–* Not only can logarithmic potentials be generated artificially, e.g., with properly engineered laser fields [@Schleich10PRA82], electrophoretic traps [@Cohen05PRL94] or with charged wires, but they also occur naturally in various situations: For example logarithmic potentials govern the motion of stars in elliptic galaxies [@Stoica], determine the interaction of vortices in flow fields [@Onsager], and of probe particles in driven fluids [@Levine05EPL70]. Log-oscillators recently received much attention in regard to their anomalous diffusion properties [@Dechant11JSP145; @Dechant11PRL107; @Hirschberg11PRE84; @Hirschberg12JSM]. The present work is complementary to these studies [@Dechant11JSP145; @Dechant11PRL107; @Hirschberg11PRE84; @Hirschberg12JSM] in the sense that our focus is on the dynamics of the particles surrounding the log-oscillator, whereas their focus is on the dynamics of the log-oscillator itself.
One of the earliest thermostats was proposed by Andersen [@Andersen80JCP72]. In the method of Andersen the system evolves according to Hamiltonian equations of motion until, at some random time $\tau$, the velocity of a randomly chosen particle in the system is instantaneously assigned a new value drawn from a Maxwell distribution with the desired temperature. The system then continues it Hamiltonian motion until the next random event occurs, and so on. Our method can be seen as a fully deterministic version of Andersen thermostat, where the times at which the collisions occur and the newly imparted velocities are not drawn randomly, but follow deterministically from the total system dynamics.
In many studies thermal baths are modeled as *infinite* collections of harmonic oscillators or free particles. In the present method this infinite collection is replaced by a single log-oscillator. It has therefore the evident advantage of not involving any thermodynamic limit while retaining the Hamiltonian structure. Roughly speaking, *the thermodynamic limit is lumped in the singularity of the log-potential*. At variance with infinite thermal baths whose temperature is given by the bath’s energy per degree of freedom, log-oscillator thermostats contain the temperature as a parameter in the total Hamiltonian. This opens the possibility, for example, to study the response of a system to a varying temperature, and take advantage from the non-equilibrium statistical mechanical machinery dealing with time dependent Hamiltonians [@Campisi83RMP11].
Another advantage of our method is that, because the Hamiltonian is written in the standard physical system+bath+interaction form: $H=H_S+H_B+h$, it provides a direct way to control the strength of the interaction $h$, allowing also to simulate thermalization to generalized Gibbs states occurring when the system-bath coupling is not weak [@Campisi09PRL102], which can be a relevant case for small systems.
*Conclusions.–* We demonstrated that log-oscillators possess infinite heat capacity, i.e., they are ideal thermal baths. As such they have a thermostating influence on the dynamics of many-particle systems. The resulting deterministic Hamiltonian dynamics are distinct from the Nosé-Hoover dynamics. Unlike previously reported generalized Hamiltonian formulations of Nosé-Hoover dynamics, our Hamiltonian (i) produces thermostated dynamics irrespective of the energy value and (ii) presents the kinetic terms in standard form. Consequently it is amenable to experimental realization. Its most promising practical use is as an analog thermostat simulator for the experimental investigation of the thermodynamics of small systems, e.g., atomic clusters.
*Acknowledgments.–* The authors thank Sergey Denisov for comments and Nianbei Li for technical advice. This work was supported by the cluster of excellence Nanosystems Initiative Munich (P.H.), the Volkswagen Foundation project No. I/83902 (P.H., M.C.), and the DFG priority program SPP 1243 (P.H., F.Z.).
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A stellarator is said to be omnigeneous if all particles have vanishing average radial drifts. In omnigeneous stellarators, particles are perfectly confined in the absence of turbulence and collisions, whereas in non-omnigeneous configurations, particle can drift large radial distances. One of the consequences of omnigeneity is that the unfavorable inverse scaling with collisionality of the stellarator neoclassical fluxes disappears. In the pioneering and influential article \[Cary J R and Shasharina S G 1997 [*Phys. Plasmas*]{} [**4**]{} 3323\], the conditions that the magnetic field of a stellarator must satisfy to be omnigeneous are derived. However, reference \[Cary J R and Shasharina S G 1997 [*Phys. Plasmas*]{} [**4**]{} 3323\] only considered omnigeneous stellarators in which all the minima of the magnetic field strength on a flux surface must have the same value. The same is assumed for the maxima. We show that omnigenenous magnetic fields can have local minima and maxima with different values. Thus, the parameter space in which omnigeneous stellarators are possible is larger than previously expected. The analysis presented in this article is only valid for orbits with vanishing radial width, and in principle it is not applicable to energetic particles. However, one would expect that improving neoclassical confinement would improve energetic particle confinement.'
address:
- '$^1$Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford, OX1 3NP, UK'
- '$^2$Culham Centre for Fusion Energy, Abingdon, OX14 3DB, UK'
- '$^3$Laboratorio Nacional de Fusión, CIEMAT, 28040 Madrid, Spain'
- '$^4$Max-Planck-Institut für Plasmaphysik, 17491 Greifswald, Germany'
- '$^5$University of Maryland, College Park, MD 20742, USA'
author:
- 'Felix I. Parra$^{1,2}$'
- 'Iván Calvo$^{1,3}$'
- Per Helander$^4$
- Matt Landreman$^5$
title: Less constrained omnigeneous stellarators
---
Introduction
============
Charged particles are not necessarily confined in the three dimensional stellarator magnetic fields. Particles can move long distances away from the flux surface in which they started, causing large neoclassical transport. For this reason, stellarators must be optimized using sophisticated codes [@gori99; @subbotin06; @kasilov13] to reduce neoclassical transport to acceptable levels [@wobig93; @nuhrenberg95; @anderson95; @neilson02]. The pioneering work of Cary and Shasharina [@cary97a; @cary97b] gave the conditions that the magnetic field on a flux surface has to satisfy to reduce the average particle radial drift to zero. A flux surface that satisfies the conditions given in [@cary97a; @cary97b] is said to be omnigeneous. It will have neoclassical particle and energy fluxes comparable to those in a tokamak and consequently, negligible compared to turbulent fluxes.
Cary and Shasharina [@cary97a; @cary97b] did not consider all possible classes of omnigenenous stellarators. References [@cary97a; @cary97b] correctly show that the local minima of the magnetic field strength $B$ on a magnetic field line have the same value as the closest local minima in the contiguous magnetic field lines on the same flux surface. The same happens to local maxima. However, in addition to this condition, a large part of the discussion in references [@cary97a; @cary97b] assumes that all these minima must have the same value on a given flux surface. Similarly, it is assumed that all the maxima must have the same value. In other words, it is assumed that there cannot be local minima larger than the global minimum, and there cannot be local maxima smaller than the global maximum (the situations that references [@cary97a; @cary97b] consider and ignore are sketched in figure \[fig:carycondition\]). However, it is not necessary for omnigeneity that all the minima in the flux surface have the same value and that all the maxima are the same, as we proceed to show by constructing an omnigeneous magnetic field that does not satisfy this condition. We first review the arguments given in [@cary97a; @cary97b], and we then use them to construct a solution with local minima and maxima of $B$ that do not have the same value.
Note that this article relies heavily on the work in [@cary97a; @cary97b], and only extends it. It does not detract from the importance of the original work. However, we believe that this correction is needed because the assumption that the value of all the local minima and maxima of the magnetic field in an omnigeneous stellarator only depends on the flux surface is considered to be true by a large part of the community.
Conditions for an omnigeneous magnetic field {#sec:conditions}
============================================
Since to lowest order particles move only along magnetic field lines, it is convenient to use coordinates that clearly distinguish between motion across and along the magnetic field. We use a radial coordinate $\rho$ to label flux surfaces, an angle $\alpha$ to label magnetic field lines on a given flux surface (we give an exact definition of $\alpha$ below in [(\[eq:alphadef\])]{}), and the arc length of the magnetic field line to locate a particle along a magnetic field line once $\rho$ and $\alpha$ are given. To lowest order, the magnitude of the velocity $v$ and the pitch-angle-like variable $\lambda = v_\bot^2/v^2 B$ are constants of the motion, making the particle parallel velocity depend only on the magnitude of the magnetic field, $$v_{||} = \sigma v \sqrt{1 - \lambda B(\rho, \alpha, l)},$$ where $\sigma = \pm 1$ is the sign of the parallel velocity. Particles with $\lambda < B_\mathrm{max}^{-1}$, where $B_\mathrm{max} (\rho)$ is the maximum value of $B$ on the flux surface $\rho$, have a parallel velocity that never vanishes, and sample the entirety of the flux surfaces that the magnetic field lines cover ergodically (the number of flux surfaces in which magnetic field lines close on themselves is negligible). These passing particles always have a vanishing average radial magnetic drift. Particles with $\lambda > B_\mathrm{max}^{-1}$ are trapped between the bounce points $l_{b1} (\rho, \alpha, \lambda)$ and $l_{b2} (\rho, \alpha, \lambda)$ that satisfy $B(\rho, \alpha, l_{b1}) = \lambda^{-1} = B(\rho, \alpha, l_{b2})$. In general, trapped particles do not have vanishing average radial magnetic drift, and they can drift off flux surfaces. Because trapped particle orbits are periodic to lowest order, they must conserve the second adiabatic invariant $$J_{||} (\rho, \alpha, v, \lambda) = \oint v_{||}\, {\mathrm{d}}l = 2v \int_{l_{b1}}^{l_{b2}} \sqrt{1 - \lambda B(\rho, \alpha, l)}\, {\mathrm{d}}l$$ when they drift away from the magnetic field line where they started. These particles move to another flux surface if there is no trapped orbit in a contiguous magnetic field line on the same flux surface that has the same values of $v$, $\lambda$ and $J_{||}$. Thus, to make the radial drift of trapped particles vanish, we need to find magnetic field configurations in which $$\label{eq:dalphaJ}
\partial_\alpha J_{||} = 0.$$ This condition, which is the definition of omnigeneity, must be satisfied for all $\lambda$ in the interval $[B_\mathrm{max}^{-1}, B_\mathrm{min}^{-1}]$, where $B_\mathrm{min} (\rho)$ is the minimum value of $B$ on the flux surface $\rho$. As a result, it imposes several important constraints on the magnetic field that were first deduced in [@cary97a; @cary97b]. We proceed to discuss these constraints one by one.
1. \[prop:Bmaxmin\] Along the magnetic field line defined by $\rho$ and $\alpha$, the magnitude of the magnetic field $B$ has in general several local minima and maxima. We use $l_{m,j} (\rho, \alpha)$ and $B_{m, j} (\rho, \alpha)$ to denote the location and value of the $j$-th local minimum, and $l_{M, k} (\rho, \alpha)$ and $B_{M, k}(\rho, \alpha)$ for the location and the value $k$-th local maximum. We choose the indices $j$ and $k$ such that $B_\mathrm{min} (\rho) \leq B_{m, 1} (\rho, \alpha) \leq B_{m,2} (\rho, \alpha) \leq \ldots \leq B_{m, J} (\rho, \alpha) < B_\mathrm{max} (\rho)$ and $B_\mathrm{min} (\rho) < B_{M, 1} (\rho, \alpha) \leq B_{M,2} (\rho, \alpha) \leq \ldots \leq B_{M, K} (\rho, \alpha) \leq B_\mathrm{max} (\rho)$, where $J (\rho, \alpha)$ and $K (\rho, \alpha)$ are the number of local minima and maxima on the magnetic field line defined by $\rho$ and $\alpha$. Particles with $\lambda = B_{m,j}^{-1}$ located at $l_{m, j}$ do not move along magnetic field lines because they are completely trapped at $l = l_{m, j}$, but they move across magnetic field lines searching for a local minimum with value $B_{m,j}$. If there is no such a minimum within the flux surface defined by $\rho$, they need to move to one of the contiguous flux surfaces. Thus, to be omnigeneous, the magnetic field has to be such that the contour $B = B_{m, j}$ crosses all magnetic field line and closes on itself poloidally, toroidally or helically. This condition can be written as $$\partial_\alpha B (\rho, \alpha, l_{m,j} (\rho, \alpha) ) = 0.$$ Then, $B_{m,j}$ does not depend on $\alpha$, and we find $$B_\mathrm{min} (\rho) \equiv B_{m,1} (\rho) \leq B_{m, 2} (\rho) \leq \ldots \leq B_{m, J} (\rho) < B_\mathrm{max} (\rho).$$ However, in contrast to what is assumed in parts of [@cary97a; @cary97b], one cannot show that all the local minima are equal to $B_\mathrm{min} (\rho)$.
A similar but more sophisticated argument for particles with $\lambda$ values close to $B_{M,k}$ located at $l_{M,k}$ gives (see section III.B of [@cary97b]) $$\partial_\alpha B (\rho, \alpha, l_{M,k} (\rho, \alpha) ) = 0.$$ As a result, the contour $B = B_{M,k}$ crosses all magnetic field lines and closes on itself poloidally, toroidally or helically. The local maxima $B_{M,k}$ do not depend on $\alpha$, leading to $$B_\mathrm{min} (\rho) < B_{M,1} (\rho) \leq B_{M, 2} (\rho) \leq \ldots \leq B_{M, K} (\rho) \equiv B_\mathrm{max} (\rho).$$ Again, in contrast to what is assumed in [@cary97a; @cary97b], it is not possible to prove that all the local maxima are the same as $B_\mathrm{max} (\rho)$.
2. \[prop:Deltazeta\] Condition [(\[eq:dalphaJ\])]{} can be used to show that the distance along a magnetic field line between two points $l_{b1} (\rho, \alpha, \lambda = B^{-1})$ and $l_{b2} (\rho, \alpha, \lambda = B^{-1})$ that have the same value of $B$ and are both within the same magnetic well is independent of $\alpha$ (see section III.C of [@cary97b]), that is, $$\label{eq:Deltaldef}
l_{b2} (\rho, \alpha, \lambda = B^{-1}) - l_{b1} (\rho, \alpha, \lambda = B^{-1}) = \Delta l (\rho, B).$$ Note that the function $\Delta l$ depends on the well, as shown in figure \[fig:carycondition\](b). This condition can be easily restated in a particular type of straight field line coordinates known as Boozer coordinates [@boozer82]. We only need two properties of the Boozer coordinates. First, for $\theta$ the Boozer poloidal angle and $\zeta$ the Boozer toroidal angle, the angle $\alpha$ is $$\label{eq:alphadef}
\alpha = \theta - \iota \zeta,$$ where $\iota(\rho)$ is the rotational transform. Second, for fixed $\rho$ and $\alpha$, the relation between the arc length of the magnetic field line and the toroidal angle, ${\mathrm{d}}l/{\mathrm{d}}\zeta$, only depends on $\rho$ and the value of $B$ at $(\rho, \alpha, l)$. This property of ${\mathrm{d}}l/{\mathrm{d}}\zeta$ and equation [(\[eq:Deltaldef\])]{} give that the angular separation between the two points $l_{b1}$ and $l_{b2}$ can only depend on $\rho$ and $B$, $$\label{eq:Deltazetadef}
\zeta (\rho, \alpha, l_{b2}) - \zeta (\rho, \alpha, l_{b1}) = \Delta \zeta ( \rho, B ).$$
3. \[prop:Bmaxstraight\] Using result [(\[eq:Deltazetadef\])]{} and the fact that most flux surfaces are covered ergodically by a magnetic field line, it is possible to show that the contour $B = B_\mathrm{max}$ is a straight line in Boozer coordinates (see section III.C of [@cary97b]). Note that this is a property of the global maximum $B_\mathrm{max} (\rho)$, and not of the local maxima $B_{M,k}$.
Importantly, due to the fact that the local maxima $B_{M, k}$ satisfy conditions [(\[prop:Bmaxmin\])]{} and [(\[prop:Deltazeta\])]{}, particle transitions from barely passing orbits to barely trapped orbits due to collisions or other effects are equivalent to the same transitions in a configuration with only one global maximum. In both cases, the radial magnetic drift vanishes at the maxima so transitioning particles do not take a large radial step, and barely trapped particles are radially confined. As a result, transitions from passing to trapped do not cause an increase in the radial flux of energy or particles.
Example
=======
The properties listed above can be used to construct an omnigeneous magnetic field with more than one local minima $B_{m,i}$ and more than one local maxima $B_{M,k}$. We follow a procedure similar to the one proposed in section V of [@cary97b].
To define the function $B(\theta, \zeta)$, we use the intermediate coordinate $\eta (\theta, \zeta)$, defined such that the contours of constant $\eta$ are contours of constant $B$, $$\frac{B}{B_0} = f (\eta),$$ where $B_0$ is a normalization constant. The function $f(\eta)$ is the function that gives the number and type of local minima and maxima. To give an example with local minima and maxima that are different from $B_\mathrm{min}$ and $B_\mathrm{max}$, we focus on the function $$\label{eq:fexample}
f(\eta) = \left \{
\begin{array}{l l}
1 + 0.3 [ \cos (3\eta/4) + \cos(9\eta/4)]& \mathrm{for}\quad \eta \in [0, 4\pi/3], \\
1 + 0.6 \cos(\pi - 3\eta/2) & \mathrm{for}\quad \eta \in [4\pi/3, 2\pi],
\end{array}
\right .$$ plotted in figure \[fig:B\_eta\], with two local minima, $B_{m, 1} \equiv B_\mathrm{min} = 0.4B_0$ and $B_{m, 2} = 0.837B_0$, located at $\eta_{m, 1} = 4\pi/3$ and $\eta_{m,2} = 0.488\pi$, and three local maxima, $B_{M,1} = 1.163B_0$ and $B_{M,2} \equiv B_{M,3} \equiv B_\mathrm{max} = 1.6B_0$, located at $\eta_{M,1} = 0.845\pi$, $\eta_{M,2} = 0$ and $\eta_{M,3} = 2\pi$.
The function $\eta (\theta, \zeta)$ is chosen so that $B(\eta(\theta, \zeta))$ satisfies properties [(\[prop:Bmaxmin\])]{}, [(\[prop:Deltazeta\])]{} and [(\[prop:Bmaxstraight\])]{} in section \[sec:conditions\]. We start by choosing the shape of the contours $\eta = \eta_{M,2}$ and $\eta = \eta_{M,3}$ that correspond to $B_\mathrm{max}$. These contours must satisfy property [(\[prop:Bmaxstraight\])]{}. Without loss of generality, we choose the contour $B = B_\mathrm{max}$ to be $\zeta = 0, 2\pi$. Then, $\eta = \zeta$ at $\zeta = 0, 2\pi$ (see the sketch in figure \[fig:problemset\](a)). According to property [(\[prop:Bmaxmin\])]{}, the contours $\eta \in \{\eta_{m,1}, \eta_{m,2}, \eta_{M,1}, \eta_{M,2}, \eta_{M,3}\}$ must close on themselves. Since we have already chosen the contours $\eta = \eta_{M,2}, \eta_{M,3}$ to be $\zeta = 0, 2\pi$, and contours cannot cross each other, the contours $\eta = \eta_{m,1}, \eta_{m,2}, \eta_{M,1}$, and consequently all the contours of constant $B$, must close poloidally. Thus, the function $\eta$ is defined by $$\label{eq:zetaeta}
\zeta = \eta + G(\theta, \eta),$$ where the function $G$ vanishes for $\eta = 0, 2\pi$ and is defined such that the relation between $\zeta$ and $\eta$ is invertible.
To completely determine $G(\theta, \eta)$ we only need to know this function in certain regions of the $(\theta, \eta)$ plane. In particular, for the function $B/B_0 = f(\eta)$ in equation [(\[eq:fexample\])]{} and figure \[fig:B\_eta\], we need information in the regions highlighted in figure \[fig:problemset\](a). We need to know $G(\theta, \eta)$
- in the region $\eta \in [0, \eta_{m,2}]$, where $$\label{eq:G0etam2}
G(\theta, \eta) = g (\theta, \eta);$$
- partially, in the region $\eta \in [\eta_{M,1}, \eta_{m,1}]$, where the difference $$\label{eq:GetaM1etam1}
G(\theta, \eta) - G(\theta, \eta_{M,1}) = h (\theta, \eta)$$ must be given;
- and on one magnetic field line (for example, $\alpha = 0$), where $$\label{eq:alphaequal0}
G(\theta_{\alpha=0} (\eta), \eta) = y (\eta).$$ Here $\theta_{\alpha=0} (\eta)$ is the solution to equations [(\[eq:zetaeta\])]{} and $\alpha = \theta - \iota \zeta = 0$ for a given $\eta$.
Using equation [(\[eq:alphaequal0\])]{} we can obtain the angular difference $\Delta \zeta (B)$ that according to property [(\[prop:Deltazeta\])]{} can only depend on the value of the magnitude of the magnetic field. Since $\eta$ and $B$ are almost equivalent, we write $\Delta \zeta$ as a function of $\eta$. All the values of $B$ can be mapped to the intervals $\eta \in [\eta_{m, 2}, \eta_{M,1}]$ and $\eta \in [\eta_{m,1}, 2\pi]$. In these intervals, we define the function $$\Delta \zeta (\eta) = \eta - \eta_l(\eta) + y (\eta ) - y (\eta_l (\eta)).$$ The function $\eta_l (\eta)$ gives the location $\eta_l$ to the immediate left of $\eta$ that satisfies $f(\eta_l) = f(\eta)$. The function $\eta_l$ is sketched in figure \[fig:eta\_l\].
Once the function $\Delta \zeta (\eta)$ is known, we can use the fact that $\Delta \zeta$ only depends on the magnitude $B$ to calculate $G(\theta, \eta)$ from [(\[eq:G0etam2\])]{} and [(\[eq:GetaM1etam1\])]{}. We find $$\label{eq:finalG}
\fl G(\theta, \eta) = \left \{
\begin{array}{l l}
g (\theta, \eta) & \mathrm{for}\quad \eta \in [0, \eta_{m,2}], \\
\Delta \zeta (\eta) - \eta + \eta_l(\eta) + g(\theta - \iota \Delta \zeta(\eta), \eta_l(\eta)) & \mathrm{for}\quad \eta \in [\eta_{m,2}, \eta_{M,1}], \\
h (\theta, \eta) + G(\theta, \eta_{M,1}) & \mathrm{for} \quad \eta \in [\eta_{M,1}, \eta_{m,1}], \\
\Delta \zeta (\eta) - \eta + \eta_l(\eta) + G(\theta - \iota \Delta \zeta(\eta), \eta_l(\eta)) & \mathrm{for}\quad \eta \in [\eta_{m,1}, 2\pi].
\end{array}
\right .$$ Here $$\fl G(\theta, \eta_{M,1}) = \Delta \zeta (\eta_{M,1}) - \eta_{M,1} + \eta_l(\eta_{M,1}) + g(\theta - \iota \Delta \zeta(\eta_{M,1}), \eta_l(\eta_{M,1})).$$ Note that the function $G(\theta - \iota \Delta \zeta (\eta), \eta_l (\eta))$ that appears in the formula for $G(\theta, \eta)$ for $\eta \in [\eta_{m,1}, 2\pi]$ is known because for $\eta \in [\eta_{m,1}, 2\pi]$, $\eta_l (\eta) \in [0, \eta_{m,2}] \cup [\eta_{M,1}, \eta_{m,1}]$.
Using equation [(\[eq:finalG\])]{} in a flux surface with rotational transform $\iota = 0.2$ and with the functions $$g(\theta, \eta) = 0.3\pi \sin \theta \sin (\eta/2),$$ $$h(\theta, \eta) = 0.3\pi\sin \theta [\sin (\eta/2) - \sin(\eta_{M,1}/2)]$$ and $$y(\eta) = 0.3\pi \sin (\theta_{\alpha = 0}(\eta)) \sin (\eta/2),$$ we obtain the function $\eta(\theta, \zeta)$ given in figure \[fig:problemset\](b). The final omnigeneous magnetic field $B / B_0 = f(\eta(\theta, \zeta))$ is given in figure \[fig:Bomni\].
Discussion
==========
With the example in figure \[fig:Bomni\], we have shown that in an omnigeneous stellarator, the values of the magnetic field strength at local minima do not have to be the same on a given flux surface. The same is true for the local maxima.
An omnigenenous toroidal flux surface must satisfy several conditions. The local maxima (minima) on a magnetic field line must have the same value as the closest maxima (minima) in the contiguous magnetic field line, forming curves of constant magnetic field strength that must close poloidally, toroidally or helically. In addition to this condition, the contours corresponding to the global maximum of the magnetic field strength on the flux surface have to be straight lines in Boozer coordinates. A consequence of these conditions is that the magnetic field strength wells extend until they close on themselves poloidally, toroidally or helically. The distance along a field line between two contours with the same value of magnetic field strength on two opposite sides of a well is a function only of the flux surface and of the well.
The extra assumption that all the local minima and all the local maxima have the same value has been used for simplicity in previous work, such as [@helander09; @landreman12]. The results of these papers only need to be generalized slightly to account for local minima and maxima with values different from the global minimum and maximum on the flux surface, but the qualitative results are probably unchanged.
In certain classes of stellarators, it may be beneficial for the minima of the magnetic field strength on a flux surface to have similar values [@mynick82], but we have shown that in general this is not a necessary condition for optimized stellarators. The more general conditions for omnigeneity discussed in this article ensure that the neoclassical fluxes do not scale inversely with collisionality. These conditions have been derived for a flux surface without consideration of the neighboring flux surfaces because we have limited our analysis to particles with small radial orbit widths. The study of long term confinement of energetic particles [@lotz92; @drevlak14] requires a more careful analysis than the one performed here, but one would expect that optimizing neoclassical fluxes would improve energetic particle confinement.
In practice, the design of an omnigeneous stellarator experiment would be based upon the multiple-criteria optimization procedures analogous to those used in references [@gori99; @subbotin06; @kasilov13] to identify a 3D MHD equilibrium. As part of such a design, collisionless charged particle losses would need to be computed directly using the codes based on the guiding center drift equations, as in [@lotz92; @drevlak14]. However, due to the high dimensionality of the optimization problem, simple but robust optimization criteria are required to use the optimization codes effectively, for selecting appropriate weighted cost functions, initial configurations, and search algorithms. Our results here give such criteria. An optimization that imposed that the local minima and maxima must be the same on a given flux surface could give a stellarator close to omnigeneity, but it would have ignored a large part of the allowed parameter space, therefore missing potentially better solutions.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank J.R. Cary and S.G. Shasharina for helpful discussions. This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement number 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. This research was supported in part by the RCUK Energy Programme (grant number EP/I501045) and by grant ENE2012-30832, Ministerio de Economía y Competitividad, Spain.
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Drevlak M, Geiger J, Helander P and Turkin Y 2014 073002
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We report grazing incidence small angle neutron scattering (GISANS) and complementary off-specular neutron reflectometry (OSR) of the magnetic order in a single-crystalline epitaxial MnSi film on Si(111) in the thick film limit. Providing a means of direct reciprocal space mapping, GISANS and OSR reveal a magnetic modulation perpendicular to the films under magnetic fields parallel and perpendicular to the film, where additional polarized neutron reflectometry (PNR) and magnetization measurements are in excellent agreement with the literature. Regardless of field orientation, our data does not suggest the presence of more complex spin textures, notably the formation of skyrmions. This observation establishes a distinct difference with bulk samples of MnSi of similar thickness under perpendicular field, in which a skyrmion lattice dominates the phase diagram. Extended x-ray absorption fine structure measurements suggest that small shifts of the Si positions within the unstrained unit cell control the magnetic state, representing the main difference between the films and thin bulk samples.'
address:
- 'Physik Department, Technische Universität München, James-Franck-Strasse 1, 85748 Garching, Germany'
- 'Physik Department, Technische Universität München, James-Franck-Strasse 1, 85748 Garching, Germany'
- 'Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford, OX1 3PU, UK'
- 'Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany'
- 'Max Planck Society, Outstation at FRM-II, D-85748, Garching, Germany'
- 'Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford, OX1 3PU, UK'
- 'Physik Department, Technische Universität München, James-Franck-Strasse 1, 85748 Garching, Germany'
- 'Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany'
- 'Max Planck Society, Outstation at FRM-II, D-85748, Garching, Germany'
- 'Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany'
- 'Max Planck Society, Outstation at FRM-II, D-85748, Garching, Germany'
- 'Forschungsneutronenquelle Heinz Maier Leibnitz (FRMII), Technische Universität München, 85748 Garching, Germany'
- 'Physik Department, Technische Universität München, James-Franck-Strasse 1, 85748 Garching, Germany'
- 'Physik Department, Technische Universität München, James-Franck-Strasse 1, 85748 Garching, Germany'
- 'Physik Department, Technische Universität München, James-Franck-Strasse 1, 85748 Garching, Germany'
- 'Physik Department, Technische Universität München, James-Franck-Strasse 1, 85748 Garching, Germany'
author:
- 'B. Wiedemann'
- 'A. Chacon'
- 'S. L. Zhang'
- 'Y. Khaydukov'
- 'T. Hesjedal'
- 'O. Soltwedel'
- 'T. Keller'
- 'S. Mühlbauer'
- 'T. Adams'
- 'M. Halder'
- 'C. Pfleiderer'
- 'P. Böni'
title: Reciprocal space mapping of magnetic order in thick epitaxial MnSi films
---
The discovery of skyrmions in chiral magnets [@muhlbauer:S:09; @neubauer:PRL:09; @munzer:PRB:10; @jonietz:S:10; @yu:N:10; @yu:NM:11; @seki:S:12; @adams:PRL:12; @schulz:NP:12; @nagaosa:NN:13; @milde:S:13; @mochizuki:NM:14; @schwarze:NM:15] has generated great efforts to exploit their unusual properties technologically [@kiselev:JPDAP:11; @fert:NN:13; @hellman:RMP:17; @garst:JPDAP:17]. Initially identified in a small pocket of the phase diagram of the cubic chiral magnets MnSi, [[Fe$_{1-x}$Co$_{x}$Si]{}]{}, FeGe, and [[Cu$_{2}$OSeO$_{3}$]{}]{}, Lorentz transmission electron microscopy (LTEM) early on revealed a strong increase of the extent of the skyrmion phase with decreasing sample thickness [@yu:N:10; @yu:NM:11; @seki:S:12; @tonomura:NL:12; @yu:NL:13]. While this demonstrated that nano-scaled systems may be ideally suited to host skyrmions, the mechanisms controlling their formation in samples prepared by established deposition techniques remain unresolved.
Based on measurements of the magnetization and magneto-transport, interpreted to provide a topological Hall signal for magnetic fields applied perpendicular to the films, seminal work on epitaxial films of MnSi [@karhu:PRB:10; @wilson:PRB:12; @li:PRL:13], Mn$_{1-x}$Fe$_{x}$Si [@yokouchi:PRB:14], Fe$_{1-x}$Co$_{x}$Si [@porter:AC:13; @sinha:PRB:14] and FeGe [@porter:PRB:14; @gallagher:PRL:17; @ahmed:AC:17] claimed the formation of skyrmions. However, the phase diagrams reported for nominally the same films differ between different studies, as well as from those determined by means of LTEM in thin bulk samples. Moreover, alternative mechanisms have been proposed explaining the same data without invoking skyrmions or topological textures [@monchesky:PRL:14; @meynell:PRB:14a]. Subsequent microscopic measurements using polarized neutron reflectometry (PNR) [@karhu:PRB:11; @wilson:PRB:14; @meynell:PRB:14], while suggesting modulations perpendicular to the films, failed to resolve this situation convincingly, as PNR requires the assumption of complex scattering profiles. Similar uncertainties arise in studies of epitaxial films using LTEM, either because of parasitic signal interferences in the presence of the substrate [@monchesky:PRL:14; @li:PRL:14], or because in free-standing films removal of the substrate changes the elastic properties of the films [@ahmed:AC:17]. Similar uncertainties exist under magnetic fields parallel to the films. Here, magnetization, magneto-transport and PNR have been interpreted to provide putative evidence of complex magnetic phase diagrams supporting the formation of in-plane skyrmions [@karhu:PRB:12; @wilson:PRB:12; @wilson:PRB:13; @yokouchi:JPSJ:15; @meynell:PRB:17].
Theoretically, it was at first believed, that the increase of the skyrmion phase in thin bulk samples originates from the destabilization of the competing magnetic phases [@yu:N:10; @yu:NM:11; @han:PRB:10]. Recent studies suggest instead that free surfaces favour the formation of skyrmions energetically, driving the formation of skyrmions in surface-dominated systems [@meynell:PRB:14; @kiselev:PRL:15; @zheng:AC:17; @wild:SA:17]. Another aspect are interface induced spin-orbit coupling effects, which are dominant for heavy element substrates as used in studies of atomically thin films of Fe and FePd [@heinze:NP:2011; @romming:S:13]. Last but not least, numerous studies have considered magnetic anisotropies induced by the lattice mismatch with the substrate [@butenko:PRB:10; @leonov:AC:10; @rossler:JPCS:11; @wilson:PRB:12; @rybakov:PRB:13; @wilson:PRB:14], which, however, are found to decrease rapidly with increasing film thickness.
To establish the nature of the long-range magnetic order in thin films unambiguously, direct reciprocal space imaging by means of neutron scattering is ideally suited. Meaningful diffraction patterns are already expected for short correlation lengths. Typical features of the magnetic structures considered in epitaxial thin films of B20 compounds can be found in the supplementary information [@SOM].
A major technical constraint of neutron scattering in thin films is the tiny sample volume. For instance, a recent study on a thick MnSi film using small angle neutron scattering (SANS) in a transmission geometry has demonstrated that even for a measurement time of 20 hours the magnetic signal was barely discernable [@meynell:PRB:17]. In comparison, a clear SANS signal was observed in a stack of 32 FeGe films ($15\times 15 \,{\rm mm^2}$ corresponding to 12mg of material) illuminated parallel to the film [@kanazawa:PRB:16]. Unfortunately, this approach requires perfectly identical large films, bearing the additional risk that important details may get averaged out.
In view of these complexities an important point of reference are the differences between thin bulk samples, providing a well-defined setting, and epitaxial layers of comparable thickness. For FeGe on Si(111), where the lattice mismatch is vanishingly small, the SANS on a stack of 32 large films has provided compelling evidence that the magnetic order consists of a helical modulation perpendicular to the film without any evidence for more complex textures, notably skyrmions. The pronounced difference of the magnetic order in epitaxial films and thin bulk samples appears to be well beyond present understanding and underscores the need for further microscopic information on structural differences.
In our study we focus on the properties of epitaxial MnSi films on Si(111) since this allows for comparison with thin bulk samples of MnSi as well as the large body of literature on FeGe. Demonstrating for the first time, that GISANS and OSR can be performed on just a single film rather than a large stack [@wiedenmann:phd], we find a magnetic modulation perpendicular to the MnSi films featuring magnetic phase diagrams under parallel and perpendicular field that are highly reminiscent of those in FeGe on Si(111). Moreover, extended x-ray absorption fine structure (EXAFS) on the same thick epitaxial MnSi films reveals that the lattice mismatch with the substrate is released within a few atomic layers, such that the films display unstrained lattice parameters, however, with small shifts of the Si positions normal to the film plane [@figueroa:PRB:16]. Taken together our results suggest formation of a magnetic modulation perpendicular to the films as a generic property of thick epitaxial films of B20 compounds supported by a substrate, caused by tiny shifts of the non-magnetic atoms.
Epitaxial MnSi(111) films were grown on Si(111) substrates using molecular beam epitaxy following the established recipe for solid-phase epitaxy [@karhu:PRB:10; @SOM]. For our study we selected three MnSi film thicknesses \[$d={\SI[parse-numbers = false]}{(390\pm10)}{\AA}, {\SI[parse-numbers = false]}{(495\pm10)}{\AA}$, and [(55310)]{}[Å]{}\] that are much larger than the length of the magnetic helix of $\sim$180 Å in bulk MnSi, and well above the thickness of $\sim$120 Å where $T_\mathrm{c}$ becomes thickness-independent. Moreover, the induced uniaxial anisotropy is vanishingly small [@karhu:PRB:12; @brasse:phd]. The $\sim$550 Å thick sample displayed structural and magnetic properties consistent with data reported in the literature [@karhu:PRB:10; @karhu:PRB:11; @wilson:PRB:12; @karhu:PRB:12; @li:PRL:13; @menzel:JotKPS:13; @wilson:PRB:13; @wilson:PRB:14; @meynell:PRB:14; @lancaster:PRB:16; @zhang:AA:16]. Samples without capping, Si capping and Cu capping were prepared. The samples with Cu capping were designed as neutron resonator (waveguide) structure to enhance spin-flip scattering by non-collinear magnetic structures such as skyrmions. However, we did not detect any significant spin-flip scattering in our PNR experiments consistent with the absence of skyrmions. All samples displayed the same magnetic properties apart of small differences of the modulation length of order 10%.
Shown in Fig. \[Fig1GISANS\](a) is the GISANS setup used for our study. Measurements were carried out at SANS-1 at the Heinz Maier-Leibnitz Zentrum (MLZ), Munich [@muhlbauer:NIaMiPRSAASDaAE:16]. The sample was illuminated under an incident angle of ${\theta_i = \SI{0.65}{\degree}}$ using neutrons of wavelength ${\lambda = {\SI[parse-numbers = false]}{(5.5\pm0.5)}{\AA}}$ collimated over and recorded with a detector behind the sample. Due to the magnetic mosaicity of the sample and the small scattering angles Bragg peaks for both +$\bm{k}_h$ and -$\bm{k}_h$ could be observed simultaneously. This allowed us to better separate the specular reflection (yellow arrow) from the Bragg peaks. GISANS data was recorded for fields perpendicular as well as parallel to the film.
Typical data at ${T=\SI{15}{\K}}$ for a MnSi film (${d=\SI{553}{\AA}}$) capped with of Cu are shown in Figs. \[Fig1GISANS\](b-e) for a field parallel to the film and the neutron beam \[Fig. \[Fig1GISANS\](a)\] after subtraction of the background determined at 60 K. The direct beam in the center of the images was partially masked (blue square) to prevent saturation of the detector, yet permitting data analysis even very close to the direct beam. In zero field, a magnetic satellite peak labeled A at ${|Q_z| = \pm\SI{0.067}{\AA^{-1}}}$ is observed, characteristic of a magnetic modulation along normal to the film plane. As compared to bulk MnSi the pitch of the modulation is a factor of two smaller. In view of the bulk properties of MnSi we assume the formation of a helical modulation.
Remarkably, with increasing magnetic field the peak sharpens without changing its location and a second peak labelled B emerges above $\sim\SI{0.2}{T}$ at ${|Q_z|\!=\!\SI{0.052}{\AA^{-1}}}$ as shown in Fig. \[Fig1GISANS\](c) for ${B\!=\!\SI{0.4}{\tesla}}$. When increasing the field, the peaks at A and B sharpen and the peak at A vanishes above $\sim\SI{0.5}{T}$ \[Fig. \[Fig1GISANS\](d)\] before the peak at B, which continues to sharpen even further, vanishes above $\sim\SI{0.8}{T}$ \[Fig. \[Fig1GISANS\](e)\]. The same behavior was observed for fields parallel to the film, regardless if the fields were parallel or perpendicular to the neutron beam. Similarly, for a magnetic field perpendicular to the film a magnetic modulation perpendicular to the film of unchanged modulation length was observed for all magnetic fields up to the onset of the field-polarised state at $H_{c2}$. Considering instrumental resolution, neither the GISANS data nor any of the other data we recorded provide indications of scattering at finite $Q_y$ or $Q_x$ in the parameter regime explored. Thus the magnetic order in our MnSi film is definitely dominated by a magnetic modulation perpendicular to the film.
![ Summary of key data as collected by means of PNR and OSR on a film with a thickness $d = 553$ Å. (a) Schematic of the PNR and OSR set-up. The sample was oriented with perpendicular to the sample surface, and along the $y$-direction. For PNR, the in-plane magnetic field ${\bm B}$ was applied along the $y$-direction, while for OSR $\bm{B}$ was aligned in the forward direction parallel to . (b,c) PNR recorded at $T = 35$ K indicating a magnetic response due to a periodic modulation of the magnetization perpendicular to the film at $B_{[112]} = 30$ mT and $B_{[112]} = 440$ mT corresponding to regimes 1 and 2 (see Figs. 3 and 4), respectively. (d,e) OSR recorded at $T = 15$K for $B_{[112]} = 30$ mT and $B_{[112]} = 240$ mT, where a single modulation and two modulations are seen, respectively.[]{data-label="Fig2PNR"}](Fig2.pdf){width="1.0\columnwidth"}
To connect the GISANS data with properties reported in the literature [@karhu:PRB:11; @karhu:PRB:12] we performed PNR (${\theta_i =\theta_f}$) and OSR (${\theta_i \neq\theta_f}$) at NREX, MLZ [@Khaydukov:JLRFJ:15]. Using neutrons with a wavelength $\lambda\!=\!\SI{4.31}{\AA}$ and a polarization parallel and antiparallel to the in-plane magnetic field ${\bm B}_y$ \[Fig. \[Fig2PNR\](a)\] yields an improved $Q_z$ resolution and corroborates the GISANS results. Shown in Fig. \[Fig2PNR\](b,c) is the specular reflectivity of polarized neutrons at ${T=\SI{35}{\K}}$ for the same film investigated in Fig.\[Fig1GISANS\]. Now the field is applied along a -direction ($y$-direction) and perpendicular to the incident neutron beam \[Fig. \[Fig2PNR\](a)\]. Data at low fields (${B_{y}=\SI{0.03}{\tesla}}$) may be well accounted for by assuming a magnetic helix with a wavevector ${k_h=\SI{0.067}{\AA^{-1}}}$ parallel to $Q_z$. This result compares with typical PNR at larger fields (${B_{\hkl[11-2]}=\SI{0.44}{\tesla}}$) shown in Fig. \[Fig2PNR\](c), where the reflectivity may be fitted by assuming an anharmonic helix with a smaller wavevector ${k_h=\SI{0.052}{\AA^{-1}}}$ and the magnetic moments pointing predominantly along $\bm {B}_y$. The appearance of helices parallel to $Q_z$ inferred from our PNR data is in excellent agreement with our GISANS results, as well as with the literature [@karhu:PRB:11; @wilson:PRB:13].
Shown in Figures \[Fig2PNR\](d) and (e) are typical OSR data of the same MnSi film at ${T=\SI{15}{\K}}$ under an in-plane magnetic field ${\bm B}_{\hkl[1-10]}$ oriented in the forward direction along , where a background recorded at 60K was subtracted; the largest field accessible was 0.24T. The vertical stripes (dark blue and red) for ${Q_x = 0}$ are due to the specular reflection of the neutrons by the sample \[cf Figs. \[Fig2PNR\](b,c)\]. Here superlattice peaks due to Bragg scattering from the helix, seen in GISANS, are buried by the strong specular signal. The arrows labeled A and B indicate weak magnetic correlations within the plane due to a small amount of roughness leading to the appearance of Bragg sheets at wavevectors ${k_h^A = \SI{0.067}{\AA^{-1}}}$ and ${k_h^B = \SI{0.052}{\AA^{-1}}}$. The position of these peaks corresponds to the helix vectors inferred from PNR as well as from GISANS. The weak intensity of the Bragg sheets is an indication for (i) the excellent quality of the samples and (ii) the excellent alignment of the helices perpendicular to the film even in large fields perpendicular to $\bm {k}_h$.
We note that PNR measurements in a MnSi film with a similar thickness of , reported in the literature [@karhu:PRB:12; @wilson:PRB:14], were satisfactorily fitted with a sinusoidal magnetization profile and a modulus of the helix vector of in zero field and a distorted sinusoidal shape together with a different periodicity for in-plane magnetic fields of 0.2T and 0.4T. This discrepancy with our data may be attributed to a reduced sample size, but reflects also the uncertainties when interpreting PNR data.
![Comparison of the $Q_z$-dependence observed in GISANS and OSR for various fields and temperatures. (a1,a2,a3) Field-dependence at 15K, and (b1,b2,b3) at 27K. The solid lines represent Gaussian fits. (c) Field dependence of the position of the magnetic satellite peaks A and B. The regimes 1 and 2 are highlighted in dark gray and white, respectively. Dotted vertical lines depict the transition from the intermediate regime to regime 2.[]{data-label="Fig3VglNrexSANS"}](Fig3.pdf){width="1.0\columnwidth"}
A comparison of the $Q_z$-dependence observed in GISANS and OSR is shown in Figure \[Fig3VglNrexSANS\]. For this comparison the GISANS data was integrated radially within the sector indicated by the red lines shown in Fig. \[Fig1GISANS\](b). The data was scaled with a single normalisation constant for the comparison with the OSR. The $Q_z$-dependence of the OSR signal was obtained by integrating the intensity on the negative side of the specular signal for momentum transfers ${Q_x \le\SI{-3e-6}{\AA^{-1}}}$ \[see Figs. \[Fig2PNR\](d,e)\]. All data (Fig. \[Fig3VglNrexSANS\]) displays a single peak at low (a1,b1) and high fields (a3,b3), respectively, and two peaks at intermediate fields (a2,b2), although the measurements have been conducted under very different resolution conditions and for two different spatial orientations with respect to the in-plane magnetic field. As no diffraction peaks or intensities are observed in the $Q_x$–$Q_y$ plane, the modulations evolve along $Q_z$ only. Even though the GISANS, OSR and PNR signals cannot be tracked all the way up to $B_{c2}$ and $T_c$, the accessible $T$ and $B$ ranges are sufficiently large to note that we do not find any evidence supporting earlier claims of in-plane skyrmions [@wilson:PRB:12; @meynell:PRB:17].
A similar reduction of the helical modulation length under magnetic field parallel to the film was previously proposed in a related study of a film with $d=267\,{\rm \AA}$ using PNR, magneto-transport and magnetometry [@wilson:PRB:13]. Our data are also consistent with an unwinding in discrete steps, also reported in Ref.[@wilson:PRB:13]. This behaviour agrees with the interplay of a perpendicular magnetic anisotropy, favouring an out-of-plane propagation, and the Zeeman energy under applied fields. It is interesting to note, that the magnetic anisotropy constant inferred in our films from torque magnetometry is vanishingly small [@brasse:phd] and consistent with estimates based on the magnetisation reported in the literature [@karhu:PRB:12]. This result suggests a more subtle origin of the propagation direction other than strain-induced magnetic anisotropies.
![$(B,T)$–phase diagrams for an epitaxial MnSi film ($d=553\,{\rm\AA}$) as inferred from GISANS, PNR, OSR and SQUID magnetometry. No magnetic signal was observed with GISANS at the gray-shaded positions. (a) Phase diagram under magnetic field parallel to the film. Two regimes may be distinguished at low fields and high fields with distinct helical modulations, $k_h^A$ and $k_h^B$, respectively. At intermediate fields both modulations coexist. (b) Phase diagram under magnetic field perpendicular to the film. A single helical modulation $k_h^A$ perpendicular to the film is observed.[]{data-label="figure4"}](Fig4.pdf){width="1\columnwidth"}
Combining our neutron data with SQUID magnetometry on the same MnSi films the magnetic phase diagram shown in Fig. \[figure4\] was constructed. The PNR data (open symbols) and GISANS/OSR data (filled circles) are fully consistent with each other. The phase boundaries between regime 2 and the field-aligned ferromagnetic phase and $T_\mathrm{c}$ were inferred from the SQUID data. The upper critical field $B_{c2} = 0.93\,{\rm T}$ (blue dots), as well as $T_\mathrm{c} = 42.5\,{\rm K}$, are significantly higher than the corresponding values in bulk MnSi, in good agreement with the literature [@li:PRL:13; @karhu:PRB:12; @wilson:PRB:12].
The magnetic order and the phase diagrams we observe are in striking similarity with the magnetic phase diagram observed in a SANS study of a stack of 32 FeGe thin films. For the FeGe films the lattice mismatch is tiny, rendering simple strain-induced magnetic anisotropies highly unlikely. For our MnSi films, EXAFS [@figueroa:PRB:16] establishes, that the lattice strain is released within a few atomic layers, while the main part of the film is unstrained apart from tiny out-of-plane shifts of the positions of the Si atoms. These shifts give way for an unexpected new mechanism controlling the appearance of an out-of-plane modulation in epitaxial MnSi films.
In summary, we present the first reciprocal space mapping of the magnetic order in single thick epitaxial MnSi films on Si(111) demonstrating the potential of GISANS and OSR, where we find a magnetic phase diagram in striking similarity with thick epitaxial FeGe films. Combining these results with EXAFS and the established properties of thin bulk samples, we conclude that the magnetic order we observe is generic, providing an entry port for unravelling the rather conflicting information on thin films in the literature as concerns the precise mechanisms that allow to tailor skyrmions in nanosystems by means of established deposition techniques.
We gratefully acknowledge technical support by the team of FRM II as well as Franz Tralmer at NREX. We also wish to thank A. Bauer, M. Brasse, D. Grundler, C. Schnarr and M. Wilde for discussions and support. Financial support through the collaborative research center TRR80 of the German Science Foundation (DFG) projects E4 and F7, as well as ERC Advanced Grant 291079 (TOPFIT) are gratefully acknowledged. BW, AC, MH and TA acknowledge financial support through the TUM graduate school. TH and SLZ acknowledge the Semiconductor Research Corporation (SRC) and EPSRC (UK).
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---
abstract: 'In this paper, we are interested in understanding the interrelationships between mainstream and social media in forming public opinion during mass crises, specifically in regards to how events are framed in the mainstream news and on social networks and to how the language used in those frames may allow to infer political slant and partisanship. We study the lingual choices for political agenda setting in mainstream and social media by analyzing a dataset of more than 40M tweets and more than 4M news articles from the mass protests in Ukraine during 2013-2014 — known as “Euromaidan” — and the post-Euromaidan conflict between Russian, pro-Russian and Ukrainian forces in eastern Ukraine and Crimea. We design a natural language processing algorithm to analyze at scale the linguistic markers which point to a particular political leaning in online media and show that political slant in news articles and Twitter posts can be inferred with a high level of accuracy. These findings allow us to better understand the dynamics of partisan opinion formation during mass crises and the interplay between mainstream and social media in such circumstances.'
author:
- 'Dmytro Karamshuk$^1$, Tetyana Lokot$^2$, Oleksandr Pryymak$^3$, Nishanth Sastry$^1$'
bibliography:
- 'biblio.bib'
title: Identifying Partisan Slant in News Articles and Twitter during Political Crises
---
Introduction
============
Social media have become a crucial communication channel during mass political or civic events by shaping “a civic and democratic discourse in a vacuum of opportunities” [@howard2010digital]. As academic debates contest the nature of social media as an alternative public sphere [@papacharissi2009virtual], it is important to study the *interrelationships between mainstream media and social networks* in shaping public opinion during mass protests, especially in regards to the origin and dissemination of news frames [@oates2013twilight]. It is also of interest to consider how *propaganda and manipulation* in the information sphere work: where partisan language and frames originate, how they spread, and whether there are certain markers that would allow to trace the distribution and paths of such content.
In this study, we analyze the role of social and mainstream media in shaping and disseminating partisan content frames during social unrest and crisis. Specifically, we focus on the mass protests in Ukraine during 2013-2014 - known as *“Euromaidan”* - in which social media played a remarkable role, helping to raise awareness, cover, and discuss ongoing events; and the post-Euromaidan Russian occupation of Crimea and the conflict between Russian, pro-Russian and Ukrainian forces in eastern Ukraine (2014-2015), periods that were characterized by a parallel information and propaganda war occurring in mainstream media and online together with military action on the ground.
We explore the extent to which lingual choices in online discourse can illuminate the *partisan confrontation* between political factions during mass crises through the analysis of the two complementary datasets of Twitter posts and news articles. More specifically, we exploit natural language processing to single out language that points to a particular political leaning and to observe whether these markers are detectable in both mainstream media language and social media posts at scale. Our contributions can be summarized as follows:
- We exploit the word embedding approach [@mikolov2013distributed; @mikolov2013efficient] to identify the indicators of partisan slant in news articles and validate it over the text corpora of around 4M news articles collected during the Ukrainian conflict. Our analysis reveals a *strong use of highly polarized partisan content frames* in news articles on both sides of the conflict.
- Next, we design a machine learning approach for *detecting the markers of partisan rhetoric* in news articles with minimal efforts required for supervision. This is achieved by a “coarse-grained” labeling of the articles based on the partisan slant of the news agencies they originate from. Our approach - trained on a collection of articles from 15 representative news agencies - is able to achieve $60$–77% accuracy in distinguishing between the news articles with pro-Ukrainian, Russian pro-government and Russian independent slants during the Ukrainian conflict.
- Finally, we study the inter-relation between traditional and social media during conflicts through an analysis of individual news sharing patterns among Twitter users and find that most of the users are *exposed to a variety of news sources but with a strong partisan focus*. Using our machine learning approach, we are also able to predict the partisan leaning of Twitter users from the content of the tweets with an accuracy up of 70%.
In summary, we demonstrate that studying the lingual choices of Twitter users and news media adds a new dimension to understanding the dynamics of information flows and partisan idea dissemination in the space between social networks and mainstream media. We also demonstrate that lingual choices-based machine learning models can be highly effective at automatically predicting the political slant of mainstream media and Twitter users, which can have serious implications for political expression in repressive and authoritarian regimes.
Related Literature
==================
Computational social scientists have given substantial attention to the mainstream and social media activity around political and social change, and to the role information shared on these platforms plays in influencing political and social agendas, protest movements and events, and public opinion. Researchers have explored the role of social media and mainstream media actors in information diffusion and protest message amplification in networks through the prism of the *collective action theory* [@gonzalez2013broadcasters], as well as the role of social networks in recruitment and mobilisation during protest, revealing connections between online networks, social contagion, and collective dynamics [@gonzalez2011dynamics].
A broad swathe of quantitative studies have focused on determining the factors that influence political leanings of social network users and metrics that allow to classify and predict this kind of political bias. Several studies have considered the *predictive power* of political news sharing habits on Twitter [@an2014sharing], the influence of partisan information sharing on political bias among Facebook users [@an2014partisan] and Twitter actors [@hegelich2016social; @conover2011predicting], and compared differences and biases in news story coverage, dissemination, and consumption among online mainstream and social media [@chakraborty2016dissemination]. Others have noted the difficulty of connecting selective exposure to political news on Facebook to partisanship levels of users [@an2013fragmented]. At the same time, researchers suggest that analysis of information consumption and distribution habits of social network users does provide data on media exposure, the relationship between various classes of media, and the diversity of media content shared on social networks [@an2011media].
Some studies have noted that reliably inferring the political orientation of Twitter users and generalizing the findings is notoriusly difficult due to it being one of the “hidden” attributes in social network data and due to differences between politically active groups of users and the general population [@cohen2013classifying]. However, other academics suggest that studying some relationships on social networks, like the co-subscriptions relationships inferred by Twitter links [@an2012visualizing], allows for some understanding of the underlying media bias — and subsequently, political bias — of social network users. Another study showed that applying machine learning techniques to classify political leanings on Twitter based on political party messages can reveal partisanship among users [@boutet2013s]. A number of studies present a comparison between the predictive power of the users’ social connections and their content sharing patterns for inferring political affiliation, ethnicity identification and detecting affinity for a particular business [@pennacchiotti2011democrats; @pennacchiotti2011machine].
While the research described above uses a fairly large spectrum of methods to study, classify and find connections between social media users’ behavior and their media and political preferences, most of the studies referenced employ social network analysis or related methods, focusing on relationships between actors or their behaviors within the network, such as sharing links, following other actors, etc. More recent studies have used computational methods to assess forms of political organization on social media [@aragon2016movement], employed machine learning models to classify rumor and misinformation in crisis-related social media content [@zeng2016unconfirmed], and used deep neural networks to identify and analyze election-related political conversations on Twitter on a longitudinal scale [@vijayaraghavan2016automatic].
We propose augmenting these approaches with a focus on the *linguistic variables* present in the data, and using natural language processing and machine learning techniques to gain further insight into how political and ideological messages travel between mainstream and social media, and how these lingual choices reflect the partisan nature of mainstream media outlets and, subsequently, social media users and their content consumption and sharing habits. Such an approach would allow for a more granular understanding of how language changes allow to detect both important events and partisan leanings in mainstream and social media data.
Background and Datasets
=======================
The Euromaidan protests and subsequent political crisis are the outcome of a continuing trend in the post-Soviet arena. The last decade or so has seen an increase in mass protest actions in the region, with protests erupting in Russia, Belarus, Ukraine, etc. From the 2004 *Orange Revolution* in Ukraine to the *Bolotnaya rallies* of 2011-2012 in Moscow to the Euromaidan protests in Ukraine, a gradual increase in the use of digital technologies and media platforms by citizens has become evident [@goldstein2007role; @oates2013twilight; @onuch2015euromaidan]. At the same time, the region is characterized by a problematic media climate, with mainstream media often co-opted or controlled by the state or the oligarchy. The interplay and mutual influence of mainstream and social media emerge as crucial for understanding the political and civic developments in the region and thus demand more scholarly attention.
\[tab:dataset-description\]
A number of researchers have already examined some of the more general aspects of Euromaidan, such as the reasons for the protest [@diuk2014euromaidan], who the protesters were [@zelinska2015were], how the protest came together and evolved [@onuch2015euromaidan]. However, few have investigated the use of civic media by Euromaidan participants beyond simply saying that social media were used in the protest as ‘tools’ for mobilization and information dissemination [@onuch2015euromaidan]. A deeper and more large-scale analysis of the relationship between mainstream media coverage of the crisis and the grassroots social media data around the political unrest, enabled by computational and big data tools and complemented by qualitative analysis, could reveal more about how the partisan agenda during the protests was formed and transformed through lingual choices and memorable memes, and who was able to exert influence on the lingual frames used by the multitudes of social media users and media outlets. Such investigation could also shed light on the reasons and mechanisms of the information war between Ukraine, Russia, and the West that gained in scale after the Euromaidan protests shifted into the crisis characterized by Russia’s annexation of Crimea and the pro-Russian uprising in eastern Ukraine.
To research these questions in this paper we analyze two complementary datasets: A *social media* dataset which consists of over $40$M tweets collected for the three most prominent hashtags during and after the Euromaidan protests - \#euromaidan \#ukraine \#kyiv (and their Ukrainian and Russian equivalents) via Twitter Streaming API and a dataset of more than $4$M *news articles* collected from a large Russian news aggregator (Rambler.ru) and its Ukrainian counterpart (smartphone app Kobzi). All three datasets were collected in the period between December 2013 and July 2014. The parameters of the datasets are summarized in [Fig. \[tab:dataset-description\]]{}.
Exemplar indicators of partisan slant
=====================================
Methodology
-----------
Our initial interest in exploring the *lingual choices* made by mainstream media sources and social media users was sparked by observing the emergence of a number of memes and buzzwords introduced during the events. In this section, we aim to measure the presence of these keywords in the rhetoric of the parties involved in the conflict and explore the extent to which this analysis can be automated.
To analyze the lingual choices of online media sources during the Ukrainian conflict we first exploit the *word embedding* methodology proposed in [@mikolov2013distributed; @mikolov2013efficient]. The proposed approach devises vector representations of words by analyzing the textual context in which they appear. This is achieved by training a model which for a given word, represented by a vector $X_i$, aims to infer the most likely $2 \times j$-surrounding words vectors which constitute the lingual context where the word was used [^1], e.g, $f(X_i) = (X_{i-j}, ..., X_{i-1}, X_{i+1}, ..., X_{i+j})$. Intuitively, semantically closed words are expected to appear in similar contexts and so should produce similar outcomes when applied as the arguments of the function $f(X)$. Thus, if trained on a significantly large text corpus the algorithm is able to assign close-by vectors to the words with similar meanings, thereby providing a powerful framework for analyzing the semantics of the word choices.
Equipped with the Word2Vec implementation[^2], we build word representations for the textual corpora extracted from our datasets. Note that we mainly focus on vector spaces extracted from the news corpora (News-UA and News-RU) as Twitter’s limit of $140$ characters significantly constrains the applicability of this approach. In pre-processing, we remove rare words with less than $10$ occurrences in each of the corpora and end up with a dictionary of $87$K words trained from a content corpus of $600$M words. Note that while the Ukrainian media space is generally bilingual, in our analysis we only focus on Russian-language news articles, in order to allow for a fair cross-comparison of the results, i.e., we focus on the Russian-language media sphere as it presents a sufficient diversity of news sources, with political views spread across the spectrum, ranging from pro- to anti-Kremlin and from pro- to anti-Ukraine, including both Ukrainian and Russian media outlets.
Mining semantics of word choices
--------------------------------
In [Table \[tab:word2vec-examples\]]{} we present several examples of word associations which were mined from both our news corpora. We pick several loaded terms which were the prominent indicators of partisan rhetoric during the conflict and match these words to the most similar ones according to the trained Word2Vec dictionaries (as measured by the cosine similarity). The results illustrate that the trained model corresponds to our understanding of the semantics of chosen terms. For instance, the word *referendum (референдум)*, frequently used in the context of the Russian-backed annexation referendum in Crimea, lies very close to its synonyms, e.g., ‘plebiscite’, ‘voting’, etc., in the devised vector space in both the News-RU and News-UA corpora, from both sides of the conflict.
On the other hand, we also notice that some of the synonyms discovered by Word2Vec reflect the propagandistic rhetoric of official Russia and Ukraine during the conflict. For instance, in the News-UA corpus (but not in the News-RU corpus), the word *referendum (референдум)* is associated with the words ‘non-legitimate’ and ‘fake referendum’, and reflect the official Ukrainian state’s position on the plebiscite. Similarly, the word *aggressor (агрессор)* as used in association with Russia’s actions in eastern Ukraine and Crimea is associated with the words ‘cynical’ and ‘unprovoked’ and captures the attitude of the Ukrainian government in regards to the events.
A similar partisan rhetoric is also observed in Word2Vec word representations mined from the Russian news corpus (News-RU). For instance, we observe that the loaded term *junta (хунта)*, which was extensively used by some Russian media sources to demonize the transitional government formed in Ukraine after the Euromaidan protests, is strongly associated with the words ‘neo-fascist’ and ‘pro-Ukrainian’ in the Russian news corpus (News-RU). Similarly, the word *punisher (каратель)* which was used by some Russian media sources to label the Ukrainian Armed Forces and their operations in the conflict in eastern Ukraine is closely associated with ‘fascism’ and ‘terrorize’, as well as with Stepan Bandera, a controversial figure who has been acknowledged by some in the new Ukrainian government.
Such biased lingual choices are in alignment with recent findings that politically charged rhetoric and biased language were central to the discourse around the Euromaidan protests in Ukraine and the subsequent conflict in eastern Ukraine, both of which featured interference by Russian political forces [@szostek2014media; @zhukov2016reporting].
[ | l | c | l | c |]{}\
&\
бандера \[Bandera [^3]\] & 0.79 & восстание \[uprising\] & 0.76\
безоружный \[unarmed\] & 0.79 & проукраинский \[pro-Ukrainian\] & 0.72\
расстреливать \[shoot\] & 0.77 & неофашист \[neo-fascist\] & 0.72\
терроризировать \[terrorize\] & 0.77 & неонацист \[neo-Nazi\] & 0.71\
оккупант \[invader\] & 0.76 & путчист \[coupist\] & 0.71\
фашист \[fascist\] & 0.76 & самозванец \[imposter\] & 0.71\
\
\
&\
псевдо-референдум \[fake referendum\] & 0.72 & неспровоцированный \[unprovoked\] & 0.72\
плебисцит \[plebiscite\] & 0.71 & враг \[enemy\] & 0.70\
голосование \[voting\] & 0.67 & циничный \[cynical\] & 0.66\
автономный \[autonomous\] & 0.66 & развязанный \[launched\] & 0.66\
нелегетимный \[non-legitimate\] & 0.64& извне \[external\] & 0.65\
присоединение \[attachment\]& 0.63 & оккупационный \[invasive\] & 0.65\
отсоединение \[separation\] & 0.62 & террор \[terror\] & 0.64\
Identifying Slant of News Stories
=================================
Inspired by the observations from the previous section we next question the power of the word choices to characterize the difference in partisan media at scale, i.e. are the linguistic choices of partisan media substantially different such that we can automatically differentiate them? We address this question by developing a classification machine learning model and conducting an extensive validation of the model over our news datasets.
Methodology
-----------
Using the Word2Vec word representations from the previous section, we train a supervised learning algorithm to find the best indicative words which characterize the language style of a given party. To this end, we first manually classify the Top-30 most popular Russian news agencies[^4] as having a strong pro-government or opposition slant and complement this list by the Top-5 Russian-language sources from the Ukrainian internet segment. We achieve this by manually examining 20 or more articles per each news source for qualitative signs of slant, and by investigating public ownership records and publicly available info about the media outlets, their owners and affiliations. While granular and manual, such an approach can be replicated in other studies using media sources, as public ownership records are usually available and provide enough context for classification, while manual examination for signs of slant is based on a designated set of relevant keywords. We also remove all neutral sources from our analysis, i.e those that have not indicated a particular partisan slant. Finally, we only consider news articles related to the Ukrainian unrest and conflict[^5]. Our classification results in three categories of media outlets: UA, RU-ind, and RU-gov, exhibiting pro-Ukrainian, Russian-independent and Russian-pro-government points of view on the Ukraine unrest and conflict respectively.
Next, we cluster the Word2Vec vectors obtained from the previous section, and use the produced clusters as a feature space to describe the content of each article. In more detail, we apply k-means clustering with $N = 1000$ clusters to the word vectors of the combined News-UA and News-RU corpus. Then, for each article we calculate a $1000$-items long feature vector $X$ that represents the normalized frequencies of occurrences of words from each cluster that occur in the article, and train a function $f(X) \rightarrow \{RU-gov, Ru-ind, UA\}$ to identify the partisan slant of the article (e.g., whether coming from Russian pro-government (Ru-gov), Russian independent (RU-ind) or Ukrainian (UA) news source[^6]).
**word** **$\rho_w^s$** **$\bar{\rho_w^s}$** **News source**
------------------------- ---------------- ---------------------- -----------------
ъ \[Ъ\] 0.38 1.0 Ъ-Kommersant
m24.ru \[m24.ru\] 0.30 1.00 Moscow 24
известиям \[izvestiam\] 0.31 0.93 Izvestia
господин \[gentlemen\] 0.29 0.79 Ъ-Kommersant
подробнее \[more\] 0.36 0.69 Ъ-Kommersant
рбк \[RBC\] 0.27 0.68 RBC
: **Exemplar markers of individual news sources** as identified by the words with the highest relative frequencies $\bar{\rho_w^s}$ across all words and news sources.[]{data-label="tab:source-bias"}
Reducing news source bias
-------------------------
One crucial aspect to account for in the proposed approach is its ability to learn linguistic patterns that generalise across all news sources of a given partisan slant, rather than markers of individual news outlets. Since the training data in the above method is sourced from a selection of few news sources, without a generalisable approach, the classifier could simply learn to label the partisan slant of an article based on unique words, or *news source markers*, that are specific to a particular biased news source.
For example, it is common that the name of the news source or its correspondents are explicitly mentioned in the byline or text of the article, making it easily distinguishable among other texts. Supposing the training data contains a biased Russian pro-government news source $B$ whose name appears in every article from $B$, we might learn a model that the word $B$ is indicative of a Pro Russian-government partisan slant. Although this is useful to classify other articles from $B$, it does not help identify other pro-russian news sources or articles. We therefore need to adapt the method to learn labels that generalise to all news sources by ignoring news source markers.
### Description of the problem.
To show that news source markers are indeed widespread among the news articles in our dataset, we measure the relative frequencies of words appearing in articles from each individual news source. More formally, we define the frequency $\rho_w^s = \frac{N_w^s}{N_s}$ of word $w$ in news source $s$ as a share of all articles $N^s$ from news source $s$ in which word $w$ appears at least once and compare it to the sum[^7] $\sum_{s \in S}{\rho_w^s}$ of the observed frequencies of $w$ across all news sources $s \in S$, i.e., we define the ratio $\bar{\rho_w^s} = \frac{\rho_w^s}{\sum_{s \in S}{\rho_w^s}}$ to identify words which are highly unique to particular news sources.
The top news source markers are extracted as the words with the highest ratio $\bar{\rho_w^s}$ across all words and all news sources. Table \[tab:source-bias\] shows the top few markers. As expected, we observe that articles from some of the news sources - such as Ъ-Kommersant, Moscow 24, Izvestia, and RBC - contain very vivid word markers of that news source. For instance, the words “Ъ” and “m24.ru” used as abbreviations of the Ъ-Kommersant and Moscow 24 news papers appear only within the news articles originating from these two sources. More interestingly, we observe a very high relative frequency of mentioning other general words such as “господин \[gentlemen\]” and “подробнее \[more\]” in articles originating from Ъ-Kommersant, suggesting that there might be other word markers – beyond just the obvious names of newspapers – which indicate the writing style of an individual news source.
[ |l | c | c | c |]{} **Metric** &
----------------
**RU-gov** vs.
**RU-ind**
----------------
: **Predicting partisan slant in news articles.** The results of a supervised machine learning experiment to identify whether news articles were published by either a Ukrainian (UA), a Russian pro-government (RU-gov) or Russian independent (RU-ind) news agency. []{data-label="tab:articles-prediction-performance"}
&
----------------
**RU-gov** vs.
**UA**
----------------
: **Predicting partisan slant in news articles.** The results of a supervised machine learning experiment to identify whether news articles were published by either a Ukrainian (UA), a Russian pro-government (RU-gov) or Russian independent (RU-ind) news agency. []{data-label="tab:articles-prediction-performance"}
&
-----------------------
**RU-gov** vs.
**RU-ind** vs. **UA**
-----------------------
: **Predicting partisan slant in news articles.** The results of a supervised machine learning experiment to identify whether news articles were published by either a Ukrainian (UA), a Russian pro-government (RU-gov) or Russian independent (RU-ind) news agency. []{data-label="tab:articles-prediction-performance"}
\
Precision & 0.66 & 0.78 & 0.57\
Recall & 0.65 & 0.77 & 0.60\
Accuracy & 0.65 & 0.77 & 0.60\
### Suggested solution.
To eliminate the aforementioned news source bias in our prediction model we develop the following approach[^8]. We use Random Forest classifiers known for a good performance on modeling high dimensional data and modify the underlying mechanism for constructing individual decision trees. By default, the trees of the Random Forest algorithm are constructed via a greedy search for the optimal split of the training data $D$ which minimizes the entropy of the label classes (i.e., parties in the conflict), i.e.,
$$\begin{aligned}
\underset{split}{\text{minimize }} H_t(D_{left}) + H_t(D_{right})\end{aligned}$$
where the entropy $H_t(D) = - \sum_{t \in T}{\rho_t^D log(\rho^D_t)}$ is defined on the label spaces $t \in T$ in the left $D_{left}$ and the right $D_{right}$ branches of the split ($\rho^D_t$ indicates the share of instances of class $t$ in the dataset $D$). In principle, an optimal split by this definition may be found around the word markers specific to individual news outlets (e.g., the one from Table \[tab:source-bias\]). To penalize this unwanted behavior of the algorithm we introduce the entropy of a news source $s$ as $H_s(D) = - \sum_{s \in S}{\rho_s^D log(\rho^D_s)}$ which - unlike the entropy defined on labels $H_t(D)$ - characterizes the purity of the split in terms of news sources $s$ rather than parties $t$ (i.e., $\rho^D_s$ indicates the share of instances from news source $s$ in the dataset $D$). Intuitively, we aim for a split that discriminates by the party $t$ but not by the news source $s$ and, so, we aim to find the split that minimizes the entropy $H_p(D)$ while preserving a high entropy of individual news sources within the party $H_s(D)$, i.e.:
$$\begin{aligned}
\underset{split}{\text{minimize }} H_t(D_{left}) + H_t(D_{right}) - \alpha (H_s(D_{left}) - H_s(D_{right}))
\label{eq:entropy}\end{aligned}$$
where $\alpha$ is a constant controlling the effect of the proposed adjustment[^9].
Cross-source validation
-----------------------
In summary, we represent each article as a 1000-long feature vector $X$ based on relative word frequencies corresponding to N=1000 clusters induced by a Word2Vec representation of the entire news corpus. We then learn a function $f(X) \rightarrow \{RU-gov, Ru-ind, UA\}$ that labels the partisan slant (Russian pro-government (Ru-gov), Russian independent (RU-ind) or Ukrainian (UA) news source), whilst at the same time ensuring (using Eq. \[eq:entropy\]) that the model $f(\cdot)$ generalises beyond learning to distinguish markers specific to individual sources.
To properly validate the proposed approach we conduct a *cross-source validation* as follows. From the news agencies labeled in the previous step we select all with at least 329 articles resulting in a dataset of five agencies from each party. The number of articles from each news agency is balanced by down-sampling the over-represented agencies. We further conduct a five-fold cross validation such that at each step we pick four news agencies from each class to train the classifier and use the remaining one for testing. Since we test the model over a news agency which has not been used for training we are able to control for over-fitting to the writing style and markers of specific outlets.
In [Table \[tab:articles-prediction-performance\]]{} we report the average values of accuracy, precision, and recall of the proposed cross-source validation. The results in the table indicate a strong prediction performance of the algorithm (accuracy of $77$%) in classifying the content of the news articles as coming from Russian pro-government or Ukrainian news sources. This result confirms a sharp difference in the linguistic choices characteristic for the content of Russian and Ukrainian news articles as observed in Table \[tab:word2vec-examples\] of the previous section. More interestingly, the accuracy is also high (i.e., 66%) for the more difficult problem of distinguishing between Russian pro-government and Russian independent news sources which often shared a common view on individual episodes of the conflict (e.g., the annexation of Crimea). For the more general problem of discriminating between all three groups of news agencies, the algorithm is able to achieve an accuracy of 60%. Note that this is significantly better than a naïve baseline of randomly guessing between the three classes (with expected accuracy of 33%) signifying the presence of a sharp partisan slant in traditional media.
Understanding partisan slant in Twitter
=======================================
Having studied the difference in linguistic choices characteristic for news agencies during the Ukrainian conflict we now switch to the analysis of the related discourse in social media. The focus of our analysis is on understanding the interrelation between the level of exposure to different news sources among Twitter users and the linguistic choices in their posts.
Methodology
-----------
To analyze the level of exposure to various news sources among Twitter users, we rely on an established approach in the literature [@an2014sharing; @boutet2013s] and look at the news sharing patterns. To this end, we identify $248$K Russian-language tweets which retweet or mention articles from one of the $Y = 22$ most popular news agencies classified in one of the three groups in our dataset (e.g., whether coming from Russian pro-government (RU-gov), Russian independent (RU-ind) or Ukrainian (UA) news sources). Next, we measure the user focus on a partisan media and a specific news agency by computing the share of the articles he/she shared from his/her most preferred agency/party, correspondingly. We note that a similar approach has been previously used to analyze geographic bias of content access in social media [@brodersen2012youtube].
More formally, for each user, we measure the number of times $n_y$ she/he shared an article from a news source $y \in Y$ and calculate the user’s *news focus* as the share of all times he/she has shared any news article from any news source in $Y$, i.e., $\beta = max_{y \in Y}\left(\frac{n_{y}}{\sum_{y \in Y}{n_{y}}}\right)$. The larger $\beta$ is, the larger the fraction of the user’s shares that come from a single source. Similarly, we measure the user’s *party focus* $\beta_{party}$ as the fraction of news articles shared from one of the three “parties”: Russian pro-government (RU-gov), Russian independent (RU-ind) or Ukrainian (UA) news sources. Finally, we measure the *diversity of news sources* with which the user expressed alignment by computing the cardinality of the subset $Y_{n_y > 0}$ of all sources in $Y$ from which a user has shared at least one article, i.e., $\gamma = |Y_{n_y > 0}|$.
Patterns of news sharing
------------------------
The results of the analysis are presented in [Fig. \[ref:fig-twitter-partisanship\]]{}. Firstly, we note that for the majority of users, less than half of all their shares come from a single news source ([Fig. \[ref:fig-twitter-partisanship\]]{}, left) and that the majority of users express alignment with more than $6$ news sources ([Fig. \[ref:fig-twitter-partisanship\]]{}, middle). At the same time, we note that although more active users tend to focus more of their shares on a single news source, they also (occasionally) share a bigger number of news sources (as indicated by the higher diversity values for the users with more than 50 and 100 tweets in [Fig. \[ref:fig-twitter-partisanship\]]{}.middle).
However, this diversity of news sources is not seen when we look at the more coarse-grained picture at the level of ’parties’ ([Fig. \[ref:fig-twitter-partisanship\]]{}, right): users’ shares tend to be heavily focused on just one party – on average, more than $85$% ($90$% for the very active users) of the shares of a user are for news sources in alignment with that user’s main ‘focus’ party. In other words, although users exhibit sophisticated behaviours such as a relatively high level of diversity in sharing from multiple news sources, most of these sources have a single partisan alignment, whether Independent Russian, Pro-Russian, or Pro-Ukrainian Government. Furthermore, both the variety of news sources and the partisan focus increase as user activity levels (number of tweets made) increase.
Predicting political slant from Twitter posts
---------------------------------------------
Next, we attempt to draw the link between news exposure among Twitter users - as inferred by the news articles they share - and the language they use in their posts. From the results of the analysis in the previous section, we build on the fact that the vast majority of users have a strong partisan focus in the article they share and use that as a label of their political slant. To understand whether and to which extent the language choices in Twitter posts can indicate the political slant of the users we formulate a machine learning classifier to infer the latter from the former and validate it over our Twitter dataset.
To this end, we focus on the $5.9$K users with at least $5$ tweets, at least $3$ of which contain news sources and balance the dataset by choosing an equal number of users from each party. To predict political slant of a Twitter user we formulate a supervised machine learning problem where we model language in a user’s tweets with a feature vector $X$, using a methodology similar to the one described in the previous section, and train binary classifiers $f(X) \rightarrow \{RU-gov, RU-ind\}$ and $f(X) \rightarrow \{RU-gov, UA\}$ to identify whether a user is predominantly exposed to Russian pro-government (RU-gov), Russian independent (RU-ind) or Ukrainian (UA) news sources. Note that we remove all tweets that contain headlines of news and all retweets when constructing the language model of a user $X$. This ensures that we concentrate on the lingual choices in the tweets originating from the user, rather than the messages/sources that he/she (re)tweets. Also, as in the previous section, we only consider Russian-language tweets from Russian-speaking Twitter users, to ensure a fair comparison across all sides of the conflict. However, note that a large number of Ukrainian tweets are also in Russian, and our dataset contains representation from all three parties.
[ |l | c | c | c |]{} **Metric** &
----------------
**RU-gov** vs.
**RU-ind**
----------------
: **Predicting partisan slant in Twitter.** The results of a supervised machine learning experiment to identify partisan slant among Russian-speaking Twitter users during the Ukrainian conflict.[]{data-label="tab:twitter-prediction-performance"}
&
----------------
**RU-gov** vs.
**UA**
----------------
: **Predicting partisan slant in Twitter.** The results of a supervised machine learning experiment to identify partisan slant among Russian-speaking Twitter users during the Ukrainian conflict.[]{data-label="tab:twitter-prediction-performance"}
&
-----------------------
**RU-gov** vs.
**RU-ind** vs. **UA**
-----------------------
: **Predicting partisan slant in Twitter.** The results of a supervised machine learning experiment to identify partisan slant among Russian-speaking Twitter users during the Ukrainian conflict.[]{data-label="tab:twitter-prediction-performance"}
\
Precision & 0.66 & 0.71 & 0.52\
Recall & 0.66 & 0.70 & 0.52\
Accuracy & 0.66 & 0.70 & 0.52\
[Table \[tab:twitter-prediction-performance\]]{} presents the averaged results of the 10-fold cross-validation of the proposed model. We note that the model has good performance, achieving an average accuracy of $70$% and $66$% in distinguishing between the users with RU-gov vs. UA slant and between the users with RU-gov vs. RU-ind slant, correspondingly. Comparing these results with the results for predicting the slant of news articles (Table \[tab:articles-prediction-performance\]), we note that the performance of *RU-gov vs. RU-ind* classifier is comparable between the two cases whereas the performance of *RU-gov vs. UA* classifier as well as the three-class classifier (accuracy of 52% in comparison to expected 33% for a random baseline) is slightly lower for the Twitter case. This can be probably attributed to the fact that the size of the text piece in an average news article is larger than in a collection of ten 140-character-long posts collected for a median Twitter user in our training and testing sets and, so, inferring political leaning from Twitter posts seems to be a harder problem than it is for news articles.
Conclusions and Discussion
==========================
In this study we investigated the linguistic indicators of partisan confrontation in mainstream and social media during times of political upheaval by analyzing a dataset of more than 40M tweets and more than 4M news articles from the mass protests in Ukraine during 2013-2014 — known as “Euromaidan” — and the post-Euromaidan conflict between Russian, pro-Russian and Ukrainian forces in eastern Ukraine and Crimea. We designed a natural language processing algorithm to analyze at scale the linguistic markers which point to a particular political leaning in online media and showed that partisan slant in news articles can be automatically inferred with an accuracy of $60$–77%. This difference in language in traditional news media is reflected in the word choices made by the supporters of similar partisan bent on Twitter — those who tweet news sources of a particular slant can be identified with an up to $70$% accuracy based on their lingual choices in tweets other than the retweets of particular news sources. Our results have two implications:
First, our results contribute to the debate on the role of lingual choices in traditional and social media in fostering political frames and partisan discourse during political crises, and confirm that both traditional news sources and users on social media are identifiably partisan. It would be interesting to conduct a more general study into whether the reinforcing partisan nature of the discourse and the political divisions we observe arise from a general lack of empathy and trust in conflict situations, or whether this is specific to the Ukraine conflict.
Second, the results reveal the extent to which partisan rhetoric and political leanings can be automatically inferred from lingual choices, which has implications for the use of social media as a safe platform for free speech in dangerous conflicts. Furthermore, we are able to infer all this with only a “coarse-grained” approach: The party labels (i.e., pro-Ukrainian, Russian pro-government or Russian independent) are assigned to the news articles (or social media profiles) based on the polarity of the news agencies they originate from/retweet. The advantage of this approach is that it requires minimal manual efforts — it only requires to label the political slant of a number of news agencies (such as the 15 considered in this paper). However, the polarity and partisan rhetoric of articles may also vary between different topics and authors within a single news source, which we do not directly account for. We partially address this concern in the current paper by focusing only on the articles related to the Ukrainian events which are known for highly polarized rhetoric in both Russian and Ukrainian online media [@szostek2014media; @zhukov2016reporting]. Although we settled on the coarse-grained approach as a proof-of-concept model, it is clear that a more fine-grained approach could allow for greater accuracy in identifying partisan tweets and political leanings of users and news media.
Acknowledgements {#acknowledgements .unnumbered}
================
The ‘A Shared Space and A Space for Sharing’ project (Grant No. ES/M00354X/1) is one of several funded through the EMoTICON network, which is funded through the following cross-council programmes: Partnership for Conflict, Crime and Security Research (led by the Economic and Social Research Council (ESRC)), Connected Communities (led by the Arts and Humanities Research Council (AHRC)), Digital Economy (led by the Engineering and Physical Sciences Research Council (EPSRC)). We would also like to thank the developers of the Kobzi application for providing the News-UA dataset.
[^1]: We used the Skip-Gram model as it provided more interpretable results.
[^2]: Gensim natural language processing library https://radimrehurek.com/gensim/
[^3]: Stepan Bandera - the leader of the western Ukrainian resistance during World War II, accused by Soviets of collaborating with Nazis.
[^4]: as ranked by the Medialogia rating agency http://goo.gl/JNvx0Y
[^5]: This is achieved through filtering the corpora by the relevant keywords, i.e., “kyiv”, “ukraine”, “donbass”, “maidan”, “crimea”, “luhansk”, “dnr” and “lnr”. Adding a wider set of keywords had little effect on improving the recall of filtering.
[^6]: We refer to each of these three classes as ‘party’ or ’parties’ in the rest of this paper.
[^7]: Note that this is equivalent to using the mean $\frac{\sum_{s \in S}{\rho_w^s}}{|S|}$, since the sum for all words is computed over the same set of sources $S$.
[^8]: We note that a straightforward approach of removing the most prominent news source markers – as measured by the relative frequency introduced in the previous section – has proved to be inefficient for the considered classification problem. In contrast, the method we introduce in the rest of this section provides a more nuanced approach in estimating the relevance of each classification feature.
[^9]: The proposed approach was implemented by adapting the internal implementation of the Random Forest algorithm from the open source scikit-learn libary.
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} |
---
abstract: 'In a simple model of propagation of asymmetric Gaussian beams in nonlinear waveguides, described by a reduction to ordinary differential eqautions of generalized nonlinear Schrödinger equations (GNLSEs) with cubic-quintic (CQ) and saturable (SAT) nonlinearities and a graded-index profile, the beam widths exhibit two different types of beating behavior, with transitions between them. We present an analytic model to explain these phenomena, which originate in a $1:1$ resonance in a 2 degree-of-freedom Hamiltonian system. We show how small oscillations near a fixed point close to $1:1$ resonance in such a system can be approximated using an integrable Hamiltonian and, ultimately, by a single first order differential equation. In particular, the beating transitions can be located from coincidences of roots of a pair of quadratic equations, with coefficients determined (in a highly complex manner) by the internal parameters and initial conditions of the original system. The results of the analytic model agree with numerics of the original system over large parameter ranges, and allow new predictions that can be verified directly. In the CQ case we identify a band of beam energies for which there is only a single beating transition (as opposed to $0$ or $2$) as the eccentricity is increased. In the SAT case we explain the sudden (dis)appearance of beating transitions for certain values of the other parameters as the grade-index is changed.'
author:
- |
David Ianetz$^{1,2,3}$ and Jeremy Schiff$^{1,4}$\
$^1$Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel\
$^2$Holon Institute of Technology (HIT), Holon 5810201, Israel\
$^3$E-mail: [email protected]\
$^4$E-mail: [email protected]
bibliography:
- 'w.bib'
title: Analytic Methods to Find Beating Transitions of Asymmetric Gaussian Beams in GNLS equations
---
Introduction
============
In the sequence of papers [@Ianetz2010PRA_1; @Ianetz2010PRA_2; @Ianetz2013PRA] a variational approach was taken to investigate the propagation of asymmetric (elliptic) Gaussian beams in nonlinear waveguides, with cubic-quintic and saturable nonlinearities and a parabolic graded-index (GRIN) profile, as described by suitable generalized nonlinear Schrödinger equations (GNLSEs). The beam widths in the two transverse directions to the direction of propagation were found to obey a set of ordinary differential equations which can be identified as the equations of motion of a point particle in certain rather complicated, but tractable, 2d potentials. Numerical analysis of these equations revealed “beating” phenomena: in addition to fast oscillations, the beam widths exhibit a (relatively) slow periodic variation. Furthermore, two types of beating were identified: In type I beating the amplitude of oscillation of the beam width in one direction remains greater than the amplitude of oscillation in the other direction, whereas in type II, there is an interchange between the widths in the two transverse directions. The type of beating depends on the parameters of the system and initial eccentricity of the beam. Remarkably, as the initial eccentricty or other parameters are changed, there can be a transition between types, and this transition is characterized by a singularity in the ratio of the periods of the beating and of the fast oscillatory motion.
The intention of the current paper is to provide a theoretical analysis of the beating phenomena and, in particular, to present an approximate analytic method to find the transitions between types. The relevant tool is the analysis of small oscillations in $2$ degree-of-freedom Hamiltonian systems near a fixed point which is close to $1:1$ resonance. The fact that resonance is the source of “beating” or “energy transfer” phenomena in mechanical systems is well known. A classic example can be found in the paper of Breitenburger and Mueller [@BM1] on the elastic pendulum, which the authors describe as a “paradigm of a conservative, autoparametric system with an internal resonance”. The paper [@BM1] has other features in common with our work (such as the use of action-angle variables and the fact that the analytic approximation used is a single elliptic function equation) but it is in the much simpler context of $1:2$ resonance. For other examples of autoparametric resonance see, for example, [@vpapr; @Hallerbook]. The most widely used tool for analysis of systems near resonance is the mutliple time scale method, see for example [@kc; @ManMan] for thorough presentations and many examples. For a typical modern application see [@VS1; @VS2]. However, averaging techniques present an alternative [@SVM], and in the context of Hamiltonian systems, working in action-angle coordinates has substantial advantages [@gensap]. A typical study of a system near resonance will involve looking at the bifurcations of special solutions. In this context much attention has been paid to the definition and identification of [*nonlinear normal modes*]{} — see [@mikhbar] for a review, and [@RPV] for an example in the context of $1:1$ resonance.
The $2$ degree-of-freedom Hamiltonian systems we study have a discrete symmetry, and are approximated by a family of systems with $1:1$ resonance studied nearly $40$ years ago by Verhulst [@Verhulst1]. Verhulst showed the existence of an approximate second integral and used this to study bifurcations of special solutions and their stability. Our work differs from that of Verhulst and other works on $1:1$ resonance in several regards. The bifurcation question we pose depends not only on the internal parameters of the system, but also on the initial conditions. The question is not only one of identifying different types of solutions of the system, but also seeing how the type of solution changes as both the initial condition and internal system parameters are varied. We have not seen a similar study in the highly complex context of $1:1$ resonance. Our methodology uses action-angle variables and canonical transformations (though in an appendix we show how to apply standard two time scale techniques). Unlike in most existing studies, it is necessary to compute the relevant canonical transformation to [*second*]{} order. However, this does not affect the result that once the correct canonical transformation has been applied, the resulting approximating Hamiltonian depends only on a single combination of the angle variables and is integrable. The equations of motion for the integrable Hamiltonian can be reduced to a single first order differential equation, and the rich bifurcation structure of the systems we study can reduces to understanding the bifurcations of roots of a pair of quadratic polynomials, with coefficients that depend (in a complex, nonexplicit manner) on the internal parameters of the systems and the initial conditions. Comparison with numerical results shows our method gives high-quality results in a significant region of parameter space, and allows a variety of interesting new predictions.
The structure of this paper is as follows. In the next section we review the relevant models from nonlinear optics and the collective variable approximation to obtain equations for the propagation of beam widths, and present the main findings of papers [@Ianetz2010PRA_1; @Ianetz2010PRA_2; @Ianetz2013PRA] and some further numerical results. In section 3, we develop our method of integrable approximation for small oscillations in a $2$ degree-of-freedom Hamiltonian system near a fixed point close to $1:1$ resonance. In section 4 we describe the application of this method to the specific systems relevant to beam propagation, confirming existing numerical results and presenting new predictions. In section 5 we summarize and conclude. Appendix A completes some technical details omitted from the main text, and Appendix B describes an alternate method of approximation of the full equations using a two time expansion. This is a more [*ad hoc*]{} approach than the one explained in section 3, but we include it as it is more commonly used in the literature, and for certain values of parameters it gives better results.
Before closing this introduction we mention a number of points concerning the relevance of the work in this paper to optical solitons. We will describe in next section the manner in which we use ordinary differential equations (ODEs) to study the behavior of solutions of GNLSEs. The use of ODEs to study GNLSEs is widespread, see for example [@Skarka2006PRL; @Skarka2008JOA; @Skarka2012PS; @hemalomed; @Skarka2014PRA; @Aleksic2015PRA] In particular, the last two papers use ODE methods in the study of rotating solitons. Our work extends the catalog of interesting bifurcations that can be observed in the context of GNLSEs; for another example; see the papers [@gmc1; @gmc2] for a case of a saddle-loop bifurcation. Finally, we mention that we neglect dispersive terms in the GNLSEs we study. This is justifiable in the context of new optical materials [@Kim2002OL; @Moon2008JNCS; @JU2011OE; @gold; @noptrev] characterized by Kerr coefficients of the order $10^{-11}$–$10^{-12}$ ${\rm cm}^2/{\rm W}$, making the critical intensity for self-focusing small enough that it can be reached using microsecond pulses and possibly even continuous wave (CW) laser beams.
Models, the collective variable approach and numerical results
==============================================================
We consider beam propagation in a nonlinear, graded-index fiber, as described by one of the following GNLSEs: $$\begin{aligned}
2i\psi_z + \psi_{xx} + \psi_{yy} + \left( |\psi|^2 - Q|\psi|^4 - g(x^2+y^2) \right) \psi &=& 0\ ,
\label{glnse1}\\
2i\psi_z + \psi_{xx} + \psi_{yy} + \left( \frac{|\psi|^2}{1+\alpha^2|\psi|^2} - g(x^2+y^2) \right) \psi &=& 0\ .
\label{glnse2} \end{aligned}$$ Here, modulo suitable normalizations [@Kivshar2003book; @Ianetz2013PRA], $\psi$ is the strength of the electric field, $z$ is the longitdinal coordinate, $x,y$ are transverse coordinates, and $Q,\alpha,g$ are parameters. The first equation is the case of cubic-quintic nonlinearity (CQ), the second is the case of saturable nonlinearity (SAT). In the low intensity limit these models are similar, but for higher intensity they display different physical properties. In both cases, the higher order nonlinearity prevents beam collapse associated with the standard Kerr nonlinearity [@Chen2004PRE; @Kivshar2003book]. The term $-g(x^2+y^2)\psi$ reflects the graded-index nature of the fibre, that the refractive index $n$ falls with distance $r$ from the center of the fibre according to the law $n^2 = n_0^2 - Gr^2$; the physical significance of this is explained in [@Sodha1977book; @Ghatak1978book; @Ianetz2013PRA].
The collective variable approximation (CVA), introduced for the study of self-focusing beams in [@Anderson1979PF; @Anderson1979PF2; @Anderson1983PRA], is a variational technique to approximate solutions of nonlinear Schrödinger-type equations which has been used and validated in many different situations [@Malomed2002PO]. The method replaces partial differential equations such as (\[glnse1\]) and (\[glnse2\]) by a system of ordinary differential equations for the coefficients of an ansatz for the full solution. The GNLSEs (\[glnse1\]) and (\[glnse2\]) are variational equations for action principles based on the Lagrangian densities $$\begin{aligned}
{\cal L}_{\rm CQ} &=& i\left( \psi\psi^*_z-\psi^*\psi_z \right) + \left|\psi_{x}\right|^2 + \left|\psi_{y}\right|^2
- \frac12 |\psi|^4 + \frac{Q}{3}|\psi|^6
+ g(x^2+y^2) |\psi|^2 \ , \label{lag1}\\
{\cal L}_{\rm SAT} &=& i\left( \psi\psi^*_z-\psi^*\psi_z \right) + \left|\psi_{x}\right|^2 + \left|\psi_{y}\right|^2
+ \frac{\ln\left(1 + \alpha^2|\psi|^2\right)-\alpha^2|\psi|^2}{\alpha^4}
+ g(x^2+y^2) |\psi|^2\ . \nonumber \\
&& \label{lag2}\end{aligned}$$ We assume $\psi$ takes the form of the [*trial function*]{} $$\label{trial_function}
\psi_{T}(x,y,z) = A(z)
\exp \left(i\phi(z)
-\frac {x^2} {2a_x^2(z)} + ib_x(z)x^2
-\frac {y^2} {2a_y^2(z)} + ib_y(z)y^2
\right)\ ,$$ where $A,\phi,a_x,a_y,b_x,b_y$ are currently undetermined, real functions of only the longitudinal coordinate $z$. This trial function describes an elliptic Gaussian beam with $a_x,a_y$ representing the widths of the beam in the $x,y$ directions. $b_x,b_y$ describe curvatures of the beam wavefront, $A$ is the normalized amplitude of the electric field, and $\phi$ is a longitudinal phase factor. Our choice of a Gaussian shape for the trial function is appropriate because the Gaussian is an exact solution of the linear Schrödinger equation for GRIN waveguides [@Sodha1977book; @Ghatak1978book]. Substituting the trial function in the Lagrangian densities (\[lag1\]),(\[lag2\]) and computing the integrals over the variables $x,y$ we obtain reduced densities for the functions $A,\phi,a_x,a_y,b_x,b_y$. The corresponding Euler-Lagrange equations in the CQ case are $$\begin{aligned}
\dot{A}&=& -(b_x +b_y)A \ , \\
\dot{a}_{x,y} &=& 2a_{x,y}b_{x,y}\ , \\
\dot{b}_{x,y} &=& \frac{1}{2a^4_{x,y}} - 2b^2_{x,y} - \frac{g}{2}
- \frac{A^2}{a^2_{x,y}}\left( \frac{1}{8}- \frac{QA^2}{9} \right)\ , \\
\dot{\phi} &=& - \frac{1}{2a^2_x} - \frac{1}{2a^2_y} + \left( \frac{3}{8} - \frac{5QA^2}{18} \right) A^2
\ . \end{aligned}$$ Here a dot denotes differentiation with respect to $z$. In the SAT case the equations for $A,a_x,a_y$ remain the same, but those for $b_x,b_y,\phi$ are replaced by $$\begin{aligned}
\dot{b}_{x,y} &=& \frac {1} {2a_{x,y}^4} -2b_{x,y}^2 - \frac {g} {2} +
\frac {\ln(1+ \alpha^2 A^2)+\textrm{Li}_{2}(-\alpha^2 A^2)}{2\alpha^4 A^2a_{x,y}^2}\ , \\
\dot{\phi} &=& -\frac{1}{2a_x^2} - \frac{1}{2a_y^2}
+ \frac{\alpha^2A^2-2\ln(1+\alpha^2A^2)- \textrm{Li}_2(-\alpha^2 A^2)}{2\alpha^4A^2}\ .\end{aligned}$$ Here $\textrm{Li}_2(x)=\sum_{k=1}^{\infty} \frac{x^k}{k^2}$ is the Spence or dilogarithm function [@Abramowitz1970book]. For both CQ and SAT cases we observe that $A^2 a_xa_y$ is conserved [@Ianetz2010PRA_1; @Ianetz2010PRA_2; @Ianetz2013PRA], and we write $A^2 a_x a_y = 4 E$ ($4E$ is the beam energy), and use this to eliminate $A(z)$ . Furthermore, $\phi(z)$ evidently plays no role in determining the other functions and can be computed by a simple quadrature once the other functions have been found. Furthermore, it is clear that we can write $b_x$ ($b_y$) in terms of $a_x$ ($a_y$) and its $z$-derivative. Thus we can reduce the system of $6$ equations to a pair of second order equations for $a_x,a_y$. After some more calculation it emerges that the equations are simply the equations of motion $$\ddot{a}_x = -\frac{\partial V}{\partial a_x} \ , \qquad
\ddot{a}_y = -\frac{\partial V}{\partial a_y}
\label{em}$$ for a particle in a potential $V(a_x,a_y)$, where for CQ $$V = V_{CQ} \equiv \frac12 \left( \frac1{a_x^2} + \frac1{a_y^2} \right)
- \frac{E}{a_x a_y}
+ \frac{16 QE^2}{9a_x^2a_y^2}
+ \frac{g}{2} \left( a_x^{2}+a_y^{2} \right)\ ,
\label{VCQ}$$ and for SAT $$V =
V_{SAT} \equiv \frac12 \left( \frac1{a_x^2} + \frac1{a_y^2} \right)
-\frac{a_xa_y}{4E\alpha^4} {\rm Li}_2 \left( -\frac{4E\alpha^2}{a_xa_y} \right)
+ \frac{g}{2} \left( a_x^{2}+a_y^{2} \right)\ .
\label{VSAT}$$
Thus integration of equations (\[em\]) for potentials (\[VCQ\]) and (\[VSAT\]) provides a first approximation to solutions of the GNLSEs (\[glnse1\]) and (\[glnse2\]). Full numerical solutions of GNLSEs have been given in both the SAT [@Yang2002PRE] and the CQ [@Michinel2002PRE] cases with $g=0$. In [@Michinel2002PRE] it was shown that the breathing frequencies found numerically are similar to those obtained by the CVA technique. In [@Yang2002PRE] it was shown that the shape of the beam obtained numerically for a saturable medium remains similar to Gaussian, even for an asymmetric initial condition. However, use of direct numeric methods to give an overall picture of the behavior of a GNLSE, as a function of all the various parameters, remains a computationally overwhelming task, and having an qualitatively correct analytic or semianalytic model is therefore useful for developing physical insight [@Kivshar2003book; @Agrawal2007book].
Appropriate initial conditions for (\[em\]) are $$a_x(0) = a_0 r \ , \qquad a_y(0) = \frac{a_0}{r} \ , \qquad
\dot{a_x}(0) = \dot{a_y}(0) = 0\ . \label{ic0}$$ The latter two conditions are equivalent to taking $b_x(0)=b_y(0)=0$. Note that both the CQ and the SAT system have a scaling symmetry $$\begin{array}{lllll}
a_x \rightarrow \lambda a_x \ , &
a_y \rightarrow \lambda a_y\ , &
a_0 \rightarrow \lambda a_0 \ , &
r \rightarrow r \ , &
z \rightarrow \lambda^2 z \ ,\\
Q \rightarrow \lambda^2 Q\ , &
\alpha \rightarrow \lambda \alpha\ , &
g \rightarrow \lambda^{-4} g\ , &
E \rightarrow E . &
\end{array}$$ Thus for CQ we do not need to study the dependence of solutions on the $5$ parameters $Q,g,E,a_0,r$, but only on the $4$ scale invariant quantities $Qa_0^{-2}, ga_0^4, E, r$. On occasion we will work with the scale invariant quantity $K_{\rm CQ} = 4QEa_0^{-2}$ instead of the quantity $Qa_0^{-2}$. (For SAT, replace all instances of $Q$ in the previous two sentences with $\alpha^2$, and $K_{\rm SAT}=4\alpha^2Ea_0^{-2}$.) Note that since there is symmetry in both the models between $a_x$ and $a_y$, there is a $r\rightarrow \frac1{r}$ inversion symmetry, and thus we need only study $r\le 1$ or $r\ge 1$.
In the papers [@Ianetz2010PRA_1; @Ianetz2010PRA_2; @Ianetz2013PRA] the ODE systems above were studied numerically. For appropriate choices of the parameters “beating” phenomena were observed: in addition to (relatively) fast “breathing” oscillations, the beam widths exhibit a (relatively) slow periodic variation. Two types of beating were identified: In type I beating, the amplitude of oscillation of the beam width in one direction remains greater than the amplitude of oscillation in the other direction. In type II beating, there is an interchange between the widths in the two transverse directions. This is illustrated in Figure 1, which shows solutions of the CQ system for $E=2.039$, $K_{\rm CQ}=0.71$, $ga_0^4=0.01$, and two choices of $r$: $r=1.14$ gives type I beating, whereas $r=1.16$ gives type II beating.
![\[fig:epsart\] Dependence of the ratio of periods of slow beating and fast oscillatory motion, $L_{\rm beat}/L_{\rm br}$, on the parameter $r^2$ for the CQ model, and for various values of the parameter $ga_0^4$. $E=2.039$ and $K_{\rm CQ}=0.71$ throughout. (1) (dash-dot black) $ga_0^4=0$, (2) (dashed blue) $ga_0^4=0.01$, (3) (solid red) $ga_0^4=0.02$, (4) (dotted green) $ga_0^4=0.03$. Roman numerals indicate the type of beating in case (2), $ga_0^4=0.01$.](Fig_2.pdf){width="70.00000%"}
The type of beating depends on the parameters of the system and, as evident from Figure 1, on the initial eccentricity of the beam. Remarkably, as the initial eccentricty is increased, or as other parmeters are changed, there can be a transition between types. The approach to this transition is characterized by a divergence in the ratio of the periods of the slow beating and of the fast oscillatory motion. In Figure 2 this ratio (determined from numerical simulations) is plotted as a function of $r^2$ for the CQ system, with $E=2.039$, $K_{\rm CQ}=0.71$ and $ga_0^4=0,0.01,0.02,0.03$. (The reason for the choice of the coordinate $r^2$ on the $x$-axis is simply to make the plot clearer.) For $r$ just above $1$ the beating is type II, then there is a transition to type I, and then a second transition back to type II. The dependence on the system parameters of the two critical values of $r$, which we denote collectively by $r_c$, is explored further in Figure 3. In Figure 3a the values of $r_c$ are plotted as a function of $ga_0^4$ for three different values of $K_{\rm CQ}$ and a constant value of $E$; in Figure 3b $r_c$ is plotted as a function of $ga_0^4$ for three different values of $E$ and a constant value of $K_{\rm CQ}$. In general we see that $r_c$ increases as a function of $ga_0^4$ (for fixed $E,K_{\rm CQ}$). From Figure 3b we see that since the (solid) red is above the (dashed) blue is above the (dot-dashed) black, $r_c$ also increases as a function of $E$ (for fixed $ga_0^4,K_{\rm CQ}$). But in Figure 3a we see there is difference between the upper and lower branches of $r_c$. We deduce that the higher value of $r_c$ also increases with $K_{\rm CQ}$ (for fixed $E,ga_0^4$), but the lower value decreases.
We shall see later that for other values of $ga_0^4,K_{\rm CQ},E$ there can be just a single transition or no transitions at all as $r$ is increased from $1$. Transitions between beating types are also observed in the SAT system, again with a complex dependence on the parameters $ga_0^4$, $K_{\rm SAT}$ and $E$. The aim of this paper is to provide an integrable approximation for equations (\[em\]) with potentials (\[VCQ\]) and (\[VSAT\]) which provides a theoretical model to predict where the transitions between types take place.
Small oscillations near $1:1$ resonance
=======================================
In this section we describe a general process of approximation near a $1:1$ resonance for a $2$ degree-of-freedom Hamiltonian system with Hamiltonian $$H = \frac12 \left(p_x^2 + p_y^2 \right) + V(a_x,a_y)\ .
\label{H}$$ Here $a_x,a_y$ are the coordinates, $p_x,p_y$ are the conjugate momenta, and the potential $V$ (which typically will depend on a number of parameters) is symmetric, $V(a_x,a_y)=V(a_y,a_x)$. We assume that for typical values of the parameters the potential has an isolated symmetric minimum (at $a_x=a_y=a_{\rm min}$, say) at which the system is close to $1:1$ resonance. Note that because of the symmetry, $\displaystyle{ \frac{ \partial^2V}{\partial a_x^2} (a_{\rm min}, a_{\rm min})
= \frac{ \partial^2V}{\partial a_y^2} (a_{\rm min},a_{\rm min})}$. Thus the Hessian matrix of the potential at $(a_{\rm min},a_{\rm min})$ has eigenvectors $\left( \begin{array}{c} 1 \\ \pm 1 \end{array} \right)$ with eigenvalues $\displaystyle{\frac{ \partial^2V}{\partial a_x^2} (a_{\rm min},a_{\rm min})
\pm \frac{ \partial^2V}{\partial a_xa_y} (a_{\rm min},a_{\rm min})}$. The condition for being close to $1:1$ resonance (i.e. equal eigenvalues) is therefore simply $ \frac{ \partial^2V}{\partial a_xa_y} (a_{\rm min},a_{\rm min}) \approx 0 $. For these systems we study orbits with initial conditions as given in (\[ic0\]). The process of approximating such a system with an integrable system has $3$ steps.
The [*first step*]{} is to expand in normal coordinates near the fixed point, retaining only terms up to order $4$ in the potential. Thus we write $$a_x = a_{\rm min} + \frac{\zeta_2+\zeta_1}{\sqrt{2}} \ , \qquad a_y = a_{\rm min} + \frac{\zeta_2-\zeta_1}{\sqrt{2}}$$ and expand to fourth order to obtain $$H_1 = \frac12 \left( p_1^2 + p_2^2 + \omega_1^2 \zeta_1^2 + \omega_2^2 \zeta_2^2 \right) + a_1 \zeta_1^2 \zeta_2 + a_2 \zeta_2^3
+ a_3 \zeta_1^4 + a_4 \zeta_1^2 \zeta_2^2 + a_5 \zeta_2^4
\label{H1}$$ where $p_1,p_2$ are the conjugate momenta to the coordinates $\zeta_1,\zeta_2$, and $\omega_1,\omega_2,a_1,a_2,a_3,a_4,a_5$ are constants that depend on the parameters of the original potential $V$. The Hamiltonian $H_1$ has $\zeta_1 \rightarrow -\zeta_1$ symmetry as a consequence of the symmetry of $H$, and is the general Hamiltonian with this symmetry and a quartic potential. Aspects of the behavior of this Hamiltonian at, or close to, $1:1$ resonance have been studied previously, for example, in [@Verhulst1; @montaldi1; @montaldi2; @pm3; @pm1; @pm2]. The initial conditions for this system, corresponding to (\[ic0\]) are $$\zeta_1(0) = \frac{a_0}{\sqrt{2}} \left(r-\frac1{r} \right)
\ , \qquad \zeta_2(0) = \frac1{\sqrt{2}} \left(a_0\left(r+\frac1{r} \right) - 2 a_{\rm min} \right)
\ , \qquad
{p_1}(0) = {p_2}(0) = 0 \ . \label{ic1}$$ Symmetric solutions correspond to the initial condition $\zeta_1(0)=0$. In regarding $H_1$ as an approximation for $H$ we are neglecting terms of fifth order and above.
The [*second step*]{} is to make the canonical transformation to action-angle coordinates associated with the quadratic part of the Hamiltonian $H_1$, i.e. to substitute $$\begin{array}{ll}
\zeta_1 = \sqrt{\frac{2J_1}{\omega_1}} \cos\theta_1\ , & p_1 = - \sqrt{2J_1\omega_1} \sin\theta_1 \ , \\
\zeta_2 = \sqrt{\frac{2J_2}{\omega_2}} \cos\theta_2\ , & p_2 = - \sqrt{2J_2\omega_2} \sin\theta_2 \ .
\end{array}$$ This gives $$\begin{aligned}
H_2 &=& \omega_1 J_1 + \omega_2 J_2
+ \left( \frac{2J_1a_1}{\omega_1} +\frac{3 J_2 a_2}{\omega_2} \right) \sqrt{\frac{J_2}{2\omega_2}} \cos\theta_2 \nonumber\\
&&
+ \left(\frac{J_2}{2\omega_2}\right)^{3/2} 2a_2\cos 3\theta_2
+ a_1 \frac{J_1}{\omega_1} \sqrt{\frac{J_2}{2\omega_2}}
\left( \cos(2\theta_1-\theta_2) + \cos(2\theta_1+\theta_2) \right) \nonumber\\
&& + 3a_3 \frac{J_1^2}{2\omega_1^2}
+ a_4 \frac{J_1J_2}{\omega_1\omega_2}
+ 3a_5 \frac{J_2^2}{2\omega_2^2}
+ a_4 \frac{J_1J_2}{2\omega_1\omega_2}\left( \cos(2\theta_1-2\theta_2) + \cos(2\theta_1+2\theta_2) \right)
\nonumber \\
&& \left( 2a_3 \frac{J_1^2}{\omega_1^2} + a_4 \frac{J_1J_2}{\omega_1\omega_2} \right)\cos(2\theta_1)
+ a_3 \frac{J_1^2}{2\omega_1^2} \cos(4\theta_1)
+\left(a_4\frac{J_1J_2}{\omega_1\omega_2}+ 2a_5\frac{J_2^2}{\omega_2^2} \right)\cos(2\theta_2) \nonumber\\
&& +a_5\frac{J_2^2}{2\omega_2^2}\cos(4\theta_2) \ . \label{H2} \end{aligned}$$ Here $\theta_1,\theta_2$ are the angle variables, and $J_1,J_2$ the conjugate actions. The initial conditions for the action variables are $$J_1(0) = \frac{a_0^2\omega_1}{4} \left(r-\frac1{r} \right)^2 \ , \qquad
J_2(0) = \frac{\omega_2}{4} \left(a_0\left(r+\frac1{r} \right) - 2 a_{\rm min} \right)^2\ .
\label{Jic}$$ The initial conditions for the angle variables depend on the sign of $\zeta_1(0)$ and $\zeta_2(0)$. If $\zeta_1(0)>0$ ($\zeta_2(0)>0$) then, from (\[ic1\]) we should take $\theta_1(0)=0$ ($\theta_2(0)=0$) and otherwise $\theta_1(0)=\pi$ ($\theta_2(0)=\pi$). Due to the $\zeta_1\rightarrow -\zeta_1$ symmetry of $H_1$ the Hamiltonian $H_2$ has period $\pi$ (and not $2\pi$) as a function of $\theta_1$ and thus the choice of the $\theta_1$ initial condition is irrelevant. The choice of the $\theta_2$ initial condition, however, is important. We are introducing a non-physical discontinuity in the approximation procedure when the sign of $\zeta_2(0)$ changes, i.e. when $r+\frac1{r}=\frac{2a_{\rm min}}{a_0}$. We will see the effects of this later, in our results for the SAT potential.
The [*third step*]{} involves a canonical change of coordinates $(\theta_1,\theta_2,J_1,J_2)\rightarrow (\phi_1,\phi_2,K_1,K_2)$ defined by a generating function of the second type $G_2(\theta_1,\theta_2,K_1,K_2)$ [@Goldstein], chosen to eliminate the nonresonant terms from the Hamiltonian (i.e. all the trigonometric terms of order $||J||^{3/2}$ or $||J||^2$ except the one involving $\cos(2\theta_1-2\theta_2)$.) The full change of coordinates is given by $$\begin{array}{ll}
\phi_1 = {\displaystyle{\frac{\partial G_2}{\partial K_1}}} \ , &
J_1 = {\displaystyle{\frac{\partial G_2}{\partial \theta_1}}} \ , \\[\bigskipamount]
\phi_2 = {\displaystyle{\frac{\partial G_2}{\partial K_2}}} \ , &
J_2 = {\displaystyle{\frac{\partial G_2}{\partial \theta_2}}} \ .
\end{array}
\label{ct}$$ The generating function $G_2$ should be taken in the form $$\begin{aligned}
G_2 &=& K_1\theta_1 + K_2\theta_2 + A_1 \sin\theta_2 + A_2\sin 3\theta_2 + A_3 \sin(2\theta_1-\theta_2)
+ A_4 \sin(2\theta_1+\theta_2) \\
&& + A_5 \sin2\theta_1 + A_6 \sin 4\theta_1 + A_7 \sin2\theta_2 + A_8 \sin 4\theta_2
+ A_9 \sin6\theta_2 + A_{10} \sin (2\theta_1 +2\theta_2)\\
&&
+ A_{11} \sin (2\theta_1 +4\theta_2)
+ A_{12} \sin (4\theta_1 +2\theta_2)
+ A_{13} \sin (2\theta_1 -4\theta_2)
+ A_{14} \sin (4\theta_1 -2\theta_2) \end{aligned}$$ where the coefficients $A_1,\ldots,A_{14}$ are functions of $K_1,K_2$, which are chosen to eliminate the nonresonant trigonometric terms in the Hamiltonian to required order. $A_1,A_2,A_3,A_4$ are of order $||K||^{3/2}$ and $A_5,\ldots,A_{14}$ are of order $||K||^2$. The calculations are long, but straightforward with the help of a symbolic manipulator, and the final Hamiltonian is found to be simply $$H_3 = \omega_1 K_1 + \omega_2 K_2
+ b_1 K_1^2 + b_2 K_1K_2 + b_3 K_2^2
+ (b_4 K_1^2 + b_5 K_1K_2) \cos\left( 2(\phi_1-\phi_2) \right)
\label{H3}$$ where $$\begin{aligned}
b_1 &=& \frac{3a_{{3}}}{2\omega_{1}^{2}}
- \frac{a_{{1}}^{2}\left( 8\omega_{{1}}^{2}-3\omega_{{2}}^{2} \right)}
{4\omega_1^{2}\omega_2^{2} \left( 2\omega_1-\omega_2 \right) \left( 2\omega_1 + \omega_2 \right)}
\nonumber\\
b_2 &=& \frac{a_{{4}}}{\omega_{{1}}\omega_{{2}}} - \frac{3 a_1 a_2}{\omega_1\omega_2^3}
- \frac {2a_{{1}}^2}
{\omega_{{1}}\omega_{{2}} \left( 2\omega_{{1}}-\omega_{{2}} \right) \left( 2\omega_{{1}}+\omega_{{2}} \right) }
\nonumber\\
b_3 &=& \frac{3a_{{5}}}{2\omega_{{2}}^{2}} - \frac{15 a_{{2}}^{2}}{4\omega_{{2}}^{4}}
\label{bs}\\
b_4 &=& \frac { \left( \omega_{{2}}-\omega_{{1}} \right) a_{{1}}^{2} }
{2\omega_{{1}}^{2}\omega_{{2}}^{2} \left( 2\omega_{{1}}-\omega_{{2}} \right) }
\nonumber\\
b_5 &=& \frac {a_{{4}}}{2\omega_{{1}}\omega_{{2}}}
- \frac{a_1a_2(4\omega_1^2-3\omega_1\omega_2-4\omega_2^2)}
{2\omega_{{1}}\omega_{{2}}^{3} \left( 2\omega_{{1}}-\omega_{{2}} \right)\left( 2\omega_{{1}}+\omega_{{2}} \right)}
- \frac{a_1^2}{\omega_1^2\omega_2(2\omega_1-\omega_2)}
\ . \nonumber\end{aligned}$$
[*The Hamiltonian $H_3$ given in (\[H3\]) is an integrable approximation of the original Hamiltonian $H$ given in (\[H\]).*]{} $H_3$ is a [*normal form*]{} for the “natural” Hamiltonian $H_1$ at or near $1:1$ resonance. Note that in the case of exact resonance $\omega_1=\omega_2$ the coefficient $b_4$ vanishes. Also close to resonance, the corresponding term in $H_3$ is of lower order than the other terms, and in [@pm3; @pm1; @pm2] it is omitted. However, we choose to retain it to avoid any assumption on the relative orders of magnitude of $|\omega_1 - \omega_2|$ and $||K||$. The integrability of $H_3$ is evident, as it only depends on the modified angle variables $\phi_1,\phi_2$ through the combination $\phi_1-\phi_2$. As a consequence the quantity $K_1+K_2$ is conserved, in addition to the Hamiltonian itself. We denote the value of the Hamiltonian by ${\cal E}$ and the value of $K_1+K_2$ by $P$ (these should be computed from the system parameters and initial conditions). The full equations of motion are $$\begin{aligned}
\dot{\phi}_1 = \frac{\partial H_3}{\partial K_1}
&=& \omega_1 + 2 b_1 K_1 + b_2 K_2 + (2 b_4 K_1 + b_5 K_2) \cos(2(\phi_1-\phi_2)) \ , \label{fe1}\\
\dot{\phi}_2 = \frac{\partial H_3}{\partial K_2}
&=& \omega_2 + b_2 K_1 + 2 b_3 K_2 + b_5 K_1 \cos(2(\phi_1-\phi_2)) \ , \label{fe2}\\
\dot{K}_1 = -\frac{\partial H_3}{\partial \phi_1}
&=& 2 K_1 (b_4 K_1 + b_5 K_2) \sin (2(\phi_1-\phi_2)) \ , \label{fe3}\\
\dot{K}_2 = -\frac{\partial H_3}{\partial \phi_2}
&=& -2 K_1 (b_4 K_1 + b_5 K_2) \sin (2(\phi_1-\phi_2)) \ . \label{fe4}\end{aligned}$$ Using the two conservation laws it is possible to eliminate $K_2$ and $\phi_1-\phi_2$ from the $K_1$ equation of motion to get a single equation for $K_1$: $$\begin{aligned}
\dot{K}_1^2 &=& - 4
((b_1-b_2+b_3+b_4-b_5)K_1^2+((b_2-2b_3+b_5)P-\omega_2+\omega_1)K_1+b_3P^2+\omega_2P-{\cal E}) \nonumber \\
&&
((b_1-b_2+b_3-b_4+b_5)K_1^2+((b_2-2b_3-b_5)P-\omega_2+\omega_1)K_1+b_3P^2+\omega_2P-{\cal E})\ .\nonumber \\
&& \label{K1e}\end{aligned}$$
Equation (\[K1e\]) is a central result of this paper. To solve (\[K1e\]) it is necessary to translate the initial conditions for $J_1,J_2,\theta_1,\theta_2$ into initial conditions for $K_1,K_2$. This step requires details of the canonical tranformation. Due to their length, the full equations determining the initial values of $K_1,K_2$ are given in Appendix A (equations (\[Keq1\])-(\[Keq2\])). Note there are two cases depending on whether $\theta_2(0)$ is $0$ or $\pi$. Note also that [*there is no guarantee that these equations will have a solution with real, positive $K_1,K_2$*]{}. In the case of the SAT system, for a certain range of parameter values we have experienced numerical problems with the solution of (\[Keq1\])-(\[Keq2\]), specifically for initial values of $J_2$ close to zero, close to the jump from $\theta_2=0$ to $\theta_2=\pi$. However, typically there are values of $K_1(0),K_2(0)$ close to the given values of $J_1(0),J_2(0)$.
Once the initial values of $K_1,K_2$ have been computed, the values of the constants ${\cal E}$ and $P$ can be found and equation (\[K1e\]) can be solved. The right hand side of (\[K1e\]) is the product of two quadratic factors in $K_1$, with up to $4$ real roots, and typical solutions will be oscillatory between two roots. When there is a [*double root*]{} then there is the possibility of the period of the oscillation becoming [*infinite*]{}, marking a bifurcation in the solution. There are two ways that a double root can occur, by the vanishing of the discriminant of one of the quadratic factors, or by one of the roots of the first factor coinciding with one of the roots of the second. The discriminants of the quadratic factors are $$\begin{aligned}
\Delta_1 &=&
\left( (b_2+b_5)^2 - 4b_3 (b_1 + b_4) \right) P^2
+ 2 \left( (b_2 - 2 b_3 + b_5) \omega_1 + (-2b_1 + b_2 - 2 b_4 + b_5) \omega_2 \right) P \nonumber \\
&& + 4 (b_1 - b_2 + b_3 + b_4 - b_5){\cal E} + (\omega_1 - \omega_2) ^ 2 \ , \label{Del1} \\
\Delta_2 &=&
\left( (b_2-b_5)^2 -4b_3 (b_1 - b_4) \right) P^2
+ 2 \left( (b_2 - 2 b_3 - b_5) \omega_1 + (-2b_1 + b_2 + 2 b_4 - b_5) \omega_2 \right) P \nonumber \\
&& + 4 (b_1 - b_2 + b_3 - b_4 + b_5){\cal E} + (\omega_1 - \omega_2) ^ 2 \ . \label{Del2}\end{aligned}$$ A simple algebraic manipulation shows that the first factor and second factor have coincident roots if either $\Delta_3=0$ or $\Delta_4=0$, where $$\begin{aligned}
\Delta_3 &=& b_3 P^2 + \omega_2 P - {\cal E}\ , \label{Del3} \\
\Delta_4 &=& \frac{ b_1 b_5^2 - b_2 b_4 b_5 + b_3 b_4^2}{(b_4-b_5)^2} P^2 +
\frac{b_4\omega_2-b_5\omega_1}{b_4-b_5} P - {\cal E} \ . \label{Del4}\end{aligned}$$ From (\[K1e\]), we see that the first case occurs when the repeated root is at $K_1=0$.
It should be emphasized that the occurence of a double root on the RHS of (\[K1e\]) is a [*necessary*]{} condition for a bifurcation of the solution (giving rise to a transition between types) but not a [*sufficient*]{} condition. For example, if the solution is describing an oscillation on the interval between two adjacent roots of the RHS, and the two other roots outside this interval merge, this will have no effect on the solution. We illustrate, in Figure 4, with two concrete examples of equation (\[K1e\]) emerging from the CQ system described in Section 2. In both cases $Qa_0^{-2}=0.077$ and $ga_0^4=0$; in the first case $r=1.01$ and in the second case $r=1.045$. In both cases we plot the roots of the RHS as a function of the single remaining parameter $E$. (The choice to plot the roots for fixed values of $Qa_0^{-2}$, $ga_0^4$ and $r$ and to vary $E$ is just an illustration; we could just as easilly vary any of the other parameters or a combination thereof.) In the first case there are $4$ points $P_1,P_2,P_3,P_4$ at which there are double roots; however, transitions only occur at the two points $P_1,P_4$ (marked in Figure 4 with large dots). In the second case there are $5$ points $P_1,P_2,P_3,P_4,P_5$ at which there are double roots; however, transitions only occur at the two points $P_2,P_4$. In both cases, the first transition is from type II to type I, and the second transition is from type I to type II, as indicated by Roman numerals on the plot.
The theoretical explanation of this is as follows. In the first case, $r=1.01$, there are $4$ values of $E$ for which there is a double root. The points labelled $P_3$ and $P_4$ on the diagram are associated with the vanishing of the discriminant $\Delta_2$; the point labelled $P_1$ is a double root at $0$, associated with the condition $\Delta_3=0$, and the point labelled $P_2$ is associated with the vanishing of the discriminant $\Delta_1$. The motion takes place between the root that is at $K_1\approx 0.00015$ and an adjacent root: for values of $E$ below $P_2$ the adjacent root is below, for values of $E$ above $P_2$ the adjacent root is above. Thus the double root at $P_1$ indicates a value of $E$ for which there is a bifurcation, and the period of oscillation diverges. The double root at $P_2$ is a special solution for which $K_1$ and $K_2$ are constant (looking at (\[fe3\])-(\[fe4\]) it can be seen that there are $3$ kinds of solution of this type, each corresponding to vanishing of one of the three factors on the RHS of this equation; these are related to the [*nonlinear normal modes*]{} of the system [@mikhbar; @pm1; @pm2]). The point $P_2$ does not, however, give rise to a transition in behavior of the CQ system; the beating period diverges there, but the type does not change. The double root at $P_3$ also does not mark a transition. This is precisely the case described above, in which the oscillation is on the interval between 2 roots, and the other two roots outside this interval merge. The point $P_4$, however, does mark a second transition, from type I beating back to type II.
Proceeding to the second example in Figure 4, there are now $5$ cases of a double root. $P_2$ and $P_4$ are associated with the vanishing of the discriminant $\Delta_2$, $P_3$ with the vanishing of the discriminant $\Delta_1$. $P_1$ is the case of a double root at zero associated with the condition $\Delta_3=0$, and $P_5$ is associated with the final possibility, $\Delta_4=0$. There are however only $2$ transitions, associated with the points $P_2$ and $P_4$, for similar reasons to the case described in the previous paragraph.
In this section we have explained how small oscillations of the original Hamiltonian (\[H\]) near its fixed point and near (symmetric) $1:1$ resonance can be approximated using the integrable Hamiltonian (\[H3\]) and the single differential equation (\[K1e\]). We have arrived at a simple analytic approximation for beating transitions, viz. [*a necessary condition for a transition between type I and type II beating is the vanishing of one of the four quantities $\Delta_1,\Delta_2,\Delta_3,\Delta_4$ given in (\[Del1\]),(\[Del2\]),(\[Del3\]),(\[Del4\]).*]{} It should be emphasized that this far from trivializes the original problem. There is substantial complexity hidden in the relationship between parameters and initial conditions of the original Hamiltonian and those of the integrable Hamiltonian. Also determining which of the vanishing conditions gives a physical transition can be subtle. In Section 4 we apply the approximation to the CQ and SAT models from Section 2 and validate its predictions against numerical results.
Application to the models
=========================
The CQ Model
------------
The potential of the CQ model, given by (\[VCQ\]), has an isolated minimum when $$a_x = a_y = a_{\rm min} \equiv \frac{4 E\sqrt{2Q}}{3\sqrt{E-1}} C_0$$ where $C_0>0$ is a solution of the equation $$1 - C_0^2 = \frac{1024E^4Q^2g}{81(E-1)^3} C_0^6 \ .$$ The $1:1$ resonance condition is $E= E_{\rm res}$ where $$% E_{\rm res}C_0^2 = 2(E_{\rm res}-1) \quad {\rm or} \quad E_{\rm res} = 2 - \frac{8192 E_{\rm res}^2Q^2g}{81}\ .
E_{\rm res} = \frac{4}{1 + \sqrt{1 + \frac{65536 Q^2g}{81}}} \label{rescon}$$ In the case of zero grade index, $g=0$, we have $C_0=1$ and the resonance condition is simply $E=2$. The model is valid if the parameters $E,Q,g$ are chosen so that $E\approx E_{\rm res}$ and the initial conditions (see (\[ic0\])) satisfy $a_0\approx a_{\rm min}$ and $r\approx 1$.
The relevant parameters for the quartic Hamiltonian (\[H1\]) are $$\begin{aligned}
\omega_1^2 &=& \frac{81}{512}\ \frac{(E-1)^2((2-E)C_0^2+E-1)}{E^4Q^2C_0^6} \nonumber\\
\omega_2^2 &=& \frac{81}{512}\ \frac{(E-1)^3(3-2C_0^2)}{E^4Q^2C_0^6} \nonumber\\
a_1 &=& \frac{243}{8192}\
\frac{ (E-1) ^{5/2} \left( 2(E-3) C_0^{2}-3(E-1) \right)}{ {Q}^{5/2}C_0^{7}{E}^{5} } \nonumber \\
a_2 &=& \frac{243}{8192}\
\frac { ( E-1) ^{7/2} (2C_0^{2}-5) }{ {Q}^{5/2}C_0^{7}{E}^{5}} \label{findas} \\
a_3 &=& \frac{729}{262144}\
\frac { (E-1) ^{3} ( 2(5-E)C_0^{2}+3(E-1) ) }{ C_0^{8}{E}^{6}{Q}^{3}} \nonumber\\
a_4 &=& \frac {729}{131072}\
\frac{(E-1)^3\left( 10(3-E)C_0^{2}+ 21(E-1) \right) }{C_0^{8}{E}^{6} {Q}^{3} } \nonumber\\
a_5 &=& \frac {3645}{262144}\
\frac{(E-1)^4( 7-2 C_0^{2} )}{C_0^{8}{E}^{6}{Q}^{3}} \nonumber\end{aligned}$$
The detailed recipe for checking whether a given set of parameters and initial conditions $E,Q,g,a_0,r$ might give rise to a transition is as follows:
1. Compute the coefficients $\omega_1^2,\omega_2^2,a_1,a_2,a_3,a_4,a_5$ using (\[findas\]). This is the only stage of the recipe that is model dependent. Compute the coefficients $b_1,b_2,b_3,b_4,b_5$ from (\[bs\]).
2. Compute the initial conditions $J_1(0),J_2(0)$ from (\[Jic\]) and $\theta_1(0),\theta_2(0)$ from the comments following (\[Jic\]). In the case of CQ, all the parameter values which we used gave $\theta_1(0)=0$ (we took $r>1$ throughout) and $\theta_2(0)=\pi$.
3. Compute the initial conditions $K_1(0),K_2(0)$ using (\[Keq1\])-(\[Keq2\]). This is the only stage of the recipe that is not completely explicit, and involves solving two equations in two variables. If no real solution can be found, the method fails. A suitable initial guess for the solution is $K_1(0)\approx J_1(0)$ and $K_2(0)\approx J_2(0)$.
4. Determine the value of ${\cal E}$, the constant value of the Hamiltonian $H_3$ using (\[H3\]), taking $\cos(2(\phi_1-\phi_2))=1$. Determine the value of $P=K_1+K_2$.
5. Compute $\Delta_1,\Delta_2,\Delta_3,\Delta_4$ from (\[Del1\]),(\[Del2\]),(\[Del3\]),(\[Del4\]). Values of $E,Q,g,a_0,r$ for which any of these quantities vanish are candidates for transitions.
Figure 5 displays results. Figure 5a shows numeric values and candidate analytic approximations of $r_c$ as a function of $E$ for $Qa_0^{-2}=0.077$ and $ga_0^4=0.01$. The dots denote numeric values of transitions in the original system. The solid curves show candidate analytic approximations of 3 distinct types: (1) (black) values for which $\Delta_2=0$ (a closed loop with a cusp on the axis at $r=1$), (2) (green) values for which $\Delta_3=0$ (a simple open curve) and (3) (red) values for which $\Delta_4=0$ (two crossing open curves). For the values of $Qa_0^{-2}$ and $ga_0^4$ specified, it seems there are two branches of parameter values for which there are transitions. We denote the lower branch (on the plot) by $r_{c,1}(E)$, which exists for $E$ greater than a certain value which we denote by $E_{c,1}$, and the upper branch by $r_{c,2}(E)$, which exists for $E$ greater than a certain value which we denote by $E_{c,2}$, with $E_{c,2} \approx 1.975 < E_{c,1} \approx 1.977$. On the lower branch, as $E$ increases from $E_{c,1}$, $r_{c,1}(E)$ at first follows the approximation $\Delta_2=0$, until a triple point at which the curves $\Delta_2=0$ and $\Delta_4=0$ intersect. As $E$ increases further, $r_{c,1}(E)$ follows the approximation $\Delta_4=0$. Surprisingly, this approximation stays reasonably accurate for the full range shown on the figure, even though $r_{c,1}(E)$ rises to approximately $1.12$. On the upper branch, as $E$ increases from $E_{c,2}$, $r_{c,2}(E)$ at first follows the approximation $\Delta_3=0$, until a triple point at which the curves $\Delta_2=0$ and $\Delta_3=0$ intersect. As $E$ increases further, $r_{c,2}(E)$ follows the approximation $\Delta_2=0$. However, the quality of this approximation rapidly decreases as $E$ and $r_{c,2}(E)$ increase further, with the discrepancy already visible on the plot for $r_c\approx 1.06$.
Figure 5b shows numeric values and the [*correct*]{} analytic approximation (made up of pieces of the curves $\Delta_2=0$, $\Delta_3=0$ and $\Delta_4=0$) in the cases (1) $ga_0^4 = 0$, (2) $ga_0^4 = 0.01$, (3) $ga_0^4 = 0.02$, all for $Qa_0^{-2}=0.077$. In addition, stars indicate numerical values of transitions obtained for the quartic system with Hamiltonian (\[H1\]). For small values of $r$, the numerical values for transitions for the exact Hamiltonian and the approximate quartic Hamiltonian (\[H1\]) are, as we would expect, very close. However as $r$ increases, we see that the results for the quartic Hamiltonian rapidly diverge from the results for the exact Hamiltonian, while, remarkably, the analytic approximation continues to be a reasonable approximation for the exact Hamiltonian. This may find an explanation in the fact that while the exact Hamitlonian (\[H\]), the quartic approximation (\[H1\]) and the integrable approximation (\[H3\]) all agree close to the fixed point, the global properties of the exact Hamiltonian are expected to be closer to those of the integrable approximation than the quartic approximation.
Note also in Figure 5b that the intercepts of the curves on the $E$ axis, that we have denoted above by $E_{c,1}$ and $E_{c,2}$, are very close to the values of $E$ determined by the resonance condition (\[rescon\]), which are (1) $E=2$, (2) $E \approx 1.977$, (3) $E\approx 1.954$. However, even though the intercepts for the two curves obtained for each set of parameter values are very close, they are not identical. This is something that is difficult to establish [*a priori*]{} by direct numerics for the original systems (as the beating periods, for values of $r$ close to $1$, are very long), but once the analytic approximation is available to give accurate candidate values for the transition locations, it is possible to verify them [*a posteriori*]{}. Thus [*in the small band of values $E_{c,2}<E<E_{c,1}$ there is only a single beating transition as the beam eccentricity is increased.*]{} As $r$ is increased from $1$ there is immediately type I beating, and as $r$ is increased further there is only a single transition to type II (as opposed, for example, to the sitution in Figure 2, where as $r$ is changed from $1$ type II beating is seen, and then there are two transitions). Using the analytic approximation it can be shown (see Appendix A) that the points $E_{c,1}, E_{c,2}$ are determined by the conditions $$\omega_1 - \omega_2 + P(b_2-2b_3 \mp b_5) = 0 \label{Ecconds}$$ (minus for $E_{c,1}$, plus for $E_{c,2}$) for a solution with $r=1$. (The condition $r=1$ implies $J_1(0) = K_1(0)=0$, and then equation (\[Keq2\]) gives a single equation from which to determine $K_2(0)$ from $J_2(0) =
\omega_2 \left(a_0 - a_{\rm min} \right)^2$.) Figure 6 shows the dependence of $E_{c,1}$ and $E_{c,2}$ on $ga_0^4$ for two values of $Qa_0^{-2}$, as computed by the analytic approximation, along with a few numeric values (computed [*a posteriori*]{}). In addition the value of $E_{\rm res}$ from (\[rescon\]) is shown, this being the value of $E$ for which there is exact $1:1$ resonance in the linear approximation. We see that the values of $E_{c,1}$, $E_{c,2}$ and $E_{\rm res}$ all decrease monotonically with $ga_0^4$.
The SAT Model
-------------
The potential of the SAT model, given by (\[VSAT\]), has an isolated minimum when $$a_x = a_y = a_{\rm min} \equiv a_0 \sqrt{\frac{K_{\rm SAT}}{K_0}}$$ where $K_0$, which depends on the parameters $E,K_{\rm SAT}, ga_0^4$, is a solution of the equation $$\frac{K_0^2}{4E} + {\rm Li}_2 ( -K_0) + \ln(1+K_0) - \frac{K_{\rm SAT}^2ga_0^4}{4E}
= 0 \ . \label{Hdef}$$ (Recall that the constant $K_{\rm SAT}$ is defined by $K_{\rm SAT}=4\alpha^2Ea_0^{-2}$.) The resonance condition can be written $K_0 = K_{\rm res}$ where $K_{\rm res}$ is the solution of $${\rm Li}_2 ( -K_{\rm res}) + 2\ln(1+K_{\rm res}) - \frac{K_{\rm res}}{1+K_{\rm res}} = 0 \ .$$ $K_{\rm res}$ has numerical value approximately $5.017$. We recall that for our analytic model to be most effective we need to be near resonance, and the initial conditions should be close to the minimum, i.e. $a_0\approx a_{\rm min}$, or $K_{\rm SAT}\approx K_0$, and $r\approx 1$. These conditions give $K_{\rm SAT} \approx K_{\rm res} = 5.017$ and $E \approx 6.550( 1 - ga_0^4)$. In practice we will look at a large range of values of $E$ and $K_{\rm SAT}$, but focus on this region. We also recall that in our model the sign of $\zeta_2(0)$ (as given in (\[ic1\]) plays a critical role. From (\[Hdef\]) we have $\zeta_2(0)=0$ (or equivalently $K_{\rm SAT}=K_0$) when [@Ianetz2013PRA] $$E = \frac{ -K_{\rm SAT}^2(1-ga_0^4)} { 4( {\rm Li}_2(-K_{\rm SAT})+\ln(1+K_{\rm SAT})) } \ . \label{z20sign}$$
The relevant parameters for the quartic Hamiltonian (\[H1\]) in the SAT case are $$\begin{aligned}
\omega_1^2 &=& \frac{2}{a_0^4}\left( \frac{K_0^2}{K_{\rm SAT}^2} + ga_0^4 \right)
\nonumber\\
\omega_2^2 &=& \frac{4}{a_0^4K_{\rm SAT}^2}\left( -2E\ln(1+K_0) + K_0^2 + \frac{2 E K_0}{1+K_0} \right) \nonumber\\
a_1 &=& \frac{\sqrt{2K_0}}{K_{\rm SAT}^{5/2}a_0^5} \left( 2E\ln(1+K_0) - 3K_0^2-\frac{2EK_0}{1+K_0} \right) \nonumber\\
a_2 &=& \frac{\sqrt{2K_0}}{3K_{\rm SAT}^{5/2}a_0^5} \left(-2E\ln(1+K_0) -3K_0^2 + \frac{2EK_0(1+3K_0)}{(1+K_0)^2} \right)
\label{asforSAT} \\
a_3 &=& \frac{K_0}{4K_{\rm SAT}^3a_0^6} \left( -2E\ln(1+K_0) + 5K_0^2 + \frac{2EK_0}{1+K_0} \right)
\nonumber\\
a_4 &=& \frac{K_0}{2K_{\rm SAT}^3a_0^6} \left( -2E\ln(1+K_0) + 15K_0^2 + \frac{2EK_0(1-K_0)}{(1+K_0)^2} \right)
\nonumber\\
a_5 &=& \frac{K_0}{12K_{\rm SAT}^3a_0^6} \left( 2E\ln(1+K_0) + 15K_0^2 - \frac{2EK_0(1+10K_0 + K_0^2)}{(1+K_0)^3} \right) \ .
\nonumber \end{aligned}$$
The method is identical to that given for CQ in the previous subsection, so we can immediately present results. For fixed values of $E$ and $ga_0^4$ we look for values of $r$ giving beating transitions as a function of $K_{\rm SAT}$. Both numeric and analytic results suggest there is a qualitative difference in behavior for $E$ above and below a critical threshold, and our results are consitent with the value of this threshold being approximately $6.550( 1 - ga_0^4)$, as found above. Figure 7 displays results for $ga_0^4=0$ and $E=6.3$ (below the threshold, left) and $E=6.7$ (above the threshold, right). The numeric results show that below the threshold, there are two ranges of $K_{\rm SAT}$ for which there is a single beating transition, from type I (for $r$ below $r_c$) to type II (for $r$ above $r_c$). For values of $K_{\rm SAT}$ below or above these two ranges, there is only type I beating, and for values between the two ranges there is only type II beating. The analytic approximation reproduces these results well. In this region of parameter space there are values for which $\Delta_2=0$ (indicated in black in the figure) and $\Delta_3=0$ (indicated in green). It is the latter that are physically relevant, and the values of $r_c$ predicted by the analytic model are accurate for a good range. Moving “above the threshold”, numerics show there is a range of values of $K_{\rm SAT}$ for which, as $r$ is increased from $1$, the beating is initially type II, then there is a transition to type I. For some of these values there is then a further transition back to type II for quite high values of $r$. It should be mentioned that these latter transitions were initially discovered using the analytic approximation, and confirmed numerically [*a posteriori*]{}. The analytic approximation reproduces the first transition very well, using pieces of the $\Delta_2=0$ and $\Delta_4=0$ degeneracy curves. The upper transition is not reproduced well, which is not surprising bearing in mind the values of $r$ involved. Pieces of the $\Delta_2=0$ and $\Delta_3=0$ degeneracy curves are close to some of the results, but for a small range of values of $K_{\rm SAT}$ and $r$ the model fails as there is no solution of equations (\[Keq1\])-(\[Keq2\]). Two branches of the $\Delta_3=0$ degeneracy curve come to an abrupt end (in the plot we have connected the ends with a dashed line, which is not associated with any degeneracy). The values of parameters involved are precisely those for which $\zeta_2(0)\approx 0$.
Figure 8 enlarges upon these results for different values of $E$ and $ga_0^4$. In the 4 panels here, the upper panels (a and b) show results for values of $E$ above the threshold, and the lower panels (c and d) show results for values of $E$ below the threshold. In the left panels (a and c), $ga_0^4=0$, in the right panels (b and d), $ga_0^4=0.02$. For $ga_0^4=0$, the values $E=6.3,6.4,6.5,6.7,7.0$ are shown, the first three of which are below the threshold (in panel c), and the last two above the threshhold (in panel a). For $ga_0^4=0.02$, the values $E=6.3,6.4,6.5,6.7$ are shown, the first two of which are below the threshold (in panel d), and the last two above the threshold (in panel b). Note specifically that for $ga_0^4=0$ the case $E=6.5$ is below the threshold (approximately $6.55$), while for $ga_0^4=0.02$ it is above (as the threshold drops to approximately $6.42$). Thus (for example) for $E=6.5$, $K_{\rm SAT}=5$ and $ga_0^4=0$, no beating transitions are observed as the beam eccentricity is increased; but if the grade index is changed to $ga_0^4=0.02$, there are two beating transitions. The analytic theory fully explains this phenomenon. Indeed, for all the cases shown in Figure 8, the analytic theory is in excellent quantitative agreement with numerics for lower values of $r$, and gives reasonable qualitative predictions for higher values of $r$.
Another conclusion from Figure 8 is that for values of $E$ below the threshold, we can find two values of $K_{\rm SAT}$ that give rise to a given value of $r_c$, but for $E$ above the threshold this need not be the case; furthermore the gap in $r_c$ values increases with the given value of $E$. In Figure 9 we illustrate this phenomenon more clearly. For the case $ga_0^4=0$, we show contours in the $K_{\rm SAT},E$ plane that give rises to the values $r_c = \frac1{0.95}\approx 1.053$ (black), $r_c = \frac1{0.92}\approx 1.087$ (blue) and $r_c = \frac1{0.895}\approx 1.117$ (red). It is clear that the “gap” between the two branches of each contour increases with $r$. Note that in the upper branch of each contour there is a small section denoted by a dashed line where the analytic method fails (the dashed line is a straight line between the last two points on each side for which the method works). As expected, the regions where the method fails straddle the curve (\[z20sign\]), incidicated by a dashed turquoise curve. Note further that in many cases the analytic method works well far beyond the region in which this is expected, but there are some exceptions.
Conclusions and discussion
==========================
In this paper we have described the beating phenomena observed in the equations of motion for the beam widths obtained in a collective variable approximation to solution of the GNLSEs relevant for beams in nonlinear waveguides with cubic-quintic (CQ) and saturable (SAT) nonlinearities and a graded-index profile. We have described the different types of beating, and the transitions between them. Arguing that the origin of these phenomena is in a $1:1$ Hamiltonian resonance, we have developed an approximation scheme for small oscillations in a class of 2 degree-of-freedom Hamiltonian systems with an isolated fixed point close to $1:1$ resonance. We have shown that such oscillations can be described by an integrable Hamiltonian, or, alternatively, a single first order differential equation (\[K1e\]). Understanding the bifurcations of the system, which include the beating transitions, can be reduced to looking at the bifurcations of the roots of a pair of quadratic equations. Applying our general methodology to the specific cases of the CQ and SAT models we managed to reproduce numerical results for beating transitions over a large range of parameter values. The theory allows us to map out the regions (of parameter space and beam eccentricities) where beating transitions do and do not exist. Amongst other things, in the CQ case we identified a band of beam energies for which there is only a single beating transition (as opposed to $0$ or $2$) as the beam eccentricity is increased, and in the SAT case we explained the appearance and disappearance of transitions with changes of the grade-index.
We expect our methods to have applications to related problems in nonlinear optics, for nonlinearities other than the ones studied here, for different beams, such as super-Gaussian beams [@supergauss], and for optical bullets [@Kivshar2003book; @bullets]. We are encouraged by the fact that there is some recent experimental evidence [@skarkarobust] of breathing in optical solitons, albeit in a dissipative setting. We also hope the general theory of $1:1$ resonances that we have developed will find application in the settings of nonlinear mechanics and astronomy, as well as suitable extensions for $1:1:1$ resonances in higher dimensional systems (see for example the recent papers [@JS1; @JS2]).
Further Technical Details
=========================
As explained in section 3, the Hamiltonian (\[H3\]) is in an integrable approximation to the Hamiltonian (\[H2\]), and is obtained from (\[H2\]) via a canonical transformation and neglecting higher order terms. The only need for explicit details of the canonical transformation is to compute the initial conditions of the variables $K_1,K_2$ from the initial conditions of $J_1,J_2$ given in (\[Jic\]). The equations to be solved are $$\begin{aligned}
J_1 &=& K_{{1}}
\mp {\frac {4K_{{1}}a_{{1}}}{4\omega_{{1}}^{2}- \omega_{{2}}^{2}}\sqrt {{\frac {2K_{{2}}}{\omega_{{2}}}}}}
+ \left(
\frac{a_1^2 \left( 48\omega_1^{4}-8\omega_1^{3}\omega_2-40\omega_2^{2}\omega_1^{2}+2\omega_1\omega_2^{3}+5\omega_2^{4} \right)}
{4\omega_{{2}}^{2}\omega_{{1}}^{3} \left( 2\omega_{{1}}+ \omega_{ {2}} \right)^{2} \left( 2\omega_{{1}}-\omega_{{2}}\right) ^{2}}
-{\frac {5a_{{3}}}{{2\omega_{{1}}}^{3}}}
\right) K_1^2 \nonumber \\
&& + \left( \frac { \left( 40\omega_{{1}}^{3}+28\omega_{{1}}^{2}\omega_{{2}}-6\omega_{{1}}\omega_{{2}}^{2}
-3\omega_{{2}}^{3} \right) a_{{1}}^{2} }
{\omega_{{1}}^{2}\omega_{{2}} \left( 2\omega_{{1}}-\omega_{{2}} \right) ^{2} \left( 2\omega_{{1}}+\omega_{{2}} \right)^{2}
\left( \omega_{{1}}+\omega_{{2}} \right) }
\right. \nonumber \\
&& \left.
+\frac { \left( 12\omega_{{1}}^{3}+11\omega_{{1}}^{2}\omega_{{2}}-10\omega_{{1}}\omega_{{2}}^{2}
-6\omega_{{2}}^{3} \right) a_ {{1}}a_{{2}}}
{2\omega_{{1}}^{2}\omega_{{2}}^{3} \left( \omega_{{1}}+\omega_{{2}} \right) \left( 4\omega_{{1}}^{2}-\omega_{{2}}^{2}\right) }
- \frac {\left( 3\omega_{{1}}+2\omega_{{2}} \right) a_{{4}}}
{2\omega_{{1}}^{2}\omega_{{2}} \left( \omega_{{1}}+\omega_{{2}} \right) } \right) K_1K_2 \ , \label{Keq1}\\
J_2 &=& K_{{2}}
\mp 2 \left(
\frac { \left( 2\omega_{{1}}^{2}-\omega_{{2}}^{2} \right)K_{{1}}a_{{1}}}
{\omega_{{1}}\omega_{{2}} \left( 4\omega_{{ 1}}^{2}-\omega_{{2}}^{2} \right) }
+ \frac{K_{{2}}a_{{2}}}{\omega_{{2}}^{2}}
\right) \sqrt{\frac {2K_{{2}}}{\omega_{{2}}}}
+ \left( \frac{33a_2^2}{4\omega_2^5} -\frac {5a_5}{2\omega_{{2}}^{3}} \right) K_2^2 \nonumber \\
&&
+ \frac{\left( 16\omega_{{1}}^{4}+8\omega_{{1}}^{3}\omega_{{2}} - 12\omega_{{1}}^{2}\omega_{{2}}^{2}
-2\omega_{{1}}\omega_{{2}}^{3}+3\omega_{{2}}^{4} \right) {a_{{1}}}^{2}{K_{{1}}}^{2}}
{2{\omega_{{1}}}^{2}\omega_{{2}}^{3} \left( 2\omega_{{1}}-\omega_{{2}} \right)^{2}\left(2\omega_{{1}}+\omega_{{2}}\right)^{2} }
\nonumber\\
&&
+ \left(
\frac {\left( 8\omega_{{1}}^{4}+16\omega_{{1}}^{3}\omega_{{2}}-10\omega_{{1}}^{2}\omega_{{2}}^{2}
-8\omega_{{1}}\omega_{{2}}^{3}+\omega_{{2}}^{4} \right) {a_{{1}}}^{2}}
{\omega_{{1}}^{2}\omega_{{2}}^{2} \left( 2\omega_{{1}}-\omega_{{2}} \right)^{2}
\left( 2\omega_{{1}}+2\omega_{{2}} \right) ^{2} \left( \omega_{{1}}+ \omega_{{2}} \right) }
\right. \nonumber\\
&& \left.
+\frac { \left( 40\omega_{{1}}^{3}+44\omega_{{1}}^{2}\omega_{{2}}-9\omega_{{1}}\omega_{{2}}^{2}-16\omega_{{2}}^{3} \right) a_{{1}}a_{{2}}}
{2\omega_{{1}}\omega_{{2}}^{4} \left( \omega_{{1}}+\omega_{{2}} \right)
\left( 2\omega_{{1}}-\omega_{{2}} \right)\left( 2\omega_{{1}}+\omega_{{2}} \right)
}
-\frac { \left( 2\omega_{{1}}+3\omega_{{2}} \right) a_{{4}}}
{2\omega_{{1}}\omega_{{2}}^{2} \left( \omega_{{1}}+\omega_{{2}} \right) }
\right) K_1K_2 \ . \label{Keq2}\end{aligned}$$ Here the upper signs should be taken in the square roots terms in the case $\theta_2(0)=0$ and the lower signs in the case $\theta_2(0)=\pi$.
In Section 4.1, in the study of the CQ system, we stated the conditions (\[Ecconds\]) for the value $r_c$ giving a beating transition to tend to $1$. We briefly describe the origin of these conditions. The symmetric solutions with $a_x=a_y$ of (\[em\]), arising from the initial condition $r=1$, correspond to solutions with $K_1\equiv 0$ of (\[fe1\])-(\[fe4\]). From (\[K1e\]), the values of $P$ and ${\cal E}$ for such a solution must evidently satisfy $b_3P^2 + \omega_2 P -{\cal E}=0$, which is just the condition $\Delta_3=0$, see (\[Del3\]). As explained in Section 3, a necessary condition for a beating transition is the vanishing of one of the quantites $\Delta_1,\Delta_2,\Delta_3,\Delta_4$. To determine $E_{c,1}$ in Section 4.1 we want $r_c\rightarrow 1$ for a solution of $\Delta_{2}=0$. Clearly this requires $\Delta_2=\Delta_3=0$, and some simple algebra then gives the condition $\omega_1-\omega_2 + P(b_2-2b_3-b_5)=0 $. To determine $E_{c,2}$, however, is not so straightforward, as for this we want we want $r_c\rightarrow 1$ for a solution of $\Delta_{3}=0$, and apparently we do not have two equations. The resolution of this conundrum is as follows: Although we stated above that the symmetric solutions of (\[em\]) correspond to solutions with $K_1\equiv 0$ of (\[fe1\])-(\[fe4\]), the latter in fact provide a [*blow up*]{} of the former — there is a $3$ parameter family of the latter and only a $2$ parameter family of the former. Solving (\[fe1\])-(\[fe4\]) in the case $K_1\equiv 0$, we obtain $K_2=P$ (constant), $\phi_2 = \phi_{2}(0) + (\omega_2 + 2 b_3P)z $, and that $\phi_1$ must satisfy the ODE $$\dot{\phi}_1 = \omega_1 + b_2 P + b_5 P \cos \left( 2\left( \phi_{2}(0) + (\omega_2 + 2 b_3P)z - \phi_1 \right)\right) \ .$$ This latter equation can be solved explicitly, and for a general choice of the constant of integration will give a complicated function $\phi_1(z)$. However, for a beating transition we seek a solution that is characterized by a single frequency, i.e. we need $$\phi_1(z) = \phi_1(0) + (\omega_2 + 2 b_3 P) z$$ Substituting this in the differential equation, we obtain $$\omega_2 + 2 b_3 P = \omega_1 + b_2 P + b_5 P \cos \left( 2\left( \phi_{2}(0) - \phi_1(0) \right)\right) \ .$$ Since the initial conditions $\phi_1(0),\phi_2(0)$ take the values $0$ or $\pi$, we deduce that $\omega_1 - \omega_2 + P(b_2 - 2b_3 + b_5) = 0$, as required.
A two time expansion approach
=============================
In this appendix we outline a two time expansion approach [@kc; @ManMan] which is an alternative to the procedure based on canonical transformations given in Section 3.
We wish to look at solutions of the Hamiltonian system with Hamiltonian (\[H\]) and initial conditions (\[ic0\]). We assume the system has an isolated symmetric minimum at which the system is close to $1:1$-resonance. To apply a two time technique we need to introduce a small parameter $\epsilon$ explicitly into the equations. Our systems involve a number of system parameters, for example in the CQ case, the parameters $E,Q,g$, for which the resonance condition is (\[rescon\]). We introduce a small parameter by selecting one system parameter and writing this as its value at resonance plus a small perturbation. However, for reasons described in [@Verhulst1], the “small perturbation” here should be [*quadratic*]{} in the small parameter. Thus, for example in CQ, we have to consider two possibilities, $ E = E_{\rm res} \pm \epsilon^2 $ where $E_{\rm res}$ (which depends on the other system parameters $Q,g$) is the value of $E$ at resonance. The two resulting expansions will differ just in signs. This is the counterpart in the two time method of the need to choose $\theta_2(0)$ to be $0$ or $\pi$ in Section 3 and the resulting choice of signs in equations (\[Keq1\])-(\[Keq2\]). However, we emphasize that it is not the same, so the resulting method is different, in particular, the “choice” in Section 3 involves the initial conditions as well as the system parameters.
Taking, as before, the minimum of the potential $V$ to be at $a_x=a_y=a_{\rm min}$ we now write $$a_x = a_{\rm min} + \epsilon \tilde{a}_x \ , \qquad a_y = a_{\rm min} + \epsilon \tilde{a}_y$$ and expand to 4th order in $\epsilon$. The order $0$ terms are irrelevant and can be discarded. The order $1$ terms vanish by definition of $a_{\rm min}$. In the other terms there is dependence on all the system parameters. However by making the assignment of the form $E = E_{\rm res} \pm \epsilon^2$, discarding all terms of order higher than $4$ and a suitable rescaling, we obtain an approximate potential of the form $$\begin{aligned}
\tilde{V} &=& \frac12 C_1 (\tilde{a}_x^2 + \tilde{a}_y^2)
+ \epsilon \left( C_2( \tilde{a}_x^3 + \tilde{a}_y^3) + C_3\tilde{a}_x\tilde{a}_y (\tilde{a}_x + \tilde{a}_y) \right) \\
&& + \epsilon^2 \left( C_4( \tilde{a}_x^4 + \tilde{a}_y^4) + C_5\tilde{a}_x\tilde{a}_y (\tilde{a}_x^2 + \tilde{a}_y^2)
+ C_6 \tilde{a}_x^2\tilde{a}_y^2 + C_7(\tilde{a}_x^2 + \tilde{a}_y^2) + C_8 \tilde{a}_x\tilde{a}_y \right) \ . \end{aligned}$$ Here $C_1,\ldots,C_8$ are all functions of the system parameters excluding the parameter replaced by $\epsilon$. Note that as a result of the dependence of the system parameters on $\epsilon$ there are now quadratic terms in $\tilde{a}_x,\tilde{a}_y$ in the $O(\epsilon^2)$ terms.
Following the usual two time formalism, we seek solutions of the system with potential $\tilde{V}$ in the form $$\begin{aligned}
\tilde{a}_x &=& \Lambda_1(\epsilon^2 z) \cos (\sqrt{C_1}z) + \Lambda_2(\epsilon^2 z) \sin (\sqrt{C_1}z) +
\epsilon \tilde{a}_{x,1} (z,\epsilon^2 z) + \epsilon^2 \tilde{a}_{x,2} (z,\epsilon^2 z) + \ldots \\
\tilde{a}_y &=& \Lambda_3(\epsilon^2 z) \cos (\sqrt{C_1}z) + \Lambda_4(\epsilon^2 z) \sin (\sqrt{C_1}z) +
\epsilon \tilde{a}_{y,1} (z,\epsilon^2 z) + \epsilon^2 \tilde{a}_{y,2} (z,\epsilon^2 z) + \ldots\end{aligned}$$ Here $\Lambda_1(\epsilon^2 z), \Lambda_2(\epsilon^2 z), \Lambda_3(\epsilon^2 z), \Lambda_4(\epsilon^2 z)$ are functions of the slow variable $\epsilon^2 z$. Substituting in the equations of motion and equating order-by-order, the first order terms $\tilde{a}_{x,1}, \tilde{a}_{y,1}$ can be determined, and a system of first order equations is obtained that $\Lambda_1,\Lambda_2,\Lambda_3,\Lambda_4$ must satisfy to guarantee the absence of secular terms in $\tilde{a}_{x,2}, \tilde{a}_{y,2}$. Writing $$\begin{aligned}
R_1 &=& \Lambda_1^2 + \Lambda_2^2 + \Lambda_3^2 + \Lambda_4^2 \\
R_2 &=& \Lambda_1^2 + \Lambda_2^2 - \Lambda_3^2 - \Lambda_4^2 \\
R_3 &=& \Lambda_1 \Lambda_3 + \Lambda_2 \Lambda_4 \\
R_4 &=& \Lambda_1 \Lambda_4 - \Lambda_2 \Lambda_3\end{aligned}$$ (c.f. [@RPV; @pm2; @hh]) we obtain the system $$\begin{aligned}
R_1' &=& 0 \nonumber \\
R_2' &=& 4 R_4( \gamma_1 R_1 + \gamma_2 + (\gamma_3+\gamma_4)R_3) \label{Reqs} \\
R_3' &=& -\gamma_3 R_2 R_4 \nonumber \\
R_4' &=& - R_2( \gamma_1 R_1 + \gamma_2 + \gamma_4 R_3) \nonumber\end{aligned}$$ where the constants $\gamma_1,\gamma_2,\gamma_3,\gamma_4$ are certain combinations of the constants $C_1,\ldots,C_8$. (Note $R_3^2 + R_4^2 = \frac14 ( R_1^2 - R_2^2 )$.) Thus $R_1$ is an invariant, as are the quantities $$Q_2 = R_2^2 + 4\left(1 + \frac{\gamma_4}{\gamma_3} \right)
\left( R_3 + \frac{\gamma_1 R_1 + \gamma_2}{\gamma_3+\gamma_4} \right)^2
\ ,
\qquad Q_3 = R_4^2 - \frac{\gamma_4}{\gamma_3}
\left( R_3 + \frac{\gamma_1 R_1+ \gamma_2}{\gamma_4} \right)^2 \ .$$ Note that $R_1,Q_2,Q_3$ are related by $$Q_2 + 4Q_3 = R_1^2 - \frac{4(\gamma_1R_1+\gamma_2)^2}{\gamma_4(\gamma_3+\gamma_4)}\ .$$ Using the invariants it is possible to write a single differential equation for the quantity $R_3$: $$(R_3')^2 = -4 \gamma_4 \left(\gamma_3 + \gamma_4 \right)
\left( \left( R_3 + \frac{\gamma_1 R_1 + \gamma_2}{\gamma_3+\gamma_4} \right)^2
-\frac{\gamma_3}{4(\gamma_3 + \gamma_4 )} Q_2 \right)
\left( \left( R_3 + \frac{\gamma_1 R_1+ \gamma_2}{\gamma_4} \right)^2
+ \frac{\gamma_3 Q_3}{\gamma_4} \right)\ .
\label{R3eq}$$ This has the same form as (\[K1e\]) — the right hand side is a product of two quadratic factors in $R_3$ — and similar techniques can be used to discuss bifurcations of its solutions. Specifically, there can be a double root if the discriminant of one of the factors vanishes (i.e. if $Q_2$ or $Q_3$ vanish), or if the factors have a common root. The latter happens in the two cases $$\left( (2\gamma_1\pm\gamma_4)R_1+2 \gamma_2\right)^2+ 4Q_3 \gamma_3\gamma_4 = 0\ .
\label{zzzz}$$ As in Section 3, detecting beating transitions requires translating the initial conditions to the constants of motion $R_1,Q_2,Q_3$ and checking up to $4$ conditions.
We have implemented this method for the CQ and SAT systems and found some satisfactory results which we do not report here; in certain cases the results were better than those found using the method based on canonical transformations. However there are numerous reasons to prefer the method based on canonical transformations. The two time method requires deciding how to explicitly introduce a small parameter and different ways of doing this give different results. It also requires advance knowledge of the correct relative order of magnitude of the oscillations around the fixed point and the deviation of the system parameters from their resonance values. In general, the algebraic manipulations required to implement the two time method, most of which we have omitted in our account here, are substantially more complicated than those required for the method based on canonical transformations; in particular the reduction of the system (\[Reqs\]) to a single differential equation (\[R3eq\]) is a surprise, that emerges from [*ad hoc*]{} manipulations, whereas the parallel steps in the canonical formalism are standard, based on the integrability of the Hamiltonian (\[H3\]). Finally, from our numerical experiments it emerges that while the results based on the vanishing of the discriminant of one of the factors of the right hand side of (\[R3eq\]) are good, the results based on conditions (\[zzzz\]) are poor.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We present an analysis of ground-based optical photometry and spectroscopy, and [*Rossi X-ray Timing Explorer*]{} X-ray observations of the old novae DI Lacertae and V841 Ophiuchi. Our optical photometry data (obtained with the automated photometry telescope RoboScope) comprise an almost decade-long light curve for each star, while the contemporaneous spectroscopy and X-ray observations repeatedly sampled each nova during separate intervals of $\approx45$–55 d in length. The long-term optical light curves of both novae reveal quasiperiodic variability on typical time scales of $\sim30$–50 d with amplitudes of $\Delta V\sim0.4$–0.8 mag. V841 Oph also displays a long-term, sinusoidal modulation of its optical light on a time scale of 3.5–5 yr. The optical spectra of these novae display quite different characteristics from each other, with DI Lac showing narrow Balmer emission cores situated in broad absorption troughs while V841 Oph exhibits strong single-peaked Balmer, and emission lines. We find little change between spectra obtained during different optical brightness states. The X-ray count rates for both novae were very low ($\lesssim1.5$ ct s$^{-1}$) and there was no reliable correlation between X-ray and optical brightness. The combined X-ray spectrum of DI Lac is best fit by a bremsstrahlung emission model (with $kT\sim4$ keV and $N_{\rm H} < 1.8\times10^{22}$ cm$^{-3}$); the X-ray spectrum of V841 Oph is too weak to allow model fitting. We discuss the possible origin of variability in these old novae in terms of magnetic activity on the secondary star, dwarf nova type disk instabilities, and the “hibernation” scenario for cataclysmic variable stars.
[Accepted by PASP on 28 August 2000 for the December 2000 issue.]{}
author:
- |
D. W. Hoard, Paula Szkody, R. K. Honeycutt, Jeff Robertson, Vandana Desai,\
and T. Hillwig
title: 'Long-term optical and X-ray observations of the old novae DI Lacertae and V841 Ophiuchi'
---
Introduction {#s-intro}
============
Cataclysmic variables (CVs) are semi-detached interacting binary stars composed of a white dwarf (WD) primary star and a low mass ($\lesssim0.5$M$_{\odot}$) main sequence secondary star, with typical orbital periods of $\lesssim1$ d. The Roche-lobe-filling secondary star loses mass through the inner Lagrangian point into an accretion disk formed around the WD (for non-magnetic CVs). Classical novae are a subclass of CV in which a thermonuclear runaway is triggered in a reservoir of matter that has been gradually accreted onto the WD. The resultant outburst produces a peak brightness increase of $\approx$ 6–15 mag, and releases $10^{44}-10^{46}$ erg [see review in @Warn95 ch. 5].
The outbursts of novae are often well-observed, and the long term behavior of many dwarf novae and novalike CVs is monitored by the American Association of Variable Star Observers. However, there have been few programs for monitoring the variability of novae at quiescence. One such program, using an automated telescope called RoboScope (see §\[s-robo\]), has monitored 22 old novae and 42 novalikes, most of them for over 9 years now. RoboScope has found several kinds of unusual photometric behavior. About one-third of the old novae have shown quasi-periodic variability for a year or two, interspersed with stable light curves. These variations do not appear stochastic as they repeat at similar periods when they reappear. Preliminary analysis of RoboScope light curves spanning several years for the old nova DI Lacertae suggested brightness oscillations on a time scale of $\approx35$ d [@Hone95], while the old nova V841 Ophiuchi also displayed prominent variability [@Hone94]. In order to determine if these oscillations could be related to a disk instability mechanism (as operates in dwarf novae) or to mass transfer or magnetic effects on the secondary star, we attempted a detailed study of DI Lac and V841 Oph, which are among the most active and brightest of the systems showing the oscillations.
DI Lac (= Nova Lac 1910) was a moderately fast nova that reached a maximum brightness of $m_{V}({\rm max}) = 4.3$ mag during its outburst. V841 Oph (= Nova Oph 1848) was a slow nova that reached a similar brightness, $m_{V}({\rm max}) = 4.2$ mag [@Kuka71]. The reddening for both objects has been measured from [*International Ultraviolet Explorer*]{} ($IUE$) ultraviolet and ground-based optical spectra. The range for DI Lac is $E(B-V)=0.15$–0.41 [@Bruc84; @Cass90], while for V841 Oph it is $E(B-V)=0.30$–0.58 [@Bruc84; @Cass90; @Weig94; @Verb97]. Nova shells were not detected in H$\alpha$ for either CV [@Cohe85]. In X-rays, DI Lac was undetected in the $ROSAT$ PSPC All Sky Survey, with a $2\sigma$ upper limit of 0.013 cts s$^{-1}$ in the 11–201 channel (0.1–2.0 keV) energy range during 299 s total exposure time, while V841 Oph was a marginal detection with $0.017\pm0.007$ cts s$^{-1}$ in the “hard” 0.9–2.0 keV [*ROSAT*]{} energy range ($0\pm0.025$ cts s$^{-1}$ in the full [*ROSAT*]{} energy range) during 402 s total exposure time [@Verb97].
We report here on the results of ground-based optical photometry and spectroscopy, and [*Rossi X-ray Timing Explorer*]{} ($RXTE$) X-ray observations of DI Lac and V841 Oph. The optical photometry comprises an almost decade-long light curve for each star, while the contemporaneous spectroscopy and X-ray observation repeatedly sampled each nova during separate intervals of $\approx45$–55 d in length.
Observations and Analysis
=========================
Our long-term optical photometry of DI Lac and V841 Oph was accomplished from 1990–1998, while the spectroscopic amd X-ray observations took place in 1997. These observations are summarized in Tables \[t-DI\_log\] and \[t-V8\_log\], and are discussed in detail below.
Optical Photometry {#s-robo}
------------------
Our optical photometry data were acquired by RoboScope, a 41-cm telescope in Indiana equipped for automated differential CCD stellar photometry [@HT92]. All observatory operations (including data reductions) are accomplished as fully unattended and unsupervised tasks, which makes practical the acquisition of long homogeneous data streams. Typically, RoboScope obtains one or two 4-min exposures per clear night for each of $\sim140$ program stars. The data are reduced using the method of incomplete ensemble photometry [@Hone92]. For DI Lac, 85 ensemble stars were used in 987 exposures over 9 observing seasons from 1990 November to 1998 November. For V841 Oph, 24 ensemble stars were used in 611 exposures over 8 observing seasons from 1991 May to 1998 September. The zero-points of the differential light curves were established using secondary standard stars from @Hend95 [@Hend97]. Six such secondary standards were used for V841 Oph, while 13 were employed for DI Lac. Typical $1\sigma$ uncertainties in the calibrated magnitudes are on the order of 0.01–0.05 mag. Over the entire multiyear range of the RoboScope light curves, the mean magnitudes were $V\approx14.55$ for DI Lac and $V\approx13.6$ for V841 Oph, with full ranges of variability of $\Delta V\approx0.9$ mag and $\Delta V\approx1$ mag, respectively. The RoboScope light curves of DI Lac and V841 Oph are shown in Figures \[f-DI\_lc\] and \[f-V8\_lc\].
With only limited excursions to anomalously fainter magnitudes, the typical minimum magnitude of $\approx14.7$ for DI Lac has a sharp boundary at any given epoch. However, the minimum brightness displays a slight, apparently linear, trend towards increasing brightness over the 8 years of RoboScope observation. If we ignore for the moment the small number of data points that fall below the well-defined low brightness limit seen in the top panel of Figure \[f-DI\_lc\], then the typical minimum brightness of DI Lac changes from $V\approx14.75$ mag at the beginning of the RoboScope coverage, to $V\approx14.67$ mag at the end. This corresponds to a mean increase of the minimum brightness by $\Delta V\approx -0.01$ mag yr$^{-1}$. The maximum brightness of DI Lac is much less uniform, and it is difficult to determine if the maximum brightness behaves in the same manner as the minimum brightness. A linear fit to the complete DI Lac light curve (with each data point weighted by the inverse square of its $1\sigma$ uncertainty) gives the rate of change of the mean magnitude as $< +0.001$ mag yr$^{-1}$; that is, the mean brightness of DI Lac is essentially constant. However, if we exclude the 2 excursions to fainter magnitudes at HJD 2450600 and HJD 2451000 (as well as the excursion to brighter magnitudes at HJD 2450300), then the slope of the linear fit changes to $\Delta V = -0.004$ mag yr$^{-1}$; that is, a trend towards increasing mean brightness.
The faint magnitude excursions in the light curve of DI Lac are very similar to the “dips” seen by @Hone98 in long-term RoboScope light curves of five old novae and novalike CVs. In the latter systems, the dips were found to often be paired with a preceding or following outburst, although the dips also sometimes occurred as isolated events. @Hone98 do not find a clear mechanism (e.g. disk instabilities, truncated disks, mass transfer modulation) that is responsible for the dips, but conclude that the overall photometric behavior (dips and outbursts) of the novae and novalikes that they studied is likely to be governed by either a combination of disk and mass transfer events, or by mass transfer events alone.
In contrast to the linear trend in the minimum brightness of DI Lac, the minimum brightness of V841 Oph appears to follow an almost sinusoidal trend that completes approximately 1.5–2 cycles during the 7.5 year RoboScope coverage. As seen in the top panel of Figure \[f-V8\_lc\], the minimum brightness of V841 Oph varies from $V\approx14.5$ mag to $V\approx13.8$ mag. The maximum brightness of V841 Oph, which ranges from $V\approx13.1$ mag to $V\approx13.4$ mag, displays the same behavior; that is, when the minimum brightness at a given epoch is at its faint (bright) level, the maximum brightness is also at its faint (bright) level. We note that the magnitude range spanned by the minimum brightness in V841 Oph is about twice as large as that spanned by the maximum brightness. Although a linear fit to this overall trend is perhaps not the optimum choice, the corresponding rate of mean magnitude change is $\Delta V = +0.010$ mag yr$^{-1}$. It is possible (but not required by the data!) that the apparently linear trend in the light curve of DI Lac may also be a sinusoid, but with a much longer cycle length than in V841 Oph.
We searched for periodicities in the range 10–100 days in the RoboScope light curve data in two ways: by applying the phase dispersion minimization (PDM) algorithm [@Stel78], as well as performing an independent power spectrum analysis using the [clean]{} algorithm [@Robe87]. We performed the period search on both the entire data set for each CV, as well as on the data subsets from individual observing seasons (indicated in Figures \[f-DI\_lc\] and \[f-V8\_lc\]). The most prominent period detected in each data set is listed in Table \[t-periods\] (in several cases, most notably for the combined data sets for each nova, two significant periods are listed). In general, the two period search methods gave equivalent results to within better than 1–4 days. All of the detected time scales appear to correspond to quasiperiodic behavior rather than truly periodic variability such as that resulting from, for example, an eclipsing orbit. The time scales have significant “flexibility” (on the order of 1–5 days for periods $\lesssim50$ d, 5–20 days for periods $\gtrsim50$ d) over the multi-year length of the light curves (as shown by substantially broadened dips and peaks in the PDM and [clean]{} analyses, respectively). Phase-binned light curves of both CVs folded on the shorter period found for each combined data set are shown in Figure \[f-folded\_lc\]. We note that the true amplitude of variation in a given cycle or observing season will be larger than the 0.1–0.2 mag suggested by Figure \[f-folded\_lc\], in which data from many cycles with slightly different amplitudes and periods have been averaged together. Inspection of Figures \[f-DI\_lc\] and \[f-V8\_lc\] shows amplitudes of 0.4–0.8 mag for the full range of variability within each observing season. The long-term sinusoidal trend in the light curve of V841 Oph is also well-fit by a periodicity in the range 1800–1900 d or, with slightly less agreement, 1250–1300 d. (We note that these two period ranges are approximately related by the ratio 3:2, so we are likely seeing an aliasing effect due to the fact that only 1.5–2 cycles of these long periods are contained in the light curves.)
Optical Spectroscopy {#s-optspec}
--------------------
We obtained optical spectra of both CVs using the Double Imaging Spectrograph on the Apache Point Observatory (APO) 3.5-m telescope [@Gill95][^1] during University of Washington share time. Spectra of DI Lac were obtained in July and August of 1997; spectra of V841 Oph were obtained in May of 1997. The APO spectra have a resolution of $\approx2$Å and cover simultaneous wavelength ranges of $\approx4200$–5000Å and $\approx5800$–6800Å. The raw spectrum images were reduced in the standard fashion using IRAF. The instrumental response was removed via spectra of standard stars [@Mass88] obtained on the same nights; however, slit losses due to guiding errors rendered the absolute flux calibration unreliable. Consequently, we have normalized the spectra to a constant continuum level of 1.0.
A representative spectrum of DI Lac is shown in Figure \[f-DI\_spec\].
Note the prominent Balmer absorption troughs containing narrow, central emission cores. Our other two APO spectra of DI Lac (not shown) are essentially identical to this one, with the exception that the Balmer emission cores in the August spectrum are somewhat stronger (relative to ) than in the July spectra. The $\lambda4471$ feature (see inset) is present with the same profile shape in all three of our spectra. It is suggestive of the absorption trough with central emission as seen in the Balmer lines. The feature labeled with “?” is likely an instrumental or reduction artifact since its wavelength does not match any identifiable line and it occurs in [*all*]{} of our APO spectra (those of V841 Oph included).
Additional optical spectra of both novae were obtained using the Hydra Multi-fiber Positioner + Bench Spectrograph [@Bard92; @Bard94] on the WIYN 3.5-m telescope[^2]. Spectra of DI Lac were obtained in 1997 September, and spectra of V841 Oph were obtained in 1997 May. The WIYN spectra have a resolution of $\approx1$Å and cover (usable) wavelength ranges of $\approx6200$–6800Å for DI Lac and $\approx4400$–5000Å for V841 Oph. For the WIYN spectra of DI Lac, no standard star observations were available, so it was not possible to correct for the instrumental response or perform a flux calibration. We do not show these spectra here, but note that they are qualitatively similar to the red APO spectrum shown in Figure \[f-DI\_spec\] – no differences corresponding to the different brightness levels in the RoboScope light curve are apparent. The V841 Oph WIYN data were reduced and calibrated as with the APO spectra. Again, slit losses and, additionally, the presence of cirrus made the absolute flux calibration unreliable, so we have normalized the spectra to a constant continuum level of 1.0. All of the spectra of V841 Oph are shown in Figure \[f-V8\_spec\] (the three WIYN spectra have been averaged together to improve the S/N).
There is no indication of the absorption troughs seen in the spectra of DI Lac. Additionally, the / + emission complex appears to be stronger, with broader lines, in V841 Oph than in DI Lac.
Flux-calibrated optical spectra of both DI Lac and V841 Oph are shown in @Will83, with resolutions of 5.1Å and 6.7Å, respectively, and in @Ring96, with resolution of 15–20Å for both. @Will83 obtained his spectra in 1981, while @Ring96 obtained theirs in 1990. In both papers, the spectra of both novae show steep blue continua; the observed continuum of DI Lac is bluer than that of V841 Oph (but different reddening values could eliminate or reverse this characteristic). @Ring96 fit a power law of the form $F_{\lambda} \propto \lambda^{-\alpha}$ to the continuum regions 5600–6500Å and 6650–8000Å in their spectra. These fits yielded indices of $\alpha=1.81(2)$ – dereddened to $\alpha_{0}=3.10(3)$ using $E(B-V)=0.41$ – for DI Lac and $\alpha=1.45(1)$ – dereddened to $\alpha_{0}=2.79(1)$ using $E(B-V)=0.58$ – for V841 Oph. @Cass89 performed this power law fit to the wavelength region 1200–3200Å in $IUE$ UV spectra of DI Lac and V841 Oph. Using reddenings at the low ends of the ranges given in §\[s-intro\], $E(B-V)=0.15$ for DI Lac and $E(B-V)=0.30$ for V841 Oph, they calculated dereddened slopes of $\alpha_{0}=1.5$ and $\alpha_{0}=2.0$, respectively. Comparison with the slope expected for the continuum flux distribution of a steady-state accretion disk composed of parcels radiating as blackbodies, $\alpha=2.3$ [@Prin81], suggests that the true reddening values for DI Lac and V841 Oph are near the middle of the ranges given in §\[s-intro\].
In 1936, both of these novae were observed spectroscopically by @Huma38, who noted their strong blue continua, but also noted the absence of [*any*]{} lines, emission or absorption. Since that time, the available published spectra of DI Lac and V841 Oph in @Gree60, @Will83, and @Ring96 are consistent with our current spectra [see spectroscopic histories summarized in @Ring96], with the exception that @Will83 did not report the detection of any $\lambda4686$ emission in either star.
X-ray Observations
------------------
DI Lac and V841 Oph were each observed ten times during July–September and April–June of 1997, respectively, with the Proportional Counter Array [PCA; @Jaho96] on the $RXTE$ satellite [e.g. @Brad93][^3]. The PCA consists of five xenon-methane proportional counters effective over the range 2–60 keV (with 18% energy resolution at 6 keV). Each $RXTE$ visit constituted a Good-Time-Interval on-source lasting $\approx1000$–2500 s. We extracted the net average X-ray count rate from each visit and the X-ray spectrum from the ten combined visits to each star using the FTOOLS/LHEASOFT (v5.0) software package. We utilized the faint source background models (pca\_bkgd\_faintl7\_e3v19990824.mdl and pca\_bkgd\_faint240\_e3v19990909.mdl) and an up-to-date South Atlantic Anomaly history file (pca\_saa\_history\_20000223). Extraction of the count rates and spectra was carried out as described in “The RXTE Cookbook”[^4].
The X-ray light curves constructed from the mean count rates at each visit for DI Lac and V841 Oph are shown in Figures \[f-DI\_Xray\] and \[f-V8\_Xray\], respectively, along with the corresponding sections of their RoboScope $V$ band light curves.
The light curve counts were summed in the energy range 0–15 keV. Inspection of the X-ray spectra (see below) revealed an essentially zero count rate dominated by noise at energies higher than 15 keV. The mean X-ray count rates averaged over all ten visits to each target are $1.30\pm0.32$ ct s$^{-1}$ for DI Lac and $0.56\pm0.73$ ct s$^{-1}$ for V841 Oph. Considering these very low count rates, it is not clear whether or not the variability seen in the X-ray light curves is real.
Additional analysis and model-fitting (DI Lac only) of the X-ray spectra of these novae was performed using the routine XSPEC (v11). Although we extracted the full range of available energy channels in the X-ray spectra, only the range 0–15 keV was used for fitting the models to the DI Lac data. The X-ray spectra for DI Lac and V841 Oph are shown in Figures \[f-DI\_modspec\] and \[f-V8\_Xspec\].
The X-ray spectrum of V841 Oph is shown on the same scale as that of DI Lac; the former is clearly much weaker than the latter. We attempted to extract a better spectrum of V841 Oph by using only the data from the four visits with the largest count rates (visits 3, 5, 9, and 11 – see Table \[t-V8\_log\]). Unfortunately, this spectrum was virtually identical to the total combined spectrum (i.e. flat and featureless), with the exception of a small upward offset of the mean count rate, from about 0.02 cts s$^{-1}$ keV$^{-1}$ in the total combined spectrum to about 0.04 cts s$^{-1}$ keV$^{-1}$ in the “high” count rate spectrum. Consequently, no attempt was made to fit a model to the spectrum of V841 Oph.
For DI Lac, we fit three simple models to the X-ray spectrum: (1) blackbody, (2) Raymond-Smith thermal plasma, and (3) bremsstrahlung. Each model was modified by a multiplicative component representing photoelectric (H column) absorption. More complex models were not warranted owing to the small number of counts per energy channel in the data. In all cases, the data were weighted using the XSPEC-recommended scheme appropriate for small count numbers, $W_{\rm i}=1+(N_{\rm i}+0.75)^{0.5}$ [@Gehr86]. The results of these model fits are summarized in the first three rows of Table \[t-DI\_model\]. All of the models produce nearly indistinguishable fits to the observed spectrum. The bremsstrahlung model is formally the best-fitting (with lowest reduced $\chi^{2}$ and highest null hypothesis probability) and is plotted over the observed spectrum in Figure \[f-DI\_modspec\].
All of the models were only able to constrain the hydrogen column density to an upper limit; the values of $N_{\rm H}$ quoted in the first three rows of Table \[t-DI\_model\] are the upper ends of the 1-parameter 90% confidence intervals. We used the relation $$N_{\rm H} = E(B-V) \times 5.8\times10^{21} {\rm cm}^{-2} {\rm mag}^{-1}$$ [@Bohl78] to estimate values of $N_{\rm H}$ spanning the reddening range given in §\[s-intro\]. We then re-fit the models to the X-ray spectrum of DI Lac with only two free parameters: temperature and normalization. The results are listed in the bottom six rows of Table \[t-DI\_model\]. While slightly different from those obtained from the original model fits, they are completely consistent with them. The bremsstrahlung model yields a slightly lower nominal temperature of $kT\approx4.1$–4.2 keV (vs. $kT=4.5$ keV when $N_{\rm H}$ is a free parameter) and is still the best-fitting model for both values of $N_{\rm H}$.
We note that the uncertainties quoted in Table \[t-DI\_model\] for $kT$ and $C_{\rm norm}$ are 1-parameter 90% confidence intervals; the interdependent parameter uncertainties will be slightly different. For example, for the bremsstrahlung model (with free $N_{\rm H}$), the XSPEC routine steppar[^5] gives a 2-parameter 90% confidence interval of $$kT = 4.5^{+1.9}_{-1.5} {\rm ~keV}$$ for $kT$ with respect to $N_{\rm H}$, and $$kT = 4.5^{+1.8}_{-1.7} {\rm ~keV}$$ for $kT$ with respect to $C_{\rm norm}$.
The X-ray count rates were too low to reliably fit spectrum models to any of the ten individual $RXTE$ visits to DI Lac. We constructed “low” and “high” state X-ray spectra by combining the data from visits with the lowest mean count rates (1, 2, 3, 4, 7) and from visits with the highest mean count rates (5, 6, 9) to see if there were any differences between the low and high X-ray brightness states[^6]. Other than an $\approx1$ cts s$^{-1}$ keV$^{-1}$ increase at the peak energy of the high state spectrum, the two spectra are essentially indistinguishable from each other (and from the combined spectrum from all ten visits). We conclude from this that either the spectral energy distribution of the X-ray emitting source in DI Lac does not change as the overall brightness of the X-ray source changes, or that the variability seen in the X-ray light curve of DI Lac is not real (i.e. it is a side-effect of the very low count rates).
Discussion
==========
Optical Photometric Variability {#s-optvar}
-------------------------------
Inspection of the RoboScope light curves of DI Lac and V841 Oph reveals variability on three distinct time scales. First, in the range 10–100 d, quasiperiods with typical lengths of $\sim30$–50 d are present in each of these CVs. Yet, the presence (or lack) of periodic and/or quasiperiodic variability on time scales up to a few hundred days in V841 Oph has been contested in the literature. (To the best of our knowledge, no extensive database of photometric observations of DI Lac has been published prior to this work.) Using 420 archival visual observations of V841 Oph spanning 28 yr, @Dell87 quote an average period of 51.5 d with individual cycles ranging from 45 d to 57 d. @Shar89 observed V841 Oph once per night for 31 nights during a 45 night interval in 1986 using a CCD camera + $B$ filter, and found a period of $\sim40$ d. All of these values are comparable to the $\sim35$–50 d we found for our complete RoboScope light curve of V841 Oph, as well as to the range of periods found for individual observing seasons in the RoboScope data (see Table \[t-periods\]). On the other hand, @Duer92 notes the presence of considerable variability in archival light curves of V841 Oph [including some of the same data used by @Dell87], but finds no evidence for periodicities up to 200 d.
It is quite interesting that the variability time scales in these two novae are similar to those that characterize dwarf nova outbursts [@Warn95 ch. 3]. Old novae are presumed to have mass transfer rates, $\dot{M}$, above the threshold level ($\dot{M}_{\rm crit}$) at which the accretion disk thermal instability mechanism operates in dwarf novae to produce their outbursts [@Osak96]. However, the hibernation theory of cyclic evolution between CV variability types [@Sh89 and references therein], predicts that when $\dot{M}$ eventually decreases back into the instability regime some decades after the nova outburst, then dwarf nova outbursts should resume. Both GK Persei [= Nova Per 1901; @Bian86] and V446 Herculis [= Nova Her 1960; @Hone98b] have displayed dwarf nova eruptions in their post-nova stages, and similar behavior has been suggested for a number of other old novae, commencing $\approx$ 50–200 yr after their outbursts [@Livi89; @Warn95 ch. 4]. The amplitudes of variability displayed by DI Lac and V841 Oph (see Figures \[f-DI\_lc\] and \[f-V8\_lc\]) are smaller than those typically observed for dwarf novae ($\Delta V \sim$ 2–4 mag), but we cannot rule out the possibility of a disk instability mechanism being in operation – after all, it has been 90 years since the nova outburst of DI Lac and over 150 years since that of V841 Oph. Further, the classic thermal instability mechanism is not well-explored for high-$\dot{M}$ disks, and other disk instabilities might operate as well [e.g. @Whit88; @Mine93; @Godo98].
The amplitudes of the $\sim30$–50 d oscillations in DI Lac and V841 Oph are similar to those of the spaced stunted outbursts seen in some novalike CVs [@Hone98]. One of the suggestions for stunted outbursts made in @Hone98 is that they are dwarf nova eruptions seen against a background of brighter light in the system. If that suggestion is correct, then the oscillations reported here [as well as similar 25 d oscillations reported for RW Tri, @Hone94] may be examples of dwarf nova type outburst behavior at relatively short recurrence times, $T_{\rm n}\lesssim50$ d [$T_{\rm n}$ is typically $\gtrsim10$ d to hundreds of days – even tens of years in extreme cases – for dwarf novae; @Warn95 ch. 3]. Among the true dwarf novae, the Z Camelopardalis stars have outburst recurrence times of $\approx10$–30 d, somewhat shorter than the time scales for the oscillations observed in these novae. The Z Cam stars are thought to have high mass transfer rates (near $\dot{M}_{\rm crit}$). This suggests that if the oscillations observed in novae are analogous to dwarf nova outbursts, then they may be linked to the presence of relatively high mass transfer rates that are, nonetheless, smaller than both $\dot{M}_{\rm crit}$ and $\dot{M}$ in the Z Cam stars.
Second, V841 Oph displays a distinct sinusoidal variation with a best period of 1800–1900 d (4.9–5.2 yr) or a slightly less preferred period of 1250–1300 d (3.4–3.6 yr). DI Lac does not appear to display similar sinusoidal variability, unless it occurs with a much longer period than in V841 Oph. @Bian90 reports a long-term period in V841 Oph of 3.4 yr with mean amplitude of $\sim0.3$ mag, which was determined from 420 visual observations spanning 30 yr [essentially the same early- to mid-20th century data utilized by @Duer92 see below]. @Rich94 re-analyzed the light curves presented by @Bian90 and consider the case for a multi-year periodicity in V841 Oph to be very weak. However, with our RoboScope light curve, sinusoidal variability on comparable multi-year times scales has now been observed in two separate data sets for V841 Oph spanning almost 80 years of observation. While this still does not provide firm evidence of strictly periodic behavior, it certainly points to the presence of a mechanism operating in this CV that modulates its brightness on a characteristic time scale of several years.
Photometric variability in old novae on time scales of several years has generally been attributed to solar-type magnetic cycles on the secondary star that might be able to control the rate of mass loss through the inner Lagrangian point [@Bian90; @Appl92]. Although @Rich94 find that no CV studied to date displays strictly periodic behavior over multi-year intervals, they also note that the observed amplitudes ($\sim0.2$ mag) and “apparent” time scales of variability of 5–40 yr are “plausible consequences from solar-type magnetic cycles.” We note that the shorter time scale ($\sim$30–50 d) variability discussed above might also be linked to modulation of $\dot{M}$ due to secondary star magnetic activity. If solar-type magnetic cycles can affect $\dot{M}$ on time scales of years, then starspots induced by the magnetic cycle that migrate under the $L_{1}$ point might also affect $\dot{M}$ on shorter time scales, in a manner suggested by @Livi94 to explain the very low brightness states seen in some CVs.
Third, the mean magnitudes of both CVs are changing slowly, at rates measured in a few millimagnitudes (mmag) per year. The mean brightness of DI Lac is increasing by 4 mmag yr$^{-1}$ since 1990. @Duer92 analyzed a large set of visual, photoelectric, and CCD observations of post-novae available in the literature [see references in @Duer92]. These data cover a large part of the 20th century, from the 1920’s through the 1980’s. @Duer92 summarizes the long-term behavior of DI Lac as exhibiting a steady decline of $13\pm1$ mmag yr$^{-1}$ in visual observations obtained between 1921 and 1952. Sparse photoelectric and CCD data from 1953 to 1981 suggest a possible small brightness increase, which is supported by our recent RoboScope observations. For V841 Oph, visual observations from 1919–52 and 1978–91 yield a brightness decline of $7\pm1$ mmag yr$^{-1}$, while “no definitive brightness decline” [@Duer92] was found in photoelectric data from 1954 to 1988. Our post-1991 observations of V841 Oph suggest a trend towards decreasing mean brightness comparable to that noted by @Duer92 in the archival visual observations, although the obvious sinusoidal variation that dominates our light curve at long time scales makes this conclusion somewhat suspect.
In the “classical” hibernation scenario [@Sh89], novae stay bright for $\sim$ 50–300 yr following outburst, gradually declining in brightness during this time to a hibernating state characterized by low brightness ($M_{V} \sim 10$ or fainter) and low mass accretion rate ($\dot{M} \lesssim 10^{-12}$ M$_{\odot}$ yr$^{-1}$). However, the photometric record of DI Lac from the 1920’s (a decade after its outburst) to the 1990’s does not show a simple decline in brightness. Instead, the brightness of DI Lac decreased for about four decades after outburst, but then apparently leveled off for several decades. In the last decade (covered in our RoboScope light curve), the brightness of DI Lac has been increasing. It is possible that we are seeing the influence of a mechanism (such as magnetic activity of the secondary star – see above – and/or accretion-induced irradiation of the secondary star) that modulates $\dot{M}$ (and, hence, the brightness) in DI Lac over a long-term cycle (with period on the order of many decades) that obscures the general decline predicted by the hibernation scenario.
Optical Spectrum Variability
----------------------------
The optical spectra of DI Lac and V841 Oph are quite different, and this can possibly be ascribed to their difference in post-outburst ages. The spectrum of V841 Oph is similar to those of novalike CVs [e.g. @Warn95 ch. 4] – this suggests that V841 Oph still has a high accretion rate, even $\approx150$ yr after outburst. The narrow emission components in DI Lac are suggestive of emission originating from the irradiated inner face of the secondary star, while the absorption troughs imply the presence of optically thick material in the system. The latter feature could be material ejected during the outburst of this younger post-nova; the lack of any detected H$\alpha$ emission shell [@Cohe85] does not preclude the existence of denser, non-emitting circumstellar material. The former feature offers a possible explanation for the gradual brightness increase seen in our RoboScope light curve of DI Lac (discussed in §\[s-optvar\]) if the secondary star is being slowly heated via irradiation and is, in turn, increasing the rate of mass transfer through the $L_{1}$ point.
Although the RoboScope coverage is incomplete, we infer from the adjacent data that DI Lac was faint during our July spectra and bright during our August spectrum (see Figure \[f-DI\_Xray\]). As mentioned in §\[s-optspec\], the only difference between our July and August spectra is that the emission cores were somewhat stronger in August (bright state) than July (faint state). This is consistent with the hypothesis that these narrow emission features originate on the irradiated face of the secondary star if we make the logical assumption that the irradiation increases when DI Lac is bright. It does not, however, illuminate the exact mechanism producing the irradiation (i.e. whether the “excess” flux in the high state originates near the disk center – presumably due to a disk instability producing increased accretion onto the WD – or in the outer disk – presumably due to an increase in mass transfer through the $L_{1}$ point.) If we assume that the spectrum shown in Figure \[f-DI\_spec\] (from July 27 09:01:47 UT) was obtained at an arbitrary orbital phase of 0.0, then the August 17 spectrum was obtained at a relative orbital phase of 0.9 [using $P_{\rm orb} = 0.543773$ d; @Ritt98]. Thus, these two spectra were obtained at similar orbital phases, and we do not expect the difference in the narrow emission component strength to be only due to system orientation.
The RoboScope coverage of V841 Oph during our spectroscopic (and X-ray) observations is more sparse than for DI Lac, but we can infer that the CV was returning to the faint state during our May 07 spectrum, was in its faint state during our May 14 and 17 spectra, and was likely near the bright state during our May 27 spectrum (see Figure \[f-V8\_Xray\]). (The two elevated brightness points at HJD 2450610 and HJD 2450625 suggest that V841 Oph may have returned to a bright state sometime between the low states bracketing HJD 2450590 and HJD 2450630.) If this is the case, then the somewhat stronger H$\alpha$ emission of the May 07 and May 27 spectra (see Figure \[f-V8\_spec\]) may be linked to the bright state of the CV. As with DI Lac, we calculated relative orbital phases for each of these spectra [using $P_{\rm orb} = 0.60423$ d; @Ritt98], and obtained $\phi = 0.0, 0.6, 0.75, 0.1$ for May 07, 14, 17, 27, respectively. Unfortunately, this casts some doubt on the link between H$\alpha$ emission strength and brightness state, since both of the bright state spectra (phases 0.0, 0.1) were obtained at different orbital phases than the faint state spectra (phases 0.6, 0.75). So, we cannot rule out the influence of system orientation effects in these spectra.
X-ray Variability
-----------------
Unfortunately, because of the weak X-ray emission from these CVs, little can be firmly stated about their X-ray variability. The count rates during visits 5 and 6 to DI Lac, which took place when we infer from the RoboScope light curve that the CV was in a bright state, are slightly elevated compared to the preceding visits. However, visit 7 also occurred during this optical bright state and does not show an elevated count rate. The mean X-ray count rate during each visit to V841 Oph also does not display any strong correlation with the corresponding optical state. The lack of large changes in X-ray flux or spectrum during the optical variations [as usually evident in dwarf novae; e.g. @Szko99], argues against a disk instability scenario. Alternatively, this could indicate that the optical brightness is determined by activity in the outer disk only and, therefore, is not reflected in the X-ray behavior; however, the X-ray count rates are too low (and their corresponding error bars too large) to make any firm conclusions.
Conclusions
===========
Our long-term optical light curves of the novae DI Lac and V841 Oph obtained with RoboScope reveal quasiperiodic variability with a characteristic time scale of $\sim30$–50 d in both CVs. In addition, the light curve of V841 Oph displays evidence for sinusoidal variability with a period of 3.5–5 yr. The latter cannot be said to be strictly periodic since our data set covers only $\approx1.5$ cycles; however, when this detection is added to past reports of multi-year periodicities in V841 Oph that have been reported in the literature, a strong case can be made for the presence of repeating multi-year variability with a preferred time scale in this CV. The most likely origin of such behavior is a solar-type magnetic cycle of the secondary star that modulates the rate of mass transfer through the $L_{1}$ point. If this is the case, then the shorter $\sim30$–50 d quasiperiodic variability in this system might also be related to the magnetic activity on the secondary star; for example, due to starspots that migrate under the $L_{1}$ point and temporarily throttle mass transfer. DI Lac does not show evidence for cyclic multi-year variability, unless it occurs with a much longer period than the length of our RoboScope coverage ($\gtrsim10$ yr). This casts some doubt on the origin of the $\sim30$–50 d quasiperiodic variability as a facet of the secondary star’s magnetic activity, since both V841 Oph and DI Lac show the $\sim30$–50 d quasiperiodic behavior, but only V841 Oph displays evidence for a multi-year solar-type magnetic cycle on the secondary star. On the other hand, the $\sim30$–50 d time scale is very reminiscent of that expected for dwarf nova type behavior (although the amplitude of variability in these novae is smaller than in typical dwarf nova outbursts). This raises the possibility that this variability is caused by a disk instability (either the thermal disk instability that leads to dwarf nova outbursts operating at a low level or some other form of disk instability).
The X-ray spectrum of DI Lac is fit almost equally well by the three models we tried: a simple blackbody, a Raymond-Smith thermal plasma, and bremsstrahlung emission. More complicated models (e.g. involving multiple components, lines, etc.) are unwarranted due to the low X-ray flux. In addition to being the most physically plausible X-ray emission emission mechanism in a (non-magnetic) CV, the bremsstrahlung model is formally the best-fitting. Our X-ray spectrum model fit parameters ($kT\sim4$ keV) are consistent with those obtained from $ROSAT$ X-ray spectra of a sample of 37 disk-accreting CVs [@Rich96]. Unfortunately, the X-ray count rates in both of these systems were too low for any conclusions to be made about their time-resolved X-ray behavior. X-ray observations sample the innermost disk region, and we would expect to see different time-resolved behavior as the novae go into their optically bright state if the transition is triggered by a disk instability vs. a change in $\dot{M}$ from the secondary star. Additional time-resolved X-ray observations using the more sensitive [*Chandra*]{} and/or [*XMM*]{} X-ray satellites may be necessary to illuminate the inner workings of these old novae.
PS, DWH, and VD acknowledge support from NASA grant NAG5-4791. DWH thanks the NOAO librarian Mary Guerrieri for her valuable assistance locating several papers cited herein. This research made use of NASA’s Astrophysics Data System Abstract Service and the SIMBAD database operated by CDS, Strasbourg, France.
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[llcccl]{} 1990 Nov 12 & 2448207.7 & RoboScope & P & 240 & starting date\
1997 Jul 25 10:43:44 & 2450654.94822 & RXTE & X & 2528 & ID 20037-02-01-00\
1997 Jul 27 09:01:47 & 2450654.87744 & APO & S & 900 &\
1997 Jul 27 09:16:53 & 2450654.88792 & APO & S & 737 &\
1997 Jul 30 04:16:00 & 2450659.67921 & RXTE & X & 944 & ID 20037-02-02-00\
1997 Aug 03 04:11:12 & 2450663.68424 & RXTE & X & 1536 & ID 20037-02-03-00\
1997 Aug 07 04:12:00 & 2450667.67683 & RXTE & X & 1680 & ID 20037-02-04-00\
1997 Aug 12 03:42:40 & 2450672.65668 & RXTE & X & 592 & ID 20037-02-05-00\
1997 Aug 17 03:42:18 & 2450677.65666 & APO & S & 900 &\
1997 Aug 17 03:45:36 & 2450677.65893 & RXTE & X & 1024 & ID 20037-02-06-00\
1997 Aug 23 03:46:40 & 2450683.65991 & RXTE & X & 832 & ID 20037-02-07-00\
1997 Aug 27 03:46:56 & 2450687.66024 & RXTE & X & 736 & ID 20037-02-08-00\
1997 Sep 01 08:12:48 & 2450692.84503 & RXTE & X & 1616 & ID 20037-02-09-00\
1997 Sep 03 05:56:28 & 2450694.74974 & WIYN & S & 900 & No flux calibration\
1997 Sep 04 04:33:17 & 2450695.69269 & WIYN & S & 900 & No flux calibration\
1997 Sep 06 08:08:48 & 2450697.84239 & RXTE & X & 1776 & ID 20037-02-10-00\
1998 Nov 30 & 2451147.6 & RoboScope & P & 240 & ending date
[llcccl]{} 1991 May 31 & 2448407.7 & RoboScope & P & 240 & starting date\
1997 Apr 30 16:10:57 & 2450569.17897 & RXTE & X & 1936 & ID 20037-01-01-00\
1997 May 05 12:50:25 & 2450574.03997 & RXTE & X & 1856 & ID 20037-01-02-00\
1997 May 07 06:30:32 & 2450575.77120 & APO & S & 600 &\
1997 May 10 20:59:13 & 2450579.37966 & RXTE & X & 1936 & ID 20037-01-03-00\
1997 May 14 06:37:42 & 2450582.77618 & APO & S & 600 &\
1997 May 15 21:12:33 & 2450584.38911 & RXTE & X & 1408 & ID 20037-01-04-00\
1997 May 17 09:35 & 2450585.899 & WIYN & S & 4500 & 3 combined spectra\
1997 May 20 07:58:41 & 2450588.83795 & RXTE & X & 1792 & ID 20037-01-05-00\
1997 May 26 06:28:11 & 2450594.77523 & RXTE & X & 1872 & ID 20037-01-06-00\
1997 May 27 06:22:21 & 2450595.76552 & APO & S & 600 &\
1997 May 30 05:06:57 & 2450598.71888 & RXTE & X & 1648 & ID 20037-01-07-00\
1997 Jun 04 22:57:37 & 2450604.46244 & RXTE & X & 832 & ID 20037-01-08-00\
1997 Jun 09 06:43:45 & 2450608.78613 & RXTE & X & 2160 & ID 20037-01-09-00\
1997 Jun 18 16:24:33 & 2450618.18935 & RXTE & X & 1376 & ID 20037-01-11-00\
1998 Sep 04 & 2451060.5 & RoboScope & P & 240 & ending date
[ccc]{} 1 & 40 & 35\
2 & 29 & 36\
3 & 39 & 69\
4 & 31 & 40\
5 & 38 & 42\
6 & 75,40 & 49\
7 & 80,28 & 54\
8 & 31,21 & 48\
combined & 43,37 & 49,36
[lcccccc]{} Blackbody & 0.705 & 0.937 & $\le1.2$ & $2.2\times10^{-12}$ & $1.1^{+0.1}_{-0.2}$ & $3.0^{+0.6}_{-0.2}\times10^{-5}$\
Raymond-Smith & 0.617 & 0.982 & $\le3.3$ & $2.6\times10^{-12}$ & $3.2^{+0.7}_{-1.1}$ & $3.4^{+3.5}_{-0.6}\times10^{-3}$\
Bremsstrahlung & 0.484 & 0.999 & $\le1.8$ & $2.7\times10^{-12}$ & $4.5^{+1.3}_{-1.3}$ & $1.1^{+0.7}_{-1.1}\times10^{-3}$\
Blackbody & 0.683 & 0.954 & 0.09 & $2.2\times10^{-12}$ & $1.1^{+0.1}_{-0.1}$ & $3.1^{+0.4}_{-0.3}\times10^{-5}$\
Raymond-Smith & 0.604 & 0.986 & 0.09 & $2.7\times10^{-12}$ & $2.9^{+0.9}_{-0.5}$ & $3.7^{+0.9}_{-0.9}\times10^{-3}$\
Bremsstrahlung & 0.471 & 0.999 & 0.09 & $2.7\times10^{-12}$ & $4.2^{+1.5}_{-1.0}$ & $1.2^{+0.4}_{-0.3}\times10^{-3}$\
Blackbody & 0.689 & 0.951 & 0.24 & $2.3\times10^{-12}$ & $1.1^{+0.1}_{-0.1}$ & $3.3^{+0.3}_{-0.4}\times10^{-5}$\
Raymond-Smith & 0.606 & 0.986 & 0.24 & $2.7\times10^{-12}$ & $2.9^{+0.9}_{-0.5}$ & $3.8^{+0.9}_{-0.9}\times10^{-3}$\
Bremsstrahlung & 0.475 & 0.999 & 0.24 & $2.7\times10^{-12}$ & $4.1^{+1.4}_{-1.0}$ & $1.2^{+0.4}_{-0.3}\times10^{-3}$
[^1]: also see <http://www.apo.nmsu.edu/>
[^2]: see <http://www.noao.edu/wiyn/>
[^3]: also see <http://heasarc.gsfc.nasa.gov/docs/xte/>
[^4]: <http://heasarc.gsfc.nasa.gov/docs/xte/recipes/cook_book.html>
[^5]: This routine iteratively refits the model to the data while stepping the values of selected parameters through a given range and produces a plot in parameter space of the 2-dimensional $\chi^{2}$ contours.
[^6]: Visits 8 and 10 were not used. The former was deemed to be an “intermediate” brightness state, and the latter an anomalously low state.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We develop a Gaussian-process mixture model for heterogeneous treatment effect estimation that leverages the use of transformed outcomes. The approach we will present attempts to improve point estimation and uncertainty quantification relative to past work that has used transformed variable related methods as well as traditional outcome modeling. Earlier work on modeling treatment effect heterogeneity using transformed outcomes has relied on tree based methods such as single regression trees and random forests. Under the umbrella of non-parametric models, outcome modeling has been performed using Bayesian additive regression trees and various flavors of weighted single trees. These approaches work well when large samples are available, but suffer in smaller samples where results are more sensitive to model misspecification – our method attempts to garner improvements in inference quality via a correctly specified model rooted in Bayesian non-parametrics. Furthermore, while we begin with a model that assumes that the treatment assignment mechanism is known, an extension where it is learnt from the data is presented for applications to observational studies. Our approach is applied to simulated and real data to demonstrate our theorized improvements in inference with respect to two causal estimands: the conditional average treatment effect and the average treatment effect. By leveraging our correctly specified model, we are able to more accurately estimate the treatment effects while reducing their variance.'
author:
- Abbas Zaidi
- Sayan Mukherjee
bibliography:
- 'refs.bib'
title: '**Gaussian Process Mixtures for Estimating Heterogeneous Treatment Effects**'
---
Gaussian Process Mixture Models, Treatment Effect Estimation, Bayesian Machine Learning.
Introduction
============
The estimation of treatment effects is one of the core problems in causal inference. A treatment effect is a measure used to compare interventions in randomized experiments, policy analysis, and medical trials. The treatment effect measures the difference in outcomes between units assigned to the treatment versus those assigned to the control. There have been a variety of related approaches for estimating treatment effects including those based on graphical models [@pearl2009causal] and the potential outcomes framework [@rubin1978bayesian]. In this paper, we develop methodology that builds on the potential outcomes framework as defined in [@rubin2005causal] to estimate treatment effects.
In the potential outcomes framework we compare the observed outcome to the outcome under the counterfactual, that is, what the outcome would be under a different set of treatment conditions. If the counterfactual outcome were known then the treatment effect on an individual unit is the difference between the outcome under the observed and counterfactual interventions. The *fundamental problem of causal inference* is that in general for any unit one can only observe the outcome under a single treatment condition. As a consequence unit level causal effects are not identifiable. However, population level causal effects can be identified under some standard assumptions (see Section \[former\]). An estimator of population level effects is the average treatment effect (ATE) which is a measure of the difference in the mean outcomes between units assigned to the treatment and units assigned to the control. If treatment effects are homogenous across individuals then estimators such as the ATE that consider causal effects at an aggregate level are reasonable, however such estimators will overlook subgroup or covariate-level specific heterogeneity in treatment effects. There is evidence that heterogeneity in treatment effects is more the rule than the exception [@heckman2006understanding; @green2012modeling; @xie2012estimating].
A quantity in addressing heterogeneous treatment effects is the conditional average treatment effect (CATE) which is the average treatment effect conditional on the covariate level of a unit of observation. One can consider the CATE as a difference of two regression functions – the average response given treatment at a set of covariate levels minus the the average response assuming the control condition and the same set of covariate levels. One can estimate the ATE by marginalizing the CATE over the joint distribution of the covariates. There are a number of approaches for estimating the two aforementioned causal estimands. The main approach for modeling heterogeneous treatment effects based on the CATE is conditional mean regression. Under this approach, we model the CATE as a difference between the conditional mean outcome given the treatment for particular covariate levels minus the mean outcome given the control at the same covariate levels [@ding2017causal]. The implementation of these models can be approached both parametrically and non-parametrically.
The most popular parametric methods for estimating the difference between the conditional mean outcomes include linear and polynomial regression [@pearl2009causal], along with penalized regression approaches such as least absolute subset selection operator and ridge regression [@tibshirani1996regression]. At the other end of the spectrum are non-parametric regression models to estimate the difference between the conditional means. Examples include boosting [@powers2017some], Bayesian additive regression trees (BART) [@hill2011bayesian; @hahn2017bayesian; @chipman2010bart] as well as classical regression trees [@athey2015machine; @breiman1984classification] and random forests [@wager2017estimation; @foster2011subgroup; @breiman2001random]. These methods have some limitations to their use and we provide a brief discussion of these.
The use of random forests for CATE estimation as defined in [@wager2017estimation] provides some interesting theoretical results that allow for probabilistically valid statistical inference. These methods are theorized to outperform classical methods particularly in the presence of irrelevant covariates. This technique however, has been demonstrated to be outperformed in application [@hahn2017bayesian]. In addition, without a procedure for imposing a degree of regularization, random forests are difficult to actually deploy for heterogeneous treatment effect estimation [@wendling2018comparing]. BART and its variants [@hahn2017bayesian; @hill2011bayesian] present a persuasive argument for their use in application, but there is limited work on their formal inferential properties [@wager2017estimation] for learning heterogeneous treatment effects. Specifically for BART, formal statistical analysis is hurdled by the lack of theory arguing posterior concentration around the true conditional mean function – the key quantity of interest in heterogeneous treatment effect estimation via conditional mean regression.
An alternative to modeling the difference in conditional mean outcomes is the use of transformed responses or outcome variables (TRV) [@dudik2011doubly; @beygelzimer2009offset] that is ideologically similar to concepts of *inverse probability weighting* (IPW) [@hirano2003efficient]. The TRV approach introduces a transformation for the outcome and the treatment indicator variable for which the conditional expectation given a covariate level is equivalent to the CATE. This allows it to be used with *off-the-shelf* machine learning techniques and has been applied to optimal treatment policy estimation in the same vein as ideas of *double-robustness* as reviewed in [@ding2017causal] that combine regression adjustment with weighting. More recent work on the TRV has attempted to model it as a function of the observed covariates via regression trees [@athey2015machine] and boosting [@powers2017some]. This has raised questions of estimation quality of the approach given the high variance of the procedure. We assert that this is a consequence of the properties of the TRV that have not been explicitly accounted for in the model since past work has relied on using it as a benchmark for other methods [@athey2016recursive].
In this paper we introduce a novel non-parametric Bayesian model based on Gaussian process regression [@singh2016gaussian; @rasmussen2006Gaussian] for inference of the TRV that allows us to infer a posterior distribution on the CATE. The model we propose is a finite mixture of Gaussian-processes [@rasmussen2000infinite] that leverages the distribution implied by the transformation. This formulation is aimed at improving the overall quality of inference on the treatment effects with a correctly specified model.
This approach has benefits over both conditional mean regression and other TRV based techniques. In practice, we never estimate either the treatment nor the control function perfectly and different covariate distributions for the treatment and control groups can lead to biases in the treatment effect estimation [@powers2017some]. The TRV allows for the joint modeling of information from both the treated and control groups which can help circumvent the aforementioned estimation challenge which for instance has been discussed as a specific limitation of conditional mean regression with random forests [@wager2017estimation]. This joint modeling is also an improvement over Bayesian techniques that place individual vague priors on the treatment and control outcome models since the prior on the treatment effect as the difference of the two is possibly *doubly* vague [@hahn2017bayesian]. This can make inference a challenge since it is difficult to control the degree of heterogeneity that the model adapts to. Furthermore the TRV generates unbiased estimates for the CATE [@powers2017some].
In addition to its benefits over conditional mean regression methods, the model we introduce offers four advantages over other TRV modeling approaches. First, we significantly improve the accuracy of point estimation by explicitly modeling the distribution of the transformed outcome. Second, by modeling the distribution of the transformed outcome specifically we are able to greatly reduce the variance of causal estimands i.e. the average treatment effect and the conditional average treatment effect. Reducing the variance of the estimators is crucial since this has been the main criticism of the TRV approach [@athey2015machine; @powers2017some]. This provides tighter uncertainty intervals relative to the approaches discussed in [@athey2015machine] and [@wager2017estimation]. Third, our approach is well suited for instances when the treated and control groups share information since our proposed mechanism jointly models the behavior of both via the transformation.
The methodology we introduce makes a number of significant contributions to the estimation of heterogeneous treatment effects. Our main contribution is that we improve the overall quality of inference by improving the point estimation with a correctly specified model. In addition, the proposed framework is flexible in that we do not assume a functional form for how heterogeneity of treatment effects are driven by the levels of the observed covariates. Finally, our proposed framework is easily adapted to studies where the mechanism by which individuals receive the treatment is unknown. For this problem, past work has relied on a two-stage procedure for learning this treatment assignment mechanism first and then utilizing this in the model. We instead propose an approach whereby the treatment assignment mechanism and the treatment effects are jointly learnt in a unified framework. By working under this paradigm we have a twofold gain. First, the uncertainty quantification from our proposed model reflects uncertainty from all stages of inference including the learning of the assignment mechanism, and the treatment effects. Second as a by-product of the *feedback* in between the two estimation stages, the assignment mechanism makes more complete use of the data, which can improve estimation of causal effects.
The remainder of this paper is organized as follows: in Section \[former\], we introduce the TRV, the relevant notation and the assumptions inherent to the TRV approach. We state our new model in Section \[model\]. Our approach is benchmarked against to TRV regression trees and random forests, along with non-TRV weighted tree methods as discussed in [@athey2016recursive], as well as Bayesian tree models in [@hahn2017bayesian; @hill2011bayesian] on both simulated and real data in Section \[data\]. We close with a summary of our findings and possible areas of future work.
Transformed Response Variables Framework {#former}
========================================
In order to formulate the approach of transformed outcomes, we first define some notation that we will use throughout this paper. The observed data $\mathcal{D}$ consists of a sample of size $n$ where for each unit of observation we are given a response variable $Y_{i} \in \mathbb{R}$ and a covariate vector$X_{i} \in \mathbb{R}^{p}$. In addition to the observed data, we denote as $W_{i} \in \{0, 1\}$ the treatment assignment. The corresponding treatment assignment probability is denoted as $e_{i} = \mathbb{P}(W_{i} = 1)$. Finally, the potential outcome is denoted as $Y_{i}(W_{i} = w)$.
Under the potential outcomes framework, in order to estimate treatment effects from observational data certain assumptions about the treatment assignment mechanism need to be satisfied. Briefly, these assumptions are that the treatment assignment is *individualistic* (A1), *probabilistic* (A2) and *ignorable*(A3). Details of these assumptions are left to the reader in [@imbens2015causal]. A1 and A2 are implied under the assumption that the units of observation are a simple random sample from the target population that are independent and identically distributed.
Assumptions A2 and A3 are together known as the *strong ignorability* assumption and grants the indentifying equivalence between the potential outcome and the causal conditioning, $Y(W = w) \stackrel{P}{=} Y \mid W = w$. All three of the assumptions summarized here are always satisfied in randomized trials; In observational studies the assumptions may hold to varying degrees.
For instance, A2, which is also sometimes referred to as the overlap condition can be directly assessed. However, by comparison A3 is untestable and therefore indirect techniques are needed to determine the degree to which it is satisfied most commonly via sensitivity analyses [@rosenbaum1982assessing]. These assumptions are necessary for the formal results in the transformed response variable framework to hold.
Beyond these, we make one additional assumption that allows us to simplify the statistical the model we specify in this paper: Stable Unit Treatment Value Assumption (SUTVA) — This condition assumes no interference between observations, and that there are no multiple versions of the treatment (A4). In its absence, we would need to define a different potential outcome for the unit of observation not just for each treatment received by that unit but for each combination of treatments received by every other observation in the experiment. Relaxing these assumptions will be discussed in Section \[futurework\] as an avenue that our future work will aim to explore.
The causal estimands considered are the conditional average treatment effect (CATE), that we denote as $\tau^{CATE}$ and the average treatment effect (ATE) that we denote as $\tau^{ATE}$. $\tau^{CATE}$ is the primary estimate of interest in modeling heterogeneous treatment effects and is defined as, $$\label{eq:cate}
\tau^{CATE} = \mathbb{E}_{Y}[Y(1) - Y(0) \mid X = x],$$ the ATE can be derived by integrating over the the joint distribution of the covariates $$\label{eq:ate}
\tau^{ATE} = \mathbb{E}_{X}\Big[ \mathbb{E}_{Y}[Y(1) -Y(0) \mid X =x ] \Big] =\mathbb{E}_{X}[\tau^{CATE}].$$
The idea behind the transformed response variable apporach is to define a variable $Y_i^{*}$ for which the conditional expectation with respect to the response recovers the CATE under A3 (see Appendix \[app:A\] for a proof of this result). A transformation that satisfies the above condition is,
$$\label{eq:trv}
Y^{*}_{i} = f(W_{i}, Y_{i}, e_{i})= \frac{W_{i} - e_{i}}{e_{i}(1-e_{i})} Y_{i}.$$
The transformation requires knowledge of the probability of receiving the treatment. We assume that the treatment assignment probability depends on the observed covariate levels, or $e_{i} = e_{i}(X = x_{i})$ is a propensity score. A trivial example is when the propensity score is a fixed covariate independent value, $e_{i} = e$. This is not an example commonly seen in real observational causal inference problems and is as such not considered as a part of the model presented here, albeit [@athey2015machine; @Athey:2015:MLC:2783258.2785466; @athey2016recursive] consider it as a means of model validation.
Strengths and Weaknesses of Past Work in TRV Modeling {#LMS}
-----------------------------------------------------
TRV modeling offers three main advantages when used for estimating treatment effects as demonstrated in prior studies. Foremost amongst these is that the TRV can easily be modeled with any supervised learning method. For instance, regression trees and random forests have been used [@athey2015machine; @Athey:2015:MLC:2783258.2785466; @wager2017estimation] as has boosting [@powers2017some]. This is not an exhaustive list, and there are a myriad of other methods that can be used in conjunction with the TRV to estimate heterogeneous treatment effects. Furthermore, relative to conditional mean regression, this method does not ignore the propensity score which explicitly enters the estimation via the transformation. Finally, based on the modeling approach used, we can address treatment effect heterogeneity flexibly and therefore avoid issues arising from model misspecification since it is likely that there are complex relationships between the covariates and heterogeneity of the treatment effects. Despite their usefulness, the TRVs have some key weaknesses.
First, as mentioned in [@athey2015machine] and [@powers2017some] using TRVs as CATE estimators results in high variance estimates of the causal estimands. By construction the treatment assignment probability and the assignment itself only enter the model implicitly via the transformation and are therefore only accounted for indirectly. In addition, the treatment assignment probability only appears in the denominator, and if this is close to zero or one, the variance can spike. Similar difficulties have been seen in IPW [@hirano2003efficient] estimators, that like this transformation grant more weight to tail (read: unlikely) observations. Combining supervised learning techniques with inverse-probability weighting, gives rise to double-robust estimators, which in spirit is also similar to our modeling of the transformed outcome. [@ding2017causal] summarize that the instability of the estimator due to extreme treatment assignment probabilities is even worse in this case than in inverse-probability weighting, since there are potentially two sources of model misspecification. While we can address concerns of model misspecification using flexible machine-learning models, this flexibility is a double-edged sword. When the model generates predictions that are inherently high variance such as those of regression trees, this means that the method suffers in terms of efficiency and the quality of inference is degraded.
Second, uncertainty quantification using methods built atop inverse-probability weighting in general and transformed outcomes in specific is difficult. As discussed at length earlier, there are theoretical concerns due to the the impact of extreme weights which is a limitation of the transformation. There are also practical concerns with uncertainty quantification under specific models for the TRV as it relates to generating intervals. For single regression trees as well as the other ensemble learning methods which have been used for TRV modeling, intervals have been generated using the bootstrap. Prior work [@wager2014confidence] has suggested that in certain applications the Monte Carlo error can dominate the uncertainty quantification produced. In conjunction with the high variance inherent to the aforementioned approaches, we might be unable to gather useful insights. If treatment effects are small (near zero), the conflation of the Monte Carlo noise with the underlying sampling noise may lead us to overstate the variance and therefore lower the power of our analysis. In addition note that when the sample size is small, [@powers2017some] demonstrate that the variance of the TRV is small as well – it increases with increasing sample size. Hence, in situations where bootstrapping is likely to do well for the uncertainty in the model i.e. in large samples, the high variance of the TRV is even more so an issue.
Based on these limitations, we propose the Gaussian process mixture model in Section \[model\]. Our proposed model attempts to overcome the aforementioned limitations by leveraging the mixture distribution implied by the transformation. In addition, we still aim to model the TRV flexibly and capture the complexity of treatment effect heterogeneity. We achieve gains in the quality of inference by constructing a likelihood that reflects the error structure imposed by the TRV under some basic assumptions that earlier work with this technique has ignored. The details of these findings will be discussed in greater depth in Section \[data\] where these approaches are applied to real and simulated data.
The Gaussian Process Mixture Model {#model}
==================================
We specify a non-parametric Bayesian model based on a mixture of Gaussian processes to model heterogeneous treatment effects. Our model is based on the transformed response variable framework. It is motivated by three objectives: (1) to explicitly model the distribution implied by the transformed outcome with the goal of reducing the variance of the TRV generated estimates that have hitherto been produced using non-probabilistic models, (2) model the two treatment groups jointly so we can borrow strength and therefore improve inference even relative to non-TRV based methods for estimating treatment effects, and (3) making more complete use of the data by jointly modeling the transformed response as well as the treatment assignment probabilities in a one step model. The *feedback* between the two stages in joint modeling can improve the point estimation of treatment effects and the propensity scores [@zigler2013model]. Throughout this section we assume A1-A4 are satisfied.
Model Specification {#understanding}
-------------------
A natural starting point is to consider two non-parametric regression functions for the response under treatment and control, respectively $$\begin{aligned}
Y_{i}(1) &=& f_{1}(x_{i}) + \varepsilon_{i}(1), \quad \epsilon_{i}(1) \stackrel{\mathrm{iid}}{\sim} \mathrm{N}(0, \sigma^{2}), \\
Y_{i}(0) &=& f_{0}(x_{i}) + \varepsilon_{i}(0), \quad \epsilon_{i}(0) \stackrel{\mathrm{iid}}{\sim} \mathrm{N}(0, \sigma^{2}).\end{aligned}$$
In expectation, the difference of these two non-parametric functions is the conditional average treatment effect. Substituting these non-parametric regression functions under the treatment and control cases in the definition of the TRV in yields the following mixture model,
$$\label{eq:newmodel}
Y_{i}^{*} = g(x_{i}) + \varepsilon_{i}^{*},$$
$$\varepsilon_{i}^{*} \sim e_{i} \mathrm{N}\bigg((1-e_{i}) h(x_{i}), \frac{1}{e_{i}^{2}}\sigma^{2}\bigg) + (1-e_{i})\mathrm{N}\bigg(-e_{i} h(x_{i}), \frac{1}{(1-e_{i})^{2}}\sigma^{2}\bigg).$$
where $g(\cdot)$ is interpreted as the conditional average treatment effect,
$$g(x_{i}) = f_{1}(x_{i}) - f_{0}(x_{i}).$$
while the function $h(\cdot)$, helps expresses the multi-modal nature of the error distribution that is implied by the transformation,
$$h(x_{i}) = \frac{f_{1}(x_{i})}{e_{i}} + \frac{f_{0}(x_{i})}{1-e_{i}}.$$
A detailed derivation of this model is given in Appendix \[app:B\].
The argument for specifying the TRV mixture model rather than individual models for the treatment and control is that the conditionals $Y_i \mid X_i, W_i =1$ and $Y_i \mid X_i, W_i =0$ may not be perfectly estimable. Past work has indicated that ignoring shared information between the treated and untreated groups is a potential source of bias in the treatment effect estimation [@powers2017some]. Under the Bayesian paradigm, methods that place individual vague priors on the aforementioned conditionals make it challenging to control the degree of heterogeneity the model adapts to since the implied priors on their differences is potentially extremely vague [@hahn2017bayesian].
Our model formulation can be considered under two specifications – when the treatment assignment probabilities are known and when they need to be inferred from the data. The details of each specification are given in Sections \[simspec\] and \[compspec1\] for the two cases respectively.
### Model specification with known assignment probabilities {#simspec}
We will place Gaussian process priors on both $g$ and $h$ and will specify an inverse gamma prior on $\sigma^2$ to leverage conjugacy. Therefore, for the case where the treatment probabilities are known we specify the following model,
$$\label{basicmodel}
\begin{aligned}
Y_{i}^{*} &= g(x_{i}) + \varepsilon_{i}^{*},\\
\varepsilon_{i}^{*} \sim e_{i} \mathrm{N}\bigg((1-e_{i}) h(x_{i}), \frac{1}{e_{i}^{2}}\sigma^{2}\bigg) &+ (1-e_{i})\mathrm{N}\bigg(-e_{i} h(x_{i}), \frac{1}{(1-e_{i})^{2}}\sigma^{2}\bigg),\\
g & \sim\mathrm{GP}(0, \kappa_{g}), \\
h & \sim\mathrm{GP}(0, \kappa_{h}), \\
\sigma^2 & \sim \mathrm{IG}(a,b).
\end{aligned}$$
Here $\mathrm{IG}(a, b)$ is the inverse gamma distribution with hyper-parameters $a$ and $b$ and $\mathrm{GP}(\textbf{0}, \boldsymbol{\kappa})$ denotes the Gaussian process priors on the function $g$ and $h$. Both priors are zero mean and have covariance kernels specified (1) a non-stationary linear kernel $\kappa_{g}(u, v) = s^{2}_{0} + \sum_{i=1}^{p}s^{2}_{i}(u_{i}-c_{i})(v_{i}-c_{i})$, with hyper-parameters $s^{2}_{0}, \ldots s^{2}_{p}$ on $g$ and (2) a square exponential, $\kappa_{h}(u, v) = s_{h}^{2}\exp\{- \tau^2 \| u-v\|^2\} $ with hyper-parameters $\tau, s^{2}$ on $h$. These kernels rely on the notion of similarity between data points – if the inputs are closer together than the target values of the response, in this case the TRV are also likely to be close together. Under the Gaussian process prior, the kernel functions described above formally define what is near or similar.
The hyper-parameters $s^{2}_{0}, \ldots s^{2}_{p}$ can be interpreted in the context of linear regression with $\{\mathrm{Normal}\sim(0, s^{2}_{j})\}_{j=0}^{p}$ priors on the $p+1$ regression coefficients including the intercept. The offset $\{c_{i}\}_{i=1}^{p}$ determines the $x$ coordinate of the point that all the lines in the posterior is meant to go through. This provides some insight into how these can be set for applied modeling problems. In cases where there is a large number of covariates, many of which are thought to share information, the prior variance for those dimensions can be made small, with a higher degree of mass concentrated near zero to induce more shrinkage. In contrast, where there is a small number of important covariates the prior variance can be set to make the prior more diffuse. The offset can be set to the average of each covariates observed value. This is a general overview of the strategy that we have employed.
### Model specification with unknown assignment probabilities {#compspec1}
Computing the TRV requires knowledge of the treatment assignment probabilities $\{e_i\}_{i=1}^{n}$. In the case where these are unknown we consider them as latent variables and add extra levels to the hierarchical model specified in to model the treatment assignment probabilities. We model the assignment probabilities individually so for notational ease, later in this paper we use $\pmb{e} = \{e_i\}_{i=1}^{n}$. Our specification, *apriori*, assumes that the assignment mechanism and the outcome model are independent.
#### Modeling the Propensity Score
In order to learn the treatment assignment probabilities, we specify a probit regression model that is layered onto the model defined in . $$\label{indassign}
\begin{aligned}
W_{i} &\sim\mathrm{Ber}(e_{i}),\\
e_{i} &= \Phi(X_{i} \boldsymbol{\beta}), \\
\boldsymbol{\beta} & \sim\mathrm{N}_{p+1}(0, \Psi_{p+1 \times p + 1}).
\end{aligned}$$ Where $\Phi$ denotes the standard Normal cumulative distribution function. In this paper we will only consider the above Gaussian prior on $\boldsymbol \beta$ with prior covariance $\Psi$. However, additional complexity can be added by allowing the coefficient vector $\boldsymbol \beta$ to vary via a hierarchical prior structure as may be motivated by more complex multi-stage clustered data.
Posterior Sampling with Known Assignment Probabilities {#inference}
------------------------------------------------------
Inference for the model specified in Section \[simspec\] involves sampling from a posterior distribution via straightforward Gibbs-sampling.
We define $\mathbf{g} = (g(x_{1}), \ldots, g(x_{n}))$ and $\mathbf{h} = (h(x_{1}), \ldots, h(x_{n}))$ as the values of the two regression functions on the training data. We denote the TRV as $\textbf{Y}^{*} = (Y_{1}^{*}, \ldots, Y_{n}^{*})$ . In this case the target joint posterior distribution is $$\label{eq:jd1}
\pi(\mathbf{g}, \mathbf{h}, \sigma^{2} \mid \mathcal{D}).$$
Due to prior conjugacy the conditional distributions: $\pi(\mathbf{g} \mid \mathbf{h}, \sigma^{2}, \mathcal{D})$, $\pi(\mathbf{h} \mid \mathbf{g}, \sigma^{2}, \mathcal{D})$ and $\pi(\sigma^{2} \mid \mathbf{h}, \mathbf{g}, \mathcal{D})$ all have simple forms that we can easily sample from. We first state some matrices and vectors that will enter our calculations: $\mathbf{D}$ is an $n\times n$ diagonal matrix with entries $\mathbf{D}_{ii} = \bigg( \frac{W_i }{e_{i}^{2}} \sigma^2 + \frac{1-W_i}{(1-e_{i})^{2}}\sigma^{2}\bigg)$, $\boldsymbol{\Lambda}$ is also an $n\times n$ diagonal matrix with entries $\boldsymbol{\Lambda} _{ii} = \bigg(W_{i}(1-e_{i}) + (1-W_{i})(-e_{i})\bigg)$, $\mathbf{K}$ is also an $n\times n$ diagonal matrix with entries $\mathbf{K}_{ii} = \sigma^{2}\mathbf{D}_{ii}$, and $\mathbf{m} = \Lambda \textbf{H}$. We also denote the covariance matrix $\pmb{\kappa}_{g}$ with the $ij$-th entry as taking the value $\kappa_g(x_i,x_j)$ and similarly $\pmb{\kappa}_{h}$ is a matrix with the $ij$-th entry taking the value $\kappa_h(x_i,x_j)$. We now state the conditional distributions that will enter our Gibbs sampler, $$\label{eq:fcg}
\begin{aligned}
\pi(\mathbf{g} \mid \mathbf{h}, \sigma^{2}, \mathcal{D}) &\sim \mathrm{N}\big((\pmb{\kappa}_{g}^{-1}+\mathbf{D}^{-1})^{-1}(\mathbf{D}^{-1}\textbf{Y}^{*} - \mathbf{m}\}, \{\pmb{\kappa}_{g}^{-1}+\mathbf{D}^{-1})^{-1}\big), \\
\pi(\mathbf{h} \mid \mathbf{g}, \sigma^{2}, \mathcal{D}) &\sim \mathrm{N}\bigg((\pmb{\kappa}_{h}^{-1} + \boldsymbol{\Lambda}^{T} \mathbf{D}^{-1} \boldsymbol{\Lambda})^{-1} \boldsymbol{\Lambda}^{T} \mathbf{D}^{-1} (\textbf{Y}^{*}-\mathbf{g}), (\pmb{\kappa}_{h}^{-1} + \boldsymbol{\Lambda}^{T} \mathbf{D}^{-1} \boldsymbol{\Lambda})^{-1} \bigg), \\
\pi(\sigma^{2} \mid \mathbf{h}, \mathbf{g}, \mathcal{D}) & \sim \mathrm{IG} \bigg(a+\frac{n}{2}, b + \frac{(\textbf{Y}^{*}-\mathbf{g} - \mathbf{m})^{T}\mathbf{K}^{-1}(\textbf{Y}^{*}-\mathbf{g}- \mathbf{m})}{2}\bigg).
\end{aligned}$$ The Gibbs steps that would be used to sample from these full conditional distributions are given appendix \[app:D\].
Posterior Sampling with Unknown Assignment Probabilities {#complexinference}
--------------------------------------------------------
There are two additional problems with respect to inference when the assignment probabilities are unknown: one needs to estimate the assignment probabilities $\pmb{e}$ and use these to compute the TRV $\textbf{Y}^{*}$. The following target posterior distribution corresponds to the model when the treatment probabilities are modeled as specified by the probit augmentation to the model in .
$$\begin{aligned}
&\pi(\mathbf{g}, \mathbf{h}, \textbf{Y}^{*}, \sigma^{2},\boldsymbol{e}, \boldsymbol{\beta} \mid \mathcal{D}). \label{post3}\end{aligned}$$
In this setting the joint posterior is more complicated than equation and is harder to sample from since it cannot be completely decomposed into Gibbs steps. Generating samples requires incorporating the full conditional distributions from the previous section, along with additional steps to sample the treatment assignment probability by using a Metropolis-within-Gibbs step and constructing the transformed outcome.
The Metropolis-Hastings step consists of specifying a proposal distribution $q(\boldsymbol{\beta})$, and given a candidate value $\pmb{\beta}^* \sim q(\pmb{\beta})$ is accepted with acceptance probability, $$\label{eq:mhratio}
\alpha = \min\left(1, \frac{\pi(\mathbf{g}, \mathbf{h}, \textbf{Y}^{*},\sigma^{2}, \boldsymbol{\beta}^{*}, \pmb{e} \mid \mathcal{D}) \, q(\pmb{\beta})}{\pi(\mathbf{g}, \mathbf{h}, \textbf{Y}^{*}, \sigma^{2}, \boldsymbol{\beta}, \pmb{e} \mid \mathcal{D}) \, q(\pmb{\beta}^{*})}\right).$$ where the posterior for evaluation is given in . We have used a symmetric random walk proposals[^1] in order to reduce the overall computational burden. Once we have sampled the coefficients for the probit model, we can deterministically compute the treatment assignment probability and the TRV. The complete algorithm for this sampling scheme is detailed in appendix \[app:D\].
#### Joint Bayesian modeling and the feedback problem:
The joint Bayesian model specified in this paper for learning the assignment mechanism $\pmb{e}$ and the transformed outcome $\textbf{Y}^{*}$ leads to a feedback problem of the type described in [@zigler2013model]. The treatment assignment probability $\pmb{e}$ appears in the joint posterior distribution both as a part of the transformed outcome model through as well as its own model in . Therefore its posterior samples involve information from both. In the specific context of the assignment model, this means that the posterior samples of parameters in learning $\pmb{e}$ are informed by information from the outcome stage.
Under the classical method of using $\pmb{e}$ as a dimension reduced covariate representation in the outcome stage model (an analog to our transformed outcome), [@zigler2013model] demonstrate that the estimation of causal effects is poor. There is a possibility of considerable bias due to the distortion of the causal effects. Furthermore, the usefulness of the propensity score adjustment as a replacement for the covariates is also compromised.
However, this is not the concern in the modeling scheme proposed in this paper. [@zigler2013model] show that the nature of the feedback between the two stages is altered when the outcome stage model is augmented with adjustment for the individual covariates and that this method can recover causal effects akin to when a classical two stage procedure is used. Our approach via the kernels of the Gaussian processes provides individual covariate adjustment therefore alleviating concerns created by the feedback. Therefore we reap the benefits of the joint estimation, but by means of suitably elicited priors, and individually controlled covariates, we bypass the concerns of feedback. In fact, by making more complete use of the data, we are arguably able to improve the overall quality of estimation.
Results on Simulated and Real Data {#data}
==================================
In this section we validate our Gaussian process based TRV model on simulated and real data. We use the simulations to show that our approach outperforms other techniques (both TRV as well as conditional mean regression type methods). This holds true both when the treatment assignment probabilities are known or need to be inferred from the data. We also observe on the simulated data that our model does in fact recover the causal effects in the TRV framework in the presence of feedback as theorized earlier. Our assertion is based on comparisons of mean squared error, bias and point-wise coverage of the uncertainty intervals generated by the model.
The real data analyzed here comes from a study of the causal effects of debit card ownership on household spending in Italy [@mercatanti2014debit] – we will refer to these data as the SHIW data. In the analysis of the SHIW data we jointly infer treatment effects as well as the treatment assignment probability for each individual, as these are not observed.
The most interesting aspect of our analysis of the SHIW data is that we are able to identify heterogeneity in the treatment effects. We find that the impact of debit card usage on aggregate household spending is found to vary based on income and this variability is highest at the lowest levels of income – a notion that is validated under behavioral economic theory which further lends credibility to our proposed model.
Estimands Used and Modelling Approaches Compared
------------------------------------------------
In this section we state the estimands that we will use for comparing our method to other non-parametric methods. We will also state in detail how we compute the relevant estimand for both our method and the other techniques considered. The analysis is focused on the estimation of the CATE.
#### Gaussian process mixture model:
We first specify the procedure we use to estimate the CATE for our model. The model is trained on data $(x_1,...,x_n)$ and the values of the two functions are $$\begin{aligned}
{\mathbf{g}} = (g(x_{1}), \ldots, g(x_{n})), \\
{\mathbf{h}} = (h(x_{1}), \ldots, h(x_{n})).\end{aligned}$$ We will use the function values to evaluate the accuracy of our estimators.
Depending on whether the treatment assignment probabilities are observed or not we obtain posterior samples $\left(\mathbf{g}^{(j)},\mathbf{h}^{(j)}\right)_{j=1}^K$ or $\left(\mathbf{g}^{(j)},\mathbf{h}^{(j)}, \boldsymbol{e}^{(j)}\right)_{j=1}^K$, respectively, using which we can compute posterior samples for the conditional average treatment effect at each location $x_i$, $i=1,...,n$ as $${\tau_{i}^{CATE}}^{(j)}(x_i) = g^{(j)}(x_i).$$ Given the posterior samples we can compute a posterior mean as a point estimate, $\widehat{\tau_{i}^{CATE}}$ along with its corresponding credible intervals. Where applicable, marginalizing over the values $x_{i}$ allows us to compute posterior estimates of the average treatment effect $\widehat{\tau_{i}^{ATE}}$.
Based on the quantities that we have specified above, the mean squared error, bias and coverage used for model validation are specified as follows,
$$\mathrm{Mean \>\>\> Squared \>\>\> Error} = \frac{1}{n}\sum_{i=1}^{n}(\tau_{i}^{CATE}-\widehat{\tau_{i}^{CATE}})^{2},$$
$$\mathrm{Bias} = \frac{1}{n}\sum_{i=1}^{n}(\tau_{i}^{CATE}-\widehat{\tau_{i}^{CATE}}),$$
$$\mathrm{Coverage} = \frac{1}{n}\sum_{i=1}^{n} \textbf{1}(\tau_{i}^{CATE}\in[\tau_{i}^{CATE, \>\>\> lwr}, \tau_{i}^{CATE, \>\>\> upr}]).$$
#### Summary of alternative methods used:
We will compare our proposed Gaussian process mixture model approach to other regression based methods for estimating treatment effects. We have considered random forests and single regression trees for treatment effect estimation via TRV modeling as well as *fit based trees*, *causal trees* [@athey2016recursive], and BART[@hahn2017bayesian; @hill2011bayesian] as non-TRV alternatives [^2].
None of the aforementioned methods have an obvious framework for learning the treatment assignment probabilities internally. This a crucial step in computing the CATE and ATE both via TRV and non-TRV based estimation techniques. In the case of the regression trees and random forests for TRV modeling, the TRV needs to be computed from the learnt propensity score first before any modeling can commence. The BART model uses the propensity score as an additional covariate, while causal and fit based trees use the propensity score as a weighting mechanism.
Therefore, we will use a two-step procedure where we first use the data to infer the treatment assignment probabilities and then given these estimates, apply the aforementioned regression methods to estimate the treatment effect. The treatment assignment probability vector $\pmb{e}$ is estimated via logistic regression [@rubin1996matching], a standard approach for estimating propensity scores in the causal inference literature.
Results on Simulated Data
-------------------------
The objective of the simulation studies presented in this section is to compare the performance of the Gaussian process mixture model to, BART, causal trees, fit based trees, the random forest and single regression tree models. We consider two criteria in our comparison. The first criteria is a comparison of the accuracy of the CATE, in terms of mean squared error and bias. The second criterion involves assessing how well the methods quantify uncertainty by considering the coverage of the intervals produced by all the models.
### Simulated Data Model {#simstudy}
In order to evaluate the proposed model as well as the other aforementioned approaches, we consider two simulation settings – one high dimensional case (with 40 covariates) and one low dimensional case (with 5 covariates) each with its own covariate level heterogeneity and a sample size of $n = 250$. For the remainder of this analysis, the high dimensional case is referred to as Case A, and the low dimensional case is referred to as Case B. By design neither of these simulation cases has a meaningful average treatment effect. We start with a detailed description of Case A.
In this framework, covariates $X_{1}, \ldots X_{30}$ are independent covariates, $X_{31}, \ldots X_{35}$ depend on pairs of covariates, while $X_{36}, \ldots, X_{40}$ depend on groups of three as follows,
$$\begin{aligned}
X_{k} &\sim \mathrm{Normal}(0, 1); \>\>\> k = 1, \ldots, 15,\\
X_{k} &\sim \mathrm{Uniform}(0, 1); \>\>\> k = 16, \ldots, 30,\\
X_{k} &\sim \mathrm{Bernoulli}(q_{k}); \>\>\> q_{k} = \mathrm{logit}^{-1}(X_{k-30} - X_{k-15}); \>\>\> k = 31,\ldots, 35,\\
X_{k} &\sim \mathrm{Poisson}(\lambda_{k}); \>\>\> \lambda_{k} = 5 + 0.75 X_{k-35}(X_{k-20} + X_{k-5}); \>\>\> k = 36,\ldots, 40.\\\end{aligned}$$
Next, we simulate the propensity score and the corresponding treatment assignments. This has been done as a simple linear transformation since the focus of the paper is not propensity score modeling but rather CATE modeling. The propensity scores and the treatment effects of interest for Case A are given in figure \[fig:simLarge\].
$$\begin{aligned}
p_{i} &= \mathrm{logit}^{-1}(0.3 \sum_{k = 1}^{5}X_{k} -0.5 \sum_{k = 21}^{25}X_{k} -0.0001 \sum_{k = 26}^{35}X_{k} + 0.055 \sum_{k = 36}^{40}X_{k} ),\\
W &\sim \mathrm{Bernoulli}(p_{i}).\end{aligned}$$
Finally we generate the potential outcomes and the observed outcomes. $$\begin{aligned}
f(\mathbf{X}) &= \frac{\sum_{k = 16}^{19} X_{k}\exp(X_{k+14})}{1+\sum_{k = 16}^{19}X_{k} \exp(X_{k+14})},\\
Y(0) &= 0.15 \sum_{k=1}^{5}X_{k} + 1.5 \exp(1+1.5 f(\mathbf{X}))+ \epsilon_{i},\\
Y(1) &= \sum_{k = 1}^{5}\{ 2.15 X_{k} + 2.75 X_{k}^{2} + 10 X_{k}^{3}\} + 1.25 \sqrt{0.5 + 1.5\sum_{k = 36}^{40}X_{k}} + \epsilon_{i},\\
Y &= WY(1) + (1-W)Y(0);\>\>\> \epsilon_{i} \stackrel{\mathrm{IID}}{\sim} \mathrm{Normal}(0, 0.0001).\end{aligned}$$
\
The lower dimensional case, which we have adapted from the simulation study in [@hahn2017bayesian] is presented similarly. We start by simulating the following 5 covariates, $$\begin{aligned}
X_{k} &\sim \mathrm{Normal}(0, 1); \>\>\> k = 1, \ldots, 3,\\
X_{4} &\sim \mathrm{Bernoulli}(p = 0.25), \\
X_{5} &\sim \mathrm{Binomial}(n = 2, p = 0.5).\end{aligned}$$
In this scheme, unlike Case A, all the covariates are independent. The propensity score model analogous to the previous case is a linear transformation of the covariates.
$$\begin{aligned}
p_{i} &= \mathrm{logit}^{-1}(0.1X_{1}-0.001X_{2}+.275X_{3}-0.03X_{4}),\\
W &\sim \mathrm{Bernoulli}(p_{i}).\end{aligned}$$
Finally we generate the potential outcomes and the observed outcomes. The results of this simulation are presented in figure \[fig:simSmall\].
$$\begin{aligned}
f(\mathbf{X}) &= -6 + h(X_{5}) + |X_{3}-1|,\\
h(0) &= 2, \>\>\> h(1) = -1, \>\>\> h(2) = -4,\\
Y(0) &= f(X) - 15 X_{3} + \epsilon_{i},\\
Y(1) &= f(X)+ (1 + 2X_{2}X_{3}) + \epsilon_{i},\\
Y&= WY(1) + (1-W)Y(0); \>\>\> \epsilon_{i} \stackrel{\mathrm{IID}}{\sim} \mathrm{Normal}(0, 0.0001).\end{aligned}$$
\
### Comparison of Methods {#ressim}
The first stage of our analysis compares the CATE estimation in instances when the treatment assignment probability is assumed to be known. We focus on the mean squared error, bias and coverage of the CATE under Case A and Case B along with visual analyses of model adaptability to gauge fit quality. For the proposed model the samplers were run for $K = 6, 000$ steps with $1,000$ initial steps burned off. No thinning of the samples generated was needed. Similarly, for the non-Bayesian methods, $K = 5, 000$ replications of the bootstrap were generated. The comparison of point estimates of the CATE under Case A is presented in figure \[fig:caseAComparisonKnown\] and Case B in figure \[fig:caseBComparisonKnown\] for the sub-case where the treatment assignment mechanism is known; the corresponding diagnostic measures are presented in tables \[tb:caseASummaryKnown\] and \[tb:caseBSummaryKnown\].
\
\
\
In Case A, both in terms of point estimation, as well as uncertainty quantification, we can conclude that when the treatment assignment is known, the proposed model is the overall winner. As we can see, it adapts well to the heterogeneity of the treatment effects in the data, and is able to recover the effects to a high degree as observed in figure \[fig:caseAComparisonKnown\](a). It also has the lowest mean squared error of the models presented and the point-wise coverage of its uncertainty intervals, while low relative to tree based methods, is better than BART (see table \[tb:caseASummaryKnown\]). Furthermore, the bias of the model is generally lower than causal trees, fit based trees and transformed outcome trees.
It warrants mention that BART only adapts to heterogeneity minimally. We can attribute this to the complexity of regularization in causal inference problems [@hahn2017bayesian] from the shrinkage prior as well as poor mixing of the MCMC used for BART in high dimensions [@pratola2016efficient]. We see similar behavior from transformed outcome trees, where post-estimation *pruning* can lead to regularization induced bias as well. An elaborate discussion on bias in causal inference applications from regularized models originally designed for prediction is given in [@hahn2017bayesian] and [@hahn2018regularization].
\
\
\
In Case B, the model performs well in terms of recovering the high degree of heterogeneity but it suffers in terms of mean square error and bias. The model still adapts well to the heterogeneity inherent in the data, and is able to recover the effects as observed in figure \[fig:caseBComparisonKnown\](a), albeit with a higher degree of overall noise. This noisiness translates to high mean squared error and bias, where the other alternative models perform better, with one minor caveat. Due to the piece-wise nature of the tree based models, they do not adapt to the heterogeneity as well as the proposed model and BART do. Furthermore, the model also has the highest degree of point-wise uncertainty interval coverage (see table \[tb:caseBSummaryKnown\]).
Model Type Mean Square Error Bias 95% CI Coverage
--- ----------------------------------- ------------------- -------- -----------------
1 Gaussian-Process Mixture 4191.665 13.207 0.780
2 Bayesian Additive Regression Tree 5856.135 -5.351 0.596
3 Transformed Outcome Tree 7769.077 14.374 0.876
4 Fit Based Tree 6154.396 15.633 0.928
5 Causal Tree 8390.039 21.923 0.964
6 Transformed Outcome Random Forest 4993.576 0.317 0.932
: Case A - Conditional Average Treatment Effect Summary (Known)[]{data-label="tb:caseASummaryKnown"}
We also compare the CATE estimation for both cases when the treatment assignment probabilities are unknown and need to be inferred from the data. The comparison of the point estimation is given in figures \[fig:caseAComparisonUnknown\] and \[fig:caseBComparisonUnknown\] respectively for the two cases, with the corresponding summary measurements of fit in tables \[tb:caseASummaryUnknown\] and \[tb:caseBSummaryUnknown\].
Model Type Mean Square Error Bias 95% CI Coverage
--- ----------------------------------- ------------------- -------- -----------------
1 Gaussian Process Mixture 50.262 3.174 0.988
2 Bayesian Additive Regression Tree 5.498 0.229 0.808
3 Transformed Outcome Tree 16.421 0.202 0.900
4 Fit Based Tree 15.620 0.282 0.952
5 Causal Tree 21.143 0.974 0.972
6 Transformed Outcome Random Forest 118.745 -0.582 0.816
: Case B - Conditional Average Treatment Effect Summary (Known)[]{data-label="tb:caseBSummaryKnown"}
For Case A, the performance of the model is far superior in terms of adapting to the heterogeneity, as indicated in figure \[fig:caseAComparisonUnknown\](a), in particular compared to the performance of the transformed outcome random forest and BART given in figures \[fig:caseAComparisonUnknown\](c) and \[fig:caseAComparisonUnknown\](f). The deterioration in the quality of the estimates from BART is particularly noticeable and can be attributed to the same over-regularization observed before which is even more of a concern since there is additional uncertainty from the learning of the assignment mechanism. Furthermore, while the point-wise coverage of the uncertainty interval is lower relative to the other models, the Gaussian process mixture is the clear winner in terms of the mean square error. The proposed model also outperforms the tree based models (causal and fit based trees as well as transformed outcome trees) in terms of bias (see table \[tb:caseASummaryUnknown\]) and its point-wise interval coverage is stable relative to BART, which speaks to the models overall robustness despite the added layer of complexity from learning the assignment mechanism.
\
\
\
We see that for Case B, the results of the analysis are similar to when the treatment assignment was known. The performance of the model is comparable in terms of adapting to the heterogeneity relative to the other models, as indicated in figure \[fig:caseBComparisonUnknown\](a) – albeit again with a similar degree of noisiness as earlier. However, we again out-perform transformed outcome random forests in terms of point estimation with lower mean squared error. The only aspect in which the model out performs all the other methods considered is in terms of point-wise interval coverage.
\
\
\
Our conclusion is that the model performs well when there are a large number of covariates present, and the degree of heterogeneity in the treatment effects is high. The flexibility of the mixture of Gaussian processes ensures adaptability, where tree based models fail particularly when there is shared information in the covariates (as is true in Case A) since the prior provides some degree of built-in regularization that is not as excessive as that of BART. However, when the number of covariates is small, the flexibility of the model hurts its overall performance since we observe that our estimates are generally noisier. These limitations of the model are discussed as avenues for future work in the last section of this paper.
Model Type Mean Square Error Bias 95% CI Coverage
--- ----------------------------------- ------------------- --------- -----------------
1 Gaussian Process Mixture 3916.562 13.207 0.780
2 Bayesian Additive Regression Tree 6754.058 -5.569 0.624
3 Transformed Outcome Tree 6289.891 7.061 0.880
4 Fit Based Tree 6154.396 15.633 0.932
5 Causal Tree 8390.039 21.923 0.968
6 Transformed Outcome Random Forest 12124.426 -21.958 0.960
: Case A - Conditional Average Treatment Effect Summary (Unknown)[]{data-label="tb:caseASummaryUnknown"}
Model Type Mean Square Error Bias 95% CI Coverage
--- ----------------------------------- ------------------- -------- -----------------
1 Gaussian Process 31.517 1.898 1.000
2 Bayesian Additive Regression Tree 6.259 0.118 0.776
3 Transformed Outcome Tree 16.421 0.202 0.892
4 Fit Based Tree 15.620 0.282 0.956
5 Causal Tree 19.652 0.876 0.972
6 Transformed Outcome Random Forest 115.329 -0.349 0.820
: Case B - Conditional Average Treatment Effect Summary (Unknown)[]{data-label="tb:caseBSummaryUnknown"}
Results on the Italy Survey on Household Income and Wealth (SHIW) {#real}
-----------------------------------------------------------------
Our application of the GP mixture model to a real data aimed at the estimation the causal effects of debit card ownership on household spending. A causal analysis of this question was developed in [@mercatanti2014debit] using data from the Italy Survey on Household Income and Wealth (SHIW) to estimate the population average treatment effect for the treated (PATT). The SHIW is a biennial, national population representative survey run by Bank of Italy. The subset of the SHIW data we considered consists of $n = 564$ observations with 385 untreated and 179 treated observations. The outcome variable is the monthly average spending of the household on all consumer goods. The treatment condition is whether the household possesses one and only one debit card, and the control condition is that the household does not possess *any* debit cards. The covariates we used include: cash inventory held by the household, household income, average interest rate in the province where the household resides, measurement of wealth, and the number of banks in the province in which the household resides. See [@mercatanti2014debit] for more details about the data. Our analysis of these data will consist of comparing estimates of the ATE and CATE (with respect to household income) of our GP mixture model to the same alternative models as the previous section.
Decile $Mean \quad Income$ $\widehat{\tau^{CATE}}$ $\widehat{\tau^{CATE}_{lwr}}$ $\widehat{\tau^{CATE}_{upr}}$
-------- --------------------- ------------------------- ------------------------------- ------------------------------- --
1 -1.137 0.629 0.404 0.857
2 -0.831 0.567 0.374 0.761
3 -0.638 0.558 0.381 0.734
4 -0.472 0.459 0.298 0.620
5 -0.310 0.425 0.270 0.578
6 -0.114 0.396 0.245 0.546
7 0.103 0.343 0.190 0.490
8 0.397 0.272 0.097 0.441
9 0.848 0.172 -0.050 0.389
10 2.143 -0.125 -0.513 0.251
: Conditional average treatment effect with average income by decile[]{data-label="tb:cateIncomeRealModel"}
We start with a presentation of the CATE under our model against income in \[fig:realComparison\](a).The proposed model estimates an overall downward trend in the effect of owning a debit card, i.e. as the level of income increases, the effect of owning a debit card declines. In order to summarize this effect, we consider the CATE for binned deciles of income for the proposed model in figure \[fig:realComparison\](b) and the alternative models in figure \[fig:realComparison\](c). We find that the proposed model detects a statistically meaningful effect for the first eight deciles of income, and this effect is estimated to decline in size. For the final two deciles, the model concludes that there is no statistically meaningful effect of owning a debit card. These results are summarized in table \[tb:cateIncomeRealModel\]. By comparison, the inference from the alternative approaches is not quite as clear. BART and transformed outcome trees, detect minimal heterogeneity. With BART, this flattening can be attributed to over-regularization due to the prior, as seen in the simulated data case, while for transformed outcome trees, the axis-parallel splits used to estimate the model are not always suitable for partitioning the covariates. By comparison transformed outcome random forests, transformed outcome trees and causal trees demonstrate the most heterogeneity at the highest two deciles of income. These results are summarized in table \[tb:comparisonTable\] in Appendix \[app:C\].
In order to be comprehensive and comparable to past work, we have also produced estimates of the average treatment effect in table \[tb:ateReal\]. The proposed Gaussian process mixture detects a statistically meaningful ATE. This result is consistent with the findings of [@mercatanti2014debit]. Furthermore, we also see that the uncertainty interval for the Gaussian process mixture is the tightest of the methods used here, all of which with the exception of BART generate similar inference. This result is consistent with the findings on simulated data presented in the last section since the BART model does not adapt to heterogeneity well in instances where the number of covariates is high with large contributions to the variation in the treatment effects. Again this argues that the GP mixture model may be outperforming the other methods.
Model Type $\widehat{\tau^{ATE}}$ $\widehat{\tau^{ATE}_{lwr}}$ $\widehat{\tau^{ATE}_{upr}}$
--- ----------------------------------- ------------------------ ------------------------------ ------------------------------
1 Gaussian Process Mixture 0.369 0.220 0.518
2 Transformed Outcome Tree 0.470 0.210 0.555
3 Fit Based Tree 0.378 0.214 0.608
4 Causal Tree 0.475 0.360 0.939
5 Bayesian Additive Regression Tree 0.115 -1.129 1.397
6 Transformed Outcome Random Forest 0.414 0.229 0.604
: Comparison of average treatment effects.[]{data-label="tb:ateReal"}
Based on the economic concepts of *income elasticity of demand*, *consumer choice* and *substitution effects* [@varian2014intermediate], the heterogeneity identified at the lowest levels of standardized income is a more sensible result relative to the implication of the other approaches. At the lowest levels of income, economic agents are more likely to substitute debit card use for cash in an effort to maximize spending. The debit cards act as an inflator of perceived financial resources and this effect is expected to diminish as the overall income grows. Therefore, the GP mixture model makes a more convincing case for capturing the true nature of how holding a debit card influences spending.
\
Discussion and Future Work {#futurework}
==========================
We have proposed a novel non-parametric Bayesian model to estimate heterogeneous treatment effects. Our approach combines the *transformed response variable* framework with a mixture of Gaussian-processes. The motivation for the GP mixture model was to improve the accuracy of our point estimates as well as to better quantify uncertainty relative to other models particularly those from the Bayesian non-parametrics literature. We compared the performance of our technique to a single regression tree and random forest model within the TRV framework as well as two conditional mean regression type weighted tree based methods and BART. We used simulation studies to show instances where our approach is a better estimator with respect to both point estimation and uncertainty quantification. Furthermore, our approach also has the advantage in that we can address the case where treatment assignment probabilities are unknown within our model; other methods require a two-stage process where another model is required to infer the treatment assignment probabilities. This tandem estimation provides better insight into the data generating process and also captures uncertainty from all levels of inference.
In addition, a Bayesian model of treatment effects with a single likelihood for the design and analysis stages creates concerns of feedback since the TRV depends on the assignment mechanism. We demonstrate that our model is robust to this feedback due to both our prior specification as well as individual covariate adjustment via the Gaussian process covariance functions. However, this raises the theoretical question of whether there is a weaker condition that can be satisfied and still lead to effective inference of treatment effects which is the first area that we aim to explore in future work.
There are several ways we can extend our model to be more robust and flexible. In the context of robustness, the GP prior covariance functions specified impose smoothness assumptions on the treatment effects that may not be realistic in a myriad of applied settings. Relaxing the smoothness and using non-parametric models that have been developed to model dose-response curves may result in richer and more reliable inference. Furthermore, as noted earlier inference using the TRV is sensitive to the probability of receiving the treatment and can create biases and instability when the assignment probability are close to their extremes. While we have addressed instability in the estimation of effects using a correctly specified model and indirectly improved propensity score estimation, we have not directly curbed the susceptibility of the method to extreme weights. The variance of the mixture model is still influenced by the reciprocal of the treatment assignment probability (as is the case generally with IPW estimators). Extending our model to be more insensitive to these extreme cases is vital in application.
Under the theme of model flexibility, we are currently fixing the hyper-parameter values within the kernels of the Gaussian-processes since attempting to learn these from the data creates two problems that we need to carefully study. First, learning these parameters is difficult from a sampling perspective since the target distributions are often extremely multi-modal. A promising avenue for addressing this is the use of a combination of sampling and optimization [@levine2001implementations] – this is particularly important since Bayesian non-parametric methods are known to be sensitive to prior calibration. This is crucial in instances where the degree of heterogeneity in treatment effects is small as we have seen via simulation study. Second, the scalability of Gaussian processes is very limited [@johndrow2015approximations] and hence increasing the number of parameters that we are attempting to learn hurts the scalability even more. This broadly summarizes the areas that we will explore in future work.
Proof of Equivalence {#app:A}
====================
We now show that the transformation presented in section \[former\] in expectation recovers the CATE i.e. $$\mathbb{E}_{Y}[Y^{*} \mid X = x] = \tau^{CATE}.$$
First observe that $Y_{i} = Y_{i}(W_{i}) = W_{i}Y_{i}(1) + (1 - W_{i})Y_{i}(0).$
By the definition of the TRV $$\begin{aligned}
A = \mathbb{E}_{Y}[Y^{*} \mid X = x, \mathcal{D}] &=& \mathbb{E}_{Y} \left[\frac{W-e_{i}}{e_{i}(1-e_{i})}Y \mid X = x, \mathcal{D}\right], \\
&=& \frac{1}{e_{i}(1-e_{i})}\left( \mathbb{E}_{Y}[YW \mid X = x, \mathcal{D}] - e_{i} \mathbb{E}_{Y}[Y \mid X=x, \mathcal{D}]\right).\end{aligned}$$
Due to the ignorability of the treatment assignment the following holds $$\begin{aligned}
A & = & \frac{1}{e_{i}(1-e_{i})}(e_{i}\mathbb{E}_{Y}[Y \mid W = 1, X=x, \mathcal{D}] - e_{i} \mathbb{E}_{Y}[Y \mid X = x, \mathcal{D}]) \\
&=&\frac{1}{1-e_{i}}\mathbb{E}_{Y}[Y \mid X = x, W =1,\mathcal{D}] - \frac{1}{1-e_{i}}\mathbb{E}_{Y}[Y \mid X=x, \mathcal{D}].\end{aligned}$$ By iterating expectations the following holds: $$\begin{aligned}
A &=&\frac{1}{1-e_{i}}\mathbb{E}_{Y}[Y \mid W=1, X=x, \mathcal{D}] - \frac{1}{1-e_{i}}\mathbb{E}_{W}[\mathbb{E}_{Y}[Y \mid W=1,X=x, \mathcal{D}]], \\
&=&\frac{1}{1-e_{i}}\mathbb{E}_{Y}[Y \mid W = 1,X_{i}=x, \mathcal{D}] - \frac{e_{i}}{1-e_{i}}\mathbb{E}_{Y}[Y \mid W= 1,X=x, \mathcal{D}] - \\
& &\mathbb{E}_{Y}[Y \mid W = 0, X=x, \mathcal{D}].\end{aligned}$$ Collecting the first two terms provides the desired result $$A=\mathbb{E}_{Y}[Y \mid W = 1, X= x, \mathcal{D}] - \mathbb{E}_{Y}[Y \mid W =0, X=x, \mathcal{D}].$$
Derivation of Model {#app:B}
===================
The derivation of the model presented in the paper begins with the transformation of interest given as follows, with $Y_{i}$ denoting the observed response, $W_{i}$ the assigned treatment and $e_{i} = P(W_{i} = 1)$ $$Y_{i}^{*} = \frac{W_{i} - e_{i}}{e_{i}(1-e_{i})}Y_{i}.$$ In addition, we define the two regression functions for the outcome, one under the treatment and one under the control, $$\begin{aligned}
(Y_{i}|W_{i} = 0) = f_{0}(X_{i})+\epsilon_{i}(0),\\
(Y_{i}|W_{i} = 1) = f_{1}(X_{i})+\epsilon_{i}(1).\\\end{aligned}$$ Using the transformation, and substituting the regression functions under the two cases i.e. when $W_{i} = 1$ and when $W_{i} = 0$ and assuming further that $\epsilon(1), \epsilon(0) \stackrel{IID}{\sim}\mathrm{N}(0, \sigma^{2})$, we can define with probability $e_{i}$,
$$\begin{aligned}
(Y_{i}^{*}|W_{i} = 1) &= \frac{f_{1}(X_{i}) - e_{i}f_{1}(X_{i})+e_{i}f_{0}(X_{i})}{e_{i}} + f_{1}(X_{i})-f_{0}(X_{i}) + \frac{\epsilon_{i}(1)}{e_{i}},\\
&= f_{1}(X_{i})-f_{0}(X_{i}) + (1-e_{i})\bigg(\frac{ f_{1}(X_{i})}{e_{i}}+\frac{f_{0}(X_{i})}{1-e_{i}} \bigg)+\frac{\epsilon_{i}(1)}{e_{i}},\\
&= g(X_{i})+ (1-e_{i})h(X_{i})+\frac{\epsilon_{i}(1)}{e_{i}}.\end{aligned}$$
and similarly, with probability $1-e_{i}$ that,
$$\begin{aligned}
(Y_{i}^{*}|W_{i} = 0) &= \frac{-(1-e_{i})f_{1}(X_{i}) +(1- e_{i})f_{0}(X_{i})-f_{0}(X_{i})}{e_{i}} + f_{1}(X_{i})-f_{0}(X_{i}) - \frac{\epsilon_{i}(0)}{1-e_{i}},\\
&= f_{1}(X_{i})-f_{0}(X_{i}) + (-e_{i})\bigg(\frac{ f_{1}(X_{i})}{e_{i}}+\frac{f_{0}(X_{i})}{1-e_{i}} \bigg)-\frac{\epsilon_{i}(0)}{1-e_{i}},\\
&= g(X_{i})+ (-e_{i})h(X_{i})+\frac{\epsilon_{i}(0)}{e_{i}-1}.\end{aligned}$$
This yields the mixture model model that we have presented in the paper, $$\begin{aligned}
Y_{i}^{*} &= g(X_{i}) + \varepsilon_{i},\\
\varepsilon_{i}\sim(e_{i})\mathrm{Normal}((1-e_{i})h(X_{i}), \frac{\sigma^{2}}{e_{i}^{2}})&+(1-e_{i})\mathrm{Normal}(-e_{i}h(X_{i}), \frac{\sigma^{2}}{(1-e_{i})^{2}}).\end{aligned}$$
Comparison of SHIW Data {#app:C}
=======================
This section presents comparative analysis using various methods for the CATE estimation for the SHIW data using the Gaussian process mixture in section \[real\]. Point estimates of the CATE along with $95\%$ uncertainty intervals for each decile of income, along with the average value of income in that decile are presented in table \[tb:comparisonTable\].
Sampling Algorithms for Model Specifications {#app:D}
============================================
#### Algorithm for inference with known assignment probabilities:
For the full conditional distributions specified in we can run the following Gibbs sampling procedure to generate a sequence $(\mathbf{g}^{(j)}, \mathbf{h}^{(j)}, \sigma^{((j)})_{j=1}^K$ as follows,
1. Initialize ${\mathbf h}^{(0)}$, $\sigma^{(0)}$, and ${\mathbf g}^{(0)}$;
2. For $j= 1,...,K$
1. $\mathbf{g}^{(j)} \sim \pi(\mathbf{g} \mid \mathbf{h}^{(j-1)}, \sigma^{(j-1)}, \mathcal{D})$;
2. $\mathbf{h}^{(j)} \sim \pi(\mathbf{h} \mid \mathbf{g}^{(j)}, \sigma^{(j-1)}, \mathcal{D})$ ;
3. $ \sigma^{(j)} \sim \pi(\sigma \mid \mathbf{h}^{(j)}, \mathbf{g}^{(j)}, \mathcal{D}).$
Given the sequence $(\mathbf{g}^{(j)}, \mathbf{h}^{(j)}, \sigma^{(j)})_{j=1}^K$ we discard an initial $K_0$ of the samples to address burn-in of the chain and we thin the remaining samples by a small factor $\gamma$ to obtain independent samples from the joint posterior distribution in section \[simspec\] and in equation . We will specify the burn-in and thinning settings whenever we discuss applications of the method.
#### Algorithm for inference with unknown assignment probabilities:
For the full posterior stated in equation a standard Gibbs sampling procedure of the type specified above cannot be used for sampling the treatment assignment probabilities. We use a näive approach to sampling the assignment probabilities in addition to the other model parameters with an additional Metropolis-within-Gibbs step. This results in the following procedure:
1. Initialize ${\mathbf h}^{(0)}$, $\sigma^{(0)}$, ${\mathbf g}^{(0)}$, and $\boldsymbol{\beta}^{(0)}$. Use $\boldsymbol{\beta}^{(0)}$to compute $\boldsymbol{e}^{(0)}$;
2. Compute $\mathbf{Y}^*$ from the initial $\boldsymbol{e}^{(0)}$ and data;
3. For $j= 1,...,K$
1. $\mathbf{g}^{(j)} \sim \pi(\mathbf{g} \mid \mathbf{h}^{(j-1)}, \sigma^{(j-1)}, \boldsymbol{e}^{(j-1)}, \mathbf{Y}^*, \mathcal{D})$;
2. $\mathbf{h}^{(j)} \sim \pi(\mathbf{h} \mid \mathbf{g}^{(j)}, \sigma^{(j-1)}, \boldsymbol{e}^{(j-1)}, \mathbf{Y}^*, \mathcal{D})$ ;
3. $ \sigma^{(j)} \sim \pi(\sigma \mid \mathbf{h}^{(j)}, \mathbf{g}^{(j)}, \boldsymbol{e}^{(j-1)}, \mathbf{Y}^*, \mathcal{D})$;
4. Use Metropolis-Hastings step to sample $\boldsymbol{\beta}^{(j)}$;
5. Compute $\boldsymbol{e}^{(j)}$ from $\boldsymbol{\beta}^{(j)}$ and data;
6. Compute $\mathbf{Y}^*$ from $\boldsymbol{e}^{(j)}$ and data.
Therefore using the steps in the algorithm above we simulate a sequence $(\mathbf{g}^{(j)}, \mathbf{h}^{(j)}, \sigma^{(j)}, \pmb{\beta}^{(j)},$ $\pmb{e}^{(j)},\mathbf{Y}^{*(j)})_{j=1}^K$ akin to earlier with burn-in and thinning considerations that reflects draws from the joint distribution in section \[compspec1\] in .
Decile $Mean \quad Income$ $tot$ $fit$ $ct$ $BART$ $RF$ $lwr_{tot}$ $upr_{tot}$ $lwr_{fit}$ $upr_{fit}$ $lwr_{ct}$ $upr_{ct}$ $lwr_{BART}$ $upr_{BART}$ $lwr_{RF}$ $upr_{RF}$
-------- --------------------- ------- ------- -------- -------- ------- ------------- ------------- ------------- ------------- ------------ ------------ -------------- -------------- ------------ ------------ --
1 -1.137 0.470 0.234 0.637 0.118 0.420 0.089 0.678 0.087 0.741 0.202 1.103 -1.166 1.394 0.082 0.772
2 -0.831 0.470 0.417 0.500 0.117 0.383 0.087 0.672 0.096 0.730 0.191 1.117 -1.211 1.397 0.083 0.770
3 -0.638 0.470 0.538 0.515 0.105 0.461 0.067 0.679 0.083 0.733 0.183 1.094 -1.204 1.398 0.082 0.772
4 -0.472 0.470 0.258 0.430 0.089 0.376 0.076 0.676 0.094 0.725 0.193 1.105 -1.224 1.393 0.085 0.744
5 -0.310 0.470 0.093 0.307 0.095 0.276 0.077 0.670 0.082 0.740 0.198 1.112 -1.191 1.398 0.075 0.766
6 -0.114 0.470 0.598 0.681 0.116 0.391 0.074 0.676 0.097 0.732 0.204 1.103 -1.107 1.397 0.083 0.772
7 0.103 0.470 0.471 0.552 0.096 0.407 0.081 0.660 0.086 0.748 0.212 1.112 -1.136 1.370 0.087 0.760
8 0.397 0.470 0.414 -0.103 0.122 0.269 0.086 0.682 0.092 0.744 0.192 1.104 -1.120 1.385 0.082 0.760
9 0.848 0.470 0.363 0.384 0.134 0.442 0.082 0.667 0.094 0.744 0.178 1.108 -1.119 1.400 0.087 0.776
10 2.143 0.470 0.396 0.835 0.156 0.706 0.075 0.673 0.107 0.735 0.190 1.108 -1.068 1.406 0.081 0.760
Acknowledgements
================
The authors gratefully acknowledge the support of Andrea Mercatanti (Department of Statistics, Bank of Italy) for providing data for this paper and sincerely thank Elizabeth Lorenzi (Duke University) for providing insightful commentary and expertise on the topic of causal inference.
[^1]: We generate proposals as $\pmb{\beta}^{*}\sim\mathrm{N}(\mu = \pmb{\beta}^{j-1}, \sigma^{2})$ i.e. from a Gaussian distribution that is centered at the last accepted value. The variance controls the step size of the proposals and needs to be tuned for the application.
[^2]: We use the implementations of these methods in the `R` packages `causalTree` [@causal2016tree], `rpart`[@rpart2002], `randomForest`[@rf2018] and `BART`[@bart2018]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Some properties of an exact solution due to Vaidya, describing the gravitational field produced by a point particle in the background of the static Einstein universe are examined. The maximal analytic extension and the nature of the singularities of the model are discussed. By using the Euclidean approach, some quantum aspects are analysed and the thermodynamics of this spacetime is also discussed.'
---
[**ON THE GRAVITATIONAL FIELD OF A POINT MASS IN EINSTEIN UNIVERSE BACKGROUND**]{}\
Dumitru Astefanesei\
[*Department of Physics, McGill University, Montreal, QC, H3A 2T8, Canada\
e-mail: [email protected]\
(on leave from Al. I. Cuza University, Blvd. Copou 11, Iasi, Romania)*]{}\
and\
Eugen Radu\
[*Albert-Ludwigs-Universität Freiburg, Fakultät für Physik,\
*Hermann-Herder-Straße 3, D-79104 Freiburg, Germany\
email: [email protected]\
(on leave from Institute of Technical Physics, D. Mangeron 47, Iasi, Romania)**]{}\
Introduction
============
The standard Schwarzschild solution is described in flat background. Because a black hole is a cosmological object, it is worthwhile to examine the effect of the cosmological background on the black hole properties. We recall that by considering a non-zero cosmological constant we obtain the Schwarzschild- (anti)de Sitter solution with rather different properties. Also, we would like to take into account the fact that a black hole may also be surrounded by a local mass distribution.
A model taking into account the deviation from flatness on a large scale was proposed a long time ago by Einstein and Strauss [@1]. In this model, the vacuum Schwarzschild field matches accross a spherical comoving boundary to a pressure-free Friedmann-Robertson-Walker (FRW) universe. This would provide for example a description of the effect of the cosmic expansion on the gravitational field of the Sun.
Another way to deal with the embedding of massive objects in a FRW universe is to solve Einstein’s field equations exactly or approximately, in such a way that the resulting solution can be interpreted as an embedding of some massive object in the considered background. In 1933, McVittie found solutions of Einstein’s field equations for a perfect fluid energy-tensor, representing a Schwarzschild field embedded in the FRW spacetime [@2] (see also ref. [@3; @4; @5] for an up-to-date discusion of the Mc-Vittie solutions’ properties ).
A rather different approach has been considered by Vaidya, who derived a perfect fluid solution, interpreted as the Kerr metric in the cosmological background of the Einstein space [@6; @7].
When specializing the Vaidya solution for a vanishing angular momentum, we obtain the simpler case of a Schwarzschild metric in the Einstein space background.
It is the purpose of this paper to study some properties of this spacetime, in an attempt to address the fundamental question of how the cosmological background will affect the black hole properties.
The paper is structured as follows: in section 2 we outline the basic features of the Vaidya solution, while in section 3 we analyse the global properties of this spacetime and the issue of singularities of the model. Based on these results, some thermodynamic properties are discussed in section 4. We conclude with section 5 where the results are compiled.
Einstein equations and matter content
=====================================
In the particular case of zero angular momentum, the metric proposed by Vaidya reduces to the form $$\label{metric}
ds^2 = \frac{dr^2}{1-\frac{2m}{R_{0}}\cot\frac{r}{R_0}} +
R_0^2 \sin^2(\frac{r}{R_0})(d\theta^2 +
\sin^2\theta d\phi^2)-(1-\frac{2m}{R_0}\cot\frac{r}{R_0})dt^2.$$ This metric is also a particular case of the Wahlquist solution [@8]. $R_0$ and $m$ are two positive constant; when setting $m$ equal to zero, the metric (\[metric\]) reduces to the metric of the Einstein universe; for $R_0 \to \infty$ the vacuum Schwarzschild solution is recovered. The parameter $R_0$ can be considered as a measure of the cosmological influence on the black hole properties. Therefore it is natural to interpret $m$ and $R_0$ as representing the Schwarzschild mass and the radius of the universe, respectively.
It is also interesting that the only metric of the form $$ds^2=e^{2A(r)}dr^2 + R_0^2 \sin^2(\frac{r}{R_0})(d\theta^2 + \sin^2\theta d\phi^2)
- e^{-2A(r)}dt^2,$$ satisfying the condition $$R_r^r=R_\theta^\theta=R_\phi^\phi$$ (with $R_a^b$ the Ricci tensor) is (\[metric\]) (such an universe will be isotropic around one observer, but not spatially homogeneous). When solving the Einstein equations, we find, in addition to the matter content of the Einstein universe, a perfect fluid with $\rho+3p=0$, violating in some regions the energy conditions. As opposed to McVittie’s solution, this matter content is not spatially homogeneous, since $$\begin{aligned}
\label{ro}
8\pi\rho&=&\frac{6m}{R_0^3}\cot(\frac{r}{R_0}),
\nonumber\\
8\pi p&=&-\frac{2m}{R_0^3}\cot(\frac{r}{R_0}).\end{aligned}$$ This new contribution to the total energy-momentum tensor, associated with the energy source at the origin of the radial coordinate, will spoil the homogeneous nature of the Einstein universe. We can think about the above relations as describing the perturbation of the local matter content of the Einstein cosmological background induced by a point mass. It is also tempting to suppose a quantum origin of this stress-energy tensor.
It is well known that the Einstein universe solution is unstable (see $e.g.$ [@12]). Also, in the median region $r \sim R_0\pi/2$, the local properties of the Vaidya model are well described by the Einstein universe. However, a local perturbation of the cosmic background in this region will propagate towards antipodes. Therefore, the radius $R_0$ would continue to change in the same direction once the model started to expand or contract. One may expect that the new time-dependent configuration will represent the static case of a general solution briefly noted in [@6] and interpreted as the Kerr field embedded in a FRW universe.
This raises the interesting question if this time-dependent solution has a central singularity and event horizon, and if so, how they are affected by the cosmic expansion.
These issues are currently being studied and will be discussed elsewhere.
We note also the form of the line-element (\[metric\]) in Schwarzschild coordinates $$\begin{aligned}
\label{sch}
ds^2&=&\frac{d\overline{r}^2}{(1-(\frac{\overline{r}}{R_0})^2)(1-\frac{2m}{\overline{r}}
\sqrt{1-(\frac{\overline{r}}{R_0})^2})}+\overline{r}^2((d\theta^2 + \sin^2\theta d\phi^2)\nonumber\\
& &-(1-\frac{2m}{\overline{r}}\sqrt{1-(\frac{\overline{r}}{R_0})^2})dt^2,\end{aligned}$$ where
$$\overline{r}=R_0\sin\frac{r}{R_0}.$$
This helps us to compare with the results obtained in [@9], where the gravitational field produced by a point mass in the background of the static Einstein universe is studied with many similar conclusions. For small $m/R_0$, the line element (\[sch\]) reproduces the approximate solution derived in [@9].
Maximal analitic extension
==========================
A metric of the form (2) has singularities where $e^{2A}$ vanishes or becomes infinite. However, some of these singularities can be pseudosingularities, caused by an inappropiate coordinate system. Following the usual techniques of analytic extension across pseudosingularities [@10], we obtain Kruskal-like form of the metric (\[metric\]) $$ds^2=\frac{32m^3
\exp(-\frac{r}{2m})}{[1+(\frac{2m}{R_0})^2]^2R_0\sin(\frac{r}{R_0})}(dz^2-dT^2)+
R_0^2\sin^2(\frac{r(z,T)}{R_0})(d\theta^2 + \sin^2\theta d\phi^2),$$ which is related to the original form by $$\label{m1}
z^2-T^2=\frac{R_0}{2m} \exp(\frac{r}{2m})
\sin\frac{r}{R_0}(1-\frac{2m}{R_0}\cot\frac{r}{R_0}),$$ $$\label{m2}
\frac{T}{z}=\tanh \{\frac{t}{4m}[1+(\frac{2m}{R_0})^2]\}\qquad
r>r_H,$$ $$\label{m3}
\frac{z}{T}=\tanh \{\frac{t}{4m}[1+(\frac{2m}{R_0})^2]\}\qquad
r<r_H,$$ where $r_H=R_{0} \arctan (\frac{2m}{R_0})$.
No singularities occur here, except the genuine singularities at $r=0$ and $r=\pi R_0$, which cannot be removed by any coordinate transformations.
On the manifold defined by coordinates $(z,\theta,\phi, T)$ with metric (8), we define four regions (fig.1): the region I with $z>|T|$ is isometric to the region of the metric (\[metric\]) for which $r>r_H$. There is also a region I’, defined by $z<-|T|$, which turns out to be again isometric with the same region of the metric (\[metric\]). This can be regarded as another universe on the other side of the throat $r=r_H$. The regions II and II’ are isometric with the region $r<r_H$ of the metric (\[metric\]). The surface $r=r_H$ presents all the characteristics of an event horizon. For a freely falling observer an infinite time $t$ is required to traverse the finite distance $L_0$ between an exterior point and a point on the horizon, but that destination is reached in a finite proper time. A photon would require an infinitely long time to cover the finite stretch $L_0$. Once a particle has fallen inside $r=r_H$ it cannot avoid the singularity. In Schwarzschild coordinates, the horizon is located at $\overline{r}=\frac{2m}{\sqrt{1+(\frac{2m}{R_0})^2}}$; the effect of the curvature of the universe on the radius of the event horizon is to reduce it. We note also that both $\rho$ and $p$ defined by (\[ro\]) are well defined at $r=r_H$.
Both $r=0$ and $r=\pi R_0$ are curvature singularities with corresponding infinities of the functions $\rho$, $p$. However the $r=0$ singularity is less dangerous being located inside an event horizon. The situation with an antipodal singularity at $r=\pi R_0$ is rather different. Since there is no horizon, this is a naked singularity, contradicting the cosmic censorship conjecture (however, it is hazardous to claim that the Vaidya metric (\[metric\]) describes the final state of a collapsing matter distribution). This singularity appears to be repulsive: no timelike geodesic hits them, though a radial null geodesic can. Then we can interpret the metric (\[metric\]) as describing a black hole with an antipodal naked singularity in the cosmological background of Einstein universe.
In fact, given the high symmetry and the simple matter content of the solution we have to expect the existence of an antipodal singularity. However, we suspect that this is not a generic property. We can hope that by considering a more general matter content this unpleasant feature can be avoided by the introducing an additional event horizon.
Thermodynamic properties
========================
The presence of a naked singularity makes difficult the formulation of a quantum field theory on the background of metric (\[metric\]). However, the standard arguments of the Euclidean approach for deriving the Hawking radiation seems to be valid in this case, too. We assume the possibility of extending path integral formulation of gravitational thermodynamics to the situation under consideration. This is a very strong assumption, given the existence of the naked singularity; we recall that one of the reasons to deal with the Euclideanization procedure was to avoid the space-time singularities [@11].
Proceed by making in (\[m1\]) the formal substitution $\xi=iT$ to yield $$ds^2=\frac{32m^3 \exp(-\frac{r}{2m})}{[1+(\frac{2m}{R_0})^2]^2 R_0\sin(\frac{r}{R_0})}
(dz^2+d\xi^2)+R_0^2\sin^2(\frac{r(z,\xi)}{R_0})(d\theta^2 + \sin^2\theta d\phi^2).$$
On the section on which $z,\xi$ are real, $r$ will be real and great or equal to $r_H$ (the singularity at $r=\pi R_0$ is still present). Define the imaginary time $\tau$ by $\tau=-it$. It follows from eq. (\[m2\]) that $\tau$ is periodic with period $$\label{beta}
\beta=\frac{8\pi m}{1+(\frac{2m}{R_0})^2}.$$ The periodicity in the imaginary (Euclidean) time is usually interpreted as evidence of a thermal bath of temperature $T=\frac{1}{\beta}$, so that the Hawking temperature of a Schwarzschild black hole in the Einstein universe background is identified as
$$T_H=\frac{1}{8\pi m}[1+(\frac{2m}{R_0})^2].$$
The same result can be obtained by direct calculation of the surface gravity $k$ ($T_H=\frac{k}{2\pi}$). The Hawking temperature of the system appears to be increased relative to that of Schwarzschild black hole of equal horizon area.
From the formula (\[beta\]) one can see that $\beta$ has a maximum value $2\pi R_0$ for $m=R_{0}/2$ and therefore $T_H$ has a minimum value of $T_0 =(2\pi R_0)^{-1}$ when $r_H=\pi R_0/4$. For $m>\frac{R_0}{2}$, the temperature $T_H$ increases with the mass; for a large enough $m$ we have $T_H\approx\frac{m}{2\pi R^2_0}$.
According to $"Tolman$ $relation"$, the local temperature $T(r)$ measured by a moving, accelerated detector is related to the Hawking temperature $T_H$ measured by a detector in the asymptotic region by [@12] $$T(r)=(-g_{\tau\tau})^{-\frac{1}{2}}T_H=\frac{1}{8\pi m}(1+
(\frac{2m}{R_0})^2)(1-\frac{2m}{R_0}\cot\frac{r}{R_0})^{-\frac{1}{2}}.$$
We can consider this solution as describing a “dirty” black hole [@13] since the interaction with a classical matter field is present; however, the metric (\[metric\]) is not asymptotically flat and the general formalism for analysing the thermodynamic properties of a dirty black hole cannot be applied [@13; @14].
Accordingly to Gibbons and Hawking [@11], thermodynamic functions including the entropy can be computed directily from the saddle point approximation to the gravitational partition function (namely the generating functional analytically continued to the Euclidean spacetime).
The Euclidean gravitational part of the action has the general form [@15] $$\label{action}
I_E=-\frac{1}{16\pi}\int_{M}(R-2\Lambda)\sqrt{g}d^4x +
\frac{1}{8\pi}\int_{\partial M} K\sqrt{h}d^3x.$$
In this case, due to the global structure, it is not necessary a so called “*reference action*” substraction (this substraction was needed for a Schwarzschild black hole in order to get a finite result [@11]). In the semiclassical approximation, the dominant contribution to the path integral will come from the neighborhood of saddle points of the action, that is, of classical solution; the zeroth order contribution to $\log Z$ will be $-I_E$.
The integral (\[action\]) is evaluated on the Euclidean section (with $r\ge r_H$) containing the naked singularity at $r=\pi R_0$; however it takes a finite value $$I_E=\frac{\beta m}{2}[2\frac{1-(\frac {2m}{R_0})^2}{1+(\frac {2m}{R_0})^2}-
\frac{R_0}{2m} \arctan(\frac{2m}{R_0})+\beta \pi\frac{ R_0}{4}].$$
All thermodynamic properties can be deduced from the partition function; for example the intrinsic entropy has the value $$S=\frac{\beta^2}{16\pi}\{4-\frac{[1+(\frac {2m}{R_0})^2]
[3+2(\frac {2m}{R_0})^2] }{1-(\frac {2m}{R_0})^2}\},$$ revealing a complex dependence on the parameters $m, R_0$. We are facing two well-known limits, namely, the Schwarzschild limit (a vanishing cosmological constant, $R_0 \to \infty$) with $S=4\pi m^2$ and the Einstein limit ($m \to 0$), with $S=0$ as expected. A curious result is obtained for a large enough $m$ ($\frac{m}{R_0} \gg 1 $); in this limit the asymptotic behavior of the entropy is $S\approx 8\pi m^2$.
However, in the $"extremal"$ case $r_H=\frac{\pi R_0}{4}$ ($i.e.$ $R_0=2m$) we obtain a diverging value for the entropy. This divergence is to be associated with the peculiar global structure of the solution and the existence of the naked singularity.
For $\frac{m}{R_0} \ll 1$ we obtain a set of corrections to the (asymptotically flat) Schwarzschild black hole thermodynamic quantities. In this case, a curious result is the rather benign character of the antipodal naked singularity.
Further discussions
===================
The exact solution proposed by Vaidya in [@6] is usually regarded as a possible approach to the study of black holes in a non-flat background.
We have discussed in this paper some properties of this solution. After considering the maximal analytic extension of this model, an interpretation has been proposed as describing a black hole with a antipodal naked singularity in the cosmological background of Einstein universe. Assuming the possibility of extending path integral formulation of gravitational thermodynamics to this case, the expression of the Hawking temperature, the Euclidean action and the intrinsic entropy have been derived. These relations reveal a complex dependence on the two parameters of the model $m, R_0$.
The existence of a naked singularity in the line element (\[metric\]) is certainly the most unpleasant feature of the Vaidya model.
For $\frac{m}{R_0} \ll 1$ a possibly way to deal with the naked singularity is to suppose that the line element (\[metric\]) is valid for $r<r_0$ only (for a suitable $r_0>r_H$) and to match it to a different metric which goes over into the Einstein line-element for large enough $r$. In this way the Vaydia model becomes an isolated island in a Einstein universe background, and the problems associated with the case $R_0=2m$ can be avoided too. A possible choice for $r_0$ is $r_0=\pi R_0/2$, where the line element (\[metric\]) becomes the line element of the Einstein universe. Unfortunately, a surface density distribution of matter seems to be always necessary.
A further extension of this paper could be the inclusion of a nonzero angular momentum and considering the immersion in an FRW universe along the line suggested by Vaidya in [@6]. Some features of the simplest case discussed in this paper (for example the existence of a naked singularity) are still present in the general case given the strong correlation between the particle and the universe.
The Vaidya model represents an oversimplified picture and can not be considered a live candidate for describing a physical situation, but can be a source of insight into the possibilities allowed by relativity theory.\
\
[**Acknowledgement**]{}\
The authors gratefully acknowledge the referee’s helpful comments.
[16]{}
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![Penrose diagram for the maximally extended Vaidya metric (\[metric\])](2.eps "fig:"){width="120.00000%"}\
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Compressed sensing provided a data-acquisition paradigm for sparse signals. Remarkably, it has been shown that practical algorithms provide robust recovery from noisy linear measurements acquired at a near optimal sampling rate. In many real-world applications, a signal of interest is typically sparse not in the canonical basis but in a certain transform domain, such as wavelets or the finite difference. The theory of compressed sensing was extended to the analysis sparsity model but known extensions are limited to specific choices of sensing matrix and sparsifying transform. In this paper, we propose a unified theory for robust recovery of sparse signals in a general transform domain by convex programming. In particular, our results apply to general acquisition and sparsity models and show how the number of measurements for recovery depends on properties of measurement and sparsifying transforms. Moreover, we also provide extensions of our results to the scenarios where the atoms in the transform has varying incoherence parameters and the unknown signal exhibits a structured sparsity pattern. In particular, for the partial Fourier recovery of sparse signals over a circulant transform, our main results suggest a uniformly random sampling. Numerical results demonstrate that the variable density random sampling by our main results provides superior recovery performance over known sampling strategies.'
author:
- |
Kiryung Lee, Yanjun Li , Kyong Hwan Jin,\
and Jong Chul Ye, [^1]
title: |
Unified Theory for Recovery of Sparse Signals\
in a General Transform Domain
---
Compressed sensing, analysis sparsity model, sparsifying transform, total variation, incoherence, variable density sampling.
Introduction {#sec:intro}
============
The theory of compressed sensing (CS) [@donoho2006compressed; @candes2006robust] provided a new data-acquisition paradigm for sparse signals. Remarkably, it has been shown that practical algorithms are guaranteed to reconstruct the unknown sparse signal from the linear measurements taken at a provably near optimal rate. Reconstruction algorithms with performance guarantees include modern optimization algorithms for $\ell_1$-norm-based convex optimization formulations (e.g., [@beck2009fast; @boyd2011distributed]) and iterative greedy algorithms (e.g., [@needell2009cosamp; @dai2009subspace; @blumensath2009iterative; @foucart2011hard]).
The canonical sparsity model in CS assumes that the unknown signal $f \in \cz^d$ is $s$-sparse in the standard coordinate basis. In other words, $\norm{f}_0 \leq s$, where $\norm{\cdot}_0$ counts the number of nonzero elements. The acquisition process in CS is linear and represented by a sensing matrix $A \in \cz^{m \times d}$ so that the $m$ linear measurements in $b \in \cz^m$ is given by $$b = A f + w,$$ where $w \in \cz^m$ denotes additive noise to the measurements and satisfies $\norm{w}_2 \leq \epsilon$. For certain random sensing matrices, it was shown that an estimate $\hat{f}$ given by $$\label{eq:bpdn}
\hat{f} = \argmin_{\tilde{f} \in \cz^d} \norm{\tilde{f}}_1 \quad \mathrm{subject~to} \quad \norm{b - A \tilde{f}}_2 \leq \epsilon$$ satisfies $\norm{\hat{f} - f}_2 \leq c_1 \epsilon$ with high probability, provided that $m \geq C s \log^\alpha d$ for some $\alpha \in \mathbb{N}$ and numerical constants $C$ and $c_1$. In particular, in the noiseless case ($\epsilon = 0$), the estimate $\hat{f}$ coincides with the ground truth signal $f$. For example, Candes and Tao [@candes2005decoding] showed that the above guarantees hold for a Gaussian sensing matrix $A$ whose entries are i.i.d. following $\mathcal{N}(0,1/m)$ via the restricted isometry property (RIP). Recent results with a sharper sample complexity of $m \geq 2 s \log(d/s)$ were derived using the Gaussian width of a tangent cone [@chandrasekaran2012convex; @amelunxen2014living].
In fact, the original idea of compressed sensing [@bresler1999image] was motivated by a need to accelerate various imaging modalities. The sensing matrix $A$ in these applications takes observations in a measurement transform domain, i.e. $$\label{eq:structred_sensing_mtx}
A = \sqrt{\frac{n}{m}} S_\Omega \Psi,$$ where $\Psi \in \cz^{n \times d}$ is the matrix representation of the measurement transform and the sampling operator $S_\Omega: \cz^n \to \cz^m$ takes the $m$ elements indexed by $\Omega = \{\omega_1,\omega_2,\ldots,\omega_m\}$. For example, when $\Psi$ is a discrete Fourier transform (DFT) matrix, $A$ is a partial Fourier matrix. The aforementioned near optimal performance guarantees were shown for a partial Fourier sensing matrix $A$ obtained using a random set $\Omega$ [@candes2006robust; @candes2006near; @rudelson2008sparse; @candes2011probabilistic] and generalized for the case where the rows of $\Psi \in \cz^{n \times d}$ correspond to an incoherent tight frame for $\cz^d$ [@rauhut2010compressive].
However, in numerous imaging applications, a signal of interest is not sparse in the standard coordinate basis. The theory of compressed sensing was accordingly extended to the so-called synthesis and analysis sparsity models [@candes2011compressed; @nam2013cosparse; @giryes2014greedy]. The synthesis sparsity model assumes that $f \in \cz^d$ is represented as a linear combination of few atoms in a dictionary $D \in \cz^{d \times N}$. Equivalently, $f$ is represented as $f = D u$ with an $s$-sparse coefficient vector $u \in \cz^N$. Compressed sensing with the synthesis sparsity model can be interpreted as conventional compressed sensing of $u$ using a sensing matrix $A D$ where $u$ is $s$-sparse in the standard basis. In particular when $A$ is a Gaussian matrix and $D$ is an orthogonal matrix, conventional performance guarantees carry over to the synthesis sparsity model. On the other hand, the analysis sparsity model, which is motivated from sparse representation in harmonic analysis [@mallat2008wavelet], assumes that the transform $\Phi f \in \cz^N$ of $f$ via $\Phi \in \cz^{N \times d}$ is $s$-sparse. In fact, a signal of interest in practical applications often follows the analysis sparsity model via various transforms including finite difference and wavelet [@mallat2008wavelet], contourlet [@do2005contourlet], curvelet [@starck2002curvelet], and Gabor transforms. Thus, the analysis sparsity model has been used as an effective regularizer for classical inverse problems in signal processing (e.g., denoising and deconvolution) and compressed sensing imaging (e.g., [@lustig2008compressed]).
Unlike the previous results with the canonical sparsity model, the theory of compressed sensing with the analysis model has been relatively less explored and known results are limited to specific cases. Candes et al. [@candes2011compressed] considered the recovery of $f \in \cz^d$ such that $\Phi f$ is $s$-sparse for a transform $\Phi \in \cz^{N \times d}$ satisfying $\Phi^* \Phi = I_d$, i.e. the columns of $\Phi^*$ correspond to a tight frame for $\cz^d$. They showed that $$\label{eq:bpdn_tfm}
\hat{f} = \argmin_{\tilde{f} \in \cz^d} \norm{\Phi \tilde{f}}_1 \quad \mathrm{subject~to} \quad \norm{A \tilde{f} - b}_2 \leq \epsilon$$ has the error bound given by $\norm{\hat{f} - f}_2 \leq c_1 \epsilon$, provided that the sensing matrix $A \Phi^*$ satisfies the RIP. In particular for a Gaussian sensing matrix $A$, their performance guarantee holds with high probability for $m = O(s \log(d/s))$ for any $\Phi$ satisfying $\Phi^* \Phi = I_d$. Indeed the performance guarantee by Candes et al. [@candes2011compressed] applies beyond the case of a Gaussian sensing matrix $A$. Krahmer and Ward [@krahmer2011new] showed that if $A \in \cz^{m \times d}$ and $D \in \cz^{d \times N}$ satisfy the RIP and $\varepsilon \in \mathbb{R}^d$ is a Rademacher sequence with random $\pm 1$ entries, then $A \diag(\varepsilon) D$ satisfies the RIP, where $\diag(\varepsilon) \in \cz^{d \times d}$ denotes the diagonal matrix whose diagonal entries are $\varepsilon$. However, applying a random sign before the acquisition might not be feasible in certain applications. In another line of research, it was shown [@nam2013cosparse; @giryes2014greedy] that greedy algorithms for the analysis sparsity model provide performance guarantees if the sensing matrix $A$ is a near isometry when acting on all transform-sparse $f$ such that $\Phi f$ is sparse. Again for $\Phi$ satisfying $\Phi^* \Phi = I_d$, this condition on $A$ is less demanding than the RIP of $A \Phi^*$ since the latter implies the former. However, it has not been studied how such a relaxation translates into less-demanding sample complexity.
A special analysis sparsity model associated with the finite difference transform $\Phi$ has been of particular interest in signal processing and imaging applications. The corresponding convex surrogate $\norm{\Phi f}_1$, known as the total variation (TV), has been popularly used as an effective regularizer for solving inverse problems. Needell and Ward [@needell2013stable] provided performance guarantees for TV minimization with a partial Fourier sensing matrix $A$ in terms of the RIP of $A D$ with a Haar wavelet dictionary $D$. Their result was further refined by Krahmer and Ward [@krahmer2014stable] with a clever idea of variable density sampling adopting the local incoherence parameters. Remarkably, Krahmer and Ward [@krahmer2014stable] provided performance guarantees at the sample complexity of $m = O(s \log^3 s \log^2 d)$. However, these results on TV minimization rely on a special embedding theorem that relates the Haar transform to the finite difference transform, which holds only for signals of dimension two or higher. Therefore, the results by Needell and Ward [@needell2013stable] and by Krahmer and Ward [@krahmer2014stable] do not apply to the 1D case and more importantly do not generalize to other sparsifying transforms.
Recently, inspired by analogous results for the canonical sparsity model [@chandrasekaran2012convex; @amelunxen2014living], Kabanava et al. [@kabanava2015robust] derived a performance guarantee of TV minimization with a Gaussian sensing matrix $A$ at the sample complexity of $m > d \,[1 - \{ 1-(s+1)/d \}^2/\pi]$. Cai and Xu [@cai2015guarantees] showed a similar result with $m \geq C \sqrt{s d} \log d$. Compared to the previous result by Krahmer and Ward [@krahmer2014stable], the above results [@kabanava2015robust; @cai2015guarantees] apply to the 1D case but at a significantly suboptimal sample complexity. More importantly, the use of a Gaussian sensing matrix might not be relevant to practical applications. Along a similar analysis strategy, Kabanava and Rauhut [@kabanava2015analysis] showed performance guarantees for with a Gaussian sensing matrix $A \in \mathbb{R}^{m \times d}$ and a frame analysis operator $\Phi \in \mathbb{R}^{N \times d}$ roughly at the sample complexity of $m \geq 2 \kappa s \log (2N/s)$, where $\kappa$ denotes the ratio of upper and lower frame bounds, i.e. $\kappa$ is the condition number of the frame operator $\Phi^* \Phi$. The sample complexity of this result for a tight frame is near optimal. However, their analysis is restricted to a Gaussian sensing matrix $A$ and does not generalize to other sensing matrices.
Contributions
-------------
Our main contribution is to derive performance guarantees for compressed sensing of analysis-sparse vectors by for general classes of sensing matrix $A$ given in the form of with measurement transform $\Psi$ and (redundant) analysis transform $\Phi$. Unlike the previous works, performance guarantees in this paper apply without being restricted to a particular choice of $\Phi$ and $\Psi$[^2]. The number of measurements implying these guarantees depends on certain properties of $(\Phi,\Psi)$ and this result identifies a class of measurement and sparsity models allowing recovery at a near optimal sampling rate. Moreover, when $\Psi$ and $\Phi$ have a few strongly correlated atoms, adopting the idea by Krahmer and Ward [@krahmer2014stable], we propose to acquire linear measurements using random sampling with respect to a variable density designed with the correlations between $\Psi$ and $\Phi$. This modified acquisition enables recovery at a lower sampling rate. We also extend the results to group sparsity models. This extension applies to various popular regularized recovery methods including the isotropic-total-variation minimization. In special cases where $\Psi$ is Fourier and $\Phi$ is a circulant matrix, our theory suggests a uniformly random sampling or its variation. For example, for the total variation minimization, unlike the common belief in practice, a sampling strategy that combines the acquisition of the lowest frequency and uniformly random sampling on the other frequencies provides better reconstruction than known variable density random sampling strategies.
Our main idea is inspired by a previous work on structured matrix completion by Chen and Chi [@chen2014robust]. They showed that a structured low-rank matrix (e.g., a low-rank Hankel matrix) is successfully recovered from partially observed entries by minimizing the nuclear norm. Indeed, their structured low-rank matrix completion can be interpreted as follows. The unknown structured matrix $M$ is given as the image $T x$ of the generator $x$ via a linear map $T$. Then the completion of the structured matrix $M$ with the low-rankness prior is equivalent to the completion of the generator $x$ with low-rankness in the transform domain via $T$. Similarly, compressed sensing with the analysis sparsity model is equivalent to the recovery of $x \in \cz^n$ with the prior that $T x \in \cz^N$ is $s$-sparse where the transform $T$ is given as $T = \Phi \Psi^\dagger$. Therefore the two problems are analogous to each other in the sense that their priors correspond to atomic sparsity [@chandrasekaran2012convex] in their respective transform domains. Moreover, in both the structured low-rank matrix model and the transform-domain sparsity model, $T$ is not necessarily surjective, which implies that $T T^*$ may be rank-deficient. This violates an important technical condition known as the *isotropy* property and many crucial steps in the proofs of existing performance guarantees for CS break down. To overcome this difficulty, we adopt the clever idea by Chen and Chi [@chen2014robust] through the aforementioned analogy and derive near optimal performance guarantees for the recovery of sparse signals in a transform domain without resorting to the isotropy property. However, besides this similarity, our results are significantly different from the analogous results [@chen2014robust] in the following sense. Chen and Chi [@chen2014robust] assumed that $T$ is restricted to a set of special linear operators that generate structured matrices and their analysis indeed relies critically on strong properties satisfied by such linear operators (e.g., $T$ needs to satisfy $T^* T = I_n$). Contrarily, we only assume a mild condition that $T$ is injective and the resulting performance guarantees apply to more general cases. For example, in compressed sensing with the analysis sparsity model, $T = \Phi \Psi^\dagger$ is non-unitary if the measurement transform $\Psi$ is non-unitary (e.g., the Radon transform) or the sparsifying transform $\Phi$ is non-unitary (e.g., biorthogonal wavelet and data-adaptive transforms).
We illustrate our theory through extensive Monte Carlo simulations. The variable density sampling in this paper provides an improved recovery performance over the previously suggested sampling strategies. For example, when $\Phi$ and $\Psi$ have atoms showing high correlation (e.g. Fourier and wavelet), our theory suggests to sample more densely in the lower frequencies, which enables successful recovery from fewer observations. Rather surprisingly, when $\Phi$ and $\Psi$ respectively correspond to the finite difference and the Fourier transforms, our theory suggests a special sampling density that always takes the lowest frequency and chooses the other frequency components randomly using the uniform density. This is contrary to the common belief in compressed sensing but the proposed sampling strategy turns out be more successful than previous works not only in theory but also empirically. These numerical results strongly support out new theory.
Organization
------------
The rest of this paper is organized as follows. The main results are presented in Sections \[sec:main\_result\] and \[sec:main\_result\_vd\], where the proofs are deferred to Sections \[sec:pf\_main\_result\_noiseless\] and \[sec:pf\_main\_result\_noisy\]. We extend the theory to the group sparsity models in Section \[sec:main\_result\_gs\] and study a special case of circulant transforms in Section \[sec:circulant\]. After demonstrating empirical observations in Section \[sec:numres\], which supports our main results, we conclude the paper with some final remarks in Section \[sec:concl\].
Notations
---------
For a positive integer $N$, we will use a shorthand notation $[N]$ for the set $\{1,\ldots,N\}$. For a vector $z \in \cz^N$, let $z[k]$ denote the $k$th element of $z$ for $k \in [N]$, i.e. $z = [z[1], z[2], \ldots, z[N]]^\transpose$. The Hadamard product of two vectors $x,y \in \cz^n$ is denoted by $x \odot y$. The circular convolution of two vectors $x,y \in \cz^n$ is denoted by $x \circledast y$. The Kronecker product of two matrices $A$ and $B$ is denoted by $A\otimes B$. The operator norm from $\ell_p^n$ to $\ell_q^N$ will be denoted by $\norm{\cdot}_{p \to q}$. For brevity, the spectral norm $\norm{\cdot}_{2 \to 2}$ will be written without subscript as $\norm{\cdot}$. For a matrix $A$, its Hermitian transpose and its Moore-Penrose pseudo inverse are respectively written as $A^*$ and $A^\dagger$.
For $J \subset [N]$, the coordinate projection with respect to $J$, denoted by $\Pi_J$, is defined as $$(\Pi_J z)[k] :=
\begin{dcases}
z[k] & k \in J, \\
0 & \mathrm{otherwise}.
\end{dcases}$$ The complex signum function, denoted by $\mbox{sgn}(\cdot): \cz^N \to \cz^N$, is defined as $$\label{eq:sgn}
(\mbox{sgn}(z))[k] :=
\begin{dcases}
\frac{z[k]}{|z[k]|} & z[k] \neq 0, \\
0 & \mathrm{otherwise}.
\end{dcases}$$ Let $e_1,\ldots,e_n$ denote the standard basis vectors for $\cz^n$. In other words, $e_k$ is the $k$th column of the $n$-by-$n$ identity matrix.
Recovery of Sparse Signals in a General Transform Domain {#sec:main_result}
========================================================
Let $T: \cz^n \to \cz^N$ be a linear map that we call a “transform” in this paper. Let $\Omega = \{\omega_1,\omega_2,\ldots,\omega_m\}$ denote the multi-set of $m$ sampling indices out of $[n] := \{1,\ldots,n\}$ with possible repetition of elements. Given $\Omega$, the sampling operator $S_\Omega: \cz^n \to \cz^m$ is defined so that the $j$th element of $S_\Omega x \in \cz^m$ is the $\omega_j$th element of $x \in \cz^n$ for $j = 1,\ldots,m$. We are interested in recovering an unknown signal $x \in \cz^n$ from its partial entries at $\Omega$ when the transform $T x$ is known $s$-sparse a priori, i.e. $\norm{T x}_0 \leq s$. Compressed sensing with the analysis sparsity model is an instance of this problem formulation as shown in the following. Recall that $\Psi$ is of full column rank. Let $T: \cz^n \to \cz^N$ be defined by $T = \Phi \Psi^\dagger$ where $\Psi^\dagger$ denotes the Moore-Penrose pseudo inverse of $\Psi$. Let $x \in \cz^n$ denote the vector containing fully sampled measurements, i.e. $x = \Psi f$. Since $f = \Psi^\dagger x$, we have $\Phi f = \Phi \Psi^\dagger x = T x$. Thus the recovery of $f$ from $b = \sqrt{n/m} S_\Omega \Psi f$ with the prior that $\Phi f$ is $s$-sparse is equivalent to the recovery of $x$ from $b = \sqrt{n/m} S_\Omega x$ with the prior that $T x$ is $s$-sparse.
In the noise-free scenario where partial entries of $x$ are observed exactly, we propose to estimate $x$ as the minimizer to the following optimization problem: $$\label{eq:ell1min}
\minimize_{g \in \cz^n} \, \norm{T g}_1 \quad \mathrm{subject~to} \quad S_\Omega g = S_\Omega x.$$ We provide a sufficient condition for recovering $x$ exactly by in the following theorem.
\[thm:uniqueness\] Suppose $T: \cz^n \to \cz^N$ is injective. Let $\gamma$ be defined by $$\label{eq:defgamma}
\gamma := \argmin_{\tilde{\gamma} > 0} \norm{\tilde{\gamma} T^* T - I_n},$$ where $\widetilde{T} = (T^\dagger)^*$. Let $\mu$ be given by $$\label{eq:incoherence}
\mu = \max_{k \in [n]} \max\left\{ n \norm{\gamma^{1/2} T e_k}_\infty^2, n \norm{\gamma^{-1/2} \widetilde{T} e_k}_\infty^2 \right\}.$$
Let $\Omega = \{\omega_1,\ldots,\omega_m\}$ be a multi-set of random indices where $\omega_k$s are i.i.d. following the uniform distribution on $[n]$. Suppose $T x$ is $s$-sparse. Then with probability $1 - e^{-\beta} - 3/n$, $x$ is the unique minimizer to provided $$m \geq \frac{C(1+\beta) \mu s}{1 - \norm{\gamma T^* T - I_n}} \left[ \log N
+ \log \left( \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \right) \right].$$
See Section \[sec:pf\_main\_result\_noiseless\].
All the results in this section including Theorem \[thm:uniqueness\] as well as other similar results [@candes2011probabilistic; @gross2011recovering; @chen2014robust], all derived using the golfing scheme, provide an *instance* recovery guarantee that applies to a single arbitrary instance of $x$, which is a weaker result than the *uniform* recovery guarantee that applies to the set of all transform-sparse signals. In compressed sensing with the canonical sparsity model, an instance guarantee is given from a fewer measurements than the uniform guarantee [@candes2011probabilistic] by a poly-log factor. For matrix completion problems, the RIP do not hold and known results [@gross2011recovering; @chen2014robust] only provide an instance guarantee. We suspect that this is the case with the recovery problem in this paper.
Note that $\norm{T}_{1 \to 2}$ denotes the largest $\ell_2$-norm among all columns of $T$. Similarly, $\norm{T^\dagger}_{2 \to \infty} = \norm{\widetilde{T}}_{1 \to 2}$ denotes the largest $\ell_2$-norm among all columns of $\widetilde{T}$. In a special case when $T^* T = I_n$, we have $T^\dagger = T^*$. Thus $\norm{T}_{1 \to 2} = \norm{T^\dagger}_{2 \to \infty} = 1$. In general, if $\norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} = O(N^{\alpha_1})$ for some $\alpha_1 \in \mathbb{N}$ and $$\frac{1}{1 - \norm{\gamma T^* T - I_n}} = O(\log^{\alpha_2} N)$$ for some $\alpha_2 \in \mathbb{N}$, then we get a performance guarantee at a near optimal scaling of sample complexity of $m = O(\mu s \log^\alpha N)$, where $\alpha = \alpha_1 + \alpha_2$. These are mild conditions and easily satisfied by transforms that arise in practical applications.
In the noisy scenario where partial entries of $x$ are observed with additive noise, we propose to estimate $x$ by solving the following optimization problem: $$\label{eq:ell1min_noisy}
\minimize_{g \in \cz^n} \, \norm{T g}_1
\quad \mathrm{subject~to} \quad \norm{S_{\Omega'} g - S_{\Omega'} x^\sharp}_2 \leq \epsilon,$$ where $x^\sharp$ denotes a noisy version of $x$, $\Omega'$ denotes the set of all unique elements in $\Omega$, and $S_{\Omega'}^*$ is the adjoint of the sampling operator $S_{\Omega'}$ that fills missing entries at outside $\Omega'$ with 0.
\[thm:stability1\] Suppose the hypotheses of Theorem \[thm:uniqueness\] hold. Let $\hat{x}$ be the minimizer to with $x^\sharp$ satisfying $$\label{eq:xharp_fid}
\norm{S_{\Omega'} (x - x^\sharp)}_2 \leq \epsilon.$$ Then $$\norm{\hat{x} - x}_2 \leq
\frac{\sigma_{\max}(T)}{\sigma_{\min}(T)}
\cdot
\left\{ 2 + 28 \sqrt{N} \left( 3 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} + 1 \right) \right\} \epsilon,$$
See Section \[subsec:pf:thm:stability1\].
We can tighten the upper bound on the estimation error in Theorem \[thm:stability1\] when $T^* T$ is well conditioned. To this end, we will use the following theorem, which is obtained by modifying [@lee2013oblique Theorem 3.1].
\[thm:rboplike\] Suppose $T: \cz^n \to \cz^N$ and $\widetilde{T} = (T^\dagger)^*$ satisfy with parameters $\mu$ and $\gamma$. Let $\Omega = \{\omega_1,\ldots,\omega_m\}$ be a multi-set of random indices where $\omega_k$s are i.i.d. following the uniform distribution on $[n]$. Then with probability $1-\xi$, we have $$\label{eq:rboplike}
\max_{|\widetilde{J}| \leq s} \left\| \Pi_{\widetilde{J}} \left(\frac{n}{m} T S_\Omega^* S_\Omega T^\dagger - T T^\dagger \right) \Pi_{\widetilde{J}} \right\| \leq \delta,
%\max_{|\widetilde{J}| \leq s} \left\| \Pi_{\widetilde{J}} \left(\frac{n}{m} B_\Omega - B\right) \Pi_{\widetilde{J}} \right\| \leq \delta,$$ provided $$\begin{aligned}
\label{eq:rboplike:cond1}
m {} & \geq \frac{C_1 \delta^{-2} \mu s \log^2 s \log N \log m}{1 - \norm{\gamma T^* T - I_n}},
\intertext{and}
\label{eq:rboplike:cond2}
m {} & \geq C_2 \delta^{-2} \mu s \log (\xi^{-1}).\end{aligned}$$
See Appendix \[sec:pf:thm:rboplike\].
Using Theorem \[thm:rboplike\], we provide another sufficient condition for stable recovery of sparse signals in a transform domain, which has a smaller noise amplification factor.
\[thm:stability2\] Suppose $T: \cz^n \to \cz^N$ is injective. Suppose $T$ and $\widetilde{T} = (T^\dagger)^*$ satisfy with parameters $\mu$ and $\gamma$. Let $\Omega = \{\omega_1,\ldots,\omega_m\}$ be a multi-set of random indices where $\omega_k$s are i.i.d. following the uniform distribution on $[n]$. Let $\hat{x}$ be the minimizer to with $x^\sharp$ satisfying . Then there exist numerical constants $C, c > 0$ for which the following holds. With probability $1 - N^{-4}$, we have $$\norm{\hat{x} - x}_2 \leq
\frac{\sigma_{\max}(T)}{\sigma_{\min}(T)}
\cdot
\left[ 14\sqrt{N} + \frac{n}{m} \left\{ \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} (|\Omega|-|\Omega'|) +1 \right\} \right] \epsilon,$$ provided $$\label{eq:noisy_samp_comp}
m \geq \frac{C \mu s \log^4 N}{1 - \norm{\gamma T^* T - I_n}}.$$ Furthermore, if $T^* T = I_n$, then with probability $1 - N^{-4}$, $$\norm{\hat{x} - x}_2 \leq
\frac{\sigma_{\max}(T)}{\sigma_{\min}(T)}
\cdot
\left( 14\sqrt{N} + \frac{n}{m} R \right) \epsilon,$$ where $R$ is the count of the most repeated elements in $\Omega$.
See Section \[subsec:pf:thm:stability2\].
By the definition of $\Omega'$, we have $|\Omega|-|\Omega'| \leq m-1$. Suppose that $\norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} = O(1)$. The resulting noise amplification factor by Theorem \[thm:stability2\] is $O(\sqrt{N} + n)$, which is already smaller than $O(\sqrt{N} n)$ by Theorem \[thm:stability1\]. In fact, the distribution of $|\Omega'|$ is explicitly given by $$\mathbb{P}\left( |\Omega'| = \ell \right) = \frac{\stirling{m}{\ell} n!}{n^m (n-\ell)!},$$ where $\stirling{m}{\ell}$ denotes the Stirling number of the second kind defined by $$\stirling{m}{\ell} := \frac{1}{\ell!} \sum_{j=0}^\ell(-1)^{\ell-j} {\ell \choose j} j^m.$$ It would be possible to compute a more tight probabilistic upper bound on $|\Omega|-|\Omega'|$ with its distribution. However, because of the other factor $\sqrt{N}$, regardless of $|\Omega|-|\Omega'|$, the noise amplification factor by Theorem \[thm:stability2\] cannot be improved to $O(1)$ as shown for compressed sensing with the canonical sparsity model [@candes2011probabilistic] or with a special analysis model [@krahmer2014stable]. We admit that this suboptimality in noise amplification is a limitation of our analysis. It will be interesting to see whether one can obtain near optimal noise amplification for a general transform $T$.
Incoherence-Dependent Variable Density Sampling {#sec:main_result_vd}
===============================================
In the results of the previous section, the number of measurements is proportional to the incoherence parameter $\mu$ in , which is the worst case $\ell_\infty$-norm among $\{T e_k\}_{k=1}^n$ and $\{\widetilde{T} e_k\}_{k=1}^n$. In certain scenarios, these $\ell_\infty$-norms are unevenly distributed. For example, in compressed sensing with the analysis sparsity model, $T$ is given by $T = \Phi \Psi^\dagger$ with the sensing transform $\Psi \in \cz^{n \times d}$ and the sparsifying transform $\Phi \in \cz^{N \times d}$. If $\Psi$ and $\Phi$ correspond to the DFT and DWT (discrete wavelet transform), respectively, low-frequency atoms have larger correlations. Thus there are a few $T e_k$s that dominate the others with large $\ell_\infty$-norms. Krahmer and Ward [@krahmer2014stable] proposed a clever idea of sampling measurements with respect to a variable density adapted to the local incoherence parameters, which are $\{\norm{T e_k}_\infty\}_{k=1}^n$ and $\{\norm{\widetilde{T} e_k}_\infty\}_{k=1}^n$.[^3] Then the sample complexity depends on not the worst case incoherence parameter but the average of the local incoherence parameters. In this section, adopting the idea by Krahmer and Ward [@krahmer2014stable], we extend the results in Section \[sec:main\_result\] to the case where the local incoherence parameters are unevenly distributed. The following theorem is analogous to Theorem \[thm:uniqueness\] and provides a sufficient condition for recovery of sparse signals in a transform domain when measurements are sampled according to a variable density.
\[thm:uniqueness\_vd\] Suppose $T: \cz^n \to \cz^N$ is injective. Let $\mu_k$ and $\tilde{\mu}_k$ be defined by $$\label{eq:loc_incoherenceT}
\mu_k = n \norm{\gamma^{1/2} T e_k}_\infty^2
\quad \mathrm{and} \quad
\tilde{\mu}_k = n \norm{\gamma^{-1/2} \widetilde{T} e_k}_\infty^2$$ for all $k \in [n]$, where $\gamma$ is defined in from $T$ and $\widetilde{T} = (T^\dagger)^*$. Let $\Omega = \{\omega_1, \ldots, \omega_m\}$ be a multi-set of random indices where $\omega_k$s are independent copies of a random variable $\omega$ with the following distribution: $$\label{eq:variable_density}
\mathbb{P}(\omega = k) = \frac{\sqrt{\mu_k \tilde{\mu}_k}}{\sum_{j=1}^n {\sqrt{\mu_j \tilde{\mu}_j}}}, \quad \forall k \in [n].$$ Suppose $T x$ is $s$-sparse. Then with probability $1 - e^{-\beta} - 3/n$, $x$ is the unique minimizer to provided $$m \geq \frac{C(1+\beta) \bar{\mu} s}{1 - \norm{\gamma T^* T - I_n}} \left[ \log N
+ \log \left( \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \right) \right],$$ where $\bar{\mu}$ is defined as $$\label{eq:defbarmu}
\bar{\mu} := \frac{1}{n} \sum_{k=1}^n \sqrt{\mu_k \tilde{\mu}_k}
%\bar{\mu} := \left( \frac{1}{n} \sum_{k=1}^n \mu_k \tilde{\mu}_k \right)^{1/2}.$$
Compared to Theorem \[thm:uniqueness\], Theorem \[thm:uniqueness\_vd\] provides a performance guarantee at a smaller sample complexity, where the worst-case incoherence parameter is replaced by the average incoherence parameter $\bar{\mu}$. Note that $\bar{\mu}$ is always no greater than the worst-case incoherence parameter $(\max_k \mu_k)^{1/2} (\max_k \tilde{\mu}_k)^{1/2}$. In particular when there exist dominant $\mu_k$s or $\tilde{\mu}_k$s compared to other incoherence parameters, the sample complexity of Theorem \[thm:uniqueness\_vd\] is much smaller than that of Theorem \[thm:uniqueness\].
Define $$\label{eq:defnu}
\nu := [\sqrt{\mu_1}, \ldots, \sqrt{\mu_n}]^\transpose
\quad \mathrm{and} \quad
\tilde{\nu} := [\sqrt{\tilde{\mu}_1}, \ldots, \sqrt{\tilde{\mu}_n}]^\transpose.$$ Without loss of generality, we may assume that $\mu_k$s and $\tilde{\mu}_k$s are strictly positive. Then all entries of $\nu$ and $\tilde{\nu}$ are nonzero.
Using $\nu$ and $\tilde{\nu}$, we construct a pair of weighted transforms $W$ and $\widetilde{W}$ as $$\label{eq:defW}
W = \sqrt{\bar{\mu}} T [\mbox{diag}(\nu)]^{-1}
\quad \mathrm{and} \quad
\widetilde{W} = \sqrt{\bar{\mu}} \widetilde{T} [\mbox{diag}(\tilde{\nu})]^{-1}.$$ Then, $W$ and $\widetilde{W}$ satisfy $$\label{eq:incoW}
\max_{k \in [n]} \norm{\gamma^{1/2} W e_k}_\infty \leq \sqrt{\frac{\bar{\mu}}{n}}
\quad \mathrm{and} \quad
\max_{k \in [n]} \norm{\gamma^{-1/2} \widetilde{W} e_k}_\infty \leq \sqrt{\frac{\bar{\mu}}{n}}.$$ Furthermore, we have $$\label{eq:isoW}
\begin{aligned}
\mathbb{E} \left( \frac{n}{m} \sum_{j=1}^m W S_\Omega^* S_\Omega \widetilde{W}^* \right)
{} & = \mathbb{E} \left( \frac{n}{m} \sum_{j=1}^m W e_{\omega_j} e_{\omega_j}^* \widetilde{W}^* \right) \\
{} & = \mathbb{E} \left( \frac{n}{m} \sum_{j=1}^m \bar{\mu} T [\mbox{diag}(\nu)]^{-1} e_{\omega_j} e_{\omega_j}^* [\mbox{diag}(\tilde{\nu})]^{-1} (\widetilde{T})^* \right) \\
{} & = \mathbb{E} \left( \frac{n}{m} \sum_{j=1}^m \frac{\bar{\mu}}{\sqrt{\mu_{\omega_j} \tilde{\mu}_{\omega_j}}} T e_{\omega_j} e_{\omega_j}^* T^\dagger \right) \\
{} & = \frac{1}{m} \sum_{j=1}^m T \widetilde{T}^\dagger = T T^\dagger.
\end{aligned}$$ Since $\mbox{diag}(\tilde{\nu})$ is an invertible matrix, $T$ and $W$ span the same subspace and $T T^\dagger$ is an orthogonal projection onto the span of $W$. Furthermore, $$\langle W e_{k'}, \widetilde{W} e_k \rangle = 0, \quad \forall k \neq k'.$$
Let $g = \sqrt{\bar{\mu}} [\mbox{diag}(\nu)]^{-1} g'$. Then $T g = W g'$. Thus is equivalent to $$\label{eq:ell1min_vd}
\minimize_{g' \in \cz^n} \, \norm{W g'}_1
\quad \mathrm{subject~to} \quad S_\Omega g' = \bar{\mu}^{-1/2} S_\Omega [\mbox{diag}(\nu)] x.$$
Applying Theorem \[thm:uniqueness\] to with incoherence parameter $\bar{\mu}$ completes the proof.
In the case when we are given sampled measurements corrupted with additive noise, Theorems \[thm:stability1\] and \[thm:stability2\] are modified according to the change of the distribution for choosing random sample indices.
\[thm:stability1\_vd\] Suppose the hypotheses of Theorem \[thm:uniqueness\_vd\] hold. Let $x^\sharp$ be a noisy version of $x$ that satisfies . Let $\hat{x}$ be the minimizer to $$\label{eq:ell1min_noisy_vd}
\minimize_{g \in \cz^n} \, \norm{T g}_1
\quad \mathrm{subject~to} \quad \norm{S_{\Omega'} [\rho \odot (g - x^\sharp)]}_2 \leq \epsilon,$$ where $\rho := \bar{\mu}^{-1/2} [\sqrt{\mu_1}, \dots, \sqrt{\mu_n}]^\transpose$. Then $$\norm{\hat{x} - x}_2 \leq
\frac{\sigma_{\max}(T)}{\sigma_{\min}(T)}
\cdot
\frac{\bar{\mu}}{\min_{k \in [n]} \tilde{\mu}_k}
\cdot
\left[ 2 + 28 \sqrt{N} \left( \frac{\max_{k \in [n]} \mu_k}{\min_{k \in [n]} \tilde{\mu}_k} \cdot 3 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} + 1 \right) \right] \epsilon.$$
Like Theorem \[thm:uniqueness\_vd\], Theorem \[thm:stability1\_vd\] provides a performance guarantee at a lower sample complexity (in order) when compared to Theorem \[thm:stability1\]. On the other hand, the noise amplification factor of Theorem \[thm:stability1\_vd\] is larger than that of Theorem \[thm:stability1\] by a factor that depends on the distribution of the local incoherence parameters $\{(\mu_k,\tilde{\mu}_k)\}_{k=1}^n$.
Let $W$ and $\widetilde{W}$ be defined in . Let $\nu$ and $\tilde{\nu}$ be defined in . Define $$\label{eq:defLambda}
\Lambda := \sqrt{\bar{\mu}} [\mbox{diag}(\nu)]^{-1}
\quad \mathrm{and} \quad
\widetilde{\Lambda} = \sqrt{\bar{\mu}} [\mbox{diag}(\tilde{\nu})]^{-1}.$$ Then $W = T \Lambda$ and is equivalent to $$\label{eq:ell1min_noisy_vd2}
\minimize_{g' \in \cz^n} \, \norm{W g'}_1
\quad \mathrm{subject~to} \quad \norm{S_{\Omega'} g' - S_{\Omega'} \Lambda^{-1} x^\sharp}_2 \leq \epsilon.$$
Let $\breve{x}$ denote the minimizer to . Then, the minimizer $\hat{x}$ to is represented as $\hat{x} = \Lambda \breve{x}$, i.e. $\breve{x} = \Lambda^{-1} \hat{x}$. By applying Theorem \[thm:stability1\] to , we obtain $$\label{eq:pf_thm_stability1_vd:bnd}
\begin{aligned}
\norm{T \hat{x} - T x}_2
{} & = \norm{W \breve{x} - W \Lambda^{-1} x}_2 \\
{} & \leq \left\{ 2 + 28 \sqrt{N} \left( 3 n \norm{T \Lambda}_{1 \to 2} \norm{\widetilde{\Lambda}^{-1} T^\dagger}_{2 \to \infty} + 1 \right) \right\} \epsilon \norm{W}.
\end{aligned}$$
The proof completes by applying the following inequalities to : $$\norm{\widetilde{\Lambda}^{-1} T^\dagger}_{2 \to \infty} \leq \norm{\widetilde{\Lambda}^{-1}} \norm{T^\dagger}_{2 \to \infty}
= \frac{\bar{\mu} \norm{T^\dagger}_{2 \to \infty}}{\min_{k \in [n]} \tilde{\mu}_k}$$ and $$\norm{T \Lambda}_{1 \to 2} \leq \norm{T}_{1 \to 2} \norm{\Lambda}
= \frac{\max_{k \in [n]} \mu_k \norm{T^\dagger}_{2 \to \infty}}{\bar{\mu}}.$$
\[thm:stability2\_vd\] Suppose $T: \cz^n \to \cz^N$ is injective. Suppose that $T$ and $\widetilde{T} = (T^\dagger)^*$ satisfy with parameters $\{\mu_k\}_{k=1}^n$, $\{\tilde{\mu}_k\}_{k=1}^n$, and $\gamma$. Let $\bar{\mu}$ be defined in . Let $\Omega = \{\omega_1, \ldots, \omega_m\}$ be a multi-set of random indices where $\omega_k$s are i.i.d. copies of a random variable $\omega$ with the distribution in . Let $\hat{x}$ be the minimizer to with $x^\sharp$ satisfying . Then there exist numerical constants $C, c > 0$ for which the following holds. With probability $1 - N^{-4}$, we have $$\norm{\hat{x} - x}_2 \leq
\frac{\sigma_{\max}(T)}{\sigma_{\min}(T)}
\cdot
\frac{\bar{\mu}}{\min_{k \in [n]} \tilde{\mu}_k}
\cdot
\left[ 14\sqrt{N} + \frac{n}{m} \left\{ \frac{\max_{k \in [n]} \mu_k}{\min_{k \in [n]} \tilde{\mu}_k} \cdot \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} (|\Omega|-|\Omega'|) +1 \right\} \right] \epsilon,$$ provided $$m \geq \frac{C \bar{\mu} s \log^4 N}{1 - \norm{\gamma T^* T - I_n}}.$$ Furthermore, if $T^* T = I_n$, then with probability $1 - N^{-4}$, $$\norm{\hat{x} - x}_2 \leq
\frac{\sigma_{\max}(T)}{\sigma_{\min}(T)}
\cdot
\frac{\bar{\mu}}{\min_{k \in [n]} \tilde{\mu}_k}
\cdot
\left( 14\sqrt{N} + \frac{n}{m} \cdot \frac{\max_{k \in [n]} \mu_k}{\min_{k \in [n]} \tilde{\mu}_k} \cdot R \right) \epsilon,$$ where $R$ is the count of the most repeated elements in $\Omega$.
Let $W$ and $\widetilde{W}$ be defined in . Let $\nu$ and $\tilde{\nu}$ be defined in . Let $\Lambda$ and $\widetilde{\Lambda}$ be defined in . In the proof of Theorem \[thm:stability1\_vd\], we have shown that is equivalent to . In the proof of Theorem \[thm:uniqueness\_vd\], we have shown that $W$ and $\widetilde{W}$ satisfy \[eq:incoW,eq:isoW\]. Therefore, applying Theorem \[thm:stability2\] to with incoherence parameter $\bar{\mu}$ completes the proof.
Extension to Group Sparsity Models {#sec:main_result_gs}
==================================
In Sections \[sec:main\_result\] and \[sec:main\_result\_vd\], we considered the sparsity model in a transform domain. In applications, the transform $T x$ exhibits additional structures – group sparsity. For example, in compressed sensing of 2D signals, the sparsifying transform $\Phi = [\Phi_1^\top, \Phi_2^\top]^\top$ can be the concatenation of the horizontal and vertical finite difference operators $\Phi_1$ and $\Phi_2$. Anisotropic total variation encourages the sparsity of $\Phi f$ [@krahmer2014stable]. On the other hand, one can choose isotropic total variation, assuming that $\Phi_1 f$ and $\Phi_2 f$ are jointly sparse (see the experiments section). For another example, in compressed sensing of color images or hyperspectral images, the sparse codes acquired from applying the analysis operator to different channels are usually assumed to be jointly sparse. To exploit such structures, we extend the results in the previous sections to group sparsity models in a transform domain. More specifically, we assume that the transform $T x$ of unknown signal $x$ via $T: \cz^n \to \cz^L$ is $(s,t)$ strongly group sparse in the following sense.
Let $\calG = \{\calG_1, \ldots, \calG_N\}$ be a partition of $[L]$, i.e. $\bigcup_{j \in [N]} \calG_j = [L]$ and $\calG_j \bigcap \calG_{j'} = \emptyset$ for $j \neq j'$. A vector $z \in \cz^L$ is $(s,t)$ strongly group sparse with respect to $\calG$ if there exists $J \subset [N]$ such that $$\supp{z} \subset \calG_J,
\quad
|\calG_J| \leq t,
\quad \mathrm{and} \quad
|J| \leq s,$$ where $\calG_J$ is defined by $$\calG_J := \bigcup_{j \in J} \calG_j.$$
An atomic norm for this group sparsity model is given by $$\tnorm{z}_{\calG,1} := \sum_{j \in [N]} \norm{\Pi_{\calG_j} z}_2.$$ The dual norm of $\tnorm{\cdot}_{\calG,1}$ is defined by $$\tnorm{z}_{\calG,\infty} := \sup_{\zeta \in \cz^L: \tnorm{\zeta}_{\calG,1} \leq 1} |\langle \zeta, z \rangle|,$$ which is equivalently rewritten as $$\tnorm{z}_{\calG,\infty} = \max_{j \in [N]} \norm{\Pi_{\calG_j} z}_2.$$
A subgradient $\tilde{z}$ of $\tnorm{\cdot}_{\calG,1}$ at $z$ is given by $$\label{eq:subgrad_mixed}
\Pi_{\calG_j} \tilde{z}
=
\begin{dcases}
\frac{\Pi_{\calG_j} z}{\norm{\Pi_{\calG_j} z}_2}, & \Pi_{\calG_j} z \neq 0, \\
0, & \mathrm{otherwise},
\end{dcases}
\qquad \forall j \in [N].$$
In a special case where $\calG_j = \{j\}$ for all $j \in [N]$, the strong group sparsity level reduces to the conventional sparsity model. The analogy between the two models is summarized in Table \[tab:replace\_notation\]. All the results in the previous section generalize to the strong group sparsity model according to this analogy. In the below, we state the extended results as Theorems, the proofs of which are obtained in a straightforward way by modifying the proofs of analogous theorems and lemmas for the usual sparsity model according to Table \[tab:replace\_notation\]. Thus, we do not repeat the proofs in this section. (The only exception is Lemma \[lemma:E3\] and we provide an analogous Lemma \[lemma:E3’\] in Section \[subsec:riplike\_lemmas\].)
[>p[20ex]{}|C|C]{} & Sparsity Model & Group Sparsity Model\
$\mbox{dim}(R(T))$ & $N$ & $L = \sum_{j=1}^N |\calG_j|$\
& & $\gsupp{z} = \{ j :~ \Pi_{\calG_j} z \neq 0 \}$\
& & $\supp{z} = \bigcup_{j \in \gsupp{z}} \calG_j$\
group sparsity level & $\norm{z}_0 = |\supp{z}|$ & $\tnorm{z}_{\calG,0} = |\gsupp{z}|$\
total sparsity level & $\norm{z}_0 = |\supp{z}|$ & $\norm{z}_0 = |\supp{z}|$\
atomic norm & $\norm{z}_1 = \sum_{j=1}^N |\Pi_{\{j\}} z|$ & $\tnorm{\cdot}_{\calG,1} = \sum_{j=1}^N \norm{\Pi_{\calG_j} z}_2$\
dual norm & $\norm{\cdot}_\infty = \max_{j \in [N]} |\Pi_{\{j\}} z|$ & $\tnorm{\cdot}_{\calG,\infty} = \max_{j \in [N]} \norm{\Pi_{\calG_j} z}_2$\
subgradient & &\
of atomic norm & &\
incoherence & &\
parameter & &\
local incoherence & &\
parameters & &\
When $T x$ is strongly group sparse, we propose to estimate $T x$ by solving the following optimization problem, which generalizes : $$\label{eq:mixedmin}
\minimize_{g \in \cz^n} \, \tnorm{T g}_{\calG,1}
\quad \mathrm{subject~to} \quad S_\Omega g = S_\Omega x.$$ Then the following theorem, which is analogous to Theorem \[thm:uniqueness\], provides a performance guarantee for .
\[thm:uniqueness\_gs\] Let $\calG = \{\calG_1, \ldots, \calG_N\}$ be a partition of $[L]$. Suppose $T: \cz^n \to \cz^L$ is injective. Let $\mu_\calG$ be given by $$\label{eq:incoherence_gs}
\mu_\calG = \max_{k \in [n]} \max_{j \in [N]}
\max\left\{
n \norm{\gamma^{1/2} \Pi_{\calG_j} T e_k}_2^2,
n \norm{\gamma^{-1/2} \Pi_{\calG_j} \widetilde{T} e_k}_2^2
\right\},$$ where $\gamma$ is defined in from $T$ and $\widetilde{T} = (T^\dagger)^*$. Let $\Omega = \{\omega_1,\ldots,\omega_m\}$ be a multi-set of random indices where $\omega_k$s are i.i.d. following the uniform distribution on $[n]$. Suppose $T x$ is $(s,t)$ strongly group sparse with respect $\calG$. Then with probability $1 - e^{-\beta} - 3/n$, $x$ is the unique minimizer to provided $$\label{eq:samprate_gs}
m \geq \frac{C(1+\beta) \mu_\calG s}{1 - \norm{\gamma T^* T - I_n}} \left[ \log N
+ \log \left( \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \right) \right].$$
Suppose that $\mu$ and $\mu_\calG$ are the smallest constants satisfying corresponding incoherence conditions. If $t = \ell s$ and $|\calG_j| = \ell$ for all $j \in [N]$, then $$\norm{\Pi_{\calG_j} T e_k}_2 \leq \sqrt{|\calG_j|} \norm{T e_k}_\infty, \quad \forall j \in [N], ~ \forall k \in [n].$$ Thus, we have $\mu_\calG \leq \ell \mu$ and a sufficient condition for is given by $$m \geq \frac{C(1+\beta) \mu t}{1 - \norm{\gamma T^* T - I_n}} \left[ \log N
+ \log \left( \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \right) \right].$$ In other words, the sample complexity is proportional to the total sparsity level $t$ and there is no gain from the group structure. This inequality is tight if each $\Pi_{\calG_j} T e_k$ have nonzero elements of the same magnitude. Contrarily, if nonzero elements of each $\Pi_{\calG_j} T e_k$ vary a lot in their magnitudes, $\mu_\calG$ is smaller than $\ell \mu$ and there is gain from the group sparsity structure.
In the presence of noise to measurements, we generalize the optimization formulation for recovery in as follows: $$\label{eq:mixedmin_noisy}
\minimize_{g \in \cz^n} \, \tnorm{T g}_{\calG,1}
\quad \mathrm{subject~to} \quad \norm{S_{\Omega'} g - S_{\Omega'} x^\sharp}_2 \leq \epsilon.$$
The following theorem, analogous to Theorem \[thm:stability1\], provides a performance guarantee for .
\[thm:stability1\_gs\] Suppose the hypotheses of Theorem \[thm:uniqueness\_gs\] hold. Let $\hat{x}$ be the minimizer to with $x^\sharp$ satisfying $$\norm{S_{\Omega'} (x - x^\sharp)}_2 \leq \epsilon.$$ Then $$\norm{\hat{x} - x}_2 \leq
\frac{\sigma_{\max}(T)}{\sigma_{\min}(T)}
\cdot
\left\{ 2 + 28 \sqrt{N} \left( 3 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} + 1 \right) \right\} \epsilon.$$
The results for recovery using a variable density sampling designed from local incoherence parameters generalize in a similar way. We state the results in the following theorems.
\[thm:uniqueness\_vd\_gs\] Let $\calG = \{\calG_1, \ldots, \calG_N\}$ be a partition of $[L]$. Suppose $T: \cz^n \to \cz^N$ is injective. Let $\mu_{\calG,k}$ and $\tilde{\mu}_{\calG,k}$ be given by $$\label{eq:loc_incoherenceT_gs}
\mu_{\calG,k} = \max_{j \in [N]} n \norm{\gamma^{1/2} \Pi_{\calG_j} T e_k}_2^2
\quad \mathrm{and} \quad
\tilde{\mu}_{\calG,k} = \max_{j \in [N]} n \norm{\gamma^{-1/2} \Pi_{\calG_j} \widetilde{T} e_k}_2^2,$$ where $\gamma$ is defined in from $T$ and $\widetilde{T} = (T^\dagger)^*$. Let $\Omega = \{\omega_1, \ldots, \omega_m\}$ be a multi-set of random indices where $\omega_k$s are independent copies of a random variable $\omega$ with the following distribution: $$\label{eq:variable_density_gs}
\mathbb{P}(\omega = k) = \frac{\sqrt{\mu_{\calG,k} \tilde{\mu}_{\calG,k}}}{\sum_{j=1}^n \sqrt{\mu_{\calG,j} \tilde{\mu}_{\calG,j}}}, \quad \forall k \in [n].$$ Suppose $T x$ is $(s,t)$ strongly group sparse with respect $\calG$. Then with probability $1 - e^{-\beta} - 3/n$, $x$ is the unique minimizer to provided $$m \geq \frac{C(1+\beta) \bar{\mu}_\calG s}{1 - \norm{\gamma T^* T - I_n}} \left[ \log N
+ \log \left( \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \right) \right],$$ where $\bar{\mu}$ is defined as $$\bar{\mu}_\calG := \frac{1}{n} \sum_{k=1}^n \sqrt{\mu_{\calG,k} \tilde{\mu}_{\calG,k}}.$$
\[thm:stability1\_vd\_gs\] Suppose the hypotheses of Theorem \[thm:uniqueness\_vd\_gs\] hold. Let $\hat{x}$ be the minimizer to with $x^\sharp$ satisfying $$\label{eq:xharp_fid_gs}
\norm{S_{\Omega'} x - S_{\Omega'} x^\sharp}_2 \leq \epsilon.$$ Then $$\norm{\hat{x} - x}_2 \leq
\frac{\sigma_{\max}(T)}{\sigma_{\min}(T)}
\cdot
\frac{\bar{\mu}_\calG}{\min_{k \in [n]} \tilde{\mu}_{\calG,k}}
\cdot
\left[ 2 + 28 \sqrt{N} \left( \frac{\max_{k \in [n]} \mu_{\calG,k}}{\min_{k \in [n]} \tilde{\mu}_{\calG,k}} \cdot 3 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} + 1 \right) \right] \epsilon.$$
Circulant Transforms {#sec:circulant}
====================
Many sparsifying transforms fall into the category of circulant transforms, including the identity transform. A block transform (e.g., block DCT), when applied to all overlapping patches of a signal (sliding window with stride $1$, including the wrap-around patches at the edges), is a union of circulant transforms applied to the signal [@pfister2015learning]. In this section, we consider only the case when the measurement matrix $\Psi\in\cz^{n\times n}$ is the DFT matrix, and compute the variable density sampling distribution for sparsity with respect to a circulant transform, and distribution for joint sparsity with respect to a union of circulant transforms. We show that, if the circulant transforms are injective, the distributions and correspond to the uniform distribution on $[n]$. On the other hand, some circulant transforms are not injective (e.g., the finite difference operator for 1D total variation or 2D isotropic total variation). Even in this case, we show that the “variable” density sampling distributions are a variation of the uniform distribution.
Injective Circulant Transforms
------------------------------
We say $\Phi\in \cz^{n\times n}$ is a circulant transform (circulant matrix), if $\Phi f = \phi \circledast f$ is the circular convolution of $f$ with some vector $\phi \in \cz^n$, i.e. the matrix representation of $\Phi$ is given by $$\label{eq:circulant}
\Phi = \begin{bmatrix}
\phi[1] & \phi[n] & \phi[n-1] & \cdots & \phi[2] \\
\phi[2] & \phi[1] & \phi[n] & \cdots & \phi[3] \\
\phi[3] & \phi[2] & \phi[1] & \cdots & \phi[4] \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
\phi[n] & \phi[n-1] & \phi[n-2] & \cdots & \phi[1] \\
\end{bmatrix}$$ The identity transform is a circulant transform with $\phi=e_1$.
By linearity and shift invariance of circular convolution, a circulant transform $\Phi$ can always be diagonalized by the DFT matrix $\Psi$: $$\label{eq:diagonalize}
\Phi = \Psi^* \diag(\lambda) \Psi.$$ where $\lambda = \sqrt{n}\Psi \phi$ is the (unnormalized) DFT of $\phi$. The same argument also applies to a 2D circulant transform (circulant block circulant matrix) and the 2D DFT matrix. To avoid verbosity, we use circulant transform $\Phi$ and DFT matrix $\Psi$ to denote both 1D and 2D transforms.
\[thm:circulant1\] For the DFT matrix $\Psi\in\cz^{n\times n}$ and an invertible circulant transform $\Phi\in\cz^{n\times n}$, the sampling density distribution is the uniform distribution on $[n]$.
By the diagonalization in , we have $$\begin{aligned}
& T = \Phi \Psi^\dagger = \Psi^* \diag(\lambda),\\
& \widetilde{T} = (T^\dagger)^* = \Psi^* \diag(\tilde{\lambda}),\end{aligned}$$ where $\tilde{\lambda}$ is the complex conjugate of the element-wise inverse of $\lambda$, i.e. $\tilde{\lambda}[k]=(\lambda[k]^*)^{-1}$. Then, the following choice of parameters $\mu_k$ and $\tilde{\mu}_k$ satisfy : $$\begin{aligned}
& \mu_k = n\gamma \norm{Te_k}_\infty^2 = n\gamma |\lambda[k]|^2 \norm{\Psi^*e_k}_\infty^2 = \gamma |\lambda[k]|^2,\\
& \tilde{\mu}_k = n\gamma^{-1} \norm{\widetilde{T}e_k}_\infty^2 = n\gamma^{-1} |\tilde{\lambda}[k]|^2 \norm{\Psi^*e_k}_\infty^2 = \gamma^{-1} |\lambda[k]|^{-2},\end{aligned}$$ where $\Psi^*e_k$ is the $k$th column of the discrete Fourier basis and has infinity norm $1/\sqrt{n}$. Therefore, $\mu_k\tilde{\mu}_k = 1$ for all $k\in [n]$, and the distribution in is $\mathbb{P}(\omega = k) = 1/n$, i.e. the uniform distribution on $[n]$.
Applying a sparsifying circulant transform to a signal is equivalent to passing the signal through a sparsifying filter. One may also pass the signal through a bank of sparsifying filters. The filter bank is equivalent to a union of circulant transforms, concatenated as $$\label{eq:concatenation}
\Phi = [\Phi_1^\transpose, \Phi_2^\transpose,\dots, \Phi_\ell^\transpose]^\transpose \in\cz^{L\times n},$$ where $L=\ell n$. For example, the patch transform sparsity model [@pfister2015learning] corresponds to this case. The 2D isotropic total variation model (see Section \[sec:TV\]), as another example, has an additional joint sparsity structure. For the latter example, let us consider a particular partition $\calG = \{\calG_1, \ldots, \calG_n\}$ given by $$\label{eq:partition}
\calG_k = \{(j-1)n+k\}_{j=1}^\ell, \qquad \forall k\in [n].$$ For group sparsity on this union of circulant transforms, we have a result similar to Theorem \[thm:circulant1\].
\[thm:circulant2\] For the DFT matrix $\Psi\in\cz^{n\times n}$, an injective transform $\Phi\in\cz^{L\times n}$ defined in , and the partition defined in , the sampling density distribution is the uniform distribution on $[n]$.
Similar to , we have the following factorization for the concatenated transform: $$\label{eq:diagonalize2}
\Phi = (I_\ell \otimes \Psi^*) [\diag(\lambda_1),\diag(\lambda_2),\dots, \diag(\lambda_\ell)]^\top \Psi,$$ where $\lambda_j$ is the (unnormalized) DFT of $\phi_j$, the convolution kernel of the $j$th cirulant transform $\Phi_j$. Hence $T$ and $\widetilde{T}$ are $$\begin{aligned}
& T = \Phi \Psi^\dagger = (I_\ell \otimes \Psi^*)[\diag(\lambda_1),\diag(\lambda_2),\dots, \diag(\lambda_\ell)]^\top,\\
& \widetilde{T} = (T^\dagger)^* = T(T^*T)^{-1} = (I_\ell \otimes \Psi^*) [\diag(\tilde{\lambda}_1),\diag(\tilde{\lambda}_2),\dots, \diag(\tilde{\lambda}_\ell)]^\top,\end{aligned}$$ where $\tilde{\lambda}_j[k] = \lambda_j[k]/\bigl(\sum_{j'=1}^{\ell} |\lambda_{j'}[k]|^2\bigr)$. Then, $\mu_{\calG,k}$ and $\tilde{\mu}_{\calG,k}$ in are given respectively by $$\begin{aligned}
\mu_{\calG,k} & = n\gamma \max_{k'\in[n]}\norm{\Pi_{\calG_{k'}}Te_k}_2^2 \\
& = n\gamma \max_{k'\in[n]} \norm{\Pi_{\calG_{k'}}\bigl\{\bigl[\lambda_1[k],\lambda_2[k],\dots,\lambda_\ell[k]\bigr]^\transpose \otimes (\Psi^*e_k) \bigl\} }_2^2 \\
& = n\gamma \max_{k'\in[n]} |e_{k'}^\transpose \Psi^*e_k| \norm{\bigl[\lambda_1[k],\lambda_2[k],\dots,\lambda_\ell[k]\bigr]^\transpose}_2^2
= \gamma \bigl(\sum_{j=1}^{\ell} |\lambda_{j}[k]|^2\bigr)\end{aligned}$$ and $$\begin{aligned}
\tilde{\mu}_{\calG,k} & = n\gamma^{-1} \max_{k'\in[n]}\norm{\Pi_{\calG_{k'}}\widetilde{T}e_k}_2^2 \\
& = n\gamma^{-1} \max_{k'\in[n]} \norm{\Pi_{\calG_{k'}}\bigl\{\bigl[\tilde{\lambda}_1[k],\tilde{\lambda}_2[k],\dots,\tilde{\lambda}_\ell[k]\bigr]^\transpose \otimes (\Psi^*e_k) \bigl\} }_2^2 \\
& = \gamma^{-1} \bigl(\sum_{j=1}^{\ell} |\tilde{\lambda}_{j}[k]|^2\bigr)
= \gamma^{-1} \bigl(\sum_{j=1}^{\ell} |\lambda_{j}[k]|^2\bigr)^{-1}.\end{aligned}$$ Therefore, $\mu_{\calG,k}\tilde{\mu}_{\calG,k} = 1$ for all $k\in [n]$, and the distribution in is $\mathbb{P}(\omega = k) = 1/n$, i.e. the uniform distribution on $[n]$.
Non-injective Circulant Transforms {#sec:TV}
----------------------------------
In this section, we consider non-injective circulant transforms, such as the finite difference operator for 1D total variation, or the union of vertical and horizontal finite difference operators for 2D isotropic total variation. Since the spectral responses of these transforms are zero at certain frequencies, they are invariant to changes in the corresponding frequency components. Therefore, unless these frequency components are sampled in the measurement, they cannot be recovered from $\ell_1$-norm minimization or mixed norm minimization . Assuming without loss of generality that the null frequencies are the first $n_0$ columns in $\Psi^*$ ($n_0<\min\{n,m\}$), we adopt the following two-step sampling scheme:
1. Always sample indices $\omega_k = k$ for $k\in [n_0]$.
2. Generate a multi-set of $m-n_0$ indices $\{\omega_{n_0+1},\omega_{n_0+2},\dots,\omega_m\}$, which are i.i.d. following a distribution on $\{n_0+1,n_0+2,\dots,n\}$.
We can compute the sampling density on $\{n_0+1,n_0+2,\dots,n\}$ based on and (or and ) by removing the zero columns in $T=\Phi\Psi^*$ (replacing $T$ with $T[e_{n_0+1},e_{n_0+2},\dots,e_{n}]$). Next, we state variations of Theorems \[thm:circulant1\] and \[thm:circulant2\] in these cases.
\[cor:circulant1\] Suppose circulant transform $\Phi$ in satisfies $\lambda[k] = 0$ for $k\in[n_0]$. Then the sampling density in Step 2), for $\ell_1$-norm minimization based on , is the uniform distribution on $\{n_0+1,n_0+2,\dots,n\}$.
\[cor:circulant2\] Suppose concatenated transform $\Phi$ in satisfies $\lambda_j[k] = 0$ for all $j\in[\ell]$ and $k\in[n_0]$. Then the sampling density in Step 2), for mixed norm minimization based on , is the uniform distribution on $\{n_0+1,n_0+2,\dots,n\}$.
These results are direct consequences of Theorems \[thm:circulant1\] and \[thm:circulant2\], whose proofs translate with no changes other than removing the zero columns in $T$ (the columns indexed by $[n_0]$).
Next, we specialize these results to total variation minimization. We define the 1D finite difference operator $\Phi_{\mathrm{TV},n}$ by , where $$\phi[k] = \begin{cases}
1 & k = 1\\
-1 & k = 2\\
0 & k\in\{3,4,\dots, n\}.
\end{cases}$$ Then $\norm{f}_{\mathrm{TV}} = \norm{\Phi_{\mathrm{TV},n} f}_1$ is the 1D total variation of $f$. Clearly, the circulant transform $\Phi_{\mathrm{TV},n}$ is not injective, since its null space contains the direct current (DC) component – $\Psi^*e_1$. By Corollary \[cor:circulant1\], we adopt the following sampling scheme for total variation minimization:
1. Always sample index $\omega_1 = 1$.
2. Generate a multi-set of $m-1$ indices $\{\omega_2,\omega_3,\dots,\omega_m\}$, which are i.i.d. following the uniform distribution on $\{2,3,\dots,n\}$.
Total variation is more commonly used for 2D signals (e.g., images). For a 2D signal $f$ of size $n_1\times n_2$, the finite difference operator is a concatenation of the vertical and horizontal finite difference operators: $$\Phi_{\mathrm{TV},n_1,n_2} = \begin{bmatrix}
I_{n_2} \otimes \Phi_{\mathrm{TV},n_1} \\
\Phi_{\mathrm{TV},n_2} \otimes I_{n_1}
\end{bmatrix}.$$ The anisotropic and isotropic total variations of $f$ are defined by $$\begin{aligned}
& \norm{f}_{\mathrm{TV, aniso}} = \norm{\Phi_{\mathrm{TV},n_1,n_2} f}_1,\\
& \norm{f}_{\mathrm{TV, iso}} = \tnorm{\Phi_{\mathrm{TV},n_1,n_2} f}_{\calG,1},\end{aligned}$$ where the partion $\calG=\{\calG_1,\dots,\calG_n\}$ is defined by for $\ell=2$ and $n=n_1n_2$. Let the measurement operator $\Psi$ be the 2D DFT on signals of size $n_1\times n_2$. Similar to the 1D case, the DC component $\Psi^* e_1$ belongs to the null space of $\Phi_{\mathrm{TV},n_1,n_2}$. By Corollary \[cor:circulant2\], we use the same two-step sampling scheme as in the 1D case.
Proof of Theorem \[thm:uniqueness\] {#sec:pf_main_result_noiseless}
===================================
In this section, we prove Theorem \[thm:uniqueness\], which provides a sufficient condition for exact recovery of sparse signals in a transform domain from noiseless observations. The proof of Theorem \[thm:uniqueness\] is based on the golfing scheme [@gross2011recovering], which was originally proposed for matrix completion [@gross2011recovering] and later adopted to compressed sensing [@candes2011probabilistic] and to structured matrix completion [@chen2014robust].
The golfing scheme constructs an inexact dual certificate. The notion of a dual certificate was originally proposed for compressed sensing (cf. [@candes2006near]). To reconstruct an $s$-sparse $f \in \cz^d$ from $b = A f$, it was proposed to estimate $f$ as the solution to $$\minimize_{\tilde{f} \in \cz^d} \, \norm{\tilde{f}}_1 \quad \mathrm{subject~to} \quad b = A \tilde{f}.$$ A dual certificate is a subgradient $v \in \cz^d$ of $\norm{\cdot}_1$ at $f$ such that $f$ is orthogonal to all null vectors of the sensing matrix $A$. In matrix completion, the objective function is replaced from the $\ell_1$-norm to the nuclear norm and the sensing matrix is replaced by a pointwise sampling operator. Gross [@gross2011recovering] proposed the clever golfing scheme that constructs an inexact dual certificate, which is close to the exact dual certificate, and showed a low-rank matrix is exactly reconstructed from partial entries sampled at a near optimal rate. Candes and Plan [@candes2011probabilistic] adopted the golfing scheme back to compressed sensing and showed that exact recovery is guaranteed from $m = O(s \log d)$ incoherent measurements, which improves on the previous performance guarantee with $m = O(s \log^4 d)$.
Chen and Chi [@chen2014robust] adopted the golfing scheme to structured matrix completion. Let $T: \cz^n \to \cz^{n_1 \times n_1}$ be a linear operator that maps a vector $x \in \cz^n$ to a structured matrix $T x \in \cz^{n_1 \times n_2}$ (e.g., a Hankel matrix). Chen and Chi [@chen2014robust] proposed to estimate $x$ by $$\minimize_{g \in \cz^n} \, \norm{T g}_* \quad \mathrm{subject~to} \quad S_\Omega g = S_\Omega x.$$ This can be interpreted as recovery of low-rank matrices in a special transform domain, where $T$ maps the standard basis vectors $\{e_1,\ldots,e_n\}$ to unit-norm matrices with disjoint supports. Unlike compressed sensing or matrix completion, in structured matrix completion, the dimension of the vector space where the unknown signal is rearranged as a structured low-rank matrix is larger than the dimension of the vector space where measurements are sampled. In other words, $T$ is a tall matrix and is not surjective. With this nontrivial difference, the conventional approaches [@gross2011recovering; @candes2011probabilistic] do not apply directly to structured matrix completion. Chen and Chi [@chen2014robust] cleverly modified the definition of an inexact dual certificate and the golfing scheme accordingly and provided performance guarantee at a near optimal sample complexity. We adopt their approach to recovery of sparse signals in a transform domain, where the nuclear norm is replaced by the $\ell_1$-norm[^4]. Notably, we extend the theory to the case where $T$ is not necessarily a unitary transform (e.g., $T^* T = I_n$), which is the case in various practical applications.
We first present the following lemma that extends the notion of an inexact dual certificate for recovery of sparse signals in a transform domain where the transform $T$ is not necessarily unitary.
\[lemma:uniqueness\] Suppose that $T: \cz^n \to \cz^N$ is injective. Let $J \subset [N]$ denote the support of $T x$, i.e. the elements of $J$ correspond to the locations of the nonzero elements in $T x$. Let $\Omega$ be a multi-set that consists of elements in $[n]$ with possible repetitions. Let $\Omega'$ denote the set of all distinct elements in $\Omega$. Suppose that $$\label{eq:local_isometry_rank_deficient}
\left\|
\frac{n}{m} \Pi_J T S_\Omega^* S_\Omega T^\dagger \Pi_J
- \Pi_J T T^\dagger \Pi_J
\right\|
\leq \frac{1}{2}.$$ If there exists a vector $v \in \cz^N$ satisfying $$\label{eq:dualcert_vanish}
(T T^\dagger - T S_{\Omega'}^* S_{\Omega'} T^\dagger)^* v = 0,$$ $$\label{eq:dualcert_sgn}
\norm{\Pi_J (v - \mbox{\upshape sgn}(T x))}_2 \leq \frac{1}{7 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty}},$$ and $$\label{eq:dualcert_bnd}
\norm{(I_N - \Pi_J) v}_\infty \leq \frac{1}{2},$$ then $x$ is the unique minimizer to (\[eq:ell1min\]).
See Section \[subsec:pf:lemma:uniqueness\].
The next lemma shows that such an inexact dual certificate exists with hight probability under the hypothesis of Theorem \[thm:uniqueness\].
\[lemma:existence\] Suppose the hypotheses of Theorem \[thm:uniqueness\] hold. Then, with probability $1 - e^{-\beta} - 1/n$, there exists a vector $v \in \cz^N$ satisfying \[eq:dualcert\_vanish,eq:dualcert\_sgn,eq:dualcert\_bnd\].
See Section \[subsec:pf:lemma:existence\].
Lemma \[lemma:existence\] together with Lemma \[lemma:uniqueness\] implies Theorem \[thm:uniqueness\]. Indeed, we only need to verify . By Lemma \[lemma:E1\], holds with probability at least $1 - 2/n$ provided that $m \geq 32 \mu s \max\{1/(1-\norm{T^* T - I_n}), 1/6\} \log n$. This completes the proof of Theorem \[thm:uniqueness\].
In the next section, we will introduce fundamental estimates that will be used in the proofs of Lemmas \[lemma:uniqueness\] and \[lemma:existence\].
Lemmas on fundamental estimates {#subsec:riplike_lemmas}
-------------------------------
The following lemmas provides estimates on various functions of the random matrix $T S_\Omega^* S_\Omega T^\dagger$, which are analogous to the corresponding estimates for RIPless compressed sensing [@candes2011probabilistic].
Similarly to RIPless compressed sensing, we also employ the notion of incoherence. In fact, our incoherence assumption in is analogous to a generalized version for anisotropic compressed sensing [@rudelson2013reconstruction; @lee2013oblique; @kueng2014ripless].
One important distinction from compressed sensing is that the *isotropy* property is not satisfied. Indeed, since the random indices in $\Omega$ are i.i.d. following the uniform distribution on $[n]$, the random matrix $T S_\Omega^* S_\Omega T^\dagger$ satisfies $$\frac{n}{m} \mathbb{E} T S_\Omega^* S_\Omega T^\dagger = T T^\dagger.$$ While $T T^\dagger$ is an orthogonal projection (idempotent and self-adjoint), it is not necessarily an identity operator. The rank-deficiency of $T T^\dagger$ requires new analysis in Lemmas \[lemma:uniqueness\] and \[lemma:existence\] compared to the previous results for compressed sensing with the canonical sparsity model [@candes2011probabilistic]. However, the fundamental estimates measure the deviations of functions of $T S_\Omega^* S_\Omega T^\dagger$ from their expectations and do not require the isotropy property ($T^* T = I_n$). We present the following lemmas for the fundamental estimates, whose proofs are deferred to the appendix.
\[lemma:E1\] Suppose $T: \cz^n \to \cz^N$ and $\widetilde{T} = (T^\dagger)^*$ satisfy with parameter $\mu$. Let $\gamma$ be defined in . Let $J$ be a fixed subset of $[N]$ satisfying $|J| = s$. Let $\Omega = \{\omega_1,\ldots,\omega_m\}$ where $\omega_j$s are i.i.d. following the uniform distribution on $[n]$. Then for $\delta > 0$, $$\mathbb{P}\left(\left\| \Pi_J \left(\frac{n}{m} T S_\Omega^* S_\Omega T^\dagger - T T^\dagger \right) \Pi_J \right\| \geq \delta\right)
\leq 2s \exp\left( -\frac{m}{s\mu} \cdot \frac{\delta^2/2}{1/(1-\norm{\gamma T^* T - I_n})+\delta/3} \right).$$
See Appendix \[sec:pf:lemma:E1\].
\[lemma:E2\] Suppose $T: \cz^n \to \cz^N$ and $\widetilde{T} = (T^\dagger)^*$ satisfy with parameter $\mu$. Let $\gamma$ be defined in . Let $q \in \cz^N$ be a fixed vector. Let $J$ be a fixed subset of $[N]$ satisfying $|J| = s$. Let $\Omega = \{\omega_1,\ldots,\omega_m\}$ where $\omega_j$s are i.i.d. following the uniform distribution on $[n]$. Then for each $t \leq 1/2$, $$\mathbb{P}\left(\left\|\Pi_J \left(\frac{n}{m} T S_\Omega^* S_\Omega T^\dagger - T T^\dagger \right) \Pi_J q\right\|_2 \geq t \norm{\Pi_J q}_2\right)
\leq \exp\left\{ -\frac{1}{4} \left( t \sqrt{\frac{m(1-\norm{\gamma T^* T - I_n})}{s\mu}} - 1 \right)^2 \right\}.$$
See Appendix \[sec:pf:lemma:E2\].
\[lemma:E3\] Suppose $T: \cz^n \to \cz^N$ and $\widetilde{T} = (T^\dagger)^*$ satisfy with parameter $\mu$. Let $\gamma$ be defined in . Let $q \in \cz^N$ be a fixed vector. Let $J$ be a fixed subset of $[N]$ satisfying $|J| = s$. Let $\Omega = \{\omega_1,\ldots,\omega_m\}$ where $\omega_j$s are i.i.d. following the uniform distribution on $[n]$. Then for each $t > 0$, $$\begin{aligned}
{} & \mathbb{P}\left(\left\|\Pi_{[N] \setminus J} \left( \frac{n}{m} \widetilde{T} S_\Omega^* S_\Omega T^* - \widetilde{T} T^* \right) \Pi_J q\right\|_\infty \geq t \norm{\Pi_J q}_2\right) \\
{} & \quad \leq 2N \exp\left( - \frac{m}{2\mu} \cdot \frac{t^2}{1/(1-\norm{\gamma T^* T - I_n}) + \sqrt{s}t/3} \right).\end{aligned}$$
See Appendix \[sec:pf:lemma:E3\].
\[lemma:E3’\] Suppose $T: \cz^n \to \cz^L$ and $\widetilde{T} = (T^\dagger)^*$ satisfy with parameter $\mu$. Let $\gamma$ be defined in . Let $q \in \cz^L$ be a fixed vector. Let $\calG = \{\calG_1, \ldots, \calG_N\}$ be a partition of $[L]$. Let $J$ be a fixed subset of $[N]$ satisfying $|J| = s$. Let $\calG_J = \bigcup_{j \in J} \calG_j$. Let $\Omega = \{\omega_1,\ldots,\omega_m\}$ where $\omega_j$s are i.i.d. following the uniform distribution on $[n]$. Then for each $t > 0$, $$\begin{aligned}
{} & \mathbb{P}\left( \left\|\Pi_{[L] \setminus \calG_J} \left( \frac{n}{m} \widetilde{T} S_\Omega^* S_\Omega T^* - \widetilde{T} T^* \right) \Pi_{\calG_J} q\right\|_{\calG,\infty} \geq t \norm{\Pi_{\calG_J} q}_2\right) \\
{} & \quad \leq 2 \max_{j \in [N]} |\calG_j| N \exp\left( - \frac{m}{2\mu} \cdot \frac{t^2}{1/(1-\norm{\gamma T^* T - I_n}) + \sqrt{s}t/3} \right).\end{aligned}$$
See Appendix \[sec:pf:lemma:E3’\].
Proof of Lemma \[lemma:uniqueness\] {#subsec:pf:lemma:uniqueness}
-----------------------------------
Our proof essentially adapts the arguments of Chen and Chi [@chen2014robust Appendix B] for the structured low-rank matrix completion problem. There are two key differences in the two proofs. First, the $\ell_1$-norm replaces the nuclear norm. Second, $T$ is a general injective transform, which is not necessarily unitary. These differences require nontrivial modifications of crucial steps in the proof. Furthermore, the upper bound on the deviation of $v$ from $\mbox{sgn}(T x)$ in is sharpened by optimizing parameters. This improvement also applies to the previous work [@chen2014robust].
Let $\hat{x} = x + h$ be the minimizer to (\[eq:ell1min\]). We show that $T h = 0$ in two complementary cases. Then by the injectivity of $T$, $h = 0$, or equivalently, $\hat{x} = x$.
**Case 1:** We first consider the case when $h$ satisfies $$\label{eq:case1}
\norm{\Pi_J T h}_2 \leq 3 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \norm{\Pi_{[N] \setminus J} T h}_2.$$
Since $\mbox{sgn}(\Pi_{[N] \setminus J} T h)$ and $\mbox{sgn}(T x)$ have disjoint supports, it follows that $\mbox{sgn}(T x) + \mbox{sgn}(\Pi_{[N] \setminus J} T h)$ and $\mbox{sgn}(T x)$ coincide on $J$. Furthermore, $\norm{\mbox{sgn}(T x) + \mbox{sgn}(\Pi_{[N] \setminus J} T h)}_\infty \leq 1$. Therefore, $\mbox{sgn}(T x) + \mbox{sgn}(\Pi_{[N] \setminus J} T h)$ is a valid sub-gradient of the $\ell_1$-norm at $T x$. Then it follows that $$\label{eq:pf_lemma_uniqueness:ineq1}
\begin{aligned}
\norm{T x + T h}_1
{} & \geq \norm{T x}_1 + \langle \mbox{sgn}(T x) + \mbox{sgn}(\Pi_{[N] \setminus J} T h), ~ Th \rangle \\
{} & = \norm{T x}_1
+ \langle v, T h \rangle
+ \langle \mbox{sgn}(\Pi_{[N] \setminus J} T h), ~ Th \rangle
- \langle v - \mbox{sgn}(T x), Th \rangle.
\end{aligned}$$
In fact, $\langle v, T h \rangle = 0$ as shown below. The inner product of $T h$ and $v$ is decomposed as $$\label{eq:pf_lemma_uniqueness:ip}
\langle v, Th \rangle
= \langle v, (I_N - T T^\dagger) Th) \rangle
+ \langle v, (T T^\dagger - T S_{\Omega'}^* S_{\Omega'} T^\dagger) Th \rangle
+ \langle v, T S_{\Omega'}^* S_{\Omega'} T^\dagger Th \rangle.$$ Indeed, all three terms in the right-hand side of are 0. Since $T T^\dagger$ is the orthogonal projection onto the range space of $T$, the first term is 0. The second term is 0 by the assumption on $v$ in . Since $\hat{x}$ is feasible for , $S_\Omega \hat{x} = S_\Omega x$. Thus $S_\Omega h = S_\Omega (\hat{x} - x) = 0$, i.e. $e_\omega^* h = 0$ for all $\omega \in \Omega$, which also implies $S_{\Omega'} h = 0$. Then it follows that $S_{\Omega'} T^\dagger T h = S_{\Omega'} h = 0$. Thus the third term of the right-hand side of is 0.
Since the $\mbox{sgn}(\cdot)$ operator commutes with $\Pi_{[N] \setminus J}$ and $\Pi_{[N] \setminus J}$ is idempotent, we get $$\begin{aligned}
\langle \mbox{sgn}(\Pi_{[N] \setminus J} T h), ~ T h \rangle
{} & = \langle \Pi_{[N] \setminus J} \mbox{sgn}(\Pi_{[N] \setminus J} T h), ~ T h \rangle \\
{} & = \langle \mbox{sgn}(\Pi_{[N] \setminus J} T h), ~ \Pi_{[N] \setminus J} T h \rangle \\
{} & = \norm{\Pi_{[N] \setminus J} T h}_1.\end{aligned}$$
Then implies $$\label{eq:pf_lemma_uniqueness:ineq2}
\norm{T x + T h}_1 \geq \norm{T x}_1 + \norm{\Pi_{[N] \setminus J} T h}_1 - \langle v - \mbox{sgn}(T x), Th \rangle.$$
We derive an upper bound on the magnitude of the last term in the right-hand side of given by
\[eq:pf\_lemma\_uniqueness:ineq3\] $$\begin{aligned}
| \langle v - \mbox{sgn}(T x), Th \rangle |
{} & = | \langle \Pi_J (v - \mbox{sgn}(T x)), Th \rangle + \langle \Pi_{[N] \setminus J} (v - \mbox{sgn}(T x)), Th \rangle | \nonumber \\
{} & \leq | \langle \Pi_J (v - \mbox{sgn}(T x)), Th \rangle | + | \langle \Pi_{[N] \setminus J} v, Th \rangle | \label{eq:pf_lemma_uniqueness:ineq3a} \\
{} & \leq \norm{\Pi_J (v - \mbox{sgn}(T x))}_2 \norm{\Pi_J Th}_2
+ \norm{\Pi_{[N] \setminus J} v}_\infty \norm{\Pi_{[N] \setminus J} T h}_1 \label{eq:pf_lemma_uniqueness:ineq3b} \\
{} & \leq \frac{1}{7 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty}} \norm{\Pi_J Th}_2 + \frac{1}{2} \norm{\Pi_{[N] \setminus J} T h}_1, \label{eq:pf_lemma_uniqueness:ineq3c}\end{aligned}$$
where holds by the triangle inequality and the fact that $T x$ is supported on $J$; by Hölder’s inequality; by the assumptions on $v$ in and .
We continue by applying to and get $$\begin{aligned}
\norm{T x + T h}_1
{} & \geq \norm{T x}_1 - \frac{1}{7 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty}} \norm{\Pi_J Th}_2 + \frac{1}{2} \norm{\Pi_{[N] \setminus J} T h}_1 \\
{} & \geq \norm{T x}_1 - \frac{3}{7} \norm{\Pi_{[N] \setminus J} T h}_2 + \frac{1}{2} \norm{\Pi_{[N] \setminus J} T h}_2 \\
{} & = \norm{T x}_1 + \frac{1}{14} \norm{\Pi_{[N] \setminus J} T h}_2,\end{aligned}$$ where the second step follows from .
Then, $\norm{T \hat{x}}_1 \geq \norm{T x}_1 \geq \norm{T \hat{x}}_1$, which implies $\Pi_{[N] \setminus J} T h = 0$. By (\[eq:case1\]), we also have $\Pi_J T h = 0$. Therefore, it follows that $T h = 0$.
**Case 2:** Next, we consider the complementary case when $h$ satisfies $$\label{eq:case2}
\norm{\Pi_J T h}_2 > 3 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \norm{\Pi_{[N] \setminus J} T h}_2.$$
In the previous case, we have shown that $S_\Omega h = 0$. Thus $S_\Omega T^\dagger T h = 0$. Then together with $(I_N - T T^\dagger) T = 0$, we get $$\begin{aligned}
\left( \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger \right) T h = 0,\end{aligned}$$ which implies $$\label{eq:pf_lemma_uniqueness:ineq4}
\begin{aligned}
0 {} & \geq \left\langle \Pi_J T h, \left( \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger \right) T h \right\rangle \\
{} & = \left\langle \Pi_J T h, \left( \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger \right) \Pi_J T h \right\rangle \\
{} & \quad + \left\langle \Pi_J T h, \left( \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger \right) \Pi_{[N] \setminus J} T h \right\rangle.
\end{aligned}$$
The magnitude of the first term in the right-hand side of is lower-bounded by $$\label{eq:pf_lemma_uniqueness:lb}
\begin{aligned}
{} & \left| \left\langle \Pi_J T h, \left( \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger \right) \Pi_J T h \right\rangle \right| \\
{} & = \left| \langle \Pi_J T h, \Pi_J T h \rangle \right|
- \left| \left\langle \Pi_J T h, \left( T T^\dagger - \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger \right) \Pi_J T h \right\rangle \right| \\
{} & \geq \norm{\Pi_J T h}_2^2 - \left\|\Pi_J T T^\dagger \Pi_J - \frac{n}{m} \Pi_J T S_\Omega^* S_\Omega T^\dagger \Pi_J\right\| \norm{\Pi_J T h}_2^2 \\
{} & \geq \frac{1}{2} \norm{\Pi_J T h}_2^2,
\end{aligned}$$ where the last step follows from the assumption in .
Next, we derive an upper bound on the second term in the right-hand side of . To this end, we first computes the operator norm of $T e_k e_k^* T^\dagger$ for $k \in [n]$. In fact, $\norm{T e_k e_k^* T^\dagger} = \norm{T e_k}_2 \norm{\widetilde{T} e_k}_2$, where $\widetilde{T}$ is the adjoint of $T^\dagger$. Therefore, we only need to compute $\norm{T e_k}_2$ and $\norm{\widetilde{T} e_k}_2$. First, $\norm{T e_k}_2$ is upper-bounded by $$\begin{aligned}
\max_{k \in [n]} \norm{T e_k}_2 = \norm{T}_{1 \to 2}.\end{aligned}$$ On the other hand, $\norm{\widetilde{T} e_k}_2$ is upper-bounded by $$\begin{aligned}
\max_{k \in [n]} \norm{\widetilde{T} e_k}_2
= \norm{\widetilde{T}}_{1 \to 2} = \norm{T^\dagger}_{2 \to \infty},\end{aligned}$$ where the last step holds since $\widetilde{T}$ is the adjoint operator of $T^\dagger$ and $\ell_\infty^n$ is the dual space of $\ell_1^n$. By the above upper bounds on $\norm{T e_k}_2$ and $\norm{\widetilde{T} e_k}_2$, we get $$\begin{aligned}
\norm{T e_k e_k^* T^\dagger} \leq \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty}.\end{aligned}$$ Then the operator norm of $ \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger $ is upper-bounded by $$\label{eq:pf_lemma_uniqueness:ineq5}
\begin{aligned}
\left\| \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger \right\|
{} & \leq \frac{n}{m} \left( \norm{T e_{\omega_1} e_{\omega_1}^* T^\dagger + I_N - T T^\dagger} + \sum_{j=2}^m \norm{T e_{\omega_j} e_{\omega_j}^* T^\dagger}_2 \right) \\
{} & \leq \frac{n}{m} \left( \max(\norm{T e_{\omega_1} e_{\omega_1}^* T^\dagger}, \norm{I_N - T T^\dagger}) + \sum_{j=2}^m \norm{T e_{\omega_j} e_{\omega_j}^* T^\dagger}_2 \right) \\
{} & \leq n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty},
\end{aligned}$$ where the second step follows since $(T e_{\omega_1} e_{\omega_1}^* T^\dagger)^*(I_N - T T^\dagger) = 0$ and $(I_N - T T^\dagger) (T e_{\omega_1} e_{\omega_1}^* T^\dagger)^* = 0$. The second term in the right-hand side of is then upper-bounded by $$\label{eq:pf_lemma_uniqueness:ub}
\begin{aligned}
{} & \left| \left\langle \Pi_J T h, \left( \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger \right) \Pi_{[N] \setminus J} T h \right\rangle \right| \\
{} & \leq \left\| \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger \right\| \norm{\Pi_J T h}_2 \norm{\Pi_{[N] \setminus J} T h}_2 \\
{} & \leq n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \norm{\Pi_J T h}_2 \norm{\Pi_{[N] \setminus J} T h}_2,
\end{aligned}$$ where the last step follows from .
Applying and to provides $$\begin{aligned}
0 {} & \geq \left| \left\langle \Pi_J T h, \left( \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger \right) \Pi_J T h \right\rangle \right| \\
{} & \quad - \left| \left\langle \Pi_J T h, \left( \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger + I_N - T T^\dagger \right) \Pi_{[N] \setminus J} T h \right\rangle \right| \\
{} & \geq \frac{1}{2} \norm{\Pi_J T h}_2^2 - n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \norm{\Pi_J T h}_2 \norm{\Pi_{[N] \setminus J} T h}_2 \\
{} & \geq \frac{1}{2} \norm{\Pi_J T h}_2^2 - \frac{1}{3} \norm{\Pi_J T h}_2^2 \\
{} & = \frac{1}{6} \norm{\Pi_J T h}_2^2 \geq 0,\end{aligned}$$ where the second inequality follows from . Then it is implied that $\Pi_J T h = 0$. By (\[eq:case2\]), we also have $\Pi_{[N] \setminus J} T h = 0$. Therefore, $T h = 0$, which completes the proof.
Proof of Lemma \[lemma:existence\] {#subsec:pf:lemma:existence}
----------------------------------
We construct a dual certificate $v$ using a golfing scheme. Since the isotropy is not satisfied, the original golfing scheme needs to be modified accordingly. We adopt the version for structured matrix completion [@chen2014robust].
Recall that the elements of $\Omega = \{\omega_1, \ldots, \omega_m\}$ are i.i.d. following the uniform distribution on $[n]$. We partition the multi-set $\Omega$ into $\ell$ multi-sets so that $\Omega_1$ consists of the first $m_1$ elements of $\Omega$, $\Omega_2$ consists of the next $m_2$ elements of $\Omega$, and so on, where $\sum_{i=1}^\ell m_i = m$. Then, $\Omega_i$s are mutually independent and each $\Omega_i$ consists of i.i.d. random indices.
The version of the golfing scheme in this paper generates a dual certificate $v \in \cz^N$ from intermediate vectors $q_i \in \cz^N$ for $i=0,\ldots,\ell-1$ by $$v = \sum_{i=1}^\ell \left(\frac{n}{m_i} \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^* + I_N - T T^\dagger \right) q_{i-1},$$ where $q_i$s are generated as follows: first, initialize $q_0 = \mbox{sgn}(T x)$; next, generate $q_i$s recursively by $$q_i = \Pi_J \left( \widetilde{T} T^* - \frac{n}{m_i} \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^* \right) q_{i-1}, \quad i = 1,\ldots,\ell-1.$$ Here, $\widetilde{T}$ denotes the adjoint of $T^\dagger$.
Note that $$\begin{aligned}
{} & (T T^\dagger - T S_{\Omega'}^* S_{\Omega'} T^\dagger)^*
\left( \frac{n}{m_i} \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^* + I_N - \widetilde{T} T^* \right) \\
{} & = \frac{n}{m_i} (T T^\dagger - T S_{\Omega'}^* S_{\Omega'} T^\dagger)^* \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^*
+ (T T^\dagger - T S_{\Omega'}^* S_{\Omega'} T^\dagger)^* (I_N - \widetilde{T} T^*) \\
{} & = \frac{n}{m_i} (\widetilde{T} T^* - \widetilde{T} S_{\Omega'}^* S_{\Omega'} T^*) \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^*
+ (\widetilde{T} T^* - \widetilde{T} S_{\Omega'}^* S_{\Omega'} T^*) (I_N - \widetilde{T} T^*) = 0.\end{aligned}$$ Thus it follows that $v$ satisfies .
The rest of the proof is devoted to show that $v$ satisfies and , which follows similarly to the proof of [@candes2011probabilistic Lemma 3.3]. For completeness, we verify that the arguments in [@candes2011probabilistic] are valid in our setting (with neither isotropy nor self-adjointness).
We show that $q_i$ satisfies the following two properties with high probability for each $i \in [\ell]$: first, $$\label{eq:decay_qi}
\norm{q_i}_2 \leq c_i \norm{q_{i-1}}_2$$ and, second, $$\label{eq:bnd_qi_infty}
\left\| \Pi_{[N] \setminus J} \left(\frac{n}{m_i} \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^* + I_N - \widetilde{T} T^* \right) q_{i-1} \right\|_\infty \leq t_i \norm{q_{i-1}}_2.$$
Let $p_1(i)$ (resp. $p_2(i)$) denote the probability that the inequality in (resp. ) does not hold. Since $q_{i-1}$ is independent of $\Omega_i$, by Lemma \[lemma:E2\], $p_1(i)$ is upper-bounded by $$p_1(i) \leq \exp\left( - \frac{1}{4}(c_i \sqrt{m_i(1-\norm{\gamma T^* T - I_n})/(s\mu)}-1)^2 \right).$$ Therefore, $p_1(i) \leq \frac{1}{\alpha} e^{-\beta}$ if $$\label{eq:bndmi_p1}
m_i \geq \frac{2+8(\beta + \log \alpha)}{c_i^2} \cdot \frac{\mu s}{1-\norm{\gamma T^* T - I_n}}.$$ On the other hand, note that $q_{i-1} = \Pi_J q_{i-1}$. Then it follows that $$\begin{aligned}
\left\| \Pi_{[N] \setminus J} \left(\frac{n}{m_i} \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^* + I_N - \widetilde{T} T^* \right) q_{i-1} \right\|_\infty
{} & = \left\| \Pi_{[N] \setminus J} \left(\frac{n}{m_i} \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^* + I_N - \widetilde{T} T^* \right) \Pi_J q_{i-1} \right\|_\infty \\
{} & = \left\| \Pi_{[N] \setminus J} \left(\frac{n}{m_i} \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^* - \widetilde{T} T^* \right) \Pi_J q_{i-1} \right\|_\infty.\end{aligned}$$ Again, we use the fact that $q_{i-1}$ is independent of $\Omega_i$. Then, by Lemma \[lemma:E3\], $p_2(i)$ is upper-bounded by $$p_2(i) \leq 2N \exp\left( - \frac{3t_i^2 m_i}{6\mu/(1-\norm{\gamma T^* T - I_n}) + 2 \mu \sqrt{s} t_i} \right).$$ Therefore, $p_2(i) \leq \frac{1}{\alpha} e^{-\beta}$ if $$\label{eq:bndmi_p2}
m_i \geq \left( \frac{2}{t_i^2 s (1-\norm{\gamma T^* T - I_n})} + \frac{2}{3t_i\sqrt{s}} \right) (\beta + \log(2\alpha) + \log N) s\mu.$$
We set the parameters similarly to the proof of [@candes2011probabilistic Lemma 3.3] as follows: $$\label{eq:param}
\begin{aligned}
\ell {} & = \left\lceil \frac{\log_2 s}{2} + \log_2 n
+ \log_2 \left( \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \right) \right\rceil + 3, \\
c_i {} & =
\begin{cases}
1/\lceil 2 \sqrt{\log N} ~ \rceil & i=1,2,3, \\
1/2 & 3 \leq i \leq \ell,
\end{cases} \\
t_i {} & =
\begin{cases}
1/\lceil 4 \sqrt{s} ~\rceil & i=1,2,3, \\
\log N / \lceil 4 \sqrt{s} ~\rceil & 3 \leq i \leq \ell,
\end{cases}\\
m_i {} & = \lceil 10(1+\log 6+\beta) \mu s c_i^{-2} \rceil, \quad \forall i. \\
\end{aligned}$$ By the construction of $v$, we have $$\begin{aligned}
\Pi_J v {} & = \sum_{i=1}^\ell \Pi_J \left(\frac{n}{m_i} \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^* + I_N - \widetilde{T} T^* \right) q_{i-1} \\
{} & = \sum_{i=1}^\ell \left[ \Pi_J q_{i-1} - \Pi_J \left(\widetilde{T} T^* - \frac{n}{m_i} \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^* \right) q_{i-1} \right] \\
{} & = \sum_{i=1}^\ell \left( q_{i-1} - q_i \right)
= q_0 - q_\ell \\
{} & = \mbox{sgn}(T x) - q_\ell = \Pi_J \mbox{sgn}(T x) - q_\ell.\end{aligned}$$ Therefore, implies $$\norm{\Pi_J (v - \mbox{sgn}(T x))}_2
= \norm{q_\ell}_2
\leq \prod_{i=1}^\ell c_i \norm{\mbox{sgn}(T x)}_2
\leq \frac{\sqrt{s}}{2^\ell \log N}.$$
Next, by and , we have $$\label{eq:dualcert_bnd_proof}
\begin{aligned}
\norm{\Pi_{[N] \setminus J} v}_\infty
{} & \leq \sum_{i=1}^\ell \left\| \Pi_{[N] \setminus J} \left(\frac{n}{m_i} \widetilde{T} S_{\Omega_i}^* S_{\Omega_i} T^* + I_N - \widetilde{T} T^* \right) q_{i-1} \right\|_\infty \\
{} & \leq \sum_{i=1}^\ell t_i \norm{q_{i-1}}_2 \\
{} & \leq \sqrt{s} \left(t_1 + \sum_{i=2}^\ell t_i \prod_{j=1}^{i-1} c_j\right).
\end{aligned}$$ By setting parameters as in , the right-hand side in is further upper-bounded by $$\frac{1}{4} \left( 1 + \frac{1}{2\sqrt{\log N}} + \frac{\log N}{4 \log N} + \cdots \right) < \frac{1}{2}.$$ Then, we have shown that $v$ satisfies .
It remains to show that and hold with the desired probability. From and , it follows that $$p_j(i) \leq \frac{1}{6} e^{-\beta}, \quad \forall i \in [\ell], ~ \forall j=1,2.$$ In particular, we have $$\sum_{j=1}^2 \sum_{i=1}^3 p_j(i) \leq e^{-\beta}.$$ This implies that the first three $\Omega_i$s satisfy and except with probability $e^{-\beta}$.
On the other hand, we also have $$p_1(i) + p_2(i) < \frac{1}{3}, \quad \forall i = 4,\ldots,\ell.$$ In other words, the probability that $\Omega_i$ satisfies and is at least $2/3$. The union bound doesn’t show that $\Omega_i$ satisfies and for all $i \geq 4$ with the desired probability.
As in the proof of [@candes2011probabilistic Lemma 3.3], we adopt the oversampling and refinement strategy by Gross [@gross2011recovering]. Recall that each random index set $\Omega_i$ consists of i.i.d. random indices following the uniform distribution on $[n]$. Thus $\Omega_i$s are mutually independent. In particular, we set $\Omega_i$s are of the same cardinality in . Therefore, $\Omega_i$s are i.i.d. random variables. We generate a few extra copies of $\Omega_i$ for $i = \ell+1,\ldots,\ell'+3$ where $\ell' = 3(\ell-3)$. Then, by Hoeffding’s inequality, there exist at least $\ell-3$ $\Omega_i$s for $i \geq 4$ that satisfy and with probability $1 - 1/n$. (We refer more technical details for this step to [@candes2011probabilistic Section III.B].) Therefore, there are $\ell$ good $\Omega_i$s satisfying and with probability $1 - e^{-\beta} - 1/n$, and the dual certificate $v$ is constructed from these good $\Omega_i$s. The total number of samples for this construction requires $$m \geq \frac{40(1+\log 6 +\beta)\mu s (3\log N + 3\ell)}{1 - \norm{\gamma T^* T - I_n}},$$ which can be simplified as $$m \geq \frac{C(1+\beta) \mu s}{1 - \norm{\gamma T^* T - I_n}} \left[ \log N
+ \log \left( \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \right) \right]$$ for a numerical constant $C$.
Proofs for Theorems \[thm:stability1\] and \[thm:stability2\] {#sec:pf_main_result_noisy}
=============================================================
In this section, we prove Theorems \[thm:stability1\] and \[thm:stability2\], which provide sufficient conditions for stable recovery of sparse signals in a transform domain from noisy data.
Proof of Theorem \[thm:stability1\] {#subsec:pf:thm:stability1}
-----------------------------------
Let $h := \hat{x} - x$. Since $\hat{x}$ is the minimizer to , it follows that $$\label{eq:pf_thm_stability:bnd1}
\begin{aligned}
\norm{T x}_1
{} & \geq \norm{T \hat{x}}_1 = \norm{T x + T h}_1 \\
{} & \geq \norm{T x + T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_1 - \norm{T S_{\Omega'}^* S_{\Omega'} h}_1.
\end{aligned}$$
Since $x$ and $\hat{x}$ are feasible for , it follows that $$\label{eq:noisy_feas}
\begin{aligned}
{} & \max\left( \norm{T S_{\Omega'}^* S_{\Omega'} (\hat{x} - x^\sharp)}_2, ~ \norm{T S_{\Omega'}^* S_{\Omega'} (x - x^\sharp)}_2 \right) \\
{} & \leq \norm{T} \max\left( \norm{S_{\Omega'} (\hat{x} - x^\sharp)}_2, ~ \norm{S_{\Omega'} (x - x^\sharp)}_2 \right) \leq \epsilon \norm{T}.
\end{aligned}$$ Therefore, by the triangle inequality, we have $$\label{eq:tube_cstr}
\norm{T S_{\Omega'}^* S_{\Omega'} h}_2 = \norm{T S_{\Omega'}^* S_{\Omega'} (\hat{x} - x)}_2
\leq \norm{T S_{\Omega'}^* S_{\Omega'} (\hat{x} - x^\sharp)}_2 + \norm{T S_{\Omega'}^* S_{\Omega'} (x - x^\sharp)}_2 \leq 2 \epsilon \norm{T}.$$
The rest of the proof will compute upper bounds on $\norm{T h}_2$ in two complementary cases similarly to the proof of Lemma \[lemma:uniqueness\]. Unlike the noiseless case ($x^\sharp = x$ and $\epsilon = 0$) in Lemma \[eq:bnd\_qi\_infty\], the condition in does not necessarily imply $S_\Omega h = 0$. In fact, the proof of Lemma \[lemma:uniqueness\] critically depends on the condition $S_\Omega h = 0$. Essentially, we replace $h$ by $(I_n - S_{\Omega'}^* S_{\Omega'}) h$. Then it follows that $$\label{eq:noisy_vanish}
%B_{\Omega'} (B - B_{\Omega'}) T h = 0.
S_{\Omega'} (I_n - S_{\Omega'}^* S_{\Omega'}) h = 0.$$
**Case 1:** We first consider the case when $(I_n - S_{\Omega'}^* S_{\Omega'}) h$ satisfies $$\label{eq:case1noisy}
\norm{\Pi_J T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2
\leq 3 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \norm{\Pi_{[N] \setminus J} T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2,$$ where $J$ denotes the support of $T x$.
Under , similarly to the proof of Lemma \[lemma:uniqueness\], we have $$\label{eq:pf_thm_stability:bnd2}
\norm{T x + T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_1
\geq \norm{T x}_1 + \frac{1}{14} \norm{\Pi_{[N] \setminus J} T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2.$$ Combining and provides $$\label{eq:pf_thm_stability:bnd3}
\norm{\Pi_{[N] \setminus J} T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2
\leq 14 \norm{T S_{\Omega'}^* S_{\Omega'} h}_1
\leq 14 \sqrt{N} \norm{T S_{\Omega'}^* S_{\Omega'} h}_2 \leq 28\sqrt{N} \epsilon \norm{T}.$$ On the other hand, implies $$\label{eq:pf_thm_stability:bnd4}
\begin{aligned}
\norm{T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2
{} & \leq \norm{\Pi_{[N] \setminus J} T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2 + \norm{\Pi_J T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2 \\
{} & \leq (1 + 3 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty}) \norm{\Pi_{[N] \setminus J} T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2.
\end{aligned}$$ Therefore, combining \[eq:tube\_cstr,eq:pf\_thm\_stability:bnd3,eq:pf\_thm\_stability:bnd4\] provides $$\label{eq:pf_thm_stability:bnd5}
\norm{T h}_2 \leq \left\{ 2 + 28 \sqrt{N} \left( 3 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} + 1 \right) \right\} \epsilon \norm{T}.$$
**Case 2:** Next, we consider the complementary case when $(I_n - S_{\Omega'}^* S_{\Omega'}) h$ satisfies $$\label{eq:case2noisy}
\norm{\Pi_J T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2
> 3 n \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} \norm{\Pi_{[N] \setminus J} T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2.$$
Again, similarly to the proof of Lemma \[lemma:uniqueness\], we get $(I_n - S_{\Omega'}^* S_{\Omega'}) h = 0$. Therefore, $$\norm{T h}_2 \leq \norm{T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2 + \norm{T S_{\Omega'}^* S_{\Omega'} T h}_2 \leq 2 \epsilon \norm{T},$$ which is smaller than the upper bound on $\norm{T h}_2$ in the previous case.
By applying $\norm{h}_2 \leq \norm{T h}_2 / \sigma_{\min}(T)$ to , we obtain the desired upper bound on $\norm{h}_2$. This completes the proof.
Proof of Theorem \[thm:stability2\] {#subsec:pf:thm:stability2}
-----------------------------------
By Theorem \[thm:rboplike\], the condition in implies that $\frac{n}{m} T S_\Omega^* S_\Omega T^\dagger$ satisfies the condition in with constant $\delta = 1/3$. Note that the estimates in Lemmas \[lemma:E1,lemma:E2,lemma:E3\] are implied by . Therefore, the rest of the proof will be identical to that of Theorem \[thm:stability2\] except that we compute a tighter upper bound on $\norm{\Pi_J T (I_n - S_{\Omega'}^* S_{\Omega'}) h}_2$ as follows.
First, we decompose $\Pi_J T (I_n - S_{\Omega'}^* S_{\Omega'}) h$ as $$\label{eq:pf_thm_stability2:decomp}
\begin{aligned}
\Pi_J T (I_n - S_{\Omega'}^* S_{\Omega'}) h
{} & = \Pi_J \left(T T^\dagger - \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger\right) T h \\
{} & \quad + \Pi_J \frac{n}{m} T (S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'}) h \\
{} & \quad + \Pi_J \left( \frac{n}{m} - 1 \right) T S_{\Omega'}^* S_{\Omega'} h.
\end{aligned}$$
Let $J_1$ correspond to the indices of the $s$-largest coefficients of $\Pi_{[N] \setminus J} T h$; $J_2$ to the indices of the next $s$-largest coefficients of $\Pi_{[N] \setminus J} T h$, and so on. Then the $\ell_2$-norm first term in the right-hand side of is upper-bounded by $$\label{eq:pf_thm_stability2:bnd1}
\begin{aligned}
{} & \left\|\Pi_J \left(T T^\dagger - \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger\right) T h\right\|_2 \\
{} & \leq \left\|\Pi_J \left(T T^\dagger - \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger\right) \Pi_{J \cup J_1} T h\right\|_2 \\
{} & \quad + \sum_{i \geq 2} \left\|\Pi_J \left(T T^\dagger - \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger\right) \Pi_{J_i} T h\right\|_2.
\end{aligned}$$ By , the first term in the right-hand side of is upper-bounded by $$\label{eq:pf_thm_stability2:bnd2}
\left\|\Pi_J \left(T T^\dagger - \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger\right) \Pi_{J \cup J_1} T h\right\|_2
\leq \frac{1}{3} \norm{\Pi_{J \cup J_1} T h}_2
\leq \frac{1}{3} \norm{T h}_2.$$
By , the second term in the right-hand side of is upper-bounded by $$\label{eq:pf_thm_stability2:bnd3}
\begin{aligned}
{} & \sum_{i \geq 2} \left\|\Pi_J \left(T T^\dagger - \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger\right) \Pi_{J_i} T h\right\|_2 \\
{} & \leq \frac{1}{3} \sum_{i \geq 2} \norm{\Pi_{J_i} T h}_2
\leq \frac{1}{3\sqrt{s}} \sum_{i \geq 1} \norm{\Pi_{J_i} T h}_1
= \frac{1}{3\sqrt{s}} \norm{\Pi_{[N] \setminus J} T h}_1.
\end{aligned}$$ Since $\hat{x}$ is the minimizer to , we have the so called “cone” constraint: $$\norm{T x}_1 \geq \norm{T \hat{x}}_1 \geq \norm{Tx + T h}_1 \geq \norm{\Pi_J T x}_1 - \norm{\Pi_J T h}_1 + \norm{\Pi_{[N] \setminus J} T h}_1 - \norm{\Pi_{[N] \setminus J} T x}_1,$$ which implies $$\norm{\Pi_{[N] \setminus J} T h}_1 \leq 2 \norm{\Pi_{[N] \setminus J} T x}_1 + \norm{\Pi_J T h}_1 = \norm{\Pi_J T h}_1,$$ where the last step follows since $T x$ is supported on $J$. Therefore, implies $$\label{eq:pf_thm_stability2:bnd4}
\sum_{i \geq 2} \left\|\Pi_J \left(T T^\dagger - \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger\right) \Pi_{J_i} T h\right\|_2
\leq \frac{1}{3\sqrt{s}} \norm{\Pi_{[N] \setminus J} T h}_1
\leq \frac{1}{3\sqrt{s}} \norm{\Pi_J T h}_1
\leq \frac{1}{3} \norm{\Pi_J T h}_2.$$
Plugging and to provides $$\label{eq:pf_thm_stability2:bnd5}
\left\|\Pi_J \left(T T^\dagger - \frac{n}{m} T S_\Omega^* S_\Omega T^\dagger\right) T h\right\|_2
\leq \frac{2}{3} \norm{T h}_2.$$
Next, we derive an upper bound on the $\ell_2$-norm of the second term in the right-hand side of . Since $\Omega'$ consists of distinct elements in $\Omega$, it follows that $$S_{\Omega'}^* S_{\Omega'} S_\Omega^* S_\Omega
= \sum_{k \in \Omega} S_{\Omega'}^* S_{\Omega'} e_k e_k^*
= \sum_{k \in \Omega} e_k e_k^*
= S_\Omega^* S_\Omega.$$ Furthermore, $S_{\Omega'}^* S_{\Omega'}$ is idempotent. Therefore, we obtain the following identity: $$\label{eq:pf_thm_stability2:id1}
S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'} = (S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'}) S_{\Omega'}^* S_{\Omega'}.$$ From , we get $$\label{eq:pf_thm_stability2:bnd6}
\begin{aligned}
\norm{T (S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'}) h}_2
{} & = \norm{T (S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'}) S_{\Omega'}^* S_{\Omega'} h}_2 \\
{} & = \norm{T (S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'}) T^\dagger T S_{\Omega'}^* S_{\Omega'} h}_2 \\
{} & \leq \norm{T (S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'}) T^\dagger} \norm{T S_{\Omega'}^* S_{\Omega'} h}_2 \\
{} & \leq \max_k \norm{T e_k e_k^* T^\dagger} (|\Omega|-|\Omega'|) \norm{T S_{\Omega'}^* S_{\Omega'} h}_2 \\
{} & \leq \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} (|\Omega|-|\Omega'|) \norm{T S_{\Omega'}^* S_{\Omega'} h}_2.
\end{aligned}$$ Then the $\ell_2$-norm of the second term in the right-hand side of is upper-bounded by $$\label{eq:pf_thm_stability2:bnd7}
\begin{aligned}
\left\|\Pi_J \frac{n}{m} T (S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'}) h\right\|_2
{} & \leq \frac{n}{m} \norm{T (S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'}) h}_2 \\
{} & \leq \frac{n}{m} \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} (|\Omega|-|\Omega'|) \norm{T S_{\Omega'}^* S_{\Omega'} h}_2.
\end{aligned}$$
By applying and to , then combining the result with and , we get $$\begin{aligned}
\norm{T h}_2
\leq 2 \epsilon \norm{T} + 28 \sqrt{N} \epsilon \norm{T} + \frac{2}{3} \norm{T h}_2
+ 2 \frac{n}{m} \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} (|\Omega|-|\Omega'|) \epsilon \norm{T}
+ 2 \left( \frac{n}{m} - 1 \right) \epsilon \norm{T},\end{aligned}$$ which implies $$\norm{T h}_2 \leq 6 \left[ 14\sqrt{N} + \frac{n}{m} \left( \norm{T}_{1 \to 2} \norm{T^\dagger}_{2 \to \infty} (|\Omega|-|\Omega'|) + 1 \right) \right] \epsilon \norm{T}.$$
In the case when $T^* T = I_n$, $T e_k e_k^* T^\dagger$s correspond to orthogonal projections onto mutually orthogonal one-dimensional subspaces. Recall that the summands in $T S_\Omega^* S_\Omega T^\dagger$ repeat at most $R$ times. Then the summands in $T S_\Omega^* S_\Omega T^\dagger - T S_{\Omega'}^* S_{\Omega'} T^\dagger$ repeat at most $R - 1$ times. Therefore, we get a sharper estimate than that in given by $$\label{eq:pf_thm_stability2:bnd8}
\begin{aligned}
\norm{T (S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'}) h}_2
{} & \leq \norm{T (S_\Omega^* S_\Omega - S_{\Omega'}^* S_{\Omega'}) T^\dagger} \norm{T S_{\Omega'}^* S_{\Omega'} h}_2 \\
{} & \leq (R - 1) \norm{T S_{\Omega'}^* S_{\Omega'} h}_2.
\end{aligned}$$ This completes the proof for the second claim.
Numerical Results {#sec:numres}
=================
In this section, we conduct numerical experiments of solving and , with partial Fourier measurements and several different sparsifying transforms. We compare different sampling schemes in Monte-Carlo experiments, and observe that the variable density sampling schemes proposed in and yield superior recovery results in terms of success rate.
The optimization problems and are solved using Alternating Direction Method of Multipliers (ADMM) [@boyd2011distributed]. For example, for $T=\Phi\Psi^*\in\cz^{n\times n}$, is rewritten in the following form with two linear constraints, and solved by ADMM with two additional terms in the augmented Lagrangian. $$\begin{array}{ll}
\displaystyle \minimize_{y \in \cz^n} & \norm{y}_1 \\
\mathrm{subject~to} & T g = y,\\
& S_\Omega g = S_\Omega x.
\end{array}$$ In all experiments, the ADMM algorithm runs for 1,000 iterations.
1D Signals
----------
For 1D signals, we use the DFT $\Psi\in\cz^{n\times n}$, and two sparsifying transforms $\Psi_{\mathrm{TV}, n}, \Psi_{\mathrm{W}, n, \ell}\in \cz^{n\times n}$, denoting the finite difference operator and the discrete Haar wavelet [@mallat2008wavelet] at level $\ell$, respectively. For the finite difference operator $\Psi_{\mathrm{TV}, n}$, we use the two-step scheme in Section \[sec:TV\]. In Step 2), we either use the uniform density according to Corollary \[cor:circulant1\] (see Fig. \[fig:1d\_a\]), or the distribution restricted to $\{2,3,\dots, n\}$ computed from the other transform $\Psi_{\mathrm{W}, n, \ell}$ (see Fig. \[fig:1d\_b\]). For the discrete Haar wavelet $\Psi_{\mathrm{W}, n, \ell}$, we test two distributions on $[n]$ – the uniform distribution (see Fig. \[fig:1d\_a\]) and the variable density distribution (see Fig. \[fig:1d\_b\]).
In the numerical experiments, we choose $n = 512$, and synthesize signals $f$ at sparsity levels $s = 16, 32, 48, \dots, 128$ with respect to the above two sparsifying transforms. We use the Haar wavelet at level $\ell = 6$. To make the experiments more realistic, we synthesize the sparse signals by thresholding a real-world signal (see Fig. \[fig:1d\_sig\]) in the transform domains. For the discrete Haar wavelet, we analyze the signal using $\Psi_{\mathrm{W}, n, \ell}$, zero out small coefficients to achieve a certain sparsity level, and synthesize the signal using $\Psi_{\mathrm{W}, n, \ell}^*$. Due to the non-injectivity of the finite difference operator, we replace the last row of $\Psi_{\mathrm{TV}, n}$ by $[1,1,\dots, 1]^\top \in\cz^n$ in the signal analysis and synthesis steps.
For every sampling scheme, we repeat the experiment for $m = 16, 32, 48, \dots, 512$. We run 50 Monte-Carlo experiments for each setting. An instance is declared a success if the following reconstruction signal-to-noise ratio (RSNR) of the solution $\hat{g}$ to exceeds $60$dB. $$\mathrm{RSNR} = -20\log_{10}\left(\frac{\norm{\hat{g}-x}_2}{\norm{x}_2}\right).$$ The success rate is computed for every pair $(s, m)$ and shown in Fig. \[fig:1d\_c\] – \[fig:1d\_f\].
For signal recovery using the Haar wavelet, the variable density sampling scheme yields higher success rate than the uniform density sampling scheme (see Fig. \[fig:1d\_c\] and \[fig:1d\_d\]). For 1D TV, the density computed using 1D TV, which is uniform, yields higher success rate than that computed from the Haar wavelet (see Fig. \[fig:1d\_e\] and \[fig:1d\_f\]). These experiments show that one can recover sparse signals more successfully using the sampling density computed from the specific sparsifying transform used in the recovery, which is adaptive to local incoherence between the transform and the measurement. It is commonly believed among practitioners that sampling more densely in the low frequency region, where the energy of a natural signal is concentrated, always provides a better reconstruction. However, that turns out not be the case when one cares about perfect recovery of exactly sparse signals in a certain transform domain from noise-free measurements.
\
\
\
2D Signals
----------
We also run numerical experiments on a 2D signal, the modified Shepp-Logan phantom (see Fig. \[fig:2d\_sig\]). We minimize the anisotropic and isotropic 2D total variations, which correspond to solving and for the 2D finite difference operator $\Phi_{\mathrm{TV},n_1,n_2}$.
For the two types of total variations, we use the two-step scheme in Section \[sec:TV\]. In Step 2), we use the densities in and for anisotropic and isotropic total variations, respectively. As a comparison, we use the variable density proposed by Krahmer and Ward [@krahmer2014stable], which is computed with the 2D separable Haar wavelet $\Psi_{\mathrm{W}, n_2, \log_2n_2}\otimes \Psi_{\mathrm{W}, n_1, \log_2n_1}$.
We use a phantom of size $n = n_1\times n_2 = 256\times 256$, and repeat the experiments for $m= n/32, n/16,3n/32,\dots, n/4$. We run 50 Monte-Carlo experiments for each setting, and compute the success rate for every chioce of $m$, as shown in Fig. \[fig:2d\_d\] and \[fig:2d\_e\].
For both anisotropic TV minimization and isotropic TV minimization , the success rates using sampling density computed from TV are higher than those using sampling density computed from separable Haar wavelet. Although the performances of two sampling schemes are relatively close for anisotropic TV minimization (see Fig. \[fig:2d\_d\]), the advantage of the density computed from TV over that computed from wavelet is more pronounced for isotropic TV minimization (see Fig. \[fig:2d\_e\]). Contrary to common belief, sampling low frequencies more densely does not always lead to superior recovery. We suggest using sampling densities and tailored to the specific sparsifying transform and the measurement operator.
Conclusion {#sec:concl}
==========
In this paper, we established a unified theory for recovery of sparse signals in a transform domain. Our theory guaranteed robust recovery from noisy measurements by convex programming and apply without relying on a particular choice of measurement and sparsifying transforms. We quantified the sufficient sampling rate using functions of the two transforms, and this result identifies a class of measurement and sparsity models enabling recovery at a near optimal sampling rate. We also proposed a variable sampling density designed with incoherence parameters of the two transforms, which provided recovery guarantee at a lower sampling rate than previous works in various scenarios. Furthermore, we extended the result to the group-sparsity models so that it also applies to the popular isotropic total variation minimization. In particular, for the partial Fourier recovery of sparse signals over a circulant transform, our theory suggests a uniformly random sampling or its variation. Our numerical results showed that our variable density random sampling strategy outperforms other known sampling strategies in various scenarios. This suggests that our new theory is indeed universally useful.
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors thank referees for their valuable comments and suggestions.
Bernstein inequalities
======================
\[thm:mtx\_bernstein\_ineq\] Let $\{X_j\} \in \cz^{d \times d}$ be a finite sequence of independent random matrices. Suppose that $\mathbb{E} X_j = 0$ and $\norm{X_j} \leq B$ almost surely for all $j$ and $$\max\left( \left\| \sum_j \mathbb{E} X_j X_j^* \right\|,~ \left\| \sum_j \mathbb{E} X_j^* X_j \right\| \right) \leq \sigma^2.$$ Then for all $t \geq 0$, $$\mathbb{P} \left( \left\| \sum_j X_j \right\| \geq t \right)
\leq 2d \exp \left( \frac{-t^2/2}{\sigma^2 + Bt/3} \right).$$
\[thm:vec\_bernstein\_ineq\] Let $\{v_j\} \in \cz^d$ be a finite sequence of independent random vectors. Suppose that $\mathbb{E} v_j = 0$ and $\norm{v_j}_2 \leq B$ almost surely for all $j$ and $\mathbb{E} \sum_j \norm{v_j}_2^2 \leq \sigma^2$. Then for all $0 \leq t \leq \sigma^2/B$, $$\mathbb{P} \left( \left\| \sum_j v_j \right\|_2 \geq t \right)
\leq \exp \left( - \frac{t^2}{8 \sigma^2} + \frac{1}{4} \right).$$
Proof of Lemma \[lemma:E1\] {#sec:pf:lemma:E1}
===========================
Define $$X_j := \Pi_J (n T e_{\omega_j} e_{\omega_j}^* T^\dagger - T T^\dagger) \Pi_J, \quad \forall j \in [m].$$ Then, $X_j$ satisfies $\mathbb{E} X_j = 0$ and $\norm{X_j} \leq \mu s$ for all $j$. Since $$\begin{aligned}
X_j^* X_j
{} & = n^2 \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^* \Pi_J T e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J \\
{} & \quad - n \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^* \Pi_J T T^\dagger \Pi_J \\
{} & \quad - n \Pi_J T T^\dagger \Pi_J T e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J \\
{} & \quad + \Pi_J T T^\dagger \Pi_J T T^\dagger \Pi_J,\end{aligned}$$ it follows that $$\begin{aligned}
\mathbb{E} X_j^* X_j
{} & = \mathbb{E} n^2 \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^* \Pi_J T e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J - \Pi_J T T^\dagger \Pi_J T T^\dagger \Pi_J \\
{} & \leq \mathbb{E} n^2 e_{\omega_j}^* T^* \Pi_J T e_{\omega_j} \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J \\
{} & = \mathbb{E} n^2 \norm{\gamma^{1/2} \Pi_J T e_{\omega_j}}_2^2 \gamma^{-1} \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J \\
{} & \leq \mathbb{E} n \mu s \gamma^{-1} \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J \\
{} & = \mu s \gamma^{-1} \Pi_J \widetilde{T} T^\dagger \Pi_J.\end{aligned}$$ By symmetry, we also have $$\mathbb{E} X_j X_j^* \leq \mu s \gamma \Pi_J T T^* \Pi_J.$$ Therefore, $$\max\left( \left\| \sum_{j=1}^m \mathbb{E} X_j X_j^* \right\|,~ \left\| \sum_{j=1}^m \mathbb{E} X_j^* X_j \right\| \right)
\leq m \mu s \max( \norm{\gamma^{-1} \widetilde{T} T^\dagger}, \norm{\gamma T T^*} )
\leq \frac{m \mu s}{1 - \norm{\gamma T^* T - I_n}}.$$ Applying the above results to Theorem \[thm:mtx\_bernstein\_ineq\] with $t = m \delta$ completes the proof.
Proof of Lemma \[lemma:E2\] {#sec:pf:lemma:E2}
===========================
Define $$v_j := \Pi_J (n T e_{\omega_j} e_{\omega_j}^* T^\dagger - T T^\dagger) \Pi_J q, \quad \forall j \in [m].$$ Then, $v_j$ satisfies $\mathbb{E} v_j = 0$ and $$\begin{aligned}
\norm{v_j}_2
{} & \leq \norm{\Pi_J n T e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J q}_2 + \norm{\Pi_J T T^\dagger \Pi_J q}_2 \\
{} & \leq (\norm{\gamma^{1/2} \Pi_J \sqrt{n} T e_{\omega_j}}_2 \norm{\gamma^{-1/2} \Pi_J \sqrt{n} \widetilde{T} e_{\omega_j}}_2 + 1) \norm{\Pi_J q}_2 \\
{} & \leq (s\mu+1) \norm{\Pi_J q}_2.\end{aligned}$$ Furthermore, $$\begin{aligned}
\mathbb{E} \norm{v_j}_2^2
{} & = \mathbb{E} n^2 q^* \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^* \Pi_J T e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J q
- q^* \Pi_J T T^\dagger \Pi_J q \\
{} & \leq n \norm{\gamma^{1/2} \Pi_J T e_{\omega_j}}_2^2 \mathbb{E} n \gamma^{-1} q^* \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J q \\
{} & \leq \mu s \gamma^{-1} \mathbb{E} n q^* \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J q \\
{} & = \mu s \gamma^{-1} q^* \Pi_J \widetilde{T} T^\dagger \Pi_J q \\
{} & \leq \frac{\mu s \norm{\Pi_J q}_2^2}{1 - \norm{\gamma T^* T - I_n}},\end{aligned}$$ where the second inequality follows by the incoherence property. Applying the above results to Theorem \[thm:vec\_bernstein\_ineq\] completes the proof.
Proof of Lemma \[lemma:E3\] {#sec:pf:lemma:E3}
===========================
Let $i \in [n] \setminus J$ be arbitrarily fixed. Define $$w_j := \langle e_i, (n T e_{\omega_j} e_{\omega_j}^* T^\dagger - T T^\dagger) \Pi_J q \rangle, \quad \forall j \in [m].$$ Then, $w_j$ satisfies $\mathbb{E} w_j = 0$ and $$\begin{aligned}
|w_j|
{} & \leq |e_j^* n T e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J q| + \norm{T T^\dagger \Pi_J q}_\infty \\
{} & \leq (\norm{\gamma^{1/2} \sqrt{n} T e_{\omega_j}}_\infty \norm{\gamma^{-1/2} \sqrt{n} \Pi_J \widetilde{T} e_{\omega_j}}_2 + 1) \norm{\Pi_J q}_2 \\
{} & \leq (\sqrt{s} \mu+1) \norm{\Pi_J q}_2.\end{aligned}$$ Furthermore, $$\begin{aligned}
\mathbb{E} |w_j|^2
{} & = \mathbb{E} n^2 q^* \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^* e_i e_i^* T e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J q
- q^* \Pi_J T T^\dagger e_i e_i^* T T^\dagger \Pi_J q \\
{} & \leq n \norm{\gamma^{1/2} T e_{\omega_j}}_\infty^2 \gamma^{-1} \mathbb{E} n q^* \Pi_J \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_J q \\
{} & \leq \mu \gamma^{-1} q^* \Pi_J \widetilde{T} T^\dagger \Pi_J q \\
{} & \leq \frac{\mu \norm{\Pi_J q}_2^2}{1 - \norm{\gamma T^* T - I_n}}.\end{aligned}$$ Applying the above results to Theorem \[thm:mtx\_bernstein\_ineq\] gives $$\mathbb{P}\left(\left| \left\langle e_i, \left(\frac{n}{m} B_\Omega^* - B\right) \Pi_J q \right\rangle \right| \geq t \norm{\Pi_J q}_2\right)
\leq \exp\left( - \frac{m}{2\mu} \cdot \frac{t^2}{1/(1-\norm{\gamma T^* T - I_n}) + \sqrt{s}t/3} \right).$$ Combine this for $i \in [N]$ with the union bound completes the proof.
Proof of Lemma \[lemma:E3’\] {#sec:pf:lemma:E3'}
============================
Let $i \in [n] \setminus J$ be arbitrarily fixed. Define $$v_j := \Pi_{\calG_i} (n T e_{\omega_j} e_{\omega_j}^* T^\dagger - T T^\dagger) \Pi_{\calG_J} q, \quad \forall j \in [m].$$ Then, $v_j$ satisfies $\mathbb{E} v_j = 0$ and $$\begin{aligned}
\norm{v_j}_2
{} & \leq \norm{\Pi_{\calG_i} n T e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_{\calG_J} q}_2 + \norm{\Pi_{\calG_i} T T^\dagger \Pi_{\calG_J} q}_2 \\
{} & \leq (\norm{\gamma^{1/2} \sqrt{n} \Pi_{\calG_i} T e_{\omega_j}}_2 \norm{\gamma^{-1/2} \sqrt{n} \Pi_{\calG_J} \widetilde{T} e_{\omega_j}}_2 + 1) \norm{\Pi_{\calG_J} q}_2 \\
{} & \leq (\sqrt{s} \mu_\calG + 1) \norm{\Pi_{\calG_J} q}_2.\end{aligned}$$ Furthermore, $$\begin{aligned}
\mathbb{E} v_j^* v_j
{} & = \mathbb{E} n^2 q^* \Pi_{\calG_J} \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^* \Pi_{\calG_i} T e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_{\calG_J} q
- q^* \Pi_{\calG_J} T T^\dagger \Pi_{\calG_i} T T^\dagger \Pi_{\calG_J} q \\
{} & \leq n \norm{\gamma^{1/2} \Pi_{\calG_i} T e_{\omega_j}}_2^2 \gamma^{-1} \mathbb{E} n q^* \Pi_{\calG_J} \widetilde{T} e_{\omega_j} e_{\omega_j}^* T^\dagger \Pi_{\calG_J} q \\
{} & \leq \mu_\calG \gamma^{-1} q^* \Pi_{\calG_J} \widetilde{T} T^\dagger \Pi_{\calG_J} q \\
{} & \leq \frac{\mu_\calG \norm{\Pi_{\calG_J} q}_2^2}{1 - \norm{\gamma T^* T - I_n}}.\end{aligned}$$ Note that $$v_j v_j^* \leq v_j^* v_j I_L, \quad \forall j \in [m].$$ Applying the above results to Theorem \[thm:vec\_bernstein\_ineq\] gives $$\mathbb{P}\left(\left\| \Pi_{\calG_i} \left(\frac{n}{m} B_\Omega^* - B\right) \Pi_{\calG_J} q \right\|_2 \geq t \norm{\Pi_{\calG_J} q}_2\right)
\leq \exp\left( - \frac{m}{2\mu} \cdot \frac{t^2}{1/(1-\norm{\gamma T^* T - I_n}) + \sqrt{s}t/3} \right).$$ Combine this for $i \in [N]$ with the union bound completes the proof.
Proof of Theorem \[thm:rboplike\] {#sec:pf:thm:rboplike}
=================================
Theorem \[thm:rboplike\] is analogous to [@lee2013oblique Theorem 3.1]. It has been shown that if $T$ is of full row rank, then the deviation of $\frac{n}{m} T S_\Omega^* S_\Omega T^\dagger$ from $T T^\dagger = I_N$ is small with high probability [@lee2013oblique Theorem 3.1]. On the contrary, Theorem \[thm:rboplike\] assumes that $T$ is of full column rank and shows that the deviation of $\frac{n}{m} T S_\Omega^* S_\Omega T^\dagger$ from $T T^\dagger$, which is not necessarily $I_N$, is small with high probability.
The proof of Theorem \[thm:rboplike\] is obtained from that of [@lee2013oblique Theorem 3.1] by replacing $I_N$ by $T T^\dagger$. For example, the isotropy condition $$\frac{n}{m} \mathbb{E} T S_\Omega^* S_\Omega T^\dagger = I_N$$ is replaced by $$\frac{n}{m} \mathbb{E} T S_\Omega^* S_\Omega T^\dagger = T T^\dagger.$$ For a matrix $M \in \cz^{N \times N}$, the term $\theta_s(M)$, previously defined by in [@lee2013oblique] $$\theta_s(M) := \max_{|\widetilde{J}| \leq s} \norm{\Pi_{\widetilde{J}} (M - I_N) \Pi_{\widetilde{J}}}$$ is replaced by $$\theta_s(M) := \max_{|\widetilde{J}| \leq s} \norm{\Pi_{\widetilde{J}} (M - T T^\dagger) \Pi_{\widetilde{J}}}$$ Clearly, $T T^\dagger = I_N$ if $T$ has full row rank. However, in the hypothesis of Theorem \[thm:rboplike\], $T$ has full column rank and $T T^\dagger$ may be rank deficient. Since the modifications are rather straightforward, we omit the details of the proof and refer them to [@lee2013oblique Appendix E].
By modifying [@lee2013oblique Theorem 3.1] and its proof as shown above, is implied by and $$m \geq C_1 \delta^{-2} K_T \mu s \log^2 s \log N \log m,$$ where the factor $K_T$ is given by $$\begin{aligned}
K_T
{} & = \left\{
\left(2 + \max_{|\widetilde{J}| \leq s} \left\|\Pi_{\widetilde{J}} (\gamma T T^* - T T^\dagger) \Pi_{\widetilde{J}}\right\|\right)^{1/2}
+
\left(2 + \max_{|\widetilde{J}| \leq s} \left\|\Pi_{\widetilde{J}} (\gamma^{-1} \widetilde{T} T^\dagger - T T^\dagger) \Pi_{\widetilde{J}}\right\|\right)^{1/2}
\right\}^2 \\
{} & \leq 4 + 2 \max\left(
\max_{|\widetilde{J}| \leq s} \norm{\Pi_{\widetilde{J}}(\gamma T T^* - T T^\dagger)\Pi_{\widetilde{J}}}
,~
\max_{|\widetilde{J}| \leq s} \norm{\Pi_{\widetilde{J}}(\gamma^{-1} \widetilde{T} T^\dagger - T T^\dagger)\Pi_{\widetilde{J}}}
\right) \\
{} & \leq 4 + 2 \max\left(
\norm{\gamma T T^* - T T^\dagger}
,~
\norm{\gamma^{-1} \widetilde{T} T^\dagger - T T^\dagger}
\right) \\
{} & = 4 + 2 \max\left(
\norm{\gamma T^* T - I_n}
,~
\norm{\gamma^{-1} T^\dagger \widetilde{T} - I_n}
\right).\end{aligned}$$ Finally, we verify that $$\norm{\gamma^{-1} T^\dagger \widetilde{T} - I_n} \leq \frac{1}{\norm{\gamma T^* T - I_n}},$$ where the upper bound dominates $\norm{\gamma T^* T - I_n}$. This completes the proof.
[^1]: This work was supported in part by the National Science Foundation under grants IIS 14-47879 and Korea Science and Engineering Foundation under grants NRF-2013M3A9B2076548 and NRF-2016R1A2B3008104. K. Lee is with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail: [email protected]). Y. Li is with the Coordinated Science Laboratory and the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA (e-mail: [email protected]). K.H. Jin is with the Biomedical Imaging Group, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland (e-mail: [email protected]). J.C. Ye is with the Department of Bio and Brain Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejon 305-701, Korea (e-mail: [email protected]).
[^2]: A Gaussian sensing matrix is also obtained in the form of by choosing $\Psi$ as a Gaussian matrix.
[^3]: Krahmer and Ward [@krahmer2014stable] considered orthonormal $\Phi$ and $\Psi$ respectively corresponding to the DFT and the Haar DWT. In this case, $T^\dagger = T^*$. Thus, $\widetilde{T} = T$.
[^4]: A subset of the authors [@ye2016compressive] sharpened the original analysis of the completion of structured low-rank matrices by Chen and Chi [@chen2014robust] particularly on the noise propagation in the recovery. In this paper, we generalize the improved version [@ye2016compressive].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Supersonic jet noise reduction is important for high speed military aircraft. Lower acoustic levels would reduce structural fatigue leading to longer lifetime of the jet aircraft. It is not solely structural aspects which are of importance, health issues of the pilot and the airfield personnel are also very important, as high acoustic levels may result in severe hearing damage. It remains a major challenge to reduce the overall noise levels of the aircraft, where the supersonic exhaust is the main noise source for near ground operation. Fluidic injection into the supersonic jet at the nozzle exhaust has been shown as a promising method for noise reduction. It has been shown to speed up the mixing process of the main jet, hence reducing the kinetic energy level of the jet and the power of the total acoustic radiation. Furthermore, the interaction mechanism between the fluidic injection and the shock structure in the jet exhaust plays a crucial role in the total noise radiation. In this study, LES is used to investigate the change in flow structures of a supersonic (M=1.56) jet from a converging-diverging nozzle. Six fluidic actuators, evenly distributed around the nozzle exit, inject air in a radial direction towards the main flow axis with a total mass flow ratio of 3%. Steady injection is compared with flapping injection. With flapping injection turned on, the injection angle of each injector is varied sinusoidally in the nozzle exit plane and the variation is the same for all injectors. This fluid dynamics video is submitted to the APS DFD Gallery of Fluid Motion 2013 at the 66 the Annual Meeting of the American Physical Society, Division of Fluid Dynamics (24-26 November, Pittsburgh, PA, USA).'
---
\
Haukur Hafsteinsson$^{1}$, Lars-Erik Eriksson$^{1}$, Niklas Andersson$^{1}$\
Daniel Cuppoletti$^{2}$, Ephraim Gutmark$^{2}$\
Erik Prisell$^{3}$\
\
\
Video Description {#video-description .unnumbered}
=================
First, a general picture is brought up to make the audience acquainted with the application. A simplified sharp throat converging-diverging nozzle in a model scale, is attached to a full size aircraft to show its actual location in real a application. Then, a slice through the full three-dimensional computational domain is showed. The domain reaches approximately 70 nozzle exit diameters downstream of the nozzle exit plane and about 4 nozzle exit diameters in the upstream direction. The flow field is obtained by solving the compressible Navier-Stokes equations using an in-house finite volume LES solver based on the G3D family of codes originally developed by Eriksson [@eriksson:95]. The computational grid used for the simulations consists of approximately 20 million cells and the simulations are done on 80 CPU’s using MPI. Three cases are shown in the video; first, the baseline supersonic case without injection is shown, second a case with a steady injection at the nozzle exit is shown and finally a case with a flapping injection. For all three cases, the nozzle is operated at a nozzle pressure ratio (NPR) of 4.0, which gives a jet-exit Mach number of $M=1.56$. For all the cases a slice through the domain colored by $\nabla^2 p$ is showed. This quantity effectively shows sudden spatial pressure variations, such as those that occur across shocks. Inside the nozzle a stationary supersonic flow field is formed. It can be noticed that a shock is formed at the sharp throat. This is a conical shock which reaches further downstream, reflects on its self towards the nozzle wall at the jet center axis. Upon reaching the nozzle wall it reflects again this time passing through the nozzle exit towards the jet center axis where it reflects radially outwards. When interacting with the shear layer it reflects back again towards the jet center axis and so on generating a set of quasi-stationary compression- and expansion waves within the jet plume. Another similar set of shocks is formed at the nozzle lip. These two shocks generate a double shock cell structure which dissipates downstream. Plotting the $M=1.0$ iso-contour reveilles the location of the boundary between the supersonic jet-core region and its subsonic surroundings. A close look at the nozzle exit shows a relatively stable shear layer which quickly unfolds into circumferential vortex cores, as the flow transitions to high turbulence levels due to steep axial velocity gradient in the radial direction.
The steady-state injection consists of 6 evenly distributed actuators around the nozzle exit. The injection angle is normal to the nozzle inner wall and is therefore directed radially inward towards the jet center axis. The total mass flow of all six injectors ($\dot{m}_\mathrm{i}$) compared to the mass flow through the nozzle throat ($\dot{m}_\mathrm{j}$) is $\dot{m}_\mathrm{i}/\dot{m}_\mathrm{j}=3\,\%$, which is considered as relatively high mass flow for practical applications. The injection has a profound effect on the jet dynamics. The $M=1$ iso-surface shows how the fluidic injectors penetrate into the shear layer and create axial vortices which are convected downstream by the main jet flow. These vortices result in increased mixing of the jet plume with its ambient air and hence the length of the potential core is reduced. Since the injectors penetrate rather deepl into the jet, the main jet flow senses a blockage and its path is forced in between the injectors. Therefore, the radial location of the $M=1$ iso-contour increases in between the injectors. Thereafter, iso-contours of $\nabla^2 p$ is showed to visualize spectacular stationary bow-shocks formations upstream of each injector.
Continuing from the animation of the stationary bow-shocks formed due to the steady injection, the injection is switched to a flapping mode which shows how the position of the bow-shocks start to follow the flapping angle. The flapping amplitude is $\pm 60^{\circ}$, the frequency of the flapping is $f=1000\mathrm{Hz}$ and as mentioned earlier, the flapping angle is in the nozzle exit plane. Shifting back to a view of the $M=1$ iso-contour, shows how the mixing-rate is dramatically increased as the flapping injectors introduce a spinning motion to the shear layer. Furthermore, looking at the slice-through the domain showing $\nabla^2 p$ along the jet axis, an interesting motion of the double shock cell structure may be noticed. The flapping injection introduces a shock motion which can be referred to as “shock pumping movement”, i.e. the two shock-cells keep more or less their original structure but the distance between them shifts back and forth. This results in a constructive and destructive shock superposition. This phenomenon is thought to be responsible for strong undesirable acoustic screech harmonics observed in the far-field as shown by Hafsteinsson et al. [@Hafsteinsson].
A final view along the jet-axis towards the nozzle exit, shows how the flapping injection creates stunning spinning-shock formations with highly complex three dimensional shock interactions.
[9]{}
L.-E. Eriksson, *“Development and Validation of Highly Modular Flow Solver Versions in G2DFLOW and G3DFLOW”*, Volvo Aero Corporation, Sweden, Internal report, 9970-1162, 1995
Haukur E. Hafsteinsson, Lars-Erik Eriksson, Niklas Andersson, Daniel R. Cuppoletti, Ephraim J. Gutmark, and Erik Prisell *“Supersonic Jet Noise Reduction Using Steady Injection and Flapping Injection”* 19th AIAA/CEAS Aeroacoustics Conference, Berlin Germany, 2013
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this work we investigate this problem in the asynchronous model by luminous robots. In this model a light is attached to the robots, which serves as externally visible bits stored in that light encoded by a color. First, we present an algorithm solving the asynchronous Filling problem with robots having $1$ hop visibility range, $O(\log\Delta)$ bits of persistent storage, and $\Delta+3$ colors, where $\Delta$ is the maximum degree of the graph. Then we show, how the number of colors can be reduced to $O(1)$ at the cost of the running time. After this we show, how the running time can be improved by robots with visibility range of $2$ hops, $O(\log \Delta)$ bits of persistent memory, and $\Delta + 3$ colors. We show, that in the fully synchronous case, the running time of this algorithm is $O(n)$. Finally, we show how to extend our asynchronous solution to the $k$-Door case, $k\geq 2$, by using $\Delta + k + 3$ colors. Finally, we present simulation results that match very well the theoretical bounds.'
author:
- Attila Hideg
- Tamás Lukovszki
bibliography:
- 'bibliography.bib'
title: Asynchronous Filling by Luminous Robots
---
Introduction
============
Multi-robot systems can achieve great scalability, fault tolerance, and cost efficiency in certain problems. Instead of having an expensive robot with redundant hardware and software, the multi-robot system usually consists of rather simple and cheap robots. The robots cooperatively can solve different problems, as gathering, flocking, pattern formation, dispersing, filling, coverage, and exploration (e.g. [@Albers2002; @Albers2000; @Amir19; @Augustine2018; @Barrameda2008; @Barrameda2014; @Brass2011; @Cohen2005; @Hsiang2002; @LukovszkiH14; @Scheideler19]; see [@Flocchini2012; @Flocchini2019] for recent surveys).
The Filling problem was introduced by Hsiang et al. [@Hsiang2002], where the robots enter an a priori unknown but connected area and have to disperse. The area is subdivided into pixels and at the end of the dispersion each pixel has to be occupied by a robot, hence the name Filling.
A fundamental question is how ‘weak’ those robots can be in terms of hardware requirements with still being able to solve the problem, which was initiated by Barrameda et al. [@Barrameda2008; @Barrameda2014].
**Model:** The area which has to be filled is represented by a connected graph. A special vertex is an entry point, which is called the *Door*. When the Door vertex becomes empty, a new robot is placed there immediately. For simplicity we assume the degree of the Door vertex is $1$. Otherwise, we introduce an auxiliary vertex of Degree $1$ connected only to the Door, which takes the role of the original Door (this models the two sides of a doorstep). We assume that (as in [@hideg2018]), for each vertex $v$, the adjacent vertices are arranged in a fixed cyclic order. This cyclic order is only visible for robots at $v$ and it does not change during the dispersion.
The robots act according to the common Look-Compute-Move (LCM) model. In this model, their actions are decomposed into three phases: a *Look* phase, where they take a snapshot of their surrounding, a *Compute* phase, in which they perform calculations, and a *Move* phase, where they move to a neighboring vertex, or stay at place.
Based on the activation times of the robots, there are three main synchronization models studied in the literature: the fully synchronous (FSYNC), the semi-synchronous (SSYNC), and the asynchronous (ASYNC). In the FSYNC model, all robots are activated at the same time and they perform their Look, Compute, and Move phases synchronously at the same time, which is ensured by a global clock. In the SSYNC model, some robots might decide to ’skip’ an LCM cycle, and stay inactive. In the ASYNC model, there is no common notion of time available: the robots activate independently after a finite but arbitrary long time, and perform their LCM cycles. Moreover, their LCM cycle length is not fixed, it also can be arbitrarily long.
The robots are *autonomous*, i.e. no central coordination is present, *homogeneous*, i.e. all the robots have the same capabilities and behaviors, *anonymous*, i.e. they cannot distinguish each other, and *silent*, i.e. they have no communication capabilities and cannot directly talk to one another. However, *luminous* robots can communicate indirectly by using a light. Such robots have a light attached to them, which is externally visible by every robot in their visibility range. They can use a finite set of colors (including the color when the light is off) representing the value of a state variable. The robots are allowed to change these colors in their Compute phase. We denote the availability of lights using a superscript representing the number of colors. In particular, we denote by $X^i$ the model $X \in \{$ASYNC, SSYNC, FSYNC$\}$ when every robot is enhanced by a light with $i > 1$ colors. In the ASYNC$^{O(1)}$ model the robots use constant number of colors (see, e.g. [@das2012]).
**Related Work:** In [@hideg2017; @hideg2018] the Filling problem has been investigated in the FSYNC model. In [@hideg2017] the authors gave a solution for the orthogonal Filling problem by using robots with $1$ hop visibility range and $O(1)$ bits of persistent memory for both the Single and Multiple Door case.
In [@hideg2018] a method for a general Filling problem has been presented, where the area is represented by an arbitrary connected graph. The robots required $1$ hop visibility range and $O(\Delta)$ bits of persistent memory, where $\Delta$ is the degree of the graph. For the $k$-Door case, the memory requirement is $O(\Delta \cdot \log k)$. The general method is called the Virtual Chain Method (VCM), which is leader-follower method. In the VCM, the robots form a chain, and filled the area mimicking a DFS-like traversal of the graph. The algorithms presented in [@hideg2017] and [@hideg2018] are intensively utilizing the synchronous nature of the model to avoid collisions and backtracking. These algorithms do not work in the ASNYC model. In this paper we solve the Filling problem in the ASYNC$^{O(1)}$ model by light-attached robots, a.k.a. luminous robots. Light-attached robots were first investigated by Das et al. [@das2012; @Flocchini16]. They considered the case, where the robots can move in the continuous Euclidean plane and they proved that the asynchronous model with a constant number of colors ASYNC$^{O(1)}$ is strictly more powerful than the semi-synchronous model SSYNC, i.e. ASYNC$^{O(1)}$ $>$ SSYNC. Das et al. [@Flocchini16] also prove that there are problems that robots cannot solve without lights, even if they are fully synchronous, but can be solved by asynchronous luminous robots with $O(1)$ colors.
D’Emidio et al. [@Navarra16] have shown that on graphs one task can be solved in the fully synchronous model FSYNC but not in the asynchronous lights-enhanced model, while for other tasks, the converse holds. We show that the Filling problem can be solved in both models.
**Our Contribution:** In this work we present solutions for the Filling problem by luminous robots on graphs in the ASYNC$^{O(1)}$ model. First, we describe a method, called PACK, which solves the problem by robots with $1$ hop visibility range, $O(\log \Delta)$ bits of persistent memory, and $\Delta + 3$ colors for the single Door case. Then we show how the number of colors can be reduced to $O(1)$ at the cost of running time.
Assuming that all the robots are active all the time until having been switched to Finished state and that the length of the LCM cycles of robot $r_i$, $i=1,\dots,n$ is $t_i$, then the running time is $O(nT)$, where $T=\sum_{i=1}^n {t_i}$. For the fully synchronous case, when all LCM cycles have unit lenght, the running time of this algorithm is $O(n^2)$.
After this, in the ASYNC model we show how the running time can be significantly improved by robots with visibility range of $2$ hops, with no communication, $O(\log \Delta)$ bits of persistent memory, and $\Delta + 3$ colors, by presenting the algorithm called BLOCK. In the fully synchronous case, the running time of this algorithm is $O(n)$.
Then we extend the BLOCK algorithm for solving the $k$-Door Filling problem, $k\geq 2$, by using $\Delta + 3 + k$ colors and $O(\log \Delta)$ bits of memory. The visibility range of $2$ is optimal for the $k$-Door case (a counter example when this problem cannot be solved in the ASYNC model with a visibility range of $1$ hop was presented in [@Barrameda2008], also holds for the ASYNC$^{O(1)}$ model).
Beside the theoretical results, we have implemented our algorithms and performed simulations. The simulation results match very well the theoretical bounds.
PACK Algorithm
==============
Now we describe the PACK algorithm to solve the Filling problem for an area represented by a connected graph of $n$ vertices. PACK is based on the Virtual Chain Method described in [@hideg2018], in which the robots filled the area in a DFS-like dispersion.
The robots are allowed to be in one of the following states: None, Follower, Leader, Finished. They are initialized with None state when placed at the Door. The first robot becomes the Leader and moves to a vertex which has never been occupied before (these vertices are called *unvisited* vertices). The rest of the robots will become Followers and follow the Leader, until the Leader becomes stuck (i.e. no unvisited neighbors available). Then the robot behind the Leader, called the *successor* robot becomes a new Leader and moves if possible. The previous Leader switches to Finished state. The algorithm terminates when each robot is in Finished state.
The name virtual chain comes from the fact that the robots form a chain which is defined as the path from the current Leader to the Door. The chain contains only visited vertices, which can be occupied by the Followers. Each Follower follows its *predecessor*, which is the previously placed robot. We define a new state for the algorithm:
*Packed:* the state of the chain, when each Follower is behind its predecessor, i.e. each vertex in the path of the Leader is occupied by a robot. In this state none of the robots can move except the Leader. Therefore, only the Leader has to know this state.
**The concept:** The Leader moves to unvisited vertices until there is no such neighboring vertex. Before each movement, the Leader waits for packed state, thus it cannot collide with other robots, and the Leader can decide which vertex is unvisited. When the Leader has no neighboring unvisited vertex, it switches to Finished state and does not move anymore. Its successor then becomes the Leader, and the new Leader moves to other unvisited vertices. The robots use $\Delta + 3$ colors. The first $\Delta$ colors show the direction of the target vertex (for each vertex, the adjacent vertices are arranged in a fixed cyclic order), we refer to them as *DIR* colors. Furthermore, we use two colors, denoted by *CONF* and *CONF2* colors, for confirming that a robot has seen a DIR color of the predecessor, which allows the predecessor to move. For this purpose, the CONF color is sufficient, when the predecessor is a Follower robot. However, when the predecessor robot is the Leader and it must change the target vertex after the Packed state is reached (details are provided later) or the predecessor becomes the Leader and it chooses an unvisited target vertex, it indicates the new direction with a new DIR color. Then the CONF2 color is needed for ensuring that the successor has seen the lastly shown DIR color. Furthermore, we use an additional color, called *MOV* color to indicate that a robot is on the way to its target vertex.
Now we describe the rules followed by the robots in different states.
*Leader:* Can only move to an unvisited vertex. When it wants to move, it shows the direction it wants to go to by setting the corresponding DIR color, then it waits until its successor gives a confirmation that it can move by setting its CONF color. During moving the Leader shows the MOV color. When its successor sets its CONF color, the chain is in *Packed* state. This means, each not occupied vertex is also an unvisited vertex (as each vertex in the path of the Leader is occupied by a robot). If the Leader is still on the Door vertex, therefore, it does not have a successor, it can move without waiting the CONF color.
*Follower:* Follows its predecessor. The Follower robot $r$ sets the CONF color if and only if $i$) the predecessor of $r$ is showing its direction, and $ii$) the successor of $r$ – if exists – have set its CONF color (i.e. the successor knows in which direction $r$ will move). This allows the predecessor $r'$ of $r$ to move to its destination knowing: $i$) all the robots behind $r'$ have set CONF color, and $ii$) the robots behind $r'$ will not move until the predecessor of $r$ moved. When $r'$ is the Leader, the chain is in Packed state.
*None:* The robots are initialized with None state when they are placed at the Door. If the robot $r$ in None state has no neighboring robot, then $r$ changes its state to Leader, chooses the unique neighboring vertex as target vertex, sets the MOV color and starts moving there. Otherwise, if the robot $r$ in None state has one neighboring robot, then $r$ becomes a Follower and sets the neighbor to its predecessor.
There are three special situations, where we need the following additional rules:
*Leader target change*: It might happen that the Leader $r$ chooses a target vertex $v$, which is empty at the moment when $r$ performs its Look operation, however, when the successor of $r$ sets the CONF color and $r$ could start to move to $v$, another robot already moved to $v$. In such case, the Leader $r$ has to choose a new target, and the successor of $r$ has to know about this choice. Assume first, that $r$ has an unvisited neighboring vertex. Then $r$ sets the corresponding DIR color and waits until its successor sets the CONF2 color. Finally, the Leader moves to the target. Note that the chain is in Packed state when the successor of the Leader $r$ sets the CONF color. In this moment only the Leader can move. Consequently, the Leader can change the target vertex only once between two movements. If $r$ does not have any unvisited neighboring vertex after $r$ sees the CONF color of the successor then $r$ can not move anymore and the successor must take the leadership (see the rule below). The robot $r$ sets the $\Delta$ direction color, which has special meaning. The successor $r'$ confirms this by seting the CONF2 color. Then $r$ turn off its light $r'$ becomes the Leader. (Note that it would be possible to omit the Leader target change rule by introducing a new color for signaling the Packed state. Then the Leader would only show its direction once the Packed state is achieved, which could be acknowledged with the CONF color.)
*Taking the leadership*: When the Leader $r$ cannot move anymore, its successor has to become the new Leader. The Leader $r$ indicates that it does not have any unvisited neighboring vertex by setting its direction color to $\Delta$. I.e. this color has a special meaning: it indicates that the Leader cannot move anymore and wants to switch to Finished state and the leadership must be taken by its successor. When this is detected by the successor $r'$, it sets its CONF color, waits for the previous Leader to turn off its light, then $r'$ becomes the Leader. Afterwards, $r'$ tries to move to an unvisited vertex.
*Setting movement color*: Before performing the movement the robots have to set their color to MOV. Keeping the old color color could lead to an error. E.g., consider the following situation. 1. The Leader sets a DIR color. 2: The Follower confirms it by the CONF color. 3: The Leader moves by keeping the DIR color. 4: The Follower shows the corresponding DIR color, receives a CONF, and follows the Leader. 5: The Follower reaches its target, sees the old DIR color of the Leader and sets the CONF color, before the Leader chouses the new target. In order to prevent such situations, the moving robots set their color to MOV and keep this color until the target is reached and a new target is determined. After the movement, the robot sets the previous position as its *Entry* vertex Pseudocode of the PACK algorithm is provided in the Appendix.
Analysis
--------
Leader only moves to unvisited vertices. \[lemma:m1\_unvisited\]
An unvisited vertex means no robot has occupied it before. As the Leader can only move when the chain is in Packed state, each vertex not occupied by a robot is an unvisited vertex. Therefore, each unoccupied vertex, which can be chosen by the Leader, is an unvisited vertex.
There can be at most one Leader at any time. \[lemma:m1\_one\_leader\]
Recall the rule *taking the leadership*. When a Leader $r$ becomes stuck, $r$ signals this with a special color $\Delta$ and switches to Finished state after the successor $r'$ sets the CONF color. Then $r'$ becomes the new Leader, and acts accordingly.
The first robot placed becomes the Leader, and from that time each robot can become a Leader after the previous one became Finished. Therefore, at most one Leader can exist at any time during the dispersion.
Robots cannot collide. \[lemma:m1\_collision\]
When placed, except the first robot, each robot has only one neighbor robot. That will be its predecessor, which the robot will follow during the algorithm. Before the predecessor moves away, it shows which direction it moves. Therefore, the Follower can always follow it. As each Follower robot has one predecessor they cannot collide with each other.
However, the Leader does not follow its predecessor (as it does not have any). It is required to move to unvisited vertices in order to avoid collisions with Followers (which only move to already visited vertices). As there is only one Leader and it always moves to unvisited vertices, collisions are not possible.
PACK fills the area represented by a connected graph. \[lemma:m1\_fills\]
For contradiction, assume the area is not filled when the algorithm terminates. As the area is connected, there is a vertex $v$ which is not occupied and has a neighboring robot in Finished state. If $v$ is unvisited, let $r$ be the last neighboring robot of $v$ which became Finished. However, $r$ cannot switch to Finished state since there is an unvisited vertex neighboring to it. This contradicts the assumption that $v$ remains unoccupied.
Assume now $v$ is unoccupied but it has been visited during the algorithm. Let $t$ be the last time $v$ was occupied by a robot $r$. After $r$ moves from $v$, its successor will occupy $v$. This contradicts the assumption that $t$ was the last time of occupation of $v$. This proves the claim the area is filled when the algorithm terminates.
Algorithm PACK fills a connected area represented by a connected graph in the ASYNC model by robots having a visibility range of $1$ hop, $O(\log \Delta)$ bits of persistent storage and $\Delta + 3$ colors. \[thm:m1\]
As the area is filled (by Lemma \[lemma:m1\_fills\]), and collisions are not possible (by Lemma \[lemma:m1\_collision\]), the area will be filled without collisions. The robots require $O(\log \Delta)$ bits of memory to store the following: *$State$* (4 states – 2 bits), *$Target$* (direction of the target vertex – $\lceil \log \Delta \rceil$ bits), *$NextTarget$* (direction of the vertex, where the robot needs to move after the vertex $Target$ is reached – $\lceil \log \Delta \rceil$ bits). Regarding the number of colors, the robots use $\Delta$ colors to show the direction where the target of the robot is. There are two additional colors (CONF and CONF2) for confirming the robot saw the signaled direction of the predecessor and one color (MOV) during the movement.
**Remark**: The ASYNC model allows a robot to be inactive between two LCM cycles. Since the inactive phase allowed to be finite but arbitrarily long, the runtime of the algorithm can also be arbitrarily long. In the case, where we do not allow inactive intervals between the LCM cycles and assume that each cycle of the robot $r_i$ takes $t_i$ time, $i=1, \dots , n$, we can give a stronger upper bound for the running time.
Assume that the robots have no inactive intervals between two LCM cycles and each cycle of the robot $r_i$ takes $t_i$ time, $i=1, \dots , n$. Then the running time of the algorithm PACK is $O(n\cdot T)$, where $T=\sum_{i=1}^n {t_i}$. \[thm:m1b\]
Assume a chain containing $r_1$, $r_2$, $\dots$, $r_i$ (where $r_1$ is the active Leader, and $r_2$, $\dots$, $r_i$ are on the path from the Leader to the Door), and assume the chain is in Packed state.
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(1.2,3.8) rectangle (2,4.2); (2.6,2.8) rectangle (3.2,3.2); (3.2,1.8) rectangle (3.5,2.2); (3.7,0.8) rectangle (4.1,1.2); (6.6,3.8) rectangle (7.4,4.2); (5.4,2.8) rectangle (6.0,3.2); (4.8,1.8) rectangle (5.1,2.2); (4.2,0.8) rectangle (4.6,1.2); (7.5,2.8) rectangle (8.1,3.2); (8.4,3.8) rectangle (9.2,4.2);
Assume first that the Leader $r_1$ has an unvisited neighboring vertex. Denote by $T'$ the time between two consecutive movements of the Leader. We divide $T'$ into three time intervals: $T'=T_1 + T_2 + T_3$. $T_1$ starts with the movement of the Leader, it includes the time, when all robots in the chain making one step forwards, and the time for placing a new robot at the Door. $T_2$ starts after placing a new robot at the Door. In $T_2$ the robots, starting from the Door, set their CONF color one by one. This CONF color is ’propagated’ to the Leader, meaning the Leader recognizes the Packed state, and can move again. $T_3$ starts after the Leader recognizes the CONF color of the successor, i.e. after achieving the Packed state. Then the Leader might find its target occupied by another robot. In this case the *Leader target change* rule will be used.
In $T_1$ the Leader $r_1$ moves to its target, then $r_1$ sets its direction color. After $t_1$ time (the length of the LCM cycle of $r_1$), the current LCM cycle of $r_2$ has to be finished, then $r_2$ recognizes the movement at the beginning of its next LCM cycle (i.e. after at most $t_2$ time). In that cycle it also moves and sets its direction color, this takes $t_2$ time. Therefore, after the movement of $r_1$ the robot $r_2$ moves within $2\cdot t_2$ time. This argument can be repeated to all robots until the last one $r_i$ moves. Therefore, the new robot $r_{i+1}$ will be placed after $T_1 \leq 2\sum_{j=1}^{i}{t_j}$ time (see \[fig:movement\_step\]). Now the second phase $T_2$ starts.
In $T_2$ the new robot $r_{i+1}$ sets its color to CONF. The previous one recognizes it in the its next LCM cycle (within $t_i$ time) and sets its color to CONF in that LCM cycle (which takes $t_i$ time). Therefore, after the placement of $r_{i+1}$, the robot $r_i$ sets its CONF color in $2t_i$ time. Repeating this argument for $r_{i-1},\dots,r_2$, $T_2 \leq 2\sum_{j=2}^{i+1}{t_j}$.
In $T_3$ if the target vertex $v$ of the Leader is unoccupied the Leader can move immediately. Otherwise, if $v$ is occupied, the *Leader target change* protocol is performed, i.e. 1: the Leader chooses a new unoccupied neighboring vertex and shows the corresponding DIR color, 2: its successor sets its color to CONF2. This takes at most $2(t_1+t_2)$ time. After this the Leader can move.
Let $T=\sum_{j=1}^n {t_j}$ be the sum of the lengths of the LCM cycles of all robots. Then $T' = T_1 + T_2 + T_3 \leq 6 T$.
Assume now that $r_1$ has no unvisited neighboring vertex. Then $r_1$ sets its $\Delta$ color. The robot $r_2$ recognizes it in its next cycle and sets its CONF color (i.e. within $2t_2$ time). The robot $r_1$ sees it in its next cycle, $r_1$ turns its light off and switches to Finished state (takes $\leq 2t_1$ additional time). The robot $r_2$ sees it in its next cycle, $r_2$ becomes the new Leader, and checks if there is a neighboring unvisited vertex. If so, $r_2$ sets the corresponding DIR color (takes $\leq 2t_2$ additional time), otherwise the leadership has to be taken by the successor of $r_2$.
When a Leader can move, it occupies an unvisited vertex within $6T$ time. Otherwise, its successor takes the leadership. Since the leadership is taken at most once by each robot during the whoole algorithm, and there are $n$ robots in the filled graph, at most $6T$ time is used for all ’leadership taking’ altogether. Thus, after at most $6nT+6T=O(nT)$ time all vertices of the graph become filled.
In the fully synchronous model, the running time of the PACK algorithm $O(n^2)$ LCM cycles.
Filling of graphs using constant number of colors
-------------------------------------------------
The PACK algorithm uses $\Delta+3$ colors. We can reduce the number of colors to $O(1)$ at the cost of the running time, as follows. We encode the $L=\Delta+3$ colors by a sequence of $\lceil \log L \rceil$ bits and transmit this sequence by emulating the Alternating Bit Protocol (ABP), also referred to as Stop-and-wait ARQ (see, e.g. [@Tannenbaum10]). This protocol uses a sequence number from $\{0,1\}$ alternately to transmit the bits. The sender has four states corresponding to the transmitted bit $b\in \{0,1\}$ and the sequence number. The receiver has two states that represent which sequence number is awaited. The data bits are accepted with alternating sequence numbers. This protocol ensures the correct transmission of the bit sequence without duplicates.
We emulate the ABP by using six different colors, one for each of the four states of the sender and one for each of the two states of the receiver. Seeing the current color of the sender, the receiver can decode the sequence number and the data bit. When a color corresponding the correct sequence number is seen, the receiver sets its color indicating that it waits for the next bit. When the sender sees the changed color of the receiver it sets its color corresponding to the next data bit and next sequence number. This modification leads to the following Theorem.
The modified Algorithm PACK fills an area represented by a connected graph in the ASYNC model by robots having a visibility range of $1$, $O(\log \Delta)$ bits of persistent storage and $O(1)$ colors.
BLOCK Algorithm
===============
The PACK algorithm solves the Filling problem in arbitrary connected graphs by robots with a visibility range of 1 hop. An important property of the PACK algorithm is that the Leader can only move when the chain has reached the Packed state. When the visibility range of the robots is only 1 hop and the Packed state of the chain is not required before the Leader moves, the Leader cannot distinguish between unvisited vertices and those that are unoccupied but part of the chain (those will be occupied and the Leader might collide with them). To handle this problem we consider robots with a visibility range of $2$ hops. Then the robots see each robot, that potentially could choose the same target vertex. The idea is that the Leader only chooses a vertex $v$ as target, if the $1$ hop neighborhood of $v$ does not contain any other robot with light turned on, except when the light showing direction $\Delta$ (i.e. the robot will not move anymore, it wants to switch to Finished state, and waiting to the confirmation of the successor). A vertex neighboring to a robot with its light on (except the color $\Delta$) is considered as *blocked* vertex for the Leader.
We introduce the following additional rules for the robots:
*Leader*: The Leader must not choose a blocked vertex as target. As the visibility range of the robots is 2 hops, the Leader can identify the blocked neighbors. When only blocked or occupied vertices surround the Leader, it chooses to terminate its actions (sets the color $\Delta$ and after the confirmation of the successor it switches to Finished state) and the leadership will be taken by its successor.
*Follower*: Follower robots ’block’ all their unoccupied neighboring vertices. As a result, all unoccupied vertices that are part of the chain are blocked: Before a Follower $r$ would move from a vertex $v$, it sets the DIR color corresponding to the target and blocks all of its unoccupied neighboring vertices. In particular, it blocks the target vertex. Thus the Leader cannot choose the same target. Then $r$ waits until the successor $r'$ sets its CONF color and $r$ moves from $v$. During the movement the MOV color is set, which keeps the same unoccupied vertices blocked. When $r$ leaves $v$, the vertex $v$ is blocked by $r'$. These rules ensure that each vertex on the chain is either occupied or blocked. Consequently, the Leader only moves to unvisited vertices. Pseudocode of the BLOCK algorithm is provided in the Appendix.
Analysis
--------
Leader only moves to unvisited vertices. \[lemma:m2\_unvisited\]
Consider a visited vertex $v$ neighboring to the Leader. Let $r$ be the last robot, that occcupied $v$. When $r$ left $v$, the successor of $r$ blocks $v$. Thus, the Leader cannot move to $v$.
There can be at most one Leader at a time. \[lemma:m2\_one\_leader\]
The arguments of Lemma \[lemma:m1\_one\_leader\] can be repeated as the rule for taking of the leadership did not change.
Robots cannot collide. \[lemma:m2\_collision\]
The arguments of Lemma \[lemma:m1\_collision\] can be repeated as Lemma \[lemma:m2\_one\_leader\] only allows one Leader, which only can move to unvisited vertices (Lemma \[lemma:m2\_unvisited\]).
BLOCK fills the area represented by a connected graph. \[lemma:m2\_fills\]
We use similar arguments to those in the proof of Lemma \[lemma:m1\_fills\]. Assume that all the robots are in Finished state and there is an unoccupied vertex $v$, such that $v$ has at least one occupied neighbor. Additionally, to the cases considered in the proof of Lemma \[lemma:m1\_fills\], we have to consider the case when $v$ is blocked, and all neighboring robots become Finished. Let $t$ be last time when a robot $r$, neighboring to $v$, switches to Finished state. Since all other neighboring robots of $v$ are in Finished state at time $t$, they do not block $v$. Therefore, at time $t$ the robot $r$ can move to $v$ instead of switching to Finished. Thus, we have a contradiction.
Algorithm BLOCK fills the area represented by a connected graph in the ASYNC model by robots having a visibility range of $2$ hops, $O(\log \Delta)$ bits of persistent storage and using $\Delta + 3$ colors.
We can use the arguments of the proof of Theorem \[thm:m1\] as the area is filled (by Lemma \[lemma:m2\_fills\]), and collisions are not possible (by Lemma \[lemma:m2\_collision\]), the area will be filled without collisions. The robots store the same data in their persistent storage as in Theorem \[thm:m1\] and use the same set of colors.
The runtime improves compared to the PACK algorithm. We provide runtime analysis of the BLOCK algorithm in the fully synchronous model.
In the fully synchronous model, the BLOCK algorithm fills the area represented by a connected graph in $O(n)$ LCM cycles. \[thm:m2r\]
Assume a chain containing $r_1$, $r_2$, $\dots$, $r_j$ (where $r_1$ is the active Leader, and $r_2$, $\dots$, $r_j$ are on the path from the Leader to the Door), and assume that the Leader $r_1$ occupied its position and its successor $r_2$ is arrived to the previous position of $r_1$. When the first robot is placed at the Door, it detects in its first LCM cycle if it is a Leader or a Follower. If the only neighbor is unoccupied, it becomes a Leader and moves in the first cycle. In the next cycle the next robot is placed at the Door.
Assume, a robot $r_i$, $i<j$, is either a Leader or a Follower and has its successor $r_{i+1}$ at its previous vertex. If $r_i$ is Leader, we additionally assume that it has an unblocked and unoccupied neighboring vertex. Then $r_i$ sets the corresponding DIR color in that LCM cycle. We denote this LCM cycle by $t$. In the LCM cycle $t+1$ the robot $r_{i+1}$ sets its color to the CONF color, allowing $r_i$ to move in $t+2$. Consequently, the robot $r_i$ can move in $3$ cycles if the successor is on its previous vertex.
Then, in cycle $t+3$ the robot $r_{i+1}$ detects that $r_i$ left the neighboring vertex $v$ and $r_{i+1}$ sets the DIR color indicating the target $v$. In cycle $t+4$ the robot $r_{i+2}$ – if exists – confirms it by setting its color to CONF. Therefore, $r_{i+1}$ can move in cycle $t+5$. (If $r_{i+1}$ is at the Door and $r_{i+2}$ does not exists, $r_{i+1}$ does not have to wait for the confirmation before the movement.) Now, as $r_{i+1}$ is on the previous vertex of $r_i$, $r_i$ can move in $3$ cycles, i.e. in cycle $t+8$. Therefore, the robot $r_i$ moves in every $6^{th}$ cycle if $r_i$ is a Follower or it is a Leader with an unblocked and unoccupied neighbor.
Assume now that $r_i$ is Leader, its successor $r_{i+1}$ is at its previous vertex, and all neighboring vertices of $r_i$ are blocked or occupied in cycle $t$. Then $r_i$ sets its $\Delta$ color to show the successor that it has to switch to Finished state. The successor $r_{i+1}$ confirms it in cycle $t+1$. In cycle $t+2$ the robot $r_i$ becomes Finished and $r_{i+1}$ becomes the new Leader. Therefore, the leadership is taken in $3$ LCM cycles. In $t+3$ the new Leader $r_{i+1}$ shows its new target if there is an unblocked and unoccupied neighboring vertex, or it sets the $\Delta$ to show the successor that it has to switch to Finished state.
When a Leader can move, it occupies an unvisited unblocked vertex in every $6^{th}$ cycle. Otherwise, its successor takes the leadership. Since the leadership is taken at most once by each robot during the whoole algorithm, and there are $n$ robots in the filled graph, at most $3n$ cycles used for all ’leadership taking’. Therefore, after $6n+3n$ cycles all vertices of the graph become filled.
Multiple Door
=============
For the multiple Door Filling, there is a situation which cannot be solved by the above methods: Let $v$ be an unvisited vertex, which is neighboring to (at least) two Leaders $r_1$ and $r_2$. In order to fill the graph, exactly one of the Leaders, $r_1$ or $r_2$, has to move to vertex $v$. If one of the robots, say $r_1$, has been activated earlier, then $r_1$ sets the direction color corresponding to $v$ and it prevents $r_2$ to move to $v$ ($r_1$ blocks $v$ from $r_2$). However, if the activation times of $r_1$ and $r_2$ are exactly the same, then they would set the direction color at the same time, meaning they mutually block each other from moving to $v$. If $r_1$ or $r_2$ has no other unvisited vertex in their neighborhood, then none of them could move, and particularly, none of them would occupy $v$.
We propose a protocol, which uses a strict priority order between the Leaders originating from different doors.
**Priority protocol**: The robots have $k$ additional different colors corresponding to the door they used for entering the area, where $k$ is the number of doors. We define a strict total order between these colors, called priority order. We call these $k$ colors priority colors. After showing the direction to the successor and after the successor has confirmed it, the Leader sets its color to the corresponding priority color (instead of the MOV color) and starts its movement. It arrives to its target showing its priority color. We modify the blocking rule for the Leader in the following way: If there is a robot with a direction color (except the special color $\Delta$), or confirmation color, or MOV color, or priority color with higher priority than $r$, then its neighbors are considered as blocked. Since there is a strict total order between the priority colors, in such situation exactly one of them is allowed to move there.
We slightly change the rule *taking the leadership*: when the successor robot $r$ notices that the Leader is switching to Finished state (by setting the direction color to $\Delta$), $r$ confirms it by setting its color to the priority color of the old Leader.
Priority protocol does not allow collisions. \[lemma:m3\_collision\]
Assume $v$ is an unvisited vertex which is neighboring to two Leaders $r_1$ and $r_2$, i.e. both $r_1$ and $r_2$ could move to $v$. Let $t_1$ (resp. $t_2$) be the first activation time of $r_1$ (resp. $r_2$) after it has arrived at its current position.
If $t_1 \neq t_2$, then one of them, which was activated before the other one, will block the other one and they cannot collide. If $t_1 = t_2$, then they will see each others priority color. Then they can decide which robot has higher priority. The robot with higher priority will block the other one. Consequently, Leaders cannot collide with each other.
Now we show that the collision with a Follower is also not possible. When the Follower $r$ would move to a vertex $v$ it has its CONF color set allowing the predecessor $r'$ to leave $v$. This blocks $v$ for all Leaders. The predecessor $r'$ also blocks $v$ until $r$ occupies it. Therefore, $r$ cannot collide with a Leader.
The BLOCK algorithm extended with the Priority protocol fills the connected graph. \[lemma:m3\_fills\]
We can repeat the arguments of the proof of Lemma \[lemma:m2\_fills\].
Algorithm BLOCK extended with the Priority protocol solves the $k$-Door Filling problem, $k\geq 2$, in the ASYNC model in finite time, with $2$ hops of visibility, $O(\log \Delta)$ bits of memory and using $\Delta + k + 3$ colors.
We can use the arguments of the proof of Theorem \[thm:m1\] as the area is filled (by Lemma \[lemma:m3\_fills\]), and collisions are not possible (by Lemma \[lemma:m3\_collision\]), the area will be filled without collisions. The robots store the same data in their persistent storage as in Theorem \[thm:m1\] and use $\Delta + k + 3$ colors.
Simulation Results
==================
We have implemented our algorithms for the single Door case and conducted simulations on different graph topologies. The tested topologies are Line graphs, Stars, and Delaunay triangulations with vertices uniformly randomly distributed in a square area. For the runtime, we assumed the FSYNC model (i.e. all robots are active in every LCM-cycle), therefore the runtimes are better comparable.
Line graph
----------
A line graph consist of $n\geq 1$ vertices $V=\{v_1,...,v_n\}$ and edges $E=\{(v_i,v_{i+1}): 1\leq i< n\}$. The Door vertex is at $v_1$ in the end of the line. In this case there are no branching vertices, the robots move on a unique path.
table \[x=a, y=b, col sep=comma\] [line\_PACK.csv]{}; table \[x=a, y=b, col sep=comma\] [line\_BLOCK.csv]{};
The line graph exhibits a worst case input for the PACK algorithm, since one Leader traverse the whole line and between two consecutive steps of the Leader all robots must form a Packed chain. This results in a quadratic running time. The BLOCK algorithm runs in linear time, which is also confirmed by the simulations. (Figure \[fig:line\_simulation\]).
Star graph
----------
A star graph of $n\geq 1$ consist of one central vertex $v_1$, which is connected to all other vertices $\{v_2,...,v_n\}$ by an edge. All vertices in $\{v_2,...,v_n\}$ are only conneted to $v_1$ by an edge. The Door is placed at one of the degree 1 vertices. In this topology the Leader first moves to the central vertex, then to one of the degree 1 nodes, and becomes Finished. The leadership is taken by its Follower occuying $v_1$. Then the new Leader moves to one of the leaves and the leadership is taken by its Follower occuying $v_1$, etc... In this case the lenghts of the chain behind the current Leader is at most 2 and the Packed state is achieved in constant number of LCM-cycles. Therefore, the PACK algorithm runs in linear time on the star.
table \[x=a, y=b, col sep=comma\] [star\_PACK.csv]{}; table \[x=a, y=b, col sep=comma\] [star\_BLOCK.csv]{};
The results can be seen in \[fig:star\_simulation\], which shows that the runtime of the PACK and the BLOCK algorithm is exactly the same in both cases, both runtimes are linear in the number of vertices.
Random Delaunay triangulation
-----------------------------
In these test cases the graphs are generated by using the following method. $i$) In a square area we select $n$ points independently, uniformly at random, where $n$ is the size of the graph. $ii$) Using the first $n-1$ points we compute a Delaunay triangulation. Then we add $n$-th point as Door vertex as an auxiliary vertex to the closest random vertex.
table \[x=a, y=b, col sep=comma\] [rand\_PACK.csv]{}; table \[x=a, y=b, col sep=comma\] [rand\_BLOCK.csv]{}; table \[x=a, y=b, col sep=comma\] [reference\_x2.csv]{};
For this simulation, we generated $50$ random Delaunay graphs using the described method for each vertex set size, $n=3,\dots, 200$. Then, for each input graph, we measured the number of LCM-cycles performed by both the PACK and the BLOCK algorithms. Then we computed the average runtimes of the 50 runs of both algorithms for each input size, $n=3,\dots, 200$. The simulation results (Figure \[fig:simulation\_del\]) for the PACK algorithm suggest quadratic growth: In the plot with log-scaled $x$- and $y$-axis the runtime curve $x\rightarrow x^2$, which suggests quadratic growth. The simulations also confirm linear runtime for the BLOCK algorithm.
Multiple Doors
--------------
For this simulation, for $n=1000$ vertices and $k=1,\dots, 200$ Doors, we generate 50 random Delaunay graphs as follows: $i$) In a square area we select $n$ points independenty, uniformly at random, where $n$ is the size of the graph. $ii$) Using the first $n-k$ points we compute a Delaunay triangulation. Then we add the remaining $k$ points as Door vertices and join each of them with the closest Delaunay vertex. The purpose of this simulation was to test the speed-up of the algorithm in case there are multiple entry points (Doors). For each $k=1,\dots, 200$ we plotted the average runtime on the 50 randomly generated Delaunay triangulations. The simulation results in Figure \[fig:simulation\_md\] indicate that runtime of the $k$-Door BLOCK algorithm is proportional to $n/k$ for this simulation setting.
table \[x=a, y=b, col sep=comma\] [multi\_BLOCK.csv]{};
Summary {#sec:summary}
=======
In this work we have presented solutions for the Filling problem by luminous robots in the ASYNC$^{O(1)}$ model. We have presented a method, called PACK, which solves the problem by robots with $1$ hop visibility range, $O(\log \Delta)$ bits of persistent memory, and $\Delta + 3$ colors for the single Door case. We have shown how the number of colors can be reduced to $O(1)$ at the cost of running time. Assuming that all the robots are active all the time until switching in Finished state and that the length of the LCM cycles of robot $r_i$, $i=1,\dots,n$ is $t_i$, then the running time is $O(nT)$, where $T=\sum_{i=1}^n {t_i}$. For the fully synchronous case, when all LCM cycles have unit length, it implies an $O(n^2)$ running time.
After this, in the ASYNC model we have shown how the running time can be significantly improved by robots with visibility range of $2$ hops, $O(\log \Delta)$ bits of persistent memory, and $\Delta + 3$ colors, by presenting the algorithm called BLOCK. In the fully synchronous case the running time of this algorithm is $O(n)$.
We have extended the BLOCK algorithm for solving the $k$-Door Filling problem, $k\geq 2$, by using $\Delta + 3 + k$ colors and $O(\log \Delta)$ bits of memory. The visibility range of $2$ is optimal for the $k$-Door case (a counter example when this problem cannot be solved in the ASYNC model with a visibility range of $1$ hop was presented in [@Barrameda2008], also holds for the ASYNC$^{O(1)}$ model).
Beside the theoretical results, we have implemented our algorithms and performed simulations. The simulation results match very well the theoretical bounds.
Appendix
========
If $r$.$State$ is Follower:\
If $r$.$NextTarget$ is not set:\
If $r$.$Predecessor$ shows DIR color:\
Store shown DIR as $r$.$NextTarget$\
Else If $r$.$Predecessor$ shows DIR color $\Delta$:\
$r$ switches to Leader state\
Else:\
If $r$.$Color$ is not set to CONF:\
If $r$.$Entry$ is not set:\
Set $r$:$Color$ to CONF\
Else If $r$.$Entry$ is occupied and $r$.$Successor$ has light set to a CONF color:\
Set $r$:$Color$ to CONF\
Else If $r$.$Target$ is unoccupied:\
$r$ sets $r$.$Color$ to MOV\
$r$ moves to $r$.$Target$ and\
$r$ sets $r$.$Color$ sets light to match $r$.$NextTarget$\
$r$ sets $r$.$Target$ to $r$.$NextTarget$\
Else If $r$.$Predecessor$ shows DIR color $\Delta$:\
$r$ switches to Leader state\
Else If $r$.$NextTarget$ has been set:\
If $r$.$Predecessor$ shows different direction:\
$r$ sets $r$.$NextTarget$ to new shown direction\
$r$ sets light to CONF2 color
If $r$.$State$ is Leader:\
If $r$.$Target$ is not set:\
$r$ sets $r$.$Target$ to first empty neighbor\
If $r$.$Target$ is not set (no empty neighbor found):\
$r$ sets DIR color $\Delta$ and becomes Finished\
Else:\
If $r$ is waiting for CONF:\
If $r$.$Successor$ has light set to a CONF color:\
If $r$.$Target$ is unoccupied:\
$r$ sets $r$.$Color$ to MOV\
$r$ moves to $r$.$Target$\
$r$ clears $r$.$Target$\
Else If $r$ has unoccupied neighbor $v$:\
$r$ sets $r$.$Target$ and set DIR color to match the direction of $v$\
$r$ is now waiting for CONF2\
Else:\
$r$ sets DIR color $\Delta$ and becomes Finished\
Else:\
Waits for $r$.$Successors$ to set CONF color\
Else If $r$ is waiting for CONF2:\
If $r$.$Successor$ has light set to a CONF2 color:\
$r$ sets $r$.$Color$ to MOV\
$r$ moves to $r$.$Target$\
$r$ clears $r$.$Target$\
Else:\
Waits for $r$.$Successors$ to set CONF color\
If $r$.$State$ is None:\
$r$ sets $r$.$Target$ to neighbor\
If $r.Target$ does not contain robot:\
$r$ becomes the Leader\
$r$ sets $r$.$Color$ to MOV\
$r$ moves to $r$.$Target$\
$r$ clears $r$.$Target$\
Else:\
$r$ becomes a Follower\
If $r$.$State$ is Follower:\
If $r$.$NextTarget$ is not set:\
If $r$.$Predecessor$ shows DIR color:\
Store shown DIR as $r$.$NextTarget$\
Set $r$.$Color$ to CONF\
If $r$.$Predecessor$ shows DIR color $\Delta$:\
$r$ switches to Leader state\
Else:\
If $r$.$Color$ is set to CONF or CONF2:\
If $r$.$Target$ is unoccupied:\
Set $r$.$Color$ to DIR to match $r$.$Target$\
Else:\
If $r$.$Predecessor$ shows DIR color $\Delta$:\
$r$ switches to Leader state\
Else If $r$.$NextTarget$ has been set:\
If $r$.$Predecessor$ shows different direction:\
$r$ sets $r$.$NextTarget$ to new shown direction\
$r$ sets light to CONF2 color Else:\
If $r$.$Entry$ is not set or $r$.$Entry$ is occupied\
and $r$.$Successor$ has light set to a CONF color:\
Else If $r$.$Target$ is unoccupied:\
$r$ sets $r$.$Color$ to MOV\
$r$ moves to $r$.$Target$\
sets light to match $r$.$NextTarget$\
$r$ sets $r$.$Target$ to $r$.$NextTarget$\
Else If $r$.$Predecessor$ shows DIR color $\Delta$:\
$r$ switches to Leader state\
$r$ is now waiting for CONF2\
If $r$.$State$ is Leader:\
If $r$.$Target$ is not set:\
$r$ sets $r$.$Target$ to first empty and not blocked neighbor\
If $r$.$Target$ is not set (no empty neighbor found):\
$r$ sets DIR color $\Delta$ and becomes Finished\
Else:\
If $r$ is waiting for CONF:\
If $r$.$Entry$ is not set:\
$r$ sets $r$.$Target$ to first empty and not blocked neighbor\
If $r$.$Target$ is not set (no empty neighbor found):\
$r$ sets DIR color $\Delta$ and becomes Finished\
Else:\
$r$ sets $r$.$Color$ to MOV\
$r$ moves to $r$.$Target$\
$r$ clears $r$.$Target$\
If $r$.$Entry$ is occupied\
If $r$.$Successor$ has light set to a CONF or CONF2 color:\
If $r$.$Target$ is occupied:\
$r$ sets $r$.$Target$ to first empty and not blocked neighbor\
If $r$.$Target$ is not set (no empty neighbor found):\
$r$ sets DIR color $\Delta$ and becomes Finished\
Else:\
$r$ sets $r$.$Color$ to MOV\
$r$ moves to $r$.$Target$\
$r$ clears $r$.$Target$\
Else:\
$r$ sets $r$.$Target$ to first empty and not blocked neighbor\
If $r$.$Target$ is not set (no empty neighbor found):\
$r$ sets DIR color $\Delta$ and becomes Finished\
Else If $r$ is waiting for CONF2:\
If $r$.$Successor$ has light set to a CONF2 color:\
$r$ sets $r$.$Color$ to MOV\
$r$ moves to $r$.$Target$\
$r$ clears $r$.$Target$\
Else:\
Waits for $r$.$Successors$ to set CONF2 color\
If $r$.$State$ is None:\
$r$ sets $r$.$Target$ to neighbor\
If $r.Target$ does not contain robot:\
$r$ becomes the Leader\
$r$ sets light to MOV\
$r$ moves to $r$.$Target$\
$r$ clears $r$.$Target$\
Else:\
$r$ becomes a Follower\
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The analytic forms of the asymptotic quasinormal frequencies of a coupled scalar field in the Garfinkle-Horowitz-Strominger dilaton spacetime is investigated by using the monodromy technique proposed by Motl and Neitzke. It is found that the asymptotic quasinormal frequencies depend not only on the structure parameters of the background spacetime, but also on the coupling between the scalar fields and gravitational field. Moreover, our results show that only in the minimal couple case, i.e., $\xi$ tends zero, the real parts of the asymptotic quasinormal frequencies agrees with the Hod’s conjecture, $T_H\ln{3}$.'
author:
- Songbai Chen
- Jiliang Jing
title: '[**Asymptotic quasinormal modes of a coupled scalar field in the Garfinkle-Horowitz-Strominger dilaton spacetime**]{}'
---
Introduction
============
It is well known that quasinormal modes possess a discrete spectra of complex characteristic frequencies which are entirely fixed by the structure of the background spacetime and are irrelevant of the initial perturbations[@Chandrasekhar]. Thus, one can directly identify a black hole existence through comparing the quasinormal modes with the gravitational waves observed in the universe. Meanwhile, it is generally believed that the study of the quasinormal modes may lead to a deeper understanding of black holes and quantum gravity because that the quasinormal frequency spectra is related to the AdS/CFT correspondence, string theory and loop quantum gravity[@Hod98]-[@Zerilli70]. Therefore, much attention has been devoted to the study of the quasinormal modes in the recent thirty years[@Detweiler]-[@Bachelot]. In the Schwarzschild spacetime, one found that the asymptotic quasinormal frequencies of high overtones are described by $$\begin{aligned}
\frac{ 2\pi\omega}{\kappa}=\ln{3}+i(2n+1)\pi,\ \ \ n\rightarrow \infty, \label{w1}
\end{aligned}$$ where $\kappa $ is the surface gravity constant of the black hole. Formula (\[w1\]) was derived numerically[@Nollert] and subsequently confirmed analytically[@Andersson][@Motl03].
Hod[@Hod98] first conjectured that the real parts of the asymptotic quasinormal frequencies of a Schwarzschild black hole can be expressed as $\omega_R=T_H\ln{3}$. Together with Bohr’s correspondence principle, the first law of black hole thermodynamics and the asymptotic quasinormal modes, he also obtained some new information about the quantization of area at a black hole event horizon. Using of Hod’s conjecture, Dreyer[@Dreyer03] found that the quasinormal modes can entirely fixed the Barbero-Immirzi parameter[@Immirzi57], which was introduced as an indefinite factor by Immirzi to obtain the right form of the black hole entropy in the loop quantum gravity. Most significantly, the presence of $\ln{3}$ also means that the gauge group in the loop quantum gravity should be $SO(3)$ rather than $SU(2)$. Thus, one suggested that Hod’s conjecture maybe create a new way to probe the quantum properties of black hole.
However, the question whether Hod’s conjecture applies to more general black holes still remain open. Recently, we probed the asymptotic quasinormal modes of a massless scalar field in the Garfinkle-Horowitz-Strominger dilaton spacetime[@04] and find that the frequency spectra formula satisfies Hod’s conjecture. Cardoso and Abdalla [@Cardoso][@Abdalla] found the asymptotic quasinormal frequencies in the Schwarzschild de Sitter and Anti-de Sitter spacetimes depend on the cosmological constant. Only under the condition that the cosmological constant vanishes, the real parts of the asymptotic quasinormal frequencies returns to $T_H\ln{3}$. For the Reissner-Nordström black hole, L. Motl and A.Neitzke[@Motl03] obtained the asymptotic quasinormal frequencies is relevant of the electric charge $Q$. It is unfortunately that the asymptotic quasinormal frequencies do not return returns $T_H\ln{3}$ as the black hole charge $Q$ tends to zero. Thus, some authors[@Berti] suggested that Hod’s conjecture should be modified in some way. However, how does the correct modification look like? It is an interesting subject need to be study more deeply in the future. At present, it is necessary and important to study the asymptotic quasinormal modes in the more general background spacetimes.
In this paper, our main purpose is to investigate the asymptotic quasinormal modes of a coupled scalar field in the Garfinkle-Horowitz-Strominger dilaton spacetime. We find that besides dependence on the structure parameters of the background spacetime, the asymptotic quasinormal frequencies also are relevant of the couple constant $\xi$. The plan of the paper is as follows. In Sec.II, we derive analytically the asymptotic quasinormal frequency formula of a coupled scalar field in the Garfinkle-Horowitz-Strominger dilaton spacetime by making use of the monodromy method[@Motl03]. At the last, a summary and some discussions are presented.
Asymptotic quasinormal frequencies formula of a coupled scalar field in the Garfinkle-Horowitz-Strominger dilaton spacetime
===========================================================================================================================
In standard coordinates, the metric for the Garfinkle-Horowitz-Strominger dilaton black hole spacetime can be expressed as [@Garfinkle91] $$\begin{aligned}
ds^2=-\left(1-\frac{2M}{r'}\right)dt^2+\left(1-\frac{2M}{r'}\right)^{-1}
dr^2+r'(r'-2a)d\Omega^2,\label{gem1}
\end{aligned}$$ $$\begin{aligned}
e^{-2\phi}&=&e^{-2\phi_0}\left(1-\frac{2a}{r'}\right),\nonumber
\end{aligned}$$ where $M$ represents the black hole mass and $a$ is a parameter related to dilaton field. The dilaton field is given by $e^{-2\phi}=e^{-2\phi_0}(1-\frac{Q^2}{Mr'})$, where $\phi_0$ is the dilaton value at $r'\rightarrow \infty$ and $Q$ is the electric charge carried by this black hole. The relationship among mass $M$, the charge $Q$ and $a$ is described as $a=\frac{Q^2}{2M}$. This black hole has an event horizon at $r'=2M$ and two singular points at $r'=0$ and $r'=2a$. The Hawking temperature $T_H=\frac{1}{8\pi M}$ is the same as that of the Schwarzschild spacetime.
In order to simplify the calculation, we introduce a coordinate change $$\begin{aligned}
r=\sqrt{r'(r'-2a)}.
\end{aligned}$$ Then the metric (\[gem1\]) can be rewritten as $$\begin{aligned}
ds^2=-\left(1-\frac{2M}{a+\sqrt{a^2+r^2}}\right)dt^2+
\left(1-\frac{2M}{a+\sqrt{a^2+r^2}}\right)^{-1}\frac{r^2}
{r^2+a^2}dr^2+r^2d\Omega^2.\label{gem2}
\end{aligned}$$ The event horizon of the black hole is now located at $r=2\sqrt{M(M-a)}$ and the Hawking temperature is still described by $T_H=\frac{1}{8\pi M}$. By means of the quantity $R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}$, we find that the point $r=0$ is a curvature singular point.
The general perturbation equation for a coupled massless scalar field in the dilaton spacetime is given by [@Frolov] $$\begin{aligned}
\frac{1}{\sqrt{-g}}\partial_\mu(\sqrt{-g}g^{\mu\nu}
\partial_\nu)\psi-\xi R\psi=0,\label{eq1}
\end{aligned}$$ where $\psi$ is the scalar field and $R$ is the Ricci scalar curvature. The coupling between the scalar field and the gravitational field represented by the term $\xi R\psi$, where $\xi$ is a numerical couple factor.
After adopting WKB approximation $\psi=\frac{e^{-i\omega
t}\phi(r)}{r}Y(\theta,\varphi)$, introducing a tortoise coordinate $$\begin{aligned}
x=\sqrt{a^2+r^2}-a+2M \ln{\left[\frac{\sqrt{a^2+r^2}-(2M-a)}
{2(M-a)}\right]},\label{x2}
\end{aligned}$$ and substituting Eqs.(\[gem2\]) and (\[x2\]) into Eq.(\[eq1\]), we know that the radial perturbation equation for a coupled scalar field in the Garfinkle-Horowitz-Strominger dilaton spacetime can be expressed as $$\begin{aligned}
\frac{d^2\phi}{dx^2}+(\omega^2-V[r(x)])\phi=0,\label{e3}
\end{aligned}$$ where $$\begin{aligned}
V[r(x)]&=&\left(1-\frac{2M}{a+\sqrt{a^2+r^2}}\right)\times\nonumber\\ &&\left[\frac{l(l+1)}{r^2}+
\frac{2M(a^2+r^2)^{3/2}-2\sqrt{a^2+r^2}a^3+2Ma^3-2a^4-r^2a^2}
{r^4(a+\sqrt{r^2+a^2})^2}+\xi R\right]\label{v},
\end{aligned}$$ and $$\begin{aligned}
R=\frac{2a^2(r^2+2aM-2M\sqrt{a^2+r^2})}{r^6}.
\end{aligned}$$ It is well known that the quasinormal modes consist of the solutions of the perturbation equation (\[e3\]) with the boundary conditions appropriate for purely ingoing waves at the event horizon and purely outgoing waves at infinity, namely, $$\begin{aligned}
\phi &=& e^{+i\omega x},\ \ \ \ \ \ x\rightarrow -\infty, \nonumber\\
\phi &=& e^{-i\omega x},\ \ \ \ \ \ x\rightarrow +\infty.
\end{aligned}$$ In general, we just consider the perturbation equation (\[e3\]) in the physical region $r\geq 2\sqrt{M(M-a)}$ in the Garfinkle-Horowitz-Strominger dilaton black hole. However, in the monodromy method, it is fundamental to extend analytically Eq.(\[e3\]) to the whole complex $r$-plane. In the process of analytical extension, we find that both the tortoise coordinate $x(r)$ and the wave function $\phi(r)$ are multivalued around the singular points $r=0$ and $r=2\sqrt{M(M-a)}$ . This multivaluedness plays an important and essential role in our analysis. As in Ref.[@Motl03], we can put branch cuts in the complex $r$-plane from $r=0$ to $r=2\sqrt{M(M-a)}$ in order to avoid dealing with multivalued functions. The monodromy of $\phi(r)$ can be defined by the discontinuity across the cut. Finally, by comparing the local and global monodromy of $\phi(r)$ along the selected contour $L$ around the point $r=2\sqrt{M(M-a)}$, we can obtain the asymptotic quasinormal frequency spectra in the Garfinkle-Horowitz-Strominger dilaton black hole spacetime.
From Eq.(\[x2\]), we find that $x$ is not uniquely defined as a function of $r$. However, it is very fortunate that we can determine the sign of $Re(x)$ in the complex $r$-plane. The regions for the different sign of $Re(x)$ are shown in the Figure 1.
{width="6cm"}
\[fig1\]
As in Ref.[@Motl03], in order to compute conveniently, we may introduce the variable $z=x-\frac{\pi i}{2\kappa}$. For $r=0$, we have $z=0$. To fix the angle of the variable $z$ at the point $r=0$, we define the branch $n=0$ for $\ln{(-1)}$.
Now, we must define the boundary condition at $r=\infty$. Similarly to Ref.[@Motl03], we can analytically continue $\phi(r)$ via “Wick rotation" to the line $Im(\omega x)=0$. For the highly damped modes, i.e., $\omega$ are almost purely imaginary, the line $Im(\omega x)=0$ is just slightly sloped off the line $Re(x)=0$. Assuming initially that $Re(\omega)>0$ and then making use of the condition $Im(\omega x)=0$, we can obtain $x=+\infty$ is rotated to $\omega x=+\infty$. Thus, on the line $Re(x)=0$, the boundary condition at $r=+\infty$ actually becomes $$\begin{aligned}
\phi(r)\sim e^{-i\omega x},\ \ \ \ \omega x\rightarrow +\infty. \label{b1}
\end{aligned}$$
Let us now compute the local monodromy around the singular point $r=2\sqrt{M(M-a)}$. This can be done by matching the asymptotic along the line $Re(x)=0$, i.e., the contour $L$ shown in the Fig. 1. When we start at point $A$ and move along the contour $L$ towards interior, the $\phi(x)$ can be look as the plane waves because the term $\omega^2$ dominates the potential in Eq.(\[e3\]) away from the origin point. At the vicinity of the point $r=0$, we have $$\begin{aligned}
z\sim -\frac{r^2}{2(M-a)},
\end{aligned}$$ and the behaviors of the Ricci scalar curvature and the potential are $$\begin{aligned}
R\sim \frac{2a(a-M)}{r^4},
\end{aligned}$$ and $$\begin{aligned}
V[r(z)]\sim-\frac{1-2\xi}{4z^2}.\label{v4}
\end{aligned}$$ We make the identification $j=\sqrt{2 \xi}$, and then the perturbation equation (\[e3\]) can be rewritten as $$\begin{aligned}
(\frac{d^2}{dz^2}+\omega^2+\frac{1-j^2}{4z^2})\phi(z)=0.
\end{aligned}$$ From the Ref.[@handbook], we find it can be exactly solved in terms of the Bessel function and the general solution near the origin point can be expressed as $$\begin{aligned}
\phi(z)=A_+c_+\sqrt{\omega z}J_{+j/2}(\omega z)+
A_-c_-\sqrt{\omega z}J_{-j/2}(\omega z).\label{J1}
\end{aligned}$$ Now, let us look for the asymptotic forms of the solution (\[J1\]) away from the origin. After considering the asymptotic behavior of $J_{\pm j/2}(\omega z)$ as $\omega z\rightarrow \infty$, we can select the normalization factors $c_{\pm}$ in (\[J1\]) so that we can write the asymptotic forms as $$\begin{aligned}
c_{\pm}\sqrt{\omega z}J_{\pm \frac{j}{2}}(\omega z)\sim 2\cos{(\omega z-\alpha_{\pm})},
\label{J2}
\end{aligned}$$ as $\omega z\rightarrow \infty$ , where $\alpha_{\pm}=\frac{\pi}{4}(1\pm j)$. From Eqs.(\[J1\]), (\[J2\]) and the boundary condition (\[b1\]), we have $$\begin{aligned}
A_+e^{-i\alpha_+}+A_-e^{-i\alpha_-}=0,
\end{aligned}$$ and $$\begin{aligned}
\phi(z)\sim (A_+e^{i\alpha_+}+A_-e^{i\alpha_-})e^{-i\omega z}.
\end{aligned}$$ To follow the contour $L$ and approach to the point $B$, we have to turn an angle $\frac{3\pi}{2}$ around the origin $r=0$, corresponding to $3\pi$ around $z=0$. From the Bessel function behavior near the origin point $$\begin{aligned}
J_{\pm \frac{j}{2}}(\omega z)=z^{\pm \frac{j}{2}}\varphi(z),
\end{aligned}$$ where $\varphi(z)$ is an even holomorphic function, we find that after the $3\pi$ rotation the asymptotic are $$\begin{aligned}
c_{\pm}\sqrt{\omega z}J_{\pm \frac{j}{2}}(\omega z)\sim e^{6i\alpha_{\pm}}
2\cos{(-\omega z-\alpha_{\pm})},
\label{J3}
\end{aligned}$$ as $\omega z\rightarrow -\infty$. Thus the asymptotic at the point $B$ $$\begin{aligned}
\phi(z)\sim (A_+e^{5i\alpha_+}+A_-e^{5i\alpha_-})e^{-i\omega z}+
(A_+e^{7i\alpha_+} +A_-e^{7i\alpha_{-}})e^{i\omega z},\ \ \ \ \omega z\rightarrow -\infty.
\label{J4}
\end{aligned}$$ Finally, we can come back from the point $B$ to the point $A$ along the large semicircle in the right half-plane. In this region, because the term $\omega^2 $ dominates the potential $V[r(x)]$, we can approximate the solutions of the perturbation equation as plane waves. When we return to the point $A$, the coefficient of $e^{-i\omega z}$ remains unchange. While the coefficient of $e^{i\omega z}$ makes only an exponentially small contribution to $\phi(z)$ in the right plane. Finally, we find that the monodromy around the contour $L$ must multiply the coefficient of $e^{-i\omega z}$ by a factor $$\begin{aligned}
\frac{A_+e^{5i\alpha_+}+A_-e^{5i\alpha_-}}{A_+e^{i\alpha_+} +A_-e^{i\alpha_{-}}}
=\frac{e^{6i\alpha_+}-e^{6i\alpha_-}}{e^{2i\alpha_+} -e^{2i\alpha_{-}}}=-(1+2\cos{\pi
j}).
\label{11}
\end{aligned}$$
Now, let us calculate the global monodromy around the contour $L$. Since the only singularity of $\phi(r)$ or $e^{-i\omega z}$ inside the contour occurs at the point $r=2\sqrt{M(M-a)}$, according to the boundary condition of the quasinormal modes, we can obtain the monodromy of $\phi(r)$ or $e^{-i\omega z}$ at this point. After a full clockwise round trip, $\phi(r)$ acquires a phase $e^{\frac{\pi\omega}{\kappa}}$, while $e^{-i\omega z}$ acquires a phase $e^{-\frac{\pi\omega}{\kappa}}$. So the coefficient of $e^{-i\omega z}$ in the asymptotic of $\phi(r)$ must be multiplied by $e^{\frac{2\pi\omega}{\kappa}}$. Substituting $j=\sqrt{2\xi}$ into Eq.(\[11\]) and comparing the local monodromy with the global one, we find the asymptotic quasinormal frequencies of a coupled scalar field in the Garfinkle-Horowitz-Strominger dilaton black hole spacetime satisfy $$\begin{aligned}
e^{\frac{2\pi\omega}{\kappa}}=-[1+2\cos{(\sqrt{2\xi} \pi)}]\label{22}.
\end{aligned}$$ Making a simple operation on Eq.(\[22\]), we obtain easily the formula $$\begin{aligned}
\frac{2\pi\omega}{\kappa}=\ln{[1+2\cos{(\sqrt{2\xi} \pi)}]}+i(2n+1)\pi,\ \ \ \
n \rightarrow \infty.\label{12}
\end{aligned}$$ It is interesting to note that the right-hand side of the formula (\[12\]) contains the couple factor $\xi$. This means that the asymptotic quasinormal modes depend not only on the structure parameters of the background spacetimes, but also on the coupling between the matter fields and gravitational field. Furthermore, we find only when $\xi=0$, namely, in the minimal couple case, the real part of the right-hand side of the formula (\[12\]) becomes $\ln{3}$, which is consistent with Hod’s conjecture.
Summary and discussion
======================
We have investigated the analytical forms of the asymptotic quasinormal frequencies for a coupled scalar field in the Garfinkle-Horowitz-Strominger dilaton spacetime by adopting the monodromy technique. It is shown that the asymptotic quasinormal frequencies depend not only on the structure parameters of the background spacetime, but also on a couple constant $\xi$. The fact tells us that the interaction between the matter fields and gravitational field will affect the frequencies spectra formula of the asymptotic quasinormal modes. It is a novel property of the quasinormal modes in Garfinkle-Horowitz-Strominger dilaton spacetime. In the Schwarzschild and Reissner-Nordström spacetimes, we find the asymptotic quasinormal frequencies do not possess this behavior because the curvature scalar $R$ in both the spacetimes is equal to zero and then the coupled term in Eq.(\[eq1\]) vanishes. Moreover, we find that Hod’s conjecture, the real parts of the asymptotic quasinormal frequencies equals to $T _H \ln{3}$, is valid only for the minimal couple case in the Garfinkle-Horowitz-Strominger dilaton spacetime, i.e., $\xi$ becomes zero. It implies that there maybe exist a more general form of the Hod’s conjecture.
It should be pointed out that although the formula (\[12\]) is not related to the dilaton field parameter $a$ obviously, it does not means that the quasinormal modes are independent of the dilaton. The reason is that we just consider the contribution of the leading term $r^{-4}$ in the potential $V$ to the local monodromy of the wave function $\phi(z)$. If we consider the contribution of the lower order terms in the potential $V(r)$, the corrected term to the asymptotic quasinormal frequencies will depend on the parameter $a$ of the dilaton field. For example, as $a$ is very small, the main corrected term is roughly in proportion to $\frac{(1-i)}{\sqrt{n+1/2}}[
l(l+1)+1-\frac{2M}{3(M+a)}]$, which shows that the quasinormal frequencies increase as $a$ increases in the Garfinkle-Horowitz-Strominger dilaton spacetime.
This work was supported by the National Natural Science Foundation of China under Grant No. 10275024; the FANEDD under Grant No. 200317; and the Hunan Provincial Natural Science Foundation of China under Grant No. 04JJ3019.
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| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
\
Institut für Physik, Humboldt-Universität zu Berlin, D-12489 Berlin, Germany\
E-mail:
title: 'Exploring Gauge-Invariant Vacuum Wave Functionals for Yang-Mills Theory'
---
Introduction
============
The Hamiltonian formulation of Yang-Mills (YM) theory in the Schrödinger picture, although not particularly efficient in the perturbative domain, offers considerable benefits when addressing nonperturbative issues. Among its attractive features are the explicit representation of the vacuum state which invokes quantum mechanical intuition [@fey81], the ability to treat genuine real-time problems (including non-equilibrium processes) as well as the transparent description of topological effects [@jac90]. In particular, however, it makes gauge theories accessible to a variational treatment [@fey81; @varYM], i.e. to one of the few approximation schemes currently available for strongly coupled quantum field theories.
Variational calculations in Yang-Mills theories are often performed in a fixed gauge, most notably in Coulomb gauge [@sze04]. In the following we will report on our complementary explorations [@for10] of a manifestly gauge-invariant formulation of the variational problem [@kog95]. This framework renders fundamental infrared (IR) physics, including dimensional transmutation and the generation of a mass gap, particularly transparent. Moreover, it preserves the full topological structure of the gauge group. The latter is particularly relevant since topological properties are likely robust enough to survive limitations of the restricted trial functional basis which keeps the approach analytically manageable. Another attractive feature of the gauge-invariant formulation is that the infrared dynamics can be re-expressed in terms of gauge-invariant collective fields which subsume contributions from whole gauge-field orbit families. After performing an IR improved variational analysis [@for10], we will make use of this feature to identify gauge-invariant and universal IR degrees of freedom of the gauge dynamics [@for06]. More details can be found in Refs. [@for10; @for06].
Gauge-invariant vacuum wave functionals {#ginv}
=======================================
Starting from an approximate and hence typically gauge-dependentcore functional $\psi _{0}\left[ \vec{A}%
\right] $ of the static gauge fields (i.e. of half of the canonical variables), we impose gauge invariance by averaging over the gauge group. The result is a trial vacuum wave functional (VWF) of the form$$\Psi _{0}\left[ \vec{A}\right] =\sum_{Q\in Z}e^{iQ\theta }\int D\mu \left[
U^{\left( Q\right) }\right] \psi _{0}\left[ \vec{A}^{U^{\left( Q\right) }}%
\right] =:\int DU\psi _{0}\left[ \vec{A}^{U}\right] \label{ginvvwf}$$where $d\mu $ is the Haar measure of the SU$\left( N_{c}\right) $ gauge group, $Q$ the topological (homotopy) charge of the group element $U^{\left(
Q\right) }$, and $\theta $ the vacuum angle. Since the vacuum wave functional is nodeless [@fey81], one may write $\psi _{0}\left[ \vec{A}%
\right] =\mathcal{N}^{-1}\exp \left( -\Phi \left[ \vec{A}\right] \right) $ and expand the real functional $\Phi $ into a power series. The constant term is absorbed into $\mathcal{N}$ and the term linear in $A$ is generally discarded (coherent gluon vacuum states are known to be unstable [@leu81]).
The next term is quadratic in $A$ and plays several crucial roles. First, it removes the ambiguity in $\Psi _{0}$ [@zar98] due to the invariance of the Haar measure in Eq. (\[ginvvwf\]) under group transformations. Furthermore, this term can incorporate asymptotic freedom and thus render the VWF exact in the ultraviolet. Finally, and from the practical perspective most importantly, functionals resulting from a quadratic term can be integrated over $A$ analytically. Hence one generally truncates the series for $\Phi $ after the quadratic term, which leads to the squeezed core functional $$\psi _{0}^{\left( G\right) }\left[ \vec{A}\right] =\frac{1}{\mathcal{N}_{G}}%
\exp \left[ -\frac{1}{2}\text{ }\int d^{3}x\int d^{3}yA_{i}^{a}\left( \vec{x}%
\right) G_{ij}^{-1ab}\left( \vec{x}-\vec{y}\right) A_{j}^{b}\left( \vec{y}%
\right) \right] \label{ga}$$with the normalization factor $\mathcal{N}_{G}^{-1}=\left[ \det \left(
G/2\right) \right] ^{-1/4}$ and a real covariance $G^{-1}$.
Gluon dispersion: asymptotic freedom and IR generality {#guv}
------------------------------------------------------
We now have to specify the properties of the function $G^{-1}$ in the trial functional family (\[ga\]). Translational invariance was already anticipated in Eq. (\[ga\]). We will further restrict ourselves to a purely transverse covariance with the Fourier transform [@kog95] $$G_{ij}^{-1,ab}\left( k\right) =\delta _{ij}\delta ^{ab}G^{-1}\left( k\right)
\label{gspat}$$(cf. Ref. [@for10] for a discussion of this choice and Refs. [@dia98; @bro98] for the impact of longitudinal contributions). The normalizability of physical wave functionals then demands $G^{-1}\left( k\right) >0$ and further ensures vacuum stability and a positive energy spectrum. In order to implement the correct UV behavior, we factorize the core functionals (\[ga\]) as $\psi
_{0}^{\left( G\right) }\left[ \vec{A}\right] =\psi _{0}^{\left( G_{<}\right)
}\left[ \vec{A}_{<}\right] \psi _{0}^{\left( G_{>}\right) }\left[ \vec{A}_{>}%
\right] $ by splitting the $\vec{k}$ integration domain in their exponentials into soft/hard momentum regions with $\left\vert \vec{k}%
\right\vert \gtrless \mu $. The separation scale $\mu $ will be determined below. Asymptotic freedom requires $G$ to approach the non-interacting, massless static vector propagator $G_{0}\left( k\right) =1/k$ for $%
k\rightarrow \infty $. As long as $\mu \gg \Lambda _{\text{YM}}$ (where $%
\Lambda _{\text{YM}}$ is the Yang-Mills scale) perturbative hard-mode corrections remain small, which allows us to approximate $$G_{>}^{-1}\left( k\right) =G_{0}^{-1}\left( k\right) =k. \label{gm1uv}$$The unknown IR covariance $G_{<}^{-1}\left( k\right) $, on the other hand, will be determined variationally. In Ref. [@kog95] the minimal one-parameter trial function $G_{<,\text{KK}}^{-1}\left( k\right) =\mu $ was adopted. We have implemented a far more comprehensive parametrization [for06]{}, based on the under reasonable analyticity assumptions general and controlled gradient expansion$$G_{<}^{-1}\left( \vec{x}-\vec{y}\right) =m_{g}\left[ 1+c_{1}\frac{%
\partial _{x}^{2}}{\mu ^{2}}+c_{2}\left( \frac{\partial _{x}^{2}}{\mu ^{2}}%
\right) ^{2}+c_{3}\left( \frac{\partial _{x}^{2}}{\mu ^{2}}\right) ^{3}+...%
\right] \delta _{<}^{3}\left( \vec{x}-\vec{y}\right) . \label{gm1}$$Eq. (\[gm1\]) can be efficiently truncated to maintain an analytically manageable trial basis for the soft-mode physics. Besides $\mu $, the variational parameter space now contains the IR gluon mass $m_{g}>0$ and a few low-momentum constants $c_{i}$ which characterize dispersive gluon properties in the vacuum. The regularized delta function $\delta
_{<}^{3}\left( \vec{x}-\vec{y}\right) :=\int d^{3}k/\left( 2\pi \right)
^{3}\theta \left( \mu ^{2}-\vec{k}^{2}\right) e^{i\vec{k}\left( \vec{x}-\vec{%
y}\right) }$ encodes the slow variation $\left\Vert \partial
A_{<}\right\Vert /\left\Vert A_{<}\right\Vert \leq \mu $ of the soft modes and ensures that the higher-order terms in Eq. (\[gm1\]) are parametrically suppressed.
As a consequence of $G^{-1}\left( k\right) >0$, the low-momentum constants are subject to the bounds $c_{1}<1,$ $c_{2}>-1$, etc. (for $m_{g}>0$). Requiring continuity of $G^{-1}\left( k\right) $ at the matching point $%
k=\mu $, furthermore, fixes $m_{g}$ as a function of the other variational parameters. When truncating to $c_{i\geq 2}=0$, for example, one has$$m_{g}\left( \mu ,c_{1}\right) =\frac{\mu }{1-c_{1}}. \label{mgc}$$Note that the requirement of a non-negative IR gluon mass restricts the $%
c_{1}$ domain to $c_{1}<1$, in agreement with the above bound from $%
G^{-1}\left( k\right) >0$. The VWF (\[ginvvwf\]) together with the core functional (\[ga\]) and the covariance (\[gm1uv\]), (\[gm1\]) (possibly with perturbative corrections) appears to be the richest gauge-invariant trial functional family whose matrix elements can be calculated analytically by currently available techniques.
Variational analysis
====================
One the basis of the trial functional family (\[ginvvwf\]) discussed above, the variational analysis amounts to minimizing the expectation value $$\left\langle \mathcal{H}\left( A,E\right) \right\rangle =\frac{\int D\vec{A}%
\Psi _{0}^{\ast }\left[ \vec{A}\right] \mathcal{H}\left( \vec{A}^{a},\vec{E}%
^{a}\right) \Psi _{0}\left[ \vec{A}\right] }{\int D\vec{A}\Psi _{0}^{\ast }%
\left[ \vec{A}\right] \Psi _{0}\left[ \vec{A}\right] } \label{hexp}$$ of the Yang-Mills Hamiltonian density $$\mathcal{H}=\frac{1}{2}\left( E_{i}^{a}E_{i}^{a}+B_{i}^{a}B_{i}^{a}\right)
\label{HYM}$$ (in temporal gauge, with $\vec{E}^{a}=i\delta /\delta \vec{A}^{a}$) with respect to the parameters appearing in $G^{-1}$. After inserting the wave functional (\[ginvvwf\]) into Eq. (\[hexp\]) and interchanging the order of integration over fields and group elements, the gauge invariance of the $%
\vec{A}$ integral allows to factor out a gauge group volume. Eq. (\[hexp\]) can thus be rewritten as $$\left\langle \mathcal{H}\left( A,E\right) \right\rangle =\frac{\int DU\int D%
\vec{A}\psi _{0}\left[ \vec{A}^{U}\right] \mathcal{H}\left( \vec{A}^{a},%
\frac{i\delta }{\delta \vec{A}^{a}}\right) \psi _{0}\left[ \vec{A}\right] }{%
\int DU\int D\vec{A}\psi _{0}\left[ \vec{A}^{U}\right] \psi _{0}\left[ \vec{A%
}\right] }$$(where $DU$ is the functional SU$\left( N_{c}\right) $ measure as defined in Eq. (\[ginvvwf\])). After evaluating the functional derivatives contained in $\mathcal{H}$, the Gaussian integration over $A$ can be performed exactly, resulting in$$\left\langle \mathcal{H}\right\rangle =\frac{\int DU\left\langle
\left\langle \left\langle \mathcal{H}\right\rangle \right\rangle
\right\rangle \exp \left\{ -\Gamma _{b}\left[ U\right] \right\} }{%
\int DU\exp \left\{ -\Gamma _{b}\left[ U\right] \right\} }$$where we introduced the notation $$\left\langle \left\langle \left\langle \vec{A}...\vec{A}...\vec{E}...\vec{E}%
\right\rangle \right\rangle \right\rangle \exp \left\{ -\Gamma _{b}%
\left[ U\right] \right\} \equiv \int D\vec{A}\psi _{0}\left[ \vec{A}^{U}%
\right] \vec{A}...\vec{A}...\frac{i\delta }{\delta \vec{A}}...\frac{i\delta
}{\delta \vec{A}}\psi _{0}\left[ \vec{A}\right]$$for matrix elements between $U$-rotated and unrotated core VWFs. The above expression defines, in particular, the effective bare action $\Gamma _{b}
\left[ U\right] =-\ln \int D\vec{A}\,\ \psi _{0}^{\ast }\left[ \vec{A}^{U}%
\right] \psi _{0}\left[ \vec{A}\right] $ which describes dynamical correlations generated by the gauge projection. This action gathers all those gauge-field contributions to the generating functional whose approximate vacua $\psi _{0}$ at $t=\pm \infty $ differ by a relative gauge orientation $U$. Hence the gauge-invariant variable $U$ represents the contributions from all such gluon field orbits to the vacuum overlap. Explicitly, one finds [@for10] $$\Gamma _{b}\left[ U\right] =\frac{1}{2g_{\text{b}}^{2}}\int
d^{3}x\int d^{3}yL_{i}^{a}\left( \vec{x}\right) \mathcal{D}^{ab}\left( \vec{x%
}-\vec{y}\right) L_{i}^{b}\left( \vec{y}\right) \label{gamb}$$with $L_{i}=U^{\dagger }\partial _{i}U=:L_{i}^{a}\lambda ^{a}/\left(
2i\right) $ and $\mathcal{D}^{ab}=\left[ \left( G+G^{U}\right) ^{-1}\right]
^{ab}\simeq \frac{1}{2}G^{-1}\delta ^{ab}+...$ where $G^{U}=G^{ab}\left(
\vec{x}-\vec{y}\right) $ $U^{\dagger }\left( \vec{x}\right) \left( \lambda
^{a}/2\right) U\left( \vec{x}\right) \otimes U\left( \vec{y}\right) \left(
\lambda ^{b}/2\right) U^{\dagger }\left( \vec{y}\right) $.
After splitting $U\left( \vec{x}\right) =U_{<}\left( \vec{x}\right)
U_{>}\left( \vec{x}\right) $ with $U_{>}\left( \vec{x}\right) =\exp \left(
-ig\phi ^{a}\left( \vec{x}\right) \lambda ^{a}/2\right) $ into hard- and soft-mode contributions and integrating over the hard modes $\phi ^{a}$ perturbatively [@kog95], furthermore, one arrives at$$\left\langle \mathcal{H}\right\rangle =\frac{\int DU_{<}\int D\phi
\left\langle \left\langle \left\langle \mathcal{H}\right\rangle
\right\rangle \right\rangle \exp \left\{ -\Gamma _{b}\left[ \phi
,U_{<}\right] \right\} }{\int DU_{<}\int D\phi \exp \left\{ -\Gamma _{b}
\left[ \phi ,U_{<}\right] \right\} }.$$With the additional definition$$\left\langle \left\langle \mathcal{O}\right\rangle \right\rangle \exp
\left\{ -\Gamma _{<}\left[ U_{<}\right] \right\} :=\int D\phi \left\langle
\left\langle \left\langle \mathcal{O}\right\rangle \right\rangle
\right\rangle \exp \left\{ -\Gamma _{b}\left[ \phi ,U_{<}\right]
\right\} , \label{o2}$$which contains the effective soft-mode action $$\Gamma _{<}\left[ U_{<}\right] :=-\ln \int D\phi \exp \left\{-\Gamma _{b}
\left[ \phi ,U_{<}\right] \right\} \label{gsm0}$$ (i.e. the RG evolved bare action (\[gamb\])), we can finally rewrite the matrix element (\[hexp\]) solely in terms of the $U_{<}$ field dynamics, i.e.$$\left\langle \mathcal{H}\right\rangle =\frac{\int DU_{<}\left\langle
\left\langle \mathcal{H}\right\rangle \right\rangle \exp \left\{ -\Gamma _{<}%
\left[ U_{<}\right] \right\} }{\int DU_{<}\exp \left\{ -\Gamma _{<}\left[
U_{<}\right] \right\} }. \label{hsmvev}$$ Since the reduced (i.e., fixed $U_{<}$) matrix element $\left\langle \left\langle \mathcal{H}\right\rangle
\right\rangle $ is a nonlocal functional of the soft modes $U_{<}$, the evaluation of Eq. (\[hsmvev\]) amounts to calculating (equal-time) soft-mode correlation functions [@for10].
Vacuum phases
-------------
In integrals over the $U_{<}$ fields (such as those in Eq. (\[hsmvev\])) the unitarity constraint $U_{<}^{\dagger }U_{<}=1$ can be resolved by inserting a delta functional which is then written as an additional integral over a hermitean auxilary field $\Sigma $. In Eq. (\[hsmvev\]) the integration over the then unconstrained $U_{<}$ becomes Gaussian and can be done analytically. In the mean-field approximation, the expression for $%
\left\langle \left\langle \mathcal{H}\right\rangle \right\rangle $ is then evaluated at the saddle point $\bar{\Sigma}=:\left( \mu \bar{\xi}\right)
^{2} $ of the $\Sigma $ integral, i.e. at the minimal-action solution of the gap equation $$\left\langle U_{<,AB}^{\dagger }\left( \vec{x}\right) U_{<,BC}\left( \vec{x}%
\right) \right\rangle =\delta _{AC} \label{geq2}$$which reintroduces unitarity in the mean. After adopting the one-loop Yang-Mills coupling $\gamma \left( \mu \right) =g_{\text{YM}}^{2}\left( \mu
\right) N_{c}/\pi ^{2}\overset{N_{c}=3}{=}24/\left( 11\ln \mu /\Lambda _{%
\text{YM}}\right) $, the solutions $\bar{\xi}$ of Eq. (\[geq2\]) depend on two variational parameters, the RG scale $\mu \geq 0$ and $c_{1}<1$. The critical line $\mu _{c}\left( c_{1}\right) $, i.e. the parameter subspace where the (dis-)order parameter $\bar{\xi}\left( \mu _{c}
\left( c_{1}\right) ,c_{1}\right) $ vanishes and the phase transition takes place, can be found analytically as the combination of the two curves $$\frac{\mu _{c,1,2}\left( c_{1}\right) }{\Lambda _{\text{YM}}}=\exp %
\left[ \frac{48}{11}\frac{\left( 1-c_{1}\right) \left[ 1-\tilde{\imath}%
\left( c_{1}\right) \right] \left[ 1+\left( 1-c_{1}\right) \tilde{\imath}%
\left( c_{1}\right) \right] }{\left( 1-c_{1}\right) \tilde{\imath}\left(
c_{1}\right) \pm \sqrt{5\tilde{\imath}^{2}\left( c_{1}\right) \left(
1-c_{1}\right) ^{2}-4\left( 1-c_{1}\right) \left[ 1-c_{1}\tilde{\imath}%
\left( c_{1}\right) \right] }}\right] \label{mu}$$($\tilde{\imath}\left( c_{1}\right) :=\mathrm{arctanh}\sqrt{c_{1}}/\sqrt{%
c_{1}}$). We plot this closed phase boundary in Fig. \[cl\]. It limits the parameter ranges to $0.5\lesssim \frac{\mu _{\text{c}}}{\Lambda _{\text{YM}}}%
\lesssim 8.86$ and $-0.48\lesssim c_{1}<1\ $and thus prevents the minimal-energy solution $\bar{\xi}^{\ast }$ from attaining unacceptably large values of $\mu $ and $\left\vert c_{1}\right\vert $. Nonzero solutions of the gap equation exist only when the gauge coupling exceeds a critical value, i.e. for $g^{2}\left( \mu \right) >g_{\text{c}}^{2}\left(
c_{1}\right) ,$ as expected on physical grounds. The (dis-)order parameter goes to zero continuously, furthermore, i.e. the disorder-order transition is of second order (which may be an artefact of the mean-field approximation [@for10]).
Vacuum energy density {#vend}
---------------------
Working with the Poincaré-invariant trial states (\[ginvvwf\]) and taking only one-loop corrections from the hard modes into acccount, it is sufficient to regularize Eq. (\[hexp\]) by a momentum cutoff $\Lambda _{%
\text{UV}}$ [@kog95]. Separating the complete vacuum energy density $%
\varepsilon =E/V=\left\langle \mathcal{H}_{\text{YM}}\right\rangle $ into hard and soft contributions,$$\varepsilon \left( \mu ,c_{1},\zeta ;\bar{\xi}\right) =\left\langle \mathcal{%
H}_{\text{YM}}\right\rangle =\varepsilon _{>}\left( \mu \right) +\varepsilon
_{<}\left( \mu ,c_{1},\zeta ;\bar{\xi}\right) , \label{etot}$$($\zeta \equiv m_{g}/\mu $) the cutoff dependence resides solely in$$\varepsilon _{>}\left( \mu \right) =\frac{N_{c}^{2}-1}{8\pi ^{2}}\left(
\Lambda _{\text{UV}}^{4}-\mu ^{4}\right) . \label{epsh}$$As expected, this is the (regularized) zero-point energy density of two transverse, *massless* vector modes in the adjoint representation of SU($N_{c}$) with energy $\omega \left( k\right)
=k$. Simple normal-ordering thus subtracts the $\Lambda _{\text{UV}}$ dependent term. For $c_{i\geq 2}=0$ one then finds the total energy density $%
\bar{\varepsilon}\left( \mu ,c_{1}\right) :=$ $\varepsilon \left( \mu
,c_{1},\zeta _{\text{ct}}\left( c_{1}\right) ;\bar{\xi}\left( \mu
,c_{1}\right) \right) $ in the disordered phase as$$\begin{aligned}
\bar{\varepsilon}\left( \mu ,c_{1}\right) =& -\frac{N_{c}^{2}}{4\pi ^{2}}\mu
^{4}\left[ \frac{4c_{1}^{3}+10c_{1}^{2}-50c_{1}+30}{30c_{1}^{2}\left(
1-c_{1}\right) }-\frac{1-c_{1}}{c_{1}^{2}}\frac{\mathrm{arctanh}\sqrt{c_{1}}%
}{\sqrt{c_{1}}}\right. \notag \\
& +\left. \frac{\tilde{\imath}_{2}-2c_{1}\tilde{\imath}_{3}+c_{1}^{2}\tilde{%
\imath}_{4}+2\gamma c_{1}\left( 1-c_{1}\right) \tilde{\imath}_{2}\left(
\tilde{j}_{3}-2c_{1}\tilde{j}_{4}+c_{1}^{2}\tilde{j}_{5}\right) }{1-c_{1}}%
\right] \label{e}\end{aligned}$$where the integrals $\tilde{\imath}_{n}\left( \xi ,c_{1}\right) ,$ $\tilde{j}%
_{n}\left( \xi ,c_{1}\right) $ are defined in Ref. [@for10] and evaluated at $\bar{\xi}\left( \mu ,c_{1}\right) $. This energy density is plotted in Fig. \[ed\].
![The energy density $\bar{\protect\varepsilon}\left(\protect\mu%
,c_{1}\right)$ of the vacuum field solution $\bar{\protect\xi}\left(\protect%
\mu,c_{1}\right) $ in the disordered phase. (The plot shows the parameter ranges $\protect\mu\in\left\{4,9\right\} \Lambda_{\text{YM}}$ and $c_{1}\in\left\{-0.5,0.8\right\}.$) Note the minimum of the energy surface at $c_{1}\simeq0.15$.[]{data-label="ed"}](endens38.eps){width=".6\textwidth"}
In the ordered phase, i.e. for $\mu \ggg \Lambda _{\text{YM}}$ where $%
g^{2}\left( \mu \right) \ll 1$, the energy density can be calculated perturbatively (in $g^{2}$). Since fluctuations $\varphi _{<}^{a}$ around $%
U_{<}\sim 1$ are small in this phase, one may approximate $U_{<}=\exp \left(
ig\varphi _{<}^{a}\lambda ^{a}\right) =1+ig\varphi _{<}^{a}\lambda
^{a}+O\left( g^{2}\right) .$ After adding the hard-mode contribution ([epsh]{}) and discarding the zero-point contribution, this results in $$\varepsilon \left( \mu ,c_{1}\right) =\frac{N_{c}^{2}-1}{4\pi ^{2}}\mu ^{4}%
\frac{1-c_{1}}{c_{1}^{2}}\left[ -\frac{c_{1}^{3}+15c_{1}^{2}-50c_{1}+30}{%
30\left( 1-c_{1}\right) ^{2}}+\frac{\mathrm{arctanh}\sqrt{c_{1}}}{\sqrt{c_{1}%
}}\right] .$$It is reasonable to expect that this perturbative result remains qualitatively reliable down to the phase transition at $\mu _{c}$ [@kog95]. The singularity of the energy density at $c_{1}\rightarrow 1$ encodes the vacuum instability for $c_{1}\geq 1$ and thus automatically ensures that the wave functional remains normalizable during the variational analysis.
The most important lesson of the above analysis is that $\varepsilon \left(
\mu ,c_{1}\right) $ *increases* monotonically with $\mu $ and $c_{1}$ (for $-2<c_{1}<1$) in the ordered phase while the energy density (\[e\]) in the strongly-coupled disordered phase monotonically *decreases* with $\mu $ and $c_{1}$, up to the phase transition. This indicates that the vacuum energy density becomes minimal at the phase boundary in the disordered phase, i.e. at $\bar{\xi}=0_{+}$ (where the number of massless particles becomes maximal [@for10]). The precise minimum, $\bar{%
\varepsilon}\left( \mu ^{\ast },c_{1}^{\ast }\right) \simeq -210.59\Lambda _{%
\text{YM}}^{4},$ is reached at $c_{1}^{\ast }\simeq 0.15$ with $\mu ^{\ast
}=\mu _{c}\left( c_{1}^{\ast }\right) =8.61\Lambda _{\text{YM}}.$ These values justify the perturbative treatment of the hard modes and of the $4U$ contributions. The $c_{1}$ corrections reduce the vacuum energy density by about 11% and provide a rather substantial improvement of the wave functional.
Gluon condensate and quasigluon kinetic mass
--------------------------------------------
At the physical parameter values, i.e. at the border of the disordered phase where the energy is minimal, the gluon condensate becomes $$\left\langle F^{2}\right\rangle =-\frac{N_{c}^{2}-1}{\pi ^{2}}\mu ^{4}\left[
\frac{7c_{1}^{3}-20\gamma ^{\ast }c_{1}^{3}+15c_{1}^{2}+20\gamma ^{\ast
}c_{1}^{2}-50c_{1}+30}{30c_{1}^{2}\left( 1-c_{1}\right) }-\left( \frac{%
1-c_{1}}{c_{1}^{2}}+\frac{2\gamma ^{\ast }}{3}\right) \frac{\mathrm{arctanh}%
\sqrt{c_{1}}}{\sqrt{c_{1}}}\right] \label{f}$$($\gamma ^{\ast }=g^{2}\left( \mu ^{\ast }\right) N_{c}/\pi ^{2}\simeq 1.012$). Numerically, this implies $$\left\langle \frac{\alpha }{\pi }F^{2}\right\rangle =20.87\Lambda _{\text{YM}%
}^{4}\simeq 0.011\text{ GeV}^{4} \label{gc}$$(for $\Lambda _{\text{YM}}\simeq 0.15$ GeV), i.e. an about 25% larger value than in the uncorrected $c_{1}=0$ case. The result (\[gc\]) lies comfortably within the standard range $\left\langle \left( \alpha /\pi
\right) F^{2}\right\rangle =0.0080-0.024$ GeV$^{4}$ obtained from QCD sum rules [@for05].
Our finite and positive result for $c_{1}$ has further interesting consequences since it reshapes the composition and dispersion of the vacuum field population. Indeed, the attractive IR interactions generated by $%
c_{1}>0$ deplete the density of ultralong-wavelength $k\rightarrow 0$ modes and populate the $k\sim \mu $ modes more strongly. This is consistent with the expected average wavelength $\lambda \sim \Lambda _{\text{YM}}^{-1}$ of the vacuum fields. Since $G^{-1}\left( k\right) $ describes the dispersion relation $\omega \left( k\right) $ of quasigluon modes in the vacuum, furthermore, one may relate $c_{1}$ to the modulus of the dimensionless quasigluon group velocity $\vec{v}\left( \vec{k}\right) =\partial G_{<}^{-1}\left( \vec{k}\right)
/\partial \vec{k}$ at $k=\mu $ [@for10],$$\left\vert c_{1}\right\vert =\frac{v\left( \mu \right) }{v\left( \mu \right)
+2}.$$For $0>c_{1}>1$ (as in our case), furthermore, the effective *kinetic* gluon mass $\overline{m}_{\text{g%
}}$, which relates velocity and momentum as $\vec{k}=\overline{m}_{\text{g}}%
\vec{v}$, is negative. Hence $\vec{v}$ is opposite to the momentum, causing the quasigluons in the vacuum to decelerate when an external force is applied. (Such dispersions are encountered in several condensed-matter systems and are in stark contrast to the behavior of free gluons.) Hence quasigluons (with their small scattering amplitudes) may show a negative differential color resistance.
Infrared degrees of freedom {#IRSPs}
===========================
Our above representation of the vacuum dynamics in terms of the *gauge-invariant* low-energy fields $U_{<}$ provides the opportunity to search for specific $U_{<}$ which may play a particularly important or even dominant role in the generating functional (and hence universally in all low-energy amplitudes) [@cor11]. If such fields exist, they can be regarded as universal infrared degrees of freedom (IRdofs). In contrast to other proposed IRdof candidates (e.g. classical gauge-field solutions like instantons [@sch98], or monopole and vortex configurations), the IRdofs expressed in terms of $U_{<}$ are gauge invariant and contain crucial quantum effects (e.g. those which stabilize the instanton size, see below). From a practical perspective, these IRdofs will be useful as well since many technical problems encountered when dealing with gauge-dependent fields are avoided from the outset. Below we will show that large classes of such IRdofs indeed exist and review how their stability and topology emerges. We then construct important IRdof classes explicitly and discuss their properties and physical interpretation.
Gauge-invariant saddle point expansion {#spex}
--------------------------------------
We start from the vacuum overlap matrix element, i.e. the functional integral $$Z=\int DU_{<}\exp \left( -\Gamma \left[ U_{<}\right] \right) \label{zsoft}$$over the soft modes, with the action $\Gamma $ given by Eq. (\[gsm0\]).(Sources can be included when needed.) A steepest descent approximation for $Z$ can be set up by expanding $U_{<}$ around the saddle point fields $%
\bar{U}_{i}\left( \vec{x}\right) $, i.e. the local minima of the soft-mode action (\[gsm0\]) which solve$$\left. \frac{\delta \Gamma \left[ U_{<}\right] }{\delta U_{<}\left( \vec{x}%
\right) }\right\vert _{U_{<}=\bar{U}_{i}^{\left( Q\right) }}=0.
\label{spaeq}$$(Different topological charges (see below) are summarily denoted by $Q$ since the action is varied in each topological sector separately.) To leading order, the saddle point expansion for $Z$ is then a weighted sum (or integral – the symbolic label $i$ becomes continuous when the saddle points form continuous families) over the contributions from all relevant solutions $\bar{U}_{i}^{\left( Q\right) }$, $$Z\simeq \sum_{Q\in Z,i}F_{i}\left[ \bar{U}_{i}^{\left( Q\right) }\right]
\exp \left( -\Gamma \left[ \bar{U}_{i}^{\left( Q\right) }\right] \right) ,
\label{zspa}$$where nontrivial pre-exponential factors $F_{i}$ are typically generated by zero-mode contributions.
For the general analysis and explicit solution of Eq. (\[spaeq\]) we adopt the parametrization $U_{<}\left( \vec{x}\right) =\exp \left[ \phi \left(
\vec{x}\right) \hat{n}^{a}\left( \vec{x}\right) \lambda ^{a}/\left(
2i\right) \right] $ of the SU$\left( N_{c}\right) $ elements and work directly with the $N_{c}^{2}-1$ independent degrees of freedom of $U_{<}$, i.e. the unit vector field $\hat{n}^{a}$ and the spin-0 field $\phi $. For simplicity, we will also specialize to $N_{c}=2$ and use the first two terms in the expansion (\[gm1\]) of the inverse finite-mass gluon propagator $%
G^{-1}\left( k\right) =\sqrt{k^{2}+\mu ^{2}}$ as a template for the covariance [@for06]. The soft-mode Lagrangian can can then be written as a sum of two- and four-derivative terms, $$\mathcal{L}\left( U_{<}\right) =\mathcal{L}_{2d}\left( \phi ,\hat{n}\right) +%
\mathcal{L}_{4d}\left( \phi ,\hat{n}\right) . \label{l24d}$$(For the explicit expressions see Ref. [@for06].) The saddle point equation (\[spaeq\]), when specialized to variations with respect to $%
\phi $ and $\hat{n}^{a}$, becomes a system of four nonlinear partial differential equations. Its localized solutions can be shown to be stable under scale transformations, due to the virial theorem $\Gamma _{2d}\left(
1\right) =\Gamma _{4d}\left( 1\right) $ [@for06]. (Clearly the four-derivative term $\Gamma _{4d}$ is crucial here – truncation of the gradient expansion (\[gm1\]) to two powers of $\partial U_{<}/\mu $ is therefore the minimal approximation which supports stable saddle points.)The origin of this stability can be traced to the mass scale $\mu $ emerging from the out-integrated short-wavelength quantum fluctuations.
An already mentioned, crucial benefit of the gauge-projected wave functionals (\[ginvvwf\]) is that they fully implement the nontrivial topology of the gauge group and fields. The $U_{<}$ fields thereby inherit three integer topological quantum numbers [@for06]: a winding number $Q%
\left[ U_{<}\right] $ (characterizing the homotopy group $\pi _{3}\left(
S^{3}\right) =Z$), a monopole-type degree $q_{m}\left[ \hat{n}\right] $ based on $\pi _{2}\left( S^{2}\right) =Z$ and finally a linking number $q_{H}%
\left[ \hat{n}\right] $ in the Hopf bundle $\pi _{3}\left( S^{2}\right) =Z$ which classifies knot solutions. This topology entails two lower action bounds [@for06] of Bogomol’nyi type,$$\Gamma \left[ U_{<}\right] \geq \frac{12\pi ^{2}}{g^{2}\left( \mu \right) }%
\left\vert Q\left[ U_{<}\right] \right\vert ,\text{\ \ \ \ \ }\Gamma \left[
\phi _{k}=\left( 2k+1\right) \pi ,\hat{n}\right] \geq \frac{%
2^{9/2}3^{3/8}\pi ^{2}}{g^{2}\left( \mu \right) }\left\vert q_{H}\left[ \hat{%
n}\right] \right\vert ^{3/4}, \label{bb}$$which ensure that contributions to soft amplitudes from saddle points in high charge sectors can generally be neglected. This allows for practicable truncations of the saddle-point expansion. (Saturation of the first bound requires the fields to solve the Bogomol’nyi-type equation $\partial
_{i}L_{j}=\mp \mu \varepsilon _{ijk}L_{k}$, incidentally, which can be considered as the analog of the self-(anti)-duality equation in Yang-Mills theory.)
Important examples of gauge-invariant infrared degrees of freedom
-----------------------------------------------------------------
In general, the saddle-point solutions have to be found numerically. Among the exceptions are the translationally invariant vacuum solutions $%
U_{c}=const.$ (which are the absolute action minima $\Gamma \left[ U_{c}%
\right] =0$) and several nontrivial solution classes which can be found analytically. In addition, important and sufficiently symmetric solutions classes can often be obtained by solving substantially simplified field equations [@for06]. (The typically smaller action values of solutions with higher symmetry generate a stronger impact on the matrix elements, furthermore.)
As an example for nontrivial analytical solutions, we consider $U_{<}$ fields with constant $\hat{n}^{a}$ for which the saddle point equation becomes linear: $\partial ^{2}\left( \partial ^{2}\phi -2\mu ^{2}\phi
\right) =0.$ The general solution does not carry any topological charge and was found in Ref. [@for06]. The subset of spherically symmetric solutions with finite action, in particular, is$$\bar{\phi}^{\left( \hat{n}=c\right) }\left( r\right) =c_{1}+\frac{c_{2}}{%
\sqrt{2}\mu r}\left( 1-e^{-\sqrt{2}\mu r}\right) \label{phineqc}$$with the action $\Gamma \left[ \phi ^{\left( \hat{n}=c\right) },\hat{n}_{c}%
\right] =c_{2}^{2}\pi /\left( \sqrt{2}g^{2}\left( \mu \right) \right) $. Since Eq. (\[phineqc\]) is not subject to topological bounds, it continuously turns into one of the vacuum solutions for $c_{2}\rightarrow 0$.
A particularly important saddle-point solution class consists of topological solitons of hedgehog type, $$\hat{n}^{a}\left( \vec{x}\right) =\hat{x}^{a},\text{ \ \ \ \ \ }\phi \left(
\vec{x}\right) =\phi ^{\left( hh\right) }\left( r\right) \label{hh}$$($\hat{x}^{a}\equiv \vec{x}/r$, $r\equiv \left\vert \vec{x}\right\vert $). Well-defined hedgehog fields must satisfy the boundary condition $\phi
^{\left( hh\right) }\left( 0\right) =2k_{1}\pi $ (regularity at the origin further requires $\phi ^{\prime \prime }\left( 0\right) =0$) and finite-action fields additionally have $\phi ^{\left( hh\right) }\left(
\infty \right) =2k_{2}\pi $ where $k_{1,2}$ and the charge $Q=k_{1}-k_{2}$ are integers. The more general boundary conditions $$\phi ^{\left( hh\right) }\left( 0\right) =n\pi ,\text{ \ \ \ \ }\phi
^{\left( hh\right) }\left( \infty \right) =m\pi ,\text{ \ \ \ \ }Q\left[
\phi ^{\left( hh\right) }\right] =\frac{n-m}{2} \label{q-hh}$$($n,m$ integer) additionally admit infinite-action solutions with half-integer winding numbers $Q$ (for either $m$ or $n$ odd). All hedgehog fields further carry the monopole-type charge $q_{m}^{\left( hh\right) }:=q_{m}\left[ \hat{x%
}\right] =\pm 1$. Due to the periodicity in $\phi $, it is sufficient to consider boundary values in the range $\phi \left( 0\right) \in \left]
0,2\pi \right] $. The dynamics of $\phi \left( r\right) $ is governed by the radial Lagrangian $$\mathcal{L}^{\left( hh\right) }\left( r\right) =\frac{\pi }{g^{2}\left( \mu
\right) \mu }\left[ \frac{1}{2}\left( r\phi ^{\prime \prime }\right)
^{2}+\left( 3+\mu ^{2}r^{2}\right) \left( \phi ^{\prime }\right) ^{2}+4\mu
^{2}\left( 1-\cos \phi \right) \right] . \label{lrad}$$The hedgehog saddle points, found numerically in Ref. [@for06], turn out to comprise mainly contributions from and around the gauge orbits of the classical Yang-Mills solutions, i.e. (multi-) instantons and (multi-) merons. The potential term in Eq. (\[lrad\]) is analogous to that of a one-dimensional pendulum in a gravitational field, with stable (unstable) equilibrium positions at $\phi =\pi $ ($\phi =0$), modulo multiples of $2\pi
$.
![The 1-instanton class solution. The dashed line corresponds to the Yang-Mills instanton.[]{data-label="iclass"}](Inst3.eps){width=".35\textwidth"}
We first discuss the regular hedgehog solutions. Their three boundary conditions $\phi \left( 0\right) =2\pi ,$ $\phi ^{\prime \prime }\left(
0\right) =0$ and $\phi \left( \infty \right) =2\pi \left( 1-Q\right) $ imply that for a given $Q$ all of them can be found by varying the initial slope $%
\beta :=\phi ^{\prime }\left( 0\right) $. (For the irregular solutions with $%
\phi \left( 0\right) =\pi $ see Ref. [@for06].) The regular solutions turn out to contain one *finite-action* solution for each $Q$, denoted as the $\left\vert Q\right\vert $ (anti) instanton class, and the remaining, continuous (in $\beta $) infinite-action families, the $2\left\vert Q\right\vert $ (anti) meron classes. The 1-instanton class solution is depicted in Fig. \[iclass\]. Its relative gauge orientation $%
U=U_{-}^{-1}U_{+}$ is even quantitatively close to that of the Yang-Mills *instanton* [@sch98] (orbit) (see dashed curve in Fig. \[iclass\]). This confirms that the $\left\vert Q\right\vert $ instanton classes indeed primarily summarize Yang-Mills instanton contributions. However, they also contain crucial, dilatation-breaking quantum corrections which dynamically stabilize the instanton size at about $\rho \simeq 2\mu ^{-1}$, compatible with instanton liquid model [@sch98] and lattice [@mic95] results, and thus overcome the chronic IR instabilities of classical Yang-Mills instanton gases. Since instanton effects play important roles in Yang-Mills theory (e.g. in the $\theta $ vacuum [@cal78] and in spin-0 glueball physics [@for05; @sch98; @for08]), it is crucial that they are (at least partly) included in the vacuum functional (\[ginvvwf\]). In fact, approximate non-hedgehog solutions corresponding dominantly to ensembles of instantons and anti-instantons should also exist and play prominent roles (since they would be enhanced by a large entropy, as in phenomenologically successful instanton liquid models [@sch98]).
All remaining regular (i.e. $\phi \left( 0\right) =2\pi $) hedgehog solutions, with initial slopes $\beta $ between the discrete instanton-class values $\beta _{I,Q}$, form the $2\left\vert Q\right\vert $ (anti-) meron classes. Those approach one of the values $\phi _{M}\left( \infty \right) =\left(
2k+1\right) \pi $ at spacial infinity and therefore have infinite action, as the Yang-Mills merons [@cal78]. Moreover, solutions with $\phi \left(
\infty \right) =\left( 2k+1\right) \pi $, corresponding to an odd number of merons, carry the half-integer topological charge $Q$ of their Yang-Mills meron counterparts. In addition, quantum effects ensure that our meron-class solutions acquire a finite size and therefore remain nonsingular. Since the $%
2\left\vert Q\right\vert $-meron classes appear in continuous families (parametrized by their size $\beta
^{-1} $), furthermore, their large entropy will help to overcome their infinite-action suppression in functional integrals. As in Yang-Mills theory, merons could then play a physical role, e.g. in the confinement mechanism [@cal78].
We conclude our discussion of selected saddle-point solutions with one of the most intriguing classes, consisting of (solitonic) links and knots. Those emerge from a generalization of Faddeev-Niemi theory [@fad70],$$\mathcal{L}^{\left( \phi _{k}\right) }\left( \vec{x}\right) =\frac{\mu }{%
g^{2}\left( \mu \right) }\left[ \left( \partial _{i}\hat{n}^{a}\right) ^{2}+%
\frac{1}{\mu ^{2}}\left( \varepsilon ^{abc}\partial _{i}\hat{n}^{b}\partial
_{j}\hat{n}^{c}\right) ^{2}+\frac{1}{2\mu ^{2}}\left( \varepsilon ^{abc}\hat{%
n}^{b}\partial ^{2}\hat{n}^{c}\right) ^{2}\right] , \label{ln}$$which is included in our soft-mode Lagrangian for constant $\phi _{k}=\left(
2k+1\right) \pi $. The corresponding solution classes $U_{<}\left( \hat{n}%
\right) $ describe twists, linked loops and knots made of closed color fluxtubes. Since Eq. (\[ln\]) follows uniquely from the VWF (\[ginvvwf\]) and the Yang-Mills dynamics, our approach provides a new framework and physical interpretation for such solutions. In fact, they remerge as gauge-invariant IR degrees of freedom representing sets of gauge-field orbits with a collective Hopf charge. While the $\hat{n}$ field of the Faddeev-Niemi model is interpreted as a gauge-dependent local color direction in the vacuum, in particular, our $\hat{n}$ is manifestly gauge-invariant. This may put the tentative interpretation of such knot solutions as glueballs on a firmer basis.
Summary and conclusions
=======================
We have studied gauge-invariant wave functionals for the Yang-Mills vacuum which incorporate asymptotic freedom and an *a priori* general dispersion for the infrared gluons in Gaussian core functionals. In this at present probably richest analytically manageable and gauge-invariant trial functional basis, we have then variationally determined several vacuum properties. Dimensional transmutation, dynamical mass generation and gluon condensation emerge transparently and generate mass scales consistent with other approaches. In addition, the improved vacuum description in the infrared predicts a negative *kinetic* mass of the soft gauge-field modes and thus suggests a negative differential color resistance of the Yang-Mills vacuum.
Another benefit of the gauge-invariant framework is that the dynamics can be reformulated as an effective theory which represents sets of gluon orbits as gauge-invariant matrix fields subject to higher-gradient interactions. In this effective theory we have set up a saddle-point expansion to determine the collective fields with maximal impact on functional integrals. These saddle points play the role of gauge-invariant infrared degrees of freedom. They are stabilized by the dynamical mass generation mechanism and inherit a rich topological structure (three topological charges and action bounds of Bogomol’nyi type) from the Yang-Mills gauge group. Moreover, they provide the principal input for a systematic saddle-point expansion of soft amplitudes (such as glueball correlators).
Several of the more symmetric and important saddle-point solution classes have been found explicitly. Among them are topological solitons related to the classical Yang-Mills (multi-) pseudoparticle solutions which mediate tunneling processes in the vacuum. Those generate a gauge-invariant representation of instanton and meron effects which includes quantum fluctuations. The latter stabilize the pseudoparticle sizes dynamically, in particular, and thereby cure the notorious infrared deseases encountered in dilute instanton ensembles. Similar solutions with other types of topological charges exist as well but seem to have no obvious counterparts in classical Yang-Mills theory. One of the most intriguing solution classes, finally, consists of solitonic links and knots. Those emerge from a generalization of Faddeev-Niemi theory which turns out to be embedded in our soft-mode dynamics. Hence in our framework the knot solutions find a new and in particular gauge-invariant physical interpretation, potentially related to glueballs.
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For an alternative approach see J.M. Cornwall, J. Papavassiliou and D. Binosi, *The Pinch Technique and its Applications to Non-Abelian Gauge Theories*, Cambridge University Press, Cambridge (UK) (2011).
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We propose and investigate an optical scheme for probabilistic implementation of an arbitrary single-mode quantum operation that can be expressed as a function of photon number operator. The scheme coherently combines multiple photon addition and subtraction and is feasible with current technology. As concrete examples, we demonstrate that the device can perform approximate noiseless linear amplification of light and can emulate Kerr nonlinearity.'
author:
- Jaromír Fiurášek
title: Engineering quantum operations on traveling light beams by multiple photon addition and subtraction
---
Introduction
============
Quantum properties of light have attracted the attention of scientists since the early days of quantum mechanics. Arguably one of the main goals in the field of quantum optics is to realize highly nonlinear interactions at the few-photon level which would enable *e.g.* to generate various highly nonclassical states of light and implement advanced schemes for quantum information processing. Unfortunately, nonlinear coupling between single photons mediated by common material media is extremely weak so other approaches have to be pursued. One of the most promising techniques appears to be that of the measurement induced nonlinearities. As shown by Knill, Laflamme, and Milburn [@KLM01], using passive linear optics, ancilla single photons and photon counting measurements one can emulate nonlinear coupling between single photons and implement all-optical quantum CNOT gate. During recent years, the feasibility of this approach has been corroborated by numerous experiments [@KokRMPreview]. Moreover, these ideas have been extended also to the so-called continuous variable regime, where quantum information is encoded into states of optical modes instead of single photons. It was shown that linear canonical transformations of quadrature operators such as squeezing or quantum non-demolition coupling can be implemented with passive linear optics, ancilla squeezed vacuum states, homodyne detection and feedforward [@Filip05; @Yoshikawa07; @Yoshikawa08]. Going beyond Gaussian operations, the basic techniques available are the addition [@Zavatta04] and subtraction [@Ourjoumtsev06] of a single photon. These operations enable to generate highly non-classical superpositions of coherent states [@Neergaard-Nielsen06; @Wakui07; @Ourjoumtsev09], and distill and concentrate continuous-variable entanglement [@Opatrny00; @Furusawa09]. If these operations are combined with coherent displacement, it is possible to prepare an arbitrary single-mode state of light from initial vacuum or squeezed state [@Dakna99; @Fiurasek05].
As both single-photon subtraction and addition have been successfully demonstrated experimentally, it is interesting to investigate whether these elementary transformations can be combined to engineer some more complex useful quantum operations on the states of traveling light beams. In this paper we propose and analyze scheme for approximate probabilistic realization of an arbitrary operation that can be expressed as a function of photon number operator $\hat{n}$. This class of transformations includes for instance the Kerr nonlinearity described by a unitary operation $\hat{U}=\exp(-i\phi \hat{n}^2)$, or a noiseless linear amplifier [@Ralph08; @Xiang09; @Marek09] $\hat{Z}=g^{\hat{n}}$, where $g >1$ is the amplification gain. Our proposal is inspired by Ref. [@Kim08], where a scheme that coherently combines photon subtraction and addition has been devised for the purpose of direct verification of the bosonic commutation relations for creation and annihilation operators. We show that this scheme can be generalized to realize arbitrary operation $f(\hat{n})$. An essential advantage of our approach is that we preserve the relative simplicity of the setup discussed in Ref. [@Kim08]. The scheme involves only a single Mach-Zehnder interferometer and a single nonlinear crystal where parametric down-conversion occurs. The precision of the approximation of a given operation $f(\hat{n})$ is controlled by the number of photons $N$ counted by two photo-detection blocks that form a part of the setup. In contrast, other proposals for emulation of operations $f(\hat{n})$ require either serial [@Clausen03] or parallel [@Ralph08; @Xiang09] implementation of many basic building blocks. Our approach is thus appealing from the experimental point of view, especially given that the core scheme has already been successfully realized experimentally by Zavatta, Bellini and coworkers [@Parigi07; @Zavatta09].
The rest of the present paper is organized as follows. In Sec. II we describe the proposed scheme and show how to determine its parameters for a given target operation $f(\hat{n})$. In Sec. III we consider approximate realization of a noiseless amplification. In. Sec. IV we study the emulation of Kerr nonlinearity. Finally, conclusions are drawn in Sec. V.
Setup description
=================
The scheme for probabilistic implementation of an arbitrary transformation $f(\hat{n})$ that can be expressed as a function of photon number operator $\hat{n}$ is depicted in Fig. 1. As already mentioned in the introduction, this scheme is very similar to the setup proposed in Ref. [@Kim08] for experimental testing of the bosonic commutation relation for annihilation and creation operators, $[\hat{a},\hat{a}^\dagger]=1$. Here we consider a generalized version of that setup where $N$ photons are added to and also $N$ photons are subtracted from the input state.
{width="0.98\linewidth"}
The photon addition is achieved by feeding the state into the input signal port of a nonlinear crystal NLC where pairs of correlated signal and idler photons are generated in the process of noncollinear parametric down-conversion pumped by a strong coherent laser beam P [@Zavatta04]. The photodetector D$_1$ counts the number of emitted idler photons $N$ which is equal to the number of photons added into the signal mode. Photon counting can be achieved e.g. by employing the time-multiplexed photon number resolving detector [@Achilles03; @Fitch03; @Micuda08] where the input pulse is divided into a sequence of many pulses by highly unbalanced interferometers. The time-separated pulses then impinge onto avalanche photodiodes (APD) and the total number of clicks of APDs gives the number of counted photons N. The main limitations of this detection scheme are the finite number of time bins which does not allow for complete photon number resolution, and the non-unit detection efficiency of the APDs. However, these limitations are not very restrictive for the present application where the down-conversion should be operated in a regime of low gain. The probability that $N+1$ photons are present in the output idler beam is then much lower than the probability that $N$ photons are present. If $N$ photons are counted by D$_1$ then, with high probability, $N$ photons were indeed emitted into the idler mode in NLC. In this regime, the main effect of the less-than-unit detection efficiency of APDs is a reduction of success probability of the scheme.
Mathematically, the state transformation after propagation through the crystal is described by the two-mode squeezing operator, $$\hat{V}=e^{\lambda \hat{a}^\dagger \hat{c}^\dagger}(1-\lambda^2)^{\frac{1}{2}(\hat{n}_A+\hat{n}_C+1)}e^{-\lambda \hat{a} \hat{c}}.
\label{Usqueezing}$$ Here $\hat{a}$ and $\hat{c}$ denote the annihilation operators of signal and idler modes, respectively, and $\lambda=\tanh s$ where $s$ is the squeezing constant. Assuming that the idler mode is initially in the vacuum state and that the detector D$_1$ registers $N$ photons, the conditional transformation of the state of signal mode reads $$\hat{A}_{N}=\frac{\lambda^N}{\sqrt{N!}} \left(1-\lambda^2\right)^{(1+\hat{n}-N)/2}\hat{a}^{\dagger N}.
\label{AN}$$ In the limit of low parametric gain, $\lambda \rightarrow 0$, we obtain $\hat{A}_N \propto \hat{a}^{\dagger N}$ as desired.
The photon subtraction is performed by splitting a tiny part of the signal beam off a highly unbalanced beam splitter followed by detection of the number of reflected photons [@Ourjoumtsev06; @Neergaard-Nielsen06; @Wakui07]. If $k$ photons are reflected from a beam splitter with amplitude transmittance $t$ and reflectance $r$ then the conditional transformation of the signal-mode state can be expressed as follows, $$S_{k}=\frac{r^k}{\sqrt{k!}} t^{\hat{n}}\hat{a}^k.$$ A crucial feature of the scheme in Fig. 1 is that the photon subtraction may occur either before or after the photon addition. The photons reflected off beam splitters BS$_1$ and BS$_2$ are recombined on a polarizing beam splitter PBS such that their polarization states are orthogonal but all other degrees of freedom are made indistinguishable to guarantee maximum visibility of multiphoton interference. The detector D$_2$ represents a detection block that is capable of making projection onto an $N$-photon polarization state $$|\phi\rangle_{B}=\sum_{k=0}^N b_k |k\rangle_{B,H}|N-k\rangle_{B,V},
\label{phiB}$$ where H and V denote the horizontally and vertically linearly polarized modes, respectively. Such a projection can be performed e.g. by reversing a linear optical scheme for preparation of the $N$-photon two-mode states put forward in Ref. [@Fiurasek02]. The input light beam is divided into $N$ spatial modes and a polarization analysis is performed on each mode by an elementary detection block consisting of quarter- and half-wave plates, polarizing beam splitter and a pair of single photon detectors. If a single photon is detected by each detection block, and a photon in the $j$-th detection block is projected onto a polarization state $$\cos\theta_j |H\rangle_B +\sin\theta_j e^{i\phi_j}|V\rangle_B,$$ then the whole detector projects onto $N$-photon polarization state [@Fiurasek02; @Zou04] $$\prod_{j=1}^N \left(\cos\theta_j \hat{b}_H^\dagger+\sin\theta_j e^{i\phi_j} \hat{b}_V^\dagger\right) |0\rangle_{B,H}|0\rangle_{B,V}.
\label{phifactorized}$$ Here $\hat{b}_H^\dagger$, $\hat{b}_V^\dagger$ denote creation operators of the horizontally and vertically polarized modes, respectively. Since every $N$-photon polarization state can be expressed in the factorized form (\[phifactorized\]), this detection scheme can project onto any state (\[phiB\]).
If $N$ photons in total are subtracted by BS$_1$ and BS$_2$ and if addition of N photons is heralded by the detector D$_1$, then the operation on signal mode A and the $N$-photon polarization state of auxiliary spatial mode B are given by, $$\sum_{k=0}^N \hat{S}_{k}\hat{A}_{N}\hat{S}_{N-k}\otimes |k\rangle_{B,H}|N-k\rangle_{B,V}.
\label{SAS}$$ With a slight abuse of notation, we use in Eq. (\[SAS\]) the tensor product symbol $\otimes$ to separate operator acting on the Hilbert space of the signal mode and the $N$-photon state prepared in the auxiliary mode B. After projecting the mode B onto the N-photon state (\[phiB\]) we finally obtain the operation on signal mode, $$\begin{aligned}
\hat{W}_N =C_N h^{\hat{n}}
\sum_{k=0}^N b_k \sqrt{{N \choose k}}\left(\frac{h}{t}\right)^{k-N} \hat{a}^k \hat{a}^{\dagger N} \hat{a}^{N-k},
\label{WNformula}\end{aligned}$$ where $C_N=\sqrt{1-\lambda^2}\,(r\lambda)^N/N!$ and $h=t^2\sqrt{1-\lambda^2}$. With the help of the expression $$\begin{aligned}
\hat{a}^k \hat{a}^{\dagger N}\hat{a}^{N-k}&= &\frac{(\hat{n}+k)!}{(\hat{n}+k-N)!}\equiv\prod_{j=k-N+1}^k (\hat{n}+j),
\label{anpolynomial}\end{aligned}$$ we can rewrite $\hat{W}_N$ as follows, $$\hat{W}_N=C_N h^{\hat{n}} P_N(\hat{n}),
\label{WNfinal}$$ where $P_N(\hat{n})$ is a polynomial of $N$-th order in $\hat{n}$, $$P_N(\hat{n})=
\sum_{k=0}^N b_k \sqrt{{N \choose k}}\left(\frac{h}{t}\right)^{k-N} \frac{(\hat{n}+k)!}{(\hat{n}+k-N)!}.
\label{PNpolynomial}$$
We now prove that by properly tailoring the state (\[phiB\]), arbitrary polynomial $P_N(\hat{n})$ can be obtained. We start from Eq. (\[WNformula\]) where we normally order the creation and annihilation operators using the formula, $$\hat{a}^k \hat{a}^{\dagger N}=\sum_{j=0}^k {N \choose j} \frac{k!}{(k-j)!} \hat{a}^{\dagger N-j}\hat{a}^{k-j}.$$ In this way we obtain $$P_N(\hat{n})= \sum_{j=0}^N d_j \hat{a}^{\dagger N-j} a^{N-j},
\label{PNnormal}$$ where $d_j=\sum_{k=j}^N M_{jk} b_k$, and $$M_{jk}= \sqrt{{N \choose k}} {N \choose j} \frac{k!}{(k-j)!} \left(\frac{h}{t}\right)^{k-N}, \quad k\geq j.$$ By setting $M_{jk}=0$, $j>k$, we can express the relation between $d_j$ and $b_k$ in a matrix form, $$\bm{d}=\bm{M} \bm{b},
\label{dMc}$$ where $\bm{b}=(b_0,b_1,\ldots,b_N)^T$, $\bm{d}=(d_0,d_1,\ldots,d_N)^T$. The system of equations (\[dMc\]) for $b_k$ can be always solved. Since $M_{jj}>0$, $\forall j$ and $M_{jk}=0$, $j>k$, we have $\det \bm{M} >0$, $\bm{M}^{-1}$ exists and we can write $\bm{b}=\bm{M}^{-1}\bm{d}$. Any polynomial in $\hat{n}$ can be recast into the form (\[PNnormal\]) by normally ordering the annihilation and creation operators and the coefficients $d_k$ thus unambiguously specify the polynomial. The amplitudes $b_k$ yielding the required $d_j$ can be calculated as described above which completes the engineering of the operator polynomial $P_N(\hat{n})$.
To summarize our findings so far, we have shown that with the scheme shown in Fig. 1 we can implement an arbitrary operator that can be expressed as a product of a polynomial in $\hat{n}$ and an attenuation factor $h^{\hat{n}}$ (recall that $h \leq 1$). The scheme can approximate any given operator $f(\hat{n})$ to arbitrarily high degree $N$ determined by the number of photon subtractions and additions. More specifically, the first $N+1$ terms in Taylor series expansion in $\hat{n}$ of $f(\hat{n})$ and $\hat{W}_N$ can be made equal up to a constant prefactor. However, the success probability of the scheme will decrease approximately exponentially with increasing $N$. Practical implementations of the proposed scheme in a near future would be thus most probably limited to $N=1$ and $N=2$.
In this low-approximation regime the influence of the attenuation factor $h^{\hat{n}}$ could be particularly significant. This effective attenuation can be suppressed by employing highly unbalanced beam splitters, $t\rightarrow 1$, and working in the regime of very low parametric gain, $\lambda\rightarrow 0$. Then we get $\hat{W}_N=C_N P_N(\hat{n})$. However, in this limit the success probability of the scheme becomes very low and vanishes when $t=1$ or $\lambda=0$. In an experiment, optimal balance between success rate and performance of the scheme can be sought by tuning $\lambda$ and $t$.
Noiseless amplifier
===================
As a first application of the scheme let us consider probabilistic implementation of a noiseless linear amplifier [@Ralph08; @Xiang09], $$\hat{Z}(g)=g^{\hat{n}},
\label{Zamplifier}$$ where $g>1$ denotes an amplitude gain of the amplifier. The amplifier (\[Zamplifier\]) acts as a non-unitary quantum filter that enhances amplitudes of the Fock states $|n\rangle$ by a factor $g^n$. Clearly, this transformation cannot be implemented exactly with a finite probability because the operator $\hat{Z}(g)$ is unbounded for any $g>1$. We can nevertheless approximately realize the transformation to $N$-th order in $\hat{n}$ by the scheme described in a previous section. In essence in this approach the amplification works well for low Fock states $|n\rangle$, $n<N$, but it fails for large Fock states $n \gg N$.
For the sake of simplicity, in what follows we shall assume that $h=1$, i.e. the limit of a low reflectance of beam splitters BS$_1$, BS$_2$ and low gain of the nonlinear crystal. The influence of $h<1$ on the performance of the amplifier is that it reduces the effective gain, $g \rightarrow hg$. This can be compensated by substitution $g \rightarrow g/h$ when determining the parameters of the scheme.
{width="0.8\linewidth"}
{width="0.8\linewidth"}
We can construct a polynomial approximation to the operator $\hat{Z}(g)$ by expanding it in Taylor series in $\hat{n}$ and keeping the first $N+1$ terms. In this way we obtain a polynomial $$\hat{Z}_N=\sum_{k=0}^N \frac{ d^k }{k!}\hat{n}^k,
\label{ZNapproximation}$$ where $d=\ln g$. An important feature of the amplifier (\[Zamplifier\]) is that it preserves the structure of coherent states and only amplifies their amplitude, $\hat{Z}|\alpha\rangle \propto |g\alpha\rangle$. It turns out that the gain of the approximate amplifier (\[ZNapproximation\]) depends on $|\alpha|$ and we can define an effective gain of the amplifier, $$g_{\mathrm{eff}}=\frac{1}{\alpha}\frac{\langle \alpha|\hat{Z}^{\dagger}_N \hat{a}\hat{Z}_N|\alpha\rangle}{\langle \alpha|\hat{Z}^{\dagger}_N\hat{Z}_N|\alpha\rangle}.
\label{geffdefinition}$$ It is natural to require that $g_{\mathrm{eff}}=g$ in the limit $|\alpha|\rightarrow 0$. This can be achieved if we set $d$ to be equal to a root of a polynomial equation $$\sum_{k=0}^N \frac{d^k}{k!}=g.$$ As a concrete example we now study in more detail the cases $N=1$ and $N=2$ that are experimentally feasible with current technology. For $N=1$ we find that $d=g-1$ and the approximate polynomial reads $$\hat{Z}_1=(g-1)\hat{n}+1.
\label{Z1amplifier}$$ For $N=2$ we must solve a quadratic equation for $d$ which yields $d=\sqrt{2g-1}-1$ and $$\hat{Z}_2=(g-\sqrt{2g-1})\hat{n}^2+(\sqrt{2g-1}-1)\hat{n}+1.
\label{Z2amplifier}$$ On inserting the operators (\[Z1amplifier\]) and (\[Z2amplifier\]) into Eq. (\[geffdefinition\]) we can derive analytical formulas for the effective gain. For $N=1$ we obtain $$g_{\mathrm{eff}}=1+\frac{(g-1)[1+(g-1)|\alpha|^2]}{1+(g^2-1)|\alpha|^2+(g-1)^2|\alpha|^4}.$$ The expression for $N=2$ is rather long and unwieldy and is not reproduced here. Instead, we plot in Fig. 2 the dependence of $g_{\mathrm{eff}}$ on $|\alpha|$ for $N=1$, $N=2$ and for five different nominal values of the gain. We can see that the effective gain decreases with increasing $|\alpha|$. Also, the performance of the scheme improves when higher-order approximation is employed, and we find that for $N=2$ the effective gain $g_{\mathrm{eff}}$ is closer to $g$ than for $N=1$. Besides the effective gain, we also calculate the fidelity of the amplified weak coherent state with the ideal target state, $$F=\frac{|\langle g\alpha|\hat{Z}_N|\alpha\rangle|^2}{\langle \alpha|\hat{Z}^{\dagger}_N\hat{Z}_N|\alpha\rangle}.$$ We plot the fidelity as a function of coherent state amplitude in Fig. 3. For the considered range of parameters $g \leq 1.5$ and $|\alpha|\geq 1$ the fidelity exceeds $0.95$ for $N=1$ and surpasses $0.986$ for $N=2$. The scheme is thus very suitable for probabilistic error-free quantum cloning of weak coherent states via noiseless amplification.
{width="0.85\linewidth"}
Another important application of a noiseless amplifier is the concentration of continuous variable entanglement [@Ralph08]. Consider a two-mode squeezed vacuum state $$|\Psi\rangle_{AB}=\sum_{n=0}^\infty c_n |n\rangle_A|n\rangle_B,
\label{PsiAB}$$ where $c_n=\sqrt{1-\xi^2} \xi^n$ and $\xi$ is the two-mode squeezing constant. Local noiseless amplification of one mode increases the two-mode squeezing, $\xi \rightarrow g \xi$, and thus enhances the entanglement of the state. We have investigated the effect of the approximate amplifications (\[Z1amplifier\]) and (\[Z2amplifier\]) on the entangled state (\[PsiAB\]) by means of numerical calculations. In the simulations we assume the limit $h\rightarrow 1$. Generally, the attenuation factor $h^{\hat{n}}$ can be accounted for by replacing $\xi$ with rescaled squeezing constant $\xi'=h\xi$.
The results are presented in Fig. 4 that shows the dependence of fidelity and entanglement of the amplified state on gain $g$ for $\xi=\frac{1}{3}$ which corresponds to $3$ dB of two-mode squeezing. We can see that the amplification enhances the state entanglement as expected, while preserving well the structure of the state as witnessed by high fidelity. This is further illustrated in Fig. 5 where we plot the Fock-state probability amplitudes $c_n$ of a state locally amplified by operations $\hat{Z}_1$ and $\hat{Z}_2$ with $g=1.75$, as well as the amplitudes of the initial two-mode squeezed vacuum state and the ideal amplified two-mode squeezed vacuum state with squeezing constant $g\xi$.
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Kerr nonlinearity
=================
The Kerr effect is a consequence of a nonlinear response of a medium whose index of refraction depends on the intensity of the light field. Quantum mechanically, this can be described by a Hamiltonian proportional to the square of photon number operator. The resulting unitary transformation reads, $$\hat{U}=e^{i\phi \hat{n}^2},
\label{U}$$ where $\phi$ is a dimensionless nonlinear phase shift. Kerr nonlinearity can be used to generate highly non-classical states of light beams and, together with Gaussian operations, it is sufficient for universal quantum computing [@Lloyd99]. With the scheme described in Sec. II we can implement an approximate truncated version of the transformation (\[U\]), $$\hat{U}_{2N}=\sum_{k=0}^{N}\frac{(i\phi)^k}{k!} \hat{n}^{2k}.
\label{U2N}$$ We can generalize this expression to compensate for the attenuation factor $h^{\hat{n}}$ up to $2N$-th order by taking truncated Taylor series of operator $e^{i\phi \hat{n}^2} h^{-\hat{n}}$, $$\hat{U}_{2N}(h)=\sum_{k=0}^{2N} H_k\left(\frac{-\ln h}{2\sqrt{-i\phi}}\right) \frac{(\sqrt{-i\phi}\,\hat{n})^k}{k!},
\label{U2Nhcompensation}$$ where $H_k(y)$ denotes Hermite polynomial. In what follows we shall assume the limit $h\rightarrow 1$ for the sake of simplicity. Emulation of Kerr nonlinearity is more resource-demanding than noiseless amplification, because already the first non-trivial approximation requires subtraction and addition of two photons, $$\hat{U}_2=1+i\phi \hat{n}^2.
\label{U2}$$ The unitary matrix $\hat{U}$ is diagonal in Fock state basis, with matrix elements $U_{nn}=e^{i\phi n^2}$. The matrix representing the approximate operation (\[U2\]) is also diagonal in Fock state basis but we find that $(U_2)_{nn}=\sqrt{1+\phi^2 n^4}e^{i\Phi_n}$, where $\Phi_{n}=\arctan(\phi n^2)$. The nonlinear phase shift $\Phi_n$ does not increase quadratically with $n$ as it should but becomes saturated, $\Phi_{n\rightarrow \infty}=\frac{\pi}{2}$. Another aspect of the approximation (\[U2\]) is that the Fock state amplitudes are modulated by the factors $A_n=\sqrt{1+\phi^2n^4}$ so the transformation acts also as an amplifier. These features are illustrated in Fig. 6.
![(Color online) Plotted are the nonlinear phase shift $\Phi_{n}$ (blue triangles) and amplitude modulation factors $A_n$ (green squares) induced by the transformation (\[U2\]) emulating Kerr nonlinearity with $\phi=\frac{\pi}{100}$. Also shown for comparison is the phase shift $\phi n^2$ corresponding to the true Kerr nonlinearity (\[U\]) (red circles).](amplifierfig6.eps){width="0.95\linewidth"}
We quantify the performance of the approximate operation (\[U2\]) by the quantum process fidelity. Since the Hilbert space of a field mode is infinite dimensional, we shall consider fidelity $F_N$ of operation restricted to a finite dimensional subspace spanned by the first $N+1$ Fock states $|0\rangle,\ldots,|N\rangle$. We define maximally entangled state on this subspace $|\Psi_N\rangle=\frac{1}{\sqrt{N+1}}\sum_{n=0}^N |n\rangle|n\rangle$ and we have $$F_N=\frac{|\langle \Psi_N| (\hat{U}^\dagger\otimes \openone) (\hat{U}_2 \otimes \openone) |\Psi_N\rangle|^2}
{\langle \Psi_N| (\hat{U}_2^\dagger \otimes \openone)(\hat{U}_2 \otimes \openone)|\Psi_N\rangle},$$ where $\openone$ denotes the identity operation. Figure 7 shows the dependence of the fidelities $F_2$, $F_3$ and $F_4$ on the phase shift $\phi$. We can see that with growing $\phi$ it is increasingly more and more difficult to emulate $\hat{U}$ by $\hat{U}_2$ and the fidelities $F_N$ decrease rapidly.
{width="0.95\linewidth"}
This problem of fidelity decrease can be remedied if we need to emulate the Kerr nonlinearity only on a finite dimensional subspace spanned by the first $N+1$ Fock states. More generally, any operation $f(\hat{n})$ can be perfectly probabilistically implemented on such subspace if we construct the polynomial approximation to this operation as follows, $$\hat{f}'_N=\sum_{k=0}^N \frac{f(k)}{h^k} \prod_{j=0,j\neq k}^N \frac{\hat{n}-j}{k-j} .
\label{fprimeN}$$ Note that this expression includes compensation for the attenuation factor $h$. In particular, on inserting $f(k)=e^{i\phi k^2}$ and $N=2$ into Eq. (\[fprimeN\]) we obtain the following polynomial that perfectly emulates Kerr nonlinearity on a three-dimensional subspace spanned by Fock states $|0\rangle$, $|1\rangle$, and $|2\rangle$, $$\hat{U}_2^\prime=\left(1-\frac{2e^{i\phi}}{h}+\frac{e^{4i\phi}}{h^2}\right)\frac{\hat{n}^2}{2}
-\left(3-\frac{4e^{i\phi}}{h}+\frac{e^{4i\phi}}{h^2}\right)\frac{\hat{n}}{2}+1.$$
Discussion and conclusions
==========================
An important practical characteristics of the proposed device that we have not addressed in detail yet is its success probability. A simple order-of-magnitude estimate reads $p_{\mathrm{succ}}\approx (r\lambda \eta)^{2N}$, where $\eta$ denotes the detection efficiency of single-photon detectors. The quantum efficiencies of APDs exceed $50\%$. However, in the experiment, the optical beams typically need to be spatially and spectrally filtered before detection which reduces the overall detection efficiency to $10\%$ or even less. Another important characteristics is the attenuation factor $h=t^2\sqrt{1-\lambda^2}$. We want to maximize $p_{\mathrm{succ}}$ while keeping $h$ as close to $1$ as possible. To give a concrete example, we can set $t^2=0.95$, $\lambda^2=0.05$, and $\eta=0.1$. For $N=1$ we obtain $p_{\mathrm{succ}} \approx 2.5\times 10^{-5}$ and $h=0.926$. Pumping the scheme with a pulsed Ti:sapphire laser with a repetition rate of $80$ MHz would then yield approximately $2000$ events per second. Second-order approximation, $N=2$, is much more challenging. For the above parameters we obtain success rate $0.05$ Hz, i.e. about $3$ events per minute which may be still acceptable for proof-of-principle experiments. Moreover, this rate can be significantly increased either by improving the detection efficiency or by using beam splitters with higher transmittance $r$. For instance, with $t^2=0.9$ and $\eta=0.2$ we obtain success rate $3$ Hz and $h=0.877$. We stress that the calculated success rates are only order-of-magnitude estimates and would depend on the input state of the scheme as well as on details of the implemented transformation. Nevertheless, these calculations clearly confirm the experimental feasibility of the scheme with single photon addition and subtraction and show that scheme with two photon additions and subtractions may be within the reach of present technology.
In summary, we have described and analyzed a scheme that allows to probabilistically implement arbitrary single-mode operation that can be expressed as a function of photon number operator. The method relies on coherent combination of multiple photon subtraction and addition and is experimentally feasible with present-day technology [@Parigi07; @Zavatta09]. As concrete examples, we have shown that the scheme can work as a probabilistic noiseless amplifier and can emulate Kerr nonlinearity. The scheme is very versatile and can find many applications in quantum optics and quantum information processing such as implementation of quantum gates, probabilistic error-free quantum cloning, or entanglement distillation.
The author would like to thank R. Filip for stimulating discussions. This work was supported by MSMT under projects LC06007, MSM6198959213, and 7E08028, by GACR under project No. GA202/08/0224, and also by the EU under the FET-Open project COMPAS (212008).
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The Fridman invariant, which is a biholomorphic invariant on Kobayashi hyperbolic manifolds, can be seen as the dual of the much studied squeezing function. We compare this pair of invariants by showing that they are both equally capable of determining the boundary geometry of a bounded domain if their boundary behaviour is apriori known.'
address:
- 'PM: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India'
- 'KV: Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India'
author:
- 'Prachi Mahajan, Kaushal Verma'
title: A comparison of two biholomorphic invariants
---
Introduction
============
Recall that the squeezing function associated to a bounded domain is a measure of the largest euclidean ball contained in all possible holomorphic embeddings of the given domain into the unit ball in ${\mathbb}C^n$. More precisely, for a bounded domain $D \subset {\mathbb}C^n$ and $p \in D$, let ${\mathcal}F$ be the family of all injective holomorphic maps $ f $ from $D$ to the unit ball ${\mathbb}B^n \subset {\mathbb}C^n$ that map $p$ to the origin. Let $S_D(p, f)$ be the supremum of those $r > 0$ for which the image $f(D)$ contains $B^n(0, r)$, the euclidean ball of radius $r$ around the origin in $ \mathbb{C}^n $. The squeezing function $s_D : D \mapsto (0, 1]$ is defined as $$s_D(p) = \sup \{S_D(p, f) : f \in {\mathcal}F \}.$$ That this is a biholomorphic invariant follows from its definition and when $D = {\mathbb}B^n$, it can be checked that $s_D \equiv 1$. Various aspects of $s_D$ have been studied of late but among those that are directly relevant to this note are its boundary behaviour on some classes of domains (for example [@FW], [@KZ], [@N] and [@NA]) and conversely, its efficacy in determining some geometric properties of the boundary of the domain if its boundary behaviour is a priori known – for example, [@SK] and [@Z2].
It is interesting to note that another biholomorphic invariant, that is dual to the squeezing function in much the same way as the Carathéodory and Kobayashi metrics are, was defined by Fridman in [@Fr1], [@Fr2]. Let us recall its construction: for $X$ a Kobayashi hyperbolic complex manifold of dimension $n$, let $B_X(p, r)$ be the Kobayashi ball around $p \in X$ of radius $r > 0$. Let ${\mathcal}R$ be the set of all $r > 0$ such that there is an injective holomorphic map $f : {\mathbb}B^n {\rightarrow}X$ with $B_X(p, r) \subset f({\mathbb}B^n)$. Note that ${\mathcal}R$ is non-empty. Indeed, the hyperbolicity of $X$ implies that the intrinsic topology on it is equivalent to that induced by the Kobayashi metric. Hence, for small $r >0$, the ball $B_X(p, r)$ is contained in a coordinate chart and this shows that there is an injective holomorphic map from the ball into $X$ whose image contains $B_X(p, r)$. The Fridman invariant is $$h_X(p) = \inf_{r \in {\mathcal}R} \frac{1}{r}$$ which is a non-negative real-valued function on $X$. This is a biholomorphic invariant since the Kobayashi metric is itself preserved by such maps and among other things, Fridman showed that (i) $h_X$ is continuous and that (ii) if $h_X(p) = 0$ for some $p \in X$, then $X$ is biholomorphic to ${\mathbb}B^n$ in [@Fr1]. Other aspects of this invariant such as its boundary behaviour were studied in [@MV].
The purpose of this note is (i) to show that much like the squeezing function, the Fridman invariant can also determine the nature of the boundary of a given domain if its boundary behaviour is a priori known and (ii) to localize and provide a different proof of some of the results in [@N] and [@SK]. While both (i) and (ii) use the methods of scaling, they rely on an observation made in [@MV] namely, the convergence of the integrated Kobayashi distance on each scaled domain to that in the limiting domain. More specifically, refer to Lemma 5.2 and 5.7 of [@MV].
Let $D \subset {\mathbb}C^n$ be a bounded convex domain with $C^{\infty}$-smooth boundary. Then $\partial D$ is strongly pseudoconvex if $h_D(z) {\rightarrow}0$ as $z {\rightarrow}{\partial}D$.
The corresponding statement for the squeezing function $s_D(z)$ is already known – see [@Z2] for instance. Note that the boundary ${\partial}D$ can apriori be of infinite type near $p^0$ in this theorem.
A similar result holds for $h$-extendible boundary points. Before proceeding further, recall that $ p^0 \in \partial D $ is said to be an $h$-extendible boundary point if $ \partial D $ is smooth pseudoconvex finite type near $ p^0 $ and the Catlin and the D’Angelo multitypes at $ p^0 $ coincide. The class of $h$-extendible points includes smooth pseudoconvex finite type boundary points in $ \mathbb{C}^2 $, convex finite type points in $ \mathbb{C}^n $ and pseudoconvex finite type boundary points in $ \mathbb{C}^n $ with Levi-form having at most one degenerate eigenvalue.
Let $D \subset {\mathbb}C^n$ be a bounded domain with $p^0 \in \partial D$. Assume that ${\partial}D$ is $C^{\infty}$-smooth and $h$-extendible near $p^0$. Then $\partial D$ is strongly pseudoconvex near $p^0$ if either $h_D(z) {\rightarrow}0$ or $s_D(z) {\rightarrow}1$ as $z {\rightarrow}p^0$.
Here, it turns out that $\partial D$ is strongly pseudoconvex near $p^0$ if either $h_D(p^j) {\rightarrow}0$ or $s_D(p^j) {\rightarrow}1$ for a sequence $ p^j $ in $ D $ converging to $ p^0 $ only along the inner normal to $ \partial D $ at $ p^0 $. This will be evident from the proof of this theorem.
For a domain $ D $ in $ \mathbb{C}^n $, $ F_D $ denotes its Kobayashi-Royden infinitesimal metric and $ d_D $ its integrated Kobayashi distance.
Before proving these theorems, we begin with:
Two Examples
============
The Fridman invariant for the unit polydisc $ \Delta^n $ is given by $$h_{\Delta^n}(z) = 2 \left(\log \left( \frac{\sqrt{n} + 1}{\sqrt{n} - 1} \right) \right)^{-1}$$ for every $z \in \Delta^n$.
Since $ \Delta^n $ is homogeneous and the function $ h_{\Delta^n} \left( \cdot \right) $ is a biholomorphic invariant, it is enough to compute the explicit formula for $ h_{\Delta^n } $ at the origin. If $ f: \mathbb{B}^n \rightarrow \Delta^n $ is a holomorphic imedding such that $ f(0) = 0 $ and $$\Delta^n \left( 0, \frac{e^{2r} -1}{e^{2r} + 1}\right) = B_{\Delta^n} (0,r) \subset f(\mathbb{B}^n),$$ then it follows from [@Alexander] that $$\frac{e^{2r} -1}{e^{2r} + 1} \leq \frac{1}{\sqrt{n}},$$ or equivalently that, $$\frac{1}{r} \geq 2 \left( \log \left( \frac{ \sqrt{n} + 1}{ \sqrt{n} -1} \right)\right)^{-1},$$ which implies that $$\label{k14}
h_{\Delta^n} \left(0 \right) \geq 2 \left( \log \left( \frac{ \sqrt{n} + 1}{ \sqrt{n} -1} \right)\right)^{-1}.$$ On the other hand, consider $ \psi_0 = i \circ \psi $, where $ \psi $ is an automorphism of $ \mathbb{B}^n $ that preserves the origin and $ i : \mathbb{B}^n \rightarrow \Delta^n $ is the inclusion map. Then $ \psi_0$ is an imbedding of $ \mathbb{B}^n $ into $ \Delta^n $ satisfying $ \psi_0(0) = 0 $ and $$\Delta^n \left( 0, \frac{1}{\sqrt{n}} \right) = B_{\Delta^n} \left(0, \frac{1}{2} \log \left( \frac{ \sqrt{n} + 1}{ \sqrt{n} -1} \right) \right)
\subset \psi_0(\mathbb{B}^n),$$ and hence $$\label{k15}
h_{\Delta^n} \left(0 \right) \leq 2 \left( \log \left( \frac{ \sqrt{n} + 1}{ \sqrt{n} -1} \right)\right)^{-1}.$$ Combining the inequalities (\[k14\]) and (\[k15\]) yields the desired expression for $ h_{\Delta^n} $.
Let $\{p^j\}$ be a sequence in the punctured disc $\Delta {\setminus}\{0\}$ that converges to the origin. Then $$\label{E9}
L_j \le h_{\Delta {\setminus}\{0\}}(p^j) \le U_j$$ where $$L_j^{-1} = \log \left( 2 \left( -\frac{\pi}{\log \vert p^j \vert} \right)^2 + 1 + \frac{2 \pi}{\log \vert p^j \vert} \sqrt{\left( -\frac{\pi}{\log \vert p^j \vert} \right)^2 + 1} \right)$$ and $$U_j^{-1} = \log \left( \left(-\frac{\pi}{\log \vert p^j \vert} \right) + \sqrt{\left(-\frac{\pi}{\log \vert p^j \vert}\right)^2 + 1 } \right).$$
Since $ h_{\Delta {\setminus}\{0\}} (\cdot) $ is a biholomorphic invariant, after composing with an appropriate automorphism of $ \Delta \setminus \{0\} $, we may assume that each $ p^j $ lies in the open interval $ (0,1) $. Consider the slit disc $ \Delta \setminus (-1, 0] $, which is a simply connected domain. Choose a conformal map $ f^j $ from the unit disc $ \Delta $ onto the slit domain $ \Delta \setminus (-1,0] $ such that $ f^j (0) = p^j $. Then $$B_{ \Delta \setminus \{0\} } \left( p^j, r (p^j) \right) \subset \Delta \setminus (-1,0],$$ where $$\label{E8}
r(p^j) = \log \left( - \frac{\pi}{\log p^j} + \sqrt{ \left( - \frac{\pi}{\log p^j}\right)^2 + 1} \right).$$ To establish this claim, it suffices to show that $$\label{E7}
d_{\Delta \setminus \{0\} } \left( p^j, (-1,0) \right) : = \inf_{q \in (-1,0)} d_{\Delta \setminus \{0\} } \left( p^j, q \right) = r(p^j).$$ To verify this, first recall that the upper half-plane $ \mathbb{H} $ is the universal covering space of the punctured disc $ \Delta \setminus \{0\} $, the projection being given by the map $$\mathbb{H} \ni z \mapsto \exp( \iota z ) \in \Delta \setminus \{0\}.$$ Hence, for each $ q $ in $ (-1,0) $, $$d_{\Delta \setminus \{0\} } \left( p^j, q \right) = \inf_{\tilde{q}} d_{\mathbb{H}} \left( - \iota \log p^j, \tilde{q} \right),$$ where the infimum is taken over all preimages $ \tilde{q} $ of $ q $ under the covering map. Furthermore, the preimage of the interval $ (-1,0) $ is the vertical line $ \Re z = \pi $ which is a geodesic in $ \mathbb{H} $. It follows that $$d_{\Delta \setminus \{0\} } \left( p^j, (-1,0) \right) = d_{\mathbb{H}} \left( - \iota \log p^j, \{ z \in \mathbb{H} : \Re z = \pi \} \right).$$ To calculate the right-hand side, observe that there is a unique geodesic (namely, the half-circle centred at $ \pi $ and radius $ | - \iota \log p^j - \pi | $) through $ - \iota \log p^j $ and orthogonal to the line $ \{ \Re z = \pi \} $. Moreover, $ d_{\mathbb{H}} \left( - \iota \log p^j, \{ z: \Re z = \pi \} \right) $ is the distance from $ - \iota \log p^j $ to $ \{ \Re z = \pi \} $ measured along this half-circle. I.e., $$d_{\mathbb{H}} \left( - \iota \log p^j, \{ z \in \mathbb{H} : \Re z = \pi \} \right) = d_{\mathbb{H}} \left( - \iota \log p^j, \pi + \iota | \iota \log p^j + \pi | \right).$$ The Kobayashi distance between two points $ z , w $ on the upper half-plane is given by $$\label{E14}
d_{\mathbb{H}} (z,w) = \log \left( \frac{ |z - \bar{w}| + |z-w| }{|z- \bar{w}| - |z-w| }\right).$$ Using the above formulation of the Kobayashi distance on $ \mathbb{H} $, it can be seen that $$d_{\mathbb{H}} \left( - \iota \log p^j, \pi + \iota | \iota \log p^j + \pi | \right) = \log \left( - \frac{\pi}{\log p^j} + \sqrt{ \left( - \frac{\pi}{\log p^j}\right)^2 + 1} \right),$$ and consequently that, $$d_{\Delta \setminus \{0\} } \left(p^j, (-1,0) \right) = \log \left(- \frac{\pi}{\log p^j} + \sqrt{ \left( - \frac{\pi}{\log p^j}\right)^2 + 1} \right),$$ thereby verifying the equation (\[E7\]). To summarize, there is a biholomorphic imbedding $ f^j : \Delta \rightarrow \Delta \setminus \{0\} $ with $ f^j(0) = p^j $ and $$B_{ \Delta \setminus \{0\} } \left( p^j, r (p^j) \right) \subset f^j(\Delta) = \Delta \setminus (-1,0],$$ where $ r(p^j) $ is as defined by equation (\[E8\]). It follows that $$h_{\Delta \setminus \{0\} } (p^j) \leq 1/r(p^j),$$ which gives the upper estimate (\[E9\]).
For the lower estimate, the following observations will be needed. Firstly, the punctured disc is complete hyperbolic and hence taut. Moreover, for each $j $, $ h_{\Delta \setminus \{0\} } (p^j) > 0 $, and hence there exist a biholomorphic imbedding $ f^j : \Delta \rightarrow \Delta \setminus \{0\} $ with $ f^j(0) = p^j $ and $$\label{E10}
B_{ \Delta \setminus \{0\} } \left( p^j, \frac{1}{h_{\Delta \setminus \{0\} } (p^j) } \right) \subset f^j(\Delta) \subset \Delta \setminus \{0\}.$$ Secondly, consider the circle centred at the origin and radius $ p^j$, $$C^j = \{ w \in \mathbb{C} : |w| = p^j\},$$ and compute $$\sup_{ q \in C^j} d_{\Delta \setminus \{0\} } \left(p^j, q \right).$$ It turns out that $$\label{E12}
s(p^j) := \sup_{ q \in C^j} d_{\Delta \setminus \{0\} } \left(p^j, q \right) =
\log \left( 2 \left( - \frac{\pi}{\log p^j } \right)^2 + 1+ \frac{2 \pi}{\log p^j } \sqrt{ \left( - \frac{\pi}{\log p^j}\right)^2 + 1} \right)$$ Grant this for now. It follows that the circle $ C^j $ is contained in the closure of the Kobayashi ball $ B_{ \Delta \setminus \{0\} } \left( p^j, s(p^j) \right) $. This forces that $$\label{E13}
\frac{1}{h_{\Delta \setminus \{0\} } (p^j)} \leq s(p^j).$$ Indeed, assume on the contrary that the above inequality does not hold, i.e., there is an $ \epsilon_0 > 0 $ such that $$s(p^j) + \epsilon_0 < \frac{1}{h_{\Delta \setminus \{0\} } (p^j) }.$$ Then it is immediate that $$\label{E11}
C^j \subset B_{ \Delta \setminus \{0\} } \left( p^j, s(p^j) + \epsilon_0 \right) \subset
B_{ \Delta \setminus \{0\} } \left( p^j, \frac{1}{h_{\Delta \setminus \{0\} } (p^j) } \right).$$ Combining (\[E10\]) and (\[E11\]) gives $$C^j \subset f^j(\Delta) \subset \Delta \setminus \{0\}.$$ But $ f^j(\Delta) $ is a simply connected sub-domain of the punctured disc and hence it cannot contain any circle centered at the origin. Hence we arrive at a contradiction, thereby proving the inequality (\[E13\]).
The final step is to establish equation (\[E12\]). It follows from the definition that $$s(p^j) := \sup_{ q \in C^j} d_{\Delta \setminus \{0\} } \left(p^j, q \right) =
\sup_{ q \in C^j} \inf_{\tilde{q}} d_{\mathbb{H} } \left(- \iota \log p^j, \tilde{q} \right),$$ where the infimum is taken over all preimages $ \tilde{q} $ of $ q $ under the covering projection $ z \mapsto \exp (\iota z) $. Write $ q = p^j \exp ( \iota \theta) $ for $ \theta \in [0, 2\pi) $, so that the right hand side above equals $$\sup_{\theta \in [0, 2\pi)} \inf_{ k \in \mathbb{Z}} d_{\mathbb{H}} \left( - \iota \log p^j, 2 \pi k + \theta - \iota \log p^j \right)$$ A direct computation using the explicit expression (\[E14\]) for $ d_{\mathbb{H}} (\cdot, \cdot) $ shows that $$\inf_{ k \in \mathbb{Z}} d_{\mathbb{H}} \left( - \iota \log p^j, 2 \pi k + \theta - \iota \log p^j \right) =
\log \left( \frac{ \theta^2 + 2 (\log p^j)^2 + \theta \sqrt{\theta^2 + 4 (\log p^j)^2} }{2 (\log p^j)^2}\right),$$ so that $$\begin{aligned}
{3}
\sup_{\theta \in [0, 2\pi)} \inf_{ k \in \mathbb{Z}} d_{\mathbb{H}} \left( - \iota \log p^j, 2 \pi k + \theta - \iota \log p^j \right)
= & \log \left( \frac{ 4 \pi^2 + 2 (\log p^j)^2 + 2 \pi \sqrt{4 \pi^2 + 4 (\log p^j)^2} }{2 (\log p^j)^2}\right) \\
= & \log \left( 2 \left( - \frac{\pi}{\log p^j } \right)^2 + 1+ \frac{2 \pi}{\log p^j } \sqrt{ \left( - \frac{\pi}{\log p^j}\right)^2 + 1} \right),
\end{aligned}$$ thereby verifying (\[E12\]).
Note that both $L_j, U_j {\rightarrow}+\infty$ as $p^j {\rightarrow}0$ which is expected. Thus $h_{\Delta {\setminus}\{0\}}$ blows up near the origin. On the other hand, that $h_{\Delta {\setminus}\{0\}} (p^j) {\rightarrow}0$ if $\vert p^j \vert {\rightarrow}1$ can be seen from the following:
Let $D \subset {\mathbb}C$ be a bounded domain with $p^0 \in {\partial}D$. Assume that ${\partial}D$ is $C^2$-smooth near $p^0$. Then $h_D(z) {\rightarrow}0$ as $z {\rightarrow}p^0$.
Let $ \rho $ be a $ C^2$-smooth local defining function for $ \partial D $ near $ p^0 $ and $ \{ p^j \} $ be a sequence of points in $ D $ converging to $ p^0 $. Consider the dilations $$T^j(z) = \frac{z-p^j}{- \rho(p^j)}$$ and note that the scaled domains $ D^j = T^j (D)$ are given by $$\{ z \in \mathbb{C}: - 1 + 2 \Re \left( \partial \rho(p^j) z \right) - \psi(p^j) O(1) < 0 \}.$$ near $ T^j(p^0) $. It follows that the sequence of domains $ D^j $ converge in the Hausdorff sense to the half-plane $$D_{\infty} = \{ z \in \mathbb{C}: 2 \Re \left( \partial \rho(p^0) z \right) - 1 < 0 \}.$$ Note that $D$ supports a local holomorphic peak function at $p_0$ since the boundary ${\partial}D$ is $C^2$-smooth near it and hence the proof of Theorem 1.1 of [@MV] can be adapted to show that $$h_{D} (p^j) \rightarrow h_{D_{\infty}} (0).$$ But $ h_{D_{\infty}} (\cdot) \equiv 0 $ as $ D_{\infty} $ is biholomorphically equivalent to $ \mathbb{B}^n $. Hence $ h_{D} (p^j) \rightarrow 0 $ as $ j \rightarrow \infty $.
Proof of Theorem 1.1
====================
Let $p^0 \in {\partial}D$. We will study the behaviour of $h_D(z)$ as $z {\rightarrow}p^0$. The proof of Theorem 1.1 divides into two parts:
1. $ \partial D $ is of finite type near $ p^0 $, or
2. $ \partial D $ is of infinite type near $ p^0 $.
It turns out that $ p^0 \in \partial D $ cannot be of infinite type, thereby, ruling out case(ii).
[*Case (i):*]{} let $ p^j $ be a sequence of points in $ D $ converging to $ p^0 $ along the inner normal to $ \partial D $ at $ p^0 $. Since $ \lim_{j \rightarrow \infty} h_{D} (p^j) = 0 $ by assumption, there exists a sequence of positive real numbers $ R_j \rightarrow \infty $ and a sequence of biholomorphic imbeddings $ F^j: \mathbb{B}^n \rightarrow D $ satisfying $ F^j (0)= p^j $ and $ B_D(p^j, R_j) \subset F^j( \mathbb{B}^n) $.
Before going further, let us briefly recall the scaling technique from [@Mcneal-1994]. Here and in the sequel, we write $ z = (z_1, z_2, \ldots, z_{n-1}, z_n ) = ('z,z_n) \in \mathbb{C}^n $ for brevity. By [@Yu-1992] there exists a local coordinate system $ \Phi $ in a neighbourhood of $ p^0 $ such that $ \Phi(p^0) = ('0, 0) $, $
\Phi (p^j) = ('0, - \| p^0 - p^j \|) $ for each $ j$ and the domain $ \Phi(D) $ near origin can be written as $$\{ ('z,z_n) \in \mathbb{C}^n : 2 \Re z_n + P_0('z) + R(z) < 0 \},$$ where $P_0$ is a nondegenerate weighted homogeneous polynomial of degree $1$ with respect to the weights $ \mathcal{M}(\partial D,
0) $, the multitype of $ \partial D $ near the origin, and $ R $ denotes terms of degree at least two. Define a dilation of coordinates by $$T^j( z_1, z_2, \ldots, z_{n-1}, z_n ) = \left( \delta_j^{-1} z_1, \delta_j^{-1} z_2,
\ldots, \delta_j^{-1} z_{n-1}, \delta_j^{-1} z_n \right),$$ where $ \delta_j = \|p^0 - p^j \| $ for each $j$. Note that $ T^j \big( ('0, - \delta_j) \big) = ('0, -1) $ for all $j$ and the scaled domains $ D^j = T^j \circ \Phi(D) $ converge in the Hausdorff sense to $$D_{ \infty} = \big\{ z \in \mathbb C^n : 2 \Re z_n + P_{0}('z) < 0 \big\}.$$ Furthermore, it follows from Theorem 1.1 of [@Mcneal-1992] that $ D_{\infty} $ is complete hyperbolic and hence $ D_{\infty} $ is taut.
Consider the dilated maps $$\psi^j := T^j \circ \phi \circ F^j : \mathbb{B}^n \rightarrow D^j.$$ Note that $ T^j \circ \phi \circ F^j \big( ('0,0) \big) = ('0,-1)$ for each $ j $. Since the domains $ D^j $ are contained in the intersections of certain half spaces (see [@Gaussier-1997] for details), it follows that the sequence $ \{ T^j \circ \phi \circ F^j \} $ admits a subsequence, that will still be denoted by the same indices, that converges uniformly on compact sets of $ \mathbb{B}^n $ to a holomorphic mapping $ \psi : \mathbb{B}^n \rightarrow D_{\infty} $.
Then $ \psi $ is a biholomorphism from $ \mathbb{B}^n $ onto $ D_{\infty} $. To establish this, first note that for each $ \epsilon > 0 $, $$B_{D_{\infty}} \left( ('0,-1), R - \epsilon \right) \subset B_{D^j} \left( ('0,-1), R \right)
\label{k9}$$ for all $ R > 0 $ and all $j$ large and this will follow from $$\displaystyle\limsup_{j \rightarrow \infty} d_{D^j} \left( ('0,-1), \cdot \right)
\leq d_{D_{\infty}} \left( ('0,-1), \cdot \right).$$ To verify the above inequality, fix $ q \in D_{\infty}$ and let $ \gamma : [0,1]
\rightarrow D_{\infty} $ be a piecewise $C^1$-smooth path in $
D_{\infty} $ such that $ \gamma(0) = ('0,-1), \gamma(1) = q$ and $$\int_0^1 { F_{D_{\infty}} \big( \gamma(t), \dot{\gamma}(t) \big)
dt} \leq d_{D_{\infty}} \left( ('0,-1), q \right) + \epsilon/2.$$ Since the trace of $ \gamma $ is relatively compact in $
D_{\infty}$, it follows that the trace of $\gamma $ is contained uniformly relatively compactly in $ D^j
$ for all large $j$. It follows from Lemma 6.2 of [@MV] that $$\int_0^1 {F_{D^j} \big( \gamma(t), \dot{\gamma}(t) \big) dt}
\leq \int_0^1 { F_{D_{\infty}} \big( \gamma(t), \dot{\gamma}(t)
\big) dt} + \epsilon/2 \leq d_{D_{\infty}} \left( ('0,-1), q \right) + \epsilon.$$ Consequently, $$d_{D^j} \left( ('0,-1), q \right) \leq \int_0^1 {F_{D^j} \big( \gamma(t),
\dot{\gamma}(t) \big) dt} \leq d_{D_{\infty}} \left( ('0,-1), q \right) +
\epsilon$$ which implies that $$\displaystyle\limsup_{j \rightarrow \infty} d_{D^j} \left( ('0,-1), \cdot \right)
\leq d_{D_{\infty}} \left( ('0,-1), \cdot \right).$$
Note that $ B_{ D } ( p^j, R_j) \subset F^j(
\mathbb{B}^n)$. Since $ T^j \circ \phi $ are biholomorphisms and hence Kobayashi isometries, it follows that $$\begin{aligned}
B_{ D^j} \left( ('0,-1), R_j \right) \subset T^j \circ \phi \circ F^j (\mathbb{B}^n). \label{4.0}\end{aligned}$$ Since $ ( D_{\infty}, d_{D_{\infty}}) $ is complete, it is possible to write $$\begin{aligned}
D_{\infty} = \displaystyle \bigcup_{ \nu = 1} ^{\infty}
B_{D_{\infty}} \big( ('0,-1), \nu \big) \label{k10}\end{aligned}$$ which is an exhaustion of $ D_{\infty}$ by an increasing union of relatively compact domains. Now, consider $$\theta^j := \big( T^j \circ \phi \circ F^j \big)^{-1}: T^j \circ \phi \circ F^j (\mathbb{B}^n) \rightarrow
\mathbb{B}^n$$ These mappings are evidently defined on an arbitrary compact subset of $ D_{\infty} $ for large $j$ (cf. (\[k9\]), (\[4.0\]) and (\[k10\])) and hence some subsequence of $ \{ \theta^j \}$ converges to $ \theta: D_{\infty} \rightarrow
\overline{\mathbb{B}}^n$. Moreover, $ \theta \big( ('0,-1) \big) = ('0,0) $ together with the maximum principle shows that $ \theta :
D_{\infty} \rightarrow \mathbb{B}^n $. Finally observe that for $w$ in a fixed compact set in $ D_{\infty} $, $$\begin{aligned}
| \psi \circ \theta (w) - w | & = & | \psi \circ \theta (w) -
\psi^j \circ \theta^j(w) | \\
& = & | \psi \circ \theta (w) - \psi \circ \theta^j (w) | + |
\psi \circ \theta^j(w) - \psi^j \circ \theta^j (w)| \\
& \rightarrow & 0 \ \mbox{as } j \rightarrow \infty\end{aligned}$$ This shows that $ \psi \circ \theta = id $. Similarly, it can be proved that $ \theta \circ \psi = id$. This shows that $
D_{\infty} $ is biholomorphically equivalent to $ \mathbb{B}^n$.
By composing with a suitable Cayley transform, if necessary, we may assume that there is a biholomorphism $ \tilde{\theta} $ from $ D_{\infty}$ onto the unbounded realization of the ball, namely to $$\Sigma = \big\{ z \in \mathbb{C}^n : 2 \Re z_n + {\left\vertz_1\right\vert}^2 + {\left\vertz_2\right\vert}^2 + \ldots +
{\left\vertz_{n-1}\right\vert}^2 < 0 \big\}$$ with the property that the cluster set of $ \tilde{\theta} $ at some point $ ('0, \iota a) \in \partial D_{\infty} $ (for $a \in \mathbb{R} $) contains a point of $ \partial \Sigma $ different from the point at infinity on $ \partial \Sigma $. Then Theorem 2.1 of [@CP] ensures that $ \tilde{\theta} $ extends holomorphically past the boundary of $ D_{\infty} $ to a neighbourhood of $ ('0, \iota a) $. Furthermore, $ \tilde{\theta} $ extends biholomorphically across some point $ ('0, \iota a^0 ) \in \partial D_{ \infty} $. To prove this claim, it suffices to show that the Jacobian of $ \tilde{\theta} $ does not vanish identically on the complex plane $$L = \big \{ ('0, \iota a) : a \in \mathbb{R} \big \} \subset \partial D_{\infty}.$$ If the claim were false, then the Jacobian of $ \tilde{\theta} $ vanishes on the entire $ z_n $-axis, which intersects the domain $ D_{\infty} $. However, $ \tilde{\theta} $ is injective on $ D_{\infty} $, and consequently, has nowhere vanishing Jacobian determinant on $ D_{\infty} $. This contradiction proves the claim.
Next, note that the translations in the imaginary $ z_n $-direction leave $ D_{\infty} $ invariant. Therefore, we may assume that $ ('0, \iota a^0) $ is the origin and that $ \tilde{\theta} $ preserves the origin. Now recall that the Levi form is preserved under local biholomorphisms around a boundary point, thereby yielding the strong pseudoconvexity of $ \partial D_{\infty} $ near the origin. Equivalently, the strong pseudoconvexity of $ p^0 \in \partial D $ follows. Hence the result.
[*Case (ii):*]{} If $ p^0 $ were $ C^{\infty} $-convex of infinite type, by [@Z2], there exists a sequence $ p^j $ in $ D $ converging to $ p^0 \in \partial D $ and affine isomorphisms $ A^j $ of $ \mathbb{C}^n $ so that the domains $ A^j(D) = D^j $ converge in the local Hausdorff topology to a convex domain $ D_{\infty} $ and $ A^j(p^j) \rightarrow
0 \in D_{\infty}$. Moreover, it follows from Proposition 6.1 of [@Z1] that the limit domain $ D_{\infty} $ contains no complex affine lines and hence, it is Kobayashi complete.
Since $ \lim_{j \rightarrow \infty} h_{D} (p^j) = 0 $ by assumption, there exists a sequence of positive real numbers $ R_j \rightarrow \infty $ and a sequence of biholomorphic imbeddings $ F^j: \mathbb{B}^n \rightarrow D $ satisfying $ F^j (0)= p^j $ and $ B_D(p^j, R_j) \subset F^j( \mathbb{B}^n) $. Consider the maps $$\psi^j := A^j \circ F^j : \mathbb{B}^n \rightarrow D^j.$$ Note that $ A^j \circ F^j (0) \rightarrow 0 \in D_{\infty} $. By Proposition 4.2 of [@Z1], some subsequence of the sequence $ \{ A^j \circ F^j \} $ converges uniformly on compact sets of $ \mathbb{B}^n $ to a holomorphic mapping $ \psi : \mathbb{B}^n \rightarrow \overline{ D}_{\infty} $. Since $ \psi(0)= 0 $, it follows that $ \psi ( \mathbb{B}^n ) \subset D_{\infty} $.
Then $ \psi $ is a biholomorphism from $ \mathbb{B}^n $ onto $ D_{\infty} $. This will be done in several steps. The first of these records the stability of the infinitesimal Kobayashi metric, i.e., $$\label{E1}
F_{D^j} ( \cdot, \cdot) \rightarrow F_{D_{\infty}} (\cdot, \cdot)$$ uniformly on compact sets of $ D_{\infty} \times \mathbb{C}^n $. The key step in proving the above assertion is to understand limits of holomorphic mappings $ f^j : \Delta \rightarrow D^j $ that almost realize $ F_{D^j} (\cdot, \cdot) $. Using Proposition 4.2 of [@Z1], it is possible to pass to a subsequence of $ \{ f^j \} $ that converges to a holomorphic mapping $ f : \Delta \rightarrow D_{\infty} $ uniformly on compact sets of $ \Delta $. It follows that the limit map $ f $ provides a candidate in the definition of $ F_{D_{\infty}}( \cdot, \cdot) $.
The second step is to establish that $$\label{E21}
\displaystyle\limsup_{j \rightarrow \infty} d_{D^j} \left( A^j(p^j), \cdot \right)
\leq d_{D_{\infty}} \left( 0, \cdot \right),$$ which would imply that $$\label{k11}
B_{D_{\infty}} \left( 0, R - \epsilon \right) \subset B_{D^j} \left( A^j(p^j), R \right),$$ for all $ R > 0 $ and all $j$ large and for each $ \epsilon > 0 $. To verify (\[E21\]), fix $ q \in D_{\infty}$ as before and let $ \gamma : [0,1]
\rightarrow D_{\infty} $ be a piecewise $C^1$-smooth path in $
D_{\infty} $ such that $ \gamma(0) = 0, \gamma(1) = q$ and $$\int_0^1 { F_{D_{\infty}} \big( \gamma(t), \dot{\gamma}(t) \big)
dt} \leq d_{D_{\infty}} \left( 0, q \right) + \epsilon/2.$$ Define $ \gamma^j: [0,1] \rightarrow \mathbb{C}^n $ by $$\gamma^j(t) = \gamma(t) + A^j(p^j)(1-t).$$ Since the trace of $ \gamma $ is relatively compact in $
D_{\infty}$ and $ A^j (p^j) \rightarrow 0 $, it follows that the trace of $\gamma $ is contained uniformly relatively compactly in $ D^j
$ for all large $j$. Note that $ \gamma^j(0) = A^j(p^j) $ and $ \gamma^j(1) = q $. In addition, $ \gamma^j
\rightarrow \gamma $ and $ \dot{\gamma}^j \rightarrow \dot{\gamma}
$ uniformly on $ [0,1]$. Appealing to (\[E1\]) yields $$\int_0^1 {F_{D^j} \big( \gamma^j(t), \dot{\gamma}^j(t) \big) dt}
\leq \int_0^1 { F_{D_{\infty}} \big( \gamma(t), \dot{\gamma}(t)
\big) dt} + \epsilon/2 \leq d_{D_{\infty}} ( 0, q) + \epsilon.$$ Therefore, $$d_{D^j} \left( A^j(p^j), q \right) \leq \int_0^1 {F_{D^j} \big( \gamma(t),
\dot{\gamma}(t) \big) dt} \leq d_{D_{\infty}} \left( 0, q \right) +
\epsilon$$ which implies that $$\displaystyle\limsup_{j \rightarrow \infty} d_{D^j} \left( A^j(p^j), \cdot \right)
\leq d_{D_{\infty}} \left( 0, \cdot \right),$$ thereby, establishing (\[E21\]).
Next, recall that $ B_{ D } ( p^j, R_j) \subset F^j(\mathbb{B}^n) $. Since $ A^j $ are biholomorphisms and hence Kobayashi isometries, it follows that $$\begin{aligned}
B_{ D^j} \left( A^j(p^j) , R_j \right) \subset A^j \circ F^j (\mathbb{B}^n). \label{5.0}\end{aligned}$$ Since $ ( D_{\infty}, d_{D_{\infty}}) $ is complete, it is possible to write $$\begin{aligned}
D_{\infty} = \displaystyle \bigcup_{ \nu = 1} ^{\infty}
B_{D_{\infty}} \big( 0, \nu \big). \label{k13}\end{aligned}$$ Now, consider the mappings $$\theta^j := \big( A^j \circ F^j \big)^{-1}: A^j \circ F^j (\mathbb{B}^n) \rightarrow
\mathbb{B}^n.$$ It follows from (\[k13\]), (\[k11\]) and (\[5.0\]) that the mappings $ \theta^j$ are defined on any arbitrary compact subset of $ D_{\infty} $ for large $j$. In particular, $ \{\theta^j \} $ is normal. Let $ \theta: D_{\infty} \rightarrow
\overline{\mathbb{B}}^n$ be a holomorphic limit of some subsequence of $ \{ \theta^j \}$. Since $ \theta \big( 0 \big) = 0 $, it follows that $ \theta :
D_{\infty} \rightarrow \mathbb{B}^n $. The final step is to note that $ \psi \circ \theta = id $ and $ \theta \circ \psi = id$ as before. A consequence of all of this is that $
D_{\infty} $ is biholomorphically equivalent to $ \mathbb{B}^n$.
Now we seek an contradiction. If $ p^0 \in \partial D $ were $ C^{\infty} $-convex of infinite type, then the limit domain $ \partial D_{\infty} $ contains a non-trivial complex affine disc (cf. Proposition 6.1, [@Z1]) and hence, it follows from Theorem 3.1 of [@Z1] that $ \left(D_{\infty}, d_{D_{\infty}} \right) $ is not Gromov hyperbolic. Since $ \left( \mathbb{B}^n, d_{\mathbb{B}^n} \right) $ is Gromov hyperbolic and Gromov hyperbolicity is an isometric invariant, it follows that $ \left(D_{\infty}, d_{D_{\infty}} \right) $ and $ \left( \mathbb{B}^n, d_{\mathbb{B}^n} \right) $ cannot be isometric. This is a contradiction since $ D_{\infty} $ is biholomorphic to $ \mathbb{B}^n $. Hence, the boundary point $ p^0 $ has to be of finite type and we are in Case (i).
Proof of Theorem 1.2
====================
Proving Theorem 1.2 involves verifying the following two results:
\[B1\] Let $D \subset {\mathbb}C^n$ be a bounded domain with $p^0 \in \partial D$. Assume that ${\partial}D$ is $C^{\infty}$-smooth and $h$-extendible near $p^0$. Then $\partial D$ is strongly pseudoconvex near $p^0$ if $h_D(z) {\rightarrow}0$ as $z {\rightarrow}p^0$.
\[B2\] Let $D \subset {\mathbb}C^n$ be a bounded domain with $p^0 \in \partial D$. Assume that ${\partial}D$ is $C^{\infty}$-smooth and $h$-extendible near $p^0$. Then $\partial D$ is strongly pseudoconvex near $p^0$ if $s_D(z) {\rightarrow}1$ as $z {\rightarrow}p^0$.
*Proof of Proposition \[B1\].* Let $ p^j $ be a sequence of points in $ D $ converging to $ p^0 $ along the inner normal to $ \partial D $ at $ p^0 $. The boundary point $ p^0 $ is a local peak point by [@Yu-1994] and hence the Fridman’s invariant function can be localized near $ p^0 $ (refer Proposition 3.4 of [@MV]). It follows that $ \lim_{j \rightarrow \infty} h_{U \cap D} (p^j, \mathbb{B}^n ) = 0 $ for any neighbourhood $ U $ of $ z^0 $. As a consequence, there exists a sequence of positive real numbers $ R_j \rightarrow \infty $ and a sequence of biholomorphic imbeddings $ F^j: \mathbb{B}^n \rightarrow U \cap D $ satisfying $ F^j (0)= p^j $ and $ B_{U \cap D}(p^j, R_j) \subset F^j( \mathbb{B}^n) $.
Let us quickly recall the local geometry of $h$-extendible domains. Firstly, if $ \mathcal{M}(\partial D, p^0) = (1, m_2, \ldots, m_n) $ denotes the Catlin’s multitype of $ \partial D $ near $ p^0 $, then there is an automorphism $ \Phi $ of $ \mathbb{C}^n $ such that such that $ \Phi(p^0) = ('0, 0) $, $ \Phi (p^j) = ('0, - \| p^0 - p^j \|) $ for each $ j$ and the defining function for the domain $ \Phi(D) $ can be expanded near the origin as $$\begin{aligned}
{3}
2 \Re z_n + P('z, ' \overline{z}) + R(z), \end{aligned}$$ where $ P('z, ' \overline{z}) $ is a $ (1/m_n , 1/m_{n-1} , \ldots , 1/m_2 ) $ homogeneous polynomial of weight one which is plurisubharmonic and does not contain pluriharmonic terms and $ R $ satisfies $$|R(z)| \lesssim \left( |z_1 |^{m_n} + |z_2 |^{m_{n-1}} + \ldots + |z_n | \right)^{\gamma}$$ for some $ \gamma > 1 $. Let $ T^j $ be the dilation defined by $$T^j( z_1, z_2, \ldots, z_{n-1}, z_n ) = \left( \delta_j^{-1/{m_n}} z_1, \delta_j^{-1/{m_{n-1}}} z_2, \ldots, \delta_j^{-1/m_2} z_{n-1}, \delta_j^{-1} z_n \right),$$ where $ \delta_j = \|p^0 - p^j \| $ for each $j$. Note that $ T^j \big( ('0, - \delta_j) \big) = ('0, -1) $ for all $j$ while the domains $ D^j := T^j \circ \Phi(U \cap D) $ converge in the Hausdorff sense to $$D_{ \infty} = \big\{ z \in \mathbb C^n : 2 \Re z_n + P('z, ' \overline{z}) < 0 \big\}.$$ On the other hand, by Theorem 4.7 of [@Yu-1995], there is an $h$-extendible model $$\Omega_0 = \big\{ z \in \mathbb{C}^n : 2 \Re z_n + Q('z, ' \overline{z}) < 0 \big\},$$ where $ Q('z, ' \overline{z}) $ is a $ (1/m_n , 1/m_{n-1} , \ldots , 1/m_2 ) $ homogeneous polynomial (of weight one which is plurisubharmonic but not pluriharmonic), such that if $ U $ is a small neighbourhood of $ p^0 $, then $$\Phi(U \cap D) \subset \Omega_0.$$ Consequently, the scaled domains $ D^j $ satisfy $$\begin{aligned}
D^j = T^j \circ \Phi(U \cap D) \subset T^j(\Omega_0)\end{aligned}$$ for all $j $ large. The homogenity of $ Q('z, ' \overline{z}) $ with weight one as described above implies that $$Q \left( \delta_j^{1/{m_n}} z_1, \delta_j^{1/{m_{n-1}}} z_2,
\ldots, \delta_j^{1/m_2} z_{n-1} \right) = \delta_j Q ('z, ' \overline{z} )$$ for each $j $. In other words, the mappings $ T^j $ leave $ \Omega_0 $ invariant and hence $$\begin{aligned}
\label{E15}
D^j = T^j \circ \Phi(U \cap D) \subset \Omega_0.\end{aligned}$$ Furthermore, observe that $ D_{\infty} $ and $ \Omega_0 $ are complete hyperbolic (cf. [@Yu-1994]) and hence both $ D_{\infty} $ and $ \Omega_0 $ are taut.
Let us consider the holomorphic mappings $$\psi^j := T^j \circ \phi \circ F^j : \mathbb{B}^n \rightarrow D^j.$$ Note that $ T^j \circ \phi \circ F^j \big( ('0,0) \big) = ('0,-1)$ for each $ j $. As the scaled domains $ D^j $ are contained in a taut domain $ \Omega_0 $, the sequence $ \{ T^j \circ \phi \circ F^j \} $ admits a subsequence, that will still be denoted by the same indices, that converges uniformly on compact sets of $ \mathbb{B}^n $ to a holomorphic mapping $ \psi : \mathbb{B}^n \rightarrow D_{\infty} $.
Here, too, it turns out that $ \psi $ is a biholomorphism from $ \mathbb{B}^n $ onto $ D_{\infty} $. To establish this, first note that $$\begin{aligned}
\label{E4}
F_{D^j} ( \cdot, \cdot) \rightarrow F_{D_{\infty}} ( \cdot, \cdot) \end{aligned}$$ uniformly on compact sets of $ D_{\infty} \times \mathbb{C}^n $. The above assertion can be proved using the fact that the scaled domains $ D^j $ are all contained in $ \Omega_0 $ for large $ j $ (cf. (\[E15\])). Once the stability of the infinitesimal metric is understood (i.e., (\[E4\])), by similar arguments as in the proof of Theorem 1.1(i), it follows that for each $ \epsilon > 0 $, $$B_{D_{\infty}} \left( ('0,-1), R - \epsilon \right) \subset B_{D^j} \left( ('0,-1), R \right)
\label{E2}$$ for all $ R > 0 $ and all $j$ large.
Recall that $ B_{U \cap D } ( p^j, R_j) \subset F^j(
\mathbb{B}^n)$. Since $ T^j \circ \phi $ are biholomorphisms and hence Kobayashi isometries, it follows that $$\begin{aligned}
B_{ D^j} \left( ('0,-1), R_j \right) \subset T^j \circ \phi \circ F^j (\mathbb{B}^n). \label{E5}\end{aligned}$$ Exploiting the Kobayashi completeness of the limit domain $ D_{\infty}$, it is possible to write $$\begin{aligned}
D_{\infty} = \displaystyle \bigcup_{ \nu = 1} ^{\infty}
B_{D_{\infty}} \big( ('0,-1), \nu \big). \label{E6}\end{aligned}$$ Now, consider the mappings $$\theta^j := \big( T^j \circ \phi \circ F^j \big)^{-1}: T^j \circ \phi \circ F^j (\mathbb{B}^n) \rightarrow
\mathbb{B}^n$$ defined on an arbitrary compact subset of $ D_{\infty} $ for large $j$ (cf. (\[E2\]), (\[E5\]) and (\[E6\])) and hence some subsequence of $ \{ \theta^j \}$ converges to $ \theta: D_{\infty} \rightarrow
\overline{\mathbb{B}}^n$. Moreover, $ \theta \big( ('0,-1) \big) = ('0,0) $ together with the maximum principle shows that $ \theta :
D_{\infty} \rightarrow \mathbb{B}^n $. Then it can be checked that $ \psi \circ \theta = id $ and $ \theta \circ \psi = id$ and hence $
D_{\infty} $ is biholomorphically equivalent to $ \mathbb{B}^n$.
Finally, we are ready to prove the theorem. Using similar arguments as in the proof of Theorem 1.1(i), it is possible to show that there is a biholomorphism $ \tilde{\theta} $ from $ D_{\infty}$ onto the unbounded realization of the ball, namely to $$\Sigma = \big\{ z \in \mathbb{C}^n : 2 \Re z_n + {\left\vertz_1\right\vert}^2 + {\left\vertz_2\right\vert}^2 + \ldots +
{\left\vertz_{n-1}\right\vert}^2 < 0 \big\},$$ with the property that $ \tilde{\theta} $ extends biholomorphically past the boundary of $ D_{\infty} $ to a neighbourhood of the origin. Also, $ \tilde{\theta} \big( ('0,0) \big) =
\big( ('0,0) \big) $. Since the Levi form is preserved under local biholomorphisms around a boundary point, it follows that $ \partial D_{\infty} $ must be strongly pseudoconvex near the origin. In particular, $ m_2= \ldots = m_n =2 $ and $ P('z, ' \overline{z}) = |z_1|^2 + \ldots + |z_n|^2 $ which gives the strong pseudoconvexity of $ p^0 \in \partial D $. Hence the result.
*Proof of Proposition \[B2\].* Let $ p^j $ be a sequence of points in $ D $ converging to $ p^0 $ along the inner normal to $ \partial D $ at $ p^0 $. Since $ \lim_{j \rightarrow \infty} s_{D} (p^j) = 1 $ by assumption, there exists a sequence of positive real numbers $ R_j \rightarrow 1 $ and a sequence of biholomorphic imbeddings $ F^j: D \rightarrow \mathbb{B}^n $ satisfying $ F^j(p^j)=0 $ and $ {B}^n(0, R_j) \subset F^j( D) $.
Let us adapt here the method of uniform scaling. Let $ U $ be a neighbourhood of $ p^0 \in \partial D $ and the mappings $ T^j \circ \Phi $ be as described in the the proof of Proposition \[B1\]. So that the domains $ D^j = T^j \circ \Phi (U \cap D) $ converge in the Hausdorff sense to $$D_{ \infty} = \big\{ z \in \mathbb C^n : 2 \Re z_n + P('z, ' \overline{z}) < 0 \big\}.$$ Here, $ P('z, ' \overline{z}) $ is a $ (1/m_n , 1/m_{n-1} , \ldots , 1/m_2 ) $ homogeneous polynomial of weight one that coincides with the polynomial of same degree in the homogeneous Taylor expansion of the defining function for $ \Phi(D) $ near the origin and $ (1, m_2, \ldots, m_n) $ is the Catlin’s multitype of $ \partial D $ near $ p^0 $.
Consider the maps $$\psi^j := F^j \circ \left( T^j \circ \phi \right)^{-1} : D^j \rightarrow F^j(D) \subset \mathbb{B}^n.$$ Note that $ F^j \circ \left( T^j \circ \phi \right)^{-1} \big( ('0,-1) \big) = ('0,0)$ for each $ j $. These mappings are defined on an arbitrary compact subset of $ D_{\infty} $ for large $j$ and hence some subsequence of $ \{ \psi^j \}$ converges to $ \psi: D_{\infty} \rightarrow
\overline{\mathbb{B}}^n$. Since $ \psi \big( ('0,-1) \big) = ('0,0) $, it follows that $ \psi (D_{\infty}) \subset \mathbb{B}^n $.
The claim is that $ \psi $ is a biholomorphism from $ D_{\infty} $ onto $ \mathbb{B}^n $. To begin with, let $ K $ be an arbitrary compact set of $ \mathbb{B}^n $ containing the origin. Since $ {B}^n (0, R_j) \subset F^j(D) $ and $ R_j \rightarrow 1 $, the mappings $ (F^j)^{-1} $ are defined on $ K $ for $ j $ large. Since $ (F^j)^{-1} \big(('0,0) \big) = p^j \rightarrow p^0 \in \partial D $ and the boundary point $ p^0 $ is a local peak point, it follows from the attraction property (see for instance Lemma 2.1.1 of [@Gaussier-1999]) that $$(F^j)^{-1} (K) \subset U \cap D$$ for all $ j $ large. Therefore $$\left( T^j \circ \phi \right) \circ \left(F^j \right)^{-1} (K) \subset \left( T^j \circ \phi \right) (U \cap D) = D^j$$ and hence the scaling sequence $$\theta^j := \left( T^j \circ \phi \right) \circ \left(F^j \right)^{-1}$$ maps $ K $ injectively into $ D^j $ for all $ j $ large. On the other hand, by (\[E15\]), there is a taut domain $ \Omega_0 $ such that $ D^j \subset \Omega_0 $ for all $j $ large. It follows that $ \{ T^j \circ \phi \circ \left(F^j \right)^{-1}|_K \} $ forms a normal family. Let $ \theta : K \rightarrow
\overline{D}_\infty $ be a holomorphic limit of some subsequence of $ \{ T^j \circ \phi \circ \left(F^j \right)^{-1} \} $. Since $ K $ is an arbitrary compact subset of $ \mathbb{B}^n $, $ \theta $ is defined on whole of $ \mathbb{B}^n $. As noted earlier, $ T^j \circ \phi \circ \left(F^j \right)^{-1}
\big( ('0,0) \big) = ('0,-1)$ for each $ j $ and so $ \theta \big( ('0,0) \big) = ('0, -1) $. But $ ('0, -1) \in D_{\infty} $ and $ D_{\infty} $ is open, so $ \theta (\mathbb{B}^n ) \subset D_{\infty} $.
Now an argument similar to the one employed in Theorem 1.1 shows that $ \psi \circ \theta = id $ and $ \theta \circ \psi = id$, which in turn implies that $ D_{\infty} $ is biholomorphically equivalent to $ \mathbb{B}^n$. As in the proof of Proposition \[B1\], the strong pseudoconvexity of $ p^0 \in \partial D $ follows.
[*Concluding Remarks:*]{} While the construction of $h_D$ and $s_D$ are completely dual to each other, the exact relation between them is unclear from their definitions. It would be interesting and useful to clarify this.
The ratio of the Carathéodory–Eisenmann and the Kobayashi volume forms is another biholomorphic invariant that has been studied in the past. It is known that it is at most $1$ everywhere and that if it equals $1$ at an interior point, then the domain is biholomorphic to ${\mathbb}B^n$. Is there an analog of this result for the ratio $s_D/h_D$?
[*Acknowledgements:*]{} The authors would like to thank A. Zimmer for pointing out a gap in an earlier version of this article.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The protection of quantum states is challenging for non-orthogonal states especially in the presence of noises. The recent research breakthrough shows that this difficulty can be overcome by feedback control with weak measurements. However, the state-protection schemes proposed recently work optimally only for special quantum states. In this paper, **by applying different weak measurements, we extend the idea of the state-protection scheme to protect general states.** We calculate numerically the optimal parameters and discuss the performance of the scheme. Comparison between this extended scheme and the earlier scheme is also presented.'
author:
- 'Y. Yang$^1$, X. Y. Zhang$^1$, J. Ma$^{1,2}$, and X. X. Yi$^1$'
title: Extended Techniques for Feedback Control of A Single Qubit
---
In classical physics, it is possible in principle to acquire all information about the state of a classical system by precise measurements. Namely, the state of a single classical system can be precisely determined by measurements. This ensures the measurement-based classical feedback control and makes the feedback control beneficial to the manipulation of classical system.
For a quantum system, however, this is not possible: If the system is prepared in one of several non-orthogonal states, no measurement can determine determinately which state the system is really in. Furthermore, Heisenberg’s uncertainty principle imposes a fundamental limit on the amount of information obtained from a quantum system, and the act of measurement necessarily disturbs the quantum system [@1; @2; @3; @4; @5] in an unpredictable way. This means when extend the measurement-based classical control theory to quantum system, we need careful examinations of the control scheme. The extension of the classical feedback to quantum systems can be used not only in quantum control [@6; @7; @8; @9; @10; @11], but also in quantum information processing, for example in the quantum key distribution [@12] and quantum computing, as well as in other practical quantum technologies [@13].
Recent works in this field [@14; @15; @16] suggested that we can balance the information gain from a measurement and the disturbance caused by the measurement via weak measurement. To be specific, in Ref.[@15] Branczyk [*et al.*]{} investigated the use of measurement and feedback control to protect the state of a qubit. The qubit is prepared in one of two non-orthogonal states **in the $x-z$ plane of the Bloch sphere and subjected to noise.** The authors shown that, in order to optimize the performance of the state protection, one must use non-projective measurements to balance the trade-off between information gain and disturbance. **The measurement operators used in[@15] are among the $y-$ axis and the subsequent correction is a rotation about the $z-$axis**. This scheme was realized recently [@14], where the stabilization of non-orthogonal states of a qubit against dephasing was experimentally reported. It is shown that the quantum measurements applied in the experiment play an important role in the feedback control. **We should notice that the measurements used in[@14] are different to those in [@15], namely, its measurement operators are along the $z-$axis and the correction is about the $y-$ axis. Geometrically, for initial states in the $x-z$ plane, the dephasing noise can not map the initial states out of the $x-z$ plane, then all states including the initial states, the states passed the noise and measurements as well as the final stats are in the $x-z$ plane in[@14], this is the difference between [@14] and [@15] from the geometric viewpoint. We will modify the measurement operators in [@14] and use it in this paper.**
[With these knowledge in quantum information science [@17; @18; @19], one may wonder if the weak measurement used in the scheme is also the best one for the protection of general states? I.e., $M_+ =\cos(\chi/2)|0\rangle\langle 0| +
\sin(\chi/2)|1\rangle\langle1|\,, $ and $M_- =
\sin(\chi/2)|0\rangle\langle 0| +\cos(\chi/2)|1\rangle\langle 1|\,,$ are these measurements best for the protection of general states? Are there other measurements that can better the performance of the scheme for general states? In this paper, we shall shed light on this issue by introducing different measurements for the feedback control. We find that the scheme can be extended to protect general quantum states with the new weak measurement. We derive the performance and give the parameters best for the performance, a discussion on this extended scheme is also presented.]{}
Consider two non-orthogonal states that we want to protect from noise, $$\label{initial state} {\vert\psi_{\pm}\rangle}{=}
\cos\frac{\theta}{2}{\vert+\rangle}{\pm}e^{i\phi}\sin\frac{\theta}{2}{\vert-\rangle},$$ with ${\vert\pm\rangle}{=}\frac{1}{\sqrt{2}}({\vert0\rangle}\pm{\vert1\rangle})$, the corresponding density matrices are given by $\rho_\pm{=}{\mbox{$|\psi_\pm\rangle\langle \psi_\pm|$}}$. Note that $|\psi_+\rangle$ and $|\psi_-\rangle$ are non-orthogonal and are more general than the states in [@14; @15], the overlapping of the two states is independent of $\phi$, but depends on $\theta$, $\langle\psi_+|\psi_-\rangle=\cos\theta.$ In fact, ${\vert\psi_{\pm}\rangle}$ are rotated about the $x$-axis with respect to the Branczyk’s one, this may offer a chance to improve the fidelity given by the previous proposals [@14; @15] for general states of a qubit.
The qubit is subjected to dephasing noises [@14; @15]. We shall use $\{{\vert0\rangle},{\vert1\rangle}\}$ as the basis of the qubit Hilbert space, and define the Pauli operator $Z$ as $Z{\vert0\rangle}{=}{\vert0\rangle},
Z{\vert1\rangle}{=}{-}{\vert1\rangle}$, similar definitions are for Pauli matrices $X$ and $Y$. The dephasing noise can be described by a phase flip $Z$ with probability $p$ and with probability $1-p$ that the system remains unchanged. The density matrix of the qubit passed through the noisy channel is, $$\rho^{'}_{\pm}= (1-p) \rho_{\pm}+p Z \rho_{\pm} Z.\label{noise}$$
The purpose of this paper is to find better measurements and controls to send the qubit back as close as possible to its initial state. For this purpose, we use a quantum operation $\mathcal{C}$ as a map acting on the single qubit to describe the controls and measurements, $$\mathcal{C}(\rho^{\prime})=Y_{+\eta}M^{\prime}_{+}
\rho^{\prime}M_{+}^{\prime\dagger}Y_{+\eta}^{\dagger}+Y_{-\eta}M^{\prime}_{-}
\rho^{\prime}M_{-}^{\prime\dagger}Y_{-\eta}^{\dagger}.$$ The notations of $Y$ and $M^{\prime}$ will be given later. To quantify the performance of $\mathcal{C}$, we use the average fidelity $\overline{F}$ [@14; @15] between the noiseless input state and the corrected output state as a measure, $$\begin{aligned}
\overline{F}&=\tfrac{1}{2}[{\langle\psi_+\vert}\mathcal{C}(\rho'_+){\vert\psi_+\rangle}
+{\langle\psi_-\vert}\mathcal{C}(\rho'_-){\vert\psi_-\rangle}]\nonumber\\
&=\tfrac{1}{2}(F_{\psi_+}+F_{\psi_-})\,.\end{aligned}$$ This measure quantifies the performance well when ${\vert\psi_+\rangle}$ and ${\vert\psi_{-}\rangle}$ are sent into the control with equal probability.
![Illustration of the scheme. The meter qubit was entangled with the qubit (for protection) which has passed through the noise channel. After the measurement and correction, we can get a signal. Then we compare the signal with the initial state and apply a feedback control to the qubit. An average fidelity is define and used to determine the parameters in the feedback control. In the earlier scheme, the authors use a ${\vert\varphi\rangle}{=}\cos\frac{\chi}{2}{\vert+\rangle}{+}\sin\frac{\chi}{2}{\vert-\rangle}$ as the meter-qubit state, while in the present scheme a complex phase factor is introduced, i.e. the meter state is, ${\vert\varphi\rangle}{=}\cos\frac{\chi}{2}{\vert+\rangle}{+}e^{i\beta}
\sin\frac{\chi}{2}{\vert-\rangle}.$[]{data-label="FIG:1"}](yyfig1){width="8cm" height="4cm"}
To find a good control procedure, we must first find the appropriate measurement which has to have the following two features. First, it must be a weak measurement, that is, it can not completely disturb the system. Second, it has to be strength-dependent, such that we can adjust the strength of the measurement as we need. This family of weak measurements in the logical basis $\{{\vert0\rangle},{\vert1\rangle}\}$ can be written as, $$\begin{aligned}
\label{eq:M'_{+}}
M'_+ &= \cos(\chi/2)|0\rangle\langle 0| + e^{i\beta}\sin(\chi/2)|1\rangle\langle 1|\,, \\
\label{eq:M-} M'_- &= e^{i\beta}\sin(\chi/2)|0\rangle\langle 0| +
\cos(\chi/2)|1\rangle\langle 1|\,.\end{aligned}$$ In contrast to the measurements used in Ref.[@14; @15; @16], $M_+
=\cos(\chi/2)|0\rangle\langle 0| + \sin(\chi/2)|1\rangle\langle1|\,,
$ $M_- = \sin(\chi/2)|0\rangle\langle 0|
+\cos(\chi/2)|1\rangle\langle
1|\,,$ a new parameter $\beta$ was introduced in this weak measurement [@20; @21]. Here $\chi$ ranges from 0 to $\pi/2$ [@20], we can change the value of the parameter $\chi$ to adjust the strength of measurement. The corresponding positive measurement operators are given by $\Pi_\pm{=}M^{'\dag}_\pm M'_\pm
{=}[\mathbbm{1}\pm\cos{(\chi)}Z]/2$, with $\mathbbm{1}$ being the identity operator. Clearly, $\chi=0$ describes the projective measurement, while $\chi=\frac{\pi}{2}$, do nothing. At first glance, this proposal is trivial, i.e., the initial states (the state sent into protection) are rotated about $x-$axis in the Bloch sphere with respect to that in Ref.[@15], by properly choosing $\beta$, the next measurements $M_+^{\prime}$ and $M_-^{\prime}$ may send them back, then the resulting states will return to that in the earlier proposal, and the performance can not be improved. We will show later that this is not the case.
Our main task is to figure out how the parameter $\beta$ affects the results of the control, and if the parameter $\beta$ can better the performance. The correction performed in this paper is the same as that in [@14], i.e., $Y_{\pm\eta} = \exp(\pm
i\tfrac{1}{2}\eta Y)$ representing a rotation with an angle $\eta$ around the $y-$axis of the Bloch sphere. All parameters should be optimized for the performance of the control.
Straightforward calculation show that the average fidelity of the control is a function of $\theta, \phi, \eta, \chi, \beta$ and $p$, $$\begin{gathered}
\label{fertility}
\overline{F'}(\theta,p,\chi,\eta,\phi,\beta)=\\
\tfrac{1}{2}\left[1+\cos\theta\cos\chi\sin\eta+
\cos\eta\cos^{2}\phi\sin^{2}\theta\right.\\
+\tfrac{(1-2p)}{2}\sin\chi\left(2\cos\beta
(\cos\eta\cos^{2}\theta+\sin^{2}\theta\sin^{2}\phi)\right.\\
\left.\left.-\sin\beta\sin\eta\sin^{2}\theta\sin2\phi\right) \right]
\,.\end{gathered}$$ [For each $\theta,$ $\phi$ and $p$, there are an optimum measurement strength $\chi,$ correction angle $\eta,$ and measurement parameter $\beta$, which maximizes the average fidelity. First we start with $\eta$.]{} The $\eta$ which optimizes the average fidelity can be given by, $$\begin{gathered}
\label{N}
\eta_{\rm opt}(\theta, p, \chi,\phi,\beta)\\
=\arctan{\frac{\cos\theta\cos\chi-\tfrac{1}{2}(1-2p)
\sin\beta\sin^{2}\theta\sin2\phi\sin\chi}{\cos^{2}\phi\sin^{2}\theta
+(1-2p)\cos^{2}\theta\sin\chi\cos\beta}
} \,.\\\end{gathered}$$ [Substituting the optimum $\eta_{opt}$ into the average fidelity, we have,]{} $$\begin{gathered}
\label{fertility'}
\overline{F'}(\theta,p,\chi,\phi,\beta)=
\tfrac{1}{2}+\tfrac{1}{2}(1-2p)\cos\beta\sin^{2}\theta\sin^{2}\phi\sin\chi\\
+\frac{1}{2}\left[(\cos\theta\cos\chi
-\tfrac{1}{2}(1-2p)\sin\beta\sin^{2}\theta\sin2\phi\sin\chi)^{2}\right.\\
+(\cos^{2}\phi\sin^{2}\theta
\left.+(1-2p)\cos^{2}\theta\sin\chi\cos\beta)^{2}\right
]^{\tfrac{1}{2}}\,.\end{gathered}$$ We can see that when $\phi=0$, $\overline{F'}(\theta,p,\chi,\phi,\beta)$ reduces to $$\begin{aligned}
&\ &\overline{F'}|_{\phi=0}=\frac 1 2 +\frac 12\left
[\cos^2\theta\sin^2\chi\right. \nonumber\\
&\
&\left.+(\sin^2\theta+(1-2p)\cos^2\theta\sin\chi\cos\beta)^2\right
]^{\frac 12}.\end{aligned}$$ Obviously, $\beta=0$ maximize the average fidelity $\overline{F'}$, this is exactly the case discussed in Ref. [@14; @15]. So, for the initial states lying in the $xz-$ plane of the Bloch sphere, the weak measurements with $\beta=0$ already maximize the performance.
![This figure shows how much our scheme improve the performance of the state protection for general qubit states. The improvement is quantified by $\delta_{F}$, which is plotted as a function of $\theta$ and $\phi$. For different $p$, the improvement is different, as (a), (b), (c) and (d) show. (a)$p=0.10;$ (b)$p=0.20;$ (c)$p=0.30;$ (d)$p=0.40.$[]{data-label="FIG:2"}](yyfig2){width="8.5cm" height="6cm"}
To find the optimal feedback control for $\phi\neq 0$, we follow the procedure in [@14]. Here again $\theta$ and $\phi$ are related to the initial state of the qubit, while $p$ characterizes the noise and is regarded as a fixed value, $\chi$ and $\beta$ are related to the measurement procedure, $\eta$ denotes the correction parameter.
![The $\beta$ which maximizes $\delta_{F}$ as a function of $\theta$ and $\phi$ for different $p$, (a)$p=0.10;$ (b)$p=0.20;$ (c)$p=0.30;$ (d)$p=0.40.$[]{data-label="FIG:3"}](yyfig3){width="8.5cm" height="6cm"}
By the same procedure as in the earlier works, we maximize the fidelity of the control over the remaining parameters $\chi$, $\theta, \phi, \beta$ and $p$. The analytical expression for the fidelity is complicated, so we choose to find the optimal parameters by numerical simulations. As aforementioned, we have already had the relations between the average fidelity and the initial parameters $\theta$ and $\phi$. We shall use $\delta_{F}{=}F'_{opt}{-}F_{opt}$ to quantify the improved fidelity due to the parameter $\beta$, select results are presented in Fig.\[FIG:2\], where $F'_{opt}$ denotes the optimal fidelity in our paper, while $F_{opt}$ denotes that by the scheme in Ref.[@14; @15], i.e., with $\beta=0.$ The optimized $\beta$ would depend on $\theta$ and $\phi$ and is shown in Fig.\[FIG:3\].
![The fidelity difference $\delta_{F}$ versus $p$.[]{data-label="FIG:4"}](yyfig4){width="8cm" height="5cm"}
Fig.\[FIG:2\] plots the improvement of the average fidelity as a function of the original states (characterized by $\theta$ and $\phi$) with different amount of noise (characterized by $p$). We note that there are no improvement for the following cases. If $p=0$, there is no noise and so the state is not perturbed, in this case the fidelity is 1 for all original states including $\phi=0$ and the measurement strength is $\chi=\frac{\pi}{2}$ (do nothing). When $\theta=\frac{\pi}{2}$, the state $|\psi_+\rangle$ and $|\psi_-\rangle$ are orthogonal, the earlier scheme gives unit fidelity, hence there is no room to improve the performance. When $\theta=0$ the two states are equal and point along the $x-$axis, these states are also the same as that in the earlier scheme, leading to zero improvement. If $\phi=\frac{\pi}{2}$, nothing should change since the two states would interchange by this control. Finally, when $\phi=0,$ the initial states return to the earlier scheme. Fig.\[FIG:3\] shows the parameter $\beta$, which maximize the average fidelity as a function of the original states and the amount of noise $p$. As expected, non-zero maximal $\beta_{opt}$ exists. To show clearly the dependence of the improvement on the noise strength, we plot $\delta_F$ in Fig. \[FIG:4\] as a function of $p$. The maximal improvement arrived at about p=0.1800, the corresponding improvement is $\delta_{F}$=0.0102.
![Bloch vectors of the original states(blue-solid), the resulting states by our scheme (red-dashed) and the resulting states in [@15] (green-dotted). The parameters chosen are, $p=0.18, \eta=0.7913, \chi=0.8583, \beta=5.8905, \theta=1.0155,
\phi=0.8976.$ All parameters except $p$ are in units of arc. []{data-label="FIG:7"}](yyfig7){width="6.5cm" height="5.5cm"}
For developing an intuitive picture, we now take a snapshot for the states going through the control and measurement. Suppose the initial state is $|\psi_+\rangle$ with $\phi=\frac{\pi}{4}$, and let $\{|0\rangle, |1\rangle\}$ be a basis for the Hilbert space. In terms of density matrix, the initial state is $$\rho_{+}=\frac 1 2 \mathbbm{1}+\frac 1 2\cos\theta\cdot\sigma_{x}
-\frac{\sqrt{2}}{4}\sin\theta\cdot\sigma_{y}+\frac{\sqrt{2}}{4}\sin\theta\cdot\sigma_{z},$$ This state lies in the $z=-y$ plane and points along the direction with an angle $\theta$ from the $x-$axis. The state passed the noisy channel is $\rho^{'}_{+},$ $$\begin{aligned}
\rho^{'}_{+}&=&\frac 1 2\mathbbm{1}\nonumber\\
&+&(\frac 1 2-p)\cos\theta\cdot\sigma_{x} \nonumber\\
&+&\frac{\sqrt{2}}{2}(p-\frac 1 2)\sin\theta\cdot\sigma_{y}\nonumber\\
&+&\frac {\sqrt{2}} {4}\sin\theta\cdot\sigma_{z},\end{aligned}$$ we see that the $z-$component of the Bloch sphere remains unchanged, while the $x-$ and $y-$ components are shortened by $(1-2p)$ times due to the noise. The resulting state (unnormalized) immediately after the measurement is denoted by $\rho^{m}_{+},$ and it takes, $$\begin{aligned}
\rho^{m}_{+}&=&\frac 1 2
(1+\frac{\sqrt{2}}{2}\cos\chi\sin\theta)\cdot\mathbbm{1}\nonumber\\
&+&\frac 1
2(1-2p)(\cos\beta\cos\theta+\frac{\sqrt{2}}{2}\sin\beta\sin\theta)\sin\chi\cdot\sigma_{x}\nonumber\\
&+&\frac 1 2
(1-2p)(\cos\theta\sin\beta-\frac{\sqrt{2}}{2}\cos\beta\sin\theta)\sin\chi\cdot\sigma_{y}\nonumber\\
&+&\frac 1 2
(\cos\chi+\frac{\sqrt{2}}{2}\sin\theta)\cdot\sigma_{z}.\label{statem}\end{aligned}$$ Finally after the correction $Y_{+\eta}$, the unnormalized states has been mapped into,
$$\begin{aligned}
\rho^{c}_{+}&=&\frac 1 2
(1+\frac{\sqrt{2}}{2}\cos\chi\sin\theta)\cdot\mathbbm{1}\nonumber\\
&+&\frac 1 2 \left(\sin\eta(\cos\chi+\frac{\sqrt{2}}{2}\sin\theta)
+(1-2p)\cos\eta\sin\chi(\cos\beta\cos\theta+\frac{\sqrt{2}}{2}\sin\beta\sin\theta)\right )
\cdot\sigma_{x}\nonumber\\
&+&\frac 1 2
(1-2p)(\cos\theta\sin\beta-\frac{\sqrt{2}}{2}\cos\beta\sin\theta)\sin\chi\cdot\sigma_{y}\nonumber\\
&+&\frac 1 2 \left(\cos\eta(\cos\chi+\frac{\sqrt{2}}{2}\sin\theta)+
(-1+2p)\sin\eta\sin\chi(\cos\beta\cos\theta+\frac{\sqrt{2}}{2}\sin\beta\sin\theta)\right
)\cdot\sigma_{z}.\end{aligned}$$
Note that this state is also unnormalized. For a specific set of $\theta$, $\phi$ and $p$, the resulting state together with the resulting state in Ref.[@14] are illustrated in Fig. \[FIG:7\]. This shows clearly that our resulting states are more close to the initial state than that given by the proposal with $\beta=0$. As shown, the new measurements can do better than the earlier one for general quantum states. This suggests that we can apply the new set of measurements to the feedback control. Now we examine how much this new scheme improves the fidelity with respect to the schemes with measurements “do nothing” and “strong measurement” (Helstrom).
Before processing, we briefly review the two special cases of the schemes, which differ from each other at the measurements: In the zero strength measurement, $\cos\chi=0$, namely, no measurement is applied. So the state protection with this measurement is called “do nothing” (DN) control scheme; The projective measurement is applied with maximum strength ($\cos\chi=1$), with which the protection scheme had already been named as “Helstrom” (H) scheme[@22]. In fact, DN control is actually not a measurement-based control because of no application of measurement to quantum states. And H scheme is not what we need, because it makes an unnecessary correction to the system. To quantify the fidelity difference between these schemes, we define $$F_{imp}=F_{opt}^{\prime}-max\{F_{DN},F_{H}\}$$ as a measure to quantify the difference, where $F_{DN}$ is the fidelity of DN control scheme, while $F_H$ represents the fidelity of the H scheme.
![$F_{imp}$ versus $\theta$ and $\phi$ with different $p$,(a)$p=0.10$; (b)$p=0.20$; (c)$p=0.30$; (d)$p=0.40.$ This figure shows the improvement of our scheme over the DN and H schemes.[]{data-label="FIG:5"}](yyfig5){width="8cm" height="5cm"}
We have performed numerical simulations for $F_{imp}$, selective results are presented in Fig.\[FIG:5\] and Fig. \[FIG:6\]. In Fig.\[FIG:5\], we present $F_{imp}$ as a function of $\theta$ and $\phi$ for different $p$. A common feature is that $F_{imp}$ reach its maximum at around $\theta=\pi/4$ and $\phi=\pi/4$. For different $p$, the improvement in the fidelity is different.
![$F_{imp}$ as a function of $p$. In this figure, $F_{imp}$ is numerically optimized over $\theta$ and $\phi$ for each $p$. $p$ runs from 0 to 0.5, covering all possible choices.[]{data-label="FIG:6"}](yyfig6){width="8cm" height="5cm"}
To show the dependence of $F_{imp}$ on $p$ clearly, we plot the maximum $F_{imp}$ versus the parameter $p$ in Fig. \[FIG:6\] with different $\theta$ and $\phi$. As the figure shows, when $p=0.2501$, $F_{imp}$ reaches the maximum value 0.0662. Although the improvement is small, it can work under most conditions and it does improve the state protection over other schemes with different measurements[@16]. This tells that the scheme without the parameter $\beta$ is not the best scheme for state protection of general states.
[It is illustrative to view the difference between our scheme (see Fig.\[fig8\](Right top)) and the scheme (Fig.\[fig8\](Left top)) in Ref.[@14] on the Bloch sphere. In Fig.\[fig8\](Left top), we can see that the original states $|\psi_+\rangle$ and $|\psi_-\rangle$ (green) are shorten by the noise, but the $z-$component of the Bloch vector remains unchanged (pink vector on the Bloch sphere, i.e., $\rho^{\prime}_{\pm}$). The measurements lengthen the Bloch vectors(blue, i.e., $M_+^{\prime}\rho^{\prime}M_+^{\prime\dagger}$) and diminish the angle between the Bloch vector and the $z-$ axis. We should remind that the Bloch vectors remains in the $xz-$ plane in the whole process of measurements and controls, this is the core difference between the scheme in [@14] and ours. This difference offers us a room to improve the performance of the control.]{}
In our scheme, the original states are rotated about the $x-$axis with respect to the earlier scheme, see Fig.\[fig8\](Right top). The effect of the noise is not only to shorten the length of the Bloch vector of the states, but also map the Bloch vector out of the plane of the original states. When the measurement is made, two things happen, as Fig.\[fig8\] (Right top) shows. (1) The Bloch vector is lengthened, in other words, the state become more pure, see also Eq.(\[statem\]). (2) The $x$ and $y$ components of the Bloch vector is mixed, in contrast to the proposal with $\beta=0$. As a consequence, the next rotation $Y_{\pm\eta}$ about the $y-$axis may make the resulting states (red vector) more close to the original states with respect to the earlier scheme.
Both control schemes in [@14] and [@15] are optimal for depolarizing noise and states lying in the $x-z$ plane, the depolarizing noise keeps these particular states in the $x-z$ plane and maintains the trace distance between the two states. If the original states are not in the $x-z$ plane, the depolarizing noise can not maintain the trace distance between the two states and causes the plane in which the two states lie to rotate as the states pass through the depolarizing channel. The optimal control scheme will depend on the orientation of the post-noise states. **From the optimality proof in Ref. [@15], we find that one optimal scheme is to use measurement operators to prolong the Bloch vectors of the post-noise states, and the correction is to bring the post-measurement states to the initial states as close as possible. The measurements and the correction are closely connected for a high performance.** In the present scheme, the optimal scheme is to use measurement operators that can map the two post-noise states as close as possible to the cone formed by the initial states. Specifically, the Bloch vectors of the initial state, the post-noise state and the post-measurement state form three cones(see the bottom figure of Fig. \[fig8\]), these cones share an axis: the $y-$axis, which pass perpendicularly through the centers of the bases. The three cones have a common apes, i.e., the origin of the Bloch sphere. One optimal scheme is to use measurement operators that map the two post-noise states very close to the initial-state-cone. The correction is a rotation about the $y-$axis, which would rotate the post-measurement states as close as possible to the initial states. This analysis simply consider the rotations of the axes of the Bloch sphere, to have a good performance, the length of the post-measurement state should be taken into account, this makes the optimization of $\beta$ complicated.
It is worth emphasizing that the angle rotated of our initial states is $\phi$. One may suspect that when the measurement cancels this rotation and send the states back to the $x-z$ plane, i.e., $\beta=\phi$, the optimal performance can be obtained. This intuition comes from the optimality proof in Ref.[@15], however this is not true as we shall show below.
[By using the average fidelity $\overline{F'}(\theta,p,\chi,\eta,\phi,\beta)$ in Eq.(\[fertility\]), we can calculate $\frac{\partial\overline{F'}}{\partial \beta}.$ From $\frac{\partial\overline{F'}}{\partial \beta}|_{\beta=\beta_c}=0,$ $\beta_c$ follows, which maximize the average fidelity $\overline{F'}$ and takes, $$\tan\beta_c=-\frac12\frac{\sin\eta\sin^2\theta\sin2\phi}
{\cos\eta\cos^2\theta+\sin^2\theta\sin^2\phi}.$$]{} [Clearly, the $\beta$ that maximize the performance depends not only on $\phi$ and $\theta$, but also on $\eta$, namely, it connects closely with the correction $Y_{\pm\eta}$. When $\phi=0$, $\beta_c=0$, returning back to the earlier scheme. This observation can be understood as follows. We denote $U$ the rotation about the $x-$axis, which sends the initial state back to the $xz-$plane, i.e., $\rho_{\pm}=U\tilde{\rho}_{\pm}U^{\dagger}.$ Here, $\tilde{\rho}_{\pm}=\rho_{\pm}|_{\phi=0}$. Then the resulting state $\mathcal{C}(\rho^{\prime})$ can be written as, $$\mathcal{C}(\rho^{\prime})=U\left(
\tilde{Y}_{+\eta}\tilde{M}^{\prime}_{+}
\tilde{\rho}^{\prime}\tilde{M}_{+}^{\prime\dagger}\tilde{Y}_{+\eta}^{\dagger}+
\tilde{Y}_{-\eta}\tilde{M}^{\prime}_{-}
\tilde{\rho}^{\prime}\tilde{M}_{-}^{\prime\dagger}\tilde{Y}_{-\eta}^{\dagger}\right
) U^{\dagger},\label{Cnew}$$ where $\tilde{\rho}^{\prime}=(1-p)\tilde{\rho}
+p\tilde{Z}\tilde{\rho}\tilde{Z},$ and $\tilde{(...)}=U^{\dagger}(...)U.$ This suggests that when the initial states are written as the same as that in the earlier scheme, the noise, measurement and the correction all need to change. Since $X$, $Y$ and $Z$ do not commute with each other, these changes are not trivial.]{} We should emphasize that the effect of the noise given in Eq. (\[noise\]) is to spoil the off-diagonal elements of the density matrix, or to shorten the $x-$ and $y-$component of the Bloch vector for any state, not only for the states lie in the $xz-$plane, so the aim of our scheme is to protect states against the same noise as that in the earlier scheme.
In conclusion, we introduce new measurements to better the state protection for a qubit. The average fidelity is calculated and discussed. [Numerical optimizations over these parameters show that the new measurements can extend the state protection scheme from special states to general states. This scheme works for a wide range of initial states and generalize the scheme in the earlier works.]{} The construction of the new proposal has several advantages. First, the initial states are more general, namely the corresponding Bloch vectors are allowed to lie outside the $xz-$plane, this extends the range of state protection and makes the scheme more realistic. The effect of the noise is to shorten the $x$ and $y$ components of the Bloch sphere, hence the noise is of dephasing. Second, we made use of a measurement which allow us to mix the $x-$ and $y-$components of the Bloch sphere, offering a room to improve the performance of the state protection. Finally, we note that the key elements to our scheme have already been experimentally demonstrated [@14], we expect that this extension of the earlier quantum control scheme is within reach of current technologies. \
\
This work is supported by the NSF of China under Grants Nos 61078011, 10935010 and 11175032.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A class of compact, isolated high–velocity clouds which plausibly represents a homogeneous subsample of the HVC phenomenon in a single physical state was objectively identified by Braun and Burton (1999). Six examples of the CHVCs, unresolved in single–dish data, have been imaged with the Westerbork Synthesis Radio Telescope. The high–resolution imaging reveals the morphology of these objects, including a core/halo distribution of fluxes, signatures of rotation indicating dark matter, and narrow linewidths constraining the kinetic temperature of several opaque cores. In these regards, as well as in their kinematic and spatial deployment on the sky, the CHVC objects are evidently a dynamically cold ensemble of dark–matter–dominated clouds accreting onto the Local Group in a continuing process of galactic evolution.'
author:
- 'W.B. Burton'
- 'R. Braun'
title: 'Morphological Characteristics of Compact High–Velocity Clouds Revealed by High–Resolution WSRT Imaging'
---
psfig.sty
Introduction
============
The criteria applied by Braun and Burton (1999) to the Leiden/Dwingeloo Survey of Hartmann & Burton (1997) and the HVC catalog of Wakker & van Woerden (1991) led to a catalog of 65 confirmed examples of compact, isolated high–velocity clouds. The selection criteria excluded the Magellanic Stream and all other HVC complexes. The catalog is more likely to represent a single phenomenon than would a sample which included the major HVC complexes. The CHVC objects plausibly originated under common circumstances, have shared a common evolutionary history, are arguably in a single physical state, and have not (yet) been strongly influenced by the radiation field of the Milky Way or of M31, or by a gravitational encounter with one of these major systems. In this context, the extended HVC complexes would be the nearby objects currently undergoing accretion onto the Galaxy, while the more compact, isolated ones would be their distant counterparts in the Local Group environment. The positional and kinematic characteristics of the compact HVCs are similar in many regards to those of the Local Group galaxies. The sample is distributed quite uniformly over the sky, and defines a well–organized kinematic system. The kinematic signature of this system suggests an in–falling population associated with the Local Group gravitational potential.
Minimized Velocity Dispersion of the CHVC Ensemble\
as an Indication of Local Group Deployment
===================================================
The possibility of the extragalactic nature of high–velocity clouds has been considered in various contexts by, among others, Oort (1966, 1970, 1981), Verschuur (1975), Eichler (1976), Einasto et al. (1976), Giovanelli (1981), Bajaja et al. (1987), Burton (1997), Wakker and van Woerden (1997), Braun and Burton (1999), and Blitz et al. (1999). Nevertheless, no direct distance determination is yet available for any of the individual CHVC objects. Distances are, of course, required to establish the values of important physical parameters: linear size varies as $D$, mass as $D^2$, and density as $D^{-1}$.
Several aspects of the topology of the class are difficult to account for if the CHVCs are viewed as a Milky Way population, in particular if they are viewed as consequences of a galactic fountain. The amplitude of the horizontal motions of these “bullets" is comparable to that of the vertical motions. The vertical motions are larger than expected for free fall onto the Milky Way from material returning in a fountain flow. The topology shows no preference for the terminal–velocity locus, where motions from violent events leading to a fountain would be expected to be most common. Unlike the situation if the major HVC complexes are considered, the CHVCs are scattered rather uniformly across the sky. The CHVCs show no tendency to accumulate in the lower halo of the Milky Way.
On the other hand, regarding both their spatial and kinematic distributions, the CHVCs show substantial similarities with the distributions of the galaxies comprising the Local Group. Since our CHVC sample has both a substantial size and a rather uniform distribution on the sky (see Fig. 1 of Braun and Burton 1999) it is appropriate to use the sample itself to define a best–fitting velocity reference system. Braun and Burton (1999) showed that the velocity dispersion of the ensemble is minimized in a reference frame consistent with the Local Group Standard of Rest. The solar apex which follows directly from a minimization of the velocity dispersion of the CHVC system, namely, $(l_\odot,b_\odot,v_\odot) = (88^\circ,-19^\circ,+293$ km s$^{-1}$), agrees within the errors with that which defines the Local Group Standard of Rest, $(l_\odot,b_\odot,v_\odot) =
(93^\circ,-4^\circ,+316$ km s$^{-1}$), found by Karachentsev & Makarov (1996). The velocity dispersion of the CHVC system in this reference frame is only $\sigma_{\rm XSR}=69$ km s$^{-1}$, while there is a mean in–fall of $v_{\rm LGSR}=v_{\rm XSR}=-100$ km s$^{-1}$.
Figure 1 shows histograms of the velocities in these reference frames. The dispersion of the velocities decreases in a progression from the $v_{\rm LSR}$ reference frame, for which $\sigma_{\rm LSR} = 175$ km s$^{-1}$, via the $v_{\rm GSR}$ ($\sigma_{\rm GSR} = 95$ km s$^{-1}$) and the $v_{\rm LGSR}$ $\sigma_{\rm
LGSR} = 88$ km s$^{-1}$ frames, to the minimum of $\sigma_{\rm
XSR}=69$ km s$^{-1}$ for the $v_{\rm XSR}$ frame. This minimization provides a quantitative demonstration of Local–Group deployment and, in addition, that the CHVC ensemble is dynamically quite cold.
Imaging with the WSRT
=====================
The CHVC objects catalogued by Braun and Burton (1999) are not spatially resolved in the single–dish data of the Leiden/Dwingeloo Survey. Insofar as the objects have not been resolved in angle, it has remained ambiguous if the single-dish linewidth refers to the intrinsic characteristics of a single entity or to the collective behavior of blended features. High–resolution imaging is required to reveal such kinematic properties as intrinsic linewidths and opacity information, or indications of rotational support requiring a higher total mass than available in the alone, as well as the resolved structural morphology. If the CHVCs are in fact a population of primordial clouds scattered throughout the Local Group, then they might reveal some morphological characteristics which would not be consistent with the expectations of other suggested scenarios, in particular for objects ejected by a galactic fountain (e.g. Shapiro and Field 1976, Bregman 1980) or located within the Galactic halo (Moore et al. 1999). High–resolution imaging is also necessary to provide specific targets for deep optical observations. Such optical probes would help clarify the distinction between the CHVCs and (sub–)dwarf galaxies, and any indication of a stellar population would offer a distance indication.
Of the sample of 65 compact, isolated HVCs catalogued by Braun and Burton (1999), only two had been subject to interferometric imaging. Wakker and Schwarz (1991) used the Westerbork array to show that both CHVC114$-$10$-$430 and CHVC111$-$06$-$466 are characterized by a core/halo morphology, with only about 40% of the single–dish flux recovered on angular scales of tens of arcmin, and, furthermore, that the linewidths of the single–dish spectra of these two sources were resolved into components of some 5 width or less. Both of the imaged systems display systematic velocity gradients along the major axis of an elliptical distribution, which Wakker and Schwarz judged to be suggestive of rotation in self–gravitating systems at Local Group distances.
Braun and Burton (2000) have used the Westerbork array to image an additional six objects of the CHVC class. Although only six CHVC sources were imaged in our program, the sources are distributed widely in galactic coordinates, span radial velocities of $-275<v_{\rm LSR}<+165$ , vary in single-dish linewidth from 6 to 95 , and in line flux from 25 to 300 Jy. One twelve–hour integration was obtained for each field in the standard WSRT array configuration, having a shortest baseline of 36 meters. The angular resolution retained for data presentation was about one arcmin, depending on the flux of the emission. The effective velocity resolution was 1.2 times the channel spacing of 2.06 km s$^{-1}$. Examples of the WSRT imaging discussed below reveal a characteristic core/halo arrangement of fluxes, narrow linewidths, and, in several cases, a signature of rotation.
Indications of a Core/Halo Morphology
=====================================
Moment images of the integrated emission, together with several representative spectra, are shown in Figures 2, 3, 4, and 6 for four of the CHVCs observed with the WSRT. In each case the emission is dominated by one or more bright knots, embedded in less intense, more diffuse gas. The linewidths in the cores are narrower than in the halos.
CHVC069+04$-$223, represented in Fig. 2, illustrates the characteristic morphology. The distribution is dominated by a bright elliptical concentration (clump A) of some 15 arcmin extent; several smaller clumps to the South are connected by tenuous emission. (Regarding the tenuous emission, we note that while structures extending over as much as 10 arcmin in a single spectral channel were adequately recovered in the WSRT images, there were also indications in the Leiden/Dwingeloo single–dish data of more diffuse features which could not be adequately imaged by the interferometer. A straightforward attempt was made to correct the images for the weak response of the interferometer to diffuse emission features, making use of the total flux measured with the Dwingeloo telescope. The integrated flux detected in the reconstructed images after primary beam correction varied from less than 1% to as much as 55% of that detected in the Leiden/Dwingeloo Survey.) The velocity dispersion of the emission from the bright core is less than that from the diffuse background. The velocity field of CHVC069+04$-$223 shows a systematic velocity gradient, oriented along the major axis of clump A from about $-$230 to $-$240 and extending over some 10 arcmin.
The basic data for CHVC115+13$-$275, represented in Fig. 3, similarly shows a core/halo morphology. This source had been completely unresolved in the single–dish data, and had shown the broadest linewidth, namely 95 , of all the CHVCs catalogued by Braun and Burton (1999). The WSRT imaging shows that this FWHM value was contributed by several separate knots of emission, each with an intrinsically much narrower velocity FWHM, amounting to about 10 . The collection of substructures, each between 1 and 10 arcmin in size, is distributed over a region of about 30 arcmin extent. Each of the cores has a distinct centroid velocity so that the collection spans the velocity interval from $-$300 to $-$220 . Detailed examination of the WSRT data from the individual cores reveals that several of them have significant velocity gradients, oriented preferentially along their long axes.
Figure 4 shows the basic WSRT data for CHVC204+30+075. Several relatively bright clumps are scattered over an extent of some 35 arcmin. The spectra toward the more compact local maxima have column densities of a few times $10^{20}$ cm$^{-2}$, and FWHM widths of less than 15 . A substantial diffuse component was reconstructed for this field, reaching values of a few times $10^{19}$ cm$^{-2}$ over some 30 arcmin. The large elliptical feature in the South–central part of the CHVC204+30+075 field (clump A) shows a well–defined velocity gradient running from about 55 to 80 over some 12 arcmin. Similarly, the elliptical feature (clump B) in the North–East region has a velocity gradient running from 45 to 75 over 20 arcmin.
Indications of Rotational Support
=================================
The velocity gradients mentioned above as shown by some of the CHVC cores constitute a noteworthy morphological characteristic of these objects. The gradients are oriented along the major axis of roughly elliptical distributions, and vary in amplitude from 0.5 to 2 arcmin$^{-1}$. Examples of the signature of rotation which are particularly well resolved are clump A of CHVC069+04$-$223 and clumps A and B of CHVC204+30+075. These are reminiscent in form and amplitude of the “spider” diagrams portraying the kinematics of some dwarf galaxies. Consequently we carried out standard tilted–ring fits to assess the extent to which the CHVC kinematics could be modeled by rotation in a flattened disk system.
Figure 5 shows that robust solutions for circular rotation pertain for the three cases. The fits indicate rotation velocity rising slowly and continuously with radius, to some 15 in the case of CHVC069+04$-$223A, and flattening out to values of 15 and 20 for CHVC204+30+075A and B, respectively. An estimate of the total mass supporting this rotation follows from $M_{\rm dyn}=Rv^2/{\rm G}=2.3 \times
10^5R_{\rm kpc}v_{\rm km/s}^2$. At an assumed distance of 0.7 Mpc, these cores have $M_{\rm dyn}=10^{7.1}$, $10^{6.5}$, and $10^{6.9}$ M$_\odot$, respectively. The mass of the gas, assuming a 40% contribution by helium, follows from $M_{\rm gas}=1.4
M_{\rm HI}=3.2 \times 10^5 S D_{\rm Mpc}^2$, where $S$ is the integrated flux in units of Jy. For these three cores, $M_{\rm
gas}=10^{7.1}$, $10^{6.5}$, and $10^{6.9}$, values which, compared to the dynamical masses, correspond to dark–to–visible mass ratios of 10, 36, and 29.
Considerations of CHVC Stability
================================
The very high total linewidth of CHVC115+13$-$275 was found to arise from the range of line–of–sight velocities contributed by the distinct individual clumps which make up this source. It is unlikely that this is a chance superposition of unrelated components, in view of the spatial and kinematic isolation of this CHVC. Braun and Burton (2000) consider the stability of these collection of clumps under several circumstances. If the collection of clumps were located at a distance of 5 kpc, the collection would have a diameter of 44 pc and would double in size on the implausibly short dynamical timescale of only $5 \times 10^5$ years. On the other hand, if this source were located at a distance of 0.7 Mpc and were self–gravitating, then the angular radius of 15 arcmin and the velocity half–width of 35 could be used to calculate a dynamical mass of $10^{8.93}$ M$_\odot$. The corresponding mass of gas at this distance, assuming 40% to be helium, is $10^{7.22}$ M$_\odot$, and the dark–to–visible mass ratio is 51 and scales as 1/D. Consistent distance and dark–to–visible mass ratios are found for other CHVC objects.
Indications of Cold, Opaque Cores
=================================
The object CHVC125+41$-$207 is particularly interesting. Of the 65 CHVCs catalogued by Braun and Burton (1999), this one showed the narrowest FWHM, amounting to 5.9 in the single–dish data. The WSRT image shown in Fig. 6 could be displayed at higher angular resolution because of the intense brightness of this source. Like the other CHVCs imaged, this one also shows a complex morphology with several compact cores. The spectrum toward the brightest of these cores is remarkable in having a linewidth which is completely unresolved with the effective resolution of 2.47 . The velocity channels adjacent to the line peak have intensities down to about 20% of the maximum value. Such a width is one of the narrowest ever measured in emission.
The narrow linewidth provides an opportunity (rare in work, where linewidths are commonly dominated by line blending and mass motions) to derive a kinetic temperature. An upper limit to the intrinsic FWHM of 2 corresponds to an upper limit to the kinetic temperature of 85 K. This limit assumes additional importance because the observed brightness temperature of this core is 75 K. Thus the temperature is tightly constrained, and a lower limit to the opacity follows from $T_{\rm
B}=T_{\rm s}(1-e^{-1})$, if $T_{\rm s}=T_{\rm k}$, yielding $\tau \geq 2$. In addition to constraining the temperature, this observation also constrains any broadening which might be due to turbulence to be less than 1 .
The narrow linewidth observed for CHVC125+41$-$207 provides an opportunity to estimate its distance. Wolfire et al. (1995a, 1995b) show that a cold, stable phase of is expected if a sufficient column of shielding gas is present and if the thermal pressure is sufficiently high. Calculations of equilibrium conditions with the Local Group environment have kindly been made available to us by Wolfire, Sternberg, Hollenbach, and McKee, for two bracketing values of the shielding column density, namely 1 and $10\times10^{20}$ cm$^{-2}$. Figure 7 shows that the corresponding equilibrium volume densities for the observed value of $T_{\rm k}=85$ K are 3.5 and 0.65 cm$^{-3}$, respectively. The distance to CHVC125+41$-$207 follows from the volume densities thus indicated, and from the measured column depths and the angular size of the source, according to $D=N_{\rm H}/(n_{\rm H}\theta)$. Consideration in this way of several compact cores in this source allow a distance determination in the range 0.5 to 1 Mpc.
Summary: Parameters of Several of the Individual\
Objects in the CHVC Ensemble
=================================================
The compact high–velocity clouds which we have imaged using the Westerbork array share several morphological characteristics. In each case we detected a number of compact cores which range in size from a few arcmin to about 15 arcmin, with peak column densities in the range $10^{19}$ to $10^{20}$ cm$^{-2}$, and FWHM linewidths of about 2 . Each core is characterized by its own line–of–sight velocity; it is the range in these core velocities, not the much narrower intrinsic linewidths, which determined the range of total widths in the single–dish catalog of CHVCs. The modest linewidths seen in all of the cores indicates that the in these structures is in the form of the Cool Neutral Medium (CNM) with typical equilibrium temperatures in the range 50 to 200 K.
In all of the cases we have imaged, as well as in the two cases imaged earlier by Wakker and Schwarz (1991), the CHVC cores are embedded in an extended halo of emission. The halos have typical extents of about $1^\circ$, column densities within the inner 30 arcmin between $2 \times
10^{18}$ and $2 \times 10^{19}$ cm$^{-2}$, and velocity dispersions consistent with the 8000 K equilibrium temperature of the Warm Neutral Medium (WNM). The cores account for some 40% of the flux, while covering some 15% of the area. Thus the data at hand suggest that a two–phase structure characterizes the CHVC morphology, with cold, opaque cores of CNM shielded by a halo of the WNM.
The compact clouds provide several indications of the crucial distance parameter. The velocity dispersion of the entire ensemble is minimized in a reference frame consistent with the Local Group Standard of Rest. Three additional measures regarding distance are given by the imaging of individual CHVCs. $(i)$ Arguments based on the dynamical stability of CHVC115+13$-$275 support a distance of about 0.7 Mpc. $(ii)$ The exceptionally narrow linewidths observed in the principal cores of CHVC125+41$-$207 lead to a measure of the volume density and that, with the measured angular size and column depth, yields a distance in the range 0.5 to 1 Mpc. $(iii)$ The kinematic gradients detected along the long axis of some of the CHVC cores resemble in form and amplitude the velocity fields of dwarf galaxies. The solutions for dynamical mass based on this comparison indicate a distance of about 0.7 Mpc. The ratio of dark–to–visible mass of the objects is in the range 10 to 50.
In agreement with the conclusions reached for the ensemble, the distances and physical properties of the individual CHVCs are suggestive of a population which has as yet had little interaction with the more massive Local Group members. At distances of about 0.5 to 1 Mpc these objects would have sizes of about 15 kpc, gas masses of about 10$^7$ M$_\odot$, and total masses of a few times $10^8$ M$_\odot$. We tentatively estimate the total CHVC ensemble to harbor some 200 objects; in that case, the total gas mass involved would be a few times 10$^9$M$_\odot$. In view of the net in–fall motion observed for the ensemble, a source of dark matter and low–metallicity gas is indicated for the continuing growth and evolution of the major Local Group galaxies. The CHVCs may be the missing Local Group satellites predicted by hierarchical growth scenarios (e.g. Klypin et al. 1999).
These CHVC parameter values correspond to those of (sub–)dwarf galaxies. Indeed, it would be difficult to distinguish these CHVC properties from those derived from the signature of a typical dwarf galaxy. If, on the other hand, the objects were produced relatively locally by an energetic mechanism responsible for a galactic fountain, then their properties would be expected to show large linewidths, motions not ordered by rotation, and a structural form other than a core/halo one. The distinction between CHVCs and dwarf galaxies with very weak star formation remains to be made and is an important challenge. The CHVCs may represent still–pristine examples of collapsed objects, with only a small amount of internal star formation and enrichment. As such, they should provide insight into the process of galaxy and structure formation.
The WSRT imaging is discussed in more detail by Braun and Burton (2000).
We are grateful to M.G. Wolfire, A. Sternberg, D. Hollenbach, and C.F. McKee for providing the equilibrium temperature curves shown in Fig. 7. The Westerbork Synthesis Radio Telescope is operated by the Netherlands Foundation for Research in Astronomy, under contract with the Netherlands Organization for Scientific Research.
Bajaja E., Morras R., Pöppel W.G.L. 1987, Pub.Astr.Inst.Czech.Ac.Sci. 69, 237 Blitz L., Spergel D.N., Teuben P.J., Hartmann D., Burton W.B. 1999, ApJ, 514, 818 Braun R., Burton W.B. 1999, A&A, 341, 437 Braun R., Burton W.B. 2000, A&A, submitted Bregman J.N. 1980, ApJ, 236, 577 Burton W.B. 1997, in The Physics of the Galactic Halo, eds. Lesch H., Dettmann R.-J., Mebold U., Schlickeiser R., Berlin: Akademie Verlag, 15 Eichler D. 1976, ApJ, 208, 694 Einasto J., Haud U., Jôeveer M., Kaasik A. 1976, MNRAS, 177, 357 Giovanelli R. 1981, AJ, 86, 1468 Hartmann D., Burton W.B. 1997, “Atlas of Galactic Neutral Hydrogen”, Cambridge: Cambridge University Press Karachentsev I.D., Makarov D.A. 1996, AJ, 111, 794 Klypin A., Kravtsov A.V., Valenzuela O., Prada F. 1999, ApJ, 522, 82 Moore B., Ghigna S., Governato G., Lake G., Quinn T., Stadel J., Tozzi P. 1999, ApJ, 524, 19 Oort J.H. 1966, Bull.Astr.Inst.Netherlands, 18, 421 Oort J.H. 1970, A&A, 7, 381 Oort J.H. 1981, A&A, 94, 359 Shapiro P.R., Field G.B. 1976, ApJ, 205, 762 Verschuur G.L. 1975, ARA&A, 13, 257 Wakker B.P., Schwarz U. 1991, A&A, 250, 484 Wakker B.P., van Woerden H. 1991, A&A, 250, 509 Wakker B.P., van Woerden H. 1997, ARA&A, 35, 217 Wolfire M.G., Hollenbach D., McKee C.F., Tielens A.G.G.M., Bakes E.L.O. 1995a, ApJ, 443, 152 Wolfire M.G., McKee C.F., Hollenbach D., Tielens A.G.G.M. 1995b, ApJ, 453, 673
| {
"pile_set_name": "ArXiv"
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---
abstract: 'We compute the geometric part of algebraic cobordism over Dedekind domains of mixed characteristic after inverting the positive residue characteristics and prove cases of a Conjecture of Voevodsky relating this geometric part to the Lazard ring for regular local bases. The method is by analyzing the slice tower of algebraic cobordism, relying on the Hopkins-Morel isomorphism from the quotient of the algebraic cobordism spectrum by the generators of the Lazard ring to the motivic Eilenberg-MacLane spectrum, again after inverting the positive residue characteristics.'
author:
- Markus Spitzweck
bibliography:
- 'ma.bib'
title: Algebraic Cobordism in mixed characteristic
---
Introduction
============
Algebraic cobordism is a theory for smooth schemes over a base scheme $S$ defined by a motivic ring spectrum ${\mathsf{MGL}}_S$ in the stable motivic homotopy category ${\mathsf{SH}}(S)$. It is the motivic counterpart of complex cobordism ${\mathsf{MU}}$. A famous Theorem of Quillen states that the natural map from the Lazard ring $L_*$ classifying formal group laws to the coefficients of ${\mathsf{MU}}$ is an isomorphism, moreover $L_* \cong {{\mathbb Z}}[x_1,x_2,x_3,\ldots]$ with $\deg(x_i)=i$ (here we divide the usual topological grading by $2$).
For an oriented motivic ring spectrum $E$ the geometric part $E_{(2,1)*}$ of the coefficients also carries a formal group law constructed in the exact same way as in topology by evaluating the theory on ${{\mathbb P}}^\infty$ and using that ${{\mathbb P}}^\infty$ is naturally endowed with a multiplication.
Thus there is a classifying map $L_* \to E_{(2,1)*}$. It is known that for $E={\mathsf{MGL}}_k$ for a field $k$ of characteristic $0$ this map is an isomorphism using the Hopkins-Morel isomorphism, see [@hoyois.hopkins-morel Proposition 8.2]. More generally in [@levine.comparison] it is shown that over such fields the Levine-Morel algebraic cobordism $\Omega^*(-)$ is isomorphic to ${\mathsf{MGL}}_k^{(2,1)*}(-)$ on smooth schemes over $k$. If the base field $k$ has positive characteristic the map $L_* \to {\mathsf{MGL}}_{(2,1)*}$ becomes at least an isomorphism after inverting the characteristic, see again [@hoyois.hopkins-morel Proposition 8.2].
The main ingredient in the proof is that the Hopkins-Morel isomorphism yields a computation of the slices of ${\mathsf{MGL}}_S$ with respect to Voevodsky’s slice filtration, that ${\mathsf{MGL}}_S$ is complete with respect to this filtration and that the slices have a simple form, namely they are shifted twists of the motivic Eilenberg-MacLane spectrum.
The facts about the slices of ${\mathsf{MGL}}_S$ hold more generally true over spectra $S$ of Dedekind domains of mixed characteristic (after inverting the positive residue characteristics), using the motivic Eilenberg-MacLane spectrum introduced in [@spitzweck.em]. The main new input of this note is that in this case ${\mathsf{MGL}}_S$ is also complete with respect to the slice filtration (Corollary \[gfrerz\]), a consequence of the fact that ${\mathsf{MGL}}_S$ is connective with respect to the homotopy sheaves, see Proposition \[gdrttt\].
This yields a computation of the geometric part of the homotopy groups of ${\mathsf{MGL}}_S$ (Theorem \[grfe5z4\]), again after inverting the residue characteristics. In our formulation we always assume a Hopkins-Morel isomorphism for the given coefficients, hoping that the Hopkins-Morel isomorphism will be settled completely in the future.
We prove cases of a Conjecture of Voevodsky ([@voevodsky.icm Conjecture 1]), see Theorem \[hte4234t\], comparing the Lazard ring to $({\mathsf{MGL}}_S)_{(2,1)*}$ for $S$ the spectrum of a regular local ring.
We also give applications to some homotopy groups or sheaves of ${\mathsf{MGL}}_S$ outside the geometric diagonal, see section \[dzu634e\], and discuss generalizations of our results to motivic Landweber spectra.
.4cm
We note that the observation that the Hopkins-Morel isomorphism yields the computation of the zero-slice of the sphere spectrum (after inverting suitable primes), see Theorem \[dhu5t3\], was independently made by Oliver Röndigs.
.7cm
[**Acknowledgements:**]{} I would like to thank Peter Arndt, Christian Häsemeyer, Marc Hoyois, Moritz Kerz, Marc Levine, Niko Naumann, Oliver Röndigs, Manfred Stelzer, Florian Strunk, Jörg Wildeshaus and Paul Arne [Ø]{}stv[æ]{}r for very helpful discussions and suggestions on the subject.
Preliminaries
=============
By a base scheme we always mean a separated Noetherian scheme of finite Krull dimension. For a base scheme $S$ we let ${\mathsf{SH}}(S)$ be the stable motivic homotopy category.
We let ${\mathsf{M}\mathbb{Z}}_S \in {\mathsf{SH}}(S)$ be the motivic Eilenberg-MacLane spectrum over $S$ constructed in [@spitzweck.em]. Also we let ${{\mathcal M}}(r) \in {{\mathsf D}}({\mathrm{Sh}}({\mathrm{Sm}}_{S,{\mathit{Zar}}},{{\mathbb Z}}))$ (for notation see [@spitzweck.em]) be the motivic complexes of weight $r \in {{\mathbb Z}}$, so as a ${\mathbb{G}_{m,S}}$-spectrum ${\mathsf{M}\mathbb{Z}}_S$ has ${{\mathcal M}}(r)[r]$ in level $r$. If $S$ is the spectrum of a Dedekind domain of mixed characteristic we note that ${{\mathcal M}}(0)= S^0 \underline{{{\mathbb Z}}}$, thus for $X \in {\mathrm{Sm}}_S$ we have $H^{0,0}(X,{{\mathbb Z}})={{\mathbb Z}}^{\pi_0(X)}$. Also ${{\mathcal M}}(1) \cong {{\mathcal O}}^*[-1]$, so $H^{1,1}(X,{{\mathbb Z}}) \cong {{\mathcal O}}^*(X)$ and $H^{2,1}(X,{{\mathbb Z}}) \cong {\mathrm{Pic}}(X)$. We have ${{\mathcal M}}(r) \cong 0$ for $r < 0$.
For general $S$ we denote by ${\mathsf{MGL}}_S \in {\mathsf{SH}}(S)$ the algebraic cobordism spectrum. There is a natural map $L_* \to ({\mathsf{MGL}}_S)_{2*,*}$, where $L_*$ denotes the Lazard ring. Fixing generators $x_i \in L_i$ there is a map $$\Phi_S \colon {\mathsf{MGL}}_S/(x_1,x_2,\ldots){\mathsf{MGL}}_S \to {\mathsf{M}\mathbb{Z}}_S,$$ see [@spitzweck.em §11.1], which is an isomorphism after inverting all positive residue characteristics of $S$, see [@spitzweck.em Theorem 11.3].
For any ring or abelian group $R$ we let $M_R \in {\mathsf{SH}}(S)$ be the Moore spectrum on $R$ and ${\mathsf{M}R}_S$ the version of ${\mathsf{M}\mathbb{Z}}_S$ with $R$-coeffcients.
Slices
======
For $i \in {{\mathbb Z}}$ denote by $f_i$ resp. $l_i$ the $i$-th colocalization resp. localization functor for Voevodsky’s motivic slice filtration on ${\mathsf{SH}}(S)$. For any $E \in {\mathsf{SH}}(S)$ and $k \ge n$ we set $E\left<n,k \right>:=l_{k+1}(f_n(E))$. Thus we have exact triangles $$f_{k+1}(E) \to f_n(E) \to E\left<n,k \right> \to f_{k+1}(E)[1]$$ and $s_n(E) = E\left<n,n \right>$.
We note that all these functors commute with homotopy colimits.
\[dhu5t3\] Let $X$ be an essentially smooth scheme over a Dedekind domain of mixed characteristic and $R$ a localization of ${{\mathbb Z}}$ such that $\Phi_X \wedge M_R$ is an isomorphism (e.g. if every positive residue characteristic of $X$ is invertible in $R$). Then $$s_0 M_R \cong s_0 ({\mathsf{MGL}}_X \wedge M_R) \cong {\mathsf{M}R}_X.$$ More generally $$s_n ({\mathsf{MGL}}_X \wedge M_R) \cong \Sigma^{2n,n} {\mathsf{M}R}_X \otimes L_n.$$
The first isomorphism of the first line follows from [@spitzweck.rel Corollary 3.3]. From the assumption that $\Phi_X \wedge M_R$ is an isomorphism it follows that the map ${\mathsf{MGL}}_X \wedge M_R \to {\mathsf{M}R}_X$ induces an isomorphism on zero-slices and that ${\mathsf{M}R}_X$ is effective. Moreover $l_1 {\mathsf{M}\mathbb{Z}}_X \cong {\mathsf{M}\mathbb{Z}}_X$, since negative weight motivic cohomology vanishes in our situation. Thus the second isomorphism of the first line follows. The second line is a version of [@spitzweck.rel Theorem 4.7] with $R$-coefficients.
It is then also possible to determine the slices of motivic Landweber spectra with $R$-coeffcients, see [@spitzweck.slices], for example of ${\mathsf{KGL}}_X \wedge M_R$.
Subcategories of the stable motivic homotopy category
=====================================================
Fix a base scheme $S$. We let ${\mathsf{SH}}(S)_{\ge n}$ be the $\ge n$ part (in the homological sense) of ${\mathsf{SH}}(S)$ with respect to the homotopy $t$-structure, see e.g. [@hoyois.hopkins-morel §2.1]. Thus ${\mathsf{SH}}(S)_{\ge n}$ is generated by homotopy colimits and extensions by the objects $\Sigma^{p,q} \Sigma^\infty_+ X$ for $X \in {\mathrm{Sm}}_S$ and $p-q \ge n$.
For each $E \in {\mathsf{SH}}(S)$ we let ${\underline{\pi}}^{\mathrm{pre}}_{p,q}(E)$ be the presheaf $$X \mapsto {\mathrm{Hom}}_{{\mathsf{SH}}(S)}(\Sigma^{p,q} \Sigma^\infty_+ X,E)$$ on ${\mathrm{Sm}}_S$. Let ${\underline{\pi}}_{p,q}(E)$ be the sheafification of ${\underline{\pi}}^{\mathrm{pre}}_{p,q}(E)$ with respect to the Nisnevich topology. We also set $\pi_{p,q}(E):={\underline{\pi}}^{\mathrm{pre}}_{p,q}(E)(S)=E_{p,q}$.
We let ${\mathsf{SH}}(S)_{h \ge n}$ be the full subcategory of ${\mathsf{SH}}(S)$ of objects $E$ such that ${\underline{\pi}}_{p,q}(E)=0$ for $p-q< n$.
The categories ${\mathsf{SH}}(S)_{h \ge n}$ are closed under homotopy colimits and extensions in ${\mathsf{SH}}(S)$.
Th functors ${\underline{\pi}}_{p,q}$ respect sums. Moreover the long exact sequences of homotopy sheaves associated to an exact triangle in ${\mathsf{SH}}(S)$ show that ${\mathsf{SH}}(S)_{h \ge n}$ is closed under cofibers and extensions. This shows the claim.
\[gegrsy\] Let $i \colon Z \hookrightarrow S$ be a closed inclusion of base schemes. Then $i_*({\mathsf{SH}}(Z)_{h \ge n}) \subset {\mathsf{SH}}(S)_{h \ge n}$.
Let $E \in {\mathsf{SH}}(Z)_{h \ge n}$. Let $Y$ be the spectrum of the henselization of a local ring of a scheme from ${\mathrm{Sm}}_S$. Then $Y_Z:=Y \times_S Z$ is also the spectrum of a henselian local ring, and ${\underline{\pi}}^{\mathrm{pre}}_{p,q}(i_*E)(Y) \cong {\underline{\pi}}^{\mathrm{pre}}_{p,q}(E)(Y_Z)=0$ for $p-q < n$ (the first isomorphism holds since $i_*$ commutes with homotopy colimits).
We let ${\mathsf{SH}}^{{S^1}}_s(S)$ be the homotopy category of presheaves of ${{S^1}}$-spectra on ${\mathrm{Sm}}_S$ localized with respect to the Nisnevich topology, and ${\mathsf{SH}}^{{S^1}}(S)$ the further ${{\mathbb A}}^1$-localization of that category.
We let ${\mathsf{SH}}^{{S^1}}_s(S)_{\ge n}$ be the $\ge n$ part (in the homological sense) of ${\mathsf{SH}}^{{S^1}}_s(S)$ with respect to the standard $t$-structure, and for $E \in {\mathsf{SH}}^{{S^1}}_s(S)$ we let $E_{\ge n}$ and $E_{\le n}$ be the corresponding truncations. We let $E_{=n} := (E_{\ge n})_{\le n}$.
As above for $E \in {\mathsf{SH}}^{{S^1}}_s(S)$ we have the presheaves ${\underline{\pi}}^{\mathrm{pre}}_p(E)$ and the sheaves ${\underline{\pi}}_p(E)$. For $E \in {\mathsf{SH}}^{{S^1}}_s(S)$ we have $E \in {\mathsf{SH}}^{{S^1}}_s(S)_{\ge n}$ if and only if ${\underline{\pi}}_k(E)=0$ for $k < n$.
Note that ${\mathsf{SH}}^{{S^1}}_s(S)_{\ge n}$ is generated by homotopy colimits and extensions by the objects $\Sigma^n \Sigma^\infty_+ X$, $X \in {\mathrm{Sm}}_S$, thus the canonical functor $\sigma \colon {\mathsf{SH}}^{{S^1}}_s(S) \to {\mathsf{SH}}(S)$ sends ${\mathsf{SH}}^{{S^1}}_s(S)_{\ge n}$ to ${\mathsf{SH}}(S)_{\ge n}$.
We have ${\mathsf{SH}}(S)_{h \ge n} \subset {\mathsf{SH}}(S)_{\ge n}$. If $S$ is the spectrum of a field then the inclusion is an equality.
Let $E \in {\mathsf{SH}}(S)_{h \ge n}$. For any $i \in {{\mathbb N}}$ let $E_i$ be the image of $\Sigma^{i,i} E$ in ${\mathsf{SH}}^{{S^1}}_s(S)$. By assumption we have $E_i \in {\mathsf{SH}}^{{S^1}}_s(S)_{\ge n}$. Thus $\Sigma^{-i,-i} \sigma(E_i) \in {\mathsf{SH}}(S)_{\ge n}$. The proof of the first statement concludes by noting that $E \cong {\mathrm{hocolim}}_{i \to \infty} \Sigma^{-i,-i} \sigma(E_i)$.
The second statement is [@hoyois.hopkins-morel Theorem 2.3].
\[hterge\] Let $E \in {\mathsf{SH}}^{{S^1}}_s(S)$. Then $E \to {\mathrm{holim}}_{n \to \infty} E_{\le n}$ is an isomorphism.
We show that for all $n \in {{\mathbb Z}}$ we have ${\underline{\pi}}_n(E) \cong {\underline{\pi}}_n({\mathrm{holim}}_{k \to \infty} E_{\le k})$. Fix $n \in {{\mathbb Z}}$ and let $X \in {\mathrm{Sm}}_S$ be of dimension $d$. We are ready if we show $${\underline{\pi}}_n(E)|_{X_{\mathit{Nis}}} \cong {\underline{\pi}}_n({\mathrm{holim}}_{k \to \infty} E_{\le k})|_{X_{\mathit{Nis}}} \; \; (*).$$ For $m > n+d$ we have ${\underline{\pi}}^{\mathrm{pre}}_n(E_{=m}[j])(Y)=0$ for $Y \in X_{\mathit{Nis}}$ and $j \ge 0$, so homing out of $\Sigma^\infty_+ Y$ into the exact triangle $$E_{=m} \to E_{\le m} \to E_{\le (m-1)} \to E_{=m}[1]$$ shows that ${\underline{\pi}}^{\mathrm{pre}}_n(E_{\le m})(Y) \cong {\underline{\pi}}^{\mathrm{pre}}_n(E_{\le (m-1)})(Y)$. Using the Milnor exact sequence this shows that $${\underline{\pi}}^{\mathrm{pre}}_n({\mathrm{holim}}_{k \to \infty} E_{\le k})|_{X_{\mathit{Nis}}} \cong {\underline{\pi}}^{\mathrm{pre}}_n(E_{\le m})|_{X_{\mathit{Nis}}}$$ for $m \ge n+d$. Sheafifiying proves $(*)$.
\[dwrzjj\] Let $$\cdots \to E_{i+1} \to E_i \to E_{i-1} \to \cdots \to E_1 \to E_0$$ be an inverse system of objects in ${\mathsf{SH}}(S)$. Suppose for each $n \in {{\mathbb N}}$ there is an $N \in {{\mathbb N}}$ such that $E_i \in {\mathsf{SH}}(S)_{h \ge n}$ for $i \ge N$. Then ${\mathrm{holim}}_{i \to \infty} E_i \cong 0$.
Fix $q \in {{\mathbb Z}}$ and let $F_i$ be the image of $\Sigma^{q,q} E_i$ in ${\mathsf{SH}}^{{S^1}}_s(S)$. We are ready if we show ${\mathrm{holim}}_{i \to \infty} F_i \cong 0$. By assumption for every $n \in {{\mathbb N}}$ there is a $N \in {{\mathbb N}}$ such that $F_i \in {\mathsf{SH}}^{{S^1}}_s(S)_{\ge n}$ for each $i \ge N$. By Lemma \[hterge\] we have $F_i \cong {\mathrm{holim}}_{k \to \infty} (F_i)_{\le k}$. Thus $${\mathrm{holim}}_{i \to \infty} F_i \cong {\mathrm{holim}}_i {\mathrm{holim}}_k (F_i)_{\le k} \cong {\mathrm{holim}}_k {\mathrm{holim}}_i (F_i)_{\le k}
\cong {\mathrm{holim}}_k 0 \cong 0.$$
We also have the
Let $E \in {\mathsf{SH}}(S)_{h \ge n}$ and $X \in {\mathrm{Sm}}_S$ of dimension $d$. Then $${\underline{\pi}}^{\mathrm{pre}}_{p,q}(E)(X)=0$$ for $p-q<n-d$.
\[gr455\] Let $$\cdots \to E_{i+1} \to E_i \to E_{i-1} \to \cdots \to E_1 \to E_0$$ be an inverse system of objects in ${\mathsf{SH}}(S)_{h \ge n}$. Suppose for each $p,q \in {{\mathbb Z}}$ and $d \in {{\mathbb N}}$ there is an $N \in {{\mathbb N}}$ such that for $X \in {\mathrm{Sm}}_S$ of dimension $d$ the map $${\underline{\pi}}^{\mathrm{pre}}_{p,q}(E_{i+1})(X) \to {\underline{\pi}}^{\mathrm{pre}}_{p,q}(E_i)(X)$$ is an isomorphism for all $i \ge N$. Then ${\mathrm{holim}}_{i \to \infty} E_i \in {\mathsf{SH}}(S)_{h \ge n}$. (Here the latter homotopy limit is computed in ${\mathsf{SH}}(S)$.)
Let $p,q \in {{\mathbb Z}}$, $d \in {{\mathbb N}}$ and $X \in {\mathrm{Sm}}_S$ of dimension $d$. Choose $N \in {{\mathbb N}}$ such that for any $Y \in {\mathrm{Sm}}_S$ of dimension $\le d$ the map $${\underline{\pi}}^{\mathrm{pre}}_{p,q}(E_{i+1})(Y) \to {\underline{\pi}}^{\mathrm{pre}}_{p,q}(E_i)(Y)$$ is an isomorphism for all $i \ge N$. We claim that $${\underline{\pi}}_{p,q}({\mathrm{holim}}_k E_k)|_{X_{\mathit{Nis}}} \cong {\underline{\pi}}_{p,q}(E_i)|_{X_{\mathit{Nis}}}$$ for all $i \ge N$. For every $Y \in X_{\mathit{Nis}}$ we have the Milnor short exact sequence $$0 \to \text{$\lim_i$}^1 {\underline{\pi}}^{\mathrm{pre}}_{p+1,q}(E_i)(Y) \to {\underline{\pi}}^{\mathrm{pre}}_{p,q}({\mathrm{holim}}_i E_i)(Y) \to \lim_i {\underline{\pi}}^{\mathrm{pre}}_{p,q}(E_i)(Y) \to 0.$$ The $\lim^1$-term vanishes because the inverse system of abelian groups stabilizes by assumption. Sheafifying we see that ${\underline{\pi}}_{p,q}({\mathrm{holim}}_k E_k)|_{X_{\mathit{Nis}}} \cong {\underline{\pi}}_{p,q}(E_i)|_{X_{\mathit{Nis}}}$ for $i \ge N$, in particular ${\underline{\pi}}_{p,q}({\mathrm{holim}}_k E_k)|_{X_{\mathit{Nis}}} = 0$ in case $p-q<n$. Since this is true for all $X \in {\mathrm{Sm}}_S$ we conclude ${\underline{\pi}}_{p,q}({\mathrm{holim}}_k E_k) =0$ for $p-q < n$.
Connectivity of algebraic cobordism
===================================
\[vftjt\] Let $X$ be a smooth scheme over a Dedekind domain of mixed characteristic or over a field. Then for any abelian group $A$ we have ${\mathsf{M}A}_X \in {\mathsf{SH}}(X)_{h \ge 0}$.
This follows from the fact that the motivic complexes ${{\mathcal M}}(r)$ have vanishing $i$-th cohomology sheaf for $i>r$, see [@geisser.dede Corollary 4.4].
\[fhtrth\] Let $S$ be the spetrum of a discrete valuation ring of mixed characteristic, $j \colon \eta \to S$ the inclusion of the generic point. Then for any abelian group $A$ we have $j_* {\mathsf{M}A}_\eta \in {\mathsf{SH}}(S)_{h \ge 0}$.
Let $i \colon s \to S$ be the inclusion of the closed point. We have an exact triangle $$i_!i^! {\mathsf{M}A}_S \to {\mathsf{M}A}_S \to j_* {\mathsf{M}A}_\eta \to i_!i^! {\mathsf{M}A}_S[1]$$ and an isomorphism $i^! {\mathsf{M}A}_S \cong {\mathsf{M}A}_s(-1)[-2] \in {\mathsf{SH}}(s)_{h \ge -1}$, see [@spitzweck.em Theorem 7.4]. We conclude with Proposition \[gegrsy\] and Lemma \[vftjt\].
\[gfewet\] Let the situation be as in Proposition \[fhtrth\]. Then $$j_*{\mathsf{MGL}}_\eta\left<0,n\right> \wedge M_A \in {\mathsf{SH}}(S)_{h \ge 0}$$ for all $n \ge 0$.
We can assume $A={{\mathbb Z}}$. Since $\eta$ is of characteristic $0$ we have $s_n {\mathsf{MGL}}_\eta \cong \Sigma^{2n,n} {\mathsf{M}\mathbb{Z}}\otimes L_n$. Moreover we have exact triangles $$s_n {\mathsf{MGL}}_\eta \to {\mathsf{MGL}}_\eta\left<0,n\right> \to {\mathsf{MGL}}_\eta \left<0,n-1 \right> \to s_n {\mathsf{MGL}}_\eta [1].$$ Applying $j_*$ to these triangles and using Proposition \[fhtrth\] one concludes by induction on $n$.
\[rergr\] Let the situation be as in Proposition \[fhtrth\]. Let $p,q \in {{\mathbb Z}}$ and $X \in {\mathrm{Sm}}_S$ of dimension $d$. Then $${\underline{\pi}}^{\mathrm{pre}}_{p,q}(j_* {\mathsf{MGL}}_\eta\left< 0,n+1 \right>)(X) \to {\underline{\pi}}^{\mathrm{pre}}_{p,q}(j_* {\mathsf{MGL}}_\eta\left< 0,n \right>)(X)$$ is an isomorphism for $n \ge p-q+d$.
Consider the exact triangle $$j_* s_{n+1} {\mathsf{MGL}}_\eta \to j_* {\mathsf{MGL}}_\eta \left< 0,n+1 \right> \to j_* {\mathsf{MGL}}_\eta \left< 0,n \right> \to s_{n+1} {\mathsf{MGL}}_\eta [1].$$ We have $${\underline{\pi}}^{\mathrm{pre}}_{p,q}(j_* s_{n+1} {\mathsf{MGL}}_\eta)(X) = H_{\mathrm{mot}}^{2(n+1)-p,n+1-q}(X_\eta, L_{n+1}).$$ The latter group vanishes for $2(n+1)-p> n+1 -q+d$, showing the claim.
\[eweggg\] Let the situation be as in Proposition \[fhtrth\]. Then $j_* {\mathsf{MGL}}_\eta \in {\mathsf{SH}}(S)_{{\mathsf{h}}\ge 0}$.
Consider the inverse system $$\cdots \to j_* {\mathsf{MGL}}_\eta \left<0,n+1 \right> \to j_* {\mathsf{MGL}}_\eta \left< 0,n \right> \to \cdots \to j_* s_0 {\mathsf{MGL}}_\eta$$ in ${\mathsf{SH}}(S)$. Since $j_*$ preserves homotopy limits the homotopy limit over this system is $j_* {\mathsf{MGL}}_\eta$, using [@hoyois.hopkins-morel Corollary 2.4 and Lemma 8.10 or Theorem 8.12]. By Lemma \[gfewet\] every object of this system is in ${\mathsf{SH}}(S)_{h \ge 0}$. Moreover by Lemma \[rergr\] the assumptions of Proposition \[gr455\] are satisfied. Thus this Proposition implies the claim.
\[greerg\] Let the situation be as in Proposition \[fhtrth\] and let $i \colon s \to S$ be the inclusion of the closed point. Then $i^! {\mathsf{MGL}}_S \in {\mathsf{SH}}(s)_{\ge -1}$.
Note first that $i^*$ sends ${\mathsf{SH}}(S)_{\ge 0}$ to ${\mathsf{SH}}(s)_{\ge 0}$. We have ${\mathsf{MGL}}\in {\mathsf{SH}}(S)_{\ge 0}$ and by Lemma \[eweggg\] also $j_* {\mathsf{MGL}}_\eta \in {\mathsf{SH}}(S)_{h \ge 0}
\subset {\mathsf{SH}}(S)_{\ge 0}$. Applying $i^*$ to the exact triangle $$i_!i^! {\mathsf{MGL}}_S \to {\mathsf{MGL}}_S \to j_* {\mathsf{MGL}}_\eta \to i_!i^! {\mathsf{MGL}}_S [1]$$ shows the claim.
\[gferghd\] Let $S$ be the spectrum of a discrete valuation ring of mixed characteristic. Then ${\mathsf{MGL}}_S \in {\mathsf{SH}}(S)_{h \ge -1}$.
Let the notation be as above. The claim follows from the exact triangle $$i_!i^! {\mathsf{MGL}}_S \to {\mathsf{MGL}}_S \to j_* {\mathsf{MGL}}_\eta \to i_!i^! {\mathsf{MGL}}_S [1],$$ Lemma \[eweggg\], Proposition \[greerg\] and Proposition \[gegrsy\].
\[gdrttt\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Then ${\mathsf{MGL}}_S \in {\mathsf{SH}}(S)_{h \ge -1}$.
The henselization of a local ring of a scheme in ${\mathrm{Sm}}_S$ lies over a local ring of $S$, thus the claim follows from Lemma \[gferghd\].
Compare the following result to [@voevodsky.open Conjecture 15].
\[gfrerz\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $R$-module $A$ we have $$f_n {\mathsf{MGL}}_S \wedge M_A \cong {\mathrm{holim}}_{k \to \infty} {\mathsf{MGL}}_S \left< n,k \right> \wedge M_A.$$
Under the assumption we have ${\mathsf{f}}_k {\mathsf{MGL}}_S \wedge M_A \in {\mathsf{SH}}(S)_{h \ge k-1}$, since this is a homotopy colimit of objects of the form $\Sigma^{2i,i} {\mathsf{MGL}}_S \wedge M_A$ with $i \ge k$, see the proof of [@spitzweck.rel Theorem 4.7], using Proposition \[gdrttt\]. Thus by Corollary \[dwrzjj\] we have ${\mathrm{holim}}_{k \to \infty} f_k {\mathsf{MGL}}_S \wedge M_A \cong 0$ implying the claim.
A similar result holds for motivic Landweber spectra using the same argument as in the proof of [@hoyois.hopkins-morel Lemma 8.11]. For example ${\mathsf{KGL}}_S \wedge M_A$ is complete with respect to the slice filtration.
The geometric part of algebraic cobordism
=========================================
\[jewet45\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Let $p,q \in {{\mathbb Z}}$ and $X \in {\mathrm{Sm}}_S$. Then for any $R$-module $A$ the inverse system of abelian groups $({\underline{\pi}}^{\mathrm{pre}}_{p,q}({\mathsf{MGL}}_S \left< 0,k \right> \wedge M_A)(X))_k$ eventually becomes constant for $k \to \infty$.
This follows from the exact triangle $$s_k {\mathsf{MGL}}_S \wedge M_A \to {\mathsf{MGL}}_S \left<0,k \right> \wedge M_A \to {\mathsf{MGL}}_S \left< 0,k-1 \right> \wedge M_A \to s_k {\mathsf{MGL}}_S \wedge M_A [1]$$ and $s_k {\mathsf{MGL}}_S \wedge M_A \cong \Sigma^{2k,k} {\mathsf{M}A}\otimes L_k$ since ${\underline{\pi}}^{\mathrm{pre}}_{p,q}(\Sigma^{2k+j,k} {\mathsf{M}A})(X)=0$, $j \ge 0$, for $k$ big enough.
\[dsgtjt\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Let $p,q \in {{\mathbb Z}}$ and $X \in {\mathrm{Sm}}_S$. Then for any $R$-module $A$ the canonical map $${\underline{\pi}}^{\mathrm{pre}}_{p,q}({\mathsf{MGL}}_S \wedge M_A)(X) \to \lim_k {\underline{\pi}}^{\mathrm{pre}}_{p,q}({\mathsf{MGL}}_S \left<0,k \right> \wedge M_A)(X)$$ is an isomorphism.
This follows from Corollary \[gfrerz\], the Milnor short exact sequence and Lemma \[jewet45\].
\[xfhtjt\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Let $n \in {{\mathbb Z}}$. Then for $k \ge n+1$ and any $R$-module $A$ the natural map $$\pi_{2n,n} {\mathsf{MGL}}_S \left<n,k+1 \right> \wedge M_A \to \pi_{2n,n} {\mathsf{MGL}}_S \left<n,k \right> \wedge M_A$$ is an isomorphism.
This follows from the exact sequence $$\pi_{2n,n} s_{k+1} {\mathsf{MGL}}_S \wedge M_A \to \pi_{2n,n} {\mathsf{MGL}}_S \left<n,k+1 \right> \wedge M_A \to$$ $$\pi_{2n,n} {\mathsf{MGL}}_S \left<n,k \right> \wedge M_A \to
\pi_{2n-1,n} s_k {\mathsf{MGL}}_S \wedge M_A$$ and the fact the the two outer terms are $0$ for $k \ge n+1$.
\[sgtjtr\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $R$-module $A$ the canonical map $$\pi_{2n,n} {\mathsf{MGL}}_S \wedge M_A \to \pi_{2n,n} {\mathsf{MGL}}_S \left< n,n+1 \right> \wedge M_A$$ is an isomorphism.
This follows from Corollary \[dsgtjt\] and Lemma \[xfhtjt\].
\[grfe5z4\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for every $n \in {{\mathbb Z}}$ and $R$-module $A$ there is a canonical isomorphism $$\pi_{2n,n} {\mathsf{MGL}}_S \wedge M_A \cong L_n \otimes A \oplus L_{n+1} \otimes \mathrm{Pic}(S) \otimes A.$$
We have the exact sequence $$\pi_{2n+1,n} s_n {\mathsf{MGL}}_S \wedge M_A \to \pi_{2n,n} s_{n+1} {\mathsf{MGL}}_S \wedge M_A \to \pi_{2n,n} {\mathsf{MGL}}_S \left<n,n+1 \right> \wedge M_A$$ $$\to \pi_{2n,n} s_n {\mathsf{MGL}}_S \wedge M_A \to \pi_{2n-1,n} s_{n+1} {\mathsf{MGL}}_S \wedge M_A.$$ The two outer terms are $0$. Also $\pi_{0,0} \Sigma^{2,1} {\mathsf{M}A}_S \cong \mathrm{Pic}(S) \otimes A$. Moreover there is a canonical map $L_n \otimes A \to \pi_{2n,n} {\mathsf{MGL}}_S \wedge M_A$ splitting the resulting short exact sequence, whence the claim follows form Corollary \[sgtjtr\].
Let $S$ be the spectrum of a Dedekind domain of mixed characteristic and $R$ the localization of ${{\mathbb Z}}$ obtained by inverting all positive residue characteristics of $S$. Then $$(\pi_{2n,n} {\mathsf{MGL}}_S) \otimes R \cong (L_n \oplus L_{n+1} \otimes \mathrm{Pic}(S)) \otimes R.$$
We have the following case of a Conjecture of Voevodsky (see [@voevodsky.icm Conjecture 1]):
\[hte4234t\] Let $S={\mathrm{Spec}}(R)$, where $R$ is a (regular) Noetherian local ring which is regular over some discrete valuation ring of mixed characteristic. Then the natural map $$L_* \to ({\mathsf{MGL}}_S)_{2*,*}$$ becomes an isomorphism after inverting the residue characteristic of the closed point of $S$.
By Popescu’s Theorem $R$ is a filtered colimit of smooth algebras over a discrete valuation ring $V$ of mixed characteristic. Thus we are reduced to the case where $R$ is the local ring of a scheme $X \in {\mathrm{Sm}}_{{\mathrm{Spec}}(V)}$ by a colimit argument. Let $p$ be the residue characteristic of the closed point of ${\mathrm{Spec}}(V)$. By the same type of argument as above and the vanishing of $(p,q)$-motivic cohomology of such local rings for $p>q$ we have $$(MGL_S)_{2n,n} [1/p] \cong (s_n {\mathsf{MGL}}_S)_{2n,n} [1/p] \cong L_n [1/p],$$ using that for a fixed dimension only a fixed finite number of slices of ${\mathsf{MGL}}_S[1/p]$ contribute to the value of $\pi^{\mathrm{pre}}_{2n,n}({\mathsf{MGL}}_S [1/p])$ on schemes of that dimension.
More generally we have
Let $S$ be as in the previous Theorem and $E \in {\mathsf{SH}}(S)$ a motivic Landweber spectrum modelled on $E^{\mathrm{top}}_{2*}$. Then the natural map $$E^{\mathrm{top}}_{2*} \to E_{2*,*}$$ is an isomorphism after inverting the residue characteristic of the closed point of $S$.
This follows from the definition of motivic Landweber spectrum.
Some other parts of algebraic cobordism {#dzu634e}
=======================================
We have the following vanishing result:
Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $p,q \in {{\mathbb Z}}$ and $R$-module $A$ we have ${\underline{\pi}}_{p,q} {\mathsf{MGL}}_S \wedge M_A \cong 0$ for $p< 2q$ or $p<q$. In particular we have ${\mathsf{MGL}}_S \wedge M_R \in {\mathsf{SH}}(S)_{h \ge 0}$.
Let $p,q \in {{\mathbb Z}}$ satisfying the condition of the statement. Let $d \in {{\mathbb N}}$. Then there is a $N \ge q$ such that for any scheme of dimension $\le d$ and $k \ge N$ the map $${\underline{\pi}}^{\mathrm{pre}}_{p,q}({\mathsf{MGL}}_S \wedge M_A)(X) \to
{\underline{\pi}}^{\mathrm{pre}}_{p,q}({\mathsf{MGL}}_S\left< 0,k \right> \wedge M_A)(X)$$ is an isomorphism. The assertion then follows by an induction argument on $i$ showing that ${\underline{\pi}}_{p,q}({\mathsf{MGL}}_S \left< q,q+i \right> \wedge M_A)=0$.
Generalizing the argument given in the last proof we get
\[dh5t43\] Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $p,q \in {{\mathbb Z}}$ and $R$-module $A$ we have $${\underline{\pi}}_{p,q}({\mathsf{MGL}}_S \wedge M_A) \cong \lim_k {\underline{\pi}}_{p,q}({\mathsf{MGL}}_S \left<0,k \right> \wedge M_A)
\cong {\underline{\pi}}_{p,q}({\mathsf{MGL}}_S \left<\max(0,q),n \right> \wedge M_A)$$ for $n \ge p-q$ or $n \ge p-2q$.
Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $R$-module $A$ and $n \in {{\mathbb Z}}$ we have ${\underline{\pi}}_{n,n}({\mathsf{MGL}}_S \wedge M_A) \cong \underline{K}^M_{-n} \otimes A$, where $\underline{K}^M_{-n}$ is the $(-n)$-th Milnor-$K$-theory sheaf defined via the degree $(-n,-n)$-motivic cohomology.
Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $R$-module $A$ and $n \in {{\mathbb Z}}$ we have ${\underline{\pi}}_{2n,n}({\mathsf{MGL}}_S \wedge M_A) \cong \underline{L}_n \otimes A$, where the latter sheaf is the constant sheaf on $L_n \otimes A$.
Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $R$-module $A$ and $n \in {{\mathbb Z}}$ we have ${\underline{\pi}}_{2n+1,n}({\mathsf{MGL}}_S \wedge M_A) \cong {{\mathcal O}}^* \otimes L_{n+1} \otimes A$.
By Lemma \[dh5t43\] we have $${\underline{\pi}}_{2n+1,n}({\mathsf{MGL}}_S \wedge M_A) \cong
{\underline{\pi}}_{2n+1,n}({\mathsf{MGL}}_S \left< n,n+1 \right> \wedge M_A).$$ The long exact sequence of sheaves associated to the exact triangle $$s_{n+1} {\mathsf{MGL}}_S \wedge M_A \to {\mathsf{MGL}}_S \left< n,n+1 \right> \wedge M_A \to s_n {\mathsf{MGL}}_S \wedge M_A
\to s_{n+1} {\mathsf{MGL}}_S \wedge M_A [1]$$ together with $${\underline{\pi}}_{2n+1,n}(s_n {\mathsf{MGL}}_S \wedge M_A [-1])= {\underline{\pi}}_{2n+1,n}(s_n {\mathsf{MGL}}_S \wedge M_A)=0$$ and $${\underline{\pi}}_{0,0}(\Sigma^{1,1} {\mathsf{M}A}_S) \cong {{\mathcal O}}^* \otimes A$$ gives the result.
Similarly we get
Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $R$-module $A$ and $n \in {{\mathbb Z}}$ there is an exact sequence $$\underline{K}^M_{1-n} \otimes A \to {\underline{\pi}}_{n+1,n}({\mathsf{MGL}}_S \wedge M_A) \to {{\mathcal H}}_{\mathrm{mot}}^{-n-1,-n}(-,A) \to 0,$$ where the latter group denotes the motivic cohomology sheaf in degees $(-n-1,-n)$ and $A$-coefficients.
We also have
Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Then for any $R$-module $A$ and $n \in {{\mathbb Z}}$ there is an exact sequences $$H^{3,2}(S) \otimes A \otimes L_{n+2} \to \pi_{2n+1,n} {\mathsf{MGL}}_S \to H^{1,1}(S,A) \otimes L_{n+1} \to 0.$$ If $A$ is torsionfree the first map is also injective.
Let $S$ be the spectrum of a Dedekind domain of mixed characteristic. Let $R$ be a localization of ${{\mathbb Z}}$ such that $\Phi_S \wedge M_R$ is an isomorphism. Let $X$ be an essentially smooth scheme over $S$. Then for any $R$-module $A$, $n \in {{\mathbb Z}}$ and $i \ge 2$ we have ${\mathsf{MGL}}_S^{2n+i,n}(X,A)=0$.
This follows from the above considerations and the fact that for $X \in {\mathrm{Sm}}_S$ we have $H^{p,q}(X)=0$ for $p \ge 2q+2$, since motivic cohomology is computed as hypercohomology over $S$ of the Bloch-Levine cycle complexes.
We leave assertions about the groups $\pi_{2n-1,n} {\mathsf{MGL}}_S \wedge M_A$, $\pi_{n,n} {\mathsf{MGL}}_S \wedge M_A$ and $\pi_{n-1,n} {\mathsf{MGL}}_S \wedge M_A$ to the interested reader.
Fakult[ä]{}t f[ü]{}r Mathematik, Universit[ä]{}t Osnabrück, Germany.\
e-mail: [email protected]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The notion of duality between the hydrodynamic and kinetic (ghost) variables of lattice kinetic formulations of the Boltzmann equation is introduced. It is suggested that this notion can serve as a guideline in the design of matrix versions of the lattice Boltzmann equation in a physically transparent and computationally efficient way.'
author:
- 'R. Adhikari$^{1,2}$ and S. Succi$^3$'
bibliography:
- 'ghost.bib'
title: Duality in matrix lattice Boltzmann models
---
Introduction
============
In the last decade, the lattice Boltzmann (LB) method has developed into a very flexible and effective numerical technique for the simulation of a large variety of complex, fluid dynamical and non-equilibrium transport phenomena [@Succi:2001]. The LB method is based on a stream-and-collide microscopic dynamics of fictitious particles, which stream with a discrete set of velocities and interact according to local collision rules that drive the system towards a local equilibrium [@Chen:1998; @Wolf-Gladrow:2000; @Succi:2001]. Mathematically, this is formulated as the lattice Boltzmann equation (LBE) $$\label{dbe}
\partial_t f_i + {\bf c}_i\cdot\nabla f_i = -\sum_j L_{ij}(f_j - f_j^0)$$ where $f_{i}({\bf x},t)$ is the mean number of particles at position ${\bf x}$ and time $t$, moving along the lattice direction defined by the discrete velocity ${\bf c}_{i}, (i, j =1,...,N $). In the above, $$f_i^0 = w_i\left(\rho + {\rho{\bf v\cdot c}_i\over c_s^2} + {\rho v_{\alpha}v_{\beta}Q_{i\alpha\beta}\over 2c_s^4}\right)$$ is a local equilibrium distribution, the discrete analogue of a Maxwellian distribution in continuum kinetic theory truncated to second order in the mean flow velocity ${\bf v}$, the $w_i$ are a set of weights which satisfy $\sum_i w_i = 1$, and $c_s$ is the speed of sound in the LB fluid. Greek indices denote Cartesian directions and the summation convention is implied. The low order velocity moments of the distribution function are related to the densities of mass, momentum and the deviatoric stress, $\{\rho, \rho v_{\alpha}, S_{\alpha\beta}\} = \sum_i f_i \{1, c_{i\alpha}, Q_{i\alpha\beta}\}$ where $S_{\alpha\beta} + \rho c_s^2\delta_{\alpha\beta}= \Pi_{\alpha\beta} $ is the Eulerian momentum flux, and $Q_{i\alpha\beta} + c_s^2\delta_{\alpha\beta}=c_{i\alpha}c_{i\beta}$. The higher moments of the distribution are related to the densities of rapidly relaxing kinetic degrees of freedom, variously called ghost or kinetic variables. Finally, $L_{ij}$ is a scattering matrix whose eigenvalues control the relaxation of the kinetic modes to their local equilibrium values. The null eigenvalues correspond to the eigenvectors associated with the conserved mass and momentum densities, while the leading non-zero eigenvalue associated with $Q_{i\alpha\beta}$ controls the viscosity of the LB fluid.
Historically, the LBE in matrix form was derived as a Boltzmann approximation to the dynamics of lattice gas cellular automata [@Higuera:1989]. It was then understood that the equilibria and collision matrix could be constructed independently of the underlying cellular automata microdynamics [@Higuera:1989a], and the lattice Boltzmann approach came into being. The collision matrix was reduced to the simplest possible form consistent with the macroscopic hydrodynamics in [@Qian:1992; @Chen:1992], where the Bhatnagar-Gross-Krook (BGK) [@Bhatnager:1954] form of the collision term was implemented on the lattice with $L_{ij} =\tau^{-1}\delta_{ij}$. In the lattice BGK (LBGK) model, the fluid viscosity, which is the only transport parameter of interest, is given by $ \nu = c_s^2(\tau-1/2)$. Even though it has always been clear that this simplification entails a crude approximation to the relaxation process (all modes relax at the same rate $\tau$), the LBGK equation, since its introduction, has held the mainstream in LB applications. In a parallel development, the collision matrix version of LBE [@Higuera:1989a; @Benzi:1992] has been revisited, optimized, and renamed the MTR (Multiple Time Relaxation) [@dHumieres:1992; @dHumieres:2002], in contrast to the single relaxation time implied by the LBGK equation. A number of authors have also made a strong case for the superiority of the MTR version over LBGK in terms of numerical accuracy and stability[@Lallemand:2000; @McCracken:2005]. Yet, lattice BGK remains by far the most popular form of LBE to date.
The limited of popularity of the MTR approach inspite of its superiority compared to the lattice BGK method, may be due to the lack of a general guiding criterion for the spectral decomposition of the collision matrix. In other words, it has not been clear *a priori* how to choose the eigenvectors of the matrix $L_{ij}$ which span the kinetic space of the discrete populations $f_i$. This ambiguity arises because the conservation laws of mass and momentum fix only the hydrodynamic and transport subset of the eigenvectors, leaving the kinetic subset unspecified (see below). As a consequence, MTR models are dependent on both spatial dimension and the choice of the discrete velocity set ${\bf c}_i$, while the LBGK model is identical across both spatial dimension and choice of velocity set. Further, the notion of orthogonality of eigenvectors in the kinetic space can itself be defined in two distinct ways: the definition followed in [@Benzi:1992] using a weighted inner product, and that in [@dHumieres:1992] using an unweighted inner product. For example, recent work on the shallow water equation [@Dellar:2002a], the fluctuating lattice Boltzmann equation [@Adhikari:2005], and on multireflection boundary conditions use a set of eigenvectors which are orthogonal under the weighted inner product [@Chun:2007]. Clearly, it is important to understand how this non-uniqueness in the kinetic space arises and to provide a guiding principle in chosing the eigenvectors. In this work we shall propose such a guiding rule, by introducing the notion of [*duality*]{} in the kinetic space of the LB.
In the next section we follow the notation in [@Behrend:1994] to highlight the structure of the kinetic space spanned by the eigenvectors and how a change of basis from populations to the moments reveals the dynamics of the various modes. We then introduce and illustrate the idea of duality with concrete examples. We show how a model in which all the ghost degrees of freedom are relaxed at the same rate [@Behrend:1994; @Ladd:1994] may provide the best compromise between full MTR models where every mode has a separate relaxation time and the LBGK model. We end with a discussion on how the present work is relevant to the algorithmic improvement of the LB method.
Spectral representation of the collision matrix
===============================================
Eigenvectors and eigenvalues
----------------------------
For a general athermal $DdQn$ LB model with $n$ velocities in $d$ space dimensions, the $n\times n$ collision matrix $L_{ij}$ has $d + 1$ null eigenvectors corresponding to the density and $d$ components of the conserved momentum, $d(d+1)/2$ eigenvectors corresponding to the stress modes, and $n - (d + 1) - d(d + 1)/2$ eigenvectors corresponding to the ghost modes[@Behrend:1994; @Adhikari:2005]. The choice of the null and stress eigenvectors $\{1, c_{i\alpha}, Q_{i\alpha\beta}\}$ follows directly from the physical definition of the densities associated with them. Without specifying the exact analytical expression for the remaining eigenvectors, let us label a linearly independent set of the eigenvectors of the scattering matrix by $\{A_i^a\}$, where $a = 1 \ldots n$ labels the eigenvector, and $ i = 1 \ldots n$ labels the component of the eigenvector in along the $i-th$ velocity direction. Then, we can define densities associated with the eigenvector $A_i^a$ as moments of the populations by $$\psi^a({\bf x}, t) = \sum_i f_i({\bf x}, t) A_i^a$$ For $A^a_i =\{1, c_{i\alpha}, Q_{i\alpha\beta}\}$ the densities are the mass, momentum and stress. The ghost eigenvectors are higher polynomials of the discrete velocities [@Benzi:1992] . The discreteness of the kinetic space implies that, unlike in the continuum, only a finite number of polynomials can be linearly independent, being equal to the number of discrete velocities. For a model with $n$ discrete velocities, the choice of the $n$ linearly independent polynomials is thus not unique, but defined only upto a similarity transformation. Thus, the reason for the non-uniqueness in the spectral decomposition can be traced to the discreteness of the velocity space itself . Independent of the precise choice, the distribution function itself can be expanded in a linearly independent set eigenvectors which are polynomials of the discrete velocities $$\label{fexpansion}
f_i({\bf x}, t) = w_i\sum_a \psi^a({\bf x}, t){ A_i^a\over N^a}$$ Consistency between the above two equations implies that the set of polynomials $A_i^a$ are both orthogonal and complete, $$\begin{aligned}
&\sum_i & w_i A_i^a A_i^b = N^a\delta^{ab},\\
&\sum_a & A_i^a A_j^a/N^a = \delta_{ij}.\end{aligned}$$ Crucially, with the definitions above [@Adhikari:2005], the eigenvectors $A_i^a$ form an orthogonal set under an inner product $(A^a, A^b) =\sum_i w_i A_i^a A_i^b $. This inner product is identical to that introduced by Benzi *et al* [@Benzi:1992], but distinct from the unweighted inner product $(A^a, A^b) =\sum_i A_i^a A_i^b $ used by d’Humieres and co-workers[@dHumieres:1992; @dHumieres:2002]. The advantages of the present choice are discussed below. As indicated before, a useful categorisation of the polynomials consists of the $d+1$ polynomials $\{1, c_{i\alpha}\}$ corresponding to the mass and momentum, the $d(d+1)/2$ quadratic polynomials $Q_{i\alpha\beta}$ corresponding to the stress, and the remaining $n - (d+1) - d(d+1)/2$ cubic and higher order polynomials corresponding to the ghost variables. Correspondingly, the distribution function can be separated into contributions from the hydrodynamic, transport, and ghost moments $$f_i = f_i^H + f_i^T + f_i^G.$$ This motivates the introduction of projection operators [@Behrend:1994] which project the distribution function onto the hydrodynamic, transport, and ghost subspaces, $$\begin{aligned}
\sum_j P^H_{ij} f_j &=& f_i^H = w_i\left(\rho + {\rho{\bf v}\cdot{\bf c}_i\over c_s^2} \right)\\
\sum_j P^T_{ij} f_j &=& f_i^T = w_i{S_{\alpha\beta}Q_{i\alpha\beta}\over 2 c_s^4}\\
\sum_j P^G_{ij} f_j &=& f_i^G = w_i\sum_{a\in G} \psi^a A_i^a/N^a\\\end{aligned}$$ The explicit form of the projection operators are $$\begin{aligned}
%P^H_{ij} &=& w_i(\delta_{ij} + {\bf c}_i\cdot{\bf c}_j/c_s^2)\\
P^H_{ij} &=& w_i(1 + {\bf c}_i\cdot{\bf c}_j/c_s^2)\\
P^T_{ij} &=& w_iQ_{i\alpha\beta}Q_{j\alpha\beta}/2c_s^4 \\
P^G_{ij} &=&\sum_{a\in G} w_i A_i^a A_j^a/N^a\end{aligned}$$ The discrete Maxwellian is a nonlinear (quadratic) function of the distribution function, and thus Eq.\[dbe\] is a only apparently linear, the nonlinearity being concealed in $f_i^0$. An useful linearisation of the LB equation consists of neglecting the quadratic term in the discrete Maxwellian to yield a local equilibrium $h_i^0$ which is linear in the mean velocity, $$h_i^{0} = w_i\left(\rho + {\rho{\bf v\cdot c}_i\over 2 c_s^2}\right) = \sum_j P_{ij}^H f_j$$ In the linearised approximation for the equilibrium distribution, we have $f_i^0 = h_i^0 = (P^H f)_i$ and so the linearised LBE can now be written as, $$\partial_t f_i + {\bf c}_i\cdot\nabla f_i = -\sum_j L_{ij}[f_j - (P^H f)_j]= -\sum_j L_{ij}^Rf_j$$ where $L_{ij}^R = \sum_k L_{ik}(1 - P^H)_{kj}$ is a right-projected collision matrix. In this form, it is clear that $L_{ij}^R$ by construction has eigenvectors of mass and momentum with zero eigenvalues. The form of the matrix, by itself, places no constraint on the eigenvalues of transport and ghost sectors. However, the requirments of an extended range of hydrodynamic behaviour, stability and isotropy motivate an optimal construction of $L_{ij}^R$. As explained in the Introduction, the simplest possible model consists of a diagonal collision matrix $L_{ij} = \delta_{ij}/\tau$ which implies that all the non-conserved modes relax at the same rate $1/\tau$. This is the very popular LBGK approximation used in the literature. In the hydrodynamic regime, a scale separation exists between the relaxation of the conserved and non-conserved variables: the mass and momentum densities relax slowly, the stress and ghost variables relax rapidly. One variant of a model used by Ladd [@Ladd:1994] uses adjustable relaxation times for the stress modes, and identical unit relaxation times for the ghost modes, i.e. the ghosts are ‘projected’ out. One advantage of this approach is that the precise form of the ghost modes, which in general differ both in number and in form between LB models, need not be known. A generalisation of this model, with two relaxation times [@Behrend:1994] reads, $$L_{ij}^R = \lambda P_{ij}^T + \sigma P_{ij}^G = \sigma(1 - P_{ij}^H) + (\lambda -\sigma)P_{ij}^T$$ where the last follows from the completeness relations $P^H + P^T + P^G = 1$. Since the precise form of the ghost projection operator, and hence the ghost eigenvectors is never needed in this formulation, it is clear that the linearised dynamics in this two-relaxation time model cannot depend on the precise choice of the ghost mode eigenvectors. The only way this model may be optimised is to tune the relaxation rate of the ghost modes in comparision to the stress modes. However, a model which allows separate relaxation times for each individual ghost mode has a greater flexibility and may be optimised to yield the best range of hydrodynamic behaviour [@dHumieres:2002]. It needs careful analysis to see if the gain is enough to justify the loss of simplicity and generality that one obtains from the two relaxation model. Hydrodynamic behaviour is obtained when there are two propagating modes with a dispersion relation $\omega = c_s k + i\nu_L k^2$, $\nu_L=\nu + 3/2 \nu_{bulk}$ being the longitudinal viscosity, and $d-1$ diffusive modes with a dispersion relation $\omega = i\nu k^2$. Both the speed of sound and the viscosities are assumed to be constant.
Linear mode structure
---------------------
The hydrodynamic behaviour of the linearised LBE is most conveniently analysed in the absence of boundaries when a Fourier mode decomposition is possible [@Das:1993; @Behrend:1994; @Lallemand:2000; @Dellar:2002a]. It is important to note that the departure from hydrodynamic behaviour can arise from two distinct sources. The first is the choice of eigenvectors and relaxation times of the discrete velocity (but space and time continuous) LBE. This is the category of error arising from discretisation in velocity space. The second is that arising from the numerical integration of the LBE. This is the category of error arising from discretisation in space and time. The physical and numerical behaviour of the fully discretised LBE dynamics is a combination of both these sources of error. The present work, focussing as it does only on the kinetic space, has direct implications for errors arising out of discretisation of velocity space. The errors arising out of discretisation of space and time are relatively well understood from the numerical analysis of the hyperbolic differential equations. In particular, it is known that an Euler integration step of size $\Delta t$ produces numerical diffusion, and thereby renormalises the viscosity to $\nu = c_s^2(\tau - \Delta t/2)$ [@Chen:1998].
To derive the dispersion relation we Fourier transform the linearised LBE to get $$\label{linlbe}
\partial_t f_i + i{\bf k}\cdot{\bf c}_i f_i = -\sum_j L_{ij}^Rf_j$$ At ${\bf k=0}$, the eigenmodes of the dynamics are the same as the eigenmodes of $L^R$. However, away from ${\bf k=0}$, neither the eigenmodes nor the eigenvalues are identical. For small $k$, an analytical expression for the eigenvalues may be obtained perturbatively [@Behrend:1994] . For arbitrary $k$, a numerical solution is necessary. The dispersion relation is obtained by a Fourier transform in time, $$-i\omega({\bf k})f_i = -\sum_j [i{\bf k}\cdot{\bf c}_i \delta_{ij} + L_{ij}^R]f_j$$ Thus we need to obtain the eigenvectors and eigenvalues of the matrix $$M_{ij} = i{\bf k}\cdot{\bf c}_i \delta_{ij} + L_{ij}^R.$$ The dynamics in Eq.\[linlbe\] can equally well be written in terms of the densities using Eq.\[fexpansion\] as $$\partial_t \psi^a = -\sum_b[\Gamma^{ab} + \lambda^a\delta^{ab}]\psi^b$$ where matrix coupling the different modes is $$N^a\Gamma^{ab} = i{\bf k}\cdot\sum_i w_i A^a_i A^b_i {\bf c}_i$$ It is worth noting that the linearised LB dynamics can be written in either of the forms $$\begin{aligned}
\partial_t f_i &=& -\sum_j (\mathcal{A + C})_{ij}f_j\\
\partial_t \psi^a &=& -\sum_b (\mathcal{A + C})^{ab}\psi^b\end{aligned}$$ The dynamical equation in the $f_i$ basis diagonalises the advection operator $\mathcal{A}_{ij} = i{\bf k\cdot c}_i \delta_{ij}$ , while the dynamical equation in the $\psi^a$ basis diagonalises the collison operator $\mathcal{C}_{ij} = L_{ij}^R$. The eigenvectors of the dynamics are a combination of the $f_i$ and the $\psi^a$. The dispersion relation equation can be conveniently non-dimensionalised by measuring time in units of the inverse of the relaxation rate for the stress modes $\tau = \lambda^{-1}$, and distance in units of $c_s\tau$. The non-dimensionalised dispersion equation then takes the form $$-i\Omega({\bf q})f_i = -\sum_j [i{\bf q}\cdot{{\bf c}_i\over c_s} \delta_{ij} + {\cal L}_{ij}^R]f_j$$ where $\Omega = \omega\tau$ is a non-dimensionalised frequency and ${\bf q} = {\bf k}c_s\tau$ is a non-dimensionalised wavevector. It should be noted that the non-dimensionalised collision matrix ${\cal L}^R = L^R/\tau$ now depends only on the ratio $\sigma/\lambda$ of the relaxation rates of the ghost and stress eigenvectors. We shall use this non-dimensionalised form of the dispersion relation to obtain the numerical eigenspectrum of one of the LBE models presented below.
Duality in lattice kinetic theory
=================================
The symmetry principle of duality, which relates two different mathematical representations of the same physical theory, is a powerful tool in many areas of physical science. Duality is often use to map strongly interacting degrees of freedom to weakly interacting ones, thus facilitating an approximate, and often, even an exact solution of the problem. A celebrated example is the solution of Kramers and Wannier for the critical temperature of the Ising model [@Kramers:1941]. To the best of our knowledge, dual symmetries do not appear to have played any major role in kinetic theory. In the context of the lattice Boltzmann schemes, we introduce duality not as an exact symmetry, but as a requirement on the structure of the kinetic space of the theory. Specifically, we require that the structure of the ghost subspace should mirror that of the hydrodynamic subspace, and consist of scalar densities and associated vector currents. In our notation, a LB kinetic space is exactly dual if each ghost field corresponds to a hydrodynamic field and a suitable transformation converts the ghost degrees of freedom into hydrodynamic degrees of freedom. If this exact correspondence is broken, but the ghost subspace still consists of sets of scalar densities and vector currents we say that the kinetic space is quasi-dual. The scalar densities and vector currents are taken to be even and odd functions of the discrete velocities respectively. Thus introduced, duality is a *normative* principle on the structure of the kinetic space of the LBE. The duality principle, as we show with several examples below, allows us to choose the eigenvectors of the collision matrix in a way which is both transparent and unique.
Two dimensions
--------------
Let us first consider the standard $D2Q9$ model with the usual set of velocities connecting the four nearest neighbours and the four next-nearest neighbours of the square lattice. Thus there are four velocities with unit modulus, another four with modulus two, which together with the zero velocity give the nine dynamical populations of the $D2Q9$ model. The kinetic space is spanned by eigenvectors corresponding to the mass and momentum, $\{A^{0}_i, A^{1}_i, A^{2}_i\} = \{1_i, c_{ix}, c_{iy}\}$. The next three natural eigenvectors associated with stress tensor are $\{A^{3}_i, A^4_i, A^5_i\}$ = $\{Q_{ixx}, Q_{ixy}, Q_{iyy}\} \equiv \{ c_{ix}^2-c_s^2, c_{ix}c_{iy}, c_{iy}^2-c_s^2\}$. All of these are recognized as discrete velocity analogues of tensor Hermite polynomials [@He:1997e] . Without any physical considerations to guide us, the choice of three higher-order eigenvectors, associated with the ghost modes remains open. An obvious choice is the next series of tensor Hermite polynomials, that is $Q_{ixx} c_{ix}$, $Q_{ixx} c_{iy}$, $Q_{iyy} c_{ix}$, $Q_{iyy} c_{iy}$. It is immediately seen that due the identity $c_{ix}^3 = c_{ix}$, holding for the $D2Q9$ lattice, only two of these are linearly independent. This lack of linear independence, as we mentioned earlier, is due to the discrete nature of the velocities, giving identities like $c_{ix}^3 = c_{ix}$, which are absent in the continuum. To complete the kinetic space, one more eigenvector is required. It is immediately checked that, as a consequence of the $D2Q9$ identity $c_{ia}^4 = c_{ia}^2$, $a=x,y$, out the five Hermite polynomials of order 4, only one is linearly independent, which we chose as $Q_{ixx} Q_{iyy}$. This then completes the construction of the remaining three ghost eigenvectors.
In a very illuminating paper, Dellar [@Dellar:2002a] proposes a different decomposition, based on the notion of ghost densities introduced in [@Benzi:1992]. The first ghost eigenvector is of the form $$G^{0}_i \equiv g_i=(1,-2,-2,-2,-2,4,4,4,4)$$ and the remaining two are simply the corresponding ‘currents’, that is $$G^{1}_i \equiv g_i c_{ix},
G^{2}_i \equiv g_i c_{iy}$$ The physical meaning of this choice is best highlighted by expressing $g_i$ in analytical form, that is $$g_i= { c_i^4\over 2c_s^4} -{5 c_i^2\over 2 c_s^2} + 1_i$$ where $c_i^2 = c_{ix}^2 + c_{iy}^2$. It is easily checked that the basis $A^{0} \dots A^{5}, A^6 = G^{0}, A^7 = G^{1}, A^8 = G^{2}$ is orthogonal under the weighted scalar product $(A^a,A^b) = \sum_i w_i A_i^a A_i^b$, where $w_0=4/9$, $w_{1-4}=1/9$ and $w_{5-8}=1/36$ are the usual $D2Q9$ weights. It is also to be noted that, owing to the D2Q9 identities, the ghost eigenbasis can also be written as
$$\begin{aligned}
G_i^0= c_{ix}^2 c_{iy}^2 - (3/2) (c_{ix}^2+c_{iy}^2) + 1\\
G_i^1 = c_{ix} c_{iy}^2 - (3/2) (c_{ix}+c_{iy}^2 c_{ix} ) + c_{ix}=-(1/2) c_{ix}(1+c_{iy}^2)\\
G_i^2 = c_{iy} c_{ix}^2 - (3/2) (c_{iy}+c_{ix}^2 c_{iy} ) + c_{iy}=-(1/2) c_{iy}(1+c_{ix}^2)\end{aligned}$$
Surprisingly, then, $G^0_i = g_i$ is a fourth order lattice Hermite polynomial, while $G_i^1 = g_i c_{ix}$ and $G_i^2 = g_i c_{iy}$ instead of being fifth order lattice Hermite polynomials turn out to be third order lattice Hermite polynomials. This fulfils exactly the duality principle introduce above: the kinetic space is decomposed into a set of eigenvectors corresponding to conserved, transport and ghost moments; the ghost degrees of freedom correspond to an even scalar density and two odd vector currents and are in one-to-one correspondence with the hydrodynamic degrees of freedom; and as we show below, the ghost and hydrodynamic degrees of freedom are related by a suitable transformation.
The duality in the decomposition is beautifully illustrated by the diamond structure of the $D2Q9$ eigenvectors shown in Table 1. The density and the two momenta are matched by a ghost density and two ghost currents. The dynamical behaviour of these degrees of freedom are of course quite different, as is revealed by displaying the LBE dynamics in the basis of moments. The kinetic moments associated with the present choice of eigenvectors is $$\{\rho, \rho v_{\alpha}, S_{\alpha\beta}, \rho^{\prime}, j^{\prime}_{\alpha}\} = \sum_i f_i \{1, c_{i\alpha}, Q_{i\alpha\beta}, g_i, g_i c_{i\alpha}\}$$ The primed quantities correspond to ghost density and its currents. The decompostion of $f_i$ as the sum of a hydrodynamic, transport and ghost components is $$\begin{aligned}
f^H_i &=& w_i( \rho +{{\bf j\cdot c}\over c_s^2})\\
f^T_i &=& w_i( {S_{\alpha\beta} Q_{i\alpha\beta} \over 2c_s^4})\\
f^G_i &=& {1\over 4} w_i g_i({\rho^{\prime}} + {{\bf j}^{\prime}\cdot{\bf c}\over c_s^2})\end{aligned}$$ From the above expressions it is clear that, to within a scale factor, the ghost sector is transformed into the conserved sector under the duality transformation $1_i \leftrightarrow g_i$.
To physically interpret the above decomposition, we first note that the combination $w_i^\prime = w_i g_i$ may be interpreted as the weight associated with the ghost degrees of freedom. Then, the weights of the hydrodynamic modes sum to unity $\sum w_i = 1$, while the weights of the ghost modes sum to zero $\sum_i w_i^\prime = 0$. This last result combined with the fact that ghost density is even in the velocities $g_i = g_{i^{\star}}$, where ${\bf c}_{i^{\star}} = -{\bf c}_{i}$, indicates that the ghosts correspond to oscillatory eigenvectors familiar in quantum and statistical mechanics, where they represent excitations above the ground state or above equilibrium. The ghost degrees of freedom are thus non-equilibrium excitations carried by even, oscillatory eigenvectors. The even and odd character of eigenvectors can be exploited to classify the entire set of moments into two categories: even moments representing densities, and odd moments representing currents. Odd moments, representing currents, vanish at global equilibrium by symmetry. The even moments are not constrained to vanish by symmetry arguments. However, since the kinetic modes have no projection onto the global equilibrium distribution function $f_i^0 = w_i\rho$, they can be conveniently chosen to vanish at equilibrium. This is one of the principal advantages of using a set of eigenvectors which are orthogonal under the weighted inner product.
By interpreting $w_i$ as ‘masses’ of the hydrodynamic modes, the $w^{\prime}_i$ can be identified with ‘masses’ of ghost modes. By construction, since they sum up to zero, some of these masses ought to be negative. For instance, the ghost density can be rewritten as an alternating sum of the populations associated with the three energy levels $c_j^2 =0,1,2$, that is $\rho^{\prime} = f_0 - 2(f_1 + f_2 + f_3 + f_4) + 4(f_5 + f_6 + f_7 + f_8) = \rho_0 -2 \rho_1 + 4 \rho_2$, where $j=0,1,2$ refer to the $j$-th energy level. Being the sum of populations, each of the partial densities is strictly non-negative at all times, but the combination of alternating coefficients is a signed quantity, $\rho^{\prime}$, which is zero only at equilibrium. The duality is made even more apparent by defining the reduced distribution function $\phi_i \equiv f_i/w_i$, thus writing $\rho = \sum_i w_i \phi_i$ and $\rho^{\prime} = \sum_i w_i^{\prime} \phi_i$. Since $w_i$ is the lattice analogue of global equilibrium distribution, $w_i^{\prime}$ may also be interpreted as a measure of the global departure from equilibrium.
The ghost currents are a measure of the skewness of the kinetic distribution function, which is non zero only out of equilibrium. Being based on this equilibrium $\leftrightarrow $ non-equilibrium duality, the ghost decomposition shows that the higher-order excitations, keeping the system away from equilibrium, can be structured exactly like their hydrodynamic counterparts. It should be appreciated the duality is structural and not dynamical: it is broken at various levels, starting with the prefactors defining the ghost density and current, because the norm of the dual hydrodynamic versus ghost eigenvectors is not the same. In particular, this implies that the ghost kinetic tensor is not isotropic, as one can easily check by a direct calculation: $P_{xx}^{\prime} = 4 P_{xx}$ and $P_{xy}^{\prime} = - 4 P_{xy}$. This is not surprising, since equilibrium and non-equilibrium are [*not*]{} physically equivalent.
However, a dynamical transformation in time, $\lambda \leftrightarrow 1/\lambda$ turns perfectly conserved modes (infinite-lifetime) into perfectly [*non*]{}-conserved ones (zero-lifetime). What this means is that the distinction between equilibrium and non-equilibrium modes is not dictated by the structure of the kinetic space, but only by the actual values of the lifetimes of the excitations supported by this equation. In this respect, we expect a signature of this dynamical duality in the form of a mirror symmetry ${\lambda/\sigma}\leftrightarrow {\sigma/\lambda}$ in the dispersion relation for the two-relaxation time model introduced earlier. Numerical dispersion relations presented in the next section do show evidence of such a symmetry.
-- -- ---- ----- ------ ----- ---- -- ---
1 C
x y C
xx xy yy T
xyx xyy G
xxyy G
-- -- ---- ----- ------ ----- ---- -- ---
: The arrangement of the eigenvectors of the $D2Q9$ lattice in a diamond structure. There is an exact duality about the transport sector (T) with the conserved hydrodynamic (C) and ghost (G) sectors, transforming into each other under the interchange of weights (see text).
The structural duality of the kinetic space is broken dynamically by the different eigenvalues assigned to the hydrodynamic and ghost modes. The hydrodynamic sector sustains itself even in the absence of ghost (standard macroscopic hydrodynamics), whereas the ghosts, because of finite lifetimes assigned to them, do not survive without a forcing from the hydrodynamic modes. Indeed, in the absence of the hydrodynamic feedback on $P^{\prime}_{\alpha \beta}$, the ghost sector would rapidly estinguish, because neither its density nor its current are conserved in time. Of course, such dynamical asymmetry can always be removed by choosing the ghost eigenvalues equal to be zero. In fact, there is even some evidence that long-lived ghosts may prove beneficial to the numerical stability of short-scale hydrodynamics in fluid turbulence [@Sbragaglia:2006a]. This is a prescription to keep the Boltzmann distribution away from equilibrium for an indefinitely long time, leading to anomalous relaxation. This prescription clearly violates the normal ordering between slow, hydrodymic and fast, kinetic modes, as it corresponds to enforcing additional conservation laws with no counterpart in the real molecular world. Hence, such a procedure can only be justified as an effective interaction between collective degrees of freedom, as for example in lattice kinetic equations for turbulent flows.
Higher order lattices in two dimensions
---------------------------------------
In two dimensions, lattices with more than nine velocities are used in thermal LBE and in applications in microfluidics and multiphase flow. Duality can be used to generate an optimal kinetic space for these higher order lattices as well. Let us denote by $D(s)$ the lattice corresponding to a hierarchical tree of eigenvectors, with $2s + 1$ levels and symmetric about the $s+1$th level.(see Table 2 and 3). Clearly, this hierarchy contains $(s + 1)^2$ independent moments. With this definition, $D(0)$ corresponds to the lattice with a single zero-speed rest particle, $D(1)$ to the $D2Q4$ lattice, and $D(2)$ with the $D2Q9$ lattice. $D(3)$, which represents a higher order lattice in the present terminology, consists of $16$ velocities with four velocities each of modulus $1, 2, 4$ and $8$. For this rather complicated lattice, the duality prescription proceeds by first chosing the usual eigenvectors corresponding to the conserved and transport sectors. Proceeding to the $3rd$ level, the eigenvectors of the type $c_{i\alpha}Q_{i\beta\gamma}$ (and permutations) now turn out to be linearly independent. The remaining moments are constructed in a top-down fashion, beginning with a sixth-order scalar density $\rho_{6i} = A c_i^6 - B c_i^4 + C c_i^2 - 1_i$ from which two currents $\rho_{6i}c_{i\alpha}$ and three tensorial densities $\rho_{6i}Q_{ixx}, \rho_{6i}Q_{ixy}, \rho_{6i}Q_{iyy}$ can be used to complete the hierarchy. The expansion coefficients $A, B, C$ for the scalar density can be computed by requiring orthogonality to the lower eigenvectors.
----- ------------ ----------- ------------ ----------- ------------ ----- --- -----------
1 C lattice
x y C Hermite
xx xy yy T expansion
xxx xxy xyy yyy T
$\rho_6$xx $\rho_6$xy $\rho_6$yy G
$\rho_6$x $\rho_6$y G duality
$\rho_6$ G
----- ------------ ----------- ------------ ----------- ------------ ----- --- -----------
: The hierarchical tree of moments for the 16 speed dual lattice $D(3)$ defined in the text. The first four levels are constructed using the usual lattice Hermite polynomials. The remaining three levels are completed using the duality prescription starting with an even, sixth-order scalar density
The next member of the hierarchy is $D(4)$, corresponding to the lattice of $25$ speeds which has recently been shown to have $8$th order isotropy in its spatial behaviour. As with $D(3)$, the eigenvectors upto the $3rd$ level are the conserved, transport, and tensorial Hermite polynomials $c_{i\alpha}Q_{i\beta\gamma}$ and permutations. Again, on the $25$ velocity lattice, the permutations give rise to independent eigenvectors. The remaining eigenvectors constructed bottom-up starting from an even, scalar eighth-order density $\rho_{8i} = A c_i^8 - B c_i^6 + C c_i^4 - Dc_i^2 +1_i$, from which two currents $\rho_{8i}c_{ix}$, s $\rho_{8i}c_{iy}$, three tensorial densities $\rho_{8i}Q_{ixx}$, $\rho_{8i}Q_{ixy}$, $\rho_{8i}Q_{iy}$, and four tensorial currents $\rho_{8i}t_{ixxxx}$, $\rho_{8i}t_{ixxyy}$, $\rho_{8i}t_{ixyyy}$, $\rho_{8i}t_{iyyyy}$ can derived, thus completing the list of $25$ independent eigenvectors.
-- ------ ------------- ------------ ------------- ------------ ------------- ------------ ------------- ------ --- -----------
1 C lattice
x y C Hermite
xx xy yy T expansion
xxx xxy xyy yyy T
xxxx xxxy xxyy xyyy yyyy T
$\rho_8$xxx $\rho_8$xxy $\rho_8$xyy $\rho_8$yyy G
$\rho_8$xx $\rho_8$xy $\rho_8$yy G duality
$\rho_8$x $\rho_8$y G
$\rho_8$ G
-- ------ ------------- ------------ ------------- ------------ ------------- ------------ ------------- ------ --- -----------
Both of the above examples show that the duality prescription offers a transparent method of choosing and ordering the set of eigenvectors in lattices which are more complicated than the most commonly used $D2Q9$ lattice in two dimensions. It has recently been shown that the $16$ and $25$ speed lattice with a proper choice of weights, provide $6$th and $8$th order isotropy respectively. [@Sbragaglia:2007; @Shan:2006]. Since the choice of weights is intimately related to both weighted inner product and the duality prescription, it is possible that there is fundamental link between isotropy and the duality prescription.
Three dimensions
----------------
In three dimensions, the model which is potentially exactly dual is the $D3Q14$ model, which has the usual $4$ hydrodynamic degrees of freedom, $6$ transport degrees of freedom, leaving $4$ ghost degrees of freedom to match the hydrodynamic ones. However, since the $D3Q14$ model is not used in practice, we pass on instead to the analysis of the most common $D3Q19$ model. Here, of course, the kinetic space can only be quasi-dual, since there are $9$ ghost degrees of freedom. To choose them according to the duality prescription, several possibilities can be explored. With one ghost density quartic in the velocity, and three associated currents which are quintic, we obtain $4$ ghost eigenvectors, leaving $5$ free. This allows three independent components of the ghost momentum-flux tensor ($xy,xz,yz)$, plus another two, which must necessarily come from a higher Hermite level. This seems to be a rather obscure and unpromising avenue. A better possibility is to select [*two*]{} ghost densities along with their currents, leaving the third one ‘naked’, *i.e.* without independent degrees of freedom for the current. Retaining [*three*]{} quartic ghost densities with their respective currents is unviable, for it gives a total of twelve eigenvectors, three too many. From these considerations, it appears that the $2$ (dressed) plus $1$ (naked) density representation comes closest to fulfillingl the duality program. It is interesting to note that, apart from the third, naked, density, this is precisely the early decomposition adopted in [@Benzi:1992], based on the 3d projection of 4d face-centered hypercube (24 speeds in $d=4$, 18 in $d=3$). The explicit form of the chosen eigenvectors is given in the Appendix.
Numerical results
=================
We now present a numerical calculation of the dispersion relation of the two-relaxation time lattice Boltzmann model with the duality-prescribed choice of eigenvectors. The dispersion relation is obtained by numerically computing the eigenvalues and eigenvectors of the matrix $M_{ij}$. As explained previously, with a suitable rescaling, the only parameter in the problem is $\sigma/\lambda$, the ratio of the relaxation rates of the kinetic and stress degrees of freedom. For $\sigma = \lambda$ the collision term reduces to the LBGK diagonal collision operator. The imaginary parts of the eigenvalues for the D$2$Q$9$ model are shown in Fig.1 , clearly showing the presence of hydrodynamic modes (relaxation rates vanish as wavenumber goes to zero) as well as non-hydrodynamic modes (relaxation rates remain finite as wavenumber goes to zero). The scale separation between the relaxation rates of the hydrodynamic and non-hydrodynamic modes becomes progressively smaller with increasing wavenumber, and there is considerable overlap at around $q = \pi$. This is fairly plausible, since $q = \pi$ is the value at which the wavelength becomes comparable with the mean free path $c_s \tau$, so that the distinction between hydrodynamics and kinetic modes fades away. This lack of scale separation is responsible for the poor range of hydrodynamic behaviour of the LBGK models, a fact that was correctly noted earlier [@Lallemand:2000]. With $\sigma = {1\over 2}\lambda$, the overlap between the hydrodynamic and non-hydrodynamic modes in Fig.2 is even greater, indicating a further reduced range of hydrodynamic behavior, compared to the LBGK models. On the other hand, for $\sigma = 2\lambda$, we see in Fig.3 a clean separation between the hydrodynamic and kinetic degrees of freedom, and it is in this range of parameters that we expect the best hydrodynamic behavior of the two-relaxation time LB model. An interesting qualitative feature that emerges from comparing Fig.2 and Fig.3, is that the eigen-frequencies are almost ‘dual’ to each other, in the sense that the dispersion curves are approximately identical after a reflection about the ordinate and a rescaling by $\lambda\over\sigma$. The structural duality reflects itself in the dynamical behaviour if the relaxation times are chosen appropriately.
This supports our earlier assertion that the structure of the hydrodynamic and non-hydrodynamic modes are dual to each other. It also illuminates the physical behavior of ghost modes, whose dynamics appears to be characterized by a competition between global decay as characterised by $\sigma$, and instabilities driven by negative diffusion as indicated by the negative curvature of the ghost dispersion relations. Thus the ghost modes decay globally, but driven by the negative diffusion, concentrate around thinner and thinner regions of space, and can thereby undermine the high frequency high wavenumber stability of the system. Good hydrodynamic behavior is thus expected whenever global decay proceeds sufficiently fast to deplete the ghost energy before this energy has time to cascade to high frequencies. This picture suggests a number of interesting questions for future studies. First, it would be interesting to explore whether the use entropic methods [@Succi:2002] simply accelerates the ghost decay, or rather turns ghosts into stable modes. Second, following [@Sbragaglia:2006a], it would be interesting to study whether the dual-decomposition can help designing the ghost dynamics in such a way as to absorb energy bursts from the hydrodynamic component, as they occur in a turbulent flow (intermittency). This could be achieved, for instance, by promoting ghost eigenvalues to dynamical fields responding self-consistently to the local dynamics of the turbulent flow, as it is currently done with the transport eigenvalues $\tau$ in the kinetic modeling of fluid turbulence [@Chen:2003]. Finally, we note that the long-wavelength dynamics obtained with the present choice of eigenvectors is, by construction, isotropic at order $k^2$ and Galilean invariant.
Conclusion
==========
In this paper we have developed the notion of duality between the hydrodynamic and ghost sectors of lattice kinetic equations, as a guiding criteria to resolve the ambiguities which arise in the practical construction of LB models in matrix form. Our main prescription is that the ghost sector should be constructed, in analogy with the hydrodynamic sector, to consist of density-current pairs. This prescription is exactly realised in the $D2Q9$ model, where in addition, the ghost and hydrodynamic sectors can be interchanged by a suitable swapping of weights. For higher order lattice in and in higher dimensions the kinetic degrees of freedom are more numerous then the hydrodynamic ones thereby ruling out an exact correspondence between the two. However, the duality prescription still provides an useful ordering of the eigenvectors into a quasi-dual kinetic space. The duality principle presented in this paper has been used previously in constructing the kinetic space of the fluctuating lattice Boltzmann equation [@Adhikari:2005]. It has also been recently used to compare the accuracy of multireflection boundary conditions with both weighted and un-weighted eigenvectors [@Chun:2007]. We hope the duality principle as introduced here will provide an impetus to further developments in the matrix formulation of the lattice Boltzmann method.
Appendix
========
For easy reference we present the eigenvectors of the $D2Q9$ and $D3Q19$ models chosen according to the duality prescription with weighted inner product. The table is arranged according to the conserved (C), transport (T) and ghost (G) sectors. $$A^T = \left[
\begin{array}{l|rrrrrrrrr}
A^{\rho}&1&1&1&1&1&1&1&1&1\\
A^{j_x}&0&1&0&-1&0&1&-1&-1&1\\
A^{j_y}&0&0&1&0&-1&1&1&-1&-1\\
A^{Q_{xx}}&-1&2&-1&2&-1&2&2&2&2\\
A^{Q_{xy}}&0&0&0&0&0&1&-1&1&-1\\
A^{Q_{yy}}&-1&-1&2&-1&2&2&2&2&2\\
A^{\rho^{\prime}}&1&-2&-2&-2&-2&4&4&4&4\\
A^{j^{\prime}_x}&0&-2&0&-2&0&4&-4&-4&4\\
A^{j^{\prime}_y}&0&0&-2&0&2&4&4&-4&-4\\
\end{array}
\right]$$
$$A^T = \left[
\begin{array}{l|rrrrrrrrrrrrrrrrrrr}
A^{\rho}& 1 & 1 & 1 & 1 & 1 & 1 & 1 &
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
A^{j_x}& 0 & 1 & -1 & 0 & 0 & 0 & 0 &
1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 0 & 0 & 0 & 0\\
A^{j_y}& 0 & 0 & 0 & 1 & -1 & 0 & 0 &
1 & -1 & 1 & -1 & 0 & 0 & 0 & 0 & 1 & 1 & -1 & -1\\
A^{j_z} & 0 & 0 & 0 & 0 & 0 & 1 & -1 &
0 & 0 & 0 & 0 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1\\
A^{Q_{xx}}& -1 & 2 & 2 & -1& -1 & -1 & -1 &
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & -1 & -1 & -1 & -1\\
A^{Q_{yy}}& -1 & -1 & -1 & 2& 2 & -1 & -1 &
2 & 2 & 2 & 2 & -1 & -1 & -1 & -1 & 2 & 2 & 2 & 2\\
A^{Q_{zz}}& -1 & -1 & -1 & -1& -1 & 2 & 2 &
-1 & -1 & -1 & -1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2\\
A^{Q_{xy}}& 0 & 0 & 0 & 0& 0 & 0 & 0 &
1 & -1 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
A^{Q_{yz}}& 0 & 0 & 0 & 0& 0 & 0 & 0 &
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & -1 & 1\\
A^{Q_{zx}}& 0 & 0 & 0 & 0& 0 & 0 & 0 &
0 & 0 & 0 & 0 & 1 & -1 & -1 & 1 & 0 & 0 & 0 & 0\\
A^{\rho^{\prime}} & 0 & 1 & 1 & 1 & 1 & -2 & -2 &
-2 & -2 & -2 & -2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
A^{j^{\prime}_x}& 0 & 1 & -1 & 0& 0 & 0 & 0 &
-2 & -2 & 2 & 2 & 1 & 1 & -1 & -1 & 0 & 0 & 0 & 0\\
A^{j^{\prime}_y}& 0 & 0 & 0 & 1& -1 & 0 & 0 &
-2 & 2 & -2 & 2 & 0 & 0 & 0 & 0 & 1 & 1 & -1 & -1\\
A^{j^{\prime}_z}& 0 & 0 & 0 & 0& 0 & -2 & 2 &
0 & 0 & 0 & 0 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1\\
A^{\rho^{\prime\prime}}& 0 & 1 & 1 & -1& -1 & 0 & 0 &
0 & 0 & 0 & 0 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1\\
A^{j^{\prime\prime}_x}& 0 & 1 & -1 & 0& 0 & 0 & 0 &
0 & 0 & 0 & 0 & -1 & -1 & 1 & 1 & 0 & 0 & 0 & 0\\
A^{j^{\prime\prime}_y}& 0 & 0 & 0 & -1& 1 & 0 & 0 &
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & -1 & -1\\
A^{j^{\prime\prime}_z}& 0 & 0 & 0 & 0& 0 & 0 & 0 &
0 & 0 & 0 & 0 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1\\
A^{\rho^{\prime\prime\prime}}& 1 & -2 & -2 & -2& -2 & -2 & -2 &
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
\end{array}
\right]$$
SS wishes to acknowledge the Physics Department of the University of Edinburgh for financial support and kind hospitality. Both authors wish to thank Prof M. E. Cates for valuable discussions and a critical reading of the manuscript. RA was funded in part by EPSRC GR/S10377.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'An axiomatisation of Hurkens’s paradox in dependent type theory is given without assuming any impredicative feature of said type theory.'
author:
- Arnaud Spiwack
bibliography:
- 'library.bib'
title: 'Notes on axiomatising Hurkens’s Paradox'
---
Hurkens’s paradox [@Hurkens1995] is a very economic, though rather hard to understand, paradox of the $\mbox{\textrm{U}}^-$ impredicative type theory, described in Section \[latex\_lib\_label\_2\], whose main characteristic is to feature to nested impredicative sorts. Its terseness makes it the weapon of choice to derive inconsistencies from logical principle or experimental language features of your favourite proof assistant. Or, rather, embedding $\mbox{\textrm{U}}^-$ in some way is the weapon of choice, Hurkens’s paradox serves as a way to turn this into a proof of false.
It may sound like a futile game to play: if you are the ideal mathematician you will never implement inconsistent feature in your proof assistant. Unfortunately, you are not, and deriving contradiction will happen to you from time to time. Having a tool for that may turn out to be of tremendous help. As a bonus, the inconsistency of $\mbox{\textrm{U}}^-$ can serve to derive potentially useful principles, such as the fact that if the principle of excluded middle holds in an impredicative sort, then types in that sort have the proof irrelevance property (see Section \[latex\_lib\_label\_6\]).
The downside in all that is that there does not seem to be a good way to express, within dependent type theory, the existence of an impredicative sort. Coquand [@Coquand86] gave a sufficient condition, albeit much stronger, to derive contradictions in a generic way. His proof was based on Girard’s [@Girard1972] paradox rather than Hurkens’s one (which came out ten years later). Geuvers [@Geuvers2007] later gave a proof based more directly on Hurkens’s one and relying on a single impredicative sort, but this proof wasn’t very generic. The result was that Hurkens’s proof was included *twice* in the distribution of the Coq proof assistant [@Coq]: Geuvers’s proof, and a variant due to Hugo Herbelin to prove slightly different results.
This situation is certainly unsatisfactory, as adapting Hurkens’s proof for every little variation around the same theme is significantly more work than describing an encoding of $\mbox{\textrm{U}}^-$. It prevents good people from finding perfectly good proof of contradictions: it isn’t fair to assume that everyone is an expert in Hurkens’s proof.
As it happens, however, there is a perfectly good axiomatisation of $\mbox{\textrm{U}}^-$ in your favourite dependently typed proof assistant (in actuality, a sufficient subsystem). And the corresponding proof of contradiction is, *mutatis mutandis*, Geuvers’s, where conversion rules are replaced by equalities.
Axiomatic Hurkens’s paradox {#latex_lib_label_1}
===========================
The trick, so to speak, of the axiomatix presentation of $\mbox{\textrm{U}}^-$ is generally attributed to Martin-Löf: a universe is given by an type describing the types in the universe, and an decoding function ${\rightarrow}$ describing, for each type in the elements of that types. Sorts are to be encoded as such universes. System $\mbox{\textrm{U}}^-$ has two of these, commonly called *large* and *small*, together with rules to combine them. Each of these rules take the form of a product formation rule (see Barendregt’s presentations of *pure type systems*, formerly known as *generalised type systems* [@Barendregt1991][@Barendregt Section 5.2]). Instead of the usual presentation where there is a single dependent product with a number of formation rules, we will have a distinct dependent product – with its own introduction rule (${\lambda}$-abstraction) and elimination rule (application) – for each of the formation rule. For each pair ${\lambda}$-abstraction & application, there may be a ${\beta}$-equivalence rule, modelled as an equality; only the ${\beta}$-equivalence rules which are effectively used in the proof are axiomatised.
Axiomatic $\mbox{\textrm{U}}^-$ {#latex_lib_label_2}
-------------------------------
The full axiomatic presentation appears below, in Coq syntax. It is also part of Coq’s distribution and can be found, at the time these notes are being written, in the file .
##### Large universe
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{U1}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{Type}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{El1}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{U1}}}~${\rightarrow}$~{\usefont{T1}{pag}{b}{n}{\small{Type}}}\symbol{46}}}$$
The large universe is closed by dependent products over types in . The definition of dependent product and ${\lambda}$-abstraction are defined using the function space of the dependent type theory. Notations are defined for dependent product, ${\lambda}$-abstraction and application. As usual, an arrow notation is used when the dependent product has a constant range.
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{Forall1}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U1}}}\symbol{44}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{El1}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{U1}}}\symbol{41}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{U1}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{b}{n}{\small{Notation}}}~\symbol{34}\symbol{39}${\forall}$$_1$\symbol{39}~{\usefont{T1}{pag}{m}{n}{\small{x}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{B}}}\symbol{34}~$\coloneqq $~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{Forall1}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{fun}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}~${\Rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{B}}}\symbol{41}\symbol{41}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{b}{n}{\small{Notation}}}~\symbol{34}{\usefont{T1}{pag}{m}{n}{\small{A}}}~\symbol{39}${\longrightarrow}$$_1$\symbol{39}~{\usefont{T1}{pag}{m}{n}{\small{B}}}\symbol{34}~$\coloneqq $~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{Forall1}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{fun}}}~$\_$~${\Rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{B}}}\symbol{41}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{lam1}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}~{\usefont{T1}{pag}{m}{n}{\small{B}}}\symbol{44}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{El1}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{El1}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{41}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{El1}}}~\symbol{40}${\forall}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{b}{n}{\small{Notation}}}~\symbol{34}\symbol{39}${\lambda}$$_1$\symbol{39}~{\usefont{T1}{pag}{m}{n}{\small{x}}}~\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{34}~$\coloneqq $~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{lam1}}}~$\_$~$\_$~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{fun}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}~${\Rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{41}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{app1}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}~{\usefont{T1}{pag}{m}{n}{\small{B}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{El1}}}~\symbol{40}${\forall}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{El1}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{41}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{El1}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{b}{n}{\small{Notation}}}~\symbol{34}{\usefont{T1}{pag}{m}{n}{\small{f}}}~\symbol{39}${\cdot}$$_1$\symbol{39}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{34}~$\coloneqq $~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{app1}}}~$\_$~$\_$~{\usefont{T1}{pag}{m}{n}{\small{f}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{beta1}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}~{\usefont{T1}{pag}{m}{n}{\small{B}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{El1}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{El1}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{41}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{44}\\
\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\symbol{40}${\lambda}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{y}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{f}}}~{\usefont{T1}{pag}{m}{n}{\small{y}}}\symbol{41}~${\cdot}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{x}}}~\symbol{61}~{\usefont{T1}{pag}{m}{n}{\small{f}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{46}}}$$
The large universe is made impredicative by a dependent product with large domain. The standard presentation would use a sort , of which is a member; the dependent product would then have, as a domain, some . This would be unnecessary complexity as is so restricted that the only interesting type in it would be . So, instead, we simply restrict the domain of the product to be .
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{ForallU1}}}~\symbol{58}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{U1}}}${\rightarrow}${\usefont{T1}{pag}{m}{n}{\small{U1}}}\symbol{41}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{U1}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{b}{n}{\small{Notation}}}~\symbol{34}\symbol{39}${\forall}$$_2$\symbol{39}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{F}}}\symbol{34}~$\coloneqq $~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{ForallU1}}}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{fun}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~${\Rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{F}}}\symbol{41}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{lamU1}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{F}}}\symbol{44}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U1}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{El1}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{F}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}\symbol{41}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{El1}}}~\symbol{40}${\forall}$$_2$~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{F}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{b}{n}{\small{Notation}}}~\symbol{34}\symbol{39}${\lambda}$$_2$\symbol{39}~{\usefont{T1}{pag}{m}{n}{\small{x}}}~\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{34}~$\coloneqq $~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{lamU1}}}~$\_$~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{fun}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}~${\Rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{41}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{appU1}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{F}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{El1}}}\symbol{40}${\forall}$$_2$~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{44}{\usefont{T1}{pag}{m}{n}{\small{F}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U1}}}\symbol{41}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{El1}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{F}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{b}{n}{\small{Notation}}}~\symbol{34}{\usefont{T1}{pag}{m}{n}{\small{f}}}~\symbol{39}${\cdot}$$_1$\symbol{39}~\symbol{91}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~\symbol{93}\symbol{34}~$\coloneqq $~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{appU1}}}~$\_$~{\usefont{T1}{pag}{m}{n}{\small{f}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{betaU1}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{F}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U1}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{El1}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{F}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}\symbol{41}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{44}\\
\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\symbol{40}${\lambda}$$_2$~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{f}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}~${\cdot}$$_1$~\symbol{91}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~\symbol{93}~\symbol{61}~{\usefont{T1}{pag}{m}{n}{\small{f}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{46}}}$$
##### Small universe
The small universe is an element of the larger one. Therefore we need an and is taken to be rather than a variable.
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{u0}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{U1}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Notation}}}~{\usefont{T1}{pag}{m}{n}{\small{U0}}}~$\coloneqq $~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{El1}}}~{\usefont{T1}{pag}{m}{n}{\small{u0}}}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{El0}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{U0}}}~${\rightarrow}$~{\usefont{T1}{pag}{b}{n}{\small{Type}}}\symbol{46}}}$$
The small universe is closed by dependent products in . The definitions are symmetric to the corresponding ones of . Notice, however, the lack of ${\beta}$-rule, which is unnecessary to derive a contradiction.
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{Forall0}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U0}}}\symbol{44}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{El0}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{U0}}}\symbol{41}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{U0}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{b}{n}{\small{Notation}}}~\symbol{34}\symbol{39}${\forall}$$_0$\symbol{39}~{\usefont{T1}{pag}{m}{n}{\small{x}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{B}}}\symbol{34}~$\coloneqq $~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{Forall0}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{fun}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}~${\Rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{B}}}\symbol{41}\symbol{41}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{b}{n}{\small{Notation}}}~\symbol{34}{\usefont{T1}{pag}{m}{n}{\small{A}}}~\symbol{39}${\longrightarrow}$$_0$\symbol{39}~{\usefont{T1}{pag}{m}{n}{\small{B}}}\symbol{34}~$\coloneqq $~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{Forall0}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{fun}}}~$\_$~${\Rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{B}}}\symbol{41}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{lam0}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}~{\usefont{T1}{pag}{m}{n}{\small{B}}}\symbol{44}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{El0}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{El0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{41}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{El0}}}~\symbol{40}${\forall}$$_0$~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{b}{n}{\small{Notation}}}~\symbol{34}\symbol{39}${\lambda}$$_0$\symbol{39}~{\usefont{T1}{pag}{m}{n}{\small{x}}}~\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{34}~$\coloneqq $~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{lam0}}}~$\_$~$\_$~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{fun}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}~${\Rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{41}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{app0}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}~{\usefont{T1}{pag}{m}{n}{\small{B}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{El0}}}~\symbol{40}${\forall}$$_0$~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{El0}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{41}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{El0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{b}{n}{\small{Notation}}}~\symbol{34}{\usefont{T1}{pag}{m}{n}{\small{f}}}~\symbol{39}${\cdot}$$_0$\symbol{39}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{34}~$\coloneqq $~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{app0}}}~$\_$~$\_$~{\usefont{T1}{pag}{m}{n}{\small{f}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{46}}}$$
The small universe is made impredicative by a dependent whose range is in . Contrary to the impredicative product of , the range cannot be restricted to be only . Here again, the ${\beta}$-rule is not needed.
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{ForallU0}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U1}}}\symbol{44}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{El1}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}${\rightarrow}${\usefont{T1}{pag}{m}{n}{\small{U0}}}\symbol{41}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{U0}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{b}{n}{\small{Notation}}}~\symbol{34}\symbol{39}${\forall}$$_0$$^1$\symbol{39}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{U}}}~\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{F}}}\symbol{34}~$\coloneqq $~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{ForallU0}}}~{\usefont{T1}{pag}{m}{n}{\small{U}}}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{fun}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~${\Rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{F}}}\symbol{41}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{lamU0}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{U}}}~{\usefont{T1}{pag}{m}{n}{\small{F}}}\symbol{44}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{El1}}}~{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{El0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{F}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}\symbol{41}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{El0}}}~\symbol{40}${\forall}$$_0$$^1$~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{F}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{b}{n}{\small{Notation}}}~\symbol{34}\symbol{39}${\lambda}$$_0$$^1$\symbol{39}~{\usefont{T1}{pag}{m}{n}{\small{x}}}~\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{34}~$\coloneqq $~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{lamU0}}}~$\_$~$\_$~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{fun}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}~${\Rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{41}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{appU0}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{U}}}~{\usefont{T1}{pag}{m}{n}{\small{F}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{El0}}}\symbol{40}${\forall}$$_0$$^1$~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{44}{\usefont{T1}{pag}{m}{n}{\small{F}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{El1}}}~{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{41}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{El0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{F}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{b}{n}{\small{Notation}}}~\symbol{34}{\usefont{T1}{pag}{m}{n}{\small{f}}}~\symbol{39}${\cdot}$$_0$\symbol{39}~\symbol{91}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~\symbol{93}\symbol{34}~$\coloneqq $~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{appU0}}}~$\_$~$\_$~{\usefont{T1}{pag}{m}{n}{\small{f}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}\symbol{46}}}$$
Proof of contradiction
----------------------
From there, we can proceed to use Hurkens’s argument to derive a contradiction. Let’s be precise: we shall prove that every type in is inhabited. It will only be an actual contradiction if contains the empty type. For this purpose, let’s assume a type in , we will then prove it is inhabited.
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{F}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U0}}}\symbol{46}}}$$
The proof will require simplifying ${\beta}$-redexes. We provide tactics to that effect.
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Ltac}}}~{\usefont{T1}{pag}{m}{n}{\small{simplify}}}~$\coloneqq $\\
\hphantom{ }\hphantom{ }\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{{\itshape repeat}}}}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape rewrite}}}}~\symbol{63}{\usefont{T1}{pag}{m}{n}{\small{beta1}}}\symbol{44}~\symbol{63}{\usefont{T1}{pag}{m}{n}{\small{betaU1}}}\symbol{41}\symbol{59}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape lazy}}}}~{\usefont{T1}{pag}{m}{n}{\small{beta}}}\symbol{46}\\[0.5\baselineskip]
{\usefont{T1}{pag}{b}{n}{\small{Ltac}}}~{\usefont{T1}{pag}{m}{n}{\small{simplify\_in}}}~{\usefont{T1}{pag}{m}{n}{\small{h}}}~$\coloneqq $\\
\hphantom{ }\hphantom{ }\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{{\itshape repeat}}}}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape rewrite}}}}~\symbol{63}{\usefont{T1}{pag}{m}{n}{\small{beta1}}}\symbol{44}~\symbol{63}{\usefont{T1}{pag}{m}{n}{\small{betaU1}}}~{\usefont{T1}{pag}{b}{n}{\small{in}}}~{\usefont{T1}{pag}{m}{n}{\small{h}}}\symbol{41}\symbol{59}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape lazy}}}}~{\usefont{T1}{pag}{m}{n}{\small{beta}}}~{\usefont{T1}{pag}{b}{n}{\small{in}}}~{\usefont{T1}{pag}{m}{n}{\small{h}}}\symbol{46}}}$$
These tactics are rather brute-force, in that they will ${\beta}$-reduce as much as possible without any particular strategy. On the other hand, they, crucially, don’t unfold Coq definitions so that we can give them hints by manually unfolding the appropriate terms to be simplified. Allowing the simplification tactics to unfold Coq definitions turns out to be intractable.
It is traditional to regard as the type of datatypes and as the type of proposition. This view is justified by the fact that is not equipped with ${\beta}$-conversion rules. In the proof, following Geuvers [@Geuvers2007], data is explicitly given, while propositions are proved with tactics. Here are the data definitions (I’m playing a bit loose here, since I consider propositions to be data, they are according to the above definition at least):
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Definition}}}~{\usefont{T1}{pag}{m}{n}{\small{V}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{U1}}}~$\coloneqq $~${\forall}$$_2$~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{44}~\symbol{40}\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{A}}}~${\longrightarrow}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{u0}}}\symbol{41}~${\longrightarrow}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{A}}}~${\longrightarrow}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{u0}}}\symbol{41}~${\longrightarrow}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{A}}}~${\longrightarrow}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{u0}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Definition}}}~{\usefont{T1}{pag}{m}{n}{\small{U}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{U1}}}~$\coloneqq $~{\usefont{T1}{pag}{m}{n}{\small{V}}}~${\longrightarrow}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{u0}}}\symbol{46}\\[0.5\baselineskip]
{\usefont{T1}{pag}{b}{n}{\small{Definition}}}~{\usefont{T1}{pag}{m}{n}{\small{sb}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{z}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{El1}}}~{\usefont{T1}{pag}{m}{n}{\small{V}}}\symbol{41}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{El1}}}~{\usefont{T1}{pag}{m}{n}{\small{V}}}~$\coloneqq $~${\lambda}$$_2$~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{44}~${\lambda}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{r}}}\symbol{44}~${\lambda}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{a}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{r}}}~${\cdot}$$_1$~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{z}}}${\cdot}$$_1$\symbol{91}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{93}${\cdot}$$_1${\usefont{T1}{pag}{m}{n}{\small{r}}}\symbol{41}~${\cdot}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{a}}}\symbol{46}\\[0.5\baselineskip]
{\usefont{T1}{pag}{b}{n}{\small{Definition}}}~{\usefont{T1}{pag}{m}{n}{\small{le}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{i}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{El1}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{U}}}${\longrightarrow}$$_1${\usefont{T1}{pag}{m}{n}{\small{u0}}}\symbol{41}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{El1}}}~{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{41}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{U0}}}~$\coloneqq $\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{x}}}~${\cdot}$$_1$~\symbol{40}${\lambda}$$_2$~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{44}~${\lambda}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{r}}}\symbol{44}~${\lambda}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{a}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{i}}}~${\cdot}$$_1$~\symbol{40}${\lambda}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{v}}}\symbol{44}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{sb}}}~{\usefont{T1}{pag}{m}{n}{\small{v}}}\symbol{41}~${\cdot}$$_1$~\symbol{91}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{93}~${\cdot}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{r}}}~${\cdot}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{a}}}\symbol{41}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Definition}}}~{\usefont{T1}{pag}{m}{n}{\small{le}}}\symbol{39}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{El1}}}~\symbol{40}\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{U}}}${\longrightarrow}$$_1${\usefont{T1}{pag}{m}{n}{\small{u0}}}\symbol{41}~${\longrightarrow}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{U}}}~${\longrightarrow}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{u0}}}\symbol{41}~$\coloneqq $~${\lambda}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{i}}}\symbol{44}~${\lambda}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{le}}}~{\usefont{T1}{pag}{m}{n}{\small{i}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Definition}}}~{\usefont{T1}{pag}{m}{n}{\small{induct}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{i}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{El1}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{U}}}${\longrightarrow}$$_1${\usefont{T1}{pag}{m}{n}{\small{u0}}}\symbol{41}\symbol{41}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{U0}}}~$\coloneqq $\\
\hphantom{ }\hphantom{ }${\forall}$$_0$$^1$~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{le}}}~{\usefont{T1}{pag}{m}{n}{\small{i}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}~${\longrightarrow}$$_0$~{\usefont{T1}{pag}{m}{n}{\small{i}}}~${\cdot}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{46}\\[0.5\baselineskip]
{\usefont{T1}{pag}{b}{n}{\small{Definition}}}~{\usefont{T1}{pag}{m}{n}{\small{WF}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{El1}}}~{\usefont{T1}{pag}{m}{n}{\small{U}}}~$\coloneqq $~${\lambda}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{z}}}\symbol{44}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{induct}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{z}}}${\cdot}$$_1$\symbol{91}{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{93}~${\cdot}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{le}}}\symbol{39}\symbol{41}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Definition}}}~{\usefont{T1}{pag}{m}{n}{\small{I}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{El1}}}~{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{41}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{U0}}}~$\coloneqq $\\
\hphantom{ }\hphantom{ }\symbol{40}${\forall}$$_0$$^1$~{\usefont{T1}{pag}{m}{n}{\small{i}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U}}}${\longrightarrow}$$_1${\usefont{T1}{pag}{m}{n}{\small{u0}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{le}}}~{\usefont{T1}{pag}{m}{n}{\small{i}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}~${\longrightarrow}$$_0$~{\usefont{T1}{pag}{m}{n}{\small{i}}}~${\cdot}$$_1$~\symbol{40}${\lambda}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{v}}}\symbol{44}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{sb}}}~{\usefont{T1}{pag}{m}{n}{\small{v}}}\symbol{41}~${\cdot}$$_1$~\symbol{91}{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{93}~${\cdot}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{le}}}\symbol{39}~${\cdot}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{41}~${\longrightarrow}$$_0$~{\usefont{T1}{pag}{m}{n}{\small{F}}}\\
\symbol{46}}}$$
The proofs follow Geuvers [@Geuvers2007] as well. The main difference is that we must explicitly call to where conversion was used implicitly and that standard Coq tactics calls to the and tactics are generally replaced by tactics of the form ${\lambda}$$_0$ $\_$ and ${\cdot}$$_0$$\_$ respectively.
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Lemma}}}~{\usefont{T1}{pag}{m}{n}{\small{Omega}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{El0}}}~\symbol{40}${\forall}$$_0$$^1$~{\usefont{T1}{pag}{m}{n}{\small{i}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U}}}${\longrightarrow}$$_1${\usefont{T1}{pag}{m}{n}{\small{u0}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{induct}}}~{\usefont{T1}{pag}{m}{n}{\small{i}}}~${\longrightarrow}$$_0$~{\usefont{T1}{pag}{m}{n}{\small{i}}}~${\cdot}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{WF}}}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Proof}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape refine}}}}~\symbol{40}${\lambda}$$_0$$^1$~{\usefont{T1}{pag}{m}{n}{\small{i}}}\symbol{44}~${\lambda}$$_0$~{\usefont{T1}{pag}{m}{n}{\small{y}}}\symbol{44}~$\_$\symbol{41}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape refine}}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{y}}}${\cdot}$$_0$\symbol{91}$\_$\symbol{93}${\cdot}$$_0$$\_$\symbol{41}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape unfold}}}}~{\usefont{T1}{pag}{m}{n}{\small{le}}}\symbol{44}{\usefont{T1}{pag}{m}{n}{\small{WF}}}\symbol{44}{\usefont{T1}{pag}{m}{n}{\small{induct}}}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{simplify}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape refine}}}}~\symbol{40}${\lambda}$$_0$$^1$~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{44}~${\lambda}$$_0$~{\usefont{T1}{pag}{m}{n}{\small{h0}}}\symbol{44}~$\_$\symbol{41}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{simplify}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape refine}}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{y}}}${\cdot}$$_0$\symbol{91}$\_$\symbol{93}${\cdot}$$_0$$\_$\symbol{41}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape unfold}}}}~{\usefont{T1}{pag}{m}{n}{\small{le}}}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{simplify}}}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape unfold}}}}~{\usefont{T1}{pag}{m}{n}{\small{sb}}}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape at}}}}~1\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{simplify}}}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape unfold}}}}~{\usefont{T1}{pag}{m}{n}{\small{le}}}\symbol{39}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape at}}}}~1\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{simplify}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape exact}}}}~{\usefont{T1}{pag}{m}{n}{\small{h0}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Qed}}}\symbol{46}}}$$
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Lemma}}}~{\usefont{T1}{pag}{m}{n}{\small{lemma1}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{El0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{induct}}}~\symbol{40}${\lambda}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{I}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{41}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Proof}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape unfold}}}}~{\usefont{T1}{pag}{m}{n}{\small{induct}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape refine}}}}~\symbol{40}${\lambda}$$_0$$^1$~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{44}~${\lambda}$$_0$~{\usefont{T1}{pag}{m}{n}{\small{p}}}\symbol{44}~$\_$\symbol{41}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{simplify}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape refine}}}}~\symbol{40}${\lambda}$$_0$~{\usefont{T1}{pag}{m}{n}{\small{q}}}\symbol{44}$\_$\symbol{41}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape assert}}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{El0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{I}}}~\symbol{40}${\lambda}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{v}}}\symbol{44}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{sb}}}~{\usefont{T1}{pag}{m}{n}{\small{v}}}\symbol{41}${\cdot}$$_1$\symbol{91}{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{93}${\cdot}$$_1${\usefont{T1}{pag}{m}{n}{\small{le}}}\symbol{39}${\cdot}$$_1${\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{41}\symbol{41}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape as}}}}~{\usefont{T1}{pag}{m}{n}{\small{h}}}\symbol{46}\\
\hphantom{ }\hphantom{ }\{~{\usefont{T1}{pag}{m}{n}{\small{{\itshape generalize}}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{q}}}${\cdot}$$_0$\symbol{91}${\lambda}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{I}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{93}${\cdot}$$_0${\usefont{T1}{pag}{m}{n}{\small{p}}}\symbol{41}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{simplify}}}\symbol{46}\\
\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape intros}}}}~{\usefont{T1}{pag}{m}{n}{\small{q}}}\symbol{39}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape exact}}}}~{\usefont{T1}{pag}{m}{n}{\small{q}}}\symbol{39}\symbol{46}~\}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape refine}}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{h}}}${\cdot}$$_0$$\_$\symbol{41}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape refine}}}}~\symbol{40}${\lambda}$$_0$$^1$~{\usefont{T1}{pag}{m}{n}{\small{i}}}\symbol{44}$\_$\symbol{41}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape refine}}}}~\symbol{40}${\lambda}$$_0$~{\usefont{T1}{pag}{m}{n}{\small{h}}}\symbol{39}\symbol{44}~$\_$\symbol{41}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape generalize}}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{q}}}${\cdot}$$_0$\symbol{91}${\lambda}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{y}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{i}}}~${\cdot}$$_1$~\symbol{40}${\lambda}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{v}}}\symbol{44}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{sb}}}~{\usefont{T1}{pag}{m}{n}{\small{v}}}\symbol{41}${\cdot}$$_1$\symbol{91}{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{93}~${\cdot}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{le}}}\symbol{39}~${\cdot}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{y}}}\symbol{41}\symbol{93}\symbol{41}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{simplify}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape intros}}}}~{\usefont{T1}{pag}{m}{n}{\small{q}}}\symbol{39}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape refine}}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{q}}}\symbol{39}${\cdot}$$_0$$\_$\symbol{41}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape clear}}}}~{\usefont{T1}{pag}{m}{n}{\small{q}}}\symbol{39}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape unfold}}}}~{\usefont{T1}{pag}{m}{n}{\small{le}}}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape at}}}}~1~{\usefont{T1}{pag}{b}{n}{\small{in}}}~{\usefont{T1}{pag}{m}{n}{\small{h}}}\symbol{39}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{simplify\_in}}}~{\usefont{T1}{pag}{m}{n}{\small{h}}}\symbol{39}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape unfold}}}}~{\usefont{T1}{pag}{m}{n}{\small{sb}}}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape at}}}}~1~{\usefont{T1}{pag}{b}{n}{\small{in}}}~{\usefont{T1}{pag}{m}{n}{\small{h}}}\symbol{39}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{simplify\_in}}}~{\usefont{T1}{pag}{m}{n}{\small{h}}}\symbol{39}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape unfold}}}}~{\usefont{T1}{pag}{m}{n}{\small{le}}}\symbol{39}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape at}}}}~1~{\usefont{T1}{pag}{b}{n}{\small{in}}}~{\usefont{T1}{pag}{m}{n}{\small{h}}}\symbol{39}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{simplify\_in}}}~{\usefont{T1}{pag}{m}{n}{\small{h}}}\symbol{39}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape exact}}}}~{\usefont{T1}{pag}{m}{n}{\small{h}}}\symbol{39}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Qed}}}\symbol{46}}}$$
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Lemma}}}~{\usefont{T1}{pag}{m}{n}{\small{lemma2}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{El0}}}~\symbol{40}\symbol{40}${\forall}$$_0$$^1${\usefont{T1}{pag}{m}{n}{\small{i}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U}}}${\longrightarrow}$$_1${\usefont{T1}{pag}{m}{n}{\small{u0}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{induct}}}~{\usefont{T1}{pag}{m}{n}{\small{i}}}~${\longrightarrow}$$_0$~{\usefont{T1}{pag}{m}{n}{\small{i}}}${\cdot}$$_1${\usefont{T1}{pag}{m}{n}{\small{WF}}}\symbol{41}~${\longrightarrow}$$_0$~{\usefont{T1}{pag}{m}{n}{\small{F}}}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Proof}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape refine}}}}~\symbol{40}${\lambda}$$_0$~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{44}~$\_$\symbol{41}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape assert}}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{El0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{I}}}~{\usefont{T1}{pag}{m}{n}{\small{WF}}}\symbol{41}\symbol{41}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape as}}}}~{\usefont{T1}{pag}{m}{n}{\small{h}}}\symbol{46}\\
\hphantom{ }\hphantom{ }\{~{\usefont{T1}{pag}{m}{n}{\small{{\itshape generalize}}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{x}}}${\cdot}$$_0$\symbol{91}${\lambda}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{I}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{93}${\cdot}$$_0${\usefont{T1}{pag}{m}{n}{\small{lemma1}}}\symbol{41}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{simplify}}}\symbol{46}\\
\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape intros}}}}~{\usefont{T1}{pag}{m}{n}{\small{q}}}\symbol{46}\\
\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape exact}}}}~{\usefont{T1}{pag}{m}{n}{\small{q}}}\symbol{46}~\}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape refine}}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{h}}}${\cdot}$$_0$$\_$\symbol{41}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape clear}}}}~{\usefont{T1}{pag}{m}{n}{\small{h}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape refine}}}}~\symbol{40}${\lambda}$$_0$$^1$~{\usefont{T1}{pag}{m}{n}{\small{i}}}\symbol{44}~${\lambda}$$_0$~{\usefont{T1}{pag}{m}{n}{\small{h0}}}\symbol{44}~$\_$\symbol{41}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape generalize}}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{x}}}${\cdot}$$_0$\symbol{91}${\lambda}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{y}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{i}}}${\cdot}$$_1$\symbol{40}${\lambda}$$_1$~{\usefont{T1}{pag}{m}{n}{\small{v}}}\symbol{44}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{sb}}}~{\usefont{T1}{pag}{m}{n}{\small{v}}}\symbol{41}${\cdot}$$_1$\symbol{91}{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{93}${\cdot}$$_1${\usefont{T1}{pag}{m}{n}{\small{le}}}\symbol{39}${\cdot}$$_1${\usefont{T1}{pag}{m}{n}{\small{y}}}\symbol{41}\symbol{93}\symbol{41}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{simplify}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape intros}}}}~{\usefont{T1}{pag}{m}{n}{\small{q}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape refine}}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{q}}}${\cdot}$$_0$$\_$\symbol{41}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape clear}}}}~{\usefont{T1}{pag}{m}{n}{\small{q}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape unfold}}}}~{\usefont{T1}{pag}{m}{n}{\small{le}}}~{\usefont{T1}{pag}{b}{n}{\small{in}}}~{\usefont{T1}{pag}{m}{n}{\small{h0}}}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{simplify\_in}}}~{\usefont{T1}{pag}{m}{n}{\small{h0}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape unfold}}}}~{\usefont{T1}{pag}{m}{n}{\small{WF}}}~{\usefont{T1}{pag}{b}{n}{\small{in}}}~{\usefont{T1}{pag}{m}{n}{\small{h0}}}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{simplify\_in}}}~{\usefont{T1}{pag}{m}{n}{\small{h0}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape exact}}}}~{\usefont{T1}{pag}{m}{n}{\small{h0}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Qed}}}\symbol{46}}}$$
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Theorem}}}~{\usefont{T1}{pag}{m}{n}{\small{paradox}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{El0}}}~{\usefont{T1}{pag}{m}{n}{\small{F}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Proof}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape exact}}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{lemma2}}}${\cdot}$$_0${\usefont{T1}{pag}{m}{n}{\small{Omega}}}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Qed}}}\symbol{46}}}$$
The takeaway insight is that because the paradox does not actually make use of the reduction rules in propositions of , using equality to model conversion in these propositions doesn’t raise any obstacle to the completion of the proof.
Nothing in this proof is particularly specific to Coq: it could be done in any variant of Martin-Löf type theory, provided that an identity type is available. Of course, the support of Coq for rewriting significantly helps, if your favourite proof assistant doesn’t have a similar feature it may be painful to port this generic paradox.
Applications
============
In this section we will see a few instances of the generic axiomatisation of Hurkens’s proof can help derive contradictions. They come from the file of the Coq distribution (version 8.5).
Sorts {#latex_lib_label_3}
-----
A common implementation of universes is to use a sort of the dependent type theory for a universe of $\mbox{\textrm{U}}^-$. In that case. is just the identity.
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{U}}}~$\coloneqq $~{\usefont{T1}{pag}{b}{n}{\small{Type}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{El}}}~$\coloneqq $~{\usefont{T1}{pag}{b}{n}{\small{fun}}}~{\usefont{T1}{pag}{m}{n}{\small{X}}}~${\Rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{X}}}\symbol{46}}}$$
For universes defined this way, small products and their ${\lambda}$-abstraction, application and ${\beta}$-rule are defined straightforwardly ( is Coq’s witness of reflexivity of equality).
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{Forall}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{B}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{A}}}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{41}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{U}}}~$\coloneqq $~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\\
{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{lam}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}~{\usefont{T1}{pag}{m}{n}{\small{B}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{44}{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}~$\coloneqq $~{\usefont{T1}{pag}{m}{n}{\small{f}}}\\
{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{app}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}~{\usefont{T1}{pag}{m}{n}{\small{B}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{44}{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}~$\coloneqq $~{\usefont{T1}{pag}{m}{n}{\small{f}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\\
{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{beta}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}~{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{f}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{f}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}~\symbol{61}~{\usefont{T1}{pag}{m}{n}{\small{f}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}~$\coloneqq $~{\usefont{T1}{pag}{m}{n}{\small{eq\_refl}}}}}$$
Impredicative sort {#latex_lib_label_4}
------------------
Impredicativity, for a sort , can also be characterised to some degree. The idea is that there must be a bigger sort which can be projected onto . See, for example, the bracketing construction in [@HerbelinSpiwack2013]. This projection could be implemented, for instance, for Coq’s impredicative sort as ${\Rightarrow}$ ${\rightarrow}$${\rightarrow}$.
The signature of Section \[latex\_lib\_label\_3\] is extended with the constraint that is bigger than and a projection.
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{39}~$\coloneqq $~{\usefont{T1}{pag}{b}{n}{\small{Type}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{39}~$\coloneqq $~{\usefont{T1}{pag}{b}{n}{\small{Type}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{proj}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{39}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{46}}}$$
With the following laws.
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Hypothesis}}}~{\usefont{T1}{pag}{m}{n}{\small{proj\_unit}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{39}\symbol{41}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{proj}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Hypothesis}}}~{\usefont{T1}{pag}{m}{n}{\small{proj\_counit}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{F}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U}}}${\rightarrow}${\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{41}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{proj}}}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{44}{\usefont{T1}{pag}{m}{n}{\small{F}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}~${\rightarrow}$~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{44}{\usefont{T1}{pag}{m}{n}{\small{F}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Hypothesis}}}~{\usefont{T1}{pag}{m}{n}{\small{proj\_coherent}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{F}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U}}}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{F}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{41}\symbol{44}\\
\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{proj\_counit}}}~$\_$~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{proj\_unit}}}~$\_$~{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{41}~{\usefont{T1}{pag}{m}{n}{\small{x}}}~\symbol{61}~{\usefont{T1}{pag}{m}{n}{\small{f}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{46}}}$$
The law expresses that if generally diminishes the ability to distinguish between elements of , it does not lose elements. We don’t have a way back from to in general, but forms a monad. The law expresses a small variation on this latter remark.
These properties are sufficient to show that is closed by large product. The ${\beta}$-rule, omitted, is easily derived from .
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{ForallU}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{F}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U}}}${\rightarrow}${\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{41}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{U}}}~$\coloneqq $~{\usefont{T1}{pag}{m}{n}{\small{proj}}}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{F}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{lamU1}}}~{\usefont{T1}{pag}{m}{n}{\small{F}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{F}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{proj}}}\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{F}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}$\coloneqq $~{\usefont{T1}{pag}{m}{n}{\small{proj\_unit}}}~$\_$~{\usefont{T1}{pag}{m}{n}{\small{f}}}\\
{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{appU1}}}~{\usefont{T1}{pag}{m}{n}{\small{F}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{proj}}}\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{F}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U}}}\symbol{41}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{F}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~$\coloneqq $~{\usefont{T1}{pag}{m}{n}{\small{proj\_counit}}}~$\_$~{\usefont{T1}{pag}{m}{n}{\small{f}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{46}}}$$
We can exploit Coq’s universe polymorphism (form version 8.5) to turn this section into a generic definition of impredicative sort. Indeed, under the polymorphic interpretation represents an arbitrary type, including the impredicative sort , which is indeed impredicative in the above sense.
Generalising Geuvers’s proof {#latex_lib_label_5}
----------------------------
Geuvers [@Geuvers2007] proves that an impredicative sort cannot be a retract of an . His proof is made for $\mbox{{\usefont{T1}{pag}{m}{n}{\small{U1}}}}=\mbox{{\usefont{T1}{pag}{b}{n}{\small{Prop}}}}$, but we can instantiate the proof of Section \[latex\_lib\_label\_1\] to obtain the same result for any sort which is impredicative sort in the sense of Section \[latex\_lib\_label\_4\].
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{U2}}}~$\coloneqq $~{\usefont{T1}{pag}{b}{n}{\small{Type}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{U1}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U2}}}~$\coloneqq $~{\usefont{T1}{pag}{b}{n}{\small{Type}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{U0}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U1}}}\symbol{46}}}$$
Where is impredicative over as in Section \[latex\_lib\_label\_4\]. The retraction is given by the following functions. Only a weak form of retraction is needed were types in which are “logically equivalent” are considered equal.
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{U0}}}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{U1}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{inj0}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{U1}}}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{U0}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Hypothesis}}}~{\usefont{T1}{pag}{m}{n}{\small{inj0\_unit}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{b}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U1}}}\symbol{41}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{b}}}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{inj0}}}~{\usefont{T1}{pag}{m}{n}{\small{b}}}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Hypothesis}}}~{\usefont{T1}{pag}{m}{n}{\small{inj0\_counit}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{b}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U1}}}\symbol{41}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{inj0}}}~{\usefont{T1}{pag}{m}{n}{\small{b}}}\symbol{41}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{b}}}\symbol{46}}}$$
From this (weak) retraction we can define and corresponding products for despite the fact that is not necessarily a sort.
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{El0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U0}}}\symbol{41}~$\coloneqq $~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\\
{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{Forall0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U0}}}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{B}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{U0}}}\symbol{41}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{U0}}}~$\coloneqq $~{\usefont{T1}{pag}{m}{n}{\small{inj0}}}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{41}\\
{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{Lambda0}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}~{\usefont{T1}{pag}{m}{n}{\small{B}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{41}\\
\hphantom{ }\hphantom{ }\hphantom{ }\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{inj0}}}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{41}\symbol{41}~$\coloneqq $~{\usefont{T1}{pag}{m}{n}{\small{inj0\_unit}}}~$\_$~{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{app0}}}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}~{\usefont{T1}{pag}{m}{n}{\small{B}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{inj0}}}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{41}\symbol{41}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{41}\\
\hphantom{ }\hphantom{ }\hphantom{ }\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}~$\coloneqq $~{\usefont{T1}{pag}{m}{n}{\small{inj0\_counit}}}~$\_$~{\usefont{T1}{pag}{m}{n}{\small{f}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}}}$$
Large products are define much the same: $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{Forall0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U1}}}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{B}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{u}}}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{U0}}}\symbol{41}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{U0}}}~$\coloneqq $~{\usefont{T1}{pag}{m}{n}{\small{inj0}}}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{41}\\
{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{Lambda0}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}~{\usefont{T1}{pag}{m}{n}{\small{B}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{41}\\
\hphantom{ }\hphantom{ }\hphantom{ }\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{inj0}}}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{41}\symbol{41}~$\coloneqq $~{\usefont{T1}{pag}{m}{n}{\small{inj0\_unit}}}~$\_$~{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{app0}}}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}~{\usefont{T1}{pag}{m}{n}{\small{B}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{inj0}}}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{41}\symbol{41}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{41}\\
\hphantom{ }\hphantom{ }\hphantom{ }\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{B}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}~$\coloneqq $~{\usefont{T1}{pag}{m}{n}{\small{inj0\_counit}}}~$\_$~{\usefont{T1}{pag}{m}{n}{\small{f}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}}}$$
From this, the paradox is set up, so we can deduce that every proposition of is “inhabited” in that $\mbox{{\usefont{T1}{pag}{m}{n}{\small{El0}}}~{\usefont{T1}{pag}{m}{n}{\small{P}}}}=\mbox{{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~{\usefont{T1}{pag}{m}{n}{\small{P}}}}$ is inhabited, and therefore, that every proposition of is inhabited since is “inhabited” in the sense of , *i.e.* is inhabited, then concludes.
Since is an instance of the signature of Section \[latex\_lib\_label\_4\], we prove, like Geuvers, that is not a retract of a proposition .
Excluded middle and proof irrelevance {#latex_lib_label_6}
-------------------------------------
Geuvers proof, from Section \[latex\_lib\_label\_5\], helps proving a result, by Coquand [@Coquand1989], that excluded middle, in an impredicative sort makes it proof irrelevant, *i.e.* every type in that sort have at most one element. This proof appear in the Coq distribution in the file , presumably written by Hugo Herbelin. It uses Geuvers result and was mostly unmodified with the new proof of said result. With the characterisation of Section \[latex\_lib\_label\_4\], this could be done in an arbitrary impredicative sort, but the Coq proof is done only for the impredicative sort , and we will present it that way for simplicity.
The basic idea is that excluded middle: $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{em}}}\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{Prop}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{A}}}${\lor}$${\lnot}${\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{46}}}$$ turns the sort into a boolean universe with only two elements. So assuming a proposition with two *distinct* values $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{U0}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{Prop}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Variables}}}~{\usefont{T1}{pag}{m}{n}{\small{t}}}~{\usefont{T1}{pag}{m}{n}{\small{f}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{U0}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Hypothesis}}}~{\usefont{T1}{pag}{m}{n}{\small{not\_eq\_t\_f}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{t}}}~${\ne}$~{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{46}}}$$ we can reflect into proposition as in Section \[latex\_lib\_label\_5\]. Where is reflected as and as , as the names suggest.
This is formalised as a retraction given by: $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{inj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{Prop}}}\symbol{41}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{U0}}}~$\coloneqq $~{\usefont{T1}{pag}{m}{n}{\small{or\_ind}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~\symbol{40}${\lnot}${\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}~{\usefont{T1}{pag}{m}{n}{\small{U0}}}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{fun}}}~$\_$~${\Rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{t}}}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{fun}}}~$\_$~${\Rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{em}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U0}}}\symbol{41}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{Prop}}}~$\coloneqq $~{\usefont{T1}{pag}{m}{n}{\small{t}}}~\symbol{61}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{46}}}$$ Where ${\rightarrow}$ ${\rightarrow}$ ${\rightarrow}$ ${\rightarrow}$ ${\lor}$ ${\rightarrow}$ is the elimination principle of disjunction.
We are left to prove the unit and counit laws of and to satisfy the premisses of the paradox in Section \[latex\_lib\_label\_5\]. The unit law is direct: $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Lemma}}}~{\usefont{T1}{pag}{m}{n}{\small{inj0\_unit}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{Prop}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{inj0}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Proof}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape intros}}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape unfold}}}}~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{inj0}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape destruct}}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{em}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape as}}}}~\symbol{91}{\usefont{T1}{pag}{m}{n}{\small{h}}}~\hspace{0.1em}\rule[-0.6ex]{1.sp}{1.\baselineskip}~{\usefont{T1}{pag}{m}{n}{\small{h}}}\symbol{93}\symbol{46}\\
\hphantom{ }\hphantom{ }\symbol{43}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape reflexivity}}}}\symbol{46}\\
\hphantom{ }\hphantom{ }\symbol{43}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape contradiction}}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Qed}}}\symbol{46}}}$$ The counit law is the step that makes a crucial use of the hypothesis: $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Lemma}}}~{\usefont{T1}{pag}{m}{n}{\small{inj0\_counit}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{Prop}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{inj0}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Proof}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape intros}}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~{\usefont{T1}{pag}{m}{n}{\small{h}}}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape unfold}}}}~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{inj0}}}~{\usefont{T1}{pag}{b}{n}{\small{in}}}~$*$\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape destruct}}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{em}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape as}}}}~\symbol{91}{\usefont{T1}{pag}{m}{n}{\small{l}}}~\hspace{0.1em}\rule[-0.6ex]{1.sp}{1.\baselineskip}~{\usefont{T1}{pag}{m}{n}{\small{l}}}\symbol{93}\symbol{46}\\
\hphantom{ }\hphantom{ }\symbol{43}~{\usefont{T1}{pag}{m}{n}{\small{{\itshape apply}}}}~{\usefont{T1}{pag}{m}{n}{\small{l}}}\symbol{46}\\
\hphantom{ }\hphantom{ }\symbol{43}~{\usefont{T1}{pag}{m}{n}{\small{absurd}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{t}}}\symbol{61}{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{41}\symbol{46}\\
\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }$*$~{\usefont{T1}{pag}{m}{n}{\small{{\itshape apply}}}}~{\usefont{T1}{pag}{m}{n}{\small{not\_eq\_t\_f}}}\symbol{46}\\
\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }$*$~{\usefont{T1}{pag}{m}{n}{\small{{\itshape apply}}}}~{\usefont{T1}{pag}{m}{n}{\small{h}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Qed}}}\symbol{46}}}$$
Section \[latex\_lib\_label\_5\] then yields a contradiction. Since is arbitrary we have: ${\lnot}$${\lnot}$. A last application of the excluded middle yields the expected result: $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{forall}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{Prop}}}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{x}}}~{\usefont{T1}{pag}{m}{n}{\small{y}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{61}{\usefont{T1}{pag}{m}{n}{\small{y}}}}}$$
Variants of Prop {#latex_lib_label_7}
----------------
A (monadic) modality on is given by a mapping: $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{M}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{Prop}}}~${\rightarrow}$~{\usefont{T1}{pag}{b}{n}{\small{Prop}}}\symbol{46}}}$$ Together with the following laws: $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Hypothesis}}}~{\usefont{T1}{pag}{m}{n}{\small{unit}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{Prop}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{M}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Hypothesis}}}~{\usefont{T1}{pag}{m}{n}{\small{join}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{Prop}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{M}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{M}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{M}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Hypothesis}}}~{\usefont{T1}{pag}{m}{n}{\small{incr}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~{\usefont{T1}{pag}{m}{n}{\small{B}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{Prop}}}\symbol{44}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{A}}}${\rightarrow}${\usefont{T1}{pag}{m}{n}{\small{B}}}\symbol{41}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{M}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{M}}}~{\usefont{T1}{pag}{m}{n}{\small{B}}}\symbol{46}}}$$ Such a modality is automatically equipped with a distribution property over arbitrary conjunctions: $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Lemma}}}~{\usefont{T1}{pag}{m}{n}{\small{strength}}}\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{P}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{A}}}${\rightarrow}${\usefont{T1}{pag}{b}{n}{\small{Prop}}}\symbol{41}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{M}}}\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{44}{\usefont{T1}{pag}{m}{n}{\small{P}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}~${\rightarrow}$~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{44}{\usefont{T1}{pag}{m}{n}{\small{M}}}\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{P}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Proof}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape eauto}}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Qed}}}\symbol{46}}}$$
With a modality we can define the type of modal propositions, where the law is actually an equivalence (modalities are closure operators, by the law, so the type of modal propositions is the image of up to logical equivalence). $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Definition}}}~{\usefont{T1}{pag}{m}{n}{\small{MProp}}}~$\coloneqq $~\{~{\usefont{T1}{pag}{m}{n}{\small{P}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{Prop}}}~\hspace{0.1em}\rule[-0.6ex]{1.sp}{1.\baselineskip}~{\usefont{T1}{pag}{m}{n}{\small{M}}}~{\usefont{T1}{pag}{m}{n}{\small{P}}}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{P}}}~\}\symbol{46}}}$$
Despite not being a sort, can be seen as a subtype of and, therefore, as a universe in the sense of Section \[latex\_lib\_label\_2\]. $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Definition}}}~{\usefont{T1}{pag}{m}{n}{\small{El}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{P}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{MProp}}}\symbol{41}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{Prop}}}~$\coloneqq $~{\usefont{T1}{pag}{m}{n}{\small{proj1\_sig}}}~{\usefont{T1}{pag}{m}{n}{\small{P}}}\symbol{46}}}$$
Because of , the universe is closed by products of arbitrary types. The keyword makes it possible to populate by giving the proposition (*first projection*) explicitly and discharging the proof that ${\rightarrow}$ to tactics. $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Program}}}~{\usefont{T1}{pag}{b}{n}{\small{Definition}}}~{\usefont{T1}{pag}{m}{n}{\small{Forall}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{Type}}}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{F}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{A}}}${\rightarrow}${\usefont{T1}{pag}{m}{n}{\small{MProp}}}\symbol{41}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{MProp}}}~$\coloneqq $\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{El}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{F}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Next}}}~{\usefont{T1}{pag}{b}{n}{\small{Obligation}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape intros}}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~{\usefont{T1}{pag}{m}{n}{\small{F}}}~{\usefont{T1}{pag}{m}{n}{\small{h}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape apply}}}}~{\usefont{T1}{pag}{m}{n}{\small{strength}}}~{\usefont{T1}{pag}{b}{n}{\small{with}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{x}}}$\coloneqq ${\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}~{\usefont{T1}{pag}{b}{n}{\small{in}}}~{\usefont{T1}{pag}{m}{n}{\small{h}}}\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape destruct}}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{F}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{46}~{\usefont{T1}{pag}{m}{n}{\small{cbn}}}~{\usefont{T1}{pag}{b}{n}{\small{in}}}~$*$\symbol{46}\\
\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{{\itshape eauto}}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Qed}}}\symbol{46}}}$$
Definitions of $\mbox{\textrm{U}}^-$ products, small and large, for follow immediately: $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{Forall1}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{MProp}}}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{F}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{El}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{MProp}}}\symbol{41}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{MProp}}}~$\coloneqq $~{\usefont{T1}{pag}{m}{n}{\small{Forall}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{El}}}~{\usefont{T1}{pag}{m}{n}{\small{u}}}\symbol{41}~{\usefont{T1}{pag}{m}{n}{\small{F}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{ForallU1}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{F}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{MProp}}}${\rightarrow}${\usefont{T1}{pag}{m}{n}{\small{MProp}}}\symbol{41}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{MProp}}}~$\coloneqq $~{\usefont{T1}{pag}{m}{n}{\small{Forall}}}~{\usefont{T1}{pag}{m}{n}{\small{MProp}}}~{\usefont{T1}{pag}{m}{n}{\small{F}}}\symbol{46}}}$$ Because $\mbox{{\usefont{T1}{pag}{m}{n}{\small{El}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{Forall}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~{\usefont{T1}{pag}{m}{n}{\small{F}}}\symbol{41}}=\mbox{{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{F}}}}$, introduction, elimination and ${\beta}$-rules for the products are immediate.
Just like in Section \[latex\_lib\_label\_5\], a retraction of into a modal proposition can be used to trigger Hurkens’s paradox. This is an example of instance of Hurkens’s paradox where neither of the universes are sorts of the system.
$$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{U0}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{MProp}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{U0}}}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{MProp}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{inj0}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{MProp}}}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{U0}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Hypothesis}}}~{\usefont{T1}{pag}{m}{n}{\small{inj0\_unit}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{MProp}}}\symbol{41}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{El}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{El}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{inj0}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Hypothesis}}}~{\usefont{T1}{pag}{m}{n}{\small{inj0\_counit}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{MProp}}}\symbol{41}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{El}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{inj0}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}\symbol{41}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{El}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{46}}}$$
Following the the proof of Section \[latex\_lib\_label\_5\], we conclude from this context that every modal proposition is inhabited. This is not necessarily a contradiction, as falsity need not be modal. For instance the trivial modality, whose only modal proposition in . $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Definition}}}~{\usefont{T1}{pag}{m}{n}{\small{M}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{Prop}}}\symbol{41}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{Prop}}}~$\coloneqq $~{\usefont{T1}{pag}{m}{n}{\small{True}}}}}$$
A more interesting modality is, for a given : $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Definition}}}~{\usefont{T1}{pag}{m}{n}{\small{M}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{Prop}}}\symbol{41}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{Prop}}}~$\coloneqq $~{\usefont{T1}{pag}{m}{n}{\small{A}}}${\lor}${\usefont{T1}{pag}{m}{n}{\small{X}}}}}$$ for such a modality exhibiting a retraction into a modal proposition only prove ${\lnot}$: it is always the case that the smallest modal proposition is .
Weak excluded middle and proof irrelevance
------------------------------------------
In this section we will be concerned with the double-negation modality, whose modal propositions are also called *negative propositions*: $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Definition}}}~{\usefont{T1}{pag}{m}{n}{\small{M}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{Prop}}}\symbol{41}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{Prop}}}~$\coloneqq $~${\lnot}$${\lnot}${\usefont{T1}{pag}{m}{n}{\small{A}}}}}$$ and will use the paradox from Section \[latex\_lib\_label\_7\], to prove that the weak principle of excluded middle $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Hypothesis}}}~{\usefont{T1}{pag}{m}{n}{\small{wem}}}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{Prop}}}\symbol{44}~${\lnot}$${\lnot}${\usefont{T1}{pag}{m}{n}{\small{A}}}~${\lor}$~${\lnot}${\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{46}}}$$ entails a weak form of proof irrelevance. This is a new proof I added to and is available from version 8.5.
Looking closely at it becomes clear that it claims decidability of exactly the negative propositions. $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Remark}}}~{\usefont{T1}{pag}{m}{n}{\small{wem}}}\symbol{39}~\symbol{58}~{\usefont{T1}{pag}{b}{n}{\small{forall}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{MProp}}}\symbol{44}~{\usefont{T1}{pag}{m}{n}{\small{El}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}~${\lor}$~${\lnot}${\usefont{T1}{pag}{m}{n}{\small{El}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{46}}}$$
The proof, therefore, proceeds just like the proof of Section \[latex\_lib\_label\_6\]. We begin by postulating a proposition with two proofs. $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Variable}}}~{\usefont{T1}{pag}{m}{n}{\small{U0}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{Prop}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Variables}}}~{\usefont{T1}{pag}{m}{n}{\small{t}}}~{\usefont{T1}{pag}{m}{n}{\small{f}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{U0}}}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Hypothesis}}}~{\usefont{T1}{pag}{m}{n}{\small{not\_eq\_t\_f}}}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{t}}}~${\ne}$~{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{46}}}$$
Notice that is negative, since has a proof, in particular ${\lnot}$${\lnot}$${\rightarrow}$ holds. So we only need to construct a retraction into . The retraction is given by and which are, *mutatis mutandis* the same as in Section \[latex\_lib\_label\_6\]: double negations have to be inserted for propositions which need to be negative, and proofs of negativity have to be provided when building negative propositions. $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{inj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{MProp}}}\symbol{41}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{U0}}}~$\coloneqq $\\
\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{or\_ind}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{El}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}~\symbol{40}${\lnot}${\usefont{T1}{pag}{m}{n}{\small{El}}}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}~{\usefont{T1}{pag}{m}{n}{\small{U0}}}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{fun}}}~$\_$~${\Rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{t}}}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{fun}}}~$\_$~${\Rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{f}}}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{wem}}}\symbol{39}~{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}\symbol{46}\\
{\usefont{T1}{pag}{b}{n}{\small{Let}}}~{\usefont{T1}{pag}{m}{n}{\small{proj0}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{U0}}}\symbol{41}~\symbol{58}~{\usefont{T1}{pag}{m}{n}{\small{MProp}}}~$\coloneqq $\\
\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }\hphantom{ }{\usefont{T1}{pag}{m}{n}{\small{exist}}}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{fun}}}~{\usefont{T1}{pag}{m}{n}{\small{P}}}${\Rightarrow}$${\lnot}$${\lnot}${\usefont{T1}{pag}{m}{n}{\small{P}}}~${\rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{P}}}\symbol{41}~\symbol{40}${\lnot}$${\lnot}$\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{t}}}~\symbol{61}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{fun}}}~{\usefont{T1}{pag}{m}{n}{\small{h}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}~${\Rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{h}}}~\symbol{40}{\usefont{T1}{pag}{b}{n}{\small{fun}}}~{\usefont{T1}{pag}{m}{n}{\small{k}}}~${\Rightarrow}$~{\usefont{T1}{pag}{m}{n}{\small{k}}}~{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{41}\symbol{41}\symbol{46}}}$$
The unit and counit laws follow and we eventually derive a contradiction. That is, since is arbitrary a proof that: $$\makebox[1.\linewidth]{\makebox[0.1\linewidth]{}\parbox{0.9\linewidth}{{\usefont{T1}{pag}{b}{n}{\small{forall}}}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{58}{\usefont{T1}{pag}{b}{n}{\small{Prop}}}\symbol{41}~\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{x}}}~{\usefont{T1}{pag}{m}{n}{\small{y}}}\symbol{58}{\usefont{T1}{pag}{m}{n}{\small{A}}}\symbol{41}\symbol{44}~${\lnot}$${\lnot}$\symbol{40}{\usefont{T1}{pag}{m}{n}{\small{x}}}\symbol{61}{\usefont{T1}{pag}{m}{n}{\small{y}}}\symbol{41}}}$$
Contrary to to the case of (strong) excluded middle, we cannot eliminate this last double-negative. So proof irrelevance doesn’t follow from weak excluded middle. However, this section proves that weak excluded middle is incompatible with any sort of proof relevance principle. In particular, in Coq lingo, weak excluded middle cannot hold in impredicative , that is an impredicative sort with strong elimination.
Conclusion
==========
The axiomatisation of Hurkens’s paradox presented in Section \[latex\_lib\_label\_1\] is very versatile. It can be used, mostly, to prove that some combination of logical principles are incompatible, but also to detect bugs in a dependent-type-theory implementation. Which is a completely fair and healthy activity if you ask this author.
It is, certainly, an improvement over a situation where each paradox would need a careful redesign of Hurkens’s proof to fit the specific premises. In practice it meant that paradoxes were not derived, because the brave paradox-finder didn’t have the energy or expertise to translate Hurkens’s paradox.
As per the axiomatisation itself. It has the pleasant property of requiring only a subset of $\mbox{\textrm{U}}^-$ where the “proofs” of “propositions” don’t require ${\beta}$-rules or any kind of equality rule. So something was learned. Adapting the proof to the axiomatisation doesn’t present any new difficulty, except from controlling rewriting a little. It wasn’t discovered before solely by the virtue of nobody looking. The reader who enjoyed this axiomatisation can celebrate the bout of optimism which made me look the right way, and the night I lost over it.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Models of hadronization of hard jets in QCD are often presented in terms of Feynman-graph structures that can be thought of as effective field theory approximations to dynamical non-perturbative physics in QCD. Such models can be formulated as a kind of multiperipheral model. We obtain general constraints on such models in order for them to be self-consistent, and we relate the constraints to the space-time structure of hadronization. We show that appropriate models can be considered as implementing string-like hadronization. When the models are put in a multiperipheral form, the effective vertices and/or lines must be momentum non-conserving: they take 4-momentum from the external string-like field.'
author:
- John Collins
- 'Ted C. Rogers'
bibliography:
- 'jcc.bib'
title: Graphical Structure of Hadronization and Factorization in Hard Collisions
---
Introduction {#sec:intro}
============
An important topic in QCD is to properly understand the interface between non-perturbative hadronization and factorization physics (with its perturbative content). This is especially important with the current widespread interest on the details of partonic interactions in hadronic and nuclear physics. An immediate motivation for the work described in this paper is the need for incorporating non-perturbative polarization effects in Monte-Carlo event generators. (See Ref. [@Sjostrand:2016bif] for an up-to-date overview of MCEGs.)
The polarization effects at issue are those responsible for the much studied Sivers and Collins functions and related quantities. The primary complications concerning event generators arise because event generators are formulated in terms of probabilistic processes for the different components of a reaction. In contrast, interesting polarization effects involve quantum-mechanical entanglement between different parts of the reaction. A simple example is given by the azimuthal correlation between back-to-back pairs of hadrons in $e^+e^-$ annihilation [@Artru:1995zu], where the correlation (via polarized dihadron fragmentation functions for a primary quark-antiquark pair) arises because the quark and antiquark are in an entangled spin state. Such entanglement implies that the fragmentation of the two jets is not literally independent even if the cross section is written as the product of two fragmentation functions.
Work proposing implementations of non-perturbative polarization effects in event generators is found in [@Kerbizi:2017rpp; @Matevosyan:2016fwi; @Bentz:2016rav; @Kerbizi:2018qpp]. To treat polarization in a way that is fully consistent with the underlying principles of quantum theory, the formulations are made in terms of Feynman graph structures. These structures can be treated as manifestations of an effective field theory that usefully approximates the true non-perturbative behavior of QCD in the relevant kinematic regime. One notable feature of this framework is the determination of the most general polarization structure in the elementary splitting functions [@Matevosyan:2016fwi].
The work in @Kerbizi:2017rpp provides in a preliminary version a phenomenologically successful Monte-Carlo simulation of the hadronization of the jet produced by a transversely polarized quark. Of the model’s 5 free parameters, 4 concern unpolarized fragmentation, and were fitted to unpolarized data in semi-inclusive deeply inelastic lepton-hadron scattering (SIDIS). The single remaining parameter needed for the polarized process was obtained from data on the Collins asymmetry in $e^+e^-$ annihilation to hadrons. The model then successfully predicted the Collins asymmetry in SIDIS data from the COMPASS experiment [@Adolph:2014zba], with 18 data points, as well as the dihadron asymmetry in the same experiment. The model was based on a string-model formulation by Artru and Belghobsi [@Artru:2012zz; @Artru:2010st; @Artru:in2p3-00953539]. While the success is encouraging, the method is yet to be incorporated in a full event generator. In addition, the effects of primary vector meson production were not incorporated, so the non-perturbative mechanisms implemented are not the whole story.
Now factorization theorems do incorporate non-perturbative effects in the form of parton densities and fragmentation functions. However, as one of us explained in [@Collins:2016ztc], there is a mismatch between the measured properties of hadronization of hard jets and the order-by-order asymptotic properties used in existing factorization proofs. In the asymptotics of individual graphs, there arises strong ordering of kinematics between different parts of the graphs, and the resulting large rapidity differences are used in an essential way in the proofs, notably to factor out the effects of soft gluon subgraphs from collinear subgraphs. But in reality, hadronization gives rise to approximately uniform distributions in rapidity with no large gaps. Event generators need to model such event structures.
Related to this is the following conceptual mismatch. Intuitively one explains the process in a space-time picture. There is first a hard collision in which a state of some number of partons is generated over a small distance and time scale. Only at distinctly later times does the partonic state turn into observed hadrons. In contrast, factorization proofs as well as actual calculations are made in terms of ordinary momentum-space Feynman graphs.
The primary purpose of this paper is therefore to provide an analysis of the structures in models of hadronization that are presented in Feynman-graph form, such as those found in or implied by [@Kerbizi:2017rpp; @Matevosyan:2016fwi; @Bentz:2016rav; @Kerbizi:2018qpp], and to determine which kinds of model are appropriate and which are not. We will concentrate exclusively on the application to $e^+e^-$ annihilation to hadrons in the 2 jet case, although the principles are more general.
There are two areas that form essential background to our treatment.
The first concerns standard string models of hadronization models such as were introduced long ago by Artru and Mennessier [@Artru:1974hr] and by Field and Feynman [@Field:1977fa]. A particularly attractive qualitative description in the context of QCD was given in [@Field:1977fa] for the case of two-jet production in $e^+e^-$ annihilation: The high-energy outgoing quark and antiquark form a tube or string of color flux between them. Quark-antiquark pairs are generated in the strong color field, and then reassemble themselves into color-singlet hadrons. A detailed quantitative dynamical description in semi-classical form in space-time was provided in [@Artru:1974hr]. Later elaborations [@Andersson:1983ia; @Artru:1979ye; @Andersson:1998tv] led to the Lund string model, used in the PYTHIA event generator [@Sjostrand:2006za; @Sjostrand:2007gs]. Closely related are cluster models [@Field:1982dg; @Webber:1983if] of hadronization. See Fig. \[fig:string\] for the space-time structure.
![String model [@Artru:1979ye; @Andersson:1983ia] for hadronization of quark-antiquark pair in $e^+e^-$ annihilation, pictured in space-time. A string (or flux tube) is created between the outgoing quark and antiquark. Quark-antiquark pairs are created in the color field in the flux tube, and then combine into color singlet hadrons. The formation of the pairs occurs dominantly near a space-like hyperbola an invariant distance of order 1 $\textrm{fm}/c$ to the future of the approximately trajectories of the original quark and antiquark.[]{data-label="fig:string"}](figures/string-space-time)
General features of string models include the chain decay ansatz, independent fragmentation, and an iterative scheme for generating final state hadrons [@Field:1977fa]. However, these classic hadronization models leave certain interesting issues, notably hadronization of polarized partons, unaddressed. This has led to recent proposals for extending hadronization models to include spin effects [@Kerbizi:2017rpp; @Matevosyan:2016fwi; @Bentz:2016rav; @Kerbizi:2018qpp].
A second background area to our work stems from the experimental observation that the particles observed in the final states of jets are approximately uniformly distributed in rapidity, e.g., [@Adolph:2014zba], with the number of hadrons per unit rapidity depending weakly on the high-energy scale $Q$ or $\sqrt{s}$.
In the case of soft/minimum-bias hadron-hadron collisions, a simple and natural initial candidate model in Feynman-graph form is a multiperipheral model (MPM) [@ELOP]. Applied to $e^+e^-$ annihilation, using the structure of the MPM would give Fig.\[fig:mpm\]. Here, a quark-antiquark pair is generated from a virtual photon; the quark and antiquark go outwards. In the Feynman graph, the quark line is formed into a loop, and the hadrons are generated along that line, at momentum-conserving quark-hadron-quark vertices (which should be effective vertices in QCD). Because the line exchanged in the vertical channel is of a spin-half field, a single graph of fixed order is power suppressed as $Q$ increases. But the typical number of particles produced is intended to increase with $Q$ proportionally to $\ln(Q^2/m^2)$, and in a strong-coupling situation this allows the result to be unsuppressed. The core idea of the MPM is a hypothesis [@Kogut:1973fn] that the relevant interactions only occur between quanta with nearby kinematics.
![Elementary multiperipheral model for hadronization of quark-antiquark pair in $e^+e^-$ annihilation.[]{data-label="fig:mpm"}](figures/mpm-basic)
We emphasize the MPM because it corresponds to graphical structures that appear to be used in the work on polarized event generators [@Kerbizi:2017rpp; @Matevosyan:2016fwi; @Bentz:2016rav; @Kerbizi:2018qpp]. In the case of the work by Artru, Kerbizi and collaborators [@Artru:2010st; @Artru:in2p3-00953539; @Artru:2012zz; @Kerbizi:2018qpp], MPM graphs are explicitly given. In the case of Matevosyan et al. [@Matevosyan:2016fwi], the connection is less direct. They present their work as involving a sequence of splitting vertices, and in earlier work from the same group, [@Bentz:2016rav] and especially [@Ito:2009zc], the splitting vertices are treated as vertices in an effective theory. Assembling the vertices to make a model for the amplitude for the whole process gives MPM graphs. In both cases, these appearances are misleading.
In this paper, we first show that taking the MPM, Fig. \[fig:mpm\], literally does not work. We show that the loop momentum can be deformed out of the intended region of kinematics into a region where short-distance pQCD physics is valid and the model, with its effective non-perturbative vertices is invalid. We then show that a motivated, minimal modification that does work is Fig. \[fig:mpm2\] with extra gluon lines; this matches both cluster hadronization and models of the Lund-string type, but now in Feynman-graph form. We will point out that models of this form could be considered a MPM, but with the quark-quark-hadron vertices and the associated connecting quark lines no longer being momentum conserving. Instead they should be considered as absorbing energy and momentum from the color flux tube. This dramatically changes the space-time structure.
![Modified multiperipheral model. It is modified from Fig. \[fig:mpm\] by allowance for the emission of gluons. Note that the gluons may arise not only directly from the initial quark and antiquark lines but also, for example, from a splitting, as shown.[]{data-label="fig:mpm2"}](figures/mpm-gauge)
Consequently, the interpretation of apparently MPM-like structures in work on hadronization like Refs. [@Kerbizi:2017rpp; @Matevosyan:2016fwi; @Bentz:2016rav; @Kerbizi:2018qpp] must be modified to match string physics. Careful examination of these papers shows that this is indeed the case: Implementation of the kinematics of multiple splitting involves momentum non-conservation at the level of MPM graphs that corresponds to implementations of a string model.
Many elements of our work can be found in the literature, e.g., a graph like Fig. \[fig:mpm2\] for cluster hadronization. Just before QCD was formulated, Kogut, Sinclair, and Susskind [@Kogut:1973fn] showed that multi-peripheral models cannot be applicable to hadronization in deep-inelastic scattering and $e^+e^-$ annihilation. Shortly afterwards, Casher, Kogut, and Susskind [@Casher:1973uf; @Casher:1974vf], still before the advent of QCD, proposed what can now be called a string model. Our argument is a modernized version of those old arguments taking account of the much better knowledge we now have of QCD, and putting it in a new context.
The importance of formulating a hadronization model in terms of Feynman graphs (with effective vertices and propagators) is that it automatically obeys general principles of quantum mechanics and quantum field theory, of causality and of Lorentz invariance. They are natural arenas for consistently incorporating spin effects, especially with entangled spin states, which are hard to formulate purely semi-classically.
There is interesting work on effective field theory for chiral symmetry breaking [@Schweitzer:2012hh]. A better understanding of the imperatives of Feynman graph implementations of hadronization should lead to suggestions as to appropriate ways of applying the chiral models to hadronization.
The elementary MPM {#sec:mpm}
==================
Basic Setup {#sec:basic}
-----------
Consider the graph shown in Fig. \[fig:mpm\], as a possible model for production of hadronic final states in $e^+e^-$ annihilation. In accordance with data, we assume that a small number of particles (around three pions[^1]) is produced per unit rapidity, and with a limited transverse momentum all with respect to a jet axis (e.g., the thrust axis). The typical transverse momentum is perhaps 0.3 or 0.4 GeV.
We will work in a center-of-mass frame, and label the particles in order of rapidity along the jet axis. To define the $z$-axis, we choose hadron 1 to have zero transverse momentum and positive rapidity. Let the center-of-mass energy be $Q$. In light-front coordinates $(+,-,T)$, the virtual photon’s momentum is $q =
(Q/\sqrt2,Q/\sqrt2,0_T)$. Then we write the hadron momenta in terms of rapidity and transverse momentum: $$\label{eq:pj}
p_j = \left( \frac{E_{j,T}}{\sqrt2} e^{y_j}, \frac{E_{j,T}}{\sqrt2} e^{-y_j}, {\boldsymbol{p}}_{j,T}\right),$$ where $E_{j,T} = \sqrt{p_{j,T}^2 + m_h^2}$ and $m_h$ is the mass of the hadron. We have chosen ${\boldsymbol{p}}_{1,T}=0$. The momenta must obey momentum conservation: $\sum_{j=1}^N p_j = q$.
Given the assumptions of the model, a rough estimate of the hadron rapidities is given by $$\label{eq:yj}
y_j \sim \frac{(N+1)-2j}{N-1} \ln \frac{Q}{m},$$ with the number of produced particles $N$ being approximately proportional to $\ln(Q^2/m^2)$, with a coefficient of around 3, as already mentioned. (The coefficient could be rather less if the primary hadrons are mostly vector mesons instead of pions.)
For the purposes of the model, we assume that the internal loop line of Fig. \[fig:mpm\] is for a quark, and that the hadrons are pions. But we will not use that assumption in any detail, since our concern is only with analytic properties, as determined by propagator denominators.
We define the origin of the loop momentum $l$ by writing the quark momenta from the electromagnetic vertex as $k_A+l$ and $k_B-l$, and defining $$\label{eq:kA.kB}
k_A = (Q/\sqrt2, 0, {\boldsymbol{0}}_T ),
\qquad
k_B = (0, Q/\sqrt2, {\boldsymbol{0}}_T ),$$ which would be the quark and antiquark momenta if they were free and massless. Thus $l$ parameterizes the deviation of quark kinematics from parton-model values.
The momentum $k_j$ of the line between hadron $j$ and hadron $j+1$ is $$\begin{aligned}
k_j ={}& l + k_A - \sum_{i=1}^j p_i
\nonumber\\
={}& l- k_B + \sum_{i=j+1}^N p_i .\end{aligned}$$ Hence $$\label{eq:kj+}
k_j^+ = l^+ + \sum_{i=j+1}^N p_i^+
~ = ~ l^+ + \Theta(m e^{y_j})$$ and $$\label{eq:kj-}
k_j^- = l^- - \sum_{i=1}^j p_i^-
~ = ~ l^- - \Theta(m e^{-y_j}).$$ In the last term in each equation, estimates are given with the aid of Knuth’s notation [@Knuth:asymptotic] $\Theta(\dots)$ rather than the conventional order notation $O(\dots)$ to emphasize that the estimates are to within a finite factor. The usual “big $O$" notation would allow the actual result to be arbitrarily much smaller, which is not the case here. The estimates arise as follows: Because of the approximately uniform distribution of hadrons in rapidity, the sum in the equations is of an approximately geometrical series. Then the sums are dominated by the few terms whose index $i$ is nearest to $j$.
In analyzing the properties of the graph, we find it useful to consider first a baseline case given by $l=0$. The power counting for all lines, i.e., the sizes of their propagator denominators is then established by the above estimates. Each $k_j^+$ is strictly positive and each $k_j^-$ is strictly negative. This implies that the hadrons get their (mostly large) plus-momentum component from the upper quark line, and their minus momentum from the lower line. These conditions, and the sizes calculated, ensure that the virtualities of all the lines are of order $\Theta(m^2)$, as is appropriate for a model of non-perturbative physics in QCD.
Integral over loop momentum $l$
-------------------------------
Within the above description, the virtualities in the graph in Fig. \[fig:mpm\] appear to be consistent with what one might expect for a model of non-perturbative physics, at least if $l=0$. However, $l$ is an integration variable.
Consider the process as having evolution in space-time from a quark-antiquark pair at short distances to a hadronic state at large distance. Then the conversion to hadrons occurs with unit probability. So the power law for making the quark-antiquark pair is the same as power counting for the full $e^+ e^- \to \text{hadrons}$ cross section. That is, hadronization is a leading power effect. This also implies that the size of the non-perturbative effective quark-quark-hadron vertex must be such as to self-consistently give exactly the unit probability of hadronization.
Furthermore, for the MPM to implement the intended non-perturbative physics, $k_A + l$ and $k_B - l$ must have a normal non-perturbative virtuality, $$(k_A+l)^2 = \Theta(m^2) \, , \qquad (k_B-l)^2 = \Theta(m^2) \, .$$ This also indicates that hadronization occurs long after the hard vertex.
If the integral were to stay in the region relevant for the assumed non-perturbative physics, then the transverse components of $l$ would be $\Theta(m)$, while the longitudinal components would be $\Theta(m^2/Q)$, i.e., $l$ would be in the Glauber region: $$l = \left( \Theta\left( \frac{m^2}{Q} \right) , \Theta\left(\frac{m^2}{Q} \right), \Theta({\boldsymbol{m}}_T) \right) \, .$$ This is simply because the virtualities of the $k_A + l$ and $k_B - l$ lines are: $$(k_A + l)^2 = 2 k_A^+ l^- + l^2\, , \qquad (k_B - l)^2 = -2 k_B^- l^+ + l^2 \, ,$$ and $k_A^+$ and $k_B^-$ are of order $Q$. If, however, $l$ is merely normal-soft (i.e., $l = (O(m), O(m), O({\boldsymbol{m}}_T) )$) then $$(k_A + l)^2 = \Theta\left(m Q \right) \, , \qquad (k_B - l)^2 = \Theta\left(m Q\right) \, . \label{eq:l.soft}$$ These are far off-shell, and therefore in a relatively short-distance region where standard weak coupling pQCD physics applies.
We will now show that we can apply a contour deformation that takes the integration out of the Glauber region, so that $l$ is at least soft and Eq. is fulfilled. This is a standard result in the theory of factorization, but the detailed demonstration is slightly modified. The modification takes care of the fact that we have an array of hadrons filling in a rapidity range with no big gaps, whereas the usual arguments in factorization theory have configurations only with widely separated momenta.
To exhibit the contour deformation conveniently, we write each $k_j$ in the form $q_j+l$, where $q_j$ is the value obtained by computing $k_j$ from the external momenta when $l=0$, i.e., $$q_j = k_A - \sum_{i=1}^j p_i = - k_B + \sum_{i=j+1}^N p_i .$$ We deform the contour by adding an imaginary part to $l^{\pm}$: $$\label{eq:deform}
l^+ = l_R^+ - i \Delta(l_R),
\qquad
l^- = l_R^- + i \Delta(l_R),$$ where $l_R^{\pm}$ are the real parts. The deformation is implemented by increasing the real number $\Delta$ from zero to a positive value. In general, we will allow $\Delta$ to depend on $l_R$, and we can also allow a different imaginary part of the two components of $l$. But initially, for use in the Glauber region, we will assume the 2 imaginary parts to be approximately equal and constant. Once $l_R^{\pm}$ is well outside the Glauber region, different amounts of deformation are allowed than in the Glauber region. But here we are only concerned with the deformation in the Glauber region, since that is the region in which the MPM is intended to be applied as a useful approximation to non-perturbative strong interactions.
[c]{} ![Singularity structure and contour deformations on $l^+$ and $l^-$ when $l$ is in the Glauber region ($|l^+l^-| \ll
m^2$). On the deformed contour, some propagators are off-shell by order $\Theta(m Q)$—see Eqs. –.[]{data-label="fig:contourdef"}](figures/contourdefa.pdf "fig:")\
\
![Singularity structure and contour deformations on $l^+$ and $l^-$ when $l$ is in the Glauber region ($|l^+l^-| \ll
m^2$). On the deformed contour, some propagators are off-shell by order $\Theta(m Q)$—see Eqs. –.[]{data-label="fig:contourdef"}](figures/contourdefb.pdf "fig:")
The denominator $k_j^2-m_q^2+i\epsilon = (q_j+l)^2-m_q^2+i\epsilon$ is $$2 (q_j^+ + l^+)(q_j^- + l^-) - ({\boldsymbol{q}}_{j,T}+ {\boldsymbol{l}}_T)^2-m_q^2 + i\epsilon,$$ whose real part is $$\label{eq:re.denom}
2 (q_j^+ + l_R^+)(q_j^- + l_R^-) - ({\boldsymbol{q}}_{j,T}+ {\boldsymbol{l}}_T)^2-m_q^2 + 2\Delta^2,$$ and whose imaginary part is $$\begin{gathered}
\label{eq:im.denom}
\epsilon + 2(q_j^+ - q_j^- + l_R^+ - l_R^-) \Delta
\\
=
\epsilon + 2\left[ \Theta(m\cosh(y_j)) + l_R^+ - l_R^- \right] \Delta, \end{gathered}$$ given the estimates of $q_j^{\pm}$ that follow from and . Also, the imaginary parts of the propagator denominators $(k_A + l)^2 - m^2 + i \epsilon$ and $(k_B - l)^2 - m^2 + i \epsilon$ are, respectively, $$\begin{aligned}
\label{eq:im.denom.ka}
2 \Delta \left( k_A^+ - l^+_R + l_R^- \right) + \epsilon &{}= 2 \Delta \left( \Theta(Q) - l_R^+ + l_R^- \right) \, , \\
\label{eq:im.denom.kb}
2 \Delta \left(k_B^- - l^+_R + l_R^- \right) + \epsilon &{}= 2 \Delta \left( \Theta(Q) - l_R^+ + l_R^- \right) \, . \end{aligned}$$
When $l_R^{\pm}$ are both smaller in size than $m$, as in the Glauber region, the imaginary part of every denominator is positive, so we have a successful deformation.
In the integration over $l_R$, once $l_R^+$ or $l_R^-$ gets to be bigger than order $m$ in size (and negative or positive, respectively), the imaginary part no longer retains its sign. For these larger values of $l_R^{\pm}$, a different deformation is needed; this traps the integral over $l^{\pm}$ in a region where the two components are of order $m$. But this is a region far beyond where the MPM was intended to be appropriate, for at least one quark line goes far off-shell, with a virtuality of order $Qm$ or more.
When $l_R^{\pm}$ is smaller than $m$, and especially much smaller, the allowed deformation enables us to get an imaginary part for $l^{\pm}$ that is of order $m$, by taking $\Delta$ of order $m$. It therefore follows that everywhere on the (deformed) integration contour, at least one of the longitudinal components of $l$ is at least of order $m$, and that we have propagators that are far off-shell, with virtuality of order $Qm$. (See Fig. \[fig:contourdef\].)
It might be naturally supposed that a model for non-perturbative physics would have some appropriate cut offs to remove any contribution from far off-shell propagators. But the cut offs should still obey standard relativistic causal and analytic properties. Therefore we can still deform out of the Glauber region into a region where the model is inappropriate.
Notice that most of the other quark lines also go far off-shell, since on the deformed contour $(q_j+l)^2-m_q^2 = \Theta(m^2 \cosh y_j)$. Only the propagators for the low rapidity lines stay at low virtuality. Thus we get a strong suppression of the graph, compared with the power-counting estimate obtained from the power-counting that would be appropriate for the Glauber region.
It follows from the above arguments that the unadorned MPM does not adequately model the phenomena that it was intended to describe.
Note that this objection does not apply to the MPM applied to soft hadron-hadron collisions. To allow the contour deformation we needed a loop momentum that circulated through the hard scattering vertex; but a relevant loop does not exist in the case of hadron-hadron scattering.
String-like MPM {#sec:string}
===============
In reality, a gluon field is created between the outgoing quark-antiquark pair. To allow for this, and for the creation of quark antiquark pairs in the gluon field, a simple model has the structure of Fig. \[fig:mpm2\]. Here gluons are emitted from the quark and antiquark, and then we have attached one gluon to each of what in the MPM were quark lines joining neighboring hadrons. We will show that the integration is trapped in a region where the explicitly shown lines in Fig. \[fig:mpm2\] all have virtuality of order $m^2$. We will also show that in this region there is a radical change in the directions of the flow of longitudinal momentum on these quark lines, compared with the simple MPM. The quark lines are given different arrows than in the MPM; this is a mnemonic to indicate an important flow of positive components of momentum that is very different than in the simple MPM. There will remain lines that are far off shell, but these are in the shaded blob in Fig. \[fig:mpm2\].
To get these results, it is not necessary that exactly one gluon attach to each quark-line segment between hadrons; our results will apply also if multiple gluons attach to a segment, or if not too high a proportion of the segments have no gluon. The particular case in Fig. \[fig:mpm2\], with one gluon per segment, simply provides one specific case to illustrate the principles.
It should be observed that not only can the diagrammatic structure of Fig. \[fig:mpm2\] be considered as implementing the string model, but that it is also related to a diagrammatic formulation of the cluster-hadronization model [@Corcella:2000bw; @Bahr:2008pv]. Cluster hadronization is the other major hadronization model used in Monte-Carlo event generators.
A possible set of $N$ independent loop variables are the momentum of the upper quark, $l_0$, and the momentum $l_j$ of each of the $N-1$ gluon lines that connect to the (now modified) multiperipheral ladder. In accordance with the general features of the hadron kinematics, we assume that transverse momenta of these lines are of order $m$, and that the rapidity of $l_j$ is similar to the rapidity $y_j$ of a hadron near where it connects. That is, each gluon connects to a part of the ladder with similar rapidity to that of the gluon. These assumptions are appropriate to the non-perturbative physics we wish to model. We will term this region of kinematics the canonical region for the model. Integration outside the canonical region puts some propagators in the ladder much further off shell than $m^2$ and corresponds to different physics. The important issue is now to determine whether or not the integration is trapped in the canonical region, and we will find that it is indeed trapped.
![Component of string-like MPM.[]{data-label="fig:component"}](figures/component)
We label the quark-line momenta as follows: $k_j$ is the momentum entering the vertex for hadron $j$ from above, and $k'_j$ is the momentum entering from below — Fig. \[fig:component\]. By momentum conservation, $$\begin{aligned}
p_j &= k_j+k'_j; \qquad &1 < j < N \\
l_j &= k'_j+k_{j+1}; \qquad &0 < j < N \\
p_1&= l_0 + k_1' \\
p_N&= l_N + k_N \, ,\end{aligned}$$ where $$l_N := q - \sum_{j=0}^{N-1} l_j \, .$$
To further analyze the kinematics of the canonical region and to locate the conditions for the integration to be trapped there, we find it useful to change variables. We define fractional momentum variables at each quark-hadron vertex by: $$x_j = k_j^+/p_j^+; \qquad 1 < j \leq N \, ,$$ $$y_j = {k'}_j^-/p_j^-; \qquad 1 \leq j < N \, .$$ (The use of $k'_j$ instead of $k_j$ in the second equation is to give a kind of symmetry under exchange of the roles of the initial quark and antiquark.) We define $x_1$ and $y_N$ by writing $$l_0^+ = x_1 p_1^+, \qquad l_N^- = q^- - \sum_{j=0}^{N-1} l_j^- = y_N p_N^- \, . \label{eq:l0lN}$$ For the middle hadrons, $1 < j < N$, $$\begin{gathered}
k_j^+ = x_j p_j^+, \quad k_j^- = (1-y_j) p_j^-,
\\
{k'}_j^+ = (1-x_j) p_j^+, \quad {k'}_j^- = y_j p_j^-,\end{gathered}$$ and hence $$\begin{aligned}
l_j^+ = {}& (1-x_j) p_j^+ + x_{j+1} p_{j+1}^+,
\\
l_j^- = {}& y_j p_j^- + (1-y_{j+1}) p_{j+1}^- ,\end{aligned}$$ while for $j = 1$ and $j = N$, we have
[k’]{}\_1\^+ &= p\_1\^+ - l\_0\^+ = (1 - x\_1) p\_1\^+; & [k’]{}\_1\^- &= y\_1 p\_1\^-\
k\_N\^+ &= x\_N p\_N\^+; & k\_N\^- &= (1 - y\_N) p\_N\^- .
(Note that there is no $k_1$ and no $k'_N$.)
The momentum on the bottom quark line has plus component $$\begin{aligned}
l_N^+ = q^+-\sum_{j=0}^{N-1} l_j^+ = {} & p_N^+ (1-x_N) \, , \label{eq:bottomplus} \end{aligned}$$ while the momentum of the top quark line has minus component $$\begin{aligned}
l_0^- = p_1^- (1-y_1) \label{eq:topminus} \, .\end{aligned}$$
The integration over the longitudinal components of the $l_j$ can be changed to integration over $x_j$ and $y_j$ with a simple Jacobian: $$\begin{aligned}
\prod_{j=0}^{N-1} ( dl_j^+ dl_j^- )
= {}&
\prod_{j=1}^N ( p_j^+ p_j^- ) \prod_{j=1}^N ( dx_j dy_j )
\nonumber\\
= {}&
\prod_{j=1}^N \left( \frac{E_{j,T}^2}{2} \right) \prod_{j=1}^N ( dx_j dy_j ).\end{aligned}$$
As in the previous section, we use the term “canonical region" to refer to the momentum region that is consistent with the spacetime picture of hadronization. Thus, in the canonical region, Eqs. – mean that all the transverse momenta are of order $m$ and the $x_j$ and $y_j$ variables of order unity, and hence all the propagator denominators are of order $m^2$.
What we would mean if the integration were not trapped would be the following: There would be an allowed contour deformation such that everywhere on the contour at least one of the quark lines shown in Fig. \[fig:mpm2\] is much further off-shell than order $m^2$. We will find that in fact no such deformation is possible.
To specify an allowed deformation, we parameterize the surface of integration by the real parts of the values of the integration variables. We restrict our attention to the longitudinal momentum components, and work in terms of the fractional momenta $x_j$ and $y_j$. We use a real value $\lambda$ taking values in the range $0\leq \lambda \leq
1$ to parameterize the deformation starting from real momenta. We therefore write $$\begin{aligned}
x_j &= x_{R,j} + i \lambda x_{I,j}(x_R,y_R) ,
\\
y_j &= y_{R,j} + i \lambda y_{I,j}(x_R,y_R) .\end{aligned}$$
By Cauchy’s theorem (generalized to multiple variables), the value of the integral is independent of $\lambda$ provided that the deformation is allowed, i.e., that no poles are crossed by the contour when $\lambda$ is increased from 0 to 1. There must be inserted in the integral a factor of the Jacobian for the transformation of variables from $x_{R,j}$ and $y_{R,j}$ to the complex variables $x_j$ and $y_j$.
Let us consider the region of integration where $x_R$ and $y_R$ are between $0$ and $1$. Then, from Eqs. –, the contour of integration passes through the canonical region if $\lambda = 0$. We will show that the integral is trapped in this region, and to do this we must show that there exists no choice of $x_{I,j}(x_R,y_R)$ and $y_{I,j}(x_R,y_R)$ such that we can increase $\lambda$ from 0 to 1, without crossing any poles, and such that the deformations obey $x_{I,j}\gg1$ and $y_{I,j}\gg1$ for at least some values of $j$. If such a deformation were to exist it would take the integration outside the canonical region. We will also require that the sizes of the derivatives, $|\partial x_{I,j}/ \partial x_{R,k}|$ etc all stay bounded, say below 1, so that strongly varying structures in the imaginary parts don’t exist, and the Jacobian remains of order unity.
There do indeed exist non-trivial allowed deformations, but these all have $x_{I,j}$ and $y_{I,j}$ of order unity at most, and therefore stay in the canonical region.
Consider the relevant denominators: $$\begin{aligned}
\label{eq:kj.denom}
k_j^2-m_q^2+i\epsilon
={}& 2 x_jp_j^+ (1-y_j) p_j^- - k_{j,T}^2 - m_q^2 + i\epsilon
\nonumber\\
={}& E_{j,T}^2 x_j(1-y_j) - k_{j,T}^2 - m_q^2 + i\epsilon ,
\\
\label{eq:kjprime.denom}
{k'}_j^2-m_q^2+i\epsilon
={}& 2 (1-x_j)p_j^+ y_j p_j^- - {k'}_{j,T}^2 - m_q^2 + i\epsilon
\nonumber\\
={}& E_{j,T}^2 (1-x_j)y_j - {k'}_{j,T}^2 - m_q^2 + i\epsilon ,
\\
\label{eq:lj.denom}
l_j^2-m_g^2+i\epsilon
={}& 2 [(1-x_j)p_j^+ + x_{j+1}p_{j+1}^+] \times
\nonumber\\ & \hspace*{-2cm}
\times [y_jp_j^- + (1-y_{j+1})p_{j+1}^-]
- l_{j,T}^2 - m_g^2 + i\epsilon.\end{aligned}$$ (Here, $m_g$ is a mass scale of order $\Lambda_{\rm QCD}^2$ representing the effects of confinement in cutting off soft gluons.) The transverse momenta and masses are of order $m$, as are the products of $p_j^+$ and $p_{j+1}^+$ with $p_j^-$ and $p_{j+1}^-$.
If we could find an allowed deformation out of the canonical region, at least one of the denominators would need to be much larger than $m^2$ on the deformed contour. As mentioned above, we are considering the part of the integration region where the real parts $x_{R,j}$ and $y_{R,j}$ are between zero and one. Then all of $x_{R,j}$, $1-x_{R,j}$, $y_{R,j}$, and $1-y_{R,j}$ are positive. Both of the momenta $k_j$ and $k'_j$ are therefore future-pointing as regards both their (real) plus- and minus-components, which is unlike corresponding momenta in the pure MPM of Fig. \[fig:mpm\].
Furthermore, to give an easy demonstration the non-existence of this hypothesized deformation, we restrict to the case that all of $x_{R,j}$, $1-x_{R,j}$, $y_{R,j}$, and $1-y_{R,j}$ are order unity rather than some being much less than unity.
The large size of $x_j$ or $y_j$ is achieved by the imaginary part, i.e., $|x_{I,j}|\gg1$ and/or $|y_{I,j}|\gg1$. The imaginary parts of the denominators for $k_j$ and $k'_j$ are $$\begin{aligned}
\label{eq:kj.I}
& \Im(k_j^2-m_q^2+i\epsilon)
\nonumber\\ & \hspace{1cm}
= \lambda E_{j,T}^2 [ x_{I,j} (1-y_{R,j}) - y_{I,j} x_{R,j} ] + \epsilon ,
\\
\label{eq:kj.prime.I}
& \Im({k'}_j^2-m_q^2+i\epsilon)
\nonumber\\ & \hspace{1cm}
= \lambda E_{j,T}^2 [ - x_{I,j} y_{R,j} + y_{I,j} (1-x_{R,j}) ] + \epsilon .\end{aligned}$$ Since $\epsilon > 0$, then, starting at $\lambda=0$, to be able to deform off the real axis without crossing a pole the coefficients of $\lambda$ must be positive when the real parts of the momenta are at the corresponding pole. At the pole on the $k_j$ propagator, with $\lambda=0$, we have $1-y_{R,j} = (k_{j,T}^2+m_q^2)/(E_{j,T}^2
x_{R,j})$. So positivity of the imaginary part of the propagator for $k_j$, as we deform the contour off the real axis requires that $$(k_{j,T}^2+m_q^2) \frac{x_{I,j}}{x_{R,j}} - E_{j,T}^2 x_{R,j} y_{I,j}$$ be positive for all $x_{R,j}$ between zero and one. Hence $x_{I,j}>0$ and $y_{I,j}<0$. But exactly the opposite condition applies to get a positive imaginary part for the denominator for the other line, $k'_j$. That is, at the pole for $k'_j$ with $\lambda = 0$, we have $1-x_{R,j} = ({k'}_{j,T}^2+m_q^2)/(E_{j,T}^2 y_{R,j})$. The condition for doing a deformation without crossing a pole is $$({k'}_{j,T}^2+m_q^2) \frac{y_{I,j}}{y_{R,j}} - E_{j,T}^2 y_{R,j} x_{I,j} > 0\, .$$ This requires $x_{I,j}<0$ and $y_{I,j}>0$. Thus, allowed deformations for $k_j$ lines clash with allowed deformations for $k'_j$ lines. So no deformation far out of the canonical region is possible for any set of $x_{I,j}$ or $y_{I,j}$.
The poles on the two lines $k_j$ and ${k'}_j$ are at different locations in $x_j$ and $y_j$. We could therefore imagine avoiding them by changing the sign of the imaginary part of one or more integration variables between the poles. But since to get a large denominator the imaginary part of the variable has to be much larger than unity, the derivative with respect to the real part would also be much larger than unity, which violates the smoothness requirement adopted earlier.
We have found that examination of the $k_j$ and $k'_j$ denominators is sufficient to show that the integration is trapped in the canonical region, with the denominators being of order $m^2$. In this region, the denominators for the gluon lines $l_j$ are also of order $m^2$. In fact these denominators also participate in the trapping. For example, if we have the $x_{I,j}$ positive and the $y_{I,j}$ negative to avoid the poles of the $k_j$ lines, as above, then the imaginary part of the $l_j$ denominator (\[eq:lj.denom\]) does not have a fixed sign, and so the hypothesized deformation is not allowed.
Notice that getting a situation where the contour is trapped in the canonical region depends on the gluons being able to inject appropriate amounts of momentum into the would-be multiperipheral ladder. Then about half the plus- and minus- momentum components on the quark lines are reversed in sign compared with the elementary MPM.
Discussion {#sec:discuss}
==========
We have shown that in the elementary MPM for non-perturbative hadronization in $e^+e^-$ annihilation, the loop integration can be deformed far out of the momentum region appropriate to its hypothesized validity. A minimal requirement for an appropriate non-perturbative model of a Feynman-graph kind, is that it incorporate the effects of gluon emission in a string-like way, as in Fig.\[fig:mpm2\].
There are dramatic differences in the directions of momentum flow and in the space-time structure between that graph and the elementary MPM Fig. \[fig:mpm\]. In the MPM, quark lines, like $k_A+l$, are far off-shell after the contour deformation. So the fastest hadrons are formed first. This is in complete contrast [@Casher:1974vf] to string-like models, where the fastest hadrons are formed last, on an appropriate time-dilated scale.
The importance of these results is for the formulation and interpretation of models of non-perturbative hadronization of hard jets. For example, the successful string-inspired model of Refs.[@Kerbizi:2017rpp; @Artru:2012zz; @Artru:2010st; @Artru:in2p3-00953539] is formulated in terms of multiperipheral graphs. The use of a Feynman graph formulation allows the systematic and consistent incorporation of spin effects. This is in contrast to purely semi-classical formulations of a string hadronization, where the use of the ideas of classical, non-quantum physics makes it much harder to see how to incorporate the intrinsically quantum mechanical phenomenon of quark spin. The results in the present paper indicate that the graphs must be interpreted as containing momentum-non-conserving vertices or propagators in an external gluon field. The consequences can be seen in Ref. [@Kerbizi:2018qpp], where there is an explicit allowance for transfer of energy and momentum between the quarks and the string, with a non-trivial identification of the momentum to use in the vertices of the multi-peripheral graph.
The results in Ref. [@Matevosyan:2016fwi; @Bentz:2016rav] are developed from earlier work [@Ito:2009zc] that used a chiral model of the Nambu-Jona-Lasinio (NJL) [@Nambu:1961tp; @Nambu:1961fr] type applied to a lowest order graph for fragmentation of a quark into a pion plus a quark. After iteration to apply to multiple splittings, such a model gives graphical structures like that of the MPM. Calculations such as those in [@Ito:2009zc] applied to a splitting $q_j\to\pi_j+q_{j+1}$, have the final quark on-shell, and the initial quark off-shell. But to iterate [@Matevosyan:2016fwi] the splitting, the final quark, $q_{j+1}$ is changed to be off-shell. This is formulated according to the Field-Feynman structure; it amounts to a kind of momentum non-conservation appropriate to implementing a string model. Another useful resources is Ref. [@Accardi:2009qv], which addresses the spacetime structure of hadronization in the Lund string model in Sec. 2.4. Such methods might help further clarify the relationship between Feynman graph structures and hadronization.
Chiral models can be argued to be useful as effective low-energy approximations to QCD. These models are important because they aim to capture the properties of QCD associated with chiral symmetry breaking. The results in this paper suggest an important direction for enhancing models such as the one in Ref. [@Schweitzer:2012hh] to apply to high-energy dynamical processes like the non-perturbative hadronization of hard jets. This is to formulate the theories to apply to processes that occur in the background of an non-vacuum state that corresponds to a gluonic flux-tube. There are probably some useful tools to relate strings and field theory in the paper by @Artru:1986vc on the quantization of the string by a sum-over-histories method.
This work was supported in part by the U.S. Department of Energy under Grant No. DE-SC0013699. T. Rogers’s work was supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under Award Number DE-SC0018106. This work was also supported by the DOE Contract No. DE- AC05-06OR23177, under which Jefferson Science Associates, LLC operates Jefferson Lab. We acknowledge useful discussions with M. Diefenthaler and with H. Matevosyan.
Estimate of number of particles per unit rapidity {#sec:TASSO}
=================================================
In Ref. [@Braunschweig:1990yd], the TASSO experiment reported results on the distribution of charged hadrons in jets in $e^-e^-$ annihilation at $Q$ between 14 and 44 GeV. In Table 9 are shown values for the normalized cross section $(1/\sigma_{\text{tot}})d\sigma/d\ln(1/x)$. Here $x=2p/Q$, where $p$ is the momentum of the detected particle; to a leading approximation, $x$ corresponds to the fragmentation variable $z$. We now roughly extract from this data the number of hadrons per unit rapidity in a jet.
For hadrons of high rapidity, Eq. (\[eq:pj\]) gives a corresponding $x$ value: $$x = \frac{E_{j,T}}{Q} e^{y_j}.$$ Hence the rapidity distribution in fragmentation is given by $dN/dy =
(1/2\sigma_{\text{tot}})d\sigma/d\ln(1/x)$. The extra factor of $1/2$ is to compensate the fact that the $x$ distribution gets a contribution from each jet. The most common particles in jets are pions. Isospin and charge conjugation symmetry show that the neutral pion fragmentation function in a $u$ or $d$ quark is half the charged pion fragmentation function: $f_{\pi^0/u} = (f_{\pi^+/u}+f_{\pi^-/u})/2$, etc. Thus the total hadronic number distribution is $3/2$ times the charged hadron distribution. Hence $dN^{\text{all}}/dy =
(1/2\sigma_{\text{tot}})d\sigma^{\text{ch.}}/d\ln(1/x)$. In Table 9 of Ref.[@Braunschweig:1990yd], the values of this quantity first increase as $\ln(1/x)$ is increased from 0 (which corresponds to $x=1$ or $z=1$). They reach a peak and then decrease. Now, once the rapidity is lower than a unit or two, it is not appropriate to apply the above approximations. So we take the peak value as the relevant one. Furthermore, as $Q$ increases, perturbative gluon emission becomes more important; this increases the hadron multiplicity beyond what it appears appropriate to attribute to purely non-perturbative QCD. So we take the peak in $(1/\sigma_{\text{tot}})d\sigma/d\ln(1/x)$ at $Q=\unit[14]{GeV}$ as the relevant one to estimate $dN/dy \simeq 3$. This is the number of hadrons per unit rapidity in the interior of the string.
[^1]: This estimate can be roughly deduced from measurements by the TASSO collaboration [@Braunschweig:1990yd] — see App. \[sec:TASSO\].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The decay $K_S^0 \to \pi e \nu$ has been observed by the CMD-2 detector at the $e^{+}e^{-}$ collider VEPP-2M at Novosibirsk. Of 6 million produced $K_L^0K_S^0$ pairs, $75 \pm 13$ events of the $K_S^0 \to \pi e \nu$ decay were selected. The corresponding branching ratio is B($K_S^0 \to \pi e \nu$)=$(7.2 \pm 1.4)\times10^{-4}$. This result is consistent with the evaluation of B($K_S^0 \to \pi e \nu$) from the $K_L^0$ semileptonic rate and $K_S^0$ lifetime assuming $\Delta S=\Delta Q$ .'
author:
- 'R.R.Akhmetshin, E.V.Anashkin, M.Arpagaus, V.M.Aulchenko,'
- 'V.Sh.Banzarov, L.M.Barkov, S.E.Baru, N.S.Bashtovoy,'
- 'A.E.Bondar, D.V.Bondarev, A.V.Bragin, D.V.Chernyak,'
- 'A.G.Chertovskikh, A.S.Dvoretsky, S.I.Eidelman, G.V.Fedotovich,'
- 'N.I.Gabyshev, A.A.Grebeniuk, D.N.Grigoriev, P.M.Ivanov,'
- 'S.V.Karpov, B.I.Khazin, I.A.Koop, L.M.Kurdadze, A.S.Kuzmin,'
- 'I.B.Logashenko, P.A.Lukin, K.Yu.Mikhailov, I.N.Nesterenko,'
- 'V.S.Okhapkin, E.A.Perevedentsev, A.S.Popov, T.A.Purlatz,'
- 'N.I.Root, A.A.Ruban, N.M.Ryskulov, A.G.Shamov,'
- 'Yu.M.Shatunov, A.I.Shekhtman, B.A.Shwartz, V.A.Sidorov,'
- 'A.N.Skrinsky, V.P.Smakhtin, I.G.Snopkov, E.P.Solodov,'
- |
P.Yu.Stepanov, A.I.Sukhanov, V.M.Titov, Yu.V.Yudin, S.G.Zverev\
\
J.A.Thompson\
title: 'Observation of $K_S^0$ semileptonic decays with CMD-2 detector. [^1]'
---
Introduction
============
While semileptonic decays of the K$_L^0$ have been well measured, the information on similar decays of K$_S^0$ is scarce. The only measurement of the K$_S \to \pi^{\pm} e^{\mp} \nu$ performed long ago assumed $\Delta S=\Delta Q$ and has low accuracy [@aubert]. The Review of Particle Physics evaluates the corresponding decay rate indirectly, using the K$_L^0$ measurements and assuming that $\Delta$S=$\Delta$Q so that\
$\Gamma(K_S^0 \to \pi^{\pm} e^{\mp} \nu) =
~\Gamma(K_L^0 \to \pi^{\pm} e^{\mp} \nu)$ [@pdg].\
We present results of the direct measurement of the branching ratio for the $K_S^0 \to \pi e \nu$ using the unique opportunity to study events containing a pure $K_L^0K_S^0$ state produced in the reaction $e^{+}e^{-} \to \phi \to K_L^0K_S^0$. The data were collected during the period of 1993-1998 with the CMD-2 detector [@cmd2; @cmd3].
The CMD-2 is a general purpose detector consisting of a drift chamber (DC) and proportional Z-chamber (ZC) used for the trigger, both inside a thin $(0.4 X_0)$ superconducting solenoid with a field of 1 T. Outside the field, there is a barrel (CsI) calorimeter and a muon range system. The CsI calorimeter covers polar angles from 0.8 to 2.3 radian. The vacuum beam pipe with a radius of 1.8 cm is placed inside the DC and $K_S^0$ mesons with the decay length $\lambda =0.6$ cm decay within it. The DC has momentum resolution of 3% for 200 MeV/c charged particles. The CsI calorimeter with $6\times6\times15$ $ cm^3$ crystals is placed at a distance of 40 cm from the beam axis and about a half of $K_L^0$ mesons with the decay length $\lambda =3.3$ m has interactions within CsI crystals. The energy resolution for photons in the CsI calorimeter is about 8%. Charged particles from the neutral kaon decays have momenta less than 290 MeV/c and stop within the CsI crystals.
Analysis
========
$K_S^0$ decays can be tagged using the presence of the second vertex with two charged particles at a distance from the $e^+e^-$ interaction region or the CsI cluster from $K_L^0$ interactions in CsI. The most probable decay channel $K_S^0 \to \pi ^+ \pi ^-$ was used for the normalization of the semileptonic $K_S^0 \to \pi e \nu$ decay. Both channels have a vertex with two charged particles near the beam axis.
To identify electrons in the decay under study, we are using the difference between measured momentum and energy loss in the detector material for stopped particles. The basic parameter used for charged particle identification was $$DPE =P_{particle} - E_{loss} - E_{cluster},$$ where $P_{particle}$ is the particle momentum measured in the DC, $E_{loss} $ is the average ionization energy loss (about 10 MeV) in the material in front of the CsI calorimeter, $E_{cluster}$ is the energy deposition in the CsI cluster matched with a particle track. CsI clusters which do not match any track are further referred to as photons. Electrons must have DPE=0 if the resolution of the detector is ideal and the leakage of showers in the CsI calorimeter is negligible. On the other hand, pions and positive muons have a broad distribution displaced from zero. Negative muons have a sharp peak displaced from zero as the energy of the CsI cluster is equal to the difference between the muon kinetic energy and $E_{loss}$. Pions from the decay $K_S^0 \to \pi ^+ \pi^-$ were used to obtain the distribution over this parameter for charged pions in the momentum range 160 - 200 MeV/c. This distribution together with the fitting function is shown in Fig.\[picrys3\]. For electrons and muons the distribution over this parameter was obtained from experimental data for reactions $e^+e^- \to e^+e^-, \mu ^+\mu ^-$ at the beam energy of 195 MeV. At this energy particle momenta are 195 MeV/c for electrons and 164 MeV/c for muons. The DPE distribution for electrons as well as the fitting function are shown in Fig.\[picrys1\]. The same distribution for muons (Fig.\[picrys2+\] and Fig.\[picrys2-\]) overlaps with the distribution for pions and this is properly taken into account.
![DPE distribution for collinear electrons with momentum 195 MeV/c.[]{data-label="picrys1"}](pict_pr3.eps){width="1.0\linewidth"}
![DPE distribution for collinear electrons with momentum 195 MeV/c.[]{data-label="picrys1"}](pict_pr1.eps){width="1.0\linewidth"}
![DPE distribution for negative muons with momentum 164 MeV/c.[]{data-label="picrys2-"}](pict_pr2+.eps){width="1.0\linewidth"}
![DPE distribution for negative muons with momentum 164 MeV/c.[]{data-label="picrys2-"}](pict_pr2-.eps){width="1.0\linewidth"}
Some kinematic features for the decay mode $K_S^0 \to \pi e \nu$ are :
- The opening angle between two tracks is between 0 and $\pi$
- The momenta of charged particles are less than 290 MeV/c
- The total energy of charged particles (assuming that both particles are charged pions) is between 330 and 550 MeV.
The same parameters for the decay mode $K_S^0 \to \pi^+ \pi^-$ are :
- The opening angle between two tracks is more than 2.6 radians
- The pion momenta are between 160 and 270 MeV/c
- The total energy of charged particles (assuming that both particles are charged pions) is equal to the beam energy (between 508 and 512 MeV).
The selection criteria for both modes of $K_S^0$ decay were:
- One or two vertices are found in the event
- Two minimum ionizing tracks with the opposite charge sign are reconstructed from the first vertex (nearest to the beam) and there is no other track with distance to the beam less than 1.4 cm
- The distance from the first vertex to the beam is between 0.2 cm and 1.4 cm. This cut rejects background from the beam region and material of the beam pipe
- The distance from the first vertex to the interaction point along the beam direction is less than 7 cm
- Each charged particle at the first vertex has a momentum between 90 and 270 MeV/c since particles with a momentum less than 90 MeV/c can not reach the CsI calorimeter in the magnetic field of the detector
- Each track from the first vertex crosses all sensitive layers in the DC in the radial direction and therefore has a polar angle $\theta$ between 0.87 and 2.27 radians
- Each charged particle at the first vertex fires the ZC and does not fire the muon range system
- The azimuthal angle difference between two tracks at the first vertex ($\Delta \phi $) is between 0.17 and 2.97 radians
- The azimuthal angle difference ($\Delta \phi $) between the plane “the first vertex ($K_S^0$) — the beam axis” and a photon with the energy greater than 50 MeV (supposedly the $K_L^0$ cluster) or the second vertex in DC (supposedly the $K_L^0$ decay in DC) is within $\pm 0.5$ radian
- There are no photons with the energy greater than 15 MeV outside the direction between “the first vertex — the beam axis” $\pm 1$ radian in the $\phi $-plane. This cut rejects background from processes with the neutral pions.
To select the decay mode $K_S^0 \rightarrow \pi e \nu$ the following criteria for the first vertex were additionally used:
- The opening angle between two tracks is between 0.35 and 2.50 radians
- The total energy of charged particles (assuming that both particles are charged pions) is between 300 and 470 MeV
- The DPE parameter corresponding to the charged particle at the first vertex is included into a histogram when this particle track matches the CsI cluster independently of the matching conditions of the other track
- The invariant mass squared of the assumed neutrino is greater than $-10000$ $MeV^2 /c^4$ and less than 6000 $MeV^2 /c^4$.
To select the decay mode $K_S^0 \rightarrow \pi^+ \pi^-$ the following criteria for the first vertex were additionally used:
- The opening angle between two tracks is more than 2.55 radians
- The total energy of charged particles (assuming that both particles are charged pions) is between 480 and 540 MeV
- The pion momenta are between 140 and 270 MeV/c
- The average momentum of two charged pions is between 190 and 230 MeV/c
- The ratio of the smaller momentum to the larger one is more than 0.58
- The angle between the vector sum of momenta and the direction “the beam axis $\rightarrow$ the first vertex” is less than $\pi$/2
- Each charged particle at the first vertex has a matched CsI cluster
- The invariant mass squared of the assumed photon is greater than $-10000$ $MeV^2 /c^4$ and less than 6000 $MeV^2 /c^4$.
Results and discussion
======================
The DPE distribution for events selected as candidates for the decay $K_S^0 \to \pi e \nu$ is shown in Fig.\[picrys4\]. The data were fit using the DPE distribution of e, $\mu$ and $\pi$ measured in experiment. The result of the fit for the number of the electrons is $N_e = 83.5 \pm 12.7$. The number of mesons is equal to $N_m = 354 \pm 21$. The distribution over the distance between the vertex and beam axis for these events is consistent with that for $K_S^0 \to \pi^+ \pi^-$ decays.
![DPE distribution for charged particles in $K_L^0$ decays.[]{data-label="picrys5"}](pict_pr4.eps){width="1.0\linewidth"}
![DPE distribution for charged particles in $K_L^0$ decays.[]{data-label="picrys5"}](pict_pr5.eps){width="1.0\linewidth"}
The main background for the $K_S^0 \rightarrow \pi e \nu$ decay mode after applying the above cuts comes from the decays $K_S^0 \rightarrow \pi^+ \pi^- \gamma$, $K_S^0 \rightarrow \pi \mu \nu$ and $K_L^0\rightarrow \pi e \nu$. The former two processes are taken into account while fitting the histogram over DPE (the fit has two free parameters - the number of electrons $N_e$ and that of mesons $N_{\pi}$ + $N_{\mu}$). To take into account the background from the latter process, the same procedure was applied to events with a distance from the first vertex to the beam axis between 3 and 7 cm. The resulting number of electrons for these events is $24.2 \pm 6.1$. Taking into account the dependence of the efficiency of vertex reconctruction on the distance from the beam axis as well as the ratio of the distance intervals for $K_L^0$- and $K_S^0$- decays it was found that the contribution of the $K_L^0$-decays is equal to $8.6 \pm 2.2$. Thus, the number of electrons and correspondingly the number of events of the $K_S^0 \to \pi e \nu$ decay, is equal to $$N_e = 75 \pm 13 .$$
To illustrate the correctness of the identification based on the DPE parameter, we applied the same procedure to look for events of the decay $K_L^0 \to \pi e \nu$. Events were selected in which there were a $K_S^0 \to \pi^+ \pi^-$ decay near the beam axis and a second vertex in DC at a long distance from the beam axis. The ratio of the number of electrons $N_e$ to the number of pions and muons $N_{\mu} +N_{\pi}$ can be calculated from the branching ratios of the main decay modes of the $K_L^0$ and should be equal to 0.33 [@pdg]. The DPE distribution for events selected as candidates for the decays of $K_L^0$ meson is shown in Fig.\[picrys5\]. The results of the fit are: $N_e = 300 \pm 22$, $N_m = 887 \pm 34$ and their ratio is 0.34 $\pm$ 0.03 in agreement with the estimate above.
Under the applied cuts the number of $K_S^0 \rightarrow \pi^+ \pi^-$ detected decays equals $N_{\pi\pi}=178110$ (of about 6 million produced $K_S^0K_L^0$ pairs). Applying the selection criteria above to the events from simulation, one obtains that the ratio of the detection efficiency for the normalization process to one for the process under study should be $\varepsilon _{rel}=2.49 \pm 0.12$. The simulation of the $K_S^0 \to \pi e \nu$ decay was performed using the same Dalitz plot as for the $K_L^0$ decay. Then one would expect for the branching ratio $$B(K_S^0 \to \pi e \nu)=(N_e\cdot \varepsilon_{rel}/N_{\pi\pi})\cdot
B(K_S^0 \to \pi^+ \pi^-).$$ From 75 $\pm$ 13 events observed by us the following result was obtained for the branching ratio:
$$B(K_S^0 \to \pi e \nu)=(7.2 \pm 1.4)\times10^{-4}.$$
The quoted error contains the statistical error and the systematic uncertainty (5% from the simulation detection efficiency and 5% from the selection criteria) added in quadrature. This result is consistent with the previous determination of B($K_S^0 \to \pi e \nu$) [@aubert] as well as with the Review of Particle Physics evaluation [@pdg].
Conclusions
===========
The possibility to perform measurements using events containing the pure $K_L^0K_S^0$ state is available at the VEPP-2M collider in Novosibirsk. This circumstance permits to have tagged $K_S^0$ mesons in contrast to experiments with the neutral kaons, produced by the charge exchange of charged kaons at a target. Using the difference between measured momentum and energy loss in the detector material for stopped particles, the electrons from $K_S^0$ decays were identified. The branching ratio is B($K_S^0 \to \pi e \nu$)=$(7.2 \pm 1.4)\times10^{-4}$. It is the first direct measurement of this quantity.
The value of the branching ratio does not contradict to that, calculated from $K_L^0$ semileptonic rates and $K_S^0$ lifetime assuming $\Delta S=\Delta Q$.
[**Acknowledgements**]{}
We would like to thank the technical personnel of VEPP-2M for the machine and detector support during experimental runs.
[99]{} B.Aubert et al., Phys. Lett. [**17**]{} (1965) 59. C.Caso et al., Eur. Phys. J. [**C3**]{} (1998) 1. G.A.Aksenov et al., Preprint BudkerINP 85-118. Novosibirsk, 1985. E.V.Anashkin et al., ICFA Instr. Bulletin [**5**]{} (1988) 18.
[^1]: This work is supported in part by the Russian Foundation for Basic Research under grant RFBR-98-02-17851 and the US DOE grant DEFG0291ER40646.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we investigate the block that has an abelian defect group of rank $2$ and its Brauer correspondent has only one simple module. We will get an isotypy between the block and its Brauer correspondent. It will generalize the result of Kessar and Linckelmann ([@KL]).'
author:
- Xueqin Hu
date: 'School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China'
title: '**Blocks with abelian defect groups of rank $2$ and one simple module**'
---
Introduction
============
Let $p$ be a prime and $\mathcal{O}$ a complete discrete valuation ring having an algebraically closed residule field $k$ of characteristic $p$ and a quotient field $\mathcal{K}$ of characteristic $0$. We will always assume that $\mathcal{K}$ is big enough for the finite groups below.
Let $G$ be a finite group and $b$ a block of $\mathcal{O}G$ with a defect group $P$. Denote by $\mathrm{Irr}_\mathcal{K}(G,b)$ and $\mathrm{IBr}(G,b)$ the set of irreducible ordinary characters in $b$ and the set of irreducible Brauer characters in $b$ respectively. Set $l_G(b)=|\mathrm{IBr}(G,b)|$. Let $c$ be the Brauer correspondent of $b$ in $N_G(P)$. In [@KL], Kessar and Linckelmann investigated the block $b$ under the assumptions that $l_{N_G(P)}(c)=1$ and $P$ is elementary abelian of rank $2$. They showed that the inertial quotient of $b$ is abelian and there is an isotypy between $b$ and $c$ all of whose signs are positive.
In this note, we will generalize these results to the blocks with defect groups of rank $2$.
\[MT\] Keep the notation as above. Assume that $P$ is abelian of rank $2$ and $l_{N_G(P)}(c)=1$. Then the inertial quotient of $b$ is abelian and there is an isotypy between $b$ and $c$.
These results are well-known when either $p$ is $2$ or the inertial quotient of $b$ is trivial. Therefore, we may assume that $p$ is odd and the inertial quotient of $b$ is non-trivial throughout this paper.
The structure of the block $c$
==============================
Keep the notation as above. In this section, we will investigate the structure of the inertial quotient of $b$ and irreducible ordinary characters of the block $c$.
Given a positive integer $a$, denote by $C_a$ the cyclic group of order $a$. We will use $[-\,\ ,\,\ -]$ to represent the commutator. Assume that $P=C_{p^n}\times C_{p^m}$ for some positive integers $n,m$. We will fix a maximal $b$-Brauer pair $(P,b_P)$. For any $Q\leq P$, denote by $(Q,b_Q)$ the unique $b$-Brauer pair contained in $(P,b_P)$. Let $E$ be the inertial quotient of $b$ associated with $(P,b_P)$, namely, $E=N_G(P,b_P)/C_G(P)$.
\[E is abelian\] The inertial quotient $E$ is abelian if $l_{N_G(P)}(c)=1$.
Let $\Phi(P)$ be the Frattini subgroup of $P$. So $P/\Phi(P)$ is $C_p\times C_p$. Set $H$ to be $N_G(P,b_P)$. Then $\Phi(P)\unlhd H$ and denote $H/\Phi(P)$ by $\bar{H}$. For any subset $X$ of $\mathcal{O}H$, $\bar{X}$ denotes the image of $X$ under the canonical map $\mathcal{O}H\longrightarrow\mathcal{O}\bar{H}$.
Since $l_H(b_P)=1$, $l_{\bar{H}}(\bar{b}_P)=1$ and $\bar{b}_P$ is a block of $\bar{H}$ with defect group $\bar{P}=C_p\times C_p$. Let $\hat{C}$ be the subgroup of $H$ such that $\hat{C}/\Phi(P)=C_{\bar{H}}(\bar{P})$. Hence, $\hat{C}=\{x\in H~|~[P,x]\subseteq\Phi(P)\}$. It is clear that $P=[P,\hat{C}]\times C_P(\hat{C})$. So $P=C_P(\hat{C})$ since $[P,\hat{C}]\leq\Phi(P)$. This means $C_{\bar{H}}(\bar{P})=\bar{C}_G(P)$. Hence, $(\bar{P},\bar{b}_P)$ is a maximal $\bar{b}_P$-Brauer pair of $\mathcal{O}\bar{H}\bar{b}_P$. By [@KL Proposition 5.2], $N_{\bar{H}}(\bar{P},\bar{b}_P)/C_{\bar{H}}(\bar{P})$ is abelian. It is evident that $E$ is isomorphic to $N_{\bar{H}}(\bar{P},\bar{b}_P)/C_{\bar{H}}(\bar{P})$. We are done.
By [@DJ Lemma 2] and the structure of blocks with normal defect groups, $E$ is a direct product of two isomorphic groups. Next, we will show that $E$ acts diagonally on $P$. This can be deduced from the following general fact.
\[diagonal action\] Let $D$ be an abelian $p$-group of rank $2$ and $F\leq\mathrm{Aut}(P)$ an abelian $p^\prime$-group which is a direct product of two isomorphic subgroups. Then we have the decompositions $F=F_1\times F_2$ and $D=D_1\times D_2$ such that $F_1$ acts faithfully on $D_1$ and centralises $D_2$ and $F_2$ acts faithfully on $D_2$ and centralises $D_1$ and $F_1\cong F_2$. In particular, $F_1$ and $F_2$ are cyclic groups of order dividing $(p-1)$.
We will exhibit it by induction on $|D|$. When $D$ is elementary abelian, it is actually done in [@KL Proposition 5.3]. We may assume that $n\geq 2$ or $m\geq 2$. Let $\Phi(D)$ be the Frattini subgroup of $D$. So $D/\Phi(D)$ is $C_p\times C_p$. Let $\pi$ be the canonical map from $F$ to $\mathrm{Aut}(D/\Phi(D))$. For any subset $X$ of $F$, $\bar{X}$ denotes the image of $X$ under $\pi$. It is clear that $\pi$ is injective. So there exist two subgroups $F_1$ and $F_2$ of $F$ and two subgroups $D_1$ and $D_2$ of $D$ containing $\Phi(D)$ satisfying the properties $\bar{F}=\bar{F}_1\times \bar{F}_2$ and $D/\Phi(D)=D_1/\Phi(D)\times D_2/\Phi(D)$ and $\bar{F}_1$ acts faithfully on $D_1/\Phi(D)$ and centralises $D_2/\Phi(D)$ and $\bar{F}_2$ acts faithfully on $D_2/\Phi(D)$ and centralises $D_1/\Phi(D)$ and $\bar{F}_1\cong \bar{F}_2$. Hence, $D_1$ and $D_2$ are $F$-stable and they fulfill
\(i) $D_1=[D_1,F_1]\cdot\Phi(D)$ and $[D_1,F_2]\subseteq\Phi(D)$ and $F_1$ acts faithfully on $D_1$;
\(ii) $D_2=[D_2,F_2]\cdot\Phi(D)$ and $[D_2,F_1]\subseteq\Phi(D)$ and $F_2$ acts faithfully on $D_2$;
\(iii) $D_1\cap D_2=\Phi(D)$ and $D_1/\Phi(D)\cong C_p\cong D_2/\Phi(D)$ and $D=D_1\cdot D_2$.
Suppose that $\Phi(D)$ is cyclic. Then $D=C_p\times C_{p^m}$ with $m\geq 2$ and $\Phi(D)=C_{p^m-1}$. Since $D_2=[D_2,F_1]\times C_{D_2}(F_1)$ and $[D_2,F_1]\subseteq\Phi(D)$, $\Phi(D)=[D_2,F_1]\times C_{\Phi(D)}(F_1)$. Then either $[D_2,F_1]=1$ or $C_{\Phi(D)}(F_1)=1$ by the assumption that $\Phi(D)$ is cyclic. If $[D_2,F_1]=1$, then $\Phi(D)\leq D_2\leq C_D(F_1)$. Clearly, $D=[D,F_1]\times C_D(F_1)$ and $D_2$ is a maximal subgroup of $D$. Thus, $D_2=C_D(F_1)$ and $[D,F_1]=[D_1,F_1]$. Since $F_1$ and $F_2$ commute with each other and $[D_1,F_2]\subseteq\Phi(D)\subseteq C_D(F_1)$, $[[D_1,F_1],F_2]=1$. So $[D_1,F_1]\leq C_P(F_2)$. Since $F_1$ acts faithfully on $D_1$ and $D_1=[D_1,F_1]\times C_{D_1}(F_1)$, $F_1$ acts faithfully on $[D_1,F_1]$. Thus, the decompositions $F=F_1\times F_2$ and $D=[D_1,F_1]\times D_2$ are what we want. We may assume that $C_{\Phi(D)}(F_1)=1$. Then $\Phi(D)=[D_2,F_1]$ and $D_2=\Phi(D)\times C_{D_2}(F_1)$. If $C_{\Phi(D)}(F_2)=1$, we can get $\Phi(D)=[D_1,F_2]$ and $D_1=\Phi(D)\times C_{D_1}(F_2)$ similarly. Then $D=C_{D_1}(F_2)\times C_{D_2}(F_1)$ which is impossible. So $C_{\Phi(D)}(F_2)\neq1$. Then replacing $D_2$ by $D_1$ in the previous argument, we can obtain the decompositions that we need.
Suppose $\Phi(D)$ is of rank $2$. Then both $D_1$ and $D_2$ are of rank $2$. Let $K$ be subgroup of $F$ consisting of automorphisms acting trivially on $D_1$. Then $D=[D,K]\times C_D(K)$ and $D_1\leq C_D(K)$. Hence, $K$ has to be trivial since $D_1$ has rank $2$. This means $F$ acts faithfully on $D_1$. By induction, we have $D_1=D_{11}\times D_{12}$ and $F=F_{11}\times F_{12}$ such that $F_{11}$ acts faithfully on $D_{11}$ and centralises $D_{12}$ and $F_{12}$ acts faithfully on $D_{12}$ and centralises $D_{11}$ and $F_{11}\cong F_{12}$. Then $D=[D,F_{11}]\times C_{D}(F_{11})$ and $D_{11}=[D_{11},F_{11}]\leq[D,F_{11}]$ and $D_{11}\leq C_{D}(F_{12})$. In particular, $C_{[D,F_{11}]}(F_{12})\neq 1$. But $[D,F_{11}]$ is cyclic. Then $[D,F_{11}]\leq C_D(F_{12})$ and moreover $C_D(F_{12})=[D,F_{11}]\times(C_D(F_{11})\cap C_D(F_{12}))$. But $C_D(F_{12})$ is also cyclic. We have $[D,F_{11}]=C_D(F_{12})$. Similarly, we can prove that $[D,F_{12}]=C_D(F_{11})$. Then the decompositions $D=[D,F_{11}]\times [D,F_{12}]$ and $F=F_{11}\times F_{12}$ are what we want. We are done.
Hence, by Lemma \[diagonal action\], we have $E=E_1\times E_2$ and $P=P_1\times P_2$ such that
\(i) $E_1$ acts faithfully on $P_1$ and centralises $P_2$;
\(ii) $E_2$ acts faithfully on $P_1$ and centralises $P_1$;
\(iii) $E_1\cong E_2$ are cyclic groups of order $l$, which $l$ is a positive integer dividing $(p-1)$.
We can easily describe the source algebra of the block $c$ by the structure theory of blocks with normal defect groups and the structure of inertial quotient $E$. It is well-known that there exists a central extension
$$\xymatrix@C=0.5cm{
1 \ar[r] & Z \ar[r] & \tilde{E} \ar[r] & E \ar[r] & 1 }$$ with $Z$ cyclic $p^\prime$-group such that there is an irreducible ordinary character $\theta$ of $Z$ which is covered by a unique irreducible character of $\tilde{E}$. Let $e_\theta\in\mathcal{O}Z$ be the central idempotent corresponding to $\theta$. Set $N=P\rtimes\tilde{E}$. Then $\mathcal{O}Ne_\theta$ is the source algebra of the block $c$. Note that $e_\theta$ is still a block of $C_N(R)$ for any $R\leq P$. The following lemma gives some information about the degrees and number of irreducible ordinary characters of $\mathcal{O}Ne_\theta$, which is similar with [@KL Proposition 5.3]. We will skip the proof.
\[irreducible character of c\] Set $A$ to be $\mathcal{O}Ne_\theta$. Then the degree of an element of $\mathrm{Irr}_\mathcal{K}(A)$ is either $l$ or $l^2$ and $\mathrm{Irr}_\mathcal{K}(A)$ has $p^n+p^m-1$ elements of degree $l$ and $\frac{p^n-1}{l}\cdot\frac{p^m-1}{l}$ elements of degree $l^2$.
The extension of local system
=============================
Keep the notation as above. In this section, we will use the so-called $(G,b)$-local system introduced by Puig and Usami in [@PU] to prove the main theorem.
First, let us recall some notation and state the definition of $(G,b)$-local system under our setting (see [@PU]).
Let $\mathcal{CF}_\mathcal{K}(G)$ be the vector space of $\mathcal{K}$-valued class functions of $G$ and $\mathcal{BCF}_\mathcal{K}(G)$ be the vector space of $\mathcal{K}$-valued class functions on the set $G_{p^\prime}$ of $p^\prime$-elements of $G$. It is clear that the set of irreducible ordinary characters of $G$ is a $\mathcal{K}$-basis of $\mathcal{CF}_\mathcal{K}(G)$ and the set of irreducible Brauer characters of $G$ is a $\mathcal{K}$-basis of $\mathcal{BCF}_\mathcal{K}(G)$. For $\chi,\chi^\prime\in\mathcal{CF}_\mathcal{K}(G)$, we denote by $\langle\chi,\chi^\prime\rangle$ the inner product of $\chi$ and $\chi^\prime$.
Let $u$ be a $p$-element of $G$. we have the well-known surjective $\mathcal{K}$-linear map $d_G^u:\mathcal{CF}_\mathcal{K}(G)\longrightarrow
\mathcal{BCF}_\mathcal{K}(C_G(u))$ defined by $d_G^u(\chi)(s)=\chi(us)$ for any $\chi\in\mathcal{CF}_{\mathcal{K}}(G)$ and $s\in C_G(u)_{p^\prime}$. It has a section $e_G^u:\mathcal{BCF}_\mathcal{K}(C_G(u))\longrightarrow
\mathcal{CF}_\mathcal{K}(G)$ such that for $\varphi\in\mathcal{BCF}_\mathcal{K}(C_G(u))$, $e_G^u(\varphi)(g)=0$ if the $p$-part of $g$ is not conjugate to $u$ in $G$.
For the block $b$, let $\mathcal{CF}_\mathcal{K}(G,b)$ be the subspace of $\mathcal{CF}_\mathcal{K}(G)$ generated by the elements in $\mathrm{Irr}_\mathcal{K}(G,b)$ and $\mathcal{L}_\mathcal{K}(G,b)$ the group of generalized characters in $b$. Also, let $\mathcal{CF}_\mathcal{K}^\circ(G,b)=
\mathcal{CF}_\mathcal{K}(G,b)\cap\mathrm{Ker}(d_G^1)$ and $\mathcal{L}_\mathcal{K}^\circ(G,b)=
\mathcal{L}_\mathcal{K}(G,b)\cap\mathrm{Ker}(d_G^1)$.
(Puig-Usami [@PU 3.2]) With the above notation and assumption. Let $X$ be an $E$-stable non-empty set of subgroups of $P$ and assume that $X$ contains any subgroup of $P$ containing an element of $X$. Let $\Gamma$ be a map over $X$ sending $Q\in X$ to a bijective isometry $$\Gamma_Q:\mathcal{BCF}_{\mathcal{K}}(C_N(Q),e_\theta)\longrightarrow
\mathcal{BCF}_\mathcal{K}(C_G(Q),b_Q).$$ If $\Gamma$ satisfies the following conditions, then $\Gamma$ is called a $(G,b)$-[[local system]{}]{} over $X$.
\(i) For any $Q\in X$, any $\eta\in\mathcal{BCF}_\mathcal{K}(C_N(Q),e_\theta)$ and any $s\in E$, we have $\Gamma_Q(\eta)^s=\Gamma_{Q^s}(\eta^s)$.
\(ii) For any $Q\in X$ and any $\eta\in\mathcal{L}_\mathcal{K}(C_N(Q),e_\theta)$, the sum $$\sum\limits_{u}e_{C_G(Q)}^u(\Gamma_{Q\cdot\langle u\rangle}
(d_{C_N(Q)}^u(\eta)))$$ where $u$ runs over a set of representatives $U_Q$ for the orbits of $C_E(Q)$ in $P$, is a generalized character of $C_G(Q)$.
Let $\Gamma$ be a $(G,b)$-local system over $X$. Such $\Gamma$ always exists by [@PU 3.4.2]. For any $Q\in X$, we have a map $\Delta_Q:
\mathcal{CF}_\mathcal{K}(C_N(Q),e_\theta)\longrightarrow
\mathcal{CF}_\mathcal{K}(C_G(Q),b_Q)$ defined by $$\Delta_Q(\eta)=
\sum\limits_{u\in U_Q}e_{C_G(Q)}^u(\Gamma_{Q\cdot\langle u\rangle}
(d_{C_N(Q)}^u(\eta))).$$ Then by [@PU 3.3 and 3.4] $\Delta_Q$ gives a perfect isometry between the block $e_\theta$ of $C_N(Q)$ and the block $b_Q$ of $C_G(Q)$ and $\Delta_Q(\lambda\ast\eta)=\lambda\ast\Delta_Q(\eta)$ for any $\lambda\in\mathcal{CF}_\mathcal{K}(P)^{C_E(Q)}$ and $\eta\in\mathcal{CF}_\mathcal{K}(C_N(Q))$. Here, $\mathcal{CF}_\mathcal{K}(P)^{C_E(Q)}$ denotes the set of $C_E(Q)$-stable elements of $\mathcal{CF}_\mathcal{K}(P)$ and $\ast$ denotes the $\ast$-construction of charaters due to Brou$\acute{\mathrm{e}}$ and Puig (see [@BP]). Hence, if $X$ contains the trivial subgroup $1$ of $P$, then $\Delta_1$ induces a perfect isometry between the block $e_\theta$ of $N$ and the block $b$ of $G$. Moreover, this is an isotypy in the sense of [@B] by [@WZZ Proposition 2.7].
In [@PU], Puig and Usami developed a criterion for the extendibility of the $(G,b)$-local system. With the notation above. Suppose that $1\not\in X$ and let $Q$ be a maximal subgroup of $P$ such that $Q\not\in X$. Denote by $X^\prime$ the union of $X$ and the $E$-orbit of $Q$. For any subset $Y$ of $\mathcal{O}C_N(Q)$, denote by $\bar{Y}$ the image of $Y$ under the canonical map from $\mathcal{O}C_N(Q)$ to $\mathcal{O}C_N(Q)/Q$. We have the similar notation for $\mathcal{O}C_G(Q)$. So $\bar{e}_\theta$ and $\bar{b}_Q$ are the blocks of $\bar{C}_N(Q)$ and $\bar{C}_G(Q)$ respectively. Set $\Delta_Q^\circ=
\sum\limits_{u\in U_Q-Q}e_{C_G(Q)}^u\circ\Gamma_{Q\cdot\langle u\rangle}
\circ d_{C_N(Q)}^u$ (see [@PU 3.6.2]). By [@PU Proposition 3.7 and Remark 3.8], $\Delta_Q^\circ$ induces a bijective isometry $$\bar{\Delta}_Q^\circ:
\mathcal{CF}_\mathcal{K}^\circ(\bar{C}_N(Q),\bar{e}_\theta)\cong
\mathcal{CF}_\mathcal{K}^\circ(\bar{C}_G(Q),\bar{b}_Q)$$ such that $\bar{\Delta}_Q^\circ(\mathcal{L}_\mathcal{K}^\circ(\bar{C}_N(Q),\bar{e}_\theta))=
\mathcal{L}_\mathcal{K}^\circ(\bar{C}_G(Q),\bar{b}_Q)$. Clearly, $\bar{\Delta}_Q^\circ(\lambda\ast\eta)=
\lambda\ast\bar{\Delta}_Q^\circ(\eta)$ for $\lambda\in\mathrm{Irr}_\mathcal{K}(\bar{P})^{C_E(Q)}$ and $\eta\in\mathcal{L}_\mathcal{K}^\circ(\bar{C}_N(Q),\bar{e}_\theta)$ (see [@W05 Case 2.2]) and $\bar{\Delta}_Q^\circ$ is $N_E(Q)$-stable. The following is the key criterion of extendibility.
([@PU Proposition 3.11])\[extendibility\] With the notation above, the $(G,b)$-local system $\Gamma$ over $X$ can be extended to a $(G,b)$-local system $\Gamma^\prime$ over $X^\prime$ if and only if $\bar{\Delta}_Q^\circ$ can be extended to an $N_E(Q)$-stable bijective isometry $$\bar{\Delta}_Q:\mathcal{CF}_\mathcal{K}(\bar{C}_N(Q),\bar{e}_{\theta})\cong
\mathcal{CF}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$$ such that $\bar{\Delta}_Q(\mathcal{L}_\mathcal{K}(\bar{C}_N(Q),\bar{e}_\theta))=
\mathcal{L}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$.
In order to prove Theorem \[MT\], it suffices to show that there is a $(G,b)$-local system over the set of all the subgroups $P$. Hence, by Proposition \[extendibility\], we can assume that there is a $(G,b)$-local system $\Gamma$ over $X$ such that $1\not\in X$ and $Q$ is a maximal subgroup of $P$ such that $Q\not\in X$.
\[MT’\] With the notation above and assumptions of Section $2$. Then $\bar{\Delta}_Q^\circ$ can be extended to an $N_E(Q)$-stable bijective isometry $$\bar{\Delta}_Q:\mathcal{CF}_\mathcal{K}(\bar{C}_N(Q),\bar{e}_{\theta})\cong
\mathcal{CF}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$$ such that $\bar{\Delta}_Q(\mathcal{L}_\mathcal{K}(\bar{C}_N(Q),\bar{e}_\theta))=
\mathcal{L}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$.
By the structure of $E$ and $P$, $C_E(Q)$ has only three possibilities: $1$, $E$ and $E_1$ or $E_2$. So we will divided the proof into $3$ cases.
Assume that $C_E(Q)=1$.
Then the blocks $e_\theta$ of $C_N(Q)$ and $b_Q$ of $C_G(Q)$ are nilpotent. By the same argument as in [@PU 4.4], $\bar{\Delta}_Q^\circ$ can be extended to an $N_E(Q)$-stable bijective isometry $\bar{\Delta}_Q$.
Assume that $C_E(Q)=E$.
Then $Q$ has to be trivial subgroup of $P$ and $N_E(Q)=E$. So $\bar{C}_N(Q)=N$ and $\bar{C}_G(Q)=G$ and we have a bijective isometry $$\bar{\Delta}^\circ:\mathcal{CF}_\mathcal{K}^\circ(N,e_\theta)\longrightarrow
\mathcal{CF}_\mathcal{K}^\circ(G,b)$$ such that $\bar{\Delta}^\circ(\mathcal{L}_\mathcal{K}^\circ(N,e_\theta))=
\mathcal{L}_\mathcal{K}^\circ(G,b)$.
The following technique we adopt to extend $\bar{\Delta}^\circ$ is essentially due to Kessar and Linckelmann (see [@KL Theorem 4.1]).
By Lemma \[irreducible character of c\], we have the following disjoint union $$\mathrm{Irr}_\mathcal{K}(N,e_\theta)=\Lambda_1\cup\Lambda_2,$$ where $\Lambda_1$ consists of irreducible ordinary characters of dimension $l$ and $\Lambda_2$ consists of irreducible ordinary characters of dimension $l^2$. Hence, $|\Lambda_1|=p^n+p^m-1$ and $|\Lambda_2|=\frac{p^n-1}{l}\cdot\frac{p^m-1}{l}$. We can assume that $n\geq 2$. Then $|\Lambda_1|>2$ and $|\Lambda_2|>2$. Choose an element $\psi_i\in\Lambda_i$ and set $\Lambda_i^\prime=\Lambda_i-\{\psi_i\}$ for $i=1,2$. Since $l_{N}(e_\theta)=1$, it is easy to see $$\mathcal{B}=\{\psi_1-\psi_1^\prime\,|\,\psi_1^\prime\in\Lambda_1^\prime\}
\cup\{\psi_2-\psi_2^\prime\,|\,\psi_2^\prime\in\Lambda_2^\prime\}
\cup\{\psi_2-l\psi_1\}$$ is a $\mathbb{Z}$-basis of $\mathcal{L}_\mathcal{K}^\circ(N,e_\theta)$. Since $p$ is odd, $|\Lambda_i^\prime|\geq 3$ for $i=1,2$. So by the same argument in [@PU 4.4], for any $i=1,2$, there exists a subset $\Omega_i=\{\chi_{\psi_i},\chi_{\psi_i^\prime}\,|\,
\psi_i^\prime\in\Lambda_i^\prime\}$ of $\mathrm{Irr}_\mathcal{K}(G,b)$ and $\delta_i\in\{\pm 1\}$ such that $\bar{\Delta}^\circ(\psi_i-\psi_i^\prime)=
\delta_i(\chi_{\psi_i}-\chi_{\psi_i^\prime})$. Since $\langle\psi_1-\psi_1^\prime,
\psi_2-\psi_2^\prime\rangle=0$ for any $\psi_1^\prime\in\Lambda_1^\prime$ and $\psi_2^\prime\in\Lambda_2^\prime$, $\{\chi_{\psi_1},\chi_{\psi_1^\prime}\,|\,
\psi_1^\prime\in\Lambda_1^\prime\}$ and $\{\chi_{\psi_2},\chi_{\psi_2^\prime}\,|\,
\psi_2^\prime\in\Lambda_2^\prime\}$ have trivial intersection. Denote $\psi_2-l\psi_1$ by $\mu$. Then $\langle\mu,
\psi_1-\psi_1^\prime\rangle=-l$ for all $\psi_1^\prime\in\Lambda_1^\prime$. Thus $$\label{equation E}
\begin{array}{ll}
\bar{\Delta}^\circ(\mu)=
\delta_1(a-l)\chi_{\psi_1}+
\delta_1a\sum\limits_{\psi_1^\prime\in\Lambda_1^\prime}\chi_{\psi_1^\prime}+
\Xi
\end{array}$$ for some integer $a$ and some element $\Xi\in\mathcal{L}_\mathcal{K}(N,e_\theta)$ not involving any of elements in $\Omega_1$. Since $\langle\mu,
\psi_2-\psi_2^\prime\rangle=1$ and $\bar{\Delta}^\circ(\psi_2-\psi_2^\prime)=
\delta_2(\chi_{\psi_2}-\chi_{\psi_2^\prime})$, $\Xi$ must involve one of the two characters occuring in $\bar{\Delta}^\circ(\psi_2-\psi_2^\prime)$ for any $\psi_2^\prime\in\Lambda_2^\prime$. Taking norms on both sides in equation (\[equation E\]), we have $$\label{inequation norm}
\begin{array}{ll}
&1+l^2\geq(a-l)^2+(p^n+p^m-2)a^2=
(p^n+p^m-1)a^2-2la+l^2\\
\Longleftrightarrow
&1\geq(p^n+p^m-1)a^2-2la
\end{array}$$
Suppose that $a\leq0$. Since $a$ is integer and $p^n+p^m-1,l$ are positive integers, $a$ has to be $0$.
Suppose that $a>0$. Since $p^n+p^m-1>2l$, $(p^n+p^m-1)a^2-2la>(p^n+p^m-1)(a^2-a)$. This forces $a=1$. Hence, $a=0$ or $1$. Notice that $\Xi\neq 0$. This implies (\[inequation norm\]) is a proper inequality. So $a$ must be $0$. Then equation (\[equation E\]) becomes $$\bar{\Delta}^\circ(\mu)=-\delta_1l\chi_{\psi_1}+\Xi.$$ Comparing norms, we have $\langle\Xi,\Xi\rangle=1$.
For any $\psi_2^\prime,\psi_2^{\prime\prime}\in\Lambda_2^\prime$, $$\langle\bar{\Delta}^\circ(\mu),
\delta_2(\chi_{\psi_2}-\chi_{\psi_2^\prime})\rangle=
\langle\mu,
\psi_2-\psi_2^\prime\rangle=1$$ and $$\langle\bar{\Delta}^\circ(\mu),
\delta_2(\chi_{\psi_2^{\prime}}-\chi_{\psi_2^{\prime\prime}})\rangle=
\langle\mu,
\psi_2^\prime-\psi_2^{\prime\prime}\rangle=0.$$ Then $\Xi=\delta_2\chi_{\psi_2}$. But $\bar{\Delta}^\circ(\mu)(1)=0$. This forces $\delta_1=\delta_2$. Since $\mathcal{B}$ is a $\mathbb{Z}$-basis of $\mathcal{L}_\mathcal{K}^\circ(N,e_\theta)$, $\mathrm{Irr}_\mathcal{K}(G,b)=\Omega_1\cup\Omega_2$. Hence, we get a bijective isometry $\bar{\Delta}$ from $\mathcal{L}_\mathcal{K}(N,e_\theta)$ to $\mathcal{L}_\mathcal{K}(G,b)$ mapping $\psi_i$ and $\psi_i^\prime$ to $\chi_{\psi_i}$ and $\chi_{\psi_i^\prime}$ respectively, where $i=1,2$. In particular, $l_G(b)=l_N(e_\theta)=1$. Clearly, it is an extension of $\bar{\Delta}^\circ$. Since $\bar{\Delta}^\circ$ is $E$-stable and $l_G(b)=l_N(e_\theta)=1$, $\bar{\Delta}$ is also $E$-stable.
Assume that $C_E(Q)=E_i$ for some $i=1,2$.
We can assume that $C_E(Q)=E_1$ and then $1\neq Q\leq P_2$ and $N_E(Q)=E$. It suffices to prove that $\bar{\Delta}_Q^\circ$ can extend to an $E_2$-stable bijective isometry $\bar{\Delta}_Q:\mathcal{L}_\mathcal{K}(\bar{C}_N(Q),\bar{e}_\theta)\longrightarrow
\mathcal{L}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$.
By [@W14 Theorem 1], $|\mathrm{Irr}_\mathcal{K}(\bar{C}_N(Q),\bar{e}_\theta)|=
|\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)|$ and $l_{\bar{C}_N(Q)}(\bar{e}_\theta)=l_{\bar{C}_G(Q)}(\bar{b}_Q)$ since the block $\bar{b}_Q$ of $\bar{C}_G(Q)$ has a cyclic hyperfocal subgroup. It is clear $C_N(Q)=(P_1\rtimes\tilde{E}_1)\times P_2$ and $C_N(Q)\unlhd N$, where $\tilde{E}_1$ is the preimage of $E_1$ in $\tilde{E}$. Hence, $E_1$ is the inertial quotient of the block $e_\theta$ of $C_N(Q)$ and $P_1$ is a hyperfocal subgroup with respect to $E_1$. By [@W14 Theorem 1], $l_{C_N(Q)}(e_\theta)=l$. We will claim that $N$ acts transitively on $\mathrm{IBr}(C_N(Q),e_\theta)$. Indeed, this holds because $l_N(e_\theta)=1$ by the assumption and $N/C_N(Q)\cong E_2$ is a cyclic group of order $l$.
Denote by $\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta$ the subset of $\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)$ consisting of characters covering $\theta$. Then $|\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta|=l$ and we set $\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta=\{\tau_i\,|\, i=1,2,\cdots,l\}$, which is transitively acted by $N$. Hence, we can write $\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta$ as $\{\tau^a\,|\,a\in E_2\}$ for any $\tau\in\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta$. By Clifford theorem, we have $\mathrm{Res}_Z^{\tilde{E}_1}(\tau_i)=\theta$ for any $i$ and $\mathrm{Ind}_Z^{\tilde{E}_1}(\theta)
=\sum\limits_{i=1}^l\tau_i$. Let $M$ be a representative of $\tilde{E}_1$-orbit of $\mathrm{Irr}_\mathcal{K}(P_1)-\{1_{P_1}\}$, where $1_{P_1}$ is the trivial character of $P_1$. Then $$\mathrm{Irr}_\mathcal{K}(\bar{C}_N(Q),\bar{e}_\theta)=
\{\tau_i\bar{\zeta}_j\,|\, \bar{\zeta}_j\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2),
i=1,2,\cdots,l\}\cup
\{\mathrm{Ind}_{P_1\times Z}^{P_1\rtimes\tilde{E}_1}(\xi\theta)\bar{\zeta}_j\,|\,
\xi\in M,\bar{\zeta}_j\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}.$$ We will write $\mathrm{Ind}_{P_1\times Z}^{P_1\rtimes\tilde{E}_1}(\xi\theta)$ and $\bar{\chi}\cdot 1_{\bar{P}_2}$ as $\mathrm{Ind}(\xi)$ and $\bar{\chi}$ respectively for simplicity. Here, $\bar{\chi}$ is an element of $\mathcal{CF}_\mathcal{K}(\overline{P_1\rtimes\tilde{E}_1})$. Clearly, $\mathrm{Ind}(\xi)$ is $N$ and $E_2$-stable for any $\xi\in M$. Similar to the argument of [@W05 Case 2], $$\{(\sum\limits_{i=1}^l\tau_i-\mathrm{Ind}(\xi))\bar{\zeta}\,|\,\xi\in M,
\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}\cup
\{\tau_i-\tau_i\bar{\zeta}\,|\,i=1,2,\cdots,l,
1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}$$ is a $\mathbb{Z}$-basis of $\mathcal{L}_\mathcal{K}^\circ(\bar{C}_N(Q),\bar{e}_\theta)$.
Assume that $\bar{P}_2=1$, i.e., $Q=P_2$.
Set $H=N_G(Q,b_Q)$. Then $H=C_G(Q)N_G(P,b_P)$ and $b_Q$ is still a block of $H$. Let $d$ be the Brauer correspondent of the block $b_Q$ of $H$ in $N_H(P)$. Then $l_{N_H(P)}(d)=1$ by the assumption. We claim that $l_H(b_Q)=1$.
Indeed, considering the canonical map from $\mathcal{O}H$ to $\mathcal{O}(H/Q)$, denote by $\bar{X}$ the image of $X$ under this canonical map for any subset $X$ of $\mathcal{O}H$. Then $\bar{b}_Q$ is still a block of $\bar{C}_G(Q)$ and $\bar{H}/\bar{C}_G(Q)$ is a cyclic group of order $l$. By [@KR Lemma 3.5], $\mathrm{Br}_{\bar{P}}(\bar{b}_Q)=\overline{\mathrm{Br}_P(b_Q)}$. Since $l_{N_H(P)}(d)=1$, $\bar{d}$ is still a block of $\bar{N}_H(P)$. Therefore, $\mathrm{Br}_{\bar{P}}(\bar{b}_Q)=\bar{d}$ is a block of $\bar{N}_H(P)$. Suppose that the blocks of $\bar{H}$ covering the block $\bar{b}_Q$ of $\bar{C}_G(Q)$ have the same defect group $\bar{P}$. Then $\bar{b}_Q$ is a block of $\bar{H}$ since $\mathrm{Br}_{\bar{P}}(\bar{b}_Q)$ is a block of $\bar{N}_H(P)$ and $N_{\bar{H}}(\bar{P})=\bar{N}_H(P)$. Hence, it has a defect group $\bar{P}$ which is cyclic by our assumption. In particular, we have $l_H(b_Q)=l_{\bar{H}}(\bar{b}_Q)=l_{N_H(P)}(d)=1$ since $N_{\bar{H}}(\bar{P})=\bar{N}_H(P)$. Consequently, the argument follows from the lemma below.
[*[**Lemma 3.4**]{} Let $L$ be a normal subgroup of $K$ such that $K/L$ is a cyclic $p^\prime$-group. Let $i$ be a $K$-stable block of $L$ with defect group $D$. For any block $e$ of $K$ covering $i$, $e$ has defect group $D$.*]{}
[*Proof.*]{} We will prove it by induction on $K/L$. Let $M\leq K$ such that $M$ contains $L$ and $|M/L|$ is a prime. Then $M\trianglelefteq K$ and $K/M$ is still a cyclic $p^\prime$-group. Denote by $M[i]$ the subgroup of $M$ consisting of elements acting on $\mathcal{O}Li$ as inner automorphisms. Therefore, $M[i]=M$ or $L$. Let $f$ be a block of $M$ covered by $e$. So $f$ covers the block $i$ of $L$. If $M[i]=M$, then $\mathcal{O}Mf$ and $\mathcal{O}Li$ are source algebra equivalent by [@K90 Theorem 7]. In particular, the block $f$ has defect group $D$. If $M[b]=L$, then $f=i$ by [@D Theorem 3.5] and certainly they have the same defect group. In conclusion, $D$ is a defect group of the block $f$. Let $K_f$ be the stabilizer of $f$ in $K$. Then blocks of $K_f$ covering $f$ have defect group $D$ by induction. So is $e$.
Moreover, we claim that there is a regular $E_2$-orbit of $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$, namely, $H$ acts transitively on it.
Indeed, since the block $\bar{b}_Q$ of $\bar{H}$ has a cyclic defect group, it must be nilpotent. By [@P11 Theorem 3.13], the block $\bar{b}_Q$ of $\bar{C}_G(Q)$ is basic Morita equivalent to its Brauer correspondent. Note that the block $\bar{b}_Q$ of $\bar{C}_G(Q)$ is not nilpotent since $l>1$. This implies that every irreducible Brauer character of the block $\bar{b}_Q$ of $\bar{C}_G(Q)$ can be uniquely lifted to an irreducible ordinary character by the theory of cyclic blocks.
On the other hand, since $l_{C_G(Q)}(b_Q)=l$ and $l_H(b_Q)=1$ and $H/C_G(Q)\cong E_2$ has order $l$, $H$ acts transitively on $\mathrm{IBr}(C_G(Q),b_Q)$. Combining this with the argument above, there exits a regular $H$-orbit of $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$. We are done.
Assume that $|M|=1$.
Then $\mathrm{rank}_\mathcal{O}
(\mathcal{L}_\mathcal{K}^\circ(\bar{C}_N(Q),\bar{e}_\theta))=1$ and $\mathcal{L}_\mathcal{K}^\circ(\bar{C}_N(Q),\bar{e}_\theta)=
\mathbb{Z}(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)$. Since there is a regular $E_2$-orbit of $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$, $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)=\{\chi_0\}\cap
\{\chi_1,\chi_2,\cdots,\chi_l\}$ such that $\chi_0$ is $E_2$-stable and $E_2$ acts regularly on $\{\chi_1,\chi_2,\cdots,\chi_l\}$. Then we have $$\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)=
\delta_0\chi_0-\sum\limits_{i=1}^l\delta_i\chi_i$$ for some $\delta_0,\delta_i\in\{\pm1\},i=1,2,\cdots,l$. Since $\bar{\Delta}_Q^\circ$ is $E_2$-stable, we have $\delta_1=\delta_2=\cdots=\delta_l=\delta_0$. If we write $\{\tau_1,\tau_2,\cdots,\tau_l\}$ and $\{\chi_1,\chi_2,\cdots,\chi_l\}$ as $\{\tau^a\,|\,a\in E_2\}$ and $\{\chi^a\,|\,a\in E_2\}$ respectively, then we can define a bijective isometry as below $$\bar{\Delta}_Q:\mathcal{L}_\mathcal{K}(\bar{C}_N(Q),\bar{e}_\theta)\longrightarrow
\mathcal{L}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$$ $$\mathrm{Ind}(\xi)\mapsto\delta_0\chi_0$$ $$\tau^a\mapsto\delta_0\chi^a.$$ It is evident that it is an extension of $\bar{\Delta}_Q^\circ$ and $E_2$-stable. We are done for this case.
Assume that $|M|\geq2$.
Then there are at least two different $\xi_1,\xi_2\in M$. So $\mathrm{Ind}(\xi_1)-\mathrm{Ind}(\xi_2)\in
\mathcal{L}_\mathcal{K}^\circ(\bar{C}_N(Q),\bar{e}_\theta)$ and $\langle\mathrm{Ind}(\xi_1)-\mathrm{Ind}(\xi_2),
\mathrm{Ind}(\xi_1)-\mathrm{Ind}(\xi_2)\rangle=2$. Then there exist $\chi_1\neq\chi_2\in
\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$ such that $$\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi_1)-\mathrm{Ind}(\xi_2))=
\delta(\chi_1-\chi_2)$$ for some $\delta\in\{\pm1\}$. Since $\bar{\Delta}_Q^\circ$ is $E_2$-stable, we have ${^a}(\delta\chi_1-\delta\chi_2)=\delta(\chi_1-\chi_2)$ for any $a\in E_2$. This means that $\chi_1$ and $\chi_2$ are both $E_2$-stable.
If there is a $\xi_3\in M$ different from $\xi_1$ and $\xi_2$, then there is a $\chi_3\in\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$ different from $\chi_1$ and $\chi_2$ such that $$\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi_1)-\mathrm{Ind}(\xi_3))=
\delta\chi_1-\delta\chi_3~\mathrm{or}~
-\delta\chi_2+\delta\chi_3$$ and $\chi_3$ is $E_2$-stable; then we may choose the notation in such a way that $$\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi_1)-\mathrm{Ind}(\xi_2))=
\delta(\chi_1-\chi_2)
~\mathrm{and}~
\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi_1)-\mathrm{Ind}(\xi_3))=
\delta(\chi_1-\chi_3)$$ for some $E_2$-stable elements $\chi_1,\chi_2,\chi_3$ of $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q).$
If $|M|\geq 4$, then for any $\xi\in M-\{\xi_1,\xi_2,\xi_3\}$, there is a unique $\chi\in\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)-
\{\chi_1,\chi_2,\chi_3\}$ such that $$\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi_1)-\mathrm{Ind}(\xi))=
\delta(\chi_1-\chi)$$ and $\chi$ is $E_2$-stable.
In conclusion, we have an injective isometry $$\Phi:\mathbb{Z}\{\mathrm{Ind}(\xi)\,|\,\xi\in M\}\longrightarrow
\mathcal{L}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$$ mapping $\mathrm{Ind}(\xi)$ to $\delta\chi_\xi$ such that $$\Phi(\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi^\prime))=
\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi^\prime))$$ and $\chi_\xi$ is $E_2$-stable for any $\xi,\xi^\prime\in M$.
Denote $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)-
\{\chi_\xi\,|\,\xi\in M\}$ by $\Omega$. Then $|\Omega|=l$ and $E_2$ acts on $\Omega$. Since there is a regular $E_2$-orbit of $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$, $E_2$ acts regularly on $\Omega$. This means that $\Omega$ can be represented as $\{\chi^a\,|\,a\in E_2\}$ for some $\chi\in\Omega$.
Now we fix an element $\xi$ of $M$. Suppose that $\chi$ does not get involved in $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)$. Then there is $\xi^\prime\in M$ such that $\langle\chi,\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi^\prime)-
\sum\limits_{i=1}^l\tau_i)\rangle\neq 0$ since $\{\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i\,|\,
\xi\in M\}$ is a $\mathbb{Z}$-basis of $\mathcal{L}_\mathcal{K}^\circ
(\bar{C}_N(Q),\bar{e}_\theta)$. Hence, $\chi$ has to get involved in $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)-
\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi^\prime)-\sum\limits_{i=1}^l\tau_i)$ which is $\delta(\chi_\xi-\chi_{\xi^\prime})$. This is impossible. So $\chi$ must get involved in $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)$ for any $\xi\in M$. Since $\bar{\Delta}_Q^\circ$ and $\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i$ are $E_2$-stable, $\chi^a$ has to get involved in $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)$ for any $a\in E_2$ and $\xi\in M$. Since $\langle\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i,
\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i\rangle=1+l$ and $\langle\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i,
\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi^\prime)\rangle=1$, $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)=
\delta\chi_\xi-\sum\limits_{a\in E_2}\delta_a\chi^a~\mathrm{or}~
-\delta\chi_{\xi^\prime}-\sum\limits_{a\in E_2}\delta_a\chi^a$, where $\delta_a\in\{\pm1\}$ for any $a\in E_2$. Note that the last situation can happen if and only if $|M|=2$. By switching $\chi_\xi$ and $\chi_{\xi^\prime}$ if necessary, we can assume that $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)=
\delta\chi_\xi-\sum\limits_{a\in E_2}\delta_a\chi^a$. Since $\bar{\Delta}_Q^\circ$ is $E_2$-stable and $E_2$ acts regularly on $\Omega$, $\delta_a$ is equal to $\delta$ for any $a\in E_2$. Then we can define an $E_2$-stable bijective isometry as follows $$\bar{\Delta}_Q:\mathcal{L}_\mathcal{K}(\bar{C}_N(Q),\bar{e}_\theta)\longrightarrow
\mathcal{L}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$$ $$\mathrm{Ind}(\xi)\mapsto\delta\chi_\xi$$ $$\tau^a\mapsto\delta\chi^a.$$ It is clear that $\bar{\Delta}_Q$ is an extension of $\bar{\Delta}_Q^\circ$.
$\bar{P}_2>1$, namely, $Q$ is a non-trivial proper subgroup of $P_2$.
Then $\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta}\in
\mathcal{L}_\mathcal{K}^\circ(\bar{C}_N(Q),\bar{e}_\theta)$ for any $\xi\in M$ and $1_{\bar{P}_2}\neq\bar{\zeta}\in
\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$.
Now we fix an element $\xi\in M$. Since $p$ is odd, $|\bar{P}_2|\geq 3$. Then there are at least two elements $\bar{\zeta}$ and $\bar{\zeta}^\prime$ of $\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$ different from $1_{\bar{P}_2}$. With the same argument in the first three paragraphs in Case 3.1.2, we can get a subset $\{\chi_\xi,\chi_{\bar{\zeta}}\,|\,
1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}$ of $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$ such that $$\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta})=
\delta(\chi_\xi-\chi_{\bar{\zeta}})$$ for any $1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$, where $\delta\in\{\pm1\}$.
Given any $1\neq a\in E_2$ and $1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$, ${^a}(\mathrm{Ind}(\xi)\bar{\zeta})=\mathrm{Ind}(\xi)({^a}\bar{\zeta})$. Since $\bar{\Delta}_Q^\circ$ is $E_2$-stable, this means ${^a}\chi_\xi-{^a}\chi_{\bar{\zeta}}=\chi_\xi-\chi_{{^a}\bar{\zeta}}$. Hence, we have $\chi_\xi$ is $E_2$-stable and ${^a}\chi_{\bar{\zeta}}=\chi_{{^a}\bar{\zeta}}$. On the other hand, $$(\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta})\bar{\zeta}=
(\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta}^2)-
(\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta}).$$ Since $\bar{\Delta}_Q^\circ$ is compatible with $\ast$-structure, using $\bar{\Delta}_Q^\circ$ on both sides in the above equality, we can get $$\delta(\chi_\xi-\chi_{\bar{\zeta}})\ast\bar{\zeta}=
\delta(\chi_{\bar{\zeta}}-\chi_{\bar{\zeta}^2}).$$ Therefore, $\chi_{\bar{\zeta}}=\chi_\xi\ast\bar{\zeta}$ for any $1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$.
Suppose that there is another element $\xi^\prime$ of $M$ different from $\xi$. Similarly, we can get a subset $\{\chi_{\xi^\prime}\ast\bar{\zeta}\,|\,
\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}$ of $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$ such that $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi^\prime)-
\mathrm{Ind}(\xi^\prime)\bar{\zeta})=
\delta^\prime(\chi_{\xi^\prime}-\chi_{\xi^\prime}\ast\bar{\zeta})$ for any $1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$ and $\chi_{\xi^\prime}$ is $E_2$-stable, where $\delta^\prime\in\{\pm 1\}$. Assume that $\{\chi_\xi\ast\bar{\zeta}\,|\,
\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}\cap
\{\chi_{\xi^\prime}\ast\bar{\zeta}\,|\,
\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}\neq\emptyset$. Then there is $\bar{\zeta}_0\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$ such that $\chi_\xi=\chi_{\xi^\prime}\ast\bar{\zeta}$. If $\bar{\zeta}_0=1_{\bar{P}_2}$, then $\chi_\xi=\chi_{\xi^\prime}$. This implies that $\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta}=
\pm(\mathrm{Ind}(\xi^\prime)-\mathrm{Ind}(\xi^\prime)\bar{\zeta})$ for any $1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$. This is impossible. Then $\bar{\zeta}_0$ is non-trivial. But it implies that $\chi_{\xi^\prime}\ast\bar{\zeta}_0^2=\chi_{\xi^\prime}$ since $\langle\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta}_0,
\mathrm{Ind}(\xi^\prime)-\mathrm{Ind}(\xi^\prime)\bar{\zeta}_0\rangle=0$. It is well-known that $\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\backslash\{1_{\bar{P}_2}\}$ acts freely on irreducible ordinary characters of height zero in the block $\bar{b}_Q$ of $\bar{C}_G(Q)$ (see [@R §1]). Hence, $\bar{\zeta}_0^2=1_{\bar{P}_2}$ since the defect group of the block $\bar{b}_Q$ of $\bar{C}_G(Q)$ is cyclic. But it is impossible because $p$ is odd. Then $$\{\chi_\xi\ast\bar{\zeta}\,|\,
\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}\cap
\{\chi_{\xi^\prime}\ast\bar{\zeta}\,|\,
\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}=\emptyset$$ for any different $\xi,\xi^\prime\in M$. It is clear that $\chi_\xi\ast\bar{\zeta}$ is an irreducible ordinary character in the block $\bar{b}_Q$ of $\bar{C}_G(Q)$ by [@BP Corollary]. Then we get an injective isometry $$\Psi:\mathbb{Z}\{\mathrm{Ind}(\xi)\bar{\zeta}\,|\,
\xi\in M,\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}
\longrightarrow
\mathcal{L}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$$ mapping $\mathrm{Ind}(\xi)\bar{\zeta}$ to $\delta_\xi(\chi_\xi\ast\bar{\zeta})$ such that $\Psi(\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta})=
\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta})$ and $\chi_\xi$ is $E_2$-stable for any $\xi\in M$ and $1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$, where $\delta_\xi\in\{\pm 1\}$.
At the same time, $\tau-\tau\bar{\zeta}\in
\mathcal{L}_\mathcal{K}^\circ(\bar{C}_N(Q),\bar{e}_\theta)$ for any $\tau\in\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta$ and $1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$. Take an element $\tau$ of $\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta$. With the same arguments as above, we can get an element $\chi_\tau$ of $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$ and $\delta_\tau\in\{\pm1\}$ such that $\bar{\Delta}_Q^\circ(\tau-\tau\bar{\zeta})=
\delta_\tau(\chi_\tau-\chi_\tau\ast\bar{\zeta})$ for any $1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$. Choosing any $1\neq a\in E_2$ and $1_{\bar{P}_2}\neq\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$, since $\bar{\Delta}_Q^\circ$ is $E_2$-stable, we have $$\delta_{{^a}\tau}(\chi_{{^a}\tau}-\chi_{{^a}\tau}\ast{^a}\bar{\zeta})=
\bar{\Delta}_Q^\circ({^a}\tau-{^a}\tau({^a}\bar{\zeta}))=
{^a}(\bar{\Delta}_Q^\circ(\tau-\tau\bar{\zeta}))=
\delta_\tau({^a}\chi_\tau-{^a}\chi_\tau\ast{^a}\bar{\zeta}).$$ Then $\chi_{{^a}\tau}={^a}\tau$ or $\chi_{{^a}\tau}={^a}\chi_\tau\ast{^a}\bar{\zeta}$. If $\chi_{{^a}\tau}={^a}\chi_\tau\ast{^a}\bar{\zeta}$, then ${^a}\chi_\tau=\chi_{{^a}\tau}\ast{^a}\bar{\zeta}$. Therefore, $\chi_{{^a}\tau}=\chi_{{^a}\tau}\ast{^a}(\bar{\zeta}^2)$, which is impossible. Hence, $\chi_{{^a}\tau}={^a}\chi_\tau$ and $\delta_{{^a}\tau}=\delta_\tau$ for any $a\in E_2$ since $E_2$ acts transitively on $\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta$. And we denote $\delta_\tau$ by $\delta$. By the facts that $\langle\tau-\tau\bar{\zeta},
\tau^\prime-\tau^\prime\bar{\zeta}^\prime\rangle=0$ and $\langle\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta},
\tau-\tau\bar{\zeta}^\prime\rangle=0$ for any $\tau\neq\tau^\prime\in\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta$ and $\xi\in M$ and $\bar{\zeta},\bar{\zeta}^\prime\in
\mathrm{Irr}_\mathcal{K}(\bar{P}_2)-\{1_{\bar{P}_2}\}$, we can get $$\{\chi_\tau\ast\bar{\zeta}\,|\,
\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}\cap
\{\chi_{\tau^\prime}\ast\bar{\zeta}\,|\,
\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}=\emptyset$$ and $$\{\chi_\xi\ast\bar{\zeta}\,|\,
\xi\in M, \bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}\cap
\{\chi_\tau\ast\bar{\zeta}\,|\,
\tau\in\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta,
\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)\}=\emptyset.$$ Hence, we have a well-defined $E_2$-stable bijective isometry as below $$\bar{\Delta}_Q:\mathcal{L}_\mathcal{K}(\bar{C}_N(Q),\bar{e}_\theta)\longrightarrow
\mathcal{L}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)$$ $$\mathrm{Ind}(\xi)\bar{\zeta}\mapsto\delta_\xi\chi_\xi\ast\bar{\zeta}$$ $${^a}\tau\bar{\zeta}\mapsto\delta{^a}\chi_\tau\ast\bar{\zeta}.$$ It suffices to show that $\bar{\Delta}_Q$ is an extension of $\bar{\Delta}_Q^\circ$, namely, $$\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)=
\delta_\xi\chi_\xi-\delta\sum\limits_{i=1}^l\chi_{\tau_i}$$ for any $\xi\in M$.
Choose an element $\xi$ of $M$. Since $\langle\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i,
\tau-\tau\bar{\zeta}\rangle=-1$, then at least $\chi_\tau$ and $\chi_\tau\ast\bar{\zeta}$ must get involved in $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)$ for any $\tau\in\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta$ and $\bar{\zeta}\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)-\{1_{\bar{P}_2}\}$.
Keep the notation as above. Suppose that there are $\tau$ and $\bar{\zeta}$ such that $\chi_\tau\ast\bar{\zeta}$ gets involved in $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)$. Since $\langle\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i,
\tau\bar{\zeta}-\tau\bar{\zeta}^\prime\rangle=0$ for any $\bar{\zeta}^\prime\in\mathrm{Irr}_\mathcal{K}(\bar{P}_2)$ different from $\bar{\zeta}$ and $1_{\bar{P}_2}$, $\chi_\tau\ast\bar{\zeta}$ must get involved in $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)$ for any $\bar{\zeta}$. At the same time, since $\bar{\Delta}_Q^\circ$ is $E_2$-stable and $\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i$ is $E_2$-stable, we have ${^a}(\chi_\tau\ast\bar{\zeta})$ must get involved in $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)$ for any $a\in E_2$ and $\bar{\zeta}$. Then there are at least $l\cdot(|\bar{P}_2|-1)$ different irreducible characters involved in $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)$. This is impossible since $\langle\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i,
\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i\rangle=1+l$ and $|\bar{P}_2|-1\geq 2$ and $l>1$.
So for any $\xi\in M$, $\bar{\Delta}_Q^\circ(\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i)=
a_\chi\chi-\delta\sum\limits_\tau\chi_\tau$. Here, $a_\chi\in\{\pm 1\}$ and $\chi$ is an element of $\mathrm{Irr}_\mathcal{K}(\bar{C}_G(Q),\bar{b}_Q)-
\{\chi_\tau\,|\,
\tau\in\mathrm{Irr}_\mathcal{K}(\tilde{E}_1)_\theta\}$. Since $\langle\mathrm{Ind}(\xi)-\sum\limits_{i=1}^l\tau_i,
\mathrm{Ind}(\xi)-\mathrm{Ind}(\xi)\bar{\zeta}\rangle=1$ for any $\bar{\zeta}\neq 1_{\bar{P}_2}$, we have $a_\chi=\delta_\xi$ and $\chi=\chi_\xi$. We are done.
Then the proof of Theorem \[MT\] will follow by Theorem \[MT’\] and [@PU 3.4.2].
[99]{} M. Brou$\acute{\mathrm{e}}$, Isom$\acute{\rm e}$tries parfaites, types de blocs, cat$\acute{\rm e}$gories d$\acute{\rm e}$riv$\acute{\rm e}$es, Ast$\acute{\rm e}$risque, **181-182** (1990), 61-92.
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A. Watanabe, On perfect isometries for blocks with abelian defect groups and cyclic hyperfocal subgroups, Kumamoto J. Math. **18** (2005), 85-92.
A. Watanabe, The number of irreducible Brauer characters in a $p$-block of a finite group with cyclic hyperfocal subgroup, J. Algebra **416** (2014), 167-183.
C. Wu, K. Zhang, Y. Zhou, Blocks with defect group $\mathbb{Z}_{2^n}\times\mathbb{Z}_{2^n}\times\mathbb{Z}_{2^m}$, J. Algebra **510** (2018), 469-498.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A study is presented to extend the measurements of photon-photon scattering in ultra-peripheral Pb-Pb collisions at the LHC into the mass region of the pseudoscalar resonances $\eta$ and $\eta''$. The elementary photon-photon scattering cross section is presented. The cross section for photon-photon scattering in Pb-Pb is derived by convoluting the elementary photon-photon cross section with the Pb-Pb photon luminosity. The main background to two-photon final states, arising from double $\pi^{0}$ production with two of the four decay photons escaping detection, is examined, and possible kinematical conditions are discussed to optimize the signal-to-background ratio for such measurements at mid-rapidity.'
address:
- 'Phys. Inst., Heidelberg'
- 'Inst. Nucl. Phys., PAN Krak$\acute{o}$w, Univ. of Rzesz$\acute{o}$w'
author:
- |
Rainer Schicker\
[Mariola Kłusek-Gawenda, Antoni Szczurek]{}
title: 'Photon-photon scattering in the resonance region at midrapidity at the LHC. [^1]'
---
Introduction
============
Classical electrodynamics is epitomised by Maxwell’s equations $$\partial_{\alpha}F^{\alpha\beta} = \frac{4\pi}{c}J^{\beta},\hspace{1.cm}
\partial_{\alpha}{\cal F}^{\alpha\beta} = 0 \, .
\label{Eq:Maxwell}$$
The Maxwell equations in vacuum are linear in the electromagnetic fields [**E**]{} and [**B**]{}. Two electromagnetic waves will pass through each other without scattering. The superposition principle conveniently expresses the non-interaction of electromagnetic fields at the classical level. The electromagnetic field energy carried by the electron, however, poses a conceptual challenge at the classical level. Attempts to circumvent the infinite Coulomb energy of a point charge resulted in Born-Infeld electrodynamics which affirms the linearity of Maxwell’s equations down to some length scale intrinsic to the electron, and introduces non-linear equations for smaller lengths scales [@Born]. The polarisation of the vacuum in view of Dirac’s positron theory led to the Euler-Kockel-Heisenberg Lagrangian which modifies the classical Maxwell’s equation in vacuum by leading non-linear terms [@Heisenberg]. The advent of accelerating heavy-ions at the LHC has opened up the possibility of measuring the photon-photon scattering cross section due to the large associated photon luminosity of the heavy-ion beams. Evidence of such events have been reported by the ATLAS and CMS collaborations at the LHC [@ATLAS; @CMS]. These measurements are, however, restricted to photon-photon invariant masses W$_{\gamma\gamma} >$ 5 and 6 GeV for the CMS and ATLAS data, respectively. The purpose of the analysis presented here is to study the feasibility of measuring photon-photon scattering in the range 0.4$<$W$_{\gamma\gamma}<$5 GeV. As a first step, we analyse the corresponding cross section in this range, and examine the background which is dominated by $\pi^{0}$ pair production with only two of the four decay photons being within the detector acceptance.
The elementary photon-photon scattering cross section
=====================================================
Different mechanisms contribute to the elementary $\gamma\gamma\rightarrow\gamma\gamma$ scattering.
![Mechanisms of $\gamma\gamma\rightarrow\gamma\gamma$ scattering (Fig. taken from Ref. [@Sz1]).[]{data-label="Fig:gg_elem"}](gg_gg_box.pdf "fig:"){width="3.2cm"} ![Mechanisms of $\gamma\gamma\rightarrow\gamma\gamma$ scattering (Fig. taken from Ref. [@Sz1]).[]{data-label="Fig:gg_elem"}](gg_gg_gluon.pdf "fig:"){width="4.4cm"} ![Mechanisms of $\gamma\gamma\rightarrow\gamma\gamma$ scattering (Fig. taken from Ref. [@Sz1]).[]{data-label="Fig:gg_elem"}](gg_gg_IPIR.pdf "fig:"){width="4.4cm"}
In Fig. \[Fig:gg\_elem\], the different mechanisms of $\gamma\gamma$ scattering are presented. On the left, the loop diagram for fermions, leptons and quarks, is shown. In the center, a QCD correction is displayed corresponding to a three-loop mechanism. On the right, the analogous process as expressed in the vector dominance approach is shown [@Sz1].
Photon-photon scattering in ultra-peripheral heavy-ion reactions
================================================================
The cross section for photon-photon scattering in ultra-peripheral heavy-ion collisions can be calculated by folding the cross section of the elementary mechanisms shown in Fig. \[Fig:gg\_elem\] with the equivalent photon flux [@Baur], $$N(\omega,b) = \frac{Z^{2}\alpha}{\pi^{2}}
\frac{1}{\beta^{2}b^{2}}
\bigg| \int_0^{\infty} dv\; v^{2} J_{1}(v)\frac{F_{el}(-\frac{u^{2}+v^{2}}{b^{2}})}{u^{2}+v^{2}}\bigg|^{2} \, .
\label{Eq:EPA}$$
The equivalent photon flux is defined in Eq. \[Eq:EPA\]. Here, $\omega$ denotes the energy of the photon, and b represents the transverse distance from the center of the nucleus where the photon density is evaluated. The integral in Eq. \[Eq:EPA\] represents the form factor of the charge distribution of the source.
![Differential cross section d$\sigma$/dM for photon scattering in PbPb-collisions.[]{data-label="Fig:dsigma1"}](dsigmadM_signal.png){width="6.5cm"}
The cross section for $\gamma\gamma\rightarrow\gamma\gamma$ in PbPb-collisions is calculated by convoluting the elementary cross section with the photon flux ,
$$\sigma^{EPA}_{PbPb \rightarrow PbPb\gamma\gamma} = \int\!\!\int dn^{1}_{\gamma}
\; dn^{2}_{\gamma} \; \sigma_{\gamma\gamma \rightarrow \gamma\gamma}(\omega_1,\omega_2) \, .
\label{Eq:dsigmadM_PbPb}$$
The differential cross section for photon-photon scattering in PbPb-collisions at $\sqrt{s}$=5.02 TeV resulting from the convolution of Eq. \[Eq:dsigmadM\_PbPb\] is shown in Fig. \[Fig:dsigma1\]. This cross section is derived with the box diagrams of Fig. \[Fig:gg\_elem\], and with conditions of the two final state photons being within the pseudorapidity range $|\eta|<$0.9, and having an energy E$_{phot} >$200 MeV.
Resonance signal from $\eta,\eta'$ decays
=========================================
The cross section for photoproduction of $\eta,\eta'$ is taken according to [@Budnev] $$\sigma_{\gamma\gamma\rightarrow R} = 8\pi(2J+1)\frac{\Gamma_{\gamma\gamma}\Gamma_{tot}}{(W^{2}-M^{2}_{R})^{2}+M^{2}_{R}\Gamma^{2}_{tot}} \, .
\label{Eq:sigma_res}$$
The cross section for photoproduction of $\eta,\eta'$ in PbPb-collisions is calculated according to the convolution defined in Eq. \[Eq:dsigmadM\_PbPb\].
![Cross section for photoproduction of $\eta,\eta'.$[]{data-label="Fig:dsigmadM_eta"}](dsigmadM_signaleta.png){width="6.5cm"}
The $\eta,\eta'$ cross section in PbPb-collisions at $\sqrt{s}$= 5.02 TeV multiplied by the branching ratio $\eta,\eta'\rightarrow\gamma\gamma$ is shown in Fig. \[Fig:dsigmadM\_eta\]. The values shown are derived with conditions of the two decay photons being within the pseudorapidity range $|\eta|<$0.9, and having an energy E$_{phot} >$200 MeV. The finite width of these two resonances results from the detector resolution of the photon measurements which is taken here to be $\sigma_{E_{phot}}/E$ = 0.02.
Photoproduction of $\pi^{0}\pi^{0}$ pairs
=========================================
The main background to the signal of two photons in the final state results from $\pi^{0}\pi^{0}$ production with two of the four decay photons escaping detection. The two measured photons from this $\pi^{0}\pi^{0}$ decay cannot be distinguished from the signal of photon-photon scattering.
![Mechanisms of $\pi^{0}\pi^{0}$ photoproduction (Fig. taken from [@Sz2]).[]{data-label="Fig:pion_pairs"}](pires.pdf "fig:"){width="4.cm"} ![Mechanisms of $\pi^{0}\pi^{0}$ photoproduction (Fig. taken from [@Sz2]).[]{data-label="Fig:pion_pairs"}](picoupl1.pdf "fig:"){width="4.cm"} ![Mechanisms of $\pi^{0}\pi^{0}$ photoproduction (Fig. taken from [@Sz2]).[]{data-label="Fig:pion_pairs"}](picoupl2.pdf "fig:"){width="4.cm"}
The contribution of resonance decays to the $\pi^{0}\pi^{0}$ final state is shown in Fig. \[Fig:pion\_pairs\] on the left. Here, the resonances $\sigma$(600), $f_{0}$(980), $f_{0}$(1500), $f_{0}$(1710), $f_{2}$(1270), $f^{`}_{2}$(1525), $f_{2}$(1565), $f_{2}$(1950) and $f_{4}$(2050) are considered [@Sz2]. The cross section $\gamma\gamma\rightarrow\pi^{+}\pi^{-}$ is much larger than for $\gamma\gamma\rightarrow\pi^{0}\pi^{0}$, hence even a small coupling between the charged and neutral pion channel might have an influence on the $\gamma\gamma\rightarrow\pi^{0}\pi^{0}$ cross section. Examples of processes leading to such channel couplings are shown in Fig. \[Fig:pion\_pairs\] in the middle and on the right.
Background from $\pi^{0}\pi^{0}$ decays
=======================================
The $\pi^{0}\pi^{0}$ cross section in PbPb-collisions is calculated the elementary $\pi^{0}\pi^{0}$ cross section with the photon flux according to Eq. \[Eq:EPA\]. The background from $\pi^{0}\pi^{0}$ decays is shown in Fig. \[Fig:bck\_pi0\] by the solid red line. This background cross section is derived by the conditions that exactly one decay photon from each of the two $\pi^{0}$’s is within the pseudorapidity range $|\eta| <$0.9, and that these photons have an energy E$_{phot} >$200 MeV.
![Signal and background from $\pi^{0}\pi^{0}$ decays.[]{data-label="Fig:bck_pi0"}](dsigmadM_signaletabck.png){width="6.5cm"}
Background suppression by asymmetry cuts
----------------------------------------
The two photons of the signal are of equal transverse momentum and are back-to-back in azimuth. These correlations are smeared out due to finite resolution in the measurement of photon energy and azimuthal angle. The two photons from $\pi^{0}\pi^{0}$ decay do not show these correlations. and vector ($A_{V}$) asymmetry can be defined for background suppression, $$A_S=\left|\frac{|\vec{p}_T(1)|-|\vec{p}_T(2)|}
{|\vec{p}_T(1)|+|\vec{p}_T(2)|}\right|,\hspace{1.cm}
A_V=\frac{|\vec{p}_T(1)-\vec{p}_T(2)|}
{|\vec{p}_T(1)+\vec{p}_T(2)|} \, .$$
![Background suppression by asymmetry cut A$_{S}$.[]{data-label="Fig:bck_pi0_asym"}](dsigmadM_sigbck.png){width="6.5cm"}
The background reduction for condition A$_{S}\!<$0.1 and A$_{S}\!<$0.02 is shown in Fig. \[Fig:bck\_pi0\_asym\] by the blue and red line, respectively. The condition A$_{S}<$0.02 reduces the background by about a factor $\sim$10 while keeping 98%
Background correction by sideband subtraction
---------------------------------------------
The background remaining after the cut A$_{S}\!<\!0.02$ can be subtracted by sideband correction. The signal band is given by with sideband 1 and 2 defined by $0.02\!<\!A_{s}\!<\!0.04$ and $0.04\!<\!A_{s}\!<\!0.06$, respectively. An estimator for the background in the signal region can be defined by linear extrapolation of the sidebands 1 and 2 into the signal region.
![Sideband corrected background.[]{data-label="Fig:bck_pi0_asym_corr"}](dsigmadM_sidecorr.png){width="6.5cm"}
The background remaining after sideband correction is shown by the red line. The sideband correction could be further improved by defining more than two sidebands, and by a non-linear extrapolation of the background into the signal region.
ACKNOWLEDGMENTS
===============
This work is supported by the German Federal Ministry of Education and Research under promotional reference 05P15VHCA1.
[99]{} M.Born, Nature 132:3329 (1933) 282. W.Heisenberg, H.Euler, Zeitschrift für Physik 98: 11-12 (1936) 714. ATLAS Collaboration, Nature Phys. 13 (2017) no.9, 852. CMS Collaboration, arXiv:1810.04602. M.Kłusek-Gawenda et al., Phys.Rev. C93 (2016), no.4, 044907. G.Baur et al., Phys.Rept. 364 (2002) 359. V.M.Budnev, $\!$I.F.Ginzburg, $\!$G.V.Meledin, $\!$V.G.Serbo, $\!$Phys.Rept.15, $\!$(1975) $\!$181. M.Kłusek-Gawenda, A.Szczurek, Phys.Rev. C87 (2013), no.5, 054908.
[^1]: Presented at Diffraction and Low-x Conference 2018, Reggio Calabria,
| {
"pile_set_name": "ArXiv"
} |
In the two-dimensional (2D) metal-insulator transition (MIT) regime, both the Coulomb interactions between electrons and the disorder are expected to be strong, leading to the formation of an electron glass [@Review; @Rice]. Recent experiments in 2D electron systems have revealed changes in the characteristics and the amplitude of the conduction noise as the change density of the system is varied from a high charge density metallic phase to a lower charge density insulating phase [@Popovic; @Sn; @Noise]. This has been interpreted as evidence for an onset of glassy dynamics near the insulating phase. These studies also find that the noise has a $1/f^{\alpha}$ characteristic, with $\alpha = 1.0$ in the metallic phase changing over to $\alpha \approx 1.8$ in the glassy phase. Other experiments have found similar results with $\alpha = 0.75$ in the metallic phase and $\alpha =1.3$ near the insulating phase [@N2d]. The glassy phase is signified by a large increase in the noise power along with the change in $\alpha$ [@Popovic; @Sn]. In addition, the noise power was observed to [*decrease*]{} with temperature [@Sn; @Noise], in contrast to single electron models with thermally activated trapping [@Weissman] and other models [@Voss] that predict an increase in the noise power with $T$. This suggests the importance of electron-electron interactions at the MIT. Other recent experiments near the 2D MIT have also found $1/f^{\alpha}$ noise, strong increases in noise power with decreasing charge density, and decreasing noise with increasing $T$ [@N2d; @Tourbot]. Additionally, $1/f^{\alpha}$ noise fluctuations in thin granular films have been interpreted as evidence for a glassy electron state [@Wu].
In theoretical studies, it was proposed that at the 2D MIT a freezing from an electron liquid to a partially ordered Wigner glass [@Kivelson] or a more strongly disordered electron glass [@Thakur; @Glass] may occur. Other theories suggest that an intermediate metallic glass phase appears between the liquid and insulating phases [@Pastor]. It is also possible that the metallic glass phase may consist of solid phase insulating regions coexisting with string-like liquid regions. Studies of glassy systems often find cooperative string-like motions or dynamical heterogeneities [@Glotzer]. Such motions can give rise to correlated dynamics and large fluctuations near a glass transition. It is, however, unclear what the origin of such cooperativity would be in the electron glass systems.
The strong enhancement of the noise observed when passing from a liquid to a glassy phase in the 2D charge system has noticeable similarities to the noise found in studies of vortex matter in superconductors [@Higgins]. In the vortex system, the existence of a transition from a weakly pinned liquid phase to a more strongly pinned glassy phase has been established. Noise studies in low temperature superconductors have shown that, as the glassy phase is approached from the liquid state, the voltage noise has a $1/f^{\alpha}$ characteristic with increasing $\alpha$, corresponding to an increase in the noise power [@Higgins].
The vortex system studies suggest that similar physics may be occurring in the 2D electron systems when there is a transition from a liquid like phase with low noise power to a pinned glassy phase with high noise power. This is also consistent with the theoretical prediction that the electron liquid freezes into a 2D disordered solid. In this work we propose a simple model for a classical 2D electron system consisting of interacting electrons with random disorder and temperature. We monitor the fluctuations and noise characteristics of the current as a function of electron density or temperature. The advantage of our model is that a large number of interacting electrons can be conveniently simulated, while a full quantum mechanical model of similar size would be computationally prohibitive. Despite the limitations of this model, we show that this approach captures many of the key experimental observations. Additionally, although our primary focus is to gain insight into the physics near the 2D MIT, our model is also relevant for other classical charge systems undergoing transitions from glass to liquid states, such as charged colloids interacting with random disorder. Our model is similar to previous studies of 2D classical electron systems with disorder [@Fertig; @Reichhardt]; however, these previous studies focused on the microscopics of the defects in the lattice [@Fertig] or the sliding dynamics [@Reichhardt]. In the present work we focus on the noise fluctuations in the strongly disordered phase as it changes from a liquid to a frozen state as a function of electron density for a fixed amount of disorder.
Our model consists of a 2D system of $N_{s}$ interacting electrons with periodic boundary conditions in the $x$ and $y$-directions. There are also $N_{p}$ defect sites which attract the electrons. We assume the electron motion is at finite temperature and the time evolution occurs through Langevin dynamics. The damping on the electrons comes
from their interactions with phonons or small scattering sites. The equation of motion for a electron $i$ is $$\eta {\bf v}={\bf f}_{i}=-\sum_{j}^{N_{s}}\nabla U(r_{ij}) +
{\bf f}_{i}^{s} + {\bf f}_{i}^{T} + {\bf f}_{d}$$ Here $\eta= 1$ is the damping constant and $U_{i}(r) = -q^2/r$, with $q=1$, is the electron-electron interaction potential, treated as in [@Jensen]. The term ${\bf f}_{i}^{s}$ comes from the $N_p$ randomly spaced defect sites modeled as parabolic traps of radius $r_{p}=0.2$ and strength $f_{p} = 1.0$. The thermal noise ${\bf f}_{i}^{T}$ arises from random Langevin kicks with $<f^{T}(t)> = 0$ and $<f^{T}_i(t)f^{T}_j(t^{\prime})> = 2\eta k_B T \delta_{ij}
\delta(t - t^{\prime})$. The driving term ${\bf f}_{d}=f_d \hat x$ comes from an applied voltage, and we take $f_d= 0.1$. We start at a high temperature where the charges are diffusing rapidly and cool to a lower temperature. We then wait for $10^4$ simulation time steps to reduce transient effects before appling the drive and measuring the average velocity $v$ of the electrons, which is proportional to the conduction or resistance. We do this for a series of electron densities at fixed disorder strength. We have considered samples with constant pin densities $n_{p}$ for different system sizes such that $N_{p}$ ranges from $317$ to $1200$.
We first consider the average electron velocity as a function of temperature and charge density for a system with a fixed $N_{p}=1200$. In Fig. 1 we show a series of conductance curves vs $T$ for charge density varied over nearly an order of magnitude, $0.4 \leq N_{s}/N_{p} \leq 2.67$. For high $N_{s}/N_{p}> 0.6$ the conductance is finite down to $T = 0$, while for $N_{s}/N_{p} \leq 0.6$, the electron velocity drops to zero within our resolution, indicating that all the electrons are strongly pinned in an insulating phase. As $N_{s}/N_{p}$ increases above 1.0, the downward curvature of $v$ at low $T$ decreases. These curves appear very similar
to those typically observed in 2D MIT studies [@Popovic]. One difference is that we do not find a charge density $n_{s}^{up}$ above which the slope of the velocities turns up slightly at low $T$, as in the experiments. This may be due to the fact that in our model we do not directly include phonons. In the experimental regime of interest to us here, the large noise increases occur at charge densities $n_s<n^{up}_s$, where the velocity curves bend down at low temperature.
We next consider the relative fluctuations in the velocities, $\delta v(t)=(v(t) - <v>)$ for varied $N_{s}/N_{p}$ at a fixed $T = 0.09$ for the system in Fig. 1. This analysis is similar to that performed in experiments [@Popovic; @Sn; @Noise]. In Fig. 2 we show the time traces of the relative velocity fluctuations for $N_{s}/N_{p} = 0.5, 0.7,
1.05, 1.64$ and $2.67$. Here the fluctuations increase as $N_s$ drops, in agreement with the experiments [@Popovic]. For $N_{s}/N_{p} < 0.44$, the system is pinned and there are no fluctuations. It is possible that, over a longer time interval such as that accessible experimentally, there would be even larger fluctuations at these small $N_s$ values; however, this is beyond the time scale we can access with simulations. A similar series of time traces can be obtained for $\delta v$ at fixed $N_{s}/N_{p}$ for increasing temperature (not shown). Here the fluctuations are reduced at the higher temperatures, in agreement with experiments [@Popovic].
From the fluctuations $\delta v$ we measure the power spectrum $$S(\nu) = \left|\int \delta v(t)e^{-2\pi i\nu t}dt\right|^2 .$$ In Fig. 3 we plot $S(\nu)$ for two different charge densities. At $N_{s}/N_{p} = 0.5$ (upper curve), the spectrum shows a $1/f^{\alpha}$ characteristic with $\alpha=1.37$ over a few orders of
magnitude in the frequency. In contrast, for $N_{s}/N_{p} = 1.67$ (lower curve), the noise power at lower frequencies is considerably reduced and the spectra is white with $\alpha \approx 0$. We not that it is the lower frequcies which will be most readly accesible in experiment. For fixed $N_{s}/N_{p} = 0.5$, we find that the power spectrum becomes white upon increasing $T$. We note that our results differ quantitatively from the experimental noise measurements [@Popovic] which find $1/f^{\alpha}$ noise with $\alpha = 1.0$ in the metallic phase and $\alpha = 1.8$ in the glassy regime. Our exponent $\alpha=1.37$ is close to the $\alpha = 1.3$ found in the glassy regime in other experiments [@N2d], where $\alpha=0.75$ in the metallic regime. It is possible that the exponents are not universal but depend on the details of the disorder strength; nevertheless, our results are in qualitative agreement with the experiments.
In Fig. 4(a) we show the noise power $S_{0}$ integrated over the first octave vs $N_{s}/N_{p}$ for a fixed $T = 0.09$. The noise power increases by four orders of magnitude as $N_{s}/N_p$ is reduced. At low $N_{s}/N_{p}$, the noise power decreases almost exponentially with charge density and begins to saturate at high $N_{s}/N_p$. Both these observations are in agreement with the experimental results [@Sn; @Noise]. In the inset of Fig. 4(a) we plot the noise spectrum exponent $\alpha$ vs $N_{s}/N_{p}$. A large increase in $\alpha$ occurs near $N_{s}/N_{p} = 0.7$. A similar sharp increase in the exponent is also observed in experiments [@Sn; @Noise; @N2d] as a function of charge density and has been interpreted as the glassy freezing transition. In Fig. 4(b) we plot $S_{0}$ vs $T$ for fixed $N_{s}/N_{p} = 0.5$. Here the noise power drops exponentially over four orders of magnitude with increasing $T$, which is in agreement with the experiments [@Sn; @Noise]. We note that most single electron models predict an [*increase*]{} in the noise power with temperature [@Weissman; @Voss]. Other models in the hopping regime predict a noise power that is independent of $T$ or has a power law decrease with $T$ [@Kozub]. These discrepancies
suggest that the noise in the experiment is not due to single electron hopping events, but is instead caused by correlated electron motions. In the inset of Fig. 4(b) we plot $\alpha$ vs $T$, showing that a sharp increase in $\alpha$ occurs near $T = 0.125$ at the onset of the glassy freezing.
We have also measured the non-Gaussian nature of the noise. At high $N_{s}$ and $T$ the noise fluctuations are Gaussian; however, in the regions of high noise power we find non-Gaussian noise fluctuations with a skewed distribution. Experiments have also found evidence for non-Gaussian fluctuations in the glassy regimes [@Noise].
Next we show evidence that the large noise is due to correlated regions of string like electron flow, and that within these regions the electrons move in 1D or quasi 1D channels. Because of the reduced dimensionality the electron motion is more correlated. In Fig. 5(a) we show the trajectories of the electrons for a fixed period of time for a system with $T = 0.09$ at $N_{s}/N_{p} = 1.67$. Here the electrons can flow freely throughout the sample, although there are some areas where electrons become temporarily trapped by a defect site. In Fig. 5(b) at $N_{s}/N_{p} = 1.37$, where the noise power is larger than in the system shown in Fig. 5(a), larger pinned regions appear and the electron motion consists of a mixture of 2D and 1D regions. If the trajectories are followed over longer times,
motion occurs throughout the entire sample. In Fig. 5(c) at $N_{s}/N_{p} = 0.5$, where the noise power and $\alpha$ are both maximum, the electron motion occurs mostly in the form of 1D channels that percolate through the sample. There are also regions where the electron motion occurs in small rings. The channel structures change very slowly with time, with a channel occasionally shutting off while another emerges elsewhere. It is the intermittent opening of the 1D channels which gives rise to the large noise fluctuations in this regime. When a percolating 1D channel opens, all the electrons in that channel move in a correlated fashion leading to a large increase in the conduction. Conversely, if a percolating channel closes all the electrons in that channel cease to move. It is well known that fluctuations in 1D are much more strongly enhanced than in 2D. As $T$ or $N_{s}$ is increased, the motion becomes increasingly 2D in nature and the strong correlations of the electron motion are lost. We also note that the appearance of string like motions in the large noise regions is consistent with studies in glassy systems, where dynamical heterogeneities in the form of 1D stringlike motions of particles have been observed in conjunction with large noise [@Glotzer]. In Fig. 5(d) at $N_{s}/N_{p} = 0.3$, deep in the insulating regime, there are no channels. Instead, the infrequent motion of electrons occurs only by small jumps from defect to defect.
In conclusion, we have presented a simple model for the glassy freezing of interacting electrons in 2D with random disorder. For high electron density or high temperatures, the electrons form a 2D liquid state and we find low conduction noise power with a white spectra. As the density of the electrons is lowered for fixed temperature, or conversely, as the temperature is lowered for fixed low electron density, there is a crossover to a $1/f^{\alpha}$ noise with large low frequency power and $\alpha = 1.37$. In this glassy regime, the electrons move in 1D intermittent stringlike paths which percolate throughout the sample. Similar stringlike motions are also observed in other glass forming systems. For low electron density, all the electrons are frozen by the defect sites and the motion occurs only by single electron hopping events. We find that the noise power decreases exponentially with temperature, in agreement with experiment. Many of our results are in qualitative agreement with recent experiments on 2D electron systems near the metal insulator transition.
This work was supported by the US DoE under Contract No. W-7405-ENG-36.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Billey and Braden defined a geometric pattern map on flag manifolds which extends the generalized pattern map of Billey and Postnikov on Weyl groups. The interaction of this torus equivariant map with the Bruhat order and its action on line bundles lead to formulas for its pullback on the equivariant cohomology ring and on equivariant $K$-theory. These formulas are in terms of the Borel presentation, the basis of Schubert classes, and localization at torus fixed points.'
address:
- |
Department of Mathematics\
University of Ibadan\
Ibadan, Oyo, Nigeria
- |
Department of Mathematics\
Texas A&M University\
College Station\
Texas\
USA
author:
- Praise Adeyemo
- Frank Sottile
title: |
Equivariant cohomology theories\
and the pattern map
---
Introduction {#introduction .unnumbered}
============
The (geometric) pattern map of Billey and Braden [@BB] is a map between flag manifolds that extends the generalized pattern map on Weyl groups of Billey and Postnikov [@BiPo]. We previously studied maps on cohomology and $K$-theory induced by sections of the pattern map [@AS], generalizing formulas for specializations of Schubert and Grothendieck polynomials that had been obtained in type A [@BS98; @LRS], which are generalizations of the decomposition formula [@BKTY1; @BKTY2].
The pattern map is torus-equivariant; we extend the results of [@AS] to (torus) equivariant cohomology and equivariant $K$-theory. Specifically, we give formulas for the pattern map on equivariant cohomology and equivariant $K$-theory in terms of localization, the Borel presentation, and Schubert classes. The localization formula is simply restriction, while the formula for the Borel construction involves the action of a minimal right coset representative. Most interesting is the formula for the pullback of a Schubert class. This is a positive (in the sense of Graham [@Gr] and of Anderson, Griffeth, and Miller [@AGM]) sum of Schubert classes, with coefficients certain explicit Schubert structure constants. The value of these formulas is the interplay between them, which we illustrate through examples. The pattern map, together with Pieri-type formulas [@So96; @LS07], was used in [@BS_Skew; @LRS] to obtain new formulas for (non-equivariant) Schubert classes in type $A$ cohomology and $K$-theory. The results of this paper should give similar formulas for equivariant Schubert classes, given a suitable Pieri-type formula in type $A$.
In Section \[S:flag\], we provide some background on equivariant cohomology and $K$-theory of flag manifolds and explain the localization presentation of equivariant cohomology and $K$-theory. In Section \[S:pattern\] we describe the pattern map and develop its interaction with localization. We use this to prove our main results for equivariant cohomology in Section \[S:equivariant\] and for equivariant $K$-theory in Section \[S:K-theory\]. Section \[S:Ex\] contains two examples.
Equivariant cohomology and $K$-theory of flag manifolds {#S:flag}
=======================================================
We state standard results on equivariant cohomology and $K$-theory for flag varieties.
Equivariant cohomology and homology
-----------------------------------
Let $Y$ be a —a projective variety equipped with a left action of a torus ${\Blue{T}}\simeq
({{\mathbb{C}}}^\times)^n$. We define $T$-equivariant homology and $T$-equivariant cohomology of $Y$ via the Borel construction. Let $T\hookrightarrow ET\twoheadrightarrow BT$ be the universal $T$-bundle where $T$ acts freely on the right of a contractible space $ET$ with quotient the classifying space $BT$ of $T$. The $T$-equivariant cohomology of $Y$ is $${\Blue{H^*_T(Y)}}\ :=\ H^*(ET\times_T Y, {{\mathbb{Q}}})\,.$$ We use $T$-equivariant Borel-Moore homology , where $ET$ is replaced by Totaro’s [@Tot] sequence of finite approximation; see [@Br00; @EG] for details. While $H^*_T(Y)$ is graded by the positive integers, $H^T_*(Y)$ is ${{\mathbb{Z}}}$-graded. When $Y$ is a point, , $H^*_T({{\it pt}})=H^*(BT,{{\mathbb{Q}}})$, which is the symmetric algebra (over ${{\mathbb{Q}}}$) of the character group ${\Blue{\Xi(T)}}:=\mbox{Hom}(T,{{\mathbb{C}}}^\times)$ of $T$, where a character has homological degree 2. The map ${\Blue{\rho}}\colon Y\to{{\it pt}}$ to a point gives functorial maps $\rho^*\colon H^*_T({{\it pt}})=S \to H^*_T(Y)$ and $\rho_*\colon H^T_*(Y) \to H^T_*({{\it pt}})$.
Equivariant cohomology $H^*_T(Y)$ has a natural ring structure and is an $S$-algebra through the map $\rho^*$. The cap product, , realizes $H^T_*(Y)$ a module over $H^*_T(Y)$ with $$H^a_T(Y) \otimes H^T_b(Y)\ \xrightarrow{\ \frown\ }\ H^T_{b-a}(Y)\,,$$ and thus $H^T_*(Y)$ is also an $S$-module.
A $T$-invariant subvariety $Z$ of $Y$ has an equivariant fundamental cycle, $${\Blue{[Z]^T}}\ \in\ H^T_{2\dim(Z)}(Y)\,.$$ The natural map $\frown[Y]^T\colon H^*_T(Y)\to H^T_{2\dim(Y)-*}(Y)$ is an isomorphism when $Y$ is smooth. Consequently, we may identify $H^T_{-*}({{\it pt}})$ with $S=H^*_T({{\it pt}})$. We thus have a pairing $$\langle\;,\;\rangle\ \colon\ H^*_T(Y)\otimes H^T_*(Y)\ \longrightarrow\ S\,,$$ defined by $\langle y,C\rangle:= \rho_*(y\frown C)$. This satisfies the projection formula; if $\phi\colon Z\to Y$ is a map of projective varieties and we have $y\in H^*_T(Y)$ and $C\in H^T_*(Z)$, then $$\label{eq:projection_formula}
\rho_*(\phi^*(y)\frown C)\ =\ \rho_*(y\frown \phi_*(C))\,.\smallskip$$
Flag varieties
--------------
Let be a connected and simply connected complex semisimple linear algebraic group, a Borel subgroup of $G$, and the maximal torus contained in $B$. The Weyl group ${\Blue{W}}:=N(T)/T$ of $G$ is the quotient of the normalizer of $T$ by $T$. Our choice of $B$ gives $W$ the structure of a Coxeter group with a preferred set of generators and a length function, $\ell\colon W\to\{0,1,2,\dotsc,\}$. These conventions will remain in force for the remainder of this paper. Let ${\Blue{w_o}}$ be the longest element in $W$.
As any Borel subgroup is its own normalizer and all Borel subgroups are conjugate by elements of $G$, we may identify the set ${\Blue{{{\mathcal{F}}}}}$ of Borel subgroups with the orbit $G/B$, called the . This has a left action by elements of $G$, and we write for the group $gBg^{-1}$. The inclusion $N(T)\hookrightarrow G$ gives an injection of the Weyl group $W\hookrightarrow{{\mathcal{F}}}$ as $T=N(T)\cap B$. Since for any $w\in W$, the Borel group $w.B$ contains $T$, this identifies the Weyl group $W$ with the set of $T$-fixed points of ${{\mathcal{F}}}$.
Elements of $W$ also index $B$-orbits on ${{\mathcal{F}}}$, which together form the Bruhat decomposition, $$\label{Eq:BruhatDecomposition}
{{\mathcal{F}}}\ =\ \bigsqcup_{w\in W} Bw.B/B\,.$$ The orbit $Bw.B$ is isomorphic to an affine space of dimension $\ell(w)$ and is a . Its closure is a , . Set ${\Blue{B_-}}:=w_o.B$, which is the Borel subgroup opposite to $B$ containing $T$. Let ${\Blue{X^w}}:=\overline{B_-w.B}$, which is also a Schubert variety and has codimension $\ell(w)$. The intersection $X^v\cap X_w$ is nonempty if and only if $w\geq v$ and in that case it is irreducible of dimension $\ell(w)-\ell(v)$ [@Deodhar; @Richardson]. Note that each of $X^v$, $X_w$, and $X^v\cap X_w$ is $T$-stable.
Equivariant cohomology of the flag manifold
-------------------------------------------
We use three presentations for the equivariant cohomology ring $H^*_T({{\mathcal{F}}})$ of the flag variety. Identifying $T$ with $B/[B,B]$, a character $\lambda\in\Xi(T)$ is also a character of $B$. Write for the one-dimensional $T$-module ${{\mathbb{C}}}$ where $T$ acts via $\lambda$. The $T$-equivariant line bundle ${\Blue{{{\mathcal{L}}}_\lambda}}=G\times_B\,{{\mathbb{C}}}_\lambda$ is the quotient of $G\times{{\mathbb{C}}}$ by the equivalence relation $(gb,z)\sim(g,\lambda(b)\cdot z)$ for $g\in G$, $b\in B$, and $z\in{{\mathbb{C}}}$. Equivariant bundles have equivariant Chern classes. The map associating a character $\lambda\in\Xi(T)$ to the first equivariant Chern class $c_1^T({{\mathcal{L}}}_\lambda)\in H^2_T({{\mathcal{F}}})$ induces a homomorphism of graded algebras, ${\Blue{c^T}}\colon S\to H^*_T({{\mathcal{F}}})$. The of $H^*_T({{\mathcal{F}}})$ is the isomorphism $$S\otimes_{S^W}\! S\ \simeq\ H^*_T({{\mathcal{F}}})\,,$$ which is defined by $f\otimes g\mapsto f\cdot c^T(g)$. The left copy of $S$ is the pullback $\rho^*(S)$ from a point, and the right copy is the subring generated by equivariant Chern classes of the ${{\mathcal{L}}}_\lambda$.
Schubert cells are even-dimensional and so the fundamental cycles of Schubert varieties () form a basis for $H^T_*({{\mathcal{F}}})$ over the ring $S$, $$H^T_*({{\mathcal{F}}})\ =\ \bigoplus_{w\in W} S\cdot [X_w]^T\,.$$ There is a dual basis $\{{\Blue{{{\mathfrak{S}}}_v}}\mid v\in W\}$ for $H^*_T({{\mathcal{F}}})$ with ${{\mathfrak{S}}}_v\in H^{2\ell(v)}_T({{\mathcal{F}}})$, defined by $$\rho_*( {{\mathfrak{S}}}_v\frown [X_w]^T )\ =\ \delta_{v,w}\,.$$ Elements of this dual basis are , as they are identified with Schubert cycles under the isomorphism between equivariant cohomology and equivariant homology, $$\label{Eq:duality}
{{\mathfrak{S}}}_v\frown[{{\mathcal{F}}}]^T\ =\ [X^v]^T
\qquad\mbox{and}\qquad
{{\mathfrak{S}}}_v\frown [X_w]^T\ =\ [X^v\cap X_w]^T\,.$$ They lie in the subring generated by the equivariant Chern classes.
There are equivariant ${\Blue{c^w_{u,v}}}\in S$ (of cohomological degree $2(\ell(u)+\ell(v)-\ell(w))$) defined by the identity $${{\mathfrak{S}}}_u \cdot {{\mathfrak{S}}}_v\ =\ \sum_w c^w_{u,v}\, {{\mathfrak{S}}}_w\,.$$ Graham [@Gr] showed that these are positive sums of monomials in the simple roots of $G$. Using the duality with Schubert cycles $[X_w]^T$, we have $$\label{Eq:triple-Int}
c^w_{u,v}\ =\ \rho_* ({{\mathfrak{S}}}_u\cdot{{\mathfrak{S}}}_v\frown [X_w]^T)\ =\
\rho_*({{\mathfrak{S}}}_u\frown [X^v\cap X_w]^T)\,.\smallskip$$
The inclusion $i\colon{{\mathcal{F}}}^T\hookrightarrow{{\mathcal{F}}}$ of the $T$-fixed points into ${{\mathcal{F}}}$ induces the localization map $$\label{Eq:localization}
i^*\ \colon\ H^*_T({{\mathcal{F}}})\ \longrightarrow\ H^*_T({{\mathcal{F}}}^T)\
\ =\ \bigoplus_{w\in W} S\,.$$ Chang and Skjelbred [@ChSk74] describe the image of the localization map and consequently the full equivariant cohomology ring. Let be the of ${{\mathcal{F}}}$: the set of points whose stabilizer has codimension at most one in $T$.
\[P:CS\] Let $j\colon {{\mathcal{F}}}_1\hookrightarrow{{\mathcal{F}}}$ be the inclusion of the equivariant one-skeleton of ${{\mathcal{F}}}$. Then the map $i^*\colon H^*_T({{\mathcal{F}}})\to H^*_T({{\mathcal{F}}}^T)$ is an injection and it has the same image as the map $j^*\colon H^*_T({{\mathcal{F}}}_1)\to H^*_T({{\mathcal{F}}}^T)$.
The flag manifold ${{\mathcal{F}}}$ has finitely many one-dimensional $T$-orbits. For such a space, Goresky, Kottwitz, and MacPherson [@GKM] used the result of Chang and Skjelbred to give an elegant description of the image of $\jmath^*$ and thus a description of $H^*_T({{\mathcal{F}}})$.
Identify $H^*_T({{\mathcal{F}}}^T)=\bigoplus_{w\in W} S$ with the set $\operatorname{Maps}(W,S)$ of functions $\phi\colon W\to S$: If $y\in H^*_T({{\mathcal{F}}})$ and $\phi=i^*(y)$ then $\phi(w)$ is defined to be $i^*_w(y)\in S$. Here, $i_w$ is the map $i_w\colon{{\it pt}}\to w.B\in{{\mathcal{F}}}^T$. Classes $\phi\in\operatorname{Maps}(W,S)$ lying in the image of $H^*_T({{\mathcal{F}}})$ under $i^*$ satisfy the : For a root $\alpha$ of $G$ and $u\in W$, we have $$\phi(u)\ -\ \phi(s_\alpha u)\ \in\ \langle\alpha\rangle\,,$$ where $s_\alpha\in W$ is the reflection corresponding to $\alpha$ and $\langle\alpha\rangle$ is the principal ideal of $S$ generated by $\alpha$ (as roots of $G$ are characters of $T$).
At each $T$-fixed point, there is a two-to-one correspondence between roots $\alpha$ of $G$ and $T$-invariant curves ($\alpha$ and $-\alpha$ correspond to the same curve whose stabilizer is annihilated by $\alpha$). Figure \[F:One\_Skeleta\] displays the equivariant one-skeleta of $Sp(4,{{\mathbb{C}}})/B$ and $SL(4,{{\mathbb{C}}})/B$, which have Lie types $C_2$ and $A_3$, respectively.
![Equivariant one-skeleta.[]{data-label="F:One_Skeleta"}](pictures/Moment_Graph_A3.eps){height="150pt"}
Let us describe the map $i^*\colon H_T^*({{\mathcal{F}}})\to H_T^*({{\mathcal{F}}}^T)$ in more detail. The inclusion of a $T$-fixed point $w.B$ is a map $$i_w\ \colon\ {{\it pt}}\ \longrightarrow\ w.B\ \in\ {{\mathcal{F}}}\,.$$ The Weyl group $W$ acts on characters $\Xi(T)$ of $T$ on the right via $$\label{Eq:rightAction}
w.\lambda(t)\ :=\ \lambda(\sigma^{-1} t \sigma)\,,$$ where $\sigma\in N(T)$ is any representative of the right coset $w\subset N(T)$, $T\sigma=w$. The action on $\Xi(T)$ induces a right action of $W$ on $S$. The following is standard; we include a proof to illustrate our conventions.
\[L:LB\_fixedPoint\] Let $\lambda\in\Xi(T)$. Then $i^*_w({{\mathcal{L}}}_\lambda)= {{\mathbb{C}}}_{w.\lambda}$.
The pullback $i^*_w({{\mathcal{L}}}_\lambda)$ of ${{\mathcal{L}}}_\lambda$ along $i_w$ is $w.B\times_B{{\mathbb{C}}}_\lambda$. We determine the character of the action of $T$ on $i_w^*({{\mathcal{L}}}_\lambda)$. Let $\sigma\in N(T)$ be a representative of $w$ so that $w=T\sigma$. Points of $w.B\times_B{{\mathbb{C}}}_\lambda$ have unique representatives $(\sigma,z)$ for $z\in {{\mathbb{C}}}$. Let $t\in T$. Then $$(t\sigma,z)\ =\ (\sigma \sigma^{-1}t\sigma, z)\ =\
(\sigma, \lambda(\sigma^{-1}t\sigma)\cdot z)\,,$$ which shows that $w.B\times_B{{\mathbb{C}}}_\lambda={{\mathbb{C}}}_{w.\lambda}$.
Let $f\otimes g\in S\otimes_{S^W}\! S\simeq H_T^*({{\mathcal{F}}})$ and $w\in W$. Then $$i_w^*(f\otimes g)\ =\ f\cdot (w.g)\,.$$
As $i^*_w$ is an $S$-module map, $i_w^*(f\otimes g)=f\cdot i_w^*(g)$. Since $g$ lies in the subring of $H^*_T({{\mathcal{F}}})$ generated by equivariant Chern classes of the ${{\mathcal{L}}}_\lambda$, the result follows by Lemma \[L:LB\_fixedPoint\].
Equivariant $K$-theory
----------------------
For a projective $T$-variety, let be the Grothendieck ring of $T$-equivariant vector bundles on $Y$. Pullback of vector bundles along a map of $T$-varieties $\phi\colon Y\to Z$ induces the map $\phi^*\colon K^0_T(Z)\to K^0_T(Y)$. The representation ring of $T$ is $K^0_T({{\it pt}})$, which is the group algebra ${{\mathbb{Z}}}[\Xi(T)]$. The pullback along the map $\rho\colon Y\to {{\it pt}}$ induces on $K^0_T(Y)$ the structure of an $R(T)$-algebra.
The Grothendieck group of $T$-equivariant sheaves on $Y$ is a module over $K^0_T(Y)$ via tensor product. The alternating sum of higher direct images (derived pushforward) gives a functorial map $\phi_*\colon K^T_0(Y)\to K^T_0(Z)$ for any map $\phi\colon Y\to Z$ of projective $T$-varieties. When $Y$ is smooth, the natural map $K^0_T(Y)\to K^T_0(Y)$ is an isomorphism, and we have a pairing $$\langle\cdot,\cdot\rangle\ \colon\ K^0_T(Y)\otimes_{R(T)}\! K^0_T(Y)\ \longrightarrow\ R(T)$$ defined by $\langle \xi,\zeta\rangle:= \rho_*(\xi\cdot\zeta)$. For any $T$-equivariant sheaf/vector bundle $E$ on $Y$, write for its class in the appropriate Grothendieck group.
Consider this for the flag manifold ${{\mathcal{F}}}$. The ring $K^0_Y({{\mathcal{F}}})$ admits a second map $\gamma$ from $R(T)$ induced by the map $\Xi(T)\ni\lambda\mapsto
[{{\mathcal{L}}}_\lambda]$. The analog of the Borel presentation for equivariant cohomology is due to McLeod [@McL].
The map $f\otimes g\mapsto f\cdot\gamma(g)$ induces an isomorphism $$R(T)\otimes_{R(T)^W}\!R(T)\ \longrightarrow\ K^0_T({{\mathcal{F}}})\,.$$
There are several bases of $K^0_T({{\mathcal{F}}})$ that come from the Bruhat decomposition ; we recommend [@GrKu] for details. The classes $\{[{{\mathcal{O}}}_{X_w}]\mid w\in W\}$ of structure sheaves of Schubert varieties $X_w=\overline{Bw.B}$ form an $R(T)$-basis for $K^0_T({{\mathcal{F}}})$. A different basis is given by the classes $\{[{{\mathcal{O}}}_{X^w}]\mid w\in W\}$ of the Schubert varieties $X^w=\overline{B_-w.B}$. Let ${\Blue{{{\mathcal{I}}}_w}}$ be the sheaf of ${{\mathcal{O}}}_{X_w}$-ideals defining the complement of the Schubert cell $Bw.B$ in $X_w$, then $\{[{{\mathcal{I}}}_w]\mid w\in W\}$ also forms an $R(T)$-basis for $K^0_T({{\mathcal{F}}})$. The last two are dual bases, $$\langle [{{\mathcal{O}}}_{X^v}]\,,\, [{{\mathcal{I}}}_w] \rangle\ =\ \delta_{v,w}\,.$$ The ideal sheaves may be expressed in terms of Schubert structure sheaves $$\label{Eq:ideal_sheaves}
[{{\mathcal{I}}}_w]\ =\ \sum_{v\leq w} (-1)^{\ell(w)-\ell(v)} [{{\mathcal{O}}}_{X^v}]\,.$$ There are ${\Blue{b^w_{u,v}}}\in R(T)$ defined by the identity $$[{{\mathcal{O}}}_{X^u}]\cdot[{{\mathcal{O}}}_{X^v}]\ =\ \sum_w b^w_{u,v}\, [{{\mathcal{O}}}_{X^w}]\,.$$ Griffeth and Ram [@GrRa] conjectured that these exhibit a positivity generalizing Graham’s positivity for equivariant cohomology. This was proven by Anderson, Griffeth, and Miller [@AGM]. These coefficients have a formula similar to , $$\label{eq:K-SSC}
b^w_{u,v}\ =\ \rho_*( [{{\mathcal{O}}}_{X^u}]\cdot[{{\mathcal{O}}}_{X^v}] \cdot [{{\mathcal{I}}}_w])\,.$$
Equivariant $K$-theory of the flag manifold has similar behavior with respect to localization as does the equivariant cohomology. The following proposition is due to Vezzosi and Vistoli [@VV] and to Knutson [@Ro].
Let $\jmath\colon {{\mathcal{F}}}_1\hookrightarrow{{\mathcal{F}}}$ be the inclusion of the equivariant one-skeleton of ${{\mathcal{F}}}$. Then the map $\iota^*\colon K^0_T({{\mathcal{F}}})\to K^0_T({{\mathcal{F}}}^T)$ is an injection and it has the same image as the map $\jmath^*\colon K^0_T({{\mathcal{F}}}_1)\to K^0_T({{\mathcal{F}}}^T)$.
The Grothendieck ring $K^0_T({{\mathcal{F}}}^T)$ is also similarly simple, $$K^0_T({{\mathcal{F}}}^T)\
\ =\ \bigoplus_{w\in W} R(T)\ =\ \operatorname{Maps}(W,R(T))\,,$$ and restriction to a fixed point is similar to that of equivariant cohomology, $$i_w^*(f\otimes g)\ =\ f\cdot (w.g)\,,$$ where $f,g\in R(T)$ and $f\otimes g\in R(T)\otimes_{R(T)^W}\! R(T)\simeq K^0_T({{\mathcal{F}}})$.
Classes $\phi\in\operatorname{Maps}(W,R(T))$ lying in the image of $K^0_T({{\mathcal{F}}})$ under $i^*$ satisfy analogs of the GKM relations. For a root $\alpha$ of $G$ and $u\in W$, we have $$\phi(u)\ -\ \phi(s_\alpha u)\ \in\ \langle 1-\alpha\rangle\,,$$ where $s_\alpha\in W$ is the reflection corresponding to $\alpha$ and $\langle 1-\alpha\rangle$ is the principal ideal of $R(T)$ generated by $1-\alpha$ (as roots of $G$ are characters of $T$).
Geometry of the pattern map {#S:pattern}
===========================
Billey and Braden [@BB] defined the geometric pattern map and developed its main properties. Let $G,B,T,W$ be a connected and simply connected complex semisimple linear algebraic group, a Borel subgroup, a maximal torus contained in $B$, and Weyl group as before. Let ${\Blue{\eta}}\colon{{\mathbb{C}}}^*\to T$ be a cocharacter whose image is the subgroup of $T$. Springer [@Sp Theorem 6.4.7] showed that the centralizer ${G'}:=Z_G(T_\eta)$ of $T_\eta$ in $G$ is a connected, reductive subgroup and $T$ is also a maximal torus of $G'$. Also, if $B_0\in{{\mathcal{F}}}$ is a fixed point of $T_\eta$, so that $T_\eta\subset B_0$, then $B_0\cap G'$ is a Borel subgroup of $G'$.
In type $A$, if we have $G=GL(n,{{\mathbb{C}}})$ with $W=S_n$, the symmetric group on $n$ letters, then $G'\simeq GL(n_1,{{\mathbb{C}}})\times\dotsb\times GL(n_s,{{\mathbb{C}}})$ with $W'\simeq S_{n_1}\times\dotsb\times S_{n_s}$, where $n=n_1+\dotsb+n_s$. In general, $G'$ may be any Levi subgroup of $G$. For example, any group whose Dynkin diagram is obtained from that of $G$ by deleting some nodes.
Set $B':=G'\cap B$. Let $\Blue{{{\mathcal{F}}}'}:=G'/B'$ be the flag variety of $G'$, and be the set of $T_\eta$-fixed points of ${{\mathcal{F}}}$, which retains an action of $G'$. Sending a $T_\eta$-fixed point $B_0\in{{\mathcal{F}}}^{\eta}$ to its intersection with $G'$, $B_0\cap G'$, defines a $G'$-equivariant map ${\Blue{\psi}}\colon{{\mathcal{F}}}^{\eta}\to{{\mathcal{F}}}'$. Restricting to $T$-fixed points gives a map $\psi\colon W\to W'$, where is the Weyl group of $G'$. This is the Billey-Postnikov pattern map [@BiPo] which is the unique map $\psi\colon W\to W'$ that is $W'$-equivariant in that $\psi(wx)=w\psi(x)$ for $w\in W'$ and $x\in W$, and which respects the Bruhat order in that if $\psi(x)\leq\psi(wx)$ in $W'$ with $w\in W'$ and $x\in W$, then $x\leq wx$ in $W$. Billey and Braden use this to deduce that the map $\psi$ is a $G'$-equivariant isomorphism between each connected component of ${{\mathcal{F}}}^{\eta}$ with the flag variety ${{\mathcal{F}}}'$, and also that the connected components of ${{\mathcal{F}}}^{\eta}$ are in bijection with right cosets $W'\backslash W$ of $W'$ in $W$.
Observe that $B_-\cap G'={\Blue{B_-'}}$, the Borel group opposite to $B'$ containing $T$. Let be the component of ${{\mathcal{F}}}^{\eta}$ corresponding to a coset $W'\varsigma$ with $\varsigma\in W'\varsigma$ having minimal length, and let ${\Blue{\iota_\varsigma}}\colon{{\mathcal{F}}}'\xrightarrow{\sim}{{\mathcal{F}}}^{\eta}_\varsigma$ be the corresponding section of the pattern map. (This is the unique $G'$-equivariant map sending the $T$-fixed point $e.B'\in{{\mathcal{F}}}'$ to the $T$-fixed point $\varsigma.B\in{{\mathcal{F}}}^\eta$.) We use a refined result of Billey and Braden.
Let $W'\varsigma$ be a coset of $W'$ in $W$ with $\varsigma$ of minimal length in $W'\varsigma$ and let $\iota_\varsigma\colon {{\mathcal{F}}}'\to{{\mathcal{F}}}^{\eta}$ be the corresponding section of the pattern map. For $w\in W'$ we have $$\iota_\varsigma(X'_w) \ =\ X^\varsigma\cap X_{w\varsigma}\,.$$
\[C:pushforward\_formulae\] For $w\in W'$, we have $$\begin{aligned}
\iota_{\varsigma,*}[X'_w]^T& =& [X^\varsigma\cap X_{w\varsigma}]^T\ =\
{{\mathfrak{S}}}_{\varsigma}\frown [X_{w\varsigma}]^T, \\
\iota_{\varsigma,*}[{{\mathcal{O}}}_{X'_w}]& =& [{{\mathcal{O}}}_{X^\varsigma\cap X_{w\varsigma}}]\ =\
[{{\mathcal{O}}}_{X^\varsigma}]\cdot [{{\mathcal{O}}}_{X_{w\varsigma}}]\,,\quad\mbox{and} \\
\iota_{\varsigma,*}[{{\mathcal{I}}}_w]& =& [{{\mathcal{O}}}_{X^\varsigma}\otimes{{\mathcal{I}}}_{w\varsigma}]\ =\
[{{\mathcal{O}}}_{X^\varsigma}]\cdot [{{\mathcal{I}}}_{w\varsigma}]\,.
\end{aligned}$$
The second equality for Schubert cycles is , the second equality for Schubert structure sheaves is due to Brion [@Br02 Lemma 2], and the third line is from Lemma 2.12 of [@AS].
In [@AS], we computed the pullback of equivariant line bundles.
\[P:pullbackLineBunde\] For $\lambda\in\Xi(T)$, we have $\iota^*_\varsigma({{\mathcal{L}}}_\lambda)= {{\mathcal{L}}}_{\varsigma.\lambda}$.
This is compatible with Lemma \[L:LB\_fixedPoint\]. Let $w\in W'$ so that $w.B'\in ({{\mathcal{F}}}')^T$. The pattern map $\iota_\varsigma$ sends $w.B'$ to $w\varsigma.B$. We have the commutative diagram $$\begin{picture}(87,37)(-1,1)
\put(-1,1){${{\it pt}}$}
\put(11,10){\vector(1,1){16}} \put(8,20){\scriptsize$i_w$}
\put(12,4){\vector(3,1){60}} \put(38,4){\scriptsize$i_{w\varsigma}$}
\put(30,25){${{\mathcal{F}}}'$}
\put(44,30){\vector(1,0){28}} \put(52,33){\scriptsize$\iota_\varsigma$}
\put(75,25){${{\mathcal{F}}}$}
\end{picture}$$ Then by Lemma \[L:LB\_fixedPoint\], $$i^*_w(\iota_\varsigma({{\mathcal{L}}}_\lambda))\ =\
i^*_w({{\mathcal{L}}}_{\varsigma.\lambda})\ =\
{{\mathbb{C}}}_{w.(\varsigma.\lambda)}\ =\ {{\mathbb{C}}}_{w\varsigma.\lambda}\ =\
i^*_{w\varsigma}({{\mathcal{L}}}_\lambda)\,.$$
As the pattern map $\iota_\varsigma$ is an isomorphism of $T$-varieties, it is an isomorphism of the equivariant one-skeleta of ${{\mathcal{F}}}'$ and ${{\mathcal{F}}}^\eta_\varsigma$ and a bijection on $T$-fixed points.
\[Ex:six\_cosets\] Suppose that $G=SL(4,{{\mathbb{C}}})$ and $\eta(t)=\mbox{diag}(t,t,t^{-1},t^{-1})$. Tnen ${{\mathbb{C}}}^4\simeq ({{\mathbb{C}}}_1)^2\oplus({{\mathbb{C}}}_{-1})^2$ as a ${{\mathbb{C}}}^\times$-module under $T_\eta$. Thus $G'=SL(2,{{\mathbb{C}}})\times SL(2,{{\mathbb{C}}})$ with Weyl group ${{\mathcal{S}}}_2\times{{\mathcal{S}}}_2$ in $W={{\mathcal{S}}}_4$, the symmetric group on four letters, and ${{\mathcal{F}}}'={{\mathbb{P}}}^1\times {{\mathbb{P}}}^1$, the product of two projective lines. The $T_\eta$-fixed point locus in ${{\mathcal{F}}}$ has six components. The equivariant one-skeleton of ${{\mathbb{P}}}^1\times{{\mathbb{P}}}^1$ is identified with the edges and vertices of a square/diamond. Figure \[F:six\_cosets\] shows the equivariant one-skeleton of ${{\mathcal{F}}}$ with the equivariant one-skeleton of ${{\mathcal{F}}}^\eta$ drawn in bold, where the faces corresponding to components of ${{\mathcal{F}}}^\eta$ shaded.
$$\includegraphics[height=150pt]{pictures/A3-eta.eps}$$
The pattern map in equivariant cohomology {#S:equivariant}
=========================================
We give three formulas for the pullback $\iota_\varsigma^*\colon H^*_T({{\mathcal{F}}})\to \colon H^*_T({{\mathcal{F}}}')$ of the pattern map, one for each of our three presentations of equivariant cohomology.
As both $G$ and $G'$ have the same maximal torus $T$, the Borel presentations of equivariant cohomology of ${{\mathcal{F}}}$ and ${{\mathcal{F}}}'$ are nearly identical, $$H^*_T({{\mathcal{F}}})\ =\ S\otimes_{S^{W}} S
\qquad\mbox{and}\qquad
H^*_T({{\mathcal{F}}}')\ =\ S\otimes_{S^{W'}} S\,.$$
\[Th:Borel\_Coh\] Let $\varsigma$ be the minimal length representative of the coset $W'\varsigma$ of $W'$ in $W$ and $\iota_\varsigma\colon{{\mathcal{F}}}'\hookrightarrow{{\mathcal{F}}}$ be the corresponding section of the pattern map. The functorial map $\iota^*_\varsigma$ on equivariant cohomology is induced by the map $f\otimes g\mapsto f\otimes \varsigma.g$, where $\varsigma$ acts on $S$ through its right action on $\Xi(T)~\eqref{Eq:rightAction}$.
Since the left hand copy of $S$ in the Borel presentation of $H^*_T({{\mathcal{F}}})$ is simply $\rho^*(S)$, and the same for $H^*_T({{\mathcal{F}}}')$, we must have $\iota^*_\varsigma(f\otimes 1)=f\otimes 1$.
The right hand copy of $S$ in the Borel presentation of $H^*_T({{\mathcal{F}}})$ is generated by the equivariant Chern classes of equivariant line bundles ${{\mathcal{L}}}_\lambda$ on ${{\mathcal{F}}}$ and ${{\mathcal{F}}}'$. Proposition \[P:pullbackLineBunde\] shows that $\iota^*_\varsigma({{\mathcal{L}}}_\lambda)={{\mathcal{L}}}_{\varsigma.\lambda}$, which implies the theorem.
The simplest formula is for localization, for it is essentially restriction of $S$-valued functions. Let $\iota_\varsigma\colon ({{\mathcal{F}}}')^T\to{{\mathcal{F}}}^T$ be the restriction of the section $\iota_\varsigma$ to the $T$-fixed points. For $w\in W'$, we have $\iota_\varsigma(w.B')=w\varsigma.B$. As the only map between two points is an isomorphism, we have the following.
\[P:EC\_local\_restriction\] Let $\phi\in H^*_T({{\mathcal{F}}}^T)=\operatorname{Maps}(W,S)$. Then $\iota^*_\varsigma(\phi)\in H^*_T(({{\mathcal{F}}}')^T)=\operatorname{Maps}(W',S)$ is the map whose value at $w\in W'$ is $\phi(w\varsigma)$.
Thus if $\alpha\in H^*_T({{\mathcal{F}}})$ is represented as a map $\phi\colon W\to S$ via localization, so that its value at $u\in W$ is $\phi(u)=i^*_u(\alpha)$, then $\iota^*_\varsigma(\alpha)$ is the map $W'\to S$ whose value at $w\in W'$ is $i^*_{w\varsigma}(\alpha)=i^*_w(\iota^*_\varsigma(\alpha))$.
We also compute the pattern map in the Schubert basis.
\[Th:EC\_Schubert\] Let $\varsigma$ be the minimal length representative of the coset $W'\varsigma$ of $W'$ in $W$ and $\iota_\varsigma\colon{{\mathcal{F}}}'\hookrightarrow{{\mathcal{F}}}$ be the corresponding section of the pattern map. For $u\in W$, we have $$\iota^*_\varsigma({{\mathfrak{S}}}_u)\ =\ \sum_{w\in W'} c^{w\varsigma}_{u,\varsigma}\, {{\mathfrak{S}}}_w\,.$$
As the decomposition coefficients expressing the pullback of a Schubert class in the Schubert basis are Schubert structure constants, they exhibit Graham positivity.
Let $u\in W$. As Schubert classes ${{\mathfrak{S}}}_w$ for $w\in W'$ form an $S$-basis of $H^*_T({{\mathcal{F}}}')$, there are decomposition coefficients $d_u^w\in S$ defined by the identity $$\iota^*_\varsigma({{\mathfrak{S}}}_u)\ =\ \sum_{w\in W'} d_u^w\,{{\mathfrak{S}}}_w\,.$$ Using duality and the pushforward map to a point, we have $$\begin{aligned}
d_u^w&=& \rho_*(\iota^*_\varsigma({{\mathfrak{S}}}_u)\frown [X_w]^T)\\
&=& \rho_*({{\mathfrak{S}}}_u \frown \iota_{\varsigma,*}[X_w]^T)
\ =\ \rho_*({{\mathfrak{S}}}_u \frown [X^v\cap X_w]^T)\ =\ c^w_{u,v}\,,
\end{aligned}$$ By the projection formula and .
\[R:product\] The formula for $\iota_{\varsigma}^*({{\mathfrak{S}}}_u)$ in Theorem \[Th:EC\_Schubert\] gives an algorithm to compute it. First expand ${{\mathfrak{S}}}_u\cdot{{\mathfrak{S}}}_\varsigma$ in the Schubert basis of $H^*_T({{\mathcal{F}}})$. Restrict the sum to terms of the form ${{\mathfrak{S}}}_{w\varsigma}$ with $w\in W'$, and then replace ${{\mathfrak{S}}}_{w\varsigma}$ by ${{\mathfrak{S}}}_{w}$ to obtain the expression for $\iota_{\varsigma}^*({{\mathfrak{S}}}_u)$.
As noted following Proposition \[P:EC\_local\_restriction\], the map $\iota^*_\varsigma$ is particularly simple when expressed in terms of localization; it is essentially restriction. Theorem \[Th:EC\_Schubert\] implies that if we restrict a localized Schubert class $i^*({{\mathfrak{S}}}_u)$ to $W'\varsigma$, considering it as a class in $H^*_T(({{\mathcal{F}}}')^T)$, then it will be a sum of restrictions of Schubert classes ${{\mathfrak{S}}}_w$ for $w\in W'$ with coefficients the Schubert structure constants $c^{w\varsigma}_{u,\varsigma}$.
Similarly, if we have a Schubert class ${{\mathfrak{S}}}_u$ expressed as a sum of tensors $f\otimes g$ in the ring $S\otimes_{S^W}\! S$, then its pullback $\iota^*_\varsigma({{\mathfrak{S}}}_u)$ to $H^*_T({{\mathcal{F}}}')=S\otimes_{S^{W'}}\! S$ is the same sum, but where tensors $f\otimes g$ are replaced by $f\otimes(\varsigma.g)$. Expanding this in the basis of Schubert classes in $S\otimes_{S^{W'}}\! S$, it will have coefficients the Schubert structure constants $c^{w\varsigma}_{u,\varsigma}$.
In Section \[S:Ex\] we illustrate these interactions between the three formulas for $\iota^*_\varsigma$.
The pattern map in equivariant $K$-theory {#S:K-theory}
=========================================
We give three formulas for the pullback $\iota_\varsigma\colon K^0_T({{\mathcal{F}}})\to K^0_T({{\mathcal{F}}}')$ of the pattern map in equivariant $K$-theory. The Borel presentations for $K^0_T({{\mathcal{F}}})$ and $K^0_T({{\mathcal{F}}}')$ are nearly identical $$K^0_T({{\mathcal{F}}})\ =\ R(T)\otimes_{R(T)^W}\! R(T)
\qquad\mbox{and}\qquad
K^0_T({{\mathcal{F}}}')\ =\ R(T)\otimes_{R(T)^{W'}} R(T)\,.$$
Let $\varsigma$ be the minimal length representative of the coset $W'\varsigma$ of $W'$ in $W$ and $\iota_\varsigma\colon{{\mathcal{F}}}'\to{{\mathcal{F}}}$ be the corresponding section of the pattern map. The functorial map $\iota^*_\varsigma$ on equivariant $K$-theory is induced by the map $f\otimes g\mapsto f\otimes \varsigma.g$, where $W$ acts on $R(T)$ through its right action on $\Xi(T)~\eqref{Eq:rightAction}$.
As with equivariant cohomology, the simplest formula, both in terms of statement and proof, is for localized classes.
Let $\phi\in K^0_T({{\mathcal{F}}}^T)=\operatorname{Maps}(W,R(T))$. For $w\in W'$, we have $\iota^*_\varsigma(\phi)(w)=\phi(w\varsigma)$.
We also compute the pattern map in the Schubert basis.
Let $\varsigma$ be the minimal length representative of the coset $W'\varsigma$ of $W'$ in $W$ and $\iota_\varsigma\colon{{\mathcal{F}}}'\hookrightarrow{{\mathcal{F}}}$ be the corresponding section of the pattern map. For $u\in W$, we have $$\iota^*_\varsigma([{{\mathcal{O}}}^u])\ =\ \sum_{w\in W'} b^{w\varsigma}_{u,\varsigma}\, [{{\mathcal{O}}}^w]\,.$$
These decompositions coefficients expressing the pullback in terms of the Schubert basis are positive in the sense of Anderson, Griffeth and Miller [@AGM].
Let $u\in W$. As Schubert structure sheaves form a basis for the Grothendieck group, there are decomposition coefficients $d^w_u\in R(T)$ defined by the identity $$\iota^*_\varsigma([{{\mathcal{O}}}_{X^u}])\ =\ \sum_{w\in W'} d^w_u\,[{{\mathcal{O}}}_{X^w}]\,.$$ Using the pairing on $K$-theory, the pushforward formula, and the computation in Corollary \[C:pushforward\_formulae\], we have that $$\begin{aligned}
d^w_u &=& \rho_*(\iota^*_\varsigma([{{\mathcal{O}}}_{X^u}])\cdot [{{\mathcal{I}}}_w])
\ =\ \rho_*([{{\mathcal{O}}}_{X^u}]\cdot \iota_{\varsigma,*}([{{\mathcal{I}}}_w]))\\
&=& \rho_*([{{\mathcal{O}}}_{X^u}]\cdot [{{\mathcal{O}}}_{X^\varsigma}]\cdot [{{\mathcal{I}}}_{w\varsigma}])
\ =\ b^{w\varsigma}_{u,\varsigma}\,.
\end{aligned}$$ The last equality is .
Examples {#S:Ex}
========
We discuss two examples that illustrate our results for equivariant cohomology. The first continues Example 2.10 of [@AS], illustrating Theorem \[Th:EC\_Schubert\] and Remark \[R:product\], while the second illustrates the interplay between all three formulas for equivariant cohomology, Proposition \[P:EC\_local\_restriction\], Theorem \[Th:EC\_Schubert\], and Theorem \[Th:Borel\_Coh\]. All calculations not done explicitly by hand were carried out in Kaji’s Maple package that was released with [@Kaji].
Let $G:=Sp(8,{{\mathbb{C}}})$, the symplectic group of Lie type $C_4$, which is the subgroup of $GL(8,{{\mathbb{C}}})$ preserving the form $\langle x,y\rangle = \sum_{i=1}^4 x_iy_{4+i}-x_{4+i}y_i$. Then $g\mapsto \operatorname{diag}(g,(g^T)^{-1})$ embeds $G':=GL(4,{{\mathbb{C}}})$ into $G$ with image the centralizer of the 1-parameter subgroup $$T_\eta\ :=\
\{\operatorname{diag}(t,t,t,t\,,\, t^{-1},t^{-1},t^{-1},t^{-1})
\mid t\in{{\mathbb{C}}}^\times\}\,.$$ Both $G$ and $G'$ have the same torus, $T$, which we identify with diagonal $4\times 4$ matrices in $GL(4,{{\mathbb{C}}})$. Write $t_1,\dotsc,t_4$ for the standard weights of $T$.
The Weyl group of $G$ is the group of signed permutations. These are words $a_1\,a_2\,a_3\,a_4$, where the absolute values $|a_1|,\dotsc,|a_4|$ form a permutation in ${{\mathcal{S}}}_4$, and the identity element is $1\,2\,3\,4$. The length function is $$\ell(a_1\,a_2\,a_3\,a_4)\ =\ \#\{i<j\mid a_i>a_j\}\ +\ \sum_{a_i<0} |a_i|\,.$$ If we use $\overline{a}$ to represent $-a$, then $$\ell(3\,\overline{1}\,4\,2)\ =\ 4\,,\quad
\ell(\overline{2}\,\overline{3}\,4\,1)\ =\ 7\,,\quad
\mbox{and}\quad \ell(\overline{2}\,\overline{1}\,3\,4)\ =\ 3\,.$$
The right cosets of $W'={{\mathcal{S}}}_4$ are all words obtained by permuting the absolute values without changing signs, and consequently correspond to subsets $P$ of $\{1,\dotsc,4\}$ indicating the positions of the negative entries. Here are minimal length coset representatives $$\overline{2}\,\overline{1}\,3\,4\,,\
3\, \overline{2}\,4\,\overline{1}\,,\
2\,3\,\overline{1}\,4\,,\ \mbox{ and }\
\overline{3}\,4\,\overline{2}\,\overline{1}$$ that correspond to subsets $\{1,2\}$, $\{2,4\}$, $\{3\}$, and $\{1,3,4\}$, respectively.
Write for $u\in C_4$ for equivariant Schubert classes in this type $C_4$ flag manifold ${{\mathcal{F}}}$ and ${{\mathfrak{S}}}_w$ for $w\in S_4$ for equivariant Schubert classes in the type $A_3$ flag manifold ${{\mathcal{F}}}'$. Set $\varsigma=\overline{2}\,\overline{1}\,3\,4$ and consider $\iota^*_\varsigma({{\mathfrak{C}}}_{3\,\overline{1}\,4\,2})$. Following Remark \[R:product\], we first compute ${{\mathfrak{C}}}_{3\,\overline{1}\,4\,2}\cdot{{\mathfrak{C}}}_{\overline{2}\,\overline{1}\,3\,4}$. $$\begin{gathered}
{{\mathfrak{C}}}_{3\,\overline{1}\,4\,2}\cdot{{\mathfrak{C}}}_{\overline{2}\,\overline{1}\,3\,4}
\ =\ 2(t_1^2{+}t_1t_3){{\mathfrak{C}}}_{\overline{3}\,\overline{1}\,4\,2}
\ +\ 2(t_1{+}t_3){{\mathfrak{C}}}_{\overline{1}\,\overline{3}\,4\,2}
\ +\ 2t_1{{\mathfrak{C}}}_{\overline{4}\,\overline{1}\,3\,2}
\ +\ 2(t_1{+}t_2{+}t_3){{\mathfrak{C}}}_{\overline{3}\,\overline{2}\,4\,1}\\
\ +\ 2(t_1{+}t_2){{\mathfrak{C}}}_{3\,\overline{2}\,4\,\overline{1}}
\ +\ {{\mathfrak{C}}}_{\overline{3}\,\overline{2}\,4\,\overline{1}}
\ +\ 2{{\mathfrak{C}}}_{2\,\overline{3}\,4\,\overline{1}} \\
\;\ +\;\ 2{{\mathfrak{C}}}_{\overline{4}\,\overline{3}\,1\,2}
\ +\ 2{{\mathfrak{C}}}_{\overline{2}\,\overline{3}\,4\,1}
\ +\ 2{{\mathfrak{C}}}_{\overline{1}\,\overline{4}\,3\,2}
\ +\ 2{{\mathfrak{C}}}_{\overline{4}\,\overline{2}\,3\,1}\,.\qquad
\end{gathered}$$ As only the indices of the first four and last four terms have the form $w\varsigma$, we obtain $$\begin{gathered}
\iota_{\varsigma}^*\bigl({{\mathfrak{C}}}_{3\,\overline{1}\,4\,2}\bigr)\ =\
2(t_1^2+t_1t_3){{\mathfrak{S}}}_{1342}
\ +\ 2(t_1+t_3){{\mathfrak{S}}}_{3142}
\ +\ 2t_1{{\mathfrak{S}}}_{1432}
\ +\ 2(t_1+t_2+t_3){{\mathfrak{S}}}_{2341}\\
\ +\ 2{{\mathfrak{S}}}_{3412}
\ +\ 2{{\mathfrak{S}}}_{3241}
\ +\ 2{{\mathfrak{S}}}_{4132}
\ +\ 2{{\mathfrak{S}}}_{2431}\,.\end{gathered}$$ The last four terms were computed in Example 2.10 of [@AS] as $\iota^*_\varsigma({{\mathfrak{C}}}_{3\,\overline{1}\,4\,2})$ in cohomology.
Consider the localization formulae for $\iota^*_\varsigma$ when $G$ and $G'$ are as in Example \[Ex:six\_cosets\]. Then $G=SL(4,{{\mathbb{C}}})$, $G'=SL(2,{{\mathbb{C}}})\times SL(2,{{\mathbb{C}}})$ with $W={{\mathcal{S}}}_4$ and $W'={{\mathcal{S}}}_2\times{{\mathcal{S}}}_2$ with generators the reflections corresponding to the first and third simple roots $\alpha_{12}:=t_2-t_1$ and $\alpha_{34}:=t_4-t_3$. There are six cosets $W'\backslash W$ with minimal representatives $$\label{Eq:minRtReps}
\{1234\,,\, 1324\,,\, 3124\,,\, 1342\,,\, 3142 \,,\, 3412\}\,.$$ Figure \[F:2143\] shows the localization $i^*{{\mathfrak{S}}}_{2143}$ displayed with the weak order on ${{\mathcal{S}}}_4$,
(270,260) (0,0)[![Localization of ${{\mathfrak{S}}}_{2143}$[]{data-label="F:2143"}](pictures/S4-weak.eps "fig:"){height="250pt"}]{} (128,237)[$4321$]{} (131,247)
( 70,206)[$4312$]{} ( 67,216) (128,206)[$4231$]{} (131,216) (197,206)[$3421$]{} (194,216)
( 23,165)[$4213$]{} ( 20,175) ( 81,165)[$4132$]{} ( 78,175) (128,165)[$3412$]{} (125,175) (175,165)[$3241$]{} (172,175) (232,165)[$2431$]{} (229,175)
( 9,118)[$4123$]{} ( 6,128) ( 54,118)[$3214$]{} ( 61,128) (105,118)[$3142$]{} (102,128) (148,118)[$2413$]{} (145,128) (202,118)[$2341$]{} (199,128) (247,118)[$2341$]{} (254,128)
( 23,70)[$3124$]{} ( 30,80) ( 82,70)[$2314$]{} ( 89,80) (129,70)[$2143$]{} (126,80) (177,70)[$1423$]{} (184,80) (234,70)[$1342$]{} (241,80)
( 70,30)[$2134$]{} ( 77,40) (128,30)[$1324$]{} (135,40) (196,30)[$1243$]{} (203,40)
(128,0)[$1234$]{} (135,10)
which contains the equivariant one-skeleton of ${{\mathcal{F}}}^\eta$. This is indicated by thick edges with the darker edges corresponding to the root $\alpha_{12}$ and lighter to $\alpha_{34}$.
Since ${{\mathcal{F}}}'\simeq {{\mathbb{P}}}^1\times{{\mathbb{P}}}^1$, its equivariant one-skeleton is a diamond. In Figure \[F:P1P1\], we show five copies of the equivariant one-skeleton, the first labels the $T$-fixed points, and the remaining four give the localizations $i^*{{\mathfrak{S}}}_w$ for $w=1234,2134,1243, 2143$, in order.
(410,81)(-20,-15) (-20,-5)
(70,65)(-12,-8) (0,0)[![Localizations of equivariant Schubert classes on ${{\mathcal{F}}}'\simeq{{\mathbb{P}}}^1\times{{\mathbb{P}}}^1$.[]{data-label="F:P1P1"}](pictures/Moment_Graph_A1A1_big.eps "fig:")]{} (18,59) (-14,25) (49,25) (18,-9)
(80,0)
(70,65)(-12,-8) (0,0)[![Localizations of equivariant Schubert classes on ${{\mathcal{F}}}'\simeq{{\mathbb{P}}}^1\times{{\mathbb{P}}}^1$.[]{data-label="F:P1P1"}](pictures/Moment_Graph_A1A1.eps "fig:")]{} (20,48) (-8,20) (48,20) (20,-8) (3,-22)[$i^*{{\mathfrak{S}}}_{1234}$]{}
(160,0)
(70,65)(-12,-8) (0,0)[![Localizations of equivariant Schubert classes on ${{\mathcal{F}}}'\simeq{{\mathbb{P}}}^1\times{{\mathbb{P}}}^1$.[]{data-label="F:P1P1"}](pictures/Moment_Graph_A1A1.eps "fig:")]{} (16,49) (-13,21) (48,20) (20,-9) (3,-22)[$i^*{{\mathfrak{S}}}_{2134}$]{}
(240,0)
(70,65)(-12,-8) (0,0)[![Localizations of equivariant Schubert classes on ${{\mathcal{F}}}'\simeq{{\mathbb{P}}}^1\times{{\mathbb{P}}}^1$.[]{data-label="F:P1P1"}](pictures/Moment_Graph_A1A1.eps "fig:")]{} (16,49) (-8,20) (40,21) (20,-9) (3,-22)[$i^*{{\mathfrak{S}}}_{1243}$]{}
(320,0)
(70,65)(-12,-8) (0,0)[![Localizations of equivariant Schubert classes on ${{\mathcal{F}}}'\simeq{{\mathbb{P}}}^1\times{{\mathbb{P}}}^1$.[]{data-label="F:P1P1"}](pictures/Moment_Graph_A1A1.eps "fig:")]{} (9,49) (-8,20) (48,20) (20,-9) (3,-22)[$i^*{{\mathfrak{S}}}_{2143}$]{}
Each of the six components of ${{\mathcal{F}}}^\eta$ gives the localization of a pullback $\iota^*_\varsigma{{\mathfrak{S}}}_{2143}$ for $\varsigma$ a minimal representative . The diamond for $\varsigma=1234$ is simply the localization $i^*{{\mathfrak{S}}}_{2143}$ in Figure \[F:P1P1\]. This agrees with Remark \[R:product\], as ${{\mathfrak{S}}}_\varsigma=1$. Inspecting the diamond for $\varsigma=1324$, we see that $\iota^*_{1324}{{\mathfrak{S}}}_{2143}={{\mathfrak{S}}}_{2143}$, as $2413=2143\cdot 1324$. We also compute $$\label{Eq:1342}
\iota^*_{1342}\, i^*{{\mathfrak{S}}}_{2143}\ =\
\raisebox{-28pt}{\begin{picture}(80,65)(-25,-8)
\put(0,0){\includegraphics{pictures/Moment_Graph_A1A1.eps}}
\put(9,49){\scriptsize{$\alpha_{12}\alpha_{14}$}}
\put(-20,21){\scriptsize{$\alpha_{12}\alpha_{14}$}} \put(48,20){\scriptsize{$0$}}
\put(20,-9){\scriptsize{$0$}}
\end{picture}}
\ =\ \alpha_{14}\, i^*{{\mathfrak{S}}}_{2134}\,.$$ Most interestingly, $\iota^*_{3412}\, i^*{{\mathfrak{S}}}_{2143}$ is $$\begin{gathered}
\raisebox{-29pt}{\begin{picture}(83,65)(-20,-8)
\put(0,0){\includegraphics{pictures/Moment_Graph_A1A1.eps}}
\put(16,49){\scriptsize{$\alpha_{14}^2$}}
\put(-20,21){\scriptsize{$\alpha_{13}\alpha_{14}$}} \put(35,21){\scriptsize{$\alpha_{14}\alpha_{24}$}}
\put(9,-9){\scriptsize{$\alpha_{13}\alpha_{24}$}}
\end{picture}}
\ =\ \alpha_{13}\alpha_{24} \cdot
\raisebox{-29pt}{\begin{picture}(68,65)(-10,-8)
\put(0,0){\includegraphics{pictures/Moment_Graph_A1A1.eps}}
\put(20,48){\scriptsize{$1$}}
\put(-8,20){\scriptsize{$1$}} \put(48,20){\scriptsize{$1$}}
\put(20,-8){\scriptsize{$1$}}
\end{picture}}
\ +\ \alpha_{13} \cdot
\raisebox{-29pt}{\begin{picture}(75,65)(-13,-8)
\put(0,0){\includegraphics{pictures/Moment_Graph_A1A1.eps}}
\put(15,49){\scriptsize{$\alpha_{12}$}}
\put(-13,21){\scriptsize{$\alpha_{12}$}} \put(48,20){\scriptsize{$0$}}
\put(20,-9){\scriptsize{$0$}}
\end{picture}}
\ +\ \alpha_{24}\cdot
\raisebox{-29pt}{\begin{picture}(70,65)(-10,-8)
\put(0,0){\includegraphics{pictures/Moment_Graph_A1A1.eps}}
\put(16,49){\scriptsize{$\alpha_{34}$}}
\put(-8,20){\scriptsize{$0$}} \put(42,21){\scriptsize{$\alpha_{34}$}}
\put(20,-9){\scriptsize{$0$}}
\end{picture}}
\\
\ +\
\raisebox{-28pt}{\begin{picture}(64,65)(-8,-8)
\put(0,0){\includegraphics{pictures/Moment_Graph_A1A1.eps}}
\put(9,49){\scriptsize{$\alpha_{12}\alpha_{34}$}}
\put(-8,20){\scriptsize{$0$}} \put(48,20){\scriptsize{$0$}}
\put(20,-9){\scriptsize{$0$}}
\end{picture}}
\ \ =\ \alpha_{12}\alpha_{24}\cdot i^*{{\mathfrak{S}}}_{1234}\ +\
\alpha_{13}\cdot i^*{{\mathfrak{S}}}_{2134}\ +\
\alpha_{24}\cdot i^*{{\mathfrak{S}}}_{1243}\ +\ i^*{{\mathfrak{S}}}_{2143}\,,
\end{gathered}$$ as $\alpha_{13}\alpha_{24}+\alpha_{13}\alpha_{12}=\alpha_{13}(\alpha_{12}+\alpha_{24})=\alpha_{13}\alpha_{14}$ and similarily, $\alpha_{13}\alpha_{24}+\alpha_{24}\alpha_{34}=\alpha_{14}\alpha_{24}$, and we have $$\alpha_{13}\alpha_{24} + \alpha_{13}\alpha_{12} +\alpha_{24}\alpha_{34}+\alpha_{12}\alpha_{34}
\ =\
(\alpha_{13}+\alpha_{34})(\alpha_{12}+\alpha_{24})\ =\ \alpha_{14}^2\,.$$
We compare this computation to the formula of Theorem \[Th:EC\_Schubert\] as explained in Remark \[R:product\]. Here are the relevant products, $$\begin{aligned}
{{\mathfrak{S}}}_{2143}\cdot{{\mathfrak{S}}}_{1324} &=& {{\mathfrak{S}}}_{2413}\ +\
\Red{{{\mathfrak{S}}}_{4123}+{{\mathfrak{S}}}_{3142}+{{\mathfrak{S}}}_{2341}}\,,\\
{{\mathfrak{S}}}_{2143}\cdot{{\mathfrak{S}}}_{1342} &=& \alpha_{14}{{\mathfrak{S}}}_{2341}\ \ +\ \
\Red{\alpha_{24}{{\mathfrak{S}}}_{3142}+{{\mathfrak{S}}}_{3241}}\,,\ \mbox{and}\\
{{\mathfrak{S}}}_{2143}\cdot{{\mathfrak{S}}}_{3412} &=& \alpha_{13}\alpha_{24}{{\mathfrak{S}}}_{3412}+ \alpha_{13}{{\mathfrak{S}}}_{3421}
+ \alpha_{24}{{\mathfrak{S}}}_{4312} + {{\mathfrak{S}}}_{4321}\,.
\end{aligned}$$ Only the first term in ${{\mathfrak{S}}}_{2143}{{\mathfrak{S}}}_{1324}$ has index of the form $w\cdot 1324$, and for it, $w=2143$, so this agrees with the observation that $\iota^*_{1324}{{\mathfrak{S}}}_{2143}={{\mathfrak{S}}}_{2143}$. Similarly, only the first term in ${{\mathfrak{S}}}_{2143}{{\mathfrak{S}}}_{1342}$ has index of the form $w\cdot 1342$, and for it $w=2134$ and thus we have $\iota^*_{1342}{{\mathfrak{S}}}_{2143}=\alpha_{14}{{\mathfrak{S}}}_{2134}$, which agrees with . Finally, all terms in the product ${{\mathfrak{S}}}_{2143}{{\mathfrak{S}}}_{3412}$ contribute to $\iota^*_{3412}$, and they agree with our computation using localization.
We now look at these same pullbacks in the Borel formulation. Let us work with $G=GL(4,{{\mathbb{C}}})$ and $G'=GL(2,{{\mathbb{C}}})\times GL(2,{{\mathbb{C}}})$, and so $T\simeq({{\mathbb{C}}}^\times)^4$ are $4\times 4$ diagonal matrices (this has no effect on the geometry). Write $e_i(x)$ for the degree $i$ elementary symmetric polynomial in its arguments. Then we have $$H^*_T({{\mathcal{F}}})\ =\ {{\mathbb{Q}}}[t_1,\dotsc,t_4,z_1,\dotsc,z_4]/\langle e_i(t)-e_i(z)\mid i=1,\dotsc,4\rangle\,.$$ Here $t_i$ are characters of the torus and the equivariant Chern classes generate the image of ${{\mathbb{Q}}}[z_1,\dotsc,z_4]$. Similarly, $$H^*_T({{\mathcal{F}}}')\ =\ {{\mathbb{Q}}}[t_1,\dotsc,t_4,z_1,\dotsc,z_4]/
\langle e_i(t_1,t_2)-e_i(z_1,z_2)\,,\, e_i(t_3,t_4)-e_i(z_3,z_4)\mid i=1,\dotsc,2\rangle\,.$$ In $H^*_T({{\mathcal{F}}}')$ we have $${{\mathfrak{S}}}_{1234}\ =\ 1\,,\
{{\mathfrak{S}}}_{2134}\ =\ z_1{-}t_1\,,\
{{\mathfrak{S}}}_{1243}\ =\ z_3{-}t_3\,,\ \mbox{ and }\
{{\mathfrak{S}}}_{2143}\ =\ (z_1{-}t_1)(z_3{-}t_3)\,.$$ Observe that $z_1-t_1=t_2-z_2$, $z_3-t_3=t_4-z_4$ and $z_1z_2=t_1t_2$.
We have in $H^*_T({{\mathcal{F}}})$, $${{\mathfrak{S}}}_{2143}\ =\ (z_1-t_1)(z_1+z_2+z_3-t_1-t_2-t_3)\ =\ (z_1-t_1)(t_4-z_4)\,.$$ Then $$\begin{aligned}
\iota^*_{1324}{{\mathfrak{S}}}_{2134} &=&(z_1-t_1)(t_4-z_4)\ \ =\ {{\mathfrak{S}}}_{2134}\\
\iota^*_{1342}{{\mathfrak{S}}}_{2134} &=&(z_1-t_1)(t_4-z_2)\ \ =\ (z_1-t_1)t_4 - z_1z_2+z_2t_1
\ =\ (z_1-t_1)t_4 - t_1t_2+z_2t_1\\
&=& (z_1-t_1)t_4 - t_1(t_2-z_2) \ =\ (t_4-t_1)(z_1-t_1) =\ \alpha_{14}{{\mathfrak{S}}}_{2134}\\
\iota^*_{3412}{{\mathfrak{S}}}_{2134} &=& (z_3-t_1)(t_4-z_2)\ =\
(z_3-t_3 \;+\; t_3-t_1)(t_4-t_2\;+\; t_2-z_2)\\
&=& \bigl((t_3-t_1)+(z_3-t_3)\bigr)\bigl((t_4-t_2)+(z_1-t_1))\bigr)\\
&=&\alpha_{13}\alpha_{24}{{\mathfrak{S}}}_{1234}+ \alpha_{13}{{\mathfrak{S}}}_{2134}
+ \alpha_{24}{{\mathfrak{S}}}_{1243} + {{\mathfrak{S}}}_{2143}\,,
\end{aligned}$$ which agrees with our previous computations.
[10]{}
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