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+arXiv:2301.03959v1  [astro-ph.SR]  10 Jan 2023
+Astronomy & Astrophysics manuscript no. 45149corr
+©ESO 2023
+January 11, 2023
+Discovery of magnetic fields in five DC white dwarfs
+Andrei V. Berdyugin1, Vilppu Piirola1, Stefano Bagnulo2, John D. Landstreet2, 3, and Svetlana V. Berdyugina4, 5, 6
+1 Department of Physics and Astronomy, FI-20014 University of Turku, Finland; e-mail: [email protected]
+2 Armagh Observatory & Planetarium, College Hill, Armagh BT61 9DG, UK;
+3 Department of Physics & Astronomy, University of Western Ontario, London, Ontario N6A 3K7, Canada;
+4 Leibniz-Institut für Sonnenphysik (KIS), Schöneckstr 6, Freibirg, Germany;
+5 IRSOL Istituto Ricerche Solari “Aldo e Cele Daccò", Faculty of Informatics, Università della Svizzera italiana, Via Patocci 57,
+Locarno, Switzerland;
+6 Euler Institute, Faculty of Informatics, Università della Svizzera italiana, Via la Santa 1, 6962 Lugano, Switzerland
+Received October 7, 2022; accepted October 27, 2022
+ABSTRACT
+About half of white dwarfs (WDs) evolve to the DC state as they cool; the others become DQ or (temporarily?) DZ WDs. The recent
+magnetic survey of the local 20 pc volume has established a high frequency of magnetic fields among WDs older than 2–3 Gyr,
+demonstrating that in low- and average-mass WDs, the effects of magnetism become more common as they age, and the fields on
+average become stronger. However, the available statistics of WDs older than about 5 Gyr do not clearly establish how fields evolve
+beyond this age. We are carrying out a survey to clarify the occurrence of magnetism in DC-type WDs in order to better understand this
+late evolution. We use broadband filter polarimetry, arguably the most efficient way to detect magnetic fields in featureless WDs via
+continuum circular polarization. Here we report the discovery of a magnetic field in five DC WDs (of 23 observed), almost doubling
+the total sample of known magnetic WDs belonging to the DC spectral class.
+Key words. White dwarfs – Stars: magnetic fields – polarization
+1. Introduction
+Single stars of M <∼ 8M⊙ evolve to become white dwarfs (WDs).
+The descendants of these single stars of intermediate mass pro-
+vide most of the population of WDs, concentrated around the
+mean mass of 0.6M⊙. A smaller fraction of current WDs were
+also formed from close binary systems. Some of these systems
+eventually merged to form a single collapsed remnant, frequently
+of a significantly larger mass; others ended their nuclear lifetimes
+as double WD binaries.
+Once formed, the evolution of a WD is normally to cool
+slowly over several gigayears. Cooling is a fairly complex process
+even for single-star evolution, both to observe and to understand.
+Observationally, young hot WDs usually show strong spectra of
+H (DA WDs), He (DB WDs), or sometimes C (DQs). As they
+cool, spectral lines of the dominant elements H or He become
+weaker: He lines vanish at about 11000 K, H lines around 5000 K.
+In parallel with this general evolution, WDs may (temporarily)
+show lines of metals such as Mg, Si, Ca, and/or Fe (DZ, DAZ,
+and DZA stars), and some have spectra dominated by C. Below
+about 5000 K, about one-quarter of WDs have very weak Hα,
+another quarter show spectral lines of metals (especially Ca ii) or
+of C2, and the remaining half show essentially featureless spectra
+(DC WDs; Bagnulo & Landstreet 2021, Table 1). It appears that
+the dominant element(s) in the atmosphere can change as cooling
+occurs, for example due to gravitational diffusion, development
+of convection, and accretion of circumstellar planetary debris.
+One of the physical effects adding complexity to our ef-
+forts to understand WD evolution is that a significant frac-
+tion, about 20-25%, of WDs in the local volume near the Sun
+(Bagnulo & Landstreet 2021) possess detectable surface mag-
+netic fields (this high frequency was already suggested on the
+basis of literature reports of fields in nine WDs in the 13 pc vol-
+ume by Kawka et al. 2007). The fields observed at the surface
+range in strength, measured by the mean surface field ⟨|B|⟩, from
+tens of kG to hundreds of MG. Such fields can significantly affect
+WD evolution by altering or suppressing surface convection and
+internal shear, and by transferring angular momentum between
+internal layers or during accretion or mass loss (see for example
+Tremblay et al. 2015). The fields may also introduce additional
+forces into envelope and atmosphere layers, altering their hy-
+drostatic structure from that expected when magnetic effects are
+absent (Landstreet 1987).
+For WDs formed by single-star evolution, which generates
+most of the large populationsof WDs with masses around 0.6M⊙,
+it has become clear that recently formed WDs (with cooling
+ages of less than, say, 1 Gyr) are very rarely detectably mag-
+netic, and when they are magnetic, the fields are usually very
+weak (Bagnulo & Landstreet 2022). As WDs cool, fields begin
+to appear more frequently and usually become stronger. In WDs
+older than 3 or 4 Gyr, megagauss-scale fields are not uncommon
+(Bagnulo & Landstreet 2021).
+The observed evolution in magnetic field frequency and
+strength of normal-mass WDs for the first few gigayears of cool-
+ing may be understood as a slow emergence – as a result of field
+relaxation to the stellar surface – of the internal fields present
+in the degenerate cores of the WD precursors. An additional
+contribution to observed surface fields may be due to magnetic
+fields generated during cooling by a dynamo that acts during
+the period when the core of the WD is crystallising (Isern et al.
+2017; Gentile Fusillo et al. 2018). Beyond the end of crystalli-
+Article number, page 1 of 6
+
+A&A proofs: manuscript no. 45149corr
+sation, the only identified evolution mechanisms are continued
+field relaxation and Ohmic decay.
+Observationally, however, after about 5 Gyr of WD cooling,
+we have very limited information with which to guide and con-
+front theory. Only small survey samples constrain observed field
+evolution on cool WDs, such as DQ WDs, where C2 bands show
+no polarization in strong fields (Berdyugina et al. 2007). Partic-
+ularly little is known about the magnetic fields of DC WDs, in
+which no spectral features are seen at all, leading to the ques-
+tions of whether field strength begins to decay Ohmically and
+whether the frequency of surface fields continues to increase.
+Data that could help us answer these questions are very limited.
+For WDs within 20 pc of the Sun (the 20 pc volume sample),
+Bagnulo & Landstreet (2021) showed that of 31 DC WDs, only 4
+are magnetic white dwarfs (MWDs), and that only 4 of 24 WDs
+of any spectral class older than ∼ 6 Gyr are magnetic. These
+data are obviously too limited to clearly describe the evolution of
+fields in these old WDs.
+Previous surveys have provided almost no information about
+magnetic fields in DC WDs. Fields in such stars cannot be de-
+tected through the magnetic splitting of spectral lines. They can
+only be detected via the observation of continuum circular po-
+larisation (CCP; Kemp 1970), a method of observation hardly
+employed since the 1970s (Angel et al. 1981). Remarkably, most
+CCP observations have led to the discovery of magnetic fields
+in stars that are not featureless but in which the magnetic field
+is strong enough to shift and broaden spectral lines in a such
+a way as to make their intensity spectra unrecognisable. Only
+seven featureless DC WDs are presently known to be mag-
+netic. Five of them were discovered only in the last couple of
+years (Bagnulo & Landstreet 2020; Berdyugin et al. 2022). Be-
+fore these results, the only known magnetic DC stars were G195-
+19 and G111-49, discovered respectively by Angel & Landstreet
+(1971) and Putney (1995).
+To improve our knowledge of the magnetic fields in the latest
+stages of stellar evolution, we have started a volume-limited sur-
+vey of DC stars in the local 33 pc volume,which is about4.5 times
+larger than the previously explored 20 pc volume and should have
+a correspondingly larger sample of DCs and DC MWDs. With
+this sample we expect to find enough DC MWDs to delineate the
+evolution of their magnetic fields, both in the WDs with He-rich
+atmospheres that become DCs as soon as their effective tempera-
+tures reach about 11 000 K (‘young’ DCs), and in DC WDs with
+Teff below about 5000 K, with cooling ages of around 4 Gyr or
+more (‘old’ DCs).
+2. Observations
+Almost all known MWDs have been discovered via the mag-
+netic (Zeeman) splitting of spectral lines, observed in stellar
+flux spectra, or via the Zeeman polarisation of spectral features
+(Ferrario et al. 2015). Using these methods, fields of a few kG
+up to 1 GG can be reliably detected. However, these techniques
+cannot be used to measure fields in WDs that lack spectral lines.
+For such stars, it is necessary to rely on continuum polarisation,
+which Kemp (1970) showed should occur in radiation from a
+magnetized emitter. The value of this effect was confirmed by the
+discovery of a very strong field in the bright WD Grw+70 8247
+= WD 1900+705 through the detection of broadband circular po-
+larisation (BBCP) by Kemp et al. (1970).
+Broadband circular polarisation is a relatively weak effect.
+Bagnulo & Landstreet(2020) have estimated that a field of ⟨Bz⟩ ∼
+15 MG is required to produce BBCP of order 1 % in optical
+radiation from a cool WD. However, with a sensitive polarimeter,
+especially one with a very stable and well-established zero point,
+it is possible in principle to detect polarisation of 10−4 or less,
+corresponding to ∼ 100 kG fields in ‘sufficiently bright’ WDs.
+To detect and measure broadband continuum polarisation,
+one uses either spectropolarimetry or filter polarimetry with
+broad, photometry-like filters. It is very difficult to establish the
+zero point with sufficient accuracy below polarisation levels of
+the order of 10−3 in spectropolarimetric measurements ofthe con-
+tinuum (Fossati et al. 2007; Siebenmorgen et al. 2014); therefore,
+in DC WDs, only megagauss-scale fields can be detected in this
+way (Bagnulo & Landstreet 2020). In contrast, broadband filter
+polarimeters can be very stable, and instrumental polarisation
+can be calibrated at the 10−5 level, so detections with such instru-
+ments of fields of hundreds of kilogauss are in practice limited
+by the telescope aperture and WD brightness (Berdyugin et al.
+2022).
+The search for magnetic fields of a fraction of 1 MG or
+stronger in DC WDs that is reported here was carried out with
+the DIPol-UF broadband filter polarimeter (Piirola et al. 2020)
+mounted on the 2.5 m Nordic Optical Telescope (NOT) at the
+Observatorio del Roque de los Muchachos on the island of La
+Palma, in the Canaries. This instrument obtains simultaneous
+circular polarisation (normalized Stokes V/I) measurements in
+three filter bands isolated by dichroic mirrors. The passbands are
+centred at about 4450 Å (the B′ band), 5400 Å (the V′ band), and
+6400Å (the R′ band) with full widths at half maximum (FWHMs)
+of 1140, 750, and 960 Å , respectively. With this instrument on
+the NOT, we can detect a polarisation degree at the 3σ level of
+∼ 10−4 for the Gaia G-band magnitude G ∼ 12, down to a degree
+of ∼ 10−3 at G ∼ 17. This instrument and the filter system are
+discussed in more detail in our previous paper, which reports the
+results of the first part of our search for magnetic fields in DC
+WDs (Berdyugin et al. 2022).
+Here we report observations of 23 DC WDs and discovery
+of 5 new DC MWDs. We note that our survey includes nine
+young DCs of He-rich atmospheres with 11000 <∼ Teff <∼ 5000 K
+and 14 old WDs with Teff <∼ 5000 K and ages τ >∼ 4 Gyr. The
+stars are selected from available classifications and with help
+from Gentile Fusillo et al. (2021). Our new observations were
+obtained between June 27 and July 5, 2022.
+2.1. Instrumental polarisation and alignment of the
+polarimetric optics
+During our observing run we obtained seven observations of
+seven different bright nearby stars, which are believed to have
+zero circular polarisation, to check for instrumental polarisation.
+These observations are reported in Table 1.
+As in our previous run in July 2021, the high S/N measure-
+ments of non-polarised stars yield the instrumental polarisation
+to a precision better than 10−5. In the B′V′R′ bands, the values
+of Stokes V/I are 0.0121 ± 0.0004 %, 0.0109 ± 0.0005 %, and
+0.0084 ± 0.0004 %, respectively. These are very close to the
+values obtained in 2021. The instrumental polarisation was sub-
+tracted from the observed polarisation of all targets, including
+the measurements of the standard stars reported in Table 1.
+In addition, we obtained one measurement of the well-known
+MWD WD 1900+705, which appears to show a signal of cir-
+cular polarisation that is nearly constant with time (see e.g.
+Bagnulo & Landstreet2019). Our new measurementis compared
+in Table 1 to one of the same star that we made during the July
+2021 run. The agreement is very satisfactory and demonstrates
+that we can obtain measurements that are precise at the 0.02 %
+Article number, page 2 of 6
+
+Andrei V. Berdyugin et al.: Discovery of magnetic fields in five DC white dwarfs
+Table 1. Observing log of bright non-polarised standard stars and the highly polarised MWD WD 1900+705. Polarisation values are given assuming
+as instrumental polarisation the values of 0.0121±0.0004 %, 0.0109±0.0005 %, and 0.0084±0.0004 % in the B′, V′, and R′ filters, respectively. For
+comparison, we report the polarisation values of WD 1900+705 measured in our 2021 and 2022 runs.
+STAR
+G
+DATE
+UT
+JD –
+Exp.
+VI (%)
+yyyy-mm-dd
+hh:mm
+2400000
+(s)
+B′
+V′
+R′
+HD 107146
+6.9
+2022-06-27
+21:23
+59758.391
+1680
+0.0005±0.0007
+−0.0006±0.0007
+−0.0001±0.0006
+HD 115043
+6.7
+2022-06-28
+21:21
+59759.390
+1520
+0.0005±0.0008
+−0.0001±0.0012
+−0.0004±0.0004
+HD 122652
+7.0
+2022-06-29
+21:18
+59760.387
+1520
+−0.0012±0.0009
+−0.0015±0.0015
+−0.0019±0.0014
+HD 122676
+7.1
+2022-06-30
+21:17
+59761.387
+1520
+0.0000±0.0011
+0.0003±0.0010
+0.0018±0.0008
+HD 124694
+7.0
+2022-07-01
+21:16
+59762.386
+1520
+−0.0012±0.0008
+0.0014±0.0011
+0.0002±0.0007
+HD 135891
+6.9
+2022-07-02
+21:18
+59763.387
+1520
+0.0014±0.0008
+0.0006±0.0010
+−0.0004±0.0007
+HD 117860
+7.2
+2022-07-03
+21:16
+59764.386
+1520
+−0.0004±0.0010
+−0.0002±0.0012
+0.0005±0.0006
+WD 1900+705
+13.2
+2021-07-02
+22:22
+59398.432
+640
+3.756±0.016
+3.604±0.016
+3.827±0.019
+2022-07-02
+00:25
+59762.518
+3.789±0.016
+3.602±0.019
+3.838±0.018
+Table 2. Programme stars and their main physical features. Star names in boldface identify WDs in which fields were discovered during the
+observations reported in this paper (see Table 3).
+STAR
+G
+d
+Teff
+log g
+M
+Age
+Atmosphere and ref.
+(pc)
+(K)
+c.g.s.
+(M⊙)
+(Gyr)
+WD 0005+395
+LP 240-30
+16.6
+34.4
+4680
+6.77
+0.08
+1.40
+DC, H ProbWD 0.70 (1,3)
+WD 0010+543
+LSR J0013+5437
+18.0
+32.3
+4123
+7.77
+0.46
+7.08
+DC, (2, H assumed)
+WD 0028+035
+PB 6002
+16.1
+27.8
+6548
+8.14
+0.68
+2.40
+DC, (2, H assumed)
+WD 1251+366
+LP 267-311
+17.2
+28.5
+4445
+7.62
+0.37
+3.78
+DC, He (1)
+WD 1315+222
+LP 378-956
+16.7
+31.8
+6235
+8.21
+0.71
+3.61
+DCH, He (1)
+WD 1346+121
+LP 498-66
+17.8
+28.3
+4150
+7.88
+0.50
+6.58
+DCH, He (1)
+WD 1425+495
+CSO 649
+16.7
+33.9
+6895
+8.41
+0.85
+3.77
+DC, (2, H assumed)
+WD 1427−238
+LP 857-45
+17.4
+32.6
+4866
+7.90
+0.52
+5.40
+DC, (2, H assumed)
+WD 1434+437
+LP 221-217
+17.2
+27.2
+4685
+7.93
+0.54
+6.30
+DC, H-He (1)
+WD 1533+469
+LP 176-60
+17.8
+30.8
+4310
+7.83
+0.48
+6.45
+DC?, H (1)
+WD 1601−073
+LP 684-16
+17.9
+26.9
+4920
+8.55
+0.94
+9.83
+DCH, (2, H assumed)
+WD 1612+092
+LSPM J1614+0906
+17.2
+27.9
+4775
+7.90
+0.52
+5.57
+DC, H (1)
+WD 1702−016
+LP 626-29
+17.3
+28.3
+4700
+7.94
+0.54
+6.50
+DC, (2, H assumed)
+WD 1737+798
+LP 24-66
+16.9
+26.8
+5535
+8.28
+0.75
+5.72
+DC, He (1)
+WD 1746+450
+GD 366
+15.5
+29.9
+9331
+8.47
+0.90
+1.72
+DC, (2, H assumed)
+WD 1800+508
+LP 139-38
+17.4
+31.0
+4635
+7.85
+0.48
+5.12
+DC, He-H (1)
+WD 1853+775
+LP 25-7
+17.0
+30.5
+4850
+7.74
+0.43
+3.63
+DCH, He (1)
+WD 2058+550
+LSR J2059+5517
+17.1
+22.7
+4415
+7.93
+0.53
+7.15
+DC, H-He (1)
+WD 2109−295
+EC 21096-2934
+15.1
+32.8
+9260
+7.98
+0.57
+0.78
+DC, He-H (3)
+WD 2152−280
+LP930-61
+16.3
+23.5
+5220
+7.85
+0.48
+3.68
+DC, He (1)
+WD 2211+372
+LP 287-35
+16.8
+29.2
+6345
+8.47
+0.88
+4.56
+DC?H, He (1)
+WD 2215+368
+LP 287-39
+16.8
+20.3
+4485
+7.92
+0.53
+6.80
+DC, H (1)
+WD 2311−068
+G 157-34
+15.3
+25.9
+7360
+7.97
+0.56
+1.31
+DC, He (1)
+Key to references: 1: Blouin et al. (2019); 2: Gentile Fusillo et al. (2021); 3: Bergeron et al. (2021). Where not found in these
+references, ages have been interpolated using the tables from Bédard et al. (2020).
+level for a G = 13.2 star with about 10 minutes of exposure time.
+This shows that the alignment of our polarimetric optics is stable
+over a few years.
+2.2. Results
+The WDs observed during our 2022 June-July run are listed in
+Table 2, with their G magnitudes, distances, physical parameters,
+cooling ages, and some comments. Physical parameters were
+obtained from various studies, cited in the table’s notes; cooling
+ages are interpolated from the online cooling data provided by
+the Montreal group (Bédard et al. 2020).
+The observations are described in the log in Table 3, which
+gives dates, integration times, and the polarisation data in the
+three filter bands for each WD observation. We list measured
+BBCP values in boldface if non-zero polarisation is detected at
+above the 3σ level. We consider that real polarisation has been
+detected if a consistent picture of detection is found across the
+bands, and we highlight star names of WDs in which polarisation
+is convincingly detected in boldface in Tables2 and 3. Of the 23
+stars observed, BBCP has been definitely detected in 5 WDs. The
+data for these stars are plotted in Fig. 1.
+We observed three of the five WDs in which fields were de-
+tected in order to fully confirm the weak field detections and to
+check for possible variability. No variability is detected with con-
+fidence. There are in addition two further WDs, WD 1434+437
+and WD 1533+469, in which marginal polarisation detections
+have been obtained; these WDs await further observation to con-
+Article number, page 3 of 6
+
+A&A proofs: manuscript no. 45149corr
+Table 3. Observing log of WDs. Detections are marked in boldface.
+STAR
+DATE
+UT
+JD –
+Exp.
+VI (%)
+yyyy-mm-dd hh:mm
+2400000
+(s)
+B′
+V′
+R′
+WD 0005+395
+2022-07-06
+04:38
+59766.693 3900 −0.017±0.063 −0.082±0.056
+0.028±0.044
+WD 0010+543
+2022-07-05
+04:12
+59765.675 7100 −0.028±0.140 −0.011±0.129
+0.094±0.067
+WD 0028+035
+2002-07-06
+03:39
+59766.652 3300 −0.084±0.045 −0.051±0.050 −0.010±0.058
+WD 1251+366
+2022-06-27
+22:41
+59758.445 5200 −0.155±0.065
+0.006±0.063
+0.039±0.038
+WD 1315+222
+2022-06-28
+22:25
+59759.435 4200
+0.104±0.045
+0.182±0.052
+0.215±0.064
+2022-07-01
+22:24
+59762.434 4200
+0.084±0.040
+0.253±0.061
+0.199±0.044
+WD 1346+121
+2022-07-02
+22:43
+59763.446 6500 −0.508±0.093
+0.044±0.091 −1.074±0.054
+2022-07-04
+22:34
+59765.44
+6500 −0.691±0.109
+0.222±0.098 −1.256±0.062
+WD 1425+495
+2022-06-29
+22:22
+59760.432 4200
+0.012±0.045 −0.002±0.058
+0.018±0.038
+WD 1427-238
+2022-06-30
+23:11
+59761.466 5600 −0.038±0.114
+0.102±0.078 −0.044±0.061
+WD 1434+437
+2022-06-30
+00:04
+59760.503 5200
+0.231±0.077 −0.019±0.067 −0.019±0.059
+WD 1533+469
+2022-07-01
+01:02
+59761.543 6600 −0.065±0.127 −0.273±0.110 −0.195±0.052
+2022-07-03
+00:42
+59763.529 6600
+0.139±0.098 −0.300±0.090 −0.019±0.042
+WD 1601-073
+2022-07-05
+22:52
+59766.453 6800 −0.484±0.111 −0.386±0.106
+1.597±0.054
+WD 1612+092
+2022-06-28
+00:59
+59758.541 5100
+0.087±0.069
+0.033±0.052
+0.097±0.046
+WD 1702-016
+2022-06-30
+01:59
+59760.582 5300
+0.215±0.091
+0.087±0.086 −0.104±0.053
+WD 1737+798
+2022-06-29
+01:34
+59759.566 4500 −0.014±0.082 −0.085±0.093 −0.071±0.056
+WD 1746+450
+2022-06-28
+02:12
+59758.592 2600
+0.012±0.035 −0.049±0.032 −0.074±0.035
+WD 1800+508
+2022-07-05
+00:59
+59765.541 5600 −0.157±0.073 −0.047±0.061
+0.051±0.053
+WD 1853+775
+2002-07-06
+01:15
+59766.552 4800 −0.098±0.066 −0.680±0.068 −0.492±0.039
+WD 2058+550
+2022-07-02
+04:15
+59762.677 5000
+0.068±0.075 −0.052±0.070
+0.064±0.045
+WD 2109-295
+2022-07-01
+03:51
+59761.660 2200
+0.011±0.020
+0.036±0.034 −0.039±0.037
+WD 2152-280
+2022-06-28
+03:59
+59758.666 3600
+0.007±0.040 −0.055±0.041 −0.043±0.022
+WD 2211+372
+2022-07-02
+02:10
+59762.590 4400
+1.254±0.041
+0.703±0.054
+0.446±0.044
+2022-07-03
+04:24
+59763.683 4400
+1.333±0.038
+0.623±0.044
+0.285±0.040
+WD 2215+368
+2022-06-30
+04:12
+59760.675 4400
+0.187±0.083
+0.133±0.068
+0.101±0.044
+WD 2311-068
+2022-06-29
+04:38
+59759.693 2400 −0.011±0.028
+0.011±0.039
+0.032±0.032
+firm (or not) the fields that may have been detected. However,
+a single pair of measurements does not probe all the possible
+timescales of variation; in particular, our measurements require
+integration of the order of one hour and so cannot probe all the
+rotation periods that might result from the formation of a MWD
+from a close binary.
+With these new discoveries, we almost double the number of
+DC WDs in which magnetic fields have been detected. One of
+the new MWDs discovered, LP 684-16 = WD 1601–073, is quite
+massive compared to most of the rest of the DC WDs observed.
+Therefore, because of its relatively small radius, it has cooled
+quite slowly, reaching only Teff = 4920 K, but has a computed
+cooling time of 9.8 Gyr. It is probably the oldest magnetic WD of
+any spectral type discovered so far. For comparison, according to
+the parameters listed by Bagnulo & Landstreet (2021), the oldest
+MWD in the 20 pc volume, in which such old MWDs are most
+likely to be discovered, is WD 1008+290 = LHS 2229. It is a
+DQpec star with an age of about 7.9 Gyr, almost 2 Gyr younger
+than LP 684-16.
+We note that no really large polarisation signals, such as that
+exhibited by WD 1900+705 (see Table 1), are found. However,
+the observed level of polarisation in three of the five definite
+detections reaches the range 1.2 to 1.6%, so some of the fields
+detected are probably quite strong.
+3. Discussion and conclusions
+We continue to detect MWDs in roughly one-fifth of the DC
+sample observed. Considering that only relatively strong fields
+can be detected in featureless stars, our results suggest that the
+frequency of the occurrence of magnetic fields in older WDs
+may be as high as 25 or even 30 %, consistent with the frequency
+suggested by Bagnulo & Landstreet (2021).
+Polarisation levels in the seven DC MWDs discovered
+by Berdyugin et al. (2022) and in this paper range from
+about 0.1 to 1.6 %. Using the order-of-magnitude estimator of
+Bagnulo & Landstreet (2020) of a longitudinal field of 15 MG,
+which leads to BBCP of the order of 1%, inferred fields ⟨Bz⟩
+are thus estimated to lie between perhaps 1 and 30 MG. From
+this result, the fields ⟨|B|⟩ that we detect likely lie in the range of
+roughly 3 to 200 MG.
+Some of the fields produce a polarisation with the same sign
+in the three filter bands, while in other stars we detect a polar-
+isation that reverses sign between one filter band and another
+(Fig. 1). Similar behaviour was found in our earlier survey data
+(Berdyugin et al. 2022) as well as in other strongly magnetic old
+WDs (Angel & Landstreet 1971; Angel et al. 1974, 1975; Putney
+1995).
+Article number, page 4 of 6
+
+Andrei V. Berdyugin et al.: Discovery of magnetic fields in five DC white dwarfs
+Fig. 1. Wavelength dependence of circular polarisation detected for five
+targets (> 3σ confidence level). A wide variety of polarisation behaviour
+is observed. Horizontal bars in the bottom panel show the FWHM of the
+B′V′R′ filter passbands.
+We carried out a second observation for three of our five new
+discoveries and for one suspected candidate. The confirming ob-
+servations were obtained between one and three days after the
+discovery observations. For each of these four stars, the repeated
+observation confirms the presence of the magnetic field first de-
+tected, except for one R′ observationof WD 1533+469.In no case
+do we detect clearly significant variability; we note, however, that
+our repeated measurements probe only a very limited range of
+timescales.
+We find that, in practice, a magnetic field can be reliably
+detected from BBCP at the polarization levels of approximately
+0.2% with our broadband filter polarimeter on a 2.5 m telescope
+in a DC WD of G ∼ 17. We would gain about a factor of 3 in
+precision, or a 2.5 magnitude increase in limiting magnitude, by
+going to an 8 m telescope.
+To summarise the current statistical situation, we com-
+bine the results from the 20 pc volume magnetic field survey
+(Bagnulo & Landstreet 2021), our exploratory observing run
+(Berdyugin et al. 2022), and this work. We have collected lit-
+erature data on and surveyed 30 young DC stars, of which 7 have
+been found to host magnetic fields, and 43 old DCs, of which 6
+are magnetic.
+As more detections of DC MWDs are made, especially in
+the context of volume-limited surveys such as ours, comparisons
+with magnetic data for other types of WDs will at first be ham-
+pered by our current inability to assign precise field strength
+values to magnetic DC WDs. However, a first comparison can be
+made between the overall level of polarisation observed in young
+and old DC MWDs, which may be taken as an indicator of the
+evolution of the overall field strength with cooling age between
+these two populations. In the currently small sample, it appears
+that larger polarisation levels (say, above Stokes V/I >∼ 1 %) ap-
+pear to be at least as common in the old DC MWD population
+as in the young group. There is no clear signal of Ohmic field
+strength decay. However, the available sample is still very small,
+and as our sample increases we can hope to obtain a statistically
+more significant constraint and carry out modelling to provide
+more accurate field strength estimates corresponding to observed
+polarisation. Such surveys need to be carried on until a clearer
+statistical view of the magnetism in DC WDs is obtained.
+Acknowledgements. Based on observations made with the Nordic Optical Tele-
+scope, owned in collaboration by the University of Turku and Aarhus University,
+and operated jointly by Aarhus University, the University of Turku and the Uni-
+versity of Oslo, representing Denmark, Finland and Norway, the University of
+Iceland and Stockholm University at the Observatorio del Roque de los Mucha-
+chos, La Palma, Spain, of the Instituto de Astrofisica de Canarias. DIPol-UF
+is a joint effort between University of Turku (Finland) and Leibniz Institute for
+Solar Physics (Germany). We acknowledge support from the Magnus Ehrnrooth
+foundation and the ERC Advanced Grant HotMol ERC-2011-AdG-291659. JDL
+acknowledges the financial support of the Natural Sciences and Engineering
+Research Council of Canada (NSERC), funding reference number 6377-2016.
+4. Data Availability
+All raw data and calibrations are available on request from the
+authors.
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+

39AzT4oBgHgl3EQfffxn/content/tmp_files/2301.01453v1.pdf.txt

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+1
+Information-Theoretic Secure Key Sharing for Wide-Area Mobile
+Applications
+Guyue Li, Member, IEEE, Hongyi Luo, Jiabao Yu, Aiqun Hu, Senior Member, IEEE and Jiangzhou Wang, Fellow, IEEE
+With the rapid growth of handheld devices in the internet
+of things (IoT) networks, mobile applications have become
+ubiquitous in everyday life. As technology is developed, so do
+also the risks and threats associated with it, especially in the
+forthcoming quantum era. Existing IoT networks, however, lack a
+quantum-resistant secret key sharing scheme to meet confidential
+message transmission demands in wide-area mobile applications.
+To address this issue, this article proposes a new scheme, channel
+reciprocity (CR) based quantum key distribution (QKD) CR-
+QKD, which accomplishes the goal of secret key sharing by
+combining emerging techniques of QKD and CR-based key
+generation (CRKG). Exploiting laws of quantum physics and
+properties of wireless channels, the proposed scheme is able
+to ensure the secrecy of the key, even against computationally
+unbounded adversaries. The basic mechanism is elaborated for
+a single-user case and it is extended into a multi-user case by
+redesigning a multi-user edge forwarding strategy. In addition, to
+make CR-QKD more practical, some enhancement strategies are
+studied to reduce the time delay and to improve the secret key
+generation rate in a secure manner. A prototype of CR-QKD
+is demonstrated in a metropolitan area network, where secret
+keys are shared between two remote IoT devices that are roughly
+fifteen kilometers apart from each other. The experimental results
+have verified that CR-QKD allows a secret key rate of 424 bits
+per second with a retransmission rate of 2.1%.
+Index Terms—Secret key generation, physical layer security,
+quantum key distribution, wide-area mobile applications, Inter-
+net of Things
+I. INTRODUCTION
+Recent years have witnessed a remarkable growth in the
+number and variety of mobile devices and applications in
+the Internet of Things (IoT) networks. The flourish of IoT,
+however, has resulted in the generation of a substantial amount
+of private messages exchanged over public channels, which
+has grabbed one’s attention. Unfortunately, IoT devices are
+susceptible to various threats and security challenges, which
+pose hazards for the advancement of IoT in sensitive fields,
+such as smart homes, unmanned vehicles, e-health, and mili-
+tary networks [1]. In order to avoid being revealed to a third
+Guyue Li and Hongyi Luo are with the School of Cyber Science
+and Engineering, Southeast University, Nanjing 210096, China (e-mail:
+[email protected]; [email protected]).
+Jiabao Yu is with the Purple Mountain Laboratories, Nanjing 210096, China
+(e-mail:[email protected]).
+Aiqun Hu is with National Mobile Communications Research Laboratory,
+Southeast University, Nanjing 210096, China (e-mail: [email protected]).
+Jiangzhou Wang is with the School of Engineering, University of Kent,
+Canterbury CT2 7NT, U.K. Email: (e-mail: [email protected]).
+Guyue Li and Aiqun Hu are also with Purple Mountain Laboratories,
+Nanjing 210096, China.
+party, a message is usually encrypted using a secret key shared
+among the communicating devices. Thus, a key prerequisite
+of achieving IoT network security is secret key sharing that
+avoids eavesdropper interception [2].
+In a classic cryptographic scheme, two legitimate parties,
+namely Alice and Bob use the public-key cryptosystem (PKC)
+for key distribution. It is extremely difficult for a third party,
+namely Eve, to derive the private key or message compu-
+tationally, due to the intractability of certain mathematical
+problems used in encryption algorithms. However, the emerg-
+ing quantum computing technology has the potential to make
+some previously-intractable problems tractable [3]. Thus, the
+security of computational security-based key distribution will
+be rendered insecure by substantial progress in quantum
+computing in the coming years, which necessitates the study of
+alternative solutions that do not rely on computational security.
+In this context, much attention has been paid to emerging
+techniques, such as quantum key distribution (QKD) [4] and
+channel reciprocity-based key generation (CRKG) [5], which
+can provide secret key sharing service with information-
+theoretic security, also known as unconditional security or
+physical security.
+• QKD is a well-known quantum-resistant mechanism,
+which distributes secret keys to distant parties by trans-
+mitting single photon through a quantum channel [6].
+Employing the laws of quantum physics, QKD can de-
+tect eavesdroppers during the key generation process,
+in which unauthorized observation of quantum commu-
+nication induces a discernible increase of errors. This
+sensitivity to eavesdropping makes QKD possible to en-
+sure the secrecy of the key, even against computationally
+unbounded adversaries.
+• CRKG is built on the basis of channel reciprocity, which
+means that the channel responses of the forward and
+backward communication links are very similar in a time
+division duplex (TDD) system. In addition, the dynamic
+and complex wireless communication environment makes
+the channel responses change over time and hard to
+predict. Therefore, legitimate users can share a pair of
+common randomness from their radio channel measure-
+ments. Since CRKG does not require assistance from a
+third party nor expensive infrastructure, it has recently
+emerged as a new paradigm that provides a lightweight
+and information-theoretic secure key sharing solution for
+decentralized or device-to-device sensor applications [7].
+Table I summarizes these typical secret key distribution
+arXiv:2301.01453v1  [cs.IT]  4 Jan 2023
+
+2
+TABLE I
+A SUMMARY AND COMPARATION OF TYPICAL SECRET KEY DISTRIBUTION METHODS
+Method
+Metric
+Security Level
+Mobility Support
+Distribution Distance
+User Cost
+PKC
+Computational secure
+Middle
+Long
+Middle
+QKD
+Information theoretically secure
+Weak
+Long
+High
+CRKG
+Information theoretically secure
+Strong
+Short
+Low
+CR-QKD
+Information theoretically secure
+Strong
+Long
+Low
+methods, and identify their characteristics from perspectives of
+security level, mobility support, distribution distance and user
+cost. We find that although the separate construction of QKD
+and CRKG can be supported in the physical layer, there is no
+investigation of a secret key sharing scheme for the security
+demands from remote mobile devices. Although point-to-point
+connections are suitable to form a backbone quantum core
+network to bridge long distances, they are less suitable to
+provide the last-mile service needed to give a multitude of
+users access to this QKD infrastructure [6]. Similarly, despite
+many research efforts in the field of CRKG, its widespread
+application is unfortunately hindered by the short distance
+between transceivers. With a rapid growth of handheld devices,
+wide-area mobile applications, such as remote environmental
+and elderly monitoring, have become an inseparable part of
+IoT networks. A new architecture needs to be developed
+where end-users between two access networks are connected
+to a metro network, thus realizing unconditionally secure key
+sharing in a more cost-effective and flexible manner.
+In this article, we introduce and experimentally demonstrate
+the concept of a ‘channel reciprocity-aided quantum key
+distribution (CR-QKD)’ based on simple and cost-effective
+telecommunication technologies. This scheme can expand the
+scope of QKD to IoT networks and therefore vastly broaden
+users’ appeal. The contributions of this article are three-fold:
+• We introduce a novel secret key sharing architecture, re-
+ferred to as CR-QKD, which bridges a backbone quantum
+core network and IoT users by exploiting the technique
+of CRKG to provide the last-mile service. CR-QKD is
+information-theoretically secure and it does not require
+IoT users to be equipped with expensive quantum infras-
+tructures for exchanging secret keys, thereby significantly
+reducing the hardware requirements.
+• We propose a multi-user mechanism to realize the con-
+cept of CR-QKD with an elaborate design of key align-
+ment. We also identify challenges that arise due to the
+hybrid architecture of CR-QKD from the perspective
+of feasibility and security, respectively. Countermeasures
+have been studied to reduce the time delay and to improve
+the secret key generation rate in a secure manner.
+• We implement a prototype CR-QKD system in a
+metropolitan area network, in which secret keys are
+shared between two remote IoT devices that are roughly
+fifteen kilometers apart from each other. The experimental
+results have verified that CR-QKD can provide a secret
+key rate of 424 bits per second with a retransmission rate
+of 2.1%.
+II. AN OVERVIEW OF THE CR-QKD ARCHITECTURE
+In this section, we first introduce QKD and CRKG, and then
+discuss their combination modes to realize secure communi-
+cation in wide-area mobile applications.
+A. QKD
+QKD protocols exploit a quantum communication channel
+and an authenticated classical channel to ensure the exchange
+of a cryptographic key between two remote parties with proven
+security. Since its inception in [8], QKD protocol design
+and analyses have flourished as a field yielding numerous
+protocols, security analyses, and practical implementation
+methodologies. Although QKD research has made remarkable
+progress, these developments have been largely focused on
+securing large-scale infrastructures using long distance fiber
+transmission and free space transmission between fixed ter-
+minals. Some efforts have been made toward handheld free-
+space QKD by exploiting a beam-steering module, which
+compensates for hand movement of the QKD module at
+the transmitter [9, 10]. However, these schemes have limited
+transmission range and their QKD receiver is currently difficult
+to be miniaturized. In other words, they can not provide a bi-
+directional transmission and are thus not applicable to the case
+of distributing a quantum key from a core network to an end-
+user. In this article, QKD is exploited to form a backbone
+quantum core network to bridge long distances.
+B. CRKG
+CRKG exploits wireless channels between transceivers as
+random sources for key generation, and these keys can be
+replenished dynamically as wireless channels vary over time.
+Eavesdroppers in such situations experience physical channels
+independent of those of the legitimate users as long as they
+are a few wavelengths away from these legitimate parties,
+which is generally the case in wireless networks. So far,
+the CRKG field has yielded fruitful results from aspects of
+theoretical exploration, modeling, protocol design, and proto-
+type implementation in various IoT platforms [11]. However,
+these developments have been largely focused on wireless
+communication technologies for short-range applications, such
+as ZigBee, ultra-wideband, Bluetooth and WiFi. When the
+distance is in the order of a few kilometers, the signal-to-
+noise ratio is small and the time delay between uplink and
+downlink packets becomes large. Therefore, CRKG at a long
+distance is challenging to meet the requirement of high cor-
+relation between channel parameter measurements for secret
+key generation [11]. Due to these reasons, CRKG is more
+suitable for secret key sharing between wireless transceivers
+
+3
+Alice
+Bob
+K𝑪𝟏
+K𝑪𝟏
+KQ
+KQ
+K𝐂𝟐
+K𝐂𝟐
+QAP1
+QAP2
+Fig. 1. An illustration of combining QKD and CRKG to realize secure communication between two remote users.
+that are within one kilometer apart and thus exploited in this
+article to complete the last-mile secret key distribution task
+from quantum access points (QAP) to IoT users.
+C. The Combination Mode of QKD and CRKG
+Neither QKD nor PKG is applicable to long-range IoT
+networks, therefore, a critical problem is how to combine their
+advantages to apply to the new scenario. Fig. 1 describes the
+system model and illustrates one possible combination mode.
+Alice and Bob are two distant wireless users, who do not have
+direct links with each other. QAP1 and QAP2 are two quantum
+nodes that are connected through long-distance optical fibers,
+or ground-to-satellite free-space links. QAP1 and QAP2 have
+a wireless link to Alice and Bob, respectively.
+To complete the secret key distribution between Alice and
+Bob, three keys are first shared between Alice and QAP1 (link
+1), QAP1 and QAP2 (link 2), and QAP2 and Bob (link 3).
+Channel keys are generated from wireless links 1 and 3 by us-
+ing the technique of CRKG, while quantum key is distributed
+from QAP1 to QAP2, or in the reverse direction, with mature
+QKD techniques. Next, the quantum key is securely delivered
+to Alice and Bob by encrypting it with channel keys. In other
+words, Alice and Bob share a unified key, which is then used to
+encrypt and to decrypt the message in the data transmissions.
+Therefore, this mode is also abbreviated as unified-key mode.
+Notably, Alice and Bob are free to choose wireless and Internet
+routes for message transmission. This consideration is due to
+the following reasons. First, due to the limited rate of the
+quantum link, its message transmission rate is relatively small.
+Second, as Alice and Bob are mobile devices, they are more
+likely to use communication routes that are different from
+those in the key distribution process. Finally, the unified-key
+requires less time delay for message transmission as it only
+needs one time of message encryption and decryption. The
+essential process to obtain unified quantum keys is referred to
+as CR-QKD, which is elaborated in the following section.
+III. CONCEPTUAL DESIGN OF CR-QKD
+In this section, we will first introduce the basic mechanism
+of CR-QKD and then study the key aligment, efficiency and
+security issues that exist in CR-QKD.
+A. Mechanism description
+As shown in Fig. 1, Alice and Bob intend to share quantum
+key with the help of QAP1 and QAP2, against an adversarial
+eavesdropper, Eve, tapping on the quantum channel and lis-
+tening to all the exchanges on the classical channels. Similar
+to most existing QKD and CRKG protocols, the classical
+communication channels are assumed to be authenticated,
+in which the identities of the communicating parties have
+been verified and the integrity of the transmitted messages
+is promised.
+The CR-QKD protocol comprises three main phases, i.e,
+QKD 1, CRKG and edge forwarding, which will be elaborated
+below.
+• QKD phase: First, QAP1 prepares and sends to QAP2
+a set of random qubits via a single-photon signal over a
+quantum channel. These qubits are selected from a set of
+four states with two bases. For every incoming state from
+QAP1, QAP2 randomly chooses one of the two bases to
+measure and record the results. Once quantum commu-
+nication has finished, QAP2 starts base reconciliation by
+announcing the position of the detected bits and the basis
+used to QAP1 over a classic channel. Then, QAP1 and
+QAP2 retain the bits with a coincident basis and discard
+the rest. After that, QAP1 publishes a subset of these bits
+to QAP2 for eavesdropping detection. If the error rate
+between what QAP2 detects and what QAP1 has sent
+is high, the eavesdropping is detected and these shared
+bits will be invalid. Otherwise, QAP1 and QAP2 perform
+information reconciliation and privacy amplification over
+the rest of the bits that have not been made public. At
+last, QAP1 and QAP2 check whether they obtain the same
+result via key verification. If so, they retain the pair of
+bits as quantum key KQ, otherwise, they discard both of
+them.
+• CRKG phase: A CRKG protocol typically contains four
+stages, i.e., channel probing, quantization, information
+reconciliation, and privacy amplification. Alice and QAP1
+first carry out channel probing, which involves bidirec-
+tional measurements within a channel coherence time.
+They then convert the analog measurements into digital
+1Our study is not bound to specific QKD protocols, and we choose the
+BB84 protocol as a representative to introduce the CR-QKD mechanism.
+
+4
+Pair
+𝐴𝑀1-𝐵1
+...
+𝐴𝑀1-𝐵1
+...
+𝐴𝑀1-𝐵𝑀2
+Number
+1
+...
+෍
+𝑖=1
+𝑀𝑖
+𝑁𝐴𝑖 +1
+...
+𝑁𝐴𝑀1
+𝐾𝑄
+1110…010
+...
+0011…010
+...
+1011…111
+Pair
+𝐴2-𝐵1
+...
+𝐴2-𝐵1
+...
+𝐴2-𝐵𝑁
+Number
+𝑁𝐴1+
+...
+𝑁𝐴2,𝐵1
+...
+𝑁𝐴2
+𝐾𝑄
+1110…010
+...
+0011…010
+...
+1011…110
+𝐴1
+𝑲𝑸
+𝑲𝑸
+𝑄𝐴𝑃1
+𝑄𝐴𝑃2
+Quantum 
+Key Buffer
+Quantum 
+Key Buffer
+𝐴2
+𝐴𝑀1
+...
+...
+𝐵1
+𝐵2
+𝐵𝑀2
+...
+...
+Request
+Request
+Pair
+𝐴1-𝐵1
+...
+𝐴1-𝐵1
+...
+𝐴1-𝐵𝑀2
+Number
+1
+...
+𝑁𝐴1,𝐵1
+...
+𝑁𝐴1
+𝐾𝑄
+1010…010
+...
+1011…010
+...
+1011…011
+Pair
+𝐴2-𝐵1
+...
+𝐴2-𝐵1
+...
+𝐴2-𝐵𝑁
+Number
+𝑁𝐴1+
+...
+𝑁𝐴2,𝐵1
+...
+𝑁𝐴2
+𝐾𝑄
+1110…010
+...
+0011…010
+...
+1011…110
+Pair
+𝐴1-𝐵1
+...
+𝐴1-𝐵1
+...
+𝐴1-𝐵𝑀2
+Number
+1
+...
+𝑁𝐴1,𝐵1
+...
+𝑁𝐴1
+𝐾𝑄
+1010…010
+...
+1011…010
+...
+1011…011
+Pair
+𝐴1-𝐵𝑀2
+𝐴2-𝐵𝑀2
+...
+𝐴𝑀1-𝐵𝑀2
+Number
+𝑁𝐴1
+𝑁𝐴1 + 𝑁𝐴2
+...
+෍
+𝑖=1
+𝑀1 𝑁𝐴𝑖
+𝐾𝑄
+1011…011
+1111…010
+...
+1011…111
+Pair
+𝐴𝑀1-𝐵1
+...
+𝐴𝑀1-𝐵1
+...
+𝐴𝑀1-𝐵𝑀2
+Number
+1
+...
+෍
+𝑖=1
+𝑀𝑖
+𝑁𝐴𝑖 +1
+...
+𝑁𝐴𝑀1
+𝐾𝑄
+1110…010
+...
+0011…010
+...
+1011…111
+Pair
+𝐴2-𝐵1
+...
+𝐴2-𝐵1
+...
+𝐴2-𝐵𝑁
+Number
+𝑁𝐴1+
+...
+𝑁𝐴2,𝐵1
+...
+𝑁𝐴2
+𝐾𝑄
+1110…010
+...
+0011…010
+...
+1011…110
+Pair
+𝐴2-𝐵1
+...
+𝐴2-𝐵1
+...
+𝐴2-𝐵𝑁
+Number
+𝑁𝐴1+
+...
+𝑁𝐴2,𝐵1
+...
+𝑁𝐴2
+𝐾𝑄
+1110…010
+...
+0011…010
+...
+1011…110
+Pair
+𝐴1-𝐵1
+...
+𝐴1-𝐵1
+...
+𝐴1-𝐵𝑀2
+Number
+1
+...
+𝑁𝐴1,𝐵1
+...
+𝑁𝐴1
+𝐾𝑄
+1010…010
+...
+1011…010
+...
+1011…011
+Fig. 2. An illustration of CR-QKD in a multi-user scenario: edges distribute quantum keys to users according to their needs, where NAi,Bj represents the
+number of key groups required by Ai − Bj user pair and NAi represents the total number of key groups required by user Ai.
+binaries. There will probably be a mismatch between
+these binaries, hence information reconciliation has to be
+adopted to correct the mismatch. To avoid information
+leakage, privacy amplification is employed to distill the
+reconciliated binaries. Finally, after key verification, Alice
+and QAP1 retain the pair of bits as channel key KC1 and
+Bob and QAP2 retain the pair of bits as channel key KC2.
+• Edge forwarding phase: In the last phase, previous
+quantum keys shared between QAP1 and QAP2 are
+forwarded to Alice and Bob, completing the ultimate task
+of secret key sharing. Security is the primary concern
+here, as eavesdroppers should not learn any information
+about the quantum key through this forwarding process.
+With the help of channel keys, it is possible for edges
+to encrypt quantum keys with them using the One-Time-
+Pad (OTP) encryption algorithm and then forward the
+ciphertext to users. So far, the secret key sharing task is
+completed.
+Although CR-QKD provides a potential solution, it still faces
+some challenges to be implemented in practice. We divide
+these challenges into three categories and discuss along coun-
+termeasures below.
+B. Key alignment
+OTP is a well-known example of encryption scheme that
+provides “perfect secrecy”, however, one challenge here is that
+the channel key used for OTP must be at least as long as
+the quantum key to be encrypted. As channel keys, KC1 and
+KC2, are generated from different wireless channels, their key
+generation rates are likely to be different from each other, and
+that of the quantum key KQ. As a result, the quantum keys
+distributed to Alice and Bob may be disordered.
+We address the key alignment issue by segmenting quantum
+key and channel key into groups and numbering them before
+edge forwarding. Each group has a fixed bit number of LG.
+Those quantum key and channel key bits belonging to the
+same group are encrypted through a binary XOR operation.
+Then, the ciphertext is forwarded, together with the group
+number. Alice and Bob eventually obtain the quantum key
+by decrypting the ciphertext using their corresponding channel
+keys. Here, the trade-off between overhead and real-time must
+be taken into account in the selection of the group size.
+If the group size is small, the group number will occupy
+a field length comparable to that of the ciphertext, and the
+communication overhead will become significant. Otherwise,
+if the group size is large, the communication overhead is
+reduced but it will take a long time to accumulate sufficient
+keys for forwarding.
+Next, we extend the key alignment issue into a multi-
+user scenario, where A = {A1, A2, · · · , AM1} and B =
+{B1, B2, · · · , BM2} are two sets of IoT users at the service
+range of QAP1 and QAP2, respectively. Users in A desire to
+share secret keys with users in B. When CR-QKD is applied
+to this case, a new problem arises, i.e., how to distribute
+quantum keys from the edge to multiple users, who have
+different requirements and channel conditions. In this article,
+we introduce a multi-user edge forwarding strategy, which
+distributes quantum keys to each user according to its needs.
+Fig. 2 illustrates one round of the quantum key distribution
+process using this strategy.
+To start with, users in A broadcast the name of their
+target users for key sharing and the number of required
+key groups. After receiving these requests, QAP1 shares the
+information with QAP2. Then, QAP2 broadcasts it over the
+air and the relevant users in B record them locally. Next,
+quantum key sequences are shared between QAP1 and QAP2
+through the above QKD phase. These quantum key sequences
+are segmented into groups and numbered, each having LG
+bits. QAP1 allocates quantum key groups for each user pair
+according to their requests. The mapping relationship of user
+pairs and the key group number is transmitted to QAP2. This
+allocation information is saved in a quantum key buffer. In this
+way, the quantum keys are synchronized at QAP1 and QAP2.
+Next, they yield channel keys with these demanding users,
+respectively. For each user, the CRKG process is performed
+multiple times until it has accumulated sufficient number of
+key groups. Finally, QAP1 and QAP2 use these CRKG keys
+to encrypt the corresponding quantum keys and broadcast the
+
+5
+ciphertext together with the user pairs and group number to
+end-users. Each end-user obtains quantum keys by decrypting
+the related ciphertext with its own CRKG keys. Finally, these
+quantum keys are divided into each user pair for message
+encryption and this round of quantum key distribution has
+come to an end.
+C. Efficiency improvement
+The basic CR-QKD mechanism is time-consuming as it
+interacts heavily to obtain identical keys in both QKD and
+CRKG phases. This situation becomes more severe in a multi-
+user case. For each round of multi-user key distribution, in a
+time division multiple access (TDMA) system, the time delay
+is the sum of the time spent on yielding quantum keys and
+channel keys plus the time used for key forwarding. The time
+spent on quantum keys is calculated by dividing the number
+of quantum key bits by the quantum key generation rate. The
+time spent on channel keys is equal to the larger one of QAP1
+and QAP2. For each QAP, its time delay is the sum of that
+used for yielding channel keys between it and all users. One
+approach to reducing the time delay is to make QKD and
+CRKG processes work in parallel. However, its reduction ratio
+is less than 50% due to the positive forwarding time and the
+maximum operation.
+Another solution to further reduce the time delay is to
+improve the secret key generation rate. In practice, key
+generation rates are largely subject to the long time delay
+caused by information reconciliation, which exchanges parity
+information or syndromes over classic channels to detect and
+correct errors in the preliminary key material. According to
+OTP with un-identical keys [12], we propose a simplified CR-
+QKD mechanism that abolishes the sophisticated information
+reconciliation step in the CRKG phase and forwards quantum
+keys using non-reconciled channel keys. The challenge is to
+decrypt the quantum keys correctly when the non-reconciled
+channel keys of two parties are different but highly correlated.
+We deem the XOR encryption and decryption modules along
+with the physical channel as an equivalent cascade channel.
+Then, the tiny differences between keys can be seen as part of
+the transmission error, and thus can be corrected by the off-
+the-shelf channel coding with a stronger correction capability.
+Fig. 3 plots performance improvement ratios of the simplified
+CR-QKD mechanism compared with the paralleled CR-QKD
+mechanism in terms of time delay and upper bound of secret
+key generation rate in a typical WiFi scenario. As shown in
+the left panel, the proportion of delay reduction decreases with
+the rise of LG, still achieving a reduction ratio above 20%
+at LG ≤ 1024. The reduction of HT-Mixed mode is more
+remarkable than Non-HT mode, as the former has a larger
+time overhead than the latter. The right panel shows that the
+growth of the upper bound of the secret key generation rate
+is more remarkable when the bit disagreement ratio between
+quantized channel measurements gets larger, while it has a
+slight fall with the rise of LG. When LG = 1024 and ϵq = 0.1,
+the proportion of delay reduction and upper bound of secret
+key rate growth are roughly 20% and 10%, respectively. These
+simulation results verify the effectiveness of the proposed
+simplified CR-QKD mechanism.
+32
+64
+128
+256
+512
+1024
+LG/bits
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+The Proportion of Delay Reduction
+Non-HT
+HT-Mixed
+0
+0.05
+0.10
+0.15
+0.20
+q
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+Upper Bound of Secret Key Generation Rate Growth 
+LG=32bit
+LG=256bit
+LG=1024bit
+Fig. 3. Performance improvements of time delay and secret key generation
+rate in a typical WiFi scenario: the transmission distance is set as 150 meters
+and the bandwidth is set as 20 MHz. The fixed overhead of a WiFi frame
+under the Non-HT (Non-High Throughput) and HT-Mixed mode is 20 us and
+40 us, respectively.
+D. Security enhancement
+Another challenge of CRKG lies in the increased security
+risks caused by its hybrid architecture, as security is only
+as strong as its weakest link. We assume that the terminal
+security of QAP1 and QAP2 is guaranteed by techniques
+such as trusted computing. Operations that are relevant to
+secret keys are run in a trusted execution environment, thereby
+attackers can read neither quantum keys nor channel keys
+from the hybrid interface on QAP1 and QAP2. Since the
+edge forwarding phase employs the OTP encryption scheme,
+its security depends on the key used for OTP. The security
+of existing CRKG approaches, however, heavily relies on
+the channel variation and thus suffers from vulnerabilities
+in slowly varying environments [13]. When users have low
+mobility, e.g. in a wireless sensor network, there exist in-
+evitable and unknown temporal correlations between adjacent
+channel samples, resulting in a large proportion of repeated bit
+segments in the quantized bit sequences. Several solutions can
+be used to facilitate the practical usage of CRKG in slowly
+varying environments. One solution is to introduce helper de-
+vices, e.g., relays and reconfigurable intelligent surface (RIS)
+to boost the key generation rate and randomness [14]. How-
+ever, this solution encounters some practical problems, such
+as the unavailability of trust relays and additional hardware
+overheads of RIS devices. Another idea is to scramble these
+bits segments through some permutation or interleaving tech-
+niques. However, the security of the key may be compromised
+when the permutation information is public. [15] has proposed
+a new physical-layer secret key generation approach with
+channel obfuscation, which improved the dynamic property
+of channel parameters based on random filtering and random
+antenna scheduling, which have mutual remedying parameters
+in hiding the obfuscation information.
+
+6
+YUHUATAI 
+DISTRICT
+JIANGNING 
+DISTRICT
+GULI 
+RESIDENTIAL 
+DISTRICT
+Jiangjunshan 
+Tourism  
+Scenic Area
+Fangshan  
+Scenic Area
+XISHANQIAO 
+RESIDENTIAL 
+DISTRICT
+TIEXINQIAO 
+RESIDENTIAL 
+DISTRICT
+Qinhuai New River
+QAP2
+Total length: 15km
+QAP1
+Alice
+QAP2
+Bob
+Bob
+Quantum Key
+Eve
+Fig. 4. An illustration of the CR-QKD prototype platform in a metropolitan area network at Nanjing, which is the capital of Jiangsu Province, East-central
+China. One is located in Yuhuatai District and the other is at Chinese Network Valley in Jiangning District.
+IV. CASE STUDY: AN IMPLEMENTATION OF CR-QKD
+To realize the concept of CR-QKD, we implement a single-
+user confidential transmission prototype system in a metropoli-
+tan area network.
+A. Experimental Setup
+As shown in the left panel of Fig. 4, QAP1 and QAP2 are
+two quantum access points at a distance of fifteen kilometers.
+Alice and Bob are two remote IoT users in the wireless
+service ranges of QAP1 and QAP2, respectively. Without loss
+of generality, we zoom in on the wireless access network
+at Chinese Network Valley, as depicted in the right panel
+of Fig. 4. Here, QAP2 is composed of a QKD terminal
+under the series of QKDM-POL40-S for yielding quantum
+keys 2, a USRP N210 SDR device embedded with the CBX
+daughterboards for providing a wireless connection service,
+and a computer under the trusted execution environment for
+yielding wireless channel keys and distributing quantum keys.
+Both the QKD terminal and USRP N210 are connected to
+the computer via the ethernet cable in QAP2. The end-user,
+Bob, and a passive eavesdropper, Eve, are realized through two
+USRP N210 SDR devices, respectively. We design a TDD
+frame for channel sounding, which consists of a sinusoidal
+sequence for synchronization and an M-sequence for channel
+estimation. The signal operates at 2.605GHz and 20MHz
+bandwidth to avoid collisions with ubiquitous 2.4GHz signals
+such as WiFi. Once Bob receives the channel sounding signal
+from AP2, it will immediately switch to TX mode and send
+the same channel sounding signal. By using the same channel
+sounding signal for channel estimation, the amplitude part of
+the CSI is further preprocessed and quantized to generate the
+wireless keys.
+B. Performance Results
+Considering the comparable experimental scenarios and
+results of QAP1 and QAP2, we only take QAP2 as an example
+for performance analysis.
+2The quantum keys meet strict key randomness, as they conform to the
+specification of the GM/T 0005-2012.
+Table II summarizes the secret key sharing results from
+QAP2 to Bob and Eve in three typical indoor scenarios,
+namely office, hall and corridor. First, the measured key
+generation rates (KGRs) of the channel keys between QAP2
+and Bob in above scenarios are 315.4, 424.7 and 383.7 bits per
+second (bps), respectively. They are sufficient for traditional
+symmetric encryption algorithms (such as AES) to update
+256-bit keys every second for secure communications. In
+the random test, we examined a bit sequence of length 3.4
+million bits that was obtained at the output of the quantization
+stage without further processing. The generated channel keys
+passed 14 NIST statistical tests, indicating their randomness.
+However, while the simplified CR-QKD mechanism leads to
+high KGRs and high randomness, removing the complicated
+information reconciliation step also results in relatively high
+key disagreement rates (KDRs) of 8.1%, 4.7% and 5.8%
+between QAP2 and Bob, respectively. The number of person-
+nel, the frequency of movement, and the switching time of
+USRP affect the reciprocity of uplink and downlink channels,
+eventually leading to KDR differences in the noisy office,
+occasionally infested corridor, and empty hall. Meanwhile,
+along with forwarding quantum keys using non-reconciled
+channel keys based on channel error correction coding, the
+need arises to retransmit quantum keys when unsuccessfully
+decoded. The corresponding retransmission rates (RRs) using
+Polar codes from QAP2 to Bob are 11.6%, 2.1%, and 6.7%,
+respectively, which are proportional to the KDRs.
+To demonstrate the security of our proposed scheme, we
+also evaluate the quantum key cracking performance of the
+near-end eavesdropper Eve in terms of KDR and cracking rate
+(CR). The KDRs between QAP2 and Eve under these three
+scenarios are all around 50%, where the line of sight in the
+straight corridor contributes to a relatively lower KDR but is
+still above 45%. What’s more, the experimental results show
+that the CRs of Eve in the three scenarios are all zero, which
+means that none of the quantum keys have been cracked.
+V. CONCLUSION AND FUTURE DIRECTIONS
+Integrating QKD into IoT networks is beneficial for QKD’s
+practical deployment and end-user’s security enhancement.
+
+(0)目7
+TABLE II
+THE QUANTUM KEY WIRELESS DISTRIBUTION PERFORMANCE IN THREE
+INDOOR SCENARIOS
+Scenario
+QAP2 - Bob
+QAP2 - Eve
+Metrics
+KGR/bps
+NIST
+KDR
+RR
+KDR
+CR
+Office
+315.4
+14
+8.1%
+11.6%
+48.1%
+0%
+Hall
+424.7
+14
+4.7%
+2.1%
+49.2%
+0%
+Corridor
+383.7
+14
+5.8%
+6.7%
+45.3%
+0%
+This article proposed a framework of CR-QKD over IoT
+networks. QKD and CRKG assembly were adopted for se-
+cret key sharing over backbone core networks and the last-
+mile wireless access networks in CR-QKD, respectively. The
+demonstration of CR-QKD prototype represented a major step
+towards real-world information theoretically security for wide-
+area mobile applications, such as confidential VoLTE and
+confidential VoWiFi.
+Some open issues in future work are given below.
+• Device Authentication: Considering the hybrid archi-
+tecture of CR-QKD, it is more vulnerable to spoofing
+attacks from either user’s side or QAP’s side. However,
+neither QKD nor CRKG provides a means to authenticate
+the transmission source. Therefore, source authentication
+in CR-QKD should be further studied by using asym-
+metric cryptography techniques or emerging physical-
+layer techniques, such as radio frequency fingerprinting
+identification and physical unclonable function [1].
+• Untrusted QAPs: The proposed CR-QKD scheme relies
+on the trust of the intermediate QAPs. In this paper, we
+use techniques of trust computing to ensure that the the
+information stored in QAP is protected from external
+software attacks. When a trusted platform is not available,
+designing a scheme that relaxes this assumption could
+also be a very good future research direction.
+• Performance Optimization: In this article, we presented
+a multi-user edge forwarding strategy, in which quantum
+keys were allocated as needed. Unfortunately, its perfor-
+mance metrics, e.g., delay, secret key generation rate, and
+energy efficiency, are limited by those user pairs with
+weak channel reciprocity. How to optimize these perfor-
+mance metrics by allocating power or spectrum resources
+among different user pairs becomes an interesting topic
+and needs to be investigated.
+• System Integration and Compatibility: Our prototype
+was built on the USRP platform, which was different
+from commercial-off-the-shelf (COTS) devices. It is un-
+known whether these performances are still achievable on
+existing communication standards and whether CR-QKD
+will affect the network efficiency. More studies should be
+done on its system integration and compatibility issues,
+including frame format design, key management scheme
+and efficiency evaluation in practical communication sys-
+tems.
+VI. ACKNOWLEDGMENT
+We thank our colleagues Prof. Linning Peng, Mr. Yanjun
+Ding, Dr. Dong Wang, Mr. Siyun Wu and Dr. Xuyang Wang
+from the Purple Mountain Laboratories, for their help with
+the experimental platform. This work was supported in part
+by the National Key Research and Development Program of
+China under Grant 2020YFE0200600 and 2022YFB2902202,
+in part by the National Natural Science Foundation of China
+under Grant 62171121, in part by the Natural Science Foun-
+dation of Jiangsu Province under Grant BK20211160 and in
+part by Jiangsu Provincial Key Laboratory of Network and
+Information Security under Grant BM2003201.
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+

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+Normal and anomalous diffusion in a bouncing ball over an irregular surface
+Ana Laura Boscoloa,∗, Valdir Barbosa da Silva Juniora, Luiz Antonio Barreiroa
+aSão Paulo State University (Unesp), Institute of Geosciences and Exact Sciences,
+Physics Department, CEP 13506-900, Rio Claro, São Paulo, Brazil
+Abstract
+The problem of a bouncing ball on a non-planar surface is investigated. We discovered that surface undulation adds
+a horizontal component to the impact force, which acquires a random character. Some aspects of Brownian motion
+are found in the horizontal distribution of the particle. On the x-axis, normal and super diffusion are observed. For
+the probability density’s functional form, a scaling hypothesis is presented.
+Keywords:
+Scaling Hypothesis, Anomalous Diffusion, Brownian Motion
+1. Introduction
+Diffusion is a common natural phenomenon and generally occurs when a system moves toward the equilibrium
+state [1]. Many domains employ the notion of diffusion, including physics (particle diffusion), chemistry, biology,
+sociology, economics, and finance [2, 3, 4]. They all represent the fundamental concept of diffusion, which asserts
+that a substance or collection expands away from a point or location where that material or collection is more
+concentrated. In a diffusion process in a set of moving elements - energy, linear momentum, atoms, molecules, cells,
+animals, etc - each element performs a random trajectory. As a result of this highly irregular individual movement,
+the ensemble diffuses. Many non-linear systems also present a diffusive behavior in your phase space. Modeling
+such a dynamic system has become one of the most challenging subjects among scientists. The modeling helps to
+understand in many cases how the system evolves in time [5, 6, 7].
+On a macroscopic level, the average collective behavior, in contrast to the microscopic individual movement,
+shows great regularity and follows well-defined dynamic laws. The non-linear dynamic formulation of these transport
+phenomena, as well as the diffusion equation, are two ways to describe the diffusion phenomena. The form of the
+temporal dependence of the mean squared distance (MSD),
+�
+x2�
+∝ t2µ, or, equivalently, of the variance, allows
+classifying the type of diffusion. For µ = 1/2 we have the usual or normal diffusion, which can be described by
+Fick’s laws. Otherwise, we have an anomalous diffusion (or non-Fickian diffusion). When µ > 1/2 the case is
+classified as superdiffusive [8, 9] and for µ < 1/2 we have the subdiffusive case [10, 11]. Indeed, a wide diversity of
+systems presents a non-linear growth of the mean squared displacement.
+In this work, we explore the diffusive behavior that occurs in a free-falling particle colliding with a non-planar
+surface. Compared to a flat surface, on which the falling particles maintain their velocity in the horizontal direction,
+a non-planar surface introduces changes in the horizontal component of velocity after each collision. This creates
+a spread in the absolute value of the horizontal component of velocity as well as in its signal. Thus, in section
+2 we study the dynamics of the model, in which the equations of motion are established, and how the iterative
+process takes place. Some special points are explored in 2.3, for which no diffusion is observed. In section 3, the
+randomness of the horizontal component of the collision force is studied. Also, the diffusion in the signal of the
+horizontal component of velocity and its relation to the random walk problem is explored. Section 4devoted to
+discussing the behavior of the mean square displacement in relation to the initial collision point and the Probability
+Distribution Function (PDF) numerically and analitically. In section 5, the conclusions and final considerations
+about the problem addressed are presented.
+2. The Model
+We now discuss how to construct the equations of the mapping that describe the dynamics of the particles.
+The model under study consists of an ensemble of non-interacting classical particles of mass m travelling in the
+∗Corresponding author
+Email address: [email protected] (Ana Laura Boscolo)
+Preprint submitted to Elsevier
+January 3, 2023
+arXiv:2301.00275v1  [nlin.CD]  31 Dec 2022
+
+2.1
+The Map
+2
+presence of a constant gravitational field g and colliding with a non-flat ground via elastic collisions. The ground
+is parametrically described by:
+x(p) = α p
+y(p) = β [1 + cos (p)] ,
+(1)
+The figure 1 shows an example of a ground.
+Figure 1: Graph obtained from equations (1) using the parameters α = 0.01 and β = 0.001.
+Here it is worth noting that if the β parameter is null then the floor becomes flat, recovering the traditional
+Bouncer model [12] with a static floor. However, different from the traditional Bouncer model, if β ̸= 0, the particles
+gain an extra degree of freedom, with movement in the x-direction too. Also, as in the Bouncer model, the action of
+the constant gravitational field g is responsable for the return mechanism of the particle for the next collision with
+the floor. The conservation of energy during the collision is controlled by a parameter which is called the coefficient
+of restitution and it is denoted by γ. Indeed γ plays a key role in our model. For γ = 1 the conservative dynamics
+is observed. However, if 0 < γ < 1 we found a dissipative behavior.
+2.1. The Map
+We now move on to the study of the temporal evolution of particles, obtaining the position coordinates of
+the collision points and their respective velocities. The dynamic evolution of the particle can be described by the
+Newton’s equation of motion
+mdv
+dt = F grav + F col,
+(2)
+where F grav = mg is the gravitational force acting on the particle and F col represents the instantaneous force of
+collision with the ground. We will assume that the collision force only changes the velocity component orthogonal
+to the surface. It is also an acceptable assumption that during the collision process the force F col has an extremely
+rapid variation.
+A typical path taken by the particles is shown in the figure 2. After the nth collision at the point defined by the
+parameter pn, the particle travels in the gravitational field until it collides at the point pn+1. This journey takes a
+δtn,n+1 time and continues incessantly if no dissipation is taken into account.
+Figure 2: Schematic drawing of the trajectory of a particle, with its collision points and the respective normal vectors.
+The normal vectors at each collision point are also shown. The unit normal and tangent vectors at the point pn
+can be written in terms of the Cartesian vectors as
+ˆnn = (−λn i + j)
+�
+1 + λ2n
+and ˆtn = (i + λn j)
+�
+1 + λ2n
+(3)
+
+0.015
+0.010
+0.005
+0.000
+0.10
+0.05
+0.00
+0.05
+0.10
+3.ina
+fin+1
+Stn,n+1
+Pn
+Pn+12.1
+The Map
+3
+where λn is the local inclination of the ground, which for the functions in (1), is given by
+λn = (dy/dp)pn
+(dx/dp)pn
+= −β
+αsin (pn) .
+(4)
+Since motion in the gravitational field is a well-known problem, the fundamental question in determining the
+dynamic evolution of the particle will be to find the points of collision with the ground. To proceed with this
+determination, we define the following two functions
+GX(p, t) = x (p) −
+�
+x (pn) + v(r)
+xn t
+�
+GY (p, t) = y (p) −
+�
+y (pn) + v(r)
+yn t − g
+2t2�
+,
+(5)
+where
+�
+v(r)
+xn , v(r)
+yn
+�
+is the velocity of the particle after it collides at point pn. The next point pn+1 and the travel time
+δtn,n+1 = (tn+1−tn) spent by the particle between pn and pn+1 are obtained by solving the system of transcendental
+equations
+�
+GX(pn+1, δtn,n+1) = 0
+GY (pn+1, δtn,n+1) = 0.
+(6)
+In such a way, if the particles make a trip with N collisions, the total time spent will be
+tN =
+N
+�
+n=1
+δtn−1,n with t0 = 0.
+(7)
+In our model, we assume that only the component of the velocity normal to the surface at the collision point is
+altered (inverted) [13]. Then, at the instant of collision, the law of reflection relating the incident velocity vector
+v(i)
+n to the reflected velocity vector v(r)
+n
+is,
+v(r)
+n
+=
+�
+v(i)
+n · ˆtn
+�
+ˆtn − γn
+�
+v(i)
+n · ˆnn
+�
+ˆnn.
+(8)
+Obviously, the velocity vector, incident at a point pn+1, is related to the velocity vector reflected at the previous point pn as
+v(i)
+n+1 = v(r)
+xn i +
+�
+v(r)
+yn − g δtn,n+1
+�
+j.
+Now we can define the following dimensionless variables ¯x(p) = x(p)/gt2
+N, ¯y(p) = y(p)/gt2
+N, ¯v(r)
+n
+= v(r)
+n /gtN and φn = tn/tN
+, where tN is defined in (7). Therefore, the dimensionless velocity vector components in (8) take the form
+¯v(r)
+xn+1 =
+�
+1 − γn+1λ2
+n+1
+�
+¯v(r)
+xn + λn+1 (1 + γn+1)
+�
+¯v(r)
+yn − δφn,n+1
+�
+1 + λ2
+n+1
+¯v(r)
+yn+1 =
+λn+1 (1 + γn+1) ¯v(r)
+xn +
+�
+λ2
+n+1 − γn+1
+� �
+¯v(r)
+yn − δφn,n+1
+�
+1 + λ2
+n+1
+.
+(9)
+and the system (6) becomes
+�
+�
+�
+�
+�
+�
+�
+�
+�
+pn+1 = pn + ¯v(r)
+xn
+¯α δφn,n+1
+cos (pn+1) = cos (pn) + ¯v(r)
+yn
+¯β δφn,n+1 − 1
+2¯β δφ2
+n,n+1
+(10)
+where ¯α = α/gt2
+N and ¯β = β/gt2
+N. Given the values of pn, ¯v(r)
+xn and ¯v(r)
+yn of the nth iteraction, the set of equations
+(10) produce the values of pn+1 and the travel time δφn,n+1 which allows us to find ¯v(r)
+xn+1 and ¯v(r)
+yn+1through (9).
+After that, the iteractive process restart.
+The last ingredient is the dimensionless energy
+¯En = 1
+2
+��
+¯v(r)
+xn
+�2
++
+�
+¯v(r)
+yn
+�2�
++ ¯yn
+(11)
+which is used to establish the initial conditions to be chosen so that all particles in the ensemble have the same
+initial energy.
+
+2.2
+Conservative case
+4
+2.2. Conservative case
+We shall only consider the conservative scenario, when γn=γn+1 = 1. Since we choose ¯β ≪ 1, it is appropriate
+to consider that the point of collision with the ground has a height ¯y(pn) ≃ ¯y(pn+1) ≃ 0 , but with local slope not
+necessarily zero. This approach avoids transcendental equations and simplifies the calculation. As a consequence,
+the second of the equations in (10) yields δφn,n+1 = φn,n+1 − φn = 2¯v(r)
+yn . Finally, a simplified form of the map
+equations used to explain motion is expressed as
+¯v(r)
+xn+1 =F1
+�
+¯v(r)
+xn , ¯v(r)
+yn , pn
+�
+¯v(r)
+yn+1 =
+���F2
+�
+¯v(r)
+xn , ¯v(r)
+yn , pn
+����
+pn+1 =F3
+�
+¯v(r)
+xn , ¯v(r)
+yn , pn
+�
+(12)
+where
+F1
+�
+¯v(r)
+xn , ¯v(r)
+yn , pn
+�
+=
+�
+1 − ¯λ2
+n
+�
+¯v(r)
+xn − 2¯λn¯v(r)
+yn
+1 + ¯λ2n
+F2
+�
+¯v(r)
+xn , ¯v(r)
+yn , pn
+�
+=2¯λn¯v(r)
+xn +
+�
+1 − ¯λ2
+n
+�
+¯v(r)
+yn
+1 + ¯λ2n
+F3
+�
+¯v(r)
+xn , ¯v(r)
+yn , pn
+�
+=pn + 2
+α ¯v(r)
+xn ¯v(r)
+yn
+(13)
+and were defined
+¯λn = λn+1 =
+∂ ¯YS/∂p
+∂ ¯
+XS/∂p
+����
+pn+ 2
+α ¯v(r)
+xn ¯v(r)
+yn
+= −
+¯β
+¯αsin
+�
+pn + 2
+α ¯v(r)
+xn ¯v(r)
+yn
+�
+.
+(14)
+The ground was assumed to be flat, as a consequence there is a small possibility of the particle presenting a negative
+value for ¯v(r)
+y . This non-physical situation is bypassed by introducing the modulus in the second equation of (12).
+This means that if such a case happens, the particle is reinjected back to the dynamics with the same velocity but
+with a positive direction.
+2.2.1. Jacobian Matrix
+The Jacobian matrix for this dynamical system may be simply calculated using equations (12-14),
+J =
+�
+�
+�
+∂F1
+∂vx
+∂F1
+∂vy
+∂F1
+∂p
+∂F2
+∂vx
+∂F2
+∂vy
+∂F2
+∂p
+∂F3
+∂vx
+∂F3
+∂vy
+∂F3
+∂p
+�
+�
+�
+leading to1
+∂F1
+∂vx
+=
+¯α4 + 4 ¯βvy
+2 cos
+�
+p +
+2vxvy
+¯
+α
+� �
+¯α2 − ¯β2 sin2 �
+p +
+2vxvy
+¯
+α
+��
+− ¯β4 sin4 �
+p +
+2vxvy
+¯
+α
+�
+− 4¯α ¯β2vxvy sin
+�
+2p +
+4vxvy
+¯
+α
+�
+�
+¯α2 + ¯β2 sin2 �
+p +
+2vxvy
+¯
+α
+��2
+∂F1
+∂vy
+=
+2 ¯β
+�
+¯α
+�
+−2 ¯βvx
+2 sin
+�
+2p +
+4vxvy
+¯
+α
+�
++ ¯α2 sin
+�
+p +
+2vxvy
+¯
+α
+�
++ ¯β2 sin3 �
+p +
+2vxvy
+¯
+α
+��
++ 2vxvy cos
+�
+p +
+2vxvy
+¯
+α
+� �
+¯α2 − ¯β2 sin2 �
+p +
+2vxvy
+¯
+α
+���
+�
+¯α2 + ¯β2 sin2 �
+p +
+2vxvy
+¯
+α
+��2
+∂F1
+∂p
+=
+2¯α ¯β cos
+�
+p +
+2vxvy
+¯
+α
+� �
+¯α2vy − ¯β sin
+�
+p +
+2vxvy
+¯
+α
+� �
+¯βvy sin
+�
+p +
+2vxvy
+¯
+α
+�
++ 2¯αvx
+��
+�
+¯α2 + ¯β2 sin2 �
+p +
+2vxvy
+¯
+α
+��2
+1In Jacobian expressions, we utilize (vx, vy, p) rather than (¯v(r)
+xn , ¯v(r)
+yn , pn) to simplify notation.
+
+2.3
+Periodic points
+5
+∂F2
+∂vx
+=
+−
+2 ¯β
+�
+¯α
+�
+2 ¯βvy
+2 sin
+�
+2p +
+4vxvy
+¯
+α
+�
++ ¯α2 sin
+�
+p +
+2vxvy
+¯
+α
+�
++ ¯β2 sin3 �
+p +
+2vxvy
+¯
+α
+��
++ 2vxvy cos
+�
+p +
+2vxvy
+¯
+α
+� �
+¯α2 − ¯β2 sin2 �
+p +
+2vxvy
+¯
+α
+���
+�
+¯α2 + ¯β2 sin2 �
+p +
+2vxvy
+¯
+α
+��2
+∂F2
+∂vy
+=
+¯α4 − ¯β
+�
+4vx
+2 cos
+�
+p +
+2vxvy
+¯
+α
+� �
+¯α2 − ¯β2 sin2 �
+p +
+2vxvy
+¯
+α
+��
++ ¯β3 sin4 �
+p +
+2vxvy
+¯
+α
+�
++ 4¯α ¯βvxvy sin
+�
+2p +
+4vxvy
+¯
+α
+��
+�
+¯α2 + ¯β2 sin2 �
+p +
+2vxvy
+¯
+α
+��2
+∂F2
+∂p
+=
+−
+2¯α ¯β cos
+�
+p +
+2vxvy
+¯
+α
+� �
+¯β sin
+�
+p +
+2vxvy
+¯
+α
+� �
+2¯αvy − ¯βvx sin
+�
+p +
+2vxvy
+¯
+α
+��
++ ¯α2vx
+�
+�
+¯α2 + ¯β2 sin2 �
+p +
+2vxvy
+¯
+α
+��2
+∂F3
+∂vx
+=
+2vy
+¯α
+∂F3
+∂vy
+=
+2vx
+¯α
+∂F3
+∂p
+=
+1
+This Jacobian matrix’s determinant is equal to one, confirming that the system is indeed conservative.
+2.3. Periodic points
+We can anticipate the occurrence of some exceptional points using the physics of the problem. These are known
+as fixed points, to which the dynamical system returns after one iteration (period-one fixed point), two iterations
+(period-two fixed point), or n iterations (period-n fixed point). The figure 3 illustrates two fixed points: (a) Fixed
+points for period one and (b) Fixed points for period two.
+Figure 3: Examples of fixed points: (a) Fixed point of period one. The dynamical system returns to the point in phase space at each
+iteration and (b) the system returns to the point after 2 iterations.
+2.3.1. Period-one Point
+It is evident that period-one fixed points, as shown in portion (a) of the figure, have a zero local slope. So, as long
+as the x component of the initial velocity is zero, the system will not experience any diffusion in the horizontal axis.
+A period-one point is obtained by solving the following equations: ¯v(r)
+xn+1 = ¯v(r)
+xn = 0, ¯v(r)
+yn+1 = ¯v(r)
+yn and pn+1 = pn
+with ¯λn = 0 (zero slope). We can verify the fact considering first eq (14)
+¯λn = 0 ⇒ sin
+�
+pn + 2
+α ¯v(r)
+xn ¯v(r)
+yn
+�
+= 0
+⇒
+¯v(r)
+xn =0
+pn = mπ,
+where m is a integer. These points indicate the locations of the peaks and valleys in Figure 1 - part (a). Thus
+¯v(r)
+xn+1
+=
+F1
+�
+0, ¯v(r)
+yn , mπ
+�
+= 0
+¯v(r)
+yn+1
+=
+F2
+�
+0, ¯v(r)
+yn , mπ
+�
+= ¯v(r)
+yn
+(15)
+pn+1
+=
+F3
+�
+0, ¯v(r)
+yn , mπ
+�
+= mπ
+We have the following physical situation: If a particle is chosen whose horizontal component of velocity is zero, in
+a zero slope point, clearly the x-coordinate of the particle will never change and the particle does not scatter in the
+x-direction.
+
+P
+Pn = Pn+1
+pn
+Pn+1
+x
+T元
+X
+(a)
+(b)2.3
+Periodic points
+6
+2.3.2. Period-two Points
+We now consider points with non-zero slope. In general, the particle gains a non-zero horizontal component
+to the velocity and then diffuses along the horizontal axis. Nevertheless, depending on the initial conditions, it is
+possible for the particle to strike the surface at point pn with velocity ⃗vn, reflect there, then it reaches point pn+1
+with velocity ⃗vn+1, where it will then reflect again and go back to point pn with velocity ⃗vn. Part (b) of Fig. 3
+depicts an illustration of this kind.. Inspired by the figure, consider points connected by ¯v(r)
+xn+2 = −¯v(r)
+xn+1 = ¯v(r)
+xn ,
+¯v(r)
+yn+2 = ¯v(r)
+yn+1 = ¯v(r)
+yn , pn+2 = pn, ¯y(pn+1) = ¯y(pn) and opposite local slopes λn+1 = −λn.
+Taking into account the figure 3 portion (b) , the points pn and pn+1 must be connected by
+�
+pn = −π − χ
+pn+1 = π + χ
+with 0 < χ < π
+where we are solely concerned with the most straightforward solution. Then, with the help of equations (10), we
+can write
+¯v(r)
+xn ¯v(r)
+yn = ¯α (π + χ) .
+(16)
+In addition, the first of the equations (13) yields
+¯v(r)
+yn =
+¯α
+¯βsin (χ) ¯v(r)
+xn .
+(17)
+These results allow us to determine both ¯v(r)
+xn and ¯v(r)
+yn as functions of χ. So the period two fixed point is written as
+¯v(r)
+xn
+=
+±
+�
+¯β (π + χ) sin (χ)
+¯v(r)
+yn
+=
+¯α (π + χ)
+�¯β (π + χ) sin (χ)
+,
+pn
+=
+∓ (π + χ)
+Figure 4 illustrates these fixed points. The middle points in black in this picture indicate the period-1 fixed
+points. The graphic also illustrates the effect of the β−parameter on the formation of period-2 fixed points. The
+points are calculated by altering the value of χ from 0 to π, and each color indicates a β parameter value: red
+(β = 0.00001), green (β = 0.00002),..., purple (β = 0.0001). α = 0.001 is used for all points.
+Figure 4: Period one fixed points are represented by the black dots in the center of the line. The other points are the period 2 fixed
+points.The gray dots at the end of the curves are the points obtained with the value χ = π/2.
+
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+8.03
+-0.02
+-0.01
+0.00
+0.01
+0.02
+0.03
+Vx7
+The choice χ = π/2 is used to calculate the gray dots in the figure 4. Each curve is divided into two branches by
+these points. The points that make up the branches we name external have χ > π/2, whereas the points that make
+up the branches we term internal have χ < π/2 . Consider the eigenvalues of the Jacobian matrix to categorize
+the stability of these points. The external points (χ ≥ π/2) can be classified as node-type stable points since the
+modules of their Jacobian matrix eigenvalues are all equal to 1. On the other hand, because all of the eigenvalues
+are real with one positive and the others negatives, the internal points (χ < π/2) are categorized as unstable points
+of the saddle type. Therefore, the gray dots in the phase space represent saddle-node bifurcations [14].
+Many additional sorts of fixed points may exist, but the purpose of this paper is to study the diffusion process
+along the x-axis.
+3. Diffusion Proccess
+3.1. The stochastic character of force
+Clearly, unless we are in some special initial point, the particles must diffuse in the x-direction. This diffusion is
+caused by the collision force with the ground. Due to the irregular nature of the ground, the collision force F col has
+components in both horizontal and vertical directions. It is intuitive to notice that the horizontal component presents
+different magnitudes and directions at each collision. To understand the behavior of this horizontal component of
+the collision force, we can describe it as
+¯Fcolx(φn) = ∆¯v
+¯τ
+����
+pn
+= ¯v(r)
+xn − ¯v(i)
+xn
+¯τ
+= ¯v(r)
+xn − ¯v(r)
+xn−1
+¯τ
+where ¯τ is the dimensionless collision time, which is extremely small. We will also assume that the collision force
+is approximately constant during the collision time and a typical example of what this force looks like is shown in
+figure 5.
+Figure 5: Typical behavior of the horizontal component of the collision force. Here we have used ¯α = 0.01 and ¯β = 0.005. The graph
+has two regions with different scales. On the left we have the region magnified between φ = 0.000 and φ = 0.020 and on the right, after
+a cut in the graph, the normal scale from φ = 0.5 to φ = 1.0 is shown.
+The width of each rectangle represents the collision time and despite the dynamics being well known and
+the irregularities in the ground having a periodicity, the numerical results presented show that the effects of the
+horizontal component of this force has a behavior comparable to a stochastic force. It is actually extremely difficult
+to tell whether a sequence is random or chaotic, but there are some proposed procedures to distinguish between
+these two behaviors. In this work we will make use of the permutation entropy (PE) method [15, 16] to establish the
+randomness of the time series produced by the collision force. Denoting the time series as {St}t=1,...,T the method
+consists in defining subsets of order O, forming the set S = {{S1, S2, . . . , SO}, {S2, S3, . . . , SO+1}, . . . , {ST −O+1, . . . ,
+ST −1, ST }}. We then compare consecutive values from each subset to establish the associated permutation. For
+example, {S1 < S2 < . . . < SO} represents the permutation {1, 2, ..., O}, while {S2 < S1 < . . . < SO} represents
+
+1.5
+1.0
+0.5
+0.0
+-0.5
+-1.0
+-1.5
+0.0
+0.005
+0.010
+0.015
+0.020
+0.6
+0.8
+1.08
+the permutation 2, 1, ..., O and so on, yielding the set of all permutations associated with the sequence S, named
+Π(S). Then, the set of all O! possible permutations πi of the numbers {1, 2, ..., O} are constructed. The relative
+frequency of each permutation πi can be calculated by counting the number of times the permutation πi is found
+in the set Π(S) divided by the total number of sequences,
+Pi = Number of times that πi appears in Π(S)
+T − O + 1
+.
+(18)
+and the normalized permutation entropy function is written as,
+PEO = −
+1
+log2(O!)
+O!
+�
+i=1
+Pi log2(Pi).
+(19)
+Formulas (18) and (19) were applied to the temporal sequences of collision forces for three different initial
+conditions and also different orders O. The table 1 shows the results obtained.
+floor parameters
+initial condiction
+O = 3
+O = 4
+O = 5
+O = 6
+α = 0.01
+β = 0.05
+p0 = −0.033
+0.998569
+0.995189
+0.981222
+0.92671
+p0 = 0.032
+0.999633
+0.995120
+0.982245
+0.925946
+α = 0.01
+β = 0.0005
+p0 = −0.033
+0.998874
+0.994082
+0.986440
+0.935262
+p0 = 0.032
+0.999501
+0.996295
+0.984878
+0.934281
+Table 1: The initial conditions are chosen in order to vary the initial point (x(p0), y(p0)) and keeping the energy ¯E = 4 constant.
+The smaller the PEO is, the more regular and more deterministic the time series is. Contrarily, the closer to 1
+the value of PEO is, the more noisy and random the time series is. The results allow us to assume that the force
+is random.
+4. Probability Distribution Function (PDF)
+This section’s major purpose is to establish the probability distribution function (PDF) Ψ(x, t), which provides
+us the probability of the particle being on the coordinate x at time t, and what it has to do with normal and
+superdiffusive processes. Among the various diffusive processes, Brownian motion is the prototype for the description
+of non-equilibrium dynamical systems. Due to the stochastic behavior of the collision force, the jumps performed
+by the particles also reproduce characteristics of random walk. We can comprehend this by calculating the chance
+of each particle going to the right. After each impact, we obtain the x−component of the velocity. Then, by
+examining the sign of these velocities and associating +1 for vx > 0 and 0 for vx < 0, we can count the number of
+jumps to the right and derive the evolution of this probability as the number of jumps increases. It is appropriate
+at this point to introduce an index that specifies the initial condition (ν), which is used to compute the Probability
+Density Function (PDF) for the complete ensemble. So, starting with an initial state labeled by ν, the probability
+of jumping to the right after n jumps is calculated as follows:
+Pr−jump(n, ν) = 1
+n
+n
+�
+i=1
+SgnPlus(v(ν)
+x,i ) where SgnPlus(v(ν)
+x,i ) =
+�
+1
+if vx > 0
+0
+if vx < 0
+Figure 6, on the left, shows examples of the time progression of individual particle jumps for four distinct initial
+conditions and two ground parameter adjustments, as well as the corresponding PDFs Ψ(x, t). With time evolution,
+the left/right jump probabilities for a ground with α = 0.01 and β = 0.005 tend to be 0.5 very quickly as we can
+see into upper graphic on the left. However, if the beta parameter is set to β = 0.0005 the graph indicates an initial
+oscillation, but the probability ultimately tends to reach 0.5.
+The coordinates of the collision points and the travel time between one point and the next are obtained from
+the mapping given in equations (9) and (10). It is obvious that the travel time varies between jumps. However, for
+our analysis, it is critical to obtain the particle’s position as a function of time with equal time intervals. This is
+simple because the particle moves in a gravitational field g, and we can easily calculate its position as a function of
+time. The time is then normalized so that the maximum time equals one. So, to get the probability distribution,
+for all iterative processes, we begin by subtracting the starting position of the particles. As a result, all of the
+particles in the ensemble start from the same position. In our scenario, we have 2000 particles performing 4000
+
+9
+Figure 6: The first graphic of each column contains time evolution examples for the likelihood of a single particle jumping to the right.
+The difference is in the β parameter value, which is lowered to one-tenth and one-hundredth of its initial value in the columns on the
+left. The evolution of the 4-particle leaps (4 initial conditions) is explored in the graphs. The different initial conditions for the particles
+are obtained by changing the initial parameter p in the functions x(p) and y(p) in Eq (1) and keeping the energy ¯E = 4 constant. The
+selected p-parameters are shown in the figures. The respective contour plots for the probability distributions are shown on the right.
+leaps, totaling 8 million collision points, but it is clear that the number of points as a function of time depends on
+the choice of interval dt and can be much higher. To demonstrate the procedure, the simulation is configured so
+that each particle in the ensemble has an energy of E = 4. The outcomes for two different types of grounds are
+shown in Figure 6 on the right.
+The first PDF graph was obtained with the parameters α = 0.01 and β = 0.005, and shows a probability density
+region following a format very similar to a Gaussian distribution. The second pdf, obtained with the parameters
+α = 0.01 and β = 0.0005, has an extremely anomalous diffusion in the early part of its time evolution, however
+when the time evolution takes place, the PDF apparently starts to show a Gaussian behavior. In order to have a
+better understanding of this behavior, we studied the moments associated with each distribution. Inspired by the
+Gaussian form of normal diffusion, with an anomalous diffusion we make a scaling hypothesis [17] so that we can
+express the anomalous distribution as
+Ψµ(x, t) =
+� a
+π
+1
+tµ exp
+�
+−a
+� x
+tµ
+�2�
+.
+(20)
+The associated moments are easily obtained as
+⟨|x(t)|m⟩ =
+∞
+ˆ
+−∞
+xmΨµ(x, t) dx =
+1
+√
+amπ Γ
+�m + 1
+2
+�
+tmµ.
+(21)
+The result shows a behavior of MSD as
+�
+x2�
+∝ t2µ, therefore normal distribution has a scale parameter µ = 1/2.
+If µ < 1/2 we have a subdifussive process and for µ > 1/2 we found a superdiffusive behavior. Figure 7 shows
+the results of the moments calculations for two different grounds. We can observe that at left we obtain the scale
+µ = 0.5 and at right we obtain µ = 0.65. So, we have two distinct behaviors: at left a normal diffusion and at right
+we have a superdiffusive behavior.
+
+0.8 
+500
+0.00200
+.0.6
+x-coordinate
+0.00175
+0.00150
+α = 0.01
+0.00125
+pue
+0.4
+0.00100
+β = 0.005
+p=2.82156
+0.00075
+0.2
+p=-2.50841
+0.00050
+p=2.19525
+500
+0.00025
+0.0 
+p=-1.88209
+0
+1000
+2000
+3000
+4000
+number of iteractions
+0.2
+0.4
+0.6
+0.8
+1.0
+time
+1.0F
+4000
+0.8
+2000
+0.00030
+X-coordinate
+0.00025
+α= 0.01
+0.00020
+and
+0.4
+p=-2.82156
+0.00015
+β=0.0005
+p=-2.50841
+2000
+0.00010
+0.2
+p=-2.19525
+0.00005
+0.0E
+p=1.88209
+4000
+0
+1000
+2000
+3000
+4000
+number of iteractions
+0.2
+0.4
+0.8
+0.8
+1.0
+time10
+Figure 8: PDF’s rescaled by the factor ξ = x2/
+�
+x(t)2�
+for four distinct times. The theoretical prediction given in Eq. (22) with an
+a = 3.75 × 10−8 is shown by the black dot-dashed line.
+Figure 7: The red lines represent equations of lines with powers mµ.
+At left we have a normal diffusion and at right we have a
+superdiffusive diffusion. We can see that almost half of the time evolution has passed before the superdiffusive behavior with scale
+µ = 0.65 manifests.
+The scaling hypothesis is carried forward using equation (21) to obtain t2µ = a√π
+�
+x(t)2�
+/Γ (3/2), which
+enables us to specify the subsequent function
+F(ξ) = tµΨµ(x, t) =
+� a
+π exp
+�
+−Γ (3/2)
+√π
+ξ
+�
+(22)
+where ξ = x2/
+�
+x(t)2�
+. Using the PDF data for the superdiffusive process (µ = 0.65) we obtain F(ξ) numerically
+and the results for t = 0.76, t = 0.765, t = 0.89 and t = 0.995 are presented in the figure 8 . The only parameter
+that can be adjusted in the theoretical forecast stated in Eq (22) is the value of a. We get a remarkable agreement
+with the simulation findings when we choose a = 3.75 × 10−8. The black dot-dashed line on the graph denotes the
+theoretical result obtained in equation (22). We observe that the theoretical modeling and the simulation outcome
+start to diverge for periods of time less than 76.5% of the overall duration of the iterative procedure. Rescaling
+the data, all simulation points for times more than this amount lie exactly on the same curve. This was already a
+foregone conclusion if we look at the second PDF in the figure 6, which shows quite anomalous behavior for times
+less than 0.8. Before this time has elapsed the particles display a strongly anomalous diffusion with a scale that
+must rely on the moment being estimated, ⟨|x|m⟩ ∝ tmµ(m), [18].
+5. Conclusions and Outlook
+In this work, we have studied a falling particle in the gravitational field colliding with a non-plane surface. We
+could observe that the horizontal component of the collision force presented a stochastic behavior. This was verified
+
+α=0.01andβ=0.005
+α = 0.01and β
+3=0.0005
+1016
+1016
+<1 x(t) /*)~ ++x0.65
+<1 x(t) 1*)~t+x0.5
+1012
+1012
+<1 x() 3)~ 3-0.65
+<ul(a)xI)
+<1x()13)~ f3x0.5
+<ul(0)xI)
+108
+(1 (t) 13)~(20.65
+00
+108
+(1 (t)/3)~f20.5
+104
+sol(1(0)x1)
+<1 (t) ~10.65
+00000000
+104
+0000000
+0000000000000
+1
+0.01
+0.05
+0.10
+0.50
+1
+0.01
+0.05
+0.10
+0.50
+1
+time
+time1.1 × 10-4
+1.0 × 10-4
+9.0× 10-5
+.
+t=0.76
+t=0.765
+8.0×10-5
+o t=0.89
+7.0×10-5
+.95.
+t=0.995
+2.0
+6.0× 10-5
+5.0× 10-5
+0.0
+0.5
+1.0
+1.5REFERENCES
+11
+by using the entropy permutation method applied to the collision force time series. Additionally, we established
+that the jumps to the right and left follow a distribution whose probabilities tend toward 0.5 while the particle’s
+temporal development takes place. It can be seen that the convergence to the factor 0.5 occurs significantly more
+quickly using the ground with parameters α = 0.01 and β = 0.005 than with β = 0.0005. We assume that a surface
+with more pronounced undulations produces a horizontal component of the force that swiftly alters the particle’s
+horizontal motion, causing the probability of jumps to fast converge to 0.5. The first case implied a diffusion process
+that follows Einstein’s famous relationship so that the horizontal mean square displacement is proportional to time,
+�
+x(t)2�
+∼ t. The system begins to become superdiffusive as the ground gets smoother. In fact, it is observed
+that the system with β = 0.0005 exhibits a strongly anomalous mean squared deviation with temporal increase
+over the earliest portion of its temporal history. Subsequently, the movement becomes "standard superdiffusive".
+To comprehend this behavior, we assumed that the probability density’s functional form must take on a Gaussian
+form of normal diffusion, with the exception that the distribution’s time dependence is scaled by tµ. We get a
+remarkable consistency between the theoretical expression and the simulation results using this approach. Future
+works are being developed including changes in the function that describes the floor, introduction of dissipation
+and oscillations in the ground, among other works, [19].
+Acknowledgments
+The authors would like to thank Prof.
+Edson Denis Leonel for the observations and comments, as well as
+Coordination for the Improvement of Higher Education Personnel (Capes) for the financial support.
+—————–
+References
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+Statistical Mechanics and its Applications 140 (1-2) (1986) 212–218. doi:10.1016/0378-4371(86)90224-4.
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+[6] S. H. Strogatz, Nonlinear dynamics and chaos with student solutions manual (sep 2018).
+doi:10.1201/
+9780429399640.
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+physics, biology, chemistry, and engineering, Computers in Physics 8 (5) (1994) 532. doi:10.1063/1.4823332.
+[8] T. Geisel, J. Nierwetberg, A. Zacherl, Accelerated diffusion in josephson junctions and related chaotic systems,
+Physical Review Letters 54 (7) (1985) 616–619. doi:10.1103/physrevlett.54.616.
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+[11] P. Massignan, C. Manzo, J. A. Torreno-Pina, M. F. GarcÃa-Parajo, M. Lewenstein, G. J. L. Jr, Nonergodic
+subdiffusion from brownian motion in an inhomogeneous medium, Phys. Rev. Lett. 112, 150603 (2014) (Jan.
+2014). arXiv:1401.6110, doi:10.1103/PhysRevLett.112.150603.
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+slowing mechanism for fermi acceleration, Physical Review E 86 (3) (sep 2012). doi:10.1103/physreve.86.
+036203.
+
+REFERENCES
+12
+[13] E. D. Leonel, M. V. C. Galia, L. A. Barreiro, D. F. M. Oliveira, Thermodynamics of a time-dependent and
+dissipative oval billiard: A heat transfer and billiard approach, Physical Review E 94 (6) (2016) 062211.
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+systems, Physical Review Research 4 (2) (2022) 023192. arXiv:2206.06786, doi:10.1103/PhysRevResearch.
+4.023192.
+URL https://ui.adsabs.harvard.edu/abs/2022PhRvR...4b3192C
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+[19] A. L. Boscolo, V. B. da Silva Junior, L. A. Barreiro, Work in progress (2022).
+

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+CyberLoc: Towards Accurate Long-term Visual
+Localization
+Liu Liu ⋆, Yukai Lin ⋆, Xiao Liang ⋆, Qichao Xu ⋆, Miao Jia, Yangdong Liu, Yuxiang
+Wen, Wei Luo, Jiangwei Li
+Cyberverse Dept, Huawei Cloud Computing Technologies Co., Ltd.
+Abstract. This technical report introduces CyberLoc, an image-based
+visual localization pipeline for robust and accurate long-term pose es-
+timation under challenging conditions. The proposed method comprises
+four modules connected in a sequence. First, a mapping module is ap-
+plied to build accurate 3D maps of the scene, one map for each refer-
+ence sequence if there exist multiple reference sequences under different
+conditions. Second, a single-image-based localization pipeline (retrieval–
+matching–PnP) is performed to estimate 6-DoF camera poses for each
+query image, one for each 3D map. Third, a consensus set maximiza-
+tion module is proposed to filter out outlier 6-DoF camera poses, and
+outputs one 6-DoF camera pose for a query. Finally, a robust pose re-
+finement module is proposed to optimize 6-DoF query poses, taking can-
+didate global 6-DoF camera poses and their corresponding global 2D-3D
+matches, sparse 2D-2D feature matches between consecutive query im-
+ages and SLAM poses of the query sequence as input. Experiments on
+the 4seasons dataset show that our method achieves high accuracy and
+robustness. In particular, our approach wins the localization challenge
+of ECCV 2022 workshop on Map-based Localization for Autonomous
+Driving (MLAD-ECCV2022).
+Keywords: autonomous driving, image-based localization, image re-
+trieval, image matching, multiple maps, multi-session PnP, consensus
+set maximization, pose graph optimization, bundle adjustment, slam
+1
+Introduction
+This technical report studies the problem of image-based localization with re-
+spect to a pre-built 3D map. This problem has attracted considerable attention
+recently due to the widespread potential applications, such as in autonomous
+driving [1], robotics [2] and VR/AR [3]. It aims to estimate the 6-DoF global
+pose for a query image given a pre-built 3D map. Although visual localization
+has progressed rapidly in the past few years, how to achieve a robust and ac-
+curate localization under long-term challenging conditions still remains to be
+solved.
+⋆ Equal contributions
+arXiv:2301.02403v1  [cs.CV]  6 Jan 2023
+
+2
+SfM
+SLAM
+Dense Mapping
+Disparity
+Session 1
+Session i
+Session C
+P1,1
+P1,i
+P1,C
+Pk,1
+Pk,i
+Pk,C
+PK,1
+PK,i
+PK,C
+Frame 1
+Frame k
+Frame K
+Pose Refinement & Polish
+SLAM or VO
+Query
+Mutilple Reference Maps
+2D-2D matches
+relative poses
+Global absolute poses
+Global 2D-3D matches
+...
+...
+...
+...
+...
+...
+...
+...
+Fig. 1: The overall framework of our method. A scene can be visited multiple times,
+resulting in multiple reference maps under different conditions (weather, lighting, etc.).
+For each reference sequence, we perform stereo SfM, SLAM, dense mapping, and dispar-
+ity to reconstruct four 3D maps for robustness. Given a query frame, we first perform
+localization with respect to each session map and obtain multiple global poses corre-
+sponding to sessions (one global absolute pose per session). We then use a consensus
+set maximization module to select the best poses for query frames (one pose per query).
+We further use a SLAM module to obtain 2D-2D matches and relative poses between
+consecutive query frames. Finally, with 2D-2D matches, relative pose, global absolute
+poses and their corresponding global 2D-3D matches, we use a pose refinement module
+to optimize the query poses. A pose polish module can be optionally used for better
+performance.
+
+CyberLoc
+3
+The main challenges are: 1) how to create accurate 3D maps that are robust
+to environmental changes, and 2) how to use the pre-built maps to accurately
+localize the camera.
+To address the first challenge, we propose to use a stereo-camera rig with
+GPS-IMU to mapping the world. Given stereo image sequences with ground-
+truth 6-DoF poses, we separately reconstruct four 3D maps using four methods:
+1) stereo Structure from Motion (SfM) with sparse 2D image features; 2) stereo
+SLAM; 3) stereo SfM with dense image matching; and 4) stereo disparity for
+each frame. Our motivation of using the above four 3D methods is to build a
+robust 3D map with respect to scene changes.
+To address the second challenge, we introduce two modules, namely the con-
+sensus set maximization and pose refinement module. Given multiple global
+poses corresponding to multiple reference maps, the consensus set maximization
+aims to select the best pose for each query, resulting in one global pose and one
+set of global 2D-3D matches per query image. For query image sequences, the
+pose refinement module utilizes the global information (i. e., the global poses
+and 2D-3D matches), and the local information (i. e., SLAM poses and 2D-2D
+matches between consecutive query images), to further optimize query poses.
+The entire localization pipeline of CyberLoc is given in Figure 1.
+The main contributions of this technical report are:
+1. We propose a visual localization pipeline consisting of four consecutive mod-
+ules that help to achieve high accuracy and robustness under long-term en-
+vironmental changes;
+2. We show that using multiple reference maps helps to overcome failed lo-
+calization caused by long-term scene changes. This is achieved by a new
+consensus set maximization module that identifies the best query pose with
+respect to multiple reference maps;
+3. We introduce a robust pose refinement method, combining global informa-
+tion from pre-built maps and local information from SLAM, to refine query
+poses.
+The proposed method is validated on the 4seasons dataset [1] and achieves state-
+of-the-art performance. In the following sections, we will give details of the pro-
+posed four modules. We first give our mapping pipeline in Sec. 2, to reconstruct
+3D maps for multiple reference sessions. Next, we present our single image local-
+ization in Sec. 3, to obtain multiple global poses for each query image. We then
+give the proposed consensus set maximization method in Sec. 4, to select the
+best query pose for each query image. Finally, in Sec. 5, we provide the proposed
+robust pose refinement method.
+2
+Mapping
+2.1
+Image Pre-processing
+In this section, we introduce some image pre-processing steps before using images
+for sparse 3D map reconstruction.
+
+4
+Low light image enhancement. For images captured in the night, we perform low
+light image enhancement using LLFlow [4]. An example is given in Figure 2. We
+find that using enhanced images would help to extract better 2D local features
+and global feature vectors from images.
+(a) low light image
+(b) enhanced image
+Fig. 2: An example of low light image enhancement.
+Semantic segmentation. When working in highly dynamic environment, both
+SfM and SLAM methods would perform poorly due to interference from dynamic
+objects [5]. Feature points on dynamic objects are either unrepeatable when
+performing 3D reconstruction or ‘fool’ the feature tracking in SLAM. Following
+common practice [6], we perform semantic segmentation using SegFormer [7] to
+mask out dynamic objects (e.g., car, cyclist, etc.). Furthermore, pixels belong to
+the car shield are also masked out.
+Resizing. Before using images to extract features, we resize images while keeping
+their original aspect ratio. The reason is that the performance of feature extrac-
+tion networks is vulnerable to image size. To extract a global feature vector from
+an image, we use a size of 800 (long side). To extract 2D local features from an
+image, we use a size of 2000. The two values are used as they perform best on
+the validation dataset.
+2.2
+Feature Extraction
+Global feature vector. Global feature vectors of images are used by image retrieval
+to, 1) find co-visible images for an image in the mapping (i. e., SfM) stage; or
+2) find a local map for each query image in the localization stage. To extract
+discriminative global feature vectors, we use an internal neural network trained
+on open street-view images and internal datasets. The comparison with respect
+to state-of-the-art NetVLAD [8], SARE [9], SFRS [10] on typical benchmarks
+and the 4seasons dataset is given in Table 1.
+Assembling multiple Global feature vectors from multiple networks is a com-
+mon practice to improve the performance of image retrieval 1. However, we found
+1 Please refer to the Google landmark retrieval challenge for more information
+
+CyberLoc
+5
+Table 1: Comparison of our image retrieval method with respect to state-of-the-art
+methods. We use the same distance threshold (25m) and evaluation metric as [8] to
+determine whether an image is successfully localized. Our model consistently outperform
+other methods on standard benchmarks and the 4seasons dataset.
+Methods
+Pitts250k
+Tokyo 24/7
+St Lucia
+Cityloop
+Oldtown
+Parkinggarage
+R@1
+R@5
+R@1
+R@5
+R@1
+R@5
+R@1
+R@5
+R@1
+R@5
+R@1
+R@5
+NetVLAD[8]
+86.0
+93.2
+73.3
+82.9
+-
+-
+-
+-
+-
+-
+-
+-
+SARE[9]
+89.0
+95.5
+79.7
+86.7
+-
+-
+-
+-
+-
+-
+-
+-
+SFRS[10]
+90.7
+96.4
+85.4
+91.1
+86.1
+93.5
+73.3
+82.4
+32.8
+44.8
+97.2
+99.3
+Ours
+93.0
+97.5
+89.5
+94.6
+99.6
+99.9
+95.9
+97.2
+67.7
+74.7
+99.7
+100.
+that the assembling technique does not work on the 4seasons dataset. The reason
+is that our single model outperforms other state-of-the-art methods by a large
+margin on the 4seasons dataset.
+Local features. 2D local features are used to perform sparse 3D reconstruction
+of the environment. We extract around 2000 SuperPoint [11] feature points for
+an image.
+2.3
+SfM
+We have pre-recorded database images with ground-truth 6-DoF camera poses
+in a world coordinate system. We perform image retrieval to find Top-K (K=40)
+similar database images. For each database image pair, we perform sparse feature
+matching and find 2D-2D matches. We further prune out wrong 2D-2D matches
+by enforcing the epipolar constraint using the ground-truth poses. With 2D-
+2D matches and ground-truth poses, we triangulate 2 3D points and perform
+structure-only bundle adjustment 3 to refine the positions of 3D points using
+colmap [12].
+Remark 1. Given Superpoint feature points, though SGMNet [13], ClusterGNN
+[14] and ELA [15] successfully reduce the computation complexity, their per-
+formance is worse than the pre-trained SuperGlue 4. Recently, our proposed
+FGCNet [18] achieves 4x speedup than SuperGlue, while maintaining competi-
+tive performance with respect to the pre-trained SuperGlue.
+Remark 2. For a triangulated 3D point, its precision depends on point-camera
+distances, number of visible cameras, view angles, etc. Though MegLoc [19] pro-
+poses to prune out 3D points with large uncertainty 5, we found the uncertainty
+threshold is hard to tune and we decided not to use this filtering step.
+2 SVD with post non-linear refinement.
+3 Fixing camera poses.
+4 See results on the YFCC100M [16] and FM-Bench [17] datasets.
+5 Please refer to Sec.5.1 of [20].
+
+6
+2.4
+SLAM
+Given ground-truth 6-DoF camera poses of images, with SuperPoint feature
+points, we perform SLAM using the ptam [21]. Note that we skip the pose
+estimation stage in the SLAM and only optimize 3D points in the local map
+optimization stage.
+Remark. The main difference between 3D points from SfM and SLAM is that
+we use ALL images in SLAM rather than keyframes in SfM. 3D points are
+triangulated and updated sequentially, rather than in a bundle.
+2.5
+Dense Mapping
+Using the same retrieved pairs in Sec.2.3, we use dense feature matching method
+QTA [22] to build sparse 2D-2D matches and then triangulate 3D points in the
+same way as Sec.2.3. Since dense feature matching methods do not detect sparse
+2D feature points and can generate unrepeatable 2D-2D matches 6, we use the
+same quantization technique in Patch2Pix [23] to alleviate this problem.
+Remark. The main difference between 3D points from Dense Mapping and SfM
+is that we use dense feature matching method QTA [22] rather than sparse fea-
+ture matching method SuperGlue [24] to build 2D-2D matches. The motivation
+is that QTA [22] can generate more matches than SuperGlue [24] for texture-
+less image pairs. More 3D points can be triangulated, resulting in a robust 3D
+representation for textureless areas.
+2.6
+Disparity
+For each image pair, we can estimate a dense disparity map using the pre-
+trained LEAStereo [25], thus obtaining a local 3D map for each database image.
+An example of the estimated disparity map is given in Figure 3.
+3
+Localization
+For each query image, we retrieve a set of candidate database keyframes using
+the same global feature vectors described in Sec.2.2. Since we have four pre-
+reconstructed 3D maps from SfM, SLAM, dense mapping, and disparity, we first
+perform four independent localization steps and then combine these localization
+results.
+Specifically, for maps from SfM and SLAM, we first find 2D-2D matches using
+SuperGlue for each query and candidate database pair. Since we have recorded
+6 Given 2D-2D matches for image pair A-B and A-C, 2D feature points in image A
+are different.
+
+CyberLoc
+7
+Fig. 3: An example of disparity estimation. Top: a database image; Bottom: the corre-
+sponding disparity image.
+the matching 3D point (if have) of each database 2D feature point, a set of 2D-
+3D matches are automatically obtained for each query and candidate database
+image pair.
+For maps from dense mapping, the only difference is that we use the dense
+feature matching method QTA [22] to find 2D-3D matches and all other com-
+ponents are the same.
+For maps from disparity, we also use QTA [22] to find 2D-2D matches. For
+each 2D feature point from the database image, we triangulate 3D points using
+the disparity map. With 2D-2D matches and 3D points for database feature
+points, we obtain a set of 2D-3D matches.
+Finally, the 2D-3D matches from the top-K (K = 40) candidate database
+images and above four maps are combined to perform PnP 7 to estimate a 6-
+DoF camera pose in the world coordinate system.
+Remark 1. The motivation of using 2D-3D matches from different maps is to uti-
+lize their complimentary characteristics to improve the robustness of our method
+with respect to large scene changes.
+Remark 2. Directly combining all 2D-3D matches from top-K (K = 40) can-
+didate database keyframes and above four maps would generate a large set of
+2D-3D matches with low inlier ratio, posing significant challenge for the subse-
+quent RANSAC-PnP step. To improve the inlier ratio, for each map, we perform
+RANSAC-PnP to obtain a set of inlier 2D-3D matches, and remove duplicated
+2D-3D matches 8 before combing 2D-3D matches from four maps.
+7 Generalized camera model, aka, Pl¨ucker lines [26].
+8 One 2D feature point from the query image may be matched to multiple different
+3D points. We only keep one 2D-3D pair with minimal reprojection error.
+
+8
+4
+Multi-session Consensus Set Maximization
+In a scene with multiple reference sequences such as the 4seasons[1] dataset,
+multi-session maps can be generated for the scene. In this section, we introduce
+a consensus set maximization method to fuse localization results using these
+multi-session maps.
+A simple method to use multi-session maps is to combine 2D-3D matches
+from multi-session maps to perform RANSAC-PnP. Although it works for com-
+plementary multi-session results, the performance is limited since the best lo-
+calization result from one-session map can be worsened by 2D-3D matches from
+other-session maps. Using combined 2D-3D matches for RANSAC-PnP would
+prone to produce averaged (or one dominant) 6-DoF pose, for poses from multi-
+session maps.
+Another method to use multi-session maps is to find the best combination
+of multi-session maps using trial-and-error [27]. It works for multi-session maps
+captured in different times of a day, in small-scale room environment. For the
+4seasons dataset, the large-scale multi-session maps are captured in different
+times of a year, making it hard to find the best combination.
+The key idea of our method is finding the best 6-DoF pose from multi-session
+maps for each query image through an optimization process, rather than com-
+bining 2D-3D matches from multi-session maps to perform RANSAC-PnP. The
+proposed method consistently produces optimal results across different datasets.
+4.1
+Problem Definition
+Let IK = {Ik|k = 1, · · · , K} denotes a query image sequence up to timestamp
+K. For each query image Ik, we have C candidate 6-DoF poses Pk = {Pk,i =
+(Rk,i, tk,i)|i = 1, · · · , C}, one from a reference map.
+We aim to find the most accurate 6-DoF pose Pk,i∗ for Ik.
+4.2
+Consensus Set Maximization
+For query images Im and In, we can obtain a set of 2D-2D matches Qm,n via
+feature tracking in SLAM. Given query poses Pm,i and Pn,j, we can compute the
+number of inlier 2D-2D matches subject to Pm,i and Pn,j, by thresholding the
+Sampson errors [28] of 2D-2D matches Qm,n. We denote the number of inlier 2D-
+2D matches as Sm,i,n,j and use it to measure the compatibleness of poses Pm,i
+and Pn,j. We assume the best poses would generate the largest score � Sm,i,n,j,
+i. e., finding the largest consensus set. Though the problem is non-convex [29]
+and the original Qm,n contains outlier matches, we found the score � Sm,i,n,j
+describes the correctness of poses well. The consensus set maximization problem
+is solved by a integer programming process.
+
+CyberLoc
+9
+Timestamp
+Session
+Pk,C
+Pk,i
+Pk,1
+Pm,C
+Pm,j
+Pm,1
+Pn,C
+Pn,l
+Pn,1
+n
+m
+k
+C
+1
+ek,i,j
+em,j,l
+Fig. 4: We use three frames k, m, n to describe our integer programming process. Given
+poses Pk, Pm, Pn from multi-session maps at timestamps k, m, n, respectively, we aim
+to find the best poses connected by the red edges. Edges ek,i,j ∈ {0, 1} denotes the
+connectivity between nodes (poses) Pk,i and Pm,j. ek,i,j equals 1 only when both nodes
+Pk,i and Pm,j are selected (deemed as the best poses). For the sake of clarity, we
+only draw edges connecting the node Pm,j. Note that all nodes are fully connected for
+consecutive timestamps.
+4.3
+Integer Programming
+We use three frames k, m, n to describe our integer programming process, as
+is given in Figure 4. We first build a densely connected graph for consecutive
+timestamps. For nodes Pk,i and Pm,j, there is an edge ek,i,j ∈ {0, 1} connecting
+them. The score of edge ek,i,j is Sk,i,m,j (abbrev. Sk,i,j for clarity), which is
+the number of inlier 2D-2D matches using poses Pk,i and Pm,j. Our integer
+programming process is given by,
+arg min
+E={ek,i,j}
+K−1
+�
+k=1
+C
+�
+i=1
+C
+�
+j=1
+−ek,i,jSk,i,j,
+(1)
+s.t.
+C
+�
+i=1
+C
+�
+j=1
+ek,i,j = 1,
+(2)
+C
+�
+i=1
+ek,i,j =
+C
+�
+l=1
+em,j,l.
+(3)
+Eq.(2) enforces that exactly one edge should be selected for the connectiv-
+ity of the graph. Eq.(3) enforces the following relationship: 1) if node Pm,j is
+selected, there must be an edge connecting Pm,j to other nodes in consecutive
+frames; 2) if node Pm,j is NOT selected, all edges connecting Pm,j to other
+nodes in consecutive frames should be removed.
+
+10
+Table 2: Result of the proposed consensus set maximization on the oldtown dataset.
+We report localization recalls with respect to different translation error thresholds. Both
+merging 2D-3D matches from multi-session maps and our integer programming method
+achieve the best performance.
+Trans.Err.
+Ref 0
+Ref 1
+Ref 2
+2D-3D Merge
+Int. Prog.
+0.05m
+36.60%
+39.80%
+43.20%
+51.00%
+49.80%
+0.1m
+60.80%
+68.40%
+69.20%
+78.30%
+78.40%
+0.2m
+79.30%
+87.00%
+85.60%
+94.10%
+93.20%
+0.5m
+89.20%
+93.50%
+92.30%
+96.60%
+96.50%
+1.0m
+92.00%
+95.50%
+94.10%
+97.80%
+97.80%
+3.0m
+93.70%
+96.50%
+95.00%
+98.30%
+98.30%
+Table 3: Result of the proposed consensus set maximization on the cityloop dataset. Our
+integer programming method achieves the best performance. In contrast, simply merging
+2D-3D matches from multi-session maps does not work. Its performance is even worse
+than single-session localization.
+Trans. Err.
+Ref 0
+Ref 1
+2D-3D Merge
+Int. Prog.
+0.05m
+69.20%
+74.20%
+73.30%
+79.20%
+0.1m
+91.80%
+88.00%
+87.30%
+92.30%
+0.2m
+98.00%
+94.70%
+94.20%
+97.70%
+0.5m
+99.60%
+99.20%
+99.10%
+99.70%
+1.0m
+99.90%
+99.50%
+99.60%
+100.00%
+3.0m
+100.00%
+99.70%
+99.70%
+100.00%
+There are a maximum 9 of (K − 1) × C × C edges (optimization variables) in
+our integer programming. The solution can be found very efficiently by off-the-
+shelf toolboxes. After optimization, nodes (poses) connected by edges ek,i,j = 1
+are deemed as the best poses.
+4.4
+Experimental Result
+We validate the proposed consensus set maximization method on the oldtown
+and cityloop datasets, from the 4seasons dataset. The number of image sequences
+with ground-truth poses is four and three, for the oldtown and cityloop dataset,
+respectively. For each dataset, one sequence is used for testing and the rest se-
+quences are used for mapping. The localization results are separately given in
+Table 2 and Table 3. The results show that the proposed consensus set max-
+imization method is robust and consistently shows good performance on the
+two datasets. In contrast, the simple method of merging 2D-3D matches does
+not work on the cityloop dataset. The reason is that localization results using
+multi-session maps are not compatible, on the cityloop dataset.
+We run our method on a PC equipped with a 2TB RAM and an AMD
+EPYC 7742 CPU. The running time of our method is given in Table 4. The
+9 If one map session fails to estimate a camera pose, the number of edges between
+consecutive timestamps is smaller than C × C.
+
+CyberLoc
+11
+Table 4: Running time of the proposed consensus set maximization. For the oldtown
+dataset, we have three reference sequences (C = 3), and each of the sequence has 3296
+frames (K = 3296). For the cityloop dataset, C = 2 and K = 10224.
+Dataset
+Total time
+Single frame time
+Oldtown
+24.4s
+7.4ms
+Cityloop
+70.7s
+6.9ms
+implementation is based on Python and uses CBC solver for optimization. The
+running time can be significantly reduced using C++.
+5
+Pose Refinement
+In this section, we show how to combine global information from Sec. 4 and local
+information from SLAM to refine query poses.
+5.1
+Problem Definition
+For each query image Ik, we have obtained its best global pose Pk from the multi-
+session localization step (Sec. 4). Considering that the accuracy (even being an
+outlier) of Pk varies for different frames, we associate a weight variable wk ∈ [0, 1]
+for each Pk, describing the weight of Pk in our optimization process. We also
+retrieve 2D-3D matches Fk corresponding to Pk, and denote the reprojection
+error at pose Xk as π (Xk, Fk).
+For consecutive frames, we can obtain their local relative poses Zk−1,k and
+2D-2D matches Qk−1,k through SLAM. Given query poses Xk−1 and Xk, we
+denote the two-view matching (Sampson) error as ρ (Xk−1, Xk, Qk−1,k).
+Given query images IK = {Ik|k = 1, · · · , K} up to timestamp K, we aim to
+solve query poses XK = {Xk|k = 1, · · · , K} and latent weights WK = {wk|k =
+1, · · · , K}. The overall framework of our optimization problem is given in Fig. 5.
+Our objective function is given by,
+arg min
+XK,WK
+�
+k
+wk
+�
+∥Pk − Xk∥2 + λ1π (Xk, Fk)
+�
++
+∥h (Xk−1, Xk) − Zk−1,k∥2 + λ2ρ (Xk−1, Xk, Qk−1,k) ,
+(4)
+where ∥·∥2 denotes the pose error, and h (Xk−1, Xk) denotes the relative pose
+between Xk−1 and Xk, λ1 and λ2 are two weights to balance the 2D-3D repro-
+jection error and 2D-2D Sampson error.
+Remark 1 If no global pose can be obtained at frame k, one can simply remove
+Fk, Pk, and wk.
+Remark 2 One can also add skip constrains Zm,n and Qm,n (n ̸= m + 1).
+
+12
+X1
+X2
+Xk−1
+Xk
+XK
+P1
+w1
+F1
+Pk
+wk
+Fk
+Zk−1,k
+Qk−1,k
+Z1,2
+Q1,2
+...
+...
+Fig. 5: The framework of our pose refinement step. {X1, ..., Xk, XC} denotes query
+poses. Pk denotes the best pose from multi-session localization (Sec. 4). Fk denotes the
+2D-3D matches corresponding to the best pose Pk. wk ∈ [0, 1] separately denotes the
+weight of Pk. Zk−1,k denotes the relative pose from SLAM. Qk−1,k denotes the 2D-2D
+matches from SLAM. Green lines denote a pose-graph in our framework. We aim to
+solve for red variables.
+5.2
+Robust Optimization
+Inspired by [30], we use Expectation-Maximization (EM) algorithm to solve
+Eq.(4). The latent weights WK and query poses XK are alternatively optimized
+until convergence (the change of WK is smaller than a threshold).
+Expectation Step We aim to find the best WK while fixing XK, the Expectation
+step becomes,
+arg min
+WK
+�
+k
+wk
+�
+∥Pk − Xk∥2 + λ1π (Xk, Fk)
+�
+− U 2 (log wk − wk)
+�
+��
+�
+Regularization
+,
+(5)
+where the regularization term is added to avoid a trivial solution of WK = 0 and
+U is a constant.
+Eq.(5) is convex, and the minimum can be found by differentiating it with
+respect to wk and setting the gradient to zero. The updating of wk at the t-th
+iteration using query pose Xt
+k is given by,
+wt+1
+k
+=
+U 2
+U 2 + ∥Pk − Xt
+k∥2 + λ1π (Xt
+k, Fk)
+.
+(6)
+Maximization Step We aim to find the best XK while fixing WK, the updating
+of XK at the t-th iteration using weights Wt
+K is given by,
+X t+1
+K
+= arg min
+XK
+�
+k
+wt
+k
+�
+∥Pk − Xk∥2 + λ1π (Xk, Fk)
+�
++
+∥h (Xk−1, Xk) − Zk−1,k∥2 + λ2ρ (Xk−1, Xk, Qk−1,k) .
+(7)
+We solve (7) using ceres [31].
+
+CyberLoc
+13
+Table 5: Result of pose refinement on the oldtown datasets. Method PGBA achieves the
+best performance.
+Trans. Err.
+Baseline
+PGO
+PGO 2D2D
+PGO 2D3D
+PGBA
+0.05m
+49.80%
+55.10%
+56.20%
+56.80%
+56.90%
+0.1m
+78.40%
+83.80%
+84.30%
+85.20%
+85.30%
+0.2m
+93.30%
+96.80%
+96.80%
+97.20%
+97.30%
+0.5m
+96.50%
+98.50%
+98.40%
+98.40%
+98.40%
+1.0m
+97.80%
+99.00%
+99.00%
+99.00%
+99.00%
+3.0m
+98.30%
+99.30%
+99.30%
+99.30%
+99.30%
+Initialization To initialize the iterative E-M updating process, we select a good
+subset of poses from PK as seeds and initialize other query poses using their
+most recent seeds and relative SLAM poses, resulting in X 1
+K. A query pose with
+the number of 2D-3D matches (Fk) larger than a pre-defined threshold is deemed
+as a seed.
+5.3
+Experimental Result
+Based on the global localization results of Sec. 4.4, we run the pose refinement
+step. Our refinement objective function has four terms (Eq. (4)) and we conduct
+following ablation studies to validate the effectiveness of each term. Specifically,
+1. Pose Graph Optimization (PGO). By removing global 2D-3D reprojection
+term π (Xk, Fk) and 2D-2D Sampson term ρ (Xk−1, Xk, Qk−1,k), the objec-
+tive function becomes PGO.
+2. PGO 2D2D. By removing global 2D-3D reprojection term π (Xk, Fk), the
+objective function becomes PGO with 2D-2D sampson term.
+3. PGO 2D3D. By removing 2D-2D Sampson term ρ (Xk−1, Xk, Qk−1,k), the
+objective function becomes PGO with 2D-3D reprojection term.
+4. PGBA. We keep all four terms and denote the objective function as PGBA
+(PGO with Bundle Adjustment).
+The results of the above four methods are given in Table 5 and 6. For the oldtown
+dataset, both PGO 2D2D and PGO 2D3D outperforms PGO, showing the ef-
+fectiveness of adding global 2D-3D reprojection and 2D-2D Sampson terms. The
+best performance is obtained by combining the two terms, resulting in PGBA.
+For the cityloop dataset, all methods have similar performance probably because
+this dataset is saturated.
+We show estimated query positions before and after consensus set maximiza-
+tion and pose refinement steps in Fig. 6. It clearly shows that wrong estimated
+query positions are refined to correct positions, resulting in a smooth trajectory
+after refinement.
+Our pose refinement step is implemented using C++, and the processing time
+is given in Table 7. For time-critical applications, methods PGO and PGO 2D3D
+are good candidates.
+
+14
+Table 6: Result of pose refinement on the cityloop dataset. Method PGBA achieves the
+best performance.
+Trans. Err.
+Baseline
+PGO
+PGO 2D2D
+PGO 2D3D
+PGBA
+0.05m
+79.20%
+81.00%
+81.00%
+81.30%
+81.40%
+0.1m
+92.30%
+93.00%
+93.00%
+93.10%
+93.00%
+0.2m
+97.70%
+98.20%
+98.20%
+98.60%
+98.60%
+0.5m
+99.70%
+99.70%
+99.70%
+99.70%
+99.70%
+1.0m
+100.00%
+100.00%
+100.00%
+100.00%
+100.00%
+3.0m
+100.00%
+100.00%
+100.00%
+100.00%
+100.00%
+Table 7: Running time of pose refinement. We have 3296 and 10224 query frames for
+the oldtown and cityloop dataset, respectively. Since initial poses of the cityloop dataset
+are more accurate than those of oldtown dataset, fewer EM iterations are performed
+for cityloop, resulting in its smaller running time.
+Dataset
+PGO
+PGO 2D2D
+PGO 2D3D
+PGBA
+Oldtown
+26.7s
+2486.7s
+102.0s
+2245.0s
+Cityloop
+21.0s
+2263.8s
+93.9s
+2677.8s
+(a) Oldtown
+(b) Cityloop
+Fig. 6: Query positions before and after refinement. Outlier positions are refined to
+correct ones, resulting in smooth trajectories.
+
+Groundtruth
+Stereo localization
+After CSM & PGBACyberLoc
+15
+Table 8: Results of pose polish on the oldtown and cityloop datasets. With the second
+round of consensus set maximization and pose refinement, we get more accurate poses.
+Oldtown
+Cityloop
+Trans. Err.
+Baseline
+Polish
+Baseline
+Polish
+0.05m
+56.90%
+59.70%
+81.40%
+82.10%
+0.1m
+85.30%
+85.20%
+93.00%
+93.30%
+0.2m
+97.30%
+97.20%
+98.60%
+98.60%
+0.5m
+98.40%
+98.50%
+99.70%
+99.70%
+1.0m
+99.00%
+99.00%
+100.00%
+100.00%
+3.0m
+99.30%
+99.30%
+100.00%
+100.00%
+6
+Pose Polishing
+Note that the global pose Pk used in the pose refinement (Sec.5) step is from
+one of the multi-sessions localization results. Using optimized query poses XK
+from Sec.5, we can unify information from multi-sessions to obtain a better Pk,i,
+followed by another round of consensus set maximization and pose refinement
+as described in Sec.4 and Sec.5. This final pose polishing step is performed
+(optionally) only once.
+As discussed in Sec.4, simply combining all 2D-3D matches from multi-session
+maps would produce an averaged (or one dominant) pose over multi-session
+poses. Using the accurate XK as an anchor, we first compute reprojection errors
+of all 2D-3D matches from multi-session maps and then prune out outlier 2D-3D
+matches. Remaining 2D-3D matches are used to compute the final global pose
+Pki for the final round of pose fusion and refinement. This guided pose polish
+step could further improve the final pose accuracy for better application.
+6.1
+Experimental Result
+Based on estimated poses from Sec. 5.3, we test the pose polish module. We use
+refined poses XK to filter 2D-3D matches and recompute the pose for each session.
+A second round of consensus set maximization and pose refinement is performed
+on these poses. The results are given in Table 8. For both datasets, polishing
+helps to further improve pose accuracies, as the [email protected] increases.
+7
+Conclusions
+We have proposed a method, named CyberLoc, for robust and accurate visual lo-
+calization under challenging conditions. The key idea is to build a robust map for
+each reference sequence, find the best global camera pose with respect to multi-
+session maps, and combine global localization and local SLAM to refine camera
+poses. The proposed robust mapping and localization, consensus set maximiza-
+tion and pose refinement facilitates the success of our method, to be used in
+scenes with illumination and environmental changes. Extensive experiments on
+4seasons datasets demonstrate the high accuracy and robustness of our method.
+
+16
+References
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+autonomous driving.
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+arXiv:2301.04863v1  [math.ST]  12 Jan 2023
+Choosing observation operators to mitigate
+model error in Bayesian inverse problems
+Nada Cvetkovi´c1, Han Cheng Lie2, Harshit Bansal1, and Karen
+Veroy–Grepl1
+1Centre for Analysis, Scientific computing and Applications, Eindhoven University of Technology,
+Groene Loper 3, 5612 AE Eindhoven, the Netherlands
+2 Institut f¨ur Mathematik, Universit¨at Potsdam, Campus Golm, Haus 9, Karl-Liebknecht-Straße 24–25,
+Potsdam OT Golm 14476, Germany
+Abstract
+In Bayesian inverse problems, ‘model error’ refers to the discrepancy between the parameter-
+to-observable map that generates the data and the parameter-to-observable map that is
+used for inference. Model error is important because it can lead to misspecified likelihoods,
+and thus to incorrect inference. We consider some deterministic approaches for accounting
+for model error in inverse problems with additive Gaussian observation noise, where the
+parameter-to-observable map is the composition of a possibly nonlinear parameter-to-state
+map or ‘model’ and a linear state-to-observable map or ‘observation operator’. Using local
+Lipschitz stability estimates of posteriors with respect to likelihood perturbations, we bound
+the symmetrised Kullback–Leibler divergence of the posterior generated by each approach
+with respect to the posterior associated to the true model and the posterior associated to the
+wrong model. Our bounds lead to criteria for choosing observation operators that mitigate
+the effect of model error on the posterior.
+Keywords: Model error, Bayesian inverse problems, experimental design, misspecified like-
+lihood, posterior error bounds
+1. Introduction
+In many applications, one considers an inverse problem where the data is a noisy observation of
+the true state of some phenomenon of interest, where the true state is the output of a parameter-
+to-state mapping or ‘model’ corresponding to an unknown parameter. That is, for the unknown
+true parameter θ† and the true model M†, the true state is u† = M†(θ†), and the data y is a
+realisation of the random variable
+Y := O ◦ M†(θ†) + ε
+(1.1)
+for an observation operator O and additive noise ε. The inverse problem consists in inferring
+the data-generating parameter θ† from the data y.
+We consider Bayesian inverse problems with finite-dimensional data, centred Gaussian obser-
+vation noise, and linear observation operators. In this setting, ε has the normal distribution
+1
+
+N(mε, Σε) for mε = 0 ∈ Rn and positive definite covariance Σε ∈ Rn×n, and the observation
+operator O is a linear mapping from the ‘state space’ U of candidate states to the ‘data space’
+Y = Rn. One fixes a possibly infinite-dimensional parameter space Θ consisting of candidate
+values of θ†, and describes the unknown true parameter θ† using a random variable θ with prior
+law µθ supported on Θ. Under certain assumptions on the prior µθ, the observation operator O,
+and the model M†, the posterior measure µy,†
+θ
+of θ|y that corresponds to O ◦M† is well-defined
+and admits the following likelihood with respect to the prior µθ:
+Θ ∋ θ′ �→ dµy,†
+θ
+dµθ
+(θ′) =
+exp(− 1
+2∥y − O ◦ M†(θ′)∥2
+Σ−1
+ε )
+�
+Θ exp(− 1
+2∥y − O ◦ M†(ˆθ)∥2
+Σ−1
+ε )dµθ(ˆθ)
+.
+(1.2)
+See e.g. [24, Theorem 4.1] for sufficient conditions for well-definedness in the case of a Gaussian
+prior µθ.
+In practice, the true model M† : Θ → U is not available, so an approximate model M : Θ → U
+is used instead. Alternatively, M† may be available but impractical or costly to evaluate: in
+the context of multifidelity or reduced order models, M† may be the element of a collection M
+of models that yields the most accurate predictions of state, and M may be a reduced-order
+model, an emulator, or a surrogate, i.e. a model that yields less accurate predictions but can
+be evaluated more cheaply and quickly than M†.
+We shall refer to the difference δ† := M† − M as the ‘model error of the appproximate
+model’, or simply the ‘model error’. Model error is also known as ‘model inadequacy’, ‘model
+discrepancy’, or ‘structural uncertainty’, for example; see [14, 4]. We do not use the term ‘model
+misspecification’, since this term is used in the statistics literature to refer to the distinct problem
+where the parameter space Θ does not contain the true parameter θ†; see e.g. [9, Section 8.5].
+In the context of inverse problems, model error is important because it may lead to a wrong
+or ‘misspecified’ likelihood, which in turn may lead to incorrect inference. The negative effects
+may persist even after applying or approximating common limits from statistics. For example,
+numerical experiments in [4, Section 3] show how ignoring the model error results in posterior
+distributions that do not converge to the true data-generating parameter as the number of
+observations grows larger. An analytical example in [1, Section 8.1] considers the problem of
+inferring the initial condition of an initial value problem on the time interval [0, T] from a noisy
+observation of the state at time T, and shows that the posterior density contracts around the
+wrong initial condition in the limit of small observation noise and T → 0.
+This raises the
+question of how to mitigate the effect of model error in Bayesian inverse problems.
+We approach this question from the point of view of selecting an appropriate observation
+operator O. Using that O is a linear mapping and substituting M† with M + δ† in (1.2) yields
+Θ ∋ θ′ �→
+exp(− 1
+2∥y − O ◦ M(θ′) − O ◦ δ†(θ′)∥2
+Σ−1
+ε )
+�
+Θ exp(− 1
+2∥y − O ◦ M(ˆθ) − O ◦ δ†(ˆθ)∥2
+Σ−1
+ε )dµθ(ˆθ)
+.
+The posterior µy,A
+θ
+that uses the approximate model M and ignores δ† has the likelihood
+Θ ∋ θ′ �→ dµy,A
+θ
+dµθ
+(θ′) =
+exp(− 1
+2∥y − O ◦ M(θ′)∥2
+Σ−1
+ε )
+�
+Θ exp(− 1
+2∥y − O ◦ M(ˆθ)∥2
+Σ−1
+ε )dµθ(ˆθ)
+.
+Thus, the presence of model error δ† leads to a misspecified likelihood if and only if O ◦ δ†(θ′)
+is nonzero with positive µθ-probability. This suggests that a possible approach to mitigate the
+2
+
+effect of model error in Bayesian inverse problems of the type given by (1.1) is to choose an
+observation operator O so that O ◦ δ†(θ′) is close to zero with high µθ-probability.
+The approach of choosing observation operators suggests a connection with experimental
+design. In Bayesian experimental design, the main task is to select observations in order to
+maximise information about the parameter to be inferred. To quantify the information gain,
+one may use the Kullback–Leibler divergence of the posterior with respect to the prior, for
+example. In contrast, we control the Kullback–Leibler divergence between pairs of posteriors
+defined by the same prior but different likelihoods, by using the L1
+µθ difference between the pair
+of negative log-likelihoods or ‘misfits’. The main task is then to select observations in order to
+minimise this L1
+µθ difference. This approach can be seen as experimental design for reducing
+the impact of likelihood misspecification due to model error.
+Contributions.
+In this paper, we consider three deterministic approaches for accounting for
+model error in Bayesian inference: the ‘enhanced error approach’ of [13]; the ‘joint approach’
+that infers both θ† and δ†; and the marginalisation approach, which integrates out the model
+error component of the posterior from the joint inference approach.
+For the first two ap-
+proaches, we compute upper bounds for the L1
+µθ difference between the misfit of each approach
+and the misfit of the best posterior µy,†
+θ
+defined by (1.2).
+These upper bounds on the L1
+µθ
+difference between misfits yield upper bounds for the symmetrised Kullback–Leibler divergence
+max{dKL(µy,•
+θ ∥µy,†
+θ ), dKL(µy,†
+θ ∥µy,•
+θ )} between the posterior µy,•
+θ
+that results from each approach
+and the best posterior µy,†
+θ
+defined by (1.2).
+We repeat the procedure for the approximate
+posterior µy,A
+θ
+instead of µy,†
+θ . For each approach, we express the upper bounds on the L1
+µθ
+differences in terms of the model error δ† and the objects that each approach uses to account
+for model error.
+To prove these bounds, we rely on the assumption of additive Gaussian noise, in the form of a
+lemma concerning the difference of two quadratic forms that are weighted by different matrices;
+see Lemma A.2. We prove the upper bounds on the symmetrised Kullback–Leibler divergence
+between the posteriors by combining the upper bounds on the L1
+µθ differences between the
+misfits with a local Lipschitz stability estimate of posteriors from [23]. An important advantage
+of this estimate is that it holds for the Kullback–Leibler topology under the mild assumption
+of L1
+µθ-integrable misfits; see Theorem 3.1.
+The upper bounds on the symmetrised Kullback–Leibler divergence with respect to the best
+posterior µy,†
+θ
+provide sufficient conditions on the observation operator O, the model error δ†,
+and the approach, in order for the resulting posterior µy,•
+θ
+to closely approximate µy,†
+θ .
+In
+contrast, the upper bounds on the symmetrised Kullback–Leibler divergence with respect to
+the approximate posterior µy,A
+θ
+provide necessary conditions for the resulting posterior µy,•
+θ
+to
+differ from µy,A
+θ
+. The first and second set of upper bounds give respectively a set of ‘positive’
+and ‘negative’ criteria by which to choose observation operators for Bayesian inverse problems
+in the presence of model error.
+1.1. Overview of related work
+The importance of accounting for the model error is well-documented in the literature on
+Bayesian inverse problems; see e.g. [14, 12, 13, 4] and the references therein. The ‘Bayesian
+approximation error’ and ‘enhanced error’ approaches due to [13] and [12] respectively have
+been applied in various contexts. For example, the enhanced error approach has been applied
+with premarginalisation to electrical impedance tomography [19], diffuse optical tomography
+3
+
+[15], and inversion for coefficient fields in the presence of uncertain conductivities [18].
+Various methods have been developed to estimate or account for model error in Bayesian
+inference. For example, the work [5] presented an iterative algorithm to update an estimate of
+the model error in a model order reduction context, and proved geometric convergence of the
+algorithm. The authors of [20, 21] take a different perspective: instead of viewing model errors
+as additive perturbations to an approximate model, they incorporate these model errors into
+parametrisations of some phenomenon of interest, and use polynomial chaos expansions. Infor-
+mation theory has been used to quantify model error uncertainty or model bias in goal-oriented
+inference settings [11] and by exploiting concentration inequalities [10]. Optimal transport was
+applied to tackle problems due to model errors in [22].
+In the context of Bayesian optimal experimental design, model error is also referred to as
+‘model uncertainty’. The work [16] considers A-optimal designs for inverse problems in the
+presence of so-called ‘irreducible’ model uncertainties, i.e. uncertainties that cannot be reduced
+by collecting more data. In contrast, the work [3] considers reducible uncertainties, and describes
+an A-optimality criterion that involves marginalising out this reducible uncertainty. The work
+[2] combines the Laplace approximation and the Bayesian approximation error approach to find
+A-optimal designs for nonlinear Bayesian inverse problems.
+As far as we are aware, the work that is most closely related to our paper consists in [6, 8]. The
+work [6] considers Bayesian inverse problems where the observation operator may be nonlinear
+and the model is approximated by a neural network. In particular, [6, Theorem 1] bounds the
+Kullback–Leibler divergence between the original and approximated posterior in terms of an Lp
+norm for p ≥ 2 of the model error itself. In contrast, we consider linear observation operators,
+do not focus on any specific class of approximate models, and bound the Kullback–Leibler
+divergence in terms of an L1 norm of the observed model error. In [8], the main stability estimate
+[8, Theorem 3.4] bounds the expected utility of a design in terms of a sequence of likelihoods.
+The focus of [8] is not model error, but on the convergence of the utility of approximate optimal
+designs corresponding to a convergent sequence of likelihoods. In contrast, we focus on choosing
+observation operators to mitigate the effect of model error on Bayesian inference instead of
+‘classical’ experimental design, and compare pairs of likelihoods instead of sequences.
+1.2. Outline
+We describe the notation that we use in Section 1.3. In Section 2 we define the posteriors
+that we analyse in this paper and state our main assumptions. In Section 3 we bound the L1
+µθ
+differences between misfits, and the symmetrised Kullback–Leibler divergences between their
+associated posterior measures. We first consider posteriors defined only on parameter space
+in Section 3.1, before we consider the misfit and posterior obtained from jointly inferring the
+parameter and model error in Section 3.2. We use the Kullback–Leibler bounds to identify
+conditions on observation operators so as to mitigate the effect of model error on parameter
+inference. We conclude in Section 4. Appendix A contains the proofs of lemmas and results
+that we do not write in the main text.
+1.3. Notation
+Let P be the probability measure of a probability space that serves as a common domain for
+all random variables of interest. Given a measurable space (E, E) and an E-valued random
+variable ξ : (Ω, F) → (E, E), we denote the law of ξ by µξ and write ξ ∼ µξ.
+Given a metric space (E, dE) with Borel σ-algebra B(E), let M1(E) denote the set of all
+4
+
+probability measures µ on (E, B(E)). Thus, for µ ∈ M1(E) and an E-valued random variable
+ξ, the expression ξ ∼ ν means that µξ = ν as measures. Given two measures µ and ν on a
+common measurable space (E, E), we denote the absolute continuity of µ with respect to ν by
+µ ≪ ν and the mutual equivalence of µ and ν by µ ∼ ν. For µ ∈ M1(E), p ≥ 1, and d ∈ N,
+Lp
+µ(E; Rd) := {f : E → Rd : ∥f∥Lp
+µ < ∞},
+∥f∥p
+Lp
+µ :=
+�
+E
+|f(x)|pdµ(x).
+We will denote a Gaussian measure with mean m and covariance operator Σ by N(m, Σ).
+The notation a ← b indicates the replacement of a using b.
+We denote the standard Euclidean inner product and norm on Rd by ⟨·, ·⟩ and ∥·∥. For d ∈ N
+and a symmetric positive semidefinite matrix L ∈ Rd×d, the matrix-weighted inner product and
+norm are
+⟨a, b⟩L := a⊤Lb = ⟨L1/2a, L1/2b⟩,
+∥a∥L := ⟨a, a⟩1/2
+L
+= ∥L1/2a∥.
+(1.3)
+Below, Θ will denote the parameter space, M will denote the space of models, O will denote
+the space of observation operators, and D will denote the space of model errors.
+Given normed vector spaces (Vi, ∥ · ∥Vi) for i = 1, 2, L (V1, V2) denotes the space of bounded
+linear operators from V1 to V2, and V ∗
+i
+denotes the continuous dual space of Vi. We denote
+the kernel, range, and adjoint of L ∈ L (V1, V2) by ker(L), ran(L), and L∗. In particular, for a
+matrix A ∈ Rm×n, ∥A∥ denotes the norm of A : Rn → Rm where both Rn and Rm are equipped
+with the Euclidean norm.
+2. Description of assumptions and posterior measures
+Setup and assumptions
+Fix a space (Θ, ∥ · ∥Θ) of admissible unknown parameters or ‘pa-
+rameters’ θ, which we take to be a measurable subset of a Banach space. Fix a Banach space
+(U, ∥ · ∥U) of ‘states’ u. We shall refer to a measurable mapping M : Θ → U as a ‘model’. Let
+M denote a fixed collection of admissible models. We shall refer to M as the ‘model space’.
+In many inverse problems, the state space U is a Banach space of functions on a fixed, bounded
+spatial or spatiotemporal domain D that take values in a common Euclidean space, e.g. U may
+be the space (C(D), ∥ · ∥L∞) of continuous functions on a bounded domain, equipped with the
+supremum norm, or a Sobolev space (Hk(D), ∥ · ∥Hk) for some k > 0. The parameter space
+is also a function space. The model M is often described implicitly, by an ordinary or partial
+differential equation where one or more coefficients are determined by the parameter θ.
+Assumption 2.1. There exists a unique best model M† ∈ M and a unique best parameter
+θ† ∈ Θ such that among all model-unknown pairs (M′, θ′) ∈ M × Θ, the corresponding ‘best
+state’ u† := M†(θ†) describes the phenomenon of interest most accurately.
+The assumption of a unique θ† is commonly made in the context of frequentist statistics,
+where θ† is often referred to as the ‘true parameter’. However, in the context of experimental
+design for statistical inverse problems where observations are assumed to have the form (1.1),
+the question of whether a parameter is the true parameter makes sense only when a parameter-
+to-state map or model has been specified. Hence, for θ† to have the interpretation of the ‘true
+parameter’, we must also fix a unique ‘true model’ M†.
+Remark 2.2. In this paper, we shall consider the setting where the best model M† in Assump-
+tion 2.1 is unknown. However, one may also consider the best model M† to be a model that is
+known but is too expensive to use ‘frequently’, where the meaning of ‘frequently’ depends on the
+5
+
+context or the target application. The model M can be considered as an emulator, a surrogate,
+or a reduced order model; M ideally has the property that it approximates M† reasonably well
+and is cheaper to evaluate than M†.
+Assumption 2.3. There is a fixed collection of measurement functions (ℓi)i∈I indexed by a
+countable set I ⊂ N, where each ℓi is a continuous linear mapping from U to Rd for some d ∈ N.
+In addition, every admissible observation operator O has the form
+O : U → RNd,
+u �→ (ℓi1(u), . . . , ℓiN (u)),
+ij ∈ I,
+(2.1)
+where the (ℓij)N
+j=1 are distinct. Associated to the collection (ℓij)N
+j=1 of measurement functions in
+(2.1) is a collection of random variables (εij)N
+j=1 that are independent and identically N(0, Σ0)-
+distributed, where Σ0 ∈ Rd×d is positive definite.
+The random variables (εij)N
+j=1 represent
+additive measurement noise.
+If U = C(D, ∥ · ∥L∞), then pointwise evaluation functionals of the form ℓij(u) := u(xij) for
+some xij ∈ D give an example of the measurement functions in (2.1).
+An important consequence of the assumptions on the measurement noise (εij)N
+j=1 in Assump-
+tion 2.3 is that the RNd-valued random variable satisfies
+ε := (εi1, . . . , εiN ) ∼ N(0, Σε),
+Σε = diag(Σ0, . . . Σ0) ∈ RNd×Nd.
+The last equation above means that Σε is a block-diagonal matrix with identical blocks that
+do not depend on O. In particular, Σε depends on the choice of O only via the number N
+of observations. One could achieve greater generality by allowing the (εij)N
+j=1 in (2.1) to be
+statistically correlated or to have different distributions. This generality would allow Σε to
+depend not only on N but on O itself.
+In the remainder of this section, we will describe the posterior measures that we shall analyse
+in this paper. For a measurable misfit Φ : Θ → R, we will write
+Z(Φ) :=
+�
+Θ
+exp(−Φ(θ′))dµθ(θ′)
+to denote the normalisation constant that makes θ′ �→ exp(−Φ(θ′))Z(Φ)−1 a probability density
+function with respect to µθ, whenever the normalisation constant belongs to (0, ∞).
+Approximate posterior
+Given O as in (2.1) and some model M ∈ M , we assume that an
+observation y is a noisy observation of state, i.e. a realisation of the Y := RNd-valued random
+variable
+Y := O ◦ M(θ†) + ε.
+(2.2)
+Given Assumption 2.3, the observation model (2.2), a prior µθ on θ, the data y and Bayes’ law,
+we obtain the approximate misfit Φy,A and the approximate posterior
+Φy,A(θ′) := 1
+2∥y − O ◦ M(θ′)∥2
+Σ−1
+ε ,
+(2.3a)
+dµy,A
+θ
+(θ′) := exp(−Φy,A(θ′))
+Z(Φy,A)
+dµθ(θ′).
+(2.3b)
+6
+
+Best posterior
+Recall from Assumption 2.1 that u† := M†(θ†) best describes the phenomenon
+of interest. For an arbitrary O ∈ O, the ‘best model’ is given by replacing M with M† in (2.2):
+Y = O ◦ M†(θ†) + ε.
+The corresponding best misfit Φy,† and best posterior µy,†
+θ
+are defined by
+Φy,†(θ′) := 1
+2∥y − O ◦ M†(θ′)∥2
+Σ−1
+ε ,
+(2.4a)
+dµy,†
+θ (θ′) := exp(−Φy,†(θ′))
+Z(Φy,†)
+dµθ(θ′).
+(2.4b)
+Given M ∈ M , the corresponding ‘model error’ is given by the difference
+δ† := M† − M ∈ D,
+D := M† − M .
+(2.5)
+We refer to D as the ‘model error space’. If the model space M is a vector space, then the
+model error space D and M coincide. If M† is not known or too expensive to evaluate, then
+so is δ†. For the unique, fixed θ† in Assumption 2.1, define the corresponding ‘state error’
+δ†(θ†) = M†(θ†) − M(θ†) ∈ U.
+Rewriting (2.5) as M† = M + δ†, substituting the latter equation into the best observation
+model, and using the linearity of O, we obtain
+Y = O ◦ M(θ†) + O ◦ δ†(θ†) + ε.
+The observation model (2.2) thus corresponds to the assumption of zero observed state error
+O ◦ δ†(θ†).
+Enhanced noise posterior
+One approach that aims to account for the observed state error
+is to group the observed state error O ◦ δ†(θ†) with the noise ε to obtain O ◦ δ†(θ†) + ε, and
+to model this random variable with an ‘enhanced noise’ random variable [12, 13]. This is also
+known as the ‘pre-marginalisation’ approach, e.g. [15]. We approximate the unknown state
+error δ†(θ†) with a random variable u, and make the following assumption.
+Assumption 2.4. The random variable u that approximates the unknown state error δ†(θ†) is
+Gaussian with mean mu and covariance Σu, and is independent of θ ∼ µθ and ε ∼ N(0, Σε).
+Given the distributional assumptions in Assumption 2.4 and the linearity assumption on O in
+Assumption 2.3, it follows from the properties of Gaussian random variables that the enhanced
+noise random variable Ou + ε has the law N(Omu, Σε + OΣuO∗). This yields the enhanced
+noise observation model
+Y = O ◦ M(θ†) + Ou + ε,
+which yields the enhanced noise misfit and enhanced noise posterior
+Φy,E(θ′) := 1
+2∥y − O ◦ M(θ′) − Omu∥2
+(Σε+OΣuO∗)−1,
+(2.6a)
+dµy,E
+θ
+(θ′) := exp(−Φy,E(θ′))
+Z(Φy,E)
+dµθ(θ′).
+(2.6b)
+7
+
+Joint parameter-error posterior
+In the enhanced noise approach presented in (2.6), we account
+for the uncertainty due to the state error δ†(θ†) by approximating it using a random variable u.
+The only unknown that we aim to infer is θ†. In the joint parameter-error inference approach,
+one aims to infer (θ†, δ†) jointly, by using a random variable (θ, δ) with prior µθ,δ and using
+Bayes’ formula.
+Assumption 2.5. The joint prior on the random variable (θ, δ) is a product measure of the
+form µθ ⊗ µδ, for µθ ∈ M1(Θ) and µδ ∈ M1(D).
+The assumption that the prior µθ,δ on (θ, δ) has product structure is equivalent to the as-
+sumption that θ and δ are independent random variables.
+Under the observation model
+Y = O ◦ M(θ) + O ◦ δ(θ) + ε
+and under the distributional assumptions on ε in Assumption 2.3, we have the joint misfit and
+joint posterior
+Φy,J(θ′, δ′) := 1
+2∥y − O ◦ M(θ′) − O ◦ δ′(θ′)∥2
+Σ−1
+ε ,
+(2.7a)
+dµy,J
+θ,δ(θ′, δ′) := exp(−Φy,J(θ′, δ′))
+Z(Φy,J)
+dµθ ⊗ µδ(θ′, δ′).
+(2.7b)
+One important disadvantage of jointly inferring the parameter and model error is that the di-
+mension of the space on which one performs inference increases; this tends to make the inference
+task more computationally expensive. It is also known that the problem of identifiability may
+arise, but we shall not consider the problem of identifiability here. On the other hand, jointly
+inferring the parameter and model error is consistent with the Bayesian approach of treating
+all unknowns as random variables and updating these distributions using the data. In addition,
+the joint inference approach offers the possibility to improve a possibly incorrect model M ∈ M
+by posterior estimates of δ†, and thus also the possibility of obtaining better estimates of both
+the parameter θ† as well as the state u† = M†(θ†).
+Marginal posterior
+The marginal approach involves first using the joint inference approach to
+obtain the joint posterior µy,J
+θ,δ on (θ, δ), and then integrating over all δ′ ∈ D:
+µy,M
+θ
+(S) =
+�
+S×D
+dµy,J
+θ,δ(θ′, δ′),
+S ∈ B(Θ).
+(2.8)
+The marginal posterior µy,M
+θ
+in (2.8) can be approximated by using Monte Carlo integration
+of the joint posterior µy,J
+θ,δ over δ′ ∈ D. The marginal approach inherits the problem of high
+computational cost from the joint inference approach. On the other hand, it has an advantage
+over the enhanced noise approach, namely that it involves a Bayesian update of the distribution
+of δ.
+3. Error bounds on misfits and posteriors
+In this section we compare the approaches presented above, by using some local Lipschitz
+stability bounds with respect to the Kullback–Leibler divergence. Recall that the Kullback–
+Leibler divergence between two probability measures µ and ν on a metric space (E, dE) is given
+8
+
+by
+dKL(µ∥ν) :=
+��
+E log dµ
+dν dµ
+µ ≪ ν
++∞
+otherwise.
+(3.1)
+Given µ ∈ M1(E) and Φ ∈ L1
+µ(E; R), define µΦ ∈ M1(E) by
+dµΦ
+dµ (x′) = exp(−Φ(x′))
+Z(Φ)
+,
+Z(Φ) :=
+�
+E
+exp(−Φ(x′))dµ(x′).
+For a measure µ on some measurable space (E, E) and a measurable function f : E → R,
+ess infµf denotes the essential infimum of f with respect to µ.
+The following local Lipschitz stability result is due to [23].
+Theorem 3.1. Let µ ∈ M1(E), Φ(1) ∈ L1
+µ(E; R≥0), and Φ(2) ∈ L1
+µ(E; R).
+Assume that
+ess infµΦ(1) = 0. Then
+dKL(µΦ(1)∥µΦ(2))
+≤ 2 exp
+�
+− min{ess infµΦ(2), 0} + ∥Φ(1)∥L1µ + ∥Φ(1) − Φ(2)∥L1µ
+�
+∥Φ(1) − Φ(2)∥L1µ,
+and thus µΦ(1) is absolutely continuous with respect to µΦ(2). In particular,
+max{dKL(µΦ(1)∥µΦ(2)), dKL(µΦ(2)∥µΦ(1))}
+≤ 2 exp
+�
+− min{ess infµΦ(2), 0} + 2∥Φ(1)∥L1µ + 2∥Φ(2)∥L1µ
+�
+∥Φ(1) − Φ(2)∥L1µ,
+and thus µΦ(1) is mutually equivalent to µΦ(2).
+Proof. The first statement follows by combining [23, Theorem 11] and [23, Proposition 6]. By
+the triangle inequality,
+max{∥Φ(1)∥L1µ + ∥Φ(1) − Φ(2)∥L1µ, ∥Φ(2)∥L1µ + ∥Φ(1) − Φ(2)∥L1µ} ≤ 2∥Φ(1)∥L1µ + 2∥Φ(2)∥L1µ,
+and thus the second statement follows from the first.
+Below, we will use Theorem 3.1 to bound the Kullback–Leibler error between pairs of poste-
+riors, for the posteriors defined in Section 2.
+Recall that the Hellinger metric between µ, ν ∈ M1(E) is defined by
+d2
+H(µ, ν) :=
+�
+E
+��
+dµ
+dλ −
+�
+dν
+dλ
+�2
+dλ,
+where λ is any measure such that both µ ≪ λ and ν ≪ λ. The definition of dH(µ, ν) does not
+depend on the choice of ν. The Hellinger metric and Kullback–Leibler divergence satisfy
+d2
+H(µ, ν) ≤ dKL(µ∥ν),
+see e.g. [25, Lemma 2.4]. Hence, the bounds on the Kullback–Leibler error that we present
+below imply bounds with respect to the Hellinger metric.
+9
+
+3.1. Kullback–Leibler error of posteriors on parameter space
+3.1.1. Error with respect to the best posterior
+Error of approximate posterior with respect to best posterior
+In Lemma 3.2 below, we bound
+the L1
+µθ error between the approximate misfit Φy,A and the best misfit Φy,† defined in (2.3a)
+and (2.4a) respectively. We express the bound in terms of the average observed model error
+O ◦ δ†.
+Lemma 3.2. Suppose Φy,† ∈ L1
+µθ. If Φy,A ∈ L1
+µθ, then
+∥Φy,† − Φy,A∥L1µθ ≤ 2−1/2∥∥O ◦ δ†∥2
+Σ−1
+ε ∥1/2
+L1µθ
+�
+∥Φy,†∥1/2
+L1µθ + ∥Φy,A∥1/2
+L1µθ
+�
+.
+(3.2)
+where
+∥∥O ◦ δ†∥2
+Σ−1
+ε ∥L1µθ ≤ 21/2�
+∥Φy,†∥1/2
+L1µθ + ∥Φy,A∥1/2
+L1µθ
+�
+.
+(3.3)
+See Appendix A.1.1 for the proof of Lemma 3.2.
+Remark 3.3. Combining (3.2) and (3.3) yields
+∥Φy,† − Φy,A∥L1µθ ≤
+�
+∥Φy,†∥1/2
+L1µθ + ∥Φy,A∥1/2
+L1µθ
+�2.
+Since
+∥Φy,†∥L1µθ + ∥Φy,A∥L1µθ ≤
+�
+∥Φy,†∥1/2
+L1µθ + ∥Φy,A∥1/2
+L1µθ
+�2,
+it follows that (3.2) and (3.3) together are not optimal: they yield a worse bound on ∥Φy,† −
+Φy,A∥L1µθ than the bound we could obtain using the triangle inequality. However, the bound
+(3.2) is useful, because it bounds ∥Φy,† − Φy,A∥L1µθ in terms of the average observed model error
+∥∥O ◦ δ†∥2
+Σ−1
+ε ∥1/2
+L1µθ and quantities that are assumed to be finite.
+Proposition 3.4. If Φy,A, Φy,† ∈ L1
+µθ(Θ, R), then
+max{dKL(µy,A
+θ
+∥µy,†
+θ ), dKL(µy,†
+θ ∥µy,A
+θ
+)} ≤C∥∥O ◦ δ†∥2
+Σ−1
+ε ∥1/2
+L1µθ
+for
+C = C(∥Φy,A∥L1µθ , ∥Φy,†∥L1µθ ) := 21/2 exp(2∥Φy,†∥L1µθ + 2∥Φy,A∥L1µθ )
+�
+∥Φy,†∥1/2
+L1µθ + ∥Φy,A∥1/2
+L1µθ
+�
+.
+The constant C in Proposition 3.4 is not optimal. The main value in defining C is to show
+that, given the hypotheses, the constant C is finite.
+Proof of Proposition 3.4. By the definition (2.3a) of Φy,A, it follows that ess infµΦy,A ≥ 0, so
+min{ess infµθΦy,A, 0} = 0. Given that Φy,A, Φy,† ∈ L1
+µθ, we may apply Lemma 3.2, and also the
+second statement of Theorem 3.1 with Φ(1) ← Φy,†, Φ(2) ← Φy,A, and µ ← µθ, which yields
+max{dKL(µy,A
+θ
+∥µy,†
+θ ), dKL(µy,†
+θ ∥µy,A
+θ
+)}
+≤2 exp
+�
+2∥Φy,A∥L1µθ + 2∥Φy,†∥L1µθ
+�
+∥Φy,† − Φy,A∥L1µθ
+≤21/2 exp
+�
+2∥Φy,A∥L1µθ + 2∥Φy,†∥L1µθ
+��
+∥Φy,†∥1/2
+L1µθ + ∥Φy,A∥1/2
+L1µθ
+�
+∥∥O ◦ δ†∥2
+Σ−1
+ε ∥1/2
+L1µθ .
+This completes the proof of Proposition 3.4.
+10
+
+If M† is not completely known, then neither are Φy,† nor δ†. Furthermore, it may be difficult
+to compute ∥Φy,A∥L1µθ exactly. Thus, it will in general be difficult to compute the constant C
+in Proposition 3.4.
+Proposition 3.4 shows that the Kullback–Leibler divergences of the approximate posterior
+µy,A
+θ
+with respect to the best posterior µy,†
+θ
+and vice versa are controlled by the average ob-
+served model error ∥∥O ◦δ†∥2
+Σ−1
+ε ∥1/2
+L1µθ . In order for dKL(µy,A
+θ
+∥µy,†
+θ ) or dKL(µy,†
+θ ∥µy,A
+θ
+) to be small,
+Proposition 3.4 suggests that one could choose the observation operator O such that δ† takes
+values in or near ker(O) with high µθ-probability. In particular, if the observation operator O
+satisfies
+P(O ◦ δ†(θ) = 0) = 1
+(3.4)
+then the corresponding approximate posterior and the best posterior coincide.
+This is not
+surprising, since if (3.4) holds then Φy,† = Φy,A coincide µθ-almost everywhere, by Lemma 3.2,
+and hence µy,†
+θ
+and µy,A
+θ
+coincide.
+The condition (3.4) can be useful for guiding the choice of observation operator O even if δ† is
+not fully known. For example, if the state space U is a Hilbert space, and if one can determine
+a priori that δ† takes values in some proper subspace V of U without knowing δ† exactly, then
+any choice of observation operator O such that V ⊆ ker(O) will yield an approximate posterior
+µy,A
+θ
+that agrees with the best posterior µy,†
+θ . The problem then consists in choosing O so that
+ker(O) is as small as possible, while satisfying the constraint that the model error takes values
+in ker(O) µθ-almost surely. The payoff in choosing O in this way is that Bayesian inference with
+µy,A
+θ
+will be as good as Bayesian inference with µy,†
+θ . Thus, the key idea in choosing observation
+operators to mitigate the effect of model error on Bayesian inference is to exploit all available
+knowledge about the model error.
+Remark 3.5. Recall from Remark 2.2 that we may also interpret M as a reduced-order model
+or surrogate for a more accurate but costly model M†. The preceding discussion then implies
+that, for suitably chosen observation operators, Bayesian inference with µy,A
+θ
+will be as good as
+µy,†
+θ , and have smaller computational cost.
+Error of enhanced noise posterior with respect to best posterior
+Lemma 3.6 below bounds
+the L1
+µθ error between the misfits Φy,† and Φy,E from (2.4a) and (2.6a) respectively. The bound
+indicates the importance of the shifted observed model error term O ◦ (δ† − mu) and difference
+Σ−1
+ε
+− (Σε + OΣuO∗)−1 of covariance matrices.
+Define the scalar
+CE := ∥Σ−1/2
+ε
+(Σε + OΣuO∗)1/2∥.
+(3.5)
+By Lemma A.1, CE satisfies
+∥z∥Σ−1
+ε
+≤ CE∥z∥(Σε+OΣuO∗)−1,
+z ∈ Rd.
+Lemma 3.6. Suppose Φy,† ∈ L1
+µθ. If Φy,E ∈ L1
+µθ, then for CE as in (3.5),
+∥Φy,† − Φy,E∥L1µθ ≤2−1/2∥∥O ◦ (δ† − mu)∥2
+Σ−1
+ε ∥1/2
+L1µθ
+�
+∥Φy,†∥1/2
+L1µθ + CE∥Φy,E∥1/2
+L1µθ
+�
+(3.6)
++ 2−1∥∥y − O ◦ M − Omu∥2
+Σ−1
+ε
+−(Σε+OΣuO∗)−1∥L1µθ .
+Furthermore,
+∥∥O ◦ (δ† − mu)∥2
+Σ−1
+ε ∥1/2
+L1µθ ≤21/2�
+∥Φy,†∥1/2
+L1µθ + CE∥Φy,E∥1/2
+L1µθ
+�
+,
+∥∥y − O ◦ M − Omu∥2
+Σ−1
+ε
+−(Σε+OΣuO∗)−1∥L1µθ ≤(CE + 1)∥2Φy,E∥L1µθ .
+11
+
+See Appendix A.1.1 for the proof of Lemma 3.6.
+Proposition 3.7. If Φy,E, Φy,† ∈ L1
+µθ(Θ, R), then
+max{dKL(µy,†
+θ ∥µy,E
+θ
+), dKL(µy,E
+θ
+∥µy,†
+θ )}
+≤C
+�
+∥∥O ◦ (δ† − mu)∥2
+Σ−1
+ε ∥1/2
+L1µθ + ∥∥y − O ◦ M − Omu∥2
+Σ−1
+ε
+−(Σε+OΣuO∗)−1∥L1µθ
+�
+for C = C(∥Φy,E∥L1µθ , ∥Φy,†∥L1µθ , CE), where
+C := exp
+�
+2∥Φy,†∥L1µθ + 2∥Φy,E∥L1µθ
+�
+max
+�√
+2
+�
+∥Φy,†∥1/2
+L1µθ + CE∥Φy,E∥1/2
+L1µθ
+�
+, 1
+�
+.
+As with Proposition 3.4, the importance of the constant C above is that C is finite under the
+hypotheses of Proposition 3.7.
+Proof of Proposition 3.7. By the definition (2.6a) of Φy,E, it follows that ess infµΦy,E ≥ 0, so
+min{ess infµθΦy,E, 0} = 0. Given that Φy,E, Φy,† ∈ L1
+µθ, we may apply Theorem 3.1 with Φ(1) ←
+Φy,†, Φ(2) ← Φy,E, and µ ← µθ, to obtain
+max{dKL(µy,†
+θ ∥µy,E
+θ
+), dKL(µy,E
+θ
+∥µy,†
+θ )}
+≤2 exp
+�
+2∥Φy,E∥L1µθ + 2∥Φy,†∥L1µθ
+�
+∥Φy,† − Φy,E∥L1µθ
+≤ exp
+�
+2∥Φy,E∥L1µθ + 2∥Φy,†∥L1µθ
+�
+max{
+√
+2
+�
+∥Φy,†∥1/2
+L1µθ + CE∥Φy,E∥1/2
+L1µθ
+�
+, 1}
+×
+�
+∥∥O ◦ (δ† − mu)∥2
+Σ−1
+ε ∥1/2
+L1µθ + ∥∥y − O ◦ M − Omu∥2
+Σ−1
+ε
+−(Σε+OΣuO∗)−1∥L1µθ
+�
+where the second inequality follows from the bound (3.6) of Lemma 3.6.
+The significance of Proposition 3.7 is similar to that of Proposition 3.4. The main differences
+follow from the fact the L1
+µθ error between the enhanced noise misfit Φy,E and the best misfit
+Φy,† — and hence also the Kullback–Leibler error between µy,E
+θ
+and µy,†
+θ
+— is now controlled
+by the sum
+∥∥O ◦ (δ† − mu)∥2
+Σ−1
+ε ∥1/2
+L1µθ + ∥∥y − O ◦ M − Omu∥2
+Σ−1
+ε
+−(Σε+OΣuO∗)−1∥L1µθ .
+(3.7)
+Recall from Section 2 that the enhanced noise approach consists in modelling the unknown
+state correction term δ†(θ†) ∈ U by a Gaussian random variable u ∼ N(mu, Σu). Since Σ−1
+ε
+is
+invertible by Assumption 2.3, the first term in (3.7) vanishes if and only if δ† − mu ∈ ker(O)
+µθ-almost surely. This condition differs from the sufficient condition for µy,†
+θ
+= µy,A
+θ
+that was
+implied by Lemma 3.2, namely, that δ† ∈ ker(O) µθ-almost surely. The difference consists in
+the mu term.
+By recalling (1.3), the second term in (3.7) vanishes if and only if
+P
+�
+y − O ◦ M(θ) − Omu ∈ ker
+�
+Σ−1
+ε
+− (Σε + OΣuO∗)−1�
+= 1.
+(3.8)
+By a rearrangement of the Woodbury formula that we obtained from [7, Eq. (3)], we have
+Σ−1
+ε
+− (Σε + OΣuO∗)−1 = Σ−1
+ε OΣuO∗Σ−1
+ε (Σε + OΣuO)−1Σ−1
+ε .
+The equation above holds for non-invertible OΣuO∗. If OΣuO∗ = 0, then both sides of the
+equation above vanish, and thus the condition (3.8) follows immediately. Since Σu describes the
+12
+
+covariance of the U-valued random model u of the state error δ†(θ†), the condition OΣuO∗ = 0
+has the equivalent formulation that the RNd-valued random variable Ou is constant P-almost
+surely. Since Ou = O(u − mu) + Omu, the latter condition is equivalent to u − mu ∈ ker(O)
+P-almost surely. More generally, if OΣuO∗ is nonzero but has non-trivial kernel, then Σ−1
+ε
+−
+(Σε + OΣuO∗)−1 also has a non-trivial kernel, and it may be possible for (3.8) to be satisfied.
+If one knows a priori that the image of Θ under δ† is contained in some affine subspace x + V
+of U for a linear subspace V of U, then one can exploit this information. For example, given
+the enhanced noise model N(mu, Σu), one should choose the observation operator O so that
+V ⊆ mu + ker(O) and u takes values in mu + ker(O) P-almost surely. In this case, the enhanced
+noise posterior µy,E
+θ
+and the best posterior µy,†
+θ
+will coincide.
+3.1.2. Error of enhanced noise posterior with respect to approximate posterior
+Lemma 3.8. Suppose Φy,A ∈ L1
+µθ. If Φy,E ∈ L1
+µθ, then for CE as in (3.5),
+∥Φy,A − Φy,E∥L1µθ ≤2−1/2∥Omu∥Σ−1
+ε
+�
+∥Φy,A∥1/2
+L1µθ + CE∥Φy,E∥1/2
+L1µθ
+�
+(3.9)
++ 2−1∥∥y − O ◦ M − Omu∥2
+Σ−1
+ε
+−(Σε+OΣuO∗)−1∥L1µθ
+where ∥∥y − O ◦ M − Omu∥2
+Σ−1
+ε
+−(Σε+OΣuO∗)−1∥L1µθ satisfies the bound in Lemma 3.6.
+For the proof of Lemma 3.8, see Appendix A.1.2.
+Proposition 3.9. If Φy,A, Φy,E ∈ L1
+µθ(Θ, R), then
+max{dKL(µy,A
+θ
+∥µy,E
+θ
+), dKL(µy,E
+θ
+∥µy,A
+θ
+)}
+≤C
+�
+∥Omu∥Σ−1
+ε
++ ∥∥y − O ◦ M − Omu∥2
+Σ−1
+ε
+−(Σε+OΣuO∗)−1∥L1µθ
+�
+for C = C(∥Φy,E∥L1µθ , ∥Φy,A∥L1µθ , CE), where
+C := exp
+�
+2∥Φy,A∥L1µθ + 2∥Φy,E∥L1µθ
+�
+max
+�√
+2
+�
+∥Φy,A∥1/2
+L1µθ + CE∥Φy,E∥1/2
+L1µθ
+�
+, 1
+�
+.
+Proof of Proposition 3.9. By Theorem 3.1 and Lemma 3.8,
+max{dKL(µy,A
+θ
+∥µy,E
+θ
+), dKL(µy,E
+θ
+∥µy,A
+θ
+)}
+≤2 exp
+�
+2∥Φy,E∥L1µθ + 2∥Φy,A∥L1µθ
+�
+∥Φy,A − Φy,E∥L1µθ
+≤ exp
+�
+2∥Φy,E∥L1µθ + 2∥Φy,A∥L1µθ
+�
+max{
+√
+2
+�
+∥Φy,A∥1/2
+L1µθ + CE∥Φy,E∥1/2
+L1µθ
+�
+, 1}
+×
+�
+∥Omu∥Σ−1
+ε
++ ∥∥y − O ◦ M − Omu∥2
+Σ−1
+ε
+−(Σε+OΣuO∗)−1∥L1µθ
+�
+.
+This completes the proof of Proposition 3.9.
+Proposition 3.9 implies that if, given an enhanced noise model with mean mu and covariance
+Σu, one chooses the observation operator O so that
+∥Omu∥Σ−1
+ε
+= ∥∥y − O ◦ M − Omu∥2
+Σ−1
+ε
+−(Σε+OΣuO∗)−1∥L1µθ = 0,
+then the enhanced noise posterior µy,E
+θ
+and the approximate posterior µy,A
+θ
+coincide. Equiva-
+lently, Omu = 0 and (3.8) together imply that µy,E
+θ
+= µy,A
+θ
+. Thus, such a choice of observation
+13
+
+operator yields an enhanced noise posterior µy,E
+θ
+that does not account for model error, in which
+case it would be simpler to use the approximate posterior µy,A
+θ
+instead. By the contrapositive
+statement, if µy,E
+θ
+and µy,A
+θ
+differ, then either Omu does not vanish, or (3.8) does not hold. If
+OΣuO∗ = 0, then (3.8) does not hold if and only if
+P(y − O ◦ M(θ) − Omu ̸= 0) > 0.
+We expect that in many cases, the latter condition will hold — and thus that µy,A
+θ
+and µy,E
+θ
+will
+differ — for a large collection of observation operators.
+3.2. Kullback–Leibler error of joint parameter-error posterior
+Recall from (2.7) that the joint misfit and the joint posterior are defined by
+Φy,J(θ′, δ′) := 1
+2∥y − O ◦ M(θ′) − O ◦ δ′(θ′)∥2
+Σ−1
+ε ,
+dµy,J
+θ,δ(θ′, δ′) := exp(−Φy,J(θ′, δ′))
+Z(Φy,J)
+dµθ ⊗ µδ(θ′, δ′).
+In this section, we shall compare µy,J
+θ
+∈ M1(Θ×D) with the other posterior measures considered
+so far. To do this, we need to redefine these posteriors as measures on Θ × D.
+For • ∈ {A, †, E}, define the lifted misfit and lifted posterior by
+Φy,•(θ′, δ′) := Φy,•(θ′),
+(3.11a)
+dµy,•
+θ,δ(θ′, δ′) := exp(−Φy,•(θ′, δ′))
+Z(Φy,•)
+dµθ ⊗ µδ,
+(3.11b)
+where the definition (3.11b) follows from Assumption 2.5, namely that the joint prior is a
+product measure.
+Note that in (3.11a), we use the notation Φy,• to refer to two functions defined on different
+domains. The following lemma shows that this abuse of notation is not problematic.
+Lemma 3.10. Let µθ, µδ, Φy,• and µy,•
+θ,δ be as in (3.11). Then for • ∈ {A, †, E},
+dµy,•
+θ,δ(θ′, δ′) = dµy,•
+θ (θ′) ⊗ µδ(δ′).
+(3.12)
+If Φy,• : Θ → R belongs to L1
+µθ, then its lifted version Φy,• defined in (3.11a) belongs to L1
+µθ⊗µδ,
+and for every q > 0,
+∥Φy,•∥Lq
+µθ⊗µδ = ∥Φy,•∥Lq
+µθ .
+(3.13)
+Proof. The second statement follows immediately from the definition (3.11a).
+For the first
+statement, observe that
+�
+Θ×D
+exp(−Φy,•(θ′, δ′))dµθ ⊗ µδ(θ′, δ′) =
+�
+Θ
+exp(−Φy,•(θ′))dµθ(θ′).
+Thus, the normalisation constant Z(Φy,•) for the lifted posterior µy,•
+θ,δ in (3.11b) agrees with the
+corresponding normalisation constant for the posterior µy,•
+θ . This implies (3.12).
+14
+
+Notation:
+Recall from Section 1.3 that P denotes the probability measure in the probability
+space on which we define all random variables. In this subsection, we will sometimes write the
+random variables θ and δ explicitly, and take Lp-norms with respect to P instead of µθ ⊗ µδ.
+For example,
+∥∥O ◦ (δ† − δ)(θ)∥2
+Σ−1
+ε ∥L1
+P =
+�
+Θ×D
+∥O ◦ (δ† − δ′)(θ′)∥2
+Σ−1
+ε dµθ ⊗ µδ(θ′, δ′).
+See Lemma 3.11 below for an example where we use this notation.
+Error with respect to lifted best posterior
+Lemma 3.11. Let Φy,† be defined as in (3.11a) with • = †. If Φy,† ∈ L1
+µθ and Φy,J ∈ L1
+µθ⊗µδ,
+then
+∥Φy,† − Φy,J∥L1
+µθ⊗µδ ≤ 2−1/2∥∥O ◦ (δ† − δ)(θ)∥2
+Σ−1
+ε ∥1/2
+L1
+P
+�
+∥Φy,J∥1/2
+L1
+µθ⊗µδ
++ ∥Φy,†∥1/2
+L1µθ
+�
+.
+(3.14)
+Furthermore,
+∥∥O ◦ (δ† − δ)(θ)∥2
+Σ−1
+ε ∥1/2
+L1
+P ≤ 21/2�
+∥Φy,J∥1/2
+L1
+µθ⊗µδ
++ ∥Φy,†∥1/2
+L1µθ
+�
+.
+Proposition 3.12. Let Φy,† and Φy,J be as in Lemma 3.11 and let µy,†
+θ,δ be defined as in (3.11b)
+with • = †. Then
+max{dKL(µy,†
+θ,δ∥µy,J
+θ,δ), dKL(µy,J
+θ,δ∥µy,†
+θ,δ)} ≤ C∥∥O ◦ (δ† − δ)(θ)∥2
+Σ−1
+ε ∥1/2
+L1
+P ,
+where
+C = 21/2 exp
+�
+2∥Φy,J∥L1
+µθ⊗µδ + 2∥Φy,†∥L1µθ
+��
+∥Φy,J∥1/2
+L1
+µθ⊗µδ
++ ∥Φy,†∥1/2
+L1µθ
+�
+.
+See Appendix A.2 for the proofs of Lemma 3.11 and Proposition 3.12.
+By Proposition 3.12, a sufficient condition for µy,J
+θ,δ to coincide with µy,†
+θ,δ is
+P
+�
+(δ† − δ)(θ) ∈ ker(O)
+�
+= 1.
+(3.15)
+For example, suppose that one knows a priori that there exist a vector x ∈ U and a linear
+subspace V of U such that
+{δ†(θ′) : θ′ ∈ Θ} ⊆ x + V.
+Suppose that (D, ⟨·, ·⟩D) is a Hilbert space and δ ∼ µδ = N(mδ, Σδ). Then mδ + Σ1/2
+δ
+D is the
+Cameron–Martin space of µδ, and the support supp(µδ) of µδ is the closure of mδ +Σ1/2
+δ
+D with
+respect to ∥ · ∥D. Assume there exists y ∈ U and a closed linear subspace W ⊆ U such that
+{δ′(θ′) : θ′ ∈ Θ, δ′ ∈ supp(µδ)} ⊆ y + W.
+If for every v ∈ V and w ∈ W it holds that (x+v)−(y +w) ∈ ker(O), then the condition (3.15)
+holds.
+The preceding example suggests that, in general, it may be difficult to choose µδ and O so
+that µy,†
+θ,δ and µy,J
+θ,δ coincide. This is to be expected, since the lifted best posterior measure µy,†
+θ,δ
+is a product measure with δ-marginal equal to µδ, whereas for many reasonable choices of µδ
+the joint posterior measure µy,J
+θ,δ will not have δ-marginal equal to µδ.
+15
+
+Error with respect to lifted approximate posterior
+Lemma 3.13. Let Φy,A be defined as in (3.11a) with • = A. If Φy,A ∈ L1
+µθ and Φy,J ∈ L1
+µθ⊗µδ,
+then
+∥Φy,A − Φy,J∥L1
+µθ⊗µδ ≤ 2−1/2∥∥O ◦ δ(θ)∥2
+Σ−1
+ε ∥1/2
+L1
+P
+�
+∥Φy,J∥1/2
+L1
+µθ⊗µδ
++ ∥Φy,A∥1/2
+L1µθ
+�
+.
+(3.16)
+Furthermore,
+∥∥O ◦ δ(θ)∥2
+Σ−1
+ε ∥1/2
+L1
+P ≤ 21/2�
+∥Φy,J∥1/2
+L1
+µθ⊗µδ
++ ∥Φy,A∥1/2
+L1µθ
+�
+.
+Proposition 3.14. Let Φy,A and Φy,J be as in Lemma 3.13 and let µy,A
+θ,δ be defined as in (3.11b)
+with • = A. Then
+max{dKL(µy,A
+θ,δ ∥µy,J
+θ,δ), dKL(µy,J
+θ,δ∥µy,A
+θ,δ )} ≤ C∥∥O ◦ δ(θ)∥2
+Σ−1
+ε ∥1/2
+L1
+P ,
+where
+C = 21/2 exp
+�
+2∥Φy,J∥L1
+µθ⊗µδ + 2∥Φy,A∥L1µθ
+��
+∥Φy,J∥1/2
+L1
+µθ⊗µδ
++ ∥Φy,A∥1/2
+L1µθ
+�
+.
+See Appendix A.2 for the proofs of Lemma 3.13 and Proposition 3.14.
+By Proposition 3.14, a sufficient condition for µy,A
+θ,δ and µy,J
+θ,δ to coincide is that δ(θ) ∈ ker(O),
+P-almost surely. By the same reasoning that we used when considering µy,†
+θ,δ and µy,J
+θ,δ, we do not
+expect µy,A
+θ,δ and µy,J
+θ,δ to coincide, since the former is a product measure with δ-marginal equal
+to µδ, and the latter will not have this property in general.
+Connection with parametrised background data-weak approach
+Recall the definition (2.1) of
+the observation operator as O := (ℓi1, . . . , ℓiN ). Suppose the state space U is a Hilbert space and
+d = 1, i.e. suppose that the measurement functions (ℓij)N
+j=1 are continuous linear functionals
+on U with Riesz representatives denoted by (Rℓij)N
+j=1 ⊂ U. Then the condition δ(θ) ∈ ker(O) is
+equivalent to δ(θ) ∈ (span(Rℓij, j = 1, . . . , N))⊥. We may reformulate the necessary condition
+for µy,A
+θ,δ and µy,J
+θ,δ to differ, namely that δ(θ) /∈ ker(O) with positive P-probability, as δ(θ) ∈
+span(Rℓij, j = 1, . . . , N) with positive P-probability.
+Given the interpretation of δ(θ) as a
+state correction term, the condition that δ(θ) ∈ span(Rℓij, j = 1, . . . , N) closely resembles the
+‘variational update’ from the parametrised background data-weak approach for state inference;
+see e.g. [17, Section 2.3].
+It is possible to state and prove the analogues of the preceding bounds for the lifted enhanced
+noise posterior µy,E
+θ,δ . These bounds are not relevant for the main goal of this paper. However,
+for the sake of completeness, we state them in Appendix A.2.
+3.3. Kullback–Leibler error of marginal posterior
+Recall from (2.8) that the marginal posterior is defined by
+µy,M
+θ
+(S) =
+�
+S×D
+dµy,J
+θ,δ(θ′, δ′),
+S ∈ B(Θ).
+In Section 3.2, we bounded the Kullback–Leibler error of the joint posterior µy,J
+θ,δ with respect
+to the lifted posteriors µy,•
+θ,δ for • ∈ {A, †, E} that were defined in (3.11b), and we observed in
+Lemma 3.10 that the θ-marginal of the lifted posterior µy,•
+θ,δ is exactly µy,•
+θ .
+16
+
+In this section, we shall bound the Kullback–Leibler error of the marginal posterior µy,M with
+respect to the θ-marginals of the lifted posteriors that we considered in Section 3.2, i.e. with
+respect to µy,†
+θ , µy,A
+θ
+, and µy,E
+θ
+.
+Below, µy,•
+δ|θ denotes the regular version of the posterior distribution of δ conditioned on θ, for
+• ∈ {A, †, E, J}. We assume that such regular conditional distributions exist and are unique up
+to sets of measure zero.
+Proposition 3.15. Let Φy,J ∈ L1
+µθ⊗µδ and • ∈ {A, †, E}. Suppose Φy,• : Θ → R≥0 belongs to
+L1
+µθ. Then
+dKL(µy,M
+θ
+∥µy,•
+θ ) =dKL(µy,J
+θ,δ∥µy,•
+θ,δ) −
+�
+Θ
+dKL(µy,J
+δ|θ∥µy,•
+δ|θ)dµθ,
+dKL(µy,•
+θ ∥µy,M
+θ
+) =dKL(µy,•
+θ,δ∥µy,J
+θ,δ) −
+�
+Θ
+dKL(µy,•
+δ|θ∥µy,J
+δ|θ)dµθ.
+In particular,
+max{dKL(µy,M
+θ
+∥µy,•
+θ ), dKL(µy,•
+θ ∥µy,M
+θ
+)}
+≤ max{dKL(µy,J
+θ,δ∥µy,•
+θ,δ), dKL(µy,•
+θ,δ∥µy,J
+θ,δ)} − min
+� �
+Θ
+dKL(µy,J
+δ|θ∥µy,•
+δ|θ)dµθ,
+�
+Θ
+dKL(µy,•
+δ|θ∥µy,J
+δ|θ)dµθ
+�
+.
+Proof. By nonnegativity of the Kullback–Leibler divergence, the first statement implies the
+second statement.
+The first statement follows by recalling the chain rule for the Kullback–Leibler divergence,
+see e.g. [26, Exercise 3.2]:
+dKL(µy,J
+θ,δ∥µy,•
+θ,δ) =dKL(µy,M
+θ
+∥µy,•
+θ ) +
+�
+Θ
+dKL(µy,J
+δ|θ∥µy,•
+δ|θ)dµy,M
+θ
+,
+(3.17)
+dKL(µy,•
+θ,δ∥µy,J
+θ,δ) =dKL(µy,•
+θ ∥µy,M
+θ
+) +
+�
+Θ
+dKL(µy,•
+δ|θ∥µy,J
+δ|θ)dµy,•
+θ .
+(3.18)
+Above, we used the definition (2.8) of the marginal posterior µy,M
+θ
+, and the fact expressed in
+(3.12), namely that the θ-marginal of µy,•
+θ,δ is µy,•
+θ .
+Proposition 3.15 is important for the following reasons. First, it implies that the Kullback–
+Leibler error of the marginal posterior µy,M
+θ
+with respect to any of the above-mentioned poste-
+riors on Θ cannot be larger than the Kullback–Leibler error of the joint posterior µy,J
+θ,δ and the
+corresponding lifted version of the posterior on Θ. In other words, marginalisation can only
+reduce the Kullback–Leibler error. As a result, for • ∈ {A, †, E}, the Kullback–Leibler error
+of the marginal posterior µy,M
+θ
+with respect to µy,•
+θ
+satisfies the same bounds as the Kullback–
+Leibler error of the joint posterior µy,J
+θ,δ with respect to the lifted posterior µy,•
+θ,δ. Thus, the same
+statements regarding sufficient conditions for the coincidence of the posteriors on Θ × D that
+were made after Proposition 3.12, Proposition 3.14, and Proposition A.4, also apply to the
+marginalised versions of the posteriors in the above-mentioned propositions.
+4. Conclusion
+We considered Bayesian inverse problems in the presence of model error in the following set-
+ting: the data is finite-dimensional; the noise is additive, Gaussian, and independent; and the
+17
+
+parameter-to-observable map is the composition of a possibly nonlinear model with a linear
+observation operator. We assumed that there exists a unique best model and best parameter,
+such that the resulting best state most accurately describes the phenomenon of interest. The
+‘model error’ is then the difference between the model that one uses and the best model.
+We described some existing deterministic approaches for accounting for model error and
+used the local Lipschitz stability property of posteriors with respect to perturbations in the
+likelihood to bound the symmetrised Kullback–Leibler error between pairs of posteriors. These
+bounds have two important properties: first, they control the Kullback–Leibler error in terms of
+quantities that depend on the observation operator and the objects used to account for model
+error; and second, the other terms in the bounds are finite under mild hypotheses, namely
+L1-integrability of the misfits with respect to the prior.
+The bounds yield sufficient conditions on the observation operator and the model error-aware
+approach to yield a posterior that performs almost as well as the best posterior that uses the
+best model. They also yield necessary conditions for a model error-aware approach to yield
+a posterior that differs from the posterior yielded by the model error-agnostic approach. A
+recurring theme in the sufficient conditions is the importance of the kernel of the observation
+operator. We expect that the design of observation operators that approximately or exactly
+satisfy the sufficient conditions will be challenging and highly problem-specific. It would be
+of interest to study the setting of nonlinear observation operators and other approaches for
+accounting for model error.
+Acknowledgements
+The research of HCL has been partially funded by the DFG — Project-ID 318763901 —
+SFB1294.
+A. Technical lemmas
+Let d ∈ N and Mi ∈ Rd×d, i = 1, 2 be symmetric and positive definite.
+For convenience, we state the following lemma.
+Lemma A.1. Suppose that M1, M2 ∈ Rd×d are symmetric, that M1 is positive definite, and that
+M2 is nonnegative definite. Then M1+M2 is symmetric positive definite and M−1
+1 −(M1+M2)−1
+is symmetric nonnegative definite. In addition,
+∥z∥(M1+M2)−1
+∥(M1 + M2)−1/2M1/2
+1
+∥
+≤ ∥z∥M−1
+1
+≤ ∥M−1/2
+1
+(M1 + M2)1/2∥∥z∥(M1+M2)−1,
+z ∈ Rd.
+(A.1)
+Proof. Let λmin(A) denote the smallest eigenvalue of a matrix A. By the assumptions on M1
+and M2, M1 + M2 is positive definite.
+As the difference of two symmetric matrices, M−1
+1
+− (M1 + M2)−1 is symmetric. To show
+that it is nonnegative, we use the following rearrangement of the Woodbury formula, which we
+take from [7, Equation (3)]:
+(M1 + M2)−1 =M−1
+1
+− M−1
+1 M2(I + M−1
+1 M2)−1M−1
+1
+⇐⇒ M−1
+1
+− (M1 + M2)−1 =M−1
+1 M2(I + M−1
+1 M2)−1M−1
+1 .
+18
+
+Invertibility of I+M−1
+1 M2 = M−1
+1 (M1+M2) follows from the invertibility of M−1
+1
+and M1+M2.
+From the second equation above, M−1
+1
+− (M1 + M2)−1 inherits the nonnegative definiteness of
+M2.
+Recall the notation (1.3) for a matrix-weighted norm. The first inequality of (A.1) follows by
+∥z∥(M1+M2)−1 = ∥(M1 + M2)−1/2z∥ = ∥(M1 + M2)−1/2M1/2
+1
+M−1/2
+1
+z∥
+≤ ∥(M1 + M2)−1/2M1/2
+1
+∥∥M−1/2
+1
+z∥
+= ∥(M1 + M2)−1/2M1/2
+1
+∥∥z∥M−1
+1 .
+The second inequality of (A.1) follows by
+∥z∥M−1
+1
+= ∥M−1/2
+1
+z∥ = ∥M−1/2
+1
+(M1 + M2)1/2(M1 + M2)−1/2z∥
+≤ ∥M−1/2
+1
+(M1 + M2)1/2∥∥(M1 + M2)−1/2z∥
+= ∥M−1/2
+1
+(M1 + M2)1/2∥∥z∥(M1+M2)−1.
+This completes the proof of Lemma A.1.
+We will use the following bound in the proofs below.
+Lemma A.2. Let (E, dE) be a metric space, µ ∈ M1(E), and f and g be Rd-valued measurable
+functions on E. Let M1 ∈ Rd×d be symmetric and positive definite and M2 ∈ Rd×d be symmetric
+nonnegative definite. Then
+∥∥f∥2
+M−1
+1
+− ∥f + g∥2
+M−1
+1 ∥L1µ ≤ ∥∥g∥2
+M−1
+1 ∥1/2
+L1µ
+�
+∥∥f + g∥2
+M−1
+1 ∥1/2
+L1µ + ∥∥f∥2
+M−1
+1 ∥1/2
+L1µ
+�
+.
+(A.2)
+More generally,
+∥∥f∥2
+M−1
+1
+− ∥f + g∥2
+(M1+M2)−1∥L1µ
+≤∥∥g∥2
+M−1
+1 ∥1/2
+L1µ
+�
+∥M−1/2
+1
+(M1 + M2)1/2∥ · ∥∥f + g∥2
+(M1+M2)−1∥1/2
+L1µ + ∥∥f∥2
+M−1
+1 ∥1/2
+L1µ
+�
+(A.3)
++ ∥∥f + g∥2
+M−1
+1
+−(M1+M2)−1∥L1µ,
+where ∥f + g∥2
+M−1
+1
+−(M1+M2)−1 := ∥f + g∥2
+M−1
+1
+− ∥f + g∥2
+(M1+M2)−1. In addition,
+∥∥g∥2
+M−1
+1 ∥1/2
+L1µ ≤ ∥∥f∥2
+M−1
+1 ∥1/2
+L1µ + ∥M−1/2
+1
+(M1 + M2)1/2∥∥∥f + g∥2
+(M1+M2)−1∥1/2
+L1µ
+(A.4)
+∥∥f + g∥2
+M−1
+1
+−(M1+M2)−1∥L1µ ≤
+�
+1 + ∥M−1/2
+1
+(M1 + M2)1/2∥
+�
+∥∥f + g∥2
+(M1+M2)−1∥L1µ.
+(A.5)
+Proof. We claim that for arbitrary a, b ∈ Rd, symmetric positive definite M1 ∈ Rd×d, and
+symmetric nonnegative definite M2 ∈ Rd×d,
+∥a∥2
+M−1
+1
+− ∥a + b∥2
+(M1+M2)−1 = −⟨b, 2a + b⟩M−1
+1
++ ∥a + b∥2
+M−1
+1
+−(M1+M2)−1.
+(A.6)
+Recall that (M1 + M2)−1 exists and is positive definite, by Lemma A.1.
+19
+
+Using the matrix-weighted inner product and norm notation from (1.3),
+∥a∥2
+M−1
+1
+− ∥a + b∥2
+M−1
+1
+=a⊤M−1
+1 a − (a + b)⊤M−1
+1 (a + b)
+=a⊤M−1
+1 a − (a⊤M−1
+1 a + 2a⊤M−1
+1 b + b⊤M−1
+1 b)
+= − b⊤M−1
+1 (2a + b)
+= − ⟨b, 2a + b⟩M−1
+1 .
+This implies (A.6), since
+∥a∥2
+M−1
+1
+− ∥a + b∥2
+M−1
+1
++ ∥a + b∥2
+M−1
+1
+− ∥a + b∥2
+(M1+M2)−1
+= − ⟨b, 2a + b⟩M−1
+1
++ ∥a + b∥2
+M−1
+1
+− ∥a + b∥2
+(M1+M2)−1.
+Now let f, g, and µ be as in the statement of the lemma. Then by (A.6) and the triangle
+inequality,
+∥∥f∥2
+M−1
+1
+− ∥f + g∥2
+(M1+M2)−1∥L1µ =∥ − ⟨g, 2f + g⟩M−1
+1
++ ∥f + g∥2
+M−1
+1
+−(M1+M2)−1∥L1µ
+≤∥⟨g, 2f + g⟩M−1
+1 ∥L1µ + ∥∥f + g∥2
+M−1
+1
+−(M1+M2)−1∥L1µ.
+(A.7)
+Next,
+∥⟨g, 2f + g⟩M−1
+1 ∥L1µ ≤∥∥g∥M−1
+1 ∥2f + g∥M−1
+1 ∥L1µ
+≤∥∥g∥M−1
+1 ∥L2µ∥∥2f + g∥M−1
+1 ∥L2µ
+≤∥∥g∥M−1
+1 ∥L2µ
+�
+∥∥f + g∥M−1
+1 ∥L2µ + ∥∥f∥M−1
+1 ∥L2µ
+�
+=∥∥g∥2
+M−1
+1 ∥1/2
+L1µ
+�
+∥∥f + g∥2
+M−1
+1 ∥1/2
+L1µ + ∥∥f∥2
+M−1
+1 ∥1/2
+L1µ
+�
+.
+(A.8)
+The first and second inequalities follow by applying the Cauchy–Schwarz inequality with respect
+to ⟨·, ·⟩M−1
+1
+and ∥·∥L1µ respectively. The third inequality and the equation follow from the ∥·∥L2µ-
+triangle inequality and the definition of the Lp
+µ norm for p = 1, 2. By (A.8), we bound the first
+term on the right-hand side of (A.7). By using M2 ← 0, the second term on the right-hand side
+of (A.7) vanishes. Thus (A.2) follows from (A.7).
+Next, we bound the first term inside the parentheses on the right-hand side of (A.8). By
+(A.1),
+∥∥f + g∥2
+M−1
+1 ∥1/2
+L1µ ≤ ∥M−1/2
+1
+(M1 + M2)1/2∥∥∥f + g∥2
+(M1+M2)−1∥1/2
+L1µ .
+Using the above bound yields (A.3).
+To prove (A.4),
+∥∥g∥2
+M−1
+1 ∥1/2
+L1µ =∥∥ − f + (f + g)∥M−1
+1 ∥L2µ
+≤∥∥f∥M−1
+1 ∥L2µθ + ∥∥f + g∥M−1
+1 ∥L2µ
+≤∥∥f∥2
+M−1
+1 ∥1/2
+L1µθ + ∥M−1/2
+1
+(M1 + M2)1/2∥∥∥f + g∥2
+(M1+M2)−1∥1/2
+L1µ
+20
+
+where the last inequality uses (A.1). To prove (A.5), observe that
+∥∥f + g∥2
+M−1
+1
+−(M1+M2)−1∥L1µ =∥∥f + g∥2
+M−1
+1
+− ∥f + g∥2
+(M1+M2)−1∥L1µ
+≤∥∥f + g∥2
+M−1
+1 ∥L1µ + ∥∥f + g∥2
+(M1+M2)−1∥L1µ
+≤(∥M−1/2
+1
+(M1 + M2)1/2∥ + 1)∥∥f + g∥2
+(M1+M2)−1∥L1µ
+where the first and second inequality follow from the triangle inequality and (A.1) in Lemma A.1.
+This completes the proof of Lemma A.2.
+A.1. Proof of lemmas in Section 3.1
+A.1.1. Proofs for Section 3.1.1
+Proof of error of approximate posterior with respect to best posterior
+Lemma 3.2 bounds
+∥Φy,†−Φy,A∥Lq
+µθ in terms of the observed model error O◦δ†, under the hypothesis that Φy,† ∈ L1
+µθ
+and Φy,A ∈ L1
+µθ. The bound is given in (3.2):
+∥Φy,† − Φy,A∥L1µθ ≤ 2−1/2∥∥O ◦ δ†∥Σ−1
+ε ∥L2µθ
+�
+∥Φy,†∥1/2
+L1µθ + ∥Φy,A∥1/2
+L1µθ
+�
+.
+Proof of Lemma 3.2. Recall from (2.4a) and (2.3a) that Φy,†(θ′) = 1
+2∥y − O ◦ M†(θ′)∥2
+Σ−1
+ε
+and
+Φy,A(θ′) = 1
+2∥y − O ◦ M(θ′)∥2
+Σ−1
+ε
+respectively. By these definitions,
+∥2 · 1
+2∥y − O ◦ M†∥2
+Σ−1
+ε ∥1/2
+L1µθ = ∥2Φy,†∥1/2
+L1µθ =
+√
+2∥Φy,†∥1/2
+L1µθ
+(A.9)
+and similarly
+∥∥y − O ◦ M∥2
+Σ−1
+ε ∥1/2
+L1µθ =
+√
+2∥Φy,A∥1/2
+L1µθ .
+(A.10)
+Now set f ← y − O ◦ M†, g ← O ◦ δ†, µ ← µθ, M1 ← Σε, and M2 ← 0. By (2.5), we have
+f + g = y − O ◦ (M† − δ†) = y − O ◦ M. Hence ∥f∥2
+M−1
+1
+= 2Φy,† and ∥f + g∥2
+M−1
+1
+= 2Φy,A.
+Applying (A.2) from Lemma A.2 with these choices yields
+2∥Φy,† − Φy,A∥L1µθ ≤∥∥O ◦ δ†∥2
+Σ−1
+ε ∥1/2
+L1µθ
+�
+∥∥y − O ◦ M†∥2
+Σ−1
+ε ∥1/2
+L1µθ + ∥∥y − O ◦ M∥2
+Σ−1
+ε ∥1/2
+L1µθ
+�
+=∥∥O ◦ δ†∥2
+Σ−1
+ε ∥1/2
+L1µθ
+√
+2
+�
+∥Φy,†∥1/2
+L1µθ + ∥Φy,A∥1/2
+L1µθ
+�
+,
+where we used (A.9) and (A.10) for the equation. This proves (3.2). The bound on ∥∥O ◦
+δ†∥2
+Σ−1
+ε ∥1/2
+L1µθ in the statement of Lemma 3.2 follows from (A.4) in Lemma A.2 with the choices
+above:
+∥∥O ◦ δ†∥2
+Σ−1
+ε ∥1/2
+L1µθ ≤
+√
+2
+�
+∥Φy,†∥1/2
+L1µθ + ∥Φy,A∥1/2
+L1µθ
+�
+.
+This completes the proof of Lemma 3.2.
+Proof of error of enhanced noise posterior with respect to best posterior
+Lemma 3.6 bounds
+∥Φy,† −Φy,E∥L1µθ in terms of the observed model error O◦δ† and the Gaussian model N(mu, Σδ)
+21
+
+of δ†(θ†).
+In particular, under the hypotheses that Φy,† ∈ L1
+µθ and Φy,E ∈ L1
+µθ, then for
+CE := ∥Σ−1/2
+ε
+(Σε + OΣuO∗)1/2∥ as in (3.5), the bound (3.6)
+∥Φy,† − Φy,E∥L1µθ ≤2−1/2∥∥O ◦ (δ† − mu)∥2
+Σ−1
+ε ∥1/2
+L1µθ
+�
+∥Φy,†∥1/2
+L1µθ + CE∥Φy,E∥1/2
+L1µθ
+�
++ 2−1∥∥y − O ◦ M − Omu∥2
+Σ−1
+ε
+−(Σε+OΣuO∗)−1∥L1µθ ,
+holds, and all terms on the right-hand side are finite.
+Proof of Lemma 3.6. In the same way that we proved (A.9), we can use the definition (2.6a) of
+Φy,E to prove
+∥∥y − O ◦ M − Omu∥2
+(Σε+OΣuO∗)−1∥1/2
+L1µθ =
+√
+2∥Φy,E∥1/2
+L1µθ .
+(A.11)
+Let f ← y − O ◦ M†, g ← O ◦ (δ† − mu), µ ← µθ, M1 ← Σε, and M2 ← OΣuO∗. Then
+f + g = y − O ◦ (M† − δ†) − Omu = y − O ◦ M − Omu,
+∥f∥2
+M−1
+1
+= 2Φy,†, and ∥f + g∥2
+(M1+M2)−1 = 2Φy,E. Applying (A.3) from Lemma A.2 yields
+2∥Φy,† − Φy,E∥L1µθ ≤∥∥O ◦ (δ† − mu)∥2
+Σ−1
+ε ∥1/2
+L1µθ
+�
+CE∥2Φy,E∥1/2
+L1µθ + ∥2Φy,†∥1/2
+L1µθ
+�
+(A.12)
++ ∥∥y − O ◦ M − Omu∥2
+Σ−1
+ε
+−(Σε+OΣuO∗)−1∥L1µθ .
+This proves (3.6). By (A.4) and CE as in (3.5),
+∥∥O ◦ (δ† − mu)∥2
+Σ−1
+ε ∥1/2
+L1µθ ≤ ∥2Φy,†∥1/2
+L1µθ + CE∥2Φy,E∥1/2
+L1µθ .
+By (A.5) from Lemma A.2,
+∥∥y − O ◦ M − Omu∥2
+Σ−1
+ε
+−(Σε+OΣuO∗)−1∥L1µθ ≤ (CE + 1)∥2Φy,E∥L1µθ .
+This completes the proof of Lemma 3.6.
+A.1.2. Proofs for Section 3.1.2
+Lemma 3.8 asserts that, under the hypotheses that Φy,A ∈ L1
+µθ and Φy,E ∈ L1
+µθ, then for
+CE := ∥Σ−1/2
+ε
+(Σε + OΣuO∗)1/2∥ as in (3.5), the bound (3.9)
+∥Φy,A − Φy,E∥L1µθ ≤2−1/2∥Omu∥Σ−1
+ε
+�
+∥Φy,A∥1/2
+L1µθ + CE∥Φy,E∥1/2
+L1µθ
+�
++ 2−1∥∥y − O ◦ M − Omu∥2
+Σ−1
+ε
+−(Σε+OΣuO∗)−1∥L1µθ ,
+holds, and all terms on the right-hand side are finite.
+Proof of Lemma 3.8. Let f ← y −O ◦M, g ← −Omu, M1 ← Σ−1
+ε , M2 ← OΣuO∗, and µ ← µθ.
+Then ∥f∥2
+M−1
+1
+= 2Φy,A and ∥f + g∥2
+(M1+M2)−1 = 2Φy,E. Applying (A.3) from Lemma A.2 yields
+2∥Φy,A − Φy,E∥L1µθ
+≤∥Omu∥Σ−1
+ε
+√
+2
+�
+∥Φy,A∥1/2
+L1µθ + CE∥Φy,E∥1/2
+L1µθ
+�
++ ∥∥y − O ◦ M − Omu∥2
+Σ−1
+ε
+−(Σε+OΣuO∗)−1∥L1µθ ,
+for CE := ∥Σ−1/2
+ε
+(Σε + OΣuO∗)1/2∥ as in (3.5). This proves (3.9). Next, (A.5) in Lemma A.2
+yields
+∥∥y − O ◦ M − Omu∥2
+Σ−1
+ε
+−(Σε+OΣuO∗)−1∥L1µθ ≤ 2
+�
+CE + 1)∥Φy,E∥L1µθ .
+This completes the proof of Lemma 3.8.
+22
+
+A.2. Proofs for Kullback–Leibler error of joint parameter-error posterior
+Proof of error of joint parameter-error posterior with respect to lifted best posterior
+In
+Lemma 3.11, one assumes that Φy,† as defined in (2.4a) belongs to L1
+µθ, and also that Φy,J ∈
+L1
+µθ⊗µδ. The resulting bound (3.14) is
+∥Φy,† − Φy,J∥L1
+µθ⊗µδ ≤ 2−1/2∥∥O ◦ (δ† − δ)(θ)∥2
+Σ−1
+ε ∥1/2
+L1
+P
+�
+∥Φy,J∥1/2
+L1
+µθ⊗µδ
++ ∥Φy,†∥1/2
+L1µθ
+�
+.
+Proof of Lemma 3.11. Let f ← y − O ◦ M†(θ), g ← O ◦ (δ† − δ)(θ), M1 ← Σε, M2 ← 0, and
+µ ← P. Then ∥f∥2
+M−1
+1
+= 2Φy,†(θ, δ) and ∥f + g∥2
+M−1
+1
+= 2Φy,J(θ, δ). Using the same argument
+that we used to prove (A.9), it follows from (2.7a) that
+∥∥y − O ◦ M(θ) − O ◦ δ(θ)∥2
+Σ−1
+ε ∥1/2
+L1
+P =
+√
+2∥Φy,J∥1/2
+L1
+µθ⊗µδ
+.
+(A.13)
+By (A.9) and (3.13) from Lemma 3.10,
+∥∥y − O ◦ M†(θ)∥2
+Σ−1
+ε ∥1/2
+L1
+P =
+√
+2∥Φy,†∥1/2
+L1µθ =
+√
+2∥Φy,†∥1/2
+L1
+µθ⊗µδ
+.
+(A.14)
+Applying (A.2) Lemma A.2 with these choices yields (3.14):
+2∥Φy,A − Φy,J∥L1
+µθ⊗µδ ≤∥∥O ◦ (δ† − δ)(θ)∥2
+Σ−1
+ε ∥1/2
+L1
+P
+√
+2
+�
+∥Φy,J∥1/2
+L1
+µθ⊗µδ
++ ∥Φy,†∥1/2
+L1µθ
+�
+.
+Next,
+∥∥O ◦ (δ† − δ)(θ)∥2
+Σ−1
+ε ∥1/2
+L1
+P ≤
+√
+2
+�
+∥Φy,J∥1/2
+L1
+µθ⊗µδ
++ ∥Φy,†∥1/2
+L1µθ
+�
+,
+follows from (A.13), (A.14), and (A.4) of Lemma A.2 with the choices stated above.
+Proof of Proposition 3.12. Both ess infµθ⊗µδΦy,J and ess infµθ⊗µδΦy,† are nonnegative, by (2.7a)
+and (2.4a). Applying Theorem 3.1 and Lemma 3.11 yields
+max{dKL(µy,†
+θ,δ∥µy,J
+θ,δ), dKL(µy,J
+θ,δ∥µy,†
+θ,δ)}
+≤2 exp
+�
+2∥Φy,J∥L1
+µθ⊗µδ + 2∥Φy,†∥L1
+µθ⊗µδ
+�
+∥Φy,† − Φy,J∥L1
+µθ⊗µδ
+≤21/2 exp
+�
+2∥Φy,J∥L1
+µθ⊗µδ + 2∥Φy,†∥L1
+µθ⊗µδ
+��
+∥Φy,J∥1/2
+L1
+µθ⊗µδ
++ ∥Φy,†∥1/2
+L1
+µθ⊗µδ
+�
+× ∥∥O ◦ (δ† − δ)(θ)∥2
+Σ−1
+ε ∥1/2
+L1
+P .
+Using the definition of C given in the statement of Proposition 3.12 and (A.14) completes the
+proof.
+Proof of error of joint parameter-error posterior with respect to lifted approximate posterior
+In Lemma 3.13, one assumes that Φy,A as defined in (2.3a) belongs to L1
+µθ, and also that
+Φy,J ∈ L1
+µθ⊗µδ. The resulting bound (3.16) is
+∥Φy,A − Φy,J∥L1
+µθ⊗µδ ≤ 2−1/2∥∥O ◦ δ(θ)∥2
+Σ−1
+ε ∥1/2
+L1
+P
+�
+∥Φy,J∥1/2
+L1
+µθ⊗µδ
++ ∥Φy,A∥1/2
+L1µθ
+�
+.
+The proof of Lemma 3.13 is very similar to the proof of Lemma 3.11 above.
+23
+
+Proof of Lemma 3.13. Let f ← y −O ◦M(θ), g ← O ◦(−δ)(θ), M1 ← Σε, M2 ← 0, and µ ← P.
+Then ∥f + g∥2
+M−1
+1
+= 2Φy,J(θ, δ) and ∥f∥2
+M−1
+1
+= 2Φy,A(θ, δ). Analogously to (A.14), we have by
+(2.3a) and (3.13) of Lemma 3.10 that
+∥∥y − O ◦ M(θ)∥2
+Σ−1
+ε ∥1/2
+L1
+P =
+√
+2∥Φy,A∥1/2
+L1µθ =
+√
+2∥Φy,A∥1/2
+L1
+µθ⊗µδ
+.
+(A.15)
+Applying (A.2) of Lemma A.2 with the choices above yields (3.16):
+2∥Φy,A − Φy,J∥L1
+µθ⊗µδ ≤∥∥O ◦ (−δ)(θ)∥2
+Σ−1
+ε ∥1/2
+L1
+P
+√
+2
+�
+∥Φy,J∥1/2
+L1
+µθ⊗µδ
++ ∥Φy,A∥1/2
+L1µθ
+�
+.
+Next,
+∥∥O ◦ (−δ)(θ)∥2
+Σ−1
+ε ∥1/2
+L1
+P ≤
+√
+2
+�
+∥Φy,A∥1/2
+L1µθ + ∥Φy,J∥1/2
+L1
+µθ⊗µδ
+�
+follows from (A.13), (A.15), and (A.4) of Lemma A.2.
+Proof of Proposition 3.14. Both ess infµθ⊗µδΦy,J and ess infµθ⊗µδΦy,A are nonnegative, by (2.7a)
+and (2.3a). Applying Theorem 3.1 and Lemma 3.13 yields
+max{dKL(µy,A
+θ,δ ∥µy,J
+θ,δ), dKL(µy,J
+θ,δ∥µy,A
+θ,δ )}
+≤2 exp
+�
+2∥Φy,J∥L1
+µθ⊗µδ + 2∥Φy,A∥L1
+µθ⊗µδ
+�
+∥Φy,A − Φy,J∥L1
+µθ⊗µδ
+≤21/2 exp
+�
+2∥Φy,J∥L1
+µθ⊗µδ + 2∥Φy,A∥L1
+µθ⊗µδ
+��
+∥Φy,J∥1/2
+L1
+µθ⊗µδ
++ ∥Φy,A∥1/2
+L1
+µθ⊗µδ
+�
+× ∥∥O ◦ (−δ)(θ)∥2
+Σ−1
+ε ∥1/2
+L1
+P .
+Using the definition of C given in the statement of Proposition 3.14 and (A.15) completes the
+proof.
+Proof of error of joint parameter-error posterior with respect to lifted enhanced noise
+posterior
+For the sake of completeness, we compare the joint posterior with the lifted enhanced
+noise posterior.
+Lemma A.3. Let Φy,E be defined as in (3.11a) with • = E. If Φy,E ∈ L1
+µθ and Φy,J ∈ L1
+µθ⊗µδ,
+then
+∥Φy,E − Φy,J∥L1
+µθ⊗µδ ≤2−1/2∥∥O ◦ (δ(θ) − mu)∥2
+Σ−1
+ε ∥1/2
+L1
+P
+�
+CE∥Φy,E∥1/2
+L1µθ + ∥Φy,J∥1/2
+L1
+µθ⊗µδ
+�
+(A.16)
++ 2−1∥∥y − O ◦ M(θ) − Omu∥2
+Σ−1
+ε
+−(Σε+OΣuO∗)−1∥L1
+P.
+Furthermore,
+∥∥O ◦ (δ(θ) − mu)∥2
+Σ−1
+ε ∥1/2
+L1
+P ≤
+√
+2
+�
+CE∥Φy,E∥1/2
+L1µθ + ∥Φy,J∥1/2
+L1
+µθ⊗µδ
+�
+∥∥y − O ◦ M(θ) − Omu∥2
+Σ−1
+ε
+−(Σε+OΣuO∗)−1∥L1
+P ≤ 2(CE + 1)∥Φy,E∥L1µθ .
+Proof of Lemma A.3. Let f ← y − O ◦ (M(θ) + δ(θ)), g ← O ◦ (δ(θ) − mu), M1 ← Σε,
+M2 ← OΣuO∗, and µ ← P. Then ∥f∥2
+M−1
+1
+= 2Φy,J(θ, δ) and ∥f + g∥2
+(M1+M2)−1 = 2Φy,E(θ, δ).
+Analogously to (A.14), we have by (2.6a) and (3.13) of Lemma 3.10 that
+∥∥y − O ◦ M(θ) − Omu∥2
+(Σε+OΣuO∗)−1∥1/2
+L1
+P =
+√
+2∥Φy,E∥1/2
+L1µθ =
+√
+2∥Φy,E∥1/2
+L1
+µθ⊗µδ
+.
+(A.17)
+24
+
+Applying (A.3) of Lemma A.2 with the choices above yields (A.16):
+2∥Φy,E − Φy,J∥L1
+µθ⊗µδ ≤∥∥O ◦ (δ(θ) − mu)∥2
+Σ−1
+ε ∥1/2
+L1
+P
+�
+CE∥2Φy,E∥1/2
+L1µθ + ∥2Φy,J∥1/2
+L1
+µθ⊗µδ
+�
++ ∥∥y − O ◦ M(θ) − Omu∥2
+Σ−1
+ε
+−(Σε+OΣuO∗)−1∥L1
+P.
+Next, we apply (A.4) and (A.5) from Lemma A.2, and use (A.17):
+∥∥O ◦ (δ(θ) − mu)∥2
+Σ−1
+ε ∥1/2
+L1
+P ≤
+√
+2
+�
+CE∥Φy,E∥1/2
+L1µθ + ∥Φy,J∥1/2
+L1
+µθ⊗µδ
+�
+∥∥y − O ◦ M(θ) − Omu∥2
+Σ−1
+ε
+−(Σε+OΣuO∗)−1∥L1
+P ≤ 2(CE + 1)∥Φy,E∥L1µθ .
+This completes the proof of Lemma A.3.
+Proposition A.4. Let Φy,E and Φy,J be as in Lemma A.3, and let µy,E
+θ,δ be as in (3.11b) with
+• = E. Then
+max{dKL(µy,E
+θ,δ ∥µy,J
+θ,δ), dKL(µy,J
+θ,δ∥µy,E
+θ,δ )} ≤ C∥∥O ◦ (mu − δ)(θ)∥2
+Σ−1
+ε ∥1/2
+L1
+P ,
+where
+C = exp
+�
+2∥Φy,J∥L1
+µθ⊗µδ + 2∥Φy,E∥L1µθ
+�
+max{21/2�
+CE∥Φy,E∥1/2
+L1
+µθ⊗µδ
++ ∥Φy,J∥1/2
+L1
+µθ⊗µδ
+�
+, 1}.
+Proof of Proposition A.4. Both ess infµθ⊗µδΦy,J and ess infµθ⊗µδΦy,E are nonnegative, by (2.7a)
+and (2.6a). Applying Theorem 3.1 and Lemma A.3 yields
+max{dKL(µy,E
+θ,δ ∥µy,J
+θ,δ), dKL(µy,J
+θ,δ∥µy,E
+θ,δ )}
+≤2 exp
+�
+2∥Φy,J∥L1
+µθ⊗µδ + 2∥Φy,E∥L1
+µθ⊗µδ
+�
+∥Φy,E − Φy,J∥L1
+µθ⊗µδ
+≤ exp
+�
+2∥Φy,J∥L1
+µθ⊗µδ + 2∥Φy,E∥L1
+µθ⊗µδ
+�
+max{21/2�
+CE∥Φy,E∥1/2
+L1
+µθ⊗µδ
++ ∥Φy,J∥1/2
+L1
+µθ⊗µδ
+�
+, 1}
+×
+�
+∥∥O ◦ (mu − δ)(θ)∥2
+Σ−1
+ε ∥L1
+P + ∥∥y − O ◦ M(θ) − Omu∥2
+Σ−1
+ε
+−(Σε+OΣuO∗)−1∥L1
+P
+�
+.
+Using the definition of C given in the statement of Proposition A.4 and (3.13) from Lemma 3.10
+completes the proof.
+References
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+certainty quantification in chaotic and geometric numerical integration, Stat. Comput. 30
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+Omar Ghattas, Residual-based error correction for neural operator accelerated infinite-
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+[8] Duc-Lam Duong, Tapio Helin, and Jose Rodrigo Rojo-Garcia, Stability estimates for the
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+[9] Subhashis Ghosal and Aad van der Vaart, Fundamentals of Nonparametric Bayesian In-
+ference, vol. 44, Cambridge: Cambridge University Press, 2017.
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+biased is your model? Concentration inequalities, information and model bias, IEEE Trans.
+Inform. Theory 66 (2020), no. 5, 3079–3097.
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+27
+

8tAzT4oBgHgl3EQf-v68/content/tmp_files/2301.01939v1.pdf.txt

@@ -0,0 +1,1976 @@
+Thomsen-type parameters and attenuation coefficients for constant-Q
+transverse isotropy
+Qi Haoa,∗, Ilya Tsvankinb
+aCollege of Geoexploration Science and Technology, Jilin University, Changchun, 130026, P. R. China
+bDepartment of Geophysics, Colorado School of Mines, Golden, 80401, USA
+Abstract
+Transversely isotropic (TI) media with the frequency-independent quality-factor elements (also called “constant-Q”
+transverse isotropy) are often used to describe attenuation anisotropy in sedimentary rocks. The attenuation coef-
+ficients in constant-Q TI models can be conveniently defined in terms of the Thomsen-type attenuation-anisotropy
+parameters.
+Recent research indicates that not all those parameters for such constant-Q media are frequency-
+independent. Here, we present concise analytic formulae for the Thomsen-type attenuation parameters for Kjar-
+tansson’s constant-Q TI model and show that one of them (δQ) varies with frequency. The analytic expression
+for δQ helps evaluate the frequency dependence of the normalized attenuation coefficients of P- and SV-waves by
+introducing the newly defined “dispersion factors”. Viscoacoustic constant-Q transverse isotropy is also discussed as
+a special case, for which the elliptical condition and simplified expressions for the parameters δ and δQ are derived.
+Our results show that in the presence of significant absorption the attenuation coefficients of the “constant-Q”
+model vary with frequency for oblique propagation with respect to the symmetry axis. This variation needs to be
+taken into account when applying the spectral-ratio method and other attenuation-analysis techniques.
+Keywords:
+seismic, attenuation, anisotropy, wave, Q
+1. Introduction
+The frequency-independent quality factor (called “constant-Q” for brevity) provides a useful phenomenologi-
+cal description of seismic attenuation in rocks and is widely used in seismic attenuation analysis. Among such
+constant-Q dissipative models are those proposed by [1] and [2]. For isotropic media, the Kjartansson model pro-
+duces the constant-Q factor for all frequencies, whereas the Kolsky model leads to nearly constant Q-values. The
+complex moduli for the Kolsky model represent the first-order Maclaurin series expansion with respect to 1/Q of
+the corresponding moduli for the Kjartansson model [3, 4].
+As an extension of non-dissipative transverse isotropy, the constant-Q TI model can be used to process seismic
+attenuation data for most sedimentary rocks, such as shale formations.
+The constant-Q assumption facilitates
+estimation of the quality factor and attenuation anisotropy [e.g., 5, 6, 7, 8]. Ultrasonic measurements demonstrate
+that attenuation anisotropy generally is stronger than velocity anisotropy for rock samples [9, 10, 11].
+Velocity anisotropy for TI media can be efficiently described by the Thomsen anisotropy parameters [12, 13].
+Likewise, attenuation anisotropy for dissipative TI media is convenient to study using the Thomsen-type nota-
+tion introduced by [14]. The combination of the velocity- and attenuation-related Thomsen-type parameters [14]
+completely defines the complex stiffness matrix at a specified frequency for a general dissipative TI model with
+a vertical symmetry axis (VTI medium). The generic Thomsen velocity parameters depend on the real parts of
+the stiffness coefficients (cij), whereas the Thomsen-type attenuation parameters are defined by both the real and
+imaginary parts of cij. [15] found that some Thomsen-type parameters in constant-Q VTI media are frequency
+∗Corresponding author
+Email addresses: [email protected] (Qi Hao), [email protected] (Ilya Tsvankin)
+January 6, 2023
+arXiv:2301.01939v1  [physics.geo-ph]  5 Jan 2023
+
+dependent. This phenomenon is not entirely surprising, because all stiffness coefficients of constant-Q TI media,
+which are involved in the definition of the Thomsen-type parameters, are functions of frequency. Investigating
+the frequency variations of these parameters can facilitate understanding of such key signatures in TI media as
+velocities, traveltimes, attenuation coefficients, and polarization vectors. However, to our knowledge, there are no
+analytic expressions for the frequency-dependent Thomsen-type attenuation parameters in constant-Q dissipative
+VTI media.
+Here, we derive analytic formulae for the Thomsen-type parameters using Kjartansson’s constant-Q VTI model.
+A set of the corresponding reference parameters is defined at a specified frequency and used to obtain those
+parameters for the entire frequency range. We also present a formula for the frequency-dependent anellipticity
+and define the condition for elliptical anisotropy in constant-Q TI media. The newly proposed formulae for the
+Thomsen-type parameters allow us to study the normalized plane-wave attenuation coefficients in constant-Q media
+with weak attenuation anisotropy and define the “dispersion factors” for P- and SV-waves. Numerical examples
+are used to analyze the accuracy of the obtained expressions for the Thomsen-type parameters, the validity of the
+elliptical condition, and the frequency dependence of the attenuation coefficients.
+2. Thomsen-type parameters of constant-Q VTI media
+2.1. Constant-Q dissipative VTI model
+Referring to [14] and [16], the complex stiffness (or modulus) matrix M for viscoelastic VTI media is given by:
+M =
+�
+�
+�
+�
+�
+�
+�
+�
+M11
+M11 − 2M66
+M13
+0
+0
+0
+M11 − 2M66
+M11
+M13
+0
+0
+0
+M13
+M13
+M33
+0
+0
+0
+0
+0
+0
+M55
+0
+0
+0
+0
+0
+0
+M55
+0
+0
+0
+0
+0
+0
+M66
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(1)
+where Mij = M R
+ij − i sgn(f)M I
+ij denote the complex stiffness coefficients for the frequency f, and the minus sign in
+front of i sgn(f)M I
+ij follows from the definition of the Fourier transform in [17] and [3]. Both the real and imaginary
+parts of Mij generally are frequency dependent.
+For the [2] model (also called the constant-Q model), the nonzero independent elements in equation 1 are
+expressed as:
+Mij =
+˜
+M R
+ij
+cos(πγij)
+�
+−i f
+f0
+�2γij
+,
+(2)
+with
+γij = 1
+π tan−1
+� 1
+Qij
+�
+,
+(3)
+where Qij ≡ M R
+ij /M I
+ij, f0 is the reference frequency, and ˜
+M R
+ij denote the real parts of Mij at f0: ˜
+M R
+ij = Re(Mij)|f=f0.
+By design, the quality-factor elements Qij for the Kjartansson model are independent of frequency. As follows from
+equation 2, the complex stiffness coefficients Mij for a given frequency can be expressed in terms of ˜
+M R
+ij and Qij.
+2.2. Thomsen-type parameterization
+[14] and [18] show that dissipative VTI media can be conveniently parameterized by the Thomsen-type attenu-
+ation parameters. The [12] velocity parameters [see 13] are defined in the nonattenuative reference VTI medium.
+The parameter VP 0 is the vertical velocity of P-waves:
+VP 0 ≡
+�
+M R
+33
+ρ ,
+(4)
+where ρ denotes density.
+2
+
+The parameter VS0 is the vertical velocity of S-waves:
+VS0 ≡
+�
+M R
+55
+ρ .
+(5)
+The parameter ϵ is approximately equal to the fractional difference between the horizontal and vertical velocities
+of P-waves:
+ϵ ≡ M R
+33 − M R
+11
+2M R
+33
+.
+(6)
+The parameter δ determines the second derivative of the P-wave phase velocity at vertical incidence and is given
+by:
+δ ≡
+�
+M R
+13 + M R
+55
+�2 −
+�
+M R
+33 − M R
+55
+�2
+2M R
+33(M R
+33 − M R
+55)
+.
+(7)
+The parameter γ is approximately equal to the fractional difference between the horizontal and vertical velocities
+of SH-waves:
+γ ≡ M R
+66 − M R
+55
+2M R
+55
+.
+(8)
+The Thomsen-type attenuation parameters [14] can be used to define the normalized phase attenuation coefficient
+A ≡ |kI|/|kR| for P-, SV-, and SH-waves, which is generally supposed to be frequency-independent in constant-Q
+models. For more details about A, see the section “Plane-wave attenuation in constant-Q VTI media” below.
+The parameter AP 0 is the vertical attenuation coefficient of P-waves:
+AP 0 ≡ Q33
+��
+1 +
+1
+Q2
+33
+− 1
+�
+≈
+1
+2Q33
+.
+(9)
+The parameter AS0 is the vertical attenuation coefficient of S-waves:
+AS0 ≡ Q55
+��
+1 +
+1
+Q2
+55
+− 1
+�
+≈
+1
+2Q55
+.
+(10)
+The parameter ϵQ is close to the fractional difference between the horizontal and vertical attenuation coefficients
+of P-waves:
+ϵQ ≡ Q33 − Q11
+Q11
+.
+(11)
+The parameter δQ controls the second derivative of the P-wave attenuation coefficient at vertical incidence and
+is expressed as [14]:
+δQ ≡
+Q33−Q55
+Q55
+M R
+55
+(M R
+13+M R
+33)
+2
+M R
+33−M R
+55
++ 2 Q33−Q13
+Q13
+M R
+13(M R
+13 + M R
+55)
+M R
+33(M R
+33 − M R
+55)
+.
+(12)
+The parameter γQ is close to the fractional difference between the horizontal and vertical attenuation coefficients
+of SH-waves:
+γQ ≡ Q55 − Q66
+Q66
+.
+(13)
+3
+
+3. Analytic description of Thomsen-type parameters
+In this section, we represent the Thomsen velocity parameters and Thomsen-type attenuation parameters in
+terms of their reference values defined at f = f0:
+˜VP 0 = VP 0|f=f0, ˜VS0 = VS0|f=f0, ˜ϵ = ϵ|f=f0, ˜δ = δ|f=f0,
+˜
+AP 0 = AP 0|f=f0,
+˜
+AS0 = AS0|f=f0, ˜ϵQ = ϵQ|f=f0, and ˜δQ = δQ|f=f0. These parameters are used to find the
+real parts of the reference stiffness coefficients ( ˜
+M R
+ij ), the quality-factor elements Qij (see Appendix A), and the
+frequency-dependent stiffness matrix M.
+According to equations 4–7 and 12, the Thomsen-type parameters involve the coefficients M R
+ij , where ij=11, 13,
+33, 55 and 66. Using equations 2 and 3, M R
+ij are approximately expressed as:
+M R
+ij ≈ ˜
+M R
+ij
+�
+1 + 2
+π Q−1
+ij ln
+����
+f
+f0
+���� + 2
+π2 Q−2
+ij ln2
+����
+f
+f0
+����
+�
+,
+(14)
+where we use the approximation tan−1(Q−1
+ij ) ≈ Q−1
+ij because typically Qij ≫ 1.
+Substitution of equation 14 into equations 4–7 and 12 allows us to derive approximate expressions for the
+frequency-dependent Thomsen-type parameters, which are discussed in the following two subsections.
+3.1. Velocity parameters
+The second-order approximations for the Thomsen velocity parameters with respect to ln|f/f0| are given by:
+VP 0 = ˜VP 0
+�
+1 + 1
+π Q−1
+33 ln
+����
+f
+f0
+���� +
+1
+2π2 Q−2
+33 ln2
+����
+f
+f0
+����
+�
+,
+(15)
+VS0 = ˜VS0
+�
+1 + 1
+π Q−1
+55 ln
+����
+f
+f0
+���� +
+1
+2π2 Q−2
+55 ln2
+����
+f
+f0
+����
+�
+,
+(16)
+ϵ = ˜ϵ + 1
+π (1 + 2˜ϵ) Q−1
+33 ˜ϵQ ln
+����
+f
+f0
+���� + 1
+π2 (1 + 2˜ϵ) Q−2
+33 ˜ϵ2
+Q ln2
+����
+f
+f0
+���� ,
+(17)
+δ = ˜δ + 1
+π Q−1
+33 ˜δQ ln
+����
+f
+f0
+���� + 1
+π2 Q−2
+33 ζQ ln2
+����
+f
+f0
+���� ,
+(18)
+γ = ˜γ + 1
+π (1 + 2˜γ) Q−1
+55 ˜γQ ln
+����
+f
+f0
+���� + 1
+π2 (1 + 2˜γ) Q−2
+55 ˜γ2
+Q ln2
+����
+f
+f0
+���� ,
+(19)
+where the P- and S-wave inverse vertical quality factors Q33 and Q55 (respectively) are:
+Q−1
+33 =
+˜
+AP 0
+2(1 − ˜
+A2
+P 0)
+,
+(20)
+Q−1
+55 =
+˜
+AS0
+2(1 − ˜
+A2
+S0)
+.
+(21)
+The coefficient ζQ in equation 18 is defined as:
+ζQ = d0 (1 − gQ)2 + d1 (1 − gQ) ˜δQ + d2 ˜δ2
+Q ,
+(22)
+with
+gQ ≡ Q33
+Q55
+,
+(23)
+4
+
+d0 =
+g(1 − g + χ)2 �
+(1 + 2˜δ) χ − (1 + 2˜δ) g + (1 + ˜δ) g2�
+(1 − g)2(χ − g)χ2
+,
+(24)
+d1 =
+2g
+�
+1 + 2˜δ + χ − (2 + ˜δ + χ) g + g2�
+(χ − g)χ2
+,
+(25)
+d2 =
+2χ − g
+2(1 + 2˜δ − g)(χ − g)
+;
+(26)
+g ≡
+˜V 2
+S0
+˜V 2
+P 0
+,
+(27)
+χ =
+�
+(1 − g)(1 + 2˜δ − g) .
+(28)
+In equations 15–19, the first-order terms with respect to ln|f/f0| are scaled by Q−1
+33 or Q−1
+55 , whereas the second-
+order terms by Q−2
+33 or Q−2
+55 . Because Q33 and Q55 typically are much greater than unity, the frequency dependence
+of the velocity parameters is mostly determined by the first-order terms. Equations 15–19 indicate that: (1) VP 0
+and VS0 always monotonically increase with frequency; (2) ϵ, δ and γ also monotonically increase with f, if ˜ϵQ > 0,
+˜δQ > 0, and ˜γQ > 0, respectively. Overall, the frequency dependence of VP 0, VS0, ϵ, δ, and γ for realistic values of
+Q33 and Q55 (Q33 ≫ 1 and Q55 ≫ 1) remains weak, as illustrated by the numerical examples below.
+Note that phase and group velocities in strongly dissipative TI media are influenced by attenuation and do
+not represent the same functions of the Thomsen parameters as in purely elastic models [14, 13]. For sedimentary
+formations, both gQ and g vary within a limited range.
+In particular, according to [9], for relatively shallow
+sedimentary rocks 0.5 < gQ ≤ 3 (Figure 1).
+gQ= 1
+2
+gQ=1
+gQ=2
+gQ=3
+0
+20
+40
+60
+80
+100
+0
+50
+100
+150
+Q55
+Q33
+Figure 1: Vertical quality factors Q33 and Q55 in dissipative VTI rocks. The black dots are the data from Table 3
+of [9]; gQ ≡ Q33/Q55.
+Using equations 17 and 18 for ϵ and δ, the anellipticity parameter η [19] can be approximately obtained as:
+η ≡ ϵ − δ
+1 + 2δ = η0 + η1 Q−1
+33 ln
+����
+f
+f0
+���� + η2 Q−2
+33 ln2
+����
+f
+f0
+����,
+(29)
+5
+
+where Q33 is given by equation 20, and
+η0 = ˜η = ˜ϵ − ˜δ
+1 + 2˜δ
+,
+(30)
+η1 =
+1 + 2˜ϵ
+(1 + 2˜δ)2
+�
+(1 + 2˜δ)˜ϵQ − ˜δQ
+�
+,
+(31)
+η2 = 1 + 2˜ϵ
+1 + 2˜δ
+�
+r0 +
+r1
+1 + 2˜δ
++
+r2
+(1 + 2˜δ)2
+�
+,
+(32)
+with
+r0 = ˜ϵ2
+Q,
+(33)
+r1 = −ζQ − 2˜ϵQ ˜δQ,
+(34)
+r2 = 2˜δ2
+Q.
+(35)
+The parameter η controls (along with the zero-dip normal-moveout velocity) all P-wave time-domain signatures for
+laterally homogeneous VTI media above a horizontal or dipping target reflector [19, 13].
+3.2. Attenuation parameters
+The following Thomsen-type attenuation parameters are expressed directly through the elements Qij and, there-
+fore, are frequency-independent in constant-Q VTI media:
+AP 0 = ˜
+AP 0,
+(36)
+AS0 = ˜
+AS0,
+(37)
+ϵQ = ˜ϵQ,
+(38)
+γQ = ˜γQ.
+(39)
+The attenuation parameter δQ, however, also depends on the coefficients M R
+ij (equation 12), which vary with
+frequency. The second-order approximation for δQ with respect to ln |f/f0| is:
+δQ = ˜δQ + 2
+π Q−1
+33 ζQ ln
+����
+f
+f0
+���� + 2
+π2 Q−2
+33 ξQ ln2
+����
+f
+f0
+����,
+(40)
+where ζQ is defined in equation 22, and
+ξQ = s0(1 − gQ)3 + s1(1 − gQ)2 ˜δQ + s2(1 − gQ) ˜δ2
+Q + s3 ˜δ3
+Q.
+(41)
+The explicit expressions for the coefficients si are given in Appendix B.
+Because for Q33 ≫ 1 the influence of the second-order term in equation 40 is insignificant, the frequency variation
+of δQ is largely controlled by the coefficient ζQ. For ζQ > 0, δQ monotonically increases with frequency. As follows
+from equations 22 and 24–28, ζQ is a function of g (equation 27), gQ (equation 23), ˜δ, and ˜δQ.
+3.3. Numerical analysis
+Here, we analyze the above expressions for the Thomsen-type parameters numerically. The reference frequency
+is set as f0 = 40 Hz and the frequency range as [1, 200] Hz for all examples below.
+First, we test the accuracy of the equations 15 and 16 for the vertical velocities and their first-order versions
+(i.e., those without the second-order term with respect to ln|f/f0|). As demonstrated by Figure 2, the first-order
+approximations for VP 0 and VS0 are sufficiently accurate even for strong attenuation in a wide frequency range.
+Overall, the frequency dependence of the vertical velocities is almost negligible, except for very low frequencies.
+6
+
+exact
+1st
+2nd
+0
+50
+100
+150
+200
+2.92
+2.94
+2.96
+2.98
+3.00
+3.02
+3.04
+f (Hz)
+vP0 (km/s)
+(a)
+exact
+1st
+2nd
+0
+50
+100
+150
+200
+1.42
+1.44
+1.46
+1.48
+1.50
+1.52
+1.54
+f (Hz)
+vS0 (km/s)
+(b)
+Figure 2: Frequency-dependent vertical velocities (a) VP 0 and (b) VS0. “Exact” in the legend refers to the exact
+values, whereas “1st” and “2nd” denote the first- and second-order approximations with respect to ln |f/f0|, re-
+spectively. On plot (a), ˜VP 0 = 3.0 km/s and Q33 = 40 ( ˜
+AP 0 = 0.0125); on plot (b), ˜VS0 = 1.5 km/s and Q55 = 20
+( ˜
+AS0 = 0.025).
+Table 1: Medium parameters for two constant-Q VTI models at the reference frequency f0 = 40 Hz.
+[H]
+Model
+˜VP 0
+˜VS0
+˜ϵ
+˜δ
+˜γ
+˜
+AP 0 (Q33)
+˜
+AS0 (Q55)
+˜ϵQ
+˜δQ
+˜γQ
+1
+3.0
+1.5
+0.3
+-0.1
+0.1
+0.0125 (40)
+0.0167 (30)
+-0.3
+-1.91
+0.5
+2
+3.0
+1.5
+0.3
+-0.1
+0.2
+0.0250 (20)
+0.0333 (15)
+0.3
+0.98
+-0.2
+Figures 3, 4 and 5 show that the first-order versions of equations 17–19 can accurately describe the variations of
+the anisotropy parameters ϵ, δ, and γ with frequency. Comparison of Figures 3, 4, and 5 confirms that the reference
+parameters ˜ϵQ, ˜δQ, and ˜γQ govern the frequency dependence of ϵ, δ, and γ. For example, if ˜ϵQ > 0, ϵ increases
+with frequency. As is the case for VP 0 and VS0, the anisotropy coefficients vary with frequency primarily in the
+low-frequency range.
+exact
+1st
+2nd
+0
+50
+100
+150
+200
+0.295
+0.300
+0.305
+0.310
+0.315
+f (Hz)
+ϵ
+(a)
+exact
+1st
+2nd
+0
+50
+100
+150
+200
+0.27
+0.28
+0.29
+0.30
+0.31
+f (Hz)
+ϵ
+(b)
+Figure 3: Variation of the Thomsen parameter ϵ with frequency for (a) Model 1 and (b) Model 2 from Table 1.
+The legend is the same as in Figure 2.
+Next, we investigate the only frequency-dependent attenuation-anisotropy parameter, δQ, by comparing the
+exact equation for δQ with its first- and second-order approximations. The first-order equation accurately models
+δQ in a wide frequency range, whereas contribution of the second-order term is practically negligible (Figure 6).
+As mentioned above, the coefficient ζQ in equation 40 is largely responsible for the frequency variation of δQ for
+7
+
+exact
+1st
+2nd
+0
+50
+100
+150
+200
+-0.12
+-0.11
+-0.10
+-0.09
+-0.08
+-0.07
+f (Hz)
+δ
+(a)
+exact
+1st
+2nd
+0
+50
+100
+150
+200
+-0.16
+-0.14
+-0.12
+-0.10
+-0.08
+f (Hz)
+δ
+(b)
+Figure 4: Same as Figure 3 but for the parameter δ.
+exact
+1st
+2nd
+0
+50
+100
+150
+200
+0.075
+0.080
+0.085
+0.090
+0.095
+0.100
+0.105
+0.110
+f (Hz)
+γ
+(a)
+exact
+1st
+2nd
+0
+50
+100
+150
+200
+0.190
+0.195
+0.200
+0.205
+0.210
+0.215
+0.220
+f (Hz)
+γ
+(b)
+Figure 5: Same as Figure 3 but for the parameter γ.
+exact
+1st
+2nd
+0
+50
+100
+150
+200
+-2.3
+-2.2
+-2.1
+-2.0
+-1.9
+-1.8
+-1.7
+-1.6
+f (Hz)
+δQ
+(a)
+exact
+1st
+2nd
+0
+50
+100
+150
+200
+0.80
+0.85
+0.90
+0.95
+1.00
+1.05
+f (Hz)
+δQ
+(b)
+Figure 6: Frequency-dependent Thomsen-type attenuation parameter δQ for (a) Model 1 and (b) Model 2 from
+Table 1. The legend is the same as in Figure 2.
+8
+
+a specified value of Q33. Equation 22 shows that ζQ is a function of the parameters g = ˜V 2
+S0/ ˜V 2
+P 0, gQ = Q−1
+55 /Q−1
+33 ,
+˜δ and ˜δQ. Using the results from Figure 1, we restrict gQ to the range 0.5 ≤ gQ ≤ 3. Figures 7 and 8 show that
+the smallest absolute value of ζQ corresponds to gQ = 1, and |ζQ| increases with the deviation of gQ from unity.
+As a result, the parameter δQ is almost independent of frequency for gQ = 1 (Figure 9). Overall, the frequency
+dependence of δQ becomes noticeable for large |gQ − 1| (e.g., gQ = 3; Figure 9), but it is also influenced by the
+parameters ˜δ and ˜δQ. For the most common values of gQ considered here, the parameter δQ significantly varies
+with f only for low frequencies.
+-0.1 0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+δ˜
+δ˜
+Q
+ζQ
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+22.5
+(a)
+-0.1 0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+δ˜
+δ˜
+Q
+ζQ
+2
+4
+6
+8
+10
+(b)
+-0.1 0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+δ˜
+δ˜
+Q
+ζQ
+0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+(c)
+-0.1 0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+δ˜
+δ˜
+Q
+ζQ
+0
+1
+2
+3
+4
+5
+6
+(d)
+Figure 7: Contour plots of the coefficient ζQ as a function of ˜δ and ˜δQ. The parameter g = ˜V 2
+S0/ ˜V 2
+P 0 = 0.3. The
+parameter gQ = Q−1
+55 / ˜Q−1
+P 0 is defined as (a) gQ = 3, (b) gQ = 2, (c) gQ = 1, and (d) gQ = 0.5.
+9
+
+g=0
+g=0.1
+g=0.3
+g=0.5
+0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
+-10
+-5
+0
+5
+10
+15
+gQ
+ζQ
+(a)
+g=0
+g=0.1
+g=0.3
+g=0.5
+0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
+-6
+-4
+-2
+0
+2
+4
+6
+8
+gQ
+ζQ
+(b)
+g=0
+g=0.1
+g=0.3
+g=0.5
+0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
+0
+10
+20
+30
+40
+50
+gQ
+ζQ
+(c)
+g=0
+g=0.1
+g=0.3
+g=0.5
+0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
+0
+10
+20
+30
+gQ
+ζQ
+(d)
+Figure 8: Variation of the coefficient ζQ with gQ for different values of g. (a) ˜δ = −0.2 and ˜δQ = −0.6; (b) ˜δ = −0.2
+and ˜δQ = 0; (c) ˜δ = 0.2 and ˜δQ = −0.4; (d) ˜δ = 0.2 and ˜δQ = 0.98.
+gQ=3
+gQ=2
+gQ=1
+gQ= 1
+2
+0
+50
+100
+150
+200
+-1.0
+-0.8
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+f (Hz)
+δQ
+(a)
+gQ=3
+gQ=2
+gQ=1
+gQ= 1
+2
+0
+50
+100
+150
+200
+-0.2
+-0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+f (Hz)
+δQ
+(b)
+gQ=3
+gQ=2
+gQ=1
+gQ= 1
+2
+0
+50
+100
+150
+200
+-0.8
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+f (Hz)
+δQ
+(c)
+gQ=3
+gQ=2
+gQ=1
+gQ= 1
+2
+0
+50
+100
+150
+200
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+f (Hz)
+δQ
+(d)
+Figure 9: Variation of the attenuation parameter δQ with frequency for different gQ and g = 0.3. The parameters
+˜δ and ˜δQ are the same as in Figure 8.
+10
+
+4. Viscoacoustic constant-Q transverse isotropy
+4.1. Simplified parameter expressions
+Next, we consider the so-called “viscoacoustic” constant-Q media described by the Thomsen-type notation. The
+acoustic approximation is implemented by setting ˜VS0 = AS0 = 0 in equations 18, 40, and 29 [20, 21, 22] . The
+parameters δ, η, and δQ then reduce to:
+δ = ˜δ + 1
+π Q−1
+33 ˜δQ ln
+����
+f
+f0
+���� + 1
+π2 Q−2
+33
+˜δ2
+Q
+1 + 2˜δ
+ln2
+����
+f
+f0
+����,
+(42)
+η = η0 + η1 Q−1
+33 ln
+����
+f
+f0
+���� +
+�
+˜ϵQ −
+˜δQ
+1 + 2˜δ
+�
+η1 Q−2
+33 ln2
+����
+f
+f0
+����,
+(43)
+δQ = ˜δQ + 2
+π Q−1
+33
+˜δ2
+Q
+1 + 2˜δ
+ln
+����
+f
+f0
+���� + 2
+π2 Q−2
+33
+˜δ3
+Q
+(1 + 2˜δ)2 ln2
+����
+f
+f0
+����.
+(44)
+Setting η (equation 43) to zero, which requires η0 = η1 = 0 (see equations 30 and 31), we obtain the elliptical
+conditions:
+˜ϵ = ˜δ,
+(45)
+˜ϵQ =
+˜δQ
+1 + 2˜δ
+.
+(46)
+Equations 45 and 46 make the parameters of viscoacoustic constant-Q media satisfy the same conditions at all
+frequencies:
+ϵ = δ,
+(47)
+ϵQ =
+δQ
+1 + 2δ ,
+(48)
+which follows from equations 17, 31, 42, and 44. Equation 47 implies that the elliptical conditions at the reference
+frequency ensure that η = 0 at all frequencies.
+For viscoelastic constant-Q media discussed earlier, equation 47 remains approximately valid (i.e., the model is
+elliptical at all frequencies), if equations 45 and 46 are satisfied (see equations 29–31).
+4.2. Numerical validation
+Here, we verify the elliptical conditions (equations 45 and 46) by computing the anellipticity parameter η. The
+exact η is calculated using equations 6, 7, 11 and 12 along with equations 2 and 3 under the acoustic approximation
+( ˜VS0 = 0 and Q−1
+55 = 0). The first-order approximation for η is given by equation 43 without the second-order term
+with respect to ln|f/f0|.
+Figure 10 shows that for models that satisfy equations 45 and 46 the exact anellipticity parameter is negligibly
+small for all frequencies (on the order of 10−7 for both models), which confirms that the elliptical conditions at the
+reference frequency lead to equation 47. In addition, our testing confirms that the difference between the left and
+right sides of equation 48 is negligible, if equations 45 and 46 are satisfied.
+11
+
+exact
+1st
+2nd
+0
+50
+100
+150
+200
+-4.×10-7
+-2.×10-7
+0
+2.×10-7
+4.×10-7
+f (Hz)
+η
+(a)
+exact
+1st
+2nd
+0
+50
+100
+150
+200
+-5.×10-7
+0
+5.×10-7
+1.×10-6
+f (Hz)
+η
+(b)
+Figure 10: Variation of the anellipticity parameter η with frequency under the elliptical conditions (equations 45
+and 46). The P-wave quality factor and reference vertical velocity at f0 = 40 Hz are Q33 = 40 and ˜VP 0 = 3 km/s.
+The parameters ˜ϵ and ˜ϵQ are (a) ˜ϵ = 0.3 and ˜ϵQ = −0.33; (b) ˜ϵ = 0.2 and ˜ϵQ = 0.4.
+5. Plane-wave attenuation in constant-Q VTI media
+In this section, we apply the obtained expressions for the Thomsen-type parameters to study the normalized
+plane-wave attenuation coefficients in constant-Q VTI media.
+The normalized phase attenuation coefficient is
+defined as A ≡ |kI|/|kR|, where kR and kI denote the real and imaginary parts of the complex wave vector [14].
+The words “phase” and “normalized” are omitted below for brevity. The angle between kR and kI is called the
+“inhomogeneity” angle, which is not defined in plane-wave propagation (i.e., it is a free parameter that can vary
+within certain bounds). The coefficient A corresponding to kR ∥ kI is approximately equal to the group attenuation
+coefficient, which can be estimated from seismic data, for a wide range of “inhomogeneity” angles [23, 18].
+5.1. Attenuation coefficients
+[14] and [18] show that the approximate attenuation coefficients of plane waves in viscoelastic constant-Q VTI
+media are given by:
+AP = AP 0 (1 + δQ sin2 θ cos2 θ + ϵQ sin4 θ),
+(49)
+ASV = AS0 (1 + σQ sin2 θ cos2 θ),
+(50)
+ASH = AS0 (1 + γQ sin2 θ),
+(51)
+where the subscripts P, SV, and SH denote the wave types, and θ is the phase angle measured from the vertical.
+The quantity σQ in equation 52 is defined as [14]:
+σQ = 2V 2
+P 0
+V 2
+S0
+�Q33
+Q55
+− 1
+�
+(ϵ − δ) + V 2
+P 0 Q55
+V 2
+S0 Q33
+(ϵQ − δQ).
+(52)
+Equations 49–51 are derived under the assumption of weak attenuation and weak anisotropy (in both velocity
+and attenuation). Note that the effective quality factor, assumed to be frequency-independent in constant-Q TI
+media, is proportional to the inverse of the attenuation coefficient [14].
+Substitution of the Thomsen parameters from equations 15–19 and 36–40 into equations 49–52 allows us to sepa-
+rate the frequency-dependent parts of the attenuation coefficients. The approximate P-wave attenuation coefficient
+then becomes (only the linear term in ln |f/f0| is retained):
+AP = ˜
+AP 0
+�
+1 + ˜δQ sin2 θ cos2 θ + ˜ϵQ sin4 θ + RP ln
+����
+f
+f0
+����
+�
+,
+(53)
+12
+
+where RP controls the derivative of AP with respect to ln |f/f0|,
+RP = 1
+π
+˜
+AP 0 ζQ sin2 θ cos2 θ;
+(54)
+ζQ is defined in equation 22.
+For SV-waves,
+ASV = ˜
+AS0
+�
+1 + ˜σQ sin2 θ cos2 θ + RSV ln
+����
+f
+f0
+����
+�
+,
+(55)
+with
+˜σQ = 2˜σ (gQ − 1) +
+1
+g gQ
+(˜ϵQ − ˜δQ),
+(56)
+RSV = 1
+π
+˜
+AS0 σ′
+Q sin2 θ cos2 θ,
+(57)
+σ′
+Q = 2(1 − gQ)
+g g2
+Q
+�
+(1 − gQ)(˜ϵ − ˜δ) − ˜δQ + (1 + ˜ϵ)˜ϵQ
+�
+− ζQ
+g g2
+Q
+,
+(58)
+where g and gQ are given by equations 27 and 23, respectively. The factor RSV controls the derivative of ASV with
+respect to ln |f/f0|.
+The terms ˜
+AP 0 RP and ˜
+AS0 RSV define the rate of the P- and SV-wave attenuation-coefficient change (increase
+or decrease) with respect to ln |f/f0|.
+The larger RP and RSV are, the stronger is the dispersion (frequency
+dependence) of AP and ASV .
+Therefore, RP and RSV can be called the P- and SV-wave dispersion factors,
+respectively.
+The SH-wave attenuation coefficient (equation 51) is independent of frequency, with γQ = ˜γQ:
+ASH = ˜
+AS0(1 + ˜γQ sin2 θ).
+(59)
+5.2. Numerical dispersion analysis
+Here, we evaluate the frequency dependence of the attenuation coefficients of P- and SV-waves, starting with the
+dispersion factors RP and RSV (equations 54 and 57). As before, we restrict gQ to the realistic range 0.5 < gQ ≤ 3
+(Figure 1). Figures 11 and 12 show that gQ = 1 yields the smallest values of RP and RSV ; the dispersion factors
+and the magnitude of their variation with angle increase with the deviation of gQ from unity.
+Next, we use the medium parameters from Figures 11d and 12d to calculate the exact attenuation coefficients
+for P- and SV-waves (respectively) at three frequencies. For the reference frequency f0 = 40 Hz, the term ln |f/f0|
+in equations 54 and 57 is close to −1 at f = 15 Hz and 1 at f = 109 Hz. In agreement with equations 53 and 55,
+the variation of AP with ln |f/f0| between 15 Hz and 40 Hz (and 40 Hz and 109 Hz) is approximately proportional
+to RP , and the corresponding variation of ASV is approximately proportional to RSV .
+Figures 13 and 14 show that the frequency dependence of the P- and SV-wave attenuation coefficients AP and
+ASV is generally mild. However, they may become noticeable for propagation angles close to 45◦ as illustrated in
+Figures 15 and 16. Both AP and ASV exhibit a more significant variation with frequency for strongly attenuative
+media (Q33=Q55=20) when gQ ≥ 2 (for P-waves) and gQ ≤ 0.5 (for SV-waves).
+13
+
+gQ=3
+gQ=2
+gQ=1
+gQ= 1
+2
+0
+20
+40
+60
+80
+0
+1
+2
+3
+4
+5
+6
+θ (degrees)
+RP (%)
+(a)
+gQ=3
+gQ=2
+gQ=1
+gQ= 1
+2
+0
+20
+40
+60
+80
+0
+1
+2
+3
+4
+5
+θ (degrees)
+RP (%)
+(b)
+gQ=3
+gQ=2
+gQ=1
+gQ= 1
+2
+0
+20
+40
+60
+80
+0
+1
+2
+3
+4
+5
+6
+θ (degrees)
+RP (%)
+(c)
+gQ=3
+gQ=2
+gQ=1
+gQ= 1
+2
+0
+20
+40
+60
+80
+0
+1
+2
+3
+4
+5
+θ (degrees)
+RP (%)
+(d)
+Figure 11: Variation of the P-wave dispersion factor RP (equation 54) with the phase angle for different gQ. The
+reference parameters defined at f0 = 40 Hz are ˜VP 0 = 3.0 km/s, g = 0.3, ˜ϵ = 0.2, ˜
+AP 0 = 0.0125 (corresponding to
+Q33 = 40), ˜ϵQ = −0.1 and ˜δQ = −0.2. (a) ˜δ = 0.1 and ˜δQ = −0.2; (b) ˜δ = 0.1 and ˜δQ = 0.2; (c) ˜δ = −0.1 and
+˜δQ = −0.2; (d) ˜δ = −0.1 and ˜δQ = 0.2.
+gQ=3
+gQ=2
+gQ=1
+gQ= 1
+2
+0
+20
+40
+60
+80
+-2.5
+-2.0
+-1.5
+-1.0
+-0.5
+0.0
+θ (degrees)
+RSV (%)
+(a)
+gQ=3
+gQ=2
+gQ=1
+gQ= 1
+2
+0
+20
+40
+60
+80
+-7
+-6
+-5
+-4
+-3
+-2
+-1
+0
+θ (degrees)
+RSV (%)
+(b)
+gQ=3
+gQ=2
+gQ=1
+gQ= 1
+2
+0
+20
+40
+60
+80
+-2.0
+-1.5
+-1.0
+-0.5
+0.0
+θ (degrees)
+RSV (%)
+(c)
+gQ=3
+gQ=2
+gQ=1
+gQ= 1
+2
+0
+20
+40
+60
+80
+-8
+-6
+-4
+-2
+0
+θ (degrees)
+RSV (%)
+(d)
+Figure 12: Variation of the SV-wave dispersion factor RSV (equation 57) with the phase angle for different gQ. The
+reference parameters defined at f0 = 40 Hz are ˜VP 0 = 3.0 km/s, g = 0.3, ˜ϵ = 0.2, ˜
+AS0 = 0.0125 (corresponding to
+Q55 = 40), ˜ϵQ = −0.1 and ˜ϵQ = −0.2. The parameters ˜δ and ˜δQ are the same as in Figure 11.
+14
+
+f=15 Hz
+f=40 Hz
+f=109 Hz
+0
+20
+40
+60
+80
+0.0115
+0.0120
+0.0125
+0.0130
+0.0135
+θ (degrees)
+AP
+(a)
+f=15 Hz
+f=40 Hz
+f=109 Hz
+0
+20
+40
+60
+80
+0.0115
+0.0120
+0.0125
+0.0130
+θ (degrees)
+AP
+(b)
+f=15 Hz
+f=40 Hz
+f=109 Hz
+0
+20
+40
+60
+80
+0.0115
+0.0120
+0.0125
+0.0130
+θ (degrees)
+AP
+(c)
+f=15 Hz
+f=40 Hz
+f=109 Hz
+0
+20
+40
+60
+80
+0.0115
+0.0120
+0.0125
+0.0130
+θ (degrees)
+AP
+(d)
+Figure 13: Variation of the P-wave normalized phase attenuation coefficient with the phase angle at different
+frequencies. The medium parameters are the same as in Figure 11d, and (a) gQ = 3; (b) gQ = 2; (c) gQ = 1; (d)
+gQ = 0.5.
+f=15 Hz
+f=40 Hz
+f=109 Hz
+0
+20
+40
+60
+80
+0.009
+0.010
+0.011
+0.012
+θ (degrees)
+ASV
+(a)
+f=15 Hz
+f=40 Hz
+f=109 Hz
+0
+20
+40
+60
+80
+0.0095
+0.0100
+0.0105
+0.0110
+0.0115
+0.0120
+0.0125
+θ (degrees)
+ASV
+(b)
+f=15 Hz
+f=40 Hz
+f=109 Hz
+0
+20
+40
+60
+80
+0.0100
+0.0105
+0.0110
+0.0115
+0.0120
+0.0125
+θ (degrees)
+ASV
+(c)
+f=15 Hz
+f=40 Hz
+f=109 Hz
+0
+20
+40
+60
+80
+0.0110
+0.0115
+0.0120
+0.0125
+0.0130
+θ (degrees)
+ASV
+(d)
+Figure 14: Variation of the SV-wave attenuation coefficient with the phase angle at different frequencies. The
+medium parameters are the same as in Figure 12d, and (a) gQ = 3; (b) gQ = 2; (c) gQ = 1; (d) gQ = 0.5.
+15
+
+gQ=3
+gQ=2
+gQ=1
+gQ= 1
+2
+0
+50
+100
+150
+200
+0.0120
+0.0125
+0.0130
+0.0135
+f (Hz)
+AP
+(a)
+gQ=3
+gQ=2
+gQ=1
+gQ= 1
+2
+0
+50
+100
+150
+200
+0.024
+0.025
+0.026
+0.027
+0.028
+0.029
+f (Hz)
+AP
+(b)
+Figure 15: Variation of the P-wave attenuation coefficient with frequency at θ = 45◦ for different gQ. Except for
+˜
+AP 0, the medium parameters are the same as in Figures 11d and 13. On plot (a), ˜
+AP 0 = 0.0125 (corresponding to
+Q33 = 40); on plot (b), ˜
+AP 0 = 0.025 (corresponding to Q33 = 20).
+gQ=3
+gQ=2
+gQ=1
+gQ= 1
+2
+0
+50
+100
+150
+200
+0.009
+0.010
+0.011
+0.012
+0.013
+0.014
+f (Hz)
+ASV
+(a)
+gQ=3
+gQ=2
+gQ=1
+gQ= 1
+2
+0
+50
+100
+150
+200
+0.018
+0.020
+0.022
+0.024
+f (Hz)
+ASV
+(b)
+Figure 16: Variation of the SV-wave attenuation coefficient with frequency at θ = 45◦ for different gQ. Except for
+˜
+AS0, the medium parameters are the same as in Figures 12d and 14. On plot (a), ˜
+AS0 = 0.0125 (corresponding to
+Q55 = 40); on plot (b), ˜
+AS0 = 0.025 (corresponding to Q55 = 20).
+16
+
+6. Conclusions
+We obtained concise analytic expressions for the Thomsen-type parameters of constant-Q TI media. All Thomsen
+velocity parameters (VP 0, VS0, ϵ, δ and γ) are frequency dependent, with the reference attenuation parameters ˜
+AP 0
+(proportional to 1/Q33) and ˜
+AS0 (proportional to 1/Q55) controlling the dispersion (frequency dependence) of the
+vertical velocities VP 0 and VS0, respectively. The reference attenuation parameters ˜ϵQ, ˜δQ, and ˜γQ govern the
+variations of the anisotropy parameters ϵ, δ, and γ with frequency.
+However, the frequency dependence of all
+Thomsen velocity parameters is weak in a wide frequency range, even for strong attenuation. In viscoacoustic
+constant-Q TI media, the elliptical conditions at the reference frequency ensure that the anellipticity parameter η
+vanishes for all frequencies.
+Despite the fact that all Qij elements in constant-Q TI media are frequency independent, one of the Thomsen-
+type attenuation parameters (δQ) does vary with frequency. The frequency dependence of δQ is controlled by the
+newly defined coefficient ζQ and can be substantial when ζQ has a large magnitude. As a result, the frequency
+variation of the P- and SV-wave attenuation coefficients may be non-negligible at oblique propagation angles with
+the symmetry axis. That variation is highly sensitive to the ratio of the vertical quality factors gQ = Q33/Q55.
+Both attenuation coefficients are insensitive to frequency for gQ = 1, whereas their frequency dependence is most
+substantial for gQ ≥ 3 (for P-waves) and gQ ≤ 0.5 (for SV-waves). In contrast, the SH-wave attenuation coefficient
+in constant-Q TI media is frequency-independent.
+The constant-Q assumption is often made in attenuation analysis because the effective attenuation coefficients
+estimated from seismic data (e.g., using the spectral-ratio method) become linear functions of frequency. However,
+our results show that this linear dependence may not hold for constant-Q TI models, which can cause errors in the
+inversion for the attenuation parameters.
+Appendix A. Appendix A: Complex stiffness coefficients expressed in terms of the Thomsen-type
+parameters
+The stiffness coefficients for the constant-Q dissipative VTI model (equations 1–3) can be found at the reference
+frequency as Mij|f=f0 = ˜
+M R
+ij (1 − i/Qij). Using the parameter definitions in equations 4–12, we express ˜
+M R
+ij and
+Qij in terms of the reference Thomsen-type parameters as follows:
+˜
+M R
+33 = ρ ˜V 2
+P 0,
+(A.1)
+˜
+M R
+55 = ρ ˜V 2
+S0,
+(A.2)
+˜
+M R
+11 = ρ ˜V 2
+P 0(1 + 2˜ϵ),
+(A.3)
+˜
+M R
+66 = ρ ˜V 2
+S0(1 + 2˜γ),
+(A.4)
+˜
+M R
+13 = −ρ ˜V 2
+S0 + ρ
+�
+( ˜V 2
+P 0 − ˜V 2
+S0)
+�
+(1 + 2˜δ) ˜V 2
+P 0 − ˜V 2
+S0
+�
+,
+(A.5)
+Q−1
+33 =
+2 ˜
+AP 0
+1 − ˜
+A2
+P 0
+,
+(A.6)
+Q−1
+55 =
+2 ˜
+AS0
+1 − ˜
+A2
+S0
+,
+(A.7)
+Q−1
+11 = Q−1
+33 (1 + ˜ϵQ),
+(A.8)
+Q−1
+66 = Q−1
+55 (1 + ˜γQ)
+(A.9)
+Q−1
+13 = ˜Q−1
+33
+�
+1 + ˜δQf1 + f2
+�
+− Q−1
+55 f2,
+(A.10)
+17
+
+with
+f1 =
+˜
+M R
+33 ( ˜
+M R
+33 − ˜
+M R
+55)
+2 ˜
+M R
+13( ˜
+M R
+13 + ˜
+M R
+55)
+,
+(A.11)
+f2 =
+˜
+M R
+55 ( ˜
+M R
+13 + ˜
+M R
+33)2
+2 ˜
+M R
+13( ˜
+M R
+13 + ˜
+M R
+55)( ˜
+M R
+33 − ˜
+M R
+55)
+.
+(A.12)
+Appendix B. Appendix B: Explicit expressions for sn
+Here, we provide explicit expressions for the coefficients sn in equation 41.
+The coefficient s0 is given by:
+s0 = g(1 − g + χ)2(h0 + h1g + h2g2 + h3g3 + h4g4 + h5g5)
+(1 − g)3χ3(g − χ)2
+,
+(B.1)
+where
+h0 = −(1 + 2˜δ)2χ,
+(B.2)
+h1 = (1 + 2˜δ)(5 + 10˜δ + 2χ),
+(B.3)
+h2 = (1 + 2˜δ)(2(˜δ − 3)χ − 13˜δ − 14),
+(B.4)
+h3 = ˜δ(7˜δ + 9χ + 30) + 7χ + 15,
+(B.5)
+h4 = −˜δ2 − 2(˜δ + 1)χ − 11˜δ − 8,
+(B.6)
+h5 = 2(1 + 2˜δ).
+(B.7)
+For the coefficient s1 we have:
+s1 = 3g(g − χ − 1)(k0 + k1g + k2g2 + k3g3 + k4g4)
+2(1 − g)χ3(g − χ)2
+,
+(B.8)
+where
+k0 = −2(1 + 2˜δ)2,
+(B.9)
+k1 = 2
+�
+1 + χ + 4˜δ(˜δ + χ + 1)
+�
+,
+(B.10)
+k2 = 2(χ + 1) − ˜δ(χ + 3),
+(B.11)
+k3 = −(˜δ + 2χ + 4),
+(B.12)
+k4 = 2;
+(B.13)
+Finally, the coefficient s2 has the form:
+s2 =
+3g
+�
+3 + 6˜δ + 2χ − 3g(3˜δ + 2χ + 3) + 3g2(˜δ + χ + 3) − 3g3�
+2χ3(g − χ)2
+;
+(B.14)
+s3 = (1 − g)2(4χ − 3g)
+4χ3(g − χ)2
+.
+(B.15)
+The quantities g and χ are defined in equations 27 and 28, respectively.
+18
+
+References
+[1] H. Kolsky, The propagation of stress pulses in viscoelastic solids, Philosophical magazine 1 (8) (1956) 693–710.
+[2] Kjartansson, Constant Q-wave propagation and attenuation, Journal of Geophysical Research 84 (1979) 4737–
+4748.
+[3] Q. Hao, S. Greenhalgh, Nearly constant Q models of the generalized standard linear solid type and the corre-
+sponding wave equations, Geophysics 86 (4) (2021) T239–T260.
+[4] Q. Hao, S. Greenhalgh, Nearly constant Q dissipative models and wave equations for general viscoelastic
+anisotropy, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477 (2251)
+(2021) 20210170.
+[5] J. Behura, I. Tsvankin, Estimation of interval anisotropic attenuation from reflection data, Geophysics 74 (6)
+(2009) A69–A74.
+[6] B. Shekar, I. Tsvankin, Estimation of shear-wave interval attenuation from mode-converted data, Geophysics
+76 (6) (2011) D11–D19.
+[7] B. Shekar, I. Tsvankin, Anisotropic attenuation analysis of crosshole data generated during hydraulic fracturing,
+The Leading Edge 31 (5) (2012) 588–593.
+[8] J. Behura, I. Tsvankin, E. Jenner, A. Calvert, Estimation of interval velocity and attenuation anisotropy from
+reflection data at coronation field, The Leading Edge 31 (5) (2012) 580–587.
+[9] A. I. Best, J. Sothcott, C. McCann, A laboratory study of seismic velocity and attenuation anisotropy in
+near-surface sedimentary rocks, Geophysical Prospecting 55 (5) (2007) 609–625.
+[10] Y. Zhu, I. Tsvankin, P. Dewangan, K. van Wijk, Physical modeling and analysis of p-wave attenuation
+anisotropy in transversely isotropic media, Geophysics 72 (1) (2007) D1–D7.
+[11] A. Zhubayev, M. E. Houben, D. M. Smeulders, A. Barnhoorn, Ultrasonic velocity and attenuation anisotropy
+of shales, Whitby, United Kingdom, Geophysics 81 (1) (2016) D45–D56.
+[12] L. Thomsen, Weak elastic anisotropy, Geophysics 51 (10) (1986) 1954–1996.
+[13] I. Tsvankin, Seismic signatures and analysis of reflection data in anisotropic media, Elsevier Science Ltd., 2001.
+[14] Y. Zhu, I. Tsvankin, Plane-wave propagation in attenuative transversely isotropic media, Geophysics 71 (2)
+(2006) T17–T30.
+[15] Q. Hao, S. Greenhalgh, X. Huang, H. Li, Viscoelastic wave propagation for nearly constant Q transverse
+isotropy, Geophysical Prospecting 70 (7) (2022) 1176–1192.
+[16] J. M. Carcione, Wave fields in real media: Theory and numerical simulation of wave propagation in anisotropic,
+anelastic, porous and electromagnetic media: Handbook of Geophysical exploration (3rd ed.), Elsevier, 2014.
+[17] V. ˇCerven´y, I. Pˇsenc´ık, Perturbation hamiltonians in heterogeneous anisotropic weakly dissipative media,
+Geophysical Journal International 178 (2) (2009) 939–949.
+[18] I. Tsvankin, V. Grechka, Seismology of azimuthally anisotropic media and seismic fracture characterization,
+Society of Exploration Geophysicists, 2011.
+[19] T. Alkhalifah, I. Tsvankin, Velocity analysis for transversely isotropic media, Geophysics 60 (5) (1995) 1550–
+1566.
+19
+
+[20] Q. Hao, T. Alkhalifah, An acoustic eikonal equation for attenuating transversely isotropic media with a vertical
+symmetry axis, Geophysics 82 (1) (2017) C9–C20.
+[21] Q. Hao, T. Alkhalifah, An acoustic eikonal equation for attenuating orthorhombic media, Geophysics 82 (4)
+(2017) WA67–WA81.
+[22] Q. Hao, T. Alkhalifah, Viscoacoustic anisotropic wave equations, Geophysics 84 (6) (2019) C323–C337.
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+(2009) WB177–WB191.
+20
+

ANFJT4oBgHgl3EQfrC3Z/content/tmp_files/2301.11607v1.pdf.txt

@@ -0,0 +1,1627 @@
+Work flux and efficiency at maximum power of a triply squeezed engine
+Manash Jyoti Sarmah and Himangshu Prabal Goswami∗
+Department of Chemistry, Gauhati University, Jalukbari, Guwahati-781014, Assam, India
+(Dated: January 30, 2023)
+We explore the effects of quantum mechanical squeezing on the nonequilibrium thermodynamics
+of a coherent heat engine with squeezed reservoirs coupled to a squeezed cavity. We observe that
+the standard known phenomenon of flux- optimization beyond the classical limit with respect to
+quantum coherence is destroyed in presence of squeezing. Under extreme nonequilibrium conditions,
+the flux is rendered independent of squeezing. The efficiency at maximum power (EMP) obtained
+by optimizing the cavity’s squeezing parameter is greater than what was predicted by Curzon and
+Ahlborn even in the absence of reservoir squeezing. The EMP with respect to the either of reservoirs’
+squeezing parameters is surprisingly equal and linear in ηC with a slope unequal to the universally
+accepted slope, 1/2. The slope is found to be proportional to the dissipation into the cavity mode
+and an intercept equal to a specific numerical value of the engine’s efficiency.
+I.
+INTRODUCTION
+One or more quantum systems that operate between
+two separate reservoirs make up a Quantum Heat En-
+gine (QHE). QHEs have the primary function of convert-
+ing heat into work [1–7]. Apart from traditional thermal
+reservoirs, the use of non-thermal baths, which are con-
+structed reservoirs with correlated characteristics, have
+provided a thorough setting for examining the relation-
+ship between quantum effects and thermodynamic quan-
+tities [8–12].
+Squeezed states or non-canonical initial
+states [13–15] are such non-thermal baths which allow
+additional control over any quantum systems’ dynamics
+garnering tremendous interest off late in the context of
+open quantum systems [11, 15, 16].
+Current technologies permit experimental realization
+of such states [17] and its effects on the thermodynamics
+are experimentally realizable through recently designed
+experimental quantum heat engines (QHE)[18–22]. In-
+tense efforts have been made to interrogate QHEs on
+the role of coherence, correlations or entanglement on
+the underlying dynamics [23–26].
+It has already been
+demonstrated that certain quantum resources can be ex-
+ploited to bend the limits of classical thermodynamics
+[16, 27, 28]. Coherence enhanced power and efficiency
+and optimization of the flux via quantum coherences in
+QHEs are well studied and established phenomena [7, 29–
+32].
+Squeezed thermal baths too have proven crucial,
+especially in the light of a proof-of-concept experiment
+based on a nanobeam heat engine[18]. Efficiency greater
+than that of Carnot has also been predicted [20].
+On the theoretical front, quantum thermodynamic
+analysis of QHEs s are performed by combining principles
+from quantum optics and nonequilibrium statistical me-
+chanics [9, 33–35]. In quantum optics, squeezing [36, 37]
+generally leads to less observation of quantum noise than
+thermal states [38].
+Squeezing alters the entropy flow
+associated with the heat exchanged with the system and
+∗ [email protected]
+introduces an additional term proportional to the second-
+order coherences which determines the asymmetry in the
+second-order moments of the mode quadratures, which
+takes into account both the relative variance shape and
+the relative optical phase space displacements[33]. This
+manifests in an increased efficiency, even surpassing the
+Carnot bound [10, 18, 23, 39, 40]. To account for a re-
+alistic performance of such QHEs, usually a finite time
+assessment is performed by evaluating the efficiency at
+maximum power (EMP), originally introduced in a clas-
+sical context [41]. From a nonequilibrium quantum sta-
+tistical point of view, the near equilibrium EMP is univer-
+sally accepted to be ηC/2 [42], with ηC being the stan-
+dard Carnot efficiency of a classical heat engine.
+Re-
+cently, this robust expression has been showed to be in-
+valid if the engine is locally optimized [43]. The EMP
+has been shown to be modified into several forms as
+one keeps changing or introducing or optimizing addi-
+tional system parameters [39, 44]. One particularly in-
+teresting form of the EMP has been predicted recently
+which holds in the presence of squeezed reservoirs, be-
+ing equal to η2
+m/(ηm − (1 − ηm) ln(1 − ηm)). Here, ηm
+being a squeezing-dependent effective Carnot efficiency
+[11].
+However, the validity of such robust thermody-
+namic expressions remains questionable when engines op-
+erate in presence of both quantum coherences and quan-
+tum squeezing since the general framework on which such
+studies were based didn’t take such effects into account.
+The current work is motivated on this latter aspect.
+In this work, we address how the thermodynamics of
+a QHE coupled to squeezed cavity respond to reservoir
+squeezing in presence of coherences using a quantum mas-
+ter equation technique.
+Such a technique is standard
+and has already been used in nonequilibrium quantum
+transport studies with squeezed reservoirs [45–47]. Un-
+squeezed dynamics of the engine that we cosider has
+also been well studied [7, 31, 48].
+In Sec.(II), we in-
+troduce our triple squeezed QHE model and its dynam-
+ics. In Sec.(III), we explore the effects of squeezing on
+the flux into the cavity mode, which we call the work-
+flux. In Sec.(IV), we evaluate the EMP with respect to
+three squeezing parameters and a system parameter after
+arXiv:2301.11607v1  [quant-ph]  27 Jan 2023
+
+2
+which we conclude.
+II.
+SQUEEZED ENGINE DYNAMICS
+The QHE model consists of four quantum levels cou-
+pled asymmetrically to two squeezed baths with the up-
+per two levels coupled to a squeezed unimodal cavity as
+shown schematically in Fig.(1a). Experimentally, similar
+QHEs have been realized in cold Rb and Cs atoms us-
+ing magneto optical traps [21, 49]. The squeezed density
+matrices of the QHE can be written as[46, 50],
+¯ρℓ = 1
+Zℓ
+exp{−βℓ ˆSℓ ˆHℓ ˆS†
+ℓ},
+(1)
+¯ρν = 1
+Zν
+exp{−βν ˆSν ˆHν ˆS†
+ν}, ν = h, c,
+(2)
+with βz = (kBTz)−1, z = ℓ, h, c being the inverse temper-
+atures of the cavity, hot and cold reservoirs respectively.
+ˆS( ˆSν) is the squeezing operator on the squeezed cavity’s
+ 
+ 
+ 
+ 
+0
+1
+2
+3
+4
+5
+6
+0.1
+0.2
+0.3
+0.4
+0.5
+xc
+Ρ12
+ss
+0
+1
+2
+3
+4
+5
+6
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+xh
+Ρ12
+ss
+0
+1
+2
+3
+4
+5
+6
+0.1
+0.2
+0.3
+0.4
+0.5
+x
+Ρ12
+ss
+a
+b
+d
+c
+Th    
+    
+Tc
+xh
+xc
+x
+(a)
+(b)
+(c)
+(d)
+FIG. 1. (Color online) a) Level scheme of the model quan-
+tum heat engine. A pair of degenerate levels |1⟩ , |2⟩ is reso-
+nantly coupled to two excited levels |a⟩ and |b⟩ by two ther-
+mally populated squeezed field modes with hot (Th) and cold
+(Tc) temperatures.
+Levels |a⟩ and |b⟩ are coupled through
+a squeezed cavity mode of frequency νℓ . Emission of pho-
+tons into this squeezed cavity is the work done by the QHE.
+The engine parameters are fixed through out the manuscript
+at E1 = E2 = 0.1, Eb = 0.4, Ea = 1.5, g = 1, r = 0.7 and
+τ = 0.5 in the unit of kB → 1 and ¯h → 1. b) The solid (dot-
+ted) curves represent the steadystate coherence, ρss
+12 (solved
+by setting the RHS of Eq.(8)=0) as a function of the b) cold
+bath squeezing parameter xc evaluated at different values of
+xh = 0, 0.5, 1, 2, bottom to top with x = 1 (x = 0), c) hot
+squeezing parameter, xh with xc = 0, 0.5, 1, 2, bottom to top
+and x = 1 (x = 0), d) cavity squeezing,x with the solid curves
+(bottom to top) evaluated at xh = 0, xc = 0, 0.5, 1, 2. The
+dotted ones represent xc = 0, xh = 0, 0.5, 1, 2.
+mode (reservoirs’ modes) given by :
+ˆSℓ = e
+1
+2 (xˆa†2
+ℓ −h.c),
+(3)
+ˆSν =
+�
+k
+e
+1
+2 (λ∗
+kνˆa†2
+kν−h.c),
+(4)
+λkν = xkνeiθkν, xkν > 0.
+(5)
+θkν and xkν are the squeezing parameters of the reservoirs
+and x is the squeezing parameter [46, 47, 50, 51]. ˆHℓ =
+ϵℓˆa†
+ℓˆaℓ is the Hamiltonian for the cavity mode and ˆHν =
+�
+k ϵkνˆa†
+kνˆakν is the Hamiltonian for the ν-th reservoir.
+The total Hamiltonian of the four level QHE is ˆHT =
+�
+ν = 1,2,a,b Eν|ν⟩⟨ν|+ ˆHℓ+ ˆHν+ ˆVsb+ ˆVsc, with the system-
+reservoir and system-cavity coupling Hamiltonians given
+by,
+ˆVsb =
+�
+k ∈ h.c
+�
+i = 1,2
+�
+x = a,b
+rikˆak|x⟩⟨i| + h.c
+(6)
+ˆVsc = gˆa†
+ℓ|b⟩⟨a| + h.c.
+(7)
+ϵk, ϵℓ and Eν denote the energy of the kth mode of the
+two thermal reservoirs, the unimodal cavity and system’s
+νth energy level respectively. The system-reservoir cou-
+pling of the ith state with the kth mode of the reservoirs
+is denoted by rik.
+ˆa†(ˆa) are the bosonic creation (an-
+nihilation) operators.
+The radiative decay originating
+from the transition |a⟩ → |b⟩ is the work done by the
+engine.
+Unsqueezed version of such a QHE has been
+thoroughly studied using a Markovian quantum mas-
+ter equation [7, 30, 31, 48, 52]. Following such a stan-
+dard procedure to derive of a quantum master equation
+[46, 48] for the matrix elements of the reduced density
+matrix ρ (supplementary information) has four popula-
+tions, ρii, i = 1, 2, a, b coupled to the real part of a co-
+herence term, ρ12. The coherence ρ12 between states |1⟩
+and |2⟩ arise due to interactions with the hot and the
+cold baths. This thermally induced coherence couples to
+populations due to transition involving the states |1⟩ and
+|2⟩. Under the symmetric coupling regime, we can now
+write down five coupled first order differential equations
+describing the time-evolution of the four populations and
+the coherence (under symmetric coupling, r), given by
+˙ρ12 = −ry
+2 ρ11 − ry
+2 ρ22 + rph ˜Nhρaa + rpc ˜Ncρbb
+− r(n + τ)ρ12
+(8)
+˙ρii = −rnρii + r ˜Nhρaa + ˜Ncρbb − ryρ12, i = 1, 2 (9)
+˙ρbb = rNcρ11 + rNcρ22 + g2 ˜Nℓρaa
+− (g2Nℓ + 2r ˜Nc)ρbb + 2rpcNcρ12
+(10)
+˙ρaa = rNhρ11 + rNhρ22 − (g2 ˜Nℓ + 2r ��Nh)ρaa
++ g2Nℓρbb + 2rphNhρ12
+(11)
+with, �
+i ρii = 1, i = 1, 2, a, b and n = Nc + Nh, y =
+Ncpc + Nhph, with the reorganized occupation factors
+
+3
+ 
+ 
+0.0 0.5 1.0 1.5 2.0 2.5 3.0
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+t
+Ρij
+�a�
+Ρ11,Ρ22�black�
+Ρaa�green�
+Ρbb�blue�
+Ρ12�brown�
+0
+1
+2
+3
+0.10
+0.15
+0.20
+0.25
+0.30
+x
+Ρij
+ss
+x
+0
+1
+2
+3
+1.00
+1.01
+1.02
+1.03
+1.04
+1.05
+1.06
+1.07
+x
+Ρbb
+ss �Ρaa
+ss
+x
+0
+1
+0.96
+0.97
+0.98
+0.99
+1.00
+1.01
+1.02
+1.03
+ph
+j�jo
+ph
+�d�
+(b)
+(c)
+(a)
+(b)
+(d)
+(c)
+FIG. 2.
+a) The solid (dotted) curves represent time evo-
+lution of ρij with, x = 2 (without, x = 0) squeezing ob-
+tained by solving Eq.(8-11) for Th = 2, Tc = 0.5, Tl = 0.9.
+b) Steady state values as a function of the squeezing pa-
+rameters for the same parameters as (a) c) Ratio of the
+steady state values between states |b⟩ and |a⟩ reaching unity
+highlighting the equipopulated nature under high squeezing;
+pc = 0.2, 0.3, 0.5, 0.7, 0.8 from the top to the bottom curves.
+(d) Optimization of the flux ratio as a function of hot coher-
+ence parameter, ph for different squeezing parameters under
+far from equilibrium conditions and pc = 1 (top to bottom:
+x = 0, π/6, π/π/2, 2π/3, 5π/6, π, 3π/2). Other parameters are
+same as Fig.(1a).
+given by
+Nz = cosh(2xz)(nz + 1
+2) − 1
+2, z = h, c,
+(12)
+Nℓ = cosh(2x)(nℓ + 1
+2) − 1
+2.
+(13)
+Here, nc, nh andnl are the Bose-Einstein distributions
+for the cold reservoir, hot reservoir and the cavity re-
+spectively. These factors are now squeezing dependent
+via the dimensionless parameters, xh, xc and x represent-
+ing the extent of squeezing in the hot, cold reservoirs
+and the cavity respectively. pν = | cos φν|, ν = h, c are
+two dimensionless parameters that governs the strength
+of coherences and whose values are dictated by the an-
+gles of relative orientation (φν) of the ν−th bath induced
+transition in the system [7, 48, 52]. A phenomenological
+dimensionless rate τ has been added to take care of the
+dephasing. Setting ˙ρ = 0, at the steady state, we can
+solve for the steady state values of ρaa, ρbb, ρ11, ρ22, and
+ρ12 and obtain these analytically (supplementary text).
+The steadystate value of the coherence term ρss
+12 as a
+function of the squeezing parameters, xh, xc and x are
+shown in Fig.(1b,c,d)) for different engine parameters.
+The different curves in Fig.(1b) represent ρss
+12 evaluated
+for different xh and x values as a function of xc. The
+solid (dotted) lines represent ρss
+12 when xh ̸= 0(xh = 0)
+and x = 0(x ̸= 0). At high xc values, the coherence is re-
+duced and saturates to a lower value in comparison to ρss
+12
+values of lower xc. At high xh values (black curve), ρss
+12
+steadily increases and reaches a maximum value around
+some intermediate xc value and then sharply drops as
+0
+1
+0.90
+0.92
+0.94
+0.96
+0.98
+1.00
+j� jo
+ph
+0
+1
+1
+ph
+j� jo
+ph
+FIG. 3.
+Failure of coherence to optimize the flux beyond
+classical values (j/j0 > 1) under high squeezing as given by
+Eq.(17). Inset: Linear dependence of the flux ratio on p − j
+under high squeezing (x ≫ 0) and Tl ≫ 0 given by Eq.(18)
+evaluated at pc = 1, Tc = 0.5, Th = 1.
+The square boxes
+represent linear fit.
+xc keeps increasing. This behavior is however absent for
+lower xh values. Fig.(1c) represent ρss
+12 evaluated for dif-
+ferent xc and x values as a function of xh.
+The solid
+(dotted) lines represent ρss
+12 when xc ̸= 0(xc = 0) and
+x = 0(x ̸= 0). At high xh values, the steady state values
+of the coherence term increases and saturates to a higher
+value in comparison to coherence at lower xh values. We
+can rationalize that, xc(xh) tend to reduce (increase) the
+steadystate values of the coherences as we keep squeez-
+ing the baths more and more. The same however cannot
+be said for ρss
+12 vs x as seen from Fig. (1d). The solid
+(dotted) lines represent the behavior at xh = 0(xh ̸= 0)
+for finite xc values.
+The time evolution of each of the equations (Eq.(8-11))
+for various engine parameters for xh = xc = 0 and x = 2
+is shown in Fig.(2a). In Fig.(2b), the steadystate values
+of the populations as a function of x is shown where solid
+(dotted) curves represent cavity-squeezed, x ̸= 0 (cavity-
+unsqueezed, x = 0) evolutions.
+Note that under high
+squeezing of the cavity mode, the steady state values,
+ρss
+aa and ρss
+bb equipopulate giving,
+lim
+x→∞
+ρss
+bb
+ρss
+aa
+= 1
+(14)
+and is shown numerically in Fig.(2c) for different values
+of the hot coherence parameter, ph. The analytical ex-
+pressions for the steadystate values are provided in the
+supplementary information.
+III.
+WORK FLUX
+We interprete the emission of photons into the
+squeezed cavity as the work done by the engine. This
+photon exchange process between the levels |a⟩, |b⟩ with
+the squeezed cavity is quantified by the rate of photon
+exchange with the cavity which we refer to as the work
+flux, j =
+d
+dt⟨a†
+ℓaℓ⟩, where the trace is with respect to
+
+4
+0
+1
+0.6
+0.7
+0.8
+0.9
+1.0
+ph
+j�jo
+ph
+�a�
+0.7
+0.8
+0.9
+1
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+pc
+ph
+�
+Tc�Colour
+0.1�Pink
+0.2�Blue
+0.3�Red
+0.4�Orange
+0.5�Blue
+1.0�Black
+pc
+�b�
+0
+1
+2
+3
+0.6
+1
+1.5
+x
+ph
+�
+x
+�c�
+0
+1
+2
+3
+0.2
+0.4
+0.6
+0.8
+1.0
+x
+Ζ�Ζx�0
+Tl�10
+Tl�0.5
+Tl�1
+Tl�2.5
+x
+�d�
+FIG. 4.
+a) Loss of optimization of flux as a function
+of ph for different squeezing parameters, near equilibrium
+(Tc = 0.9, Th = 1, Tl = 10). (b) Loss of linear dependence of
+pc on the optimal value p∗
+h as given by Eq.(21). The topmost
+curve represents Eq.(22). (c) Plot showing breakdown of the
+coherent optimization of the flux as a function of squeezing
+parameter. The shaded region is not allowed since the maxi-
+mum possible value of p∗
+h is unity. Under far from equilibrium
+condition p∗
+h exists which saturates (bottom curve) at higher
+values of x given by Eq.(21). The top curve shows the behav-
+ior of p∗
+h near equilibrium which is nonexistent after a certain
+squeezing value. (d) (d) Lowering of thermodynamic affinity
+as a function of squeezing evaluated at Tc = 0.1, Th = 2 and
+xc = xh = 0.
+the squeezed cavity density matrix.
+Following a stan-
+dard procedure to second order in the coupling as devel-
+oped in[30, 48] we get, j = g2( ˜Nℓρss
+aa − Nℓρss
+bb). We can
+substitute the values of the steadystate populations to
+obtain an analytical expression for the flux (supplemen-
+tary information). When, the hot and the cold coherence
+parameters individually go to zero (pc = ph = 0), the
+coherence vanishes (ρss
+12=0) and we obtain a coherence -
+unaffected value of the flux, which we denote as jo. Note
+that, jo depends on the squeezing parameters x, xh and
+xc. In the absence of squeezing (xh = xc = x = 0), jo
+shall be denoted by j0
+o, which we refer to as the classical
+value of the flux. There are no effects of coherence or
+squeezing on j0
+o. It is a well known phenomena that, in
+absence of squeezing, j > jo can be achieved as a func-
+tion of coherence parameter, ph [7, 29]. We plot the ratio
+j/jo in shown in Fig.(2d) for different squeezing values
+of the cavity for xc = xh = 0. As the cavity squeezing
+parameter is increased the optimal value of the flux grad-
+ually decreases and the ph value that optimizes the ratio
+(denoted as p∗
+h) shifts towards larger ph values. We now
+attempt to explore the dependence of the flux in presence
+of squeezing on the coherences in detail. Since the ana-
+lytical expressions of j and j0
+o are too lengthy we focus
+on some limiting cases.
+Under high cavity squeezing, (x → ∞), we obtain
+ρaa
+aa = ρss
+bb as seen from Eq.(14). The expression for the
+flux in this case is simply given by,
+lim
+x→∞ j = g2( lim
+x→∞ ρss
+aa),
+(15)
+which under the condition pc = 0, ph = 0 in Eq.(15) is,
+lim
+x→∞ jo = r(Nh − Nc)
+2(n + 1)
+.
+(16)
+Eq. (15), with pc = 1 can be expressed as,
+lim
+x→∞ j|pc=1 = r(Nh − Nc)
+�
+Nh
+�
+1 − p2
+h
+�
++ t
+�
+(1 − ph)fn + 2τ(n + 1)
+(17)
+with fn = 4NcNh + n(2Nh(ph + 1) + ph + 2). The RHS
+of Eq.(16) is always greater than RHS of Eq.(17) as seen
+from the numerical result in Fig (3). The physical in-
+terpretation is that the coherences are no longer able to
+increase the flux beyond the non coherence values. Un-
+der this condition, the ratio is bounded below unity as
+seen in Fig.(3). We can analytically prove this by invok-
+ing a few conditions. In Eq.(16) and (17), if τ = 0 and
+Nh = zNc, z being a positive integer), the ratio between
+the two fluxes becomes,
+lim
+x→∞ j|pc=1
+lim
+x→∞ jo
+����
+Nh=zNc
+=
+2z(ph+1)(Ncz+ Nc+ 1)
+2Nc(ph+1)z2+z(4Nc+ ph+2)+1
+(18)
+which is a rational fraction of two linear terms of ph.
+Eq.(18) can be shown to have a linear dependence on ph
+for some appropriate conditions of the coefficients which
+is graphically shown as an inset in Fig.(3). In Eq.(18),
+for z = 1 and Nh = Nc (no bias), we see a flux value that
+solely depends on only the coherence value, given by
+lim
+x→∞
+j
+jo
+|Nh=Nc = 2(1 + ph)
+(3 + ph)
+(19)
+≤ 1
+(20)
+and is linear in ph for small values as seen in the inset
+of Fig.(3) and in Fig.(4a).
+In Fig.(4a), the flux ratio
+j/jo is plotted for different squeezing parameters. The
+squeezing decreases from top to bottom. For smaller ph,
+the linearity is prominent, but for higher ph values, the
+linearity is gradually less apparent as the squeezing pa-
+rameter increases.
+It has been previously reported that p∗
+h increases lin-
+early in pc under the unsqueezed case [30]. In the current
+case, we observe that under an extremely biased scenario
+(Nh ≫ 0) and high squeezing, x ≫ 0, the linear depen-
+dence is lost as shown graphically in Fig.(4b) and the
+dependence of p∗
+h on the cold coherence parameter, pc is
+given by the nonlinear function,
+p∗
+h| =
+�
+(1−p2c) (4N 2c (1−p2c)+4Nc+1)+2Nc
+�
+p2
+c+1
+�
++1
+4Ncpc + pc
+(21)
+which reduces to unity when pc = 1 as seen in the Fig.
+(4b). The nonlinear dependence takes a simplistic form
+when Tc → 0, where the above expression reduces to,
+p∗
+h|Tc=0 = 1 −
+�
+1 − p2c
+pc
+(22)
+
+5
+which is shown as the topmost curve in Fig.(4b). The
+RHS of Eq.(21) also has a strange dependence on the cav-
+ity squeezing parameter. p∗
+h increases as a function of x
+and saturates at higher x values as shown in the bottom-
+most curve of Fig.(4c). However under extremely biased
+conditions, p∗
+h sharply rises beyond unity and goes to the
+shaded region. The shaded region is not allowed as the
+maximum value of p∗
+h is unity. Since an analytical expres-
+sion of p∗
+h as a function of x is beyond the scope of sim-
+plistic analysis, the exact identification of this numerical
+fallout range is not possible. We simply speculate that
+such a breakdown happens when the cavity temperature
+Tℓ is set to be very high. Since nℓ is a function of Tℓ, the
+numerics blows when there is competition between x and
+Tℓ to dominate the behavior. The upper dashed curve in
+the shaded portion also corresponds to an unrealistic p∗
+h
+evaluated at a high cavity temperature. In Fig.(4d), we
+plot the thermodynamic force as a function of squeezing.
+The force can be identified from the analytical expression
+of the flux (supplementary text) and is given by,
+ζ =
+˜Nc ˜
+NℓNh
+Nc ˜NhNℓ
+.
+(23)
+When ζ > (<)1, j > (<)1. In Fig.(4d), we plot the ratio
+between the thermodynamic forces in presence and ab-
+sence of squeezing for different cavity temperatures. As
+squeezing increases, the ratio decreases for a fixed set
+of engine parameters and then saturates. This leads to
+lower magnitude of the flux in comparison to the un-
+squeezed case and is more prominent when the cavity
+temperature is low.
+In Fig.(5a,b and c), we plot the ratio between the total
+flux j and the classical flux j0
+o as a function of xc, xh and
+x respectively for the same parameters as Fig.(2).
+As
+a function of both the baths’ squeezing parameters, the
+increase of the total flux is quite large in comparison to
+the classical case. All of the curves show saturation be-
+havior. Particularly interesting is the ratio’s dependence
+on xh where the saturation value of the ratio is always
+greater than unity.
+We now focus on an extreme biased case (Th ≫ Tc, a
+limit which we invoke by taking Th → ∞ and Tc → 0),
+a scenario when the temperature gradient is very high.
+This case is different from a standard extreme nonequilib-
+rium case where the thermodynamic force must be very
+high (ζ ≫ 0). Under the high temperature gradient sce-
+nario, the steadystate populations of the upper two states
+are given by,
+lim
+Th≫Tc ρss
+aa =
+�
+p2
+h + 1
+� �
+g2Nℓ + 2r
+�
+g2 (4Nℓ + p2
+h + 1) − 2 (p2
+h − 3) r
+(24)
+lim
+Th≫Tc ρss
+bb =
+g2 ˜Nℓ
+�
+p2
+h + 1
+�
+g2 (4Nℓ + p2
+h + 1) − 2 (p2
+h − 3) r,
+(25)
+which no longer depends on the squeezing parameters of
+the two baths. Using these above values the flux can be
+(a)
+(b)
+(d)
+(c)
+FIG. 5. Giant increase of the total flux (in presence of squeez-
+ing as well as coherence) in comparison to the classical case.
+The solid (dotted) lines represent the ratio between the to-
+tal flux j and the classical flux j0
+o as a function of a) xc
+evaluated at x = 0(1), xh = 0, 0.5, 1, 2, b) xh evaluated at
+x = 0(1), xc = 0, 0.5, 1, 2. c) Solid (dotted) curves indicate the
+total flux ratio as a function of cavity squeezing x evaluated
+at Tc = 0.5(0.1) with {xh, xc} = {0.5, 0.1}, {0.1, 0.5}, {0, 0}
+(top to bottom). (d) Change in the sign of the thermody-
+namic affinity, A = log ζ as function of cavity squeezing pa-
+rameter evaluated at{xh, xc} = {1, 0.1} (upper curve) and
+{0.1, 1} (lower curve). The sign change happens at x∗ given
+by Eq.(32).
+recast as,
+lim
+Th≫Tc j =
+2g2r ˜Nℓ(1 + p2
+h)
+g2(1 + 4Nl + p2
+h) − 2r(p2
+h − 3)
+(26)
+while the coherence-unaffected value of the flux is simply,
+lim
+Th≫Tc jo =
+2g2 ˜Nℓr
+g2(1 + 4Nℓ) + 6r
+(27)
+It is interesting to note that, in this highly biased sce-
+nario, the flux expression (RHS of Eq.(26)) doesn’t de-
+pend on the cold coherence parameter any more. In the
+above two expressions, if we invoke the high squeezing
+scenario (x → ∞), we can write down the ratio between
+the two fluxes as,
+lim
+x→∞
+lim
+Th≫Tc j
+lim
+Th≫Tc jo
+= (1 + p2
+h)
+(28)
+Note that, the above expression is bound, 1 ≤ 1 + p2
+h ≤
+2.
+In this limit with ph = 1(pc ̸= 1), coherences can
+double the value of the flux from its zero coherence value.
+Likewise, the ratio between the flux in this limit and the
+classical value of the flux can be written as,
+lim
+x→∞
+lim
+Th≫Tc j
+lim
+Th≫Tc j0
+o
+= (1 + p2
+h)(1 + 6r − 3g2
+4g2˜nℓ
+)
+(29)
+≥ 1.
+(30)
+As long as r > g2/2 and pc ̸= ph, within the high bias
+scenario and maximal cavity-squeezing, the flux is always
+greater than unity in comparison to the classical case.
+
+20
+15
+15
+10
+10
+5
+5
+0
+0
+0
+1
+2
+3
+4
+5
+6
+0
+1
+2
+3
+4
+5
+6
+Xc
+Xh
+3.0
+1.5
+2.5
+1.0
+0, 2.0
+0.5
+A
+0.0
+1.0
+0.5
+0.5
+0
+1
+2
+3
+4
+5
+6
+0.0 0.5 1.0 1.5 2.0 2.5 3.0
+X
+X6
+0
+0.7 1.2
+2
+2.5
+3
+1.00
+1.05
+1.10
+1.15
+1.20
+1.25
+1.30
+x
+W�Wo
+Tl�0.5
+Tl�0.4
+Tl�1.0
+Tl�10
+x
+�a�
+0
+0.7 1.2
+2
+2.5
+3
+�1
+0
+1
+2
+3
+x
+W�Wo
+Tc�0.1
+Tc�2.0
+Tc�0.4
+Tc�0.7
+x
+�b�
+0
+0.2
+0.7
+1
+0.990
+0.995
+1.000
+1.005
+ph
+ΗEa
+� �Ηo
+�
+x�0.5
+x�0
+x�1
+x�2Π
+�c�
+ph
+0
+1
+2
+3
+1.0
+1.1
+1.2
+1.3
+1.4
+1.5
+x
+ΗEa
+� �Ηo
+�
+Tl�0.9
+Tl�1.0
+Tl�1.2
+Tl�1.7
+x
+�d�
+FIG. 6.
+(a) Squeezing induced increase of the work done
+beyond classical limits (Tc = 0.1, Th = 1). The increase is
+larger when the cavity temperature is lower.
+(b) Negative
+work done as a function of squeezing for different Tc( Th =
+2, Tl = 1). (c) EMP with respect to Ea as a function of ph
+for different squeezing values. (d) EMP with respect to Ea
+for the range of squeezing at different cavity temperatures
+(pc = 0.1, ph = 1).
+IV.
+EFFICIENCY AT MAXIMUM POWER
+We now move to perform a thorough analysis on the
+efficiency at maximum power (EMP or η∗). In a stan-
+dard context, the EMP is calculated by maximizing the
+efficiency with respect to a system parameter.
+In our
+QHE model, the efficiency is defined as η = W/Qh with
+Qh = (Ea −E1), and the useful work done (W) is defined
+as,
+W = Ea − Eb − WdissTc,
+(31)
+with Wdiss = kBln
+˜
+Nℓ
+Nℓ is the dissipation into the cavity
+mode [30, 48]. W doesn’t depend on the squeezing pa-
+rameters of the two squeezed reservoirs or the noise in-
+duced coherences. In Fig. (6a), we show the variation of
+W/Wo (Wo being the useful work in absence of squeez-
+ing, x = 0) as a function of x for several values of the
+cavity temperature, Tl. As can be seen, the work done
+increases as Tl is lowered and saturates at higher values
+of x and is always greater than unity as long as Tc > Tℓ.
+When Tc < Tℓ (Fig.(6)b), the work done is negative. In
+general, the work changes its sign at x = x∗, given by
+x∗ = 1
+2ℜ
+�
+cosh−1
+�
+˜NcNh + Nc ˜Nh
+(2nℓ + 1)(Nc − Nh)
+��
+.
+(32)
+Although W and η are independent of coherences and
+the reservoir squeezing parameters, the EMP however de-
+pends on these parameters. The EMP obtained by max-
+imizing P with respect to any system parameter puts an
+implicit dependence via the optimized value of the chosen
+parameter. We choose the three squeezing parameters
+xc, xh, x and Ea to optimize the EMP and denote these
+by η∗
+xc, η∗
+xh, η∗
+x and η∗
+Ea respectively. The squeezing unaf-
+fected values of the EMP are denoted by η∗
+o. In Fig.(6c),
+ 
+ 
+0
+0.2
+0.8
+1
+0.990
+0.995
+1.000
+1.005
+1.010
+ph
+Ηx
+��Ηo
+�
+�a�
+0.6
+0.8
+0.40
+0.45
+0.50
+0.55
+0.60
+Ηx
+�
+�b� r � g
+Η�2
+��2��
+ΗC
+Ηc
+0.6
+0.75
+0.9
+0.40
+0.45
+0.50
+0.55
+0.60
+Ηx
+�
+�c� r � g
+��2��
+ΗC
+Ηc
+0.6
+0.7
+0.8
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65
+0.70
+ΗEa
+�
+�d�
+ΗC
+��2��
+(c)
+(a)
+(b)
+(b)
+(d)
+FIG. 7. (Color online)(a) EMP with respect to squeezing as a
+function of ph fr various pc. In a), b) and c), the black curves
+(overlayed with red color) represent the evaluated EMP of
+our QHE. The green dashed curve is the upper bound on
+the EMP, η∗∗. The brown dashed line represents ηCA. The
+dotted line represent ηL. (b) and (c) EMP with respect to x
+as a function of ηC with r = 0.7, g = 1) and r = 0.1, g = 3
+respectively. When r ≈ g, η∗
+x > ηCA as seen in (b). (d) EMP
+with respect to Ea as a function of ηC with r = 0.7, g = 1).
+Here, η∗
+Ea > ηCA with x = 1(xc = xh = 0).
+we show the dependence of the ratio η∗
+Ea/η∗
+o as a function
+of ph for several x-values evaluated at xc = xh = 0 and
+pc = 0.9. The dependence of this ratio on ph is extremely
+nonlinear and is unity at ph = 0.8 where effects of coher-
+ence vanish. At lower (higher) squeezing values, the ratio
+decreases (increases) to unity and then sharply increases
+beyond unity as a function of ph. We can theorize that,
+lower ph values (under the condition ph < pc), smaller
+values of cavity squeezing favor increasing the EMP be-
+yond classical values while for larger ph (ph > pc), high
+squeezing favor increase of the EMP beyond classical val-
+ues. In Fig.(6d), we plot the same ratio as a function of
+cavity squeezing parameter for different cavity tempera-
+tures, Tl. There is an optimization of the EMP at lower
+values of x and the hump keeps shifting leftward to even
+smaller values as Tl is increased and the EMP ratio keeps
+decreasing. From Fig.(6d), we can conclude that lower
+values of Tℓ yield very high values of EMP with respect
+to Ea under moderate squeezing conditions of the cavity.
+In Fig. (7a), we plot η∗
+x as a function of ph for differ-
+ent combinations of xc and xh for a fixed pc value (0.5).
+Here, for a fixed set of engine parameters, when xc < xh
+leads to a larger optimized value (around ph = 0.5) of the
+EMP with respect to x (blue curve in the figure). How-
+ever as ph approaches unity, there is a sharper fall in the
+EMP and goes below unity. For the case when xc = xh,
+the behavior is similar (dotted curve) but the increase is
+not as high as the previous case. When squeezed to the
+limits, xc → ∞, xh → ∞, the EMP with respect to x no
+longer depends on the coherence (dashed curve). This
+is due to the fact that, under this scenario, the power
+cannot be optimized with respect to x and the maximum
+value occurs at x = 0.
+In general, the EMP has a universally accepted for-
+mula, the Curzon-Ahlborn EMP, ηCA = 1 − √1 − ηC
+
+7
+ 
+ 
+0
+0.3
+0.6 0.8
+1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Ηc
+Ηxc
+�
+Ηc
+0
+0.3
+0.6 0.8
+1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Ηc
+Ηxh
+�
+Ηc
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.55
+0.60
+0.65
+0.70
+0.75
+Tc
+Η,Η�
+0.0
+0.2
+0.4
+0.6
+0.8
+0.65
+0.70
+0.75
+0.80
+0.85
+Ηc
+ΗEa
+�
+(a)
+(b)
+(d)
+(c)
+FIG. 8.
+Linear dependence of η∗
+xc(a) and η∗
+xh(b) as a func-
+tion of ηC, governed by Eq.(33) evaluated at x = ∞, 1 and 0
+(top to bottom ). Note that η∗
+xh = η∗
+xc with the upper (mid-
+dle) curves having a slope of m = 0.02(0.19) and intercept
+of c = 0.76(0.59). c) Solid line represents the EMP, given by
+Eq.(34) while the dotted line is simply the normal efficiency,
+η = W/Qh. d) Appearance of a quadratic term and an in-
+tercept for η∗
+Ea as a function of ηC, evaluated at x = 1.5(∞)
+denoted by lower (upper) curves. The fit parameters for the
+upper (lower) curves are a1 = 0.85(0.76), a2 = 1.7(0.75), a3 =
+0.05(−0.28), a4 = 0.07(−0.28).
+[41, 53] and is represented by the dashed curves in
+Fig.(7b,c and d). As a function of ηC, the EMP is bound
+between ηC/2 ≤ η∗ ≤ η∗∗, where the upper bound is
+η∗∗ =
+ηC
+2−ηC [54]. In Fig.(7b,c and d), we show the be-
+havior of our engine’s EMP as a function of the Carnot
+efficiency, ηC.
+The solid (topmost green) curve repre-
+sent the upper bound η∗∗. The EMP of the QHE op-
+timized with respect to x for xc = xh = 0 is repre-
+sented by the solid line highlighted with red dots.
+In
+Fig.(7b,c), η∗
+x ≥ (<)ηCA is observed under the condition
+r ≥ (<)g.
+Values of EMP larger than ηCA has been
+previously reported with squeezed reservoirs [17, 20]. In
+our case, one can have EMP more than the predicted
+ηCA just by squeezing the cavity even in the absence
+of squeezed reservoirs. In Fig.(7d), for nonzero values of
+cavity-squeezing, η∗
+Ea > ηCA is shown (solid black curve).
+This result is valid irrespective of r and g values. The
+upper bound is always obeyed in presence of squeezing
+as evident from Fig.(8b,c and d). The EMP of the QHE
+is always lower than the upper dashed curve (η∗∗). Note
+that the universal slope of 1/2 (any EMP = ηC/2 near
+equilibrium)[42] is maintained in all the curves for smaller
+values of ηC when maximized with respect to x.
+We now move to discuss a rather interesting finding
+observed when the EMP is maximized with respect to
+a reservoir squeezing parameter. As can be seen from
+Fig.(8a and b), both η∗
+xh and η∗
+xc are found to be linear
+in ηC with a slope which is not equal to the universally
+predicted value of 1/2[53]. By a linear curve fitting tech-
+nique, we infer that the EMP with respect to xc or xh is
+dictated by the equation,
+η∗
+xh = η∗
+xc = mηC + c.
+(33)
+Our numerical results reveal that the slope, m is equal
+to the numerical value of Wdiss/Qh and the intercept, c
+being given by the numerical value of the quantity, (Eab−
+Wdiss)/Qh. This intercept is interestingly the efficiency
+of the engine albeit with Tc = 1. Note that, η∗
+xc = η∗
+xh
+and is shown as two identical plots in Fig.(8a,b). In these
+two figures. The numerical plots reveal that the m ̸=
+1/2. Such a breakdown of the universality of the linear
+coefficient has also been observed in presence of geometric
+phaselike effects [52, 55].
+Since Wdiss > 1, the EMP
+increases as x is increased (for fixed Tℓ) to a maximum
+value of Eab/Qh at ηC = 1. The efficiency of the QHE,
+η = W/Qh is always less than η∗
+ν and is shown as a
+function of Tc in Fig.(8c).
+This linear dependence doesn’t exist for η∗
+Ea for finite
+x as seen from the numerical results in Fig.(8d) for x = 1
+and x → ∞. It has been previously reported that such
+a nonlinear dependence of the EMP on the squeezing
+parameter x takes the form η∗
+∗ = 1 −
+�
+sech(2x)√1 − ηC
+[56]. We assess the validity if this expression by defining
+two curve fitting equations,
+η∗
+Ea ≈ a1 −
+�
+sech(a2x)√a3 − a4ηC
+(34)
+≈ a5ηC + a6η2
+C + c
+(35)
+that can best represent the EMP with respect to the sys-
+tem parameter Ea. Here, ai-s are fit parameters. We
+observe that a1 ̸= a3 ̸= a4 ̸= 1 and a3 ̸= 2 result-
+ing in η∗
+Ea ̸= η∗
+∗ and is shown in Fig.(8d). Further, in
+Eq.(35), a5 ̸= 1/2 and a6 ̸= 1/8. In this engine, it is al-
+ready known that the quadratic coefficient is not 1/8 [30].
+Both the above equations are good fits (solid curves) on
+the numerically evaluated η∗
+Ea (dots) as function of ηC
+as seen in Fig.(8d). It is interesting to note that the in-
+tercept of η∗
+Ea as a function of ηC in Eq.(35) is the same
+numerical value of the engine’s efficiency of the engine,
+η = W/Qh similar to what was observed in Eq.(33). This
+lets us rationalize that Eq.(35) is a better representation
+of η∗
+Ea vs ηC than Eq.(34). At ηC = 1, η∗
+Ea again reaches
+a maximum value of Eab/Qh. For x = 0, m = 1/2 is
+recovered. Further for x = 0, the intercept in Eq.(34)
+also vanishes by mixing with the quadratic term. Since
+we cannot derive analytical expressions for these coeffi-
+cients, we demonstrated it this numerically shown as the
+bottom-most dotted line in Fig.(7d)).
+The EMP also has other interesting logarithmic
+expressions[43, 57, 58], one particularly claimed to be
+valid for squeezed states[11], η∗
+L = η2
+m/{1−(1−ηm) ln(1−
+ηm)}. ηm is a modified Carnot efficiency given by ηm =
+1−Tc/T m
+h . T m
+h is a modified but fictitious reservoir tem-
+perature and is directly proportional to the energy of the
+squeezed mode and inversely proportional to the logarith-
+mic ratio of the squeezed mode’s occupation factor. By
+an analogy with this previous work [11], we can express
+the modified temperature in our QHE to be,
+T m
+h = Ea − E1
+ln 1+Nh
+Nh
+.
+(36)
+
+8
+0
+0.3
+0.6
+0.8
+1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Ηc
+ΗEa
+�
+Ηc
+FIG. 9.
+Disagreement between the QHE’s EMP optimized
+with respect to Ea and the predicted EMP, η∗
+L for the same
+parameters. η∗
+L is evaluated using the definition in Eq.(36).
+The dotted (dashed) curves represent η∗
+Ea(η∗
+L). Parameters
+used are Th = 3, xc = xh = 0.1, x = 0.6 (top dotted), Th =
+4, xc = xh = 0.2, x = 0.5 (middle dotted) and Th = 6, xc =
+xh = 0.2, x = 2π.
+We numerically evaluate ηE∗
+a for different squeezing pa-
+rameters and Th values and plot it in Fig.(9) along side
+the corresponding η∗
+L values. As can be seen, η∗
+Ea ̸= η∗
+L.
+Further since η∗
+xh and η∗
+xc is found to be linear in ηC, these
+anyway don’t agree with the predicted value η∗
+L.
+Un-
+der extremely low squeezing conditions of the hot bath,
+ηm
+C → ηC in the expression for η∗
+L. Under this condition,
+η∗
+L has been high lighted as dotted curves in Fig.(7b,c
+and d) and is seen to be unequal to η∗
+x.
+V.
+CONCLUSION
+By deriving a coherence-population coupled quantum
+master equation, we carried out a comprehensive study
+of the thermodynamics of quantum heat engine coupled
+to two squeezed reservoirs and a squeezed unimodal cav-
+ity.
+We showed that the steadystate value of the co-
+herence term of the density matrix vanishes (saturates)
+under maximal squeezing of the cold (hot) bath. Under
+high squeezing conditions of the cavity, the two upper
+states of the engine equipopulate. We showed that under
+high squeezing of the cavity, the quantum coherence can
+no longer optimize the flux beyond the classical values.
+We also showed how the flux can be linearized with re-
+spect to coherences under high squeezing conditions and
+equal Bose-Einstein distributions for the hot and cold
+baths. We also showed that larger EMP favors lower val-
+ues of cavity temperatures and lower values of squeezing.
+The EMP can be increased beyond the Curzon-Ahlborn
+limit by squeezing the cavity alone even if the baths are
+unsqueezed.
+We also show a linear dependence of the
+EMP with respect to the reservoirs’ squeezing parameters
+which we identify analytically with a slope proportional
+to the dissipation into the cavity mode. The EMP with
+respect to a system parameter, Ea doesn’t obey the uni-
+versal slope of 1/2 for finite squeezing and is not equal to
+a recently proposed general form of the EMP in presence
+of squeezed reservoirs [11].
+ACKNOWLEDGMENTS
+MJS and HPG acknowledge the support from Science
+and Engineering Board, India for the start-up grant,
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+

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AtAzT4oBgHgl3EQfTPzA/content/tmp_files/2301.01247v1.pdf.txt

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+Rate Adaptive Autoencoder-based Geometric Constellation 
+Shaping 
+ 
+Ognjen Jovanovic, Metodi P. Yankov, Francesco Da Ros and Darko Zibar 
+Department of Electrical and Photonics Engineering, Technical University of Denmark, Kgs. Lyngby, 2800, Denmark 
+[email protected] 
+ 
+Abstract: An autoencoder is used to optimize bit-to-symbol mappings for geometric constellation 
+shaping. The mappings allow for net rate adaptivity without additional hardware complexity, while 
+achieving up to 300km of transmission distance compared to uniform QAM.  © 2023 The Author(s) 
+ 
+1. Introduction 
+State of the art coherent optical communications need to be deployed in dynamic network scenarios, which require 
+a certain degree of adaptivity to varying channel conditions [1]. Classically, this is handled by varying the modulation 
+format size, which 1) often produces a coarse granularity in the rate with steps of 2 bits/symbol; and 2) requires the 
+transceiver to support bit to symbol mapping and demapping to/from constellations of different size, increasing the 
+complexity. The probabilistic amplitude shaping (PAS) scheme [1] has emerged as an efficient architecture that 
+provides a solution to the first problem, at the cost of rate matcher and dematcher, which increases complexity. 
+Autoencoders (AEs) are becoming a popular tool to optimize the signaling constellation of digital communication 
+transceivers [2, 3]. In optical communications, AEs have been employed for geometric constellation shaping (GCS) 
+[4], bit labeling [5], mostly with the target of mitigating the impact of fiber nonlinearities, as well as transceiver 
+impairment mitigation [6, 7]. GCS is beneficial over PAS because it does not require explicit matcher and dematcher 
+blocks. However, rate adaptivity with GCS typically requires a rate-flexible forward error correction (FEC), which 
+may increase the complexity of the digital logic. Further, GCS does not allow for straight-forward Gray labeling to be 
+performed, which may lead to sub-optimality when combined with conventional bit-metric decoders as in standard 
+coherent optical communications [8]. 
+In this paper, an AE is used to 1) find optimal bit mappings; and 2) find optimized constellation points for a variety 
+of net rates while maintaining a fixed FEC and demapper logic (i.e. log-likelihood ratio (LLR) computation). The 
+system therefore allows for shaping gain to be achieved with a finer granularity without a complexity increase w.r.t. 
+conventional bit-interleaved coded modulation (BICM).   
+2. System description 
+The AE-based GCS training setup is given in Fig. 1 a). The 
+mapping function is learned using the AE architecture from [5] to 
+jointly optimize bit labeling and constellation position on the I/Q 
+plane. During the AE training, the transmitter and receiver employ 
+neural networks for mapping of bits and demapping to LLRs, 
+respectively. The transmitter of the proposed system is given in Fig 
+1. b). During testing, the mapping and demapping functionalities are 
+replaced by a look-up table (LUT) and conventional Gaussian bit-
+metric receiver [8], respectively. 
+The AE requires that the modulation format size is known and fixed. For a fixed modulation format size, the 
+generalized mutual information (GMI) typically is penalized w.r.t. the mutual information, as the signal to noise ratio 
+(SNR) decreases. This is due to the penalty in the demapper function related to the inability to resolve constellation 
+points with similar likelihoods at the receiver. An AE implicitly addresses this problem by ‘merging’ such points 
+closer together and assigning (more than 2) labels with a very small Hamming distance to virtually the same point [7], 
+i.e. a many-to-one mapping (MOM) of bits-to-symbols is produced. Here, we exploit this fact to achieve rate adaptivity 
+in the following way. The GMI of the MOM is analyzed and the bit levels which are ambiguous are not used for data. 
+Instead, they are assigned dummy bits in order to maintain the bit flow and logic at the transmitter and receiver. For a 
+system with an FEC rate of ������������ = ������������/������������, a code length of N, information block length of K and a modulation format size 
+of M, the net data rate per dual polarization channel may be calculated as ������������′ = (2������������ − ������������������������)������������, where ������������ = ������������������������������������2 ������������ is 
+the number of bit carried by the constellation and ������������������������ is the number of dummy bits in the labeling. If ������������������������ is even, each 
+polarization gets the same number of dummy bits, whereas if it is odd, the allocation is done such that the GMI per 
+
+a)
+bits-
+>.Channeli
+LLRS
+nd@
+b)
+LUT
+FEC
+Channel
+bits-
+S/P
+K/N
+2m - nd
+Fig. 1. a) Training setup; b) Testing setup with
+dummy bit insertionpolarization is similar for both polarization. In this paper, the ������������������������ bit positions are selected by sorting their per-bit GMI 
+and choosing as many as required from the lowest ones. For example, when the SNR decreases, ������������������������ can be increased. 
+The performance degradation of the large-size constellation at low SNR is compensated for by the increased Euclidean 
+distance of the effectively smaller constellation achieved via MOM.  The receiver architecture does not change since 
+K, N, and M are fixed. The only addition w.r.t. BICM is the additional LUTs for mapping depending on the channel 
+conditions.  
+3. Results 
+A wavelength division multiplexing system is optimized using the nonlinear interference noise model from [9] 
+with dual polarization, 5 channels, 100km spacing, FEC of rate 3/4 and modulation size ������������ = 256. Fig. 2 a) shows the 
+maximum distance for a given data rate at which the GMI is above the target rate for the AE-based MOM and uniform 
+QAM (red dotted line, ������������������������ = 0) for the central channel. The AE-optimized MOM achieve an extra span of transmission 
+distance with respect to both 256QAM and 128QAM. Here, due to the SNR reduction with the increase of distance, 
+the AE learns a MOM allowing the system to be rate adaptive through insertion of dummy bits without losing the 
+shaping gain or changing the modulation format. Fig. 2 b) shows the constellation learned for 8 spans transmission 
+which does not achieve MOM. In Fig. 2 c), the constellation learned for 20 spans that achieves MOM is shown. In the 
+latter, the AE “merged” some of the points together as shown in Fig. 2 d). The red box shows the bits that are not 
+shared by the 4 points. These 2 bits effectively carry no information and can be assigned dummy bits. The system then 
+exploits the resulting increased Euclidean distance of the constellation to improve the performance. 
+ 
+Fig. 2. a) Net rate per dual polarization channel w.r.t. distance for uniform QAM and AE-based GCS with 3/4 FEC; 
+GCS learned at: b) 8 spans; c) 20 spans; d) Zoomed in point to show that 4 points collapsed to each other. 
+4. Conclusion 
+An autoencoder (AE) is used to optimize rate dependent bit-to-symbol mappings for QAM of fixed size and FEC 
+of fixed rate. Rate adaptivity is achieved through the AE-optimized mapping functions as a result of many-to-one 
+mapping. It achieves shaping gain for a variety of net rates without changing the receiver architecture, the modulation 
+format size or requiring a distribution matcher and dematcher, resulting in a hardware friendly flexible architecture. 
+Acknowledgement: This work was financially supported by the ERC-CoG FRECOM project (grant no. 771878), 
+the Villum Young Investigator OPTIC-AI project (grant no. 29334), and DNRF SPOC, DNRF123.  
+References 
+[1] G. Böcherer, et. al, “Bandwidth efficient and rate matched low-density parity-check coded modulation,” IEEE Trans. on Comm., 2015. 
+[2] T. O’Shea and J. Hoydis, “An Introduction to Deep Learning for the Physical Layer,” IEEE Trans. on Cognitive Comm. and Net., 2017. 
+[3] B. Karanov, et. al., “End-to-End Deep Learning of Optical Fiber Communications,” JLT, 2018. 
+[4] R. T. Jones, et. al., “Deep Learning of Geometric Constellation Shaping Including Fiber Nonlinearities,” in ECOC, 2018. 
+[5] R. T. Jones, et. al., “End-to-end learning for GMI optimized geometric constellation shape,” in ECOC, 2019. 
+[6] J. Song, et.al., “Model-Based End-to-End Learning for WDM Systems With Transceiver Hardware Impairments,” IEEE JOSTQE, 2022. 
+[7] O. Jovanovic, et. al.” Geometric Constellation Shaping for Fiber-Optic Channels via End-to-End Learning,” arXiv:2211.04311, 2022. 
+[8] G. Böcherer, et. al., "Probabilistic Shaping and Forward Error Correction for Fiber-Optic Communication Systems," JLT, 2019. 
+[9] R. Dar, et. al., “Accumulation of nonlinear interference noise in fiber-optic systems,” Optics Express, 2014. 
+
+0.71
+256QAM
+12
+Xn
+=0
+10101100
+10111100
+Xn
+3/4 FEC
+11
+256GS
+b)
+0.7
+128QAM
+-2
+0
+10.5
+= 2
+2
+10
+Xn_=3
+10110
+00
+9.5
+0.69
+1
+10100100
+ 64QAM
+n
+(e 6
+c)
+d)
+-2
+0
+5
+10
+15
+20
+-2
+0
+2
+1.125
+1.135
+1.145
+Number of spans

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf,len=138
+page_content='Rate Adaptive Autoencoder-based Geometric Constellation  Shaping  Ognjen Jovanovic, Metodi P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' Yankov, Francesco Da Ros and Darko Zibar  Department of Electrical and Photonics Engineering, Technical University of Denmark, Kgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' Lyngby, 2800, Denmark  ognjo@dtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='dk  Abstract: An autoencoder is used to optimize bit-to-symbol mappings for geometric constellation  shaping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' The mappings allow for net rate adaptivity without additional hardware complexity, while  achieving up to 300km of transmission distance compared to uniform QAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' © 2023 The Author(s)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' Introduction  State of the art coherent optical communications need to be deployed in dynamic network scenarios, which require  a certain degree of adaptivity to varying channel conditions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' Classically, this is handled by varying the modulation  format size, which 1) often produces a coarse granularity in the rate with steps of 2 bits/symbol;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' and 2) requires the  transceiver to support bit to symbol mapping and demapping to/from constellations of different size, increasing the  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' The probabilistic amplitude shaping (PAS) scheme [1] has emerged as an efficient architecture that  provides a solution to the first problem, at the cost of rate matcher and dematcher, which increases complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='  Autoencoders (AEs) are becoming a popular tool to optimize the signaling constellation of digital communication  transceivers [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' In optical communications, AEs have been employed for geometric constellation shaping (GCS)  [4], bit labeling [5], mostly with the target of mitigating the impact of fiber nonlinearities, as well as transceiver  impairment mitigation [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' GCS is beneficial over PAS because it does not require explicit matcher and dematcher  blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' However, rate adaptivity with GCS typically requires a rate-flexible forward error correction (FEC), which  may increase the complexity of the digital logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' Further, GCS does not allow for straight-forward Gray labeling to be  performed, which may lead to sub-optimality when combined with conventional bit-metric decoders as in standard  coherent optical communications [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='  In this paper, an AE is used to 1) find optimal bit mappings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' and 2) find optimized constellation points for a variety  of net rates while maintaining a fixed FEC and demapper logic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' log-likelihood ratio (LLR) computation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' The  system therefore allows for shaping gain to be achieved with a finer granularity without a complexity increase w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='  conventional bit-interleaved coded modulation (BICM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' System description  The AE-based GCS training setup is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' 1 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' The  mapping function is learned using the AE architecture from [5] to  jointly optimize bit labeling and constellation position on the I/Q  plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' During the AE training, the transmitter and receiver employ  neural networks for mapping of bits and demapping to LLRs,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' The transmitter of the proposed system is given in Fig  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' During testing, the mapping and demapping functionalities are  replaced by a look-up table (LUT) and conventional Gaussian bit-  metric receiver [8], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='  The AE requires that the modulation format size is known and fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' For a fixed modulation format size, the  generalized mutual information (GMI) typically is penalized w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' the mutual information, as the signal to noise ratio  (SNR) decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' This is due to the penalty in the demapper function related to the inability to resolve constellation  points with similar likelihoods at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' An AE implicitly addresses this problem by ‘merging’ such points  closer together and assigning (more than 2) labels with a very small Hamming distance to virtually the same point [7],  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' a many-to-one mapping (MOM) of bits-to-symbols is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' Here, we exploit this fact to achieve rate adaptivity  in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' The GMI of the MOM is analyzed and the bit levels which are ambiguous are not used for data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='  Instead, they are assigned dummy bits in order to maintain the bit flow and logic at the transmitter and receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' For a  system with an FEC rate of ������������ = ������������/������������, a code length of N, information block length of K and a modulation format size  of M, the net data rate per dual polarization channel may be calculated as ������������′ = (2������������ − ������������������������)������������, where ������������ = ������������������������������������2 ������������ is  the number of bit carried by the constellation and ������������������������ is the number of dummy bits in the labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' If ������������������������ is even, each  polarization gets the same number of dummy bits, whereas if it is odd, the allocation is done such that the GMI per  a)  bits-  >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='Channeli  LLRS  nd@  b)  LUT  FEC  Channel  bits-  S/P  K/N  2m - nd  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' a) Training setup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' b) Testing setup with  dummy bit insertionpolarization is similar for both polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' In this paper, the ������������������������ bit positions are selected by sorting their per-bit GMI  and choosing as many as required from the lowest ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' For example, when the SNR decreases, ������������������������ can be increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='  The performance degradation of the large-size constellation at low SNR is compensated for by the increased Euclidean  distance of the effectively smaller constellation achieved via MOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' The receiver architecture does not change since  K, N, and M are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' The only addition w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' BICM is the additional LUTs for mapping depending on the channel  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' Results  A wavelength division multiplexing system is optimized using the nonlinear interference noise model from [9]  with dual polarization, 5 channels, 100km spacing, FEC of rate 3/4 and modulation size ������������ = 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' 2 a) shows the  maximum distance for a given data rate at which the GMI is above the target rate for the AE-based MOM and uniform  QAM (red dotted line, ������������������������ = 0) for the central channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' The AE-optimized MOM achieve an extra span of transmission  distance with respect to both 256QAM and 128QAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' Here, due to the SNR reduction with the increase of distance,  the AE learns a MOM allowing the system to be rate adaptive through insertion of dummy bits without losing the  shaping gain or changing the modulation format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' 2 b) shows the constellation learned for 8 spans transmission  which does not achieve MOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' 2 c), the constellation learned for 20 spans that achieves MOM is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' In the  latter, the AE “merged” some of the points together as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' 2 d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' The red box shows the bits that are not  shared by the 4 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' These 2 bits effectively carry no information and can be assigned dummy bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' The system then  exploits the resulting increased Euclidean distance of the constellation to improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' a) Net rate per dual polarization channel w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' distance for uniform QAM and AE-based GCS with 3/4 FEC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='  GCS learned at: b) 8 spans;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' c) 20 spans;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' d) Zoomed in point to show that 4 points collapsed to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' Conclusion  An autoencoder (AE) is used to optimize rate dependent bit-to-symbol mappings for QAM of fixed size and FEC  of fixed rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' Rate adaptivity is achieved through the AE-optimized mapping functions as a result of many-to-one  mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' It achieves shaping gain for a variety of net rates without changing the receiver architecture, the modulation  format size or requiring a distribution matcher and dematcher, resulting in a hardware friendly flexible architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='  Acknowledgement: This work was financially supported by the ERC-CoG FRECOM project (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' 771878),  the Villum Young Investigator OPTIC-AI project (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' 29334), and DNRF SPOC, DNRF123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
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+page_content=' Dar, et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
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+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='71  256QAM  12  Xn  =0  10101100  10111100  Xn  3/4 FEC  11  256GS  b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='7  128QAM  2  0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='5  = 2  2  10  Xn_=3  10110  00  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='69  1  10100100  64QAM  n  (e 6  c)  d)  2  0  5  10  15  20  2  0  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='125  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='135  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}
+page_content='145  Number of spans' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfTPzA/content/2301.01247v1.pdf'}

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+Nuclear shape fluctuations in high-energy heavy ion collisions
+Aman Dimri,1, ∗ Somadutta Bhatta,1 and Jiangyong Jia1, 2, †
+1Department of Chemistry, Stony Brook University, Stony Brook, NY 11794, USA
+2Physics Department, Brookhaven National Laboratory, Upton, NY 11976, USA
+(Dated: January 10, 2023)
+Atomic nuclei often exhibit a quadrupole shape that fluctuates around some average profile.
+We investigate the impact of nuclear shape fluctuation on the initial state geometry in heavy ion
+collisions, particularly its eccentricity ε2 and inverse size d⊥, which can be related to the elliptic flow
+and radial flow in the final state. The fluctuation in overall quadrupole deformation enhances the
+variances and modifies the skewness and kurtosis of the ε2 and d⊥ in a controllable manner. The
+fluctuation in triaxiality reduces the difference between prolate and oblate shape for any observable,
+whose values, in the large fluctuation limit, approach those obtained in collisions of rigid triaxial
+nuclei. The method to disentangle the mean and variance of the quadrupole deformation is discussed.
+PACS numbers: 25.75.Gz, 25.75.Ld, 25.75.-1
+I.
+INTRODUCTION
+Ultra-relativistic heavy ion physics aims to understand the dynamics and properties of the Quark-Gluon Plasma
+(QGP) created in collisions of atomic nuclei at very high energy [1].
+Achieving this goal is currently limited by
+the lack of understanding of the initial condition, i.e. how the energy is deposited in the overlap region before the
+formation of QGP [2]. The energy deposition process is not calculable from first principles and is often parameterized
+via phenomenological approaches with multiple free parameters [3]. On the other hand, heavy atomic nuclei are
+well-studied objects exhibiting a wide range of shapes and radial profiles [4], which are often characterized by a few
+collective nuclear structure parameters such as quadrupole, triaxial, and octupole deformations, nuclear radius and
+skin thickness. One can leverage species with similar mass numbers but different structures, such as isobars, to directly
+probe the energy deposition mechanism and hence constrain the initial condition. The efficacy of this approach has
+been investigated recently [5–7].
+One good example demonstrating this possibility is the 96Ru+96Ru and 96Zr+96Zr collisions, recently carried out
+by the STAR Collaboration at the relativistic heavy ion collider [8, 9]. Ratios of many bulk observables between
+the isobars, such as harmonic flow vn, charged particle multiplicity Nch, and average transverse momentum ⟨pT⟩,
+have been measured, which show significant and observable- and centrality-dependent deviation from unity. Model
+studies show that these ratios are insensitive to final-state effects and are controlled mainly by the differences of the
+collective nuclear structure parameters between 96Ru and 96Zr [10]. Comparing the calculations with experimental
+data, Refs. [5, 11] have estimated structure parameters that are broadly consistent with general knowledge from low
+energy. However, these studies also suggest a sizable octupole collectivity for Zr, not predicted by mean field structure
+models [12]. The rich and versatile information from isobar or isobar-like collisions provides a new constraint on the
+heavy ion initial condition and a new way to probe nuclear structure at high energy [13].
+However, it is important to point out that atomic nuclei in the ground state often do not have a static shape, but
+can fluctuate due to interplay between collective modes and single-particle states [14]. The potential energy surface of
+such species usually has shallow minimums as a function of deformation parameters, such as quadruple deformation
+β and triaxiality γ. The ground state nuclear wave function is often treated as a mixture of configurations with
+different (β,γ) values [4, 15]. Then there are the phenomena of shape coexistence, which happens when the same
+nuclei can have multiple low-lying states with widely different shapes but small energy differences [16]. From the
+nuclear structure side, the quadrupole fluctuations can be estimated from the sum rules of matrix elements of various
+moments of quadrupole operators that can be measured experimentally [17, 18]. From the heavy ion collision side,
+the shape fluctuations can be accessed using multi-particle correlations, which probe moments of the nucleon position
+in the initial condition [19]. For instance, the elliptic flow v2 in each event is proportional to the elliptic eccentricity
+ε2, v2 = kε2 calculable from participating nucleons [20]. Therefore, the fluctuations of flow are related to fluctuations
+of quadruple deformation via their respective moments: ⟨vm
+2 ⟩ = km ⟨εm
+2 ⟩ ∝ ⟨βm⟩,m = 2,4... In principle, one could
+∗Electronic address: [email protected]
+†Electronic address: [email protected]
+arXiv:2301.03556v1  [nucl-th]  9 Jan 2023
+
+2
+constrain the mean and variance of quadrupole fluctuations from the ⟨β2⟩ and ⟨β4⟩, which in terms can be determined
+from ⟨v2
+2⟩ and ⟨v4
+2⟩.
+This paper extends our previous study to investigate the influence of fluctuations of quadruple deformation param-
+eters (β,γ) to several selected two-, three- and four-particle heavy-ion observables. We first derive simple analytical
+relations between these observables and the means and variances of (β,γ). We then perform a more realistic Glauber
+model simulation, assuming Gaussian fluctuations, to quantify the region of validity of these relations. We discuss
+the sensitivity of these observables on the nuclear shape, as well as the prospect of separating the average shape from
+shape fluctuations.
+II.
+EXPECTATION AND MODEL SETUP
+We consider the eccentricity vector ϵ2 and inverse transverse size d⊥, which are estimators for elliptic flow V2 ≡
+v2e2iΨ2 and average transverse momentum ⟨pT⟩ or radial flow, calculated from the transverse position of nucleon
+participants in each event,
+ϵ2 = −
+⟨r2
+⊥ei2φ⟩
+⟨r2⊥⟩
+, d⊥ =
+√
+Npart/⟨r2⊥⟩,
+(1)
+where r⊥ is the transverse radius and Npart is the number of participating nucleons. Following the heuristic argument
+from Ref. [21], for collisions of nuclei with small quadrupole deformation, the eccentricity vector and d⊥ in a given
+event have the following leading-order form:
+δd⊥
+d⊥
+≈ δd + p0(Ωp,γp)βp + p0(Ωt,γt)βt , ϵ2 ≈ ϵ0 + p2(Ωp,γp)βp + p2(Ωt,γt)βt,
+(2)
+where the scalar δd and vector ϵ0 are valued for spherical nuclei, and we are considering the general situation where
+the projectile and target, denoted by subscripts “p” and “t”, have different deformation values. The p0 and p2 are
+phase space factors, which depend on γ and the Euler angles Ω.
+Since the fluctuations of δd (ϵ0) are uncorrelated with p0 (p2), an average over collisions with different Euler angles
+is expected to give the following leading-order expressions for the variances, skewness, and kurtosis of the fluctuations
+c2,ϵ{2} ≡ ⟨ε2
+2⟩ = ⟨ε2
+0⟩ + ⟨p2(γp)p∗
+2(γp)⟩β2
+p + ⟨p2(γt)p∗
+2(γt)⟩β2
+t
+cd{2} ≡ ⟨(δd⊥
+d⊥
+)
+2
+⟩ = ⟨δ2
+d⟩ + ⟨p0(γp)2⟩β2
+p + ⟨p0(γt)2⟩β2
+t
+Cov ≡ ⟨ε2
+2
+δd⊥
+d⊥
+⟩ = ⟨ε2
+0δd⟩ + ⟨p0(γp)p2(γp)p2(γp)∗⟩β3
+p + ⟨p0(γt)p2(γt)p2(γt)∗⟩β3
+t
+cd{3} ≡ ⟨(δd⊥
+d⊥
+)
+3
+⟩ = ⟨δ3
+d⟩ + ⟨p0(γp)3⟩β3
+p + ⟨p0(γt)3⟩β3
+t
+c2,ϵ{4} ≡ ⟨ε4
+2⟩ − 2⟨ε2
+2⟩
+2 = ⟨ε4
+0⟩ − 2⟨ε2
+0⟩
+2 + (⟨p2
+2p2∗
+2 ⟩⟨β4⟩ − 2⟨p2p∗
+2⟩2 ⟨β2⟩
+2)
+p + (⟨p2
+2p2∗
+2 ⟩⟨β4⟩ − 2⟨p2p∗
+2⟩2 ⟨β2⟩
+2)
+t .
+(3)
+These quantities relate directly to the final state observables, ⟨v2
+2⟩, ⟨(δpT/⟨pT⟩)2⟩, ⟨v2
+2
+δpT
+⟨pT⟩⟩, ⟨(δpT/⟨pT⟩)3⟩ and
+⟨v4
+2⟩ − 2⟨v2
+2⟩
+2, respectively.
+Previous studies have demonstrated that the moments ⟨p2
+0⟩, ⟨p2p∗
+2⟩, and ⟨p2
+2p2∗
+2 ⟩ are independent of γ, while
+⟨p0p2p∗
+2⟩ and ⟨p3
+0⟩ have leading order dependence on γ: c + bcos(3γ). Here, c ≪ b for ⟨p0p2p∗
+2⟩, whereas c ≲ b for
+⟨p3
+0⟩ [21]. In the presence of quadrupole fluctuations, we also need to average these quantities over “independent”
+
+3
+fluctuations for projectile and target, giving,
+⟨(δd⊥
+d⊥
+)
+2
+⟩ = a0 + b0
+2 (⟨β2
+p⟩ + ⟨β2
+t ⟩) = a0 + b0 ⟨β2⟩
+⟨ε2
+2⟩ = a1 + b1
+2 (⟨β2
+p⟩ + ⟨β2
+t ⟩) = a1 + b1 ⟨β2⟩
+⟨ε2
+2
+δd⊥
+d⊥
+⟩ = a2 − 1
+2 (⟨(c2 + b2 cos(3γp))β3
+p⟩ + ⟨(c2 + b2 cos(3γt))β3
+t ⟩) = a2 − ⟨(c2 + b2 cos(3γ))β3⟩
+⟨(δd⊥
+d⊥
+)
+3
+⟩ = a3 + 1
+2 (⟨(c3 + b3 cos(3γp))β3
+p⟩ + ⟨(c3 + b3 cos(3γt)β3
+t ⟩) = a3 + ⟨(c3 + b3 cos(3γ))β3⟩
+⟨ε4
+2⟩ − 2⟨ε2
+2⟩
+2 = a4 + b4
+2 (⟨β4
+p⟩ + ⟨β4
+t ⟩) − c4
+2 (⟨β2
+p⟩
+2 + ⟨β2
+t ⟩
+2) = a4 + b4 ⟨β4⟩ − c4 ⟨β2⟩
+2 ,
+(4)
+where the averages are performed over fluctuations in β and γ, and the coefficients an, bn and cn are centrality-
+dependent positive quantities satisfying c2 ≪ b2 and c3 ≲ b3 [21]. The second part of these equations is obtained by
+assuming that the fluctuations of the projectile and target are sampled from the same probability density distributions.
+For a more quantitative estimation, we consider the liquid-drop model where the nucleon density distribution has
+a sharp surface. For head-on collisions with zero impact parameter, it predicts the following simple relations [21],
+δd⊥
+d⊥
+=
+√
+5
+16π β2 (cosγD2
+0,0 + sinγ
+√
+2
+[D2
+0,2 + D2
+0,−2]) , ϵ2 = −
+√
+15
+2π β2 (cosγD2
+2,0 + sinγ
+√
+2
+[D2
+2,2 + D2
+2,−2]) ,
+(5)
+where the Dl
+m,m′(Ω) are the Wigner matrices. The analytical results obtained for various cumulants are listed in
+Table I. They provide approximate estimates for the values of bn in most central collisions.
+⟨(δd⊥/d⊥)2⟩
+⟨(δd⊥/d⊥)3⟩
+⟨(δd⊥/d⊥)4⟩ − 3 ⟨(δd⊥/d⊥)2⟩
+2
+1
+32π ⟨β2⟩
+√
+5
+896π3/2 ⟨cos(3γ)β3⟩
+−
+3
+14336π2 (7 ⟨β2⟩
+2 − 5 ⟨β4⟩)
+⟨ε2
+2⟩
+⟨ε4
+2⟩ − 2 ⟨ε2
+2⟩
+2
+(⟨ε6
+2⟩ − 9 ⟨ε4
+2⟩ ⟨ε2
+2⟩ + 12 ⟨ε2
+2⟩
+3)/4
+3
+4π ⟨β2⟩
+−
+9
+112π2 (7 ⟨β2⟩
+2 − 5 ⟨β4⟩)
+81
+256π3 [⟨β2⟩
+3 − 45
+14 ⟨β4⟩ ⟨β2⟩ − 1175
+6006 ⟨β6⟩ +
+25
+3003 ⟨cos(6γ)β6⟩]
+⟨ε2
+2(δd⊥/d⊥)⟩
+⟨ε2
+2(δd⊥/d⊥)2⟩ − ⟨ε2
+2⟩ ⟨(δd⊥/d⊥)2⟩
+⟨ϵ2
+2ϵ∗
+4⟩
+−
+3
+√
+5
+112π3/2 ⟨cos(3γ)β3⟩
+−
+3
+1792π2 (7 ⟨β2⟩
+2 − 5 ⟨β4⟩)
+45
+56π2 ⟨β4⟩
+TABLE I: The leading-order results of various cumulants calculated for the nucleus with a sharp surface via Eq. 5. The two
+nuclei are placed with zero impact parameter and results are obtained by averaging over random orientations.
+To make further progress, we consider the case where the fluctuations of β and γ are independent of each other.
+The observables in Eq. 4 and Table I can be expressed in terms of central moments. Assuming Gaussian fluctuations
+with means ¯β or ¯γ and variances σβ or σγ, Eq. 4 becomes
+⟨ε2
+2⟩ = a1 + b1(¯β2 + σ2
+β) ,⟨(δd⊥
+d⊥
+)
+2
+⟩ = a0 + b0(¯β2 + σ2
+β)
+⟨ε2
+2
+δd⊥
+d⊥
+⟩ = a2 − (b2e−
+9σ2
+γ
+2 cos(3¯γ) + c2)¯β(¯β2 + 3σ2
+β)
+⟨(δd⊥
+d⊥
+)
+3
+⟩ = a3 + (b3e−
+9σ2
+γ
+2 cos(3¯γ) + c3)¯β(¯β2 + 3σ2
+β)
+⟨ε4
+2⟩ − 2⟨ε2
+2⟩
+2 = a4 + b4(¯β4 + 6σ2
+β ¯β2 + 3σ4
+β) − c4(¯β2 + σ2
+β)2 .
+(6)
+where we have used the well-known expression for Gaussian smearing of an exponential function, ⟨einγ⟩ = e−
+n2σ2
+γ
+2
+ein¯γ.
+
+4
+If the fluctuations of β and γ are non-Gaussian, one should also consider the higher cumulants of β. For example,
+⟨β3⟩ = ¯β(¯β + 3σ2
+β) + k3,β and ⟨β4⟩ = ¯β4 + 6σ2
+β ¯β2 + 3σ4
+β + 4¯βk3,β + k4,β, where k3,β = ⟨(β − ¯β)3⟩ and k4,β = ⟨(β − ¯β)4⟩ −
+3⟨(β − ¯β)2⟩
+2 are the skewness and kurtosis of the β fluctuation. The expectation value of cos(nγ) can be expressed
+via the cumulant generating function of γ. Keeping the cumulants km,γ up to leading order correction in skewness
+and kurtosis, k3,γ = ⟨(γ − ¯γ)3⟩ and k4,γ = ⟨(γ − ¯γ)4⟩ − 3⟨(γ − ¯γ)2⟩
+2, we have,
+⟨cos(nγ)⟩ = 1
+2 (⟨ein¯γ⟩ + ⟨e−in¯γ⟩) = 1
+2 (exp(
+∞
+∑
+m=1
+κm,γ
+(in)m
+m!
+) + exp(
+∞
+∑
+m=1
+κm,γ
+(−in)m
+m!
+))
+= exp(
+∞
+∑
+m=1
+κ2m,γ
+(−1)m(n)2m
+2m!
+)[cos(
+∞
+∑
+m=1
+κ2m+1,γ
+(−1)m(n)2m+1
+(2m + 1)!
++ n¯γ)]
+≈ e−
+n2σ2
+γ
+2
++
+n4k4,γ
+24
+cos(n¯γ + n3
+6 k3,γ) ≈ e−
+n2σ2
+γ
+2
+[cos(n¯γ) + sin(n¯γ)n3
+6 k3,γ](1 + n4
+24k4,γ).
+(7)
+Clearly, the net effect of skewness is a rotation of ¯γ by k3,γn2/6, and the net effect of kurtosis is to increase or decrease
+the overall variation with γ depending on its sign.
+For a more realistic estimation of the influences of shape fluctuations, we perform a Monte-Carlo Glauber model
+simulation of 238U+238U collisions. The setup of the model and the data used in this analysis are the same as those
+used in our previous work [19]. We simulate ultra-central collisions with zero impact parameter, where the impact of
+nuclear deformation reaches maximum. The nucleon distribution is described by a deformed Woods-Saxon function
+ρ(r,θ,φ) =
+ρ0
+1 + e[r−R(θ,φ)/a] , R(θ,φ) = R0 (1 + β[cosγY2,0(θ,φ) + sinγY2,2(θ,φ)]),
+(8)
+where the nuclear surface R(θ,φ) is expanded into spherical harmonics Y2,m in the intrinsic frame. Each nucleus is
+assigned a random (β,γ) value, sampled from Gaussian distributions with means (¯β, ¯γ) and variances (σβ,σγ). The
+nucleus is then rotated by random Euler angles before they are set on a straight line trajectory towards each other
+along the z direction. Furthermore, three quark constituents are generated for each nucleon according to the quark
+Glauber model from Ref. [22]. From this, the nucleons or the constituent quarks in the overlap region are identified,
+which are used to calculate the ε2 and d⊥ defined in Eqs. (1), and the results are presented as a function of deformation
+parameters.
+For the study of the β fluctuation, we fix the γ = 0 (prolate nucleus) and choose 11 values each for ¯β2 and
+σ2
+β with 0, 0.01,...,0.09, 0.1. So a total of 11 × 11 = 121 simulations have been performed. For the study of the
+γ fluctuation, we fix β = 0.28 and choose seven ¯γ and seven σγ values: cos(3¯γ) = 1,0.87,0.5,0,−0.5,0.87,−1 and
+σγ = 0,π/18,2π/18,...,6π/18, so a total of 7 × 7 = 49 simulation have been performed. For each case, about 50 Million
+events were generated and all the observables were calculated. Our discussion is mainly based on the nucleon Glauber
+model, and the results from the quark Glauber model are included in the Appendix.
+III.
+IMPACT OF TRIAXIALITY FLUCTUATION
+Due to the three-fold symmetry of nuclei shape in triaxiality, the γ dependence of a given observable can be
+generally expressed as a0 +∑∞
+n=1 [an cos(3n¯γ) + bn sin(3n¯γ)]e−
+n2σ2
+γ
+2
+. If we further impose the condition that a random
+fluctuation for a triaxial nucleus does not impact the value of the observable, which is found to be true in our analysis.
+This leads to the γ dependence of the form a0 + ∑∞
+n=1 [an(cos(3n¯γ) − cos(3n π
+6 )) + bn(sin(3n¯γ) − sin(3n π
+6 ))]e−
+n2σ2
+γ
+2
+.
+We first discuss the impact of triaxiality fluctuation on three-particle observables ⟨ε2
+2
+δd⊥
+d⊥ ⟩ and ⟨(δd⊥/d⊥)3⟩. We first
+subtract them by the values for the undeformed case, to isolate the second term in Eq. 4 containing the triaxiality.
+Figure 1 show the results obtained for different values of cos3¯γ as a function of σγ. The values for triaxial nucleus
+cos(3¯γ) = 0 are indeed independent of σγ. The fluctuation of γ reduces the difference between the prolate ¯γ = 0 and
+the oblate ¯γ = π/3 shape. This reduction is largely described by e−
+9σ2
+γ
+2 cos(3¯γ), except for a small asymmetry between
+¯γ = 0 and ¯γ = π/3, clearly visible for ⟨(δd⊥/d⊥)3⟩.
+We account for this small asymmetry by including higher-order terms in the fit function allowed by symmetry.
+
+5
+0
+0.2
+0.4
+0.6
+0.8
+1
+γ
+σ
+0.2
+−
+0.15
+−
+0.1
+−
+0.05
+−
+0
+0.05
+0.1
+0.15
+0.2
+3
+−
+10
+×
+ 
+=0
+β
+ Cov  -  Cov
+1.00 
+0.50 
+-0.50 
+-1.00 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+-5
+10
+×
+ = -0.85
+0
+a
+-5
+10
+×
+) = (-19.98,-0.03)
+2
+,a
+1
+(a
+U+U Glauber Model
+-5
+10
+×
+) = (0.28,0.08)
+2
+,b
+1
+(b
+ = 0.28
+β
+b = 0 fm, 
+)  Data  Fit
+γ
+Cos(3
+0
+0.2
+0.4
+0.6
+0.8
+1
+γ
+σ
+2
+−
+0
+2
+4
+6
+8
+6
+−
+10
+×
+ 
+=0
+β
+{3}
+d
+ - C
+{3}
+d
+C
+0.87 
+0.00 
+-0.87 
+ 
+ 
+ 
+ 
+ 
+ 
+-6
+10
+×
+ = 3.19
+0
+a
+-6
+10
+×
+) = (5.18,-0.18)
+2
+,a
+1
+(a
+U+U Glauber Model
+-6
+10
+×
+) = (0.27,0.04)
+2
+,b
+1
+(b
+ = 0.28
+β
+b = 0 fm, 
+)  Data  Fit
+γ
+Cos(3
+FIG. 1:
+The dependence of ⟨ε2
+2
+δd⊥
+d⊥ ⟩ (left) and ⟨(δd⊥/d⊥)3⟩ (right) on smearing in triaxiality σγ for different values of ¯γ. The
+lines indicate a simultaneous fit to Eq. 10 with the parameter values displayed on the plot.
+Keeping leading and subleading terms, we have,
+⟨ε2
+2
+δd⊥
+d⊥
+⟩ − ⟨ε2
+2
+δd⊥
+d⊥
+⟩
+β=0
+= [a′
+0 + (a′
+1 cos(3¯γ) + b′
+1 [sin(3¯γ) − 1])e−
+9σ2
+γ
+2
++ (a′
+2 [cos(6¯γ) + 1] + b′
+2 sin(6¯γ))e−
+36σ2
+γ
+2 ] ¯β3
+(9)
+= a0 + (a1 cos(3¯γ) + b1 [sin(3¯γ) − 1])e−
+9σ2
+γ
+2
++ (a2 [cos(6¯γ) + 1] + b2 sin(6¯γ))e−
+36σ2
+γ
+2
+.
+(10)
+The same fit function is also used to describe ⟨(δd⊥/d⊥)3⟩. The parameters in the first line and those in the second
+line differ by a scale factor ¯β3 = 0.283 = 0.021. From the values of parameters displayed in Fig. 1, we concluded that
+the magnitude of the high-order order terms is less than 2% of the magnitude of a1 for ⟨ε2
+2
+δd⊥
+d⊥ ⟩ but reaches up to 5%
+for ⟨(δd⊥/d⊥)3⟩.
+Figure 1 shows that the signature of triaxiality in heavy ion collisions is greatly reduced for large value of σγ, often
+found in γ-soft nuclei. A twenty-degree fluctuation in triaxiality, for example, reduces the signal by nearly 50%. It
+would be difficult to distinguish between static rigid triaxial nuclei and nuclei with large fluctuations around ¯γ = π/6
+using heavy ion collisions. In particular, nuclei that fluctuate uniformly between prolate and oblate shapes would give
+the same three-particle correlation signal as rigid triaxial nuclei! Such strong smearing also degrades the prospects of
+using higher-order cumulants of ε2 to infer the value of σγ.
+For the other three observables, ⟨ε2
+2⟩, ⟨(δd⊥/d⊥)2⟩ and ⟨ε4
+2⟩ − 2⟨ε2
+2⟩
+2, γ dependence are known to be very weak [19].
+Nevertheless, up to a few percent dependence is observed, which can also be parameterized by Eq. 9, except that
+we should change ¯β3 to ¯β2 for the variances and to ¯β4 for ⟨ε4
+2⟩ − 2⟨ε2
+2⟩
+2. However, since ¯β is fixed at 0.28, all these
+observables can be parameterized by Eq. 10. The data and the results of the fits are shown in Fig. 2. First, we observe
+that the parameter a0, representing the baseline contribution associated with ¯β is by far the largest, and the other
+terms only cause a few percent of modulation. Secondly, while the ⟨ε2
+2⟩ and ⟨ε4
+2⟩ − 2⟨ε2
+2⟩
+2 can be largely described
+by including the cos(3¯γ) term, the description of ⟨(δd⊥/d⊥)2⟩ requires the inclusion of sin(3¯γ), cos(6¯γ) and sin(6¯γ)
+terms with comparable magnitudes. Lastly, all three observables have no sensitivity to ¯γ at large σγ.
+IV.
+IMPACT OF QUADRUPLE DEFORMATION FLUCTUATION
+Next, we consider the impact of β fluctuations.
+For this purpose, we shall fix the γ to be prolate shape, e.g
+cos(3γ) = 1. Figure 3 displays the finding for two-particle observables ⟨ε2
+2⟩ and ⟨(δd⊥/d⊥)2⟩, again they are corrected
+by the undeformed baseline. Although approximately-linear dependencies on ¯β2 are observed for both observables,
+the slopes of the data points also vary with σβ. To describe this feature, we include two higher-order terms,
+
+6
+0
+0.2
+0.4
+0.6
+0.8
+1
+γ
+σ
+15.8
+16
+16.2
+16.4
+16.6
+16.8
+3
+−
+10
+×
+=0
+β〉 
+2
+2
+ε 〈
+ - 
+〉 
+2
+2
+ε 〈
+1.00 
+0.50 
+-0.50 
+-1.00 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+-5
+10
+×
+ = 1621.87
+0
+a
+-5
+10
+×
+) = (52.46,-0.29)
+2
+,a
+1
+(a
+U+U Glauber Model
+-5
+10
+×
+) = (-0.79,-0.53)
+2
+,b
+1
+(b
+ = 0.28
+β
+b = 0 fm, 
+)  Data  Fit
+γ
+Cos(3
+0
+0.2
+0.4
+0.6
+0.8
+1
+γ
+σ
+0.6
+0.62
+0.64
+0.66
+0.68
+3
+−
+10
+×
+=0
+β
+{2}
+d
+  - C
+{2}
+d
+C
+0.87 
+0.00 
+-0.87 
+ 
+ 
+ 
+ 
+ 
+ 
+-5
+10
+×
+ = 65.83
+0
+a
+-5
+10
+×
+) = (-1.89,-1.31)
+2
+,a
+1
+(a
+U+U Glauber Model
+-5
+10
+×
+) = (1.48,0.08)
+2
+,b
+1
+(b
+ = 0.28
+β
+b = 0 fm, 
+)  Data  Fit
+γ
+Cos(3
+0
+0.2
+0.4
+0.6
+0.8
+1
+γ
+σ
+0.105
+−
+0.1
+−
+0.095
+−
+0.09
+−
+0.085
+−
+3
+−
+10
+×
+=0
+β
+{4}
+ε
+2,
+ - c
+{4}
+ε
+2,
+c
+-5
+10
+×
+ = -9.76
+0
+a
+-5
+10
+×
+) = (1.03,0.10)
+2
+,a
+1
+(a
+U+U Glauber Model
+-5
+10
+×
+) = (-0.13,-0.01)
+2
+,b
+1
+(b
+ = 0.28
+β
+b = 0 fm, 
+FIG. 2:
+The dependence of ⟨ε2
+2⟩ (left), ⟨(δd⊥/d⊥)2⟩ (middle), and ⟨ε4
+2⟩ − 2 ⟨ε2
+2⟩
+2 (right) on σγ for different values of ¯γ. The
+dashed lines indicate a simultaneous fit to Eq. 10, with fit results are displayed on the plot.
+⟨ε2
+2⟩ − ⟨ε2
+2⟩β=0 or ⟨(δd⊥
+d⊥
+)
+2
+⟩ − ⟨(δd⊥
+d⊥
+)
+2
+⟩
+β=0
+= c1 ⟨β2⟩ + c2 ⟨β3⟩ + c3 ⟨β4⟩
+= c1(¯β2 + σ2
+β) + c2 ¯β(¯β2 + 3σ2
+β) + c3(¯β4 + 6σ2
+β ¯β2 + 3σ4
+β)
+(11)
+The fits including only the leading term and all three terms are shown in the first row and the last row of Fig. 3,
+respectively. The fits in the middle row include the c1 and c3 terms for ⟨ε2
+2⟩, while they include c1 and c2 terms for
+⟨(δd⊥/d⊥)2⟩. Clearly, the behavior of ⟨(δd⊥/d⊥)2⟩ at large ¯β or σβ requires the presence of the ⟨β3⟩ term in Eq. 11
+with a negative coefficient c2 < 0. In general, a large fluctuation σβ tends to reduce the slope of the dependence on
+¯β2.
+For the three-particle correlators, we include three terms in the fitting function as
+⟨ε2
+2
+δd⊥
+d⊥
+⟩ − ⟨ε2
+2
+δd⊥
+d⊥
+⟩
+β=0
+or ⟨(δd⊥
+d⊥
+)
+3
+⟩ − ⟨(δd⊥
+d⊥
+)
+3
+⟩
+β=0
+= c1 ⟨β3 cos(3γ)⟩ + c2 ⟨β4 cos(3γ)⟩ + c3 ⟨β5 cos(3γ)⟩
+= [c1 ¯β(¯β2 + 3σ2
+β) + c2(¯β4 + 6σ2
+β ¯β2 + 3σ4
+β) + c3(¯β5 + 10σ3
+β ¯β2 + 15¯βσ4
+β)]cos(3γ)
+(12)
+The fitting results are shown in Fig. 4 as a function of ¯β3 for the prolate case cos(3γ) = 1. Inclusion of the high-order
+terms, mostly contribution from the ⟨β5⟩ component, improves the description of ⟨ε2
+2
+δd⊥
+d⊥ ⟩ in the region of large σβ.
+However, they are not sufficient to describe the ⟨(δd⊥/d⊥)3⟩ in the region of large ¯β and σβ. In particular, the fit also
+misses all points at ¯β = 0. We checked that the fit can be systematically improved by including more higher moment
+terms, albeit only very slowly.
+Lastly, we consider the four-particle observable c2,ε{4} = ⟨ε4
+2⟩ − 2⟨ε2
+2⟩
+2. According to findings in Fig. 3, the Taylor
+expansion of ⟨ε2
+2⟩ should give the first two terms as c1 ⟨β2⟩ + c2 ⟨β4⟩, similarly the first few terms of ⟨ε4
+2⟩ has the form
+of a0 ⟨β2⟩
+2 + a1 ⟨β4⟩ + a2 ⟨β6⟩. Therefore, the natural expression for c2,ε{4} up to second order correction should be
+c2,ε{4} − c2,ε{4}β=0 = a0 ⟨β2⟩
+2 + a1 ⟨β4⟩ + a2 ⟨β6⟩ − (c1 ⟨β2⟩ + c2 ⟨β4⟩)2 ≈ a1 ⟨β4⟩ − b1 ⟨β2⟩
+2 + a2 ⟨β6⟩ − b2 ⟨β2⟩⟨β4⟩
+= a1(¯β4 + 6¯β2σ2
+β + 3σ4
+β) − b1(¯β2 + σ2
+β)2 + a2(¯β6 + 15¯β4σ2
+β + 45¯β2σ4
+β + 15σ6
+β) − b2(¯β2 + σ2
+β)(¯β4 + 6¯β2σ2
+β + 3σ4
+β)
+(13)
+with b1 = c2
+1 − a0 and b2 = 2c1c2. The leading order correction includes the first two terms with a1 and b1, while the
+remaining two terms are the subleading-order corrections.
+The results from the Glauber model and the fit to Eq. 13 are shown in the left panel of Fig. 5. The strong variation
+of c2,ε{4} with both ¯β and σβ is captured nicely by the fit. For small values of σβ, the deformation has a negative
+contribution to c2,ε{4} that is proportional to ¯β4.
+For large values of σβ, c2,ε{4} becomes positive.
+A previous
+study shows that the centrality fluctuation also tends to give a positive value of c2,ε{4} [23]. Therefore, a negative
+c2,ε{4} which decreases further in central collisions would be an unambiguous indication for a large static quadrupole
+deformation of the colliding nuclei.
+The values of the fit parameters show some interesting relations, i.e. b1 ≈ 3/2a1 and b2 ≈ 2a2. This means that the
+
+7
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+2
+β
+0
+5
+10
+15
+20
+25
+30
+35
+40
+3
+−
+10
+×
+0,0
+〉 
+2
+2
+ε 〈
+ - 
+〉 
+2
+2
+ε 〈
+0.00
+0.01
+0.02
+0.03
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+U+U Glauber Model
+-1
+10
+×
+ = 1.93
+1
+c
+b = 0 fm
+     Data  Fit
+2
+β
+σ
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+2
+β
+0
+0.2
+0.4
+0.6
+0.8
+1
+3
+−
+10
+×
+0,0
+{2}
+d
+  - C
+{2}
+d
+C
+U+U Glauber Model
+-3
+10
+×
+ = 5.94
+1
+c
+b = 0 fm
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+2
+β
+0
+5
+10
+15
+20
+25
+30
+35
+40
+3
+−
+10
+×
+0,0
+〉 
+2
+2
+ε 〈
+ - 
+〉 
+2
+2
+ε 〈
+0.04
+0.05
+0.06
+0.07
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+U+U Glauber Model
+-1
+10
+×
+) = (2.15,-0.68)
+3
+,c
+1
+(c
+b = 0 fm
+     Data  Fit
+2
+β
+σ
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+2
+β
+0
+0.2
+0.4
+0.6
+0.8
+1
+3
+−
+10
+×
+0,0
+{2}
+d
+  - C
+{2}
+d
+C
+U+U Glauber Model
+-3
+10
+×
+) = (8.89,-5.98)
+2
+,c
+1
+(c
+b = 0 fm
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+2
+β
+0
+5
+10
+15
+20
+25
+30
+35
+40
+3
+−
+10
+×
+0,0
+〉 
+2
+2
+ε 〈
+ - 
+〉 
+2
+2
+ε 〈
+0.08
+0.09
+0.10
+ 
+ 
+ 
+ 
+ 
+ 
+U+U Glauber Model
+-1
+10
+×
+) = (2.10,0.27,-0.91)
+3
+,c
+2
+,c
+1
+(c
+b = 0 fm
+     Data  Fit
+2
+β
+σ
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+2
+β
+0
+0.2
+0.4
+0.6
+0.8
+1
+3
+−
+10
+×
+0,0
+{2}
+d
+  - C
+{2}
+d
+C
+U+U Glauber Model
+-3
+10
+×
+) = (8.74,-3.62,-3.01)
+3
+,c
+2
+,c
+1
+(c
+b = 0 fm
+FIG. 3:
+The simultaneous fit of the ⟨ε2
+2⟩ (¯β, σβ) (left column) and ⟨(δd⊥/d⊥)2⟩ (¯β, σβ) (right column) calculated in U+U
+collisions with zero impact parameter. The top row shows the fits to Eq. 11 with only the leading term and the last row
+shows the fits with all three terms. The middle row show the fits including c1 and c3 terms for ⟨ε2
+2⟩ and c1 and c2 terms for
+⟨(δd⊥/d⊥)2⟩.
+distribution can also be described by the following alternative form,
+c2,ε{4} − c2,ε{4}β=0 ≈ a1
+2 (6¯β2σ2
+β + 3σ4
+β − ¯β4) + a2(¯β4σ2
+β + 27¯β2σ4
+β + 9σ6
+β − ¯β6)
+(14)
+
+8
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+3
+β
+0.8
+−
+0.7
+−
+0.6
+−
+0.5
+−
+0.4
+−
+0.3
+−
+0.2
+−
+0.1
+−
+0
+0.1
+3
+−
+10
+×
+0,0
+Cov  - Cov
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+U+U Glauber Model
+b = 0 fm
+-3
+10
+×
+ = -6.63
+1
+c
+    Data  Fit
+2
+β
+σ
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+3
+β
+4
+−
+2
+−
+0
+2
+4
+6
+8
+10
+12
+14
+6
+−
+10
+×
+0,0
+{3}
+d
+  - C
+{3}
+d
+C
+U+U Glauber Model
+b = 0 fm
+-4
+10
+×
+ = 1.36
+1
+c
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+3
+β
+0.8
+−
+0.7
+−
+0.6
+−
+0.5
+−
+0.4
+−
+0.3
+−
+0.2
+−
+0.1
+−
+0
+0.1
+0.2
+3
+−
+10
+×
+0,0
+Cov  - Cov
+0.06
+0.07
+0.08
+0.09
+0.10
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+U+U Glauber Model
+b = 0 fm
+-3
+) = (-8.96,-0.47,5.95)*10
+3
+,c
+2
+,c
+1
+(c
+    Data  Fit
+2
+β
+σ
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+3
+β
+4
+−
+2
+−
+0
+2
+4
+6
+8
+10
+12
+14
+6
+−
+10
+×
+0,0
+{3}
+d
+  - C
+{3}
+d
+C
+U+U Glauber Model
+b = 0 fm
+-4
+) = (3.11,-0.69,-2.74)*10
+3
+,c
+2
+,c
+1
+(c
+FIG. 4:
+The simultaneous fit of the ⟨ε2
+2
+δd⊥
+d⊥ ⟩ (¯β, σβ) (left column) and ⟨(δd⊥/d⊥)3⟩ (¯β, σβ) (right column) calculated in U+U
+collisions with zero impact parameter. The top row shows the results of the fit to Eq. 12 with only the leading term and the
+second row shows the fits with all three terms. The fit results imply that the contribution from ⟨β4⟩ is negligible, though.
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+2
+β
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+0.6
+3
+−
+10
+×
+0,0
+{4}
+ε
+2,
+ - c
+{4}
+ε
+2,
+c
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+U+U Glauber Model
+b = 0 fm
+-2
+10
+×
+) = (2.78,4.20,-1.55,-2.92)
+2
+,b
+2
+,a
+1
+,b
+1
+(a
+   Data  Fit
+2
+β
+σ
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+2
+β
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+0.6
+3
+−
+10
+×
+0,0
+{4}
+ε
+2,
+ - c
+{4}
+ε
+2,
+c
+0.06
+0.07
+0.08
+0.09
+0.10
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+U+U Glauber Model
+b = 0 fm
+-2
+10
+×
+) = (2.74,-1.64)
+2
+,a
+1
+(a
+Modified Fit Function
+   Data  Fit
+2
+β
+σ
+FIG. 5:
+The fit of the c2,ε{4}(¯β, σβ) data calculated in U+U collisions with zero impact parameter to Eq. 13 (left) and Eq. 14
+(right).
+The contribution of residual terms is only a few percent. Indeed, a fit of this form describes the data very well as
+shown in the right panel of Fig. 5. This behavior provides clear intuition on how the fluctuation terms containing σβ
+compete with the terms containing only ¯β. For example, assuming ¯β = σβ, the contribution from fluctuation-related
+terms is a factor of 9 (37) times the ¯β4 (¯β6) in the leading-order (subleading order). Thus, even a relatively small
+
+9
+0
+0.005
+0.01
+4
+β
+0.5
+−
+0.4
+−
+0.3
+−
+0.2
+−
+0.1
+−
+0
+3
+−
+10
+×
+0,0
+2〉
+2
+2
+ε〈
+ - k
+〉
+2
+4
+ε〈
+-
+2〉
+2
+2
+ε〈
+ - k
+〉
+2
+4
+ε〈
+2
+β
+σ
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.10
+k=2.541
+Nucleon Glauber
+k
+2.4
+2.5
+2.6
+2.7
+5
+10
+ (ab. units)
+2
+χ
+0
+0.005
+0.01
+4
+β
+0.5
+−
+0.4
+−
+0.3
+−
+0.2
+−
+0.1
+−
+0
+3
+−
+10
+×
+0,0
+2〉
+2
+2
+ε〈
+ - k
+〉
+2
+4
+ε〈
+-
+2〉
+2
+2
+ε〈
+ - k
+〉
+2
+4
+ε〈
+2
+β
+σ
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.10
+=2.530
+0
+k
+Quark Glauber
+k
+2.4
+2.5
+2.6
+2.7
+5
+10
+ (ab. units)
+2
+χ
+FIG. 6:
+The values of ⟨ε4
+2⟩−K ⟨ε2
+2⟩
+2 for the value of K that minimize the dependence on σβ in the nucleon Glauber model (left)
+and quark Glauber model (right). The inserts shows the K dependence of χ2, which is calculated as χ2 = ∑i ∑j(fij − ¯fi)2/σ2
+i.j,
+where fij = f(¯βi, σβ,j, ¯fi = ∑j fij/ ∑j and σij is the statistical error bar on the i, j-th data point.
+fluctuation could have a stronger impact on c2,ε{4} than the modestly large ¯β. Note that the liquid-drop model
+results in Table I predict b1 = 7/5a1, slightly smaller than the Glauber model expectation.
+Experimentally, we can measure ⟨v2
+2⟩ and ⟨v2
+4⟩, which are linearly related to ⟨ε2
+2⟩ and ⟨ε4
+2⟩, respectively. Thus, it is
+natural to ask whether one could constrain the ¯β and σβ from these two quantities. So far we have learned that the
+combination in the cumulant definition c2,ε{4} = ⟨ε4
+2⟩ − 2⟨ε2
+2⟩
+2 is not sufficient to achieve such separation. Motivated
+by this fact, we tried a more general combination f(¯β,σβ;k) = ⟨ε4
+2⟩ − k ⟨ε2
+2⟩
+2, and identify the k value for which the
+f(¯β,σβ;k) have the least variation in σβ. The best value found is k = k0 = 2.541, for which the data points follow an
+approximately-linear dependence ¯β4 as shown in the left panel of Fig. 6. The right panel of Fig. 6 shows a similar
+exercise in the quark Glauber model which gives a nearly identical k0 value. The data points yet do not collapse on a
+single curve, implying a small σβ dependence remaining. The amount of spread is estimated to be about 25% relative
+for a given ¯β, corresponding to a variation of ¯β of about 1 −
+4√
+0.75 = 7%. This 7% value is the best precision for
+determining the ¯β in the Glauber model using this method. The determined ¯β value can then be plugged into Eq. 11
+(considering only the leading order is sufficient for ⟨ε2
+2⟩ as shown in Fig. 3) to determine the σβ.
+In the study of flow fluctuations in heavy ion collisions, it is often desirable to calculate the normalized quantities
+between high-order cumulant and low-order cumulants, which have the advantage of canceling the final state effects.
+Here we study the following three quantities following the convention from Ref. [21],
+ρ =
+⟨ε2
+2
+δd⊥
+d⊥ ⟩ − ⟨ε2
+2
+δd⊥
+d⊥ ⟩
+β=0
+(⟨ε2
+2⟩ − ⟨ε2⟩β=0)
+√
+⟨( δd⊥
+d⊥ )
+2
+⟩ − ⟨( δd⊥
+d⊥ )
+2
+⟩
+β=0
+, ncd{3} =
+⟨( δd⊥
+d⊥ )3⟩ − ⟨( δd⊥
+d⊥ )3⟩
+β=0
+(⟨( δd⊥
+d⊥ )
+2
+⟩ − ⟨( δd⊥
+d⊥ )
+2
+⟩
+β=0
+)
+3/2 , ncε{4} = c2,ε{4} − c2,ε{4}β=0
+(⟨ε2
+2⟩ − ⟨ε2
+2⟩β=0)
+2
+(15)
+Since 96Zr has little quadruple deformation βZr ≈ 0, these quantities can be constructed from measurements in
+96Ru+96Ru and 96Zr+96Zr collisions.
+The results from our Glauber model calculation are shown in Fig. 7 for prolate nuclei cos(3γ) = 0. The values of ρ
+are nearly independent of ¯β and have a weak dependence on σβ. In the large ¯β region, ρ quickly converges to a value
+around −0.62 nearly independent of σβ. In the moderate ¯β region say ¯β ∼ 0.2, the ρ first decreases quickly to a value
+around -0.6, but then increases gradually with σβ. The values of ncd{3} have similar convergence trends towards large
+¯β around 0.4, but much more slowly compare to ρ. The ncε{4} has a negative and nearly constant value when σβ = 0,
+while it increases rather quickly with σβ. Even for a value of σ2
+β = 0.01, the ncε{4} stays positive until ¯β2 > 0.06. For
+larger values of σ2
+β, the ncε{4} decreases with increasing ¯β2, but always remains positive over the range of ¯β studied.
+V.
+SUMMARY
+We studied the impact of the fluctuations of nuclear quadrupole deformation on the heavy ion observables in a Monte
+Carlo Glauber model. In particular, we focus on eccentricity ε2 and inverse size d⊥ in each event, which can be related
+
+10
+2
+β
+0
+0.05
+0.1
+ρ
+0.6
+−
+0.5
+−
+0.4
+−
+0.3
+−
+   
+2
+β
+σ
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.10
+2
+β
+0
+0.05
+0.1
+{3}
+d
+nc
+0
+0.2
+0.4
+0.6
+0.8
+) = 0
+γ
+cos(3
+2
+β
+0
+0.05
+0.1
+{4}
+ε
+nc
+0
+0.5
+1
+FIG. 7:
+The normalized three-particle correlators, ρ (left) and ncd{3} (middle) and normalized four-particle correlator ncε{4}
+(right) defined in Eq. 11 as a function of ¯β2 for different values of σ2
+β.
+to the event-wise elliptic flow and mean transverse momentum in the final state. The triaxiality γ has a strong impact
+on three-particle correlators, but the impact diminishes for larger σγ. In particular, when σγ is large, the observables
+do not distinguish between prolate deformation and oblate deformation, i.e. the values of all observables approach
+those obtained in collisions of rigid triaxial nuclei with the same β. The mean and variance of quadrupole fluctuations,
+¯β and σ2
+β, have a strong influence on all observables. The influence on two-particle observables ⟨ε2
+2⟩ and ⟨(δd⊥/d⊥)2⟩ is
+proportional to ⟨β2⟩ = ¯β2 +σ2
+β, however, the ⟨(δd⊥/d⊥)2⟩ also has a sizable subleading order term proportional to ⟨β3⟩.
+The three-particle observables to the leading order are proportional to ⟨cos(3γ)β3⟩ = cos(3γ)¯β(¯β + 3σ2
+β), whereas the
+four-particle observables to the leading order are proportional to ⟨β4⟩ = ¯β4 + 6σ2
+β ¯β2 + 3σ4
+β. Hence, the variance of β
+fluctuation has a stronger impact than ¯β for these higher-order observables.
+By combining two and four-particle cumulant of ε2, we have constructed a simple formula to constrain parameters ¯β
+and σβ simultaneously. Such separation becomes less effective when σβ is comparable or larger than ¯β. In the future,
+it would be interesting to carry out a full hydrodynamic model simulation to quantify the efficacy of this method on
+the final state flow observables.
+This research is supported by DOE DE-FG02-87ER40331.
+Appendix
+The default results in this paper are obtained with the nucleon Glauber model. We have repeated the analysis for
+the quark Glauber model and compared it with the nucleon Glauber model results in Figs. 8 and 9 for the impact of
+γ fluctuation and β fluctuation, respectively. The trends are mostly very similar. A few exceptions are observed. In
+particular, the results of the two models are shifted vertically from each other in Fig. 8. In the case of β fluctuation
+in Fig. 9, the variance cd{2} and skewness cd{3} are systematically different between the two models in the high ¯β
+region.
+
+11
+0
+0.2
+0.4
+0.6
+0.8
+1
+2
+γ
+σ
+0.2
+−
+0.15
+−
+0.1
+−
+0.05
+−
+0
+0.05
+0.1
+0.15
+0.2
+3
+−
+10
+×
+=0
+β
+Cov  - Cov
+Nucleon Glauber
+Quark Glauber
+U+U Glauber Model
+ = 0.28
+β
+b = 0 fm, 
+0
+0.2
+0.4
+0.6
+0.8
+1
+2
+γ
+σ
+2
+−
+0
+2
+4
+6
+8
+6
+−
+10
+×
+=0
+β
+{3}
+d
+  - C
+{3}
+d
+C
+Nucleon Glauber
+Quark Glauber
+U+U Glauber Model
+ = 0.28
+β
+b = 0 fm, 
+0
+0.2
+0.4
+0.6
+0.8
+1
+2
+γ
+σ
+0.12
+−
+0.11
+−
+0.1
+−
+0.09
+−
+0.08
+−
+3
+−
+10
+×
+=0
+β
+{4}
+ε
+2,
+ - c
+{4}
+ε
+2,
+c
+Nucleon Glauber
+Quark Glauber
+U+U Glauber Model
+ = 0.28
+β
+b = 0 fm, 
+0
+0.2
+0.4
+0.6
+0.8
+1
+2
+γ
+σ
+15.8
+16
+16.2
+16.4
+16.6
+16.8
+3
+−
+10
+×
+=0
+β〉 
+2
+2
+ε 〈
+ - 
+〉 
+2
+2
+ε 〈
+)
+γ
+Cos(3
+1.00
+0.87
+0.50
+0.00
+-0.50
+-0.87
+-1.00
+Nucleon Glauber
+Quark Glauber
+U+U Glauber Model
+ = 0.28
+β
+b = 0 fm, 
+0
+0.2
+0.4
+0.6
+0.8
+1
+2
+γ
+σ
+0.54
+0.56
+0.58
+0.6
+0.62
+0.64
+0.66
+3
+−
+10
+×
+=0
+β
+{2}
+d
+  - C
+{2}
+d
+C
+Nucleon Glauber
+Quark Glauber
+U+U Glauber Model
+ = 0.28
+β
+b = 0 fm, 
+FIG. 8:
+Comparison of the five observables between nucleon Glauber model (symbols) and quark Glauber model (lines with
+matching colors) as a function of σγ for different values of ¯γ.
+!
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+2
+β
+0.8
+−
+0.7
+−
+0.6
+−
+0.5
+−
+0.4
+−
+0.3
+−
+0.2
+−
+0.1
+−
+0
+3
+−
+10
+×
+0,0
+Cov  - Cov
+Nucleon Glauber
+Quark Glauber
+U+U Glauber Model
+b = 0 fm
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+2
+β
+2
+−
+0
+2
+4
+6
+8
+10
+12
+6
+−
+10
+×
+0,0
+{3}
+d
+  - C
+{3}
+d
+C
+Nucleon Glauber
+Quark Glauber
+U+U Glauber Model
+b = 0 fm
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+2
+β
+0.1
+−
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+3
+−
+10
+×
+0,0
+{4}
+ε
+2,
+ - c
+{4}
+ε
+2,
+c
+Nucleon Glauber
+Quark Glauber
+U+U Glauber Model
+b = 0 fm
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+2
+β
+0
+5
+10
+15
+20
+25
+30
+35
+40
+3
+−
+10
+×
+0,0
+〉 
+2
+2
+ε 〈
+ - 
+〉 
+2
+2
+ε 〈
+2
+β
+σ
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.10
+Nucleon Glauber
+Quark Glauber
+U+U Glauber Model
+b = 0 fm
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+2
+β
+0
+0.2
+0.4
+0.6
+0.8
+1
+3
+−
+10
+×
+0,0
+{2}
+d
+  - C
+{2}
+d
+C
+Nucleon Glauber
+Quark Glauber
+U+U Glauber Model
+b = 0 fm
+FIG. 9:
+Comparison of the five observables between nucleon Glauber model (symbols) and quark Glauber model (lines with
+matching colors) as a function of ¯β2 for different values of σβ.
+
+12
+[1] W. Busza, K. Rajagopal, and W. van der Schee, Ann. Rev. Nucl. Part. Sci. 68, 339 (2018), arXiv:1802.04801 [hep-ph] .
+[2] J. E. Bernhard, J. S. Moreland, S. A. Bass, J. Liu,
+and U. Heinz, Phys. Rev. C 94, 024907 (2016), arXiv:1605.03954
+[nucl-th] .
+[3] G. Giacalone, (2022), arXiv:2208.06839 [nucl-th] .
+[4] M. Bender, P.-H. Heenen, and P.-G. Reinhard, Rev. Mod. Phys. 75, 121 (2003).
+[5] J. Jia and C.-J. Zhang, (2021), arXiv:2111.15559 [nucl-th] .
+[6] G. Nijs and W. van der Schee, (2021), arXiv:2112.13771 [nucl-th] .
+[7] J. Jia, G. Giacalone, and C. Zhang, (2022), arXiv:2206.10449 [nucl-th] .
+[8] M. Abdallah et al. (STAR), Phys. Rev. C 105, 014901 (2022), arXiv:2109.00131 [nucl-ex] .
+[9] Haojie Xu and Chunjian Zhang (STAR Collabration), Constraints on neutron skin thickness and nuclear deformations using
+relativistic heavy-ion collisions from STAR, “https://indico.cern.ch/event/895086/contributions/4724887/,https:
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+

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+1
+Super-resolution with Sparse Arrays: A Non-
+Asymptotic Analysis of Spatio-temporal Trade-offs
+Pulak Sarangi, Mehmet Can H¨uc¨umeno˘glu, Robin Rajam¨aki, and Piya Pal
+Abstract—Sparse arrays have emerged as a popular alternative to
+the conventional uniform linear array (ULA) due to the enhanced
+degrees of freedom (DOF) and superior resolution offered by
+them. In the passive setting, these advantages are realized by
+leveraging correlation between the received signals at different
+sensors. This has led to the belief that sparse arrays require
+a large number of temporal measurements to reliably estimate
+parameters of interest from these correlations, and therefore
+they may not be preferred in the sample-starved regime. In
+this paper, we debunk this myth by performing a rigorous non-
+asymptotic analysis of the Coarray ESPRIT algorithm. This
+seemingly counter-intuitive result is a consequence of the scaling
+of the singular value of the coarray manifold, which compensates
+for the potentially large covariance estimation error in the limited
+snapshot regime. Specifically, we show that for a nested array
+operating in the regime of fewer sources than sensors (S = O(1)),
+it is possible to bound the matching distance error between the
+estimated and true directions of arrival (DOAs) by an arbitrarily
+small quantity (ϵ) with high probability, provided (i) the number
+of temporal snapshots (L) scales only logarithmically with the
+number of sensors (P), i.e. L = Ω(ln(P)/ϵ2), and (ii) a suitable
+separation condition is satisfied. Our results also formally prove
+the well-known empirical resolution benefits of sparse arrays,
+by establishing that the minimum separation between sources
+can be Ω(1/P 2), as opposed to separation Ω(1/P) required by
+a ULA with the same number of sensors. In addition to the
+array geometry, our sample complexity expression reveals the
+dependence on other key model parameters such as Signal to
+Noise Ratio (SNR) and the dynamic range of the source powers.
+This enables us to establish the superior noise-resilience of nested
+arrays both theoretically and empirically. 1
+Index Terms—Sparse Arrays, Nested Sampling, Super-resolution,
+Toeplitz Covariance Matrix, Non-Asymptotic Guarantees.
+I. INTRODUCTION
+The problem of source localization arises in different contexts
+ranging from target detection in sonar and radar, hybrid
+mmWave channel estimation, and DOA estimation in array
+signal processing [1]–[3]. Traditionally, these applications
+consider ULAs, which are known to resolve up to S = O(P)
+sources with P sensors. However, deterministic sparse array
+geometries, such as nested and coprime arrays [1], [2], have
+recently gained significant attention primarily due to two
+attractive properties. Firstly, sparse arrays are able to identify
+up to S = O(P 2) uncorrelated sources using only P sensors.
+Secondly, sparse arrays enjoy a performance gain showcased
+by lower Cram´er-Rao bound and higher angular resolution [4]–
+[8]. Both of these properties can be attributed to the enhanced
+spatial DOF enabled by the so-called difference coarray, which
+can be as large as Θ(P 2).
+1This work was supported by Grants ONR N00014-19-1-2256, ONR
+N00014-19-1-2227, NSF 2124929, and NSF CAREER ECCS 1700506.
+The enhanced DOF of the coarray are realized by comput-
+ing temporal correlations between the spatial measurements
+and constructing an augmented covariance matrix called the
+“coarray covariance matrix”, whose size is determined by the
+size of the difference coarray. Following the construction of
+the coarray covariance matrix, it is possible to fully harness
+the power of the difference coarray and identify the unknown
+source directions using classical subspace techniques, such
+as MUSIC, ESPRIT or the matrix pencil method [9]–[11].
+Despite the success of coarray-based algorithms, a common
+belief is that they require a large number of temporal snapshots
+to fully utilize the number of DOFs provided by the coarray.
+The root of this belief mainly lies in the inadequacy of
+existing performance analyses, which are primarily based on
+characterizing the asymptotic Mean Squared Error (MSE) of
+the Coarray MUSIC [5] and Coarray ESPRIT algorithms [12].
+In particular, such asymptotic results primarily rely on the
+first-order perturbation analysis framework proposed in [13],
+which leaves two key questions unanswered regarding the
+performance of coarray algorithms. Firstly, the perturbation
+framework fails to theoretically explain the improvement in
+resolution offered by sparse arrays over the ULA—a phe-
+nomenon that has been extensively observed in numerical ex-
+periments [5], [14]. Secondly, the analysis does not adequately
+reveal the dependence of temporal snapshots on key model
+parameters such as the array geometry, number of sensors,
+SNR and dynamic range of the source powers.
+The aforementioned shortcomings are partially addressed in
+[15], which adapts recent advances in the theory of super-
+resolution [16], [17] to the coarray setting. The analysis,
+which is based on Total-Variational norm minimization, is
+indeed non-asymptotic. However, it is possible to show that
+the snapshot requirement in this setting scales quadratically
+(rather than linearly) with the number of sensors P, which
+is undesirable. In a parallel line of work using a grid-based
+model, we recently showed that Ω(P 2) snapshots are sufficient
+for ensuring exact support recovery with high probability
+even for closely-spaced sources, where the smallest source
+separation scales as Ω(1/P 2) [18]. Although the analysis is
+applicable for scenarios where S > P (more sources than
+sensors), the sample complexity Ω(P 2) is still conservative
+when S ≤ P. In [19], an atomic norm formulation is adopted
+to exploit the Toeplitz structure of the coarray covariance
+matrix. The analysis provides a characterization of the covari-
+ance matrix estimation error, but not of the sample complexity
+required to achieve a desired DOA estimation error, which is
+often the main quantity of interest. Indeed, common folklore
+arXiv:2301.01734v1  [eess.SP]  4 Jan 2023
+
+2
+suggests that the benefits of sparse arrays necessarily come
+at the cost of a large number of snapshots, since the coarray
+covariance matrix, which typically needs to be estimated, is of
+size Θ(P 2). Hence, one might be tempted to falsely conclude
+that sparse arrays are at a disadvantage compared to ULAs.
+In this paper, our goal is to dispel this belief by providing
+new non-asymptotic results on the performance of Coarray
+ESPRIT with a focus on nested arrays in the regime S ≤ P.
+Our analysis is motivated by contemporary applications such
+as autonomous sensing and mmWave channel estimation [3],
+[14], where identifying more sources than sensors may not be
+necessary, and the number snapshots may be restricted either
+due to coherent multipaths or a rapidly varying environment.
+While subspace-based algorithms have been around for several
+decades and actively used in practice, performance guarantees
+characterizing their precise resolution limit were obtained only
+recently [20]–[24]. This analysis has also been extended to
+multi-snapshot setting in [25]. The key factor enabling these
+guarantees is the characterization of the smallest singular value
+of Vandermonde matrices [24]. However, all the aforemen-
+tioned results are only applicable to the ULA. Furthermore,
+no statistical assumptions are made on the source signals, and
+hence, the coarray perspective is missing. The key difference
+between deterministic and random sources is that in the latter
+case, the perturbation to the subspace of interest is a con-
+sequence of both noise as well as finite-snapshot covariance
+estimation error. Therefore, extending the analysis in [20], [25]
+to the stochastic case requires non-trivial modifications.
+Contributions: Our first main contribution is to probabilis-
+tically characterize the coarray covariance matrix estimation
+error due to finite snapshots. Our second main contribution is a
+non-asymptotic performance analysis for the Coarray ESPRIT
+algorithm in terms of the matching distance error metric.
+Specifically, we characterize the number of temporal snapshots
+(sample complexity) required to bound the matching distance
+error by a specified parameter. To the best of our knowledge,
+our sample complexity expression (in terms of snapshots) is
+the first to explicitly bring out the dependence on key model
+parameters such as the array geometry, SNR and dynamic
+range of the source powers. Furthermore, we establish that it
+is possible to bound the matching distance error with an arbi-
+trarily small quantity for both the nested array and ULA, using
+the (order-wise) same number of snapshots L = Ω(ln P).
+However, a nested array can achieve this in a much smaller
+separation regime ∆min = Ω(1/P 2) compared to the ULA, for
+which ∆min = Ω(1/P). Our analysis dispels the widely-held
+belief that sparse arrays require significantly more snapshots
+compared to ULAs when the number of sources is less than
+the number of sensors, and at the same time establishes the
+superior resolution capabilities of nested arrays. In addition
+to advancing the theoretical understanding, this analysis could
+also serve as a guiding principle for practitioners to determine
+suitable operating conditions. Notations: Symbol ⊙ represents
+the Khatri-Rao (columnwise Kronecker) product, whereas ∥·∥2
+and ∥·∥F denote the spectral and Frobenius norm of a matrix.
+Moreover, σi(A) is the i-th largest singular value of A. For
+a set real numbers {p1, p2, . . . , pK}, pmin and pmax denote
+the minimum and maximum numbers in the set, respectively.
+The symbol T := [0, 1) denotes the torus. For a sub-Gaussian
+random variable X, ∥X∥ψ2 denotes its sub-Gaussian norm
+defined as ∥X∥ψ2 := inf{t > 0 | E[exp X2/t2] ≤ 2}.
+II. BACKGROUND ON SPARSE ARRAYS
+Consider a sparse linear array (SLA) with P sensors located
+at {dpλ/2}P
+p=1, where λ is the wavelength of the incoming
+far-field narrow-band source signals and dp belongs to an
+integer set S (|S| = P). Suppose S sources with distinct
+DOAs θ = {θ1, θ2, · · · , θS} impinge on the array where
+θi ∈ (−π/2, π/2] for i = 1, . . . , S. The signal received at
+the P sensors at time instance t is given by:
+y(t) = AS(θ)x(t) + n(t),
+t = 1, . . . , L.
+(1)
+The
+matrix
+AS(θ)
+=
+[aS(θ1), aS(θ2), . . . , aS(θS)]
+∈
+CP ×S
+is the array manifold matrix where: aS(θi)
+=
+[ejπd1 sin(θi), ejπd2 sin(θi)
+. . . ejπdP sin(θi)]⊤, represents the
+steering vector corresponding to the direction θi, L denotes
+the total number of temporal snapshots, x(t) ∈ CS is the tth
+temporal snapshot of the source signal vector and n(t) ∈ CP
+is an additive noise term. We define the normalized spatial
+frequencies (which we refer to as normalized DOAs) as
+ωi = sin(θi)/2. Throughout this paper, we make the following
+statistical assumptions on the source signals and noise:
+[A1] Uncorrelated Gaussian Sources: The source signals
+x(t) are assumed to be uncorrelated white circularly sym-
+metric Gaussian CN(0, P) where P = diag(p1, p2, . . . , pS)
+represents a diagonal covariance matrix of source powers.
+[A2] Gaussian Noise: The noise n(t) follows a zero-mean
+circularly symmetric complex Gaussian distribution n(t) ∼
+CN(0, σ2I), and is uncorrelated with x(t).
+Under assumptions [A1-A2], the measurements follow y(t) ∼
+CN(0, Ry), where Ry is given by:
+Ry = AS(θ)PAH
+S (θ) + σ2IP ∈ CP ×P .
+(2)
+By vectorizing Ry, we obtain the “virtual measurements”:
+ry = (AS(θ)∗ ⊙ AS(θ))p + σ2i, where i = vec(IP ) and
+p = [p1, . . . , pS]T . The matrix AS(θ)∗ ⊙ AS(θ) can be
+viewed as a “virtual array” with sensor locations given by
+the difference set of the SLA.
+Definition
+II.1
+(Difference
+Set).
+Given
+a
+SLA
+S
+=
+{d1, d2, · · · , dP },
+its
+difference
+set
+DS
+is
+defined
+as:
+DS = {dm − dn|dm, dn ∈ S}.
+The difference set DS of S is also called its virtual difference
+coarray. Let Mca > 0 be the largest integer such that the
+set US := {0, 1, . . . , Mca} satisfies US ⊆ DS. This set
+US denotes the largest contiguous non-negative segment of
+the difference set and is essentially a ULA with Mca + 1
+sensors. By harnessing the structure of US, sparse arrays enjoy
+enhanced degrees of freedom over the physical SLA. An
+array is called hole-free if its difference set is a ULA, i.e.,
+DS = {−Mca, · · · , Mca}. We now introduce the notation for
+a “generalized nested array”, which is a special hole-free array.
+
+3
+Definition II.2 (Nested array). A generalized nested array
+S(N1,N2) with N1 ≥ N2 > 0, is defined as: S(N1,N2) =
+{n}N1
+n=1 ∪ {m(N1 + 1)}N2
+m=1.
+It can be shown that any nested array S(N1,N2) is hole-free,
+i.e., US = {0, 1, · · · , Mca} with Mca = N2(N1 + 1) − 1.
+Furthermore, Sula = S(P −1,1), i.e., choosing N1 = P − 1 and
+N2 = 1, yields a ULA with P sensors. For a given P, if
+N1 = ⌈ P
+2 ⌉, N2 = ⌊ P
+2 ⌋, then Mca + 1 = ⌊ P
+2 ⌋(⌈ P
+2 ⌉ + 1). It can
+be verified that for P ≥ 3, we have:
+P 2/5 ≤ Mca + 1 ≤ P 2.
+(3)
+Therefore, Mca = Θ(P 2) is indeed achievable. Next, we
+introduce an important quantity that is essential for describing
+correlation-based processing.
+Definition II.3 (Weight Function). Consider a hole-free array
+S. For every i ∈ DS, its weight function is defined as |Ωi|:
+Ωi = {(m, n)|dm − dn = i, 1 ≤ m, n ≤ P} where the set Ωi
+essentially captures all pairs (dm, dn) of sensor locations that
+generate the difference of i = dm − dn.
+Due to symmetry, it can be verified that |Ωi| = |Ω−i|. Next,
+we review the widely-used “redundancy averaging” technique
+used for correlation-domain processing. Following [1], [5],
+[26], the virtual ULA measurements are given by:
+t = Favry,
+(4)
+where t = [t−Mca, · · · , t−1, t0, t1, · · · , tMca]⊤ and Fav is the
+redundancy averaging matrix given by:
+[Fav]i+Mca+1,m+P (n−1) =
+�
+1
+|Ωi|
+If dm − dn = i
+0
+Otherwise,
+(5)
+with −Mca ≤ i ≤ Mca and 1 ≤ m, n ≤ P. The element
+ti is obtained by averaging all entries [Ry]m,n whose indices
+(m, n) generate a difference of i, i.e., dm − dn = i. Define
+a Toeplitz operator TMca : C2Mca+1 → CMca+1×Mca+1 as:
+[TMca(z)]m,n = zMca+1+m−n, 1 ≤ m, n ≤ Mca + 1.
+If
+the
+vector z ∈ C2Mca+1 is conjugate symmetric, i.e., zMca+1+i =
+z∗
+Mca+1−i, i = 0, 1, . . . , Mca, then TMca(z) is a Hermitian
+matrix. Using the virtual measurement t, an augmented vir-
+tual co-array covariance matrix Tca ∈ C(Mca+1)×(Mca+1) is
+constructed as follows:
+Tca := TMca(t) = AUS(θ)PAUS(θ)H + σ2IMca+1.
+(6)
+Once this virtual coarray covariance matrix has been obtained,
+any subspace-based algorithm [9], [10] applied to Tca can
+exactly recover the source DOAs provided Mca ≥ S. Hence,
+this also reveals that by efficiently designing sparse arrays,
+we can resolve up to Θ(P 2) sources with only P sensors. In
+the next section, we describe how the correlation processing
+is modified in the finite snapshot setting.
+A. Finite-Snapshot Coarray Covariance Estimation
+Let �Ry be the sample covariance matrix given by:
+�Ry := 1
+L
+L
+�
+t=1
+y(t)y(t)H.
+(7)
+With a finite L, all the operations on the true covariance matrix
+are replaced by operations on the sample covariance matrix.
+First, we apply the redundancy averaging on ˆry:
+ˆt := Favˆry, where ˆry := vec( �Ry).
+(8)
+Here ˆt
+=
+[ˆt−Mca, · · · , ˆt−1, ˆt0, ˆt1, · · · , ˆtMca]⊤ with ˆti
+=
+1
+|Ωi|
+�
+dm−dn=i[ �Ry]m,n. Next, the estimated coarray covari-
+ance matrix is obtained by constructing a Toeplitz Hermitian
+matrix from ˆt as follows:
+�Tca = TMca(ˆt).
+(9)
+For a hole-free sparse array S, from (4), the elements of the
+matrix Ry are given by:
+[Ry]m,n = tdm−dn 1 ≤ m, n ≤ P.
+(10)
+Similarly, using the estimated coarray covariance matrix �Tca,
+we define matrix Rav ∈ CP ×P as
+[Rav]m,n := �tdm−dn, 1 ≤ m, n ≤ P.
+(11)
+This essentially maps the entries ˆti into a P × P matrix with
+the assignments specified by the difference set of the array
+S. Since the sample covariance matrix �Ry is imperfect, the
+estimate �Tca also incurs an error due to a finite number of
+snapshots. We denote the covariance estimation error as:
+EL = Tca − �Tca.
+(12)
+The error in estimating the coarray covariance matrix naturally
+causes errors in DOA estimation as well. Since subspace based
+algorithms are typically applied to this estimated covariance
+matrix �Tca, it becomes crucial to probabilistically characterize
+the estimation error EL and how it affects the DOA estimation
+error. This paper provides such a rigorous theoretical charac-
+terization of the DOA estimation error with limited snapshots.
+B. Review of Existing Performance Analysis of Coarray-Based
+Angle Estimation
+The existing performance analyses for coarray-based algo-
+rithms are largely asymptotic in nature. In particular, they rely
+on the first-order perturbation analysis framework proposed
+in [13], which has been used to obtain expressions for the
+mean square error (MSE) of coarray MUSIC [5], and coarray
+ESPRIT [12]. Consider the eigen decomposition Tca
+=
+UΓsU + U⊥ΓnUH
+⊥, where U ∈ CMca+1×S and U⊥ ∈
+CMca+1×Mca+1−S denote the eigenvectors corresponding to
+the signal and noise subspaces, respectively. The correspond-
+ing perturbed matrices are denoted as �Tca = Tca + ∆Tca,
+�U⊥ = U⊥ + ∆U⊥ and �Γn = Γn + ∆Γn. The perturbed
+matrices satisfy: (Tca + ∆Tca)(U⊥ + ∆U⊥) = (U⊥ +
+∆U⊥)(Γn+∆Γn). The perturbation analysis in [5] hinges on
+(i) the perturbations being “small enough” and (ii) ignoring
+the higher order perturbation terms such as ∆Tca∆U⊥ etc.
+One of the key drawbacks of this analysis is that a rigorous
+characterization of an upper bound on the “small enough
+perturbation” ∥∆Tca∥2 ≤ ϵ1 has not been provided explicitly.
+Secondly, [5, Theorem 1] makes a critical assumption that “the
+signal subspace and the noise subspace are well-separated”.
+
+4
+This assumption leaves open the possibility of problematic
+(unidentifiable) source configurations, which have not been
+explicitly addressed in their analysis. We address both of the
+aforementioned issues by adopting a non-asymptotic analysis
+framework that is free from any approximations. Our analysis
+also explicitly characterizes source configurations that ensure
+separation between the so-called signal and noise subspaces. In
+[18], the first rigorous non-asymptotic probabilistic guarantees
+were provided for support recovery using a grid-based model.
+Although their analysis is valid for S > P, the sample
+complexity L = Ω(P 2) is conservative when S < P as our
+analysis in Section IV will show.
+III. PERFORMANCE ANALYSIS OF COARRAY ESPRIT
+WITH FINITE SNAPSHOTS
+The Coarray ESPRIT algorithm, an adaptation of ESPRIT in
+the coarray domain, was introduced in [12]. It applies ESPRIT
+on the estimated coarray covariance matrix �Tca as opposed to
+covariance matrix �Ry of the physical measurements. For a
+self-contained exposition, we review the Coarray ESPRIT al-
+gorithm and point out certain invariance properties of Coarray
+ESPRIT. We describe Coarray ESPRIT for the ideal coarray
+covariance matrix Tca. The extension to the sample covariance
+estimate is straightforward.
+A. The Coarray ESPRIT Algorithm
+The coarray signal subspace is defined as the span of
+the steering vectors: Sca := R (AUS(θ)) . Matrix T0 :=
+AUS(θ)PAUS(θ)H is positive semi-definite and permits the
+following eigendecompostion: T0 = BΓBH, where the di-
+agonal of Γ comprises of the eigenvalues ordered in non-
+increasing fashion and B is a unitary matrix. We can partition
+B as B = [U, U⊥], where the columns of U ∈ C(Mca+1)×S
+denote the eigenvectors of T0 corresponding to its non-zero
+eigenvalues. Following this decomposition, we write Tca as:
+Tca = T0 + σ2IMca+1 = B(Γ + σ2IMca+1)BH.
+(13)
+If Mca ≥ S, the Vandermonde structure of AUS(θ) allows us
+to argue that rank(AUS(θ)PAUS(θ)H) = S, and hence:
+Sca = R(AUS(θ)) = R(AUS(θ)PAUS(θ)H) = R(U). (14)
+As a result of (14), ∃ an invertible Q ∈ CS×S such that
+U = AUS(θ)Q.
+(15)
+Let U0 ∈ CMca×S and U1 ∈ CMca×S denote the submatrices
+corresponding to the first and last Mca rows of U. Similarly,
+let V0, V1 ∈ CMca×S be the submatrices corresponding to the
+first and last Mca rows of AUS(θ). Due to the Vandermonde
+structure of AUS(θ), the following holds: V1 = V0D, where
+D = diag(ejπ sin(θ1), ejπ sin(θ2), . . . , ejπ sin(θS)). By (15), ma-
+trices U0 and U1 satisfy:
+U0 = V0Q,
+U1 = V0DQ.
+(16)
+Now, consider the matrix
+Ψ = U†
+0U1 ∈ CS×S.
+(17)
+Since U0 has full column rank (17) implies U†
+0 = Q−1V†
+0.
+Plugging this in (17) and combining with (16), we have:
+Ψ = Q−1DQ. Hence, the DOAs can be inferred from the
+eigenvalues of Ψ. Since L is finite, we do not have access to
+Tca and Coarray ESPRIT is instead applied on its estimate
+�Tca defined in (9). If we can ensure that the error EL is
+small enough (which we will rigorously specify using Weyl’s
+inequality), �Tca will be at least rank-S. Let ˆU be the matrix
+of eigenvectors corresponding to the largest S eigenvalues of
+�Tca (which is well-defined). We can consider �U as a basis of
+the perturbed coarray signal space ˆSca. From �U, we compute
+the matrices �U0, �U1, �Ψ following the same construction as
+U0, U1 and Ψ. Let ˆλi = riej �φi be the polar representation
+of the eigenvalues of the matrix ˆΨ. The estimated normalized
+frequencies ˆΩ = {�ωi}S
+i=1 are then given by �ωi =
+�φi
+2π.
+B. Basis Invariance Property of ESPRIT
+In the previous section, ESPRIT is performed using the basis
+given by the singular vectors �U (U) of �Tca (Tca). However,
+the following Lemma shows that the output of ESPRIT is
+invariant to the choice of the basis for the subspace.
+Lemma 1. Let �U ∈ C(Mca+1)×S be another basis for R( �U).
+Then, the matrix �Ψ := �U†
+0 �U1 is similar to the matrix �Ψ, i.e.,
+�Ψ and �Ψ share the same eigenvalues.
+Proof. Since R( �U) = R( �U), there exists an invertible matrix
+W ∈ CS×S such that �U :=
+�UW. Thus, the following
+holds: �U0 = �U0W, �U1 = �U1W. Since W is an invertible
+matrix, �U†
+0 = W−1 �U†
+0 and �Ψ = �U†
+0 �U1 = W−1 �U†
+0 �U1W =
+W−1 �ΨW. This completes the proof.
+C. Covariance Estimation Error
+In this section, we obtain tail bounds on ∥EL∥2 in terms of
+array parameters in a finite snapshot setting. Such a bound
+brings out the effect of the array geometry on the estimation
+error. Our analysis leverages recent results derived in [27]
+which we specialize for complex Toeplitz Hermitian matrices.
+Some of our intermediate steps depart from [27] by invoking
+a result on the bounding the supremum of a certain spectral
+function from [28]. We first introduce the key quantities
+and intermediate results on bounding the spectral norm of a
+Toeplitz Hermitian matrix from [28], [29].
+Let M
+∈
+CN×N be any Hermitian symmetric Toeplitz
+matrix. Such a matrix can be completely described by only
+its first column. Consider the “spectral function” associated
+with m = [m−(N−1), . . . , m−1, m0, m1, . . . , mN−1]⊤ [29]:
+fm(θ) = �N−1
+k=−(N−1) mk exp(−jkθ), where mk = Mk+1,1
+and m−k = m∗
+k as a result of the Hermitian Toeplitz structure.
+Evidently, the spectral function is a trigonometric polynomial
+of order N − 1 [28] whose coefficients are determined by the
+vector m. This spectral function fm(θ) can be used to bound
+∥M∥2 as indicated by the following lemma from [27]–[29]:
+Lemma 2. Let M∈CN×N be a Hermitian symmetric Toeplitz
+matrix and fm be the associated spectral function. Then,
+∥M∥2 ≤ supθ∈[−π,π] |fm(θ)|.
+
+5
+Lemma 2 indicates that the spectral norm of a Hermitian
+symmetric Toeplitz matrix can be bounded by the supre-
+mum of its associated spectral function. Note that the co-
+variance estimation error EL = Tca − �Tca is a Toeplitz
+Hermitian matrix, satisfying EL
+=
+T (e), where e
+=
+[e−Mca, . . . , e−1, e0, e1, . . . , eMca]T is conjugate symmetric
+and ei = ti − ˆti. Therefore, to bound ∥EL∥2 using Lemma 2,
+we need to investigate the spectral function fe(θ):
+fe(θ) :=
+Mca
+�
+k=−Mca
+ek exp(−jθk).
+(18)
+Towards this purpose, define Λ(θ), for 1 ≤ m, n ≤ P,
+[Λ(θ)]m,n =
+1
+|Ωdm−dn| exp(j(dm − dn)θ),
+(19)
+and Ey := Ry − Rav, where Ry and Rav are defined in (10)
+and (11), respectively. The elements of Ey are given by:
+[Ey]m,n = tdm−dn − �tdm−dn = edm−dn, 1 ≤ m, n ≤ P. (20)
+Proposition 1 provides a compact representation of fe(θ).
+Proposition 1. Let fe(θ) be the spectral function defined in
+(18). Then, the following equality holds: fe(θ) = tr (EyΛ(θ))
+where Λ(θ) and Ey are defined in (19) and (20), respectively.
+Proof.
+tr (EyΛ(θ)) =
+P
+�
+m,n=1
+[Ey]m,n[Λ(θ)]n,m =
+P
+�
+m,n=1
+edm−dn
+e−j(dm−dn)θ
+|Ωdm−dn|
+=
+Mca
+�
+s=��Mca
+�
+m,n
+dm−dn=s
+es
+exp(−jsθ)
+|Ωs|
+=
+Mca
+�
+s=−Mca
+es exp(−jsθ).
+We introduce a quantity referred to as “Redundancy coeffi-
+cient” that will play an important role in bounding ∥EL∥2.
+Definition III.1 (Redundancy Coefficient). Given a hole-free
+sparse array S, let Mca be the largest element in its differ-
+ence set DS. The redundancy coefficient ∆(S) is defined as:
+∆(S) := �Mca
+i=0
+1
+|Ωi|, where set Ωi is defined in Definition II.3.
+The quantity ∆(S) is controlled by the redundancy pattern of
+the sparse array S, i.e., the number of times an element repeats
+in the difference set. We provide an illustrative example to
+show how the quantity ∆(S) grows as a function of P.
+Lemma 3. Given a generalized nested array S(N1,N2)
+nest
+with
+P := N1 + N2 ≥ 3 sensors, the following holds: ln(P) ≤
+∆(S(N1,N2)
+nest
+)
+≤
+2 ln(P), if N2
+=
+1, and P 2/16
+≤
+∆(S(N1,N2)
+nest
+) ≤ P 2, if N1 = ⌈P/2⌉ and N2 = ⌊P/2⌋ ≥ 2.
+Proof. Case I (N2 = 1): The choice N2 = 1 corresponds to
+a ULA, with P = N1 + 1 sensors and |Ωi| = P − i, i ≥ 0.
+Therefore, ∆(S(P −1,1)
+nest
+) = �P −1
+i=0
+1
+P −i. Such a harmonic sum
+can be bounded as ln(P) ≤ �P −1
+i=0
+1
+P −i ≤ 1+ln(P) [30]. For
+P ≥ 3, we get the desired bound since 1 + ln(P) ≤ 2 ln(P).
+Case II (N2 = ⌊P/2⌋ ≥ 2): The differences between the
+elements of the outer and inner ULA which are of the form
+k = i(⌈P/2⌉ + 1) − j, 2 ≤ i ≤ ⌊P/2⌋ and 1 ≤ j ≤
+⌈P/2⌉, satisfy |Ωk| = 1. Therefore, we have ∆(S(N1,N2)
+nest
+) ≥
+⌈P/2⌉⌊P/2⌋/2 ≥ (P 2/8 − P/8) ≥ P 2/16, where the first
+inequality follows from ⌊P/2⌋ − 1 ≥ ⌊P/2⌋/2 and the last
+inequality uses P ≤ P 2/2 for P ≥ 2. Since S(N1,N2)
+nest
+is hole
+free, it implies 1/|Ωi| ≤ 1 for all 0 ≤ i ≤ Mca. Therefore, we
+can bound ∆(S(N1,N2)
+nest
+) ≤ Mca + 1 ≤ P 2.
+As the following Theorem will show, ∆(S) determines the
+sample complexity for controlling the covariance estimation
+error. Therefore, with the same number of sensors, two differ-
+ent array geometries could require drastically different sample
+complexity for ensuring that the covariance estimation error is
+bounded by the same quantity with high probability.
+Theorem 1. Consider the measurement model (1) obeying
+assumptions [A1-A2], where S is a hole-free sparse array with
+redundancy coefficient ∆(S). Let Tca ∈ CMca+1×Mca+1 be
+the coarray covariance matrix defined in (6) and �Tca be its
+estimate given by (9). For any ϵ ≥ 0, we have
+P
+�
+∥Tca − �Tca∥2 ≥ ϵ
+�
+≤ 8Mca exp
+�
+−c1L min
+�
+c2ϵ2
+∥Ry∥2
+2∆(S) ,
+ϵ
+∥Ry∥2
+�
+∆(S)
+��
+,
+where c1 and c2 are a positive universal constants.
+Proof. The proof is in Appendix A-B.
+D. Frequency/Angle Estimation Error of Coarray ESPRIT
+We next bound the DOA estimation error in terms of the
+covariance estimation error EL. Finally, we will combine this
+bound with the probabilistic bounds on ∥EL∥2 in Theorem 1
+to obtain the main sample complexity result (in Theorem 3).
+We will use the matching distance metric, defined as follows
+[20]:
+md(θ, ˆθ) := min
+Π∈P max
+j
+min
+k∈Z |ˆωΠ(j) − ωj + k|
+(21)
+where ωi (ˆωi) are the normalized DOAs and P denotes the
+set of all possible permutations on {1, 2, · · · , S}.
+For our analysis, we will use an additional assumption that
+will be invoked whenever suitable:
+[A3] The number of sources S = O(1), i.e., S is held
+constant and does not grow with P.
+Eigen Gap condition: Define:
+β := pminσ2
+S(AUS(θ)) − σ2.
+(22)
+Henceforth, we will refer the condition β > 0 as the “eigen
+gap condition” and it will play an important role in our analy-
+sis. Recall, from the definition of Tca = AUS(θ)PAUS(θ)H +
+σ2, β > 0 ensures that there is a margin between the smallest
+singular value of AUS(θ)PAUS(θ)H and the (S+1)th singular
+value of Tca (determined by the noise σ) as pminσ2
+S(AUS(θ))
+is a lower bound on σS(Tca). The following theorem relates
+the DOA estimation error in terms of matching distance to the
+covariance estimation error EL, provided the latter is upper
+bounded by a suitable quantity.
+Theorem 2. Let S be a hole-free sparse linear array with P
+sensors. Let Tca ∈ CMca+1×Mca+1 be the coarray covariance
+
+6
+matrix defined in (6) and �Tca be its estimate given by (9).
+If assumption [A3] holds and the following conditions are
+satisfied:
+β > 0
+and ∥EL∥2 ≤ CSβ
+(23)
+then the matching distance error of ESPRIT algorithm satisfies
+md(θ, ˆθ) ≤ q∥EL∥2
+(24)
+where
+EL,
+β
+are
+defined
+in
+(12),
+(22),
+q
+=
+(C′
+S
+√Mca + 1)/(βσS(AUS(θ))). Quantities CS, C′
+S
+are
+dependent only on S which is assumed to be O(1).
+Proof. See Appendix B-A.
+The following Lemma obtains both lower and upper bounds
+on the spectral norm ∥Ry∥2 that are valid regardless of the
+array geometry.
+Lemma 4. Consider the covariance matrix Ry given by (2),
+where S is any (sparse) array. Given a fixed S, signal powers
+p and noise power σ2, for all θ the following holds:
+pminP ≤ ∥Ry∥2 ≤ pmaxPS + σ2.
+(25)
+Proof. For any S, we can bound the spectral norm ∥Ry∥2 as:
+∥Ry∥2 = σ1(AS(θ)PAS(θ)H) + σ2 ≤ pmaxσ1(AS(θ))2 + σ2
+≤ pmaxPS + σ2
+where
+the
+last
+inequality
+follows
+from
+the
+fact
+that
+σ1(AS(θ))2
+≤
+∥AS(θ)∥2
+F
+=
+PS. Similarly, we can
+lower bound the norm ∥Ry∥2 ≥ σ1(AS(θ)PAS(θ)H) ≥
+pminσ2
+1(AS(θ)) ≥ pmin∥AS(θ)∥2
+F /S = pminP.
+Combining Theorem 1 and 2, we next present a sufficient
+condition on the number (L) of snapshots in terms of the
+model parameters (array geometry, SNR and source config-
+uration) that allows us to bound the matching distance error
+by a prescribed ϵ with probability at least 1 − δ.
+Theorem 3. Consider the measurement model (1), where S
+is a hole-free sparse array. Suppose β > 0 and the statistical
+assumptions [A1-A3] hold. Then for any 0 < δ < 1 and ϵ > 0,
+the matching distance error satisfies md(θ, ˆθ) ≤ min(ϵ, CSβq)
+with probability at least 1 − δ, provided
+L≥c3 ln
+�8Mca
+δ
+�
+max
+�
+q2
+1∆(S)
+c2ϵ2
+,q1
+�
+∆(S)
+ϵ
+,L2
+0
+c2
+,L0
+�
+. (26)
+Here q1 = q∥Ry∥2, L0 = ∥Ry∥2
+�
+∆(S)/(CSβ) and c2, c3
+are universal constants.
+Proof. See Appendix B-C
+Corollary 1. Consider the measurement model (1), where
+S is a hole-free sparse array. Suppose β
+> 0 and the
+statistical assumptions [A1-A3] hold. Then for any 0 < δ < 1
+and 0 < ϵ ≤ q min(CSβ, pminP
+�
+∆(S)/c2), the matching
+distance error satisfies md(θ, ˆθ) ≤ ϵ with probability at least
+1 − δ provided
+L ≥ c3 ln (8Mca/δ) q2
+1∆(S)/(c2ϵ2),
+(27)
+where q1, L0,c2, c3 are given in Theorem 3.
+Proof. Using the lower bound on ∥Ry∥2 from Lemma 4,
+we can see ϵ
+≤
+min(CSβq, q1
+�
+∆(S)/c2). Since β
+≥
+ϵ/(CSq),
+this
+implies
+L0
+≤
+q1
+�
+∆(S)/ϵ.
+This
+in-
+equality
+also
+implies
+L2
+0/c2
+≤
+q2
+1∆(S)/(c2ϵ2).
+Us-
+ing ϵ
+≤
+q1
+�
+∆(S)/c2, we can conclude that L0
+≤
+(q1
+�
+∆(S)/ϵ2)(q1
+�
+∆(S)/c2) = q2
+1∆(S)
+c2ϵ2 . Therefore, (27) im-
+plies (26) since max( q2
+1∆(S)
+c2ϵ2 ,
+q1√
+∆(S)
+ϵ
+, L2
+0
+c2 , L0) = q2
+1∆(S)
+c2ϵ2 , and
+the proof is completed.
+Role of redundancy coefficient in determining Temporal
+Sample Complexity: Corollary 1 indicates that if the number
+of snapshots grows proportional to the redundancy coefficient
+∆(S), then it is possible to bound the matching distance error
+by an arbitrarily small ϵ. Recall that ∆(S) is a function of
+the redundancy pattern of S and from Lemma 3 we have
+∆(Sula) = Θ(ln(P)) and ∆(Snest) = Θ(P 2). Based on this, at
+a cursory glance, one may be tempted to conclude from (27)
+that for the same number of sensors, the snapshot requirement
+for the nested array is significantly larger than for the ULA.
+This is also consistent with an existing misconception that co-
+array based processing requires a large number of snapshots.
+However, in reality the sample complexity is also controlled
+by the interaction of ∆(S) with other geometry dependent
+terms in (27) such as q1 = q∥Ry∥2, which in turn depend on
+both the physical array and coarray size. In the next section,
+we clarify this misconception regarding the seemingly higher
+snapshot requirement of nested arrays in the setting S = O(1).
+Spatiotemporal trade-offs: The snapshot requirement in
+Corollary 1 is inversely proportional to β (since q ∝
+1
+β ).
+If the array geometry and source configuration are kept
+fixed and we increase the SNR (either by increasing pmin
+or decreasing noise power σ), Corollary 1 suggests that it
+is possible to achieve the same probability of error with
+fewer snapshots. Our simulations also are consistent with
+this theoretical prediction. This SNR and geometry dependent
+snapshot characterization is another novel contribution of our
+work.
+IV. A CLOSER LOOK AT THE SEPARATION CONDITION FOR
+SUPER-RESOLUTION WITH SPARSE ARRAYS
+In order to understand the behavior of the smallest non-
+zero singular value σS(AUS(θ)), we consider the notion of
+minimum separation [20]:
+∆min(θ) = min
+i,j∈Ω
+i̸=j
+min
+k∈Z
+���ωi − ωj + k
+���
+(28)
+where ωi is the normalized spatial frequency corresponding to
+direction θi. By definition, for all θ we have 0 ≤ ∆min(θ) ≤
+1/2. Instead of analyzing an arbitrary source configuration θ,
+one can obtain a more interpretable condition by representing
+
+7
+(23) as a function of the minimum separation. The source
+configurations where ∆min(θ) is larger than some threshold
+inversely proportional to Mca +1 (i.e. ∆min(θ) >
+γ
+Mca+1, γ >
+1) will be referred to as the “well-separated” regime. We will
+inspect what this means for specific array geometries such as
+the ULA and nested array, and obtain tight bounds on L.
+A. The “Well-Separated” Case
+In this section, we turn our attention to how the eigen gap
+condition can be utilized to obtain sufficient conditions on
+SNR for different array geometries in the “well-separated”
+regime. Let V ∈ CK×S be a Vandermonde matrix, with
+[V]m,n = zm−1
+n
+where {zn}S
+n=1 are the so called “nodes”
+of the matrix. We begin by summarizing results from [21],
+[24], [31], [32] which characterize the minimum singular value
+of a Vandermonde matrix in the well-separated regime. The
+following Lemma follows from [32, Eq. (32)] which is an
+intermediate result from [32, Theorem 1].
+Lemma 5. Let V(α) ∈ CK×S be a Vandermonde matrix with
+zn = ej2παn for 1 ≤ n ≤ S and S ≤ K. If αi ∈ [0, 1) are all
+distinct and satisfy:
+min
+i,j∈Ω
+i̸=j
+min
+k∈Z
+���αi − αj + k
+��� ≥ γ
+K
+(29)
+for some constant γ > 1, then the following holds:
+σS(V(α))2 ≥ K/C′, where C′ := γ/(γ − 1).
+(30)
+From Lemma 5, for S = Sula if the source configurations θ
+satisfies ∆min(θ) ≥ γ
+P for some γ > 1 and S ≤ P then we
+have the following lower bound:
+σS(AUS(θ))2 ≥ P/C′
+(31)
+In the following Proposition, we apply Lemma 5 to character-
+ize lower bounds on σS(AUS) for the nested array.
+Proposition 2 (Well-Separated). Let S = S(N1,N2)
+nest
+be a nested
+array with N1 = ⌈P/2⌉ and N2 = ⌊P/2⌋ with P ≥ 3.
+Suppose ∆min(θ) ≥
+5γ
+P 2 for some γ > 1 and S ≤ P 2/5.
+Then, the following lower bound holds:
+σS(AUS(θ))2 ≥ P 2/C′
+n, where C′
+n = 5γ/(γ − 1).
+(32)
+Proof. For the nested array with N1 = ⌈P/2⌉ and N2 =
+⌊P/2⌋, from (3) we have Mca + 1 ≥ P 2
+5 . Hence, ∆min(θ) ≥
+5γ
+P 2 implies ∆min(θ) ≥
+γ
+Mca+1. Therefore, the condition on
+∆min(θ) in Lemma 5 holds and we have the desired lower
+bound: σS(AUS(θ))2 ≥ Mca+1
+C′
+≥ ( γ−1
+γ ) P 2
+5 = P 2
+C′n .
+Proposition 2 shows that for a nested array, the sources
+are well-separated if ∆min(θ) ≥ 5γ/P 2 and in this case,
+σS(AUS(θ)) grows as Ω(P), owing to the the larger difference
+coarray of a nested array.
+In order to highlight the dependence of sample complexity
+only on key model parameters, we define quantities to combine
+parameters that are held fixed (such as S, pmin, pmax, σ):
+Cula(S, σ, pmax) := 8C
+′2
+S C
+′3 c3
+c2
+(S +
+σ2
+pmax
+)2
+(33)
+Cnest(S, σ, pmax) := 4C
+′2
+S C
+′3
+n
+c3
+c2
+(S +
+σ2
+pmax
+)2
+(34)
+where C′, C′
+n are universal constants and CS defined in
+Theorem 2 is dependent only on S. Using Proposition 2, we
+now specialize Corollary 1 for the ULA and nested array.
+Theorem 4. Let S = Sula be a ULA with P sensors. Suppose
+the minimum angular separation between the sources, and the
+SNR satisfy the following conditions for some γ > 1:
+∆min(θ) ≥ γ/P,
+pmin/σ2 > 2C′/P, where C′ =
+γ
+γ − 1.
+Under assumptions [A1-A3], for any 0 < δ < 1 and 0 < ϵ ≤
+C1(S) := CSC′
+S, md(θ, �θ) ≤ ϵ is satisfied with probability at
+least 1 − δ, provided P ≥ 3 and
+L ≥ Cula(S, σ, pmax)
+ϵ2
+�pmax
+pmin
+�2 �
+ln
+�8P
+δ
+��2
+.
+(35)
+Proof. From Lemma 5, if ∆min(θ)
+≥
+γ/P, we have
+σ2
+S(AUS(θ)) ≥
+P
+C′ . Under the assumption on the SNR, we
+have pminσ2
+S(AUS(θ)) ≥ pmin P
+C′ > 2σ2 which ensures β >
+pminσ2
+S(AUS(θ))/2 > 0. Notice that for ULA Mca + 1 = P
+and from the fact that σ2
+S(AUS(θ)) ≥
+P
+C′ , we can obtain the
+following bound:
+q =
+C′
+S
+√
+P
+βσS(AUS(θ)) ≤
+2C′
+S
+√
+P
+pminσ3
+S(AUS(θ)) ≤
+C′′
+S
+pminP
+(36)
+where C′′
+S = 2C′
+SC′1.5. Notice that:
+CSβq = CSC′
+S
+√Mca + 1
+σS(AUS(θ))
+≥ C1(S)
+√
+P
+√
+P
+= C1(S)
+(37)
+where the inequality follows from σS(AUS)≤∥AUS∥F /
+√
+S =
+√
+P. Using the fact that β ≤ pminσ2
+S(AUS(θ)), and the above
+lower bound on σS(AUS(θ)), we obtain
+q ≥
+C′
+S
+√
+P
+pminσ3
+S(AUS(θ)) ≥
+C′
+S
+pminP .
+(38)
+Therefore,
+qpminP
+�
+∆(Sula)/c2
+≥
+C′
+S
+�
+∆(Sula)/c2
+≥
+C′
+S
+�
+ln(P)/c2, where the last inequality follows from the
+lower bound on ∆(Sula) in Lemma 3. Recall that c2 < 1
+2, and therefore for P ≥ 3,
+√
+ln P/c2 > 1. This implies that
+min(C1(S), C′
+S
+�
+ln(P)/c2) = C1(S). Combining this with
+(37), we have ϵ ≤ C1(S) = min(C1(S), C′
+S
+�
+ln(P)/c2) ≤
+min(CSβq, qpminP√∆Sula/c2), which ensures that the as-
+2The constant c2 = 3/16
+√
+2 is specified in the proof of Theorem 1 in
+Appendix A.
+
+8
+sumption on ϵ in Corollary 1 holds. From Lemma 4, we have
+∥Ry∥2 ≤ pmaxPS + σ2. Using this bound and (36), we get:
+q1
+�
+∆(Sula) ≤
+C′′
+S
+pminP (PSpmax + σ2)
+�
+2 ln(P)
+= C′′
+S(S +
+σ2
+pmaxP )
+�pmax
+pmin
+� �
+2 ln(P)
+≤ �C1(S, σ, pmax)
+�pmax
+pmin
+� �
+ln(8P/δ)
+(39)
+where �C1(S, σ, pmax) := (S +
+σ2
+pmax )
+√
+2C′′
+S. The upper bound
+follows from the observations that (S +
+σ2
+pmaxP ) ≤ (S +
+σ2
+pmax )
+for all P ≥ 1 and ln(P) ≤ ln(8P/δ) for any δ < 1. Notice
+from (33), that Cula(S, σ, pmax) = c3/c2 �C2
+1(S, σ, pmax). From
+(39), we have
+c3 ln(8P
+δ )q2
+1∆(Sula)
+c2ϵ2
+��
+c3
+c2ϵ2 �C2
+1(S, σ, pmax)(pmax
+pmin
+ln(8P/δ))2
+= Cula(S, σ, pmax)
+ϵ2
+�pmax
+pmin
+�2
+(ln(8P/δ))2 .
+Therefore, (35) implies (27) and the proof is completed by
+applying Corollary 1 since β > 0 and the conditions on ϵ and
+L required for applying the corollary are satisfied.
+Theorem 5. Let S = S(N1,N2)
+nest
+be a nested array with
+N1 = ⌈P/2⌉ and N2 = ⌊P/2⌋. Suppose the minimum
+angular separation between the sources, and the SNR satisfy
+the following conditions for some γ > 1:
+∆min(θ) ≥ 5γ
+P 2 ,
+pmin
+σ2
+> 2C′
+n
+P 2 , where C′
+n = 5γ/(γ − 1).
+Under the assumptions [A1-A3], for any δ > 0 and 0 <
+ϵ ≤ C2(S) :=
+�
+1/5CSC′
+S, md(θ, �θ) ≤ ϵ is satisfied with
+probability at least 1 − δ provided P ≥ 3 and
+L ≥ Cnest(S, σ, pmax)
+ϵ2
+�pmax
+pmin
+�2
+ln
+�8P 2
+δ
+�
+.
+(40)
+Proof. From Proposition 2, if ∆min(θ) ≥ 5γ/P 2, we have
+σ2
+S(AUS(θ)) ≥ P 2
+C′n . Following the same argument as Theo-
+rem 4, this ensures that β > 0. Using the fact that Mca + 1 ≤
+P 2 (from (3)) and the lower bound on σ2
+S(AUS(θ)), we obtain
+q ≤
+C′
+SP
+βσS(AUS(θ)) ≤
+2C′
+SP
+pminσ3
+S(AUS(θ)) ≤
+¯C′′
+S
+pminP 2
+(41)
+where
+¯C′′
+S
+:=
+2C′
+SC′1.5
+n
+. Notice that σS(AUS(θ))
+≤
+∥AUS∥F /
+√
+S = √Mca + 1 ≤ P. Hence, similar to (37),
+we can establish that CSβq
+≥
+C2(S). Using the fact
+P 2/5 ≤ Mca + 1 from (3), similar to (38) we obtain q ≥
+C′
+SP
+√
+5pminσ3
+S(AUS(θ)) ≥
+C′
+S
+√
+5pminP 2 . From Lemma 3, ∆(Snest) ≥
+P 2/16. It follows that qpminP
+�
+∆(Snest)/c2 ≥
+C′
+S
+4c2
+√
+5. Since
+4c2 < 1, it follows that min(C2(S), C′
+S/(4c2
+√
+5)) = C2(S)
+and therefore ϵ ≤ C2(S) = min(C2(S), C′
+S/(4c2
+√
+5)) en-
+sures that the assumption on ϵ in Corollary 1 holds. Using
+∆(Snest) ≤ P 2 (from Lemma 3), Lemma 4, and (41), we get:
+q1
+�
+∆(Snest) ≤ �C1(S, σ, pmax)(pmax/pmin),
+(42)
+where �C1(S, σ, pmax)=(S +
+σ2
+pmax ) ¯C′′
+S. By (42), we have
+ln(8Mca
+δ
+)c3q2
+1∆(Snst)
+c2ϵ2
+≤
+c3
+c2ϵ2 �C2
+1(S, σ, pmax) ln(8P 2/δ)(pmax
+pmin
+)2
+= Cnest(S, σ, pmax)
+ϵ2
+�pmax
+pmin
+�2 �
+ln(8P 2/δ)
+�
+.
+Therefore (40) implies (27) and the proof is again completed
+by applying Corollary 1 since β > 0 and the conditions on ϵ
+and L required for applying the corollary are satisfied.
+Note that the range of values for ϵ where Theorem 4 and
+5 are applicable differ slightly. However in the regime ϵ ≤
+min(C1(S), C2(S)) = C2(S) and P ≥ 3, we can fairly
+compare the two array geometries.
+Towards higher resolution with same snapshots: Theorem 4
+states that for a ULA, the matching distance error for Coarray
+ESPRIT can be bounded by ϵ provided (i) the snapshots
+scales only (poly)logarithmically in the dimension of the
+coarray covariance matrix and (ii) the minimum separation
+is ∆min ≥ γ/P. On the other hand, Theorem 5 guarantees
+that for a nested array with P sensors, it is possible to bound
+the matching distance error by the same ϵ with order wise the
+same number of snapshots (L = Ω(ln(P 2)), but with a relaxed
+separation condition that allows ∆min to be ∆min = Ω(1/P 2).
+This validates the superior resolution properties of nested
+arrays compared to ULA with the same budget of temporal
+snapshots. This has been empirically observed in the literature,
+but never theoretically established, until now.
+Noise Resilience of Nested Arrays: If we consider the
+separation regime ∆min = Ω(1/P) that is applicable for both
+the ULA and nested array, Theorems 4 and 5 indicate that
+the SNR (pmin/σ2) requirement for the nested array can be
+P times smaller than that of the ULA, in order to achieve the
+same DOA error bound with order-wise the same number of
+snapshots (L = Ω(ln P)). This brings out another advantage of
+nested arrays in terms of robustness against noise, especially
+in the low-SNR regime [8].
+Effect of Dynamic Range: Our analysis also reveals the chal-
+lenge posed by sources with higher dynamic range pmax/pmin
+as also observed in [22]. Theorem 5 suggests that at the
+same SNR (defined with respect to the weakest source pmin),
+more snapshots maybe needed for resolving sources with
+disproportionately varying powers (higher pmax compared to
+the fixed pmin). As will be shown, the numerical results are
+indeed consistent with the prediction made by our analysis.
+B. The Myth of Large Snapshots: Correlation Error vs. Angle
+Estimation Error
+Since nested (and other) sparse arrays realize the virtual
+difference coarray by correlation-processing, it is commonly
+believed that one needs a large number (L = Ω(P 2)) of
+temporal snapshots to estimate Θ(P 2) (cross) correlation
+values between sensor pairs. This ‘myth’ of large snapshots
+(that grows quadratically in the number of sensors P) is
+partially true, if our goal is to estimate the coarray covariance
+matrix Tca. If we only allow L to scale as L = Θ(log P)
+
+9
+(the so-called sample-starved regime), then one may indeed
+incur large error in covariance estimation. However, Theorem
+5 shows that the angle estimation error can be made arbitrarily
+small (ϵ) with high probability (1 − δ) provided L scales
+only as Ω( 1
+ϵ2 ln(8P 2/δ)), despite the possibility of the coarray
+covariance error of a nested array increasing with P in this
+snapshot-starved regime. This surprising phenomenon is due
+to the fact that the potentially large covariance estimation error
+(which can even grow with P in this regime) can actually be
+mitigated/counterbalanced by the enhanced aperture/difference
+set of the nested array that results in a large restricted smallest
+singular value σS(AUS). As long as ∆min(θ) ≥ 5γ
+P 2 , σ2
+S(AUS)
+scales as cP 2 (for some constant c), and this helps us obtain
+reliable angle estimation, although the covariance estimates
+may be unreliable.
+V. SIMULATIONS
+We numerically investigate the useful SNR regime for coarray
+processing (Section V-A), the impact of SNR and the number
+of snapshots on DOA estimation error (V-B and V-C), the
+relationship between DOA and covariance estimation error
+(V-D), and the effect of the dynamic range of source powers
+on resolving two closely spaced sources (V-E).
+A. When is Coarray-Based DOA Estimation Beneficial?
+We begin by examining under which circumstances coarray-
+based algorithms offer an advantage over more conventional
+DOA estimation methods. Specifically, in case of the ULA,
+we could apply MUSIC or ESPRIT directly to the sample
+covariance matrix �Ry in (7) instead of the averaged coarray
+covariance matrix �Tca in (9). Fig. 1 shows the matching
+distance error of coarray ESPRIT and direct ESPRIT, averaged
+over 103 Monte Carlo trials, in case of the ULA, and, for
+comparison, coarray ESPRIT in case of the nested array with
+the same number of sensors (P = 20). We consider L = 100
+snapshots, and S = 4 equipower sources equally spaced by
+∆ = 2/P. At medium to low SNR, the advantage of coarray-
+based processing is apparent. At high SNR, the situation is
+reversed, as the error of direct ESPRIT continues decreasing
+as a function of SNR, whereas the error of coarray ESPRIT
+saturates3. However, coarray-based processing—including re-
+dundancy averaging (8)—can clearly offer significant benefits
+in SNR or snapshot-limited conditions. As mostly such chal-
+lenging scenarios are of interest in many applications, we focus
+on coarray ESPRIT herein.
+B. Improving Resolution by Increasing SNR or Snapshots
+Next, we compare the probability of resolution as a function of
+the minimum separation for the nested array and ULA with
+the same number of sensors, P = 20. Coarray ESPRIT is
+employed for both array geometries. We consider two sources
+with equal power (p1 = p2) and (normalized) angles ω =
+{0.1, 0.1 + ∆}. The sources are declared to be successfully
+resolved when the estimated DOAs satisfy maxi |ˆωi − ωi| ≤
+∆/10. Fig. 2 shows the empirical probability of resolution
+(averaged over 1000 Monte-Carlo trials) for varying separation
+3This well-known and fundamental phenomenon is due to the finite-
+snapshot error of the coarray covariance matrix, see [5]–[7].
+-20
+-10
+0
+10
+20
+30
+40
+50
+SNR (dB)
+10-6
+10-4
+10-2
+100
+Average Matching Distance
+ULA (coarray ESPRIT)
+Nested (coarray ESPRIT)
+ULA (direct ESPRIT)
+Fig. 1: Comparison of ESPRIT applied to the sample covariance
+matrix (7) (direct ESPRIT) and the estimated coarray covariance
+matrix (9) (coarray ESPRIT). Coarray ESPRIT achieves lower angle
+estimation error than direct ESPRIT at medium to low SNR.
+10-3
+10-2
+10-1
+Angular Separation 
+0
+0.2
+0.4
+0.6
+0.8
+1
+Probability of resolution
+# Sensors = 20, Snapshots = 55
+= (1/P)
+= (1/P2)
+ULA (SNR = 0 dB)
+Nested (SNR = 0 dB)
+ULA (SNR = -16 dB)
+Nested (SNR = -16 dB)
+10-3
+10-2
+10-1
+Angular Separation 
+0
+0.2
+0.4
+0.6
+0.8
+1
+Probability of resolution
+# Sensors = 20, SNR = 0 dB
+= (1/P)
+= (1/P2)
+ULA (L = 55)
+Nested (L = 55)
+ULA (L = 600)
+Nested (L = 600)
+Fig. 2: Probability of resolution vs. source separation for different
+SNR levels (top) and number of snapshots (bottom). Increasing either
+improves resolution for both arrays.
+∆ and a fixed number of snapshots L = 55 and SNR = 0 and
+−16 dB. We observe that both array geometries can operate
+at a smaller separation at a higher SNR, i.e., smaller σ/pmin
+ratio. Indeed, the transition from low to high probability of
+resolution occur around ∆ ∝ 1/P for the ULA and ∆ ∝ 1/P 2
+for the nested array, as predicted by Theorems 4 and 5. It is
+also possible to enhance resolution by increasing the number
+of snapshots, as Fig. 2 demonstrates. Here, the SNR is fixed
+at 0 dB and the number of snapshots is L = 55 and L = 600,
+respectively.
+C. Snapshot and SNR Trade-off
+Section V-B showed that SNR and the number of temporal
+snapshots can be exchanged for improved resolution. We now
+study this trade-off in further detail. We consider S = 2
+equipowered sources located at ω = {0.1, 0.1 + ∆}, where
+∆ ∈ {2/P, 2/P 2} and P = 20. Fig. 3 shows the separation-
+relative matching distance error md(θ, �θ)/∆ (averaged over
+103 Monte Carlo trials) as a function of both the number
+of snapshots and SNR. Firstly, fewer snapshots are required
+at higher SNR (and vice versa) to obtain the same recovery
+error, both in case of the ULA (left column) and nested array
+(right column). This supports Theorem 3, where the match-
+
+10
+ing distance depends on the number of snapshots and SNR
+through (22) and (26), respectively. Secondly, the nested array
+displays a more advantageous trade-off between snapshots and
+SNR compared to the ULA for both source separation 2/P
+(top row) and 2/P 2 (bottom row). The benefit is especially
+apparent for ∆ = 2/P 2, where the nested array has a greatly
+larger range of operating points where the relative matching
+distance is low, as predicted by Theorem 5. Note that the gray
+pixels correspond to a relative error of approximately 10% of
+the separation, whereas white corresponds ≤ 1% error.
+Fig. 3: Relative matching distance error md(θ, �θ)/∆ as a function of
+snapshots and SNR. The nested array (right column) achieves lower
+error than the ULA (left column) for both source separation ∆ = 2/P
+(top row) and ∆ = 2/P 2 (bottom row).
+D. DOA and Covariance Estimation Error
+Next, we illustrate an intriguing benefit of coarray-based DOA
+estimation in case of the nested array. We consider the average
+DOA matching distance and average covariance estimation
+error defined as ∥Tca − �Tca∥2 for a varying number of
+sensors P and S = 4 equipower sources equally spaced by
+∆ ∈ {1/P 1.5, 1/P 2}. The number of snapshots is L = 50
+and SNR = 0 dB. Fig. 4 shows that the nested array incurs a
+larger covariance estimation error compared to the ULA with
+the same number of sensors. However, despite obtaining a
+worse estimate of the covariance matrix �Tca, the nested array
+achieves superior DOA estimation performance when coarray
+ESPRIT is applied to �Tca. In fact, when the separation is
+∆ = 1/P 2, the average matching distance no longer decays
+with P for the ULA, whereas it continues to do so for the
+nested array. This is enabled by the larger coarray aperture
+of the nested array, which offsets the effect of finite snapshot
+covariance estimation error as discussed in Section IV-B. Note
+that for a fixed number of snapshots and a growing number of
+sensors P, the entries of the coarray covariance matrix Tca
+become increasingly challenging to estimate, since the size of
+Tca is proportional to the number of coarray elements Mca,
+which is ∝ P for the ULA and ∝ P 2 for the nested array.
+E. Effect of Dynamic Range of Source Powers
+In the final experiment, we investigate the ability of coarray
+ESPRIT to resolve two sources with unequal powers. We set
+10
+15
+20
+25
+30
+35
+40
+Number of sensors (P)
+10-4
+10-3
+10-2
+10-1
+Average Matching
+Distance Error
+# Snapshot = 50, SNR = 0 dB
+Nested (
+=1/P2)
+ULA (
+=1/P2)
+Nested (
+=1/P1.5)
+ULA (
+=1/P1.5)
+10
+15
+20
+25
+30
+35
+40
+Number of sensors (P)
+101
+102
+Average Covariance
+Estimation Error
+# Snapshot = 50, SNR = 0 dB
+Fig. 4: Average matching distance (top) and covariance estimation
+error (bottom) as a function of the number of sensors P. The
+DOA estimation error of the nested array decays despite the larger
+covariance estimation error compared to the ULA.
+the dynamic range to pmax/pmin ∈ {1, 10} by fixing the power
+of the weaker source to pmin = 0.2 and varying pmax. Fig. 5
+shows that the number of snapshots required to distinguish
+two sources (separated by ∆ = 1/P) is significantly larger
+when pmax/pmin = 10 compared to pmax/pmin = 1. This
+is consistent with Theorems 4 and 5, which imply that the
+sufficient number of snapshots for resolving two sources (with
+high probability) grows with pmax if pmin and σ are held
+fixed, irrespective of the array geometry. This brings out a
+non-trivial dependence of the dynamic range pmax/pmin on
+the sample complexity. Hence, distinguishing two sources with
+greatly different powers is more challenging and requires more
+snapshots than when the powers are equal.
+101
+102
+103
+104
+Snapshots
+0
+0.2
+0.4
+0.6
+0.8
+1
+Probability of resolution
+ULA (pmax/pmin = 10)
+Nested ULA (pmax/pmin = 10)
+ULA (pmax/pmin = 1)
+Nested ULA (pmax/pmin = 1)
+Fig. 5: Effect of dynamic range of source powers on probability of
+resolution. Coarray ESPRIT requires more snapshots to detect two
+sources with larger dynamic range pmax/pmin.
+VI. CONCLUSION
+This paper investigated angle estimation error of coarray
+ESPRIT. We considered both additive noise and finite-snapshot
+covariance estimation error, which we probabilistically char-
+acterized in the case of Toeplitz covariance matrices. Our
+results show that if the number temporal snapshots scales
+logarithmically with the number of sensors, coarray ESPRIT
+achieves arbitrarily low estimation error with high probability.
+This also shows that the DOA estimation error can be small
+even though the covariance estimation error may be large.
+
+ULA,Separation=2/P
+10
+100
+Matchingdistance/separation
+5
+0
+-5
+10~1
+-15
+-20
+10~2
+101
+102
+103
+104
+SnapshotsNested,Separation=2/P
+10
+100
+Matchingdistance/separation
+5
+SNR (in dB)
+0
+-5
+10~1
+-10
+-15
+-20
+10~2
+101
+102
+103
+104
+SnapshotsULA, Separation=2/p?
+10
+100
+Matchingdistance/separation
+5
+SNR (in dB)
+0
+-5
+10~1
+-10
+-15
+-20
+10~2
+101
+102
+103
+104
+SnapshotsNested, Separation=2/p?
+10
+100
+Matchingdistance/separation
+5
+SNR (in dB)
+0
+-5
+10~1
+-10
+-15
+-20
+102
+101
+102
+103
+104
+Snapshots11
+Finally, our theoretical and simulation results demonstrate that
+sparse arrays can provide higher resolution and better noise
+resilience compared to the ULA with the same number of
+sensors and snapshots.
+APPENDIX A
+A. Intermediate Results
+We will first state the complex extension of Hanson-Wright
+inequality [33], which is obtained by applying [34, Theorem
+1.1] with the strategy described on [34, Section 3.1, Page 9].
+Lemma 6. Let A ∈ Cn×n be a fixed Hermitian matrix.
+Consider the random vector x = [x1, x2, · · · , xn]⊤ ∈ Cn with
+independent real and imaginary components Re(xi), Im(xi)
+satisfying E(Re(xi)) = E(Im(xi)) = 0, and ∥Re(xi)∥ψ2 ≤ K,
+∥Im(xi)∥ψ2 ≤ K. Then for any ϵ > 0, we have
+P(|xHAx − E(xHAx)|>ϵ) ≤ 2 exp
+�
+− c min(
+ϵ2
+2K4∥A∥2
+F
+,
+ϵ
+K2∥A∥2
+)
+�
+where c > 0 is a universal constant.
+Proof. Let z = [Re(x)⊤, Im(x)⊤]⊤ ∈ R2n and define : ˜A =
+�Re(A)
+−Im(A)
+Im(A)
+Re(A)
+�
+. It is easy to see that for any Hermitian
+A, we have the following equality xHAx = zT ˜Az. Further,
+it can be verified that ∥˜A∥F =
+√
+2∥A∥F and ∥˜A∥2 = ∥A∥2.
+Now, we can apply [34, Theorem 1.1], to obtain the desired
+probability bound.
+Lemma 7. Let wi ∈ Cn, 1 ≤ i ≤ T be i.i.d com-
+plex circularly symmetric Gaussian random variable with
+distribution CN(0, Σ). Let A ∈ Cn×n be a fixed Her-
+mitian matrix, then for any ϵ > 0 and universal constant
+c, we have P(| 1
+L
+�L
+i=1 wH
+i Awi − E[wH
+i Awi]| ≥ ϵ) ≤
+2 exp
+�
+−cL min
+�
+ϵ2
+2K4∥Σ∥2
+2∥A∥2
+F ,
+ϵ
+K2∥Σ∥2∥A∥2
+��
+.
+Proof. Since wi is a complex circularly symmetric Gaussian
+random variable distributed according to CN(0, Σ), we define
+a new transformed variable ui = Σ−1/2wi where Σ1/2 is the
+square root of the covariance matrix Σ. It can be verified that
+ui ∼ CN(0, In), i.e., it is also a complex circularly symmetric
+Gaussian
+random
+variable
+with
+independent
+real
+and
+imaginary components. Define block-wise diagonal matrices
+˜A
+=
+diag(A, . . . , A), ˜Σ1/2
+=
+diag(Σ1/2, . . . , Σ1/2)
+∈
+CnL×nL and ˜u = [uT
+1 , . . . , uT
+L]T
+∈ CnL. Next, we can
+observe that �L
+i=1 wH
+i Awi = �L
+i=1 uH
+i Σ1/2AΣ1/2ui =
+˜uH ˜Σ1/2 ˜A˜Σ1/2˜u.
+We
+have
+E(�L
+i=1 wH
+i Awi)
+=
+LE
+�
+wH
+i Aw
+�
+,
+since
+it
+is
+a
+sum
+of
+L
+i.i.d
+random
+variables. The desired probability can be re-written as:
+P(| 1
+L
+�L
+i=1 Re(wH
+i Awi) − E[Re(wH
+i Awi)]|
+≥
+ϵ)
+=
+P(|Re(˜uH ˜Σ1/2 ˜A ˜Σ1/2˜u) − E[Re(˜uH ˜Σ1/2 ˜A ˜Σ1/2˜u)]| ≥ Lϵ).
+Recall
+that
+Re(˜ui), Im(˜ui)
+are
+i.i.d
+distributed
+as
+N(0, 1/2) and hence sub-Gaussian with K
+=
+2/
+√
+3.
+Note that due to the block-diagonal structure we have
+∥ ˜Σ1/2 ˜A ˜Σ1/2∥2
+F = L∥Σ1/2AΣ1/2∥2
+F ≤ L∥A∥2
+F ∥Σ∥2
+2
+and
+∥ ˜Σ1/2 ˜A ˜Σ1/2∥2 = ∥Σ1/2AΣ1/2∥2 ≤ ∥A∥2∥Σ∥2. The proof
+is completed by applying Lemma 6 with ϵ = ϵL.
+B. Proof of Theorem 1
+From Lemma 2, we have P(∥EL∥2 ≥ ϵ)≤P(sup |fe(θ)|≥ϵ).
+In general, it is not straightforward to evaluate this supremum,
+however, we exploit the following result from [28]that bounds
+it by using the function value evaluated at a few grid points.
+Lemma 8.
+[28, Theorem 7.28, Chapter 10, Vol.2, Pg. 33]
+Let f(θ) be a trigonometric polynomial of order N. Then,
+supθ∈[−π,π] |f(θ)| ≤ 2 max1≤k≤4N |f(θk)|,
+θk = k−2N
+4N π.
+From Proposition 1, we have fe(θ) = tr(EyΛ(θ)). However,
+we want to relate it to the sample covariance matrix �Ry. In
+order to do this, we show that tr(RavΛ(θ)) = tr( �RyΛ(θ))
+where recall from (7) that �Ry is the sample covariance matrix:
+tr(RavΛ(θ)) =
+P
+�
+m=1
+P
+�
+n=1
+[Rav]m,n[Λ(θ)]n,m
+=
+Mca
+�
+s=−Mca
+�
+m,n:
+dm−dn=s
+ˆts
+exp(−jsθ)
+|Ωs|
+=
+Mca
+�
+s=−Mca
+ˆts|Ωs| exp(−jsθ)
+|Ωs|
+=
+(a)
+Mca
+�
+s=−Mca
+�
+m,n:
+dm−dn=s
+[ �Ry]m,n[Λ(θ)]n,m = Tr( �RyΛ(θ)),
+where (a) follows from the redundancy averaged estimator
+where for all m, n such that dm − dn = s, we have |Ωs|ˆts =
+�
+dm−dn=s[ �Ry]m,n. Therefore, we have the following rela-
+tion: fe(θ)=tr (EyΛ(θ))=tr ((Ry − Rav)Λ(θ)) = tr((Ry −
+�Ry)Λ(θ)) = 1
+L
+�L
+t=1
+�
+E[y(t)HΛ(θ)y(t)] − y(t)HΛ(θ)y(t)
+�
+.
+Since the snapshots are i.i.d, we can define i.i.d ran-
+dom variables {Zt(θ)}L
+t=1 as Zt(θ) ≜ y(t)HΛ(θ)y(t) −
+E(y(t)HΛ(θ)y(t)) with y(t) ∼ CN(0, Ry). Note that Λ(θ)
+is Hermitian. Hence, we can apply Lemma 7 with Σ = Ry
+and A = Λ(θ) to obtain ∀ϵ > 0,
+P
+�
+1
+L |
+L
+�
+t=1
+Zt(θ)| ≥ ϵ
+�
+≤
+(43)
+2 exp
+�
+−cL min
+�
+ϵ2
+2K4∥Ry∥2
+2∥Λ(θ)∥2
+F
+,
+ϵ
+K2∥Ry∥2∥Λ(θ)∥2
+��
+.
+We want to obtain a universal upper bound that is similar
+to (43) but not dependent on θ. Notice, ∥Λ(θ)∥2
+F =
+1
+|Ω0| +
+�Mca
+s=1
+2
+|Ωs| ≤ 2∆(S). Similarly, we can also bound ∥Λ(θ)∥2 ≤
+∥Λ(θ)∥F ≤
+�
+2∆(S). This gives us the following bound:
+P
+�
+1
+L |
+L
+�
+t=1
+Zt(θ)| ≥ ϵ
+�
+≤
+(44)
+2 exp
+�
+−cL min
+�
+ϵ2
+4K4∥Ry∥2
+2∆(S) ,
+ϵ
+K2∥Ry∥2
+�
+2∆(S)
+��
+.
+Note fe is a trigonometric polynomial of order Mca. Now,
+we will use Lemma 8 to bound the spectral function |fe(θ)|.
+P(supθ∈[−π,π] |fe(θ)| ≥ ϵ) ≤ P(2 max1≤k≤4Mca |fe(θk)| ≥
+ϵ)
+≤
+�4Mca
+k=1 P
+�
+|fe(θk)| ≥ ϵ
+2
+�
+≤
+8Mca exp
+�
+−
+c1L min
+�
+c2ϵ2
+∥Ry∥2
+2∆(S),
+ϵ
+∥Ry∥2√
+∆(S)
+��
+, where c1 = c/(2
+√
+2K2)
+(c was given in Lemma 7) and c2
+=
+1/(4
+√
+2K2)
+=
+3/(16
+√
+2) < 1. The first inequality follows due to Lemma
+8, the second inequality follows from union bound. The last
+inequality is a consequence of the bound computed in (44).
+
+12
+APPENDIX B
+A. Proof of Theorem 2
+The proof uses several results from [20]. However, unlike [20]
+the underlying subspace of interest is the coarray subspace
+and the perturbation is due to covariance estimation error and
+noise. We provide key intermediate steps to make the results
+self-contained.
+Recall that columns of U and �U are orthonormal bases
+for the subspaces R(U) and R( �U). Let the principal an-
+gles between the subspaces R(U) and R( �U) be denoted as
+Θ(R(U), R( �U)) := [ψ1, ψ2, · · · , ψS]T where 0 ≤ ψ1 ≤
+ψ2 ≤ · · · ≤ ψS ≤ π/2. Then from [35], we have cos(ψi) =
+σi(UH �U) i = 1, 2, · · · , S. Recall from Lemma 1, the output
+of ESPRIT is invariant to the choice of the basis. For ease
+of analysis, we will choose a pair of basis for R(U) and
+R( �U), which are also known as “canonical bases” [20]. Let
+the SVD of the matrix UH �U be of the form UH �U :=
+LΣcRH, L, R ∈ CS×S, where Σc = diag(σc
+1, σc
+2, · · · , σc
+S)
+where σc
+i = σi(UH �U) are arranged in descending order. The
+canonical basis U(c) and �U(c) are given by:
+U(c) := UL,
+�U(c) := �UR
+(45)
+Using the canonical basis, we define the following matrices:
+Ψ(c) :=U(c)†
+0
+U(c)
+1 , �Ψ(c) := �U(c)†
+0
+�U(c)
+1 . Since R(U)=R(U(c))
+and R( �U) = R( �U(c)), we have Θ(R(U(c)), R( �U(c))) =
+Θ(R(U), R( �U)). Notice that the canonical basis has the
+following property: cos(ψi) = σi(U(c)H �U(c)) = u(c)H
+i
+�u(c)
+i .
+We will use [20, Lemma 2] that relates the matching distance
+error to the quantity ∥ �Ψ(c) − Ψ(c)∥2 and holds universally:
+md(θ, ˆθ) ≤ π S3/2√Mca + 1
+σS(AUS(θ)) ∥ �Ψ(c) − Ψ(c)∥2.
+(46)
+B. Relating ∥ �Ψ(c) − Ψ(c)∥2 to ∥EL∥2
+Let B = A + N ∈ CM×N, where rank(A) ≥ L. Suppose ψL
+is the largest principal angle between the subspace spanned
+by L principal singular vectors (corresponding to L largest
+singular values) of A and B, respectively. If σL+1(A) ≤ α
+and σL(B) ≥ α + δ for some α ≥ 0 and δ > 0 then, Wedin’s
+Theorem [36] states that:
+sin(ψL) ≤ ∥N∥2/δ
+(47)
+Lemma
+9. Suppose σS(Tca)
+≥
+2∥EL∥2
+and β
+=
+pminσ2
+S(AUS(θ)) − σ2 > 0. Then
+sin(ψS) ≤ 2∥EL∥2/β
+(48)
+Proof. Recall that ˆTca = Tca − EL. To apply Wedin’s
+theorem, we need to characterize quantities α and δ such
+that: σS( ˆTca) ≥ δ + α,
+σS+1(Tca) ≤ α. From (13), we
+have σS+1(Tca) = σ2. We choose α = σ2. Using Weyl’s
+inequality, σS(�Tca) ≥ σS(Tca) − ∥EL∥2
+(a)
+≥ σS(Tca)/2
+(b)
+=
+(σS(AUS(θ)PAUS(θ)H) + σ2)/2, where (a) follows from
+the assumption 2∥EL∥2 ≤ σS(Tca) and (b) follows from
+(13). Combining with the preceding inequality, we obtain
+σS( ˆTca) − σ2 ≥ (σS(AUS(θ)PAUS(θ)H) − σ2)/2 ≥ β/2 >
+0, where the last term is positive due to the given condition.
+Then we can choose δ = σS(�Tca) − σ2 which satisfies
+σS(�Tca) = α + δ with δ > 0. The proof is completed by
+using (47).
+Lemma 10. If pminσ2
+S(AUS(θ)) > σ2 and
+∥EL∥2 ≤ σS(U(c)
+0 )(σS(AUS(θ)PAUS(θ)H) − σ2)
+4
+√
+2
+(49)
+then ∥Ψ(c) − ˆΨ(c)∥2 ≤
+14
+√
+2∥EL∥2
+σ2
+S(U(c)
+0 )(pminσ2
+S(AUS(θ))−σ2).
+Proof. From the definition of U(c), �U(c) we have:
+∥U(c) − �U(c)∥2
+2 = ∥(U(c) − �U(c))H(U(c) − �U(c))∥2
+= 2(1 − cos(ψS)) ≤ 2(1 − cos2(ψS)) = 2 sin2(ψS).
+(50)
+By
+the
+assumption
+of
+this
+lemma,
+2∥EL∥2
+≤
+σS(U(c)
+0 )(σS(AUS(θ)PAUS(θ)H) − σ2) and σS(U(c)
+0 ) ≤ 1,
+we have 2∥EL∥2 ≤ σS(Tca)σS(U(c)
+0 ) ≤ σS(Tca). This
+together with the assumption pminσ2
+S(AUS(θ)) > σ2 enables
+us to apply Lemma 9. Combining (48) with (50) we obtain
+the following bound:
+∥ �U(c) − U(c)∥2 ≤
+2
+√
+2∥EL∥2
+σS(AUS(θ)PAUS(θ)H) − σ2 .
+(51)
+Notice that
+∥ �Ψ
+(c) − Ψ(c)∥2 = ∥( �U(c)†
+0
+− U(c)†
+0
+) �U(c)
+1
++ U
+(c)†
+0
+( �U(c)
+1
+− U(c)
+1 )∥2
+≤ ∥ �U(c)†
+0
+− U(c)†
+0
+∥2∥ �U(c)
+1 ∥2 + ∥U(c)†
+0
+∥2∥ �U(c)
+1
+− U(c)
+1 ∥2
+≤ ∥ �U(c)†
+0
+− U(c)†
+0
+∥2 + ∥U(c)†
+0
+∥2∥ �U(c) − U(c)∥2
+(52)
+where the last inequality follows from the fact that
+�U(c)
+1 , �U(c)
+1
+− U(c)
+1
+are submatrices of �U(c) and �U(c) − U(c),
+respectively. Therefore, we have ∥ �U(c)
+1 ∥2 ≤ ∥ �U(c)∥2 = 1,
+and ∥ �U(c)
+1
+− U(c)
+1 ∥2 ≤ ∥ �U(c) − U(c)∥2. We use a result from
+[37, Theorem 3.2] which states that a matrix F with rank S,
+and its perturbed matrix �F = F + �E satisfy the following
+inequality: ∥F† − �F†∥2 ≤ 3∥�E∥2/(σS(F)(σS(F) − ∥�E∥2))
+provided ∥�E∥2 < σS(F). From (51), and using the assumption
+of the lemma we have:
+∥ �U(c)
+0
+− U(c)
+0 ∥2 ≤ ∥ �U(c) − U(c)∥2 ≤
+2
+√
+2∥EL∥2
+σS(AUS(θ)PAUS(θ)H) − σ2
+≤ σS(U(c)
+0 )/2.
+(53)
+We
+can
+use
+the
+aforementioned
+result
+by
+substi-
+tuting
+F
+with
+U(c)
+0 ,
+and
+�F
+with
+�U(c)
+0 :
+∥( �U(c)†
+0
+−
+U(c)†
+0
+)∥2 ≤
+3∥( �
+U(c)
+0 −U(c)
+0 )∥2
+σS(U(c)
+0 )(σS(U(c)
+0 )−∥ �
+U(c)
+0 −U(c)
+0 ∥2) ≤ 6∥ �
+U(c)
+0 −U(c)
+0 ∥2
+σ2
+S(U(c)
+0 )
+≤
+6∥ �
+U(c)−U(c)∥2
+σ2
+S(U(c)
+0 )
+, where the second inequality follows from (53).
+Combining this with (52), we get the final bound: ∥Ψ(c) −
+ˆΨ(c)∥2 ≤
+6∥( �
+U(c)−U(c))∥2
+σ2
+S(U(c)
+0 )
++
+1
+σS(U(c)
+0 )∥( �U(c) − U(c))∥2 ≤
+7∥( �
+U(c)−U(c))∥2
+σ2
+S(U(c)
+0 )
+≤
+14
+√
+2∥EL∥2
+σ2
+S(U(c)
+0 )(σS(AUS(θ))PAUS(θ))H)−σ2)
+≤
+14
+√
+2∥EL∥2/σ2
+S(βU(c)
+0 ).
+Next, we state the following Lemma from [20] that can be
+used to obtain a lower bound on σ2
+S(U(c)
+0 ).
+Lemma 11 (Lemma 3, [20]). Let U(a) be any orthonormal
+basis for R(AUS(θ)). Then the following holds: σ2
+S(U(a)
+0 ) ≥
+max(1 −
+S
+σ2
+S(AUS(θ)), 4−S)
+
+13
+Proof of Theorem 2. Define
+CS
+=
+2−S
+4
+√
+2
+and
+C′
+S
+=
+14π
+√
+2S3/24S. Under Assumption A3, these quantities are
+constants since S is held fixed. If β > 0 and the assump-
+tion ∥EL∥2 ≤ CSβ ensures that condition (49) holds since
+σS(U(c)
+0 ) ≥ 2−S from Lemma 11. Now, we can apply
+Lemma 10 to bound ∥Ψ(c) − ˆΨ(c)∥2. We plug this bound
+on ∥Ψ(c) − ˆΨ(c)∥2 in (46): md(θ, �θ) ≤ 14
+√
+2π S3/2q∥EL∥2
+σ2
+S(U(c)
+0 )C′
+S ≤
+q∥EL∥2, where the last inequality follows from the bound
+σ2
+S(U(c)
+0 ) ≥ 4−S in Lemma 11.
+C. Proof of Theorem 3
+We will utilize Theorem 1 and Theorem 2 to prove Theorem 3.
+One can see from Theorem 2 that under the assumptions β > 0
+and ∥EL∥2 ≤ CSβ we can bound md(θ, �θ) ≤ q∥EL∥2. For a
+given ϵ > 0, two cases arise:
+Case I (ϵ ≤ CSβq): In this case, min(CSβ, ϵ
+q) = ϵ/q. There-
+fore, ∥EL∥2 ≤ ϵ
+q ⇒ ∥EL∥2 ≤ CSβ, and from Theorem 2 the
+matching distance error is less than md(θ, �θ) ≤ ϵ. This means
+P(md(θ, �θ) ≤ ϵ) ≥ P
+�
+∥EL∥2 ≤ ϵ
+q
+�
+. From Theorem 1, we
+can obtain the following tail bound:
+P(∥EL∥2 ≤ ϵ
+q ) ≥
+1 − 8Mca exp
+�
+−c1L min
+�
+c2ϵ2
+q2∥Ry∥2
+2∆(S) ,
+ϵ
+q∥Ry∥2
+√
+∆(S)
+��
+. (54)
+Case II (ϵ > CSβq): For values of ϵ satisfying ϵ > Csβq,
+we have min(CSβ, ϵ/q) = CSβ. Therefore, if ∥EL∥2 ≤ CSβ,
+then from Theorem 2 we have md(θ, �θ) ≤ CSβq. We obtain
+the following bound on the tail probability due to Theorem 1,
+P(∥EL∥2 ≤ CSβ) ≥
+1 − 8Mca exp
+�
+−c1L min
+�
+c2C2
+Sβ2
+∥Ry∥2
+2∆(S) ,
+CSβ
+∥Ry∥2
+√
+∆(S)
+��
+.
+(55)
+If the number of snapshots L satisfy the following bound:
+L ≥ c3 ln
+� 8Mca
+δ
+�
+max
+� q2
+1∆(S)
+c2ϵ2 ,
+q1√
+∆(S)
+ϵ
+, L2
+0
+c2 , L0
+�
+, where q1 =
+q∥Ry∥2, c3 = 1/c1 and L0 =
+∥Ry∥2√
+∆(S)
+CSβ
+then combining
+(54) and (55) we obtain the following bound P
+�
+md(θ, �θ) ≤
+min(ϵ, CSβq)) ≥ 1 − δ.
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+

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@@ -0,0 +1,1182 @@
+Cross Modal Transformer via Coordinates Encoding for 3D Object Dectection
+Junjie Yan
+Yingfei Liu
+Jianjian Sun
+Fan Jia
+Shuailin Li
+Tiancai Wang
+Xiangyu Zhang
+MEGVII Technology
+Abstract
+In this paper, we propose a robust 3D detector, named
+Cross Modal Transformer (CMT), for end-to-end 3D multi-
+modal detection.
+Without explicit view transformation,
+CMT takes the image and point clouds tokens as inputs
+and directly outputs accurate 3D bounding boxes.
+The
+spatial alignment of multi-modal tokens is performed im-
+plicitly, by encoding the 3D points into multi-modal fea-
+tures. The core design of CMT is quite simple while its
+performance is impressive. CMT obtains 73.0% NDS on
+nuScenes benchmark. Moreover, CMT has a strong robust-
+ness even if the LiDAR is missing. Code will be released at
+https://github.com/junjie18/CMT.
+1. Introduction
+Multi-sensor fusion has shown its great superiority in au-
+tonomous driving system [1,8,20,24,28]. Different sensors
+usually provide the complementary information for each
+other. For instance, the camera captures information in a
+perspective view and the image contains rich semantic fea-
+tures while point clouds provide much more localization
+and geometry information. Taking full advantage of differ-
+ent sensors helps reduce the uncertainty and makes accurate
+and robust prediction.
+Sensor data of different modalities usually has large
+discrepancy in distribution, making it hard to merge the
+multi-modalities. State-of-the-art (SoTA) methods tend to
+fuse the multi-modality by constructing unified bird’s-eye-
+view (BEV) representation [20,24,28] or querying from to-
+kens [1,8]. For example, BEVFusion [28] explores a unified
+representation by BEV transformation for BEV feature fu-
+sion (see Fig. 1(a)). TransFusion [1] follows a two-stage
+pipeline and the camera images in second stage provide
+supplementary information for prediction refinement (see
+Fig. 1(b)). However, exploring a truly end-to-end pipeline
+for multi-sensor fusion remains to be a question.
+Recently, the effectiveness of end-to-end object detec-
+tion with transformer (DETR) [3, 56] has been proved in
+many perception tasks, such as instance segmentation [10,
+(a) BEVFusion
+quries
+PC
+Transformer
+image
+quries
+PC
+Transformer
+step1
+image
+Transformer
+topk
+step2
+PC
+image
+BEV
+Encoder
+VT
+(b) TransFusion
+(c) CMT (Ours)
+PE
+Figure 1.
+Comparison between BEVFusion, TransFusion, and
+our proposed CMT. (a) In BEVFusion, the camera features are
+transformed into BEV space by view transform. Two modality
+features are concatenated in BEV space and the BEV encoder is
+adopted for fusion.
+(b) TransFusion first generates the queries
+from the high response regions of LiDAR features. After that, ob-
+ject queries interact with point cloud features and image features
+separately. (c) In CMT, the object queries directly interact with
+multi modality features simultaneously. Position encoding (PE) is
+added to the multi-modal features for alignment. ”VT” is the view
+transformation from image to 3D space.
+12], multi-object tracking [30, 51] and visual 3D detec-
+tion [26, 27, 45]. The DETR architecture is simple yet ef-
+fective thanks to the object queries for representing different
+instances and bipartite matching for one-to-one assignment.
+Inspired by DETR, we aim to build an elegant end-to-
+end pipeline for multi-modal fusion in 3D object detection.
+In DETR, object queries directly interact with the image to-
+kens through cross-attention in transformer decoder. For 3D
+arXiv:2301.01283v1  [cs.CV]  3 Jan 2023
+
+0
+10
+20
+30
+40
+50
+60
+70
+80
+NDS[%]
+CMT
+w/o Cams
+w/o LiDAR
+PETR
+CMT-L
+71.9
+67.7
+43.4
+45.5
+68.6
+drop
+result
+Figure 2. CMT has a strong robustness under sensor missing con-
+dition. During inference, CMT without LiDAR achieves similar
+detection performance compared to the SoTA camera-only detec-
+tor PETR [26]. CMT without camera input only introduce a slight
+drop, compared to our LiDAR-only baseline CMT-L. (Note: we
+evaluate without any finetune process)
+object detection, one intuitive way is to concatenate the im-
+age and point cloud tokens together for further interaction
+with object queries. However, the concatenated tokens are
+disordered and unaware of their corresponding locations in
+3D space. Therefore, it is necessary to provide the location
+prior for multi-modal tokens and object queries.
+In this paper, we propose Cross-Modal Transformer
+(CMT), a simple yet effective end-to-end pipeline for high-
+performance 3D object detection (see Fig. 1(c)). First, we
+propose the Coordinates Encoding Module (CEM), which
+produces position-aware features, by encoding 3D points
+set implicitly into multi-modal tokens.
+Specifically, for
+camera images, 3D points sampled from frustum space are
+used to indicate the probability of 3D positions for each
+pixel. While for LiDAR, the BEV coordinates are simply
+encoded into the point cloud tokens. Next, we introduce
+the position-guided queries. Each query is initialized as a
+3D reference point following PETR [26]. We transform the
+3D coordinates of reference points to both image and Li-
+DAR spaces, to perform the relative coordinates encoding
+in each space. Moreover, for faster convergence, we in-
+troduce the inductive bias of locality, by extending Query
+Denoising [19] to a point-based formulation.
+The proposed CMT framework brings many advantages
+compared to existing methods. Firstly, our method is a sim-
+ple and end-to-end pipeline and can be easily extended. The
+3D positions are encoded into the multi-modal features im-
+plicitly, which avoids introducing the bias caused by ex-
+plicit cross-view feature alignment. Secondly, our method
+only contains basic operations, without the feature sam-
+pling or complex 2D-to-3D view transformation on multi-
+modal features. Thirdly, the robustness of our CMT is much
+stronger than other existing approaches. Extremely, under
+the condition of LiDAR miss, our CMT with only image
+tokens can achieve similar performance compared to those
+visual 3D object detectors [23,26] (see Fig. 2).
+To summarize, our contributions are:
+• we propose a robust 3D detector, which is a truly end-
+to-end framework without any post-process. It over-
+comes the sensor missing problem.
+• The 3D positions are implicitly encoded into the multi-
+modal tokens, without any complex operations, like
+grid sampling and voxel-pooling.
+• CMT achieves state-of-the-art 3D detection perfor-
+mance on nuScenes dataset. It provides a simple base-
+line for future research.
+2. Related Work
+2.1. Camera Based 3D Object Detection
+Camera-based 3D object detection is one of the basic
+tasks in computer vision. Early works [41, 42] mainly fol-
+low the dense prediction pipeline. They first localize the
+objects on image plane and then predict their relevant 3D at-
+tributes, such as depth, size and orientation. However, with
+the surrounding cameras, the perspective-view based design
+requires elaborate post-processes to eliminate the redundant
+predictions of the overlapping regions. Recently, 3D ob-
+ject detection under the BEV has attracted increasing atten-
+tion. The BEV representation provides a unified coordinate
+to fuse information from multiple camera views. LSS [32],
+BEVDet [15] and BEVDepth [21] predict the depth distri-
+bution to lift the image features to 3D frustum meshgrid.
+Besides, inspired by DETR [4], DETR3D [45] and BEV-
+Former [23] project the predefined BEV queries onto im-
+ages and then employ the transformer attention to model
+the relation of multi-view features. The above methods ex-
+plicitly project the local image feature from 2D perspective
+view to BEV. Different from them, PETR [26,27] and Spa-
+tialDETR [9] adopt the positional embedding that depends
+on the camera poses, allowing the transformer to implicitly
+learn the projection from image views to 3D space.
+2.2. LiDAR Based 3D Object Detection
+LiDAR-based 3D object detection aims to predict 3D ob-
+ject bounding boxes using the point clouds captured from
+LiDAR. Existing methods process the point cloud into dif-
+ferent representations. Point-based methods [22,33–36,49]
+directly extract features from raw point clouds and pre-
+dict 3D bounding boxes. PointNet [34] is the first archi-
+tecture to process the point cloud in an end-to-end man-
+ner, which preserves the spatial characteristics of the point
+cloud. Other methods project the unordered, irregular Li-
+DAR point clouds onto a regular feature space such as
+3D voxels [54], feature pillars [17, 44, 50] and range im-
+ages [11, 38]. Then the features are extracted in the BEV
+
+Figure 3. The architecture of Cross-Modal Transformer (CMT) paradigm. The multi-view images and point clouds are input to two back-
+bone networks to extract feature tokens. In coordinates encoding module, coordinates of camera rays and BEV positions are transformed
+into the image position encoding (Im PE) and point cloud position encoding (PC PE), respectively. The queries are generated by the
+position-guided query generator. In query generator, 3D anchor points are projected to different modalities and the relative coordinates are
+encoded (see the right part). Multi-modal tokens further interact with queries in the transformer decoder. The updated queries are further
+used to predict the 3D bounding boxes. To accelerate the model convergence, the point-based query denoising is introduced.
+plane using a standard 2D backbone. VoxelNet [54] first di-
+vides the raw point clouds into regular voxel grids, and then
+uses PointNet network to extract features from the points in
+each voxel grid.
+2.3. Multi-modal 3D Object Detection
+Multi-sensor fusion in 3D detection has gained great at-
+tention in recent years.
+State-of-the-art (SoTA) methods
+tend to find a unified representation for both modalities, or
+define object queries to fuse the features for further predic-
+tion. For example, BEVFusion [24, 28] applies a lift-splat-
+shoot (LSS) operation to project image feature onto BEV
+space and concatenates it with LiDAR feature. UVTR [20]
+generates a unified representation in the 3D voxel space by
+deformable attention [56]. While for query-based methods,
+FUTR3D [8] defines the 3D reference points as queries and
+directly samples the features from the coordinates of pro-
+jected planes. TransFusion [1] follows a two-stage pipeline.
+The proposals are generated by LiDAR features and further
+refined by querying the image features.
+2.4. Transformer-based Object Detection
+The pioneering work DETR [3] proposes a transformer-
+based detector paradigm without any hand-craft compo-
+nents, and has achieved state-of-the-arts in both 2D and
+3D detection [6, 23, 27, 53]. However, DETR-like meth-
+ods usually suffer from the slow convergence. To this end,
+many works [5, 16, 19, 25, 52, 53, 56] are proposed to im-
+prove the training efficiency from various aspects. Other
+improvements in 2D detection mainly focus on modifying
+the transformer layers [52,56], designing informative object
+queries [19,25,53], or exploring the label assignment mech-
+anism [5, 16].
+Deformable DETR [56] proposes the de-
+formable attention, which only attends to sampling points of
+local regions. SAM-DETR [52] presents a semantic aligner
+between object queries and encoded features to accelerate
+the matching process. To alleviate the instability of bipartite
+matching, DAB-DETR [25] formulates the object queries
+as dynamic anchor boxes, while DN-DETR [19] auxillarily
+reconstructs the ground-truths from the noisy ones. Based
+on them, DINO [53] further improves the denoising anchor
+boxes via a contrastive way.
+3. Method
+The overall architecture of the proposed CMT is illus-
+trated in Fig. 3. Multi-view images and LiDAR points are
+fed into two individual backbones to extract multi-modal
+tokens.
+The 3D coordinates are encoded into the multi-
+modal tokens by the coordinates encoding.
+The queries
+from the position-guided query generator are used to in-
+teract with the multi-modal tokens in transformer decoder
+and then predict the object class as well as the 3D bounding
+boxes. Point-based query denoising is further introduced
+to accelerate the training convergence by introducing local
+prior. The whole framework is learned in a fully end-to-
+end manner and LiDAR backbone is trained from scratch
+without pretraining.
+3.1. Coordinates Encoding Module
+The coordinates encoding module (CEM) is used to en-
+code the 3D position information into multi-modal tokens.
+It generates both the camera and BEV position encodings
+(PEs), which are added to image tokens and point cloud
+tokens respectively. With the help of CEM, multi-modal
+tokens can be implicitly aligned in 3D space.
+Let P(u, v) be the 3D points set corresponding to the
+
+feature map F(u, v) of different modalities. Here (u, v) in-
+dicates the coordinate in the feature map. Specifically, F is
+the image feature for camera while BEV feature for LiDAR.
+Suppose the output position embedding of CEM is Γ(u, v),
+its calculation can be formulated as:
+Γ(u, v) = ψ(P(u, v))
+(1)
+where ψ is a multi-layer perception (MLP) layer.
+CE for Images. Since the image is captured from a per-
+spective view, each pixel can be seen as an epipolar line
+in 3D space. Inspired by PETR [26], for each image, we
+encode a set of points in camera frustum space to per-
+form the coordinates encoding. Given the image feature
+Fim, each pixel can be formulated as a series of points
+{pk(u, v) = (u ∗ dk, v ∗ dk, dk, 1)T , k = 1, 2, ..., d} in the
+camera frustum coordinates. Here, d is the number of points
+sampled along the depth axis. The corresponding 3D points
+can be calculated by:
+pim
+k (u, v) = T l
+ciK−1
+i
+pk(u, v)
+(2)
+where T l
+ci ∈ R4×4 is the transformation matrix from the i-
+th camera coordinate to the LiDAR coordinate. Ki ∈ 4 × 4
+is the intrinsic matrix of i-th camera. The position encoding
+of pixel (u, v) for image is formulated as:
+Γim(u, v) = ψim({pim
+k (u, v),
+k = 1, 2, ..., d})
+(3)
+CE for Point Clouds.
+We choose VoxelNet [48, 54] or
+PointPillar [17] as backbone to encode the point cloud to-
+kens Fpc.
+Intuitively, the point set P in Eq. (1) can be
+sampled along the Z-axis. Suppose (u, v) is the coordi-
+nates in BEV feature map, the sampled point set is then
+pk(u, v) = (u, v, hk, 1)T , where hk indicates the height of
+k-th points and h0 = 0 as default. The corresponding 3D
+points of BEV feature map can be calculated by:
+ppc
+k (u, v) =(u ∗ ud, v ∗ vd, hk, 1)
+(4)
+where (ud, vd) is the size of each BEV feature grid. To
+simplify, we only sample one point along the height axis. It
+is equivalent to the 2D coordinate encoding in BEV space.
+The position embedding of point cloud can be obtained by:
+Γpc(u, v) = ψpc({ppc
+k (u, v),
+k = 1, 2, ..., h})
+(5)
+3.2. Position-guided Query Generator
+Following Anchor-DETR [46] and PETR [26], we firstly
+initialize the queries with n anchor points A = {ai =
+(ax,i, ay,i, az,i), i = 1, 2, ..., n} sampled from uniform dis-
+tribution between [0, 1]. Then these anchor points are trans-
+formed into 3D world space by linear transformation:
+�
+�
+�
+�
+�
+ax,i =
+ax,i∗(xmax − xmin) + xmin
+ay,i =
+ay,i∗(ymax − ymin) + ymin
+az,i =
+az,i∗(zmax − zmin) + zmin
+(6)
+box center
+anchor point
+add 
+noise
+noisy query
+decoder
+box center
+anchor point
+add 
+noise
+noisy query
+decoder
+None
+positive
+negative
+Figure 4. Illustration of the proposed point-based query denoising.
+The noise queries are generated from the box center of ground-
+truths. The positive and negative queries are split by the noise
+scale. Positive queries are used to reconstruct the ground-truths
+boxes, while negative queries predict the “no object”.
+where [xmin, ymin, zmin, xmax, ymax, zmax] is the region
+of interest (RoI) of 3D world space. After that, we project
+the 3D anchor points A to different modalities and encode
+the corresponding point sets by CEM. Then the positional
+embedding Γq of object queries can be generated by:
+Γq = ψpc(Apc) + ψim(Aim)
+(7)
+where Apc and Aim are the point set projected on BEV
+plane and image plane, respectively. The positional embed-
+ding Γq are further added with the query content embedding
+to generate the initial position-guided queries Q0.
+3.3. Point-based Query Denoising (PQD)
+For fast convergence, we extend the query denoising
+strategy in DN-DETR [19] to 3D object detection as shown
+in Fig. 4. Different from DN-DETR [19], we generate the
+noisy anchor points by center shifting since the box scale
+is not that important in 3D object detection.
+For each
+3D ground-truths box (x, y, z, w, l, h, θ), we first sample
+the random ratio λ1, λ2, λ3 within (−λ, λ), where λ is the
+hyper-parameter to control the noise scale. Since 3D space
+is sparse and unobstructed, we adopt a larger tolerance (e.g.
+λ = 1.0). Then the center noise (∆x, ∆y, ∆z) can be cal-
+culated as:
+∆x = λ1w
+2 , ∆y = λ2l
+2 , ∆z = λ3h
+2 .
+(8)
+The center noise is added to the center of ground-truths
+to obtain the noise anchor points.
+Then each noise an-
+chor point can be converted into noise query, as described
+in the last section. Inspired by DINO [53], we also intro-
+duce the negative noisy queries to predict the “no object”.
+To simplify the pipeline, the positive and negative queries
+are simply split by the random ratios λ1, λ2, λ3 and a given
+threshold ξ. For each noise query, it is a positive query if
+�
+λ2
+1 + λ2
+2 + λ2
+3 < ξ, otherwise a negative query.
+
+Table 1. Performance comparison on the nuScenes test set. “L” is LiDAR and “C” is camera.
+Methods
+Modality NDS↑ mAP↑ mATE↓ mASE↓ mAOE↓ mAVE↓ mAAE↓
+BEVDet [15]
+C
+0.488 0.424
+0.524
+0.242
+0.373
+0.950
+0.148
+DETR3D [45]
+C
+0.479 0.412
+0.641
+0.255
+0.394
+0.845
+0.133
+PETR [26]
+C
+0.504 0.441
+0.593
+0.249
+0.383
+0.808
+0.132
+CenterPoint [50]
+L
+0.673 0.603
+0.262
+0.239
+0.361
+0.288
+0.136
+UVTR [20]
+L
+0.697 0.639
+0.302
+0.246
+0.350
+0.207
+0.123
+TransFusion [1]
+L
+0.702 0.655
+0.256
+0.240
+0.351
+0.278
+0.129
+PointPainting [39]
+LC
+0.610 0.541
+0.380
+0.260
+0.541
+0.293
+0.131
+PointAugmenting [40]
+LC
+0.711 0.668
+0.253
+0.235
+0.354
+0.266
+0.123
+MVP [7]
+LC
+0.705 0.664
+0.263
+0.238
+0.321
+0.313
+0.134
+FusionPainting [47]
+LC
+0.716 0.681
+0.256
+0.236
+0.346
+0.274
+0.132
+UVTR [20]
+LC
+0.711 0.671
+0.306
+0.245
+0.351
+0.225
+0.124
+TransFusion [1]
+LC
+0.717 0.689
+0.259
+0.243
+0.359
+0.288
+0.127
+BEVFusion [28]
+LC
+0.729 0.702
+0.261
+0.239
+0.329
+0.260
+0.134
+CMT-L
+L
+0.701 0.646
+0.298
+0.242
+0.330
+0.222
+0.124
+CMT
+LC
+0.730 0.704
+0.299
+0.241
+0.323
+0.240
+0.112
+Table 2. Performance comparison on the nuScenes val set. “L” is
+LiDAR and “C” is camera.
+Methods
+modality
+NDS↑
+mAP↑
+FUTR3D [8]
+L
+0.655
+0.593
+UVTR [20]
+L
+0.676
+0.608
+TransFusion [1]
+L
+0.702
+0.655
+FUTR3D [8]
+LC
+0.683
+0.645
+UVTR [20]
+LC
+0.702
+0.654
+TransFusion [1]
+LC
+0.713
+0.675
+BEVFusion [28]
+LC
+0.714
+0.685
+CMT-L
+L
+0.686
+0.624
+CMT
+LC
+0.719
+0.694
+3.4. Decoder and Loss
+As for the decoder, we follow the original transformer
+decoder in DETR [46] and use L decoder layers. For each
+decoder layer, the position-guided queries interact with the
+multi-modal tokens and update their representations. Two
+feed-forward networks (FFNs) are used to predict the 3D
+bounding boxes and the classes using updated queries. We
+formulate the prediction process of each decoder layer as
+follows:
+ˆbi = Ψreg(Qi), ˆci = Ψcls(Qi),
+(9)
+where Ψreg and Ψcls respectively represent the FFN for
+regression and classification. Qi is the the updated object
+queries of the i-th decoder layer.
+For set prediction, the bipartite matching is applied for
+one-to-one assignment between predictions and ground-
+Figure 5. We analyze the system robustness of CMT at test period
+under three simulated sensor errors: (a) single camera miss, (b) all
+camera miss and (c) LiDAR miss.
+truths. We adopt the focal loss for classification and L1
+loss for 3D bounding box regression:
+L(y, ˆy) = ω1Lcls(c, ˆc) + ω2Lreg(b,ˆb)
+(10)
+where ω1 and ω2 are the hyper-parameter to balance the two
+loss terms. Note that for positive and negative queries in
+query denoising, the loss is calculated in the same way.
+3.5. Masked-Modal Training for Robustness
+Security is the most important concern for autonomous
+driving systems.
+An ideal system requires solid perfor-
+mance even if part of them fails, as well as not relying on
+any input of a specific modality. Recently, BEVFusion [24]
+has explored the robustness of LiDAR sensor failure. How-
+ever, the exploration is limited to restricted scan range and
+model need be retrained. In this paper, we try more extreme
+failures, including single camera miss, camera miss and Li-
+DAR miss, as shown in Fig. 5. It is consistent with the
+actual scene and ensures the safety of autonomous driving.
+
+Table 3. Quantitative results on the nuScenes val with LiDAR or camera miss. With the masked-modal training, the efficacy and robustness
+of our CMT is significantly improved, especially when the LiDAR camera is missed.
+Metric
+Vanilla training
+Masked-modal training
+CMT
+only LiDAR
+only Cams
+CMT
+only LiDAR
+only Cams
+NDS ↑
+0.716
+0.594
+0.067
+0.719 (↑0.3%)
+0.677 (↑8.3%)
+0.434 (↑36.7%)
+mAP ↑
+0.685
+0.472
+0.000
+0.694 (↑0.9%)
+0.613 (↑14.1%)
+0.386 (↑38.6%)
+To improve the robustness of the model, we propose a
+training strategy, called masked-modal training. In training
+process, we randomly use only a single modality for train-
+ing, such as camera or LiDAR, with the ratio of η1 and η2.
+This strategy ensures that the model are fully trained with
+both single modal and multi-modal. Then the model can be
+tested with single modal or multi-modal, without modify-
+ing the model weight. The experimental results show that
+masked-modal training will not affect the performance of
+our fusion model. Even if LiDAR is damaged, it can still
+achieve similar performance compared to SoTA range-view
+3D detectors [15,26].
+4. Experiments
+4.1. Datasets and Metrics
+We evaluate our method on nuScenes [2]. nuScenes is a
+large-scale multi-modal dataset, which is composed of data
+from 6 cameras, 1 LiDAR and 5 radars. The dataset has
+1000 scenes totally and is divided into 700/150/150 scenes
+as train/validation/test sets, respectively.
+Cameras. Each scene has 20s video frames with 12
+FPS. 3D bounding boxes are annotated every 0.5s. We only
+use these key frames. In each frame, nuScenes provides
+images from six cameras.
+LiDAR. NuScenes provides a 32-beam LiDAR with 20
+FPS. The key frames are also annotated every 0.5s, the same
+as cameras. We follow the common practice to transform
+the points from the past 9 frames to the current frame for
+training and evaluation.
+Metrics. We follow the nuScenes official metrics. We
+report the nuScenes Detection Score (NDS), mean Average
+Precision (mAP), mean Average Translation Error (mATE),
+mean Average Scale Error (mASE), mean Average Orien-
+tation Error(mAOE), mean Average Velocity Error (mAVE)
+and mean Average Attribute Error (mAAE).
+4.2. Implementation Details
+We use ResNet [13] or VoVNet [18] as image backbone
+to extract the 2D image features. The C5 feature is up-
+sampled and fused with C4 feature to produce P4 feature.
+We use VoxelNet [54] or PointPillars [17] as the backbone
+to extract the point-cloud features. We set the region-of-
+interest (RoI) to [−54.0m, 54.0m] for X and Y axis, and
+[−5.0m, 3.0m] for Z axis. The 3D coordinates in the world
+space are normalized to [0, 1]. All the feature dimension
+is set to 256, including the LiDAR feature, image feature
+and query embedding. Six decoder layers are adopted in
+transformer decoder. Voxel size of 0.075 and image size of
+1600 × 640 are adopted as default in our experiments.
+Our model is trained with the batch size of 16 on 8 A100
+GPUs. It is trained for total 20 epochs with CBGS [55]. We
+adopt the AdamW [29] optimizer for optimization. The ini-
+tial learning rate is 1.0×10−4 and we follow the cycle learn-
+ing rate policy [37]. The mask ratios η1 and η2 are both set
+to 0.25 for masked-modal training. The threshold ξ is set to
+0.75 to divide the noise queries into positives and negatives
+for training. The tolerance λ that controls the noise scale is
+set to 1. The GT sample augmentation is employed for the
+first 15 epochs and closed for the rest epochs. As for the
+loss weights, we follow the default setting in DETR3D [45]
+and set the ω1 and ω2 to 2.0 and 0.25, respectively.
+4.3. State-of-the-Art Comparison
+As shown in Tab. 1, CMT achieves comparable results
+compared to several state-of-the-art methods on nuScenes
+test set. Our LiDAR-only baseline, named CMT-L, achieves
+the 70.1% NDS, which is a nearly SoTA performance
+among all existing LiDAR-only methods. Our multi-modal
+method CMT achieves 73.0% NDS and 70.4% mAP and
+outperforms existing SoTA BEVFusion. Benefits from the
+large receptive field, CMT gains better results on some met-
+rics like mAVE. We also compare the performance with
+other SoTA methods on nuScenes val set (see Tab. 2).
+It shows that our proposed CMT with multi-modal fu-
+sion outperforms the BEVFusion by 0.5% NDS and 0.9%
+mAP. CMT introduces large performance improvements
+compared to our LiDAR-only CMT-L by 2.9%/5.8% and
+3.3%/7.0% NDS/mAP on test and validation set, showing
+that camera images bring complementary information.
+4.4. Strong Robustness
+We evaluate the robustness of our framework under var-
+ious harsh environments, including LiDAR miss and cam-
+era miss. Tab. 3 shows the results when the sensor miss
+occurs, by simulating the scenarios of any modality totally
+broken. The performance is compared between the vanilla
+training and masked-modal training. It validates the effect
+
+Table 4. The ablation studies of different components in the proposed CMT.
+Im
+PC NDS mAP mATE mASE mAOE
+✓
+0.595 0.554 0.515 0.258
+0.429
+✓ 0.665 0.626 0.372 0.255
+0.347
+✓
+✓ 0.669 0.641 0.377 0.254
+0.375
+(a) Position encoding for query.
+PQD
+NDS mAP mATE mASE mAOE
+0.626 0.584 0.429 0.259
+0.420
+✓
+0.669 0.641 0.377 0.254
+0.375
+(b) Point-based Query Denoise (PQD)
+NDS mAP mATE mASE mAOE
+0.075
+0.669 0.641 0.377 0.254
+0.375
+0.1
+0.671 0.638 0.378 0.252
+0.334
+0.125
+0.655 0.624 0.396 0.255
+0.397
+(c) Voxel size of LiDAR backbone.
+NDS mAP mATE mASE mAOE
+ResNet-50
+0.658 0.623 0.376 0.253
+0.399
+ResNet-101 0.664 0.629 0.383 0.254
+0.363
+VoV-99
+0.669 0.641 0.377 0.254
+0.375
+(d) Image backbone.
+NDS mAP mATE mASE mAOE
+800 × 320 0.654 0.609 0.374 0.256
+0.389
+1600 × 640 0.669 0.641 0.377 0.254
+0.375
+(e) Input size of image backbone.
+NDS mAP mATE mASE mAOE
+PointPillars 0.628 0.598 0.430 0.252
+0.455
+VoxelNet
+0.669 0.641 0.377 0.254
+0.375
+(f) Lidar backbone
+of masked-modal training. Note that the model are only
+trained with multi-modality and evaluated without any fine-
+tune process. With vanilla training, the model fails to pre-
+dict anything meaningful (only Cams with mAP=0) when
+LiDAR is missing. With masked-modal training, the ab-
+sence of LiDAR or camera modalities lead to 4.2% and
+28.5% NDS drop compared to CMT, respectively.
+It is
+observed that losing one modality still remains similar re-
+sults compared to single-modal training settings. It over-
+comes the drawback that multi-modal method usually rely
+on one major modality and performance would degrade sig-
+nificantly if losing the major modality. Especially, for the
+case of LiDAR missing, the performance is still compara-
+ble to the SoTA camera-only method PETR [26], validating
+the strong robustness of our method.
+Moreover, we also investigate the case when any one
+of cameras fails. Experimental result shows slight perfor-
+mance drop, indicating the tolerable to single camera miss
+of our method. Six sensors brings an average decrease of
+0.7% NDS, no more than 1% performance of the oracle ver-
+sion. The front and back sensor relatively play the important
+role among camera sensors, with 1.1% and 0.8% decrease
+respectively, due to their distant or large field of view. Com-
+pared to the camera-only setting, our multi-modal frame-
+work facilitate the compensation between LiDAR and im-
+age domains, thus presenting a robust performance.
+4.5. Ablations
+We present the ablation studies in Tab. 4. All the experi-
+ments are conducted for 20 epochs without CBGS [31].
+We first ablate the effect of Im PE and PC PE on the gen-
+eration of position-guided queries (see Tab. 4 (a)). It shows
+that removing PC PE introduces a 7.4%/8.70% NDS/mAP
+performance drop, which is much larger than the drop of re-
+moving Im PE 0.4%/1.5%. Next, we explore the effective-
+ness of our proposed PQD, as shown in Tab. 4(b). We can
+easily find that PQD can greatly improve the overall perfor-
+mance by 4.3%/5.7% NDS/mAP. With PQD, the training
+convergence can be boosted, which is similar to the practice
+in DN-DETR [19]. Further, Tab. 4 (c-f) illustrates the effect
+of scaling up the CMT model as well as the input size. Over-
+all, CMT can benefit from the scaling model size. Interest-
+ingly, we find increasing the voxel number (smaller voxel
+size) and image size achieves similar improvements ≈ 1.5%
+in NDS. While scaling the image size increases more mAP
+than the voxel number(+3.2% vs. +1.7%). When increasing
+the image size from 800 × 320 to 1600 × 640, we find the
+performance improvements are mainly from these small ob-
+jects, such as pedestrian and motorcycle. We also conduct
+experiments on replacing image and LiDAR backbones,
+we use VoV-99 [18] and ResNet [13] as our image back-
+bones. Experiments show that our proposed CMT can ben-
+efit from larger backbones. For image, VoV-99 backbone
+achieves the best result and outperforms the ResNet-50 by
+1.1%/1.8% in NDS/mAP. While for LiDAR, VoxelNet out-
+performs the PointPillar by 4.1%/4.3% in NDS/mAP.
+4.6. Analysis
+CMT is a direct and easy pipeline for multi-modal fu-
+sion and can be easily extended.
+Moreover, benefiting
+from DETR [3] framework and our training schedule, CMT
+shows strong robustness under sensor miss conditions. We
+present some attempted experiments in this section.
+Data extension. Multi-frame is now a common setting in
+
+Figure 6. Visualization of attention maps on multi-view images and BEV point clouds. The blue points (•) are the initialized anchor points
+while red points (•) are the corresponding centers of box predictions. It can be easily found that the high response regions of attention
+maps mainly focus on the foreground objects, closest to the anchor points.
+camera-based 3D object detection [14,23,27]. Using multi
+frames often outperforms the single frame by a clear margin
+and can solve some typical occlusion problem. We follow
+the multi-frame alignment in PETRv2 [27]. Considering
+the high memory cost of multi-frames, we conduct our ex-
+periment with a 800 × 320 image resolution. As shown in
+Tab. 5, adding image frame only improves the NDS/mAP
+by 0.2%/0.7%. More image frames rarely improve the per-
+formance with multi-modal fusion.
+Radar has advantages in long range detection and the ro-
+bustness of extreme weather. Following FUTR3D [8], we
+stack points from 5 radars together to generate the point
+cloud. We use several MLP layers to perform coordinates
+encoding on Radar features, the same as LiDAR. Tab. 5
+shows that adding Radar data to our pipeline degrades the
+performance by 0.9% NDS and 0.6% mAP.
+Visualization. For better understanding on querying from
+multi-modal tokens, we visualize the attention map of
+cross-attention on different modalities (see Fig. 6). We can
+clearly find that the attention maps have higher response on
+images and point clouds. It shows that our method can im-
+plicitly achieve the cross-modal interaction. We visualize
+the initial anchor points and the center points of predictions.
+Most anchor points focus on the closest foreground objects.
+After the interaction with queries in decoder, anchor points
+gradually access the accurate center points.
+Table 5. Results with more image frames or with Radar points.
++Frame
++ Radar
+NDS
+mAP
+0.698
+0.662
+✓
+0.700
+0.669
+✓
+0.689
+0.656
+5. Conclusions
+In this paper, we propose a fully end-to-end framework
+for multi-modal 3D object detection. It implicitly encodes
+the 3D coordinates into the tokens of images and point
+clouds. With the coordinates encoding, the simple yet effec-
+tive DETR pipeline can be adopted for multi-modal fusion
+and end-to-end learning. With masked-modal training, our
+multi-modal detector can be learned with strong robustness,
+even if one of multi-modalities are missed. We hope such
+a simple pipeline design could provide more insights on the
+end-to-end 3D object detection.
+Limitation: Though our CMT brings some advantages
+compared to those existing approaches, it also reveals some
+limitations. The computation cost is relatively larger due
+to the large number of multi-modal tokens and the global
+attention employed in transformer decoder. To solve this
+problem, some efforts in two directions maybe taken. The
+first one is to reduce the redundancy of multi-modal tokens.
+The foreground tokens can be roughly selected with another
+individual network [43]. The foreground tokens are then in-
+put to our network for high-speed inference. Another pos-
+sible solution is to replace the global attention with other
+efficient attentions, like deformable attention [56]. One can
+also employ a small set of object queries since most queries
+correspond to the empty objects.
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+

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+ 
+Switchable anomalous Hall effects in polar-stacked 2D antiferromagnet MnBi2Te4 
+Tengfei Cao*, Ding-Fu Shao*,†, Kai Huang, Gautam Gurung‡, and Evgeny Y. Tsymbal* 
+ Department of Physics and Astronomy & Nebraska Center for Materials and Nanoscience, 
+ University of Nebraska, Lincoln, Nebraska 68588-0299, USA 
+       Van der Waals (vdW) assembly allows controlling symmetry of two-dimensional (2D) materials that determines their 
+physical properties. Especially interesting is the recently demonstrated breaking inversion symmetry by polar layer stacking 
+to realize novel electronic, magnetic, and transport properties of 2D vdW materials switchable by induced electric 
+polarization. Here, based on symmetry analyses and density-functional calculations, we explore the emergence of the 
+anomalous Hall effect (AHE) in antiferromagnetic MnBi2Te4 films assembled by polar layer stacking. We demonstrate that 
+breaking 𝑃̂𝑇̂ symmetry in an MnBi2Te4 bilayer makes this 2D material magnetoelectric and produces a spontaneous AHE 
+switchable by electric polarization. We find that reversable polarization at one of the interfaces in a three-layer MnBi2Te4 
+film drives a metal-insulator transition, as well as switching between an AHE and quantum AHE (QAHE). Finally, we 
+predict that engineering an interlayer polarization in a three-layer MnBi2Te4 film allows converting MnBi2Te4 from a trivial 
+insulator to a Chern insulator. Overall, our work emphasizes the emergence of quantum-transport phenomena in 2D vdW 
+antiferromagnets by polar layer stacking, which do not exist in this material in the bulk or bulk-like thin-film forms.  
+Symmetry breaking plays an important role in generating 
+quantum states in condensed matter, and thus its control allows   
+manipulating these states and the associated transport properties, 
+promising new functionalities1. For example, time-reversal (𝑇̂) 
+symmetry breaking directly affects the spin density distribution 
+in material systems, producing spontaneous magnetic orders and 
+novel spin-dependent transport properties useful for electronic 
+applications2,3,4,5,6,7,8. The control of these states usually requires 
+an external magnetic field as the stimulus directly modulating 
+the 𝑇̂ symmetry breaking by reorienting the magnetic moments. 
+However, the use of an external magnetic field in electronic 
+devices is not efficient. On one hand, it requires substantial 
+electric currents and thus costs large energy dissipation. On the 
+other hand, an external magnetic field can hardly control the 
+magnetic order in antiferromagnets – materials that have 
+recently attracted increasing interest due to their potential 
+application for next generation spintronics9,10,11,12,13,14. It would 
+be desirable to have a method to directly control the spin-
+dependent properties of a material without changing its 
+magnetic ground state, e.g., an applied electric field, although 
+such a “nonmagnetic” method does not directly influence 𝑇̂ 
+symmetry 15,16,17.  
+In recent years, two-dimensional (2D) van der Waals (vdW) 
+materials have become an extensively studied model system for 
+exploring the new physics and potential applications at the 
+nanoscale18,19,20,21,22,23,24. A particular interest was stimulated by 
+the recent discoveries of unexpected phenomena driven by 
+interlayer stacking 25,26,27,28,29.  Despite weak interlayer coupling, 
+minor interlayer sliding26,27,28 or  small-angle interlayer 
+twisting25,30,31 can generate electronic and transport properties 
+which do not exist in the bulk-like phases. This is largely due to 
+the modification of the interlayer stacking pattern breaking 
+crystal symmetries. Specifically, a combination of interlayer 
+sliding and 180˚ twisting results in a non-centrosymmetric 
+stacking pattern accompanied by the emergence of an out-of-
+plane electric polarization reversable by an external electric 
+field 26,27,28, 32 . This approach allows the design of 2D 
+ferroelectrics out of parent nonpolar compounds to uncover new 
+functionalities not existent in the bulk phase33. 
+Such vdW polar stacking can be employed to engineer 
+recently discovered 2D antiferromagnets 34 . Breaking space 
+inversion symmetry in these materials could potentially induce 
+novel spin-dependent properties controlled by the intrinsic 
+ferroelectric polarization. Especially interesting are 2D 
+materials derived from antiferromagnet MnBi2Te4 which 
+exhibits non-trivial transport properties in the thin-film 
+form4,5,35,36,37,38,39,40. Although 𝑇̂ symmetry is globally broken 
+by magnetism of these antiferromagnetic materials, the 
+magnetism is “hidden” by 𝑇̂𝑂̂ symmetry that combines 𝑇̂ and 
+crystal symmetry 𝑂̂  such as space inversion ( 𝑃̂ ) or mirror 
+reflection (𝑀̂ ). Since the polar layer stacking affects crystal 
+symmetry 𝑂̂, it allows an indirect control of the 𝑇̂ symmetry 
+breaking in 2D magnets and thus spin-dependent properties, 
+such as the anomalous Hall effect (AHE).  
+The intrinsic AHE is governed by the Berry curvature Ω, a 
+property arising from the spin-dependent band structure of 
+magnetic systems 41. The anomalous Hall conductance (AHC) 
+is quantized in 2D magnetic topological insulators due to 
+formation of the topologically protected edge channels resulting 
+in a dissipationless quantum AHE (QAHE) 2,4,35. Ferromagnetic 
+systems naturally support the AHE due to the intrinsic 
+𝑇̂ symmetry breaking. Antiferromagnets also break 𝑇̂ symmetry, 
+but usually preserve 𝑇̂𝑂̂  symmetry that enforces the Berry 
+curvature to be antisymmetric in k-space, i.e. 𝑇̂𝑂̂𝜴(𝒌) =
+−𝜴(𝑇̂𝑂̂𝒌), prohibiting the AHE. The AHE can emerge in 
+magnetically uncompensated antiferromagnets, where the 
+combined 𝑇̂𝑂̂  symmetry is broken either by noncollinear 
+magnetic configurations or non-magnetic atoms at low 
+
+2 
+ 
+symmetry positions 6,41,42,43,44,45. In the former case, the AHE is 
+dependent on the orientation of the Néel vector, which can be 
+changed by an applied magnetic field. In the latter case, the 
+atomic structure of the non-magnetic sublattice can influence 
+the AHE; however, there are no straightforward means to 
+control this structure by external stimulus.   
+       In this letter, based on symmetry analysis and first-
+principles density functional calculations, we demonstrate that 
+the 
+AHE 
+can 
+be 
+effectively 
+induced 
+in 2D 
+vdW 
+antiferromagnets by polar layer stacking. Using recently 
+discovered antiferromagnetic topological insulator MnBi2Te4, 
+as a representative material, we show that polar layer stacking 
+breaks 𝑃̂𝑇̂ symmetry and induces a magnetoelectric effect, a 
+normal AHE, or QAHE, depending on the number of MnBi2Te4 
+layers and their stacking. These effects can be controlled by the 
+switchable electric polarization of the polar stacked MnBi2Te4, 
+thus providing a route to manipulate the spin-dependent and 
+quantum properties without an external magnetic field.  
+Results and discussion  
+Monolayer MnBi2Te4 represents a septuple layer, which can be 
+viewed as a MnTe bilayer intercalated into the center of a Bi2Te3 
+quintuple layer (Fig. 1(a)). Bulk MnBi2Te4 has 𝑃̂𝑇̂ symmetry 
+and an A-type antiferromagnetic order with colinear out-of-
+plane magnetic moments. A MnBi2Te4 film in bulk-like stacking 
+inherits the bulk A-type antiferromagnetic order and exhibits 
+quantum phenomena dependent on the number of layers 2,4,36. 
+An MnBi2Te4 film with an even number of layers, e.g., a bilayer, 
+has intrinsic 𝑃̂𝑇̂  symmetry, which not only prevents the net 
+magnetic moment, but also impedes both the AHE and QAHE. 
+This can be seen from the expression for the Berry curvature:  
+𝛺(𝒌) = −2Im ∑
+⟨𝑛|
+𝜕𝐻̂
+𝜕𝑘𝑥
+|𝑚⟩⟨𝑚|
+𝜕𝐻̂
+𝜕𝑘𝑦
+|𝑛⟩
+(𝐸𝑛𝒌−𝐸𝑚𝒌)2
+𝑚≠𝑛
+ ,  
+(1) 
+where 𝐻̂ is the Hamiltonian of the system and 𝐸𝑛𝑘 is the energy 
+of the n-th band at wave vector k. The AHC determined by the 
+integration of 𝛺(𝒌) in the Brillouin zone (BZ) as follows: 
+𝜎𝑥𝑦 = −
+𝑒2
+ℎ ∫
+𝑑2𝒌
+2𝜋
+𝐵𝑍
+𝛺(𝒌). 
+(2) 
+According to Eq. (1), the Berry curvature is odd under the 𝑃̂𝑇̂ 
+symmetry operation, i.e., 𝑃̂𝑇̂𝛺(𝒌) = −𝛺(𝒌), and thus is zero in 
+any 𝑃̂𝑇̂ -symmetric system enforcing 𝜎𝑥𝑦  to vanish. On the 
+contrary, due to the A-type antiferromagnetic order, an 
+MnBi2Te4 film with an odd number of layers does not have the 
+restriction of 𝑃̂𝑇̂  symmetry, due to an uncompensated net 
+magnetic moment. As a result, an intrinsic QAHE can occur in 
+such a film where the number of quantum states is determined 
+by the number of MnBi2Te4 layers, as was predicted 
+theoretically 4 and demonstrated experimentally 2,36,38. Here, we 
+show that a polar-stacked MnBi2Te4 provides even a broader 
+spectrum of quantum states and these states can be controlled by 
+electric polarization.  
+Using the vdW assembly approach demonstrated in refs. 26 
+and 27, a MnBi2Te4 bilayer can be engineered to be polar. This 
+requires top monolayer Ā to be deposited as mirror reflection 
+( 𝑀̂𝑧 ) of the bottom monolayer. The resulting non-
+centrosymmetric AĀ stacking (Fig. 1(b)) is unstable due to the 
+ 
+Fig. 1 (a) Atomic structure of monolayer MnBi2Te4 (top and side 
+views). (b) Unstable AĀ stacking of a MnBi2Te4 bilayer where the top 
+monolayer is vdW assembled as a mirror reflection (𝑀̂𝑧) of the bottom 
+monolayer. (c, d) A polar-stacked MnBi2Te4 bilayer with antiparallel 
+magnetic moments and electric polarization pointing down Pd (c) and 
+up Pu (d) due to the in-plane translation, −t∥ (c) or t∥ (d), of the top 
+monolayer from the AĀ stacking in (b). The bilayer structures with 
+opposite polarization are related by the  𝑀̂𝑧𝑇̂  symmetry operation or 
+equivalently by in-plane translation 2t∥. 
+ 
+Fig. 2 (a) Top: schematic of a polar-stacked MnBi2Te4 bilayer with 
+polarization pointing up (Pu) or down (Pd). Middle and bottom: an 
+AHE in a polar-stacked MnBi2Te4 bilayer switchable by ferroelectric 
+polarization. (b) Top: schematic of a MnBi2Te4 trilayer with a single 
+(top) polar interface. Bottom: switching between AHE and QAHE 
+with polarization reversal. (c) Top: schematic of a MnBi2Te4 trilayer 
+with two polar interfaces. Bottom: transition between trivial and Chern 
+insulators driven by polarization switching.   
+ 
+ 
+ 
+ 
+
+a
+b
+AA
+C
+AB
+d
+BA
+t
+M,T
+Mn"- 
+ti
+"- q
+-t
+c
+ti
+4
+4..
+1.
+Top
++M
++M
++M
+Top
++M
++M
++M
+Top
+↑
+Pu.1
+P.1
+P.1
+P
+-M
+Bottom
+-M
+Middle
+-M
+Middle
+Td
+Bulk-like stacking
+xy1
+Bottom
+Bottom
+W+
++M
+Oxy
+0
+0
+Pu Pd
+Pu Pd
+P3 
+ 
+proximity of the Te atoms across the interface. It spontaneously 
+relaxes to the lower-energy AB or BA stacking with antiparallel 
+magnetic moments (Figs. 1(c,d)). The latter is distinguished by 
+the lateral translation, −t∥ or t∥, along the [11̅0] direction of the 
+top MnBi2Te4 monolayer with respect to the bottom ( 𝒕|| =
+⅓(𝒂 − 𝒃) in Fig. 1(a)), placing the interfacial Bi atom atop the 
+Te atom. The resulting two structures, AB and BA, have 
+opposite polarizations pointing down Pd or up Pu depending on 
+the lateral translation, –t∥ or t∥, respectively (Figs. 1(c,d)). The 
+polarization can be switched by an applied electric field through 
+interlayer sliding.  
+Depending on the number of monolayers, the interlayer 
+polarization has different effects on quantum states of MnBi2Te4 
+films. In case of a bilayer (Fig. 2 (a)), polar staking breaks 
+inversion symmetry and hence 𝑃̂𝑇̂  symmetry. This lifts the 
+𝑃̂𝑇̂𝛺(𝒌) = −𝛺(𝒌) constraint on the Berry curvature of the bulk 
+phase making AHC of a polar MnBi2Te4 bilayer nonzero. The 
+two polarization states are related by the 𝑀̂𝑧𝑇̂  symmetry 
+transformation (Fig. S2(a)), which changes sign of the Berry 
+curvature, 𝑀̂𝑧𝑇̂𝛺(𝒌) = −𝛺(−𝒌), and hence sign of the AHC 
+σxy in Eq. (2). Therefore, the switchable polarization of the polar 
+MnBi2Te4 bilayer can serve as a control parameter for the AHE. 
+For a polar-stacked MnBi2Te4 trilayer, the net magnetic 
+moment is non-zero and hence the AHE is always finite due to 
+broken 𝑃̂𝑇̂  symmetry. In case of a trilayer with a polar 
+MnBi2Te4 bilayer deposited on an MnBi2Te4 monolayer with 
+bulk-like interface (Fig. 2(b)), the interlayer polarization 
+controls band bending across the layer stack and can lead to the 
+AHE or QAHE depending on the polarization orientation. In 
+case of a trilayer with two polar interfaces (Fig 2(c)), the 
+polarization of the top interface can be switched to be parallel or 
+antiparallel to the polarization of the bottom interface resulting 
+in the topological phase transition from a trivial band insulator 
+to a non-trivial Chern insulator.  
+To explicitly demonstrate the anticipated properties, we 
+apply density functional calculations, as described in 
+Supplementary Material. Fig. 3 (a) shows the calculated energy 
+profile and corresponding polarization for a polar MnBi2Te4 
+bilayer when sliding the top layer with respect to the bottom 
+along the [11̅0] direction (see Fig. S1 for the sliding energy 
+profile along the [100] direction). Zero fractional shift matches 
+to the AĀ stacking which has the highest energy and thus 
+unstable. The energy drops down with the shift and exhibits two 
+local minima separated by an energy barrier of 0.09 eV/f.u. 
+These minima occur at a fractional shift of a third and two thirds 
+the in-plane unit cell vector in the [11̅0] direction (equivalent to 
+the lateral translation, t∥ and −t∥, respectively). The two local 
+energy minima correspond to the polar structures with 
+polarization pointing up and down (Fig. 3(a)). The interlayer 
+polarization is mainly contributed by the charge transfer and 
+associated displacements of the interfacial cation Bi and anion 
+Te atoms, creating a dipole pointing from Bi to Te. The 
+calculated out-of-plane polarization magnitude is 0.01 μC/cm2. 
+This polarization induces the electrostatic potential energy drop 
+across the MnBi2Te2 bilayer of 0.05 eV as shown in Fig. S2 (b)). 
+Thus, the interlayer sliding transforms the polar structure with 
+polarization pointing up to the structure with polarization 
+pointing down or vice versa. Such sliding and the associated 
+polarization reversal can be induced by an applied out-of-plane 
+electric field of less than 1 V/nm, which is feasible in 
+experimental conditions.  
+The 𝑃̂𝑇̂ symmetry broken by polar-layer stacking breaks 
+Kramers’ degeneracy of the energy bands for the bilayer. In the 
+absence of spin-orbit coupling (SOC), the bands are exchange 
+split into spin-up (𝜎 = ↑) and spin-down (𝜎 = ↓) states (Fig. 
+S3). The spin character of the bands 𝐸𝜎(𝒌) is reversed with 
+ferroelectric polarization, as follows from 𝑀̂𝑧𝑇̂𝐸↑(𝒌) =
+𝐸↓(−𝒌). Including SOC, changes the band structure of a polar 
+MnBi2Te4 bilayer. As seen from Fig. 3(b), similar to bulk 
+MnBi2Te4, the polar bilayer represents a direct band gap 
+semiconductor with the conduction band minimum (CBM) and 
+the valence band maximum (VBM) located at the Γ point. The 
+states around the Fermi level are mainly composed of the p 
+states of Bi and Te with the d states of Mn being relatively far 
+away from the Fermi energy. Due to the presence of ferroelectric 
+polarization, the band gap of the polar bilayer is reduced to about 
+0.06 eV from 0.08 eV obtained for a non-polar bilayer and 0.16 
+eV known for bulk MnBi2Te4.   
+ 
+Fig. 3 (a) Total energy and out-of-plane electric polarization of a polar 
+MnBi2Te4 bilayer when sliding the top monolayer with respect to the 
+bottom along the [11̅0] direction. The two minima in energy profile 
+correspond to polarization pointing up (Pu) and down (Pd). (b) 
+Electronic band structure of a polar MnBi2Te4 bilayer along the high-
+symmetry directions in the 2D Brillouin zone. (c) Berry curvature of a 
+polar MnBi2Te4 bilayer in the 2D Brillouin zone calculated at E – EF = 
+0.07 eV for polarization pointing up (top) and down (bottom). (d) 
+Anomalous Hall conductance 𝜎𝑥𝑦 of a polar MnBi2Te4 bilayer as a 
+function of electron energy for polarization pointing up and down. 
+
+a
+0.00
+0.02
+C
+M
+100
+/cm2)
+0.02
+0.01
+K
+1
+AE(eV/f.u.)
+/on)
+0.04
+0.00
+rization
+0.06
+M
+-0.01
+Pola
+-0.08
+K
+0.000.170.330.500.670.841.00
+-0.02
+-100
+FractionalShift
+b
+d
+0.6
+2.5
+Pu
+0.4
+1.5
+Pd
+(eV)
+0.2
+Te(p)
+(e?/h)
+0.0
+Bi(p)
+0.0
+E
+0.2
+Mn(d)
+0.4
+1.5
+-0.6
+-2.5
+M
+K
+M
+-0.30-0.15
+0.00
+0.15
+0.30
+E-Er(eV)4 
+ 
+We note that a polar-stacked MnBi2Te4 bilayer represents a 
+magnetoelectric multiferroic. It has both a spontaneous electric 
+polarization and an antiferromagnetic order. The magnetic 
+moments of Mn atoms of about 4.47 μB are aligned antiparallel 
+on top and bottom MnBi2Te4 monolayers. Importantly, due to 
+broken 𝑃̂𝑇̂  symmetry, the antiferromagnetism of the polar 
+bilayer is not fully compensated. While the uncompensated net 
+magnetic moment is small, about 0.002 μB per formular unit, it 
+is non-vanishing and reversible by ferroelectric polarization, as 
+follows from 𝑀̂𝑧𝑇̂𝑴 = −𝑴, where 𝑴 is the net magnetization. 
+Potentially this functionality of the polar-stacked vdW 
+antiferromagnets may be useful for applications. 
+The broken  𝑃̂𝑇̂ symmetry causes the Berry curvature of a 
+polar-stacked MnBi2Te4 bilayer to be non-vanishing. Fig. 3(c) 
+shows the calculated Berry curvature 𝛺(𝒌) in the 2D Brillouin 
+zone for up and down polarization states at energy E – EF = 0.07 
+eV above the band gap. It is seen that 𝛺(𝒌) exhibits a flower-
+like pattern with alternating red and blue color of the petals (i.e. 
+alternating sign of 𝛺(𝒌)  ) reflecting the 3-fold rotational 
+symmetry of the trilayer around the z axis. This symmetry 
+combined with 𝑀̂𝑧𝑇̂𝛺(𝒌) = −𝛺(−𝒌), makes the color of the 
+petals (reflecting the sign of 𝛺(𝒌) ) independent of the 
+polarization orientation. On the contrary, a persistent color 
+around the Γ point is reversed from red to blue when polarization 
+is switched from pointing up to down. 
+ The non-vanishing Berry curvature implies finite AHC. 
+Fig. 3(d) shows the calculated AHC as a function of electron 
+energy. It is evident that the AHC is zero within the band gap of 
+an MnBi2Te4 bilayer, indicating that the bilayer represents a 
+trivial insulator (semiconductor). However, with electron or 
+hole doping, AHC becomes non-zero and, as expected, has 
+opposite sign for ferroelectric polarization pointing up and down. 
+Thus, we observe the new functionality of a polar-stacked 
+MnBi2Te4 bilayer which does not exist in bulk-like MnBi2Te4. 
+The presence of a finite AHC in a polar-stacked MnBi2Te4 
+bilayer can be understood in terms of a layer-dependent Hall 
+effect39. While a bulk-like MnBi2Te4 bilayer represents an axion 
+insulator where the top and bottom MnBi2Te4 monolayers 
+spontaneously deflect electrons in opposite directions resulting 
+in zero net AHE, the presence of spontaneous polarization in a 
+polar MnBi2Te4 bilayer breaks the AHE-compensating 
+symmetry between the top and bottom monolayers producing a 
+finite AHC. This phenomenon is analogous to the effect of 
+electric field on a bulk-like MnBi2Te4 bilayer39. However, while 
+in the case of a bulk-like MnBi2Te4 bilayer, the electric field 
+needs to be maintained to have a non-zero AHC, the AHE in a 
+polar MnBi2Te4 bilayer is spontaneous and non-volatile with the 
+sign of AHC being linked to the direction of electric polarization.  
+       The polarization controlled AHC emerges not only in a 
+MnBi2Te4 bilayer that is composed of two polar-stacked 
+MnBi2Te4 monolayers, but also in MnBi2Te4 films assembled by 
+polar stacking of two MnBi2Te4 layers formed of two or more 
+monolayers. Fig. S3 shows the results of calculations of the band 
+structure and AHC for polar stacked four- (Fig. S3(a)) and six- 
+(Fig. S3(d)) monolayer MnBi2Te4 films. It is seen that compared 
+to the MnBi2Te4 bilayer, the band gap is reduced in the four-
+monolayer MnBi2Te4 (Fig. S3(b)) and almost vanishes in the 
+six-monolayer MnBi2Te4 (Fig. S3(e)). The AHC as a function 
+of energy (Figs. S3(c,f)) exhibits trends similar to those of the 
+bilayer.  In particular, the AHC is reversed by switching the 
+interlayer polarization. Thus, in experimental conditions, vdW 
+assembly can be used to engineer polar MnBi2Te4 films with an 
+even number of monolayers to realize an AHE switchable by the 
+intrinsic ferroelectric polarization.  
+Next, we consider effects of polar stacking in three-
+monolayer MnBi2Te4 films. Due A-type antiferromagnetism, 
+such films have a large uncompensated magnetic moment of 
+about 4.47 μB per supercell. The presence of this magnetic 
+moment breaks 𝑃̂𝑇̂  symmetry of the parent bulk phase 
+producing a nonzero Berry curvature and the associated AHE. 
+Following the schematic Fig. 2(b), we assume that a three-
+monolayer MnBi2Te4 slab is composed of a polar-stacked 
+bilayer deposited on a MnBi2Te4 monolayer with the bulk-like 
+stacked bottom interface (Figs. 4(a,b) left panels)). We find a 
+small band offset (~ 0.05 eV) between the polar and non-polar 
+stacked interfaces resulting in a built-in electric field Eb across 
+the trilayer pointing down (indicated by blue arrows in Figs. 
+ 
+Fig. 4 (a,b) Three-monolayer MnBi2Te4 film with a polar top interface  
+with polarization pointing down (Pd) (a) or up (Pu) (b) and non-polar 
+bulk-like bottom interface (left panels), and corresponding layer-
+resolved density of states (DOS) as a function of energy(right panels). 
+In the left panels, blue arrows indicate a built-in electric field due to 
+the band offset between top and bottom interfaces in the trilayer. The 
+real-space distributions of the charge density at the conduction band 
+minimum are shown by yellow colored isosurfaces of 10 % of their 
+maxima. (c) Anomalous Hall conductivity as a function of electron 
+energy for Pu and Pd. (d) Surface band structure of the three-monolayer 
+MnBi2Te4 film for Pu indicating the appearance of the topologically 
+protected edge state at the Fermi energy. 
+
+a
+b
+Pd
+0.03
+0.03
+eV)
+0.00
+e
+0.00
+ites/
+tes/
+0.03
+0.03
+(Stat
+(Stat
+Eb
+0.000
+0.00
+DOSO
+0.03
+0.03
+0.00
+0.00
+0.30.00.3
+-0.30.00.3
+E-Er(eV)
+E-E (eV)
+c
+d
+0.15
+6
+0.1
+5
+1.0
+4
+(eV)
+0.05
+0.0
+3
+山
+2
+-1.0
+Pu
+山-0.05
+1
+Pd
+0
+-2.0
+-0.1
+2.
+-0.08
+3-0.040.00
+0.04
+0.08
+-0.15
+2
+E-Er(eV)
+-M
+M5 
+ 
+4(a,b)). The presence of this bias field independent of 
+polarization orientation makes the transport behavior of the 
+polar MnBi2Te4 trilayer different from that of a polar bilayer.  
+When polarization is pointing down (Pd), the associated 
+depolarizing field is directed opposite to the bias field and 
+effectively compensates it, thus maintaining the band gap across 
+the whole polar trilayer (≈ 0.07 eV), as seen from the layer-
+resolved density of states (DOS) in Fig. 4(a). Also, as evident 
+from the real space distribution of the charge density at the CBM 
+in this figure, the CBM lies away from the polar interface. It is 
+expected in this case that the transport behavior of the polar 
+trilayer in the Pd state to be reminiscent to that of the bulk-like 
+non-polar trilayer. The latter is known to represent a Chern 
+insulator exhibiting an QAHE5. Likewise, we find that the polar 
+MnBi2Te4 trilayer in the Pd state signifies a Chern insulator with 
+the Chern number equal to 1 and exhibits an QAHE with AHC  
+𝜎𝑥𝑦 = 𝑒2/ℎ  in the energy gap region (yellow curve in Fig. 4(c)). 
+The quantized AHC is associated with the non-trivial 
+topological character of the system which is reflected in the 
+topologically protected edge state shown in Fig. 4 (d).  
+On the contrary, when polarization is pointing up (Pu), the 
+associated depolarizing field is aligned in the direction of the 
+bias field which effectively doubles its effect producing strong 
+band bending across the trilayer, as seen from the layer-resolved 
+DOS in Fig. 4(b). As a result, the band gap is closed at the polar 
+interface of the trilayer, producing a non-zero charge density at 
+the Fermi energy. The metallic character of the polar MnBi2Te4 
+trilayer in the Pd state makes the Chen number equal to 0, 
+eliminates a topologically protected surface state, and changes 
+the AHC to a smaller value at the Fermi energy (red curve in Fig. 
+4(c)). Thus, the switchable electric polarization of the polar 
+MnBi2Te4 trilayer can serve as a knob to control the transition 
+between the non-trivial (associated with QAHE) and trivial 
+(associated with AHE) electronic states in the MnBi2Te4 films. 
+A different type of stacking in a MnBi2Te4 trilayer occurs 
+when a polar MnBi2Te4 bilayer is placed on a monolayer in such 
+a way that the bottom interface is also polar and has polarization 
+pointing down (Fig. 2(c)).  In this case the trilayer has two polar 
+interfaces and the polarization of the top interface can be 
+switched up (Pu) to be antiparallel (AP) or down (Pd) to be 
+parallel (P) to the polarization P of the bottom interface (Figs. 
+5(a) and 5(d), respectively). Depending on the relative 
+polarization orientation (AP or P), we find different topology of 
+the trilayer. While in both cases, we observe the presence of a 
+band gap in the band structure of AP- (Fig. 5(b)) and P- (Fig. 
+5(e)) aligned trilayers, the nature of this gap is different. By 
+comparing the band structures in Fig. 5(b) and Fig. 5(e), we see 
+for the former, that the bands exhibit conventional dispersions 
+typical for a normal semiconductor with the CBM and VBM 
+located at the Γ point. On the contrary, for the latter, the valence 
+bands reveal a “mexican-hat” shape placing the VBM away 
+from the Γ point and indicating a possible band inversion and 
+thus a non-trivial character of the band gap.  
+To elucidate these different behaviors, we explore the 
+evolution of the band gap for the AP- and P- aligned MnBi2Te4 
+trilayers as a function of SOC. Fig. 5(c) shows variation of the 
+band gap for AP- and P-aligned aligned MnBi2Te4 trilayers as a 
+function of SOC strength given in units of a fraction of the actual 
+SOC (see Figs. S5 and S6 for the associated bands structure 
+around the Γ point). It is seen that, with the increasing fraction 
+of SOC, the band gap of the AP-aligned trilayer continuously 
+reduces (yellow stars in Fig. 5(c) and Fig. S5), until it reaches 
+the smallest value of 0.06 eV at the actual value of SOC. On the 
+contrary, the band gap of the P-aligned trilayer first decreases 
+with increasing SOC, so that the band gap reduces to zero at 
+SOC = 0.93 (red circles in Fig. 5(c) and Fig. S6). A further 
+increase of SOC reopens and then enlarges the band gap. This 
+behavior indicates band inversion in the P-aligned MnBi2Te4 
+trilayer that is induced by SOC, reflecting the topologically non-
+trivial origin of the band gap. This contrasts to the AP-aligned 
+MnBi2Te4 trilayer where there is no band inversion with 
+increasing SOC, thus signaling the trivial character of the 
+system. These conclusions are confirmed by the direct 
+calculation of the Chern number. While the Chern number is 0 
+for the AP-aligned trilayer, it is 1 for the P-aligned trilayer. 
+The non-trivial character of the AP-aligned MnBi2Te4 
+trilayer is reflected in the quantized AHC exhibiting a plateau of 
+𝜎𝑥𝑦 = 𝑒2/ℎ  within the energy gap (red curve in Fig. 5(f)), 
+indicating that the trilayer represents a Chern insulator 
+exhibiting an QAHE. On the contrary,  𝜎𝑥𝑦 = 0   in the band gap 
+region of the P-aligned MnBi2Te4 trilayer (yellow curve in Fig. 
+5(f)), indicating that the trilayer represents a trivial insulator. In 
+both cases, for electron energies above and below the band gap 
+ 
+Fig. 5: (a,b,d,e) Atomic structure (a,d) and energy bands near the Γ 
+point (b,e) of a polar MnBi2Te4 trilayer with antiparallel (AP) (a,b) and 
+parallel (P) (d,e) interface polarization. (c) Variation of the band gap 
+for parallel (P) and antiparallel (AP) aligned MnBi2Te4 trilayer as a 
+function of SOC strength given in units of a fraction of the actual SOC. 
+(f) Anomalous Hall conductivity of the P and AP-aligned MnBi2Te4 
+trilayer as a function of electron energy. 
+P
+AP
+c
+f
+P
+AP
+M    ←     Γ
+→     K
+M    ←     Γ
+→     K
+b
+a
+e
+d
+
+AP-MBT
+P-MBTP-MBT
+AP-MBT0.2
+0.1
+(eV)
+-
+0.0
+0.1
+0.2·
+(d)al
+Bi(p)
+Mn(d)6 
+ 
+the AHC is non-zero, reflecting the broken  𝑃̂𝑇̂ symmetry of the 
+system, and changes with reversal of ferroelectric polarization 
+of the top interface. Therefore, a polar MnBi2Te4 trilayer can be 
+switched between the trivial insulator and Chern insulator states 
+by reversing electric polarization of the top interface resulting 
+in the antiparallel or parallel polarization alignment.   
+Conclusions 
+Overall, our results demonstrate that utilizing polar stacking of 
+a 2D vdW antiferromagnet MnBi2Te4 with switchable interface 
+polarization allows achieving new functional properties that are 
+not available in the bulk form of this material. Specifically, due 
+to broken 𝑃̂𝑇̂  symmetry, a polar-stacked MnBi2Te4 bilayer 
+exhibits an AHE, whose sign is determined by ferroelectric 
+polarization orientation, as well as the magnetoelectric effect 
+with the net magnetic moment of the bilayer switchable by 
+polarization. The reversable polarization in a single-polar-
+interface MnBi2Te4 trilayer controls the transition between 
+metallic and insulating phases associated with the AHE and 
+QAHE states, respectively. Switching between antiparallel and 
+parallel interlayer polarization states in a bi-polar MnBi2Te4 
+trilayer enforces the transition between a trivial insulator to a 
+Chern insulator carrying a topologically protected edge state and 
+exhibiting an QAHE.   
+ 
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+Acknowledgments. The work was supported by the grant DE-
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+of Science. Computations were performed at the University of 
+Nebraska Holland Computing Center. 
+ 
+† Current affiliation: Key Laboratory of Materials Physics, Institute of 
+Solid State Physics, HFIPS, Chinese Academy of Sciences, Hefei 
+230031, China 
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+‡ Current affiliation: Trinity College, University of Oxford, Oxford 
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+

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+Fermion pair radiation from accelerating classical systems
+Margarita Gavrilova,1, ∗ Mitrajyoti Ghosh,1, † Yuval Grossman,1, ‡
+Walter Tangarife,2, § and Tien-Hsueh Tsai3, ¶
+1Department of Physics, LEPP, Cornell University, Ithaca, NY 14853, USA
+2Department of Physics, Loyola University Chicago, Chicago, IL 60660, USA
+3Department of Physics, National Tsing Hua University, Hsinchu 300, Taiwan
+Abstract
+Accelerating classical systems that couple to a fermion-antifermion pair at the microscopic level can
+radiate pairs of fermions and lose energy in the process. In this work, we derive the generalization of the
+Larmor formula for fermion pair radiation. We focus on the case of a point-like classical source in an
+elliptical orbit that emits fermions through vector and scalar mediators. Ultra-light fermion emission
+from such systems becomes relevant when the mass of the mediator is larger than the frequency of the
+periodic motion. This enables us to probe regions of the parameter space that are inaccessible in on-
+shell bosonic radiation. We apply our results to pulsar binaries with mediators that couple to muons
+and neutrinos. Using current data on binary period decays, we extract bounds on the parameters of
+such models.
+∗ [email protected]
+† [email protected]
+‡ [email protected]
+§ [email protected]
+¶ [email protected]
+1
+arXiv:2301.01303v1  [hep-ph]  3 Jan 2023
+
+CONTENTS
+I. Introduction
+2
+II. Fermion pair radiation by a point-like object
+4
+A. General formalism
+4
+B. Power loss formulae
+7
+III. Discussion of the power-loss formula
+9
+A. General features of the power-loss formula
+9
+B. Asymptotic behavior for the case of circular orbits
+12
+C. Fermion-pair radiation in the SM
+14
+IV. Fermion pair radiation by pulsar binaries
+15
+A. Pulsar binaries as a classical source
+16
+B. Neutrino pair radiation by pulsar binaries in the SM
+18
+C. New physics constraints from the neutrino pair radiation by pulsar binaries
+19
+V. Conclusion
+24
+Acknowledgements
+25
+A. Derivation of the power loss formula
+25
+1. The case of a vector boson mediator
+25
+2. The case of the scalar mediator
+29
+References
+31
+I.
+INTRODUCTION
+Radiation by a classical system is a well-known phenomenon. Probably the most familiar
+example is the radiation of electromagnetic waves by an accelerating point-like particle. The
+power loss, in this case, is calculated using the famous Larmor formula [1, 2], which, in natural
+units, is given by
+Ploss = 1
+6πq2a2,
+(1.1)
+where q is the electric charge of the particle and a is its acceleration. The Larmor formula in
+Eq. (1.1) has been also generalized to other types of radiation by accelerating classical sources,
+such as radiation of massive vector and scalar bosons [3–9].
+2
+
+Generalizations of the Larmor formula to exotic types of radiation are motivated, among
+other things, by their applications to new physics searches. The basic idea is that if a new
+physics radiation accompanies an accelerating astrophysical object, the power loss effect can
+be enhanced thanks to the large number density of an object, even if the coupling between the
+new physics and the Standard Model (SM) is very small. This expected enhancement can be
+used to obtain constraints on various new physics scenarios using astrophysical observations.
+One example is the radiation of an ultra-light gauged Lµ − Lτ vector boson [10–13] by pulsar
+binaries. The measurement of the orbital period decay, when compared to the prediction due
+to the gravitational wave (GW) radiation, was used to constrain the mass of the Lµ −Lτ gauge
+boson and its couplings to the SM [5, 6, 14–18].
+In this paper, we extend the previous work and derive the generalization of the Larmor
+formula to the case of fermion-antifermion pair radiation by classical systems. The interest in
+this scenario is twofold. First, it is interesting theoretically since it is one more example of a
+case where a fermion pair behaves like a boson (other cases are Cooper pairs in superconductors
+and the mediation of forces between objects via 2-fermion forces [19–22]). Thus we can study
+the coherent radiation of fermions. The key point is that single-fermion emission changes the
+source and thus can not be treated classically. Fermion-pair emission, however, can take place
+without changing any quantum degrees of freedom of the emitting system (such as spin). Thus,
+fermion-pair emission (or emission of any even number of fermions) can be treated classically.
+The second aspect is phenomenological. In particular, we consider radiation by astrophysical
+systems. In the SM, as we show below, the effect of the fermion pair radiation is negligible.
+In beyond the SM (BSM) theories, however, such processes can be enhanced, enabling us to
+probe various new physics scenarios using astrophysical observations. In particular, fermion-
+pair radiation can become significant in models with a new light mediator (a vector or scalar
+boson) that couples to some light fermionic degrees of freedom. These fermionic degrees of
+freedom can be the well-known neutrinos or some new BSM fermions.
+The effects of this
+radiation can become relevant when the mediator is too heavy to be produced on-shell, but
+the fermions are much lighter and can be radiated out. Since fermion pairs can be produced
+via off-shell mediators, the fermion pair radiation can be used to probe broader regions of the
+parameter space of such models.
+As a particular application of our result for the fermion-pair radiation, we consider two
+models: (i) a model with a gauged Lµ − Lτ symmetry and (ii) a model with a muonophilic
+scalar that couples to the muon and the muon neutrino. We study the implications of these
+scenarios for the power loss by pulsar binaries and compare our results to the cases of on-shell
+vector boson radiation [3, 5, 6] and on-shell scalar radiation [3]. A stark difference is that
+3
+
+the emission of neutrino pairs in a particular harmonic mode of the periodic system is not
+kinematically forbidden when the mediator mass becomes larger than the frequency of that
+particular mode. In the case of on-shell bosonic radiation, radiation from a harmonic mode
+is cut off once the boson mass exceeds the frequency of that particular mode due to energy
+conservation. We use the available period decay data for pulsar binaries to demonstrate how
+neutrino pair radiation, mediated by BSM bosons, can be used to probe a broader parameter
+space than the on-shell boson emission. We, however, do not perform a comprehensive study
+of other bounds on the models we consider.
+This paper is organized as follows: In Sec. II, we discuss the general machinery required for
+calculating fermion-pair radiation from a classical system. In Sec. III, we discuss the main fea-
+tures of the power-loss formula. In Sec. IV, we perform the computation for the particular case
+where the classical system is a binary system. We then use available data to place constraints
+on the parameters of a few models. We conclude in Sec. V. The detailed calculations are shown
+in the appendix.
+II.
+FERMION PAIR RADIATION BY A POINT-LIKE OBJECT
+In this section, we outline the calculation of the power of fermion-pair radiation that accom-
+panies a non-relativistic point-like object. We formulate a general approach to the derivation of
+the power loss formula with a focus on the case of elliptical orbits. The fermion pair radiation
+is realized in our analysis via the coupling of the classical object to a massive boson: a vector,
+or a scalar, which is unstable and decays into a fermion pair. We consider the emission of Dirac
+fermions and generalize our result to the case of Weyl fermions when we discuss the application
+of our result to the SM in Section III C. While a point-like object is a purely theoretical entity,
+it is worthwhile to perform this calculation since the approximation of a radiating extended
+object as a point is valid in the limit of long-wavelength radiation.
+A.
+General formalism
+We describe a point-like object as a classical source using classical current, Jµ
+cl(x) and classical
+density, ρcl(x), which are given by
+Jµ
+cl(x) = Qδ3(x − x(t))uµ,
+(2.1)
+ρcl(x) = Nδ3(x − x(t)).
+(2.2)
+Here, Q is the total charge of the object under the symmetry of interest, N is the number of
+the relevant microscopic constituents, x(t) is its position as a function of time, t, and uµ is its
+4
+
+four-velocity.
+Assuming motion in the x − y plane, in the non-relativistic limit, the four-velocity of the
+object is given by
+uµ = (1, ˙x, ˙y, 0) .
+(2.3)
+We focus on the case of the elliptical motion in the x − y plane, which can be parametrically
+described by
+x = a(cos ξ − e),
+y = a
+√
+1 − e2 sin ξ,
+Ω t = ξ − e sin ξ,
+(2.4)
+where e is the eccentricity, a is the semi-major axis of the ellipse, and Ω is the fundamental
+frequency of revolution. One full revolution around the ellipse corresponds to changing the
+parameter ξ from 0 to 2π.
+The power loss due to the fermion-pair radiation is calculated using
+Ploss =
+�
+(ω1 + ω2) dΓ,
+(2.5)
+where ω1 and ω2 are the energies of the emitted fermion and anti-fermion, respectively, and dΓ
+is the differential rate of the fermion-pair emission. The rate depends on the type of mediator,
+i.e., a scalar or a vector, and the specific form of the classical current or density.
+In general, the acceleration is not constant. In the case of periodic orbits, the motion can
+be decomposed into harmonic modes with frequencies Ωn = nΩ, where Ω is the fundamental
+frequency of revolution. The total emission rate can then be written as a sum of emission rates
+at different harmonics n,
+dΓ =
+�
+n
+dΓn .
+(2.6)
+The sum goes over all kinematically allowed harmonics n > 2mψ/Ω, where mψ is the mass of
+the emitted fermions. The emission rate at harmonic n is found using
+dΓn =
+�
+s1,s2
+|Mn(s1, s2)|2(2π)δ(Ωn − ω1 − ω2) d3k1
+(2π)3ω1
+d3k2
+(2π)3ω2
+.
+(2.7)
+Here, k1 = (ω1, k1) and k2 = (ω2, k2) are the four-momenta of the fermion and anti-fermion
+respectively, and s1(s2) is the spin of the fermion (anti-fermion). The microscopic physics enters
+via Mn (s1, s2), which is the matrix element of the fermion-pair emission at harmonic n. At
+leading order, this matrix element is obtained from the diagram in Fig. 1. In the diagram, ⊗
+denotes the classical source, which is given by the classical current, Jµ
+cl(x), in the case of vector
+mediator and by the density, ρcl(x), in the case of the scalar mediated radiation. The total
+power loss via fermion-pair radiation is simply a sum of power losses over all harmonics
+Ploss =
+�
+n
+Pn,
+Pn =
+�
+(ω1 + ω2) dΓn.
+(2.8)
+5
+
+ψ, k1
+ψ, k2
+mediator
+FIG. 1. Feynman diagram for a fermion pair emission by a classical current.
+Here, Pn is the power loss of the nth harmonic.
+In what follows, we consider two types of mediators: a massive gauge boson and a massive
+scalar. We only consider s-channel exchange and remark on t-channel exchange at the end of
+this subsection.
+First, we consider a vector mediator, Aµ, that corresponds to a broken U(1)′ and has mass
+mA. This gauge boson couples to a classical current Jµ
+cl(x), which has charge Q under U(1)′.
+The gauge boson Aµ is unstable and decays into a fermion pair. The terms in the effective
+Lagrangian, relevant for the fermion-pair radiation via Aµ, are
+Leff ⊃ gAµJµ
+cl + gqψ ¯ψγµAµψ ,
+(2.9)
+where qψ is the U(1)′ charge of the fermion ψ, g is a dimensionless coupling constant, and Jµ
+cl(x)
+is the classical current defined in Eq. (2.1). Both the vector boson and the fermions are assumed
+to be massive with masses mA and mψ, respectively. The leading order matrix element for the
+emission, at the n−th harmonic, is given by
+Mn(s1, s2) = g2qψ ¯u(k1, s1)γµv(k2, s2) i(−ηµν + (k1 + k2)µ(k1 + k2)ν/m2
+A)
+(k1 + k2)2 − m2
+A + imAΓA
+Jν
+cl(Ωn) ,
+(2.10)
+where Jν
+cl(Ωn) is the Fourier transform of Jν
+cl(x), given by
+Jν
+cl(Ωn) = Ω
+2π
+� 2π/Ω
+0
+dt
+�
+d3x ei(nΩt−p·x)Jν
+cl(x)
+(2.11)
+with p = k1 + k2, ΓA is the decay width of the gauge boson, and 2π/Ω is the period. We
+assume that the decay into a ¯ψψ pair is the only decay channel for the gauge boson Aµ, and
+that the fermion mass mψ is negligible compared to the gauge boson mass mA. Under these
+assumptions, the decay width of Aµ is given by
+ΓA = g2q2
+ψmA
+12π
+.
+(2.12)
+The other case we consider is that of a scalar mediator, φ, for which the relevant terms in
+the Lagrangian are
+L ⊃ gφρcl + g′φ ¯ψψ,
+(2.13)
+6
+
+where g is the dimensionless coupling between the scalar φ and the classical source, g′ is the
+Yukawa coupling of the fermion ψ to the scalar φ, and ρcl(x) is the number density of relevant
+particles in the classical source. Both the scalar and the fermions are assumed to be massive
+with masses mφ and mψ, respectively. The matrix element in this case is given by
+Mn(s1, s2) = gg′¯u(k1, s1)v(k2, s2)
+iρcl(Ωn)
+(k1 + k2)2 − m2
+φ + imφΓφ
+,
+(2.14)
+where ρcl(Ωn) is the Fourier transform of ρcl(x),
+ρcl(Ωn) = Ω
+2π
+� 2π/Ω
+0
+dt
+�
+d3x ei(nΩt−p·x)ρcl(x),
+(2.15)
+and the decay width of the scalar is Γφ. As in the case of the vector mediator, we assume
+that the fermionic decay mode is the only available mode, and the fermion mass mψ can be
+neglected compared to the mass of a scalar mφ. Thus we have
+Γφ = g′2mφ
+8π
+.
+(2.16)
+So far, we have only considered the s-channel contribution to the fermion pair radiation.
+Fermion pair radiation via t−channel process mediated by a vector or scalar is also a possibility.
+Such contributions, however, are highly suppressed for mS ≫ Ω, mM, where mS is the mass
+of the particles in the source that couple to the fermion pairs ¯ψψ at the microscopic level,
+and mM is the mediator mass. Since the emitted fermions have energy of the order of Ω, the
+fundamental frequency of the system, the t-channel contribution to the momentum entering
+the propagator is of the order of mS − Ω. Thus the t-channel propagator is schematically given
+by
+Π ∼
+1
+(mS − Ω)2 − m2
+M
+.
+(2.17)
+In the case where mS is much larger than both Ω and mM, the propagator is dominated by the
+mass of the source particles, and the process is heavily suppressed. In this paper, we assume
+that the mass hierarchy mS ≫ Ω, mM and neglect the t−channel contributions to the fermion
+pair radiation everywhere.
+B.
+Power loss formulae
+Using Eqs. (2.7)–(2.14), we can calculate the power loss via fermion-pair radiation from
+a point-like object moving in an elliptical orbit. The detailed derivations are shown in Ap-
+pendix A, and here we only quote the final result. The power loss due to fermion-pair emission
+7
+
+in harmonic n > 2mψ/Ω, for the cases of the vector and scalar mediator, can be written as
+P A
+n = g4q2
+ψQ2
+12π3
+a2Ω4 BA
+n (nA, nψ, nΓ),
+(2.18)
+P φ
+n = g2g′2N 2
+12π3
+a2Ω4 Bφ
+n(nφ, nψ, nΓ).
+(2.19)
+The functions BM
+n (nA, nψ, nΓ), where M = A, φ, are given by
+BM
+n (nM, nψ, nΓ) ≡
+�
+J′
+n(ne)2 + 1 − e2
+e2
+Jn(ne)2
+� � n−nψ
+nψ
+dx F M(x, n, nM, nψ, nΓ).
+(2.20)
+Here
+nM ≡ mM/Ω,
+nψ ≡ mψ/Ω,
+nΓ ≡ ΓM/Ω,
+(2.21)
+and Jn(ne) is a Bessel function of order n with argument ne.
+The integration variable in
+Eq. (2.20) is defined by x ≡ ω1/Ω, where ω1 is the energy of one of the final-state fermion. In
+what follows, for brevity, we use the notation
+F M(x) ≡ F M(x, n, nM, nψ, nΓ),
+BM
+n ≡ BM
+n (nM, nψ, nΓ).
+(2.22)
+The functions F M(x) have the general form
+F M(x) = F M
+0 (x) + F M
+1 (x)
+nMnΓ
+�
+tan−1
+�a(x) + b(x)
+nMnΓ
+�
+− tan−1
+�a(x) − b(x)
+nMnΓ
+��
++ F M
+2 (x) tanh−1
+�
+2a(x)b(x)
+a(x)2 + b(x)2 + n2
+Mn2
+Γ
+�
+,
+(2.23)
+with a(x) and b(x) being universal for both gauge boson and scalar mediators,
+a(x) = 2n2
+ψ − n2
+M + 2nx − 2x2 ,
+b(x) = 2
+�
+x2 − n2
+ψ
+�
+(n − x)2 − n2
+ψ .
+(2.24)
+The functions F M
+0 (x), F M
+1 (x), and F M
+2 (x) are different for the two cases. For a gauge boson
+mediator, we obtain
+F A
+0 (x) = b(x)/2n ,
+F A
+1 (x) = 1
+4n
+�
+n4
+A + 4n2n2
+ψ − n2
+An2
+Γ + 2n2
+An2 − 4nxn2
+A + 4x2n2
+A
+�
+,
+F A
+2 (x) = 1
+2n
+�
+n2
+A + n2 − 2nx + 2x2�
+,
+(2.25)
+while for a scalar mediator,
+F φ
+0 (x) = −b(x)/2n ,
+F φ
+1 (x) = 1
+4n
+�
+n2
+φn2
+Γ + (n2 − n2
+φ)(n2
+φ − 4n2
+ψ)
+�
+,
+F φ
+2 (x) = 1
+4n
+�
+n2 + 4n2
+ψ − 2n2
+φ
+�
+.
+(2.26)
+8
+
+Eqs. (2.18)–(2.26) are the main results of our work. Analytical integration of F A(x) and
+F φ(x) is challenging, but it still can be performed in certain limits. In Sec. III B, we consider
+two limiting cases: the case of nM ≪ 1, which reproduces the Larmor formula, and nM ≫ 1,
+which is relevant for the fermion pair radiation in the SM. In general, however, calculating the
+power loss requires numerical analysis. We perform such an analysis in Sec. IV when we discuss
+a particular phenomenological application of our result.
+III.
+DISCUSSION OF THE POWER-LOSS FORMULA
+The power loss due to fermion-pair emission by a classical source on an elliptical orbit is
+given by Eqs. (2.18)-(2.26). Below we discuss the main features and the asymptotic behavior
+of this result.
+A.
+General features of the power-loss formula
+We start with the general features that hold for both the vector and scalar cases.
+• The radiation rate is proportional to the charge-squared; that is, the functions P A
+n and P φ
+n
+depend on Q2 and N 2, respectively. This is a manifestation of the fact that the fermion-pair
+radiation that we are considering is coherent.
+• The form of F M(x), with M = A, φ, in Eq. (2.23) is somewhat general. We show in
+Appendix A that the overall form of F M(x), at the tree level, is the same for any renormalizable
+theory that couples fermions to a classical source moving in an elliptical orbit. Note that the
+functions a(x) and b(x) defined in Eq. (2.24) are purely kinematic and thus have the same form
+for any theory of fermion pair emission, while the form of F M
+0 (x), F M
+1 (x), and F M
+2 (x) vary with
+the theory considered. For instance, considering non-renormalizable interactions would lead to
+a different momentum dependence of the matrix element that could, in principle, change the
+form of F M(x).
+• The power loss for both vector and scalar mediators behaves qualitatively the same way
+despite the different functional forms of F A
+i (x) vs. F φ
+i (x), with i = 0, 1, 2. This is not surprising
+since there is nothing fundamentally different between the matrix elements for the vector and
+scalar cases.
+• Energy conservation implies that the functions F M(x) are invariant under x → (n − x)
+exchange. The reason is that the total energy radiated in fermion pairs in the n-th harmonic
+is nΩ. The transformation x → (n − x) exchanges the energies of the emitted fermion and
+anti-fermion, and the emission rate is the same regardless of the order in which the integrals
+9
+
+are carried out. This invariance results from the fact that the fermion-antifermion emission
+from a classical system is essentially a 2-body decay. Note that this has nothing to do with the
+details of the considered model.
+• For nA < n, the power loss has a very weak dependence on nA. This is true for the
+particular models that we chose here but is not expected to be true in general. For an example
+when this is not the case, see the discussion of Proca fields in Ref. [6], where dependence on nA
+appears due to the absence of gauge symmetry.
+• There is an interplay of three energy scales: The mass of the mediator, mM, the mass of
+the fermion, mψ, and the frequency of the harmonics, nΩ. The fermions cannot be produced
+when 2mψ > nΩ. In the opposite limit, when 2mψ < nΩ, the production rate depends strongly
+on the mediator mass. For mM < 2mψ < nΩ, fermion production is strongly suppressed since
+the on-shell boson is kinematically forbidden from decaying into fermions. (Note that strictly
+speaking, our result cannot be straightforwardly applied in this case as everywhere we assume
+ΓM > 0.) For 2mψ < mM < nΩ, the fermions are produced via decay of the on-shell mediator.
+Thus the power loss in the fermion-pair radiation is equal to that of the on-shell boson radiation.
+The region of the parameter space where mM > nΩ > 2mψ is of the most interest to us, as in
+this region the fermions are kinematically allowed, the mediator is off-shell, and therefore the
+fermion pair emission is most significant.
+• As an example that illustrates the qualitative features of the power loss, consider Fig. 2.
+It shows BA
+n , defined in Eq. (2.20), as a function of nA for massless fermions for the first four
+harmonics. The most striking feature of the plots is a sharp drop at nA ∼ n. This behavior
+follows from the fact that at nA ∼ n, the radiation regime switches from the radiation dominated
+by on-shell boson production (nA < n), which is proportional to g2 to the off-shell production
+(nA > n) proportional to g4. The power loss in the regime dominated by fermion-pair radiation
+is thus suppressed by g2 compared to the power loss in the regime dominated by the on-shell
+boson radiation. The power loss in the case of the scalar mediator exhibits the same behavior.
+• Comparing our results to the cases of vector [3, 5, 6] and scalar radiation [3], we note that
+from kinematic considerations alone, boson radiation drops to zero as soon as nM = n. This
+is not what we observe for the fermion-pair emission. In the case of fermion-pair radiation,
+off-shell boson production is possible, even though there is an extra suppression by g2 for
+a vector and g′2 for a scalar compared to on-shell boson radiation. As a result, the regime
+nM > n opens up new regions of the parameter space for each harmonic n and is of particular
+phenomenological interest to us.
+• Next, we remark on the dependence of the power loss on the eccentricity in the case of
+orbits close to circular.
+For that, we note that the eccentricity only enters the power loss
+10
+
+�����
+�����
+��
+����
+��-��
+��-��
+��-��
+��
+FIG. 2. BA
+n vs nA for fixed eccentricity, e = 10−3, coupling constant g = 10−15, and massless final
+state fermions, mψ = 0. See Eqs. (2.20)-(2.25) for the definition of BA
+n .
+through the Bessel function prefactor of BM
+n in Eq. (2.20), which we denote as K(n, e),
+K(n, e) = J′
+n(ne)2 + 1 − e2
+e2
+Jn(ne)2 .
+(3.1)
+We recall that Jn(z) and J′
+n(z) behave asymptotically, in the limit z ≪ 1, as
+Jn(z) ≈ 1
+n!
+�z
+2
+�n
+,
+J′
+n(z) ≈ n
+n!
+1
+2
+�z
+2
+�n−1
+≈ n
+z Jn(z),
+z ≪ 1.
+(3.2)
+Using Eq. (3.2), we find for the eccentricity dependent prefactor K(n, e), in the limit ne ≪ 1,
+that
+K(n, e) = J′
+n(ne)2 + n2 − (ne)2
+(ne)2
+Jn(z)2 ≈ J′
+n(ne)2 +
+n2
+(ne)2Jn(z)2
+= 2n2
+z2 Jn(ne)2 = (ne)2n−2
+22n−1
+n2
+(n!)2 =
+(ne)2n−2
+22n−1 ((n − 1)!)2.
+(3.3)
+Thus we learn that in the limit ne ≪ 1, prefactor K(n, e) scales with the eccentricity as
+K(n, e) ∝ (ne)2n−2 .
+(3.4)
+This shows that for small eccentricities (and thus orbits close to circular ones), the contributions
+from higher harmonics die away very fast as n increases. For n = 1 and e ≪ 1, we have K(1, e) ≈
+1/2. For each subsequent harmonic power drops by a factor of order e2, until the factorial in
+the denominator of K(n, e) (see Eq. (3.3)) starts to dominate. Then the contributions from the
+higher harmonics start to decay away even faster. Fig. 2 illustrates the behavior of the power
+loss for the first four harmonics in the case of small eccentricity e = 10−3.
+• The case of highly eccentric orbits e ∼ 1 is significantly more involved. First, the contri-
+butions from different modes do not follow the simple hierarchy of the low eccentricity case.
+11
+
+�
+�
+��
+��
+��-��
+��-��
+��-�
+��-�
+���
+�
+�
+�
+��
+��
+��-��
+��-��
+��-��
+��-��
+��-��
+FIG. 3. Left: BA
+n as a function of n in the regime where the radiation is dominated by on-shell boson
+production. Different colors correspond to different values of eccentricity. The values of nψ, nA and g
+are fixed. Right: BA
+n as a function of n for a highly eccentric orbit with e = 0.6 in the regime where
+the radiation is dominated by off-shell boson production.
+The contributions from higher modes can be of the same order or even larger than the first
+mode depending on the values of other parameters. See the left panel of Fig. 3 to compare
+the n-dependence of BA
+n for different eccentricity values. Second, as Fig. 3 demonstrates, the
+hierarchy of modes in the on-shell dominated part of the parameter space does not carry into
+the off-shell dominated region. Consider the green line corresponding to a highly eccentric orbit
+with e = 0.6. For nA = 10−1 (left panel), the maximum contribution to the power loss comes
+from the mode with n = 2 and the first 5 modes contribute at about the same order. The
+situation is drastically different for nA = 50 (right panel). The maximum contribution to the
+power loss comes from the n = 8 mode. We learn that for e ∼ 1, generally speaking, the power
+loss per mode first increases as we increase n and then starts decreasing after reaching a certain
+value of n. Where this maximum occurs depends on other parameters.
+B.
+Asymptotic behavior for the case of circular orbits
+We now move to the discussion of the asymptotic behaviour of the power loss in two limiting
+cases mM ≪ Ω and mM ≫ Ω, where mM is the mass of the mediator, M = A, φ. In this
+subsection, for simplicity we consider the straightforward case of circular orbits (e = 0) and
+massless fermions (mψ = 0). For the eccentricity dependent part of the power loss, K(n, e), we
+have
+lim
+e → 0 K(n, e) = lim
+e → 0
+�
+J′
+n(ne)2 + 1 − e2
+e2
+Jn(ne)2
+�
+= 1
+2δn,1.
+(3.5)
+Thus the only mode that contributes to the power loss in the circular orbit limit is the mode
+with n = 1.
+12
+
+First, let us consider the regime of light mediators, mM ≪ Ω, or equivalently nM ≪ 1. In
+this limit, F M(x) defined in Eq. (2.23) is dominated by the second term. We thus neglect the
+first and the third terms of F M(x) and take the second term’s limit nM → 0. After that, the
+integral in (2.20) can be performed analytically, yielding the following asymptotic expressions
+for the power radiated via vector and scalar, respectively:
+P A(mA ≪ Ω) ≈ g2
+6πQ2a2Ω4,
+(3.6)
+P φ(mφ ≪ Ω) ≈ g2
+12πN 2a2Ω4.
+(3.7)
+The asymptotic behavior that we find for P A and P φ reproduces the known results for the
+on-shell vector [3, 5, 6] and scalar [3] radiation. This is expected as, in the regime mM ≪ Ω,
+the fermion pair radiation is dominated by on-shell boson production. Additionally, Eq. (3.6)
+also reproduces the Larmor formula for the power of the electromagnetic wave radiation given
+in Eq. (1.1). To see this, recall that the acceleration on a circular orbit is equal to aΩ2, where
+a is the radius of the orbit and Ω is the frequency of revolution.
+Next, we study the regime when on-shell boson production is kinematically forbidden, and
+the fermion pair radiation takes place through the off-shell mediator. This is the limit of heavy
+mediators, mM ≫ Ω, or equivalently nM ≫ 1. As in the case of the light mediators, we take the
+nM → ∞ limit of F M(x) and find that the resulting expression can be integrated analytically.
+Upon performing the integration, we find that the vector and scalar-mediated radiation behave
+as
+P A(mA ≫ Ω) ≈ g4q2
+ψQ2
+210π3
+a2Ω8
+m4
+A
+=
+1
+35π2
+g2q2
+ψΩ4
+m4
+A
+× P A(mA ≪ Ω),
+(3.8)
+P φ(mφ ≫ Ω) ≈ g2g′2N 2
+840π3
+a2Ω8
+m4
+φ
+=
+1
+70π2
+g′2Ω4
+m4
+φ
+× P φ(mφ ≪ Ω).
+(3.9)
+We learn that in the limit of heavy mediators, the fermion pair radiation is suppressed compared
+to on-shell boson radiation by the following factors:
+1. A factor of g2q2
+ψ or g′2, which, at the amplitude level, comes from the coupling of the
+mediator to the fermion pair.
+2. A factor of Ω4/m4
+φ, which comes from the propagator of the mediator.
+3. A phase space factor of 1/35π2 or 1/70π2, which arises from the fact that there are more
+particles in the final state in the case of the off-shell pair production than in the case of
+the on-shell boson production.
+Note that Eqs. (3.8) and (3.9) can be interpreted as integrating out the heavy mediator,
+resulting in an effective 4-Fermi interaction with a coefficient proportional to g2/m2
+A or gg′/m2
+φ.
+Thus, it is also valid for t-channel and u-channel interactions.
+13
+
+Last, we compare the results of the vector to that of the scalar mediators. Consider mA = mφ,
+Q2 = N 2 and g′ = gqψ. In this case, the power radiated via the vector mediator is greater than
+the power radiated via the scalar mediator in both radiation regimes. In particular, we have
+P A(mA ≪ Ω)
+P φ(mφ ≪ Ω) ≈ 2,
+P A(mA ≫ Ω)
+P φ(mφ ≫ Ω) ≈ 4.
+(3.10)
+These factors are related to the different number of degrees of freedom between the vector and
+scalar cases. There are two polarization states for an on-shell massless vector, while the scalar
+has only one. For the deeply off-shell mediator, the correspondence is not so clear, but it seems
+to us that it is related to the fact that off shell gauge boson, Aµ, has four degrees of freedom
+C.
+Fermion-pair radiation in the SM
+The expression in Eq. (3.8) can be used to estimate the power loss due to fermion pair
+radiation by classical sources within the SM. In this subsection, we consider neutrino pair
+radiation mediated by Z-boson. The contribution due to W-boson mediated pair emission is
+qualitatively the same as the Z-boson contribution and is expected to be of the same order.
+The main difference between the two contributions is due to the fact that W-boson mediated
+radiation is only relevant for leptons in the source while Z-boson contribution is present for all
+types of fermions.
+Consider a source made of NΨ fermions of type Ψ with the total weak charge Q = NΨqΨ.
+To apply Eq. (3.8) to the neutrino pair radiation in the SM, we need to recall that Eq. (3.8)
+was derived under the assumption of vectorial couplings, while the SM is a chiral theory. The
+relevant parts of the SM Lagrangian are different from the Lagrangian in Eq. (2.9); in particular,
+in the SM we have
+LSM ⊃ −i
+g
+2 cos θW
+�
+¯Ψγµ(cΨ
+V − cψ
+A)Ψ + ¯νγµ(cν
+V − cν
+A)ν
+�
+Zµ.
+(3.11)
+Thus Eq. (3.8) yields the following expression for the Z-boson mediated power loss due to the
+neutrino pair radiation in the SM
+P Z(mZ ≫ Ω) ≈
+1
+210π3
+g4q2
+νq2
+ΨN 2
+Ψ
+16 cos4 θW
+a2Ω8
+m4
+Z
+,
+(3.12)
+where we perform the replacement g → g/(2 cos θW) in Eq. (3.8) and define
+q2
+ψ = q2
+ν = (cν
+V )2 + (cν
+A)2,
+qΨ = cΨ
+V ,
+mA = mZ .
+(3.13)
+Note that, for the source, only vectorial coupling cΨ
+V enters the power loss. This is because we
+consider coherent radiation.
+14
+
+The expression in Eq. (3.12) can be rewritten as
+P Z(mZ ≫ Ω) ≈ G2
+effq2
+Ψq2
+νN 2
+Ψ
+a2Ω8
+210π3 ,
+(3.14)
+where Geff =
+√
+2GF and GF is the Fermi constant. When the power loss is written in the form
+of Eq. (3.14), it becomes clear that it is the same as what one would obtain by performing the
+calculation for the effective Fermi theory with the effective Lagrangian given by
+LZ
+eff ⊃ Geff[¯Ψγµ(cΨ
+V − cΨ
+Aγ5)Ψ][¯νγµ(cν
+V − cν
+Aγ5)ν].
+(3.15)
+This, of course, is not surprising as we consider radiation at the energy Ω, which is much less
+than the electroweak scale, Ω ≪ mZ. In fact, the result in Eq. (3.14) applies to any effective
+4-Fermi interaction. While we derive our results for s-channel exchange, in the limit where the
+mediator is much heavier than the orbit frequency, we do not need to distinguish between s-
+channel and t-channel. Thus, Eqs. (3.12) and (3.14) can also be used for t-channel W-exchange
+in the SM.
+Finally, we discuss the situation when there are several different types of fermions in the
+source. In this case, we need to first add all the amplitudes that correspond to the radiation
+by different fermions Ψ (for leptons, we add both Z-boson and W-boson contributions). Then,
+we square the sum of the relevant amplitudes to obtain the total emission rate.
+We end this subsection with the following remark. The power loss due to neutrino pair
+radiation in the SM was estimated in Ref. [4] to be P Z
+SM ∼ G2
+FΩ6. Using the explicit calculation,
+however, we find that P Z
+SM ∼ G2
+Fa2Ω8. That is, there is an extra factor of a2Ω2 compared to
+the estimation of Ref. [4]. In fact, our result includes the semi-major axis a as an additional
+energy scale of the system.
+IV.
+FERMION PAIR RADIATION BY PULSAR BINARIES
+We now move to discuss the phenomenological applications of our results to astrophysical
+systems. We focus on the neutrino-pair emission from pulsar binaries [23–33]. A pulsar binary
+is a binary system of a pulsar and companion. This choice is motivated by the availability of
+extensive period decay data for such systems. In particular, we apply our results to two binaries:
+Hulse-Taylor binary PSR B1913+16 [34–36] (a system of a pulsar and a neutron star) and PSR
+J1738+0333 [29, 37] (a system of a pulsar and a white dwarf). The parameters characterizing
+the two systems are summarized in Table I.
+In what follows, we first discuss the applicability of our results of Section II B to pulsar
+binaries in general. Then we estimate the contribution to the power loss due to neutrino pair
+15
+
+Binary system
+PSR B1913+16 [36]
+PSR J1738+0333 [29]
+Eccentricity e
+0.6171340(4)
+3.4(11) × 10−7
+Pulsar mass m1 (M⊙)
+1.438(1)
+1.46(6)
+Companion mass m2 (M⊙)
+1.390(1)
+0.181(8)
+Binary period Tb (GeV−1)
+4.240 × 1028
+4.657 × 1028
+Intrinsic period decay ˙Tb
+−2.398(4) × 10−12
+−2.59(32) × 10−14
+Predicted period decay due to GW ˙TGW
+−2.40263(5) × 10−12
+−2.77(19) × 10−14
+Ratio of period decays R = ˙Tb/ ˙TGW
+0.9983(16)
+0.94(13)
+Orbital frequency Ω = 2π/Tb (GeV)
+1.482 × 10−28
+1.349 × 10−28
+Semi-major axis a (GeV−1)
+9.878 × 1024
+8.77 × 1024
+TABLE I. The relevant parameters for the PSR B1913+16 and PSR J1738+0333 binary systems.
+Figures in parenthesis are the 1σ uncertainties in the last quoted digit, where all the uncertainties
+are symmetrized. M⊙ is the mass of the sun. The relative experimental error of the binary period
+Tb is ∼ 10−12 for PSR B1913+16, and ∼ 10−11 for PSR J1738+0333.
+The double line separates
+binary parameters quoted in Ref. [29, 36] and the ones we derive. Values of the semi-major axis a are
+calculated using Eq. (4.5).
+emission in the SM and show that it is negligible compared to the gravitational wave radiation.
+We then consider neutrino pair radiation in two BSM scenarios via ultralight vector and scalar
+mediators and apply our results to the pulsar binaries with the parameters in Table I.
+A.
+Pulsar binaries as a classical source
+The results for the fermion pair radiation, summarized in Eqs. (2.18)-(2.26), were derived for
+the case of classical current describing non-relativistic point-like object following an elliptical
+orbit. To justify the application of our results to pulsar binaries, we note the following:
+1. A pulsar binary can be treated as a classical source. The typical size of a pulsar binary
+can be estimated as the size of the semi-major axis which varies between 106 and 108 km,
+that is, a ∼ 1024 − 1026 GeV−1. The wavelength of the radiation is determined by the
+fundamental frequency of the orbit, and for a typical pulsar binary with periods in the
+range of 10−1 − 103 days, the wavelength is λ ∼ 1028 − 1032 GeV−1. Thus, λ ≫ a and we
+conclude that pulsar binaries can be treated as classical radiation sources.
+2. Stars of the pulsar binary can be treated as point-like objects. Typical sizes of stars
+in a binary vary from r ∼ 10 km ∼ 1019 GeV−1, for neutron stars, and r ∼ 103 km ∼
+16
+
+1021 GeV−1, for white dwarfs. Thus r ≪ a, λ and both pulsar and its companion can be
+treated as point-like objects. Moreover, r ≪ λ implies the coherence of the radiation.
+3. The motion of the pulsar and its companion in the binary system is non-relativistic. We
+can roughly estimate the orbital velocity of the stars in a binary as v ∼ aΩ, which for
+characteristic values quoted above implies v ≲ 10−2.
+4. For a wide range of pulsar binary systems, the observed power loss is such that it has no
+significant effect on the eccentricity of the orbit. Thus we can treat the orbit as elliptical
+over the time of observation.
+For example, the Hulse-Taylor binary has e ∼ 1, with
+Tb(de/dt) ≲ 10−11, where Tb is the binary period and de/dt is the time derivative of the
+eccentricity [36].
+Now that we have established that the results of Section II B can be applied to pulsar
+binaries, we proceed in two steps. First, we modify our expressions for the classical current and
+number density in Eqs. (2.1) and (2.2) to the case of two point-like objects on an elliptical orbit.
+Second, we perform the standard reduction of the two-body problem to a one-body problem.
+We write the classical current and number density as
+Jµ
+cl(x) =
+�
+b=1,2
+Qb δ3(x − xb(t))uµ
+b ,
+(4.1)
+and
+ρcl(x) =
+�
+b=1,2
+Nb δ3(x − xb(t) ,
+(4.2)
+respectively. Here, b = 1, 2 is the index that labels the stars of the binary system, xb(t) is the
+position of the b-th star at time t, and uµ
+b is its four-velocity.
+Next, we move to the binary system’s Center-of-Mass (CoM) frame. For that, we define R,
+the coordinate of center of mass, and r, the distance between the two stars,
+R =
+m1
+m1 + m2
+x1 +
+m2
+m1 + m2
+x2,
+r = x1 − x2 ,
+(4.3)
+where m1 and m2 are the masses of the two stars.
+As we are not concerned with the translational motion of the system as a whole, which is
+described by R, we can solely focus on r. This is the standard two-body to one-body problem
+reduction for central force motion. The non-relativistic classical trajectory of the stars in the
+CoM frame can thus be described by the vector r = (x, y, 0) and is given by elliptical orbits as
+in Eq. (2.3):
+x = a(cos ξ − e),
+y = a
+√
+1 − e2 sin ξ,
+Ωt = ξ − e sin ξ,
+(4.4)
+17
+
+where e is the eccentricity, a is the semi-major axis of the elliptical orbit, and the fundamental
+frequency of revolution is given by
+Ω =
+�
+GN(m1 + m2)
+a3
+.
+(4.5)
+The results of Eqs. (2.18)-(2.26) generalize to the case of binary systems via the following
+replacements that follow from the 2-body to 1-body reduction procedure:
+Q2 → M 2
+�Q1
+m1
+− Q2
+m2
+�2
+,
+N 2 → M 2
+�N1
+m1
+− N2
+m2
+�2
+,
+(4.6)
+where
+M =
+m1m2
+m1 + m2
+(4.7)
+is the reduced mass of the binary system. As a result we obtain the following expressions for
+the power loss in n-th harmonic for a vector and scalar mediators respectively:
+P A
+n = g4q2
+ψ
+12π3M 2
+�Q1
+m1
+− Q2
+m2
+�2
+a2Ω4 BA
+n (nA, nψ, nΓ),
+(4.8)
+P φ
+n = g2g′2
+12π3 M 2
+�N1
+m1
+− N2
+m2
+�2
+a2Ω4 Bφ
+n(nφ, nψ, nΓ),
+(4.9)
+where the functions BA
+n and Bφ
+n are defined in Eqs. (2.20)-(2.26).
+B.
+Neutrino pair radiation by pulsar binaries in the SM
+In the SM, for the pulsar binary, the power loss via electroweak mediators is discussed in
+Sec. III C. Here, we simply generalize it to the case of 2-body motion using Eq. (4.6). We obtain
+the following expression for the power loss in neutrino pair radiation via Z-exchange in the SM
+PSM ≈ G2
+F
+�
+cν
+V
+2 + cν
+A
+2�
+105π3 cos2 θW
+M 2a2Ω8
+�
+1
+m1
+�
+i=n,p,e,...
+ci
+V N1iQ1i − 1
+m2
+�
+i=n,p,e,...
+ci
+V N2iQ2i
+�2
+(4.10)
+where the sum goes over all microscopic constituents of binary stars, such as neutrons (n),
+protons (p), electrons (e), etc. To perform a numerical estimate, we consider a pulsar binary
+with a neutron star companion and assume that all of the neutron star mass is in the form of
+neutrons. We consider a typical pulsar-neutron star binary with
+m1,2 ∼ M⊙ ∼ 1057GeV,
+a ∼ 1025 GeV−1,
+Ω ∼ 10−28 GeV,
+(4.11)
+and non-zero dipole moment
+M 2
+�Q1
+m1
+− Q2
+m2
+�2
+∼ Q2
+1,2 ∼ 10114,
+(4.12)
+18
+
+where Qb = Nb(n)−Nb(¯n) ≈ Nb(n) ≈ M⊙/mn ≈ 1057, with b = 1, 2, are the neutron charges of
+the neutron stars, Nb(n) and Nb(¯n) are the numbers of neutrons and anti-neutrons respectively,
+mn is the neutron mass. Using cν
+V = cν
+A = 1/2, cn
+V = −1/2, and the measured values of mn,
+GF, and θW, we find the following numerical estimate for the radiated power
+PSM ∼ 10−56eV2.
+(4.13)
+To see if the above result is significant, we compare it to the power loss in the form of
+gravitational wave (GW) radiation. Using the quadrupole formula for the GW radiation [38]
+for the case of circular orbit (e = 0) we have
+PGW = 32
+5 GNM 2a4Ω6 ∼ 108 GeV2
+(4.14)
+where GN is Newton’s gravitational constant. The rough estimates in Eqs. (4.13) and (4.14)
+show that, in the SM, the fermion-pair radiation by astrophysical objects is completely negligible
+compared to the gravitational wave radiation.
+We close the subsection with one remark. Within the SM, neutron stars also emit syn-
+chrotron radiation of fermion-antifermion pairs in their self-produced magnetic fields, as shown
+in Ref. [39]. This phenomenon is different from the one we consider here. Synchrotron radiation
+is an incoherent effect. Thus, the power loss, in this case, scales as N, the number of neutrons
+in the star. In the case we are considering, the radiation is coherent and comes from the star’s
+acceleration as a whole. Then, the net power that is radiated is proportional to N 2.
+C.
+New physics constraints from the neutrino pair radiation by pulsar binaries
+Since extra radiation in the SM is negligible, any observed deviation from the gravitational
+wave radiation would be strong evidence for the physics beyond the SM. In particular, fermion-
+pair radiation can be enhanced in BSM models with light vector or scalar mediators, with
+mA,φ ≪ mZ. To explain why such light bosonic states have evaded detection so far, we must
+require that they have small couplings, thus evading all the available constraints. The smallness
+of couplings, however, still can be compensated in cases where the object has a large charge
+under the new symmetries. This can be the case for astrophysical objects. Thus, such objects
+are our prime focus in the rest of this work.
+In particular, in this subsection, we demonstrate how our results can be used to derive new
+physics bounds from the neutrino pair radiation by pulsar binaries. As we mentioned above,
+we use two distinct pulsar binary systems, the Hulse-Taylor binary PSR B1913+16 and PSR
+J1738+0333.
+The relevant properties of the two systems are summarized in Table. I. The
+19
+
+Hulse-Taylor binary is a pulsar binary with a neutron star companion, it is highly eccentric,
+and the mass ratio of the two stars is close to 1. The PSR J1738+0333, on the other hand,
+is a pulsar-white dwarf binary with an almost circular orbit and a high pulsar-to-companion
+mass ratio. For both systems, the data on the orbital period decay is shown in Table I. Both
+binaries lie within 1σ of the general relativity prediction.
+In our analysis, we exploit the fact that typical neutron stars contain a very large number of
+muons, N(µ) ∼ 1055 [40–43]. Thus, the effects of muonophilic new physics can be significantly
+enhanced. The presence of the large muon number in neutron stars is attributed to the fact
+that when the electron chemical potential, µe, is larger than the muon mass µe > mµ, it
+becomes energetically favorable for relativistic electrons at the Fermi surface to decay into
+muons via e− → µ− + ¯νµ + νe. Moreover, the muonic beta-decay n → p + µ− + ¯νµ and inverse
+beta-decay p + µ− → n + νµ reactions become energetically favorable, while the muon decay
+µ− → e− + ¯νe + νµ is forbidden by Fermi statistics.
+Being motivated by the neutron star muonic content, we consider neutrino pair emission by
+pulsar binaries via the following two types of BSM mediators:
+• U(1)Lµ−Lτ massive gauge boson with
+L ⊃ gAα (¯µγαµ − ¯τγατ + ¯νµγανµ − ¯ντγαντ) ,
+(4.15)
+• Massive muonophilic scalar with
+L ⊃ gφ¯µµ + g′φ¯νµνµ .
+(4.16)
+It is known that at least two of the SM neutrinos are massive, while the third neutrino can
+be very light or massless. This means that only one neutrino mass eigenstate can be radiated
+in the two scenarios we consider here. A realistic treatment of neutrino emission would include
+insertions of the corresponding PMNS matrix elements [44], resulting in an additional factor of
+order one. Since we already neglecting an O(1) factor coming from the estimate of the muon
+number density in the neutron stars, we also ignore any PMNS factors in the rest of this section.
+Note also that in a theory with general couplings to the left and right-handed neutrinos,
+i.e., gAα¯νγα(cV − cAγ5)ν, the results for the power loss are qualitatively similar. Moreover, in
+the case of massless neutrinos, the power loss for the case of the general coupling is the same
+as the power loss for the case of purely vectorial coupling up to g2 → g2(c2
+A + c2
+V ) replacement.
+This is why in what follows, for simplicity, we consider the case of the vectorial coupling only.
+These two BSM models imply the possibility for the neutrino pair radiation at rates enhanced
+compared to the SM. Our results from Eqs. (4.8) and (4.9) thus can be used to set bounds on
+the coupling constants and masses of the new bosons.
+20
+
+The presence of the muonophilic new physics, however, not only alters the radiation patterns
+of pulsar binaries, but it also has important implications for the neutron star’s equation of state.
+In particular, the presence of a repulsive (vector) or attractive (scalar) interaction between
+muons could affect the muon number, which depends on the coupling g to the new physics. In
+the following, we write the muon number as N(µ, g) to keep the dependence on g explicit.
+The number of muons becomes g-dependent as the interactions change the muon chemical
+potential. The muon interaction due to the Lµ − Lτ vector boson is repulsive, and thus the
+chemical potential is increased compared to its SM value by ε ∼ g2N(µ, g)/R, where R is
+the radius of the neutron star the boson mass is neglected. When the coupling g is small,
+such that ε ≪ mµ, the effect of the new interaction is insignificant, and the number of muons
+is approximately given by its value in the limit of no interaction N(µ, g = 0).
+When the
+interaction is strong, such that ε ≫ mµ, it becomes energetically less favorable to have muons
+inside the neutron star and thus N(µ, g) < N(µ, g = 0).
+Similar reasoning applies to the case of the scalar mediator. The only difference is the sign
+of the interaction. In the scalar case, the interaction between muons is attractive. Thus the
+muon chemical potential is decreased by ε. This leads to the increase of the muon number
+for larger couplings N(µ, g) > N(µ, g = 0). In both cases, the change from the regime when
+N(µ, g) ≈ N(µ, g = 0) to the situation when the interaction starts to affect the muon number
+happens for couplings such that ε ∼ mµ, or numerically g ∼ 10−18 for a typical neutron star [5].
+However, in what follows, we ignore the effect of the new physics on the muon number.
+Everywhere in our analysis, we use the muon number in the limit of no new physics interaction,
+that is we set N(µ) = N(µ, g = 0) ∼ 1055 [40–43]. In principle, g-independence of muon
+number can be achieved in models with both vector and scalar mediators with fine-tuned
+coupling constants such that the repulsive and attractive interactions cancel each other.
+To apply Eqs. (4.8) and (4.9), we define Nb(µ) and Nb(¯µ) as the number of muons and
+antimuons respectively in neutron star labeled by b = 1, 2. Then, as there are almost no tau
+leptons in neutron stars, Qb = Nb(µ) − Nb(¯µ) is the total charge of the neutron star under the
+Lµ−Lτ gauge symmetry, and Nb = Nb(µ)+Nb(¯µ) is the total number of muons and anti-muons
+in the star. Additionally, since Nb(¯µ) ≈ 0, we have Qb ≈ Nb.
+The energy lost through radiation in a binary star system can be directly probed by measur-
+ing the decay of the orbital period. Assuming that the attractive gravitational force between
+the two stars is such that their orbits stay Keplerian, the decay rate of the period of revolution
+Tb is related directly to the energy lost via radiation [6]:
+˙Tb = −6πa5/2G−3/2
+N
+(m1m2)−1(m1 + m2)−1/2 × Ploss,
+(4.17)
+21
+
+where ˙Tb is the time derivative of the binary period, GN is the gravitational constant, m1 and m2
+are the masses of the stars in the binary system, a is the semi-major axis of the elliptical orbit,
+and Ploss is the total power radiated. The decay of the period per unit of time is dimensionless
+and is measured experimentally.
+GW emission is the dominant source of power loss in a binary star system. Assuming that
+the GW emission and neutrino pair emission are the only sources of energy loss, we have
+Ploss = PGW + P¯νν,
+(4.18)
+where P¯νν is the power loss due to the neutrino pair radiation and PGW is the power loss
+due to GW emission, which, to the leading order, is given by the GW quadrupole radiation
+formula [38],
+P GW
+loss = 32
+5 GΩ6M 2a4(1 − e2)−7/2
+�
+1 + 73
+24e2 + 37
+96e4
+�
+,
+(4.19)
+where M is the reduced mass of the system, as defined in Eq. (4.7). The binary period decay
+˙Tb thus can be written as a sum of two contributions,
+˙Tb = ˙TGW + ˙T¯νν .
+(4.20)
+We next introduce the period decay ratio R as the ratio of the measured period decay to
+the theoretical prediction of the period decay due to GW radiation,
+R =
+˙Tb
+˙TGW
+= 1 +
+˙T¯νν
+˙TGW
+.
+(4.21)
+We use the measured value of R to set 2σ limits on the masses and couplings of the BSM
+mediators of neutrino pair radiation as
+˙T¯νν
+˙TGW
+≤ (R − 1) + 2σ .
+(4.22)
+The resulting constraints on the parameter space (g, mA) and (g, mφ) that we derive from
+the period decay data for the Hulse-Taylor binary and PSR J1738+033 are shown in Fig. 4.
+When deriving the constraints, we use Qb = Nb = 1055 with b = 1, 2 and qν = 1. For the gauge
+boson mediator (left panel), we calculate the period decay due to neutrino pair emission, ˙T¯νν,
+using Eqs. (4.8) and (4.17). As we take all three neutrinos to be massless, and as Lµ −Lτ boson
+couples to two neutrino types, there is an extra factor of 2 in Eq. (4.8). Similarly, for the case
+of the scalar mediator (right panel), we use Eqs. (4.9) and (4.17). As there is no symmetry
+that requires equality of g and g′ in the case of the scalar mediator, we present our results for
+the scalar case in the (g, mφ) plane for four different values of g′ that vary from 10−7 to 10−1.
+First, let us discuss the left panel of Fig. 4, which shows constraints on the mass and coupling
+of the gauge boson. For the PSR J1738+0333 (red line), whose orbit is very close to circular, the
+22
+
+��� �����+��
+��� �����+����
+��-��
+��-��
+��-��
+��-��
+��-��
+��-��
+��-��
+��-��
+��-��
+��-��
+��-�
+��-�
+PSR J1738+0333
+10-22
+10-20
+10-18
+10-16
+10-20
+10-18
+10-16
+10-14
+10-12
+10-10
+10-8
+10-6
+FIG. 4. Left: Constraints on g vs mA from the highly eccentric PSR B1913+16 (Hulse-Taylor) Bounds
+from the neutrino pair radiation (solid) and vector boson radiation (dashed) are shown such that the
+region above the curves is excluded by the measurements of the period decay. The system parameters
+are taken from Table I. Right: Constraints on g vs mφ from PSR J1738+033. The dashed gray line
+corresponds to the bound set by the emission of the scalar boson only, while the solid lines show the
+bounds from including a coupling g′ to the neutrinos.
+effect of neutrino pair radiation becomes significant for the mediator masses greater than the
+second harmonic frequency, mA > 2Ω. For the highly eccentric Hulse-Taylor binary, off-shell
+radiation dominates for mA > 85Ω. In the region mA > 2Ω (mA > 85Ω) for PSR J1738+0333
+(Hulse-Taylor binary), the boundary of the excluded region is approximately quadratic in the
+mediator mass. This is in stark contrast with the case of the on-shell boson emission discussed
+in Ref. [3, 5, 6], where the boundary of the excluded region jumps in steps at mA = nΩ, with
+n being an integer. For comparison, the dashed lines in Fig. 4 show the bounds due to the
+on-shell boson radiation.
+Finally, we comment on the right panel of Fig. 4, which shows the constraints on the mass
+mφ and coupling g for different values of g′ in the case of the scalar mediated radiation. We
+only demonstrate the constraints for PSR J1738+0333; the results for the Hulse-Taylor binary
+are qualitatively the same. Depending on the value of g′ the off-shell scalar radiation starts
+to dominate for mφ > Ω (g′ ≳ 10−4) or mφ > 2Ω (g′ ≲ 10−8). As one can see from the plot,
+g′ = 10−1 provides the strongest bound.
+We conclude this section by noting that we do not perform a detailed analysis of the bounds
+on muonophilic light states. We only remark that very strong bounds on light states are derived
+23
+
+from fifth force searches. Most of these bounds do not apply in our case as these experiments
+are done using materials made out of protons, neutrons, and electrons.
+V.
+CONCLUSION
+It is well known that fermion pairs can behave as bosons in several circumstances. In this
+work, we show that fermion pairs can also constitute classical radiation just like bosonic states
+do. We use this understanding to derive the generalization of the Larmor formula for the case
+of the fermion pair emission.
+Being motivated by the potential of applying fermion pair radiation to astrophysical objects,
+we consider the case of classical sources following elliptical orbits. The most interesting regime
+of fermion pair radiation is when the mediator is off-shell, which takes place when the mass of
+the mediator is much smaller than the frequency of the periodic motion of the source. In this
+regime, the fermion pair emission takes over from on-shell boson production. This opens up a
+window into a broader region of parameter space for various models that allow for the fermion
+pair radiation by classical sources.
+Subsequently, we apply our results to neutrino-antineutrino emission by two pulsar binary
+systems PSR B1913+16 and PSR J1738+0333. Neutrino pair emission by binary systems is
+highly suppressed in the SM compared to GW radiation, but can be significantly enhanced in
+various BSM scenarios. In particular, we consider two possibilities: light muonophilic vector
+and scalar mediators that couple to the SM neutrinos. Using period decay data for the two
+binary systems, we derive bounds on the parameters of the two models. While we did not
+perform a comprehensive study of the relevance of these bounds, the key point is that they
+provide a demonstration of the fact that fermion pair radiation can be used to enhance BSM
+probes using astrophysical data.
+There are several future directions to go from here. Here are a few that we find particularly
+interesting:
+• A thorough and detailed study of the bounds that we find on specific models is called for.
+This, however, is complicated by the large uncertainties that come from the estimates on
+the neutron star constituents. In particular, new physics interactions alter the equation
+of state of a neutron star and, currently, there is no precise quantitative understanding
+of how this affects its content.
+• It also would be interesting to see if we can find more systems to which our results can
+be applied. In particular, exotic astrophysical systems and exotic types of new physics
+models.
+24
+
+• In this work, we only consider fermion pair radiation; however, the results can be modified
+to also include bosonic pair radiation. All that needs to be done is to calculate the relevant
+matrix elements. It is expected to result in a different kinematic dependence.
+We conclude with the main message of our paper: If nature includes new light states, fermion
+pair radiation can be one more tool in our toolbox to probe them.
+ACKNOWLEDGEMENTS
+We are grateful to Kfir Blum, Jeff Dror, Toby Opferkuch, Nadav Outmezguine, Ira Rothstein,
+Ryosuke Sato, and Kohsaku Tobioka for useful discussions. The work of YG is supported in
+part by the NSF grant PHY1316222. The research of WT is supported by NSF Grant No.
+PHY-2013052.
+Appendix A: Derivation of the power loss formula
+We present below an explicit derivation of the power loss formula for the fermion pair
+radiation by a point-like classical object on an elliptical orbit. We perform the calculation
+separately for the case of vector and scalar mediators. In our calculation, we follow closely the
+analysis in Ref. [6].
+1.
+The case of a vector boson mediator
+The power loss is a sum over different harmonics, as given by Eqs. (2.7) and (2.8). The matrix
+element, at leading order, for a vector boson mediator, is given by Eq. (2.10). It includes the
+Fourier Transform of the classical current Jµ
+cl(x) defined in Eq. (2.1). We rewrite it here for
+convenience:
+Mn(s1, s2) = g2Qψ ¯u(k1, s1)γµv(k2, s2) i(−ηµν + (k1 + k2)µ(k1 + k2)ν/m2
+A)
+(k1 + k2)2 − m2
+A + imAΓA
+Jν
+cl(Ωn) ,
+(A1)
+where ηµν is the Minkowski metric tensor. Note that the contribution from the (k1 + k2)µ(k1 +
+k2)ν term vanishes by means of the Dirac equation since the fermions are on-shell, that is,
+¯u(/k1 + /k2)v = ¯u(mψ − mψ)v = 0.
+(A2)
+Squaring the amplitudes corresponding to different harmonics and summing over spins, we
+25
+
+find
+|Mn|2 =
+�
+s1,s2
+|Mn|2 =
+g4Q2
+ψ
+((k1 + k2)2 − m2
+A)2 + m2
+AΓ2
+A
+Jµ
+cl(Ωn)J∗ν
+cl (Ωn) Tr [(/k1 + mν)γµ(/k2 − mν)γν]
+=
+4g4Q2
+ψ
+((k1 + k2)2 − m2
+A)2 + m2
+AΓ2
+A
+Jµ
+cl(Ωn)J∗ν
+cl (Ωn)
+�
+k1µk2ν + k1νk2µ − 1
+2(k1 + k2)2ηµν
+�
+.(A3)
+Finally, we are ready to write the expression for the rate of energy loss due to ψ ¯ψ emission
+at harmonic n by the classical source as
+Pn =
+�dE
+dt
+�
+n
+=
+�
+Ωn dΓn
+= Ωn
+�
+d3k1
+(2π)3(2ω1)
+d3k2
+(2π)3(2ω2)(2π)δ(Ωn − ω1 − ω2)|Mn|2
+= Ωn
+�
+dΦ1dΦ2
+|k1|dω1
+2(2π)3
+|k2|dω2
+2(2π)3 (2π)δ(Ωn − ω1 − ω2)|Mn|2 ,
+(A4)
+where |k1,2| =
+�
+ω2
+1,2 − m2
+ψ, we used Ωn = ω1 + ω2 for the total energy carried away by the
+fermion pair, dΦ1,2 are the differential elements of solid angles in the fermion’s direction of
+flight, and
+��Mn
+��2 is given in Eq. (A3). The total power radiated is found by summing over all
+kinematically allowed harmonics:
+P =
+�
+n
+Pn.
+(A5)
+To calculate the power radiated in fermion pairs by a point-like source in an elliptical orbit,
+we need to evaluate the integrals in Eq. (A4), after substituting in the explicit form of Jµ
+cl(Ωn)
+in Eq. (A3). Using Eqs. (2.1) and (2.4), we find the Fourier Transform Jµ
+cl(Ωn) as:
+Ji
+cl(Ωn) = aΩQji
+n,
+J0
+cl(Ωn) = aΩQ
+�jn · p
+nΩ
+�
+,
+(A6)
+where the 3-vector jn is defined as
+jn =
+�
+−iJ′
+n(ne),
+√
+1 − e2
+e
+Jn(ne), 0
+�
+,
+(A7)
+with Jn(z) denoting a Bessel function, and p = k1 + k2.
+The terms in the numerator of |M|2 in Eq. (A3), are then given by
+(Jµ
+cl(Ωn)k1µ) (Jν∗
+cl (Ωn)k2ν) = a2Ω2Q2ji
+njj∗
+n
+� ω1ω2
+(nΩ)2pipj − ω1
+nΩpikj
+2 − ω2
+nΩki
+1pj + ki
+1kj
+2
+�
+,
+(A8)
+and
+|Jµ
+cl(Ωn)|2 = |J0
+cl(Ωn)|2 − |Jcl(Ωn)|2 = a2Ω2Q2ji
+njj∗
+n
+� pipj
+(Ωn)2 − δij
+�
+,
+(A9)
+where we used Ωn = nΩ.
+Note that all quantities above are 3-vectors with Latin indices
+i = 1, 2, 3, and a sum over i and j is implicit. The expression for (Jµ
+cl(Ωn)k2µ) (Jν∗
+cl (Ωn)k1ν) is
+obtained from Eq. (A8) via complex conjugation.
+26
+
+Next we note that the denominator of |Mn|2, see Eq. (A3), depends only on mA, ΓA, ω1,2, the
+magnitudes |k1,2| and the relative angle between the two momenta k1, and k2 that we denote
+as γ. Because of this, it is convenient to perform the change of coordinates in the integral in
+Eq. (A4) from the integration over the solid angles dΦ1dΦ2 to the integration over dΦ1dΦr
+2
+where the solid angle of the second neutrino is measured relative to the direction of k1, hence
+the super index r. (Equivalently, one can also choose to integrate over dΦr
+1dΦ2.) The Jacobian
+of this coordinate change is unity since the transformation is simply a coordinate rotation, and
+thus
+dΦ1dΦ2 = dΦ1dΦr
+2.
+(A10)
+Defining
+dΦb = sin θbdθbdφb,
+dΦr
+2 = sin γdγdδ,
+b = 1, 2 ,
+(A11)
+we find the following relations between the two sets of integration variables
+cos γ = cos θ1 cos θ2 + sin θ1 sin θ2 cos (φ2 − φ1) ,
+sin δ = sin θ2 sin (φ2 − φ1)
+sin γ
+.
+(A12)
+Since, out of all the angular variables, the denominator only depends on the relative angle γ,
+the integrals over θ1, φ1 and δ can be taken easily using the following relations
+�
+dΦ1dΦ2ki
+akj
+a =
+�
+dΦ1dΦr
+2ki
+akj
+a = δij 8π2
+3 k2
+a
+�
+sin γdγ,
+�
+dΦ1dΦ2ki
+1kj
+2 =
+�
+dΦ1dΦr
+1ki
+1kj
+2 = δij 8π2
+3 (k1 · k2)
+�
+sin γdγ,
+�
+dΦ1dΦ2 =
+�
+dΦ1dΦr
+2 = 8π2
+�
+sin γdγ .
+(A13)
+Using this and the results of Eqs. (A8) and (A9), we perform the integration over θ1, φ1 and δ
+in Eq. (A4), and find the following expression for the power radiated in harmonic n,
+Pn = g4 (nΩ)
+12π3 a2Ω2Q2
+ψQ2 |jn|2
+�
+δ(nΩ − ω1 − ω2)
+((k1 + k2)2 − m2
+A)2 + m2
+AΓ2
+A
+×
+�
+−1
+2 (k1 + k2)2 �
+(k1 + k2)2 /
+�
+(nΩ)2 − 3
+��
++ 2 ω1ω2
+(nΩ)2 (k1 + k2)2
+−2 ω1
+nΩ
+�
+k2
+2 + k1 · k2
+�
+− 2 ω2
+nΩ
+�
+k2
+1 + k1 · k2
+�
++ 2k1 · k2
+�
+×
+ω1ω2
+�
+1 − m2
+ψ
+ω2
+1
+�1/2 �
+1 − m2
+ψ
+ω2
+2
+�1/2
+sin γ dγdω1dω2 ,
+(A14)
+where the only integrals left are the integrals over γ, ω1 and ω2.
+Next, we introduce the following dimensionless variables and parameters
+x1 = ω1
+Ω ,
+x2 = ω2
+Ω ,
+nψ = mψ
+Ω ,
+nA = mA
+Ω ,
+nΓ = ΓA
+Ω .
+(A15)
+27
+
+Performing the change of variables in Eq. (A14) from (ω1, ω2) to (x1, x2), we rewrite the ex-
+pression for the power radiated in harmonic n as follows:
+Pn =
+g4
+12π3a2Ω4Q2
+ψ|jn|2
+�
+sin γ dγ dx1 dx2 δ(n − x1 − x2) F(cos γ, x1, x2) .
+(A16)
+Upon taking the integral over x2 and performing the replacement x1 → x, we obtain
+Pn =
+g4
+12π3a2Ω4Q2
+ψQ2|jn|2
+� n−nψ
+nψ
+dx
+� 1
+−1
+d(cos γ) F(cos γ, x) ,
+(A17)
+where function F(cos γ, x) is given by
+F(cos γ, x) = b(x)
+2n
+1
+2b2(x) cos2 γ + b(x)c(x) cos γ + d(x)
+(a(x) − b(x) cos γ)2 + g2
+,
+(A18)
+with
+a(x) = 2n2
+ψ + 2x(n − x) − n2
+A ,
+b(x) = 2
+�
+x2 − n2
+ψ
+�
+(n − x)2 − n2
+ψ ,
+c(x) = −
+�
+n2 + 2n2
+ψ
+�
+,
+d(x) = 2(x(n3 − 2n2x + 2nx2 − x3) + 2n2n2
+ψ + n4
+ψ),
+g2 = n2
+An2
+Γ .
+(A19)
+The variable x here is the ratio of the energy of one of the fermions to the fundamental oscillation
+frequency. It can be at least nψ or at most n − nψ, hence the limits on the integral. Also note
+that F also depends on the parameters of the problem namely nA, nψ, nΓ defined in Eq. (A15),
+but we do not write them explicitly for brevity. Lastly, note that the γ-dependence of the
+numerator of function F is through a term quadratic in cos γ and a term linear in cos γ. This
+behavior is attributed to the theory that we pick – renormalizable theories such as in the case
+considered here would only contribute at most two powers of momentum in the matrix element,
+leading to a cos γ dependence that is at most quadratic. However non-renormalizable theories
+have more momenta in the matrix element, and will give us a different cos γ dependence in the
+F.
+Now, we define
+F A(x) ≡ F A(n, x, nψ, nA, nΓ) =
+� 1
+−1
+d (cos γ) F (cos γ, x, n) ,
+(A20)
+where the superscript A denotes the vector boson mediator.
+The integral over cos γ can be taken analytically. Then, we find that the function F A(x),
+has the form:
+F A(x) = F A
+0 (x) + F A
+1 (x)
+nMnΓ
+�
+tan−1
+�a(x) + b(x)
+nMnΓ
+�
+− tan−1
+�a(x) − b(x)
+nMnΓ
+��
++ F A
+2 (x) tanh−1
+�
+2a(x)b(x)
+a(x)2 + b(x)2 + n2
+Mn2
+Γ
+�
+,
+(A21)
+28
+
+with:
+F A
+0 (x) = b(x)/2n ,
+F A
+1 (x) = 1
+4n
+�
+n4
+A + 4n2n2
+ψ − n2
+An2
+Γ + 2n2
+An2 − 4nxn2
+A + 4x2n2
+A
+�
+,
+F A
+2 (x) = 1
+2n
+�
+n2
+A + n2 − 2nx + 2x2�
+.
+(A22)
+Consequently, the power loss formula of each mode with n > 2nψ becomes
+Pn = 2g4Q2
+ψQ2
+3(2π)3 a2Ω4
+�
+J′
+n(ne)2 + 1 − e2
+e2
+Jn(ne)2
+� � n−nψ
+nψ
+dxF A(x),
+(A23)
+which gives us Eq. (2.18) for the case M = A, where we define for mediator M
+BM
+n (nM, nν, nΓ) ≡
+�
+J′
+n(ne)2 + 1 − e2
+e2
+Jn(ne)2
+� � n−nψ
+nψ
+dx F M(x, n, nM, nν, nΓ),
+(A24)
+where Jn(z) is a Bessel function of order n in the variable z.
+2.
+The case of the scalar mediator
+The derivation for the power loss in the scalar mediator is similar to the vector case, but the
+matrix element is different, as shown in Eq. (2.14). This matrix element contains the number
+density ρcl(x) of source particles, instead of a current. As such, the difference in the calculation
+in this case comes from the calculation of the squared matrix element, which in this case, is
+given by:
+�
+s1,s2
+|Mn(s1, s2)|2 =
+g2g′2
+((k1 + k2)2 − m2
+φ)2 + m2
+φΓ2
+φ
+Tr(( /k1 + mψ)( /k2 − mψ))|ρcl(Ωn)|2
+=
+4g2g′2
+((k1 + k2)2 − m2
+φ)2 + m2
+φΓ2
+φ
+(k1 · k2 − m2
+ψ)|ρcl(Ωn)|2 ].
+(A25)
+The power loss is again given by Eq. (A4).
+Using Eqs. (2.2) and (2.4), we find the Fourier Transform ρµ
+cl(Ωn) as:
+ρ0
+cl(Ωn) = aΩN
+�jn · p
+nΩ
+�
+,
+(A26)
+where, like in the vector case, we define the 3-vector ji
+n as follows:
+jn =
+�
+−iJ′
+n(ne),
+√
+1 − e2
+e
+Jn(ne), 0
+�
+,
+(A27)
+with Jn(z) denoting a Bessel function, amd p = k1 + k2.
+After performing all the steps analogous to Eqns. (A4)–(A20) in the previous section, i.e,
+after performing the angular integration, we get:
+Pn = g2g′2
+12π3 a2Ω4N 2|jn|2
+� n−nψ
+nψ
+dx
+� 1
+−1
+d cos γ F(cos γ, x) ,
+(A28)
+29
+
+where function F(cos γ, x) is given by
+F(cos γ, x) = −b(x)
+2n
+1
+2b2(x) cos2 γ + b(x)c(x) cos γ + d(x)
+(a(x) − b(x) cos γ)2 + g2
+,
+(A29)
+with
+a(x) = 2n2
+ψ + 2x(n − x) − n2
+φ ,
+b(x) = 2
+�
+x2 − n2
+ψ
+�
+(n − x)2 − n2
+ψ ,
+c(x) = (n − 2x)2
+2
+,
+d(x) = (n2
+ψ − nx + x2)(n2 − 2n2
+ψ − 2nx + 2x2),
+g2 = n2
+φn2
+Γ .
+(A30)
+Like before, we define
+F φ(x) ≡ F φ(n, x, nψ, nφ, nΓ) =
+� 1
+−1
+d (cos γ) F (cos γ, x, n) ,
+(A31)
+where the superscript φ denotes the scalar mediator.
+The integral over cos γ can be taken analytically to find a form for F φ:
+F φ(x) = F φ
+0 (x) + F φ
+1 (x)
+nMnΓ
+�
+tan−1
+�a(x) + b(x)
+nMnΓ
+�
+− tan−1
+�a(x) − b(x)
+nMnΓ
+��
++ F φ
+2 (x) tanh−1
+�
+2a(x)b(x)
+a(x)2 + b(x)2 + n2
+Mn2
+Γ
+�
+,
+(A32)
+with:
+F φ
+0 (x) = −b(x)/2n ,
+F φ
+1 (x) = 1
+4n
+�
+n2
+φn2
+Γ + (n2 − n2
+φ)(n2
+φ − 4n2
+ν)
+�
+,
+F φ
+2 (x) = 1
+4n
+�
+n2 + 4n2
+ν − 2n2
+φ
+�
+.
+(A33)
+Consequently, the power loss formula of each mode with n > 2nψ becomes
+Pn = 2g2g′2
+3(2π)3a2Ω4N 2
+�
+J′
+n(ne)2 + 1 − e2
+e2
+Jn(ne)2
+� � n−nψ
+nψ
+dxF φ(x),
+(A34)
+which gives us Eq. (2.19) for the case M = φ
+P φ
+n = g2g′2
+12π3 a2Ω4
+�N1
+m1
+− N2
+m2
+�2
+Bφ
+n(nA, nν, nΓ).
+(A35)
+We find that the form of the function F M is general for the two types of mediators, the
+difference lying in the explicit forms of the functions F M
+0 , F M
+1
+and F M
+2 . This is due to the fact
+that the cos γ dependence of the function F is the same in both cases, as in both cases, the
+theory considered is a renormalizable one. As we explained in the previous sub-section, this
+30
+
+general form of F M is not what we will have when we consider non-renormalizable theories that
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@@ -0,0 +1,1037 @@
+Trajectory Optimization on Matrix Lie Groups
+with Differential Dynamic Programming and
+Nonlinear Constraints
+Gokhan Alcana, Fares J. Abu-Dakkab and Ville Kyrkia
+Abstract— Matrix Lie groups are an important class of
+manifolds commonly used in control and robotics, and the
+optimization of control policies on these manifolds is a
+fundamental problem. In this work, we propose a novel
+approach for trajectory optimization on matrix Lie groups
+using an augmented Lagrangian based constrained dis-
+crete Differential Dynamic Programming (DDP) algorithm.
+Our method involves lifting the optimization problem to the
+Lie algebra in the backward pass and retracting back to
+the manifold in the forward pass. In contrast to previous
+approaches which only addressed constraint handling for
+specific classes of matrix Lie groups, our method provides
+a general approach for nonlinear constraint handling for a
+generic matrix Lie groups. We also demonstrate the effec-
+tiveness of our method in handling external disturbances
+through its application as a Lie-algebraic feedback control
+policy on SE(3). The results show that our approach is able
+to effectively handle configuration, velocity and input con-
+straints and maintain stability in the presence of external
+disturbances.
+I. INTRODUCTION AND RELATED WORK
+T
+HE configuration space of a physical system represents
+the set of all possible configurations. Modeling this space
+using local coordinates may lead to several potential problems.
+One issue is that these coordinate systems may suffer from
+singularities or degeneracies, where the coordinates become
+ill-defined or degenerate. For example, Euler angles can
+experience gimbal lock, where two of the angles become
+degenerate, leading to a loss of degree of freedom and making
+it difficult to represent certain configurations of the system
+[1], [2]. Similarly, quaternions can experience a similar loss
+of degree of freedom when their magnitude becomes close to
+zero and multiple different representations can describe the
+same configuration [3], although it has global representation.
+These types of singularities can make it difficult to accurately
+represent the transition between certain configurations of the
+system, and may require the use of additional techniques to
+avoid or mitigate these issues.
+This work was supported by the Academy of Finland B-REAL Project
+under Grant 328399.
+Corresponding author: Gokhan Alcan ([email protected])
+a The authors are with the Intelligent Robotics Group, Department of
+Electrical Engineering and Automation (EEA), Aalto University, 02150
+Espoo, Finland
+b Munich Institute of Robotics and Machine Intelligence, Technische
+Universit¨at M¨unchen, 80992 M¨unchen, Germany. Part of the research
+presented in this work has been conducted when Abu-Dakka, F. was at
+Intelligent Robotics Group, EEA, Aalto University, 02150 Espoo, Finland
+On the other hand, a natural way to model the geometry
+of the configuration space is through the use of matrix Lie
+groups, which offer a continuous and structured framework
+for understanding the structure and motion of the underlying
+system [4], [5]. However, it brings the difficulties to deal with
+non-flatness of manifolds in a coordinate-free manner [6].
+The use of geometric control techniques, which seamlessly
+combine differential geometry and control theory, has become
+increasingly prevalent in the field of robotics and control
+systems [7]–[9]. Optimality conditions for geometric control
+techniques are often simplified and numerical ill-conditioning
+is avoided through the use of specific details about the control
+problem.
+Differential dynamic programming (DDP) is a numerical
+method for solving optimal control problems that has gained
+widespread use in various engineering fields. Originally pro-
+posed by Mayne and Jacobson [10], DDP has been applied to
+a wide range of complex, high-dimensional systems [11]. One
+of the key advantages of DDP is its scalability, which allows
+it to handle large and complex systems with many degrees
+of freedom. In addition, DDP has a fast convergence rate,
+which allows it to quickly find near-optimal solutions to the
+control problem. Another important attribute of DDP is its
+ability to generate feedback control policies, which can be
+used to implement the optimal control solution in real-time.
+Several early works have investigated the use of DDP for
+geometric control [9], [12]. These studies focused on specific
+matrix Lie groups, including SE(3), in order to derive the
+final form of the DDP algorithm for applications in geo-
+metric control. Boutselis and Theodorou [13] extended the
+original DDP method by using quadratic expansion schemes
+for cost functions and dynamics defined on Lie groups. They
+demonstrated that DDP has significantly better convergence
+rates compared to sequential quadratic programming (SQP)
+methods. Teng et al. [18] further improved the convergence
+performance of DDP for matrix groups by designing the
+control objective in its Lie algebra. Both of these approaches
+[13], [18] formulate the trajectory optimization on matrix Lie
+groups in an unconstrained framework. In order to address
+this limitation, Liu et al. [14] extended the work [13] by
+imposing SO(3) pose constraints. However, this method is not
+generalizable to nonlinear constraints for generic matrix Lie
+groups. Table I provides a comparison of those methods in
+terms of their cost definitions and constraints.
+arXiv:2301.02018v1  [eess.SY]  5 Jan 2023
+
+TABLE I
+COMPARISON OF DDP METHODS FOR MATRIX LIE GROUPS
+Cost
+SO(3)
+Any Group
+Described in
+Constraints
+Constraints
+Boutselis et al. [13]
+manifold
+
+
+Liu et al. [14]
+manifold
+
+
+Teng et al. [18]
+tangent space
+
+
+Our method
+tangent space
+
+
+The present paper aims to solve the problem of generic
+constraints by extending the idea of Lie algebric cost definition
+[18] and developing a DDP method for matrix Lie groups
+under nonlinear constraints. The main contributions of our
+work are:
+1) Development of an augmented Lagrangian based con-
+strained DDP algorithm for trajectory optimization on
+matrix Lie groups.
+2) A principled approach for nonlinear constraint handling
+for generic matrix Lie groups unlike [14], which only ad-
+dressed constraint handling for SO(3) pose constraints.
+3) Evaluating the effectiveness of the proposed DDP
+method in handling external disturbances through its
+application in a numerical simulation as a Lie-algebraic
+feedback control policy on SE(3).
+The rest of this paper is organized as follows. In Section II,
+we provide preliminaries regarding matrix Lie groups. Section
+III defines the trajectory optimization problem for matrix
+Lie groups. We detail our proposed method in Section IV.
+In Section V, we provide numerical simulation experiments
+for SE(3) to demonstrate the effectiveness of our approach.
+Finally, the paper is concluded with potential directions for
+future work in Section VI.
+II. PRELIMINARIES
+Consider G is an n-dimensional matrix Lie group, and its
+associated Lie algebra, i.e., tangent space at the identity, is
+denoted as g, where dim g = n. Isomorphism between the
+vector space Rn and g can be defined through the following
+operators:
+(.)∧ : Rn �→ g
+(.)∨ : g �→ Rn
+(1)
+Mapping between Rn and G can be defined using the
+functions Exp(.) : Rn �→ G and Log(.) : G �→ Rn for any
+φ ∈ Rn and X ∈ G as follows:
+Exp(φ) = expm(φ∧) = X
+Log(X) = logm(X)∨ = φ
+(2)
+where expm and logm are the exponential and logarithm of
+square matrices, respectively.
+The adjoint action, denoted as AdX : g �→ g for any X ∈ G,
+is a Lie algebra isomorphism that allows change of frames.
+Given φ, η ∈ Rn and φ∧, η∧ ∈ g, the adjoint action can be
+expressed in the function form as
+AdX (φ) = Xφ∧X −1
+(3)
+or in the matrix form as
+(AdX φ)∧ = Xφ∧X −1
+(4)
+The adjoint map is the derivative of the adjoint action with
+respect to X at the identity element and is defined as
+adφ η = [φ∧, η∧]
+(5)
+where [φ∧, η∧] is the Lie bracket, calculated as
+[φ∧, η∧] = φ∧η∧ − η∧φ∧
+(6)
+III. PROBLEM DEFINITION
+We consider the systems whose states reside in the tangent
+bundle of a matrix Lie group. This encompasses a diverse
+array of systems [15] whose states can be represented as pairs
+{X, ξ∧} ∈ G × g, where X represents the configuration and
+ξ∧ represents the velocity. The continuous equations of motion
+for such systems can be written as:
+˙Xt = Xtξ∧
+t
+˙ξt = f
+�
+ξt, ut
+�
+(7)
+where ut ∈ Rm is the generalized control input and f(.) is
+the function of velocity dynamics. For a given initial state
+{X0, ξ0}, a goal state {Xg, ξg} and a time horizon N, we
+define the discrete-time constrained optimal control problem
+as
+min
+u0,...,uN−1
+ℓf(XN, ξN) +
+N−1
+�
+k=0
+ℓ(Xk, ξk, uk)
+subject to
+Xk+1 = FX (Xk, ξk),
+k = 0, ..., N − 1
+ξk+1 = Fξ(ξk, uk),
+k = 0, ..., N − 1
+umin ≤ uk ≤ umax,
+∀k,
+g(Xk, ξk, uk) ≤ 0,
+∀k,
+given
+X0, ξ0,
+(8)
+where ℓf : G × Rn �→ R and ℓ : G × Rn × Rm �→ R
+are the final cost and the running cost, respectively. FX and
+Fξ are the discretized form of the configuration and velocity
+dynamics, which can be obtained by using either a zero-
+order hold or Euler first-order integration method with a fixed
+time step of ∆t. Lastly, g is a vector of p constraints in
+the form of differentiable nonlinear functions representing the
+state constraints.
+IV. PROPOSED METHOD
+It is often difficult to solve the general problem outlined
+in Section III analytically. Additionally, finding the global
+minimum numerically can be time-consuming, particularly for
+systems with high dimensions. Therefore, we aim to propose a
+method that provides feasible solutions, even if they may not
+be globally optimal. As such, we aim to propose a method
+that yields feasible solutions, even if they may not be globally
+optimal. To accomplish this, we utilize the Differential Dy-
+namic Programming (DDP) framework [20], which iteratively
+solves sub-optimization problems in the backward pass and
+generates a new trajectory in the forward pass based on the
+found optimal policy, in order to approach a local optimum.
+
+In this work, we propose augmenting the cost function with
+multiplier and penalty terms from the augmented Lagrangian
+in order to account for constraints imposed on the system,
+whose states lie in the tangent bundle of a matrix Lie group.
+Our approach involves lifting the problem to the Lie algebra
+in the backward pass by computing the gradient of the cost
+function within the corresponding Lie algebra, and retracting
+back to the manifold in the forward pass by integrating the
+dynamics using the optimal policy obtained in the backward
+pass.
+A. Dynamics on Tangent Space
+The central concept of DDP is that, at each iteration, all
+nonlinear constraints and objectives are approximated using
+first or second order Taylor series expansions. This allows the
+approximate functions, which now operate on deviations from
+the nominal trajectory, to be solved using discrete Linear-
+Quadratic Regulator (LQR) techniques. In order to define
+the cost and constraint functions in Lie algebra, we need
+to determine the error dynamics for the configuration. To
+obtain the perturbed state dynamics, we followed the approach
+proposed by Teng et al. [17]. For completeness, we outline the
+necessary steps here. Interested readers may refer to [17], [18]
+for more information.
+Consider a perturbed state {Xp, ξ∧
+p } that is in the vicinity
+of a nominal state {X, ξ∧}. Then, the configuration error can
+be defined as
+Ψ = X −1Xp ∈ G
+(9)
+Differentiating both sides of (9) yields the configuration error
+dynamics as
+˙Ψ = X −1 d
+dt
+�
+Xp
+�
++ d
+dt
+�
+X −1�
+Xp
+= X −1 ˙Xp − X −1 ˙XX −1Xp
+= X −1Xpξ∧
+p − X −1Xξ∧X −1Xp
+= Ψξ∧
+p − ξ∧Ψ
+(10)
+Here, we can define a vector ψ in Rn such that the matrix ex-
+ponential of ψ∧ corresponds to Ψ, denoted as Ψ = expm(ψ∧).
+Using the first-order approximation of the matrix exponential,
+which states that expm(ψ∧) ≈ In + ψ∧, the dynamics of the
+configuration error in (10) can be linearized as follows:
+˙Ψ = Ψξ∧
+p − ξ∧Ψ
+= (In + ψ∧)ξ∧
+p − ξ∧(In + ψ∧)
+= ξ∧
+p + ψ∧ξ∧
+p − ξ∧ − ξ∧ψ∧
+= ξ∧
+p − ξ∧ + ψ∧ξ∧
+p − ξ∧ψ∧
+= ξ∧
+p − ξ∧ + ψ∧(ξ∧ − ξ∧ + ξ∧
+p ) − ξ∧ψ∧
+= ξ∧
+p − ξ∧ + ψ∧ξ∧ − ξ∧ψ∧ + ψ∧(ξ∧
+p − ξ∧)
+= ξ∧
+p − ξ∧ + ψ∧ξ∧ − ξ∧ψ∧
+= ξ∧
+p − ξ∧ + adψ ξ
+˙ψ = ξ∧
+p − ξ∧ − adξ ψ
+(11)
+Note that the second order term of ψ∧(ξ∧
+p − ξ∧) is also
+discarded to obtain the linear dynamics of the configuration
+error. ψ in (11) is the perturbed configuration represented in
+Lie algebra. The perturbed velocity and control input are also
+defined as
+δξ = ξp − ξ,
+and
+δu = up − u
+(12)
+The perturbed velocity dynamics then become:
+δ ˙ξt = Γtδξt + Λtδut
+(13)
+where Γt and Λt are the Jacobians of f(ξt, ut) defined in (7)
+around the nominal trajectory about ξt and ut.
+Defining the perturbed states as concatenation
+x =
+�ψ
+δξ
+�
+,
+¯u = δu
+(14)
+the perturbed state dynamics are expressed as
+˙x = h(x, u)
+˙x =
+�− adξ
+In
+0n×n
+Γt
+�
+�
+��
+�
+≜At
+x +
+�0n×m
+Λt
+�
+�
+��
+�
+≜Bt
+¯u
+(15)
+The discretized versions of the matrices At and Bt can be
+simply obtained by applying a zero-order hold or a first-order
+Euler integration method with a fixed time step of ∆t.
+B. Constraint Handling
+By decomposing the state into configuration and velocity,
+we can also decompose the constraints in vector g(Xk, ξk, uk)
+in equation (8) into two types: those that constrain the velocity
+and those that specify configurations to be avoided. This allows
+us to separately handle the constraints on velocity (cξ) and on
+configuration (cX ) as
+cξ(ξk, uk) ≤ 0
+cX (Xk, ξk) ≤ 0
+(16)
+The velocity component of the state (ξ) resides in Rn, and
+as a result, constraints involving any metric in Euler space
+produce a distance vector in Rn. Therefore, any boundary
+velocity constraint can be written as
+δξb = ξb − ξ,
+¯cξ(βδξb) ≤ 0
+(17)
+where
+β =
+�
+−1
+if ξb is upper bound,
++1
+if ξb is lower bound
+(18)
+On the other hand, the difference between two group
+elements in the configuration state produces a geodesic in
+the group. To handle this, we propose mapping the distance
+geodesic to the tangent space of the configuration at the current
+time step and addressing the constraint in that vector space.
+Configuration avoidance constraints can typically be formu-
+lated as inequality constraints using an n-spherical function,
+with the center of the n-sphere located at the configuration
+to be avoided (Xc) and the radius (rc) defining the restricted
+region. This allows us to specify a region of configurations
+that should be avoided. The distance between the nominal and
+restricted configurations in the tangent space of the nominal
+trajectory, ψc, is given by:
+ψc = logm(X −1Xc)
+(19)
+
+Then, the configuration avoidance constraint can be written as
+¯cX (ψc) = (r2
+c − ∥ψc∥2) ≤ 0
+(20)
+In this approach, we consider the same restricted region
+for each axis in the n-dimensional sphere. However, it is
+also possible to specify different radius values for each axis,
+resulting in an n-dimensional ellipsoid as the restricted region.
+Our method can accommodate these types of configuration
+constraints as well.
+In order to handle the constraints in DDP framework,
+we need the first-order approximations of them around the
+perturbed state dynamics introduced in (15) as follows:
+¯c(x + δx, u + δu) ≈ ¯c(x, u)
++ ¯cx(x, u)δx + ¯cu(x, u)δu
+(21)
+where
+¯c(x) =
+�¯cX (ψc)
+¯cξ(δξb)
+�
+(22)
+¯cx and ¯cu are the derivative of ¯c with respect to x and
+u, respectively. If the constraints are designed according to
+equations (18) and (20), the derivatives can be calculated as
+follows:
+¯cx(x, u) =
+�−2(adξ ψc)⊤
+2ψc⊤
+01×n
+β(Γδξb)⊤
+�
+¯cu(x, u) =
+�
+01×m
+β(Λ⊤δξb)⊤
+�
+(23)
+C. Constrained Differential Dynamic Programming
+Using the perturbed state dynamics defined in (15), the
+backward pass of differential dynamic programming (DDP) is
+lifted to the tangent space. The backward pass of DDP involves
+computing the cost-to-go function at each time step in a given
+trajectory. Unlike [18], our algorithm not only considers the
+objective function when calculating the cost-to-go function,
+but also takes into account any constraints on the state and
+control variables.
+An effective method for solving constrained optimization
+problems is to transform the constraints into the objective
+function and iteratively increase the penalty for violating or
+approaching them. This technique, known as penalty method,
+guarantees convergence to the optimal solution as the penalty
+terms increase indefinitely. However, this may not be prac-
+tical to implement in numerical optimization routines due
+to the limitations of finite precision arithmetic. Augmented
+Lagrangian methods [19] offer an alternative solution by
+maintaining estimates of the Lagrange multipliers associated
+with the constraints, allowing for convergence to the optimal
+solution without requiring the penalty terms to increase indef-
+initely.
+Here we obtain the augmented Lagrangian as
+LA = LN(xN) +
+N−1
+�
+k=0
+Lk(xk, uk)
+LN(xN) = ¯ℓf(xN) + (λ + 1
+2 ¯g(xN)Iµ)⊤¯g(xN)
+Lk(xk, uk) = ¯ℓ(xk, uk) + (λ + 1
+2 ¯g(xk, uk)Iµ)⊤¯g(xk, uk)
+(24)
+where ¯ℓf : R2n �→ R and ¯ℓ : R2n ×Rm �→ R are the final cost
+and the running cost functions for perturbed system dynamics,
+respectively. A typical design of such functions are
+¯ℓf(x) = 1
+2∥δx∥SV
+¯ℓ(x, u) = 1
+2∥δx∥SQ + 1
+2∥δu∥SU
+(25)
+where SV ∈ R2n×2n, SQ ∈ R2n×2n and SU ∈ Rm×m are
+the cost matrices that are specified by the user and remain
+constant throughout all iterations.
+In (24), ¯g is a vector of p constraints for perturbed state
+dynamics as introduced in (21). λ ∈ Rp is a Lagrange
+multiplier, µ ∈ Rp is a penalty weight and Iµ ∈ Rp×p is
+the penalty matrix defined as
+Iµ =
+�
+0
+if gi(.) < 0 and λi = 0,
+µi
+otherwise
+(26)
+In general, one can define time varying Lagrange multiplier
+and penalty weight, but we kept them constant for each time
+step during the same iteration for simplicity.
+We define the cost-to-go and action-value functions as
+Vk(xk) = min
+uk {Lk(xk, uk)} + Vk+1(Akxk + Bkuk)
+= min
+uk Q(xk, uk))
+(27)
+The matrices Ak and Bk represent the discretized versions
+of At and Bt in equation (15). Second order Taylor series
+expansion of cost-to-go function can be written as
+δVk(x) ≈ 1
+2δx⊤
+k Vxx,kδxk + V ⊤
+x,kδxk
+(28)
+where Vxx,k and Vx,k are the Hessian and gradient of the
+cost-to-go at time step k, respectively. Action-value function
+defined in (27) can be also approximated as a quadratic
+function as
+Q(x + δx, u + δu) ≈ Q(x, u) + Q⊤
+x δx + Q⊤
+uδu
++ 1
+2(δx⊤Qxxδx + δx⊤Qxxδx)
++ δx⊤Qxuδu
+(29)
+To compute the derivative matrices in (29):
+Qx = ¯ℓx + A⊤V ′
+x + ¯g⊤
+x (λ + Iµ¯g)
+Qu = ¯ℓu + B⊤V ′
+x + ¯g⊤
+u (λ + Iµ¯g)
+Qxx = ¯ℓxx + A⊤V ′
+xxA + ¯g⊤
+x Iµ¯gx + (V ′
+xhxx)
+Quu = ¯ℓuu + B⊤V ′
+xxB + ¯g⊤
+u Iµ¯gu + (V ′
+xhuu)
+Qux = ¯ℓux + B⊤V ′
+xxA + ¯g⊤
+u Iµ¯gx + (V ′
+xhux)
+(30)
+To simplify the notation, we have omitted the time indices
+on all variables. All variables in this expression are evaluated
+at time step k, except for those marked with ′, which are
+evaluated at time step k + 1.
+Calculating the full second-order expansion of the state
+dynamics (hxx, huu, hux), can be computationally expensive,
+particularly for systems with complex dynamics and high-
+dimensional states. DDP refers to iterative LQR by discarding
+the second-order dynamics and computing only the first-order
+
+expansion. This results in a Gauss-Newton approximation of
+the true Hessian, which reduces the local fidelity and requires
+more iterations. However, these iterations are less expensive
+to compute and often lead to a faster overall convergence
+rate. Therefore, in this work, we eliminated the second order
+dynamics as approximating the perturbed state dynamics as
+described in (11).
+Minimizing (29) with respect to δu results in an affine
+controller
+δu∗ = −Q−1
+uu(Quxδx + Qu) ≜ Kδx + d
+(31)
+Substituting δu∗ into (29) yields the derivatives of the cost-
+to-go at time step k in terms of the derivatives of the action
+value function as:
+Vx = Qx + KQu + K⊤Quud + Q⊤
+uxd,
+Vxx = Qxx + K⊤QuuK + K⊤Qux + Q⊤
+uxK
+(32)
+At the final step, Vx and Vxx can be easily computed as the
+first and second derivatives of the final cost function (¯ℓf). This
+way, the derivatives of the action-value function (30) and in
+turn the local optimal control policy (31) at each step can be
+calculated backwards starting from the final step.
+After determining the optimal control policy for each time
+step, we update the nominal trajectories by simulating the
+dynamics forward on the manifold itself starting from the
+initial state as:
+δxk =
+�logm(Xk−1 ¯
+Xk)
+¯ξk − ξk
+�
+δuk = Kkδxk + αdk
+¯uk = uk + δuk
+¯ξk = ξ + f(ξ, ¯uk)∆t
+¯
+Xk = X expm(¯ξk∆t)
+(33)
+where {Xk, ξk, uk} and { ¯
+Xk, ¯ξk, ¯uk} represent the nominal
+state-actions and the updated state-actions at time step k,
+respectively. In the above expression, 0 ≤ α ≤ 1 is a
+scaling term for simple linear search on the feedforward term.
+Practically, the parameter α is initially set to 1, but if the
+cost of the updated trajectory does not decrease, it will be
+decreased. Once the cost of the updated trajectory is decreased,
+the forward pass is succesfully completed, the new trajectory
+is accepted as nominal trajectory and the backward pass is
+triggered.
+To optimize the performance of DDP-based algorithms,
+there are a few implementation practices to consider. In the
+backward pass, Quu may need to be regularized as Quu +ρI
+if it is invertible or the former forward pass is unsuccesful, i.e.,
+the cost is not decreased. After the DDP iterations converge,
+the parameters λ and µ can be updated as:
+λ+ = max(0, λi + µi¯gi(x∗, u∗))
+µ+ = γµ,
+γ > 0
+(34)
+The DDP iterations can then be restarted until convergence is
+achieved. For more information, see reference [19].
+V. EXPERIMENTS
+The goal of this section is to devise a trajectory on SE(3)
+that satisfies both position and orientation constraints while
+avoiding unsafe configurations and obstacles.
+A. Dynamics on SE(3)
+We consider a 3D rigid body in SE(3) where the states of
+the system can be represented by a rotation matrix
+R ∈ SO(3) = {R ∈ R3×3|R⊤R = I3, det(R) = 1}
+(35)
+and position p ∈ R3. The homogeneous representation of a
+typical group element in SE(3) is
+X =
+�R
+p
+0
+1
+�
+∈ SE(3)
+(36)
+The velocity vector ξ in SE(3) is known as a “twist” and is
+composed of both angular (ω) and linear (v) velocities in body
+frame as
+ξ =
+�
+ω
+v
+�
+∈ R6,
+ξ∧ =
+�
+ω∧
+v
+0
+0
+�
+∈ se(3)
+(37)
+The forced Euler-Poincar´e equations [16] define the twist
+dynamics as
+Jb ˙ξ = ad∗
+ξ Jbξ + u
+(38)
+In this expression, Jb represents the generalized inertia matrix
+in the body fixed principal axes, while u ∈ g∗ represents
+the generalized control input forces applied to these axes.
+g∗ denotes the cotangent space and the coadjoint map is
+represented by ad∗
+ξ. The matrix representations of the adjoint
+action in the Lie algebra (adξ) and the coadjoint map (ad∗
+ξ)
+are as follows:
+adξ =
+�ω∧
+0
+v∧
+ω∧
+�
+,
+ad∗
+ξ = ad⊤
+ξ = −
+�ω∧
+v∧
+0
+ω∧
+�
+.
+(39)
+The continuous equations of motion described in (7) can be
+written for SE(3) group as
+˙X = Xξ∧,
+˙ξ = Jb
+−1�
+ad∗
+ξ Jbξ + u
+�
+(40)
+The linearized twist dynamics as given in [17]:
+˙ξ = Γtξ + Λtu + bt
+(41)
+where Γt, Λt and bt are:
+Γt := J−1
+b
+ad∗
+ξ Jb + J−1
+b
+�(Ibω)∧
+mv∧
+mv∧
+0
+�
+Λt := Jb
+−1
+bt := −Jb
+−1
+�(Ibω)∧
+mv∧
+mv∧
+0
+�
+ξ
+(42)
+assuming the inertia matrix Jb is defined as
+Jb :=
+�Ib
+0
+0
+mI3
+�
+(43)
+where Ib and m are the moment of inertia in the body frame
+and the body mass, respectively.
+
+Fig. 1. Constrained (red) and unconstrained (blue) trajectories generated by proposed DDP method.
+Fig. 2.
+Control input signals of the planned trajectories. Torques refers
+to inputs acting on angular velocity and Forces refers to inputs acting on
+linear velocity. Left column: unconstrained, Right column: constrained
+B. Simulations
+We now test the proposed algorithm for planning a safe path
+for a SE(3) rigid body. The task we have defined involves
+rotating the body to the configuration Rz(180) from the
+identity and translating it to the position {2, 2, 2} from the
+initial position {1, 1, 1} in 3 seconds with a fixed time step
+of ∆t = 0.01. To denote the rotation around the x, y, and z
+axes of the body-fixed frame, we use Rx(.), Ry(.), and Rz(.),
+respectively. During the motion, the configuration Rz(90) is
+considered to be unsafe and must be avoided. Additionally, it is
+assumed that there is a spherical obstacle at {1.25, 1.25, 1.25}
+with a radius of r = 0.5 that must be avoided as well.
+We only penalized the final state and control inputs, so we
+set the controller parameters as SV = 1000I12, SU = 0.001I6,
+and SQ = 0. In order to assess the performance of the
+proposed DDP in terms of constraint handling, we conducted
+an experiment with the same trajectory optimization task in
+both constrained and unconstrained cases, and the resulting
+trajectories are depicted in Fig. 1.
+The
+configuration
+trajectories
+were
+converted
+to
+Eu-
+ler angles (φ, θ, ψ in degrees) using the rotation order
+Rx(.)Ry(.)Rz(.). In the unconstrained case (shown in blue), it
+is shown that the ψ angle goes directly from 0 to 180 degrees
+while the φ and θ angles remain constant (Fig. 1), as the
+task only involves rotating around the z-axis. This can also be
+observed in the upper left part of Fig. 2, where only a single
+input is non-zero to achieve the desired motion, while the
+others remain zero. However, in the constrained case (shown
+in red), rotations around x and y axes (changes in φ and θ
+angles) were also observed to avoid the unsafe configuration
+of Rx(90) (Fig. 1).
+As shown in the last row of Fig. 1, the shortest paths for
+the positional states were obtained in the unconstrained case.
+However, the presence of a sphere at {1.25, 1.25, 1.25} with
+a radius of r = 0.5 forced the positional trajectories to take
+a circuitous route around obstacle by deviating along y and x
+axes.
+To evaluate the effectiveness of the proposed DDP method
+in handling external disturbances, we extend the analysis done
+in [13] to SE(3) and employ the proposed DDP method as a
+Lie-algebraic feedback control:
+ufb
+k := u∗
+k + Kk logm((X ∗
+k )−1X ϵ
+k)
+(44)
+where u∗
+k and Kk represent the (sub)optimal control sequence
+and time-varying feedback gains, respectively, which are ob-
+tained through DDP. The feed-forward term dk in (31) is
+not explicitly shown in the feedback policy (44) because it
+is already included in u∗
+k. The variables X ∗
+k and X ϵ
+k represent
+the (sub)optimal states obtained through DDP convergence and
+the perturbed states due to disturbance, respectively.
+We test the policy (44) on the following stochastic version
+of the SE(3) dynamics:
+X ϵ
+k+1 = X ϵ
+k expmSE(3)(ξϵ
+k∆t)
+ξϵ
+k+1 = ξϵ
+k + f(ξϵ
+k, uk)∆t + σωω
+(45)
+where ω is assumed to be spatially uncorrelated independent
+and identically distributed noise, drawn from a zero mean
+Gaussian distribution, ω ∼ N(06×1, I6), σω = 0.001. The
+performance of the proposed DDP method under stochas-
+tic conditions was evaluated by allowing the optimizer to
+converge on the deterministic system and then testing its
+performance on the stochastic system. Fig. 3 shows the results
+of this comparison, using 1000 sampled trajectories under
+noisy dynamics to compare the open-loop policy u∗
+k with the
+feedback policy ufb
+k
+(44). The results demonstrate that the
+use of the obtained feedback gains significantly reduces state
+variance, particularly in the vicinity of the the goal points, as
+expected.
+
+50
+0
+[6ap] 
+[deg]
+[deg]
+100
+0
+-50
+0
+-50
+0
+2.0
+2.0
+2.0
+1.5
+× 1.5
+y
+N 1.5
+1.0
+1.0
+1.0
+0
+1
+2
+3
+0
+1
+2
+3
+0
+1
+2
+3
+Time [sec]
+Time [sec]
+Time [sec]20
+Torques
+25
+0
+0
+-25
+-20
+0.5
+Forces
+0
+0.0
+-10
+-0.5
+0
+2
+0
+2
+Time [sec]
+Time [sec]Fig. 3. Evaluation of DDP’s control sequence under random disturbances. Left Column: Open-loop control, Right column: Closed-loop control
+VI. CONCLUSION
+In conclusion, the optimization of control policies on matrix
+Lie groups is an important problem in robotics and control
+with numerous applications. In this work, we have proposed a
+novel approach for tackling this problem using an augmented
+Lagrangian based constrained discrete DDP algorithm. Our
+method involves lifting the optimization problem to the Lie
+algebra in the backward pass, allowing us to compute the
+gradient of the objective and constraint functions within the
+corresponding tangent space. In the forward pass, we retract
+back to the manifold by integrating the dynamics using the
+optimal policy obtained in the backward pass.
+One of the key contributions of our work is the development
+of a general approach for nonlinear constraint handling for
+a wide range of matrix Lie groups. Previous methods were
+only able to handle constraints for specific classes of matrix
+Lie groups, such as SO(3) pose constraints. Our method, on
+the other hand, is able to handle a wide range of nonlinear
+constraints, making it a more widely applicable and flexible
+solution.
+In addition, we have demonstrated the effectiveness of our
+method in handling external disturbances through its applica-
+tion as a Lie-algebraic feedback control policy on SE(3). The
+results show that our approach is able to effectively handle
+constraints and maintain its stability in the presence of external
+disturbances.
+Overall, our proposed DDP algorithm represents a signif-
+icant step forward in the field of optimization on matrix
+Lie groups. It provides a flexible and general approach for
+nonlinear constraint handling and has the potential to be
+applied to a wide range of problems in robotics and control.
+As a future direction, it would be interesting to investigate
+the closed loop uncertainty propagation through the prediction
+horizon, as has been done for Euclidean models [11], in order
+to design a robust Model Predictive Control (MPC) framework
+for matrix Lie groups. This would allow the proposed method
+to be used in more challenging and dynamic environments.
+Additionally, further experimentation with different types of
+matrix groups and constraints would provide valuable insights
+into the capabilities and limitations of the proposed DDP
+algorithm.
+REFERENCES
+[1] E. G. Hemingway and O. M. O‘Reilly, “Perspectives on Euler angle
+singularities, gimbal lock, and the orthogonality of applied forces and
+applied moments,” Multibody Syst. Dyn., 44, 31–56, 2018.
+[2] M. D. Shuster, “A survey of attitude representations.” Navigation, 8(9),
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+[3] J. Diebel, “Representing attitude: Euler angles, unit quaternions, and
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+and R. M. Murray, Eds. Springer, New York, NY, 2015.
+[5] K. M. Lynch and F. C. Park, Modern robotics. Cambridge University
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+[6] F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems:
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+[7] M. Kobilarov, M. Desbrun, J. E. Marsden, and G. S. Sukhatme, “A dis-
+crete geometric optimal control framework for systems with symmetries,”
+in Proc. Robot., Sci. Syst.,1–8, 2008.
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+robotic vehicles,” in Proc. IEEE Int. Conf. Robot. Autom., 5523–5529,
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+[10] D. H. Jacobson and D. Q. Mayne, Differential Dynamic Programming.
+New York, NY, USA: Elsevier, 1970.
+[11] G. Alcan and V. Kyrki, “Differential Dynamic Programming With
+Nonlinear Safety Constraints Under System Uncertainties,” in IEEE
+Robotics and Automation Letters, vol. 7, no. 2, pp. 1760-1767, 2022
+
+150
+150
+Φ,e, [deg]
+100
+,e,w [deg]
+100
+50
+50
+0
+0
+-50
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+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
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+2.5
+3.0
+Time [sec]
+Time [sec]
+1.0
+1.0
+0.8
+0.8
+0.6
+0.6
+X,y,z
+z'Kx
+0.4
+0.4
+0.2
+0.2
+0.0
+0.0
+0.2
+0.2
+0.0
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+1.0
+1.5
+2.0
+2.5
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+1.0
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+2.5
+3.0
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+ming for optimal estimation,” in Proc. IEEE Int. Conf. Robot. Autom.,
+863-–869,2015
+[13] G.I. Boutselis, E. Theodorou, “Discrete-time differential dynamic pro-
+gramming on lie groups: Derivation, convergence analysis, and numerical
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+Systems, 8850-8857, 2022.
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+“Lie Algebraic Cost Function Design for Control on Lie Groups.”
+arXiv:2204.09177, 2022
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+Solver for Constrained Trajectory Optimization,” IEEE/RSJ International
+Conference on Intelligent Robots and Systems, pp. 7674-7679,2019.
+

UNA0T4oBgHgl3EQfEf9Z/content/tmp_files/load_file.txt

@@ -0,0 +1,398 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf,len=397
+page_content='Trajectory Optimization on Matrix Lie Groups  with Differential Dynamic Programming and  Nonlinear Constraints  Gokhan Alcana, Fares J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Abu-Dakkab and Ville Kyrkia  Abstract— Matrix Lie groups are an important class of  manifolds commonly used in control and robotics, and the  optimization of control policies on these manifolds is a  fundamental problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' In this work, we propose a novel  approach for trajectory optimization on matrix Lie groups  using an augmented Lagrangian based constrained dis-  crete Differential Dynamic Programming (DDP) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  Our method involves lifting the optimization problem to the  Lie algebra in the backward pass and retracting back to  the manifold in the forward pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' In contrast to previous  approaches which only addressed constraint handling for  specific classes of matrix Lie groups, our method provides  a general approach for nonlinear constraint handling for a  generic matrix Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' We also demonstrate the effec-  tiveness of our method in handling external disturbances  through its application as a Lie-algebraic feedback control  policy on SE(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' The results show that our approach is able  to effectively handle configuration, velocity and input con-  straints and maintain stability in the presence of external  disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' INTRODUCTION AND RELATED WORK  T  HE configuration space of a physical system represents  the set of all possible configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Modeling this space  using local coordinates may lead to several potential problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  One issue is that these coordinate systems may suffer from  singularities or degeneracies, where the coordinates become  ill-defined or degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' For example, Euler angles can  experience gimbal lock, where two of the angles become  degenerate, leading to a loss of degree of freedom and making  it difficult to represent certain configurations of the system  [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Similarly, quaternions can experience a similar loss  of degree of freedom when their magnitude becomes close to  zero and multiple different representations can describe the  same configuration [3], although it has global representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  These types of singularities can make it difficult to accurately  represent the transition between certain configurations of the  system, and may require the use of additional techniques to  avoid or mitigate these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  This work was supported by the Academy of Finland B-REAL Project  under Grant 328399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  Corresponding author: Gokhan Alcan (gokhan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='alcan@aalto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='fi)  a The authors are with the Intelligent Robotics Group, Department of  Electrical Engineering and Automation (EEA), Aalto University, 02150  Espoo, Finland  b Munich Institute of Robotics and Machine Intelligence, Technische  Universit¨at M¨unchen, 80992 M¨unchen, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Part of the research  presented in this work has been conducted when Abu-Dakka, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' was at  Intelligent Robotics Group, EEA, Aalto University, 02150 Espoo, Finland  On the other hand, a natural way to model the geometry  of the configuration space is through the use of matrix Lie  groups, which offer a continuous and structured framework  for understanding the structure and motion of the underlying  system [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' However, it brings the difficulties to deal with  non-flatness of manifolds in a coordinate-free manner [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  The use of geometric control techniques, which seamlessly  combine differential geometry and control theory, has become  increasingly prevalent in the field of robotics and control  systems [7]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Optimality conditions for geometric control  techniques are often simplified and numerical ill-conditioning  is avoided through the use of specific details about the control  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  Differential dynamic programming (DDP) is a numerical  method for solving optimal control problems that has gained  widespread use in various engineering fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Originally pro-  posed by Mayne and Jacobson [10], DDP has been applied to  a wide range of complex, high-dimensional systems [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' One  of the key advantages of DDP is its scalability, which allows  it to handle large and complex systems with many degrees  of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' In addition, DDP has a fast convergence rate,  which allows it to quickly find near-optimal solutions to the  control problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Another important attribute of DDP is its  ability to generate feedback control policies, which can be  used to implement the optimal control solution in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  Several early works have investigated the use of DDP for  geometric control [9], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' These studies focused on specific  matrix Lie groups, including SE(3), in order to derive the  final form of the DDP algorithm for applications in geo-  metric control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Boutselis and Theodorou [13] extended the  original DDP method by using quadratic expansion schemes  for cost functions and dynamics defined on Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' They  demonstrated that DDP has significantly better convergence  rates compared to sequential quadratic programming (SQP)  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Teng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' [18] further improved the convergence  performance of DDP for matrix groups by designing the  control objective in its Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Both of these approaches  [13], [18] formulate the trajectory optimization on matrix Lie  groups in an unconstrained framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' In order to address  this limitation, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' [14] extended the work [13] by  imposing SO(3) pose constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' However, this method is not  generalizable to nonlinear constraints for generic matrix Lie  groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Table I provides a comparison of those methods in  terms of their cost definitions and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='02018v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='SY]  5 Jan 2023  TABLE I  COMPARISON OF DDP METHODS FOR MATRIX LIE GROUPS  Cost  SO(3)  Any Group  Described in  Constraints  Constraints  Boutselis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' [13]  manifold  \x17  \x17  Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' [14]  manifold  \x13  \x17  Teng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' [18]  tangent space  \x17  \x17  Our method  tangent space  \x13  \x13  The present paper aims to solve the problem of generic  constraints by extending the idea of Lie algebric cost definition  [18] and developing a DDP method for matrix Lie groups  under nonlinear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' The main contributions of our  work are:  1) Development of an augmented Lagrangian based con-  strained DDP algorithm for trajectory optimization on  matrix Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  2) A principled approach for nonlinear constraint handling  for generic matrix Lie groups unlike [14], which only ad-  dressed constraint handling for SO(3) pose constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  3) Evaluating the effectiveness of the proposed DDP  method in handling external disturbances through its  application in a numerical simulation as a Lie-algebraic  feedback control policy on SE(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' In Section II,  we provide preliminaries regarding matrix Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Section  III defines the trajectory optimization problem for matrix  Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' We detail our proposed method in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  In Section V, we provide numerical simulation experiments  for SE(3) to demonstrate the effectiveness of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  Finally, the paper is concluded with potential directions for  future work in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' PRELIMINARIES  Consider G is an n-dimensional matrix Lie group, and its  associated Lie algebra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=', tangent space at the identity, is  denoted as g, where dim g = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Isomorphism between the  vector space Rn and g can be defined through the following  operators:  (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' )∧ : Rn �→ g  (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' )∨ : g �→ Rn  (1)  Mapping between Rn and G can be defined using the  functions Exp(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=') : Rn �→ G and Log(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=') : G �→ Rn for any  φ ∈ Rn and X ∈ G as follows:  Exp(φ) = expm(φ∧) = X  Log(X) = logm(X)∨ = φ  (2)  where expm and logm are the exponential and logarithm of  square matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  The adjoint action, denoted as AdX : g �→ g for any X ∈ G,  is a Lie algebra isomorphism that allows change of frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  Given φ, η ∈ Rn and φ∧, η∧ ∈ g, the adjoint action can be  expressed in the function form as  AdX (φ) = Xφ∧X −1  (3)  or in the matrix form as  (AdX φ)∧ = Xφ∧X −1  (4)  The adjoint map is the derivative of the adjoint action with  respect to X at the identity element and is defined as  adφ η = [φ∧, η∧]  (5)  where [φ∧, η∧] is the Lie bracket, calculated as  [φ∧, η∧] = φ∧η∧ − η∧φ∧  (6)  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' PROBLEM DEFINITION  We consider the systems whose states reside in the tangent  bundle of a matrix Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' This encompasses a diverse  array of systems [15] whose states can be represented as pairs  {X, ξ∧} ∈ G × g, where X represents the configuration and  ξ∧ represents the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' The continuous equations of motion  for such systems can be written as:  ˙Xt = Xtξ∧  t  ˙ξt = f  �  ξt, ut  �  (7)  where ut ∈ Rm is the generalized control input and f(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=') is  the function of velocity dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' For a given initial state  {X0, ξ0}, a goal state {Xg, ξg} and a time horizon N, we  define the discrete-time constrained optimal control problem  as  min  u0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=',uN−1  ℓf(XN, ξN) +  N−1  �  k=0  ℓ(Xk, ξk, uk)  subject to  Xk+1 = FX (Xk, ξk),  k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=', N − 1  ξk+1 = Fξ(ξk, uk),  k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=', N − 1  umin ≤ uk ≤ umax,  ∀k,  g(Xk, ξk, uk) ≤ 0,  ∀k,  given  X0, ξ0,  (8)  where ℓf : G × Rn �→ R and ℓ : G × Rn × Rm �→ R  are the final cost and the running cost, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' FX and  Fξ are the discretized form of the configuration and velocity  dynamics, which can be obtained by using either a zero-  order hold or Euler first-order integration method with a fixed  time step of ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Lastly, g is a vector of p constraints in  the form of differentiable nonlinear functions representing the  state constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' PROPOSED METHOD  It is often difficult to solve the general problem outlined  in Section III analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Additionally, finding the global  minimum numerically can be time-consuming, particularly for  systems with high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Therefore, we aim to propose a  method that provides feasible solutions, even if they may not  be globally optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' As such, we aim to propose a method  that yields feasible solutions, even if they may not be globally  optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' To accomplish this, we utilize the Differential Dy-  namic Programming (DDP) framework [20], which iteratively  solves sub-optimization problems in the backward pass and  generates a new trajectory in the forward pass based on the  found optimal policy, in order to approach a local optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  In this work, we propose augmenting the cost function with  multiplier and penalty terms from the augmented Lagrangian  in order to account for constraints imposed on the system,  whose states lie in the tangent bundle of a matrix Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  Our approach involves lifting the problem to the Lie algebra  in the backward pass by computing the gradient of the cost  function within the corresponding Lie algebra, and retracting  back to the manifold in the forward pass by integrating the  dynamics using the optimal policy obtained in the backward  pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Dynamics on Tangent Space  The central concept of DDP is that, at each iteration, all  nonlinear constraints and objectives are approximated using  first or second order Taylor series expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' This allows the  approximate functions, which now operate on deviations from  the nominal trajectory, to be solved using discrete Linear-  Quadratic Regulator (LQR) techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' In order to define  the cost and constraint functions in Lie algebra, we need  to determine the error dynamics for the configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' To  obtain the perturbed state dynamics, we followed the approach  proposed by Teng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' For completeness, we outline the  necessary steps here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Interested readers may refer to [17], [18]  for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  Consider a perturbed state {Xp, ξ∧  p } that is in the vicinity  of a nominal state {X, ξ∧}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Then, the configuration error can  be defined as  Ψ = X −1Xp ∈ G  (9)  Differentiating both sides of (9) yields the configuration error  dynamics as  ˙Ψ = X −1 d  dt  �  Xp  �  + d  dt  �  X −1�  Xp  = X −1 ˙Xp − X −1 ˙XX −1Xp  = X −1Xpξ∧  p − X −1Xξ∧X −1Xp  = Ψξ∧  p − ξ∧Ψ  (10)  Here, we can define a vector ψ in Rn such that the matrix ex-  ponential of ψ∧ corresponds to Ψ, denoted as Ψ = expm(ψ∧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  Using the first-order approximation of the matrix exponential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  which states that expm(ψ∧) ≈ In + ψ∧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' the dynamics of the  configuration error in (10) can be linearized as follows:  ˙Ψ = Ψξ∧  p − ξ∧Ψ  = (In + ψ∧)ξ∧  p − ξ∧(In + ψ∧)  = ξ∧  p + ψ∧ξ∧  p − ξ∧ − ξ∧ψ∧  = ξ∧  p − ξ∧ + ψ∧ξ∧  p − ξ∧ψ∧  = ξ∧  p − ξ∧ + ψ∧(ξ∧ − ξ∧ + ξ∧  p ) − ξ∧ψ∧  = ξ∧  p − ξ∧ + ψ∧ξ∧ − ξ∧ψ∧ + ψ∧(ξ∧  p − ξ∧)  = ξ∧  p − ξ∧ + ψ∧ξ∧ − ξ∧ψ∧  = ξ∧  p − ξ∧ + adψ ξ  ˙ψ = ξ∧  p − ξ∧ − adξ ψ  (11)  Note that the second order term of ψ∧(ξ∧  p − ξ∧) is also  discarded to obtain the linear dynamics of the configuration  error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' ψ in (11) is the perturbed configuration represented in  Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' The perturbed velocity and control input are also  defined as  δξ = ξp − ξ,  and  δu = up − u  (12)  The perturbed velocity dynamics then become:  δ ˙ξt = Γtδξt + Λtδut  (13)  where Γt and Λt are the Jacobians of f(ξt, ut) defined in (7)  around the nominal trajectory about ξt and ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  Defining the perturbed states as concatenation  x =  �ψ  δξ  �  ,  ¯u = δu  (14)  the perturbed state dynamics are expressed as  ˙x = h(x, u)  ˙x =  �− adξ  In  0n×n  Γt  �  �  ��  �  ≜At  x +  �0n×m  Λt  �  �  ��  �  ≜Bt  ¯u  (15)  The discretized versions of the matrices At and Bt can be  simply obtained by applying a zero-order hold or a first-order  Euler integration method with a fixed time step of ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Constraint Handling  By decomposing the state into configuration and velocity,  we can also decompose the constraints in vector g(Xk, ξk, uk)  in equation (8) into two types: those that constrain the velocity  and those that specify configurations to be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' This allows  us to separately handle the constraints on velocity (cξ) and on  configuration (cX ) as  cξ(ξk, uk) ≤ 0  cX (Xk, ξk) ≤ 0  (16)  The velocity component of the state (ξ) resides in Rn, and  as a result, constraints involving any metric in Euler space  produce a distance vector in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Therefore, any boundary  velocity constraint can be written as  δξb = ξb − ξ,  ¯cξ(βδξb) ≤ 0  (17)  where  β =  �  −1  if ξb is upper bound,  +1  if ξb is lower bound  (18)  On the other hand, the difference between two group  elements in the configuration state produces a geodesic in  the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' To handle this, we propose mapping the distance  geodesic to the tangent space of the configuration at the current  time step and addressing the constraint in that vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  Configuration avoidance constraints can typically be formu-  lated as inequality constraints using an n-spherical function,  with the center of the n-sphere located at the configuration  to be avoided (Xc) and the radius (rc) defining the restricted  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' This allows us to specify a region of configurations  that should be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' The distance between the nominal and  restricted configurations in the tangent space of the nominal  trajectory, ψc, is given by:  ψc = logm(X −1Xc)  (19)  Then, the configuration avoidance constraint can be written as  ¯cX (ψc) = (r2  c − ∥ψc∥2) ≤ 0  (20)  In this approach, we consider the same restricted region  for each axis in the n-dimensional sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' However, it is  also possible to specify different radius values for each axis,  resulting in an n-dimensional ellipsoid as the restricted region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  Our method can accommodate these types of configuration  constraints as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  In order to handle the constraints in DDP framework,  we need the first-order approximations of them around the  perturbed state dynamics introduced in (15) as follows:  ¯c(x + δx, u + δu) ≈ ¯c(x, u)  + ¯cx(x, u)δx + ¯cu(x, u)δu  (21)  where  ¯c(x) =  �¯cX (ψc)  ¯cξ(δξb)  �  (22)  ¯cx and ¯cu are the derivative of ¯c with respect to x and  u, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' If the constraints are designed according to  equations (18) and (20), the derivatives can be calculated as  follows:  ¯cx(x, u) =  �−2(adξ ψc)⊤  2ψc⊤  01×n  β(Γδξb)⊤  �  ¯cu(x, u) =  �  01×m  β(Λ⊤δξb)⊤  �  (23)  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Constrained Differential Dynamic Programming  Using the perturbed state dynamics defined in (15), the  backward pass of differential dynamic programming (DDP) is  lifted to the tangent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' The backward pass of DDP involves  computing the cost-to-go function at each time step in a given  trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Unlike [18], our algorithm not only considers the  objective function when calculating the cost-to-go function,  but also takes into account any constraints on the state and  control variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  An effective method for solving constrained optimization  problems is to transform the constraints into the objective  function and iteratively increase the penalty for violating or  approaching them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' This technique, known as penalty method,  guarantees convergence to the optimal solution as the penalty  terms increase indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' However, this may not be prac-  tical to implement in numerical optimization routines due  to the limitations of finite precision arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Augmented  Lagrangian methods [19] offer an alternative solution by  maintaining estimates of the Lagrange multipliers associated  with the constraints, allowing for convergence to the optimal  solution without requiring the penalty terms to increase indef-  initely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  Here we obtain the augmented Lagrangian as  LA = LN(xN) +  N−1  �  k=0  Lk(xk, uk)  LN(xN) = ¯ℓf(xN) + (λ + 1  2 ¯g(xN)Iµ)⊤¯g(xN)  Lk(xk, uk) = ¯ℓ(xk, uk) + (λ + 1  2 ¯g(xk, uk)Iµ)⊤¯g(xk, uk)  (24)  where ¯ℓf : R2n �→ R and ¯ℓ : R2n ×Rm �→ R are the final cost  and the running cost functions for perturbed system dynamics,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' A typical design of such functions are  ¯ℓf(x) = 1  2∥δx∥SV  ¯ℓ(x, u) = 1  2∥δx∥SQ + 1  2∥δu∥SU  (25)  where SV ∈ R2n×2n, SQ ∈ R2n×2n and SU ∈ Rm×m are  the cost matrices that are specified by the user and remain  constant throughout all iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  In (24), ¯g is a vector of p constraints for perturbed state  dynamics as introduced in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' λ ∈ Rp is a Lagrange  multiplier, µ ∈ Rp is a penalty weight and Iµ ∈ Rp×p is  the penalty matrix defined as  Iµ =  �  0  if gi(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=') < 0 and λi = 0,  µi  otherwise  (26)  In general, one can define time varying Lagrange multiplier  and penalty weight, but we kept them constant for each time  step during the same iteration for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  We define the cost-to-go and action-value functions as  Vk(xk) = min  uk {Lk(xk, uk)} + Vk+1(Akxk + Bkuk)  = min  uk Q(xk, uk))  (27)  The matrices Ak and Bk represent the discretized versions  of At and Bt in equation (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Second order Taylor series  expansion of cost-to-go function can be written as  δVk(x) ≈ 1  2δx⊤  k Vxx,kδxk + V ⊤  x,kδxk  (28)  where Vxx,k and Vx,k are the Hessian and gradient of the  cost-to-go at time step k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Action-value function  defined in (27) can be also approximated as a quadratic  function as  Q(x + δx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' u + δu) ≈ Q(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' u) + Q⊤  x δx + Q⊤  uδu  + 1  2(δx⊤Qxxδx + δx⊤Qxxδx)  + δx⊤Qxuδu  (29)  To compute the derivative matrices in (29):  Qx = ¯ℓx + A⊤V ′  x + ¯g⊤  x (λ + Iµ¯g)  Qu = ¯ℓu + B⊤V ′  x + ¯g⊤  u (λ + Iµ¯g)  Qxx = ¯ℓxx + A⊤V ′  xxA + ¯g⊤  x Iµ¯gx + (V ′  xhxx)  Quu = ¯ℓuu + B⊤V ′  xxB + ¯g⊤  u Iµ¯gu + (V ′  xhuu)  Qux = ¯ℓux + B⊤V ′  xxA + ¯g⊤  u Iµ¯gx + (V ′  xhux)  (30)  To simplify the notation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' we have omitted the time indices  on all variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' All variables in this expression are evaluated  at time step k, except for those marked with ′, which are  evaluated at time step k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  Calculating the full second-order expansion of the state  dynamics (hxx, huu, hux), can be computationally expensive,  particularly for systems with complex dynamics and high-  dimensional states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' DDP refers to iterative LQR by discarding  the second-order dynamics and computing only the first-order  expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' This results in a Gauss-Newton approximation of  the true Hessian, which reduces the local fidelity and requires  more iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' However, these iterations are less expensive  to compute and often lead to a faster overall convergence  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Therefore, in this work, we eliminated the second order  dynamics as approximating the perturbed state dynamics as  described in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  Minimizing (29) with respect to δu results in an affine  controller  δu∗ = −Q−1  uu(Quxδx + Qu) ≜ Kδx + d  (31)  Substituting δu∗ into (29) yields the derivatives of the cost-  to-go at time step k in terms of the derivatives of the action  value function as:  Vx = Qx + KQu + K⊤Quud + Q⊤  uxd,  Vxx = Qxx + K⊤QuuK + K⊤Qux + Q⊤  uxK  (32)  At the final step, Vx and Vxx can be easily computed as the  first and second derivatives of the final cost function (¯ℓf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' This  way, the derivatives of the action-value function (30) and in  turn the local optimal control policy (31) at each step can be  calculated backwards starting from the final step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  After determining the optimal control policy for each time  step, we update the nominal trajectories by simulating the  dynamics forward on the manifold itself starting from the  initial state as:  δxk =  �logm(Xk−1 ¯  Xk)  ¯ξk − ξk  �  δuk = Kkδxk + αdk  ¯uk = uk + δuk  ¯ξk = ξ + f(ξ, ¯uk)∆t  ¯  Xk = X expm(¯ξk∆t)  (33)  where {Xk, ξk, uk} and { ¯  Xk, ¯ξk, ¯uk} represent the nominal  state-actions and the updated state-actions at time step k,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' In the above expression, 0 ≤ α ≤ 1 is a  scaling term for simple linear search on the feedforward term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  Practically, the parameter α is initially set to 1, but if the  cost of the updated trajectory does not decrease, it will be  decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Once the cost of the updated trajectory is decreased,  the forward pass is succesfully completed, the new trajectory  is accepted as nominal trajectory and the backward pass is  triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  To optimize the performance of DDP-based algorithms,  there are a few implementation practices to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' In the  backward pass, Quu may need to be regularized as Quu +ρI  if it is invertible or the former forward pass is unsuccesful, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=',  the cost is not decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' After the DDP iterations converge,  the parameters λ and µ can be updated as:  λ+ = max(0, λi + µi¯gi(x∗, u∗))  µ+ = γµ,  γ > 0  (34)  The DDP iterations can then be restarted until convergence is  achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' For more information, see reference [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' EXPERIMENTS  The goal of this section is to devise a trajectory on SE(3)  that satisfies both position and orientation constraints while  avoiding unsafe configurations and obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Dynamics on SE(3)  We consider a 3D rigid body in SE(3) where the states of  the system can be represented by a rotation matrix  R ∈ SO(3) = {R ∈ R3×3|R⊤R = I3, det(R) = 1}  (35)  and position p ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' The homogeneous representation of a  typical group element in SE(3) is  X =  �R  p  0  1  �  ∈ SE(3)  (36)  The velocity vector ξ in SE(3) is known as a “twist” and is  composed of both angular (ω) and linear (v) velocities in body  frame as  ξ =  �  ω  v  �  ∈ R6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  ξ∧ =  �  ω∧  v  0  0  �  ∈ se(3)  (37)  The forced Euler-Poincar´e equations [16] define the twist  dynamics as  Jb ˙ξ = ad∗  ξ Jbξ + u  (38)  In this expression,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Jb represents the generalized inertia matrix  in the body fixed principal axes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' while u ∈ g∗ represents  the generalized control input forces applied to these axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  g∗ denotes the cotangent space and the coadjoint map is  represented by ad∗  ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' The matrix representations of the adjoint  action in the Lie algebra (adξ) and the coadjoint map (ad∗  ξ)  are as follows:  adξ =  �ω∧  0  v∧  ω∧  �  ,  ad∗  ξ = ad⊤  ξ = −  �ω∧  v∧  0  ω∧  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  (39)  The continuous equations of motion described in (7) can be  written for SE(3) group as  ˙X = Xξ∧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  ˙ξ = Jb  −1�  ad∗  ξ Jbξ + u  �  (40)  The linearized twist dynamics as given in [17]:  ˙ξ = Γtξ + Λtu + bt  (41)  where Γt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Λt and bt are:  Γt := J−1  b  ad∗  ξ Jb + J−1  b  �(Ibω)∧  mv∧  mv∧  0  �  Λt := Jb  −1  bt := −Jb  −1  �(Ibω)∧  mv∧  mv∧  0  �  ξ  (42)  assuming the inertia matrix Jb is defined as  Jb :=  �Ib  0  0  mI3  �  (43)  where Ib and m are the moment of inertia in the body frame  and the body mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Constrained (red) and unconstrained (blue) trajectories generated by proposed DDP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  Control input signals of the planned trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Torques refers  to inputs acting on angular velocity and Forces refers to inputs acting on  linear velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Left column: unconstrained, Right column: constrained  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Simulations  We now test the proposed algorithm for planning a safe path  for a SE(3) rigid body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' The task we have defined involves  rotating the body to the configuration Rz(180) from the  identity and translating it to the position {2, 2, 2} from the  initial position {1, 1, 1} in 3 seconds with a fixed time step  of ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' To denote the rotation around the x, y, and z  axes of the body-fixed frame, we use Rx(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' ), Ry(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' ), and Rz(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' ),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' During the motion, the configuration Rz(90) is  considered to be unsafe and must be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Additionally, it is  assumed that there is a spherical obstacle at {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='25}  with a radius of r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='5 that must be avoided as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  We only penalized the final state and control inputs, so we  set the controller parameters as SV = 1000I12, SU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='001I6,  and SQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' In order to assess the performance of the  proposed DDP in terms of constraint handling, we conducted  an experiment with the same trajectory optimization task in  both constrained and unconstrained cases, and the resulting  trajectories are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  The  configuration  trajectories  were  converted  to  Eu-  ler angles (φ, θ, ψ in degrees) using the rotation order  Rx(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=')Ry(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=')Rz(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' In the unconstrained case (shown in blue), it  is shown that the ψ angle goes directly from 0 to 180 degrees  while the φ and θ angles remain constant (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' 1), as the  task only involves rotating around the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' This can also be  observed in the upper left part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' 2, where only a single  input is non-zero to achieve the desired motion, while the  others remain zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' However, in the constrained case (shown  in red), rotations around x and y axes (changes in φ and θ  angles) were also observed to avoid the unsafe configuration  of Rx(90) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  As shown in the last row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' 1, the shortest paths for  the positional states were obtained in the unconstrained case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  However, the presence of a sphere at {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='25} with  a radius of r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='5 forced the positional trajectories to take  a circuitous route around obstacle by deviating along y and x  axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  To evaluate the effectiveness of the proposed DDP method  in handling external disturbances, we extend the analysis done  in [13] to SE(3) and employ the proposed DDP method as a  Lie-algebraic feedback control:  ufb  k := u∗  k + Kk logm((X ∗  k )−1X ϵ  k)  (44)  where u∗  k and Kk represent the (sub)optimal control sequence  and time-varying feedback gains, respectively, which are ob-  tained through DDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' The feed-forward term dk in (31) is  not explicitly shown in the feedback policy (44) because it  is already included in u∗  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' The variables X ∗  k and X ϵ  k represent  the (sub)optimal states obtained through DDP convergence and  the perturbed states due to disturbance, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  We test the policy (44) on the following stochastic version  of the SE(3) dynamics:  X ϵ  k+1 = X ϵ  k expmSE(3)(ξϵ  k∆t)  ξϵ  k+1 = ξϵ  k + f(ξϵ  k, uk)∆t + σωω  (45)  where ω is assumed to be spatially uncorrelated independent  and identically distributed noise, drawn from a zero mean  Gaussian distribution, ω ∼ N(06×1, I6), σω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' The  performance of the proposed DDP method under stochas-  tic conditions was evaluated by allowing the optimizer to  converge on the deterministic system and then testing its  performance on the stochastic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' 3 shows the results  of this comparison, using 1000 sampled trajectories under  noisy dynamics to compare the open-loop policy u∗  k with the  feedback policy ufb  k  (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' The results demonstrate that the  use of the obtained feedback gains significantly reduces state  variance, particularly in the vicinity of the the goal points, as  expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  50  0  [6ap]  [deg]  [deg]  100  0  50  0  50  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='5  × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='5  y  N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='0  0  1  2  3  0  1  2  3  0  1  2  3  Time [sec]  Time [sec]  Time [sec]20  Torques  25  0  0  25  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='5  Forces  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='0  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='5  0  2  0  2  Time [sec]  Time [sec]Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Evaluation of DDP’s control sequence under random disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Left Column: Open-loop control, Right column: Closed-loop control  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' CONCLUSION  In conclusion, the optimization of control policies on matrix  Lie groups is an important problem in robotics and control  with numerous applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' In this work, we have proposed a  novel approach for tackling this problem using an augmented  Lagrangian based constrained discrete DDP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Our  method involves lifting the optimization problem to the Lie  algebra in the backward pass, allowing us to compute the  gradient of the objective and constraint functions within the  corresponding tangent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' In the forward pass, we retract  back to the manifold by integrating the dynamics using the  optimal policy obtained in the backward pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  One of the key contributions of our work is the development  of a general approach for nonlinear constraint handling for  a wide range of matrix Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Previous methods were  only able to handle constraints for specific classes of matrix  Lie groups, such as SO(3) pose constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' Our method, on  the other hand, is able to handle a wide range of nonlinear  constraints, making it a more widely applicable and flexible  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  In addition, we have demonstrated the effectiveness of our  method in handling external disturbances through its applica-  tion as a Lie-algebraic feedback control policy on SE(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' The  results show that our approach is able to effectively handle  constraints and maintain its stability in the presence of external  disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  Overall, our proposed DDP algorithm represents a signif-  icant step forward in the field of optimization on matrix  Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' It provides a flexible and general approach for  nonlinear constraint handling and has the potential to be  applied to a wide range of problems in robotics and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  As a future direction, it would be interesting to investigate  the closed loop uncertainty propagation through the prediction  horizon, as has been done for Euclidean models [11], in order  to design a robust Model Predictive Control (MPC) framework  for matrix Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content=' This would allow the proposed method  to be used in more challenging and dynamic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
+page_content='  Additionally, further experimentation with different types of  matrix groups and constraints would provide valuable insights  into the capabilities and limitations of the proposed DDP  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNA0T4oBgHgl3EQfEf9Z/content/2301.02018v1.pdf'}
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UdAyT4oBgHgl3EQfuvkh/content/tmp_files/2301.00617v1.pdf.txt

@@ -0,0 +1,1834 @@
+arXiv:2301.00617v1  [math.FA]  2 Jan 2023
+SOME REMARKS ON CONVEX BODY DOMINATION
+TUOMAS P. HYTÖNEN
+Dedicated, with admiration, to the Ukrainian people.
+Abstract. Convex body domination is an important elaboration of the tech-
+nique of sparse domination that has seen significant development and applica-
+tions over the past ten years. In this paper, we present an abstract framework
+for convex body domination, which also applies to Banach space -valued func-
+tions, and yields matrix-weighted norm inequalities in this setting. We explore
+applications to “generalised commutators”, obtaining new examples of bounded
+operators among linear combinations of compositions of the form aiTbi, where
+ai, bi are pointwise multipliers and T is a singular integral operator.
+1. Introduction
+The technique of sparse domination was developed to provide a simpler approach,
+achieved by Lerner [23], to the “A2 conjecture” on sharp weighted norm inequalities
+for Calderón–Zygmund operators, which was first proved with a different machinery
+by the author [16]. However, beyond this original aim, sparse domination imme-
+diately led to significant further consequences and has by now been applied to a
+variety of new questions, of which [1, 2, 3, 7, 11] is only a sample. The method
+consists of two main steps that are largely independent of each other and essentially
+decouple the operator from the space or norm in which it should be estimated:
+(1) Dominating an operator of interest by a suitable sparse operator/form.
+(2) Estimating the sparse form with respect to relevant norms of interest.
+While sparse domination very efficiently captures the local size of an object un-
+der consideration, and this is precisely what is needed in many applications, it
+loses information about directions, which is sometimes relevant when dealing with
+vector-valued functions, and especially so, matrix-valued weights are involved. To
+extend the method to such questions, Nazarov et al. [27] developed the so-called
+convex body domination, where the numerical averages featuring in sparse dom-
+ination are replaced by convex subsets of Rn, thus containing information about
+different behaviour in different directions. Since its introduction in the context of
+Calderón–Zygmund operators and matrix A2 weights by [27] (see also [10] for an-
+other approach but based on the same key idea), convex body domination has been
+applied to matrix Ap-weight and two-weight bounds by Cruz-Uribe et al. [8], and ex-
+tended to commutators of Calderón–Zygmund operators by Isralowitz et al. [20, 21]
+and rough singular integral operators by Di Plinio et al. [13] and Muller and Rivera-
+Ríos [26]. In a recent breakthrough, Bownik and Cruz-Uribe [6] extended the Rubio
+Date: January 3, 2023.
+2010 Mathematics Subject Classification. 42B20, 46E40.
+The author is supported the Academy of Finland via the Finnish Centre of Excellence in
+Randomness and Structures “FiRST” (grant no. 346314).
+1
+
+2
+T. P. HYTÖNEN
+de Francia algorithm, and its key application to weighted extrapolation, to matrix-
+valued weights, by further development of the convex body philosophy.
+The aim of this paper is to further explore this technique, providing extensions,
+new applications and—hopefully—some additional insight into the abstract under-
+lying mechanisms. We begin by developing a somewhat general framework, but
+our claims for originality in this regard are relatively mild, as most of the ideas
+are at least implicit in the previous works in the existing literature.
+A certain
+justification for this framework comes from the observation that it applies almost
+verbatim to the case of Banach space -valued functions. To be precise, given a Ba-
+nach space E, we consider functions taking values in En, and develop a version of
+convex body domination applicable to weighted norm inequalities involving matrix
+weights W : Rd → Rn×n, acting on En in the natural way. That is, we make no
+attempt towards a fully operator-valued theory of weighted norm inequalities in
+infinite dimensions, yet the results that we obtain are still new even in this more
+modest generality. In particular, if E is a Banach space with the UMD property,
+the classical Hilbert transform extends boundedly to the matrix-weighted space
+L2(W; En) of En-valued functions; see Corollary 6.3 for the result, and Section 6
+for the relevant definitions and background. A key to this extension is the observa-
+tion that the convex bodies arising from our framework are still Rn-valued in this
+generality—and not, for instance, En-valued, as one might have (and this author
+certainly had) initially expected. Thus the powerful Euclidean machinery, most
+notably the John ellipsoid theorem, is still available in this setting.
+As for new applications of the theory, we build on a recent observation from
+Isralowitz et al. [20, 21] that convex body domination of an operator T bootstraps
+to a domination of its commutators [b, T ] = bT − T b with pointwise multipliers. As
+we will explore in Section 7, this phenomenon is far more general, and can be used
+to estimate any operators of the form
+f �→
+n
+�
+i=1
+aiT (bif),
+where an operator T satisfying convex body domination is pre- and post-composed
+with pointwise multipliers ai, bi. From this general principle, we can in particular
+recover and sharpen a recent sufficient condition [17] for the boundedness of iterated
+mixed commutators [b1, [b2, T ]] in terms of joint conditions on the pair of functions
+(b1, b2), but also obtain new examples.
+In contrast to the development of the abstract framework in the first part of the
+paper, we have not strived for the greatest generality in terms of the applications
+in the later sections. In many cases, it will be clear to an experienced reader that
+several variants and extensions could be obtained, and some of them will most likely
+be pursued in forthcoming works, by this author and others. Besides the concrete
+results contained in this paper, our aim is to hint at the many rich directions for
+the further development of the theory.
+2. Norms and convex bodies
+Let X be a real normed space. We denote by
+¯BX := {x ∈ X : ∥x∥X ≤ 1}
+
+SOME REMARKS ON CONVEX BODY DOMINATION
+3
+its closed unit ball, and by X∗ the normed dual, which is a Banach space. For
+x∗ ∈ X∗, we define, as usual,
+∥x∗∥X∗ := sup{|⟨x, x∗⟩| : x ∈ ¯BX}.
+As a consequence of the Hahn–Banach theorem, we have
+∥x∥X = sup{|⟨x, x∗⟩| : x∗ ∈ ¯BX∗} = max{⟨x, x∗⟩ : x∗ ∈ ¯BX∗};
+(2.1)
+in particular, the supremum is reached as a maximum, and we have ∥x∥X = ⟨x, x∗⟩
+for some x∗ ∈ ¯BX∗.
+For ⃗x = (xi)n
+i=1 ∈ Xn and x∗ ∈ X∗, we define the Rn-valued pairing ⟨⃗x, x∗⟩ :=
+(⟨xi, x∗⟩)n
+i=1 ∈ Rn and the set-valued “norm”
+⟨⟨⃗x⟩⟩X := {⟨⃗x, x∗⟩ : x∗ ∈ ¯BX∗} ⊂ Rn.
+2.2. Remark. The notation is adapted from Nazarov et al. [27], who introduced
+the version with X = Ł1(Q), the space L1(Q) with the normalised norm
+1
+|Q|∥ ∥1).
+The extension to X = Łp(Q) (i.e., Lp(Q) with the normalised norm
+1
+|Q|1/p ∥ ∥p)
+is due to Di Plinio et al. [13]. Although our main applications will be concerned
+with spaces of functions (living on a cube Q), we find it illuminating to develop the
+basics of the theory on a completely abstract level. Among other things, this point
+of view will make it clear that there will be essentially no difference in treating a
+space X = Lp(Q; E) of E-valued functions for an arbitrary Banach space E; for
+⃗f ∈ Xn, the corresponding ⟨⟨⃗f⟩⟩X will still be subsets of Rn and not, say, of En.
+This will allow us to make effortless use of the powerful John ellipsoid theorem from
+Euclidean geometry, even when working with functions taking values in an infinite-
+dimensional Banach space! In other applications, a choice like X = L log L(Q)
+might also be relevant.
+For ⃗a ∈ Rn and ⃗x ∈ Xn, we define the X-valued dot product
+⃗a · ⃗x := ⃗x · ⃗a :=
+n
+�
+i=1
+aixi.
+We observe the easy identities
+⃗a · ⟨⃗x, x∗⟩ = ⟨⃗a · ⃗x, x∗⟩,
+∀⃗a ∈ Rn, ⃗x ∈ Xn, x∗ ∈ X∗,
+and
+spanX(⃗x) := span{xi}n
+i=1 = {⃗a · ⃗x : a ∈ Rn} ⊂ X.
+2.3. Lemma. For each ⃗x ∈ Xn, the set ⟨⟨⃗x⟩⟩X ⊂ Rn is convex, compact, and
+symmetric about the origin.
+Proof. Symmetry, convexity and boundedness are immediate from the fact that
+¯BX∗ has these properties. For compactness in Rn, it remains to show closedness,
+so suppose that ⟨⃗x, x∗
+k⟩ → ⃗e ∈ Rn as k → ∞, where each x∗
+k ∈ ¯BX∗; we need to
+show that ⃗e ∈ ⟨⟨x⟩⟩X. For each ⃗a ∈ Rn, it follows that
+|⃗a · ⃗e| = lim
+k→∞ |⃗a · ⟨⃗x, x∗
+k⟩| = lim
+k→∞ |⟨⃗a · ⃗x, x∗
+k⟩| ≤ ∥⃗a · ⃗x∥X.
+This in turn implies that
+Λ(⃗a · ⃗x) := ⃗a · ⃗e,
+∀⃗a · ⃗x ∈ spanX(⃗x),
+
+4
+T. P. HYTÖNEN
+gives a well-defined linear functional of norm 1 on the subspace spanX(⃗x) ⊂ X. By
+the Hahn–Banach theorem, Λ is the restriction of some x∗ ∈ ¯BX∗. Hence
+⃗a · ⃗e = Λ(⃗a · ⃗x) = ⟨⃗a · ⃗x, x∗⟩ = ⃗a · ⟨⃗x, x∗⟩
+∀⃗a ∈ Rn,
+and thus limk→∞⟨⃗x, xk⟩ = ⃗e = ⟨⃗x, x∗⟩ ∈ ⟨⟨⃗x⟩⟩X, as we wanted to show.
+□
+For A, B ⊂ Rn, we define the Minkowski dot product
+A · B := {⃗a ·⃗b : ⃗a ∈ A,⃗b ∈ B} ⊂ R.
+If A, B ⊂ Rn are convex, compact and symmetric, so is A · B ⊂ R. On R, such
+sets are precisely intervals of the form [−c, c]. Hence we can, and sometimes will,
+identify A · B = [−c, c] ⊂ R with its right end-point c ∈ [0, ∞). In particular, for
+⃗x ∈ Xn and ⃗y ∈ Y n, we will use this identification when dealing with
+⟨⟨⃗x⟩⟩X · ⟨⟨⃗y⟩⟩Y = {⟨⃗x, x∗⟩ · ⟨⃗y, y∗⟩ : x∗ ∈ ¯BX∗, y∗ ∈ ¯BY ∗}
+=
+�
+n
+�
+i=1
+⟨xi, x∗⟩⟨yi, y∗⟩ : x∗ ∈ ¯BX∗, y∗ ∈ ¯BY ∗
+�
+.
+3. Bi-linear forms
+Let X, Y be real normed spaces, and suppose that we have a bilinear from
+t : X × Y → R. We define its extension acting on pairs of vectors (⃗x, ⃗y) ∈ Xn × Y n
+as follows. If ⃗e ∈ Rn and x ∈ Xn, we have ⃗x · ⃗e ∈ F by our previous convention
+about the X-valued dot product. If (ei)n
+i=1 is a fixed orthonormal basis of Rn, we
+then define
+t(⃗f,⃗g) :=
+n
+�
+i=1
+t(⃗f · ⃗ei,⃗g · ⃗ei),
+For x ∈ X, y ∈ Y , and ⃗e, ⃗u ∈ Rn, it follows that
+t(x⃗e, y⃗u) :=
+n
+�
+i=1
+t(x⃗e · ⃗ei, y⃗u · ⃗ei) = t(x, y)
+n
+�
+i=1
+(⃗e · ⃗ei)(⃗u · ⃗ei) = t(x, y)⃗e · ⃗u.
+If (⃗ui)n
+i=1 is another orthonormal basis, then
+n
+�
+i=1
+t(⃗x · ⃗ui, ⃗y · ⃗ui) =
+n
+�
+i,j,k=1
+t(⃗x · ⃗ej, ⃗y · ⃗ek)(⃗ej · ⃗ui)(⃗ek · ⃗ui)
+=
+n
+�
+j,k=1
+t(⃗x · ⃗ej, ⃗y · ⃗ek)(⃗ej · ⃗ek) =
+n
+�
+j=1
+t(⃗x · ⃗ej, ⃗y · ⃗ej) =: t(⃗x, ⃗y),
+so the definition of t(⃗f,⃗g) is independent of the chosen orthonormal basis.
+If A ∈ Rn×n is a linear transformation of Rn, acting in a natural way on F n,
+then
+t(A⃗x, ⃗y) =
+n
+�
+i=1
+t(A⃗f · ⃗ei,⃗g · ⃗ei) =
+n
+�
+i=1
+t(⃗f · At⃗ei,⃗g · ⃗ei)
+=
+n
+�
+i,j=1
+t(⃗f · ⃗ej,⃗g · ⃗ei)(⃗ej · At⃗ei)
+=
+n
+�
+i,j=1
+t(⃗f · ⃗ej,⃗g · ⃗ei)(A⃗ej · ⃗ei) =
+n
+�
+j=1
+t(⃗f · ⃗ej,⃗g · A⃗ej) = t(⃗f, At⃗g).
+
+SOME REMARKS ON CONVEX BODY DOMINATION
+5
+4. From norm bounds to convex body bounds
+The idea of the following lemma lies behind many of the existing convex body
+domination results. To isolate the key point, we state it here in an operator-free
+version, involving functions and therir norms only.
+4.1. Lemma. Let X, Y be normed spaces, and ⃗f ∈ Xn,⃗g ∈ Y n.
+Let EK be the John ellipsoid of K := ⟨⟨⃗f⟩⟩X such that
+EK ⊂ K ⊂ √nEK,
+and suppose that EK is non-degenerate (i.e., of full dimension). Let RK be a linear
+transformation such that RKEK = ¯BRn, the closed unit ball of Rn, and let (⃗ei)n
+i=1
+be an orthonormal basis of Rn. If
+fi := RK ⃗f · ⃗ei,
+gi := R−t
+K ⃗g · ei,
+i = 1, . . . , n,
+then
+n
+�
+i=1
+∥fi∥X∥gi∥Y ≤ n3/2⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y .
+Proof. If φ ∈ ¯BX∗, then
+⟨φ, RK ⃗f · ⃗ei⟩ = RK⟨φ, ⃗f⟩ · ⃗ei
+∈ RK⟨⟨⃗f⟩⟩X · ⃗ei ⊂ RK
+√nEK · ⃗ei = √n ¯BRn · ⃗ei = √n[−1, 1],
+and hence
+∥fi∥X = ∥RK ⃗f · ⃗ei∥X ≤ √n.
+If ψ ∈ ¯BY ∗, then
+⟨ψ, R−t
+K ⃗g · ⃗ei⟩ = R−t
+K ⟨ψ,⃗g⟩ · ⃗ei
+∈ R−t
+K ⟨⟨⃗g⟩⟩Y · ⃗ei ⊂ [−M, M],
+M := max{|⃗y| : ⃗y ∈ R−t
+K ⟨⟨⃗g⟩⟩Y }.
+It follows that |⟨ψ, R−t
+K ⃗g · ⃗ei⟩| ≤ M, and hence
+∥gi∥Y = ∥R−t
+K ⃗g · ⃗ei∥Y ≤ M.
+Combining the estimates, we have
+n
+�
+i=1
+∥fi∥X∥gi∥Y ≤
+n
+�
+i=1
+√nM = n3/2M.
+On the other hand,
+⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y ⊃ EK · ⟨⟨⃗g⟩⟩Y = RKEK · R−t
+K ⟨⟨⃗g⟩⟩Y = ¯BRn · R−t
+K ⟨⟨⃗g⟩⟩Y = [−M, M],
+and hence
+M ≤ ⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y ,
+using the identification of the symmetric interval ⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y with its right end-
+point in the last step. Substituting back, this completes the proof.
+□
+The following proposition contains the basic idea of bootstrapping norm bounds
+to convex body bounds.
+Unfortunately, it is a bit too simple for most actual
+applications, but we include it as an illustrative toy model for the more serious
+result to be presented after it.
+4.2. Proposition. Let X, Y be normed spaces with subspaces F ⊂ X and G ⊂ Y ,
+and let t : F × G → R be a bilinear form. Consider the following conditions:
+
+6
+T. P. HYTÖNEN
+(1) For all (f, g) ∈ F × G, we have
+|t(f, g)| ≤ C∥f∥X∥g∥Y .
+(2) For all (⃗f,⃗g) ∈ F n × Gn, we have
+|t(⃗f,⃗g)| ≤ Cn⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y .
+For each n ∈ Z+, condition (1) implies condition (2) with Cn = Cn3/2.
+Proof. Given ⃗f, consider the compact, convex, symmetric set
+K := ⟨⟨⃗f⟩⟩X,
+and denote by EK its John ellipsoid such that
+EK ⊂ K ⊂ √nEK.
+Case: EK is non-degenerate. Let RK be a linear transformation such that RKEK =
+¯BRn, the closed unit ball of Rn. Let (⃗ei)n
+i=1 be some orthonormal basis of Rn. We
+then write
+t(⃗f,⃗g) = t(R−1
+K RK ⃗f,⃗g) = t(RK ⃗f, R−t
+K ⃗g)
+=
+n
+�
+i=1
+t(RK ⃗f · ⃗ei, R−t
+K ⃗g · ⃗ei) =:
+n
+�
+i=1
+t(fi, gi),
+(4.3)
+where fi and gi are as in Lemma 4.1.
+By assumption (1) and Lemma 4.1, it follows that
+|t(⃗f,⃗g)| ≤
+n
+�
+i=1
+|t(fi, gi)| ≤
+n
+�
+i=1
+C∥fi∥X∥gi∥Y ≤ Cn3/2⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y,
+and this completes the proof in the case that EK is non-degenerate.
+Case: EK is degenerate. Suppose then that EK is degenerate; hence H := span K
+is a strict subspace of Rn. Let P denote the orthogonal projection of Rn onto H.
+For each x∗ ∈ ¯BX∗, we have ⟨⃗f, x∗⟩ ∈ K ⊂ H, hence
+⟨⃗f, x∗⟩ = P⟨⃗f, x∗�� = ⟨P ⃗f, x∗⟩,
+and thus ⃗f = P ⃗f. It follows that
+t(⃗f,⃗g) = t(P ⃗f,⃗g) = t(⃗f, P t⃗g) = t(⃗f, P⃗g),
+(4.4)
+and similarly
+⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y = P⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y = ⟨⟨⃗f⟩⟩X · P t⟨⟨⃗g⟩⟩Y = ⟨⟨⃗f⟩⟩X · ⟨⟨P⃗g⟩⟩Y .
+(4.5)
+So it is enough to prove the claim with P⃗g in place of ⃗g, and hence we may assume
+without loss of generality that also ⃗g = P⃗g. But then we can repeat the argument
+in the non-degenerate case, but with Rn replaced by its subspace H throughout;
+within this subspace, EK ⊂ H is non-degenerate, and the previous case applies to
+give the desired result.
+□
+In the following proposition, condition (1) is a typical intermediate step that
+is established in the course of proving a sparse domination result for an operator,
+while condition (2) is its convex body analogue.
+The proposition says that (1)
+in fact implies (2). It is essentially an abstraction (from L1 averages to general
+dominating norms) of an idea already present in [27, Lemma 3.2].
+
+SOME REMARKS ON CONVEX BODY DOMINATION
+7
+4.6. Proposition. Let X, Y be normed spaces with subspaces F ⊂ X and G ⊂ Y .
+Let Q0 ∈ D, and suppose that there are bilinear forms tQ : F × G → R indexed by
+all Q ∈ D(Q0). Consider the following conditions:
+(1) For all (f, g) ∈ F × G, there exist disjoint ˆQk ⊂ Q0 with �
+k | ˆQk| ≤ ε|Q0|
+and such that: whenever Qj ⊂ Q0 are disjoint, not strictly contained in
+any ˆQk, and cover all ˆQk, then
+|tQ0(f, g) −
+�
+j
+tQj(f, g)| ≤ C∥f∥X∥g∥Y .
+(2) For all (⃗f,⃗g) ∈ F n × Gn, there exist disjoint Qk ⊂ Q0 with �
+k |Qk| ≤
+εn|Q0| and such that
+|tQ0(⃗f,⃗g) −
+�
+k
+tQk(⃗f,⃗g)| ≤ Cn⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y .
+For each n ∈ Z+, condition (1) implies condition (2) with εn = nε and Cn = Cn3/2.
+Of course, the condition �
+k |Qk| ≤ εn|Q0| is only useful for εn < 1. For a fixed
+ε and εn = nε, this would only allow us to conclude (2) for boundedly many values
+of n; so in order to obtain (2) for all n ∈ N, we need (1) for arbitrarily small ε > 0.
+This is seldom a problem in concrete situations.
+Proof. As in the proof of Proposition 4.2, given ⃗f, we consider the compact, convex,
+symmetric set
+K := ⟨⟨⃗f⟩⟩X,
+and denote by EK its John ellipsoid such that
+EK ⊂ K ⊂ √nEK.
+Case: EK is non-degenerate. Let RK be a linear transformation such that RKEK =
+¯BRn, the closed unit ball of Rn. Let (⃗ei)n
+i=1 be some orthonormal basis of Rn. As
+in (4.3), we then write
+tQ0(⃗f,⃗g) = tQ0(R−1
+K RK ⃗f,⃗g) = tQ0(RK ⃗f, R−t
+K ⃗g)
+=
+n
+�
+i=1
+tQ0(RK ⃗f · ⃗ei, R−t
+K ⃗g · ⃗ei) =:
+n
+�
+i=1
+tQ0(fi, gi),
+(4.7)
+where fi and gi are as in Lemma 4.1.
+It is from this point on that the present proof requires some elaboration compared
+to the proof of Proposition 4.2. According to assumption (1), for each of the pairs
+of functions fi := RK ⃗f · ⃗ei and gi := R−t
+K ⃗g · ⃗ei, we can find disjoint ˆQi,k ⊂ Q0
+with �
+k | ˆQi,k| ≤ ε|Q0| and such that: whenever Qj ⊂ Q0 are disjoint, not strictly
+contained in any ˆQi,k, and cover all ˆQi,k, then
+|tQ0(fi, gi) −
+�
+j
+tQj(fi, gi)| ≤ C∥fi∥X∥gi∥Y .
+(4.8)
+We make the following specific choice of the cubes Qj: Let {Qj}∞
+j=1 be the maximal
+cubes among { ˆQi,k}1≤k<∞
+1≤i≤n . Then
+�
+j
+|Qj| ≤
+n
+�
+i=1
+∞
+�
+k=1
+| ˆQi,k| ≤
+n
+�
+k=1
+ε|Q0| = nε|Q0|,
+
+8
+T. P. HYTÖNEN
+and (4.8) holds with these Qj for each i = 1, . . . , n. Using (4.7), and observing that
+it also holds with Q0 replaced by Qj, it follows that
+|tQ0(⃗f,⃗g) −
+�
+j
+tQj(⃗f,⃗g)| ≤
+n
+�
+i=1
+|tQ0(fi, gi) −
+�
+j
+tQj(fi, gi)|
+≤ C
+n
+�
+i=1
+∥fi∥X∥gi∥Y ≤ Cn3/2⟨⟨⃗f⟩⟩X · ⟨⟨⃗g⟩⟩Y ,
+using Lemma 4.1 in the last step. This completes the proof under the assumption
+that EK is non-degenerate.
+Case: EK is degenerate. This follows the corresponding case in the proof of Propo-
+sition 4.2 almost verbatim. Like there, let H := span K, and let P denote the
+orthogonal projection of Rn onto H. We then have (4.4) for each t = tQ, as well as
+(4.5). So it is again enough to prove the claim with P⃗g in place of ⃗g, and hence we
+may assume without loss of generality that also ⃗g = P⃗g. But then we can repeat
+the argument in the non-degenerate case, but with Rn replaced by its subspace H
+throughout; within this subspace, EK ⊂ H is non-degenerate, and the previous case
+applies to give the desired result.
+□
+5. From single-scale bounds to global bounds
+This passage is by now a relatively routine part of the theory, but we include some
+details for completeness. The following lemma is again stated in an operator-free,
+and even function-free way, simply as a criterion for dominating a real number by
+sum over a sparse collection. A more concrete situation for applying this criterion
+is presented afterwards.
+5.1. Lemma. Consider numbers a ∈ R and aQ, cQ ∈ R indexed by dyadic cubes
+Q ∈ D, with the following properties:
+(1) There is a family Q of disjoint dyadic cubes such that
+a =
+�
+Q∈Q
+aQ.
+(2) For some δ ∈ (0, 1) and each Q ∈ D that is contained in some P ∈ Q,
+there is a family of disjoint Qk ∈ D(Q) such that
+�
+k
+|Qk| ≤ δ|Q|,
+���aQ −
+�
+k
+aQk
+��� ≤ cQ.
+(3) For some α, C ∈ [1, ∞) and each Q ∈ D that is contained in some P ∈ Q,
+we have |aQ| ≤ C|Q|α.
+Then there is a (1 − δ)-sparse family of dyadic cubes S such that
+S ⊂
+�
+Q∈Q
+D(Q),
+|a| ≤
+�
+S∈S
+cS.
+5.2. Remark. If Q = {Q0} consists of a single cube only, then condition (1) is
+automatic with a = aQ0.
+
+SOME REMARKS ON CONVEX BODY DOMINATION
+9
+Proof. Let Q ⊂ D be a disjoint collection provided by assumption (1). For each
+P ∈ Q, denote S0(P) := {P}.
+Assuming that a disjoint Sj(P) ⊂ D(P) has
+already been constructed, for each Q ∈ Sj(P), let S ′(Q) := {Qk}∞
+k=1 be the
+collection provided by assumption (2), and let Sj+1(P) := �
+Q∈Sj(P ) S ′(Q). Let
+also S (P) := �∞
+j=0 Sj(P), and S := �
+P ∈Q S (P).
+For Q ∈ S , let E(Q) := Q \ �
+R∈S ′(Q) R. From the construction it is clear that
+these sets E(Q) are pairwise disjoint, and by assumption (2) we have |E(Q)| ≥
+(1 − δ)|Q|.
+By telescoping, for each P ∈ Q, we have
+aP =
+k−1
+�
+j=0
+�
+Q∈Sj(P )
+�
+aQ −
+�
+R∈S ′(Q)
+aR
+�
++
+�
+S∈Sk(P )
+aS.
+and hence, using assumptions (2) and (3),
+|aP | ≤
+k−1
+�
+j=1
+�
+Q∈Sj(P )
+cQ +
+�
+S∈Sk(P )
+C|S|α
+By an elementary inequality and induction, we have
+�
+S∈Sk(P )
+|S|α ≤
+�
+�
+S∈Sk(P )
+|S|
+�α
+≤ (δk|P|)α,
+and hence
+|aP | ≤ lim
+k→∞
+k−1
+�
+j=1
+�
+Q∈Sj(P )
+cQ =
+�
+Q∈S (P )
+cQ.
+Substituting this into assumption (1), we obtain the claim.
+□
+5.3. Lemma. Suppose that t is a bilinear form on L∞
+c (Rd; E) × L∞
+c (Rd; H), and
+moreover bounded with respect to the norm of Lp(Rd; E) × Lq(Rd; H) for some
+exponents with 1/p+1/q ≥ 1. For (⃗f,⃗g) ∈ L∞
+c (Rd; E)n ×L∞
+c (Rd; H)n, the numbers
+a = t(⃗f,⃗g),
+aQ = t(13Q ⃗f, 1Q⃗g)
+satisfy assumptions (1) and (3) of Lemma 5.1, provided that D is a dyadic system
+without quadrants.
+Proof. Since D is without quadrants, each Q ∈ D is contained in some (large
+enough) R ∈ D that contains supp ⃗f. Thus the collection Q of maximal cubes
+that do not contain supp ⃗f form a cover of Rd.
+By maximality, it follows that
+supp ⃗f ⊂ 3Q, and hence ⃗f = 13Q ⃗f for every Q ∈ Q. On the other hand, any
+Q with ℓ(Q) < diam(supp ⃗f) cannot contain supp ⃗f; hence any Q with ℓ(Q) <
+1
+2 diam(supp ⃗f) cannot be among the maximal cubes Q, and thus every Q ∈ Q
+will have to satisfy ℓ(Q) ≥ 1
+2 diam ⃗f. Since ⃗g ∈ L∞
+c (Rd; F)n, there are only finitely
+many Q ∈ Q with 1Q⃗g ̸= 0. Hence, without any issues of convergence, we can write
+t(⃗f,⃗g) = t
+�
+⃗f,
+�
+Q∈Q
+1Q⃗g
+�
+=
+�
+Q∈Q
+t(⃗f, 1Q⃗g) =
+�
+Q∈Q
+t(13Q ⃗f, 1Q⃗g),
+which is condition (1).
+If n = 1, the assumed boundedness directly implies that
+|t(13Qf, 1Qg)| ≤ C∥13Qf∥Lp(Rd;E)∥1Qg∥Lq(Rd;F ) ≤ C3d/p∥f∥∞∥g∥∞|Q|1/p+1/q,
+
+10
+T. P. HYTÖNEN
+where α := 1/p + 1/q ≥ 1, as required for condition (3). In general, if (⃗ei)n
+i=1 is an
+orthonormal basis of Rn and ⃗f = �n
+i=1 fi⃗ei and similarly for ⃗g, we have
+|t(13Q ⃗f, 1Q⃗g)| ≤
+n
+�
+i=1
+|t(13Qfi, 1Qgi)| ≤ Cn3d/p∥⃗f∥∞∥⃗g∥∞|Q|1/p+1/q,
+using the previous bound in each component and trivial bounds like ∥fi∥∞ ≤ ∥⃗f∥∞.
+□
+We are finally ready to state a semi-generic convex body domination principle.
+Condition (1) below is a typical intermediate estimate in a number of sparse dom-
+ination proofs for different operators. The conclusion is that it is already good
+enough to conclude convex body domination as well.
+5.4. Corollary. Let E and H be Banach spaces, and suppose that t is a bilinear
+form defined on F × G := L∞
+c (Rd; E) × L∞
+c (Rd; H) and bounded with respect to
+the norm of Lp(Rd; E) × Lq(Rd; H) for some exponents with 1/p + 1/q ≥ 1, and
+suppose that
+(1) for all (f, g) ∈ F × G and all Q ∈ D, there are disjoint ˆQk ⊂ Q with
+�
+k | ˆQk| ≤ ε|Q| and such that: whenever Qj ⊂ Q are disjoint, not strictly
+contained in any ˆQk, and cover all ˆQk, then
+|t(13Qf, 1Qg) −
+�
+j
+t(13Qjf, 1Qjg)| ≤ c∥f∥X(Q)∥g∥Y (Q)|Q|
+(5.5)
+for some norms ∥ ∥X(Q) on L∞
+c (Rd; E) and ∥ ∥Y (Q) on L∞
+c (Rd; H).
+Then for all (⃗f,⃗g) ∈ F n × Gn, there is a (1 − εn)-sparse collection S ⊂ D such
+that
+|t(⃗f,⃗g)| ≤ cn
+�
+S∈S
+⟨⟨⃗f⟩⟩X(S) · ⟨⟨⃗g⟩⟩Y (S)|S|,
+where εn = nε and cn = cn3/2.
+Proof. Let us begin by considering a fixed cube Q = Q0 ∈ D. We observe that
+assumption (1) of the present corollary coincides with condition (1) of Proposition
+4.6 with
+tQ(f, g) := t(13Qf, 1Qg),
+C = c|Q|,
+X = X(Q),
+Y = Y (Q).
+Hence the said proposition, applied to each fixed Q = Q0 ∈ D at a time, implies:
+(2) For all (⃗f,⃗g) ∈ L∞
+c (Rd; E)n ×L∞
+c (Rd; H)n and all Q ∈ D, there are disjoint
+Qk ⊂ Q with �
+k |Qk| ≤ εn|Q| and such that
+|t(13Q ⃗f, 1Q⃗g) −
+�
+j
+t(13Qj ⃗f, 1Qj⃗g)| ≤ cn⟨⟨⃗f⟩⟩X(Q) · ⟨⟨g⟩⟩Y (Q)|Q|,
+where εn = nε and cn = cn3/2.
+Let us then consider a fixed pair (⃗f,⃗g) ∈ L∞
+c (Rd; E)n×L∞
+c (Rd; H)n. We observe
+that condition (2) above coincides with condition (2) of Lemma 5.1 with the choices
+aQ = t(13Q ⃗f, 1Q⃗g),
+cQ = cn⟨⟨⃗f⟩⟩X(Q) · ⟨⟨g⟩⟩Y (Q)|Q|,
+δ = εn.
+On the other hand, Lemma 5.3 shows that these same aQ, together with a :=
+t(⃗f,⃗g), also satisfy conditions (1) and (3) of Lemma 5.1. Thus, all assumptions,
+and hence the conclusions, of Lemma 5.1 are valid for the said quantities, and these
+
+SOME REMARKS ON CONVEX BODY DOMINATION
+11
+conclusions agree with the claims of the result that we are proving. The proof is
+thus complete.
+□
+To facilitate the discussion of consequences of Corollary 5.4, we give
+5.6. Definition. Suppose that a pair of normed spaces (X(Q), Y (Q)) is associated
+to every dyadic cube Q ∈ D. We say that a bilinear form t : F ×G → R satisfies the
+(X(Q), Y (Q)) convex body domination of order n ∈ N if F ⊆ X(Q) and G ⊆ Y (Q)
+for every Q ∈ D, and if for every (f, g) ∈ F n×Gn, there exists a δn-sparse collection
+S ⊂ D such that
+|t(⃗f,⃗g)| ≤ Cn
+�
+Q∈S
+|Q|⟨⟨⃗f⟩⟩X(Q) · ⟨⟨⃗g⟩⟩Y (Q).
+We say that t : F × G → R satisfies the (X(Q), Y (Q)) convex body domination if
+it satisfies this for every n ∈ N. We say that an operator T : F → G∗ satisfies these
+properties if its associated bilinear form t(f, g) := ⟨T f, g⟩ does.
+Let us now consider some examples:
+5.7. Example (Calderón–Zygmund operators). Let T be a Dini–Calderón–Zygmund
+operator, i.e., T is L2(Rd) bounded and has the representation
+T f(x) =
+ˆ
+Rd K(x, y)f(y) dy,
+x /∈ supp f,
+where |K(x, y)| ≤ c|x − y|−d and, for |x − x′| ≤ 1
+2|x − y|,
+|K(x, y) − K(x′, y)| + |K(y, x) − K(y, x′)| ≤ ω
+�|x − x′|
+|x − y|
+�
+1
+|x − y|d ,
+(5.8)
+where ω : [0, 1
+2] → [0, ∞) is increasing, subadditive, and satisfies the Dini condition
+ˆ 1/2
+0
+ω(t) dt
+t < ∞.
+Then (1) of Corollary 5.4 holds for t(f, g) = ⟨T f, g⟩ and E = H = R and X(Q) =
+Ł1(3Q), Y (Q) = Ł1(Q), even in a stronger form. Namely, on the left oif (5.5), we
+have
+���
+�
+T (13Qf), 1Qg
+�
+−
+�
+j
+⟨T (13Qjf), 1Qjg⟩
+���
+≤
+���1QT (13Qf) −
+�
+j
+1QjT (13Qjf)
+���
+L∞(Q)∥g∥L1(Q),
+(5.9)
+and even the L∞ norm here is dominated by ∥f∥Ł1(3Q), as essentially shown in [24,
+(3.4)]. (Strictly speaking, [24, (3.4)] is formally slightly weaker, but a straightfor-
+ward modification of the argument gives the desired version, as observed in [27,
+Proof of Theorem 3.4].) Thus Corollary 5.4 says that a Dini–Calderón–Zygmund
+operator satisfies (Ł1(3Q), Ł1(Q)) convex body domination, but this was of course
+already known from [27] by essentially the same argument.
+5.10. Example (Banach space -valued Calderón–Zygmund operators). Let T be as
+in Example 5.7 but now acting on the Bochner space L2(Rd; E) of Banach space E
+-valued functions, and with an operator-valued kernel K(x, y) ∈ L (E) satisfying
+the same estimates as above but for the operator norm in place of the absolute value,
+e.g., ∥K(x, y)∥L (E) ≤ c|x − y|−d. It is in general a difficult problem to check the
+
+12
+T. P. HYTÖNEN
+L2(Rd; E)-boundedness of such an operator, but we now take this as an assumption.
+For g ∈ L2(Rd; E∗), we have (5.9) with L∞(Q; E) and L1(Q; E∗) in place of L∞(Q)
+and L1(Q), and the same proof of [24, (3.4)] (with same modifications pointed
+out in [27, Proof of Theorem 3.4]) shows that the L∞(Q; E) norm is dominated
+by ∥f∥Ł1(3Q;E). Thus we find that (1) of Corollary 5.4 also holds with X(Q) =
+Ł1(3Q; E) and Y (Q) = Ł1(Q; E∗).
+The resulting sparse domination (i.e., case
+n = 1 of the conclusion of Corollary 5.4) was known before, first in [15] for a
+slightly smaller class of kernels, and since [22, discussion on page 193] in the present
+generality. However, the convex body domination in this Banach space -valued
+setting is completely new.
+5.11. Example (Operators with grand maximal function control). Let 1 ≤ q ≤ r
+and s ≥ 1. Suppose that T is a linear operator
+T : L∞
+c (Rd) → L1
+loc(Rd),
+(5.12)
+that T has weak type (q, q), and that the bi-sublinear maximal operator
+MT (f, g)(x) := sup
+Q∋x
+ 
+Q
+|T (1(3Q)cf)| · |g|
+maps boundedly MT : Lr ×Ls → Lν,∞, where 1/ν = 1/r+1/s. Then condition (1)
+of Corollary 5.4 holds for t(f, g) = ⟨T f, g⟩ and E = H = R and X(Q) = Łr(3Q),
+Y (Q) = Łs(Q). This result is essentially contained in the proof of [25, Theorem
+3.1], where it appears as an intermediate step towards the sparse domination (i.e.,
+case n = 1 of the conclusion of Corollary 5.4) for such operators. The extension
+to convex body domination was recently achieved in [26], so Corollary 5.4 only
+reproduces a known result here. A key example of concrete operators satisfying
+these assumptions consists of rough homogeneous singular integrals
+T f(x) =
+ˆ
+Rd
+Ω(y)
+|y|d f(x − y) dy,
+where Ω(y) = Ω(y/|y|) is a bounded function with vanishing average over the unit
+sphere.
+As in Example 5.10, the abstract result above, involving a priori bounds of T
+and MT , extends straightforwardly to the Banach space -valued setting; however,
+verifying these bounds for concrete operators such as the rough homogeneous sin-
+gular integrals may present a problem in this generality, since the scalar-valued
+versions depend on deep results of Seeger [30], which so far lack a Banach space
+-valued extension.
+6. Matrix-weighted inequalities for Banach space -valued operators
+A matrix weight is a locally integrable function W : Rd → Rn×n that is a.e.
+positive definite -valued. The space Lp(W) consists of all measurable ⃗f : Rd → Rn
+such that W 1/p ⃗f ∈ Lp(Rd; Rn), and ∥⃗f∥Lp(W) := ∥W 1/p ⃗f∥Lp(Rd;Rn).
+For a Banach space E, we extend this definition in a natural way: The space
+Lp(W; En) consists of all measurable ⃗f : Rd → En such that W 1/p ⃗f ∈ Lp(Rd; En),
+and ∥⃗f∥Lp(W;En) := ∥W 1/p ⃗f∥Lp(Rd;En). Here, at each x ∈ Rd, we define (W 1/p ⃗f)(x) ∈
+En as the vector with components (W 1/p ⃗f)i(x) := �n
+j=1(W 1/p(x))ijfj(x), i.e., the
+matrix multiplication on Rn is extended to En in the natural way.
+
+SOME REMARKS ON CONVEX BODY DOMINATION
+13
+We now concentrate on p = 2. For two matrix weights W, V : Rd → Rn×n, we
+define
+[W, V ]A2 := sup
+Q
+|⟨W⟩1/2
+Q ⟨V ⟩1/2
+Q |2,
+[W]A2 := [W, W −1]A2,
+where we denote the operator norm in Rn×n ≃ L (Rn) simply by | |. We denote by
+A2(Rd; Rn) the class of matrix weights W : Rd → Rn×n for which [W]A2 < ∞. We
+also define
+[W]A∞ := sup
+⃗e∈Rn[x �→ ⃗e · W(x)⃗e]A∞,
+where on the right we have A∞ “norms” of some scalar weights, defined as usual by
+[w]A∞ := sup
+Q
+1
+w(Q)
+ˆ
+Q
+M(1Qw).
+According to [27, Remark 4.4], we have
+[W]A∞ ≤ 4[W]A2.
+(6.1)
+As a consequence of the Banach space -valued convex body domination from
+Example 5.10, we obtain:
+6.2. Theorem. Let E be a Banach space, and T ∈ L (L2(Rd; E)) be a Dini–
+Calderón–Zygmund operator with L (E)-valued kernel. For any W ∈ A2(Rd; Rn),
+the operator T extends boundedly to L2(W; En) and satisfies
+∥T ∥L(L2(W;En)) ≤ cn,T ([W]A2[W]A∞[W −1]A∞)1/2 ≤ cn,T [W]3/2
+A2 .
+Note that Theorem 6.2 applies to a general Banach space E, but contains the
+(difficult) a priori boundedness hypothesis that T ∈ L (L2(Rd; E)). Concrete ex-
+amples are available in the class of UMD spaces, treated in detail in [18].
+6.3. Corollary. Let E be a UMD space, and T ∈ L (L2(Rd)) be a scalar-valued
+Calderón–Zygmund operator with a Hölder-type modulus of continuity ω(t) = ctδ,
+δ ∈ (0, 1] in (5.8). For any W ∈ A2(Rd; Rn), the operator T extends boundedly to
+L2(W; En) and satisfies
+∥T ∥L(L2(W;En)) ≤ cn,E,T ([W]A2[W]A∞[W −1]A∞)1/2 ≤ cn,E,T [W]3/2
+A2 .
+In particular, this estimate holds when T is the classical Hilbert transform.
+Proof. We reduce Corollary 6.3 to Theorem 6.2 with the help of the T (1) theorem of
+David and Journé [12], and its extension to UMD spaces by Figiel [14]. By the (easy
+half of) the David–Journé theorem, the assumptions on T imply that that T satisfies
+the so-called weak boundedness property as well as T (1), T ∗(1) ∈ BMO(Rd). Then,
+by Figiel’s theorem, an operator satisfying these conditions and the Calderón–
+Zygmund kernel assumptions extends boundedly to L2(Rd; E), for any UMD space
+E. Thus T satisfies the assumptions, and hence the conclusions, of Theorem 6.2,
+and we are done.
+□
+These results, even just for the Hilbert transform, and even in their qualitative
+form (i.e., just concluding the boundedness of T , without specifying any concrete
+bound for the norm), are completely new in the combined setting of matrix weights
+and Banach spaces. For E = R and the Hilbert transform T , the qualitative form
+of Corollary 6.3 is due to Treil and Volberg [31]. The quantitative form for E = R
+was obtained by Nazarov et al. [27], and this is the best that is known at the time
+of writing. For scalar-weights, the power 3/2 can be replaced by 1 [16], and the
+
+14
+T. P. HYTÖNEN
+product of [W]A∞ and [W −1]A∞ by their sum [19], but extending these to the
+general matrix case consists of the outstanding open “matrix A2 conjecture”.
+Turning to the proof of Theorem 6.2, we begin with:
+6.4. Remark (Without loss of generality, we assume that E is reflexive). Since
+Theorem 6.2 is about the bounded extension of an operator, it suffices to prove an a
+priori estimate on a dense subspace of functions ⃗f. In particular, we can assume that
+each component fi takes its values in a finite-dimensional subspace of E. Since any
+finite-dimensional space is reflexive, we make the standing assumption, without loss
+of generality, that E is reflexive. (Note that this is automatic in Corollary 6.3 in any
+case, since UMD spaces are reflexive [18, Theorem 4.3.3].) Under this assumption,
+we have L1(Q; E)∗ = L∞(Q; E∗) (see [18, Theorems 1.3.10 and 1.3.21]), which is
+convenient in view of calculations involving the convex bodies ⟨⟨ ⟩⟩Ł1(Q;E).
+6.5. Lemma.
+|Q|⟨⟨W 1/2 ⃗f⟩⟩Ł1(3Q;E) · ⟨⟨V 1/2⃗g⟩⟩Ł1(Q;E∗)
+≤
+ˆ �
+1Q(x)
+ 
+3Q
+|V 1/2(x)W 1/2(y)|∥⃗f(y)∥En dy
+�
+∥⃗g(x)∥ ⃗E∗n dx
+Proof. Under the standing assumption from Remark 6.4, we evaluate consider a
+generic element of the convex body on the left with φ ∈ ¯BL∞(Q;E∗) and ψ ∈
+¯BL∞(Q;E):
+|Q|
+���
+ 
+3Q
+W 1/2(y)⟨⃗f(y), φ(y)⟩ dy ·
+ 
+Q
+V 1/2(x)⟨⃗g(x), ψ(x)⟩ dx
+���
+= |Q|
+���
+ 
+Q
+ 
+3Q
+V 1/2(x)W 1/2(y)⟨⃗f(y), φ(y)⟩ · ⟨⃗g(x), ψ(x)⟩ dy dx
+���
+≤
+ˆ
+Q
+ 
+3Q
+|V 1/2(x)W 1/2(y)|∥⃗f(y)∥En∥⃗g(x)∥ ⃗E∗n dy dx.
+□
+Summing over a sparse collection, we obtain
+�
+Q∈S
+|Q|⟨⟨W 1/2 ⃗f⟩⟩Ł1(3Q;E) · ⟨⟨V 1/2⃗g⟩⟩Ł1(Q;E∗)
+≤
+ˆ � �
+Q∈S
+1Q(x)
+ 
+3Q
+|V 1/2(x)W 1/2(y)|∥⃗f(y)∥En dy
+�
+∥⃗g(x)∥ ⃗E∗n dx
+=:
+ˆ
+˜L(∥⃗f∥En)(x)∥⃗g(x)∥ ⃗E∗n dx,
+(6.6)
+where ˜L, here acting on the scalar-valued function y �→ ∥⃗f(y)∥En, is an operator
+denoted by the same symbol in [27, (5.8)]. By [27, Lemma 5.6], we have
+∥˜L∥L (L2) ≤ C([W, V ]A2[W]A∞[V ]A∞)1/2.
+(6.7)
+By duality and standard changes of variables, which present no essential differ-
+ence in the Banach space -valued setting, an estimate of the form
+∥T ⃗f∥L2(V ;En) ≤ N∥⃗f∥L2(V ;En)
+is equivalent to
+⟨T (W 1/2 ⃗f), V 1/2⃗g⟩ ≤ N∥⃗f∥L2(Rd;En)∥⃗g∥L2(Rd;E∗n).
+(6.8)
+
+SOME REMARKS ON CONVEX BODY DOMINATION
+15
+If T is an in Theorem 6.2, it satisfies the (Ł1(3Q; E), Ł1(Q; E∗)) convex body dom-
+ination by Example 5.10, which means that the left-hand side of (6.8) is dominated
+by the left-hand side of (6.6), and hence, by (6.6) and (6.7), we have
+N ≤ cn,T ([W, V ]A2[W]A∞[V ]A∞)1/2.
+This is the desired bound, and concludes the proof of Theorem 6.2.
+7. Convex domination and generalised commutators
+For an operator T and two vector functions ⃗a = (a1, . . . , an) and ⃗b = (b1, . . . , bn),
+let us consider the operator
+⃗a · T⃗b : f �→ ⃗a · T (⃗bf) =
+n
+�
+i=1
+aiT (bif).
+We are mainly interested in the boundedness on Lp(Rd), or a weighted Lp(w),
+or between two such spaces, and the case when T is a singular integral operator
+bounded on the space. However, we do not require that ai, bi ∈ L∞(Rd), and hence
+the pointwise multipliers f �→ bif and g �→ aig, and the compositions f �→ aiT (bif),
+may be unbounded operators. Nevertheless, their sum ⃗a · T⃗b may still be bounded,
+thanks to cancellation between different terms.
+A case that has been much studied in the literature consists of ⃗b = (1, b) and
+⃗a = (b, −1), in which case
+⃗a · T (⃗bf) = bT f − T (bf) = [b, T ]f
+is the commutator of b and T , whose Lp(Rd)-boundedness is characterised by b ∈
+BMO(Rd), the space of functions of bounded mean oscillation, which is strictly
+larger than L∞(Rd), and contains in particular functions like b(x) = log |x|.
+By dualising with a function g, and denoting by t(f, g) = ⟨T f, g⟩ the bilinear
+form of T , we arrive at
+⟨⃗a · T (⃗bf), g⟩ =
+n
+�
+i=1
+⟨T (bif), aig⟩ = t(⃗bf,⃗ag),
+where the action of the bilinear form is extended to vector-valued functions as
+before. To be precise, if t in defined on F × G, we should now require that
+f ∈ F⃗b := {f ∈ F : bif ∈ F for all i = 1, . . . , n},
+and g ∈ G⃗a, defined similarly. If F ⊇ L∞
+c (Rd), then clearly F⃗b contains in particular
+all f ∈ L∞
+c (Rd) with supp f ⊆ EN := {|⃗b| ≤ N} for any N ∈ N. For a.e. finite-
+valued bi, the union �
+N∈N EN covers Rd up to a null set, it is immediate that F⃗b
+is dense in any Lp(w) with finite p.
+7.1. Lemma. Suppose that T satisfies the (X(Q), Y (Q)) convex body domination.
+Then for all relevant functions, we have
+|⟨⃗a · T (⃗bf), g⟩| ≤ C
+�
+Q∈S
+|Q|⟨⟨⃗bf⟩⟩X(Q) · ⟨⟨⃗ag⟩⟩Y (Q).
+(7.2)
+Proof. This is immediate by applying definition to ⃗f = ⃗bf and ⃗g = ⃗ag.
+□
+We take a closer look at the case when X(Q) = Y (Q) = Ł1(γQ).
+
+16
+T. P. HYTÖNEN
+7.3. Lemma. For all s, t ∈ (1, ∞) and all functions in the relevant spaces, we have
+⟨⟨⃗bf⟩⟩Ł1(Q) · ⟨⟨⃗ag⟩⟩Ł1(Q) ≤ ∥(x, y) �→ ⃗a(x) ·⃗b(y)∥Ł(s,t)
+min (Q×Q)∥f∥Łt′ (Q)∥g∥Łs′(Q),
+where
+∥F∥Ł(s,t)
+min (Q×Q) :=
+
+
+
+
+
+� ffl
+Q
+� ffl
+Q |F(x, y)|s dx
+�t/s
+dy
+�1/t
+,
+if s ≤ t,
+� ffl
+Q
+� ffl
+Q |F(x, y)|t dy
+�s/t
+dx
+�1/s
+,
+if t ≤ s.
+Proof. The generic element of ⟨⟨⃗bf⟩⟩X(Q) · ⟨⟨⃗ag⟩⟩Y (Q) has the following form, where
+φ, ψ ∈ ¯BL∞(Q):
+ 
+Q
+⃗b(y)f(y)φ(y) dy ·
+ 
+Q
+⃗a(x)g(x)ψ(x) dx
+=
+ 
+Q
+ 
+Q
+(⃗a(x) ·⃗b(y))f(y)g(x)φ(y)ψ(x) dx dy,
+and hence
+⟨⟨⃗bf⟩⟩X(Q) · ⟨⟨⃗ag⟩⟩Y (Q) ≤
+ 
+Q
+ 
+Q
+|⃗a(x) ·⃗b(y)||f(y)||g(x)| dx dy
+≤ ∥(x, y) �→ a(x) · b(y)∥Z∥(x, y) �→ f(y)g(x)∥Z∗,
+for either choice of
+(Z, Z∗) ∈ {(Łs
+x(Q; Łt
+y(Q)), Łs′
+x (Q; Łt′
+y (Q))), (Łt
+y(Q; Łs
+x(Q)), Łt′
+y (Q; Łs′
+x (Q)))},
+by Hölder’s inequality for mixed-norm Lp spaces. By Fubini’s theorem, we have
+∥(x, y) �→ f(x)g(y)∥Z∗ = ∥f∥Łt′ (Q)∥g∥Łs′(Q)
+in either case, and hence, taking the minimum over the two choices of Z, we arrive
+at the factor
+min
+Z ∥(x, y) �→ b(x) · a(y)∥Z = ∥(x, y) �→ ⃗b(x) · ⃗a(y)∥Ł(s,t)
+min (Q×Q).
+□
+7.4. Proposition. Let T be an operator that satisfies the (Ł1(γQ), Ł1(γQ)) convex
+body domination. Let ⃗a,⃗b ∈ L1
+loc(Rd)n be functions such that
+As,t := sup
+Q
+∥(x, y) �→ ⃗a(x) ·⃗b(y)∥Ł(s,t)
+min (Q×Q) < ∞.
+Then ⃗a·T⃗b extends to a bounded operator on Lp(Rd) for all p ∈ (t′, s). In particular,
+if As := As,s < ∞ for some s ∈ (2, ∞), then ⃗a · T⃗b extends boundedly to L2(Rd).
+
+SOME REMARKS ON CONVEX BODY DOMINATION
+17
+Proof. Combining Lemmas 7.1 and 7.3, we find that
+|⟨⃗aT (⃗bf), g⟩| ≤ C
+�
+Q∈S
+|Q|⟨⟨⃗bf⟩⟩Ł1(γQ) · ⟨⟨⃗ag⟩⟩Ł1(γQ)
+≤ C
+�
+Q∈S
+|Q|∥(x, y) �→ a(x) · b(y)∥Ł(s,t)
+min (Q×Q)∥f∥Łt′ (Q)∥g∥Łs′(Q)
+≤ C
+�
+Q∈S
+|E(Q)|
+δ
+As,t inf
+Q Mt′f inf
+Q Ms′g
+≤ CAs,t
+δ
+�
+Q∈S
+ˆ
+E(Q)
+Mt′fMs′g ≤ CAs,t
+δ
+ˆ
+Rd Mt′fMs′g
+≤ CAs,t
+δ
+∥Mt′f∥Lp(Rd)∥Ms′g∥Lp′(Rd),
+where
+∥Mt′f∥Lp(Rd) ≲t,p ∥f∥Lp(Rd),
+∥Ms′g∥Lp′(Rd) ≲s,p ∥g∥Lp′(Rd)
+for p > t′ and p′ > s′, where the latter is equivalent to p < s.
+□
+Let us consider some examples:
+7.5. Example (Classical commutators). As we already observed, ⃗a = (b, −1) and
+⃗b = (1, b) gives rise to the usual commutator [b, T ]. In this case
+⃗a(x) ·⃗b(y) = b(x) − b(y)
+and each As,t is equivalent to ∥b∥BMO(Rd) by elementary considerations and the
+John–Nirenberg inequality. Thus Proposition 7.4 reproduces the well-known suffi-
+cient condition for the boundedness of commutators.
+7.6. Example (Iterated commutators). More generally, choosing ⃗a,⃗b so that
+⃗a(x) ·⃗b(y) = (b(x) − b(y))k =
+k
+�
+i=0
+�k
+i
+�
+b(x)k−i(−b(y))i,
+thus e.g. ai(x) =
+�k
+i
+�
+b(x)k−i and bi(y) = (−b(y))i, we reproduce the kth order
+commutator
+⃗a · T⃗b = Tk,b := [b, Tk−1,b],
+T0,b := T,
+and As,t is equivalent to ∥b∥k
+BMO(Rd) by the John–Nirenberg inequality.
+7.7. Example (Iterated commutators with different multipliers). Let us then choose
+⃗a,⃗b so that
+⃗a(x) ·⃗b(y) = (b1(x) − b1(y))(b2(x) − b2(y));
+without specifying the precise choice of ai(x) and bi(y), it is evident that such
+a choice can be easily written down, if desired. (We deliberately use superscript
+indices for bi above, since these not be the same as the components bi of ⃗b.) This
+reproduces the second order iterated commutator with two different functions,
+⃗a · T⃗b = [b1, [b2, T ]].
+It is well-known and classical that bi ∈ BMO(Rd) for both i = 1, 2 is sufficient for
+the L2(Rd) boundedness of [b1, [b2, T ]]; however, as recently observed in [17], much
+
+18
+T. P. HYTÖNEN
+weaker sufficient conditions can be given for the pair (b1, b2). Namely, in [17, (1.1)],
+it shown that the pair of conditions
+Ss := sup
+Q
+�  
+Q
+|b1(x) − ⟨b1⟩Q|s dx
+�1/s�  
+Q
+|b2(y) − ⟨b2⟩Q|s dy
+�1/s
+< ∞,
+Ts := sup
+Q
+�  
+Q
+|b1(x) − ⟨b1⟩Q|s|b2(x) − ⟨b2⟩Q|s dx
+�1/s
+< ∞,
+is sufficient for the L2(Rd) boundedness of [b1, [b2, T ]] for s > 2. On the other hand,
+by Proposition 7.4, another sufficient condition for the same conclusion is As < ∞.
+Let us compare the two. Adding and subtracting terms and multiplying out, we
+find that
+(b1(x) − b1(y))(b2(x) − b2(y))
+= [(b1(x) − ⟨b1⟩Q) − (b1(y) − ⟨b1⟩Q)][(b2(x) − ⟨b2⟩Q) − (b2(y) − ⟨b2⟩Q)]
+= (b1(x) − ⟨b1⟩Q)(b2(x) − ⟨b2⟩Q) + (b1(y) − ⟨b1⟩Q)(b2(y) − ⟨b2⟩Q)
+− (b1(x) − ⟨b1⟩Q)(b2(y) − ⟨b2⟩Q) − (b1(y) − ⟨b1⟩Q)(b2(x) − ⟨b2⟩Q).
+Taking Łs(Q × Q) and then supremum over Q on both sides, we deduce that
+As ≤ 2(Ts + Ss),
+so that the new criterion provided by Proposition 7.4 is at least as sharp as that of
+[17, (1.1)], and it seems less obvious to make any estimate in the other direction.
+Perhaps more importantly, the new condition As < ∞ arises more “naturally” as
+an instance of a general principle.
+(Let us note that there is a more general criterion [17, Theorem 3.10], where the
+Łs norms in Ss and Tt are replaced by more general Orlicz norms. On the other
+hand, it is apparent that similar generalisations could be achieved in Proposition
+7.4: what we used was the boundedness of the rescaled maximal operators Mt′ on
+Lp(Rd) for p > t′, and this could be replaced having an Orlicz maximal operator
+MA with the same mapping property. A characterisation of this property in terms
+of the so-called Bp condition on the Orlicz function A is a classical result of Pérez
+[29]; this very result is used in [17]; see [17, Proposition 3.8].)
+Let us finally consider an “exotic” example with no obvious predecessor in the
+existing literature. We begin with a lemma:
+7.8. Lemma. Suppose that 0 ≤ b ∈ BMO(Rd). If 0 ≤ α, β and α + β ≤ 1, then
+B(x, y) := b(x)αb(y)β − b(x)βb(y)α
+satisfies
+�  
+Q
+ 
+Q
+|B(x, y)|p dx dy
+�1/p
+≤ (2∥b∥BMOp(Rd))α+β.
+Proof. Let γ := min(α, β) ∈ [0, 1
+2] and δ := max(α, β) − γ ∈ [0, 1]. Then
+|B(x, y)| = b(x)γb(y)γ|b(x)δ − b(y)δ|.
+We observe the following elementary inequality:
+|uδ − vδ| ≤
+|u − v|
+max(u, v)1−δ ,
+∀u, v ≥ 0, δ ∈ [0, 1].
+(7.9)
+
+SOME REMARKS ON CONVEX BODY DOMINATION
+19
+Indeed, by symmetry and homogeneity, it is enough to consider u = 1 and v ∈ [0, 1],
+in which case we are reduced to proving that
+1 − vδ ≤ 1 − v,
+which is immediate from the fact that v ≤ vδ for v, δ ∈ [0, 1].
+Using (7.9), and noting that δ + 2γ = α + β ∈ [0, 1], it follows that
+|B(x, y)| ≤ b(x)γb(y)γ
+|b(x) − b(y)|
+max(b(x), b(y))1−δ ≤
+|b(x) − b(y)|
+max(b(x), b(y))1−δ−2γ
+=
+� |b(x) − b(y)|
+max(b(x), b(y))
+�1−δ−2γ
+|b(x) − b(y)|δ+2γ ≤ |b(x) − b(y)|α+β,
+and hence
+�  
+Q
+ 
+Q
+|B(x, y)|p dx dy
+�1/p
+≤
+�  
+Q
+ 
+Q
+|b(x) − b(y)|p dx dy
+�(α+β)/p
+≤
+��  
+Q
+|b(x) − c|p dx
+�1/p
++
+�  
+Q
+|b(y) − c|p dy
+�1/p�α+β
+for all constants c.
+□
+7.10. Corollary. Let T be an operator satisfying (Ł1(γQ), Ł1(γQ)) convex body
+domination, let 0 ≤ b ∈ BMO(Rd) and 0 ≤ α, β with α + β ≤ 1. Then
+∥bαT (bβf) − bβT (bαf)∥Lp(Rd) ≲p ∥b∥α+β
+BMO(Rd)∥f∥Lp(Rd).
+Proof. By Proposition 7.4 with s = t, the Lp(Rd) operator norm of f �→ bαT (bβf)−
+bβT (bαf) is dominated by
+As := sup
+Q
+∥(x.y) �→ b(x)αb(y)β − b(x)βb(y)α∥Łs(Q×Q)
+if p ∈ (s′, s), i.e., if s > max(p, p′). By Lemma 7.8 and the John–Nirenberg inequal-
+ity, we have
+As ≤ (2∥b∥BMOs(Rd))α+β ≲s ∥b∥α+β
+BMO(Rd),
+and fixing (say) s = 2 max(p, p′), we obtain a dependence on p only.
+□
+7.11. Remark. Aside from the examples already discussed, the generalised commu-
+tators ⃗a · T⃗b also arise in the following question studied by Bloom [4, 5]. Suppose
+that a matrix weight W is given in the diagonalised form W = U ∗ΛU, where U is
+unitary, Λ is diagonal, and the diagonal entries λk of Λ are scalar A2 weights. What
+does one need to know about U in order to conclude that W ∈ A2? (According to
+[5, Theorem 4.2], the condition that λk ∈ A2 is necessary for W ∈ A2, if in addition
+U is assumed to be continuous.)
+Let T be the Hilbert transform, or another operator whose boundedness on
+the matrix-weighted L2(W) characterises W ∈ A2.
+By connecting the L2(W)
+boundedness of T to the boundedness of the classical commutators [T, ¯uij] between
+the weighted spaces L2(λi) and L2(λk) (sic: the condition involves triplets of indices
+(i, j, k)), [4, Theorem 5.1] shows that uij ∈ BMO√
+λi/λk (a weighted BMO space,
+nowadays commonly referred to as Bloom-type BMO) is a sufficient condition. In
+the special case of 2 × 2 matrices, it is also necessary by [5, Theorem 4.3] but, over
+30 years since these contributions, the general case seems to remain open. (The
+author is grateful to Amalia Culiuc for bringing this question to his attention [9].)
+
+20
+T. P. HYTÖNEN
+Here is a possible approach to the problem. As is well known, the L2(W) bound-
+edness of T is equivalent to the (unweighted) L2 boundedness of
+W 1/2T W −1/2 = U ∗Λ1/2UT U ∗Λ−1/2U.
+Multiplication by U and U ∗ is isometric on L2, and the L2 boundedness of a matrix
+of operators is equivalent to the L2 boundedness of each of the components
+(Λ1/2UT U ∗Λ−1/2)ij =
+n
+�
+k=1
+λ1/2
+i
+uikT ¯ujkλ−1/2
+j
+= λ1/2
+i
+⃗ui · T ¯⃗ujλ−1/2
+j
+,
+where i, j = 1, . . . , n and ⃗ui = (uik)n
+k=1. These are operators of the form ⃗a · T⃗b that
+we have studied here and, up to this point, we kept an exact equivalence with the
+original question; the question then would be, whether we can give useful conditions
+on the boundedness of these operators. A further equivalent condition is of course
+the two-weight boundedness
+⃗ui · T ¯⃗uj : L2(λj) → L2(λi),
+i, j = 1, . . . , n,
+where the spaces are more complicated, but the multipliers are simply rows of the
+unitary matrix U.
+7.12. Remark. We have concentrated in this section on the application of convex
+body domination—an inherently vector-valued theory—to questions of generalised
+commutators acting on scalar-valued functions. We have made this choice for two
+reasons: to make the case that this vector-valued theory is useful even for such
+scalar-valued applications, and not to obscure the relatively simple basic philoso-
+phy behind too many technicalities of notation. This said, it is quite plain that
+the presented ideas can be immediately generalised to the case of vector-valued
+functions ⃗f and ⃗g (in place of scalar f and g) and matrix-valued multipliers A and
+B (in place of the vectors ⃗a and ⃗b). In the particular case of the classical-style
+commutator [T, B] with a matrix-valued function, this idea has been developed in
+[21].
+8. Stopping times and maximal functions involving convex bodies
+The aims of this final section are two-fold. Concretely, we establish a convex-
+body analogue of a result of Nieraeth [28], which shows that the estimation of
+sums over sparse collection that arise in the usual sparse domination is equivalent
+to the estimation of certain maximal functions. On the way of achieving this, we
+develop some convex-body versions of the typical stopping time arguments involving
+averages of scalar-valued functions; these might have some independent interest
+elsewhere.
+We begin with an estimate of a sum of convex-body “norms” over disjoint subsets.
+8.1. Lemma. Let p, q ∈ [1, ∞) and 1
+r := 1
+p + 1
+q . Let Qi ∈ D(Q0) be disjoint cubes.
+Then
+∞
+�
+i=1
+�
+⟨⟨⃗f⟩⟩Lp(Qi) · ⟨⟨⃗g⟩⟩Lq(Qi)
+�r ≤ nmax(r,1)+r/2�
+⟨⟨f⟩⟩Lp(Q) · ⟨⟨g⟩⟩Lq(Q)
+�r.
+Note that for p, q ∈ [1, ∞), we have 1
+r = 1
+p + 1
+q ≤ 1 + 1 = 2, and hence
+nmax(r,1)+r/2 =
+�
+nmax(1,1/r)+1/2�r ≤
+�
+n5/2�r.
+
+SOME REMARKS ON CONVEX BODY DOMINATION
+21
+Proof. For orientation, let us begin with the proof in the case n = 1, i.e., with ∥ ∥
+in place of ⟨⟨ ⟩⟩ throughout. By Hölder’s inequality with 1 = r
+p + r
+q, we have
+∞
+�
+i=1
+�
+∥f∥Lp(Qi)∥g∥Lq(Qi)
+�r =
+∞
+�
+i=1
+�
+∥f∥p
+Lp(Qi)
+�r/p�
+∥g∥q
+Lq(Qi)
+�r/q
+≤
+� ∞
+�
+i=1
+∥f∥p
+Lp(Qi)
+�r/p� ∞
+�
+i=1
+∥g∥q
+Lq(Qi)
+�r/q
+≤
+�
+∥f∥p
+Lp(Q0)
+�r/p�
+∥g∥q
+Lq(Q0)
+�r/q
+=
+�
+∥f∥Lp(Q0)∥g∥Lq(Q0)
+�r
+.
+In the general case of the lemma, let
+Ai := ⟨⟨⃗f⟩⟩Lp(Qi) =
+� ˆ
+Qi
+φi ⃗f : ∥φi∥Lp′(Qi) ≤ 1
+�
+,
+Bi := ⟨⟨⃗g⟩⟩Lq(Qi).
+Then we observe that
+⟨⟨⃗f⟩⟩Lp(Q) =
+� ˆ
+Q
+φ⃗f : ∥φ∥Lp′(Q) ≤ 1
+�
+⊇
+� ∞
+�
+i=1
+ai
+ˆ
+Qi
+φi ⃗f : ∥φi∥Lp′(Qi) ≤ 1, ∥(ai)∥ℓp′ ≤ 1
+�
+=
+� ∞
+�
+i=1
+aiAi : ∥(ai)∥ℓp′ ≤ 1
+�
+=:
+�
+ℓp
+Ai =: A,
+and similarly
+⟨⟨⃗g⟩⟩Lq(Q) ⊇
+�
+ℓq
+Bi =: B.
+Hence, the lemma is reduced to proving that
+∞
+�
+i=1
+�
+Ai · Bi
+�r ≤ nmax(r,1)+r/2�
+A · B
+�r,
+A :=
+�
+ℓp
+Ai,
+B :=
+�
+ℓq
+Bi.
+Let EA be the John ellipsoid of A, and let RAEA = ¯BRn.
+Since Ai · Bi =
+RAAi · R−t
+A Bi, The claim above is equivalent to a version where each Ai is replaced
+by RAAi and each Bi by R−t
+A Bi. Hence, without loss of generality, we assume that
+EA = ¯BRn to begin with, hence ¯BRn ⊆ A ⊆ √n ¯BRn. Thus
+A · B ⊃ ¯BRn · B = [−M, M],
+where
+M := max{|⃗b| : ⃗b ∈ B}.
+On the other hand, if (⃗ej)n
+j=1 is some orthonormal basis of Rn, then
+Ai · Bi = {⃗a ·⃗b : ⃗a ∈ Ai,⃗b ∈ Bi}
+=
+�
+n
+�
+j=1
+(⃗a · ⃗ej)(⃗b · ⃗ej) : ⃗a ∈ Ai,⃗b ∈ Bi} ⊆
+n
+�
+j=1
+(Ai · ⃗ej)(Bi · ⃗ej),
+or, using the identification of [−s, s] with s,
+Ai · Bi ≤
+n
+�
+j=1
+(Ai · ⃗ej)(Bi · ⃗ej).
+
+22
+T. P. HYTÖNEN
+Thus
+(Ai · Bi)r ≤
+n
+�
+j=1
+�
+(Ai · ⃗ej)(Bi · ⃗ej)
+�r,
+r ∈ (0, 1],
+and
+� ∞
+�
+i=1
+(Ai · Bi)r�1/r
+≤
+n
+�
+j=1
+� ∞
+�
+i=1
+�
+(Ai · ⃗ej)(Bi · ⃗ej)
+�r�1/r
+,
+r ∈ [1, ∞).
+In the sum over i, we use Hölder’s inequality as in the toy model in the beginning:
+∞
+�
+i=1
+�
+(Ai · ⃗ej)(Bi · ⃗ej)
+�r =
+∞
+�
+i=1
+�
+(Ai · ⃗ej)p�r/p�
+(Bi · ⃗ej)q�r/q
+≤
+� ∞
+�
+i=1
+(Ai · ⃗ej)p�r/p� ∞
+�
+i=1
+(Bi · ⃗ej)q�r/q
+= sup
+�� ∞
+�
+i=1
+aiAi · ⃗ej
+�1/r� ∞
+�
+i=1
+biBi · ⃗ej
+�1/r
+: ∥(ai)∥ℓp′ ≤ 1, ∥(bi)∥ℓq′ ≤ 1
+�
+= (A · ⃗ej)r(B · ⃗ej)r
+Here
+A · ⃗ej ⊆ √n ¯BRn · ⃗ej = [−√n, √n],
+A · ⃗ej ≤ √n,
+and clearly
+B · ⃗ej ≤ M.
+Altogether, writing s := max(r, 1), we have
+� ∞
+�
+i=1
+(Ai · Bi)r�1/s
+≤
+n
+�
+j=1
+� ∞
+�
+i=1
+(Ai · ⃗ej)r(Bi · ⃗ej)r�1/s
+≤
+n
+�
+j=1
+�
+(A · ⃗ej)r(B · ⃗ej)r�1/s
+≤ n[nr/2M r]1/s,
+and hence
+∞
+�
+i=1
+(Ai · Bi)r ≤ nsnr/2M r = nmax(1,r)+r/2(A · B)r,
+which remained to be proved.
+□
+The following lemma is a convex-body analogue of the basic principle underlying
+the simplest stopping time constructions: for a function on a cube Q0, the total
+measure of the subcubes, where the average of a function is much bigger than on
+the whole Q0, can be at most a fraction of the measure of Q0.
+8.2. Lemma. Let A, p, q ∈ [1, ∞) and let Qi ∈ D(Q0) be disjoint cubes such that
+⟨⟨⃗f⟩⟩Łp(Qi) · ⟨⟨⃗g⟩⟩Łq(Qi) ≥ A⟨⟨⃗f⟩⟩Łp(Q0) · ⟨⟨⃗g⟩⟩Łq(Q0).
+Then
+∞
+�
+i=1
+|Qi| ≤ nmax(r,1)+r/2
+Ar
+|Q0|,
+1
+r := 1
+p + 1
+q .
+
+SOME REMARKS ON CONVEX BODY DOMINATION
+23
+Proof. Directly from the definition, it is easy to extend the basic identity ∥f∥Łp(Q) =
+|Q|−1/p∥f∥Lp(Q) to convex bodies as
+⟨⟨⃗f⟩⟩Łp(Q) = |Q|−1/p⟨⟨⃗f⟩⟩Lp(Q).
+(8.3)
+From this, the assumption of the lemma can be rewritten as
+|Qi|−1/p−1/q⟨⟨⃗f⟩⟩Lp(Qi) · ⟨⟨⃗g⟩⟩Lq(Qi) ≥ A|Q0|−1/p−1/q⟨⟨⃗f⟩⟩p(Q0) · ⟨⟨⃗g⟩⟩q(Q0),
+or, rearranging,
+|Qi| ≤
+A−r|Q0|
+�
+⟨⟨⃗f⟩⟩p(Q0) · ⟨⟨⃗g⟩⟩q(Q0)
+�r
+�
+⟨⟨⃗f⟩⟩Lp(Qi) · ⟨⟨⃗g⟩⟩Lq(Qi)
+�r.
+Summing over i and using Lemma 8.1, we obtain the claim.
+□
+We now obtain the following proposition, which is a convex body analogue of
+a result of Nieraeth [28, Prop. 2.7; especially Eq. (2.7) for m = 1]. It says that
+estimating the sums over sparse collections, like those that arise from convex body
+domination, is equivalent to estimating related bi-sublinear maximal operators. In
+[28, Prop. 2.7], the result is formulated as a set of equivalent conditions for a tuple
+of weights. The formulation below has no reference to weights as such, but as soon
+as one starts asking questions about the boundedness of either side on spaces like
+Ls(W) × Ls′(W ′), the proposition guarantees that one can equally well study this
+boundedness for the other side of the equivalence.
+8.4. Proposition. For all δ ∈ (0, 1), all dimensions d, n ≥ 1, exponents p, q ∈
+[1, ∞), and functions ⃗f ∈ Lp
+loc(Rd)n, ⃗g ∈ Lq
+loc(Rd)n, we have the two-sided estimate
+sup
+S
+�
+Q∈S
+⟨⟨⃗f⟩⟩Łp(Q) · ⟨⟨⃗g⟩⟩Łq(Q)|Q| ≂
+��� sup
+Q∈D
+1Q⟨⟨⃗f⟩⟩Łp(Q) · ⟨⟨⃗g⟩⟩Łq(Q)
+���
+L1(Rd),
+where the supremum is taken over all δ-sparse collections of dyadic cubes in Rn,
+and the implied constants depend only on n, p, q, and δ.
+Proof. With ⃗f ∈ Lp
+loc(Rd)n and ⃗g ∈ Lq
+loc(Rd)n fixed, let us denote
+aQ := ⟨⟨⃗f⟩⟩Łp(Q) · ⟨⟨⃗g⟩⟩Łq(Q).
+The estimate ≲ is immediate: From δ-sparseness, we have |Q| ≤ δ−1|E(Q)| for
+some disjoint sets E(Q), and hence
+�
+Q∈S
+aQ|Q| ≤ 1
+δ
+�
+Q∈S
+aQ|E(Q)| = 1
+δ
+ˆ
+Rd
+�
+Q∈S
+aQ1E(Q) ≤ 1
+δ
+ˆ
+Rd sup
+Q∈D
+aQ1Q.
+The estimate ≳ needs a bit more. By monotone convergence, it is enough to
+consider D(Q0) in place of D. Let S0 := {Q0}. For some A > 1 to be chosen and
+Q ∈ D(Q0), let S ′(Q) consist of all maximal Q′ ∈ D(Q) such that aQ′ > AaQ. By
+maximality, the cubes Q′ ∈ S ′(Q) are disjoint. By Lemma 8.2, we have
+�
+Q′∈S ′(Q)
+|Q′| ≤ nmax(1,r)+r/2
+Ar
+|Q| ≤ (1 − δ)|Q|,
+1
+r := 1
+p + 1
+q ,
+provided that A is chosen large enough, depending on n, p, q, and δ. Hence, defining
+inductively Sj+1 := �
+Q∈Sj S ′(Q) and S := �∞
+j=0 Sj, we find that S is δ-sparse.
+
+24
+T. P. HYTÖNEN
+If Q ∈ D(Q0) and S ∈ S is the minimal stopping cube that contains Q, then
+aQ ≤ AaS by the way that the cubes S ∈ S were chosen, hence
+sup
+Q∈D(Q0)
+1QaQ ≤ sup
+S∈S
+1SAaS ≤ A
+�
+S∈S
+1SaS,
+and thus ���
+sup
+Q∈D(Q0)
+1QaQ
+���
+L1(Rd) ≤ A
+���
+�
+S∈S
+1SaS
+���
+L1(Rd) = A
+�
+S∈S
+aS|S|.
+□
+References
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+Math. (2), 175(3):1473–1506, 2012.
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+setting. J. Lond. Math. Soc. (2), 106(1):1–26, 2022.
+
+SOME REMARKS ON CONVEX BODY DOMINATION
+25
+[22] M. T. Lacey. An elementary proof of the A2 bound. Israel J. Math., 217(1):181–195, 2017.
+[23] A. K. Lerner. A simple proof of the A2 conjecture. Int. Math. Res. Not. IMRN, (14):3159–
+3170, 2013.
+[24] A. K. Lerner. On pointwise estimates involving sparse operators. New York J. Math., 22:341–
+349, 2016.
+[25] A. K. Lerner. A weak type estimate for rough singular integrals. Rev. Mat. Iberoam.,
+35(5):1583–1602, 2019.
+[26] P. A. Muller and I. P. Rivera-Ríos. Quantitative matrix weighted estimates for certain singular
+integral operators. J. Math. Anal. Appl., 509(1):Paper No. 125939, 38, 2022.
+[27] F. Nazarov, S. Petermichl, S. Treil, and A. Volberg. Convex body domination and weighted
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+Department of Mathematics and Statistics, P.O.B. 68 (Pietari Kalmin katu 5),
+FI-00014 University of Helsinki, Finland
+Email address: [email protected]
+

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+Study of the long-range transverse field Ising model with fermionic Gaussian states
+Michael P. Kaicher,1, ∗ Davide Vodola,2, † and Simon B. J¨ager3
+1Departamento de F´ısica Te´orica, Universidad Complutense, 28040 Madrid, Spain
+2Dipartimento di Fisica e Astronomia, Universit`a di Bologna, I-40129, Bologna, Italy
+3Department of Physics and Research Center OPTIMAS,
+University of Kaiserslautern-Landau, D-67663 Kaiserslautern, Germany
+(Dated: January 10, 2023)
+We numerically study the one-dimensional long-range Transverse Field Ising Model (TFIM) in the
+antiferromagnetic (AFM) regime at zero temperature using Generalized Hartree-Fock (GHF) theory.
+The spin-spin interaction extends to all spins in the lattice and decays as 1/rα, where r denotes the
+distance between two spins and α is a tunable exponent. We map the spin operators to Majorana
+operators and approximate the ground state of the Hamiltonian with a Fermionic Gaussian State
+(FGS). Using this approximation, we calculate the ground state energy and the entanglement entropy
+which allows us to map the phase diagram for different values of α. In addition, we compute the
+scaling behavior of the entanglement entropy with the system size to determine the central charge
+at criticality for the case of α > 1. For α < 1 we find a logarithmic divergence of the entanglement
+entropy even far away from the critical point, a feature of systems with long-range interactions.
+We provide a detailed comparison of our results to outcomes of Density Matrix Renormalization
+Group (DMRG) and the Linked Cluster Expansion (LCE) methods. In particular, we find excellent
+agreement of GHF with DMRG and LCE in the weak long-range regime α ≥ 1, and qualitative
+agreement with DMRG in the strong-long range regime α ≤ 1. Our results highlight the power of
+the computationally efficient GHF method in simulating interacting quantum systems.
+I.
+INTRODUCTION
+Quantum phase transitions describe the behavior of
+quantum many-body systems at zero temperature when
+tuning a non-thermal control parameter, such as an ap-
+plied magnetic field.
+The phase transition appears as
+a result of competing phases that describe the ground
+state at the corresponding parameter and typically lead
+to a fundamental change in the nature of the correlation
+present in the ground state. Quantum many-body sys-
+tems can undergo a quantum phase transition and their
+study has lead to the discovery of many exotic collective
+phenomena such as superconducting ground states [1],
+long-range topological order [2], and anyonic statistics[3].
+Close to the critical point, the properties of many differ-
+ent physical systems can be classified by a universality
+class which is independent of the system size and only
+depends on the underlying dimensions and symmetries of
+the problem. In this situation, one can in many instances
+describe the many-body problem by an interacting spin
+system [4].
+One of the paradigmatic microscopic models display-
+ing a quantum phase transition is the Transverse Field
+Ising Model (TFIM) at zero temperature [5]. This model
+is exactly solvable in the limit of short-range, nearest-
+neighbour interactions.
+However, the solution of this
+problem is much harder if one considers beyond nearest-
+∗ Correspondence to:
+michael.p.kaicher(at)gmail.com; Now at:
+BASF Digital Solutions, Next Generation Computing, Pfalz-
+grafenstr. 1, D-67056, Ludwigshafen, Germany
+† Now at: BASF Digital Solutions, Next Generation Computing,
+Pfalzgrafenstr. 1, D-67056, Ludwigshafen, Germany
+neighbour or even long-range interactions [6–10]. Long-
+range interacting systems can host exotic states of quan-
+tum matter and are therefore of large scientific interest.
+Recent advances have made effective long-range spin-
+interactions experimentally accessible [11–15].
+In such
+systems, the effective interaction extends to all spins in
+the lattice and decays as a power law 1/rα, where r is
+the distance of the spins in the lattice and α is a tunable
+algebraic exponent. In the experiments one can realize
+0 ≤ α ≤ 3 which allows one to experimentally probe the
+regime of long-range interactions in spin systems [11].
+In order to analyze the properties of a quantum many-
+body system, it is important to study large system sizes,
+which is in our case the number of spins N. The exponen-
+tial scaling of the Hilbert space dimension with N makes
+the ad-hoc diagonalization of such many-body problems
+illusive. Consequently, one demands numerical methods
+which are able to capture the qualitative behavior of the
+many-body system with a computational cost that dis-
+plays a low scaling with N. A range of many-body meth-
+ods of varying computational complexity have been ap-
+plied to study finite size long-range quantum many-body
+systems, including Quantum Monte Carlo (QMC) [16],
+stochastic series expansion QMC [17], a combination of
+QMC and renormalization group methods [18], Lanczos
+exact diagonalization [19], and Density Matrix Renor-
+malization Group (DRMG) [6, 8]. Recently, a method
+to study short-range quantum-lattice models in the ther-
+modynamic limit, the Linked-Cluster Expansion (LCE),
+has been extended to allow for the study of long-range
+systems for α > 1 [9, 10].
+In this work, we add Generalized Hartree-Fock (GHF)
+theory to this mix of methods.
+GHF is a mean-field
+method which aims to approximate the ground state of
+arXiv:2301.02939v1  [quant-ph]  7 Jan 2023
+
+2
+an interacting quantum system as a free electron gas [20],
+where the latter describes a class of variational func-
+tions known as Fermionic Gaussian States (FGS). Due
+to its mean-field nature, GHF is a method with very
+low computational cost, where the most-demanding com-
+pute operation—the evaluation of the Pfaffian Pf(A) of
+a M × M matrix A—scales at most as O(M 3) [21].
+Even though FGS describe ground or thermal states of
+quadratic fermionic Hamiltonians [22], they have been
+applied to various areas of quantum many-body physics
+with great success, most notably as ab-initio methods to
+obtain approximate ground states in electronic structure
+problems and to condensed matter systems [20, 23, 24].
+In this paper, in order to find the FGS which best ap-
+proximates the ground state of the long-range TFIM,
+we employ two physically-motivated methods which have
+been described in Ref. [23]. The first one (ITE) derives
+the ground state using Imaginary Time Evolution. The
+second one (ZT) uses a self-consistent equation for the
+FGS ground state covariance matrix. Using these two
+methods we calculate the ground state energy and the
+entanglement entropy.
+By comparison of these results
+with the ones obtained from DMRG and ZT we will show
+that GHF is able to capture the qualitative and quantita-
+tive behavior of the long-range TFIM. This highlights the
+ability of GHF in predicting physically relevant material
+properties at computationally low cost.
+This work is structured as follows. In Section II we
+discuss the GHF theory which we then apply to the
+TFIM model described in
+II A. The introduced meth-
+ods are used in Section III where we numerically study
+the ground state energy and the entanglement entropy.
+We conclude by summarizing our findings in Section IV
+and providing an outlook for future work.
+II.
+THEORY
+A.
+Long-range transverse field Ising model
+In this work we consider the TFIM Hamiltonian de-
+scribing a system of N spins with open boundary condi-
+tions
+ˆH =
+N
+�
+p=1
+hpˆσz
+p +
+N
+�
+p<q
+Jpqˆσx
+p ˆσx
+q ,
+(1)
+where we introduced the transversal magnetic field
+strength hp = cos(θ), the interaction strength Jpq =
+sin(θ)/|p − q|α and the Pauli matrices ˆσa
+p (a ∈ {x, y, z})
+for each spin indexed by p, q.
+The magnetic field and
+interactions strengths are parameterized by the angle
+θ and the algebraic scaling of the interaction range is
+given by α. In this work we furthermore focus on an-
+tiferromagnetic (AFM) couplings which implies Jpq > 0
+or θ ∈ (0, π).
+Because the Hamiltonian is symmetric
+under the simultaneous transformations ˆσz
+p �→ −ˆσz
+p and
+θ → π − θ we can restrict our study to θ ∈ (0, π/2].
+In a next step, we map the TFIM Hamiltonian onto a
+fermionic Hamiltonian. To this end, we use the Jordan-
+Wigner transformation ˆσ+
+p = ˆc†
+peiπ �p−1
+q=1 ˆc†
+qˆcq and ˆσ−
+p =
+ˆcpe−iπ �p−1
+q=1 ˆc†
+qˆcq [25]. Here, we used ˆσ±
+p = [ˆσx
+p ± iˆσy
+p]/2
+and introduced the fermionic raising and lowering opera-
+tors ˆc†
+p, ˆcp, respectively. The latter obey the canonical an-
+ticommutation relations {ˆcp, ˆcq} = 0 and {ˆcp, ˆc†
+q} = δp,q,
+where δp,q is the Kronecker delta and { ˆA, ˆB} = ˆA ˆB+ ˆB ˆA
+denotes the anticommutator of two operators ˆA, ˆB. In-
+stead of analyzing the problem in the basis of the 2 × N
+fermionic operators ˆcp, ˆc†
+p we represent the Hamiltonian
+in 2N Majorana operators ˆa2p−1 = ˆc†
+p + ˆcp and ˆa2p =
+i(ˆc†
+p−ˆcp). The latter posses the anticommutation relation
+{ˆal, ˆam} = 2δl,m (l, m = 1, 2, . . . , 2N) and the Hamilto-
+nian (1) in the Majorana representation is given by
+ˆH = − i
+N
+�
+p=1
+hpˆa2p−1ˆa2p +
+N
+�
+p<q
+(−i)q−pJpqˆa2p ˆSpqˆa2q−1.
+(2)
+Here,
+we
+introduced
+the
+string
+operator
+ˆSpq
+=
+�q−1
+k=p+1 (ˆa2k−1ˆa2k) which is the product of 2×(q −p+1)
+Majorana operators. For nearest-neighbour interactions,
+α = ∞ and Jpq = δp,q±1, this string operator becomes
+the identity, ˆSpq = 1 and ˆH becomes quadratic in the
+Majorana operators. Consequently, the model can be de-
+scribed by free fermions and is therefore exactly solved by
+a FGS [22]. In general, however, for the long-range TFIM
+we will need to include the contribution of the operator
+ˆSpq. To avoid ambiguity, we use the term long-range in
+this work for all systems with α < ∞, since the spin in-
+teraction breaks up into a sum of terms proportional to
+1/|p−q|α, where all lattice sites p, q give non-zero contri-
+butions, and not just nearest-neighbour sites p, p + 1 (as
+in the special case α → ∞). Often times, the term long-
+range is reserved in literature for an algebraic exponent
+α = σ + d in a d-dimensional system for σ < 0 (which
+in a one-dimensional system refers to the regime α < 1)
+[26, 27]. Thus, to avoid confusion, in our work we will
+refer to α < 1 as the strong long-range, and to α > 1 as
+the weak long-range regime, while the special case α = 1
+is marginal.
+B.
+Fermionic Gaussian States
+The formal definition of a FGS is given by [22],
+ˆρGS =tr
+�
+e−β ˆ
+HGS�−1
+e−β ˆ
+HGS,
+(3)
+where ˆHGS =
+i
+4ˆaT Gˆa is a Hermitian operator, β ∈ R,
+ˆa = (ˆa1, ˆa2, . . . , ˆa2N)T is a column vector of Majorana
+operators, and G is a (2N × 2N) real-valued and anti-
+symmetric matrix. FGS are fully described by the real
+
+3
+and anti-symmetric covariance matrix Γ with entries
+Γlm = i
+2tr (ˆρGS[ˆal, ˆam]) ,
+(4)
+l, m ∈ {1, 2, . . . , 2N}, and where [ ˆA, ˆB] =
+ˆA ˆB − ˆB ˆA
+denotes the commutator of two operators ˆA, ˆB. While
+Eqs. (3)-(4) describe both pure and mixed FGS, we only
+focus on pure FGS in this work, since we are interested
+in the ground state.
+Pure FGS are characterized by
+Γ2 = −12N (1k denotes the (k × k)-identity matrix),
+and eigenvalues of the covariance matrix are given by
+λ ∈ {−1, 1}. All information contained in the density
+matrix (3) of a FGS is also contained in its covariance
+matrix (4). The expectation value of a single tensor prod-
+uct of Majorana or fermionic operators can be computed
+efficiently through Wick’s theorem [22, 28],
+tr (ˆρGSˆai1ˆai2 · · · ˆai2m) =(−i)mPf
+�
+Γ|i1i2...i2m
+�
+,
+(5)
+where i1 ̸= i2 ̸= . . . ̸= i2m for ik ∈ {1, . . . , 2N} and k =
+1, . . . , 2N. The matrix Γ|i1i2...i2m denotes a (2m × 2m)-
+submatrix of Γ with the corresponding rows and columns
+i1, i2, . . . , i2m, and Pf(A) denotes the Pfaffian of a skew-
+symmetric matrix A.
+C.
+Approximating the ground state with a
+fermionic Gaussian State
+Using Wick’s theorem (5), we are able to compute the
+energy expectation value
+E(Γ) =tr
+�
+ˆρGS ˆH
+�
+,
+(6)
+which results in
+E(Γ) = −
+N
+�
+p=1
+hp
+2 (Γ2p−1,2p − Γ2p,2p−1)
++
+N
+�
+p<q
+Jpq(−1)q−pPf
+�
+Γ|2p,2p+1,...,2q−1
+�
+,
+(7)
+for the Hamiltonian given by Eq. (1). In order to ap-
+proximate the ground state of the Hamiltonian within
+the family of FGS, one has to find a covariance matrix
+Γ which minimizes E(Γ). While one can apply any con-
+strained optimization method, in the following, we will
+discuss two particular algorithms for finding the optimal
+Γ, which we will use in Section III.
+a.
+Imaginary Time Evolution (ITE)
+The first algo-
+rithm performs an Imaginary Time Evolution (ITE) un-
+der the constraint that Wick’s theorem holds through-
+out the evolution. This constraint guarantees that the
+evolved state remains a FGS, and leads to an equation
+of motion for the corresponding covariance matrix,
+dΓ
+dτ = 1
+2[Γ, [Γ, H(mf)]],
+(8)
+where τ ∈ R denotes the imaginary time.
+We derive
+Eq. (8) in Appendix A. The central quantity is hereby the
+mean-field Hamiltonian H(mf)(Γ) which is the gradient of
+the energy with respect to the covariance matrix,
+H(mf)
+lm
+= 4dE(Γ)
+dΓlm
+.
+(9)
+This term can be computed explicitly by using identities
+for the matrix derivative of a Pfaffian, which we have also
+employed in Appendix B.
+We solve Eq. (8) iteratively, by discretizing the ITE
+into small time steps ∆τ. Starting from a random ini-
+tial covariance matrix, we evolve the covariance ma-
+trix through Γ(τ + ∆τ) ≈ O(∆τ)Γ(τ)O(∆τ)T , where
+O(∆τ) = e
+1
+2[H(mf),Γ]∆τ is an orthogonal matrix. As ex-
+plicitly shown in Ref. [23], this approach preserves the
+purity of the FGS, while ensuring a monotonic decrease
+of the energy in each iteration.
+b.
+Zero Temperature (ZT)
+The second algorithm
+uses a self-consistent equation for the steady-state so-
+lution of Eq. (8). In this algorithm, for a given Γ, we
+diagonalize the mean-field matrix iH(mf) = UDU† and
+recalculate
+Γ = iUsgn(D)U†.
+(10)
+Here, U is a unitary matrix, D is a diagonal matrix con-
+taining the real eigenvalues of iH(mf), and sgn(D) is the
+sign function applied to the diagonal entries of D. From
+this Γ we recalculate iH(mf) and repeat the procedure
+until the covariance matrix is converged. One can check
+that the solution of this algorithm is also a stationary
+state of Eq. (8) with Γ2 = −1.
+In both algorithms we choose several random initial
+covariance matrices Γinit to ensure unbiased results. A
+random Γinit is generated through Γinit = OT ΩO, where
+O is a random orthogonal matrix and we defined the
+block diagonal matrix Ω = �N
+k=1(−1)rk � 0
+1
+−1 0
+�
+, where
+rk ∈ {0, 1} is chosen randomly and � denotes the direct
+sum. After convergence of the corresponding algorithm
+we achieve a stationary solution Γst. With the help of
+this solution we can then find the GHF approximation
+to the ground state energy given by E(Γst). Besides the
+energy and entanglement entropy introduced in the fol-
+lowing section, the covariance matrix also allow us direct
+access to quantum correlations.
+D.
+Entanglement entropy and central charge
+Entanglement entropy is a well-studied measure for
+the amount of quantum correlations in a pure quantum
+state [29, 30]. It is defined as SNA = −tr(ˆρA log(ˆρA)),
+where A describes a subsystem containing NA spins. The
+reduced density matrix ˆρA = trB(ˆρ) is obtained by per-
+forming a partial trace over the disjoint subsystem B,
+with NB = N − NA spins. For the spins numbered as
+
+4
+A = {1, 2, . . . , N/2} we define the corresponding Majo-
+rana operators by MA = {1, 2, . . . , N−1, N}. The entan-
+glement entropy is then fully determined by the matrix
+ΓA = Γ|MA and can be calculated with [23, 31, 32]
+SN/2 =N
+2 log(2) − 1
+2tr [(1N + iΓA) log (1N + iΓA)] .
+(11)
+For short-range 1D systems, the entanglement en-
+tropy typically follows two different scalings: for gapped
+phases, SN/2 saturates to a constant value independent of
+N and thus obeys the so-called area law [33]. For gapless
+phases, the entanglement entropy exhibits the following
+behavior [34]
+SN/2 = c
+6 log(N) + B,
+(12)
+where c is the central charge characterizing the universal-
+ity class of the system and B is a non-universal constant.
+For the nearest-neighbour TFIM at α = ∞ the value of
+c = 1/2 can be found exactly.
+For long-range systems we need to differentiate be-
+tween weak long-range interactions, α > d = 1, and the
+strong long-range interactions, α < d = 1.
+For weak long-range interactions and a non-vanishing
+energy gap we expect also an area law scaling, implying
+that SN/2 is independent of N. For the case of a vanish-
+ing gap one also finds a logarithmic divergence [33, 35–37]
+following Eq. (12).
+For strong long-range interactions in the AFM-TFIM
+we expect instead a logarithmic divergence of the entan-
+glement entropy, where SN/2 obeys Eq. (12) and one can
+find c ̸= 0 even in presence of a non-vanishing gap [38–
+41]. In this regime c is strictly speaking not a central
+charge but because of the same functional dependence of
+SN/2 in Eq. (12), we also denote c as the effective central
+charge.
+III.
+RESULTS
+A.
+Phase diagram
+In this section, we show that a computationally in-
+expensive GHF mean-field approach can reproduce the
+phase diagram of the AFM-TFIM for a wide range of
+values α, both in the weak and strong long-range regime,
+and is able to locate the point of the phase transition
+for α ≥ 1 in excellent agreement with state-of-the-art
+numerical methods.
+As a first benchmark and in the same spirit of Ref. [6]
+we map the phase diagram by calculating the entangle-
+ment entropy for a wide range of values of α, from weak to
+strong long-range interactions, and for θ ∈ (0, π/2). The
+values of SN/2 [Eq. (11)] computed with the ZT GHF
+method are visible in Fig. 1 for N = 100. For θ = 0 the
+interactions vanish and SN/2 = 0 for all values of α. This
+FIG. 1.
+We plot the entanglement entropy SN/2 from the
+covariance matrix obtained through the ZT algorithm for a
+system size N = 100, α ∈ {0.30, 0.50, 0.75, 1.00, . . . , 3.00},
+and θ ∈ (0, π/2). Black squares represent the quantum critical
+points θ∞
+c /π in the thermodynamic limit, which are listed in
+Tab. I, while the dashed line serves as a guide to the eye.
+represents the phase where all spins are uncorrelated and
+align with the external magnetic field. However, when
+θ and therefore the AFM interactions are increased, the
+minimization of the interaction energy competes with the
+external magnetic field. This is accompanied by an in-
+crease of SN/2. Dependent on α, there is a critical value
+θc(α) beyond which the spins favor an AFM order. This
+transition is highlighted in Fig. 1 by a sharp rise of SN/2.
+Our findings are in qualitative agreement with the ones
+obtained in Ref. [6] from DMRG calculations. To com-
+pare our results also quantitatively, we will now focus on
+the weak and strong long-range interactions cases sepa-
+rately.
+B.
+Weak long-range interactions
+1.
+Comparison of GHF and DMRG
+For weak long-range interactions, α ≥ 1, we show the
+ground state energy and the entanglement entropy in
+Fig. 2(a) and Fig. 2(b), respectively.
+The solid lines represent the results obtained from
+the GHF theory while hollow markers represent the re-
+sults obtained from DMRG simulations. Both simulation
+methods predict a rather smooth behavior of the energy
+in Fig. 2(a). For larger values of α ≥ 1.5 we find a max-
+imum and a decrease beyond the maximum point. The
+GHF and DMRG simulations agree perfectly.
+The entanglement entropy, visible in Fig. 2(b), shows
+for all values and both simulations methods a very quick
+increase and a pronounced singularity. The latter is an
+indicator for the phase transition point.
+Beyond this
+
+1.4
+3.0
+1.2
+2.5
+1.0
+2.0 -
+0.8
+SN/2
+a
+1.5
+0.6
+1.0
+0.4
+0.2
+0.5
+0.1
+0.2
+0.3
+0.4
+0/ π5
+(a)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+θ/π
+−100
+−95
+−90
+−85
+−80
+−75
+−70
+Energy
+ZT, α=1.0
+ZT, α=1.25
+ZT, α=1.5
+ZT, α=1.75
+ZT, α=2.0
+ZT, α=2.25
+ZT, α=2.5
+ZT, α=2.75
+ZT, α=3.0
+DMRG
+(b)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+θ/π
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+SN/2
+ZT, α=1.0
+ZT, α=1.25
+ZT, α=1.5
+ZT, α=1.75
+ZT, α=2.0
+ZT, α=2.25
+ZT, α=2.5
+ZT, α=2.75
+ZT, α=3.0
+DMRG
+FIG. 2.
+For a system of size N = 100 and exponents
+α ∈ [1, 3], we plot (a) the energy E and (b) the entanglement
+entropy SN/2 (bottom), as defined in Eqs. (6) and (11), ob-
+tained from the covariance matrix of the ZT algorithm (solid
+lines) and compare it to DMRG (hollow markers).
+point we find again a decrease of the entanglement en-
+tropy. Both methods, GHF and DMRG, are in very good
+agreement.
+2.
+Threshold and central charge
+In order to find a value for the threshold at N → ∞,
+we are performing a finite-size scaling.
+For this we
+carry out analogue simulations for a range of smaller
+system sizes N ∈ {20, 30, . . . , 100}.
+Then we find nu-
+merically the maximum of the entanglement entropy
+of the half chain Smax = SN/2(θmax) and the corre-
+sponding value θmax.
+The latter is found using the
+0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050
+1/N
+0.26
+0.28
+0.30
+0.32
+0.34
+0.36
+0.38
+θmax/π
+α =1.0, θ∞
+c /π =0.3534(4)
+α =2.0, θ∞
+c /π =0.3013(2)
+α =3.0, θ∞
+c /π =0.2760(2)
+FIG. 3.
+Example for the fit of Eq. (13) to the value of θmax
+obtained by maximization of the entanglement entropy S with
+FGS. The thresholds θ∞
+c /π are shown for the respective cases
+α ∈ {1, 2, 3}, see Tab. I for more details.
+optimizer scipy.optimize.fminbound() which is pre-
+implemented in python.
+For every value θ examined
+by the optimizer we find the optimal FGS for the cor-
+responding Hamiltonian. Optimizing SN/2 over θ can be
+achieved as FGS provide a way for calculating SN/2 poly-
+nomially in N, see Eq. (11). We then use the following
+finite-size scaling law [42]
+θmax(N) = θ∞
+c + a
+N ,
+(13)
+where θ∞
+c
+is the threshold at N → ∞ and a is a fitting
+parameter which determines the finite-size scaling. Fit-
+ting Eq. (13) to the numerically obtained data of θmax
+reveals the θ∞
+c
+in the thermodynamic limit. In Fig. 3 we
+provide examples for the fits that are used to calculate
+θ∞
+c . We perform these fits for various values of α and the
+results for the threshold are collected in Tab. I. In addi-
+tion, we have plotted the results of θmax
+c
+in Fig 1 as black
+squares which mark the sudden spike of the entanglement
+entropy. In Tab. I we compare the results obtained from
+the GHF theory with the ones obtained from LCE calcu-
+lations [10], DMRG data of Ref. [6] (labeled DMRG) and
+Ref. [8] (labeled DMRG*). We find in general very good
+agreement of the thresholds obtained from the different
+methods.
+Besides the threshold θ∞
+c
+we can also extract the scal-
+ing of the maximum entropy Smax = S(θmax). At the
+critical point we use the scaling law [34] given by Eq. (12).
+We fit Eq. (12) to the maximum values Smax as displayed
+in Fig. 4(a). From these fits we extract the central charge
+c, which is shown in Fig. 4(b) as function of α. The cen-
+tral charge is always above the result c = 1/2 expected
+from the short-range TFIM. We also compare our results
+to different DMRG results of Ref. [6, 8]. We find that
+the central charges obtained from FGS are systematically
+smaller than the values provided by Ref. [6] and larger
+
+6
+than the DMRG results of Ref. [8]. The central charge c is
+monotonically decreasing in the weak long-range regime,
+but drops at the onset of the strong long-range regime
+at α = 1. In conclusion, we found that the results of
+the GHF method are in good qualitative and quantita-
+tive agreement with state-of-the-art numerical methods
+for weak long-range interactions.
+C.
+Strong long-range interactions
+1.
+Comparison of GHF and DMRG
+We will now shift our focus to the regime of strong
+long-range interactions, α < 1. We first plot the ground
+state energy and the entanglement entropy in Fig. 5(a)
+and Fig. 5(b) for three different values of α < 1 of size
+N = 100.
+In Fig. 5(a) we obtain for all three values
+of α a monotonously increasing energy with θ. This is
+different to the case of weak long-range interactions (see
+Fig. 2(a)) where we have observed a maximum close to
+the threshold at least for sufficiently large α ≥ 1.5. We
+compare our results obtained from FGS also with the
+ones obtained from DMRG results. Here, we find that
+DMRG always predicts a lower ground state energy. The
+discrepancy of the two methods is even more striking
+in the entanglement entropy visible in Fig. 5(b). Here,
+while we still observe very good agreement for α = 0.75
+we found clear deviations for α = 0.3. The DMRG results
+predict tendentially a larger entanglement entropy than
+the FGS. This is an indicator that FGS are less well-
+suited for the description of the TFIM for very small α,
+i.e. very strong long-range interactions.
+α
+θ∞
+c /π FGS
+LCE
+DMRG DMRG*
+1.00
+0.3534(4) -
+0.3509
+-
+1.25
+0.3357(1) 0.35(5)
+-
+-
+1.50
+0.3218(1) 0.3213(5)
+0.3226
+-
+1.75
+0.3106(1) -
+-
+-
+2.00
+0.3013(2) 0.3026(8)
+0.3027
+0.3021
+2.25
+0.2932(2) 0.294(4)
+-
+-
+2.50
+0.2865(1) 0.2871(11)
+-
+-
+2.75
+0.2807(2) -
+-
+-
+3.00
+0.2760(2) 0.27722(25) 0.2782
+-
+TABLE I. The critical points θ∞
+c /π obtained from Eq. (13)
+with FGS and ZT, in comparison to LCE [10], and DMRG
+[6], DMRG*[8] results. The values are obtained for various
+exponents α and for simulations up to N = 100 spins. The
+error indicated in the FGS column in round brackets is the
+standard deviation for the intersect of a linear regression fit
+of θmax/π as a function of 1/N.
+(a)
+3.0
+3.2
+3.4
+3.6
+3.8
+4.0
+4.2
+4.4
+4.6
+log(N)
+0.72
+0.74
+0.76
+0.78
+0.80
+SN/2
+α =1.0, c =0.5455(40)
+α =2.0, c =0.5428(72)
+α =3.0, c =0.5269(73)
+(b)
+1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+2.50
+2.75
+3.00
+α
+0.50
+0.52
+0.54
+0.56
+0.58
+0.60
+c
+ZT
+FGS
+DMRG
+DMRG*
+FIG. 4.
+(a) Extracting the central charge. Using the ZT
+algorithm for various α, here exemplified by α ∈ {1, 2, 3}, we
+plot the entanglement entropy SN/2 against log(N). For each
+α we perform a linear regression fit, neglecting the system
+sizes N ∈ {20, 30, 40} to mitigate finite size effects. (b) Cen-
+tral charge c obtained from finite-size scaling up to system
+size N = 100 of FGS evolutions through the ZT algorithm
+(blue squares) for the AFM long-range TFIM. For compari-
+son, DMRG results from finite-size scaling of system sizes of
+up to N = 100 from Ref. [6] (’DMRG’, orange square) and
+[8] (’DMRG*’, green triangles) are included. The red hori-
+zontal line represents the value c = 1/2 which describes the
+Ising universality class.
+Error bars represent the standard
+deviation from the linear regression fit.
+2.
+Violations to the area law
+We will now analyze the scaling of the entanglement
+entropy with the system size. For this we calculate the
+entanglement entropy for various parameters θ and α and
+for different numbers of spins N ∈ {40, 50, . . . , 100}. We
+
+7
+(a)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+θ/π
+−100
+−90
+−80
+−70
+−60
+Energy
+ITE, α=0.3
+ITE, α=0.5
+ITE, α=0.75
+DMRG
+(b)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+θ/π
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+SN/2
+ITE, α=0.3
+ITE, α=0.5
+ITE, α=0.75
+DMRG
+FIG. 5.
+(a) Energy and (b) entanglement entropy obtained
+from the covariance matrix of the ITE algorithm (solid lines)
+and DMRG (empty markers) simulations for N = 100 and
+α ∈ {0.3, 0.5, 0.75}.
+then fit the coefficients c and B using Eq. (12) to the
+obtained values of the entanglement entropy.
+The ob-
+tained values of c are shown in Fig. 6. At this point we
+remark that the effective central charge c is calculated far
+away from the threshold in a phase with a non-vanishing
+energy gap [6].
+For the values α < 1, we find c = 0 only at θ = 0. For
+increasing θ we find a sharp increase of c. For α = 0.3
+and α = 0.5 we find a maximum and then a decrease
+again for larger values of θ. A qualitatively similar be-
+havior has also been observed in Ref. [6]. This has been
+seen as a violation to the area law since this logarithmic
+divergence does not originate from a closing gap in the
+spectrum of the system [6]. We therefore conclude that
+the FGS are able to predict this feature, although the
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+θ/π
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+0.045
+c/6
+α =0.3
+α =0.5
+α =0.75
+α =1
+FIG. 6.
+Violations to the area law: The effective central
+charge c [Eq. (12)] calculated from finite scaling of system
+sizes N ∈ {40, 50, . . . , 100} for 50 different values deep in the
+gapped region θ ∈ (0, π/4) for the GHF ITE algorithm. Error
+bars for the standard deviation are also included, but too
+small to be visible.
+quantitative values deviate from the ones obtained from
+DMRG results.
+IV.
+SUMMARY AND OUTLOOK
+This work presents an extensive study of the AFM
+long-range TFIM in both the weak and strong long-range
+regime using generalized Hartree Fock theory, a mean-
+field method with low computational cost. We validate
+our results by comparing the computed energy and entan-
+glement entropy to DMRG. We plot the phase diagram
+and provide estimates for the location of the critical point
+of the second order phase transition through finite-size
+scaling for α ∈ [1, 3] and find that they are in excel-
+lent agreement with both LCE calculations of Ref. [10]
+and DMRG simulations of Refs. [6, 8]. At the critical
+point, we compute the central charge c of the underly-
+ing conformal field theory for α ∈ {0.3, 0.5, 1}, and find
+c > 1/2 for all values of α.
+In the strong long-range
+regime we still found qualitative agreement between FGS
+and DMRG calculations. Hereby we found larger quan-
+titative deviations for smaller values of α. Remarkably,
+GHF can predict the logarithmic violations to the area
+law in the AFM-TFIM which has previously been stud-
+ied with DRMG. Based on these findings, we conclude
+that FGS provide a numerically inexpensive alternative
+to study the AFM long-range TFIM and that our results
+are in good agreement with DMRG, the current state-
+of-the-art numerical method for one-dimensional lattice
+systems.
+All simulations were carried out using a standard lap-
+top computer.
+Since the dimensionality of the system
+only appears in the Hamiltonian elements hpq and Jpq,
+
+8
+it is straightforward to apply FGS to the two- and
+three-dimensional TFIM. Therefore, it would be inter-
+esting to compare FGS simulations with methods that
+can be applied to the two-dimensional AFM-TFIM [10].
+Moreover, while we have focused on the AFM regime,
+FGS can readily be applied to the ferromagnetic regime
+θ ∈ (−π, 0). In this work we have focused on the en-
+tanglement entropy, however, pair correlation functions
+and the entanglement spectrum can be extracted from
+the covariance matrix as well.
+FGS can also be used
+to study dynamics under the evolution of the TFIM,
+with equations of motion similar to Eq. (8) [23, 24]. In
+particular, studying the dynamics of the entropy after a
+quench would offer the possibility to verify the breaking
+of conformal symmetry in the regime α < 1 [43]. From
+a numerical standpoint, more efficient calculations of the
+central quantities such as Eqs. (7) could lead to dramatic
+computational speedups. As a possible pathway, it would
+be interesting to see if sum-identities for Pfaffians such as
+provided in Refs. [44–46] could be applied to the TFIM
+Hamiltonian. Finally, one could study if different spin-
+to-fermion mappings [47–49], each resulting in a different
+form of H when expressed in fermionic operators, have
+an effect on the FGS simulations.
+ACKNOWLEDGMENTS
+The authors thank Kai Phillip Schmidt for providing
+the data for the LCE calculations from Ref.[10] and Luca
+Tagliacozzo for providing the DMRG data from Ref. [6].
+The authors also thank Giovanna Morigi for insightful
+discussions. M.K. thanks Miguel ´Angel Mart´ın-Delgado
+and Frank Wilhelm-Mauch for helpful discussions and
+support. S.B.J. acknowledges support from the Research
+Centers of the Deutsche Forschungsgemeinschaft (DFG):
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+R. Babbush, P. V. Coveney, F. Mintert, F. Wilhelm, and
+P. J. Love, International Journal of Quantum Chemistry
+115, 1431 (2015).
+[49] Z. Jiang, A. Kalev, W. Mruczkiewicz, and H. Neven,
+Quantum 4, 276 (2020).
+[50] R. M. Wilcox, Journal of Mathematical Physics 8, 962
+(1967).
+Appendix A: Derivation of the equations of motion for the ITE algorithm
+In this section, we will derive Eq. (8) which describes the imaginary-time evolution of a FGS. Eq. (8) was also
+shown in Ref. [23] for fourth-order polynomials of fermionic opertors and for an even more general case in Ref. [24].
+We start by writing down the imaginary-time time evolution for the pure FGS ˆρGS = |ΨGS⟩ ⟨ΨGS| determined by
+d
+dτ |ΨGS⟩ = −
+�
+ˆH − ⟨ΨGS| ˆH|ΨGS⟩
+�
+|ΨGS⟩ .
+(A1)
+The pure FGS can be generated by a Gaussian transformation
+|ΨGS⟩ = ˆUGS |vac⟩ ,
+(A2)
+where |vac⟩ denotes the fermionic vacuum and
+ˆUGS(ξ) =e
+i
+4 ˆaT ξˆa
+(A3)
+describes the generator of a pure FGS [22]. Here, ξ denotes a (2n × 2n) anti-symmetric and Hermitian matrix (the
+matrix elements ξkl = −ξlk are purely imaginary). To calculate the covariance matrix Γ we use
+Γ = −UξΥUT
+ξ ,
+(A4)
+where
+Υ =
+N
+�
+p=1
+�
+0
+1
+−1 0
+�
+.
+(A5)
+is the covariance of the vacuum state and where we employed the transformation
+ˆU †
+GS(ξ)ˆa ˆUGS(ξ) = Uξˆa,
+(A6)
+with
+Uξ = eiξ.
+(A7)
+Now, the idea is that we derive from Eq. (A1) a differential equation for Uξ which can then be used to calculate Γ
+using Eq. (A4). For this we treat the left- and right-hand side of Eq. (A1) separately and rewrite it as
+ˆUGS ˆL |vac⟩ = ˆUGS ˆR |vac⟩ .
+(A8)
+We then expand the operators ˆL and ˆR up to second order in terms of normal-ordered monomials of the Majorana
+operators and apply them to the vacuum state.
+a.
+Left-hand side of Eq. (A1):
+The operator ˆL is defined as
+ˆL = ˆU †
+GS
+�
+d ˆUGS(ξ)
+dτ
+�
+.
+(A9)
+
+10
+The derivative of the unitary transformation is given by
+d ˆUGS(ξ)
+dτ
+=UGS(ξ)
+� i
+4ˆaT UT
+ξ
+dUξ
+dτ ˆa
+�
+.
+(A10)
+Here, we have used Eq. (A3), the identity [50]
+de ˆ
+J(τ)
+dτ
+=
+� 1
+0
+due(1−u) ˆ
+J(τ) �
+dτ ˆJ(τ)
+�
+eu ˆ
+J(τ),
+(A11)
+and the orthogonality property UξUT
+ξ = 1. For the normal-ordered expression we therefore find
+ˆL = ˆL0 + ˆL2,
+(A12)
+with
+ˆL0 = i
+4tr
+�dUξ
+dτ UT
+ξ Γ
+�
+,
+(A13)
+ˆL2 =1
+4 : ˆaT UT
+ξ
+dUξ
+dτ ˆa :,
+(A14)
+where : ˆA : denotes the elementwise normal-ordering of ˆA.
+b.
+Right-hand side of Eq. (A4):
+The definition of ˆR is
+ˆR = − ˆU †
+GS
+�
+ˆH − ⟨ΨGS| ˆH |ΨGS⟩
+�
+ˆUGS.
+(A15)
+We can now use a modification of Wicks theorem to calculate
+ˆU †
+GS ˆH ˆUGS = ⟨ΨGS| ˆH |ΨGS⟩ + i
+4 : ˆaT UT
+ξ H(mf)Uξa : + ˜Q,
+(A16)
+where ˜Q collects all normally ordered monomials of quartic order or higher. We derive this expression in Appendix B.
+Inserting Eq. (A16) into Eq. (A15), we find for ˆR the following expression
+ˆR = ˆR2 − ˜Q,
+(A17)
+with
+ˆR2 = − i
+4 : ˆaT UT
+ξ H(mf)Uξa : .
+(A18)
+c.
+Comparing left- and right-hand side:
+We now require to match ˆL |vac⟩ and ˆR |vac⟩ up to second order. This
+is a consequence of our restriction to FGS. Therefore, we find two equations
+ˆL0 |vac⟩ =0,
+(A19)
+ˆL2 |vac⟩ = ˆR2 |vac⟩ ,
+(A20)
+from which we wish to derive the equations of motion of the ITE. We first consider Eq. (A20),
+: ˆaT UT
+ξ
+dUξ
+dτ ˆa : |vac⟩ = − i : ˆaT UT
+ξ H(mf)Uξˆa : |vac⟩ .
+(A21)
+For any normal-ordered polynomial of fermionic operators applied to the vacuum state, the only terms that do not
+vanish are polynomials which exclusively contain fermionic creation operators. Therefore, we define the vector
+r = ˆc† ⊗
+�
+1
+i
+�
+,
+(A22)
+where ˆc† = (ˆc†
+1, . . . , ˆc†
+N), and rewrite Eq. (A21) in terms of fermionic creation and annihilation operators, which leads
+to
+ˆrT UT
+ξ
+dUξ
+dτ ˆr |vac⟩ = − iˆrT UT
+ξ H(mf)Uξˆr |vac⟩ ,
+(A23)
+
+11
+where the normal-ordering “: :” may now be dropped. We would now like to compare the matrices of Eq. (A23).
+However, before doing so, we need to take into account the symmetry operations which leave the operator ˆr invariant.
+For this, we first rewrite Υ defined in Eq. (A5) as Υ = 1N ⊗
+� 0
+1
+−1 0
+�
+. Thus, the symmetry operations on the operators
+are given by −iΥˆr = ˆr and iˆrT Υ = ˆrT . The real-valued skew-symmetric solution for dUξ
+dτ
+which satisfies Eq. (A19)
+is then given by
+dUξ
+dτ
+= −1
+2ΓH(mf)Uξ − 1
+2H(mf)UξΥ.
+(A24)
+For the derivative of the covariance matrix Γ we find then
+dΓ
+dτ = − dUξ
+dτ ΥUT
+ξ − UξΥ
+dUT
+ξ
+dτ
+= −H(mf) − ΓH(mf)Γ,
+(A25)
+which is identical to Eq. (8) using Γ2 = −1.
+Appendix B: Best quadratic approximation
+In this section we will show Eq. (A16), which can be derived using Wick’s theorem. In particular, we will derive
+this formula for an arbitrary Hamiltonian which is a sum of even products of Majorana operators. By the application
+of normal-ordering onto a polynomial ˆp(k) of order k we understand the sum of normal-ordered monomials ˆm(l) of
+order l ≤ k, in other words : ˆp(k) := �k
+l=0 : ˆm(l) :. Therefore, it is sufficient to show the relation (A16) for arbitrary
+even products of Majorana operators. Without loss of generality we number the Majorana operators from 1, ..., 2n
+with n ∈ N. Using normal-ordering and Wicks theorem we can write the product ˆA = ˆa1ˆa2 . . . ˆa2n (a monomial of
+Majorana operators) in the following way
+ˆA = ⟨vac| ˆA |vac⟩ +
+�
+i<j
+(−1)i+j+1 ⟨vac| ˆAˆi,ˆj |vac⟩ : ˆaiˆaj : + ˆQ,
+(B1)
+where ˆQ collects all normal-ordered monomials which are at least quartic in the Majorana operators. We furthermore
+introduced the reduced product ˆAˆi,ˆj which emerges from ˆA by removing the operators ˆai and ˆaj. If we transform the
+vacuum state using Eq. (A2), this expansion needs to be modified according to
+ˆUGS ˆA ˆU †
+GS = ⟨ΨGS| ˆA |ΨGS⟩ +
+�
+i<j
+(−1)i+j+1 ⟨ΨGS| ˆAˆi,ˆj |ΨGS⟩ : ˆUGSˆaiˆaj ˆU †
+GS : + ˜Q,
+(B2)
+where ˜Q collects terms of order four and higher. Denoting A = Γ|1,2,...,2n and using Eq. (5), we can rewrite the above
+equation and find
+ˆa1ˆa2 . . . ˆa2n = (−i)nPf(A) +
+�
+i<j
+(−i)n−1(−1)i+j+1Pf(Aˆiˆj) : ˆUGSˆaiˆaj ˆU †
+GS : + ˜Q.
+(B3)
+Here, we have introduced the submatrices Aˆiˆj which emerge from A by canceling the ith and jth rows and columns.
+Using
+∂Pf(A)
+∂Γij
+= (−1)i+j+1
+2
+Pf(Aˆiˆj),
+(B4)
+we obtain the expression
+ˆA =(−i)nPf(A) + i
+�
+i,j
+(−i)n ∂Pf(A)
+∂Γij
+: ˆUGSˆaiˆaj ˆU †
+GS : + ˜Q
+= ⟨ΨGS| ˆA |ΨGS⟩ + i
+4 : ˆaT UT
+ξ A(mf)Uξa : + ˜Q,
+(B5)
+with matrix entries
+A(mf)
+ij
+= 4∂ ⟨ΨGS| ˆA |ΨGS⟩
+∂Γij
+.
+(B6)
+We want to remark that Eq. (B4) can also be used to efficiently calculate the derivative Eq. (9) of Eq. (7).
+

UtE0T4oBgHgl3EQf2gK9/content/tmp_files/2301.02714v1.pdf.txt

@@ -0,0 +1,1228 @@
+ 
+ 
+A Deep Reinforcement Learning-Based Controller for 
+Magnetorheological-Damped Vehicle Suspension 
+AmirReza BabaAhmadi1 , Masoud ShariatPanahi2 , Moosa Ayati3 
+1,2,3 School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran 
+Abstract 
+This paper proposes a novel approach to controller design for an MR-damped vehicle suspension 
+system. This approach is predicated on the premise that the optimal control strategy can be 
+“learned” through real-world or simulated experiments utilizing a reinforcement learning 
+algorithm with continuous states/actions. The sensor data is fed into a Twin Delayed Deep 
+Deterministic Policy Gradient (TD3) algorithm, which generates the actuation voltage required for 
+the MR damper. The resulting suspension working space  (displacement), sprung mass 
+acceleration, and dynamic tire load are calculated using a quarter vehicle model incorporating the 
+modified Bouc-Wen MR damper model. Deep RL’s reward function is based on sprung mass 
+acceleration. The proposed approach outperforms traditional suspension control strategies 
+regarding ride comfort and stability, as demonstrated by multiple simulated experiments. 
+Keywords: Magnetorheological-damped suspension, ride comfort, deep reinforcement learning 
+ 
+1. Introduction 
+As one of the most critical components of the vehicle, the suspension system significantly 
+improves ride comfort and road holding, preventing damage and reducing passenger fatigue. 
+Suspension systems are classified as passive, active, or semi-active. Due to the fixed and 
+unchanging nature of passive suspension parameters, passive suspension cannot guarantee ride 
+comfort and stability when the environment or suspension parameters vary. Thus, active and semi-
+active suspension systems with tunable parameters can compensate for the limitations mentioned 
+above of passive suspension systems. The uncertainty inherent in the suspension system’s road 
+roughness and parameter variation in real applications is unavoidable.  
+MR fluid dampers are adaptive controllable devices that have garnered significant interest due to 
+their simplicity, low power consumption, and high capacity force, among other characteristics. 
+MR dampers have found widespread application in a variety of industries, including the 
+automotive industry [1][2], civil structures [3][4], and railway vehicles [5]. MR dampers are semi-
+active devices because we can simply apply voltage to their coils rather than using a mechanism 
+for the damper to produce damping force directly. As a result, two distinct controller types are 
+required to regulate semi-active suspensions. First, the system controller calculates the required 
+
+damping force to ensure ride comfort and road holding simultaneously. The system controller 
+inputs are derived from the suspension system’s state feedback. Second, the damper controller 
+determines the voltage that should be applied to the MR-damper for its current force to track the 
+force specified by the system controller. 
+Numerous control techniques for semi-active suspension systems have been developed. 𝐻∞ 
+control [6], [7], [8], skyhook control [9], [10], [11], adaptive control based on neural networks 
+[12], robust control [13], LQG control [14], [15], and optimal PID control [16][17] are some of 
+the studies that have been conducted to develop an efficient controlling system for improving ride 
+comfort and stability. According to previous research, PID controllers have a simple structure but 
+are ineffective in semi-active suspension systems with uncertain parameters. Skyhook suspension 
+control is a classic semi-active suspension control method described in [18]. It features simple 
+structures, straightforward implementation, and acceptable performance. On the other hand, the 
+time delay significantly affects suspension performance, resulting in instability and wheel jump. 
+The authors of [19] designed an optimal LQG controller, but the control parameters were 
+calculated by ignoring uncertain factors in the system modeling. 
+When the system’s parameters change sufficiently, the system becomes unstable. The authors of 
+[20] aimed to address the issue of the blind design of the fuzzy controller, where this was, in fact, 
+a PID controller, and a PID was designed using rule description. Indeed, developing a proper fuzzy 
+control system necessitates extensive knowledge of the system. The sliding mode controller in [21] 
+shows good performance and robust behavior. On the other hand, chattering is a fatal flaw in 
+sliding mode control. Chattering may result in instability due to the controlled system’s activated 
+high-frequency modes. Although the adaptive controller in [12] improves ride comfort, it performs 
+poorly in road holding. Reference [22] proposed a fast model prediction controller (FMPC). The 
+authors of [23] proposed a robust model prediction controller (RMPC). 
+Both of the aforementioned predictive controllers utilized road models in advance of designing 
+controllers. The primary goal of developing a predictive controller is to provide highly accurate 
+information from the system, which is impossible when driving on various roads with high 
+uncertainty. In [24], it is demonstrated that neural-networked-based controllers can solve complex 
+and nonlinear problems. However, like many other supervised learning algorithms, neural 
+networks require a large number of labeled samples. When developing the control effort, only the 
+current state is considered; future states are not considered. These critical issues impose constraints 
+on the use of neural network controllers. Reference [25] proposes a novel nonlinear adaptive smart 
+controller based on the classification of road profiles. The primary disadvantage of this method is 
+that a new classifier must be constructed when the control strategy changes. 
+With the advent of Deep Learning (DL) techniques that overcome several of the significant 
+limitations of classical machine learning algorithms, some researchers attempted to apply DL 
+algorithms to the suspension control problem. The authors of [26] propose a DDPG-based 
+algorithm for a particular type of suspension system. The primary drawback of this algorithm is 
+its inability to locate the globally optimal solution to a problem. 
+
+Two types of controllers are required to control a semi-active suspension system. A system 
+controller analyzes the system’s feedback samples and predicts the desired damping force. The 
+damper controller receives the system controller’s predicted force and suspension system 
+displacement; it then predicts the applied voltage exerted on damper coils.  
+The paper proposes an improved DRL controller that combines a system controller and a damper 
+controller to provide the best ride comfort and stability possible. Notably, MR-based suspension 
+systems operate in two modes: open-loop mode (without the use of a controller) operates with a 
+constant damping coefficient, similar to passive suspension systems. Both the system and damper 
+controllers work in a closed-loop system to adjust the damping force in response to road conditions 
+by tuning the required damper’s voltage input.  
+ 
+2. MR-damped suspension model 
+A simplified quarter-vehicle model with a semi-active suspension is shown in Fig. 1, where mb 
+represents the vehicle’s body mass, mw represents the wheel mass, and xb and xw denote body 
+displacement and wheel displacement, respectively. The road profile is denoted by xr. The spring 
+stiffness of the suspension system is ks, and the tire spring stiffness is denoted by kt. We exclude 
+tire damping from this study due to its negligible value. Table 1 contains parameters taken from 
+[27]. Newton’s second law is applied to the quarter-model of the vehicle to derive the following 
+equations: 
+ 
+     
+ 
+ 
+ 
+                                           
+    𝑚𝑏𝑋̈𝑏 + 𝐾𝑠(𝑋𝑏 − 𝑋𝑤) + 𝑓 = 0 
+ 
+(1) 
+ 
+ 
+  𝑚𝑤𝑋̈𝑤 − 𝐾𝑠(𝑋𝑏 − 𝑋𝑤) + 𝐾𝑡(𝑋𝑤 − 𝑋𝑟) − 𝑓 = 0 
+(2) 
+ 
+ 
+
+m,
+oassive
+m.w
+semi-activeFig. 1. A quarter-car model with two operational modes for a semi-active suspension system 
+(passive and semi-active) 
+ 
+Where the following equation gives the damping force generated by the MR device: 
+ 
+                       
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  𝑓 = {𝐶𝑠(𝑋̇𝑏 − 𝑋̇𝑤)
+for passive suspension  
+𝑓 𝑀𝑅
+for semi − active suspension 
+ 
+(3) 
+ 
+Cs is a coefficient describing the MR passive suspension mode of operation in an MR-damper.  
+ 
+The state-space representation of the semi-active suspension system is prepared as follows [17]: 
+                                             
+                                                                                                                                     
+   𝑤̇ = 𝐴𝑤 + 𝐵𝑓𝑀𝑅 + 𝐷𝑥𝑟  
+ 
+(4) 
+ 
+ 
+𝑤 = [𝑥𝑏
+𝑥𝑤
+𝑥̇𝑏
+𝑥̇𝑤]𝑇 
+ 
+(5) 
+ 
+                                                                                                         
+    𝐴=
+[
+ 
+ 
+ 
+ 
+0
+0
+0
+0
+1
+0
+0
+1
+−
+𝐾𝑠
+𝑚𝑠
+𝐾𝑠
+𝑚𝑠
+𝐾𝑠
+𝑚𝑤
+−
+𝐾𝑠+𝐾𝑡
+𝑚𝑤
+0
+0
+0
+0]
+ 
+ 
+ 
+ 
+   
+(6) 
+ 
+ 
+                                                                                                              
+    𝐵 = [0
+0
+−
+1
+𝑚𝑠
+1
+𝑚𝑤]
+𝑇
+   
+(7) 
+ 
+                                                                                                                     
+    𝐷 =[0
+0
+0
+𝐾𝑡
+𝑚𝑤]
+𝑇
+ 
+ 
+(8) 
+ 
+ 
+
+ 
+ 
+Table 1. Semi-active suspension parameters [27]   
+Value (Unit)
+ 
+Symbol
+ 
+Parameter
+ 
+375(kg) 
+𝑚𝑏 
+Vehicle body mass 
+29.5 (kg) 
+𝑚𝑤 
+Vehicle wheel mass 
+20.58 (KN/m) 
+𝑘𝑠 
+Suspension stiffness 
+772 (Ns/m) 
+𝐶𝑠 
+Damping coefficient 
+200 (KN/m) 
+𝑘𝑡 
+Tire stiffness 
+ 
+Magnetorheological damper force is denoted by fmr depending on the time series of the external 
+voltage to its magnetic coil V and relative displacement x=xb-xw, which is known as suspension 
+working space (SWS). fmr is computed from Eq. 9, the modified Bouc-Wen model, derived from 
+[28] and incorporated into the suspension system shown in Fig. 1. 
+ 
+ 
+     
+ 
+  𝐹(𝑡) = 𝑐1𝑦̇ + 𝑘1(𝑥 − 𝑥0)   
+(9) 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+   
+ 
+    𝑧̇ = −𝛾|𝑥̇ − 𝑦̇||𝑧||𝑧|𝑛−1 − 𝛽(𝑥̇ − 𝑦̇)|𝑧|𝑛 +
+𝐴(𝑥̇ − 𝑦̇) 
+ 
+(10) 
+ 
+ 
+ 
+    𝑦̇ =
+1
+𝑐1+𝑐0 {𝛼𝑧 + 𝑘0(𝑥 − 𝑦) + 𝑐0𝑥̇}  
+(11) 
+ 
+ 
+ 
+𝛼 = 𝑎𝑎 + 𝑎𝑏𝑢  
+(12) 
+ 
+ 
+ 
+    𝑐1 = 𝑐1𝑎 + 𝑐1𝑏𝑢 
+ 
+(13) 
+ 
+ 
+ 
+    𝑐0 = 𝑐0𝑎 + 𝑐0𝑏𝑢 
+ 
+(14) 
+ 
+ 
+ 
+𝑢̇ = −𝜂(𝑢 − 𝑣) 
+(15) 
+ 
+ 
+The internal movement of the MR-damper is denoted as y. u is the output signal from a first-order 
+filter, and 𝑧 is a parameter that guarantees the hysteretic behavior of MR fluid. Accumulator 
+stiffness is denoted by kt. C0 and C1 represent viscous damping at high and low velocity for the 
+
+MR-damper, respectively. The stiffness regulator at high damper velocity is denoted by K0. MR 
+damper accumulator simulation is performed via X0. The scale and shape of the hysteresis behavior 
+of the MR-damper parameters for the simulation are 𝛾, 𝛽, 𝛿 and 𝜂, respectively.  
+ 
+ 
+Fig. 2. Modified Bouc-Wen model for the MR-damper 
+ 
+Parameters for the modified Bouc-Wen MR-damper model are presented in Table 2 (adapted from 
+[28].) 
+ 
+Table 2. Modified Bouc-Wen parameters for the MR-damper [28] 
+Parameter 
+Value 
+Parameter Value 
+𝑐0𝑎 
+784 
+( 1)
+Nsm −  
+𝛼𝑏 
+38430  
+( 1)
+( 1)
+NsV
+m
+−
+−  
+𝑐0𝑏 
+1803 
+( 1)
+( 1)
+NsV
+m
+−
+−  
+𝛽 
+2059020 
+2
+m−  
+𝑐1𝑎 
+14649 
+( 1)
+Nsm
+−  
+𝛾 
+136320 
+2
+m−  
+𝑐1𝑏 
+34622 
+( 1)
+( 1)
+NsV
+m
+−
+−  
+𝜂 
+190
+1
+s−  
+𝑘1 
+840 
+( 1)
+Nm −  
+A 
+58 
+𝑘0 
+3610 
+( 1)
+Nm −  
+n 
+2 
+𝛼𝑎 
+12441 
+( 1)
+Nm −  
+𝑥0 
+0.245
+ 
+ 
+ 
+2.1. Control Strategy based on DRL-TD3 
+As previously stated, the Twin Delayed Deep Deterministic Policy Gradient (TD3) was used as 
+the universal controller for the MR-damped suspension system. TD3 is identical to DDPG but 
+incorporates three additional features to address DDPG issues.  
+• Clipped double Q learning: In TD3, we compute the Q value using two critic networks 
+and the target value using two target networks. We used only one critic and one target 
+network in DDPG. In TD3, we use two target critic networks to compute two target Q 
+
+Bouc-Wen
+ko
+>F
+Co
+WWvalues. Then, when computing the loss function, we choose the smaller of the two. This 
+will ensure that we do not overestimate the target Q value. 
+• Delayed policy update: Unlike DDPG, we added a delay to the actor-network parameter 
+update in TD3. While the parameters of the critic networks are updated at each step of the 
+episode, the actor-network (policy network) is delayed and updated after every two steps. 
+• Target policy smoothing:  In DDPG, the algorithm generates distinct target values for 
+identical actions. As a result, the target’s variance would be high. Thus, we reduce variance 
+by adding noise to the target action. 
+ 
+This section provides additional information about Twin Delayed Deep Deterministic Policy 
+Gradients (TD3). In TD3, we employ six artificial neural networks, four of which are critic 
+networks and two of which are two-actor networks.  
+• The main critic neural network parameters are denoted by 𝜃1 and 𝜃2 
+• The two target critic neural network parameters are represented by 𝜃1′and 𝜃2′ 
+• The main actor-network parameter is denoted by 𝜙 
+• The target actor-network parameter is denoted by 𝜙′ 
+ 
+Initially, we must initialize the two main critic network parameters, 𝜃1 and 𝜃2, and the main critic 
+network parameter 𝜙 with random values. Since the target network parameter is only a copy of the 
+main network parameter, the two target critic network parameter 𝜃1′ and  𝜃2′ by copying 𝜃1 and 
+𝜃2,  are simply initialized, respectively. Similarly, we initialize the target actor parameter 𝜙′, by 
+just copying the main actor-network parameter 𝜙 . Also, the replay buffer is initialized too. Now, 
+we select action a using the actor-network: 
+( )
+a
+s
+
+
+=
+. Rather than selecting the action directly, 
+some noise  is added to ensure exploration when ~
+(0,
+)
+N
+
+
+. Accordingly, the output action is 
+expressed as follows: 
+( )
+a
+s
+
+
+=
++ (16).  
+Then, after performing action a, we move to the next state s  and obtain reward r . This transition 
+information is stored in the replay buffer. During the next step, we randomly sample a minibatch 
+of K transition ( , , , )
+s a r s   from the replay buffer. The K transition will be used for updating both 
+the critic and actor network.  
+The loss function of the critic network is expressed as follows: 
+ 
+2
+1
+(
+)
+(
+( ,
+))
+1,2
+j
+j
+i
+i
+i
+J
+y
+Q
+s a
+for j
+k
+
+
+=
+
+−
+=
+ 
+(17) 
+ 
+In the preceding equation, the following applies: 
+The action 
+ia   is the action produced by the actor-network as follows:  
+
+   
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+( )
+ia
+s
+
+
+=
+ 
+(18) 
+ 
+𝑦𝑖 is the target value of the critic, that is  
+1,2
+min
+( , )
+j
+i
+i
+j
+y
+r
+Q
+s a
+
+
+=
+
+
+=
++
+ 
+(19) 
+ 
+, and the action a   is the action produced by the target critic network:  
+( )
+~ (
+(0,
+),
+,
+)
+i
+a
+s
+where
+N
+c
+c
+
+
+
+
+
+=
++
+
+−
++
+ 
+(20) 
+ 
+After computing the loss of the critic network, the gradients 
+(
+)
+j
+j
+J
+
+
+
+ is computed, and then the 
+critic network parameters will be updated using the gradient descent method. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(
+)
+1,2
+j
+j
+j
+j
+J
+for j
+
+
+
+
+
+=
+− 
+=
+     
+(21) 
+ 
+Now, the actor network must be updated. The objective function of the actor-network is as follows: 
+       
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+1
+( )
+( , )
+i
+i
+i
+J
+Q
+s a
+k
+
+ = 
+ 
+(22) 
+ 
+It must be noted that in Eq. 22, we only use state (si) from the sampled K transitions ( , , , )
+s a r s  . 
+The action a is selected by the actor-network. 
+( )
+ia
+s
+
+
+=
+ 
+(23) 
+ 
+ In order to maximize the objective function, the gradient of the objective function 
+( )
+J
+
+
+
+  is 
+computed, and the parameters of the network are updated using the gradient ascent: 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+( )
+J
+
+
+
+
+
+=
++ 
+     
+(24) 
+ 
+We delay the update rather than updating the actor-parameters networks at each time step of the 
+episode. If the time step of the episode is denoted by t and d denotes the number of time steps 
+which we want to delay the update, it can be described as follows: 
+
+1- If t mod d=0, then: 
+1. Compute the gradient of the objective function 
+( )
+J
+
+
+
+ 
+2. Update the actor-network parameter using the gradient ascent method  (24). 
+Finally, the parameters of the target critic network 𝜃1′ and 𝜃2′and the parameters of the target 
+actor-network 𝜙′   will be updated via soft replacement:  
+                
+ 
+ 
+  
+ 
+ 
+ 
+ 
+{𝜃𝑗
+′ = 𝜏𝜃𝑗 + (1 − 𝜏)𝜃𝑗
+′ 𝑓𝑜𝑟 𝑗 = 1,2
+𝜙′ = 𝜏𝜙𝑗 + (1 − 𝜏)𝜙′
+ 
+(25) 
+ 
+There is a small change in updating the parameters of the target networks. As with the actor-
+network parameter, we delay updating it for d steps; similarly, we update the target network 
+parameters for every d step; in this case, we can say: 
+ 
+1- If t mod d=0, then: 
+1. Compute the gradient of the objective function 𝛻𝜙𝐽(𝜙) and update the actor-
+network parameter using gradient ascent. 
+𝜙 = 𝜙 + 𝛼𝛻𝜙𝐽(𝜙) 
+(26) 
+ 
+ 
+2. Update the target critic network parameter and target actor-network parameter as  
+(1
+)
+1,2
+j
+j
+j for j
+
+
+ 
+
+
+=
++
+−
+=
+ 
+(27) 
+and    
+(1
+)
+
+
+ 
+
+
+=
++
+−
+ 
+(28) 
+, respectively. 
+ 
+The previous steps for several episodes must be repeated to improve the policy. The following 
+pseudocode is prepared to understand better how TD3 works: 
+ 
+ 
+
+ 
+Fig. 3. TD3 Algorithm pseudocode 
+ 
+2.2. TD3 application in closed-loop vibration control for a semi-active suspension system 
+The main inputs for vehicle dynamics are road disturbance profile and damping force produced by 
+the MR device. The outputs are body (sprung mass) acceleration and suspension working space 
+(SWS). The RL-Agent inputs for implementing a controller for a one-quarter suspension system 
+are body acceleration, denoted by 𝑞, and a reward function, which is as follows: 
+   
+       
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+𝑟 = {0           𝑖𝑓  𝑞 = 𝑞𝑔𝑜𝑎𝑙 = 0
+−𝑘𝑞2    𝑖𝑓  𝑞 ≠ 0
+  
+ 
+(29) 
+ 
+Where k is a hyperparameter that specifies the intensity coefficient for agent punishment, the agent 
+punishments experienced in the replay buffer are also exerted in the RL-agent. The damper’s input 
+voltage must be applied to the coil via the RL-agent output (action). TD3 analyzes its performance 
+by measuring body acceleration and fine-tuning its action to road profile disturbances. The 
+
+Twin Delayed Deep Deterministic Policy Gradient Algorithm:
+Initialize critic networks Qe1, Qe2, and actor network Ts
+withrandomparameters01.02.Φ
+Initializetargetnetworks←1%←2,'←Φ
+2
+InitializereplaybufferB
+for t=1 toTdo
+3
+Selectactionwithexplorationnoisea~π(s)+E
+e~N(O,o)and observe reward r and new states
+4
+Store transition tuple (s, a,r,s in B
+Samplemini-batchofNtransitions(s,a,r,sfromB
+a ← π(s) + E, E ～ clip(N(O,), -C,c)
+5
+y ← r +mini=1.2Qe(s',a)
+Update critics i ← argming. N-1 (y-Qe, (s, a))2
+iftmoddthen
+7
+Update  by the deterministic policy gradient:
+VJ(d) = N-i VaQe (s,a)lg
+元(s)V(s)
+Updatetargetnetworks:
+8
+, ← T+ (1 - T)O
+Φ← +(1-)
+end if
+end forsurrounding environment consists of a vehicle suspension system equipped with an MR-damper 
+and a road profile. The agent is a neural network that constructs the controller part. The 
+hyperparameters listed in Table 3 pertain to TD3 agents used in closed-loop suspension control. 
+Figures 5 and 6 detail the neural networks used in the TD3 algorithm. 
+The hidden size is 400 and 300 neurons in each layer for the actor and critic network, respectively. 
+The optimization method is Adam for both actor and critic networks. The discount factor 𝛾  is 0.8. 
+The input voltage varies from 0 to 3 V, and the damping force varies from -1.5 to 1.5 KN. 
+Table 3. Neural Networks Parameters in TD3 
+Network 
+Learning Rate 
+Optimizer 
+Delay for update 
+Actor 
+0.002 
+Adam 
+2 
+Critic 
+0.002 
+Adam 
+1 
+Actor Target 
+0.006 
+- 
+2 
+Critic Target 
+0.006 
+- 
+2 
+ 
+ 
+ 
+Fig. 4. Closed-loop semi-active suspension system block diagram with DRL Agent 
+ 
+ 
+
+Road
+Profile
+qt
+F
+Vehicle
+MR Damper
+Model
+Deep
+Reinforcement
+Learning
+action by policy μ,voltage
+Suspension Working Space(SWws)
+Agent (TD3)
+Reward Function: r, = -kq
+3+(IB II'D))+= +
+Replay Buffer 
+Fig. 5. Actor-critic architecture in TD3 Agent 
+ 
+
+Agent
+Critic Network
+Actor Network
+Qe (stat)
+Random
+△Js TD error update
+△Jμ DPG update
+noise
+εE N(O,o)
+at
+Vehicle
+0
+Critic 2
+Critic 1
+at
+Model
+' Target
+target2
+target1
+Behavior
+target
+policy
+Compared target y +
+Mini batch
+qt llqgoal , at, qt+1 lqgoal , rt
+ReplayBuffer
+Parameter
+update
+Data flow
+neural network
+Global Memory
+HER 
+Fig. 6. The details of actor-critic networks in semi-active suspension control 
+ 
+3. Results and Discussion 
+Vertical body acceleration (BA), Dynamic Tire Load (DTL), and Suspension Working space are 
+three primary performance criteria used in the suspension design of a vehicle to improve ride 
+comfort and road holding, preventing the suspension system from bottoming out excessively. To 
+achieve the objectives mentioned above, we should reduce BA or SWS to improve ride comfort 
+and DTL to improve road holding and minimize suspension space distance. The BA has been 
+chosen as the objective function in this article. 
+Two types of controllers for semi-active suspension are investigated in this section: the RL-based 
+TD3 and the PID controller; their gains are tuned using the Particle Swarm Optimization (PSO) 
+algorithm from [17], as well as an uncontrolled system (MR-Passive) with no applied voltage to 
+the damper’s coil.  
+A particular type of bump-road excitation was chosen due to its resemblance to actual road profiles. 
+Additionally, the bump-road profile demonstrates the characteristics of transient response. The 
+road-bump profile has been proposed in [27] as: 
+ 
+
+Critic
+0;=1,2
+Ao  VeQe,(quHe(qt Il qgoal)
+Actor Φ
+qt,at
+Qe i-1, (qt, at)
++ μp (qt+1 Il qgoal)
+a, = μp(qt+1 Il qgoat) +
+qt Il qgoal
+A;  (t - Qe: (qt.at)?
+Replay Buffer
+Yt = r + ymini=12Qe(qt+1, at)
+(qt Il qgoal , t, qt+1, Il qgoal , rt)qt+:
+Qg i=12 (q+1 +at)
+Hp(qt+1 Il qgoal )
+qt+1 I qgoal
+at
+Critic target e't=1,2
+Actor target $'
+at = μs'(qt+1 I qgoat) + s              
+ 
+ 
+ 
+ 
+  𝑋𝑟 = {𝑎 {1 − 𝑐𝑜𝑠( 𝜔𝑟(𝑡 − 0.5))}, 𝑓𝑜𝑟0.5 ≤ 𝑡 ≤ 0.5 +
+𝑑𝑏
+𝑉𝑐
+0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
+   
+(1) 
+ 
+ 
+Where a denotes half of the bump amplitude, 
+2
+c
+r
+b
+V
+d
+
+
+=
+ , db is the bump width, and Vc  denotes 
+the vehicle velocity. In this research, 𝑎 = 0.035𝑚, 𝑑𝑏 = 0.8𝑚, 𝑉𝑐 = 0.856
+𝑚
+𝑠 , derived from [27]. 
+ 
+ 
+Fig. 7. Road-bump profile 
+ 
+Table 4 compares the results from the TD3 controller to an uncontrolled system (applied zero 
+voltage), and Table 5 compares the results from the PSO PID controller to the TD3. 
+The results demonstrate unequivocally that the DRL-based controller (TD3) algorithm 
+outperforms PSO-tuned PID. TD3 is excellent at dissipating vibrations caused by bump excitation. 
+Additionally, it reduces settling time and enhances road holding and ride comfort.   
+DRL TD3 significantly reduces body acceleration, suspension working space, and Dynamic Tire 
+Load compared to an uncontrolled suspension system by 35.8%, 68.5%, and 33.6%, respectively. 
+Simultaneously, PSO-PID reduces those criteria by 32.2%, 50%, and 12.4%, respectively, 
+compared to MR passive (uncontrolled suspension).  
+ 
+
+0.07
+0.06
+0.05
+ (m)
+0.04
+Road
+0.03
+0.02
+0.01
+0
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+Time (sec) 
+Fig. 7. Body Acceleration (BA) 
+ 
+ 
+ 
+Fig. 8. Dynamic Tire Load (DTL) 
+
+..Uncontrolled
+3
+-PSO-PID
+DRL-TD3
+Acceleration(m/s)
+Body,
+-2
+-3
+4
+Q
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+Time (sec)1500
+uncontrolled
+-PSO-PID
+1000
+DRL-TD3
+Dynamic Tyre Load (N)
+500
+-500
+-1000
+-1500
+-
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+Time (sec) 
+Fig. 9. Suspension Working Space (SWS) 
+ 
+Table 4. RMS Values for the suspension system criteria with TD3 algorithm and Uncontrolled 
+Suspension System 
+RMS-values 
+Body 
+Acceleration(𝒎/𝒔𝟐) 
+Suspension Working 
+Space(𝒎) 
+Dynamic Tire Load (𝑵) 
+DRL-TD3
+ 
+0.97
+ 
+0.0084
+ 
+381.32
+ 
+Uncontrolled
+ 
+1.5179
+ 
+0.0267
+ 
+537.7
+ 
+ 
+Table 5. Comparison of DRL-TD3 with PSO-PID 
+Controller Type 
+Body 
+Acceleration(𝒎/𝒔𝟐) 
+Suspension 
+Working 
+Space(𝒎) 
+Dynamic Tire Load (𝑵) 
+DRL-TD3 
+35.8 % 
+68.53 % 
+33.6% 
+PSO-PID 
+22.4% 
+46.1% 
+11.9% 
+Improvement 
+13.4% 
+22.43% 
+21.7% 
+ 
+4. Conclusion 
+The paper proposes the DRL-based TD3 algorithm for vibration mitigation in a semi-active 
+suspension system equipped with an MR damper. Three major criteria, BA, SWS, and DTL, 
+were considered and investigated when evaluating the proposed controller. Instead of using two 
+distinct controllers for damper control and force estimation, a single universal controller based on 
+real-time learning was used. The time-domain results were analyzed, and the effectiveness of the 
+
+0.06
+ uncontrolled
+PSO-PID
+DRL-TD3
+0.04
+0.02
+(w) s
+swS
+-0.02
+-0.04
+-0.06
+1
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+Time (sec)DRL-based controller was compared to the optimal PSO-PID controller. Finally, the results 
+demonstrate that the TD3 algorithm outperforms the PSO-tuned PID controller. 
+Declaration of competing interest 
+The authors declare that they have no known competing financial interests or personal 
+relationships that could have appeared to influence the work reported in this paper. 
+ 
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+suspension system via hardware in-the-loop simulation,” J. Dyn. Syst. Meas. Control. Trans. ASME, 
+vol. 122, no. 1, pp. 114–121, Mar. 2000, doi: 10.1115/1.482435. 
+[28] 
+C. Y. Lai and W. H. Liao, “Vibration Control of a Suspension System via a Magnetorheological 
+Fluid Damper,” J. Vib. Control, vol. 8, no. 4, pp. 527–547, Apr. 2002, doi: 
+10.1177/107754602023712. 
+ 
+ 
+

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+arXiv:2301.02264v1  [hep-th]  5 Jan 2023
+MITP/23-001
+On ε-factorised bases and pure Feynman integrals
+Hjalte Frellesvig a and Stefan Weinzierl b
+a Niels Bohr International Academy, University of Copenhagen,
+Blegdamsvej 17, 2100 København, Denmark
+b PRISMA Cluster of Excellence, Institut für Physik,
+Johannes Gutenberg-Universität Mainz,
+D - 55099 Mainz, Germany
+Abstract
+We investigate ε-factorised differential equations and purity for Feynman integrals. We are
+in particular interested in Feynman integrals beyond the ones which evaluate to multiple
+polylogarithms. We show that a ε-factorised differential equation does not necessarily lead
+to pure Feynman integrals. We also point out that a proposed definition of purity works
+locally, but not globally.
+
+1
+Introduction
+The concepts of ε-factorised differential equations [1], purity and uniform transcendental weight,
+simple poles and constant leading singularities [2–4] play a crucial role in modern techniques for
+computing Feynman integrals. These concepts are well understood for Feynman integrals which
+evaluate to multiple polylogarithms.
+However, as soon as we leave this class of function not everything is as clear as we want
+it. This is already the case for the simplest Feynman integrals beyond the class of multiple
+polylogarithms, the ones which are associated to an elliptic curve. It is therefore timely and
+appropriate to clarify several issues. The points which we discuss can be exemplified by the
+simplest elliptic Feynman integral, the two-loop sunrise integral with equal non-zero masses.
+We start with ε-factorised differential equations. A ε-factorised differential equation together
+with boundary values at a given point allows for a systematic solution in terms of iterated inte-
+grals to any order in the dimensional regularisation parameter ε. But do these iterated integrals
+have additional nice properties like a definition of transcendental weight or integrands with sim-
+ple poles only? In this paper we show that the general answer is no, but there might be bases of
+master integrals which have more of the nice properties than others.
+This occurs already for the sunrise integral: We know two bases of master integrals, which
+put the the associated differential equation into an ε-factorised form. The construction of either
+basis generalises to more complicated integrals, so it is worth examining the two bases in detail.
+The first basis is constructed along the lines of an analysis of the maximal cut [5, 6] and/or
+along the lines of prescriptive unitarity [7, 8]. Concretely this basis is constructed by the re-
+quirement that the period matrix on the maximal cut is proportional to the unit matrix [9]. For
+the sunrise integral we present a cleaned-up basis along these lines. Throughout this paper we
+denote this basis by ⃗K.
+The second basis is constructed from Picard-Fuchs operators and leads to a differential equa-
+tion with modular forms [10]. For the sunrise integral we consider the basis given in [11]. This
+approach generalises nicely to more complicated Feynman integrals [12–17]. Throughout this
+paper we denote this basis by ⃗J.
+In this paper we work out the relation between the two bases. The first question we address
+is the following: Do these bases define master integrals of uniform weight? In principle, this
+requires a definition of transcendental weight for elliptic Feynman integrals. Let us first be
+agnostic to a full and complete definition of transcendental weight. We only make the minimal
+assumption that the definition of transcendental weight in the elliptic case should be compatible
+with the restriction of the kinematic space to a sub-space. With this assumption we may restrict
+to a point in kinematic space where the elliptic curve degenerates. The master integrals reduce
+to multiple polylogarithms, for which the definition of transcendental weight is unambiguous.
+Choosing this point as the boundary point for the integration of the differential equation forces
+the boundary constants (given by special values of multiple polylogarithms) to be of uniform
+weight (in the classical sense for multiple polylogarithms). In this way we may detect master
+integrals of non-uniform weight.
+It turns out that basis ⃗K (constructed by the requirement that the period matrix on the maximal
+cut is proportional to the unit matrix) has boundary constants of non-uniform weight. Hence it is
+2
+
+not a basis of uniform weight if we require that the notion of uniform weight is compatible with
+restrictions in the kinematic space.
+The second question which we address in this paper is the relation between functions of
+uniform weight and logarithmic singularities. Functions of uniform weight are also called pure
+functions. In order to answer this question we have to adopt a definition of transcendental weight
+for elliptic Feynman integrals. A generalisation of weight, which can be applied to the elliptic
+case, has been defined in ref. [18]: Functions which satisfy a differential equation without any
+homogeneous term are called unipotent. Unipotent functions, whose total differential involves
+only pure functions and one-forms with at most simple poles are called pure. Adopting this
+definition, we investigate if basis ⃗J (i.e. the modular form basis) for the sunrise integral is of
+uniform weight in this sense. We find that this is the case locally, but not globally. The argument
+which we present applies not only to the specific example of the equal mass sunrise integral, but
+to a wide range of elliptic Feynman integrals expressible in terms of the elliptic polylogarithms
+�Γ [19].
+This paper is organised as follows: In section 2 we start with a toy example, showing that
+an ε-factorised differential equation alone does not guarantee a solution of uniform weight. The
+boundary values need to be of uniform weight as well. The toy example is entirely within the
+class of multiple polylogarithms. In section 3 we introduce the standard example of an elliptic
+Feynman integral: the two-loop sunrise integral with equal non-zero masses. We introduce the
+notation which we will use in later sections of this paper.
+In section 4 we investigate the first question: Are the known bases, which put the differential
+equation into an ε-factorised form also of uniform weight? In sub-section 4.1 we introduce three
+bases⃗I, ⃗J and ⃗K for the sunrise integral. The first one⃗I is a pre-canonical basis and serves only
+in intermediate steps. The basis ⃗J is the one appearing in [11], while the basis ⃗K is the one
+appearing in [9]. The associated differential equations are given in sub-section 4.2. For the bases
+⃗J and ⃗K, the differential equations are in ε-factorised form. In sub-section 4.3 we discuss the
+period matrix on the maximal cut for the bases ⃗J and ⃗K. By construction, the period matrix for
+the basis ⃗K is proportional to the unit matrix. In sub-section 4.4 we present the solutions for
+the master integrals for the bases ⃗J and ⃗K. We then look at the values at p2 = 0. At this point
+the elliptic curve degenerates and both solutions are given in terms of special values of multiple
+polylogarithms. We find that the basis ⃗K is not of uniform weight.
+In section 5 we investigate the second question: What is the relation between purity and
+simple poles? We start in sub-section 5.1 with recapitulating the definition of purity from the
+literature. We then show in sub-section 5.2 that this definition does fit the modular form basis
+locally, but not globally. In sub-section 5.3 we demonstrate that our argument extends to Feyn-
+man integrals expressible in terms of elliptic polylogarithms �Γ. The problem is the behaviour at
+the finite cusps. However, modular transformations, which we discuss in sub-section 5.4, allow
+us to cover the kinematic space with coordinate charts such that in each coordinate chart the
+requirement from the definition of purity holds locally. Our conclusions are given in section 6.
+In appendix A we present the q-expansions of the modular forms and Eisenstein series appearing
+in the main text. In appendix B we give the boundary constants for the sunrise integral.
+3
+
+2
+A toy example
+We start with a simple toy example, showing that an ε-factorised differential equation alone does
+not guarantee a solution of uniform weight. The boundary values need to be of uniform weight
+as well.
+Consider the two functions F1(x) and F2(x)
+F1 (x)
+=
+eεln(x)
+=
+1+εln(x)+ 1
+2ε2(ln(x))2 +O
+�
+ε3�
+,
+F2 (x)
+=
+(1+2ε)eεln(x)
+=
+1+ε[2+ln(x)]+ε2
+�
+2ln(x)+ 1
+2 (ln(x))2
+�
++O
+�
+ε3�
+.
+(1)
+F1(x) is of uniform weight (where we count algebraic numbers to be of weight zero, ln(x) to be
+of weight one, and ε to be of weight minus one), while F2(x) is not. However, both function
+satisfy the ε-factorised differential equation
+d
+dxFi (x)
+=
+ε
+xFi (x),
+i ∈ {1,2}.
+(2)
+The general solution of eq. (2) as a power series in ε reads
+Fi (x) = C(0)
+i
++
+�
+C(1)
+i
++C(0)
+i
+ln(x)
+�
+ε+
+�
+C(2)
+i
++C(1)
+i
+ln(x)+ 1
+2C(0)
+i
+(ln(x))2
+�
+ε2 +O
+�
+ε3�
+,
+(3)
+with boundary values C(j)
+i
+. For F1(x) the boundary values are
+C(0)
+1
+= 1,
+C(j)
+1
+= 0 for j ≥ 1.
+(4)
+For F2(x) the boundary values are
+C(0)
+2
+= 1,
+C(1)
+2
+= 2,
+C(j)
+2
+= 0 for j ≥ 2.
+(5)
+For a solution of uniform weight we must have that any non-zero boundary valueC(j)
+i
+is of weight
+j. This is the case for F1(x), but not for F2(x): The boundary value C(1)
+2
+is of weight zero, for a
+solution of uniform weight it is supposed to be of weight one.
+From this simple example we see that a ε-factorised differential equation alone does not
+guarantee a solution of uniform weight, we must in addition require that the boundary values
+C(j)
+i
+ε j are of uniform weight as well.
+3
+Feynman integrals and elliptic curves
+In this section we introduce the standard example of an elliptic Feynman integral: the two-loop
+sunrise integral with equal non-zero masses. This section also serves to set up the notation.
+4
+
+We consider the family of Feynman integrals
+Iν1ν2ν3 (D,x)
+=
+e2εγE �
+m2�ν123−D � dDk1
+iπ
+D
+2
+dDk2
+iπ
+D
+2
+1
+�
+−q2
+1 +m2�ν1 �
+−q2
+2 +m2�ν2 �
+−q2
+3 +m2�ν3 ,
+(6)
+with x = p2/m2, ν123 = ν1 +ν2 +ν3 and q1 = k1, q2 = k2 −k1, q3 = −k2 − p. Below we will set
+D = 2−2ε.
+The elliptic curve associated to this Feynman integral can be obtained from the maximal cut
+and is given by a quartic polynomial P(u,v) = 0:
+E
+:
+v2 −u(u+4)
+�
+u2 +2(1+x)u+(1−x)2�
+= 0.
+(7)
+We denote the roots of the quartic polynomial by
+u1 = −4,
+u2 = −
+�
+1+√x
+�2 ,
+u3 = −
+�
+1−√x
+�2 ,
+u4 = 0.
+(8)
+For 0 < x < 1 the roots are real and ordered as
+u1 < u2 < u3 < u4.
+(9)
+We set
+U1 = (u3 −u2)(u4 −u1) = 16√x,
+U2 = (u2 −u1)(u4 −u3) =
+�
+1−√x
+�3�
+3+√x
+�
+,
+U3 = (u3 −u1)(u4 −u2) =
+�
+1+√x
+�3�
+3−√x
+�
+.
+(10)
+We define the modulus and the complementary modulus of the elliptic curve E by
+k2 = U1
+U3
+=
+16√x
+(1+√x)3 (3−√x)
+,
+¯k2 = 1−k2 = U2
+U3
+= (1−√x)3(3+√x)
+(1+√x)3(3−√x)
+.
+(11)
+Our standard choice for the periods and quasi-periods is
+ψ1 = 4K (k)
+U
+1
+2
+3
+,
+ψ2 = 4iK
+�¯k
+�
+U
+1
+2
+3
+,
+φ1 = 4[K (k)−E (k)]
+U
+1
+2
+3
+,
+φ2 = 4iE
+�¯k
+�
+U
+1
+2
+3
+.
+(12)
+The geometric interpretation is as follows: For simplicity we assume that the roots u1-u4 are
+real and ordered as in eq. (9). The square root v can be taken as a single-valued and continuous
+function on C\([u1,u2]∪[u3,u4])
+v
+=
+√u−u1
+√u−u2
+√u3 −u√u4 −u,
+(13)
+5
+
+u1
+u2
+u3
+u4
+γ1
+γ2
+Figure 1: Branch cuts and cycles for the computation of the periods of an elliptic curve.
+where √x denotes the standard square root with a branch cut along the negative real axis. For the
+ordering as in eq. (9) v is positive for u ∈]u2,u3[. It is purely imaginary with positive imaginary
+part just below the branch cut [u3,u4]. Let γ1 and γ2 be two cycles which generate the homology
+group H1(E,Z). This is shown in fig. 1. We choose γ1 and γ2 such that their intersection number
+is (γ1,γ2) = +1. Note that the intersection number is anti-symmetric: (γ2,γ1) = −1. The periods
+are alternatively given by
+ψ1 =
+�
+γ1
+du
+v
+= 2
+u3
+�
+u2
+du
+v ,
+ψ2 =
+�
+γ2
+du
+v
+= 2
+u3
+�
+u4
+du
+v .
+(14)
+In the last expression the square root is evaluated below the cut [u3,u4]. Similar formulae can be
+given for the quasi-periods.
+The derivatives of the periods and quasi-periods are given for i ∈ {1,2} by
+d
+dxψi
+=
+−1
+2ψi
+d
+dx (lnU2)+ 1
+2φi
+d
+dx
+�
+ln U2
+U1
+�
+,
+d
+dxφi
+=
+−1
+2ψi
+d
+dx
+�
+ln U2
+U3
+�
++ 1
+2φi
+d
+dx
+�
+ln U2
+U2
+3
+�
+.
+(15)
+In particular we may use these relations to replace φi by dψi
+dx or vice versa. Explicitly we have
+3
+�
+1+√x
+�2 φi
+=
+4√x
+�
+2+√x
+�
+ψi −4x
+�
+1−√x
+��
+3+√x
+� d
+dxψi.
+(16)
+Replacing φi by dψi
+dx is often advantageous to eliminate the square root √x. In the following we
+will often write ∂x for d
+dx. The Legendre relation reads
+ψ1φ2 −φ1ψ2
+=
+8πi
+(1+√x)3 (3−√x)
+.
+(17)
+We denote the Wronskian by
+W
+=
+ψ1∂xψ2 −ψ2∂xψ1 = −
+6πi
+x(1−x)(9−x).
+(18)
+6
+
+Finally, we set
+τ = ψ2
+ψ1
+,
+q = e2πiτ.
+(19)
+We have
+dτ
+=
+W
+ψ2
+1
+dx
+(20)
+and
+x
+=
+9 η(τ)4η(6τ)8
+η(3τ)4η(2τ)8,
+(21)
+where η denotes Dedekind’s eta-function. The first few terms read
+x
+=
+9q−36q2 +90q3 +O
+�
+q4�
+.
+(22)
+4
+Uniform weight and ε-factorised differential equations
+In this section we investigate the question of uniform weight for bases of master integrals,
+which have ε-factorised differential equations. The two-loop sunrise integral with equal non-
+zero masses serves as an example.
+4.1
+Bases of master integrals
+We consider three bases ⃗I, ⃗J and ⃗K for the family of the sunrise integral. The first one, ⃗I, is a
+basis without any additional properties and given by
+⃗I
+=
+
+
+I110
+I111
+I211
+
+.
+(23)
+The latter two, ⃗J and ⃗K, put the differential equation into an ε-form:
+d⃗J = εB⃗J,
+d⃗K = εC⃗K,
+(24)
+where the (3×3)-matrices B and C are independent of the dimensional regularisation parameter
+ε. The basis ⃗J, appearing in [11,20–22], is defined by
+J1
+=
+ε2 I110,
+J2
+=
+ε2 π
+ψ1
+I111,
+J3
+=
+ψ2
+1
+2πiεW
+d
+dxJ2 + 1
+24
+�
+3x2 −10x−9
+��ψ1
+π
+�2
+J2.
+(25)
+7
+
+In terms of I111 and I211 the master integral J3 is given by
+J3
+=
+�
+− ε2
+24
+�
+x2 −30x+45
+� ψ1
+π − ε
+4
+�
+1+√x
+��
+3−√x
+� ψ1
+π + ε
+16
+�
+1+√x
+�3�
+3−√x
+� φ1
+π
+�
+I111
++ε
+4 (1−x)(9−x) ψ1
+π I211.
+(26)
+Note that the definition of the master integrals ⃗J involves only ψ1 and φ1 (through d
+dxψ1), but not
+ψ2 nor φ2.
+The basis ⃗K, appearing in [9], is defined by
+K1
+=
+ε2 I110,
+(27)
+K2
+=
+−ε(1+2ε)
+4π
+�
+1+√x
+��
+3−√x
+��
+ψ2 − 1
+4
+�
+1+√x
+�2 φ2
+�
+I111 + ε
+4π (1−x)(9−x)ψ2I211,
+K3
+=
++ε(1+2ε)
+4π
+�
+1+√x
+��
+3−√x
+��
+ψ1 − 1
+4
+�
+1+√x
+�2 φ1
+�
+I111 − ε
+4π (1−x)(9−x)ψ1I211.
+In the definition of the master integrals ⃗K all periods ψ1,ψ2 and all quasi-periods φ1,φ2 appear.
+The master integrals K2 and K3 are related by ψ2 ↔ ψ1, φ2 ↔ φ1 and an overall minus sign.
+4.2
+The differential equations
+The differential equation in the basis⃗I reads
+d⃗I
+=
+A⃗I,
+(28)
+with
+A
+=
+
+
+0
+0
+0
+0
+−(1+2ε)
+3
+0
+−1
+3 (1+2ε)(1+3ε)
+1+3ε
+
+ dx
+x
++
+
+
+0
+0
+0
+0
+0
+0
+ε2
+4
+1
+4 (1+2ε)(1+3ε)
+−(1+2ε)
+
+ dx
+x−1
++
+
+
+0
+0
+0
+0
+0
+0
+−ε2
+4
+1
+12 (1+2ε)(1+3ε)
+−(1+2ε)
+
+ dx
+x−9.
+(29)
+In this basis, the entries are rational dlog-forms. However, the differential equation is not in
+ε-form.
+The differential equation in the basis ⃗J reads
+d⃗J
+=
+εB⃗J,
+(30)
+8
+
+with
+B
+=
+
+
+0
+0
+0
+0
+ω2
+ω0
+ω3
+ω4
+ω2
+
+
+(31)
+and
+ω0 = 2πi dτ
+= 2πiW
+ψ2
+1
+dx,
+ω2 = − f2(τ) (2πi)dτ = dx
+2x − dx
+x−1 − dx
+x−9,
+ω3 = f3(τ) (2πi)dτ
+= − 1
+2
+ψ1
+π dx,
+ω4 = f4(τ) (2πi)dτ
+=
+(x+3)4
+48x(x−1)(x−9)
+�ψ1
+π
+�2
+dx.
+(32)
+f2, f3 and f4 are modular forms of Γ1(6). The minus sign in front of f2 is convention. In terms
+of the variable x they are given by
+f2
+=
+1
+24
+�
+3x2 −10x−9
+��ψ1
+π
+�2
+,
+f3
+=
+− 1
+24x(x−1)(x−9)
+�ψ1
+π
+�3
+,
+f4
+=
+1
+576 (3+x)4�ψ1
+π
+�4
+.
+(33)
+Their q-expansions are given in appendix A.
+The differential equation in the basis ⃗K reads
+d⃗K
+=
+εC⃗K,
+(34)
+with
+C
+=
+
+
+0
+0
+0
+C2,1
+C2,2
+C2,3
+C3,1
+C3,2
+C3,3
+
+
+(35)
+and
+C2,1
+=
+−1
+2
+ψ2
+π dx,
+(36)
+C2,2
+=
+iπ
+6
+�
+(1+x) ψ1
+π
+ψ2
+π +
+�
+3x2 −10x−9
+� ψ2
+π
+∂xψ1
+π
++2x(x−1)(x−9) ∂xψ1
+π
+∂xψ2
+π
+�
+dx,
+C2,3
+=
+iπ
+6
+�
+(1+x)
+�ψ2
+π
+�2
++
+�
+3x2 −10x−9
+� ψ2
+π
+∂xψ2
+π
++2x(x−1)(x−9)
+�∂xψ2
+π
+�2�
+dx,
+9
+
+C3,1
+=
+1
+2
+ψ1
+π dx,
+C3,2
+=
+−iπ
+6
+�
+(1+x)
+�ψ1
+π
+�2
++
+�
+3x2 −10x−9
+� ψ1
+π
+∂xψ1
+π
++2x(x−1)(x−9)
+�∂xψ1
+π
+�2�
+dx,
+C3,3
+=
+−iπ
+6
+�
+(1+x) ψ1
+π
+ψ2
+π +
+�
+3x2 −10x−9
+� ψ1
+π
+∂xψ2
+π
++2x(x−1)(x−9) ∂xψ1
+π
+∂xψ2
+π
+�
+dx.
+4.3
+Periods on the maximal cut
+In this section we investigate the period matrices on the maximal cut of the sunrise integral. On
+the maximal cut of the sunrise integral only the last two master integrals are relevant (either I2,I3
+or J2,J3 or K2,K3). The defining property for basis ⃗K is that the period matrix on the maximal
+cut is diagonal and constant.
+We denote by
+ϕX
+i ,
+X ∈ {I,J,K}, i ∈ {1,2,3},
+(37)
+the integrand of the master integral Xi in the loop-by-loop Baikov representation [23]. In the
+loop-by-loop Baikov representations we have four integration variables z1 − z4, where z1 − z3
+correspond to the three propagators and z4 to an irreducible scalar product. Let C MaxCut be
+the integration domain selecting the maximal cut, i.e. a small counter-clockwise circle around
+z1 = 0, a small counter-clockwise circle around z2 = 0 and a small counter-clockwise circle
+around z3 = 0. We set z4 = u in accordance with the notation used in eq. (7). We denote by γ1
+and γ2 the two cycles of the elliptic curve. They define the integration domain in the variable u.
+We define
+C2 = C MaxCut ∪γ1,
+C3 = C MaxCut ∪γ2.
+(38)
+We consider the period matrix
+PX
+=
+� �
+ϕX
+2 |C2
+�
+�
+ϕX
+2 |C3
+�
+�
+ϕX
+3 |C2
+�
+�
+ϕX
+3 |C3
+�
+�
+.
+(39)
+In the i-th row of this matrix we then look at the leading term in the expansion in the dimensionsal
+regularisation parameter ε for this row. We denote the order of the leading term of row i by
+jmin(i). This defines a matrix PX,leading with entries
+PX,leading
+i j
+=
+coeff
+��
+ϕX
+i |Cj
+�
+,ε jmin(i)�
+·ε jmin(i).
+(40)
+One finds
+PI,leading
+=
+−8iπ
+�
+ψ1
+ψ2
+ψ1− 1
+4(1+√x)2φ1
+(1−√x)(3+√x)
+ψ2− 1
+4(1+√x)2φ2
+(1−√x)(3+√x)
+�
+,
+10
+
+PJ,leading
+=
+2i
+�
+(2πiε)2
+(2πiε)2τ
+0
+−(2πiε)
+�
+,
+PK,leading
+=
+4πε
+� 1
+0
+0
+1
+�
+.
+(41)
+As advertised, we see that PK,leading is proportional to the unit matrix. Note that PJ,leading can be
+written as
+PJ,leading
+=
+2i
+�
+(2πiε)2
+0
+0
+−(2πiε)
+��
+1
+τ
+0
+1
+�
+.
+(42)
+This is the decomposition of the period matrix PJ,leading into a semi-simple matrix and an unipo-
+tent matrix [24].
+4.4
+Solutions
+In the basis ⃗J we may give a solution for the master integrals in terms of iterated integrals of
+modular forms.
+Let f1(τ), f2(τ), ..., fn(τ) be a set of modular forms. We define the n-fold iterated integral of
+these modular forms by
+I (f1, f2,..., fn;τ,τ0)
+=
+(2πi)n
+τ
+�
+τ0
+dτ1
+τ1
+�
+τ0
+dτ2···
+τn−1
+�
+τ0
+dτn f1 (τ1) f2 (τ2)... fn(τn).
+(43)
+With q = exp(2πiτ) we may equally well write
+I (f1, f2,..., fn;τ,τ0) =
+q
+�
+q0
+dq1
+q1
+q1
+�
+q0
+dq2
+q2
+...
+qn−1
+�
+q0
+dqn
+qn
+f1 (τ1) f2(τ2)... fn(τn),
+τ j = 1
+2πi lnqj.
+(44)
+It will be convenient to introduce a short-hand notation for repeated letters. We use the notation
+{ fi}j
+=
+fi, fi,..., fi
+�
+��
+�
+j
+(45)
+to denote a sequence of j letters fi and more generally
+{fi1, fi2,..., fin}j
+=
+fi1, fi2,..., fin,......, fi1, fi2,..., fin
+�
+��
+�
+j copies of fi1, fi2,..., fin
+(46)
+to denote a sequence of ( j ·n) letters, consisting of j copies of fi1, fi2,..., fin. For example
+{ f1, f2}3
+=
+f1, f2, f1, f2, f1, f2.
+(47)
+11
+
+Our standard choice for the base point τ0 will be τ0 = i∞, corresponding to q0 = 0. This is
+unproblematic for modular forms which vanish at the cusp τ = i∞. In this case we have for a
+single integration
+f =
+∞
+∑
+j=1
+ajqj
+⇒
+q
+�
+0
+dq1
+q1
+f =
+∞
+∑
+j=1
+aj
+j qj.
+(48)
+For modular forms which attain a finite value at the cusp τ = i∞ we employ the standard “trailing
+zero” or “tangential base point” regularisation [10,25,26]: We first take q0 to have a small non-
+zero value. The integration will produce terms with ln(q0). Let Rln(q0) be the operator, which
+removes all ln(q0)-terms. After these terms have been removed, we may take the limit q0 → 0.
+With a slight abuse of notation we set
+I ( f1, f2,..., fn;q) = lim
+q0→0Rln(q0)
+
+
+q
+�
+q0
+dq1
+q1
+q1
+�
+q0
+dq2
+q2
+...
+qn−1
+�
+q0
+dqn
+qn
+f1 (τ1) f2(τ2)... fn (τn)
+
+.
+(49)
+We define the boundary constants Bk for the sunrise integral J2 by
+lim
+q→0Rln(q)J2
+=
+e
+2
+∞
+∑
+n=2
+(−1)n
+n
+ζnεn ∞
+∑
+k=2
+εkBk.
+(50)
+The left-hand side corresponds to setting all iterated integrals to zero, including the ones which
+are proportional to powers of ln(q). The boundary values Bk are collected in appendix B. Let us
+mention that the boundary values Bk are of weight k. The right-hand side of eq. (50) is therefore
+of uniform weight.
+We may express the master integrals in the basis ⃗J to all orders in the dimensional regulari-
+sation parameter in terms of iterated integrals of modular forms. We have
+J1 = e
+2
+∞
+∑
+n=2
+(−1)n
+n
+ζnεn
+,
+J2 = e
+−εI(f2;q)+2
+∞
+∑
+n=2
+(−1)n
+n
+ζnεn
+
+
+
+�
+∞
+∑
+j=0
+�
+ε2jI
+�
+{1, f4}j ;q
+�
+− 1
+2ε2j+1I
+�
+{1, f4}j ,1;q
+���
+∞
+∑
+k=2
+εkBk
++
+∞
+∑
+j=0
+ε j+2
+⌊ j
+2⌋
+∑
+k=0
+I
+�
+{1, f4}k ,1, f3,{f2}j−2k ;q
+�
+
+
+,
+J3 = e
+−εI(f2;q)+2
+∞
+∑
+n=2
+(−1)n
+n
+ζnεn
+
+
+
+�
+∞
+∑
+j=0
+�
+ε2j+1I
+�
+{ f4,1}j , f4;q
+�
+− 1
+2ε2jI
+�
+{ f4,1}j ;q
+���
+∞
+∑
+k=2
+εkBk
+12
+
++
+∞
+∑
+j=0
+ε j+1
+⌊ j
+2⌋
+∑
+k=0
+I
+�
+{f4,1}k , f3,{ f2}j−2k ;q
+�
+
+
+.
+(51)
+The expression for J2 appeared already in [10], the expression for J3 follows from (see eq. (25))
+J3
+=
+1
+ε
+1
+2πi
+d
+dτJ2 + f2J2.
+(52)
+For the first few terms of ε-expansion we have
+J1
+=
+1+ζ2ε2 − 2
+3ζ3ε3 + 7
+10ζ2
+2ε4 +O
+�
+ε5�
+,
+J2
+=
+[B2 +I (1, f3;q)]ε2
++
+�
+B3 − 1
+2B2I (1;q)−B2I ( f2;q)−I (1, f2, f3;q)−I ( f2,1, f3;q)
+�
+ε3
++
+�
+B4 +ζ2B2 − 1
+2B3I (1;q)−B3I ( f2;q)+ 1
+2B2I (1, f2;q)+ 1
+2B2I ( f2,1;q)
++B2I (1, f4;q)+B2I (f2, f2;q)+ζ2I (1, f3;q)+I (1, f2, f2, f3;q)+I ( f2, f2,1, f3;q)
++I (1, f4,1, f3;q)+I (f2,1, f2, f3;q)
+�
+ε4 +O
+�
+ε5�
+,
+J3
+=
+εI ( f3;q)+
+�
+−1
+2B2 −I ( f2, f3;q)
+�
+ε2 +
+�
+−1
+2B3 + 1
+2B2I ( f2;q)+B2I (f4;q)
++ζ2I ( f3;q)+I ( f2, f2, f3;q)+I (f4,1, f3;q)
+�
+ε3 +
+�
+−1
+2B4 − 1
+2ζ2B2 + 1
+2B3I (f2;q)
++B3I ( f4;q)− 2
+3ζ3I ( f3;q)−B2I (f4, f2;q)−B2I ( f2, f4;q)− 1
+2B2I (f2, f2;q)
+−1
+2B2I (f4,1;q)−ζ2I ( f2, f3;q)−I ( f2, f2, f2, f3;q)−I ( f4, f2,1, f3;q)
+−I (f2, f4,1, f3;q)−I (f4,1, f2, f3;q)
+�
+ε4 +O
+�
+ε5�
+.
+(53)
+Let us also summarise the boundary values: From eq. (50) and eq. (51) we obtain
+lim
+q→0Rln(q)J1
+=
+e
+2
+∞
+∑
+n=2
+(−1)n
+n
+ζnεn
+,
+lim
+q→0Rln(q)J2
+=
+e
+2
+∞
+∑
+n=2
+(−1)n
+n
+ζnεn ∞
+∑
+k=2
+εkBk,
+lim
+q→0Rln(q)J3
+=
+−1
+2e
+2
+∞
+∑
+n=2
+(−1)n
+n
+ζnεn ∞
+∑
+k=2
+εkBk.
+(54)
+In all three cases the right-hand sides are of uniform weight.
+13
+
+Given a solution in the basis ⃗J, we easily obtain a solution in the basis ⃗K. The two bases are
+related by
+⃗K
+= U⃗J,
+(55)
+with
+U
+=
+
+
+1
+0
+0
+0
+−(1+2ε)
+2πiε −g2 ·τ
+τ
+0
+g2
+−1
+
+
+(56)
+and
+g2
+=
+1
+24
+��
+3x2 −10x−9
+� ψ1
+π +4x(1−x)(9−x) ∂xψ1
+π
+� ψ1
+π .
+(57)
+In the modular variable τ the quantity g2 is given by
+g2
+=
+f2 +2 π
+ψ1
+1
+2πi
+d
+dτ
+ψ1
+π
+=
+4
+�
+3b2
+1 −3b1b2 −6b2
+2 −e2
+�
+.
+(58)
+The modular forms b1 and b2 and the quasi-modular form e2 are defined in appendix A. The
+quantity g2 is a quasi-modular form of modular weight 2 and depth 1. For γ ∈ Γ1(6) the quantity
+g2 transforms as
+(g2|2γ)(τ)
+=
+g2(τ)+ 2
+2πi
+c
+cτ+d ,
+γ =
+�
+a
+b
+c
+d
+�
+,
+(59)
+where the operator |kγ is defined by
+( f|kγ)(τ)
+=
+(cτ+d)−k · f(γ(τ)),
+γ(τ) = aτ+b
+cτ+d .
+(60)
+For the first few terms of ε-expansion we have
+K2
+=
+1
+2πi [−B2 +I ( f3,1;q)]ε+ 1
+2πi [−B3 −2B2 +B2I ( f2;q)−I ( f2, f3,1;q)−2I (1, f3;q)
+−2g2I (1,1, f3;q)−g2I (1, f3,1;q)−g2B2I (1;q)]ε2 +O
+�
+ε3�
+,
+K3
+=
+−I ( f3;q)ε+
+�1
+2B2 +I ( f2, f3;q)+g2B2 +g2I (1, f3;q)
+�
+ε2 +O
+�
+ε3�
+.
+(61)
+Let us look at the boundary values of K2
+lim
+q→0Rln(q)K2
+=
+− B2
+2πiε− (B3 +2B2)
+2πi
+ε2 +O
+�
+ε3�
+.
+(62)
+The term
+−2B2
+2πi ε2
+(63)
+is of weight minus one and spoils the uniform weight property. Hence we conclude that the basis
+⃗K is not of uniform weight if we require that the notion of uniform weight is compatible with
+restrictions in the kinematic space.
+14
+
+5
+Purity and simple poles
+In this section we address the second main question of this paper: The relation between uniform
+weight and simple poles in the elliptic case.
+5.1
+Definition of pure functions in the literature
+We recapitulate the definitions of unipotent and pure function as given in ref. [18]:
+Definition 1. A function is called unipotent, if it satisfies a differential equation without a homo-
+geneous term.
+To give an example, the functions ln(x) and Li2(x) are unipotent
+d
+dx ln(x) = 1
+x,
+d
+dxLi2 (x) = −1
+x ln(1−x),
+(64)
+while ex is not
+d
+dxex
+=
+ex.
+(65)
+Definition 2. Unipotent functions, whose total differential involves only pure functions and one-
+forms with at most simple poles are called pure.
+The standard example are multiple polylogarithms, whose total differential is given by
+dG(z1,...,zr;y)
+=
+r
+∑
+j=1
+G(z1,..., ˆz j,...,zr;y)
+�
+d ln
+�
+z j−1 −z j
+�
+−d ln
+�
+z j+1 −z j
+��
+,
+(66)
+where we set z0 = y and zr+1 = 0. A hat indicates that the corresponding variable is omitted.
+In addition one uses the convention that for z j+1 = z j the one-form d ln(z j+1 −z j) equals zero.
+Clearly, the one forms
+d ln
+�
+z j+1 −z j
+�
+=
+dz j+1 −dz j
+z j+1 −z j
+(67)
+have only simple poles.
+5.2
+Iterated integrals of modular forms
+Let us now look at iterated integrals of modular forms, as defined in eq. (43). It is clear that
+these iterated integrals are unipotent functions, as differentiation removes one integration. We
+investigate the order of the poles of the total differential.
+We denote by
+H
+=
+{ τ ∈ C | Im(τ) > 0 }
+(68)
+15
+
+the complex upper half-plane and by
+H
+=
+H∪{i∞}∪Q
+(69)
+the extended complex upper half-plane. Under the map q = exp(2πiτ) the complex upper half-
+plane H is mapped to the punctured open disk
+D
+=
+{ q ∈ C | 0 < |q| < 1 }
+(70)
+and H is mapped to
+D
+=
+D∪{0}∪
+�
+e2πir | r ∈ Q
+�
+.
+(71)
+Let fk(τ) be a modular form of weight k for a congruence subgroup Γ and
+ωmodular
+k
+=
+2πi fk (τ)dτ.
+(72)
+For simplicity we assume that
+� 1
+1
+0
+1
+�
+∈ Γ. In this case fk has the q-expansion [27]
+fk
+=
+∞
+∑
+n=0
+anqn.
+(73)
+(In the general case fk will have an expansion in q
+1
+N′ , where N′ is the smallest positive inte-
+ger such that fk(τ + N′) = fk(τ). The general case is only from a notational perspective more
+elaborate.) In addition we will always assume that modular forms are normalised such that the
+coefficients of their q-expansion are algebraic numbers. We view ωmodular
+k
+as a differential one-
+form on D. In the variable q we have
+ωmodular
+k
+=
+∞
+∑
+n=0
+anqn−1dq.
+(74)
+This shows immediately that ωmodular
+k
+is holomorphic on D and has a simple pole at q = 0 if
+a0 ̸= 0. Thus, in a neighbourhood of q = 0 the differential one-form ωmodular
+k
+has at most simple
+poles.
+Let us now discuss if this extends globally to D. The answer will be no. We have to look at
+the other cusps. We investigate the behaviour at
+q0
+=
+e2πi(− d
+c),
+c,d ∈ Z, c ̸= 0.
+(75)
+We may derive the behaviour of ωmodular
+k
+at q0 from the modular properties of fk. We consider
+the modular transformation
+γ =
+� a
+b
+c
+d
+�
+∈ SL2(Z),
+γ−1 =
+�
+d
+−b
+−c
+a
+�
+(76)
+16
+
+and set
+τ′ = γ(τ) = aτ+b
+cτ+d ,
+q′ = e2πiτ′.
+(77)
+This maps τ = −d
+c to τ′ = i∞ and q0 to q′
+0 = 0. For the automorphic factor we have
+cτ+d
+=
+c
+2πi
+(q−q0)
+q0
++O
+�
+(q−q0)2�
+.
+(78)
+( fk|kγ−1)(τ′) has again a q′-expansion as in eq. (73)
+�
+fk|kγ−1��
+τ′�
+=
+∞
+∑
+n=0
+a′
+n
+�
+q′�n .
+(79)
+If fk is a modular form for the congruence subgroup Γ and γ ∈ Γ we have a′
+n = an, otherwise
+the coefficients need not be the same. Usually we are interested in the cusps not equivalent to
+τ = i∞, this implies γ ∈ SL2 (Z)\Γ. For a′
+0 ̸= 0 we have
+ωmodular
+k
+=
+a′
+0
+� c
+2πi
+�−k
+qk−1
+0
+dq
+(q−q0)k +O
+�
+(q−q0)−k+1�
+.
+(80)
+Thus we see that whenever fk is non-vanishing at the cusp τ0 = −d
+c, the differential one-form
+ωmodular
+k
+has a pole of order k in the variable q at q = q0. Globally, ωmodular
+k
+has poles up to order
+k on D.
+5.3
+Elliptic polylogarithms
+The discussion of the previous sub-section is not restricted to iterated integrals of modular forms
+and carries over to elliptic polylogarithms �Γ.
+Let g(k)(z,τ) denote the coefficients of the Kronecker function and set
+ωKronecker
+k
+(z,τ)
+=
+(2πi)2−k
+�
+g(k−1) (z,τ)dz+(k −1)g(k) (z,τ) dτ
+2πi
+�
+.
+(81)
+We may view ωKronecker
+k
+as a differential one-form on the two-dimensional moduli space M1,2.
+Coordinates on this moduli space are (z,τ). The elliptic polylogarithms �Γ are iterated integrals
+of ωKronecker
+k
+(z−cj,τ) along z at constant τ. It is known that the functions g(k)(z,τ) have at most
+simple poles in z and when restricted to τ = const the elliptic polylogarithms �Γ are pure functions
+in the sense of definition 2. However in the applications towards Feynman integrals it is usually
+the case that the assumption τ = const is not justified. A variation of the kinematic variables of
+the Feynman integral will imply a variation of τ and we have to consider the τ-dependence as
+well. For the argument we want to make it is sufficient to restrict to z = a + bτ with a,b ∈ Q
+and k ≥ 2. In this case the differential one-forms ωKronecker
+k
+reduce to the form of ωmodular
+k
+[24]
+and the argument from the previous sub-section applies: In this case the differential one-forms
+ωKronecker
+k
+may have poles up to order k in the variable q (or τ).
+17
+
+5.4
+Modular transformations
+We have seen that locally in the coordinate chart D ∪ {0} the basis ⃗J satisfies the criteria of
+definition 2. This coordinate chart includes the point x = 0. Let us now investigate the global
+picture. For the sunrise integral we have four singular points x ∈ {0,1,9,∞} and we may cover
+the kinematic space with four charts, such that each chart includes exactly one singular point.
+In each chart we may construct a basis, which satisfies the criteria of definition 2 locally. In
+different charts we will have different coordinates τ and τ′, but also different bases of master
+integrals ⃗J and ⃗J′. The coordinates τ and τ′ will be related by a modular transformation. The
+modular transformation induces also the transformation between ⃗J and ⃗J′.
+Let us discuss the behaviour near the cusp τ0 = −d
+c. The modular transformation γ defined
+in eq. (76) maps τ0 = −d
+c to τ′
+0 = i∞. Let fk be a modular form for a congruence subgroup Γ.
+Then by definition fk(τ) is holomorphic on H and ( fk|kγ−1)(τ′) has a q′-expansion as in eq. (79)
+for any γ ∈ SL2(Z). This suggest to change in a neighbourhood of τ = −d
+c coordinates from q to
+q′. The differential one-form
+2πi
+�
+fk|kγ−1��
+τ′�
+dτ′
+(82)
+has then a simple pole at q′ = 0 (corresponding to τ = −d
+c). However, ωmodular
+k
+as defined in
+eq. (72) does not transform under this coordinate change into eq. (82). Instead we find
+ωmodular
+k
+=
+�
+−cτ′ +a
+�k−2 ·2πi
+�
+fk|kγ−1��
+τ′�
+dτ′.
+(83)
+(−cτ′ + a) is the automorphic factor for γ−1. For k ̸= 2 this factor spoils that iterated integrals
+of modular forms transform under modular transformations into iterated integrals of modular
+forms. However, elliptic Feynman integrals transform nicely: Let us consider for
+γ(τ)
+=
+aτ+b
+cτ+d ,
+γ ∈ SL2(Z)
+(84)
+the combined transformation [21]
+⃗J′
+=
+
+
+1
+0
+0
+0
+1
+cτ+d
+0
+0
+− c
+2πiε
+cτ+d
+
+⃗J,
+τ′
+=
+aτ+b
+cτ+d .
+(85)
+One obtains
+d⃗J′
+=
+εB′⃗J′
+(86)
+with
+B′
+=
+2πi
+
+
+0
+0
+0
+0
+−( f2|2γ−1)(τ′)
+1
+( f3|3γ−1)(τ′)
+( f4|4γ−1)(τ′)
+−( f2|2γ−1)(τ′)
+
+dτ′.
+(87)
+18
+
+As Γ(6) is a subgroup of Γ1(6) we have Mk(Γ1(6)) ⊂Mk(Γ(6)) and as Γ(6) is a normal subgroup
+of SL2(Z) it follows that
+fk|kγ−1
+∈ Mk(Γ(6)).
+(88)
+We see that the entries of B′ are again differential one-forms of the form as in eq. (72). We may
+express ⃗J′ again in terms of iterated integrals of modular forms, this time in the variable q′. It can
+be shown that the boundary constants are again of uniform weight. Hence it follows that in the
+coordinate chart with coordinate q′ (or τ′) the basis ⃗J′ satisfies the criteria of definition 2 locally.
+6
+Conclusions
+For Feynman integrals which evaluate to multiple polylogarithms we have a clear understanding
+of purity: These are Feynman integrals, whose term of order j in the ε-expansion is pure of
+transcendental weight j. We are interested in extending this concept to Feynman integrals beyond
+the ones which evaluate to multiple polylogarithms.
+This is non-trivial and in this paper we discussed some subtleties: We showed that an ε-
+factorised differential equation alone does not lead necessarily lead to a solution which is pure.
+The boundary values have to be pure as well. This applies in particular to a basis constructed by
+the requirement that the period matrix on the maximal cut is proportional to the unit matrix. The
+argument we presented is agnostic to the exact definition of purity beyond the case of multiple
+polylogarithms, we only assumed that the definition of transcendental weight in the general case
+is compatible with the restriction of the kinematic space to a sub-space.
+In the second part of the paper we adopted a particular definition of purity from the literature.
+We showed that this definition works only locally – but not globally – for a particular basis of
+the two-loop equal mass sunrise integral. Of course, it might well be that this particular basis is
+not the optimal one, but another possibility is that the definition of purity needs a more refined
+definition. The modular transformation properties, which we discussed in section 5.4, point
+towards a possible modification.
+We believe that the detailed analysis we carried out in this paper will be helpful for a defi-
+nition of purity which not only includes the elliptic case, but also Feynman integrals related to
+Calabi-Yau geometries.
+Acknowledgements
+S.W. thanks the Niels Bohr Institute for hospitality and H.F. thanks the Mainz Institute of Theo-
+retical Physics for hospitality.
+H.F. is partially supported by a Carlsberg Foundation Reintegration Fellowship, and has re-
+ceived funding from the European Union’s Horizon 2020 research and innovation program under
+the Marie Sklodowska-Curie grant agreement No. 847523 “INTERACTIONS”.
+19
+
+A
+Modular forms
+In this appendix we give the q-expansions of the modular forms f2, f3 and f4, appearing in
+eq. (32) and the q-expansions of ψ1 (which is a modular form of modular weight 1). In addition,
+we define the Eisenstein series e2, which appears in eq. (58).
+We start by introducing a basis {b1,b2} for the modular forms of modular weight 1 for the
+Eisenstein subspace E1(Γ1(6)):
+b1 = E1(τ;χ1,χ−3),
+b2 = E1(2τ;χ1,χ−3),
+(89)
+where χ1 and χ−3 denote primitive Dirichlet characters with conductors 1 and 3, respectively. In
+terms of the coefficients g(k)(z,τ) of the Kronecker function we have
+b1 =
+√
+3
+6π g(1)(1
+3,τ),
+b2 = −
+√
+3
+12π
+�
+g(1)(1
+3,τ)−g(1)(1
+6,τ)
+�
+.
+(90)
+Then
+f2
+=
+−6
+�
+b2
+1 +6b1b2 −4b2
+2
+�
+,
+f3
+=
+36
+√
+3
+�
+b3
+1 −b2
+1b2 −4b1b2
+2 +4b3
+2
+�
+,
+f4
+=
+324b4
+1.
+(91)
+In terms of the coefficients g(k)(z,τ) of the Kronecker function we have
+f2
+=
+1
+2π2
+�
+3g(2)(1
+2,τ)−g(2)(1
+3,τ)+g(2)(1
+6,τ)
+�
+,
+f3
+=
+1
+4π3
+�
+15g(3)(1
+3,τ)−12g(3)(1
+6,τ)
+�
+,
+f4
+=
+1
+4π4
+�
+−18g(4)(0,τ)−27g(4)(1
+3,τ)
+�
+.
+(92)
+The q-expansions are
+f2
+=
+−1
+2 −8q−4q2 −44q3 +4q4 −48q5 −40q6 +O
+�
+q7�
+,
+f3
+=
+−3
+√
+3
+�
+q−5q2 +9q3 −11q4 +24q5 −45q6�
++O
+�
+q7�
+,
+f4
+=
+1
+4 +6q+54q2 +222q3 +438q4 +756q5 +1998q6 +O
+�
+q7�
+.
+(93)
+In addition we have
+ψ1
+π
+=
+2
+√
+3(b1 +b2)
+=
+2
+3
+√
+3
+�
+1+3q+3q2 +3q3 +3q4 +3q6�
++O
+�
+q7�
+.
+(94)
+20
+
+We define the Eisenstein series e2 by
+e2 (τ)
+=
+1
+2(2πi)2
+∑
+′
+(n1,n2)∈Z2\(0,0)
+1
+(n1 +n2τ)2.
+(95)
+The prime at the summation sign denotes the Eisenstein summation prescription defined by
+∑
+′
+(n1,n2)∈Z2
+f (z+n1 +n2τ)
+=
+lim
+N2→∞
+N2
+∑
+n2=−N2
+�
+lim
+N1→∞
+N1
+∑
+n1=−N1
+f (z+n1 +n2τ)
+�
+.
+(96)
+The q-expansion of e2 starts with
+e2 (τ)
+=
+− 1
+24 +q+3q2 +4q3 +7q4 +6q5 +12q6 +O
+�
+q7�
+.
+(97)
+The Eisenstein series e2 is a quasi-modular form.
+B
+Boundary values
+In this appendix we give the boundary values Bk. These are easily obtained from [10, 28]. We
+have
+∞
+∑
+k=0
+εkBk
+=
+3
+43−ε
+�
+h− 2πε
+3
+Γ(1+2ε)
+Γ(1+ε)2
+�
+,
+(98)
+where
+h = 1
+i
+�
+(−r3)−ε
+2F1 (−2ε,−ε;1−ε;r3)−
+�
+−r−1
+3
+�−ε
+2F1
+�
+−2ε,−ε;1−ε;r−1
+3
+��
+(99)
+and r3 = exp(2πi/3). The hypergeometric function can be expanded systematically in ε with the
+methods of [29–32]. The first few terms are given by
+2F1 (−2ε,−ε;1−ε;x) = 1+2ε2Li2 (x)+ε3 [2Li3(x)−4Li21 (x,1)]
++ε4[2Li4 (x)−4Li31 (x,1)+8Li211 (x,1,1)]+O
+�
+ε5�
+.
+(100)
+The first few boundary values are given by
+B0
+=
+0,
+B1
+=
+0,
+B2
+=
+3
+2i
+�
+Li2 (r3)−Li2
+�
+r−1
+3
+��
+,
+B3
+=
+3
+2i
+�
+−2Li21 (r3,1)−Li3(r3)+2Li21
+�
+r−1
+3 ,1
+�
++Li3
+�
+r−1
+3
+��
+21
+
+−ln(3)B2,
+B4
+=
+3
+2i
+�
+4Li211 (r3,1,1)−2Li31 (r3,1)+Li4 (r3)−4Li211
+�
+r−1
+3 ,1,1
+�
++2Li31
+�
+r−1
+3 ,1
+�
+−Li4
+�
+r−1
+3
+��
+−ln(3)B3 − 1
+2 ln2(3)B2 + 1
+3ζ2B2.
+(101)
+These can be reduced to polylogarithms of depth 1 as follows [33,34]:
+B2
+=
+3 Im Li2 (r3),
+B3
+=
+24
+5 Im Li3
+� i√
+3
+�
+− 17
+90π3 − 1
+10π(ln(3))2 ,
+B4
+=
+−63
+10Im Li4 (r3)+ 48
+5 Im Li4
+� i√
+3
+�
++ 17
+90π3 ln(3)+ 1
+30π(ln(3))3.
+(102)
+References
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+23
+

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+A Survey about Acquisition System Design for Myoelectric
+Prosthesis
+Dina Reda Eldamak
+December 2022
+1
+Introduction
+According to the World Health Organization (WHO), 30 million people are in need of prosthetic
+and orthotic devices [1]. Some people are born with this limb loss, while others lose limbs due to
+diseases such as Cancer, diabetes, and work accidents. Additionally, limb amputation is among
+the most severe and heavily reported injuries among veterans during war [2, 3]. Example of
+female with hand amputation is shown in Figure 1.
+Figure 1: Female with Prosthetic limb [4]
+The medical applications of integrated circuit technology have recently made significant ad-
+vances, thus improving human quality of life. Moreover, the use of microelectronics integration
+1
+arXiv:2301.00163v1  [eess.SP]  31 Dec 2022
+
+dominates a lot of medical applications, especially portable and wearable battery-operated de-
+vices. Bio-signals mostly arise from natural physiological processes, such as cardiac potentials
+(ECG -electro-cardiogram), potentials of the ocular tissue (EOG - electro-oculogram), potentials
+of the muscular tissues (electro-myogram -EMG), brain potential (electro-encephalogram -EEG),
+and respiratory signals , etc. Electro-myogram - EMG is an important factor for muscle disease
+diagnosis. Furthermore, it’s the key factor in connecting any amputee to a prosthetic limb. This
+can be done through extracting the EMG signal from the body using a readout electronics that
+can detect the muscles electrical activity. Consequently, the extracted signal is processed and
+used to control the prosthetic limb. Thus, the objective of this report is to provide the reader
+with the basic understanding of integrated solutions for controlling prosthetic limbs either arms
+or legs.
+The top level block diagram of a smart EMG acquisition system is shown in Fig. 2. The
+system includes a self-powered readout portable acquisition device for measuring the patient’s
+EMG signal in order to send it to a controller that can be used to emulate the right action
+to the prosthetic limb similar to the same action in a normal person. It should be noted that
+miniaturized EMG acquisition system idea, which continuously monitor muscles activity, can be
+extended to different applications such as physical rehabilitation and prosthesis.
+Figure 2: Block diagram of a general smart sEMG recorder [5]
+2
+
+Tissue Interface
+ProcessingUnit
+Sensor Interface2
+System Architecture
+An electronic system can control a prosethetic device by monitoring the EMG signals of the arm,
+and use those signals to control the prothetic arm. Moreover, the devices can be battery-free by
+being powered solely using energy harvesting from the ambient.
+Since these prosthetic devices requires precise fitting to the residual limb, pressure and tem-
+perature sensor at the skin-prosthetic interface are added to the system. Pressure sensors are
+needed for monitoring the prosthetic limb to avoid the development of regions of high pressure
+as the limb moves during walking or grasping objects. Temperature sensor are necessary as high
+temperature can accelerate tissue damage [6]. The signals from the sensor at the skin-prosthetic
+can be transmitted to the outer surface of the prosthetic socket using Near Field Communication
+(NFC) or to a smart phone using Bluetooth Low Energy (BLE) as shown in Figure 3.
+Figure 3: Illustration of sensors mounted at the skin-prosthetic interface transmitting data to
+the device at the outer surface of the prosthetic leg using NFC and to smart phone using BLE
+[6].
+3
+Block Diagram
+Three major research directions are available when designing an EMG acquisition system. The
+first is to acquire the signal from the surrounding noisy environment using a sensor interface
+3
+
+Residual
+limb
+NFC/BLE
+modules
+Multimodal
+battery-free
+sensors
+Prosthetic leg
+Prosthetic
+Portable
+socket
+NFC/BLE
+electronicdevice
+modulesFigure 4: Block diagram of the proposed smart sEMG recorder including sensors, AFE, and RF
+integrated system [5]
+circuit that’s designed in CMOS technology. The second involves reducing the form factor and
+power consumption of the acquisition system. The third is the signal conversion to the digital
+world and the interface with the digital controller. At this point, the extracted EMG signal
+is in a digital form and can be processed through FPGA or any other processor to control a
+Pprosthetic limb.
+A typical block diagram of the proposed EMG acquisition system is shown in Fig.
+4.
+The system consists of an EMG sensor, analog front end (AFE), and radio frequency (RF)
+transmission unit. The AFE is typically composed of an analog amplification, filtration, analog
+to digital converter (ADC), and controller to process the digital signal and send it to a prosthetic
+limb. The acquisition system design can be integrated on a single chip, then the digital data is
+fed to FPGA or a controller.
+In addition, because the integrated solution takes a considerable time during design, fabrica-
+tion, and testing phases, a discrete solution in parallel with the integrated one can be used as a
+proof of concept to validate the proposed methodology.
+Figure 5 shows a detailed system block diagram of the proposed smart sEMG acquisition
+system. An analog multiplexer is inserted to choose between different EMG electrodes in the
+smart sEMG recorder shown in the figure. The design of each of the building blocks involves
+4
+
+AFE+RF
+NSPU
+FlexBandFigure 5: Detailed block diagram of the proposed smart sEMG acquisition system [5]
+several design challenges requiring some research. The following section includes a list of major
+research directions that can be pursued.
+4
+Circuit Implementation
+In the following subsections, the basic system building blocks are introduced. First, the EMG
+sensor specifications are explored.
+Second, the low noise amplifier LNA design is presented.
+Third, the filter design and bandwidth are provided. Fourth, the signal conversion from analog
+to digital is presented through an ADC. Last, digital signal processing through FPGA is explored.
+4.1
+Sensor Specifications
+EMG sensor placement plays an important role in signal acquisition. According to its orientation
+and position, the EMG signal strength varies significantly. This effect is shown in Fig 6. As
+seen, by placing the sensor in the middle of muscle fiber, the maximum signal strength can be
+easily obtained. Otherwise, the signal degrades significantly when placing the sensor far away
+from the middle.
+EMG sensor can be represented in different forms. It can be in either needle that is inserted
+into the muscle or surface electrode that picks the signal from the skin. An example of surface
+5
+
+Smart sEMG Recorder
+FPGA
+16-Channel Recorder
+GBDT based NSPU
+Flexible
+16x
+GBDT Core
+Feature
+Band
+LNA
+Extractors
+00
+MUX
+PGA
+ADC
+16:1
+LNA
+Pre-
+Processing
+Result
+Model
+Digital Logic
+Data Ready| CLK_ADC
+Generator
+Loader
+nRF52
+Ping Pong Buffer
+Recorder
+TF Card
+BLE
+CIC Filter
+Interface
+Buffer A
+Buffer B
+Interface i
+X
+:=
+TF Card
+Mobile Devices
+Possible Applications
+Offline trainingFigure 6: Effect of EMG sensor position [7]
+EMG sensor specifications that have to be met through out the design are as follow shown in
+Fig. 7.
+4.2
+Low Noise Amplifier Design
+It’s the first and the major block in the EMG chain that comes after the sensor. The measurement
+sensitivity and accuracy is determined in this stage. This complicates the design and requires a
+large amount of adaptability to accommodate the input signal. The previous stage, which is the
+EMG sensor, adds large parasitic capacitance at the input of this stage, and thus reduces gain,
+bandwidth, noise performance and the sensitivity of the amplifier.
+Sources of noise and interference like flicker noise, electrodes offset, and 60 Hz power line
+noise can affect the whole acquisition procedure. The bandwidth of the EMG signal is up to
+6
+
+Raw EMG output
+Innervation Zone
+Correct Placement
+Midline of the muscle belly
+between an innervation zone
+and a myotendon junction
+Midline Offset
+Myotendon JunctionFigure 7: sensor specifications [8]
+500 Hz with amplitude that ranges from 0.1 to 5 mV and the high-frequency noise can be easily
+removed using a low pass filter. However, low-frequency noise and DC offset fall within the
+EMG bandwidth and hence require different rejection techniques. Chopping technique is one
+of the best candidates to modulate the offset and flicker noise to a higher spectrum which in
+turn enable the acquisition system to effectively suppress the interference from ambient and 1/f
+noise. Different architectures with different requirements in terms of input signal levels, BW and
+amplitudes are proposed in literature [9, 10]. Figure 8 shows the block diagram of implemented
+analog front-end for acquiring of EEG, ECG, and EMG signals [9]. The shown diagram consists
+of a chopper instrumentation amplifer in addition to capacitive coupling, filter stage to remove
+the chopping spikes, a digitally controlled variable gain amplifier.
+4.3
+Filter Design
+A Gm-C filter cab be used in the design. A standard architecture is shown in Figure 9. Offset
+from the electrodes can be canceled using current-mode DAC [11]. Power Consumption of this
+7
+
+DataLITE Wireless EMG Amplifier
+Wired EMG Amplifier
+Product Ref
+LE230FW
+SX230FW
+42 × 24 × 14 mm
+38 x 20
+Dimensions
+Two 4 mm snap connectors on 100 mm wires
+Two 4 mm snap connectors on 100 mm wires
+Mass
+17 g (excluding cable and plug)
+8g (excluding cable and plug)
+Bandwidth
+10 - 250,470, 950, 5000Hz
+20 - 460Hz
+5Hz - 480Hz
+Additional Bandwidths
+N/A
+5Hz - 1000Hz
+Contact Diameter
+Dependant on electrode size
+Contact Center Spacing
+Variable
+Electrodes
+Disposable
+CMRR @ 60 Hz (dB)
+> 96 dB (typically 110 dB)
+Full Scale
++/- 6 mV Peak to Peak
++/- 3 mV Peak to Peak
+Gain
++/- 60 microvolts to +/- 6 millivolts
+Standard unit x1000 (100 als0 available)
+Input Impedance
+>100 Mohms
+Accuracy
++/- 1.0%
++/- 2% full scale
+Noise
+<5μv
+Supply Voltage
+N/A
++3.50 to +5.5 Vdc
+Battery Life
+Up to 8 hours
+N/A
+Battery Type
+Rechargeable Li-lon Polymer
+N/A
+Wireless Transmission
+Tolerant for 100 mS
+N/A
+Data Loss
+1.25m cable
+Range from Interface
+Wireless range up to 30m
+(custom lengths available on request)
+Compatible Interfaces
+DataLITE PIONEER, ADVANCE, EXPLORE
+DataLOG, DataLINK, Amplifier or 3rd partyFigure 8: Architecture of the bio-potential readout front-end for the acquisition of EEG, ECG,
+and EMG signals [9]
+topology can also be reduced by low-voltage supply operation [10].
+Figure 9: Transistor level implementation of Gm-C filter. DDA: Differential Difference Amplifier
+[11].
+4.4
+ADC Design
+Non-uniform sampling can minimize the power consumption of ADC while digitizing activity-
+dependent biological signals. For example, a continous-time (CT) charge-based ADC that ac-
+8
+
+DC Level
+Select BW Select Gain
+Programmable
+1pF
+Gain Stage
+BW Select
+Cext2
+OTA2
+■out
+Buffer
+vin+
+IA
+C12
+OTA
+vin- 
+BW Select
+Cs
+C12= 20pF
+AC
+23
+21
+Coupled
+Chopping
+C22
+Chopped
+Spike
+C11
+IA
+Filter
+BIOPOTENTIAL
+Bias Current
+I=
+clk
+CLK Generator
+ASIC
+GeneratorVDDA
+Velectrode
+VBIAS
+on
+Vt
+IIN
+TIN
+VBODY
+rst
+Voutp
+Cint
+H
+Vref
+TcN
+VBN
+TcN
+IFBP
+IFBN
+DDA
+TL
+LT
+Hquires samples when the input crosses a specific threshold is shown in Figure 10. The ADC
+works by storing the analog equivalent of the last digitized input as a voltage across the across
+the capacitor 𝐶𝑏. Once the input signal crosses this voltage, a pulse with length 𝑇𝑃 is generated
+to charge or discharge the capacitor 𝐶𝑏 by 𝑉𝐿𝑆𝐵 using one of the current sources connected to
+the supply and ground [12]. Non-uniform sampling adapts to the instantaneous bandwidth of
+the signal, consequently the dynamic power consumption scales with the activity of the input
+signal. The FOM of the CT charge-based ADC can be improved by reducing the power supply
+further [10].
+Figure 10: Top level architecture of Continous-time (CT) charge based ADC with non-uniform
+sampling rate [12].
+4.5
+FPGA Processing
+Machine learning algorithms such as Support Vector Machine (SVM) have allowed for on-chip
+feature extraction and classification of biomedical signals [13]. Machine learning can also be
+deployed in the domain of prosthetic devices for precise control.
+Figure 11 depicts the con-
+troller of prosthetic device which can be implemented using Field Programmable Gate Array
+(FPGA). Figure 12 depicts the experimental setup for analyzing the data from high density
+EMG acquisition system using Xilinix Zedboard [14].
+9
+
+Biasing
+Pulse
++OPulseUp
+Generator
+Comparator
+Vin O
+UP
+Vb
+DOWN
+OUT
++00UT<7:0>
+Cb
+RESET
+ORESET
+Pulse
++OPulseDown
+Generator
+Conf guration
+Register
+aLkT
+OCLK
+DO
+SI 
+ISO
+VFigure 11: Top level architecture of controller of prosthetic hand including feature extraction
+and classification [14].
+Figure 12: Experimental Setup of EMG acquisition and processing using Xilinix ZedBoard [14].
+4.6
+Energy Harvesting
+The electrical power harvested from the environment (specially, thermal energy) can power
+the ultra-low-power EMG Sensor.
+We have previously developed energy harvesting systems
+from various sources and high-efficiency DC-DC converters [15, 16]. For example, the system
+architecture of power management IC for solar energy harvesting applications , designed by the
+author, and chip micrograph are shown in Figure 13.
+10
+
+class
+192 HD EMG
+decision
+channels
+Data
+Feature
+Classification
+Acquisition
+Extraction
+Embedded Prosthesis Controller
+prosthesis
+movementI92 ch. HD EMG
+ZedBoard for EMG
+electrode array
+signal processing
+Michelangelo
+HD EMG
+PC for
+hand prosthesis
+DAQ
+comm.Figure 13: System architecture of power management IC for solar energy harvesting applications,
+designed by one of the team members, and chip micrograph [15]
+5
+Conclusion
+This paper provided a survey about EMG acquisition systems for prostehtics and orthotic de-
+vices.
+References
+[1] Shirley Ryan AbilityLab. Facts about limb loss. [Online]. Available: https://www.sralab.
+org/research/labs/bionic-medicine/news/facts-about-limb-loss
+[2] UK
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+year
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+https://www.gov.uk/
+government/statistics/uk-service-personnel-amputations-financial-year-20192020/
+afghanistan-and-iraq-amputation-statistics-1-april-2015-to-31-march-2020
+[3] L. G. Stansbury, S. J. Lalliss, J. G. Branstetter, M. R. Bagg, and J. B. Holcomb,
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+3 mirr
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+2.2mm
+Startup
+VcBUF
+smtehe
+Switch Matrix
+Gapetsi
+S2
+Switch
+DXADE
+Current
+VBA
+VINDN市
+Matrix
+Ref.
+Startup
+and
+Drivers
+Dnivers
+Mp1
+Configuration Block
+Modell
+MIPP
+lectior
+VIN
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+S
+DA
+Pulse Generation Block
+Test Block
+Φ1
+Φ2
+En
+A
+PTrig
+Dvlee
+VINDN
+VBA
+AE
+Test Block
+险电
+Voltase:
+Curont
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+and systems, vol. 13, no. 6, pp. 1563–1574, 2019.
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+C. Liang, M. Patel, I. Jung, J. Kim, M. Namkoong, K. Kwon, X. Guo, C. Ogle,
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+R. Caldwell, A. Banks, S. Xu, Y. Huang, S. Fatone, and J. A. Rogers, “Wireless sensors for
+continuous, multimodal measurements at the skin interface with lower limb prostheses,”
+Science Translational Medicine, vol. 12, no. 574, p. eabc4327, 2020. [Online]. Available:
+https://www.science.org/doi/abs/10.1126/scitranslmed.abc4327
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+https://www.biometricsltd.com/
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+[9] R. F. Yazicioglu, “A 60𝜇w 60 nV/
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+13
+

XtAyT4oBgHgl3EQfWfc6/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf,len=394
+page_content='A Survey about Acquisition System Design for Myoelectric  Prosthesis  Dina Reda Eldamak  December 2022  1  Introduction  According to the World Health Organization (WHO), 30 million people are in need of prosthetic  and orthotic devices [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Some people are born with this limb loss, while others lose limbs due to  diseases such as Cancer, diabetes, and work accidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Additionally, limb amputation is among  the most severe and heavily reported injuries among veterans during war [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Example of  female with hand amputation is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  Figure 1: Female with Prosthetic limb [4]  The medical applications of integrated circuit technology have recently made significant ad-  vances, thus improving human quality of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Moreover, the use of microelectronics integration  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='00163v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='SP]  31 Dec 2022  dominates a lot of medical applications, especially portable and wearable battery-operated de-  vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Bio-signals mostly arise from natural physiological processes, such as cardiac potentials  (ECG -electro-cardiogram), potentials of the ocular tissue (EOG - electro-oculogram), potentials  of the muscular tissues (electro-myogram -EMG), brain potential (electro-encephalogram -EEG),  and respiratory signals , etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Electro-myogram - EMG is an important factor for muscle disease  diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Furthermore, it’s the key factor in connecting any amputee to a prosthetic limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' This  can be done through extracting the EMG signal from the body using a readout electronics that  can detect the muscles electrical activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Consequently, the extracted signal is processed and  used to control the prosthetic limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Thus, the objective of this report is to provide the reader  with the basic understanding of integrated solutions for controlling prosthetic limbs either arms  or legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  The top level block diagram of a smart EMG acquisition system is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' The  system includes a self-powered readout portable acquisition device for measuring the patient’s  EMG signal in order to send it to a controller that can be used to emulate the right action  to the prosthetic limb similar to the same action in a normal person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' It should be noted that  miniaturized EMG acquisition system idea, which continuously monitor muscles activity, can be  extended to different applications such as physical rehabilitation and prosthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  Figure 2: Block diagram of a general smart sEMG recorder [5]  2  Tissue Interface  ProcessingUnit  Sensor Interface2  System Architecture  An electronic system can control a prosethetic device by monitoring the EMG signals of the arm,  and use those signals to control the prothetic arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Moreover, the devices can be battery-free by  being powered solely using energy harvesting from the ambient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  Since these prosthetic devices requires precise fitting to the residual limb, pressure and tem-  perature sensor at the skin-prosthetic interface are added to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Pressure sensors are  needed for monitoring the prosthetic limb to avoid the development of regions of high pressure  as the limb moves during walking or grasping objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Temperature sensor are necessary as high  temperature can accelerate tissue damage [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' The signals from the sensor at the skin-prosthetic  can be transmitted to the outer surface of the prosthetic socket using Near Field Communication  (NFC) or to a smart phone using Bluetooth Low Energy (BLE) as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  Figure 3: Illustration of sensors mounted at the skin-prosthetic interface transmitting data to  the device at the outer surface of the prosthetic leg using NFC and to smart phone using BLE  [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  3  Block Diagram  Three major research directions are available when designing an EMG acquisition system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' The  first is to acquire the signal from the surrounding noisy environment using a sensor interface  3  Residual  limb  NFC/BLE  modules  Multimodal  battery-free  sensors  Prosthetic leg  Prosthetic  Portable  socket  NFC/BLE  electronicdevice  modulesFigure 4: Block diagram of the proposed smart sEMG recorder including sensors, AFE, and RF  integrated system [5]  circuit that’s designed in CMOS technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' The second involves reducing the form factor and  power consumption of the acquisition system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' The third is the signal conversion to the digital  world and the interface with the digital controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' At this point, the extracted EMG signal  is in a digital form and can be processed through FPGA or any other processor to control a  Pprosthetic limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  A typical block diagram of the proposed EMG acquisition system is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  The system consists of an EMG sensor, analog front end (AFE), and radio frequency (RF)  transmission unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' The AFE is typically composed of an analog amplification, filtration, analog  to digital converter (ADC), and controller to process the digital signal and send it to a prosthetic  limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' The acquisition system design can be integrated on a single chip, then the digital data is  fed to FPGA or a controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  In addition, because the integrated solution takes a considerable time during design, fabrica-  tion, and testing phases, a discrete solution in parallel with the integrated one can be used as a  proof of concept to validate the proposed methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  Figure 5 shows a detailed system block diagram of the proposed smart sEMG acquisition  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' An analog multiplexer is inserted to choose between different EMG electrodes in the  smart sEMG recorder shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' The design of each of the building blocks involves  4  AFE+RF  NSPU  FlexBandFigure 5: Detailed block diagram of the proposed smart sEMG acquisition system [5]  several design challenges requiring some research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' The following section includes a list of major  research directions that can be pursued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  4  Circuit Implementation  In the following subsections, the basic system building blocks are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' First, the EMG  sensor specifications are explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  Second, the low noise amplifier LNA design is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  Third, the filter design and bandwidth are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Fourth, the signal conversion from analog  to digital is presented through an ADC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Last, digital signal processing through FPGA is explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='1  Sensor Specifications  EMG sensor placement plays an important role in signal acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' According to its orientation  and position, the EMG signal strength varies significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' This effect is shown in Fig 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' As  seen, by placing the sensor in the middle of muscle fiber, the maximum signal strength can be  easily obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Otherwise, the signal degrades significantly when placing the sensor far away  from the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  EMG sensor can be represented in different forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' It can be in either needle that is inserted  into the muscle or surface electrode that picks the signal from the skin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' An example of surface  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Smart sEMG Recorder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='FPGA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='16-Channel Recorder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='GBDT based NSPU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Flexible  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='16x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='GBDT Core  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Feature  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Band  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='LNA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Extractors  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='00  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='MUX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
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+page_content='ADC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='16:1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='LNA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Pre-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Processing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Result  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Digital Logic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Data Ready| CLK_ADC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Generator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Loader  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='nRF52  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Ping Pong Buffer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Recorder  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='TF Card  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='BLE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='CIC Filter  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Interface  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Buffer A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Buffer B  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Interface i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='X  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=':=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='TF Card  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Mobile Devices  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Possible Applications  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Offline trainingFigure 6: Effect of EMG sensor position [7]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='EMG sensor specifications that have to be met through out the design are as follow shown in  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='2  Low Noise Amplifier Design  It’s the first and the major block in the EMG chain that comes after the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' The measurement  sensitivity and accuracy is determined in this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' This complicates the design and requires a  large amount of adaptability to accommodate the input signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' The previous stage, which is the  EMG sensor, adds large parasitic capacitance at the input of this stage, and thus reduces gain,  bandwidth, noise performance and the sensitivity of the amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  Sources of noise and interference like flicker noise, electrodes offset, and 60 Hz power line  noise can affect the whole acquisition procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' The bandwidth of the EMG signal is up to  6  Raw EMG output  Innervation Zone  Correct Placement  Midline of the muscle belly  between an innervation zone  and a myotendon junction  Midline Offset  Myotendon JunctionFigure 7: sensor specifications [8]  500 Hz with amplitude that ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='1 to 5 mV and the high-frequency noise can be easily  removed using a low pass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' However, low-frequency noise and DC offset fall within the  EMG bandwidth and hence require different rejection techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Chopping technique is one  of the best candidates to modulate the offset and flicker noise to a higher spectrum which in  turn enable the acquisition system to effectively suppress the interference from ambient and 1/f  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Different architectures with different requirements in terms of input signal levels, BW and  amplitudes are proposed in literature [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Figure 8 shows the block diagram of implemented  analog front-end for acquiring of EEG, ECG, and EMG signals [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' The shown diagram consists  of a chopper instrumentation amplifer in addition to capacitive coupling, filter stage to remove  the chopping spikes, a digitally controlled variable gain amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='3  Filter Design  A Gm-C filter cab be used in the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' A standard architecture is shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Offset  from the electrodes can be canceled using current-mode DAC [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Power Consumption of this  7  DataLITE Wireless EMG Amplifier  Wired EMG Amplifier  Product Ref  LE230FW  SX230FW  42 × 24 × 14 mm  38 x 20  Dimensions  Two 4 mm snap connectors on 100 mm wires  Two 4 mm snap connectors on 100 mm wires  Mass  17 g (excluding cable and plug)  8g (excluding cable and plug)  Bandwidth  10 - 250,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='470,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' 950,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' 5000Hz  20 - 460Hz  5Hz - 480Hz  Additional Bandwidths  N/A  5Hz - 1000Hz  Contact Diameter  Dependant on electrode size  Contact Center Spacing  Variable  Electrodes  Disposable  CMRR @ 60 Hz (dB)  > 96 dB (typically 110 dB)  Full Scale  +/- 6 mV Peak to Peak  +/- 3 mV Peak to Peak  Gain  +/- 60 microvolts to +/- 6 millivolts  Standard unit x1000 (100 als0 available)  Input Impedance  >100 Mohms  Accuracy  +/- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='0%  +/- 2% full scale  Noise  <5μv  Supply Voltage  N/A  +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='50 to +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='5 Vdc  Battery Life  Up to 8 hours  N/A  Battery Type  Rechargeable Li-lon Polymer  N/A  Wireless Transmission  Tolerant for 100 mS  N/A  Data Loss  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='25m cable  Range from Interface  Wireless range up to 30m  (custom lengths available on request)  Compatible Interfaces  DataLITE PIONEER, ADVANCE, EXPLORE  DataLOG, DataLINK, Amplifier or 3rd partyFigure 8: Architecture of the bio-potential readout front-end for the acquisition of EEG, ECG,  and EMG signals [9]  topology can also be reduced by low-voltage supply operation [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  Figure 9: Transistor level implementation of Gm-C filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' DDA: Differential Difference Amplifier  [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='4  ADC Design  Non-uniform sampling can minimize the power consumption of ADC while digitizing activity-  dependent biological signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' a continous-time (CT) charge-based ADC that ac-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='DC Level  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Select BW Select Gain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Programmable  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='1pF  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Gain Stage  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='BW Select  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Cext2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='OTA2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='■out  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Buffer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='vin+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='IA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='C12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='OTA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='vin-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='BW Select  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Cs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='C12= 20pF  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='AC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='21  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
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+page_content='C22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
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+page_content='C11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='IA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Filter  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='BIOPOTENTIAL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
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+page_content='IFBP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
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+page_content='DDA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='TL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='LT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Hquires samples when the input crosses a specific threshold is shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' The ADC  works by storing the analog equivalent of the last digitized input as a voltage across the across  the capacitor 𝐶𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Once the input signal crosses this voltage, a pulse with length 𝑇𝑃 is generated  to charge or discharge the capacitor 𝐶𝑏 by 𝑉𝐿𝑆𝐵 using one of the current sources connected to  the supply and ground [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Non-uniform sampling adapts to the instantaneous bandwidth of  the signal, consequently the dynamic power consumption scales with the activity of the input  signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' The FOM of the CT charge-based ADC can be improved by reducing the power supply  further [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  Figure 10: Top level architecture of Continous-time (CT) charge based ADC with non-uniform  sampling rate [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='5  FPGA Processing  Machine learning algorithms such as Support Vector Machine (SVM) have allowed for on-chip  feature extraction and classification of biomedical signals [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Machine learning can also be  deployed in the domain of prosthetic devices for precise control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  Figure 11 depicts the con-  troller of prosthetic device which can be implemented using Field Programmable Gate Array  (FPGA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Figure 12 depicts the experimental setup for analyzing the data from high density  EMG acquisition system using Xilinix Zedboard [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  9  Biasing  Pulse  +OPulseUp  Generator  Comparator  Vin O  UP  Vb  DOWN  OUT  +00UT<7:0>  Cb  RESET  ORESET  Pulse  +OPulseDown  Generator  Conf guration  Register  aLkT  OCLK  DO  SI  ISO  VFigure 11: Top level architecture of controller of prosthetic hand including feature extraction  and classification [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  Figure 12: Experimental Setup of EMG acquisition and processing using Xilinix ZedBoard [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='6  Energy Harvesting  The electrical power harvested from the environment (specially, thermal energy) can power  the ultra-low-power EMG Sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  We have previously developed energy harvesting systems  from various sources and high-efficiency DC-DC converters [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' For example, the system  architecture of power management IC for solar energy harvesting applications , designed by the  author, and chip micrograph are shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  10  class  192 HD EMG  decision  channels  Data  Feature  Classification  Acquisition  Extraction  Embedded Prosthesis Controller  prosthesis  movementI92 ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' HD EMG  ZedBoard for EMG  electrode array  signal processing  Michelangelo  HD EMG  PC for  hand prosthesis  DAQ  comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='Figure 13: System architecture of power management IC for solar energy harvesting applications,  designed by one of the team members, and chip micrograph [15]  5  Conclusion  This paper provided a survey about EMG acquisition systems for prostehtics and orthotic de-  vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  References  [1] Shirley Ryan AbilityLab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Facts about limb loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Available: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='sralab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  org/research/labs/bionic-medicine/news/facts-about-limb-loss  [2] UK  Ministry  of  Defence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  Uk  service  personnel  amputations:  fi-  nancial  year  2019/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  Available:  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='uk/  government/statistics/uk-service-personnel-amputations-financial-year-20192020/  afghanistan-and-iraq-amputation-statistics-1-april-2015-to-31-march-2020  [3] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Stansbury, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Holcomb,  “Amputations in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' military personnel in the current conflicts in Afghanistan and Iraq,”  Journal of Orthopaedic Trauma, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
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+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Available:  https://journals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='lww.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='com/00005131-200801000-00009  [4] iStock  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  Image  of  a  female  with  a  prosthetic  limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  Available:  https:  //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
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+page_content='com/  [5] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
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+page_content=' Lin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Yan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
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+page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
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+page_content=' Wang  11  3 mirr  VON  VLOAD  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='2mm  Startup  VcBUF  smtehe  Switch Matrix  Gapetsi  S2  Switch  DXADE  Current  VBA  VINDN市  Matrix  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  Startup  and  Drivers  Dnivers  Mp1  Configuration Block  Modell  MIPP  lectior  VIN  VLOAD  S  DA  Pulse Generation Block  Test Block  Φ1  Φ2  En  A  PTrig  Dvlee  VINDN  VBA  AE  Test Block  险电  Voltase:  Curont  Boost2et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=', “Design of a flexible wearable smart sEMG recorder integrated gradient boosting  decision tree based hand gesture recognition,” IEEE transactions on biomedical circuits  and systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
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+page_content=' 1563–1574, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content='  [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
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+page_content=' Han, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Xie, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
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+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Lee, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Avila, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Yohay, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Chen,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Liang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Patel, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Jung, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Kim, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Namkoong, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Kwon, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Guo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Ogle,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Grande, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Ryu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Madhvapathy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Liu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Park,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Caldwell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Banks, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
+page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfWfc6/content/2301.00163v1.pdf'}
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+Scalability of non-adiabatic effects in lithium-decorated graphene superconductor
+Dominik Szcz¸e´sniak
+Department of Theoretical Physics, Faculty of Science and Technology,
+Jan D�lugosz University in Cz¸estochowa, 13/15 Armii Krajowej Ave., 42200 Cz¸estochowa, Poland
+(Dated: February 1, 2023)
+The analysis is conducted to unveil how the non-adiabatic effects scale within the superconducting
+phase of lithium-decorated graphene (LiC6). Based on the Eliashberg formalism it is shown that
+the non-adiabatic effects notably reduce essential superconducting parameters in LiC6 and arise as
+a significant oppressor of the discussed phase. Moreover, nonadiabaticity is found to scale with
+the strength of superconductivity, proportionally to the phonon energy scale and inversely with
+respect to the electron-phonon coupling. These findings are partially in contrast to other theoretical
+studies and show that superconductivity in LiC6 is more peculiar than previously anticipated. In
+this context, the guidelines for enhancing superconducting phase in LiC6 and sibling materials are
+also proposed.
+I.
+INTRODUCTION
+In the literature, a number of scenarios exist on how
+to potentially induce conventional superconductivity in
+graphene, promising novel applications of this intrigu-
+ing carbon allotrope [1–12]. These strategies are mainly
+aimed at the structural modifications of graphene to
+enhance number of charge carriers at the Fermi level
+and alter intrinsic semimetallic character of this mate-
+rial. Among the resulting structures, lithium-decorated
+graphene (LiC6) is here of particular interest since it
+is suggested to host phonon-mediated superconducting
+state generated via process analogous to the intercalation
+of graphite [10]. This approach relies on the introduction
+of adatoms that break chiral symmetry of graphene and
+lifts Fermi level to the van Hove singularity [10, 13–15].
+However, superconductivity in LiC6 not only derives from
+the well-established method of synthesis but also appears
+to be actually feasible, as partially confirmed within the
+experiment [16].
+According to the above, LiC6 may be considered as a
+proving ground for the phonon-mediated superconduc-
+tivity at low-dimensions.
+Indeed, in recent years the
+superconducting phase in LiC6 received notable atten-
+tion in terms of its fundamental properties.
+The re-
+lated studies were devoted, but not limited to, the role of
+strong electron-phonon coupling [17], character of super-
+conducting gap [5], symmetry-breaking effects [13], ways
+of enhancing superconducting state [7], or the substrate
+ievmpact on superconductivity [9]. Beside listed direc-
+tions of research, LiC6 appears also to be a perfect exam-
+ple of superconducting material for studying the influence
+of non-adiabatic pairing [18]. This aspect is particularly
+intriguing since non-adiabatic effects tend to manifest
+themselves relatively rarely in conventional superconduc-
+tors. The reason for that relates to the non-comparable
+electronic and phononic energy scales in most supercon-
+ducting materials with the electron-phonon pairing mech-
+anism [19–21]. In other words, it can be often assumed
+that electrons follow adiabatically ionic oscillations and
+that the corresponding superconducting state can be de-
+scribed in a self-consistent manner [22, 23].
+However,
+this is not the case when superconducting materials ex-
+hibit shallow conduction band such as the fullerenes [24],
+fullerides [23], bismuthates [25], transition-metal-oxides
+[26] or the discussed LiC6 [18] (see [27] for the review of
+nonadiabatic superconductors).
+So far, the characteristic energy scales are one of
+the few signatures of non-adiabatic superconductivity in
+LiC6. Other than that recent theoretical studies show
+non-adiabatic effects to notably reduce magnitude of de-
+pairing interaction in the discussed material, in compar-
+ison to the adiabatic regime [18]. These findings are also
+supplemented by the considerations suggesting that non-
+adiabaticity contributes to the electron-phonon coupling
+(λ) and modulates the transition temperature (TC) in
+doped graphene [22] or two-dimensional superconductors
+in general [28]. Still, little is known about the scalability
+of non-adiabatic effects in LiC6. In the first approxima-
+tion, it can be only qualitatively argued that their impact
+changes according to the Migdal’s ratio (known also as
+the expansion ratio) given by m = λωD/EF , where EF
+is the Fermi energy and ωD denotes Debye’s frequency
+[19–21]. In fact, although m-ratio can provide some in-
+formation on the scalability problem it is mostly used to
+determine whether or not given material can be described
+within the Migdal’s theorem [29] i.e. within adiabatic or
+non-adiabatic regime. Hence, the measurable and direct
+role of the above parameters and their variations in shap-
+ing non-adiabatic superconductivity in LiC6 is somewhat
+hindered. In details, it is unknown how changes in the m-
+ratio components modify experimentally observable ther-
+modynamic properties such as the superconducting gap
+or the transition temperature. This is to say, what trends
+in thermodynamics can be expected due to the strength
+of the non-adiabatic effects. As a result, the relevancy
+of the energy scales and the electron-phonon coupling
+in the non-adiabatic limit is also not well-estimated yet.
+Therefore, addressing these aspects would be of great im-
+portance to the better understanding of superconductiv-
+ity in LiC6 and potentially other sibling low-dimensional
+materials. Moreover, it should also help in assessing im-
+pact of the external factors that can be applied to mod-
+ify the aforementioned properties and ultimately enhance
+arXiv:2301.13697v1  [cond-mat.supr-con]  31 Jan 2023
+
+2
+the superconducting state even further.
+To provide deeper insight into the scalability of non-
+adiabatic effects in LiC6, the present study analyzes be-
+havior of the discussed material in the adiabatic and
+non-adiabatic limit when the expansion ratio parame-
+ters vary. This is done within the Eliashberg formalism
+that generalizes conventional Bardeen-Cooper-Schrieffer
+(BCS) theory of superconductivity [30, 31] by incorporat-
+ing the strong-coupling, retardation and non-adiabatic
+effects [19–21, 32, 33]. As a result it allows to consider
+both regimes of interest and relate predictions on the piv-
+otal thermodynamics to the potential factors responsible
+for the m-ratio variations. Here the latter is modeled in
+reference to [7], by recalling the fact that the deforma-
+tion potential is able to simultaneously influence energy
+scales and the electron-phonon coupling in a given su-
+perconducting material. This effect is captured via the
+percentage change in graphene lattice constant given as
+δ = |a − a0| /a0 × 100%, where a0(a) is the unmodified
+(modified) lattice constant value. In what follows, sev-
+eral levels of δ are considered allowing for tracing changes
+in the m-ratio components on the same footing and in
+the direct relation to the experimentally observable case
+(δ = 0%). Based on that it is possible to unveiled how the
+non-adiabatic effects scale in LiC6 and what can be done
+to eliminate their potentially negative consequences.
+II.
+METODOLOGY
+The theoretical formalism of choice is provided here
+by following the study of Freericks et al. [34], where con-
+venient form of the generalized Eliashberg equations for
+considering the non-adiabatic superconductivity is pre-
+sented. This theoretical approach is based on the per-
+turbative theory introduced originally by Pietronero et
+al. in [19–21], which incorporates non-adiabatic effects
+via vertex corrections to the electron-phonon interac-
+tion. However, the theoretical scenario given in [34] in-
+cludes specific computational techniques for better ac-
+curacy and efficiency, such as the perturbative theory
+on the imaginary-axis and the high-frequency resum-
+mation schemes.
+In this manner, the resulting equa-
+tions provide compromise between predictive capabili-
+ties and computational requirements. Note that such ap-
+proach was already proved successful in describing non-
+adiabatic superconductivity not only in LiC6 but also
+other phonon-mediated superconductors such as lead [34]
+or bismuthates [35].
+In respect to the above, inital approximations are as-
+sumed in accordance to [34] and the character of the
+superconducting phase in LiC6.
+In particular, (i) the
+direct dependence on momentum is neglected for the
+electron-phonon matrix elements, in correspondence to
+the isotropic nature of superconducting gap in LiC6,
+(ii) the depairing correlations are modeled only by
+the first-order Coulomb pseudopotential terms, due to
+the fact that higher-order contributions are negligibly
+small for phonon-mediated superconductors, (iii) sim-
+ilarly only the lowest-order vertex corrections to the
+electron-phonon interaction are considered to describe
+the non-adiabatic effects, since the Fermi liquid picture
+in LiC6 appears to be conserved. As a results, it is possi-
+ble to derive self-consistent Eliashberg equations beyond
+Midal’s theorem within perturbation scheme. In details,
+their form on the imaginary axis for the order parameter
+function (φn = φ (iωn)) and the wave function renormal-
+ization factor (Zn = Z (iωn)) is following:
+φn = πkBT
+M
+�
+m=−M
+Kn,m − µ⋆
+m
+�
+ω2mZ2m + φ2m
+φm − Vφ,
+(1)
+Zn = 1 + πkBT
+ωn
+M
+�
+m=−M
+Kn,m
+�
+ω2mZ2m + φ2m
+ωmZm − VZ, (2)
+where, kBT is the inverse temperature, with kB denot-
+ing the Boltzmann constant.
+In what follows, ωn =
+πkBT (2n + 1) is the n-th Matsubara frequency with the
+cutoff M = 1100 for numerical stability above T = 2 K.
+Moreover, Kn,m stands for the electron-phonon pairing
+kernel given as:
+Kn,m = 2
+� ωD
+0
+dω
+ω
+ω2 + 4π2 (kBT)2 (n − m)2 α2F (ω) ,
+(3)
+with ω being the phonon frequency, α describing the av-
+erage electron-phonon coupling and F (ω) denoting the
+phonon density of states. Note that the product of the
+two latter is known as the electron-phonon spectral func-
+tion which provides most important information about
+a physical system within the Eliashberg formalism [33].
+Here, several α2F (ω) functions are considered, each of
+them corresponding to the different δ-value in order to
+analyze behavior of LiC6 when the Migdal’s parameter
+and its components very.
+For this purpose, the exact
+forms of the α2F (ω) functions are assumed after [7] for
+δ ∈ ⟨0, 3, 5, 7, 10⟩ %, where the first case corresponds to
+the experimentally observed superconducting phase of
+LiC6 whereas the remaining functions describe poten-
+tial variations from its pristine form.
+The remaining
+information is given via the Coulomb pseudopotential
+µ⋆
+n = µ⋆θ (ωc − |ωn|), with θ standing for the the Heavi-
+side function and ωc for the cut-off frequency. To consider
+all the δ cases on equal footing the conventional value of
+µ⋆ = 0.1 [36] which is close to the magnitude of Coulomb
+depairing interaction predicted for LiC6 at δ = 0% [18].
+Finally, Vφ and VZ are the lowest-order vertex correc-
+
+3
+tion terms of the following form:
+Vφ = π3 (kBT)2
+4EF
+M
+�
+m=−M
+M
+�
+m′=−M
+Kn,mKn,m′
+×
+1
+�
+(ω2mZ2m + φ2m) (ω2
+m′Z2
+m′ + φ2
+m′)
+×
+1
+�
+(ω2
+m′′Z2
+m′′ + φ2
+m′′)
+× (φmφm′φm′′ + 2φmωm′Zm′ωm′′Zm′′
+− ωmZmωm′Zm′φm′′) ,
+(4)
+and
+VZ = π3 (kBT)2
+4EF ωn
+M
+�
+m=−M
+M
+�
+m′=−M
+Kn,mKn,m′
+×
+1
+�
+(ω2mZ2m + φ2m) (ω2
+m′Z2
+m′ + φ2
+m′)
+×
+1
+�
+(ω2
+m′′Z2
+m′′ + φ2
+m′′)
+× (ωmZmωm′Zm′ωm′′Zm′′ + 2ωmZmφm′φm′′
+− φmφm′ωm′′Zm′′) .
+(5)
+Based on the above, when vertex corrections are consid-
+ered within the Eqs. (1) and (2) they are refereed here to
+as the non-adiabatic Eliashberg equations (N-E), other-
+wise, when the corrections are neglected, the formalism
+is reduced to the adiabatic Eliashberg equations (A-E).
+In what follows, by solving Eqs. (1) and (2) it is pos-
+sible to obtain estimates on the most important ther-
+modynamic properties of superconducting state in LiC6.
+Specifically, the central role in such analysis is played
+by the order parameter function that is obtained from
+Eqs. (1) and (2) as: ∆n(T) = φn/Zn. Here of special
+interest is the maximum value (m = 1) of ∆n(T) which
+contains information on the transition temperature and
+the superconducting gap half-width. The former is de-
+termined based on the relation ∆m=1(TC) = 0, whereas
+the latter is given by ∆m=1(T0), with T0 = 2 K being
+the lowest temperature assumed for calculations. Since
+the aforementioned α2F (ω) function is dependent on the
+characteristic energy scales and the electron-phonon cou-
+pling constant, the described solutions of the Eliashberg
+equations also inherit such dependence, allowing for the
+analysis of interest.
+III.
+THE RESULTS AND DISCUSSION
+In Fig.
+1, the main numerical results are presented
+as obtained by solving Eqs. (1) and (2) iteratively with
+respect to the temperature (see [25] and [34] for more
+details on the computational methods used here). In de-
+tails, Fig. 1 depicts the behavior of ∆m=1(T) function
+for T ∈ ⟨T0, TC⟩ at the assumed levels of lattice constant
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+0
+1
+2
+3
+4
+5
+6
+ 
+ 
+∆m=1 (mev)
+T (K)
+A-E:
+ δ=0%
+ δ=3%
+ δ=5%
+ δ=7%
+ δ=10%
+N-E:
+ δ=0%
+ δ=3%
+ δ=5%
+ δ=7%
+ δ=10%
+FIG. 1: The maximum value of order parameter (∆m=1(T))
+as a function of temperature in LiC6.
+The results are de-
+picted for the selected values of lattice constant deviation in
+graphene (δ) as obtained within the adiabatic (closed sym-
+bols) and non-adiabatic (open symbols) regime of the Eliash-
+berg equations. Solid lines constitute the guides for an eye.
+deviation denoted by δ. Note that the 0% case corre-
+sponds to the unaltered LiC6 material, hosting the ex-
+perimentally observable superconducting phase. On the
+other hand, the remaining cases describe situation when
+crystal lattice of graphene changes according to the al-
+ready mentioned expression: δ = |a − a0| /a0 × 100%,
+with a0(a) standing for the unmodified (modified) lattice
+constant. Moreover, as allowed by the employed formal-
+ism, the discussed thermal behavior of ∆m=1(T) func-
+tion is plotted for the adiabatic (open symbols) and non-
+adiabatic (closed symbols) regime. In all figures, symbols
+relate to the exact numerical results of the Eliashberg
+equations and solid lines constitute guides for an eye.
+The result depicted in Fig. 1 reveal several general as-
+pects of the superconducting state in LiC6. In particular,
+it can be observed that for δ > 3% the increase of the
+δ value causes notable increase of the ∆m=1(T) in the
+entire temperature range for both considered regimes.
+This trend is not conserved only when comparing re-
+sult obtained at the two lowest levels of δ, as caused
+by the δ-driven increase of charge transfer that emp-
+ties interlayer states and notably reduces the electron-
+phonon coupling constant at δ = 3% with respect to
+the 0% case [7]. Nonetheless, the observed effect means
+that above some level of δ the superconducting state
+in LiC6 is clearly enhanced.
+This observation can be
+quantified by deducing the transition temperature values
+from the obtained results. In particular, TC = 8.78 K
+at δ = 0% and TC ∈ ⟨8.48, 34.61⟩ K for δ ∈ ⟨3, 10⟩ %
+within the A-E limit, whereas TC = 7.29 K at δ = 0%
+and TC ∈ ⟨6.59, 28.46⟩ K for δ ∈ ⟨3, 10⟩ % when consid-
+ering the N-E equations. Note that these observations
+are in qualitative agreement with the previous studies
+conducted within the adiabatic limit by using the Allen-
+
+4
+Dynes formula in [7]. The difference between these data
+sets and results given in [7] is due to the fact that the as-
+sumed Eliashberg equations incorporate strong-coupling,
+retardation, and non-adiabatic effects which are missing
+in the Allen-Dynes formula. At this point, it is also in-
+structive to note that the TC value estimated at δ = 0%
+is slightly higher in comparison to the predictions made
+within the Eliashberg formalism for the experimentally
+derived electron-phonon spectral function, as presented
+in [18]. This discrepancy is obviously caused by the as-
+sumed value of µ⋆, smaller than the one suggested in
+[18]. The reason to make such assumption is to allow
+for better comparison not only with the BCS-derived re-
+sults given in [7] but also other two-dimensional super-
+conductors, which are often still hypothetical structures
+and their superconducting state is described by µ⋆ ∼ 0.1
+(see e.g. [37, 38]). Note that even if µ⋆ would be as-
+sumed here after [18], the main outcomes and findings of
+the present analysis would not change. This includes es-
+timates on the superconducting gap half-width that can
+be made based on the results plotted in Fig. 1. This is to
+say, the general behavior of ∆m=1(T0) parameter is the
+same as in the case of TC and it will not change qualita-
+tively when assuming other µ⋆ value. Specifically, in the
+present study ∆m=1(T0) = 1.39 meV at δ = 0% whereas
+∆m=1(T0) ∈ ⟨1.30, 5.55⟩ meV for δ ∈ ⟨3, 10⟩ % in the A-E
+regime, while the N-E equations yield ∆m=1(T0) = 1.22
+meV at δ = 0% and ∆m=1(T0) ∈ ⟨1.11, 4.84⟩ meV for
+δ ∈ ⟨3, 10⟩ %. Note that results on TC and ∆m=1(T0)
+can be supplemented by introducing their characteristic
+ratio, familiar in the BCS theory and given by [30, 31, 33]:
+R = 2∆m=1(T0)
+kBTC
+.
+(6)
+The Eq. (6) not only allows for additional insight into
+the considered problem but also provides yet another ob-
+servable for future comparisons with the experiment. As
+it can be expected, the obtained values of R follow the
+same trends like the TC and ∆m=1(T0) parameters. The
+values of R base on the A-E equations are R = 3.67 at
+δ = 0% and R ∈ ⟨3.55, 3.72⟩ for δ ∈ ⟨3, 10⟩ %. On the
+other hand, the N-E equations give R = 3.89 at δ = 0%
+and R ∈ ⟨3.92, 3.98⟩ for δ ∈ ⟨3, 10⟩ %. Still, both sets
+present values higher than the level suggested within the
+BCS theory and equal to 3.53. It means that the strong-
+coupling and retardation effects play relatively impor-
+tant role in shaping the superconducting state in LiC6.
+This observation is in agreement with the strength of
+the electron-phonon coupling reported in [7] and previ-
+ous findings given in [18].
+Beside the above observations, it is also crucial to note
+that the reported results suggest simultaneous changes
+in the energy scales and the electron-phonon coupling
+constant due to the variations of δ parameter. Indeed,
+all of these characteristic parameters exhibit increasing
+or decreasing trends along with the growing δ value (see
+Fig. 2 (A)). For convenience, their cumulative behavior
+is depicted in Fig. 2 (B) in terms of already introduced
+120
+140
+160
+180
+1000
+1200
+1400
+1600
+1800
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+0.04
+0.06
+0.08
+0.10
+0.12
+0.14
+0
+2
+4
+6
+8
+10
+4
+8
+12
+16
+20
+24
+28
+ 
+ωD (meV)                     EF (meV)
+(A)
+λ
+ 
+m
+ m=ωD/EF
+ m=λωD/EF
+(B)
+Dx (%)
+ x=1
+ x=2
+ x=3
+δ (%)
+(C)
+FIG. 2: The behavior of (A) the Fermi energy (EF ), Debye’s
+frequency (ωD) and electron-phonon coupling constant (λ),
+(B) the dressed (m = λωD/EF ) and bare (m = ωD/EF )
+Migdal’s ratio as well as (C) the percentage differences be-
+tween adiabatic and non-adiabatic estimates for the critical
+temperature (D1), superconducting gap half-width (D2) and
+their cumulative ratio (D3) at the selected values of the lat-
+tice deviation in graphene (δ) in the LiC6 superconductor.
+The closed symbols depicts exact results, the solid lines are
+the guides for an eye and the color arrows points to the cor-
+responding axes.
+dressed Migdal’s ratio (m = λωD/EF ) but also its bare
+
+5
+TABLE I: The parameters of superconducting state in LiC6 for the selected values of the lattice deviation in graphene (δ).
+In a consecutive order, the parameters are: the electron-phonon coupling constant (λ), the Debye’s frequency (ωd), the Fermi
+energy (EF ), the bare (ωd/EF ) and dressed (λωd/EF ) Migdal’s ratio, the transition temperature (TC), the superconducting
+gap half-width (∆m=1(T0)) as well as the thermodynamic ratio for the two last ones (R). Note that, where necessary, the
+results are presented for the in terms of the adiabatic (A − E) and non-adiabatic (N − E) regime. Moreover, the percentage
+differences between estimates in these two limits for TC (D1), ∆m=1(T0) (D2) and R (D3) are also given.
+A-E
+N-E
+δ
+λ
+ωd
+EF
+ωd/EF
+λωd/EF
+TC
+∆m=1(T0)
+R
+TC
+∆m=1(T0)
+R
+D1
+D2
+D3
+(%)
+(meV)
+(meV)
+(K)
+(meV)
+(K)
+(meV)
+(%)
+(%)
+(%)
+0
+0.61
+141.21
+1100
+0.128
+0.078
+8.78
+1.39
+3.67
+7.29
+1.22
+3.89
+18.54
+13.03
+6.37
+3
+0.47
+171.34
+1300
+0.132
+0.062
+8.48
+1.30
+3.55
+6.59
+1.11
+3.92
+25.08
+15.77
+9.91
+5
+0.49
+158.82
+1624
+0.098
+0.048
+12.61
+1.94
+3.58
+9.90
+1.68
+3.93
+24.08
+14.36
+9.32
+7
+0.55
+148.16
+1726
+0.086
+0.047
+18.86
+2.95
+3.63
+14.98
+2.56
+3.97
+22.93
+14.16
+8.95
+10
+0.73
+132.79
+1800
+0.074
+0.054
+34.61
+5.55
+3.72
+28.26
+4.84
+3.98
+20.20
+13.67
+6.75
+counterpart (m = ωD/EF ). In what follows, it is argued
+here that the scalability of non-adiabatic effects in LiC6
+can be traced with respect to the pivotal parameters en-
+tering Migdal’s ratio. In this context, first it should be
+noted that for each considered δ value the non-adiabatic
+equations yield lower ∆m=1(T) values than their adia-
+batic counterparts (see Fig. 1). This directly relates to
+the fact that the transition temperature values as well as
+the estimates of the superconducting gap are lower in the
+non-adiabatic regime when comparing to the adiabatic
+one. As a results, the percentage difference between esti-
+mates made in two considered regimes can be introduced
+as a measure of non-adiabatic effects impact on super-
+conducting phase in LiC6. This new measure is depicted
+for all considered thermodynamic parameters in Fig. 2
+(C). In details, the percentage difference between TC val-
+ues determined in the adiabatic and non-adiabatic limits
+is D1 = 18.54% at δ = 0% and D1 = ⟨25.08, 20.20⟩%
+for δ ∈ ⟨3, 10⟩ %. Similarly, the same percentage mea-
+sure but for the ∆m=1(T0) parameter is D2 = 13.03%
+for δ = 0% and D2 = ⟨15.77, 13.67⟩% when δ ∈ ⟨3, 10⟩ %.
+Finally, the cumulative ratio gives the corresponding per-
+centage D3 = 6.37% for δ = 0% and D3 = ⟨9.91, 6.75⟩%
+when again δ ∈ ⟨3, 10⟩ %. Based on these results, it is
+clear that the critical temperature is the most influenced
+by the non-adiabatic effects from all three considered pa-
+rameters. It also presents the most visible signature of
+the charge transfer from interlayer states at δ ∈ 3%. On
+the contrary, the cumulative ratio shows the smallest dis-
+crepancies. Still all three parameters exhibit the same
+qualitative behavior i.e.
+the percentage difference be-
+tween adiabatic and non-adiabatic results increases up
+to δ ∈ 3% and then starts to almost linearly decrease as
+the δ takes higher values.
+The final observations can be made when comparing
+results presented in Fig. 2 (C) with the estimates de-
+picted in Figs. 2 (A) and (B). In particular, it can quali-
+tatively argued that the behavior of percentage measures
+does not fully resemble the Migdal’s ratio dependence on
+the δ value, although the latter is considered to be the
+first approximation approach to provide information on
+the scalability of non-adiabatic effects in a superconduc-
+tors. Precisely speaking, only the bare ratio can be con-
+sidered somewhat similar in behavior to the percentage
+measures, whereas its dressed value presents almost in-
+verse character with respect to the parameters given in
+Fig. 2 (C). To inspect these discrepancies even further it
+is instructive to compare the percentage difference mea-
+sures with the characteristic component parameters of
+the Migdal’s ratio, as plotted in Figs. 2 (A). The out-
+come is that only the Debye’s energy scales qualitatively
+the same as the percentage difference measures, while the
+electron-phonon coupling gives inverse characteristic and
+the electronic energy scale is practically nowhere similar
+to the results given in Figs. 2 (C).
+IV.
+SUMMARY AND CONCLUSIONS
+In summary, the presented analysis provides new in-
+sight into the superconducting properties of LiC6 in
+terms of its non-adiabatic characteristic. It shows how
+the non-adiabatic effects scale in LiC6 with respect to the
+deviation of lattice constant in graphene, which can be
+considered as an exemplary factor that modifies strength
+of the superconducting state. In details, the discussed
+scalability is expressed here in terms of the percentage
+difference between estimates of pivotal thermodynamic
+parameters (the transition temperature, superconduct-
+ing gap and their ratio) obtained within the adiabatic
+and non-adiabatic regime, allowing for further compari-
+son with the Migdal’s expansion ratio that characterizes
+nonadiabaticity in the first approximation. For conve-
+nience, all the obtained numerical results are summarized
+in Tab. I.
+Based on the above findings it is possible to draw sev-
+eral conclusions related, but no limited to, the scalability
+of non-adiabatic effects in LiC6. In details:
+(i) The introduction of vertex corrections to the
+electron-phonon interaction causes notable changes
+in the pivotal thermodynamic parameters of the su-
+perconducting state in LiC6, in particular their de-
+crease in comparison to the adiabatic limit (see Fig.
+
+6
+2 (C)). This is to say, the superconducting state
+appears to have strongly non-adiabatic character,
+where non-adiabatic effects act as an important op-
+pressor of superconductivity in LiC6. Notably the
+superconducting state sustains its non-adiabatic
+character even when superconductivity in LiC6 is
+strongly enhanced. Still the non-adiabatic effects
+are observed to visibly vary with the strength of
+superconducting phase. Moreover, it is found that
+nonadiabaticity is supplemented by the strong-
+coupling and retardation effects, meaning that LiC6
+is a somewhat unorthodox phonon-mediated super-
+conductor.
+(ii) The considered thermodynamic parameters present
+the same qualitative behavior under the influence
+of non-adiabatic effects when the strength of super-
+conductivity is varied (see Fig. 2 (C)). Nonetheless,
+the critical temperature is suggested to be partic-
+ularly sensitive to nonadiabaticity, while supercon-
+ducting gap is showing much smaller dependence
+on the variation of the discussed effects. As a re-
+sult, this opens new prospect for increasing tran-
+sition temperature value in LiC6 according to the
+presented here findings, saying that smaller mag-
+nitude of non-adiabatic effects leads to the higher
+transition temperature (see Tab. I). Note that this
+trend is not conserved when including results for
+the unaltered LiC6, due to the decreased charge
+transfer from the interlayer states in comparison
+to other considered cases. It can be additionally
+argued that such trend may be considered general
+for other graphene-based superconductor which ex-
+hibit similar dependence on nonadiabaticity (see
+e.g. recent study on the electron-doped graphene
+[39]).
+(iii) The deeper inspection of the obtained results
+shows that the non-adiabatic effects scale with
+the strength of superconductivity, proportionally
+to the phonon energy scale and inversely to the
+electron-phonon coupling magnitude (see Figs. 2
+(A) and (C)). Note that, while the former observa-
+tion agrees with the predictions of both considered
+forms of the Migdal’s ratio, the latter is in contrast
+to what can be expected based on the dressed pa-
+rameter and previous theoretical studies consider-
+ing superconductivity in two-dimensional systems
+[22, 28]. However, the mentioned trends does not
+take into account existing interplay between all
+components of the Migdal’s ratio and the fact that
+the electron-phonon coupling constant increases al-
+most linearly with the electronic energy scale (see
+Tab. I). As a results, strong electron-phonon cor-
+relations correspond to the relatively wide conduc-
+tion band that causes suppression of nonadiabatic-
+ity. This argument is confirmed qualitatively by the
+observed here cumulative behavior of the Migdal’s
+ratio (see Figs. 2 (B)). This is to say the electron-
+phonon coupling cannot be always considered to
+be improved in graphene-based superconductors by
+the non-adiabatic effects as previously suggested in
+[22, 28].
+To sum up, the perspectives for future research can be
+given. In details, to provide better understating of the su-
+perconducting state in LiC6 the discussed non-adiabatic
+effects should be considered beyond the isotropic ap-
+proximation.
+Note that such preliminary analysis can
+be already found in [28], while the adiabatic anisotropic
+investigations are available in [5].
+The present study
+can be also extended further toward other experimen-
+tally observable thermodynamic parameters such as the
+free energy or the critical thermodynamic field, accord-
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+value of the Migdal’s ratio. In other words, the super-
+conducting state in LiC6 may appear to be more peculiar
+than previously anticipated and additional investigations
+in this directions should be of great interest.
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+The stochastic digital human is now enrolling for in
+silico imaging trials – Methods and tools for
+generating digital cohorts
+A Badano1, M Lago1, E Sizikova1, JG Delfino1, S Guan1
+and MA Anastasio2 and B Sahiner1
+1Division of Imaging, Diagnostics, and Software Reliability, Office of Science and
+Engineering Laboratories, Center for Devices and Radiological Health,
+U. S. Food and Drug Administration, Silver Spring, MD 20993
+2Department of Bioengineering, The Grainger College of Engineering, University
+of Illinois, Urbana, IL 61801
+E-mail: [email protected]
+23 January 2023
+Abstract.
+Randomized clinical trials,
+while often viewed as the highest
+evidentiary bar by which to judge the quality of a medical intervention, are
+far from perfect.
+In silico imaging trials are computational studies that seek
+to ascertain the performance of a medical device by collecting this information
+entirely via computer simulations. The benefits of in silico trials for evaluating new
+technology include significant resource and time savings, minimization of subject
+risk, the ability to study devices that are not achievable in the physical world,
+allow for the rapid and effective investigation of new technologies and ensure
+representation from all relevant subgroups.
+To conduct in silico trials, digital
+representations of humans are needed.
+We review the latest developments in
+methods and tools for obtaining digital humans for in silico imaging studies. First,
+we introduce terminology and a classification of digital human models. Second,
+we survey available methodologies for generating digital humans with healthy and
+diseased status, and examine briefly the role of augmentation methods. Finally,
+we discuss the trade-offs of four approaches for sampling digital cohorts and the
+associated potential for study bias with selecting specific patient distributions.
+Social media blur (100-w): From digital twins to other digital humans for
+in silico trials: we review methods and tools for obtaining stochastic humans for
+digital cohorts [LINK]
+Submitted to: PRGB
+arXiv:2301.08719v1  [cs.AI]  20 Jan 2023
+
+CONTENTS
+2
+Contents
+1
+Introduction
+3
+2
+Terminology
+4
+3
+Representations
+4
+4
+Individual models
+5
+4.1
+Personalized models
+. . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+4.2
+Family models
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+5
+Population models
+6
+5.1
+Image-based models
+. . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+5.1.1
+Image-based parametric models . . . . . . . . . . . . . . . . . .
+6
+5.1.2
+Image-based generative models . . . . . . . . . . . . . . . . . .
+7
+5.2
+Knowledge-based models . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+6
+Modeling disease
+9
+6.1
+Image-based models of disease . . . . . . . . . . . . . . . . . . . . . . .
+9
+6.2
+Knowledge-based models of disease . . . . . . . . . . . . . . . . . . . .
+10
+7
+Role of augmentation methods
+10
+8
+Considerations for sampling digital cohorts
+11
+9
+Summary and conclusions
+12
+
+CONTENTS
+3
+1. Introduction
+Two decades ago, in the epilogue of their seminal
+textbook on image science [1], Barrett and Myers
+pointed out that in the future, sport games might be
+played with simulated athletes. The advancement of
+computer graphics and simulation technologies sparked
+the notion that perhaps the excitement of a real-life
+sports event could be conducted in the simulation
+space with digital models of athletes.
+Since then,
+continuous advances in computer processing power and
+modeling techniques have taken place, driven primarily
+by entertainment applications [2] and quickly becoming
+a significant component of research and development
+(R&D) efforts in a variety of industries‡.
+Industries
+that have widely adopted computational modeling and
+in silico methods throughout the product life-cycle
+include automotive [3] and manufacturing [4] among
+others [5]. Medicine lags considerably behind [6] due,
+in part, to model complexity, challenging validation,
+associated potential risks for new devices and drugs,
+and lack of consensus and regulatory standards.
+Randomized clinical trials, while often viewed
+as the highest evidentiary bar by which to judge
+the quality of a medical intervention, are far from
+perfect. Common causes of failure include safety issues,
+difficulties with patient recruitment, enrollment, and
+retention [7]. In addition, clinical trials can suffer from
+under-representation of rare subpopulations [8]. These
+limitations represent a unique opportunity to develop
+in silico trials that are completed as planned, safely,
+and that include digital cohorts with a representative
+distribution of subject characteristics and numbers
+large enough for appropriate statistical power.
+As
+pointed out in [9], in silico data has the potential to
+address lack of data availability, sharing mechanisms
+and privacy challenges associated with the use of
+medical information.
+In silico imaging trials are computational studies
+that seek to ascertain the performance of a medical
+device for the intended population, collecting this
+information entirely in the digital world via computer
+simulations. The benefits of in silico imaging trials for
+evaluating new technology include significant resource
+and time savings, minimization of subject risk, and
+ethical considerations [10, 11].
+Moreover, in silico
+trials can be used to study devices that do not
+yet exist or are not practically attainable in the
+(limited) physical world, allow for the rapid and
+effective investigation of new technologies [11, 12,
+13], and facilitate representation from all relevant
+subpopulations.
+Each one of these benefits is an
+‡ To date, Superbowl games are played with physical-world
+athletes, in part due to the difficulty of conveying real-life
+personal struggle, an essential component of the entertainment
+context for sport players and teams (see, for instance, here).
+essential consideration within the context of the
+regulatory evaluation of medical technology [11].
+The realization that computational models of
+humans would take center stage in medical imaging
+system assessment is not new.
+Full optimization of
+imaging systems for specific medical tasks requires
+objects
+(physical
+or
+digital)
+that
+represent
+the
+variability seen in patients.
+For many decades,
+scientists have relied on practical and simpler versions
+of patients [14]. However, recent advances in computer
+processing power and simulation methods are now
+facilitating the development of more detailed and
+realistic patient models that are based on digital
+stochastic descriptions of the model components. For
+instance, a recent report demonstrated the feasibility of
+an in silico trial, the Virtual Imaging Clinical Trial for
+Regulatory Evaluation (VICTRE), as an alternative
+approach to establish regulatory evidence in support
+of medical imaging products [15].
+There are numerous parallels between digital-
+and physical-world trials.
+Fundamentally, in silico
+trials must include the same essential elements of
+well-designed physical-world clinical trials.
+Firstly,
+the population of subjects for whom the new device
+or technology is intended must be defined.
+The
+study design must contain clear rules for selection and
+rejection of subjects from a distribution of healthy and
+diseased subjects.
+However, in silico trials are not
+subject to effects from covariates in patient selection.
+For
+instance,
+a
+common
+problem
+in
+evaluating
+screening tests meant for asymptomatic subjects is
+that a portion of the enrolled population might be
+symptomatic [16] with the potential for verification
+bias [17]. Secondly, when there are two technologies
+that are being compared, i.e., a new, yet unproven
+technology and a comparator technology currently in
+clinical use, both must be unambiguously defined.
+A good choice for comparator technology should be
+associated with accurate representations of the device
+characteristics as supported by validation studies [18].
+Thirdly, the study requires a definition of the users
+of the device’s outcome (i.e., images in the case of an
+imaging device trial).
+These first three components
+reflect the physical intended use of the device under
+investigation, i.e., the intended populations of subjects,
+the intended device comparison, and the intended
+image interpreters that will be using the device in the
+physical world. Finally, whether physical or digital, the
+trial design must provide a definition of the primary
+outcome to be evaluated, a protocol and statistical
+analysis associated with the trial, and an analysis of
+the risk and benefits introduced by the device under
+investigation.
+Both
+physical
+and
+in
+silico
+studies
+require
+enrollment
+of
+representative
+subjects.
+In
+this
+
+CONTENTS
+4
+review, we survey the latest developments in methods
+and
+tools
+for
+generating
+the
+cohorts
+of
+digital
+humans
+for
+imaging
+studies
+that
+represent
+the
+variability of physical-world subject populations. We
+refer to the digital cohorts consisting of digital
+humans (realizations of the digital human models) as
+“stochastic humans”. Assessment of new technology
+and the regulatory evaluation of that technology
+requires establishing performance levels for intended
+populations and, therefore, necessitates computational
+models that allow sampling of the parameter space
+defining the subject population in the physical world.
+We propose to name these models digital humans as
+opposed to digital replicas or twins to avoid confusion.
+The review is organized as follows.
+First,
+we introduce terminology and representation models
+regarding
+the
+different
+types
+of
+digital
+humans
+described throughout the article. Second, we survey
+available methodologies for generating digital humans
+with healthy status and for generating diseased cases.
+Then, we briefly discuss the role of augmentation
+methods and conclude with an analysis of sampling
+techniques that may be used to generate the digital
+cohorts for evaluating the performance of imaging
+devices.
+2. Terminology
+A variety of terminologies are being used or proposed
+for describing digital representations of humans in
+medicine and other fields.
+In the literature, some
+of these are often used without the benefit of a
+clear definition and,
+in some instances,
+wrongly
+interchangeably.
+We propose to use the term stochastic digital
+human to denote digital representations of humans (or
+human body parts) generated from multiple random
+outputs by sampling known distributions for the
+model characteristics matching the variability observed
+in human populations.
+In contrast, non-stochastic
+representations are deterministic digital versions of a
+single physical exemplar (e.g., a model of a human
+body at a given time) or a group (or family) of
+physical exemplars which are differentiated by varying
+physical parameters.
+Contrary to other terms and
+concepts currently being discussed including digital
+families, avatars, chimeras, and digital twins, the
+concept of a stochastic digital human represents an
+approach for in silico trials and regulatory evaluation
+that estimates the performance of an imaging device for
+a population of subjects rather than for an individual
+patient, thus incorporating the variability observed in
+the population.
+We propose to classify all digital humans as
+either individual or population models (see Figure 1).
+Individual models are necessarily image-based while
+population models can be derived either from images
+or from knowledge of the fundamental characteristics
+that define the relevant features of a human.
+Note
+that we will use the term digital human to refer to the
+models even if the represented object is a part of the
+body or the whole body of a subject.
+3. Representations
+Physical objects (including humans) can be repre-
+sented using continuous variables.
+We consider the
+models of humans as continuous in space (r) and time
+(t) and described by a coefficient vector affecting a set
+of model characteristics:
+fm(r, t) ≈
+N
+�
+n=1
+θnφn(r, t).
+(1)
+Here, N is the dimension of the approximate finite-
+dimensional representation of the object, and the
+subscript m indicates the modeling approximation to
+differentiate from the actual object f(r, t).
+The collection of expansion functions {φn(r, t)}N
+n=1
+is employed to form fm(r, t), and θn denotes the n-th
+component of the N-dimensional expansion coefficient
+vector θ. The quantity fm(r, t) constitutes a discrete
+representation of a digital human that can be readily
+displayed on a computer or digitally processed. For the
+case where the expansion functions are defined as in-
+dicator functions that describe non-overlapping space-
+time voxels, θ can sometimes be interpreted as a digital
+image whose components θn represent the integrated
+value of the object over the support of the voxel.
+More generally, a digital human model can be es-
+tablished by integrating the continuous representation
+fm(r, t) over a collection of N voxels as
+fn =
+�
+vn
+fm(r, t) d3r dt,
+n = 1, · · · , N,
+(2)
+where vn denotes the support of the n-th spatial-
+temporal voxel and fn denotes the n-th component of
+a N-dimensional vector f that represents the digital
+human.
+As discussed below, the choice of the expansion
+functions and associated expansion coefficients can be
+specified in different ways, with the general goal of
+making fm(r, t) an accurate approximation of f(r, t).
+The expansion functions can depict geometry (e.g.,
+size, morphology), material properties (e.g., x-ray
+interaction cross-sections, elasticity) or other relevant
+features (e.g., radioactivity, blood oxygenation levels).
+For simplicity, we will consider that the stochastic
+human does not vary with time and proceed only with
+
+CONTENTS
+5
+the spatial dimension r. However, the concepts that
+follow can readily be generalized to model time-varying
+descriptions.[19]
+In practice, the coefficient vector θ can be modeled
+as a random vector and the expansion functions
+{φn(r)}N
+n=1 as random processes.
+Methodologies for
+generating large cohorts of digital stochastic models of
+humans for in silico imaging trials, including models
+for organs and tissues with appropriate variability, can
+rely on either sampling θ, φn or both from appropriate
+distributions representing the intended population. We
+can denote the cohort of digital stochastic humans as
+follows,
+{fs}S
+s=1 =
+�
+n
+θs
+nφn(r),
+(3)
+where
+s
+denotes
+a
+particular
+state
+or
+random
+realization of a digital human in a cohort of size S.
+When φn are known, analytically or numerically,
+the stochastic models are referred to as procedural.
+In this case, the modeler is left with choosing the
+coefficient vector defining the object (θ). In cases for
+which the defining characteristics are unknown, θn and
+φn can be estimated from imaging data.
+In the following sections, we review available
+methods and tools for generating digital human models
+and digital cohorts.
+We present a classification of
+available approaches in Figure 1.
+4. Individual models
+Individual models attempt to create a digital replica
+of a specific physical object. Individual models can be
+categorized as personalized and family models. These
+models are not stochastic since they are meant to
+represent individual subjects with as much detail and
+accuracy as achievable from the image data. In this
+respect, the representation introduced in Section 3
+applies only with S = 1 resulting in a single coefficient
+vector (θn) defining the individual.
+The
+digital
+representation
+in
+these
+cases
+is
+typically a multidimensional voxelized array that
+can be segmented into structures such as tissues
+and organs.
+Early attempts relied on geometrical
+volumes represented by analytical expressions altered
+to generate a wide variety of sizes and shapes. In other
+words, φn are described by quadrics and θn represent
+properties of the volumes defined by the surfaces (e.g.,
+x-ray attenuation and scattering properties).
+These
+computational models have proved useful in areas of
+quality control of imaging systems [20, 21] and in
+radiation dosimetry [22]. Even with more sophisticated
+geometrical structures [23, 24, 25] and more spatial
+detail, these approaches lack the ability to accurately
+represent the statistical variability found in humans,
+organs and tissues. While these simpler models remain
+practical and useful for some tasks, the lack of realism
+and variability makes them unsuitable for generating
+digital humans for in silico imaging trials.
+4.1. Personalized models
+Personalized models aim to capture patient-specific
+information in a digital representation [26]. Medical
+digital
+replicas
+of
+human
+subjects
+are
+in
+silico
+representations of an individual in terms of anatomy
+and physiology.
+Sometimes referred to as digital
+twins [27], these replicas are designed to simulate
+parts or the whole body of a subject for prognostic
+or predictive assessments.
+These models including digital twins can be
+continuously updated from multimodal medical if
+the data characteristics change over time§.
+Digital
+twins are of interest in the context of evaluating and
+selecting optimal medical treatments [28] or imaging
+procedures [29] within clinical practice, and can also
+be incorporated into other in silico applications [30].
+For instance, Wang [31] suggested three applications
+in the areas of medical imaging: optimal selection of
+scanning techniques (so called “virtual comparative
+scanning”), data sharing from in silico scanning of
+the digital replica to the open source community,
+and improvement of the regulatory process of image
+reconstruction algorithms. Patient image datasets can
+also be used to generate models of specific tissues and
+organs. For instance, the Visible Human project [32]
+was first made available in 1994 by the National
+Library of Medicine (NIH) to facilitate anatomy
+visualization applications and includes a detailed data
+set of cross-sectional photographs of the human body.
+4.2. Family models
+Personalized models of a small number of subjects can
+be assembled into families to generate a collection of
+a small number of digital humans spanning a common
+set of parameters, such as subjects’ body size and age.
+These models are based on image acquisitions using
+different modalities including computed tomography
+(CT), magnetic resonance imaging (MRI) and chest
+radiographs (CXR).
+An example of a family model is the Virtual
+Family [33], released by FDA ∥ in 2012. The Virtual
+Family consists of a set of detailed, anatomically
+correct whole-body models of an adult male, an adult
+female, and two children based on high-resolution
+MRI data of healthy volunteers. Organs and tissues
+§ A related concept is an avatar, an artistic and sometimes
+aspirational digital representation of the human in the digital
+world for interactivity purposes.
+∥ https://www.fda.gov/about-fda/cdrh-offices/
+virtual-family
+
+CONTENTS
+6
+Digital
+humans for
+in silico trials
+Population
+models
+(stochastic)
+Knowledge-
+based
+Image-based
+Generative
+Parametric
+Individual
+models (non-
+stochastic)
+Family
+Personalized
+Figure 1. Classification of ethods to generate digital humans for in silico clinical trials.
+are represented using computer-aided design (CAD)
+techniques where each component is a high-resolution,
+non self-intersecting mesh.
+In this case, the models
+are used for electromagnetic, thermal and acoustic
+simulations in the safety assessment of active and
+passive medical implants [34].
+Safety evaluations do
+not require full sampling of the intended population
+and can be performed with a small number of
+exemplars, provided the exemplars adequately cover
+the needed parameter space.
+Similar approaches are utilized in efforts to
+provide models of patient anatomy using patient
+images
+as
+the
+basis
+for
+development
+of
+cohorts
+including using MRI and CT images for modeling
+lungs [35] and torso [14]. More recently, image-derived
+digital and physical models of the breast have been
+proposed by Kiarashi [36] and Bliznakova [37].
+In
+this approach, a voxelized breast model is derived
+from patient images through image segmentation for
+determining the composition of each voxel [38, 39, 40,
+41, 42, 43, 44]. Patient-derived models are limited to
+the imaging characteristics of the acquisition system
+and are also affected by the imperfections of the
+segmentation methods. The resulting models can also
+be augmented with physiological features to facilitate
+imaging studies involving contrast agents [45].
+5. Population models
+Testing new imaging devices, however, requires the
+availability of large digital cohorts of stochastic digital
+humans that can be assembled to properly power
+a study not only on the aggregate (i.e., for the
+entire population), but also to analyze for specific
+subgroups
+with
+notable
+characteristics,
+including
+under-represented populations.
+In this section, we
+focus our attention on models suited for the generation
+of large cohorts of digital humans to be enrolled within
+in silico imaging trials.
+5.1. Image-based models
+Image-based
+models
+estimate
+and
+sample
+model
+components from relevant characteristics within the
+acquired
+medical
+images.
+Image-based
+models
+estimate model components φn and θn in Eq. 3
+from within the acquired medical images.
+Whether
+parametric or generative,
+all image-based models
+are
+limited
+by
+the
+quality
+of
+the
+source
+data
+(i.e.
+medical images),
+including noise,
+artifacts,
+and contrast constraints,
+and do not provide an
+unequivocal mapping to the underlying tissues.
+In
+practice, the use of image-based models should also
+acknowledge the limitation arising from the existence
+of a null space of the imaging system [46].
+The
+null space, which typically arises from the mapping
+of a continuous object to discrete data with an
+imperfect image acquisition system, results in an
+unavoidable loss of information regarding the object.
+Given that the imaging system operator is only
+partially known for most imaging systems and cannot
+represent information obscured by the null space of
+the imaging transformation, image-based models are
+limited even when imaging system models include noise
+measurement.
+5.1.1.
+Image-based parametric models
+In image-
+based parametric models, the generation of cohorts is
+achieved by creating models based on available sets
+of patient imaging data and model modification tech-
+niques including parametric deformation, morphing,
+and registration.
+Parametric models (also known as
+stylized phantoms [47]) capture a population cohort by
+a set of mathematical equations representing a series
+of surfaces (e.g., splines) defining organs that are later
+
+CONTENTS
+7
+voxelized into a volumetric model. The popular 4D ex-
+tended cardiac-torso (XCAT) phantom [48] is an exam-
+ple of an image-based parametric model, and a survey
+of other representations can be found in Kainz [49].
+One limitation of this approach is that model
+development is typically performed on a small number
+of available patient images. For instance, Erickson [39]
+presented a methodology to create a database of
+anatomically variable 3D digital breast models from
+dedicated breast CT images using a tissue classification
+and segmentation algorithm and a fuzzy C-means
+segmentation algorithm.
+The study provided a
+population of 224 breast phantoms incorporating
+a range of breast types,
+volumes,
+densities,
+and
+parenchymal patterns.
+However, using hundreds of
+images might be insufficient to properly characterize
+a population for statistically powered in silico imaging
+trials across patient variability.
+Some recently released image datasets include
+a
+larger
+number
+of
+cases.
+For
+example,
+the
+Medical Information Mart for Intensive Care (MIMIC)
+CXR dataset [50] contains 227,835 imaging studies
+from 65,379 patients presenting to the Beth Israel
+Deaconess Medical Center Emergency Department
+between 2011–2016.
+Similarly, the Medical Imaging
+and Data Resource Center (MIDRC) effort [51] is
+undertaking a large, multi-year, systematic effort to
+collect high-quality COVID data, and over 100,000
+imaging studies have been made public after 2 years
+of work and with significant funding from the NIH.
+However, data sets collected in these well-defined
+areas are likely still insufficient to capture the total
+variability in patient images and the large number of
+subgroups one may find interesting to study ¶. This
+limitation precludes the use of image-based parametric
+models for accurately creating digital cohorts for large
+scale in silico trials.
+Generation of multiple realizations of humans to
+constitute a cohort can be obtained by extending
+image-derived models to create populations in a
+statistical manner.
+For instance,
+Sturgeon [52]
+developed synthetic breast models using principal
+component analysis (PCA) to describe a small training
+set of patient images.
+In this approach,
+each
+existing patient breast CT volume was compactly
+represented by the mean image plus a weighted sum of
+eigenbreasts. The distribution of weights was sampled
+to create synthesized breast phantoms that matched
+fibroglandular density and noise power law exponent
+distributions in real images. Hence, the distribution
+of the synthetic model is determined by that of the
+training data, and, therefore, might suffer from a lack
+¶ “I cannot breed them. So help me, I have tried. We need more
+. . . than can ever be assembled. Millions, so we can be trillions
+more,” Niander Wallace in Blade Runner 2049 (see https:
+//www.imdb.com/title/tt1856101/characters/nm0001467).
+of appropriate representations of cases at the tails of
+the distribution (e.g., very large or very small, very
+dense or very glandular breasts).
+A related concept
+from the computer vision and graphics community is
+the statistical human body model, in which a vertex-
+based model of the body surface is learned, typically
+via PCA, from subjects’ input. The techniques rely on
+linear blend skinning (LBS) to constrain the surface
+vertex deformation with respect to a template bone
+skeleton [53]. Created for non-medical purposes, these
+parametric models are typically learned from training
+examples of lower resolution than what is common in
+medical imaging.
+One alternative approach is to add deformation
+morphing using an anatomic template [26]. Lee [47]
+introduce a hybrid,
+non-uniform rational B-spline
+surface (NURBS) based phantom of an infant by
+combining the expressiveness of a voxel phantom
+with the flexibility of geometric manipulation and
+organ positioning in a parametric phantom. Another
+example is the XCAT Warp [54], where AI-assisted
+unsupervised registration is used to warp XCAT to
+patient CT images to capture a more broad set
+of variations, compared to the existing organ and
+model scaling offered by XCAT. These methods are
+suitable for investigating digital-twin approaches where
+individual models reflecting the characteristics of a
+single individual are needed.
+5.1.2. Image-based generative models
+Image-based
+generative models attempt to synthesize a population
+of stochastic digital humans from information con-
+tained in medical images. Ideally this population cap-
+tures the variability in the anatomy and tissue prop-
+erties within a specified cohort of to-be-imaged sub-
+jects. Consider a collection of N-dimensional digital
+humans {fs}S
+s=1 that represents the cohort of interest
+as described by Eq. 3. This setting corresponds to a
+practical situation in which an in silico study employs
+a fully discrete representation of an imaging system in
+which a finite-dimensional approximation of an object
+is mapped to discrete image data.
+As mentioned in
+Section 3, each digital human fs can be interpreted as
+a realization of a random vector f that is characterized
+by an unknown probability density function pr(f). The
+ability to sample from pr(f) to generate large ensem-
+bles of objects for use in in silico imaging trials is, at
+least conceptually, the ultimate objective of a stochas-
+tic digital human model. Emerging generative methods
+that utilize neural networks are being actively devel-
+oped for this purpose [55]. We refer to these methods
+as generative models.
+A generative adversarial net-
+work (GAN) is a type of generative model that has
+recently been very popular for high-resolution image
+synthesis [56], image translation [57, 58] and a number
+
+CONTENTS
+8
+of generative image applications [59]. Instead of explic-
+itly modelling pr(f), which is difficult due to the high
+dimensionality of f, GANs seek to define a stochas-
+tic process for drawing samples. As such, GANs are
+categorized as implicit generative models. Specifically,
+GANs operate by mapping samples from an analyti-
+cally tractable, low-dimensional distribution pr(z) to
+the sought after samples of the high-dimensional dis-
+tribution pr(f). Typically, pr(z) is specified as an in-
+dependent and identically distributed (i.i.d.) standard
+normal distribution, and therefore, samples of the ran-
+dom vector z can be readily generated. The mapping
+is usually implemented via a deep neural network re-
+ferred to as the generator. Simultaneously with gen-
+erator training, a discriminator network is trained to
+discriminate between the real and generated examples.
+Therefore, the training process is adversarial and is
+approximately solving a min-max optimization prob-
+lem [60]. In this case, a collection of training data (typ-
+ically images) are utilized to learn how to sample from
+an empirical distribution that approximates pr(f). An
+excellent review of GAN applications for medical im-
+age generation can be found in [61]. The adversarial
+training process for GANs is inherently unstable and
+can result in a phenomenon known as mode collapse,
+in which the model fails to sample from certain re-
+gions of probability space. In addition, the generated
+samples are often of low resolution. A number of alter-
+native generative models [62, 63] have been developed
+to address these challenges in applications to medical
+imaging [64]. For example, generative latent optimiza-
+tion (GLO) [62] trains deep convolutional generators
+by minimizing a simple reconstruction loss, improving
+on GAN training instabilities. Diffusion models [63, 65]
+learn a Markov chain of diffusion steps incrementally
+adding and subtracting noise from data, significantly
+outperforming GANs in output image quality [66]. To
+date, almost all studies of deep generative models have
+focused on synthesizing images rather than object rep-
+resentations.
+Limitations
+There are several significant challenges to
+employing GANs or other types of deep generative
+models to establish stochastic human models.
+A
+fundamental and potentially limiting issue is the fact
+that a collection of objects {fs}S
+s=1 is generally not
+available. Medical images are degraded by the presence
+of measurement noise and/or reconstruction artifacts
+which are a limitation of the imaging system and
+not representative of the true underlying objects. As
+such, conventional GANs that are directly trained
+on degraded images will not learn how to sample
+from the true distribution of objects.
+In essence,
+there is a “chicken and egg problem” when seeking to
+establish stochastic human models via deep generative
+models. There are two possible ways to circumvent this
+limitation. First, one can utilize high-quality medical
+images as surrogates of the objects.
+For example,
+in certain tomographic imaging modalities and under
+specific conditions, images of object properties can
+be reconstructed and accurately approximate the true
+object properties.
+In this case, GANs are trained
+in the conventional manner, with images representing
+the training data. If these images are representative
+of the desired subject cohort, the GAN has the
+opportunity to accurately capture object variability.
+Second, one can modify the GAN training process
+to incorporate the image degradation process in
+training.
+This approach, referred to as an ambient
+GAN (AmGAN) [67], utilizes a generator network
+that is augmented with a measurement operator.
+Objects produced by the generator are mapped to
+degraded image data, which are then compared with
+experimental images by the discriminator network.
+This permits establishment of an implicit generative
+model that describes object randomness to be learned
+from indirect and noisy measurements of the objects
+themselves. In a preliminary study, the AmGAN was
+explored for establishing stochastic object models from
+imaging measurements for use in optimizing imaging
+systems [67].
+While promising,
+the use of deep generative
+models for in silico clinical trials is nascent and
+there remain important topics for future investigation.
+The objective assessment of these technologies is
+largely lacking, and there is no consensus regarding
+what statistical information can be reliably learned.
+Additionally, current models have largely been applied
+on 2D images and their extension to three-dimensions
+is an ongoing topic of research.
+Finally, as with
+any data-driven method for establishing stochastic
+human models, the presence of an imaging system null
+space will fundamentally limit the ability of GANs
+to describe certain components of the to-be-imaged
+objects.
+The extent to which the null space can be
+mitigated also remains a topic of ongoing research [67].
+5.2. Knowledge-based models
+Knowledge-based (also known as procedural) models
+are constructed by sampling a set of φn and θn in
+Eq. 3 from distributions representing the relevant
+characteristics of the models.
+The characteristics of
+the distributions are often derived from physical or
+biological measurements.
+Procedural models allow
+for an unlimited number of random realizations of
+the object, leading to the possibility of creating large
+cohorts of digital humans including the representation
+of
+rare
+cases,
+and
+at
+varying
+spatial
+resolution
+which can properly account for small structures
+that
+might
+be
+relevant
+for
+the
+specific
+imaging
+
+CONTENTS
+9
+task being studied.
+However,
+they are usually
+computationally intensive and require a large number
+of parameters to be defined and estimated based
+on prior knowledge.
+Their accuracy and realism
+depend on the parameter combinations and they can
+sometimes generate completely unrealistic outputs.
+Knowledge-based, procedural models are common
+in modeling breast anatomy for imaging studies.
+Graff [68] proposed a detailed model that begins
+with defining an outside surface using a quadratic
+hemisphere
+shell
+with
+a
+skin
+layer
+and
+nipple
+area overlaid.
+The shape of the shell is then
+adjusted for the overall breast volume and surface
+curvature.
+Using a Voronoi segmentation approach,
+the interior is randomly divided into regions of
+fat or glandular components, with each glandular
+component containing a ductal network with terminal
+duct
+lobular
+units.
+The
+volume
+is
+then
+filled
+with Cooper’s ligaments, chest muscle, and blood
+vessels.
+For the VICTRE trial [15],
+the breast
+model was sampled with a 50-µm voxel size.
+The
+implementation is initiated with a set of random seeds
+and creates random voxelized breast anatomy objects
+segmented into nine different tissue types.
+Several
+different modeling techniques are employed including
+a non-isotropic Voronoi segmentation, recursive tree
+branching algorithms to generate a ductal tree and
+vascular network, and Perlin-noise perturbed random
+spheroids to create fat lobules.
+A similar effort by Bliznakova [37] describes a 3D
+breast software model for x-ray breast imaging simula-
+tions based on a breast external shape, ductal lobular
+system, Cooper’s ligaments and pectoralis muscle. In
+this approach, a mammographic background texture
+is added to the tissue regions. Blood vessels, nerves
+and lymphatics were not modeled explicitly. A simi-
+lar, more simplistic approach, was developed by Bakic
+[69] based on two ellipsoidal regions of large scale tissue
+elements: predominantly adipose tissue and predomi-
+nantly fibro-glandular tissue. Internal tissue structures
+within these regions are approximated by a distribution
+of elements including shells, blobs, and a ductal tree.
+Similar approaches have been reported for full-body
+models [47].
+6. Modeling disease
+Disease states can be incorporated into digital cohorts
+using image-based methods or object-space models of
+the condition.
+The analogy between digital human
+models and disease models can be established if we
+consider lesions as continuous variables in space (r)
+and time (t), described by a coefficient vector affecting
+a set of lesion model characteristics. For simplicity, we
+will consider the disease independent (of the underlying
+anatomy where the disease is located) and additive.
+This assumption allows us to represent the disease
+cases as a sum of the stochastic human model and
+the disease component, an addition that is typically
+performed in the voxelized object model or directly
+within the model images. We recognize this approach
+is a known simplification, as disease processes often
+have significant impact on underlying tissues.
+Analogously to the description provided by Eq. 3,
+we can generate a set of disease models {ds} defined
+by:
+{ds}S
+s=1 =
+�
+n
+λs
+nψn(r, t),
+(4)
+where λs
+n is a disease characteristics coefficient vec-
+tor described by the function ψn over N parameters.
+Characteristics that define lesions can include geomet-
+ric functions (e.g., size, morphology), material proper-
+ties (e.g., x-ray interaction cross-sections, elasticity) or
+other relevant features (e.g., radioactivity, blood oxy-
+genation levels).
+Methodologies for generating and incorporating
+disease into cohorts of digital stochastic models rely
+on sampling λn and ψn from appropriate distributions
+representing the intended population. In some cases,
+disease models are specific to a given anatomical
+location or physiology corresponding to a digital
+human exemplar. In other cases, disease models are
+independent of the digital healthy human and are
+simply added or inserted multiple times into models
+of healthy anatomy. In both cases, diseased subjects
+are denoted by a cohort of digital stochastic humans
+with added disease components:
+{fs}S
+s=1 =
+�
+n
+θs
+nφn(r) +
+�
+n
+λs
+nψn(r),
+(5)
+where {fs}S
+s=1 is a cohort of diseased digital humans
+(for simplicity,
+and similarly as in the previous
+section,
+we choose to omit the time dimension).
+Similarly to normal models, when ψn are unknown,
+models of disease can be obtained relying on imaging.
+Alternatively, when ψn are known, analytically or
+numerically, the stochastic disease models are referred
+to as knowledge-based (also known as procedural).
+6.1. Image-based models of disease
+Similar to image-based models of the human body,
+image-based models of disease rely on imaging data
+for extracting lesion information. Various techniques
+for capturing disease characteristics, particularly for
+breast
+lesions,
+have
+recently
+been
+explored
+[70,
+71].
+Image-based neural network models for disease
+modelling have also been explored.
+For instance,
+Kadia [72] proposed a method to generate synthetic,
+infection-like patterns in the lung to create large
+
+CONTENTS
+10
+collections of 2D and 3D training examples for deep
+segmentation models.
+While image-based models
+contain features from actual patient data and thus
+may look more realistic at first glance, they suffer
+from limited resolution of the tumor model, largely
+determined by the imaging acquisition characteristics
+and limited number of available lesion morphologies,
+shapes, and sizes. In addition, image-based methods
+require an institutional review board (IRB) approval
+for obtaining and utilizing the diseased case data
+for research and development, which could delay or
+disadvantage some analysis efforts.
+6.2. Knowledge-based models of disease
+Knowledge-based models of disease are constructed
+by sampling a set of known (or assumed known)
+ψn and λn in Eq. 4 from distributions representing
+the relevant characteristics of the disease,
+where
+distributions
+are
+often
+derived
+from
+physical
+or
+biological measurements. In contrast to image-based
+models, knowledge-based models enable the generation
+of unlimited numbers of lesion shapes with variable
+resolution.
+Examples of knowledge-based models
+include de Sisternes [73] spiculated breast cancer mass
+model and Sengupta [74] growing breast mass models.
+In [74], a breast lesion growth method based on
+biological and physiological phenomena accounting for
+the stiffness of surrounding anatomical structures was
+introduced. Breast ligaments were considered as rigid
+structures with elastic moduli in the range of 8x104-
+4x105 kPa, while fat (elastic modulus varying from
+0.5 to 25 kPa) and glandular tissues (elastic modulus
+varying from 7.5 to 66 kPa) constituting the more
+elastic regions of the breast. In this approach, tumor
+cells are less likely to grow through stiffer structures
+and instead, preferentially proliferate through the more
+elastic regions of the breast. Depending on the breast
+local anatomical structures, a range of unique lesion
+morphologies can be realized, allowing lesions to blend
+naturally into the anatomical regions.
+A common simplifying assumption is to define the
+disease model independent from other human model
+components.
+For example, in VICTRE [15] and in
+Sengupta [75], breast cancer mass lesions are added
+to the normal breast models by replacing voxels in
+the breast with voxels of the lesion model, without
+modification to adjacent voxels. This approach, while
+practical, does not account for the significant effect
+of the growing tumors on its surrounding tissues,
+typically visible in x-ray images as architectural
+distortions suggestive of abnormalities.
+To consider
+these effects, Eq. 5 needs to be modified to account for
+the interaction between normal and disease models.
+7. Role of augmentation methods
+Augmentation methods start with an already-defined
+object, image or a set of defined objects, and generate
+new examples based on properties of inputs,
+as
+well as pre-defined or data-driven transformations (in
+contrast, digital human models start with only an
+object description, such as that given in Eq.
+1).
+GAN-based models (see Section 5.1.2) are similar to
+augmentation methods in that they employ complex
+transformations derived with the help of training
+data sets.
+Augmentation methods typically employ
+analytically-defined or stochastic operators that do
+not require the use of neural networks, and can be
+applied both in the object domain and in the acquired
+image domain. Techniques in the latter group generate
+examples that could be obtained through an imaging
+system applied to an object with an accompanying
+degradation (e.g., smoothing, noise, reconstruction
+artifacts).
+Geometric transformations, intensity operations,
+and spatial filtering are among the most basic types
+of augmentation methods. Geometric transformations
+redefine the spatial relationships among voxels or
+geometrical locations in an object, and include affine
+(scaling, rotation, translation, reflection and shearing),
+as well as non-affine transformations, such as non-
+linear warping and morphing [76]. Intensity operations
+modify intensity values in a grayscale image or
+channel values (e.g., RGB or CMYK) in a color
+image. Examples include operations such as a family
+of gamma corrections, linear contrast adjustments,
+and remapping voxel values using a pre-defined or
+pseudo-random remapping curve [77, 78].
+Spatial
+filtering (using a filter mask) is another possibility for
+generating a new image or object based on an existing
+one. Spatial filtering can be linear (in which case it can
+be implemented by a convolution operation) or non-
+linear (e.g., median filtering), and can be implemented
+to smooth or sharpen to emphasize certain features.
+Finally,
+all three types of augmentations can be
+combined
+using
+a
+continuous
+mapping
+from
+the
+parameter space of transformations to the image or
+object space [79].
+Noise injection is an image augmentation method
+that enhances robustness of machine learning models
+and belongs to the family of domain randomization
+(DR) methods [80].
+Although noise injection after
+data acquisition does not generate a new member
+of a patient population, it can generate a different
+representation of an object in the image domain, and
+can be useful for augmenting patient cohorts obtained
+with in silico modeling. Some earlier and non-medical
+applications of noise injection in machine learning
+sought to augment the image data sets without
+regard to the physics of image acquisition [81, 82].
+
+CONTENTS
+11
+Other works used physics-based techniques for noise
+modeling and addition, improving realism of the noise
+appearance in the augmented images [83, 84].
+The
+main benefit of noise injection in the image domain
+for in silico trials is that it may allow for the rapid
+generation of different representations of the same
+object at different noise levels, leading to comparisons
+that may require less computational power compared
+to a full implementation of image acquisition physics
+applied to a digital stochastic object model. Addition
+of texture to a model in the object domain has
+similarities to noise injection in the image domain
+in that both techniques aim at producing noise-like
+properties (e.g., using a noise power spectrum in
+modeling), but are different in that addition of texture
+in the object domain does not attempt to model the
+noise from data acquisition [85].
+Combination of objects or images is another
+popular augmentation technique.
+In the object
+domain,
+combination
+of
+an
+object
+model
+for
+a
+normal (non-diseased) patient with a lesion model
+(as described in Section 6) can be thought of as an
+example of this type of augmentation.
+Generating
+new members of a patient population based on an
+eigenspace analysis of existing patient objects, as was
+done in [52] and described in Section 5.1.1 is another
+example of augmentation in the object domain.
+In
+the image domain, researchers investigated tools for
+the extraction of image parts from one clinical image
+and then their insertion into a new location on the
+same or different image. Pezeshk [86] used an image
+blending technique based on Poisson image editing to
+insert pulmonary nodules extracted from one chest CT
+exam into another.
+Augmenting a training data set
+for a machine learning model using this technique can
+improve the model performance on independent, real
+test datasets [87]. Likewise, Ghanian [88] used a similar
+technique to insert microcalcification clusters extracted
+from one mammogram into another mammogram,
+and showed that experienced observers cannot reliably
+distinguish
+between
+computationally
+inserted
+and
+native clusters. Besides the ability to convince experts,
+desirable properties for such combination techniques
+include acceptable noise properties in the combined
+image, plausible lesion-background combinations (that
+might require the intervention of an operator during
+the augmentation process), and a sufficient range
+of
+variation
+in
+the
+combined
+images
+that
+can
+be generated,
+which are often difficult to satisfy
+simultaneously.
+The
+main
+advantage
+of
+data
+augmentation
+methods is their practicality.
+For example, existing
+images or models both for normal and diseased
+patients can be manipulated (with relative ease) with
+geometric transformations leading to expanded patient
+representations.
+When implemented in the image
+domain, augmentation methods are fast, bypassing the
+stage where a model for the imaging system is applied
+to the object to yield an image. However, important
+shortcomings accompany these advantages.
+Unless
+deliberate attention is paid, augmentation methods
+may yield objects or images that are biologically or
+physically implausible. An extreme example may be
+an intensity transformation that results in bones with
+lower Hounsfield units than soft tissue.
+Although
+this can be avoided easily by using an intensity
+transformation that is monotonically increasing, most
+augmentation
+methods
+and
+transformations
+need
+careful planning to avoid such inconsistencies, and it
+may not be possible to avoid all inconsistencies. The
+consequences of such implausible images or objects
+on the results of an in silico imaging trial should be
+carefully considered. In addition, many augmentation
+techniques do not result in an independent, new
+representation from the population, but rather in
+representations that are highly dependent on the
+original objects or images used as inputs to the
+augmentation method. For example, lesion insertion
+methods described in the previous paragraph do not
+increase the number of lesions in the augmented
+data set, but only the lesion-background combinations
+that are generated.
+Again, the consequences of this
+limitation in the range of variation of generated images
+should be an important consideration in an in silico
+imaging trial that uses augmentation.
+8. Considerations for sampling digital cohorts
+In silico studies require careful study planning and
+good
+clinical
+trial
+design.
+Even
+if
+and
+when
+methodologies for developing digital stochastic models
+of humans for imaging studies become widely available,
+generating digital cohorts needs an understanding of
+the trade-offs and potential for bias associated with
+selecting a specific distribution of study subjects. At
+the start of the design of an in silico imaging trial
+is the challenging task of scoping the population
+of the digital humans to be included in the study.
+For instance, a number of previous computational
+studies in breast imaging using procedural models used
+a uniform sampling with a desired average of 50%
+adipose and 50% fibroglandular voxels [89] with an
+uncompressed breast size of 14 cm. Another example
+of enrollment strategy can be found in the OpenVCT
+platform, where a range of size and glandularity is
+specified and then uniformly randomly sampled [90].
+A more recent in silico imaging study used sampling
+from a multi-class distribution identifying 4 different
+breast densities resulting in the characteristics of the
+intended population [15].
+
+CONTENTS
+12
+������������������������ = 0.79
+������������������������ = 0.89
+∆������������ = 0.10
+������������������������ = 0.84
+������������������������ = 0.90
+∆������������ = 0.06
+������������������������ = 0.77
+������������������������ = 0.88
+∆������������ = 0.11
+������������������������ = 0.99
+������������������������ = 1.00
+∆������������ = 0.01
+Figure 2. Effect of sampling strategies on performance assessment. Sampling is from a bimodal distribution of subjects (seen in
+3D insert in the second panel from the left) described by 2 random parameters: (from left to right) uniform, matched, simpler, and
+narrow. Only 20 samples are shown here for ease of visualization. The gray shading depicts the distribution from which samples
+are taken in each of the 4 cases. AM, AT , and ∆A refer to the lesion detection average AUC for mammography, average AUC for
+digital breast tomosynthesis, and the average AUC difference, respectively. .
+Through in silico enrollment,
+digital cohorts
+{fs}S
+s=1 are generated.
+We denote the distribution
+of the population of digital humans as fd, where d
+represents the digital world, and the distribution of
+subjects in the intended population as fi.
+In this
+context, the goal of the in silico enrollment is to
+minimize the difference ∆f = |fd − fi| between the
+digital (d) and physical-world intended distributions,
+where
+|.|
+denotes
+a
+statistical
+distance
+measure.
+Clinical trial enrollment programs in the physical world
+require strategies to ensure a reasonable ∆f given
+available sampling resources. At first approximation,
+the
+in
+silico
+enrollment
+should
+approximate
+the
+intended distribution to a greater extent than the
+corresponding physical clinical trial enrollment.
+Analysis of ∆f corresponding to a given in silico
+enrollment strategy may be needed to understand how
+the difference across study subject distributions could
+affect the outcome of the trial. Here, we discuss a test
+case (see Figure 2) that compares different enrollment
+strategies for an in silico trial comparing digital
+mammography (DM) and digital breast tomosynthesis
+(DBT) derived from the VICTRE [15] project.
+We
+assume the populations (digital and physical) consist
+of normal and diseased subjects with a prevalence
+of 0.5.
+These two classes of patients are therefore
+sampled with equal probability.
+We calculate the
+difference of performance (measured using the area
+under the receiver operating characteristic curve, or
+AUC, in the task of differentiating between normal
+and disease subjects) between mammography and
+digital breast tomosynthesis. We consider the following
+four sampling approaches.
+In the first approach
+(uniform), fi is unknown and subjects are sampled
+uniformly within a range of interest, from all possible
+combinations of the input parameters that define
+f.
+In the second approach (matched), fi is known
+and subjects are sampled from the true underlying
+distribution.
+In the third approach (simpler), fi
+is unknown, but can be approximated by another,
+simpler distribution from which samples are obtained.
+Finally, in the fourth approach (narrow), fi is known
+to be a narrow, well-defined subset of the general
+population of subjects of particular interest (e.g., rare
+diseases or very obese subjects).
+For this simplified example, let fi be a bimodal
+distribution defined by two parameters (e.g., breast
+size and glandularity).
+Using Eq. 3, we can express
+the model through two expansion functions φ1,2, each
+associated with one of the two random variables
+affected by a random parameter set given by θ1,2. As
+seen in Figure 2, one of the modes of the distribution
+has twice the amplitude and half the variance of the
+other.
+The four density plots illustrate a top view
+of the distribution contour plot with the individual
+samples drawn using the four different sampling
+strategies. The results demonstrate that the choice of
+sampling strategy can have a significant effect on the
+difference in AUC, which for this example case, ranges
+from a difference of 0.01 (almost zero) to 0.11 in terms
+of device performance.
+9. Summary and conclusions
+In silico trials are an emerging area of regulatory
+research that offer the ability to capture highly
+diverse patient distributions at a significant time and
+cost savings, compared to traditional physical clinical
+trials.
+To conduct in silico trials, realistic digital
+representations of humans are needed. In this paper,
+we reviewed and discussed existing techniques for
+generating digital humans, including disease models,
+for in silico imaging trials.
+Digital humans can
+be created using image-based or knowledge-based
+techniques.
+In summary, we favor techniques with
+object-based representations (rather than images of
+
+80
+60■CONTENTS
+13
+objects) in order to decouple the characteristics of the
+image acquisition system from the characteristics of
+the object (true representation of the physical-world
+human).
+In generating digital humans for in silico
+trials, one should consider the quality and quantity of
+the source data or knowledge used, and whether the
+models represent a single patient, a small cohort, or a
+sizable population with realistic patient variability.
+It remains a crucial next step to evaluate the
+quality of the digital human models and the images
+that can be generated with them. In particular, it is
+essential to carefully identify the patient distribution
+that the particular digital human model can and
+cannot capture, in order to prevent misuse and ensure
+patient safety.
+We need to study to what extent
+model-derived data contributes to our understanding of
+performance levels for populations with rare diseases or
+for populations underrepresented in traditional clinical
+trials.
+Future work should examine the ethical and
+safety considerations of relying on digital humans for
+clinical trials. Overall, the use of in silico imaging trials
+and in silico trials in medicine is a rapidly developing
+field and has the potential to address many of the
+emerging challenges in the regulatory evaluation of
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+
+CONTENTS
+16
+digital breast tomosynthesis,” in Medical Imaging 2018:
+Physics of Medical Imaging (J. Y. Lo, T. G. Schmidt, and
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+

_NE1T4oBgHgl3EQf8gXW/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf,len=226
+page_content='ON PERTURBATIONS RETAINING CONSERVATION  LAWS OF DIFFTRENTIAL EQUATIONS  ALEXEY SAMOKHIN  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' The paper deals with perturbations of the equation  that have a number of conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' When a small term is  added to the equation its conserved quantities usually decay at in-  dividual rates, a phenomenon known as a selective decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' These  rates are described by the simple law using the conservation laws’  generating functions and the added term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Yet some perturbation  may retain a specific quantity(s), such as energy, momentum and  other physically important characteristics of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' We intro-  duce a procedure for finding such perturbations and demonstrate  it by examples including the KdV-Burgers equation and a system  from magnetodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Some interesting properties of solutions  of such perturbed equations are revealed and discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  Keywords: conservation laws, perturbed equations, selective de-  cay, traveling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  MSC[2010]: 35Q53, 35B36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Introduction  Many physical systems are modeled using equations that have a sig-  nificant number of conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Yet when an additional (usually  dissipative) term is added to the equation its conserved quantities decay  at individual rates, which are connected to their generating functions  [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' The famous example is the KdV equation (it has infinitely many  conservation laws) and the KdV-Burgers equation (with additional,  with respect to KdV, dissipative term and only one conservation law).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  To be precise let E(u) = 0 be a system of equations describing an  ideal (unperturbed) media state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' A scalar H depending on u and its  derivatives is a conservation law if for ⟨H⟩, the integral of H over some  fixed spatial domain, ∂⟨H⟩  ∂t  ����  E  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  For the perturbed equation the quantity H is constant no more and  ∂⟨H⟩  ∂t  ̸= 0 is called the decay rate of H, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  A perturbed state usually satisfies the equation E(u) + LF(u) = 0,  where L is a small parameter diagonal matrix diag(λi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' for L = 0 we  get the ideal state equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' The decay rate depends on the additional  term L F(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' The connection between decay rate and LF(u) was called  a ’balance law’ in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='03547v1  [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='PS]  9 Jan 2023  2  ALEXEY SAMOKHIN  This law expresses ∂t⟨H⟩ in terms of scalar product of LF(u) and  the generating function g of the conserved quantity H, [1]:  ∂⟨H⟩  ∂t  = ⟨g · LF⟩  (1)  Remarks  The right-hand side of (1) is not unique: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='g, one can get a  different but equivalent form integrating by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  In the case of the integrand in the right-hand side of (1) is null or  an exact form we get the situation when the conserved quantity  ⟨H⟩ is conserved as well for the correspondent perturbed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  Let us restrict considerations to R[u], the ring of differential  polynoms of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Then all perturbations F retaining the conser-  vation law with the generating function g must satisfy  g · LFdx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' dxn ∈ Im(d),  where d : Λn−1 → Λn and Λk are k-forms of spatial variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  Of course, the intersection of the principal ideal g · R[u] with  Im(d) is huge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  A considerable difference in decay rates leads to a simple method,  first discovered by Taylor, [4], for finding quasi-stationary states of  plasma which are of great practical importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  He studied the model where the decay of energy E is monotonic but  those of momentum M and helicity are not necessarily so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Such an  inequality in decay rates leads to a distinct physical phenomenon of  ’self–organization’ or quasi–stable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  There exist a very simple procedure for finding solutions of the above  described behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' It was suggested in [4], and is known as ’Taylor  trick’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' The procedure is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  Taking into consideration their comparative decay rates, minimize  E with M as constrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Put δ(E + λM = 0), M and presumed con-  stant, λ being Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' This Euler–Lagrange equation is  not necessarily compatible with the initial equation but nevertheless it  gives a way for good approximations of self-organization phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  There is a considerable number of publication in the field, see a recent  paper [5] for recent developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  Another application of selective decay is given in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' The problem  is the behavior of the soliton which, while moving in non-dissipative  and dispersion-constant medium encounters a finite-width barrier with  varying dissipation and/or dispersion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' beyond the layer dispersion is  constant (but not necessarily of the same value) and dissipation is null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  The transmitted wave either retains the form of a soliton (though of  different parameters) or scatters a into a number of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Using the  relative decay of the KdV conserved quantities a simple algorithm to  predict the number and amplitudes of resulting solitons was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  ON PERTURBATIONS RETAINING CONSERVATION LAWS  3  In [7] the selective decay approach was applied to some well-known  equations of mathematical physics (KdV and KdV-Burgers equation,  BBM and its dissipative generalization, two-dimensional generalized  shallow water wave equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' It have showed that the Taylor trick  extremals are associated with first-order PDEs and travelling wave so-  lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  In this paper we search, for some popular equations, their low-order  perturbations which retain a chosen conservation law (in a sense that  the perturbed equation has the same conserved quantity as initial one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  Examples include KdV and its conserved energy or momentum and the  Kadomtsev-Pogutse system of equation from magnetohydrodynamics  with its three known conserved quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Some interesting properties  of solutions of such perturbed equations are revealed and discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' KdV and KdV-Burgers  The generalized KdV equation (KdV-Burgers equation) considered  here is of the form  ut = 2uux + uxxx + λuxx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  (2)  The classical KdV equation corresponds to λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  The first three conserved quantities for KdV are  m  =  � +∞  −∞  u(x, t) dx — mass,  M  =  � +∞  −∞  u2(x, t) dx — momentum,  E  =  � +∞  −∞  �  2u3(x, t) − 3(ux(x, t))2�  dx — energy,  and there are infinite number of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  The generating functions for the above conservation laws of the KdV  are, up to multiplication constants, 1, u and u2 + uxx correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  As for the equation (2), it has a form of a conservation law, ut = Fx,  the ”mass”  � +∞  −∞  u dx is a conserved quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' For a soliton this mass  is equal to 12aγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  But the impulse ⟨u2⟩ =  � +∞  −∞  u2 dx declines monotonically:  Mt = 1  2⟨u2⟩t = ⟨uut⟩ = ⟨u(u2 + uxx + λux)x⟩ = 2  3u3��+∞  −∞ − u2  x|+∞  −∞ − λ⟨u2  x⟩  =  (3)  By analogy, for the energy  4  ALEXEY SAMOKHIN  Et = ⟨  �  2u3(x, t) − 3(ux(x, t))2�  ⟩t = 6λ⟨uxx(u2 + uxx)⟩  (4)  Thus the energy does not necessary declines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Transformations of KdV that retain momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Now let  us find perturbations of the form F(u, ux, uxx) that retain momen-  tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Accordingly to the remark 2 above, the differential form λu ·  F(u, ux, uxx)dx must be exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Thus  u · F(u, ux, uxx) = Dx(A(u, ux))  (5)  for some A(u, ux).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Here  Dx = ∂  ∂x +  ∞  �  n=0  uxn+1  ∂  ∂uxn  is the operator of the full differentiation with respect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  Below we restrict the search to polynomials of u and its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  Then in (5) the polynomial Dx(A(u, ux)) is divisible by u, so A(u, ux) =  u2B(u, ux).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  On the other hand  Dx(u2B(u, ux)) = 2uuxB(u, ux) + u2(ux  ∂B  ∂u + uxx  ∂B  ∂ux  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  Hence the second order retaining momentum perturbation is defined  by  F(u, ux, uxx) = 2uxB(u, ux) + u(ux  ∂B  ∂u + uxx  ∂B  ∂ux  )  for an arbitrary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Note that F is linear in uxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  For instance, if B = ux the λ transformation of the KdV equation  ut = 2uux + uxxx + λ(2u2  x + uuxx)  (6)  retains ⟨u2⟩ as its conserved quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' This construction can be generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' If g is the gen-  erating function for some conserved quantity Cl of an one-spational  equation E, then F = g−1Dx(g2Φ) is the addendum to E which re-  tains Cl, Φ being a arbitrary function of u and its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' The equation (6) has travelling wave solutions, in par-  ticular shock waves of the form  3  2λ  �  a tanh  �a3λ2 + 3a  λ2  t + ax  �  + 1  λ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  (7)  This shock moves to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' If require u|−∞ = 0 then (7) becomes  the shock wave  3  2λ2  �  1 + tanh  � 4  λ2t + 1  λx  ��  ON PERTURBATIONS RETAINING CONSERVATION LAWS  5  with the velocity 4/λ, see figure 1, left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' The travelling wave solution  Left: for the equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Right: For the equation (9),  a = 1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' The perturbed equation has only translations in x and  t as its point symmetries, but a lot of conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Transformations of KdV that retain energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Now for energy  saving transformations of KdV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Since the generating function of energy  is, up to a constant multiplier, u2 + uxx, one must solve  (u2 + uxx) · F(u, ux, uxx, uxxx) = Dx(A(u, ux, uxx))  (8)  for some A(u, ux, uxx), to find an low-order F(u, ux, uxx), the suitable  transformation term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' By analogy to the momentum case, the one pos-  sibility is A = (u2 + uxx)2B  F(u, ux, uxx) = 2Dx(u2+uxx)B+(u2+uxx)(ux  ∂B  ∂u +uxx  ∂B  ∂ux  +uxxx  ∂B  ∂uxx  ),  for an arbitrary B = B(u, ux, uxx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' If B = u then F = 5u2ux+2uuxxx+  uxuxx  The corresponding transformed equation is  ut = 2uux + uxxx + λ(5u2ux + 2uuxxx + uxuxx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  (9)  Its point symmetries are only translations in x and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' The equation (9) has travelling wave solutions, in par-  ticular — solutons of the form of a vertically shifted soliton  u(x, t) = −6a2 tanh2(a(4a4·λt+x))+4a2 = 6a2 sech2(a(4a4·λt+x))−2a2  (10)  found by Maple, with the velocity V = 4a4λ, see figure 1, right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  6  ALEXEY SAMOKHIN  Yet it is not the whole answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Computer experiments demonstrate  that an arbitrary initial datum for this equation scatters into a number  of solitary peaks of different but constant height and velocity and a ’tail’  (see figures 2 and 3) — in a manner of the KdV itself, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Left: Initial profile 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='5 sech2(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='5x) for the  equation (9), λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  Right: Resulting profile at t = 6: single soliton-like  peak of a constant form and velocity and an oscillating  tail moving in opposite direction  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Left: Initial profile sech2(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='1x) for the equa-  tion (9), λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  Right: Resulting profile at t = 40: multiple soliton-like  peaks of a constant form and velocity and (seemingly)  no tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  The analytical description of these peaks is so far unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' The  reason is that the equation on travelling waves, u = u(x + V t), here  V u′ = 2uu′ + u′′ + λ(5u2u′ + 2uu′′′ + u′u′′  can be readily integrated introducing the new dependent variable u′ =  p(u) which leads to a linear first order ordinary differential equation on  z(u) = p(u)p′(u),  (2uλ + 1)z′ + λz = V − 5λu2 − 2u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  ON PERTURBATIONS RETAINING CONSERVATION LAWS  7  But the resulting general solution looks hopelessly implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' The likes  of (10) arise in the case of a very special combination of the arbitrary  constants entering this general solution, and such combinations are  hard to discover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Two-dimensional MHD System  Consider the Kadomtsev-Pogutse sysnem of equations  �  ∆ut + ux∆uy − uy∆ux + vy∆vx − vx∆vy  =  0  vt + uxvy − uyvx  =  0  (11)  which describes quasi-stationary states of plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' It has three conser-  vation laws,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' that is there are three non–trivial conserved densities (two  of them depending on arbitrary functions): the total energy E (mag-  netic plus kinetic energy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' generalized ’cross helicity’ Hc and mean  magnetic potential A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  E  =  1  2⟨u2  x + u2  y + v2  x + v2  y⟩  H  =  ⟨f ′(v) · (uxvx + uyvy)⟩  A  =  ⟨Φ(v)⟩  (12)  Their generating functions are,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' respective order,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  �  u  ∆v  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  �  f(v)  f ′(v)∆u  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  �  0  Φ′(v)  �  (13)  where f and Φ are arbitrary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  Let us seek transformations of (11) of the form  �  ∆ut + ux∆uy − uy∆ux + vy∆vx − vx∆vy  =  νF(u, v)  vt + uxvy − uyvx  =  ηG(u, v)  (14)  Here F, G are functions of u(x, y, t), v(x, y, t) and their derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Energy-retaining transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' In this instance  ∂⟨E⟩/∂t = 0 implies  (−νu · F − η∆v · G)dx ∧ dy = d(A(u, v)dy − B(u, v)dx) =  (DxA(u, v) + DyB(u, v))dx ∧ dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  (15)  There are a lot of solutions to (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' We restrict ourselves to some  low-order examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Ortogonal transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' One can always get zero right hand  side in equation (15): just put F = η∆ and G = −νu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' The vector  (F, G) is orthogonal to the generating function so ∂⟨E⟩/∂t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' It  works if the number of any system of equations is greater than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  8  ALEXEY SAMOKHIN  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Splitted sum transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Another solution may be obtained  assuming  − νu · F(u, v) = DxA(u, v),  η∆v · G(u, v) = DyB(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  (16)  Here again A, B are functions of u(x, y, t), v(x, y, t) and their deriva-  tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' This equations may be solved by analogy to the KdV case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  One of numerous solutions here is A = νun,  B = η(∆v)2, so F =  −νnun−2ux, G = 2η∆vy  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' {ν = η}—case transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Take A = Gvx, B = Gvy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  Then uF = vxDxG + vyDyG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' For instance, choose G = u2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' it fol-  lows that F = 2(uxvx + uyvy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Mean magnetic potential retaining transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Here  ∂⟨A⟩/∂t = 0 implies  (−ν0 · F − ηΦ′(v) · G)dx ∧ dy = d(A(u, v)dy − B(u, v)dx) =  (DxA(u, v) + DyB(u, v))dx ∧ dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  (17)  Thus F is an arbitrary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Then one possible solution is  −ηΦ′(v) · Φ(v)(αvx + βvy) = DxαΦ2 + DyβΦ2, α, β ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  That is, to retain the mean magnetic potential of (11), its first equation  may be transformed in arbitrary way and the second one by ηG =  −ηΦ(v)(αvx + βvy) for all α, β ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Cross helicity retaining transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Here ∂⟨Hc⟩/∂t = 0  implies  −νf(v)·F(u, v)−ηf ′(v)∆(u)·G(u, v) = DxA(u, v)+DyB(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' (18)  In the case ηη = ν it is not hard to find some suitable transformations  (F, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Namely, take  A = −ηf 2(v)f ′(v)ux, B = −ηf 2(v)f ′(v)uy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  It follows  F = [2f ′2(v) + f(v)f ′′(v)](vxux + uyvy), G = f ′(v)f(v)∆u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  For f(v) = v it comes to  F = −2η(vxux + uyvy) G = −ηv∆u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  ON PERTURBATIONS RETAINING CONSERVATION LAWS  9  Conclusion  The paper deals with perturbations of the equation that have a num-  ber of conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' When a small term is added to the equation  its conserved quantities usually decay at individual rates, a phenome-  non known as a selective decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' These rates are described by the simple  law using the conservation laws’ generating functions and the added  term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Yet some perturbation may retain a specific quantity(s), such  as energy, momentum and other physically important characteristics  of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' We introduced a procedure for finding such perturbations  and demonstrated it by examples including the KdV-Burgers equation  and a system from magnetodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  Our worked out examples show that the perturbed equations retain-  ing a specific conservation law frequently also retain additional alge-  braic properties such as travelling wave solutions or a presence of other  conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  Thus the present paper as well as [5] and our previous research of  the KdV solitons in nonhomogeneous media, [6], persuades that the  selective decay approach is a valid and effective instrument to obtain  qualitative approximations and estimates for behavior of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  The figures in this paper were generated numerically using Maple  PDETools package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  The mode of operation uses the default Euler  method, which is a centered implicit scheme, and can be used to find  solutions to PDEs that are first order in time, and arbitrary order in  space, with no mixed partial derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
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+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
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+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='101723  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content=' Samokhin, Taylor Trick and Travelling Wave Solutions, Lobachevskii  Journal  of  Mathematics,  2022,  43,  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  10,  2808—2815,  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='  DOI:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='1134/S1995080222130406  Institute of Control Sciences of Russian Academy of Sciences 65  Profsoyuznaya street, Moscow 117997, Russia  Email address:  samohinalexey@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}
+page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQf8gXW/content/2301.03547v1.pdf'}