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09E4T4oBgHgl3EQfzA3P/content/tmp_files/2301.05271v1.pdf.txt

@@ -0,0 +1,1024 @@
+Observation of terahertz second harmonic generation from surface states in the
+topological insulator Bi2Se3
+Jonathan Stensberg,1 Xingyue Han,1 Zhuoliang Ni,1 Xiong Yao,2, ∗ Xiaoyu
+Yuan,2 Debarghya Mallick,2 Akshat Gandhi,2 Seongshik Oh,2 and Liang Wu1, †
+1Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, U.S.A
+2Department of Physics and Astronomy, Rutgers,
+The State University of New Jersey, Piscataway, New Jersey 08854, U.S.A.
+(Dated: January 16, 2023)
+We report the observation of second harmonic generation with high conversion efficiency ∼ 0.005%
+in the terahertz regime from thin films of the topological insulator Bi2Se3 that exhibit the linear
+photogalvanic effect, measured via time-domain terahertz spectroscopy and terahertz emission, re-
+spectively.
+The features of both phenomena are found to be consistent with the characteristics
+of and attributable to the surface of Bi2Se3, which breaks both inversion symmetry and two-fold
+rotation symmetry and therefore permits second-order processes.
+Since both phenomena result
+from processes that reverse sign in oppositely-oriented domains of Bi2Se3, the observation of both
+phenomena is attributable to the presence of unequally populated twinned domains in the sample
+over millimeter length scales, confirmed by atomic force microscopy measurements. These results
+represent the first observation of intrinsic terahertz second harmonic generation in an equilibrium
+system, unlocking the full suite of both even and odd harmonic orders in the terahertz regime.
+Introduction
+Harmonic generation (HG) has been an invaluable non-
+linear optical technique since its first demonstration [1]
+and continues to power recent advances ranging from the
+imagining of microscopic magnetic domains [2, 3] to the
+development of tabletop sources of extreme ultraviolet
+and x-ray light for attosecond science [4, 5]. However,
+since nth-order nonlinear optical processes scale with the
+nth power of the optical intensity, employing HG to study
+phenomena below ∼100 meV has been severely impeded
+by the historical terahertz (THz) gap [6], traditionally ∼
+0.1-30 THz (1 THz ≈ 4.1 meV), where technical chal-
+lenges have impeded the development of intense light
+sources. Recent progress in intense THz generation [7, 8],
+however, has enabled the first applications of HG to the
+THz regime. Since its first demonstration [9], THz third
+harmonic generation (THG) has rapidly become a stan-
+dard tool for characterizing the Higgs mode [9–13] and
+other nonlinear optical processes [14–26] in a variety of
+superconductors [27–34]. Yet more recently, odd-order
+THz-HG has been reported in doped Si [35, 36] and ma-
+terials hosting Dirac fermions, namely graphene [37–40],
+Cd3As2 [41–43], and the bismuth chalcogenide family
+of topological insulators [40, 44, 45].
+The latest stud-
+ies have explored controlling and optimizing THz-HG,
+demonstrating that the nonlinear process can be effec-
+tively tuned via gating [38] and metasurfacing [39, 40].
+Despite these exciting advances, THz-HG remains
+highly constrained, limited to odd-order low harmon-
+ics. Most strikingly, intrinsic even-order THz harmonics,
+which are only generated in systems that break inversion
+∗ Current Affiliation: Ningbo Institute of Materials Technology
+and Engineering, Chinese Academy of Sciences, Ningbo 315201,
+China
+† [email protected]
+symmetry, have never been demonstrated in an equilib-
+rium material, having been observed only in supercon-
+ductors with a net propagating supercurrent [46, 47] or
+in carefully engineered devices [48].
+The lack of THz
+second harmonic generation (SHG) in the studies of the
+bismuth chalcogenides [40, 44, 45] is of particular note, as
+SHG–and even-order HG in general–originating from the
+surface has been a well-established feature of the nonlin-
+ear optical response outside of the THz regime [49–55].
+Furthermore, as the bismuth chalcogenides are prototyp-
+ical topological insulators [56–59] with a centrosymmet-
+ric bulk and inversion symmetry-breaking surfaces, the
+second-order optical response of THz-SHG offers a path-
+way to measuring the properties of the topological sur-
+face state while intrinsically avoiding the properties of
+the bulk band, without resorting to doping [45].
+Here, we report the observation of THz-SHG from
+Bi2Se3 samples exhibiting the linear photogalvanic effect
+(LPGE). With LPGE determined by THz emission
+[60] and THz-SHG measured via intense time-domain
+THz spectroscopy (TDTS) [61], thin films of Bi2Se3
+that display LPGE are found to produce THz-SHG
+that is highly efficient and independent of the sam-
+ple thickness.
+As both LPGE and SHG result from
+second-order nonlinear processes, both effects originate
+from the three-fold symmetric surface of Bi2Se3, which
+breaks both inversion symmetry and two-fold rotation
+symmetry.
+We further show that the observation of
+both LPGE and THz-SHG is dependent upon the
+presence of unequally populated twinned domains in
+the sample, since twinned (oppositely-oriented) domains
+produce oppositely-signed second-order responses in
+such a three-fold symmetry system, which tend to
+cancel out (See the supplementary information (SI)
+for the derivation).
+These results represent the first
+observation of intrinsic SHG in the THz regime for an
+equilibrium system, to our knowledge, and thereby open
+arXiv:2301.05271v1  [cond-mat.str-el]  12 Jan 2023
+
+2
+the investigation of material properties via THz-HG to
+the full suite of harmonic orders, both even and odd.
+The dependence of both the LPGE and THz-SHG upon
+the presence of untwinned domains further motivates
+the future development of techniques to preferentially
+control the orientation of crystal growth on millimeter
+scales, particularly for materials that break various
+symmetries.
+Results and Discussion
+Thin film samples of Bi2Se3 are grown via molecular
+beam epitaxy on c-axis Al2O3 substrates (10 mm x 10
+mm x 0.5 mm), following the two-step growth process [62,
+63] to prevent disorder at the sample-substrate interface
+and achieve atomically sharp interfaces. The samples are
+then capped in situ with 50 nm of Se to protect against
+damage and the effects of atmosphere [45, 49, 51, 53,
+64]. As each van der Waals unit of Bi2Se3 is formed of a
+quintuple layer (QL) of Bi2Se3 (1 QL ≈ 1 nm), samples
+with thicknesses 16 QL, 32 QL, 64 QL, and 100 QL are
+grown to form a thickness series.
+The samples of Bi2Se3 are evaluated for their room
+temperature LPGE response by measuring the THz emis-
+sion [60] of the samples under normal incidence, near in-
+frared (NIR) pumping at the center wavelength of 1530
+nm. When a single domain of Bi2Se3 is pumped with
+NIR, the LPGE produces a current across the domain
+[65, 66], which couples out to free space as a THz pulse.
+This emitted THz pulse is generated and detected by a
+THz emission spectrometer depicted schematically in Fig
+1.a and described in previous works [67, 68]. In brief, the
+sample is pumped over a spot size of order 1 mm by lin-
+early polarized, broadband 1530 nm, 50 fs pulses with a
+repetition rate of 1 kHz. A quasi-single cycle THz pulse is
+emitted from the sample in transmission geometry; col-
+lected, collimated, and focused onto a ZnTe crystal by
+a pair of off-axis parabolic mirrors in 4f geometry; and
+measured via electro-optic sampling [69]. By varying the
+optical path length of the NIR probe pulse via the delay
+stage, the electric field profile of the emitted THz pulse
+ET Hz is mapped out in the time domain.
+THz emission data is depicted in Fig 1.b,c for a typical
+100 QL Bi2Se3 sample. As shown in 1.b, a pronounced
+quasi-single cycle THz pulse is emitted upon NIR pump-
+ing, the polarity of which changes sign throughout the
+duration of the pulse when the sample is rotated az-
+imuthally by 180 degrees.
+By tracing out the peak
+value of ET Hz as the sample is rotated, as shown in
+Fig 1.c, the azimuthal angle dependence clearly follows
+Emax
+T Hz = E0 sin (3φ + φ0), where E0 is the peak electric
+field strength, φ is the azimuthal angle, and φ0 is an ar-
+bitrary angle difference between the crystalline axes and
+the lab frame for a given sample. See SI for derivation.
+This sin (3φ) dependence of the emitted ET Hz is pre-
+cisely the azimuthal angle dependence expected for THz
+emission from a single domain of Bi2Se3 due to LPGE
+under normal incidence [49–52].
+LPGE is only per-
+mitted in systems that break inversion symmetry [65].
+a
+b
+c
+FIG. 1. a. Schematic of the THz emission spectrometer. The
+NIR and THz beam paths are depicted in magenta and green,
+respectively, and the THz beam path is contained in a dry air-
+purged box. Both sample (S) and polarizer (P) are mounted
+in rotating stages to enable characterization of the azimuthal
+angle dependence of the THz emission. Labeled optical ele-
+ments include beam splitter (BS), pelical (Pel), ZnTe crystal
+(ZnTe), quarter wave plate (QWP), Wollaston prism (WP),
+photodiodes (PD), and delay stage (DS). b. Normalized elec-
+tric field profile of the emitted THz pulse from Se-capped 100
+QL Bi2Se3 obtained by electro-optic sampling mapped in the
+time domain. c. The peak normalized electric field as the
+sample azimuthal angle φ is rotated.
+As bulk Bi2Se3 is centrosymmetric, only the surface of
+Bi2Se3 breaks inversion symmetry, and hence, only the
+surface contributes to the LPGE. Since the surface of
+Bi2Se3 is three-fold symmetric and breaks two-fold ro-
+tation symmetry, the normal-incidence LPGE from the
+surface must also be three-fold symmetric with respect to
+the azimuthal angle. This yields a sin (3φ) dependence of
+the LPGE current, which when coupled out to free space,
+
+OAP
+Pel
+BS
+S
+ZnTe
+P
+WP
+DS
+QWP
+PD1.0-
+THz Field (norm.)
+0.5-
+180°
+0.0
+-0.5-
+-1.0-
+0
+1
+2
+3
+4
+Time Delay (ps)
+1.0
+Peak THz Field (norm.)
+0.5.
+0.0-
+-0.5-
+-1.0-
+0
+50
+100
+150
+200
+250
+300
+350
+Azimuthal Angle (degrees)3
+c
+d
+a
+b
+FIG. 2.
+a.
+Schematic of the intense TDTS system.
+The NIR and THz beam paths are depicted in magenta and green,
+respectively, and the THz beam path is contained in a dry air-purged box.
+Labeled optical elements include sample (S),
+LiNbO3 crystal (LiNbO3), THz filters (F1 and F2), diffraction grating (DG), beam splitter (BS), pelical (Pel), ZnTe crystal
+(ZnTe), quarter wave plate (QWP), Wollaston prism (WP), photodiodes (PD), and delay stage (DS). b. Harmonic generation
+spectra for Se-capped Bi2Se3 samples under a 0.5 THz fundamental pump with respect to a reference substrate. The change
+between spectra taken with 1.0 THz-specific and 1.5 THz-specific filters are indicated by breaks in the spectra.
+c.
+Peak
+spectral weight at the 2nd and 3rd harmonic as a function of the peak 0.5 THz pump field Epump, with fits to E2
+pump and
+E3
+pump respectively. d. Peak spectral weight at the 2nd and 3rd harmonics as function of sample thickness.
+results in the Emax
+T Hz = E0 sin (3φ) dependence of the THz
+emission observed here. However, since the spot size of
+the NIR pump (order 1 mm) vastly exceeds the domain
+size of Bi2Se3 (order 1 µm; see Fig 3.c,d), the THz emis-
+sion method measures the net LPGE produced by a large
+ensemble of Bi2Se3 domains. Since twinned domains in
+the sample produce oppositely-signed LPGE responses,
+as demonstrated in Fig 1.b, and hence cancel each other
+out, the observation of a clear LPGE signal from the sam-
+ple therefore indicates the presence of a dominant domain
+orientation over millimeter length scales.
+As SHG is limited by the same symmetry considera-
+tions as LPGE and expected to be generated from the
+surface of Bi2Se3 [49–52], the THz-HG of the samples
+is measured via intense TDTS [61] at room temperature
+as shown schematically in Fig 2.a. Intense broadband,
+quasi-single cycle THz pulses are generated from LiNbO3
+via the tilted pulse front method [70–72] by pumping
+with linearly polarized, broadband 800 nm, 35 fs pulses
+with a repetition rate of 1 kHz. The generated intense
+THz pulses are collected, directed through the sample at
+a waist of order 1 mm, and focused onto a ZnTe crys-
+tal by a quartet of OAPs in 8f geometry. Prior to the
+sample, optical filters (F1) convert the broadband pulse
+into a narrow-band few cycle pulse centered at 0.5 THz
+(spectral width ∼ 20%). After transmitting through the
+sample, the resulting THz pulse is passed through op-
+tical filters (F2) to suppress the spectral weight of the
+0.5 THz fundamental pulse and pass the frequency range
+around the harmonic to be observed: 1.0 THz for SHG
+or 1.5 THz for THG. The remaining THz that impinges
+upon the ZnTe crystal is measured by standard electro-
+optic sampling [69], allowing the electric field profile to
+be mapped out in the time domain by varying the de-
+lay stage of the probe pulse. Finally, taking the Fourier
+transform of the THz pulse in the time domain yields the
+spectral weight of the pulse as a function of frequency.
+The HG spectra for the Bi2Se3 samples shown in Fig
+
+Substr.
+Spectral Weight (norm.)
+Fund.
+16 QL
+0.1
+32 QL
+64 QL
+100 QL
+0.01
+0.001
+11
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+Frequency (THz)
+norm.,
+亚
+norm.
+20
+20
+2nd Harmonic
+TO
+3rd Harmonic
+Harmonic Peak (x10~
+15
+Harmonic Peak (x10~
+16
+2nd Harmonic
+10
+0
+3rd Harmonic
+12
+5
+8-
+0
+20
+25
+30
+35
+40
+45
+0
+20
+40
+60
+80
+100
+Pump Field (kV/cm)
+Sample Thickness (QL)LiNbO.
+BS
+DG
+F1
+OAP
+S
+F2
+OAP
+Pel
+DS
+PD
+ZnTe
+Wp
+QWP4
+a
+b
+d
+c
+FIG. 3. a,b. Comparison of THz-SHG and LPGE, respectively, for two bare 100 QL Bi2Se3 samples. The azimuthal angle in
+(b) is offset for clarity. c,d. Atomic force microscopy images of bare 100 QL Bi2Se3 for Sample 1 and Sample 2, respectively,
+where oppositely-oriented domains on the surface are highlighted with blue and red boxes.
+2.b exhibit clear THz-SHG at 1.0 THz and THz-THG
+at 1.5 THz when pumping with the 0.5 THz funda-
+mental. Three key features of the THz-THG response
+demonstrate strong agreement with the previous THz-
+HG studies [40, 44, 45] of bismuth chalcogenides: First,
+the THG conversion efficiency is ∼ 0.04% (accounting for
+the THG-specific filters), which closely matches the con-
+version efficiency in previous reports. Second, the yield of
+the THz-THG scales perturbatively as E3
+pump, as shown
+in Fig 2.c which is likewise in agreement with previous
+results and contrasting sharply with the saturation of
+harmonic yield observed in graphene [37–40] and Cd3As2
+[41, 42] at similar THz pumping field strengths. Third,
+the THz-THG yield is nearly thickness-independent, as
+shown in Fig 2.d, which is consistent with the conclu-
+sion that the dominant contribution to the THz-THG is
+the response of the topological surface state. Together,
+these features of the THz-THG reaffirm the results of the
+previous studies and demonstrate that the intrinsic non-
+linear properties of the Bi2Se3 samples measured here are
+consistent with those of the previous studies.
+Returning to Fig 2.b, however, a clear THz-SHG peak
+is observed at 1.0 THz, in addition to the THz-THG
+peak at 1.5 THz. As shown in Fig 2.c, the 1.0 THz peak
+scales according to the E2
+pump expectation for a pertur-
+bative second-order response. And since only the surface
+of Bi2Se3 breaks inversion symmetry and two-fold rota-
+tion symmetry as required for a second order process,
+the 1.0 THz peak is found to be thickness independent,
+as dictated by the symmetry and shown in Fig 2.d. This
+clear THz-SHG response from Bi2Se3, which reaches a
+high conversion efficiency of ∼ 0.005% (accounting for
+the SHG-specific filters), is consistent with HG studies
+outside of the THz regime [49–55], but contrasts sharply
+with the previous THz studies [40, 44, 45] of bismuth
+chalcogenides, which failed to report THz-SHG.
+We turn then to the question of why THz-SHG is
+observed here but not in previous studies. Since both
+LPGE and SHG are second-order processes that require
+the breaking of inversion symmetry, which only occurs
+at the Bi2Se3 surface, both processes are governed by
+the same crystal properties of the sample. Hence, both
+processes are expected to be observed in single crystals
+of Bi2Se3, but may be diminished by the presence of
+twinned domains when probing an ensemble of domains,
+as is the case for the relatively large spot sizes employed
+both here and in the previous THz-HG studies [40, 44, 45]
+of bismuth chalcogenides. Thus it may be possible that
+twinned domains suppressed the THz-SHG below the ob-
+servable level of the previous studies.
+This possibility is confirmed by comparing samples of
+Bi2Se3 that have been grown without the 50 nm Se cap-
+ping layer. Fig 3 compares the results for two 100 QL
+bare Bi2Se3 samples taken from the same batch to en-
+sure similar growth quality and similar exposure to at-
+mosphere [45, 49, 51, 53, 64].
+The two samples show
+a clear difference in both THz-SHG and LPGE, shown
+in Fig 3.a,b, respectively, where Sample 1 shows a con-
+sistently smaller second-order response than Sample 2.
+Since both samples are not capped, the orientation of
+surface domains can be determined by atomic force mi-
+croscopy (AFM). As shown in Fig 3.c,d, respectively,
+AFM clearly reveals twinned domains on the surface of
+both Sample 1 and Sample 2. A careful counting of these
+domains shows that the ratio of oppositely-oriented do-
+mains is ∼ 1.5 : 1 in Sample 1 and ∼ 1.8 : 1 in Sample 2.
+Since Sample 1 has a lesser degree of untwinned domains
+than Sample 2, it should produce a lesser degree of THz-
+SHG and LPGE, precisely as observed in these measure-
+ments. Since the ordinary growth of Bi2Se3 tends to pro-
+duce samples with twinned domains that suppresses both
+LPGE and THz-SHG, as shown here, a sufficiently high
+degree of twinned domains could suppress both effects
+below the noise level of current measurement techniques.
+
+0
+Spectral Weight (x10*
+8 
+Sample 1
+Sample 2
+6
+2
+0
+0.8
+0.9
+1.0
+1.1
+1.2
+Frequency (THz)
+Peak THz Field (norm.)
+Sample 1
+Sample 2
+1.0.
+0.5
+0.0
+-0.5
+-1.0
+0
+100
+200
+300
+Azimuthal Angle (degrees)6
+8-
+im
+5
+nm
+3
+4
+0
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+μm40
+9
+3
+8
+30
+20
+L
+15
+10
+4.92
+6.01
+2
+4
+5
+6
+8
+6
+10
+μm5
+This problem of twinned domains therefore presents one
+potential reason why that the previous studies [40, 44, 45]
+of bismuth chalcogenides failed to report THz-SHG, and
+it highlights the importance of improving control over
+crystal growth to enable more reliable experimental re-
+sults, particularly for materials that break various sym-
+metries.
+To summarize, we have observed THz-SHG from
+Bi2Se3 thin films that exhibit LPGE as measured
+via intense TDTS and THz emission,
+respectively.
+Moreover, the THz-SHG may be attributed to the
+topological surface state of the Bi2Se3 and features
+a highly efficient conversion rate of ∼ 0.005% that is
+independent of the film thickness. These results extend
+beyond previous studies [40, 44, 45] of similar topological
+insulator bismuth chalcogenides, which reported only
+odd-order harmonics, and furthermore represent the
+first demonstration of intrinsic SHG–or indeed any
+even-order HG–in the THz regime for an equilibrium
+system.
+This advance enables and motivates further
+development of HG techniques for the characterization
+of material properties and the development of useful
+devices in the THz regime.
+In particular, THz-HG
+employing circularly and elliptically polarized light
+remains in its infancy [43], despite the discovery of
+highly nonlinear dependencies [55, 73–76] in high har-
+monic generation [77, 78] studies employing mid-infrared
+fundamentals, and despite the recent demonstration of
+elliptically polarized harmonics as an effective probe of
+topological properties [55, 79, 80]. This highlights the
+need to develop higher performance and more widely
+available THz optical elements, especially waveplates
+[81, 82], which have been historically limited due to the
+broadband nature of THz techniques. Furthermore, the
+connection between untwinned domains and THz-SHG
+in Bi2Se3, a member of the broader bismuth chalcogenide
+family that serves as standard topological insulators in
+myriad studies, highlights the need to develop growth
+methods that reliably produce untwinned domains over
+millimeter scales, especially if the preferential growth
+orientation can be controlled. Altogether, these results
+vastly expand the possible range of future studies by
+unlocking even-order HG in the THz regime, open a new
+pathway to the low-energy study of topological surface
+states, and motivate further efforts to develop efficient
+THz optical elements and material growth techniques
+that yield untwinned domains.
+Acknowledgement
+We thank J. Lu for helpful discussions. This project
+was sponsored by the Army Research Office and was
+accomplished under the grants no. W911NF-20-2-0166
+and W911NF-19-1-0342. J.S. was also supported by the
+NSF EAGER grant via the CMMT programme (DMR-
+2132591) and the Gordon and Betty Moore Foundation’s
+EPiQS Initiative under the grant GBMF9212 to L.W..
+X.H. is supported by the NSF EPM program under grant
+no. DMR-2213891. Z.N. acknowledges support from the
+Vagelos Institute of Energy Science and Technology grad-
+uate fellowship and the Dissertation Completion Fellow-
+ship at the University of Pennsylvania.
+The work at
+Rutgers by X. Yao, X. Yuan, D. M., A. G. and S. O.
+was also supported by NSF DMR2004125, and the cen-
+ter for Quantum Materials Synthesis (cQMS), funded by
+the Gordon and Betty Moore Foundation’s EPiQS initia-
+tive through grant GBMF10104.
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+

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+arXiv:2301.02834v1  [quant-ph]  7 Jan 2023
+n-photon blockade with an n-photon parametric drive
+Yan-Hui Zhou1, Fabrizio Minganti2, Wei Qin2, Qi-Cheng Wu1, Junlong
+Zhao1, Yu-Liang Fang1, Franco Nori2,3∗, and Chui-Ping Yang1,4†
+1 Quantum Information Research Center, Shangrao Normal University, Shangrao 334001, China
+2 Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
+3 Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA
+4 Department of Physics, Hangzhou Normal University, Hangzhou 311121, China
+(Dated: January 10, 2023)
+We propose a mechanism to engineer an n-photon blockade in a nonlinear cavity with an n-photon
+parametric drive λ(ˆa†n + ˆan). When an n-photon-excitation resonance condition is satisfied, the
+presence of n photons in the cavity suppresses the absorption of the subsequent photons.
+To
+confirm the validity of this proposal, we study the n-photon blockade in an atom-cavity system,
+a Kerr-nonlinear resonator, and two-coupled Kerr nonlinear resonators. Our results demonstrate
+that n-photon bunching and (n + 1)-photon antibunching can be simultaneously obtained in these
+systems. This effect is due both to the anharmonic energy ladder and to the nature of the n-photon
+drive. To show the importance of the drive, we compare the results of the n-photon drive with a
+coherent (one-photon) drive, proving the enhancement of antibunching in the parametric-drive case.
+This proposal is general and can be applied to realize the n-photon blockade in other nonlinear
+systems.
+PACS numbers: 42.50.Ar, 42.50.Pq
+I.
+INTRODUCTION
+In a nonlinear cavity driven by a classical light
+field, the single-photon existence in the cavity blocks
+the
+creation
+of
+a
+second
+photon
+[1–3],
+which
+is
+known as the single-photon blockade (1PB). Due to its
+potential applications in information and communication
+technology, 1PB has been extensively studied in the
+past years [4–13].
+For example, the PB has been
+predicted in cavity quantum electrodynamics [14–16],
+quantum optomechanical system [17–20], and second
+order nonlinear system [21–25].
+Traditionally,
+realizing
+1PB
+requires
+a
+large
+nonlinearity
+to
+change
+the
+energy-level
+structure
+of the system, and 1PB can be used to create a
+single-photon source [26].
+The 1PB effect was first
+observed in an optical cavity coupled to a single trapped
+atom [27].
+Since then, many experimental groups
+have observed this strong antibunching behavior in
+different systems, including a photonic crystal [28] and
+a superconducting circuit [29]. In addition, the 1PB can
+also enable by another mechanism, i.e., the quantum
+interference [30–35], which has been recently observed
+experimentally [36, 37].
+In this paper, we are only
+concerned with the photon blockade based on energy
+level splitting due to the large nonlinearity.
+The n-photon blockade (nPB) was proposed with the
+development of 1PB. In analogy to 1PB, nPB (n ≥ 2) is
+defined by the existence of n photons in a nonlinear cavity
+∗Corresponding address: [email protected]
+†Corresponding address: [email protected]
+suppressing the creation of subsequent photons. The 2PB
+(nPB with n = 2) was studied in a Kerr-type system
+driven by a laser [38], in a strong-coupling qubit-cavity
+system [39], and in a cascaded cavity QED system [40].
+The 2PB can also be generated by squeezing [41].
+Experimentally, 2PB was realized in an optical cavity
+strongly coupled to a single atom [42], where driving the
+atom gives a larger optical nonlinearity than driving the
+cavity. nPB with n > 2 has been studied in a cavity
+strongly coupled to two atoms [43], in a cavity with
+two cascade three-level atoms [44], and in a Kerr-type
+system driven by a laser [45, 46]. Meanwhile, in analogy
+to photon blockades, the phonon blockades have been
+widely studied [47–51].
+In this paper, we theoretically propose that nPB
+can be triggered in a nonlinear cavity with n-photon
+parametric drive. For convenience, we denote “n-photon
+parametric
+drive”
+as
+nPD.
+We
+first
+give
+a
+brief
+introduction to this proposal and then confirm its validity
+by considering three examples, i.e., an atom-cavity
+system, a single mode Kerr-nonlinearity system, and
+a two-coupled-cavities Kerr-nonlinearity system.
+This
+proposal is quite general and can be extended to other
+nonlinear systems for studying nPB via nPD. The study
+of the nPB in recent decades has mainly focused on a
+coherent (i.e., single-photon) driving. Comparing with a
+proposal using a coherent driving, the use of a nPD has
+the following advantages: (i) The nonlinear systems like
+atom-cavity system will not exist nPB with a coherent
+driving to the cavity due to the bosonic enhancement
+of photon [42], while we find that the nPB will exist in
+these system with a nPD, so the proposal with the nPD is
+more general to realize a nPB. (ii) In the same nonlinear
+system, the nPD approach has a stronger (n+ 1)-photon
+
+2
+bunching than the coherent driving approach, so the nPD
+approach has a better nPB effect.
+The remainder of this paper is organized as follows. In
+Sec. II, we introduce the Proposal for nPB with nPD.
+In Sec. III, we illustrate the nPB in an atom-cavity
+system.
+In Sec. IV, we show the nPB in single-mode
+Kerr-nonlinearity system.
+In Sec. V, we study the
+2PB in two-coupled-cavities Kerr-nonlinearity system.
+Conclusion are given in Sec. VI.
+II.
+PROPOSAL FOR nPB WITH nPD
+The nPD with n = 2 has many applications, such as in
+the realization of quantum metrology [52] and cooling of
+a micromechanical mirror [53]. In the following, we will
+present our basic idea for studing the nPB via nPD on a
+nonlinear cavity.
+nPD involved in our proposal is described by ˆHd =
+λ(ˆa†ne−iωpt + ˆaneiωpt), where ˆa is the cavity annihilation
+operator, λ is the parametric driving amplitude, and
+ωp is the driving frequency. Apart from the cavity on
+which nPD is applied, an auxiliary nonlinear system (e.g.,
+an atom, a Kerr-nonlinearity medium, or an auxiliary
+cavity) is required to realize nPB. The Hamiltonian of
+the auxiliary nonlinear system and the cavity is denoted
+by ˆH0. The form of ˆH0 is not unique, and it depends
+on the type of the nonlinear system. Generally speaking,
+the Hamiltonian ˆH0 can be diagonalized and expressed
+as
+ˆH0 =
+k1
+�
+j=1
+ωj
+1|ψj
+1⟩⟨ψj
+1| +
+k2
+�
+j=1
+ωj
+2|ψj
+2⟩⟨ψj
+2| +
+· · · +
+kn
+�
+j=1
+ωj
+n|ψj
+n⟩⟨ψj
+n| + · · · ,
+(1)
+where
+ωj
+n
+is
+the
+jth
+eigenfrequency
+of
+ˆH0
+for
+the
+photon
+excitation
+number
+n,
+and
+we
+have
+assumed that the ground state energy is zero.
+The
+corresponding eigenstate |ψj
+n⟩ is constructed by the
+kn
+basis for n-photon excitation,
+where the basis
+forms a closed space.
+The set of eigenfrequencies
+{ωj
+1}, {ωj
+2} · · · , {ωj
+n}, · · ·
+are
+anharmonic
+due
+to
+the
+strong
+nonlinear
+interaction.
+Among
+these
+eigenfrequencies, {ωj
+n} (where j is from 1 to kn) is
+crucial to nPB because the corresponding eigenstate
+{|ψ⟩j
+n} includes a n-photon state. When the parametric
+drive frequency ωp is tuned to the {ωj
+n}, the parametric
+drive resonantly excites n photons in the cavity.
+As
+a result, the system occupies the state {|ψ⟩j
+n} via the
+nonlinear interaction.
+This gives rise to an important
+result for nPB. The corresponding conditions for nPB
+are
+ωp = ω1
+n,
+ωp = ω2
+n,
+· · ·
+ωp = ωkn
+n ,
+(2)
+The n-photon resonance excitation by nPD ensures that
+the n-photon blockade is triggered in the nonlinear cavity.
+To verify the validity of the above proposal, we will
+study three examples to study nPB, in: an atom-cavity
+system, a single-mode Kerr-nonlinearity system, and a
+two-coupled-cavities Kerr-nonlinearity system. In these
+systems, the analytical conditions for nPB and the
+accurate numerical results are studied, which conform
+that nPB can be triggered in a nonlinear cavity with
+nPD if the Hamiltonian ˆH0 can be diagonalized.
+The
+numerical
+confirmation
+of
+nPB
+adopts
+an
+equal-time
+correlation
+function,
+the
+equal-time
+n-order correlation function is defined as g(n)(0)
+=
+⟨ˆa†nˆan⟩/⟨ˆa†ˆa⟩n.
+The correlation function is calculated
+by numerically solving the master equation in the
+steady state.
+In order to prove nPB, it is sufficient
+to fulfill the conditions g(n)(0) ≥ 0 and g(n+1)(0) < 0
+simultaneously [42].
+III.
+ATOM-CAVITY SYSTEM
+The
+atom-cavity
+system
+is
+described
+by
+the
+Jaynes-Cummings Hamiltonian,
+where the cavity is
+driven by a nPD. In a frame rotating at the parametric
+drive frequency ωp/n, the Hamiltonian is (assuming
+ℏ = 1 hereafter)
+ˆH = ∆aˆa†ˆa + ∆eˆσ+ˆσ− + g(ˆa†ˆσ− + ˆσ+ˆa) + λ(ˆa†n + ˆan),(3)
+where ˆa is the cavity annihilation operator, ˆσ± are the
+atom raising and lowering operators, g is the coupling
+strength of the atom and the cavity mode, λ is the
+amplitude of nPD, and ∆a = ωa−ωp/n (∆e = ωe−ωp/n)
+is the detuning between the cavity frequency ωa (the
+atom frequency ωe) and the 1/n driving frequency. Here
+and below, we study the case of ωa = ωe for convenience,
+resulting in ∆a = ∆e. The Hamiltonian (3) with n = 2
+can be used to exponentially enhance the light-matter
+coupling in a generic cavity QED [54–56].
+In
+the
+absence
+of
+the
+nPD,
+the
+atom-cavity
+Hamiltonian ˆH0 (the first three terms of Eq. (3) without
+driving) is diagonalized as
+ˆH0 =
+2
+�
+j=1
+ωj
+1|ψj
+1⟩⟨ψj
+1| +
+2
+�
+j=1
+ωj
+2|ψj
+2⟩⟨ψj
+2| +
+· · · +
+2
+�
+j=1
+ωj
+n|ψj
+n⟩⟨ψj
+n| + · · · .
+(4)
+The energy eigenstates of the system are |ψ1,2
+n ⟩
+=
+1/
+√
+2(|n − 1, e⟩ ∓ |n, g⟩),
+where
+|g⟩
+(|e⟩)
+is
+the
+ground (excited) state of the atom, n denotes the
+photon excitation number.
+For a n-photon excitation,
+the basis {|n, g⟩, |n − 1, e⟩} forms a closed space.
+The corresponding eigenfrequencies with the n-photon
+excitation are ω1,2
+n
+= nωa ∓ √ng.
+The energy-level
+diagram of the system is shown in Fig. 1(a). The optimal
+conditions for nPB are calculated according to Eq. (2),
+
+3
+-20
+-10
+0
+10
+20
+Detuning
+0
+5
+10
+g(3)(0)
+g(4)(0)
+-15
+-10
+-5
+0
+5
+10
+15
+Detuning
+0
+2
+4
+6
+g(4)(0)
+g(5)(0)
+(a)
+a
+�
+a
+�
+p
+�
+p
+�
+(b)
+a
+�
+g
+3
+2
+g
+2
+2
+g
+2
+g
+0
+1
+1
+�
+2
+1
+�
+1
+2
+�
+2
+2
+�
+1
+3
+�
+2
+3
+�
+(c)
+�
+/
+�
+�
+/
+�
+FIG. 1: (Color online) (a) Schematic energy-level diagram
+explaining the occurrence of 3PB. (b) The logarithmic plot
+(of base e) of three-order correlation function g(3)(0) and
+fourth-order correlation function g(4)(0) as a function of
+detuning ∆/κ, for g/κ = 10
+√
+3, γ/κ = 0.1, and λ/κ = 0.3.
+(c) g(4)(0) and g(5)(0) as a function of ∆/κ, for g/κ = 10,
+γ/κ = 0.1, and λ/κ = 1.5.
+which are simplified as
+g = ±√n∆,
+(5)
+where ∆ = ∆a = ∆e. There is one path for the system
+to reach the state |ψ1,2
+n ⟩: the system first arrives at a
+n-photon state by nPD, then goes to the state of |ψ1,2
+n ⟩
+via the coupling g, i.e., |0g⟩
+λ
+−→ |ng⟩
+g
+−→ |ψ1,2
+n ⟩, the
+nPD and the n-photon resonance excitation make that
+the nPB is triggered.
+Next, we numerically study the nPB effect.
+The
+system dynamics is governed by the master equation
+∂ˆρ/∂t = −i[ ˆH, ˆρ]+κℓ(ˆa)ρ+γℓ( ˆ
+σ−)ρ, where κ denotes the
+decay rate of the cavity and γ is the atomic spontaneous
+emission rate. The superoperators are defined by ℓ(ˆo)ˆρ =
+ˆoˆρˆo† − 1
+2 ˆo†ˆoˆρ − 1
+2 ˆρˆo†ˆo.
+The numerical solutions of
+g(n)(0) and g(n+1)(0) are calculated by solving the master
+equation in the steady state.
+In Fig. 1(b), we study
+a 3PB by plotting g(3)(0) and g(4)(0) versus ∆/κ with
+g/κ = 10
+√
+3.
+We note that the 3PB appears on
+∆/κ = ±10 (g(3)(0) ≥ 0 and g(4)(0) < 0 simultaneously),
+which agrees well with the conditions for nPB in Eq. (5)
+with n = 3.
+The 4PB is studied in Fig. 1(c) with
+g/κ = 10, and 4PB appears on ∆/κ = ±5, which also
+agrees with Eq. (5) with n = 4. The numerical results
+confirm the analytic conditions and the corresponding
+analysis. In the above atom-cavity system, it was proved
+that the nPB will not exist with a coherent driving
+(driving the cavity) due to a consequence of the bosonic
+enhancement of photon [42], while the nPB will exist for
+this system with a nPD. So the proposal with the nPD
+is more general and the nPB will occur as long as the
+(a)
+0
+a
+�
+a
+�
+p
+�
+U
+2
+�
+/
+�
+(b)
+(c)
+a
+�
+U
+6
+�
+/
+�
+-40
+-30
+-20
+-10
+0
+Detuning
+0
+5
+10
+15
+20
+g(3)(0)
+g(4)(0)
+-40
+-30
+-20
+-10
+0
+Detuning
+0
+10
+20
+g(4)(0)
+g(5)(0)
+(b)
+(c)
+1
+1
+�
+1
+2
+�
+1
+3
+�
+0
+0.2
+0.4
+Driving λ/ κ
+-4
+-2
+0
+2
+4
+6
+8
+g(3)(0)
+g(4)(0)
+2
+3
+4
+5
+Driving F/ κ
+-2
+0
+2
+4
+6
+8
+g(3)(0)
+g(4)(0)
+(e)
+(d)
+FIG. 2: (Color online) (a) Energy spectrum of the single mode
+Kerr-nonlinearity system leading to 3PB via 3PD. (b) The
+logarithmic plot of g(3)(0) and g(4)(0) as a function of ∆/κ.
+(c) The logarithmic plot of g(4)(0) and g(5)(0) as a function of
+∆/κ. In (b, c), the parameters are U/κ = 10 and λ/κ = 0.1.
+(d) and (e) The logarithmic plot of g(3)(0) and g(4)(0) as a
+function of λ/κ (F/κ) for U/κ = 10 and ∆/κ = −20.
+analytical eigenvalues of the nonlinear Hamiltonian {ωj
+n}
+is solvable.
+IV.
+SINGLE-MODE KERR-NONLINEARITY
+SYSTEM
+The system of a single-mode cavity with a Kerr
+nonlinearity, driven by nPD with n = 2, has been
+extensively studied due to its rich physics [57–61].
+Here we investigate nPB utilizing this system.
+The
+Hamiltonian of this model in a rotating frame is written
+as [58]
+ˆH = ∆ˆa†ˆa + Uˆa†ˆa†ˆaˆa + λ(ˆa†n + ˆan),
+(6)
+where ∆a = ωa−ωp/n is the cavity detuning from the 1/n
+driving eigenfrequency, U is the Kerr nonlinear strength,
+and λ is the amplitude of the nPD.
+The
+undriven
+part
+of
+the
+Hamiltonian
+(6)
+is
+
+4
+diagonalized as
+ˆH0 = ω1
+1|ψ1
+1⟩⟨ψ1
+1| + ω1
+2|ψ1
+2⟩⟨ψ1
+2| + · · ·
++ω1
+n|ψ1
+n⟩⟨ψ1
+n| + · · · ,
+(7)
+where the eigenstate is written as the Fock-state basis
+|ψ1
+n⟩
+=
+|n⟩ (with n photons in the cavity),
+the
+corresponding eigenfrequency is ω1
+n = ωan + U(n2 − n).
+The nPB can be triggered by the n-photon-excitation
+resonance, and the |0⟩ → |n⟩ transition is enhanced. The
+condition for nPB is obtained according to Eq. (2), which
+is given by
+U = −
+∆
+n − 1.
+(8)
+Because
+of
+the
+nPD
+and
+the
+n-photon-excitation
+resonance, the n photon probability will increase when
+the condition (8) is satisfied, and the nPB is triggered.
+The master equation for the system is given by
+∂ˆρ/∂t = −i[ ˆH, ˆρ] + κℓ(ˆa)ρ.
+The energy-level diagram
+for 3PB is shown in Fig. 2(a), and the corresponding
+numerical simulation is shown in Fig. 2(b), where we plot
+g(3)(0) and g(4)(0) as a function of ∆/κ with g/κ = 10.
+These results show that 3PB can be obtained at ∆/κ =
+−20, as predicted in Eq. (8) for n = 3.
+The 4PB is
+studied in Fig. 2(c) and the 4PB appears on ∆/κ = −30,
+which also agrees with Eq. (8) with n = 4.
+We note that the studies to date on the nPB are mainly
+focused on a coherent driving F(ˆa† + ˆa), where F is the
+coherent driving strength. So we compare the 3PB based
+on the 3PD with that based on the coherent driving.
+To this end, we plot g(3)(0) and g(4)(0) versus the 3PD
+strength and coherent driving strength in Fig. 2(d, e)
+under the blockade condition of Eq. (8) (g/κ = 10,
+∆/κ = −20), respectively.
+The 3PB with the 3PD is
+obtained in a region of small λ, while the implementation
+of 3PB with coherent driving needs a larger F.
+The
+most striking feature is that the 3PB with the 3PD has
+a stronger four-photon antibunching and three-photon
+bunching.
+V.
+TWO-COUPLED-CAVITIES
+KERR-NONLINEARITY SYSTEM
+Two coupled cavities with Kerr nonlinearity were
+considered to study 1PB [62]. We define the two cavities
+as cavities a and b. The Hamiltonian in a rotating frame
+is
+ˆH = ∆ˆa†ˆa + ∆ˆb†ˆb + J(ˆa†ˆb + ˆb†ˆa) + U(ˆa†ˆa†ˆaˆa + ˆb†ˆb†ˆbˆb)
++λ(ˆa†n + ˆan),
+(9)
+where ˆa (ˆb) is the photon annihilation operator for cavity
+a (b) with frequency ωa (ωb), ∆ = ωa−ωp/n = ωb−ωp/n,
+J is the coupling strength of the two cavities, U is the
+Kerr nonlinear strength, and λ is the nPD strength.
+(a)
+00
+1
+1
+�
+2
+1
+�
+1
+2
+�
+2
+2
+�
+3
+2
+�
+U
+U
+2
+2
+2
+4
+U
+J �
+a
+�
+a
+�
+J
+2
+p
+�
+p
+�
+-15 -12.1-10
+-5
+0 2.07
+5
+Detuning
+-2
+0
+2
+4
+6 (b)
+g(2)(0)
+g(3)(0)
+-15 -12.1-10
+-5
+0 2.07
+5
+Detuning
+-5
+0
+5 (c)
+g(2)(0)
+g(3)(0)
+�
+/
+�
+�
+/
+�
+FIG. 3:
+(a) Energy spectrum for two coupled cavities with
+Kerr nonlinearity.
+(b, c) The logarithmic plot (of base e)
+of g(2)(0) and g(3)(0) as a function of ∆/κ for cavity a and
+cavity b, respectively. (b) Cavity a. (c) Cavity b. In (b, c),
+the parameters are U/κ = 10, J/κ = 5, and λ/κ = 0.5.
+The Hamiltonian for the two cavities with the Kerr
+nonlinearity (the first four terms in Eq. (9) without
+driving) is diagonalized as
+ˆH0 =
+2
+�
+j=1
+ωj
+1|ψj
+1⟩⟨ψj
+1| +
+3
+�
+j=1
+ωj
+2|ψj
+2⟩⟨ψj
+2| +
+· · · +
+n+1
+�
+j=1
+ωj
+n|ψj
+n⟩⟨ψj
+n| + · · · .
+(10)
+We find that our approach comes with its own limitations
+in this system. The eigenfrequencies {ωj
+n} are more and
+more difficult to analytically solve with the increase of
+n, so we only study the case of n = 2, the corresponding
+energy-level diagram is shown in Fig. 3(a). Now we derive
+the eigenfrequencies {ωj
+2} and the eigenstates {|ψj
+2⟩}. To
+obtain these, the Hamiltonian will be expanded with the
+two-cavity states |20⟩, |02⟩ and |11⟩ for the two-photon
+excitation, where |αβ⟩ is the Fock-state basis of the
+system with the number α (β) denoting the photon
+number in cavity a (b). The two-cavity states satisfy the
+two-photon excitation condition α+β = 2, and the states
+|20⟩, |02⟩ and |11⟩ form a closed space. Under these basis
+states, the Hamiltonian with two-photon excitation can
+be described as
+ˆH2 =
+
+
+2ωa + 2U
+√
+2J
+0
+√
+2J
+2ωa
+√
+2J
+0
+√
+2J 2ωa + 2U
+
+ .
+(11)
+The three eigenfrequencies are ω2
+2 = 2(U + ωa), and
+ω1,3
+2
+= 2ωa + U ∓
+√
+4J2 + U 2.
+The corresponding
+unnormalized eigenstates are |ψ2
+2⟩ = −|20⟩ + |02⟩, and
+|ψ1,3
+2 ⟩ = |20⟩ − [
+√
+2U ∓
+�
+2(4J2 + U 2)]/(2J)|11⟩ + |02⟩.
+The conditions for 2PB, obtained from Eq. (2), are given
+
+5
+0
+0.5
+1
+Driving λ/κ
+-4
+-2
+0
+2
+4
+6
+(a)
+g(2)(0)
+g(3)(0)
+0
+2
+4
+Driving F/κ
+-2
+0
+2
+4
+6
+8
+(a')
+g(2)(0)
+g(3)(0)
+0
+1
+2
+Driving λ/κ
+-4
+-2
+0
+2
+4
+6
+(b)
+g(2)(0)
+g(3)(0)
+0
+2
+4
+Driving F/κ
+-2
+0
+2
+4 (b')
+g(2)(0)
+g(3)(0)
+0
+0.2
+0.4
+0.6
+Driving λ/κ
+-5
+0
+5
+(c)
+g(2)(0)
+g(3)(0)
+0
+0.2
+0.4
+0.6
+Driving F/κ
+-4
+-2
+0
+2
+(c')
+g(2)(0)
+g(3)(0)
+FIG. 4:
+The logarithmic plot of g(2)(0) and g(3)(0) of cavity
+b as a function of λ/κ (F/κ) for U/κ = 10 and J/κ = 5. (a,
+a’) ∆/κ = −12.5. (b, b’) ∆/κ = −10. (c, c’) ∆/κ = 2.07.
+by
+∆ = −U,
+∆ = −U ±
+√
+4J2 + U 2
+2
+.
+(12)
+Under these resonance conditions, 2PB can be triggered,
+which enhances the transition |00⟩ → {|ψ2
+2⟩, |ψ1,3
+2 ⟩}. The
+two cavities occupy the two-photon states |20⟩ and |02⟩,
+which ensures that 2PB is simultaneously realized in the
+two cavities when the conditions (12) are satisfied.
+The numerical study of 2PB is the same as before. In
+Fig. 3(b, c), we plot g(2)(0) and g(3)(0) as a function of
+∆/κ for cavity a and cavity b, respectively. The results
+indicate that 2PB occurs for ∆/κ = −12.7, ∆/κ = −10
+and ∆/κ = 2.07, which are predicted by the three nPB
+conditions given in Eq. (12) with n = 2. The anharmonic
+distribution of the blockade points are determined by the
+anharmonic splitting of the energy levels ω1
+2, ω2
+2, and ω3
+2.
+The distance of the two blockade points on the left is
+calculated as d =
+√
+4J2 + U 2 − U, and the distance of
+the two points on the right is d =
+√
+4J2 + U 2 +U. Thus,
+it can be concluded that 2PB is simultaneously realized
+in both cavity a and cavity b due to the feature of the
+system and the NPD.
+The undriven cavity b has a better 2PB effect than
+cavity a for a smaller g(3)(0) shown in Fig. 3(b, c), so
+we compare the 2PD approach with the coherent driving
+approach for cavity b. The results are presented in Fig. 4,
+where we plot of g(2)(0) and g(3)(0) as a function of λ/κ
+(F/κ) under the three blockade conditions, respectively.
+We find that the two approaches have different blockade
+regions.
+And the same conclusion is arrived as the
+single-mode Kerr-nonlinearity system that the 2PB with
+the 2PD has a stronger three-photon antibunching and
+two-photon bunching.
+VI.
+CONCLUSION
+We have proposed that n-photon blockade can be
+realized in a nonlinear cavity with a n-photon parametric
+drive. The validity of this proposal is confirmed by three
+examples, i.e., n-photon blockade in an atom-cavity
+system, in a single-mode Kerr nonlinear device, and
+in a two-coupled-cavities Kerr-nonlinear system.
+By
+solving the master equation in the steady-state limit
+and computing the correlation functions g(n)(0) and
+g(n+1)(0), we have shown that nPB can be realized,
+and the
+optimal conditions for nPB are in good
+agreement with the numerical simulations, which clearly
+illustrates the validity of our proposal.
+This proposal
+can be extended to other nonlinear systems, as long as
+the n-photon-excitation analytical eigenvalues of the
+nonlinear Hamiltonian is solvable.
+This
+work
+is
+supported
+by
+the
+Key
+R&D
+Program
+of
+Guangdong
+province
+(Grant
+No.
+2018B0303326001),
+the
+NKRDP
+of
+china
+(Grants
+Number
+2016YFA0301802),
+the
+National
+Natural
+Science Foundation of China (NSFC) under Grants
+No.
+11965017,
+11705025,11804228,
+11774076,
+the
+Jiangxi Natural Science Foundation under Grant No.
+20192ACBL20051, the Jiangxi Education Department
+Fund under Grant No.
+GJJ180873.
+This work is
+also supported by the NTT Research, Army Research
+Office (ARO) (Grant No.
+W911NF-18-1-0358), the
+Japan Science and Technology Agency (JST) (via the
+CREST Grant No.
+JPMJCR1676), the Japan Society
+for the Promotion of Science (JSPS) (via the KAKENHI
+Grant Number JP20H00134, JSPS-RFBR Grant No.
+17-52-50023), the Grant No. FQXi-IAF19-06 from the
+Foundational Questions Institute Fund (FQXi), and a
+donor advised fund of the Silicon Valley Community
+Foundation.
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+

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+PAPER
+Offline Evaluation for Reinforcement
+Learning-based Recommendation:
+A Critical Issue and Some Alternatives
+Romain Deffayet
+Naver Labs Europe & University of Amsterdam
+France / The Netherlands
+[email protected]
+Thibaut Thonet
+Naver Labs Europe
+France
+[email protected]
+Jean-Michel Renders
+Naver Labs Europe
+France
+[email protected]
+Maarten de Rijke
+University of Amsterdam
+The Netherlands
+[email protected]
+Abstract
+In this paper, we argue that the paradigm commonly adopted for offline evaluation of se-
+quential recommender systems is unsuitable for evaluating reinforcement learning-based rec-
+ommenders. We find that most of the existing offline evaluation practices for reinforcement
+learning-based recommendation are based on a next-item prediction protocol, and detail three
+shortcomings of such an evaluation protocol. Notably, it cannot reflect the potential benefits
+that reinforcement learning (RL) is expected to bring while it hides critical deficiencies of
+certain offline RL agents. Our suggestions for alternative ways to evaluate RL-based recom-
+mender systems aim to shed light on the existing possibilities and inspire future research on
+reliable evaluation protocols.
+1
+Introduction
+Recommender systems play a major role in defining internet users’ experience due to their ubiqui-
+tous presence on, e.g., content providing and e-commerce platforms. Correct and careful evaluation
+of recommender systems is therefore critical as it directly impacts business metrics as well as user
+satisfaction – and sometimes even society as a whole.
+While recommendation accuracy (i.e., recommending relevant items) is often taken to be the
+main indicator of performance, the literature on recommender systems highlights the importance
+of additional criteria. Beyond-accuracy goals include, e.g., diversity, novelty or serendipity, fair-
+ness, and user experience in general [McNee et al., 2006]. Such criteria sometimes cannot be
+enforced in one-shot recommendation (i.e., in a single interaction between the user and the rec-
+ommender system) but they may require that we consider the longer-term experience. These
+concerns have motivated researchers and practitioners alike to acknowledge the sequential nature
+ACM SIGIR Forum
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+Vol. 56 No. 2 December 2022
+arXiv:2301.00993v1  [cs.IR]  3 Jan 2023
+
+of many recommendation engines, and to seek to optimize over whole sequences instead of one-shot
+predictions [Quadrana et al., 2018].
+Reinforcement learning (RL) formulates this problem as a Markov decision process (MDP), in
+which we wish to select appropriate actions (i.e., item recommendations) in order to maximize the
+sum of rewards (e.g., clicks, purchases, etc.) along the full sequence of user interactions with the
+recommender system. RL is a natural fit for this problem because the underlying MDP is able
+to model the long-term influence of recommendations on the user. Note that in recommendation
+scenarios, online exploration is often impossible, so the policy must be trained from a fixed dataset
+of interactions, i.e., by offline RL. While sequence optimization with offline RL is not expected
+to entirely fulfill all the desired beyond-accuracy criteria highlighted in the literature, it holds the
+promise of making some of the desired properties naturally emerge as a result of whole-sequence
+optimization. Indeed, one can expect that, given an appropriate reward function, policies that
+are effective over the entire span of the user’s experience require some of these desired properties:
+diversity, novelty, etc. Because these auxiliary metrics are embedded into the sequence’s cumula-
+tive reward, whole-sequence optimization with RL can be seen as a way to bridge the gap between
+offline and online performance.
+In this paper, we argue that the progress supposedly achieved in sequential recommendation,
+thanks to RL, lacks ecological validity [Andrade, 2018]: the trained agents are likely not to gener-
+alize to real-world scenarios, because of certain shortcomings in the current evaluation practices.
+Namely, RL-based recommender systems are often evaluated in an offline fashion, following a tra-
+ditional one-shot accuracy-oriented protocol that cannot capture the potential benefits introduced
+by the use of RL algorithms. We refer to this evaluation protocol as next-item prediction (NIP).
+More critically, we highlight that the specifics of this protocol are likely to hide the deficiencies
+of recommender systems trained by offline RL. Briefly, we argue that with the most commonly
+employed evaluation practices, we cannot verify that the RL algorithm correctly optimizes the very
+metric it is designed to optimize, i.e., expected cumulative reward. We worry that instead of
+bridging the gap between offline and online performance, it only widens it.
+We then provide
+suggestions towards a sound evaluation methodology for RL-based recommendation in order to
+help practitioners and researchers avoid common pitfalls and to inspire future research on this
+important topic.
+After contrasting our criticism with that formulated by previous studies in Section 2, we
+provide in Section 3 a definition of the next-item prediction evaluation protocol along with an
+overview of its use in sequential recommendation with RL. Section 4 dives into the three major
+issues of the NIP protocol, and their implications for the evaluation of RL-based recommender
+systems. Finally, we formulate our suggestions towards a sound evaluation methodology in RL-
+based recommendation in Section 5.
+2
+Related studies
+Deficiencies in recommender systems evaluation have been a long-standing problem in the recom-
+mendation literature. In this section we review previous studies that discuss this topic.
+Firstly, as we recalled in the introduction, McNee et al. [2006]; Jannach et al. [2016] have
+highlighted the need for recommender systems that go beyond accuracy of the proposed item, i.e.,
+which do not only consider recommendation as a matrix completion problem. This is motivated
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+
+by an observed gap between offline and online performance, sometimes rendering any conclusions
+drawn from offline evaluation obsolete [Garcin et al., 2014; Gomez-Uribe and Hunt, 2016; Jeunen,
+2019].
+Secondly, pitfalls of recommender system evaluation – including the next-item prediction pro-
+tocol for offline evaluation that we focus on in this study – have been extensively discussed in
+the past: Chen et al. [2017]; Jeunen [2019]; Ji et al. [2020]; Cremonesi and Jannach [2021]; Sun
+[2022]; Zhao et al. [2022] highlighted multiple issues resulting from data leakage and other dataset
+construction fallacies, which can lead to counter-intuitive statements. The presence of selection
+bias in the data used for evaluating recommender systems from implicit feedback has also been
+identified as a major source of inaccuracies [Gomez-Uribe and Hunt, 2016; Jannach et al., 2016;
+Chen et al., 2017; Jeunen, 2019]. In addition, and more specifically to the next-item prediction
+protocol, Krichene and Rendle [2020]; Zhao et al. [2022] have shown that sampling negative items
+at inference time in order to ease the computation of ranking metrics leads to drawing incorrect
+conclusions on the recommendation performance.
+Finally, many studies reaffirm the importance of appropriate baseline selection in order to
+ensure that progress has been made, and have shown that certain claims do not hold against
+properly tuned baselines [Ludewig et al., 2019; Ferrari Dacrema et al., 2019; Rendle et al., 2019;
+Sun et al., 2020; Zhao et al., 2022].
+The argument we formulate in this paper is specific to RL-based recommendation and while it
+has, to the best of our knowledge, never been expressed, it is not incompatible with the issues listed
+in this section. It is rather to be considered as an additional caveat when evaluating RL-based
+recommender systems.
+3
+Next-item prediction in RL-based recommendation
+We propose an (informal) definition of next-item prediction that encompasses the offline evaluation
+protocols of many sequential recommendation studies, and that we consider to be problematic
+when used to evaluate RL-based recommender systems:
+Definition 1. Next-item prediction (NIP) is an offline evaluation protocol for sequential item
+recommendation from real user feedback. The task is to ensure that the next interacted item
+is among the top items ranked by the model, given the sequence of past interactions. Model
+performance is measured according to ranking metrics (e.g., hit rate, recall, NDCG, etc).
+We propose this definition because it is representative of the evaluation setup adopted in many se-
+quential recommendation studies, e.g., GRU4REC [Hidasi et al., 2016], and also encompasses sev-
+eral variants. In particular, the choice of “next interacted item” can vary depending on the dataset
+and task at hand: the next clicked item in content recommendation (e.g., Last.fm [Last.fm]), the
+next purchased product in product recommendation (e.g., RecSys Challenge 2015 [Ben-Shimon
+et al., 2015] or RetailRocket [RetailRocket, 2016]), the next highly rated movie in movie recom-
+mendation (e.g., MovieLens [GroupLens]), the next basket in grocery shopping [Instacart, 2017],
+etc.
+How prevalent is it in RL-based recommendation? RL-based recommendation (RL4REC)
+has become increasingly popular in recent years: we counted 55 papers about RL4REC in the
+ACM SIGIR Forum
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+Vol. 56 No. 2 December 2022
+
+2017
+2018
+2019
+2020
+2021
+2022
+Year
+1
+7
+10
+11
+12
+14
+Number of papers
+Figure 1: Evolution of the number of RL-based recommendation papers published in major RecSys
+and IR conferences between 2017 and 2022.
+proceedings of major information retrieval and recommender systems (or related) conferences
+between January 2017 and October 2022. To obtain this result, we queried “reinforcement learning
+recommendation” and “reinforcement learning recommender” on DBLP1 and included papers
+published at AAAI, CIKM, ICDM, IJCAI, KDD, RecSys, SIGIR, WSDM or WWW. Figure 1
+shows the increasing trend in published RL4REC papers. Out of the 55 papers retrieved from
+DBLP, we identified 39 papers that address sequential item recommendation using RL-based
+approaches. Other tasks irrelevant to our argument included conversational recommendation or
+explainable recommendations, so we ignore papers related to these topics in this study. Among
+the 39 relevant articles, we found 24 papers performing a form of offline evaluation, including 22
+papers that followed the NIP protocol from Definition 1. The 15 other papers exclusively rely
+on online evaluation, either in production using an industrial recommendation platform or based
+on a simulator. The NIP protocol is therefore by far the most commonly adopted type of offline
+evaluation.
+4
+Three shortcomings of NIP
+Before engaging with the explanation of the issues with next-item prediction, we would like to
+recall the benefits promised by the use of RL algorithms:
+• RL aims to optimize long-term outcomes resulting from a sequence of decisions. This requires
+accounting for the effect of the recommender on the user. RL-based methods are able to
+optimize whole-sequences by assigning the credit for observed rewards to individual actions,
+thereby preventing costly search throughout the combinatorial space of action sequences.
+1https://dblp.org/
+ACM SIGIR Forum
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+Vol. 56 No. 2 December 2022
+
+• RL algorithms learn in a self-supervised manner, by maximizing scalar rewards. Doing so
+allows them to recover open-ended solutions and generate novel policies. However, training
+the agent in an offline fashion also comes with the risk of deriving policies with inaccurate
+estimation of their expected return.
+In the following, we list three major shortcomings of the NIP protocol for evaluating offline RL
+agents, and explain how they harm the ecological validity of the claims derived from this evaluation
+protocol.
+4.1
+A myopic evaluation
+Evaluating an offline RL-based recommender system using Definition 1 only accounts for short-
+term rewards and ignores the causal effect of the recommendations on the user.
+Indeed, an
+important motivation to design RL algorithms is to maximize the return (i.e., sum of rewards)
+along full trajectories, as opposed to bandit algorithms that aim to maximize the average reward
+at each timestep. When the actions (i.e., recommendations) cause the environment (i.e., user) to
+change its state, RL algorithms still have convergence guarantees, while the environment appears
+as non-stationary to bandit algorithms that fail to find the optimal policy both in theory and
+in practice. But the next-item prediction evaluation protocol only requires short-term thinking
+as it rewards one-shot prediction of the next interacted item – this is due to the offline, static
+nature of the evaluation that overlooks the causal impact of the recommendation policy of interest
+over subsequent interactions. This argument has been formulated by Lee et al. [2022], who also
+empirically verified that greedy, myopic agents achieve similar or better performance on the NIP
+protocol than long-term-aware RL agents on standard recommendation datasets. Quadrana et al.
+[2018] also warned about the limits of the NIP evaluation protocol in sequential recommendation
+when not only immediate satisfaction but also diversity or user guidance in content discovery is
+desired.
+However, in contrast to Lee et al. [2022], we additionally argue that the inclusion of delayed re-
+wards such as dwell-time in content recommendation or lifetime value in product recommendation
+would not be sufficient to solve this issue. Indeed, the long-term outcomes encoded in the delayed
+reward (e.g., was the product satisfactory over its whole lifetime?) can be orthogonal to the long-
+term outcomes encoded in the sum of rewards along the trajectory (e.g., was the trajectory diverse
+enough to avoid boring out the user?). While the former clearly seem to be important in order
+to obtain useful and enjoyable recommender systems, the latter are the ones that are modeled
+by the Markov decision process underlying the RL agent. Consequently, if we include delayed
+rewards but ignore the long-term outcomes induced by the sequential decision-making process, we
+still cannot observe the benefits brought by RL training from the NIP protocol. Note that these
+two types of long-term outcomes are not incompatible and we recommend using a reward function
+that is as close as possible to the user’s needs and satisfaction, including delayed outcomes.
+4.2
+A suboptimal target
+As explained in Section 3, in datasets commonly employed for next-item prediction, we observe
+the rewards (e.g., clicks, purchases) only on the items that the user interacted with. This incurs a
+selection bias in the evaluation protocol, caused by the application of a particular treatment to the
+ACM SIGIR Forum
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+Vol. 56 No. 2 December 2022
+
+user. This treatment can take the form of a logging policy or a mixture of logging policies when
+data is gathered from organic interactions on recommendation platforms, or the implicit effect
+of exogenous factors when the observed data is the result of active user feedback, e.g., voluntary
+movie reviews or product search. We refer to the latter kind of bias as an implicit logging policy
+for simplicity. Note that another source of sub-optimality of the interacted items is that user
+choice may also be shortsighted or reluctant to novelty, even though acting so may lead to a less
+enjoyable experience overall.
+By considering the fact that selecting the interacted item is a binary target, instead of a
+scalar reward to be maximized, the NIP evaluation incentivizes researchers and practitioners to
+build policies that are close to the (implicit) logging policy, at the expense of choosing optimal
+actions. It is a close-ended task of policy matching while RL allows for open-ended outcomes,
+i.e., generating novel policies achieving high return. There exists simpler methods to replicate the
+policy which generated the data, e.g., imitation learning [Hussein et al., 2017], and the reward
+maximization objective of RL is likely to deteriorate the results on this evaluation by selecting
+items that are different from the interacted item but incurring higher returns. Consequently, NIP
+will discard performant policies and encourage policies similar to the logging policy, even when the
+sequences in the dataset were highly suboptimal. Considering stronger signals such as purchases
+or high ratings mitigates this issue, but the selection bias that users were exposed to during data
+collection implies that some highly rewarding items are likely discarded.
+4.3
+Risky deployment
+The two previous points that we have formulated indicate that the next-item prediction evaluation
+cannot reflect the potential benefits brought by offline RL-based recommender systems.
+The
+third problematic aspect that we discuss shows that next-item prediction may also hide critical
+deficiencies of offline RL agents.
+Even though in the evaluation protocol of Definition 1 we account for the position of the next
+interacted item in the model predictions, through the use of ranking metrics, the recommender
+system will only select its most preferred item (or top-k most preferred items in slate recommenda-
+tion) when used in production, while none of the other items will be shown to the user. It therefore
+seems crucial to ensure that the top item is satisfactory, regardless of the full ranking. This is
+unfortunately not possible with a fixed dataset where only one or a few items have been shown to
+the considered user. A tacit assumption of NIP is that higher ranking metrics correlate with a top
+item causing high return. However, a gap between offline and online results has been identified
+in previous studies [Garcin et al., 2014; Gomez-Uribe and Hunt, 2016]. More importantly, it has
+been shown that even under the strong assumption that the Q-value associated to every action
+(i.e., item recommendation) can be correctly estimated in expectation (i.e, no bias), there can be
+an overestimation of the predicted offline reward with respect to the actual online reward, because
+the selected item is more likely to be one of those with an overestimated Q-value [Jeunen and
+Goethals, 2021]. This phenomenon is called the optimizer’s curse, and while its practical impact
+in certain cases can be limited, we argue that it can critically affect RL algorithms. Indeed, a
+particular set of conditions has been identified to cause a catastrophic impact of the optimizer’s
+curse and is often called the deadly triad [van Hasselt et al., 2018; Sutton and Barto, 2018]. It can
+be observed with most RL algorithms and occurs when (i) the value estimate at one state is used
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+
+to update the value estimate at the previous state, (ii) function approximation is used to build
+the estimate of the value function, and (iii) the RL agent is trained in an off-policy fashion.
+Under such conditions, small overestimations of the value function on out-of-distribution ac-
+tions can be amplified and propagated to neighboring states and actions, potentially leading to
+divergence of the value function. In that case, while the model predicts high Q-values for its policy,
+the observed return after deployment can be arbitrarily bad. The highly damaging effect of the
+deadly triad has been observed in multiple scenarios and motivated the emergence of extensive
+research on offline reinforcement learning [van Hasselt et al., 2018; Fu et al., 2019, 2020; Levine
+et al., 2020; Brandfonbrener et al., 2021; Kostrikov et al., 2021].
+Unfortunately, this harmful
+phenomenon cannot be detected in the standard next-item prediction evaluation of Definition 1:
+while the interacted item may rightfully be ranked high by the model, it is likely that at least
+one out-of-distribution item is drastically overestimated and preferred by the model. Since this
+item will be the one selected by the model, we may observe an unpredicted catastrophic failure
+at deployment time. Even worse, this probability of failure tends to increase with the size of the
+action-space [Gu et al., 2022], which can be enormous in certain recommendation scenarios.
+4.4
+Upshot
+The three shortcomings we presented in this section render offline evaluation using the NIP proto-
+col of RL-based recommender systems unreliable. They effectively widen the gap between offline
+and online metrics, where RL algorithms were actually supposed to bridge this gap. In the next
+section, we suggest potential solutions to address this issue.
+5
+Some alternatives to NIP
+The limitations of NIP make offline evaluation of RL-based recommender systems difficult. We
+detail below some partial solutions to this problem and discuss their limitations and remaining
+open questions.
+5.1
+Online evaluation in recommendation platforms
+The most obvious counter-measure to the issues raised above is to evaluate recommender systems
+online when possible, directly on the metrics we care about. This is usually done by deploying the
+policies on an actual recommendation platform. However, it is obvious that not all researchers and
+practitioners have access to an operational industrial platform, and online evaluation itself may
+include other forms of biases, e.g., through the inclusion of business rules in recommendations.
+Online evaluation clearly circumvents the three issues we highlighted in the previous section, but
+since the focus of this paper is on offline evaluation, we will not further detail it.
+5.2
+Counterfactual off-policy evaluation
+There is a large body of work on off-policy evaluation (OPE) in information retrieval, often based
+on techniques such as inverse propensity scoring [Swaminathan and Joachims, 2015; Joachims
+et al., 2017], where a propensity weight is applied to rescale the observed rewards and returns.
+ACM SIGIR Forum
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+Vol. 56 No. 2 December 2022
+
+Although OPE has mostly been tackled for the one-shot bandit problem, some studies address
+OPE of RL policies both in the RL community [Fu et al., 2021] and in the IR community [Chen
+et al., 2019], and more recently a library for off-policy evaluation of RL algorithms in IR has been
+proposed in [Kiyohara and Kawakami, 2022].
+Counterfactual methods for off-policy evaluation are attractive in that they can provide unbi-
+asedness guarantees under mild assumptions. However, we want to stress three (known) deficien-
+cies of these methods: (i) IPS suffers from a notoriously high variance which becomes exponentially
+higher when applied on sequences, because of the product of inverse propensity weights [Precup
+et al., 2000]; (ii) in non-tabular settings (i.e., when one can generalize the predictions from a
+state-action pair to another, for example with continuous spaces), generalization capabilities must
+implicitly or explicitly be assumed when the logging policy is not known, in order to compute the
+propensity [Hanna et al., 2019]; and (iii) when we train RL algorithms in an offline manner, the
+error of the off-policy training and of the off-policy evaluation are likely correlated, which means
+that counterfactual OPE may still be biased and wrongly choose certain methods above others.
+An extreme example of the latter occurs if we train and evaluate a policy-gradient recommender
+with the same propensity weights, which makes the agent appear as optimal regardless of its true
+performance. While using an ensemble of estimators might mitigate this issue, it remains unclear
+how to fully alleviate this issue. Counterfactual OPE circumvents all three shortcomings high-
+lighted in the previous section in theory, but as we have seen it comes with its own shortcomings
+which may make it unreliable in certain practical settings.
+5.3
+Simulator-based evaluation
+Simulators have proved useful to assess progress in other domains, such as robotics, games or
+industrial applications [Fu et al., 2020; Gulcehre et al., 2020; Qin et al., 2021]. While the inter-
+action with a recommender system is arguably one of the hardest problems to simulate because
+of the complexity and apparent stochasticity of human behavior, the true value of simulators lies
+in their ability to observe how recommenders react under a chosen set of assumptions on user
+behavior. Additionally, by allowing the researcher to access otherwise unobservable metrics, they
+can enlighten us on the inner workings of the systems we build.
+Many studies proposed to build semi-synthetic simulators, where the synthetic part is as limited
+as possible in order to adhere to real-world scenarios. This can for instance be done by using real
+item embeddings [Shi et al., 2019] or by extending the implicit feedback to unseen data, with
+debiasing in the missing-not-at-random case [Huang et al., 2020].
+Moreover, it is possible to
+assess the generalizability of a method by benchmarking it against a wide range of simulated
+configurations, so as to mitigate the influence of simulator design on the results. Regardless of the
+chosen setup, one should ensure that the simulator exhibits the characteristics we wish to model,
+most notably long-term influence of the recommender system on the user.
+Simulators are not sensitive to the three issues of the NIP protocol, but their ecological validity
+may clearly be limited. On top of building simulators from real data, some approaches aim to
+bridge the gap between simulation and reality, for example with domain randomization [Tobin
+et al., 2017; OpenAI et al., 2020].
+ACM SIGIR Forum
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+
+5.4
+Intermediate evaluation
+By intermediate evaluation, we refer to the offline evaluation of models, simulators or propensities
+that are used as building blocks in the final recommendation model [Huang et al., 2020; Deffayet
+et al., 2022]. In certain cases, it may be easier to evaluate these intermediate models than the final
+model, for example when they can be evaluated thanks to the availability of human annotations,
+e.g., of item relevance. By breaking down the evaluation protocol into several components, we can
+isolate and reduce the sources of bias. For instance, in top-k recommendation for cumulative click
+maximization, if the click model is correctly estimated, i.e., the relevance and propensity scores
+are correct, then only state dynamics (i.e., how a user changes in response to a recommendation)
+are left as a source of uncertainty.
+Doing so mitigates the risks associated with deploying RL agents, but does not suppress them.
+Moreover, we want to stress that offline RL agents will likely use the intermediate models outside
+of their training distribution in order to perform policy evaluation, and therefore may exploit
+inaccuracies in these high uncertainty regions if no proper countermeasure is applied [Deffayet
+et al., 2022].
+5.5
+Uncertainty-aware evaluation
+While it may not be feasible to accurately evaluate the final performance of an RL policy in a
+purely offline fashion, we argue that quantifying its performance at different levels of uncertainty
+can help assess the risks of deployment. Indeed, the value overestimation issue highlighted in
+the previous section results from the high uncertainty on out-of-distribution state-action pairs.
+We can constrain the RL algorithm to recover safe policies, that stay within the distribution of
+the logging policy, or allow exploration in order to find potentially high-return policies, at the
+cost of increasing uncertainty [Brandfonbrener et al., 2021]. By quantifying the match between
+the support of the logging policy and that of the target policy, we can assess the risk induced
+by the deployment of the target policy. In particular, if we restrict the set of available actions
+to those considered “in-support”, we can get an accurate estimate of the performance of the
+policy on those actions. Indeed, uncertainty is low inside the support of the logging policy, and
+it is anyway possible to evaluate the quality of the Q-value prediction on a held-out test set of
+the offline dataset as in, e.g., [Ji et al., 2021]. A safe policy achieving high in-support expected
+return would constitute a reliable improvement, while an unsafe policy not even achieving good
+in-support expected return can probably be discarded. This type of evaluation needs a proper
+definition of in-support and out-of-support, e.g., as in [Fujimoto et al., 2019; Gu et al., 2022],
+which is not trivial in the non-tabular setting and requires assuming a certain degree of tolerance
+to uncertainty, but Kumar et al. [2021] show that it is possible to adjust this tolerance based on
+the training curves of certain offline RL algorithms.
+This type of evaluation focuses on characterizing and mitigating the risks induced by the third
+issue we raise in Section 4.3, while potentially allowing us to detect the benefits brought by RL
+training. The main open question lies in the ability to properly define distance measures between
+the support of the logging and target policy.
+ACM SIGIR Forum
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+Vol. 56 No. 2 December 2022
+
+6
+Conclusion
+In this study, we highlighted that the most commonly employed protocol for the offline evaluation
+of RL-based recommender systems is in fact unsuitable, because it cannot reflect the benefits that
+RL supposedly brings compared to more traditional approaches and because it may hide critical
+deficiencies of offline RL agents that can lead to catastrophic deployment. These shortcomings
+can be summarized as follows: (i) a myopic protocol aimed only at measuring shortterm accuracy,
+(ii) a close-ended, suboptimal recommendation target, and (iii) sensitivity to the optimizer’s curse.
+As of now, there exists no truly satisfactory solution to the problem of evaluating RL policies
+in an entirely offline fashion. Yet, several proxies for online performance can be used to bridge
+the gap between offline metrics and online performance. Finding appropriate offline evaluation
+protocols is still an active research area in the offline RL literature, and we urge the sequential
+recommendation community to join the effort and develop protocols suitable for the recommen-
+dation scenario. Additionally, acknowledging the presence of uncertainty in the deployment of
+RL-based recommender systems paves the way towards solutions that are robust or resilient to
+such uncertainty. For instance, Oosterhuis and de Rijke [2021] propose a criterion for fallback to a
+safer policy when out-of-distribution (although in a different context, i.e., counterfactual learning
+to rank), and Ghosh et al. [2022]; Reichlin et al. [2022] propose adaptive offline RL policies that
+are able to recover from stepping in uncertain states during deployment by branching back to sup-
+ported states. We hope that future research in recommender systems will put stronger emphasis
+on these aspects and reduce the gap between offline and online performance.
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+

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+arXiv:2301.00760v1  [math.RA]  27 Nov 2022
+Extending structures for Poisson bialgebras
+Tao Zhang, Fang Yang
+Abstract
+We introduce the concept of braided Poisson bialgebras.
+The theory of cocycle bi-
+crossproducts for Poisson bialgebras is developed. As an application, we solve the extending
+problem for Poisson bialgebras by using some non-abelian cohomology theory.
+2020 MSC: 17B63, 17B62, 16W25.
+Keywords: Poisson-Hopf modules, Braided Poisson bialgebras, cocycle bicrossproduct,
+extending structure, non-abelian cohomology.
+Contents
+1
+Introduction
+1
+2
+Preliminaries
+2
+3
+Braided Poisson bialgebras
+6
+3.1
+Poisson-Hopf modules and braided Poisson bialgebras
+. . . . . . . . . . . . . .
+6
+4
+Unified product of Poisson bialgebras
+9
+4.1
+Matched pair of braided Poisson bialgebras
+. . . . . . . . . . . . . . . . . . . .
+9
+4.2
+Cocycle bicrossproduct Poisson bialgebras . . . . . . . . . . . . . . . . . . . . .
+12
+5
+Extending structures for Poisson bialgebras
+22
+5.1
+Extending structures for Poisson algebras
+. . . . . . . . . . . . . . . . . . . . .
+22
+5.2
+Extending structures for Poisson coalgebras . . . . . . . . . . . . . . . . . . . .
+27
+5.3
+Extending structures for Poisson bialgebras . . . . . . . . . . . . . . . . . . . .
+34
+1
+Introduction
+Poisson algebra is an algebra with a Lie algebra structure and a commutative associative
+algebra structure which are entwined by Leibniz rule. Poisson algebras appear in several areas
+of mathematics and mathematical physics. Pre-Poisson algebras are investigated by M.Aguiar
+in [8]. It is shown that a pre-Poisson algebra gives rise to a Poisson algebra by passing to
+the corresponding Lie and commutative algebras. Poisson bialgebra has the structure of both
+Lie bialgebra and infinitesimai bialgebra. Lie bialgebras have been studied in [13, 15, 16], and
+1
+
+infinitesimai bialgebra have been studied in [9, 18]. The concept of Poisson bialgebras was
+introduced by Ni and Bai in [10] which related to classical Yang-Baxter equation(CYBE) and
+associative Yang-Baxter equation(AYBE) uniformly.
+The theory of extending structure for many types of algebras were well developed by A. L.
+Agore and G. Militaru in [1, 2, 3, 4, 5, 6]. Let A be an algebra and E a vector space containing
+A as a subspace. The extending problem is to describe and classify all algebra structures on
+E such that A is a subalgebra of E. They show that associated to any extending structure of
+A by a complement space V , there is a unified product on the direct sum space E ∼= A ⊕ V .
+Recently, extending structures for 3-Lie algebras, Lie bialgebras, infinitesimal bialgebras, Lie
+conformal superalgebras and weighted infinitesimal bialgebras were studied in [16, 17, 18].
+As a continue of our paper [15] and [16], the aim of this paper is to study extending struc-
+tures for Poisson bialgebras. For this purpose, we will introduce the concept of braided Poisson
+bialgebras. Then we give the construction of cocycle bicrossproducts for Poisson bialgebras.
+We will show that these new concept and construction will play a key role in considering ex-
+tending problem for Poisson bialgebras. As an application, we solve the extending problem for
+Poisson bialgebras by using some non-abelian cohomology theory.
+This paper is organized as follows. In Section 2, we recall some definitions and fix some
+notations. In Section 3, we introduced the concept of braided Poisson bialgebras and proved the
+bosonisation theorem associating braided Poisson bialgebras to ordinary Poisson bialgebras.
+In section 4, we define the notion of matched pairs of braided Poisson bialgebras and construct
+cocycle bicrossproduct Poisson bialgebras through two generalized braided Poisson bialgebras.
+In section 5, we studied the extending problems for Poisson bialgebras and proof that they can
+be classified by some non-abelian cohomology theory.
+Throughout the following of this paper, all vector spaces will be over a fixed field of character
+zero.
+A Lie algebra or a Lie coalgebra is denoted by (A, [, ]) or (A, δ) and a commutative
+associative algebra or a cocommutative coassociative coalgebra is denoted by (A, ·) or (A, ∆).
+The identity map of a vector space V is denoted by idV : V → V or simply id : V → V . The
+flip map τ : V ⊗ V → V ⊗ V is defined by τ(u ⊗ v) = v ⊗ u for any u, v ∈ V .
+2
+Preliminaries
+Definition 2.1. A Poisson algebra is a triple (A, [, ], ·) where A is a vector space equipped
+with two bilinear operations [, ], · : A ⊗ A → A, such that (A, [, ]) is a Lie algebra and (A, ·)
+is a commutative associative algebra and the following compatibility condition is satisfied,
+[x, y · z] = [x, y] · z + y · [x, z],
+(1)
+for all x, y, z ∈ A .
+Sometimes, we just omit “ · ” in calculation of the following paper for convenience.
+Note that the above identities are equivalent to the following identities:
+[x, yz] = [x, y]z + y[x, z].
+(2)
+2
+
+Definition 2.2. ([10]) A Poisson coalgebra is a triple (A, δ, ∆) where A is a vector space
+equipped with two maps δ, ∆ : A → A ⊗ A, such that (A, δ) is a Lie coalgebra and (A, ∆)
+is a cocommutative coassociative coalgebra, such that the satisfy the following compatibile
+condition :
+(id ⊗ ∆)δ(x) = (δ ⊗ id)∆(x) + (τ ⊗ id)(id ⊗ δ)∆(x),
+(3)
+for all x ∈ A.
+Definition 2.3. ([10]) A Poisson bialgebra is a 5-triple (A, [, ], ·, δ, ∆) where (A, [, ], ·) is a
+Poisson algebra, (A, δ, ∆) is a Poisson coalgebra, (A, [, ], δ) is a Lie bialgebra and (A, ·, ∆) is
+a commutative and cocommutative infinitesimal bialgebra, such that the following compatible
+conditions hold:
+δ(xy) = (Ly ⊗ id) δ(x) + (Lx ⊗ id) δ(y) + (id ⊗ adx) ∆(y) + (id ⊗ ady) ∆(x),
+(4)
+∆([x, y]) = (adx ⊗id + id ⊗ adx) ∆(y) + (Ly ⊗ id − id ⊗ Ly) δ(x)
+(5)
+where Lx and adx are the left multiplication operator and the adjoint operator defined by
+Lx(y) = xy and adx(y) = [x, y] respectively.
+If we use the sigma notation ∆(x) = x1 ⊗
+x2, δ(x) = x[1] ⊗ x[2], then the above two equations (4) and (5) can be written as
+δ(xy) = x[1]y ⊗ x[2] + xy[1] ⊗ y[2] + y1 ⊗ [x, y2] + x1 ⊗ [y, x2],
+(6)
+∆([x, y]) = [x, y1] ⊗ y2 + y1 ⊗ [x, y2] + yx[1] ⊗ x[2] − x[1] ⊗ yx[2],
+(7)
+for all x, y ∈ A .
+Definition 2.4. ([14]) Let H be a Poisson algebra, V be a vector space. Then V is called
+a left H-Poisson module if there is a pair of linear maps ⊲ : H ⊗ V → V, (x, v) → x ⊲ v and
+⇀: H ⊗ V → V, (x, v) → x ⇀ v such that (V, ⇀) is a left module of (H, ·) as associative
+algebra and (V, ⊲) is a left module of (H, [, ]) as Lie algebra, i.e.,
+(xy) ⇀ v = x ⇀ (y ⇀ v),
+(8)
+[x, y] ⊲ v = x ⊲ (y ⊲ v) − y ⊲ (x ⊲ v),
+(9)
+and the following conditions hold:
+(xy) ⊲ v = x ⇀ (y ⊲ v) + y ⇀ (x ⊲ v),
+(10)
+[x, y] ⇀ v = x ⊲ (y ⇀ v) − y ⇀ (x ⊲ v),
+(11)
+for all x, y ∈ H and v ∈ V .
+The category of left Poisson modules over H is denoted by HM.
+3
+
+Definition 2.5. Let H be a Poisson coalgebra, V be a vector space. Then V is called a left
+H-Poisson comodule if there is a pair of linear maps φ : V → H ⊗ V and ρ : V → H ⊗ V such
+that (V, ρ) is a left module of (H, ∆) as coassociative coalgebra and (V, φ) is a left module of
+(H, δ) Lie coalgebra, i.e.,
+(∆H ⊗ idV ) ρ(v) = (idH ⊗ ρ)ρ(v),
+(12)
+(δH ⊗ idV )φ(v) = (idH ⊗ φ)φ(v) − τ12(idH ⊗ φ)φ(v),
+(13)
+and the following conditions hold:
+(∆H ⊗ idV ) φ(v) = τ12 (idH ⊗ φ) ρ(v) + (idH ⊗ φ)ρ(v),
+(14)
+(idH ⊗ ρ) φ(v) = (δH ⊗ idV ) ρ(v) + τ12(idH ⊗ φ)ρ(v).
+(15)
+If we denote by φ(v) = v⟨−1⟩ ⊗ v⟨0⟩ and ρ(v) = v(−1) ⊗ v(0), then the above equations can be
+written as
+∆H
+�
+v(−1)
+�
+⊗ v(0) = v(−1) ⊗ ρ(v(0)),
+(16)
+δH
+�
+v⟨−1⟩
+�
+⊗ v⟨0⟩ = v⟨−1⟩ ⊗ φ(v⟨0⟩) − τ12(v⟨−1⟩ ⊗ φ(v⟨0⟩)),
+(17)
+∆H
+�
+v⟨−1⟩
+�
+⊗ v⟨0⟩ = τ12
+�
+v(−1) ⊗ φ(v(0))
+�
++ v(−1) ⊗ φ(v(0)),
+(18)
+v⟨−1⟩ ⊗ ρ(v⟨0⟩) = δH(v(−1)) ⊗ v(0) + τ12(v(−1) ⊗ φ(v(0))).
+(19)
+The category of left Poisson comodules over H is denoted by HM.
+Definition 2.6. Let H and A be Poisson algebras. An action of H on A is a pair of linear
+maps ⊲ : H ⊗ A → A, (x, a) → x ⊲ a and ⇀: H ⊗ A → A, (x, a) → x ⇀ a such that
+(1) (A, ·, ⇀) is a left H-module algebra over (H, ·), i.e.,
+x ⇀ (ab)
+=
+(x ⇀ a)b,
+(20)
+(2) (A, [, ], ⊲) is a left H-module Lie algebra over (H, [, ]), i.e.,
+x ⊲ [a, b]
+=
+[a, x ⊲ b] + [x ⊲ a, b],
+(21)
+(3) The following conditions are satisfied:
+x ⊲ (ab)
+=
+(x ⊲ a)b + a(x ⊲ b),
+(22)
+x ⇀ [a, b]
+=
+[a, x ⇀ b] + (x ⊲ a)b,
+(23)
+for all x ∈ H and a, b ∈ A. In this case, we call (A, ⇀, ⊲) to be a left H-Poisson module
+algebra.
+Definition 2.7. Let H and A be Poisson coalgebras. A coaction of H on A is a pair of linear
+maps φ : A → H ⊗ A and ρ : A → H ⊗ A such that
+4
+
+(1) (A, ∆A, ρ) is a left H-comodule coalgebra over (H, ∆H), i.e.,
+(idH ⊗ ∆A)ρ(a) = (ρ ⊗ idA)∆A(a).
+(24)
+(2) (A, δA, φ) is a left H-comodule Lie coalgebra over (H, δH), i.e.,
+(idH ⊗ δA)φ(a) = (φ ⊗ idA)δA(a) + τ12(idA ⊗ φ)δA(a).
+(25)
+(3) The following conditions are satisfied:
+(idH ⊗ ∆A)φ(a)
+=
+(φ ⊗ idA)∆A(a) + τ12(idA ⊗ φ)∆A(a);
+(26)
+(idH ⊗ δA)ρ(a)
+=
+τ12(idA ⊗ ρ)δA(a) + (φ ⊗ idA)∆A(a).
+(27)
+If we denote by φ(a) = a⟨−1⟩ ⊗ a⟨0⟩ and ρ(a) = a(−1) ⊗ a(0), then the above equations (26) and
+(27) can be written as
+a⟨−1⟩ ⊗ ∆A
+�
+a⟨0⟩
+�
+= φ (a1) ⊗ a2 + τ12(a1 ⊗ φ(a2)),
+(28)
+a(−1) ⊗ δA(a(0)) = τ12(a[1] ⊗ ρ(a[2])) + φ(a1) ⊗ a2,
+(29)
+for all a ∈ A. In this case, we call (A, φ, ρ) to be left H-comodule Poisson coalgebras.
+Definition 2.8. Let (A, ·) be a given Poisson algebra (Poisson coalgebra, Poisson bialgebra), E
+be a vector space. An extending system of A through V is a Poisson algebra(Poisson coalgebra,
+Poisson bialgebra) on E such that V a complement subspace of A in E, the canonical injection
+map i : A → E, a �→ (a, 0) or the canonical projection map p : E → A, (a, x) �→ a is a Poisson
+algebra(Poisson coalgebra, Poisson bialgebra) homomorphism. The extending problem is to
+describe and classify up to an isomorphism the set of all Poisson algebra(Poisson coalgebra,
+Poisson bialgebra) structures that can be defined on E.
+We remark that our definition of extending system of A through V contains not only
+extending structure in [1, 2, 3] but also the global extension structure in [5].
+In fact, the
+canonical injection map i : A → E is a Poisson (co)algebra homomorphism if and only if A is
+a Poisson sub(co)algebra of E.
+Definition 2.9. Let A be a Poisson algebra (Poisson coalgebra, Poisson bialgebra), E be a
+Poisson algebra (Poisson coalgebra, Poisson bialgebra) such that A is a subspace of E and V
+a complement of A in E. For a linear map ϕ : E → E we consider the diagram:
+0
+� A
+idA �
+i
+� E
+ϕ
+�
+π
+� V
+idV �
+� 0
+0
+� A
+i′
+� E
+π′
+� V
+� 0.
+(30)
+where π : E → V are the canonical projection maps and i : A → E are the inclusion maps.
+We say that ϕ : E → E stabilizes A if the left square of the diagram (30) is commutative. Let
+5
+
+(E, ·) and (E, ·′) be two Poisson algebra (Poisson coalgebra, Poisson bialgebra) structures on
+E. (E, ·) and (E, ·′) are called equivalent, and we denote this by (E, ·) ≡ (E, ·′), if there exists a
+Poisson algebra (Poisson coalgebra, Poisson bialgebra) isomorphism ϕ : (E, ·) → (E, ·′) which
+stabilizes A. Denote by Extd(E, A) (CExtd(E, A), BExtd(E, A)) the set of equivalent classes
+of Poisson algebra(Poisson coalgebra, Poisson bialgebra) structures on E.
+3
+Braided Poisson bialgebras
+In this section, we introduce the concept of left Poisson-Hopf modules and braided Poisson
+bialgebras which will be used in the following sections.
+3.1
+Poisson-Hopf modules and braided Poisson bialgebras
+Definition 3.1. Let H be a Poisson bialgebra. A left Poisson-Hopf module over H is a vector
+space V endowed with linear maps
+⊲ : H ⊗ V → V,
+⇀: H ⊗ V → V,
+φ : V → H ⊗ V,
+ρ : V → H ⊗ V,
+which are denoted by
+⊲(x ⊗ v) = x ⊲ v,
+⇀ (x ⊗ v) = x ⇀ v,
+φ(v) =
+�
+v⟨−1⟩ ⊗ v⟨0⟩,
+ρ(v) =
+�
+v(−1) ⊗ v(0),
+such that V is simultaneously a left module, a left comodule over H and satisfying the following
+compatibility conditions
+(HM1) φ(x ⇀ v) = v⟨−1⟩x ⊗ v⟨0⟩ + v(−1) ⊗ (x ⊲ v(0)) − x1 ⊗ (x2 ⊲ v),
+(HM2) τφ(x ⇀ v) = (x ⇀ v⟨0⟩) ⊗ v⟨−1⟩ − v(0) ⊗ [x, v(−1)] − (x[1] ⇀ v) ⊗ x[2],
+(HM3) ρ(x ⊲ v) = [x, v(−1)] ⊗ v(0) + v(−1) ⊗ (x ⊲ v(0)) − x[1] ⊗ (x[2] ⇀ v),
+(HM4) ρ(x ⊲ v) = x1 ⊗ (x2 ⊲ v) + v⟨−1⟩ ⊗ (x ⇀ v⟨0⟩) − xv⟨−1⟩ ⊗ v⟨0⟩,
+for all x ∈ H and v ∈ V .
+We denote the category of left Poisson-Hopf modules over H by H
+HM.
+Definition 3.2. Let H be a Poisson bialgebra, A be simultaneously a left H-module algebra
+(coalgebra) and left H-comodule algebra (coalgebra).
+We call A to be a braided Poisson
+bialgebra, if the following conditions are satisfied
+(BB1) δA(ab) = a[1]b ⊗ a[2] + ab[1] ⊗ b[2] + b1 ⊗ [a, b2] + a1 ⊗ [b, a2]
++ (a⟨−1⟩ ⇀ b) ⊗ a⟨0⟩ + (b⟨−1⟩ ⇀ a) ⊗ b⟨0⟩ − b(0) ⊗ (b(−1) ⊲ a) − a(0) ⊗ (a(−1) ⊲ b),
+6
+
+(BB2) ∆A([a, b]) = [a, b1] ⊗ b2 + b1 ⊗ [a, b2] + ba[1] ⊗ a[2] − a[1] ⊗ ba[2]
++ a⟨0⟩ ⊗ (a⟨−1⟩ ⇀ b) + (a⟨−1⟩ ⇀ b) ⊗ a⟨0⟩ − (b(−1) ⊲ a) ⊗ b(0) − b(0) ⊗ (b(−1) ⊲ a).
+Now we construct Poisson bialgebras from braided Poisson bialgebras. Let H be a Poisson
+bialgebra, A be a Poisson algebra and a Poisson coalgebra in H
+HM. We define multiplications
+and comultiplications on the direct sum vector space E := A ⊕ H by
+[(a, x), (b, y)]E := ([a, b] + x ⊲ b − y ⊲ a, [x, y]),
+(31)
+δE(a, x) := δA(a) + φ(a) − τφ(a) + δH(x),
+(32)
+(a, x) ·E (b, y) := (ab + x ⇀ b + y ⇀ a, xy),
+(33)
+∆E(a, x) := ∆A(a) + ρ(a) + τρ(a) + ∆H(x).
+(34)
+This is called biproduct of A and H which will be denoted by A>⊳· H.
+Theorem 3.3. Let H be a Poisson bialgebra, A be a Poisson algebra and a Poisson coalgebra
+in H
+HM. Then the biproduct A>⊳· H forms a Poisson bialgebra if and only if A is a braided
+Poisson bialgebra in H
+HM.
+Proof. First, it is obvious that (A>⊳· H, [, ]) and (A>⊳· H, ·) are respectively a Lie algebra and
+a commutative associative algebra. It is easy to prove that A>⊳· H is a Poisson algebra and a
+Poisson coalgebra with the multiplications (31) and (33) and comultiplications (32) and (34).
+Now we show the compatibility conditions:
+δE((a, x) ·E (b, y)) =(a, x)[1] ·E (b, y) ⊗ (a, x)[2] + (a, x) ·E (b, y)[1] ⊗ (b, y)[2]
++ (b, y)1 ⊗ [(a, x), (b, y)2]E + (a, x)1 ⊗ [(b, y), (a, x)2]E,
+∆E([(a, x), (b, y)]E) =[(a, x), (b, y)1]E ⊗ (b, y)2 + (b, y)1 ⊗ [(a, x), (b, y)2]E
++ (b, y) ·E (a, x)[1] ⊗ (a, x)[2] − (a, x)[1] ⊗ (b, y) ·E (a, x)[2].
+By direct computations, the left hand side of the first equation is equal to
+δE((a, x) ·E (b, y))
+=
+δE(ab + x ⇀ b + y ⇀ a, xy)
+=
+δA(ab) + δA(x ⇀ b) + δA(y ⇀ a) + φ(ab) + φ(x ⇀ b) + φ(y ⇀ a)
+−τφ(ab) − τφ(x ⇀ b) − τφ(y ⇀ a) + δH(xy),
+and the right hand side is equal to
+(a, x)[1] ·E (b, y) ⊗ (a, x)[2] + (a, x) ·E (b, y)[1] ⊗ (b, y)[2]
++(b, y)1 ⊗ [(a, x), (b, y)2]E + (a, x)1 ⊗ [(b, y), (a, x)2]E
+=
+a[1]b ⊗ a[2] + (y ⇀ a[1]) ⊗ a[2] + (a⟨−1⟩ ⇀ b) ⊗ a⟨0⟩ + a⟨−1⟩y ⊗ a⟨0⟩
+−a⟨0⟩b ⊗ a⟨−1⟩ − (y ⇀ a⟨0⟩) ⊗ a⟨−1⟩ + (x[1] ⇀ b) ⊗ x[2] + x[1]y ⊗ x[2]
++ab[1] ⊗ b[2] + (x ⇀ b[1]) ⊗ b[2] + (b⟨−1⟩ ⇀ a) ⊗ b⟨0⟩ + xb⟨−1⟩ ⊗ b⟨0⟩
+7
+
+−ab⟨0⟩ ⊗ b⟨−1⟩ − (x ⇀ b⟨0⟩) ⊗ b⟨−1⟩ + (y[1] ⇀ a) ⊗ y[2] + xy[1] ⊗ y[2]
++b1 ⊗ [a, b2] + b1 ⊗ (x ⊲ b2) + b(−1) ⊗ [a, b(0)] + b(−1) ⊗ (x ⊲ b(0))
++b(0) ⊗ [x, b(−1)] − b(0) ⊗ (b(−1) ⊲ a) + y1 ⊗ [x, y2] − y1 ⊗ (y2 ⊲ a)
++a1 ⊗ [b, a2] + a1 ⊗ (y ⊲ a2) + a(−1) ⊗ [b, a(0)] + a(−1) ⊗ (y ⊲ a(0))
++a(0) ⊗ [y, a(−1)] − a(0) ⊗ (a(−1) ⊲ b) + x1 ⊗ [y, x2] − x1 ⊗ (x2 ⊲ b).
+Then the two sides are equal to each other if and only if
+(1)δA(ab) = a[1]b ⊗ a[2] + ab[1] ⊗ b[2] + b1 ⊗ [a, b2] + a1 ⊗ [b, a2] + (a⟨−1⟩ ⇀ b) ⊗ a⟨0⟩
++(b⟨−1⟩ ⇀ a) ⊗ b⟨0⟩ − b(0) ⊗ (b(−1) ⊲ a) − a(0) ⊗ (a(−1) ⊲ b),
+(2) δA(x ⇀ b) = (x ⇀ b[1]) ⊗ b[2] + b1 ⊗ (x ⊲ b2),
+(3) φ(ab) = b(−1) ⊗ [a, b(0)] + a(−1) ⊗ [b, a(0)],
+(4) τφ(ab) = a⟨0⟩b ⊗ a⟨−1⟩ + ab⟨0⟩ ⊗ b⟨−1⟩,
+(5) φ(x ⇀ b) = xb⟨−1⟩ ⊗ b⟨0⟩ + b(−1) ⊗ (x ⊲ b(0)) − x1 ⊗ (x2 ⊲ b),
+(6) τφ(x ⇀ b) = (x ⇀ b⟨0⟩) ⊗ b⟨−1⟩ − b(0) ⊗ [x, b(−1)] − (x[1] ⇀ b) ⊗ x[2].
+For the second equation, the left hand side is equal to
+∆E[(a, x), (b, y)]E
+=∆E([a, b] + x ⊲ b − y ⊲ a, [x, y])
+=∆A([a, b]) + ∆A(x ⊲ b) − ∆A(y ⊲ a) + ρ([a, b]) + ρ(x ⊲ b) − ρ(y ⊲ a)
++ τρ([a, b]) + τρ(x ⊲ b) − τρ(y ⊲ a) + ∆H([x, y]),
+and the right hand side is equal to
+[(a, x), (b, y)1]E ⊗ (b, y)2 + (b, y)1 ⊗ [(a, x), (b, y)2]E
++(b, y) ·E (a, x)[1] ⊗ (a, x)[2] − (a, x)[1] ⊗ (b, y) ·E (a, x)[2]
+=
+[a, b1] ⊗ b2 + (x ⊲ b1) ⊗ b2 + b1 ⊗ [a, b2] + b1 ⊗ (x ⊲ b2)
++[x, b(−1)] ⊗ b(0) − (b(−1) ⊲ a) ⊗ b(0) + b(−1) ⊗ [a, b(0)] + b(−1) ⊗ (x ⊲ b(0))
++[a, b(0)] ⊗ b(−1) + (x ⊲ b(0)) ⊗ b(−1) + b(0) ⊗ [x, b(−1)] − b(0) ⊗ (b(−1) ⊲ a)
++[x, y1] ⊗ y2 − (y1 ⊲ a) ⊗ y2 + y1 ⊗ [x, y2] − y1 ⊗ (y2 ⊲ a)
++ba[1] ⊗ a[2] + (y ⇀ a[1]) ⊗ a[2] − a[1] ⊗ ba[2] − a[1] ⊗ (y ⇀ a[2])
++(a⟨−1⟩ ⇀ b) ⊗ a⟨0⟩ + ya⟨−1⟩ ⊗ a⟨0⟩ − a⟨−1⟩ ⊗ ba⟨0⟩ − a⟨−1⟩ ⊗ (y ⇀ a⟨0⟩)
+−ba⟨0⟩ ⊗ a⟨−1⟩ − (y ⇀ a⟨0⟩) ⊗ a⟨−1⟩ + a⟨0⟩ ⊗ ya⟨−1⟩ + a⟨0⟩ ⊗ (a⟨−1⟩ ⇀ b)
++yx[1] ⊗ x[2] + (x[1] ⇀ b) ⊗ x[2] − x[1] ⊗ yx[2] − x[1] ⊗ (x[2] ⇀ b).
+Then the two sides are equal to each other if and only if
+(7) ∆A([a, b]) = [a, b1] ⊗ b2 + b1 ⊗ [a, b2] + ba[1] ⊗ a[2] − a[1] ⊗ ba[2]
++a⟨0⟩ ⊗ (a⟨−1⟩ ⇀ b) + (a⟨−1⟩ ⇀ b) ⊗ a⟨0⟩ − (b(−1) ⊲ a) ⊗ b(0) − b(0) ⊗ (b(−1) ⊲ a),
+(8) ∆A(x ⊲ b) = (x ⊲ b1) ⊗ b2 + b1 ⊗ (x ⊲ b2),
+(9) ∆A(y ⊲ a) = a[1] ⊗ (y ⇀ a[2]) − (y ⇀ a[1]) ⊗ a[2],
+8
+
+(10) ρ([a, b]) = b(−1) ⊗ [a, b(0)] − a⟨−1⟩ ⊗ ba⟨0⟩,
+(11) ρ(x ⊲ b) = [x, b(−1)] ⊗ b(0) + b(−1) ⊗ (x ⊲ b(0)) − x[1] ⊗ (x[2] ⇀ b),
+(12) ρ(y ⊲ a) = y1 ⊗ (y2 ⊲ a) + a⟨−1⟩ ⊗ (y ⇀ a⟨0⟩) − ya⟨−1⟩ ⊗ a⟨0⟩.
+From (2)–(4) and (8)–(10) we have that A is a Poisson algebra and a Poisson coalgebra in
+H
+HM, from (5)–(6) and (11)–(12) we get that A is a left Poisson-Hopf module over H, and (1)
+together with (7) are the conditions for A to be a braided Poisson bialgebra.
+The proof is completed.
+4
+Unified product of Poisson bialgebras
+4.1
+Matched pair of braided Poisson bialgebras
+In this section, we construct Poisson bialgebra from the double cross biproduct of a matched
+pair of braided Poisson bialgebras.
+Let A, H be both Poisson algebras and Poisson coalgebras. For a, b ∈ A, x, y ∈ H, we
+denote linear maps
+⇀: H ⊗ A → A,
+↼: H ⊗ A → H,
+⊲ : H ⊗ A → A,
+⊳ : H ⊗ A → H,
+φ : A → H ⊗ A,
+ψ : H → H ⊗ A,
+ρ : A → H ⊗ A,
+γ : H → H ⊗ A,
+by
+⇀ (x ⊗ a) = x ⇀ a,
+↼ (x ⊗ a) = x ↼ a,
+⊲(x ⊗ a) = x ⊲ a,
+⊳(x ⊗ a) = x ⊳ a,
+φ(a) =
+�
+a⟨−1⟩ ⊗ a⟨0⟩,
+ψ(x) =
+�
+x⟨0⟩ ⊗ x⟨1⟩,
+ρ(a) =
+�
+a(−1) ⊗ a(0),
+γ(x) =
+�
+x(0) ⊗ x(1).
+Definition 4.1. ([10]) A matched pair of Poisson algebras is a system (A, H, ⊳, ⊲, ↼, ⇀)
+consisting of two Poisson algebras A and H and four bilinear maps ⊳ : H ⊗ A → H, ��� :
+H ⊗ A → A, ↼: H ⊗ A → H, ⇀: H ⊗ A → A such that (A, H, ⊲, ⊳) is a matched pair of
+Lie algebras, (A, H, ⇀, ↼) is a matched pair of commutative associative algebras, and the
+following compatibility conditions is satisfied for all a, b ∈ A, x, y ∈ H:
+(AM1) x ⇀ [a, b] = [a, x ⇀ b] + (x ⊲ a)b + (x ⊳ a) ⇀ b − (x ↼ b) ⊲ a,
+(AM2) x ⊲ (ab) = (x ⊲ a)b + (x ⊳ a) ⇀ b + a(x ⊲ b) + (x ⊳ b) ⇀ a,
+(AM3) [x, y] ↼ a = [x, y ↼ a] + x ⊳ (y ⇀ a) − y(x ⊳ a) − y ↼ (x ⊲ a),
+(AM4) (xy) ⊳ a = x ↼ (y ⊲ a) + x(y ⊳ a) + y ↼ (x ⊲ a) + (x ⊳ a)y.
+9
+
+Lemma 4.2. ([10]) Let (A, H, ⊳, ⊲, ↼, ⇀) be a matched pair of Poisson algebras.
+Then
+A ⊲⊳ H := A ⊕ H, as a vector space, with the multiplication defined for any a, b ∈ A and
+x, y ∈ H by
+[(a, x), (b, y)]E := ([a, b] + x ⊲ b − y ⊲ a, [x, y] + x ⊳ b − y ⊳ a),
+(a, x) ·E (b, y) := (ab + x ⇀ b + y ⇀ a, xy + x ↼ b + y ↼ a),
+is a Poisson algebra which is called the bicrossed product associated to the matched pair of
+Poisson algebras A and H.
+Now we introduce the notion of matched pairs of Poisson coalgebras, which is the dual
+version of matched pairs of Poisson algebras.
+Definition 4.3. A matched pair of Poisson coalgebras is a system (A, H, φ, ψ, ρ, γ) consisting
+of two Poisson coalgebras A and H and four bilinear maps φ : A → H ⊗ A, ψ : H → H ⊗ A,
+ρ : A → H ⊗ A, γ : H → H ⊗ A such that (A, H, φ, ψ) is a matched pair of Lie coalgebras,
+(A, H, ρ, γ) is a matched pair of cocommutative coassociative coalgebras, and the following
+compatibility conditions is satisfied for any a ∈ A, x ∈ H:
+(CM1) a[1] ⊗ ρ(a[2]) − a⟨0⟩ ⊗ γ(a⟨−1⟩) = −τφ(a1) ⊗ a2 − τψ(a(−1)) ⊗ a(0) + τ12(a(−1) ⊗ δA(a(0))),
+(CM2) a⟨−1⟩ ⊗ ∆A(a⟨0⟩) = φ(a1) ⊗ a2 + ψ(a(−1)) ⊗ a(0) + τ12(a1 ⊗ φ(a2)) + τ12(a(0) ⊗ ψ(a(−1))),
+(CM3) x[1] ⊗ γ
+�
+x[2]
+�
++ x⟨0⟩ ⊗ ρ(x⟨1⟩) = δH(x(0)) ⊗ x(1) + τ12(x1 ⊗ ψ(x2)) + τ12(x(0) ⊗ φ(x(1))),
+(CM4) x⟨1⟩ ⊗ ∆H(x⟨0⟩) = τψ(x1) ⊗ x2 + τφ(x(1)) ⊗ x(0) + τ12(x1 ⊗ τψ(x2)) + τ12(x(0) ⊗ τφ(x(1))).
+Lemma 4.4. Let (A, H) be a matched pair of Poisson coalgebras. We define E = A ◮◭ H as
+the vector space A ⊕ H with comultiplication
+∆E(a) = (∆A + ρ + τρ)(a),
+∆E(x) = (∆H + γ + τγ)(x),
+δE(a) = (δA + φ − τφ)(a),
+δE(x) = (δH(x) + ψ − τψ)(x),
+that is
+∆E(a) =
+�
+a1 ⊗ a2 +
+�
+a(−1) ⊗ a(0) +
+�
+a(0) ⊗ a(−1),
+∆E(x) =
+�
+x1 ⊗ x2 +
+�
+x(0) ⊗ x(1) +
+�
+x(1) ⊗ x(0),
+δE(a) =
+�
+a[1] ⊗ a[2] + a⟨−1⟩ ⊗ a⟨0⟩ − a⟨0⟩ ⊗ a⟨−1⟩,
+δE(x) =
+�
+x[1] ⊗ x[2] + x⟨0⟩ ⊗ x⟨1⟩ − x⟨1⟩ ⊗ x⟨0⟩.
+Then A ◮◭ H is a Poisson coalgebra which is called the bicrossed coproduct associated to the
+matched pair of Poisson coalgebras A and H.
+The proof of the above Lemma 4.4 is omitted since it is by direct computations. In the
+following of this section, we construct Poisson bialgebra from the double cross biproduct of
+a pair of braided Poisson bialgebras. First we generalize the concept of Hopf module to the
+case of A is not necessarily a Poisson bialgebra. But by abuse of notation, we also call it
+Poisson-Hopf module.
+10
+
+Definition 4.5. Let A be simultaneously a Poisson algebra and a Poisson coalgebra. If H is
+a right A-module, a right A-comodule and satisfying
+(HM1’) ψ(x ↼ a) = (x⟨0⟩ ↼ a) ⊗ x⟨1⟩ + (x ↼ a[1]) ⊗ a[2] + x(0) ⊗ [a, x(1)],
+(HM2’) τψ(x ↼ a) = x⟨1⟩a ⊗ x⟨0⟩ + x(1) ⊗ (x(0) ⊳ a) − a1 ⊗ (x ⊳ a2),
+(HM3’) γ(x ⊳ a) = (x ⊳ a1) ⊗ a2 + (x⟨0⟩ ↼ a) ⊗ x⟨1⟩ − x⟨0⟩ ⊗ ax⟨1⟩,
+(HM4’) γ(x ⊳ a) = (x(0) ⊳ a) ⊗ x(1) − x(0) ⊗ [a, x(1)] − (x ↼ a[1]) ⊗ a[2],
+then H is called a right Poisson-Hopf module over A.
+We denote the category of right Poisson-Hopf modules over A by MA
+A.
+Definition 4.6. Let A be a Poisson algebra and Poisson coalgebra and H is a right Poisson-
+Hopf module over A. If H is a Poisson algebra and a Poisson coalgebra in MA
+A, then we call
+H a braided Poisson bialgebra over A, if the following conditions are satisfied:
+(BB1’) δH(xy) = x[1]y ⊗ x[2] − (y ↼ x⟨1⟩) ⊗ x⟨0⟩ + xy[1] ⊗ y[2] − (x ↼ y⟨1⟩) ⊗ y⟨0⟩
++ y1 ⊗ [x, y2] + y(0) ⊗ (x ⊳ y(1)) + x1 ⊗ [y, x2] + x(0) ⊗ (y ⊳ x(1)),
+(BB2’) ∆H([x, y]) = [x, y1] ⊗ y2 + (x ⊳ y(1)) ⊗ y(0) + y1 ⊗ [x, y2] + y(0) ⊗ (x ⊳ y(1))
++ yx[1] ⊗ x[2] − (y ↼ x⟨1⟩) ⊗ x⟨0⟩ − x[1] ⊗ yx[2] − x⟨0⟩ ⊗ (y ↼ x⟨1⟩).
+Definition 4.7. Let A, H be both Poisson algebras and Poisson coalgebras. If the following
+conditions hold:
+(DM1) φ(ab) = (a⟨−1⟩ ↼ b) ⊗ a⟨0⟩ + (b⟨−1⟩ ↼ a) ⊗ b⟨0⟩ + b(−1) ⊗ [a, b(0)] + a(−1) ⊗ [b, a(0)],
+(DM2) τφ(ab) = a⟨0⟩b ⊗ a⟨−1⟩ + ab⟨0⟩ ⊗ b⟨−1⟩ + b(0) ⊗ (b(−1) ⊳ a) + a(0) ⊗ (a(−1) ⊳ b),
+(DM3) ψ(xy) = x⟨0⟩y ⊗ x⟨1⟩ + xy⟨0⟩ ⊗ y⟨1⟩ + y(0) ⊗ (x ⊲ y(1)) + x(0) ⊗ (y ⊲ x(1)),
+(DM4) τψ(xy) = (y ⇀ x⟨1⟩) ⊗ x⟨0⟩ + (x ⇀ y⟨1⟩) ⊗ y⟨0⟩ − y(1) ⊗ [x, y(0)] − x(1) ⊗ [y, x(0)],
+(DM5) δA(x ⇀ b) = (x⟨0⟩ ⇀ b) ⊗ x⟨1⟩ + (x ⇀ b[1]) ⊗ b[2] − x(1) ⊗ (x(0) ⊲ b) + b1 ⊗ (x ⊲ b2),
+(DM6) δH(x ↼ b) = (x[1] ↼ b) ⊗ x[2] − (x ↼ b⟨0⟩) ⊗ b⟨−1⟩ + b(−1) ⊗ (x ⊳ b(0)) − x1 ⊗ (x2 ⊳ b),
+(DM7) φ(x ⇀ b) + ψ(x ↼ b) = (x⟨0⟩ ↼ b) ⊗ x⟨1⟩ + (x ↼ b[1]) ⊗ b[2]
++ xb⟨−1⟩ ⊗ b⟨0⟩ + b(−1) ⊗ (x ⊲ b(0)) − x1 ⊗ (x2 ⊲ b) + x(0) ⊗ [b, x(1)],
+(DM8) τφ(x ⇀ b) + τψ(x ↼ b) = x⟨1⟩b ⊗ x⟨0⟩ + (x ⇀ b⟨0⟩) ⊗ b⟨−1⟩
++ x(1) ⊗ (x(0) ⊳ b) − (x[1] ⇀ b) ⊗ x[2] − b(0) ⊗ [x, b(−1)] − b1 ⊗ (x ⊳ b2),
+(DM9) ρ([a, b]) = (a⟨−1⟩ ↼ b) ⊗ a⟨0⟩ − (b(−1) ⊳ a) ⊗ b(0) + b(−1) ⊗ [a, b(0)] − a⟨−1⟩ ⊗ ba⟨0⟩,
+(DM10) γ([x, y]) = [x, y(0)] ⊗ y(1) + y(0) ⊗ (x ⊲ y(1)) + yx⟨0⟩ ⊗ x⟨1⟩ − x⟨0⟩ ⊗ (y ⇀ x⟨1⟩),
+11
+
+(DM11) ∆A(x ⊲ b) = (x ⊲ b1) ⊗ b2 + b1 ⊗ (x ⊲ b2) + (x⟨0⟩ ⇀ b) ⊗ x⟨1⟩ + x⟨1⟩ ⊗ (x⟨0⟩ ⇀ b),
+(DM12) ∆A(y ⊲ a) = −(y ⇀ a[1]) ⊗ a[2] + a[1] ⊗ (y ⇀ a[2]) + (y(0) ⊲ a) ⊗ y(1) + y(1) ⊗ (y(0) ⊲ a),
+(DM13) ∆H(x ⊳ b) = (x ⊳ b(0)) ⊗ b(−1) + b(−1) ⊗ (x ⊳ b(0)) + (x[1] ↼ b) ⊗ x[2] − x[1] ⊗ (x[2] ↼ b),
+(DM14) ∆H(y ⊳ a) = (y1 ⊳ a) ⊗ y2 + y1 ⊗ (y2 ⊳ a) + (y ↼ a⟨0⟩) ⊗ a⟨−1⟩ + a⟨−1⟩ ⊗ (y ↼ a⟨0⟩),
+(DM15) ρ(x ⊲ b) + γ(x ⊳ b) = (x ⊳ b1) ⊗ b2 + [x, b(−1)] ⊗ b(0) + b(−1) ⊗ (x ⊲ b(0))
++ (x⟨0⟩ ↼ b) ⊗ x⟨1⟩ − x[1] ⊗ (x[2] ⇀ b) − x⟨0⟩ ⊗ bx⟨1⟩,
+(DM16) ρ(y ⊲ a) + γ(y ⊳ a) = (y(0) ⊳ a) ⊗ y(1) − y(0) ⊗ [a, y(1)] − (y ↼ a[1]) ⊗ a[2]
+− ya⟨−1⟩ ⊗ a⟨0⟩ + y1 ⊗ (y2 ⊲ a) + a⟨−1⟩ ⊗ (y ⇀ a⟨0⟩),
+then (A, H) is called a double matched pair.
+Theorem 4.8. Let (A, H) be matched pair of Poisson algebras and Poisson coalgebras, A is
+a braided Poisson bialgebra in H
+HM, H is a braided Poisson bialgebra in MA
+A. If we define the
+double cross biproduct of A and H, denoted by A ·⊲⊳· H, A ·⊲⊳· H = A ⊲⊳ H as Poisson algebra,
+A ·⊲⊳· H = A ◮◭ H as Poisson coalgebra, then A ·⊲⊳· H become a Poisson bialgebra if and only if
+(A, H) form a double matched pair.
+The proof of the above Theorem 4.8 is omitted since it is a special case of Theorem 4.16
+in next subsection.
+4.2
+Cocycle bicrossproduct Poisson bialgebras
+In this section, we construct cocycle bicrossproduct Poisson bialgebras, which is a generaliza-
+tion of double cross biproduct.
+Let A, H be both Poisson algebras and Poisson coalgebras. For a, b ∈ A, x, y ∈ H, we
+denote linear maps
+σ : H ⊗ H → A,
+θ : A ⊗ A → H,
+ω : H ⊗ H → A,
+ν : A ⊗ A → H,
+p : A → H ⊗ H,
+q : H → A ⊗ A,
+s : A → H ⊗ H,
+t : H → A ⊗ A,
+by
+σ(x, y) ∈ A,
+θ(a, b) ∈ H,
+ω(x, y) ∈ A,
+ν(a, b) ∈ H,
+p(a) =
+�
+a1p ⊗ a2p,
+q(x) =
+�
+x1q ⊗ x2q,
+s(a) =
+�
+a1s ⊗ a2s,
+t(x) =
+�
+x1t ⊗ x2t.
+A pair of bilinear maps σ, ω : H ⊗ H → A are called cocycles on H if
+12
+
+(CC1) x ⊲ ω(y, z) + σ(x, yz) = z ⇀ σ(x, y) + ω([x, y], z) + y ⇀ σ(x, z) + ω(y, [x, z]).
+A pair of bilinear maps θ, ν : A ⊗ A → H are called cocycles on A if
+(CC2) θ(a, bc) − ν(b, c) ⊳ a = θ(a, b) ↼ c + ν([a, b], c) + θ(a, c) ↼ b + ν(b, [a, c]).
+A pair of bilinear maps p, s : A → H ⊗ H are called cycles on A if
+(CC3) a⟨−1⟩ ⊗ s(a⟨0⟩) + a1p ⊗ ∆H(a2p) = p(a(0)) ⊗ a(−1) + δH(a1s) ⊗ a2s
++ τ12(a(−1) ⊗ p(a(0))) + τ12(a1s ⊗ δH(a2s)).
+A pair of bilinear maps q, t : H → A ⊗ A are called cycles on H if
+(CC4) x1q ⊗ ∆A(x2q) − x⟨−1⟩ ⊗ t(x⟨0⟩) = q(x(0)) ⊗ x(1) + δA(x1t) ⊗ x2t
++ τ12(x(1) ⊗ q(x(0))) + τ12(x1t ⊗ δA(x2t)).
+In the following definitions, we introduced the concept of cocycle Poisson algebras and
+cycle Poisson coalgebras, which are in fact not really ordinary Poisson algebras and Poisson
+coalgebras, but generalized ones.
+Definition 4.9. (i): Let σ, ω be cocycles on a vector space H equipped with multiplications
+[, ], · : H ⊗ H → H, satisfying the following cocycle associative identity:
+(CC5) [x, yz] + x ⊳ ω(y, z) = [x, y]z + z ↼ σ(x, y) + y[x, z] + y ↼ σ(x, z).
+Then H is called a cocycle (σ, ω)-Poisson algebra which is denoted by (H, σ, ω).
+(ii): Let θ, ν be cocycle on a vector space A equipped with multiplications [, ], · : A⊗A → A,
+satisfying the following cocycle associative identity:
+(CC6) [a, bc] − ν(b, c) ⊲ a = [a, b]c + θ(a, b) ⇀ c + b[a, c] + θ(a, c) ⇀ b.
+Then A is called a cocycle (θ, ν)-Poisson algebra which is denoted by (A, θ, ν).
+(iii) Let p, s be cycles on a vector space H equipped with comultiplications ∆, δ : H →
+H ⊗ H, satisfying the following cycle coassociative identity:
+(CC7) x[1] ⊗ ∆H(x[2]) + x⟨0⟩ ⊗ s(x⟨1⟩) = δH(x1) ⊗ x2 + p(x(1)) ⊗ x(0)
++ τ12(x1 ⊗ δH(x2)) + τ12(x(0) ⊗ p(x(1))).
+Then H is called a cycle (p, s)-Poisson coalgebra which is denoted by (H, p, s).
+(iv) Let q, t be cycles on a vector space A equipped with comultiplications ∆, δ : A → A⊗A,
+satisfying the following cycle coassociative identity:
+(CC8) a[1] ⊗ ∆A(a[2]) − a⟨0⟩ ⊗ t(a⟨−1⟩) = δA(a1) ⊗ a2 + q(a(−1)) ⊗ a(0)
++ τ12 ⊗ (a1 ⊗ δA(a2)) + τ12(a(0) ⊗ q(a(−1))).
+Then A is called a cycle (q, t)-Poisson coalgebra which is denoted by (A, q, t).
+13
+
+Definition 4.10. A cocycle cross product system
+is a pair of (θ, ν)-Poisson algebra A and
+(σ, ω)-Poisson algebra H, where σ, ω : H ⊗ H → A are cocycles on H, θ, ν : A ⊗ A → H are
+cocycles on A and the following conditions are satisfied:
+(CP1) [a, x ⇀ b] − (x ↼ b) ⊲ a = x ⇀ [a, b] + ω(x, θ(a, b)) − (x ⊲ a)b − (x ⊳ a) ⇀ b,
+(CP2) (xy) ⊲ a − [a, ω(x, y)] = y ⇀ (x ⊲ a) + ω(x ⊳ a, y) + x ⇀ (y ⊲ a) + ω(x, y ⊳ a),
+(CP3) x ⊲ (ab) + σ(x, ν(a, b)) = (x ⊲ a)b + (x ⊳ a) ⇀ b + a(x ⊲ b) + (x ⊳ b) ⇀ a,
+(CP4) x ⊲ (y ⇀ a) + σ(x, y ↼ a) = σ(x, y)a + [x, y] ⇀ a + y ⇀ (x ⊲ a) + ω(y, x ⊳ a),
+(CP5) [x, y ↼ a] + x ⊳ (y ⇀ a) = [x, y] ↼ a + ν(σ(x, y), a) + y(x ⊳ a) + y ↼ (x ⊲ a),
+(CP6) [x, ν(a, b)] + x ⊳ (ab) = (x ⊳ a) ↼ b + ν(x ⊲ a, b) + (x ⊳ b) ↼ a + ν(a, x ⊲ b),
+(CP7) (xy) ⊳ a − θ(a, ω(x, y)) = (x ⊳ a)y + y ↼ (x ⊲ a) + x(y ⊳ a) + x ↼ (y ⊲ a),
+(CP8) θ(a, x ⇀ b) − (x ↼ b) ⊳ a = θ(a, b)x + x ↼ [a, b] − (x ⊳ a) ↼ b − ν(b, x ⊲ a).
+Lemma 4.11. Let (A, H) be a cocycle cross product system. If we define E = Aσ,ω#θ,νH as
+the vector space A ⊕ H with the multiplication
+[(a, x), (b, y)]E =
+�
+[a, b] + x ⊲ b − y ⊲ a + σ(x, y), [x, y] + x ⊳ b − y ⊳ a + θ(a, b)
+�
+,
+(35)
+and
+(a, x) ·E (b, y) =
+�
+ab + x ⇀ b + y ⇀ a + ω(x, y), xy + x ↼ b + y ↼ a + ν(a, b)
+�
+.
+(36)
+Then E = Aσ,ω#θ,νH forms a Poisson algebra which is called the cocycle cross product Poisson
+algebra.
+Proof. First, it is obvious that (E, [, ]) and (E, ·) are respectively a Lie algebra and a com-
+mutative associative algebra. Then, we need to prove the multiplications · and [, ] satisfying
+[(a, x), (b, y) ·E (c, z)]E = [(a, x), (b, y)]E ·E (c, z) + (b, y) ·E [(a, x), (c, z)]E. By direct computa-
+tions, the left hand side is equal to
+[(a, x), (b, y) ·E (c, z)]E
+=
+[(a, x), (bc + y ⇀ c + z ⇀ b + ω(y, z), yz + y ↼ c + z ↼ b + ν(b, c))]E
+=
+�
+[a, bc] + [a, y ⇀ c] + [a, z ⇀ b] + [a, ω(y, z)] + x ⊲ (bc) + x ⊲ (y ⇀ c)
++x ⊲ (z ⇀ b) + x ⊲ ω(y, z) − (yz) ⊲ a − (y ↼ c) ⊲ a − (z ↼ b) ⊲ a
+−ν(b, c) ⊲ a + σ(x, yz) + σ(x, y ↼ c) + σ(x, z ↼ b) + σ(x, ν(b, c)),
+[x, yz] + [x, y ↼ c] + [x, z ↼ b] + [x, ν(b, c)] + x ⊳ (bc) + x ⊳ (y ⇀ c)
++x ⊳ (z ⇀ b) + x ⊳ ω(y, z) − (yz) ⊳ a − (y ↼ c) ⊳ a − (z ↼ b) ⊳ a
+−ν(b, c) ⊳ a + θ(a, bc) + θ(a, y ⇀ c) + θ(a, z ⇀ b) + θ(a, ω(y, z))
+�
+,
+14
+
+and the right hand side is equal to
+[(a, x), (b, y)]E ·E (c, z) + (b, y) ·E [(a, x), (c, z)]E
+=
+([a, b] + x ⊲ b − y ⊲ a + σ(x, y), [x, y] + x ⊳ b − y ⊳ a + θ(a, b)) ·E (c, z)
++(b, y) ·E ([a, c] + x ⊲ c − z ⊲ a + σ(x, z), [x, z] + x ⊳ c − z ⊳ a + θ(a, c))
+=
+�
+[a, b]c + (x ⊲ b)c − (y ⊲ a)c + σ(x, y)c + [x, y] ⇀ c + (x ⊳ b) ⇀ c − (y ⊳ a) ⇀ c
++θ(a, b) ⇀ c + z ⇀ [a, b] + z ⇀ (x ⊲ b) − z ⇀ (y ⊲ a) + z ⇀ σ(x, y)
++ω([x, y], z) + ω(x ⊳ b, z) − ω(y ⊳ a, z) + ω(θ(a, b), z), [x, y]z + (x ⊳ b)z
+−(y ⊳ a)z + θ(a, b)z + [x, y] ↼ c + (x ⊳ b) ↼ c − (y ⊳ a) ↼ c + θ(a, b) ↼ c
++z ↼ [a, b] + z ↼ (x ⊲ b) − z ↼ (y ⊲ a) + z ↼ σ(x, y) + ν([a, b], c) + ν(x ⊲ b, c)
+−ν(y ⊲ a, c) + ν(σ(x, y), c)
+�
++
+�
+b[a, c] + b(x ⊲ c) − b(z ⊲ a) + bσ(x, z)
++y ⇀ [a, c] + y ⇀ (x ⊲ c) − y ⇀ (z ⊲ a) + y ⇀ σ(x, z) + [x, z] ⇀ b + (x ⊳ c) ⇀ b
+−(z ⊳ a) ⇀ b + θ(a, c) ⇀ b + ω(y, [x, z]) + ω(y, x ⊳ c) − ω(y, z ⊳ a) + ω(y, θ(a, c)),
+y[x, z] + y(x ⊳ c) − y(z ⊳ a) + yθ(a, c) + y ↼ [a, c] + y ↼ (x ⊲ c) − y ↼ (z ⊲ a)
++y ↼ σ(x, z) + [x, z] ↼ b + (x ⊳ c) ↼ b − (z ⊳ a) ↼ b + θ(a, c) ↼ b + ν(b, [a, c])
++ν(b, x ⊲ c) − ν(b, z ⊲ a) + ν(b, σ(x, z))
+�
+=
+�
+[a, b]c + (x ⊲ b)c − (y ⊲ a)c + σ(x, y)c + [x, y] ⇀ c + (x ⊳ b) ⇀ c − (y ⊳ a) ⇀ c
++θ(a, b) ⇀ c + z ⇀ [a, b] + z ⇀ (x ⊲ b) − z ⇀ (y ⊲ a) + z ⇀ σ(x, y) + ω([x, y], z)
++ω(x ⊳ b, z) − ω(y ⊳ a, z) + ω(θ(a, b), z) + b[a, c] + b(x ⊲ c) − b(z ⊲ a) + bσ(x, z)
++y ⇀ [a, c] + y ⇀ (x ⊲ c) − y ⇀ (z ⊲ a) + y ⇀ σ(x, z) + [x, z] ⇀ b + (x ⊳ c) ⇀ b
+−(z ⊳ a) ⇀ b + θ(a, c) ⇀ b + ω(y, [x, z]) + ω(y, x ⊳ c) − ω(y, z ⊳ a) + ω(y, θ(a, c)),
+[x, y]z + (x ��� b)z − (y ⊳ a)z + θ(a, b)z + [x, y] ↼ c + (x ⊳ b) ↼ c − (y ⊳ a) ↼ c
++θ(a, b) ↼ c + z ↼ [a, b] + z ↼ (x ⊲ b) − z ↼ (y ⊲ a) + z ↼ σ(x, y) + ν([a, b], c)
++ν(x ⊲ b, c) − ν(y ⊲ a, c) + ν(σ(x, y), c) + y[x, z] + y(x ⊳ c) − y(z ⊳ a) + yθ(a, c)
++y ↼ [a, c] + y ↼ (x ⊲ c) − y ↼ (z ⊲ a) + y ↼ σ(x, z) + [x, z] ↼ b + (x ⊳ c) ↼ b
+−(z ⊳ a) ↼ b + θ(a, c) ↼ b + ν(b, [a, c]) + ν(b, x ⊲ c) − ν(b, z ⊲ a) + ν(b, σ(x, z))
+�
+.
+Thus the two sides are equal to each other if and only if (CP1)–(CP8) hold.
+Definition 4.12. A cycle cross coproduct system
+is a pair of (p, s)-coalgebra A and (q, t)-
+coalgebra H, where p, s : A → H ⊗ H are cycles on A, q, t : H → A ⊗ A are cycles over H such
+that following conditions are satisfied:
+(CCP1) a[1] ⊗ ρ(a[2]) − a⟨0⟩ ⊗ γ(a⟨−1⟩) = −τφ(a1) ⊗ a2 − τψ(a(−1)) ⊗ a(0)
++ τ12(a(−1) ⊗ δA(a(0))) + τ12(a1s ⊗ q(a2s)),
+(CCP2) a⟨0⟩ ⊗ ∆H(a⟨−1⟩) − a[1] ⊗ s(a[2]) = τφ(a(0)) ⊗ a(−1) + τψ(a1s) ⊗ a2s
++ τ12(a(−1) ⊗ τφ(a(0))) + τ12(a1s ⊗ τψ(a2s)),
+15
+
+(CCP3) a⟨−1⟩ ⊗ ∆A(a⟨0⟩) + a1p ⊗ t(a2p) = φ(a1) ⊗ a2 + ψ(a(−1)) ⊗ a(0)
++ τ12(a1 ⊗ φ(a2)) + τ12(a(0) ⊗ ψ(a(−1))),
+(CCP4) a⟨−1⟩ ⊗ ρ(a⟨0⟩) + a1p ⊗ γ(a2p) = δH(a(−1)) ⊗ a(0) + p(a1) ⊗ a2
++ τ12(a(−1) ⊗ φ(a(0))) + τ12(a1s ⊗ ψ(a2s)),
+(CCP5) x[1] ⊗ γ(x[2]) + x⟨0⟩ ⊗ ρ(x⟨1⟩) = δH(x(0)) ⊗ x(1) + p(x1t) ⊗ x2t
++ τ12(x1 ⊗ ψ(x2)) + τ12(x(0) ⊗ φ(x(1))),
+(CCP6) x[1] ⊗ t(x[2]) + x⟨0⟩ ⊗ ∆A(x⟨1⟩) = ψ(x(0)) ⊗ x(1) + φ(x1t) ⊗ x2t
++ τ12(x(1) ⊗ ψ(x(0))) + τ12(x1t ⊗ φ(x2t)),
+(CCP7) x⟨1⟩ ⊗ ∆H(x⟨0⟩) − x1q ⊗ s(x2q) = τψ(x1) ⊗ x2 + τφ(x(1)) ⊗ x(0)
++ τ12(x1 ⊗ τψ(x2)) + τ12(x(0) ⊗ τφ(x(1))),
+(CCP8) x⟨1⟩ ⊗ γ(x⟨0⟩) − x1q ⊗ ρ(x2q) = τψ(x(0)) ⊗ x(1) + τφ(x1t) ⊗ x2t
+− τ12(x(0) ⊗ δA(x(1))) − τ12(x1 ⊗ q(x2)).
+Lemma 4.13. Let (A, H) be a cycle cross coproduct system. If we define E = Ap,s#q,tH to
+be the vector space A ⊕ H with the comultiplication
+δE(a) = (δA + φ − τφ + p)(a),
+δE(x) = (δH + ψ − τψ + q)(x),
+∆E(a) = (∆A + ρ + τρ + s)(a),
+∆E(x) = (∆H + γ + τγ + t)(x),
+that is
+δE(a) = a[1] ⊗ a[2] + a⟨−1⟩ ⊗ a⟨0⟩ − a⟨0⟩ ⊗ a⟨−1⟩ + a1p ⊗ a2p,
+δE(x) = x[1] ⊗ x[2] + x⟨0⟩ ⊗ x⟨1⟩ − x⟨1⟩ ⊗ x⟨0⟩ + x1q ⊗ x2q,
+∆E(a) = a1 ⊗ a2 + a(−1) ⊗ a(0) + a(0) ⊗ a(−1) + a1s ⊗ a2s,
+∆E(x) = x1 ⊗ x2 + x(0) ⊗ x(1) + x(1) ⊗ x(0) + x1t ⊗ x2t,
+then Ap,s#q,tH forms a Poisson coalgebra which we will call it the cycle cross coproduct Poisson
+coalgebra.
+Proof. Due to the fact that (E, δ) and (E, ∆) are respectively a Lie coalgebra and a cocommu-
+tative coassociative coalgebra, we only need to prove (id ⊗ ∆E)δE(a, x) = (δE ⊗ id)∆E(a, x) +
+(τ ⊗ id)(id ⊗ δE)∆E(a, x).
+The left hand side is equal to
+(id ⊗ ∆E)δE(a, x)
+=
+(id ⊗ ∆E)(a[1] ⊗ a[2] + a⟨−1⟩ ⊗ a⟨0⟩ − a⟨0⟩ ⊗ a⟨−1⟩ + a1p ⊗ a2p + x[1] ⊗ x[2]
++x⟨0⟩ ⊗ x⟨1⟩ − x⟨1⟩ ⊗ x⟨0⟩ + x1q ⊗ x2q)
+=
+a[1] ⊗ ∆A
+�
+a[2]
+�
++ a[1] ⊗ ρ
+�
+a[2]
+�
++ a[1] ⊗ τρ
+�
+a[2]
+�
++ a[1] ⊗ s
+�
+a[2]
+�
++a⟨−1⟩ ⊗ ∆A
+�
+a⟨0⟩
+�
++ a⟨−1⟩ ⊗ ρ
+�
+a⟨0⟩
+�
++ a⟨−1⟩ ⊗ τρ
+�
+a⟨0⟩
+�
++ a⟨−1⟩ ⊗ s
+�
+a⟨0⟩
+�
+16
+
+−a⟨0⟩ ⊗ ∆H
+�
+a⟨−1⟩
+�
+− a⟨0⟩ ⊗ γ
+�
+a⟨−1⟩
+�
+− a⟨0⟩ ⊗ τγ
+�
+a⟨−1⟩
+�
+− a⟨0⟩ ⊗ t
+�
+a⟨−1⟩
+�
++a1p ⊗ ∆H(a2p) + a1p ⊗ γ(a2p) + a1p ⊗ τγ(a2p) + a1p ⊗ t(a2p)
++x[1] ⊗ ∆H
+�
+x[2]
+�
++ x[1] ⊗ γ(x[2]) + x[1] ⊗ τγ(x[2]) + x[1] ⊗ t(x[2])
++x⟨0⟩ ⊗ ∆A
+�
+x⟨1⟩
+�
++ x⟨0⟩ ⊗ ρ
+�
+x⟨1⟩
+�
++ x⟨0⟩ ⊗ τρ
+�
+x⟨1⟩
+�
++ x⟨0⟩ ⊗ s
+�
+x⟨1⟩
+�
+−x⟨1⟩ ⊗ ∆H
+�
+x⟨0⟩
+�
+− x⟨1⟩ ⊗ γ
+�
+x⟨0⟩
+�
+− x⟨1⟩ ⊗ τγ
+�
+x⟨0⟩
+�
+− x⟨1⟩ ⊗ t
+�
+x⟨0⟩
+�
++x1q ⊗ ∆A(x2q) + x1q ⊗ ρ(x2q) + x1q ⊗ τρ(x2q) + x1q ⊗ s(x2q),
+and the right hand side is equal to
+(δE ⊗ id)∆E(a, x) + (τ ⊗ id)(id ⊗ δE)∆E(a, x)
+=
+(δE ⊗ id)(a1 ⊗ a2 + a(−1) ⊗ a(0) + a(0) ⊗ a(−1) + a1s ⊗ a2s + x1 ⊗ x2
++x(0) ⊗ x(1) + x(1) ⊗ x(0) + x1t ⊗ x2t) + (τ ⊗ id)(id ⊗ δE)(a1 ⊗ a2
++a(−1) ⊗ a(0) + a(0) ⊗ a(−1) + a1s ⊗ a2s + x1 ⊗ x2 + x(0) ⊗ x(1)
++x(1) ⊗ x(0) + x1t ⊗ x2t)
+=
+δA (a1) ⊗ a2 + φ (a1) ⊗ a2 − τφ (a1) ⊗ a2 + p(a1) ⊗ a2 + δH
+�
+a(−1)
+�
+⊗ a(0)
++ψ(a(−1)) ⊗ a(0) − τψ(a(−1)) ⊗ a(0) + q(a(−1)) ⊗ a(0) + δA
+�
+a(0)
+�
+⊗ a(−1)
++φ
+�
+a(0)
+�
+⊗ a(−1) − τφ
+�
+a(0)
+�
+⊗ a(−1) + p(a(0)) ⊗ a(−1) + δH(a1s) ⊗ a2s
++ψ(a1s) ⊗ a2s − τψ(a1s) ⊗ a2s + q(a1s) ⊗ a2s + δH (x1) ⊗ x2
++ψ(x1) ⊗ x2 �� τψ(x1) ⊗ x2 + q(x1) ⊗ x2 + δH(x(0)) ⊗ x(1) + ψ(x(0)) ⊗ x(1)
+−τψ(x(0)) ⊗ x(1) + q(x(0)) ⊗ x(1) + δA(x(1)) ⊗ x(0) + φ(x(1)) ⊗ x(0)
+−τφ(x(1)) ⊗ x(0) + p(x(1)) ⊗ x(0) + δA(x1t) ⊗ x2t + φ(x1t) ⊗ x2t
+−τφ(x1t) ⊗ x2t + p(x1t) ⊗ x2t + τ12(a1 ⊗ δA(a2)) + τ12(a1 ⊗ φ(a2))
+−τ12(a1 ⊗ τφ(a2)) + τ12(a1 ⊗ p(a2)) + τ12(a(−1) ⊗ δA(a(0)))
++τ12(a(−1) ⊗ φ(a(0))) − τ12(a(−1) ⊗ τφ(a(0))) + τ12(a(−1) ⊗ p(a(0)))
++τ12(a(0) ⊗ δH(a(−1))) + τ12(a(0) ⊗ ψ(a(−1))) − τ12(a(0) ⊗ τψ(a(−1)))
++τ12(a(0) ⊗ q(a(−1))) + τ12(a1s ⊗ δH(a2s)) + τ12(a1s ⊗ ψ(a2s))
+−τ12(a1s ⊗ τψ(a2s)) + τ12(a1s ⊗ q(a2s)) + τ12(x1 ⊗ δH(x2))
++τ12(x1 ⊗ ψ(x2)) − τ12(x1 ⊗ τψ(x2)) + τ12(x1 ⊗ q(x2))
++τ12(x(0) ⊗ δA(x(1))) + τ12(x(0) ⊗ φ(x(1))) − τ12(x(0) ⊗ τφ(x(1)))
++τ12(x(0) ⊗ p(x(1))) + τ12(x(1) ⊗ δH(x(0))) + τ12(x(1) ⊗ ψ(x(0)))
+−τ12(x(1) ⊗ τψ(x(0))) + τ12(x(1) ⊗ q(x(0))) + τ12(x1t ⊗ δA(x2t))
++τ12(x1t ⊗ φ(x2t)) − τ12(x1t ⊗ τφ(x2t)) + τ12(x1t ⊗ p(x2t)).
+Thus the two sides are equal to each other if and only if (CCP1)–(CCP8) hold.
+Definition 4.14. Let A, H be both Poisson algebras and Poisson coalgebras. If the following
+conditions hold:
+17
+
+(CDM1) φ(ab) + ψ(ν(a, b)) = (a⟨−1⟩ ↼ b) ⊗ a⟨0⟩ + (b⟨−1⟩ ↼ a) ⊗ b⟨0⟩ + b(−1) ⊗ [a, b(0)]
++ a(−1) ⊗ [b, a(0)] + ν(a[1], b) ⊗ a[2] + ν(a, b[1]) ⊗ b[2] − b1s ⊗ (b2s ⊲ a) − a1s ⊗ (a2s ⊲ b),
+(CDM2) τφ(ab) + τψ(ν(a, b)) = a⟨0⟩b ⊗ a⟨−1⟩ + ab⟨0⟩ ⊗ b⟨−1⟩ + b(0) ⊗ (b(−1) ⊳ a) + a(0) ⊗ (a(−1) ⊳ b)
+− (a1p ⇀ b) ⊗ a2p − (b1p ⇀ a) ⊗ b2p − b1 ⊗ θ(a, b2) − a1 ⊗ θ(b, a2),
+(CDM3) ψ(xy) + φ(ω(x, y)) = x⟨0⟩y ⊗ x⟨1⟩ + xy⟨0⟩ ⊗ y⟨1⟩ + y(0) ⊗ (x ⊲ y(1)) + x(0) ⊗ (y ⊲ x(1))
++ (y ↼ x1q) ⊗ x2q + (x ↼ y1q) ⊗ y2q + y1 ⊗ σ(x, y2) + x1 ⊗ σ(y, x2),
+(CDM4) τψ(xy) + τφ(ω(x, y)) = (y ⇀ x⟨1⟩) ⊗ x⟨0⟩ + (x ⇀ y⟨1⟩) ⊗ y⟨0⟩ − y(1) ⊗ [x, y(0)]
+− x(1) ⊗ [y, x(0)] − ω(x[1], y) ⊗ x[2] − ω(x, y[1]) ⊗ y[2] − y1t ⊗ (x ⊳ y2t) − x1t ⊗ (y ⊳ x2t),
+(CDM5) δA(x ⇀ b) + q(x ↼ b) = (x⟨0⟩ ⇀ b) ⊗ x⟨1⟩ + (x ⇀ b[1]) ⊗ b[2] − x(1) ⊗ (x(0) ⊲ b)
++ b1 ⊗ (x ⊲ b2) + x1qb ⊗ x2q + ω(x, b⟨−1⟩) ⊗ b⟨0⟩ + b(0) ⊗ σ(x, b(−1)) + x1t ⊗ [b, x2t],
+(CDM6) δH(x ↼ b) + p(x ⇀ b) = (x[1] ↼ b) ⊗ x[2] − (x ↼ b⟨0⟩) ⊗ b⟨−1⟩ + b(−1) ⊗ (x ⊳ b(0))
+− x1 ⊗ (x2 ⊳ b) − ν(x⟨1⟩, b) ⊗ x⟨0⟩ + xb1p ⊗ b2p + b1s ⊗ [x, b2s] + x(0) ⊗ θ(b, x(1)),
+(CDM7) φ(x ⇀ b) + ψ(x ↼ b) = (x⟨0⟩ ↼ b) ⊗ x⟨1⟩ + (x ↼ b[1]) ⊗ b[2] + xb⟨−1⟩ ⊗ b⟨0⟩
++ b(−1) ⊗ (x ⊲ b(0)) − x1 ⊗ (x2 ⊲ b) + x(0) ⊗ [b, x(1)] + ν(x1q, b) ⊗ x2q + b1s ⊗ σ(x, b2s),
+(CDM8) τφ(x ⇀ b) + τψ(x ↼ b) = x⟨1⟩b ⊗ x⟨0⟩ + (x ⇀ b⟨0⟩) ⊗ b⟨−1⟩ + x(1) ⊗ (x(0) ⊳ b)
+− (x[1] ⇀ b) ⊗ x[2] − b(0) ⊗ [x, b(−1)] − b1 ⊗ (x ⊳ b2) − ω(x, b1p) ⊗ b2p − x1t ⊗ θ(b, x2t),
+(CDM9) ρ([a, b]) + γ(θ(a, b)) = (a⟨−1⟩ ↼ b) ⊗ a⟨0⟩ − (b(−1) ⊳ a) ⊗ b(0) + b(−1) ⊗ [a, b(0)]
+− a⟨−1⟩ ⊗ ba⟨0⟩ + θ(a, b1) ⊗ b2 − b1s ⊗ (b2s ⊲ a) + ν(b, a[1]) ⊗ a[2] − a1p ⊗ (a2p ⇀ b),
+(CDM10) γ([x, y]) + ρ(σ(x, y)) = [x, y(0)] ⊗ y(1) + y(0) ⊗ (x ⊲ y(1)) − x⟨0⟩ ⊗ (y ⇀ x⟨1⟩)
++ yx⟨0⟩ ⊗ x⟨1⟩ + (x ⊳ y1t) ⊗ y2t + y1 ⊗ σ(x, y2) + (y ↼ x1q) ⊗ x2q − x[1] ⊗ ω(y, x[2]), -
+(CDM11) ∆A(x ⊲ b) + t(x ⊳ b) = (x ⊲ b1) ⊗ b2 + b1 ⊗ (x ⊲ b2) + (x⟨0⟩ ⇀ b) ⊗ x⟨1⟩
++ x⟨1⟩ ⊗ (x⟨0⟩ ⇀ b) + σ(x, b(−1)) ⊗ b(0) + b(0) ⊗ σ(x, b(−1)) + bx1q ⊗ x2q − x1q ⊗ bx2q,
+(CDM12) ∆A(y ⊲ a) + t(y ⊳ a) = −(y ⇀ a[1]) ⊗ a[2] + a[1] ⊗ (y ⇀ a[2]) + (y(0) ⊲ a) ⊗ y(1)
++ y(1) ⊗ (y(0) ⊲ a) − [a, y1t] ⊗ y2t − y1t ⊗ [a, y2t] − a⟨0⟩ ⊗ ω(y, a⟨−1⟩) − ω(y, a⟨−1⟩) ⊗ a⟨0⟩,
+(CDM13) ∆H(x ⊳ b) + s(x ⊲ b) = (x ⊳ b(0)) ⊗ b(−1) + b(−1) ⊗ (x ⊳ b(0)) + (x[1] ↼ b) ⊗ x[2]
+− x[1] ⊗ (x[2] ↼ b) + [x, b1s] ⊗ b2s + b1s ⊗ [x, b2s] − ν(b, x⟨1⟩) ⊗ x⟨0⟩ − x⟨0⟩ ⊗ ν(b, x⟨1⟩),
+(CDM14) ∆H(y ⊳ a) + s(y ⊲ a) = (y1 ⊳ a) ⊗ y2 + y1 ⊗ (y2 ⊳ a) + (y ↼ a⟨0⟩) ⊗ a⟨−1⟩
++ a⟨−1⟩ ⊗ (y ↼ a⟨0⟩) − θ(a, y(1)) ⊗ y(0) − y(0) ⊗ θ(a, y(1)) − ya1p ⊗ a2p − a1p ⊗ ya2p,
+(CDM15) ρ(x ⊲ b) + γ(x ⊳ b) = (x ⊳ b1) ⊗ b2 + [x, b(−1)] ⊗ b(0) + b(−1) ⊗ (x ⊲ b(0)) − x⟨0⟩ ⊗ bx⟨1⟩
++ (x⟨0⟩ ↼ b) ⊗ x⟨1⟩ − x[1] ⊗ (x[2] ⇀ b) + b1s ⊗ σ(x, b2s) + ν(b, x1q) ⊗ x2q,
+(CDM16) ρ(y ⊲ a) + γ(y ⊳ a) = (y(0) ⊳ a) ⊗ y(1) − y(0) ⊗ [a, y(1)] − (y ↼ a[1]) ⊗ a[2]
+− ya⟨−1⟩ ⊗ a⟨0⟩ + y1 ⊗ (y2 ⊲ a) + a⟨−1⟩ ⊗ (y ⇀ a⟨0⟩) − θ(a, y1t) ⊗ y2t + a1p ⊗ ω(y, a2p).
+18
+
+then (A, H) is called a cocycle double matched pair.
+Definition 4.15. (i) A cocycle braided Poisson bialgebra A is simultaneously a cocycle Poisson
+algebra (A, θ, ν) and a cycle Poisson coalgebra (A, q, t) satisfying the conditions
+(CBB1) δA(ab) + q(ν(a, b)) = a[1]b ⊗ a[2] + (a⟨−1⟩ ⇀ b) ⊗ a⟨0⟩ + ab[1] ⊗ b[2] + (b⟨−1⟩ ⇀ a) ⊗ b⟨0⟩
++ b1 ⊗ [a, b2] − b(0) ⊗ (b(−1) ⊲ a) + a1 ⊗ [b, a2] − a(0) ⊗ (a(−1) ⊲ b),
+(CBB2) ∆A([a, b]) + t(θ(a, b)) = [a, b1] ⊗ b2 − (b(−1) ⊲ a) ⊗ b(0) + b1 ⊗ [a, b2] − b(0) ⊗ (b(−1) ⊲ a)
++ ba[1] ⊗ a[2] + (a⟨−1⟩ ⇀ b) ⊗ a⟨0⟩ − a[1] ⊗ ba[2] + a⟨0⟩ ⊗ (a⟨−1⟩ ⇀ b).
+(ii) A cocycle braided Poisson bialgebra H is simultaneously a cocycle Poisson algebra (H, σ, ω)
+and a cycle Poisson coalgebra (H, p, s) satisfying the conditions
+(CBB3) δH(xy) + p(ω(x, y)) = x[1]y ⊗ x[2] − (y ↼ x⟨1⟩) ⊗ x⟨0⟩ + xy[1] ⊗ y[2] − (x ↼ y⟨1⟩) ⊗ y⟨0⟩
++ y1 ⊗ [x, y2] + y(0) ⊗ (x ⊳ y(1)) + x1 ⊗ [y, x2] + x(0) ⊗ (y ⊳ x(1)),
+(CBB4) ∆H([x, y]) + s(σ(x, y)) = [x, y1] ⊗ y2 + (x ⊳ y(1)) ⊗ y(0) + y1 ⊗ [x, y2] + y(0) ⊗ (x ⊳ y(1))
++ yx[1] ⊗ x[2] − (y ↼ x⟨1⟩) ⊗ x⟨0⟩ − x[1] ⊗ yx[2] − x⟨0⟩ ⊗ (y ↼ x⟨1⟩).
+The next theorem says that we can obtain an ordinary Poisson bialgebra from two cocycle
+braided Poisson bialgebras.
+Theorem 4.16. Let A, H be cocycle braided Poisson bialgebras, (A, H) be a cocycle cross
+product system and a cycle cross coproduct system. Then the cocycle cross product Poisson
+algebra and cycle cross coproduct Poisson coalgebra fit together to become an ordinary Poisson
+bialgebra if and only if (A, H) forms a cocycle double matched pair. We will call it the cocycle
+bicrossproduct Poisson bialgebra and denote it by Ap,s
+σ,ω#q,t
+θ,νH.
+Proof. We only need to check the compatibility conditions
+δE((a, x) ·E (b, y)) =(a, x)[1] ·E (b, y) ⊗ (a, x)[2] + (a, x) ·E (b, y)[1] ⊗ (b, y)[2]
++ (b, y)1 ⊗ [(a, x), (b, y)2]E + (a, x)1 ⊗ [(b, y), (a, x)2]E,
+∆E([(a, x), (b, y)]E) =[(a, x), (b, y)1]E ⊗ (b, y)2 + (b, y)1 ⊗ [(a, x), (b, y)2]E
++ (b, y) ·E (a, x)[1] ⊗ (a, x)[2] − (a, x)[1] ⊗ (b, y) ·E (a, x)[2].
+For the first equation, the left hand side is equal to
+δE((a, x) ·E (b, y))
+=
+δE(ab + x ⇀ b + y ⇀ a + ω(x, y), xy + x ↼ b + y ↼ a + ν(a, b))
+=
+δA(ab) + δA(x ⇀ b) + δA(y ⇀ a) + δA(ω(x, y)) + φ(ab) + φ(x ⇀ b)
++φ(y ⇀ a) + φ(ω(x, y)) − τφ(ab) − τφ(x ⇀ b) − τφ(y ⇀ a) − τφ(ω(x, y))
++p(ab) + p(x ⇀ b) + p(y ⇀ a) + p(ω(x, y)) + δH(xy) + δH(x ↼ b)
++δH(y ↼ a) + δH(ν(a, b)) + ψ(xy) + ψ(x ↼ b) + ψ(y ↼ a) + ψ(ν(a, b))
+19
+
+−τψ(xy) − τψ(x ↼ b) − τψ(y ↼ a) − τψ(ν(a, b)) + q(xy) + q(x ↼ b)
++q(y ↼ a) + q(ν(a, b)),
+and the right hand side is equal to
+(a, x)[1] ·E (b, y) ⊗ (a, x)[2] + (a, x) ·E (b, y)[1] ⊗ (b, y)[2] + (b, y)1 ⊗ [(a, x), (b, y)2]E
++(a, x)1 ⊗ [(b, y), (a, x)2]E
+=
+a[1]b ⊗ a[2] + (y ⇀ a[1]) ⊗ a[2] + (y ↼ a[1]) ⊗ a[2] + ν(a[1], b) ⊗ a[2]
++(a⟨−1⟩ ⇀ b) ⊗ a⟨0⟩ + ω(a⟨−1⟩, y) ⊗ a⟨0⟩ + a⟨−1⟩y ⊗ a⟨0⟩ + (a⟨−1⟩ ↼ b) ⊗ a⟨0⟩
+−a⟨0⟩b ⊗ a⟨−1⟩ − (y ⇀ a⟨0⟩) ⊗ a⟨−1⟩ − (y ↼ a⟨0⟩) ⊗ a⟨−1⟩ − ν(a⟨0⟩, b) ⊗ a⟨−1⟩
++(a1p ⇀ b) ⊗ a2p + ω(a1p, y) ⊗ a2p + a1py ⊗ a2p + (a1p ↼ b) ⊗ a2p
++(x[1] ⇀ b) ⊗ x[2] + ω(x[1], y) ⊗ x[2] + x[1]y ⊗ x[2] + (x[1] ↼ b) ⊗ x[2]
++(x⟨0⟩ ⇀ b) ⊗ x⟨1⟩ + ω(x⟨0⟩, y) ⊗ x⟨1⟩ + x⟨0⟩y ⊗ x⟨1⟩ + (x⟨0⟩ ↼ b) ⊗ x⟨1⟩
+−x⟨1⟩b ⊗ x⟨0⟩ − (y ⇀ x⟨1⟩) ⊗ x⟨0⟩ − (y ↼ x⟨1⟩) ⊗ x⟨0⟩ − ν(x⟨1⟩, b) ⊗ x⟨0⟩
++x1qb ⊗ x2q + (y ⇀ x1q) ⊗ x2q + (y ↼ x1q) ⊗ x2q + ν(x1q, b) ⊗ x2q
++ab[1] ⊗ b[2] + (x ⇀ b[1]) ⊗ b[2] + (x ↼ b[1]) ⊗ b[2] + ν(a, b[1]) ⊗ b[2]
++(b⟨−1⟩ ⇀ a) ⊗ b⟨0⟩ + ω(x, b⟨−1⟩) ⊗ b⟨0⟩ + xb⟨−1⟩ ⊗ b⟨0⟩ + (b⟨−1⟩ ↼ a) ⊗ b⟨0⟩
+−ab⟨0⟩ ⊗ b⟨−1⟩ − (x ⇀ b⟨0⟩) ⊗ b⟨−1⟩ − (x ↼ b⟨0⟩) ⊗ b⟨−1⟩ − ν(a, b⟨0⟩) ⊗ b⟨−1⟩
++(b1p ⇀ a) ⊗ b2p + ω(x, b1p) ⊗ b2p + xb1p ⊗ b2p + (b1p ↼ a) ⊗ b2p
++(y[1] ⇀ a) ⊗ y[2] + ω(x, y[1]) ⊗ y[2] + (y[1] ↼ a) ⊗ y[2] + xy[1] ⊗ y[2]
++(y⟨0⟩ ⇀ a) ⊗ y⟨1⟩ + ω(x, y⟨0⟩) ⊗ y⟨1⟩ + xy⟨0⟩ ⊗ y⟨1⟩ + (y⟨0⟩ ↼ a) ⊗ y⟨1⟩
+−ay⟨1⟩ ⊗ y⟨0⟩ − (x ⇀ y⟨1⟩) ⊗ y⟨0⟩ − (x ↼ y⟨1⟩) ⊗ y⟨0⟩ − ν(a, y⟨1⟩) ⊗ y⟨0⟩
++ay1q ⊗ y2q + (x ⇀ y1q) ⊗ y2q + (x ↼ y1q) ⊗ y2q + ν(a, y1q) ⊗ y2q
++b1 ⊗ [a, b2] + b1 ⊗ (x ⊲ b2) + b1 ⊗ (x ⊳ b2) + b1 ⊗ θ(a, b2)
++b(−1) ⊗ [a, b(0)] + b(−1) ⊗ (x ⊲ b(0)) + b(−1) ⊗ (x ⊳ b(0)) + b(−1) ⊗ θ(a, b(0))
+−b(0) ⊗ (b(−1) ⊲ a) + b(0) ⊗ σ(x, b(−1)) + b(0) ⊗ [x, b(−1)] − b(0) ⊗ (b(−1) ⊳ a)
+−b1s ⊗ (b2s ⊲ a) + b1s ⊗ σ(x, b2s) + b1s ⊗ [x, b2s] − b1s ⊗ (b2s ⊳ a)
+−y1 ⊗ (y2 ⊲ a) + y1 ⊗ σ(x, y2) + y1 ⊗ [x, y2] − y1 ⊗ (y2 ⊳ a)
++y(0) ⊗ [a, y(1)] + y(0) ⊗ (x ⊲ y(1)) + y(0) ⊗ (x ⊳ y(1)) + y(0) ⊗ θ(a, y(1))
+−y(1) ⊗ (y(0) ⊲ a) + y(1) ⊗ σ(x, y(0)) + y(1) ⊗ [x, y(0)] − y(1) ⊗ (y(0) ⊳ a)
++y1t ⊗ [a, y2t] + y1t ⊗ (x ⊲ y2t) + y1t ⊗ (x ⊳ y2t) + y1t ⊗ θ(a, y2t)
++a1 ⊗ [b, a2] + a1 ⊗ (y ⊲ a2) + a1 ⊗ (y ⊳ a2) + a1 ⊗ θ(b, a2)
++a(−1) ⊗ [b, a(0)] + a(−1) ⊗ (y ⊲ a(0)) + a(−1) ⊗ (y ⊳ a(0)) + a(−1) ⊗ θ(b, a(0))
+−a(0) ⊗ (a(−1) ⊲ b) + a(0) ⊗ σ(y, a(−1)) + a(0) ⊗ [y, a(−1)] − a(0) ⊗ (a(−1) ⊳ b)
+−a1s ⊗ (a2s ⊲ b) + a1s ⊗ σ(y, a2s) + a1s ⊗ [y, a2s] − a1s ⊗ (a2s ⊳ b)
+20
+
+−x1 ⊗ (x2 ⊲ b) + x1 ⊗ σ(y, x2) + x1 ⊗ [y, x2] − x1 ⊗ (x2 ⊳ b)
++x(0) ⊗ [b, x(1)] + x(0) ⊗ (y ⊲ x(1)) + x(0) ⊗ (y ⊳ x(1)) + x(0) ⊗ θ(b, x(1))
+−x(1) ⊗ (x(0) ⊲ b) + x(1) ⊗ σ(y, x(0)) + x(1) ⊗ [y, x(0)] − x(1) ⊗ (x(0) ⊳ b)
++x1t ⊗ [b, x2t] + x1t ⊗ (y ⊲ x2t) + x1t ⊗ (y ⊳ x2t) + x1t ⊗ θ(b, x2t).
+If we compare both the two sides item by item, one will find all the cocycle double matched
+pair conditions (CDM1)–(CDM8) in Definition 4.14.
+For the second equation, the left hand side is equal to
+∆E([(a, x), (b, y)]E)
+=
+∆E([a, b] + x ⊲ b − y ⊲ a + σ(x, y), [x, y] + x ⊳ b − y ⊳ a + θ(a, b))
+=
+∆A([a, b]) + ∆A(x ⊲ b) − ∆A(y ⊲ a) + ∆A(σ(x, y)) + ρ([a, b]) + ρ(x ⊲ b)
+−ρ(y ⊲ a) + ρ(σ(x, y)) + τρ([a, b]) + τρ(x ⊲ b) − τρ(y ⊲ a) + τρ(σ(x, y))
++s([a, b]) + s(x ⊲ b) − s(y ⊲ a) + s(σ(x, y)) + ∆H([x, y]) + ∆H(x ⊳ b)
+−∆H(y ⊳ a) + ∆H(θ(a, b)) + γ([x, y]) + γ(x ⊳ b) − γ(y ⊳ a) + γ(θ(a, b))
++τγ([x, y]) + τγ(x ⊳ b) − τγ(y ⊳ a) + τγ(θ(a, b)) + t([x, y]) + t(x ⊳ b)
+−t(y ⊳ a) + t(θ(a, b)),
+and the right hand side is equal to
+[(a, x), (b, y)1]E ⊗ (b, y)2 + (b, y)1 ⊗ [(a, x), (b, y)2]E
++(b, y) ·E (a, x)[1] ⊗ (a, x)[2] − (a, x)[1] ⊗ (b, y) ·E (a, x)[2]
+=
+[a, b1] ⊗ b2 + (x ⊲ b1) ⊗ b2 + (x ⊳ b1) ⊗ b2 + θ(a, b1) ⊗ b2
+−(b(−1) ⊲ a) ⊗ b(0) + σ(x, b(−1)) ⊗ b(0) + [x, b(−1)] ⊗ b(0) − (b(−1) ⊳ a) ⊗ b(0)
++[a, b(0)] ⊗ b(−1) + (x ⊲ b(0)) ⊗ b(−1) + (x ⊳ b(0)) ⊗ b(−1) + θ(a, b(0)) ⊗ b(−1)
+−(b1s ⊲ a) ⊗ b2s + σ(x, b1s) ⊗ b2s + [x, b1s] ⊗ b2s − (b1s ⊳ a) ⊗ b2s
+−(y1 ⊲ a) ⊗ y2 + σ(x, y1) ⊗ y2 + [x, y1] ⊗ y2 − (y1 ⊳ a) ⊗ y2
+−(y(0) ⊲ a) ⊗ y(1) + σ(x, y(0)) ⊗ y(1) + [x, y(0)] ⊗ y(1) − (y(0) ⊳ a) ⊗ y(1)
++[a, y(1)] ⊗ y(0) + (x ⊲ y(1)) ⊗ y(0) + (x ⊳ y(1)) ⊗ y(0) + θ(a, y(1)) ⊗ y(0)
++[a, y1t] ⊗ y2t + (x ⊲ y1t) ⊗ y2t + (x ⊳ y1t) ⊗ y2t + θ(a, y1t) ⊗ y2t
++b1 ⊗ [a, b2] + b1 ⊗ (x ⊲ b2) + b1 ⊗ (x ⊳ b2) + b1 ⊗ θ(a, b2)
++b(−1) ⊗ [a, b(0)] + b(−1) ⊗ (x ⊲ b(0)) + b(−1) ⊗ (x ⊳ b(0)) + b(−1) ⊗ θ(a, b(0))
+−b(0) ⊗ (b(−1) ⊲ a) + b(0) ⊗ σ(x, b(−1)) + b(0) ⊗ [x, b(−1)] − b(0) ⊗ (b(−1) ⊳ a)
+−b1s ⊗ (b2s ⊲ a) + b1s ⊗ σ(x, b2s) + b1s ⊗ [x, b2s] − b1s ⊗ (b2s ⊳ a)
+−y1 ⊗ (y2 ⊲ a) + y1 ⊗ σ(x, y2) + y1 ⊗ [x, y2] − y1 ⊗ (y2 ⊳ a)
++y(0) ⊗ [a, y(1)] + y(0) ⊗ (x ⊲ y(1)) + y(0) ⊗ (x ⊳ y(1)) + y(0) ⊗ θ(a, y(1))
+−y(1) ⊗ (y(0) ⊲ a) + y(1) ⊗ σ(x, y(0)) + y(1) ⊗ [x, y(0)] − y(1) ⊗ (y(0) ⊳ a)
+21
+
++y1t ⊗ [a, y2t] + y1t ⊗ (x ⊲ y2t) + y1t ⊗ (x ⊳ y2t) + y1t ⊗ θ(a, y2t)
++ba[1] ⊗ a[2] + (y ⇀ a[1]) ⊗ a[2] + (y ↼ a[1]) ⊗ a[2] + ν(b, a[1]) ⊗ a[2]
++(a⟨−1⟩ ⇀ b) ⊗ a⟨0⟩ + ω(y, a⟨−1⟩) ⊗ a⟨0⟩ + ya⟨−1⟩ ⊗ a⟨0⟩ + (a⟨−1⟩ ↼ b) ⊗ a⟨0⟩
+−ba⟨0⟩ ⊗ a⟨−1⟩ − (y ⇀ a⟨0⟩) ⊗ a⟨−1⟩ − (y ↼ a⟨0⟩) ⊗ a⟨−1⟩ − ν(b, a⟨0⟩) ⊗ a⟨−1⟩
++(a1p ⇀ b) ⊗ a2p + ω(y, a1p) ⊗ a2p + ya1p ⊗ a2p + (a1p ↼ b) ⊗ a2p
++(x[1] ⇀ b) ⊗ x[2] + ω(y, x[1]) ⊗ x[2] + yx[1] ⊗ x[2] + (x[1] ↼ b) ⊗ x[2]
++(x⟨0⟩ ⇀ b) ⊗ x⟨1⟩ + ω(y, x⟨0⟩) ⊗ x⟨1⟩ + yx⟨0⟩ ⊗ x⟨1⟩ + (x⟨0⟩ ↼ b) ⊗ x⟨1⟩
+−bx⟨1⟩ ⊗ x⟨0⟩ − (y ⇀ x⟨1⟩) ⊗ x⟨0⟩ − (y ↼ x⟨1⟩) ⊗ x⟨0⟩ − ν(b, x⟨1⟩) ⊗ x⟨0⟩
++bx1q ⊗ x2q + (y ⇀ x1q) ⊗ x2q + (y ↼ x1q) ⊗ x2q + ν(b, x1q) ⊗ x2q
+−a[1] ⊗ ba[2] − a[1] ⊗ (y ⇀ a[2]) − a[1] ⊗ (y ↼ a[2]) − a[1] ⊗ ν(b, a[2])
+−a⟨−1⟩ ⊗ ba⟨0⟩ − a⟨−1⟩ ⊗ (y ⇀ a⟨0⟩) − a⟨−1⟩ ⊗ (y ↼ a⟨0⟩) − a⟨−1⟩ ⊗ ν(b, a⟨0⟩)
++a⟨0⟩ ⊗ (a⟨−1⟩ ⇀ b) + a⟨0⟩ ⊗ ω(y, a⟨−1⟩) + a⟨0⟩ ⊗ ya⟨−1⟩ + a⟨0⟩ ⊗ (a⟨−1⟩ ↼ b)
+−a1p ⊗ (a2p ⇀ b) − a1p ⊗ ω(y, a2p) − a1p ⊗ ya2p − a1p ⊗ (a2p ↼ b)
+−x[1] ⊗ (x[2] ⇀ b) − x[1] ⊗ ω(y, x[2]) − x[1] ⊗ yx[2] − x[1] ⊗ (x[2] ↼ b)
+−x⟨0⟩ ⊗ bx⟨1⟩ − x⟨0⟩ ⊗ (y ⇀ x⟨1⟩) − x⟨0⟩ ⊗ (y ↼ x⟨1⟩) − x⟨0⟩ ⊗ ν(b, x⟨1⟩)
++x⟨1⟩ ⊗ (x⟨0⟩ ��� b) + x⟨1⟩ ⊗ ω(y, x⟨0⟩) + x⟨1⟩ ⊗ yx⟨0⟩ + x⟨1⟩ ⊗ (x⟨0⟩ ↼ b)
+−x1q ⊗ bx2q − x1q ⊗ (y ⇀ x2q) − x1q ⊗ (y ↼ x2q) − x1q ⊗ ν(b, x2q).
+If we compare both the two sides term by term, one obtain all the cocycle double matched pair
+conditions (CDM9)–(CDM16) in Definition 4.14.
+This complete the proof.
+5
+Extending structures for Poisson bialgebras
+In this section, we will study the extending problem for Poisson bialgebras. We will find some
+special cases when the braided Poisson bialgebra is reduced into an ordinary Poisson bialgebra.
+It is proved that the extending problem can be solved by using of the non-abelian cohomology
+theory based on our cocycle bicrossedproduct for braided Poisson bialgebras in last section.
+5.1
+Extending structures for Poisson algebras
+First we are going to study extending problem for Poisson algebras.
+There are two cases for A to be a Poisson algebra in the cocycle cross product system
+defined in last section, see condition (CC6). The first case is when we let ⇀, ⊲ to be trivial
+and θ ̸= 0, ν ̸= 0, then from conditions (CP1) and (CP3) we get σ(x, ν(a, b)) = ω(x, θ(a, b)) = 0,
+since θ ̸= 0, ν ̸= 0 we assume σ = 0, ω = 0 for simplicity, thus we obtain the following type
+(a1) unified product for Poisson algebras.
+22
+
+Lemma 5.1. ([5]) Let A be a Poisson algebra and V a vector space. An extending datum of
+A by V of type (a1) is Ω(1)(A, V ) consisting of bilinear maps
+⊳ : V × A → V,
+θ : A × A → V,
+↼: V × A → V,
+ν : A × A → V.
+Denote by A#θ,νV the vector space E = A ⊕ V together with the multiplication given by
+[(a, x), (b, y)]
+:=
+�
+[a, b], [x, y] + x ⊳ b − y ⊳ a + θ(a, b)
+�
+,
+(37)
+(a, x) · (b, y)
+:=
+�
+ab, xy + x ↼ b + y ↼ a + ν(a, b)
+�
+.
+(38)
+Then A#θ,νV is a Poisson algebra if and only if the following compatibility conditions hold for
+all a, b ∈ A, x, y, z ∈ V :
+(A0)
+�
+↼, ν) is an algebra extending system of the associative algebra A trough V and
+�
+⊳, θ
+�
+is a Lie extending system of the Lie algebra A trough V ,
+(A1) [x, y ↼ a] = [x, y] ↼ a + y(x ⊳ a),
+(A2) [x, ν(a, b)] + x ⊳ (ab) = (x ⊳ a) ↼ b + (x ⊳ b) ↼ a,
+(A3) (xy) ⊳ a = (x ⊳ a)y + x(y ⊳ a),
+(A4) (x ⊳ a) ↼ b = θ(a, b)x + x ↼ [a, b] + (x ↼ b) ⊳ a,
+(A5) [x, yz] = [x, y]z + y[x, z].
+Note that (A1)–(A4) are deduced from (CP1)–(CP4) and by (A5) we obtain that V is
+a Poisson algebra. Furthermore, V is in fact a Poisson subalgebra of A#θ,νV but A is not
+although A is itself a Poisson algebra.
+Denote the set of all algebraic extending datum of A by V of type (a1) by A(1)(A, V ).
+In the following, we always assume that A is a subspace of a vector space E, there exists a
+projection map p : E → A such that p(a) = a, for all a ∈ A. Then the kernel space V := ker(p)
+is also a subspace of E and a complement of A in E.
+Lemma 5.2. ([5]) Let A be a Poisson algebra and E a vector space containing A as a subspace.
+Suppose that there is a Poisson algebra structure on E such that V is a Poisson subalgebra of
+E and the canonical projection map p : E → A is a Poisson algebra homomorphism. Then
+there exists a Poisson algebraic extending datum Ω(1)(A, V ) of A by V such that E ∼= A#θ,νV .
+Proof. Since V is a Poisson subalgebra of E, we have x ·E y ∈ V for all x, y ∈ V . We define
+the extending datum of A through V by the following formulas:
+⊳ : V ⊗ A → V,
+x ⊳ a
+:=
+[x, a]E − p([x, a]E),
+θ : A ⊗ A → V,
+θ(a, b)
+:=
+[a, b]E − p
+�
+[a, b]E
+�
+,
+[, ]V : V ⊗ V → V,
+[x, y]V
+:=
+[x, y]E,
+23
+
+↼: V ⊗ A → V,
+x ↼ a
+:=
+x ·E a − p(x ·E a),
+ν : A ⊗ A → V,
+ν(a, b)
+:=
+a ·E b − p
+�
+a ·E b
+�
+,
+·V : V ⊗ V → V,
+x ·V y
+:=
+x ·E y,
+for any a, b ∈ A and x, y ∈ V . It is easy to see that the above maps are well defined and
+Ω(1)(A, V ) is an extending system of A trough V and
+ϕ : A#θ,νV → E,
+ϕ(a, x) := a + x
+is an isomorphism of Poisson algebras.
+Lemma 5.3. Let Ω(1)(A, V ) =
+�
+↼, ⊳, θ, ν, ·, [, ]
+�
+and Ω′(1)(A, V ) =
+�
+↼′, ⊳′, θ′, ν′, ·′, [, ]′�
+be two
+algebraic extending datums of A by V of type (a1) and A#θ,νV , A#θ′,ν′V be the corresponding
+unified products. Then there exists a bijection between the set of all homomorphisms of Poisson
+algebras ϕ : Aθ,ν#↼,⊳V → Aθ′,ν′#↼′,⊳′V whose restriction on A is the identity map and the
+set of pairs (r, s), where r : V → A and s : V → V are two linear maps satisfying
+r(x ⊳ a) = [r(x), a],
+(39)
+[a, b]′ = [a, b] + rθ(a, b),
+(40)
+r([x, y]) = [r(x), r(y)]′,
+(41)
+s(x) ⊳′ a + θ′(r(x), a) = s(x ⊳ a),
+(42)
+θ′(a, b) = sθ(a, b),
+(43)
+s([x, y]) = [s(x), s(y)]′ + s(x) ⊳′ r(y) − s(y) ⊳′ r(x) + θ′(r(x), r(y)),
+(44)
+r(x ↼ a) = r(x) ·′ a,
+(45)
+a ·′ b = ab + rν(a, b),
+(46)
+r(xy) = r(x) ·′ r(y),
+(47)
+s(x) ↼′ a + ν′(r(x), a) = s(x ↼ a),
+(48)
+ν′(a, b) = sν(a, b),
+(49)
+s(xy) = s(x) ·′ s(y) + s(x) ↼′ r(y) + s(y) ↼′ r(x) + ν′(r(x), r(y)),
+(50)
+for all a ∈ A and x, y ∈ V .
+Under the above bijection the homomorphism of Poisson algebras ϕ = ϕr,s : A#θ,νV →
+A#θ′,ν′V to (r, s) is given by ϕ(a, x) = (a + r(x), s(x)) for all a ∈ A and x ∈ V . Moreover,
+ϕ = ϕr,s is an isomorphism if and only if s : V → V is a linear isomorphism.
+Proof. Let ϕ : A#θ,νV → A#θ′,ν′V be a Poisson algebra homomorphism whose restriction on
+A is the identity map. Then ϕ is determined by two linear maps r : V → A and s : V → V
+such that ϕ(a, x) = (a + r(x), s(x)) for all a ∈ A and x ∈ V . In fact, we have to show
+ϕ([(a, x), (b, y)]) = [ϕ(a, x), ϕ(b, y)]′,
+ϕ((a, x)(b, y)) = ϕ(a, x) ·′ ϕ(b, y).
+24
+
+For the first equation, the left hand side is equal to
+ϕ([(a, x), (b, y)])
+=
+ϕ ([a, b], x ⊳ b − y ⊳ a + [x, y] + θ(a, b))
+=
+�
+[a, b] + r(x ⊳ b) − r(y ⊳ a) + r([x, y]) + rθ(a, b),
+s(x ⊳ b) − s(y ⊳ a) + s([x, y]) + sθ(a, b)
+�
+,
+and the right hand side is equal to
+[ϕ(a, x), ϕ(b, y)]′
+=
+[(a + r(x), s(x)), (b + r(y), s(y))]′
+=
+�
+[a + r(x), b + r(y)]′, s(x) ⊳′ (b + r(y)) − s(y) ⊳′ (a + r(x))
++[s(x), s(y)]′ + θ′(a + r(x), b + r(y))
+�
+.
+For the second equation, the left hand side is equal to
+ϕ((a, x)(b, y))
+=
+ϕ (ab, x ↼ b + y ↼ a + xy + ν(a, b))
+=
+�
+ab + r(x ↼ b) + r(y ↼ a) + r(xy) + rν(a, b),
+s(x ↼ b) + s(y ↼ a) + s(xy) + sν(a, b)
+�
+,
+and the right hand side is equal to
+ϕ(a, x) ·′ ϕ(b, y)
+=
+(a + r(x), s(x)) ·′ (b + r(y), s(y))
+=
+�
+(a + r(x)) ·′ (b + r(y)), s(x) ↼′ (b + r(y)) + s(y) ↼′ (a + r(x))
++s(x) ·′ s(y) + ν′(a + r(x), b + r(y))
+�
+.
+Thus ϕ is a homomorphism of Poisson algebras if and only if the above conditions hold.
+The second case is when θ = 0, ν = 0, we obtain the following type (a2) unified product.
+Theorem 5.4. ([5]) Let A be a Poisson algebra and V a vector space. An extending datum
+of A through V of type (a1) is Ω(2)(A, V ) consisting of bilinear maps
+⊳ : V × A → V,
+⊲ : V × A → A,
+σ : V × V → A,
+↼: V × A → V,
+⇀: V × A → A,
+ω : V × V → A.
+Denote by Aσ,ω#V the vector space E = A ⊕ V together with the multiplication given by
+[(a, x), (b, y)]
+:=
+�
+[a, b] + x ⊲ b − y ⊲ a + σ(x, y), [x, y] + x ⊳ b − y ⊳ a
+�
+,
+(51)
+(a, x) · (b, y)
+:=
+�
+ab + x ⇀ b + y ⇀ a + ω(x, y), xy + x ↼ b + y ↼ a
+�
+.
+(52)
+Then Aσ,ω#V is a Poisson algebra if and only if the following compatibility conditions hold
+for all a, b ∈ A, x, y, z ∈ V :
+25
+
+(B0)
+�
+⇀, ↼, ω) is an algebra extending system of the associative algebra A trough V and
+�
+⊲, ⊳, σ
+�
+is a Lie extending system of the Lie algebra A trough V ,
+(B1) x ⊲ (ab) = (x ⊲ a) b + (x ⊳ a) ⇀ b + a (x ⊲ b) + (x ⊳ b) ⇀ a,
+(B2) x ⊳ (ab) = (x ⊳ a) ↼ b + (x ⊳ b) ↼ a,
+(B3) x ⇀ [a, b] = [a, x ⇀ b] + (x ⊳ a) ⇀ b + (x ⊲ a)b − (x ↼ b) ⊲ a,
+(B4) x ↼ [a, b] = (x ⊳ a) ↼ b − (x ↼ b) ⊳ a,
+(B5) (xy) ⊲ a = [a, ω(x, y)] + y ⇀ (x ⊲ a) + ω(x ⊳ a, y) + x ⇀ (y ⊲ a) + ω(x, y ⊳ a),
+(B6) (xy) ⊳ a = (x ⊳ a)y + y ↼ (x ⊲ a) + x(y ⊳ a) + x ↼ (y ⊲ a),
+(B7) [x, y] ⇀ a = x ⊲ (y ⇀ a) + σ(x, y ↼ a) − σ(x, y)a − y ⇀ (x ⊲ a) − ω(y, x ⊳ a),
+(B8) [x, y] ↼ a = [x, y ↼ a] + x ⊳ (y ⇀ a) − y(x ⊳ a) − y ↼ (x ⊲ a),
+(B9) σ(x, yz) = −x ⊲ ω(y, z) + z ⇀ σ(x, y) + ω([x, y], z) + y ⇀ σ(x, z) + ω(y, [x, z]),
+(B10) [x, yz] = [x, y]z + y[x, z] − x ⊳ ω(y, z) + z ↼ σ(x, y) + y ↼ σ(x, z).
+Theorem 5.5. ([5]) Let A be a Poisson algebra, E a vector space containing A as a subspace.
+If there is a Poisson algebra structure on E such that A is a Poisson subalgebra of E. Then
+there exists a Poisson algebraic extending structure Ω(A, V ) =
+�
+⊳, ⊲, ↼, ⇀, σ, ω
+�
+of A through
+V such that there is an isomorphism of Poisson algebras E ∼= Aσ,ω#V .
+Lemma 5.6. Let Ω(1)(A, V ) =
+�
+⊲, ⊳, ↼, ⇀, σ, ω, ·, [, ]
+�
+and Ω′(1)(A, V ) =
+�
+⊲′, ⊳′, ↼′, ⇀′, σ′, ω′, ·′, [, ]′�
+be two Poisson algebraic extending structures of A through V and Aσ,ω#V , Aσ′,ω′#V the asso-
+ciated unified products. Then there exists a bijection between the set of all homomorphisms of
+algebras ψ : Aσ,ω#V → Aσ′,ω′#V which stabilize A and the set of pairs (r, s), where r : V → A,
+s : V → V are linear maps satisfying the following compatibility conditions for any x ∈ A, u,
+v ∈ V :
+(M1) r([x, y]) = [r(x), r(y)]′ + σ′(s(x), s(y)) − σ(x, y) + s(x) ⊲′ r(y) − s(y) ⊲′ r(x),
+(M2) s([x, y]) = s(x) ⊳′ r(y) − s(y) ⊳′ r(x) + [s(x), s(y)]′,
+(M3) r(x ⊳ a) = [r(x), a] + s(x) ⊲′ a − x ⊲ a,
+(M4) s(x ⊳ a) = s(x) ⊳′ a,
+(M5) r(x · y) = r(x) ·′ r(y) + ω′(s(x), s(y)) − ω(x, y) + s(x) ⇀′ r(y) + s(y) ⇀′ r(x),
+(M6) s(x · y) = s(y) ↼′ r(x) + s(x) ↼′ r(y) + s(x) ·′ s(y),
+(M7) r(x ⊳ a) = r(x) ·′ a − x ⇀ a + s(x) ⇀′ a,
+26
+
+(M8) s(x ⊳ a) = s(x) ↼′ a.
+Under the above bijection the homomorphism of algebras ϕ = ϕ(r,s) : Aσ,ω#V → Aσ′,ω′#V
+corresponding to (r, s) is given for any a ∈ A and x ∈ V by:
+ϕ(a, x) = (a + r(x), s(x)).
+Moreover, ϕ = ϕ(r,s) is an isomorphism if and only if s : V → V is an isomorphism linear
+map.
+The proof of the above is similar as to the proof of Lemma 5.3, so we omit the details.
+Let A be a Poisson algebra and V a vector space.
+Two algebraic extending systems
+Ω(i)(A, V ) and Ω′(i)(A, V ) are called equivalent if ϕr,s is an isomorphism. We denote it by
+Ω(i)(A, V ) ≡ Ω′(i)(A, V ). From the above lemmas, we obtain the following result.
+Theorem 5.7. Let A be a Poisson algebra, E a vector space containing A as a subspace and
+V be a complement of A in E. Denote HA(V, A) := A(1)(A, V ) ⊔ A(2)(A, V )/ ≡. Then the
+map
+Ψ : HA(V, A) → Extd(E, A),
+Ω(1)(A, V ) �→ A#θ,νV,
+Ω(2)(A, V ) �→ Aσ,ω#V
+(53)
+is bijective, where Ω(i)(A, V ) is the equivalence class of Ω(i)(A, V ) under ≡.
+5.2
+Extending structures for Poisson coalgebras
+Next we consider the Poisson coalgebra structures on E = Ap,s#q,tV .
+There are two cases for (A, ∆A, δA) to be a Poisson coalgebra.
+The first case is when
+q = 0, t = 0, then we obtain the following type (c1) unified product for Poisson coalgebras.
+Lemma 5.8. Let (A, ∆A, δA) be a Poisson coalgebra and V a vector space.
+An extending
+datum of A by V of type (c1) is Ω(3)(A, V ) = (φ, ψ, ρ, γ, p, s, ∆V , δV ) with linear maps
+∆V : V → V ⊗ V,
+δV : V → V ⊗ V,
+φ : A → V ⊗ A,
+ψ : V → V ⊗ A,
+ρ : A → V ⊗ A,
+γ : V → V ⊗ A,
+p : A → V ⊗ V,
+s : A → V ⊗ V.
+Denote by Ap,s#V the vector space E = A ⊕ V with the linear maps δE : E → E ⊗ E ,
+∆E : E → E ⊗ E given by
+δE(a) = (δA + φ − τφ + p)(a),
+δE(x) = (δV + ψ − τψ)(x),
+∆E(a) = (∆A + ρ + τρ + s)(a),
+∆E(x) = (∆V + γ + τγ)(x),
+27
+
+that is
+δE(a) = a[1] ⊗ a[2] + a⟨−1⟩ ⊗ a⟨0⟩ − a⟨0⟩ ⊗ a⟨−1⟩ + a1p ⊗ a2p,
+δE(x) = x[1] ⊗ x[2] + x⟨0⟩ ⊗ x⟨1⟩ − x⟨1⟩ ⊗ x⟨0⟩,
+∆E(a) = a1 ⊗ a2 + a(−1) ⊗ a(0) + a(0) ⊗ a(−1) + a1s ⊗ a2s,
+∆E(x) = x1 ⊗ x2 + x(0) ⊗ x(1) + x(1) ⊗ x(0).
+Then Ap,s#V is a Poisson coalgebra with the comultiplication given above if and only if the
+following compatibility conditions hold:
+(C0)
+�
+ρ, γ, s) is an algebra extending system of the associative coalgebra A trough V and
+�
+φ, ψ, p
+�
+is a Lie extending system of the Lie coalgebra A trough V ,
+(C1) a[1] ⊗ ρ(a[2]) − a⟨0⟩ ⊗ γ(a⟨−1⟩) = −τφ(a1) ⊗ a2 − τψ(a(−1)) ⊗ a(0)
++ τ12(a(−1) ⊗ δA(a(0))) + τ12(a1s ⊗ q(a2s)),
+(C2) a⟨0⟩ ⊗ ∆V (a⟨−1⟩) − a[1] ⊗ s(a[2]) = τφ(a(0)) ⊗ a(−1) + τψ(a1s) ⊗ a2s
++ τ12(a(−1) ⊗ τφ(a(0))) + τ12(a1s ⊗ τψ(a2s)),
+(C3) a⟨−1⟩ ⊗ ∆A(a⟨0⟩) = φ(a1) ⊗ a2 + ψ(a(−1)) ⊗ a(0) + τ12(a1 ⊗ φ(a2)) + τ12(a(0) ⊗ ψ(a(−1))),
+(C4) a⟨−1⟩ ⊗ ρ(a⟨0⟩) + a1p ⊗ γ(a2p) = δV (a(−1)) ⊗ a(0) + p(a1) ⊗ a2
++ τ12(a(−1) ⊗ φ(a(0))) + τ12(a1s ⊗ ψ(a2s)),
+(C5) x[1] ⊗ γ(x[2]) + x⟨0⟩ ⊗ ρ(x⟨1⟩) = δV (x(0)) ⊗ x(1) + τ12(x1 ⊗ ψ(x2)) + τ12(x(0) ⊗ φ(x(1))),
+(C6) x⟨0⟩ ⊗ ∆A(x⟨1⟩) = ψ(x(0)) ⊗ x(1) + τ12(x(1) ⊗ ψ(x(0))),
+(C7) x⟨1⟩ ⊗ ∆V (x⟨0⟩) = τψ(x1) ⊗ x2 + τφ(x(1)) ⊗ x(0) + τ12(x1 ⊗ τψ(x2)) + τ12(x(0) ⊗ τφ(x(1))),
+(C8) x⟨1⟩ ⊗ γ(x⟨0⟩) = τψ(x(0)) ⊗ x(1) − τ12(x(0) ⊗ δA(x(1))),
+(C9) x[1] ⊗ ∆V (x[2]) + x⟨0⟩ ⊗ s(x⟨1⟩)
+= δV (x(0)) ⊗ x(1) + p(x(1)) ⊗ x(0) + τ12(x1 ⊗ δH(x2)) + τ12(x(0) ⊗ p(x(1))).
+Denote the set of all coalgebraic extending datum of A by V of type (c1) by C(3)(A, V ).
+Lemma 5.9. Let (A, ∆A, δA) be a Poisson coalgebra and E a vector space containing A as
+a subspace. Suppose that there is a Poisson coalgebra structure (E, ∆E, δE) on E such that
+p : E → A is a Poisson coalgebra homomorphism. Then there exists a Poisson coalgebraic
+extending system Ω(3)(A, V ) of (A, ∆A, δA) by V such that (E, ∆E, δE) ∼= Ap,s#V .
+Proof. Let p : E → A and π : E → V be the projection map and V = ker(p). Then the
+extending datum of (A, ∆A, δA) by V is defined as follows:
+φ : A → V ⊗ A,
+φ(a) = (π ⊗ p)δE(a),
+ψ : V → V ⊗ A,
+ψ(x) = (π ⊗ p)δE(x),
+28
+
+ρ : A → V ⊗ A,
+ρ(a) = (π ⊗ p)∆E(a),
+γ : V → V ⊗ A,
+γ(x) = (π ⊗ p)∆E(x),
+δV : V → V ⊗ V,
+δV (x) = (π ⊗ π)δE(x),
+∆V : V → V ⊗ V,
+∆V (x) = (π ⊗ π)∆E(x),
+p : A → V ⊗ V,
+p(a) = (π ⊗ π)δE(a),
+s : A → V ⊗ V,
+s(a) = (π ⊗ π)∆E(a).
+One check that ϕ : Ap,s#V → E given by ϕ(a, x) = a + x for all a ∈ A, x ∈ V is a Poisson
+coalgebra isomorphism.
+Lemma 5.10. Let
+Ω(3)(A, V ) = (φ, ψ, ρ, γ, p, s, δV , ∆V )
+and
+Ω′(3)(A, V ) = (φ′, ψ′, ρ′, γ′, p′, s′, δ′
+V , ∆′
+V )
+be two Poisson coalgebraic extending datums of (A, ∆A, δA) by V . Then there exists a bijection
+between the set of Poisson coalgebra homomorphisms ϕ : Ap,s#V → Ap′,s′#V whose restriction
+on A is the identity map and the set of pairs (r, s), where r : V → A and s : V → V are two
+linear maps satisfying
+p′(a) = s(a1p) ⊗ s(a2p),
+(54)
+φ′(a) = s(a⟨−1⟩) ⊗ a⟨0⟩ + s(a1p) ⊗ r(a2p),
+(55)
+δ′
+A(a) = δA(a) + r(a⟨−1⟩) ⊗ a⟨0⟩ − a⟨0⟩ ⊗ r(a⟨−1⟩) + r(a1p) ⊗ r(a2p),
+(56)
+δ′
+V (s(x)) + p′(r(x)) = (s ⊗ s)δV (x),
+(57)
+ψ′(s(x)) + φ′(r(x)) = s(x[1]) ⊗ r(x[2]) + s(x⟨0⟩) ⊗ x⟨1⟩,
+(58)
+δ′
+A(r(x)) = r(x[1]) ⊗ r(x[2]) − x⟨1⟩ ⊗ r(x⟨0⟩) + r(x⟨0⟩) ⊗ x⟨1⟩,
+(59)
+s′(a) = s(a1s) ⊗ s(a2s),
+(60)
+ρ′(a) = s(a(−1)) ⊗ a(0) + s(a1s) ⊗ r(a2s),
+(61)
+∆′
+A(a) = ∆A(a) + r(a(−1)) ⊗ a(0) + a(0) ⊗ r(a(−1)) + r(a1s) ⊗ r(a2s),
+(62)
+∆′
+V (s(x)) + s′(r(x)) = (s ⊗ s)∆V (x),
+(63)
+γ′(s(x)) + ρ′(r(x)) = s(x1) ⊗ r(x2) + s(x(0)) ⊗ x(1),
+(64)
+∆′
+A(r(x)) = r(x1) ⊗ r(x2) + x(1) ⊗ r(x(0)) + r(x(0)) ⊗ x(1).
+(65)
+Under the above bijection the Poisson coalgebra homomorphism ϕ = ϕr,s : Ap,s#V → Ap′,s′#V
+to (r, s) is given by ϕ(a + x) = (a + r(x), s(x)) for all a ∈ A and x ∈ V . Moreover, ϕ = ϕr,s is
+an isomorphism if and only if s : V → V is a linear isomorphism.
+Proof. Let ϕ : Ap,s#V → Ap′,s′#V be a Poisson coalgebra homomorphism whose restriction
+on A is the identity map. Then ϕ is determined by two linear maps r : V → A and s : V → V
+29
+
+such that ϕ(a + x) = (a + r(x), s(x)) for all a ∈ A and x ∈ V . We will prove that ϕ is a
+homomorphism of Poisson coalgebras if and only if the above conditions hold. First it is easy
+to see that δ′
+Eϕ(a) = (ϕ ⊗ ϕ)δE(a) for all a ∈ A.
+δ′
+Eϕ(a)
+=
+δ′
+E(a) = δ′
+A(a) + φ′(a) − τφ′(a) + p′(a),
+and
+(ϕ ⊗ ϕ)δE(a)
+=
+(ϕ ⊗ ϕ) (δA(a) + φ(a) − τφ(a) + p(a))
+=
+δA(a) + r(a⟨−1⟩) ⊗ a⟨0⟩ + s(a⟨−1⟩) ⊗ a⟨0⟩ − a⟨0⟩ ⊗ r(a⟨−1⟩) − a⟨0⟩ ⊗ s(a⟨−1⟩)
++r(a1p) ⊗ r(a2p) + r(a1p) ⊗ s(a2p) + s(a1p) ⊗ r(a2p) + s(a1p) ⊗ s(a2p).
+Thus we obtain that δ′
+Eϕ(a) = (ϕ ⊗ ϕ)δE(a) if and only if the conditions (54), (55) and (56)
+hold. Then we consider that δ′
+Eϕ(x) = (ϕ ⊗ ϕ)δE(x) for all x ∈ V .
+δ′
+Eϕ(x)
+=
+δ′
+E(r(x) + s(x)) = δ′
+E(r(x)) + δ′
+E(s(x))
+=
+δ′
+A(r(x)) + φ′(r(x)) − τφ′(r(x)) + p′(r(x)) + δ′
+V (s(x)) + ψ′(s(x)) − τψ′(s(x)),
+and
+(ϕ ⊗ ϕ)δE(x)
+=
+(ϕ ⊗ ϕ)(δV (x) + ψ(x) − τψ(x))
+=
+(ϕ ⊗ ϕ)(x[1] ⊗ x[2] + x⟨0⟩ ⊗ x⟨1⟩ − x⟨1⟩ ⊗ x⟨0⟩)
+=
+r(x[1]) ⊗ r(x[2]) + r(x[1]) ⊗ s(x[2]) + s(x[1]) ⊗ r(x[2]) + s(x[1]) ⊗ s(x[2])
+−x⟨1⟩ ⊗ r(x⟨0⟩) − x⟨1⟩ ⊗ s(x⟨0⟩) + r(x⟨0⟩) ⊗ x⟨1⟩ + s(x⟨0⟩) ⊗ x⟨1⟩.
+Thus we obtain that δ′
+Eϕ(x) = (ϕ ⊗ ϕ)δE(x) if and only if the conditions (57), (58) and (59)
+hold.
+Then it is easy to see that ∆′
+Eϕ(a) = (ϕ ⊗ ϕ)∆E(a) for all a ∈ A.
+∆′
+Eϕ(a)
+=
+∆′
+E(a) = ∆′
+A(a) + ρ′(a) + τρ′(a) + s′(a),
+and
+(ϕ ⊗ ϕ)∆E(a)
+=
+(ϕ ⊗ ϕ) (∆A(a) + ρ(a) + τρ(a) + s(a))
+=
+∆A(a) + r(a(−1)) ⊗ a(0) + s(a(−1)) ⊗ a(0) + a(0) ⊗ r(a(−1)) + a(0) ⊗ s(a(−1))
++r(a1s) ⊗ r(a2s) + r(a1s) ⊗ s(a2s) + s(a1s) ⊗ r(a2s) + s(a1s) ⊗ s(a2s).
+Thus we obtain that ∆′
+Eϕ(a) = (ϕ ⊗ ϕ)∆E(a) if and only if the conditions (60), (61) and (62)
+hold. Then we consider that ∆′
+Eϕ(x) = (ϕ ⊗ ϕ)∆E(x) for all x ∈ V .
+∆′
+Eϕ(x)
+=
+∆′
+E(r(x) + s(x)) = ∆′
+E(r(x)) + ∆′
+E(s(x))
+30
+
+=
+∆′
+A(r(x)) + ρ′(r(x)) + τρ′(r(x)) + s(r(x)) + ∆′
+V (s(x)) + γ′(s(x)) + τγ′(s(x))),
+and
+(ϕ ⊗ ϕ)∆E(x)
+=
+(ϕ ⊗ ϕ)(∆V (x) + γ(x) + τγ(x))
+=
+(ϕ ⊗ ϕ)(x1 ⊗ x2 + x(0) ⊗ x(1) + x(1) ⊗ x(0))
+=
+r(x1) ⊗ r(x2) + r(x1) ⊗ s(x2) + s(x1) ⊗ r(x2) + s(x1) ⊗ s(x2)
++x(1) ⊗ r(x(0)) + x(1) ⊗ s(x(0)) + r(x(0)) ⊗ x(1) + s(x(0)) ⊗ x(1).
+Thus we obtain that ∆′
+Eϕ(x) = (ϕ ⊗ ϕ)∆E(x) if and only if the conditions(63), (64) and (65)
+hold. By definition, we obtain that ϕ = ϕr,s is an isomorphism if and only if s : V → V is a
+linear isomorphism.
+The second case is φ = 0 and ρ = 0, we obtain the following type (c2) unified coproduct
+for coalgebras.
+Lemma 5.11. Let (A, ∆A, δA) be a Poisson coalgebra and V a vector space. An extending
+datum of (A, ∆A, δA) by V of type (c2) is Ω(4)(A, V ) = (ψ, γ, q, t, ∆V , δV ) with linear maps
+ψ : V → V ⊗ A,
+δV : V → V ⊗ V,
+q : V → A ⊗ A,
+γ : V → V ⊗ A,
+∆V : V → V ⊗ V,
+t : V → A ⊗ A.
+Denote by A#q,tV the vector space E = A⊕V with the comultiplication ∆E : E → E ⊗E, δE :
+E → E ⊗ E given by
+δE(a) = δA(a),
+δE(x) = (δV + ψ − τψ + q)(x),
+∆E(a) = ���A(a),
+∆E(x) = (∆V + γ + τγ + t)(x),
+that is
+δE(a) = a[1] ⊗ a[2],
+δE(x) = x[1] ⊗ x[2] + x⟨0⟩ ⊗ x⟨1⟩ − x⟨1⟩ ⊗ x⟨0⟩ + x1q ⊗ x2q,
+∆E(a) = a1 ⊗ a2,
+∆E(x) = x1 ⊗ x2 + x(0) ⊗ x(1) + x(1) ⊗ x(0) + x1t ⊗ x2t.
+Then A#q,tV is a Poisson coalgebra with the comultiplication given above if and only if the
+following compatibility conditions hold:
+(D0)
+�
+γ, t) is an algebra extending system of the associative coalgebra A trough V and
+�
+ψ, q
+�
+is a Lie extending system of the Lie coalgebra A trough V ,
+(D1) x[1] ⊗ γ(x[2]) = δV (x(0)) ⊗ x(1) + τ12(x1 ⊗ ψ(x2)),
+(D2) x[1] ⊗ t(x[2]) + x⟨0⟩ ⊗ ∆A(x⟨1⟩) = ψ(x(0)) ⊗ x(1) + τ12(x(1) ⊗ ψ(x(0))),
+31
+
+(D3) x⟨1⟩ ⊗ ∆V (x⟨0⟩) = τψ(x1) ⊗ x2 + τ12(x1 ⊗ τψ(x2)),
+(D4) x⟨1⟩ ⊗ γ(x⟨0⟩) = τψ(x(0)) ⊗ x(1) − τ12(x(0) ⊗ δA(x(1))) − τ12(x1 ⊗ q(x2)),
+(D5) x[1] ⊗ ∆V (x[2]) = δV (x(0)) ⊗ x(1) + τ12(x1 ⊗ δH(x2)),
+(D6) x1q ⊗ ∆A(x2q) − x⟨1⟩ ⊗ t(x⟨0⟩)
+= q(x(0)) ⊗ x(1) + δA(x1t) ⊗ x2t + τ12(x(1) ⊗ q(x(0))) + τ12(x1t ⊗ δA(x2t)).
+Note that in this case (V, ∆V , δV ) is a Poisson coalgebra.
+Denote the set of all Poisson coalgebraic extending datum of A by V of type (c2) by
+C(4)(A, V ).
+Similar to the Poisson algebra case, one show that any Poisson coalgebra structure on E
+containing A as a Poisson subcoalgebra is isomorphic to such a unified coproduct.
+Lemma 5.12. Let (A, ∆A, δA) be a Poisson coalgebra and E a vector space containing A as
+a subspace. Suppose that there is a Poisson coalgebra structure (E, ∆E, δE) on E such that
+(A, ∆A, δA) is a Poisson subcoalgebra of E. Then there exists a Poisson coalgebraic extending
+system Ω(2)(A, V ) of (A, ∆A, δA) by V such that (E, ∆E, δE) ∼= A#q,tV .
+Proof. Let p : E → A and π : E → V be the projection map and V = ker(p). Then the
+extending datum of (A, ∆A, δA) by V is defined as follows:
+ψ : V → V ⊗ A,
+φ(x) = (π ⊗ p)δE(x),
+δV : V → V ⊗ V,
+δV (x) = (π ⊗ π)δE(x),
+q : V → A ⊗ A,
+q(x) = (p ⊗ p)δE(x),
+γ : V → V ⊗ A,
+γ(x) = (π ⊗ p)∆E(x),
+∆V : V → V ⊗ V,
+∆V (x) = (π ⊗ π)∆E(x),
+t : V → A ⊗ A,
+t(x) = (p ⊗ p)∆E(x).
+One check that ϕ : A#q,tV → E given by ϕ(a, x) = a + x for all a ∈ A, x ∈ V is a Poisson
+coalgebra isomorphism.
+Lemma 5.13. Let Ω(4)(A, V ) = (ψ, γ, q, t, δV , ∆V ) and Ω′(4)(A, V ) = (ψ′, γ′, q′, t′, δ′
+V , ∆′
+V ) be
+two Poisson coalgebraic extending datums of (A, ∆A, δA) by V . Then there exists a bijection
+between the set of Poisson coalgebra homomorphisms ϕ : A#q,tV → A#q′,t′V whose restriction
+on A is the identity map and the set of pairs (r, s), where r : V → A and s : V → V are two
+linear maps satisfying
+ψ′(s(x)) = s(x[1]) ⊗ r(x[2]) + s(x⟨0⟩) ⊗ x⟨1⟩,
+(66)
+δ′
+V (s(x)) = (s ⊗ s)δV (x),
+(67)
+δ′
+A(r(x)) + q′(s(x)) = r(x[1]) ⊗ r(x[2]) − x⟨1⟩ ⊗ r(x⟨0⟩) + r(x⟨0⟩) ⊗ x⟨1⟩ + q(x),
+(68)
+γ′(s(x)) = s(x1) ⊗ r(x2) + s(x(0)) ⊗ x(1),
+(69)
+32
+
+∆′
+V (s(x)) = (s ⊗ s)∆V (x),
+(70)
+∆′
+A(r(x)) + q′(s(x)) = r(x1) ⊗ r(x2) + x(1) ⊗ r(x(0)) + r(x(0)) ⊗ x(1) + t(x).
+(71)
+Under the above bijection the Poisson coalgebra homomorphism ϕ = ϕr,s : A#q,tV → A#q′,t′V
+to (r, s) is given by ϕ(a, x) = (a + r(x), s(x)) for all a ∈ A and x ∈ V . Moreover, ϕ = ϕr,s is
+an isomorphism if and only if s : V → V is a linear isomorphism.
+Proof. The proof is similar as the proof of Lemma 5.10. Let ϕ : A#q,tV → A#q′,t′V be a
+Poisson coalgebra homomorphism whose restriction on A is the identity map. First it is easy
+to see that δ′
+Eϕ(a) = (ϕ⊗ϕ)δE(a) for all a ∈ A. Then we consider that δ′
+Eϕ(x) = (ϕ⊗ϕ)δE(x)
+for all x ∈ V .
+δ′
+Eϕ(x)
+=
+δ′
+E(r(x), s(x)) = δ′
+E(r(x)) + δ′
+E(s(x))
+=
+δ′
+A(r(x)) + δ′
+V (s(x)) + ψ′(s(x)) − τψ′(s(x)) + q′(s(x)),
+and
+(ϕ ⊗ ϕ)δE(x)
+=
+(ϕ ⊗ ϕ)(δV (x) + ψ(x) − τψ(x) + q(x))
+=
+(ϕ ⊗ ϕ)(x[1] ⊗ x[2] + x⟨0⟩ ⊗ x⟨1⟩ − x⟨1⟩ ⊗ x⟨0⟩ + q(x))
+=
+r(x[1]) ⊗ r(x[2]) + r(x[1]) ⊗ s(x[2]) + s(x[1]) ⊗ r(x[2]) + s(x[1]) ⊗ s(x[2])
+−x⟨1⟩ ⊗ r(x⟨0⟩) − x⟨1⟩ ⊗ s(x⟨0⟩) + r(x⟨0⟩) ⊗ x⟨1⟩ + s(x⟨0⟩) ⊗ x⟨1⟩ + q(x).
+Thus we obtain that δ′
+Eϕ(x) = (ϕ ⊗ ϕ)δE(x) if and only if the conditions (66), (67) and (68)
+hold.
+First it is easy to see that ∆′
+Eϕ(a) = (ϕ ⊗ ϕ)∆E(a) for all a ∈ A. Then we consider that
+∆′
+Eϕ(x) = (ϕ ⊗ ϕ)∆E(x) for all x ∈ V .
+∆′
+Eϕ(x)
+=
+∆′
+E(r(x), s(x)) = ∆′
+E(r(x)) + ∆′
+E(s(x))
+=
+∆′
+A(r(x)) + ∆′
+V (s(x)) + γ′(s(x)) + τγ′(s(x)) + t′(s(x)),
+and
+(ϕ ⊗ ϕ)∆E(x)
+=
+(ϕ ⊗ ϕ)(∆V (x) + γ(x) + τγ(x) + t(x))
+=
+(ϕ ⊗ ϕ)(x1 ⊗ x2 + x(0) ⊗ x(1) + x(1) ⊗ x(0) + t(x))
+=
+r(x1) ⊗ r(x2) + r(x1) ⊗ s(x2) + s(x1) ⊗ r(x2) + s(x1) ⊗ s(x2)
++x(1) ⊗ r(x(0)) + x(1) ⊗ s(x(0)) + r(x(0)) ⊗ x(1) + s(x(0)) ⊗ x(1) + t(x).
+Thus we obtain that ∆′
+Eϕ(x) = (ϕ ⊗ ϕ)∆E(x) if and only if the conditions (69), (70) and (71)
+hold. By definition, we obtain that ϕ = ϕr,s is an isomorphism if and only if s : V → V is a
+linear isomorphism.
+33
+
+Let (A, ∆A, δA) be a Poisson coalgebra and V a vector space. Two Poisson coalgebraic
+extending systems Ω(i)(A, V ) and Ω′(i)(A, V ) are called equivalent if ϕr,s is an isomorphism.
+We denote it by Ω(i)(A, V ) ≡ Ω′(i)(A, V ). From the above lemmas, we obtain the following
+result.
+Theorem 5.14. Let (A, ∆A, δA) be a Poisson coalgebra, E be a vector space containing A as
+a subspace and V be a A-complement in E. Denote HC(V, A) := C(3)(A, V ) ⊔ C(4)(A, V )/ ≡.
+Then the map
+Ψ : HC2
+A(V, A) → CExtd(E, A),
+Ω(3)(A, V ) �→ Ap,s#V,
+Ω(4)(A, V ) �→ A#q,tV
+is bijective, where Ω(i)(A, V ) is the equivalence class of Ω(i)(A, V ) under ≡.
+5.3
+Extending structures for Poisson bialgebras
+Let (A, ·, [, ], ∆A, δA) be a Poisson bialgebra. From (CBB1) and (CBB2) we have the following
+two cases.
+The first case is that we assume q = 0, t = 0 and ⇀, ⊲ to be trivial. Then by the above
+Theorem 4.16, we obtain the following result.
+Theorem 5.15. Let (A, ·, [, ], ∆A, δA) be a Poisson bialgebra and V a vector space. An ex-
+tending datum of A by V of type (I) is
+Ω(I)(A, V ) = (↼, ⊳, φ, ψ, ρ, γ, p, s, θ, ν, ·V , [, ]V , ∆V , δV )
+consisting of linear maps
+⊳ : V ⊗ A → V,
+θ : A ⊗ A → V,
+[, ]V : V ⊗ V → V,
+φ : A → V ⊗ A,
+ψ : V → V ⊗ A,
+p : A → V ⊗ V,
+δV : V → V ⊗ V,
+↼: V ⊗ A → V,
+ν : A ⊗ A → V,
+·V : V ⊗ V → V,
+ρ : A → V ⊗ A,
+γ : V → V ⊗ A,
+s : A → V ⊗ V,
+∆V : V → V ⊗ V.
+Then the unified product Ap,s#θ,ν V with product
+[(a, x), (b, y)] =
+�
+[a, b], [x, y] + x ⊳ b − y ⊳ a + θ(a, b)
+�
+,
+(72)
+(a, x) · (b, y) =
+�
+ab, xy + x ↼ b + y ↼ a + ν(a, b)
+�
+,
+(73)
+and coproduct
+δE(a) = δA(a) + φ(a) − τφ(a) + p(a),
+δE(x) = δV (x) + ψ(x) − τψ(x),
+(74)
+∆E(a) = ∆A(a) + ρ(a) + τρ(a) + s(a),
+∆E(x) = ∆V (x) + γ(x) + τγ(x),
+(75)
+forms a Poisson bialgebra if and only if A#θ,νV forms a Poisson algebra, Ap,s# V forms a
+Poisson coalgebra and the following conditions are satisfied:
+34
+
+(E0)
+�
+↼, ν, ρ, γ, s) is an algebra extending system of the associative algebra and coassociative
+coalgebra A trough V and
+�
+⊳, θ, φ, ψ, p
+�
+is a Lie extending system of the Lie algebra and
+Lie coalgebra A trough V ,
+(E1) φ(ab) + ψ(ν(a, b)) = (a⟨−1⟩ ↼ b) ⊗ a⟨0⟩ + (b⟨−1⟩ ↼ a) ⊗ b⟨0⟩ + b(−1) ⊗ [a, b(0)]
++ a(−1) ⊗ [b, a(0)] + ν(a[1], b) ⊗ a[2] + ν(a, b[1]) ⊗ b[2],
+(E2) τφ(ab) + τψ(ν(a, b)) = a⟨0⟩b ⊗ a⟨−1⟩ + ab⟨0⟩ ⊗ b⟨−1⟩ + b(0) ⊗ (b(−1) ⊳ a) + a(0) ⊗ (a(−1) ⊳ b)
+− b1 ⊗ θ(a, b2) − a1 ⊗ θ(b, a2),
+(E3) ψ(xy) = x⟨0⟩y ⊗ x⟨1⟩ + xy⟨0⟩ ⊗ y⟨1⟩,
+(E4) τψ(xy) = −y(1) ⊗ [x, y(0)] − x(1) ⊗ [y, x(0)],
+(E5) δV (x ↼ b) = (x[1] ↼ b) ⊗ x[2] − (x ↼ b⟨0⟩) ⊗ b⟨−1⟩ + b(−1) ⊗ (x ⊳ b(0))
+− x1 ⊗ (x2 ⊳ b) − ν(x⟨1⟩, b) ⊗ x⟨0⟩ + xb1p ⊗ b2p + b1s ⊗ [x, b2s] + x(0) ⊗ θ(b, x(1)),
+(E6) ψ(x ↼ b) = (x⟨0⟩ ↼ b) ⊗ x⟨1⟩ + (x ↼ b[1]) ⊗ b[2] + xb⟨−1⟩ ⊗ b⟨0⟩ + x(0) ⊗ [b, x(1)],
+(E7) τψ(x ↼ b) = x⟨1⟩b ⊗ x⟨0⟩ + x(1) ⊗ (x(0) ⊳ b) − b(0) ⊗ [x, b(−1)] − b1 ⊗ (x ⊳ b2),
+(E8) ρ([a, b]) + γ(θ(a, b)) = (a⟨−1⟩ ↼ b) ⊗ a⟨0⟩ − (b(−1) ⊳ a) ⊗ b(0) + b(−1) ⊗ [a, b(0)]
+− a⟨−1⟩ ⊗ ba⟨0⟩ + θ(a, b1) ⊗ b2 + ν(b, a[1]) ⊗ a[2],
+(E9) γ([x, y]) = [x, y(0)] ⊗ y(1) + yx⟨0⟩ ⊗ x⟨1⟩,
+(E10) ∆V (x ⊳ b) = (x ⊳ b(0)) ⊗ b(−1) + b(−1) ⊗ (x ⊳ b(0)) + (x[1] ↼ b) ⊗ x[2]
+− x[1] ⊗ (x[2] ↼ b) + [x, b1s] ⊗ b2s + b1s ⊗ [x, b2s] − ν(b, x⟨1⟩) ⊗ x⟨0⟩ − x⟨0⟩ ⊗ ν(b, x⟨1⟩),
+(E11) ∆V (y ⊳ a) = (y1 ⊳ a) ⊗ y2 + y1 ⊗ (y2 ⊳ a) + (y ↼ a⟨0⟩) ⊗ a⟨−1⟩
++ a⟨−1⟩ ⊗ (y ↼ a⟨0⟩) − θ(a, y(1)) ⊗ y(0) − y(0) ⊗ θ(a, y(1)) − ya1p ⊗ a2p − a1p ⊗ ya2p,
+(E12) γ(x ⊳ b) = (x ⊳ b1) ⊗ b2 + [x, b(−1)] ⊗ b(0) − x⟨0⟩ ⊗ bx⟨1⟩ + (x⟨0⟩ ↼ b) ⊗ x⟨1⟩,
+(E13) γ(y ⊳ a) = (y(0) ⊳ a) ⊗ y(1) − y(0) ⊗ [a, y(1)] − (y ↼ a[1]) ⊗ a[2] − ya⟨−1⟩ ⊗ a⟨0⟩,
+(E14) δV (xy) = x[1]y ⊗ x[2] − (y ↼ x⟨1⟩) ⊗ x⟨0⟩ + xy[1] ⊗ y[2] − (x ↼ y⟨1⟩) ⊗ y⟨0⟩
++ y1 ⊗ [x, y2] + y(0) ⊗ (x ⊳ y(1)) + x1 ⊗ [y, x2] + x(0) ⊗ (y ⊳ x(1)),
+(E15) ∆V ([x, y]) = [x, y1] ⊗ y2 + (x ⊳ y(1)) ⊗ y(0) + y1 ⊗ [x, y2] + y(0) ⊗ (x ⊳ y(1))
++ yx[1] ⊗ x[2] − (y ↼ x⟨1⟩) ⊗ x⟨0⟩ − x[1] ⊗ yx[2] − x⟨0⟩ ⊗ (y ↼ x⟨1⟩).
+Conversely, any Poisson bialgebra structure on E with the canonical projection map p : E → A
+both a Poisson algebra homomorphism and a Poisson coalgebra homomorphism is of this form.
+Note that in this case, (V, ·, [, ], ∆V , δV ) is a braided Poisson bialgebra. Although (A, ·, [, ], ∆A, δA)
+is not a Poisson sub-bialgebra of E = Ap,s#θ,ν V , but it is indeed a Poisson bialgebra and a sub-
+space E. Denote the set of all Poisson bialgebraic extending datum of type (I) by IB(I)(A, V ).
+35
+
+The second case is that we assume p = 0, s = 0, θ = 0, ν = 0 and φ, ρ to be trivial. Then
+by the above Theorem 4.16, we obtain the following result.
+Theorem 5.16. Let A be a Poisson bialgebra and V a vector space. An extending datum of
+A by V of type (II) is Ω(II)(A, V ) = (⇀, ↼, ⊲, ⊳, σ, ω, ψ, γ, q, t, ·V , [, ]V , δV , ∆V ) consisting of
+linear maps
+⊳ : V ⊗ A → V,
+⊲ : A ⊗ V → V,
+σ : V ⊗ V → A,
+[, ]V : V ⊗ V → V,
+ψ : V → V ⊗ A,
+q : V → A ⊗ A,
+δV : V → V ⊗ V,
+↼: V ⊗ A → V,
+⇀: A ⊗ V → V,
+ω : V ⊗ V → A,
+·V : V ⊗ V → V,
+γ : V → V ⊗ A,
+t : V → A ⊗ A,
+∆V : V → V ⊗ V.
+Then the unified product Aσ,ω#q,t V with product
+[(a, x), (b, y)]E =
+�
+[a, b] + x ⊲ b − y ⊲ a + σ(x, y), [x, y] + x ⊳ b − y ⊳ a
+�
+,
+(76)
+(a, x) ·E (b, y) =
+�
+ab + x ⇀ b + y ⇀ a + ω(x, y), xy + x ↼ b + y ↼ a
+�
+,
+(77)
+and coproduct
+δE(a) = δA(a),
+δE(x) = δV (x) + ψ(x) − τψ(x) + q(x),
+(78)
+∆E(a) = ∆A(a),
+∆E(x) = ∆V (x) + γ(x) + τγ(x) + t(x),
+(79)
+forms a Poisson bialgebra if and only if Aσ,ω#V forms a Poisson algebra, A#q,tV forms a
+Poisson coalgebra and the following conditions are satisfied:
+(F0)
+�
+⇀, ↼, ω, γ, t) is an algebra extending system of the associative algebra and coassociative
+coalgebra A trough V and
+�
+⊲, ⊳, σ, ψ, q
+�
+is a Lie extending system of the Lie algebra and
+Lie coalgebra A trough V ,
+(F1) ψ(xy) = x⟨0⟩y ⊗ x⟨1⟩ + xy⟨0⟩ ⊗ y⟨1⟩ + y(0) ⊗ (x ⊲ y(1)) + x(0) ⊗ (y ⊲ x(1))
++ (y ↼ x1q) ⊗ x2q + (x ↼ y1q) ⊗ y2q + y1 ⊗ σ(x, y2) + x1 ⊗ σ(y, x2),
+(F2) τψ(xy) = (y ⇀ x⟨1⟩) ⊗ x⟨0⟩ + (x ⇀ y⟨1⟩) ⊗ y⟨0⟩ − y(1) ⊗ [x, y(0)] − x(1) ⊗ [y, x(0)]
+− ω(x[1], y) ⊗ x[2] − ω(x, y[1]) ⊗ y[2] − y1t ⊗ (x ⊳ y2t) − x1t ⊗ (y ⊳ x2t),
+(F3) δA(x ⇀ b) + q(x ↼ b) = (x⟨0⟩ ⇀ b) ⊗ x⟨1⟩ + (x ⇀ b[1]) ⊗ b[2] − x(1) ⊗ (x(0) ⊲ b)
++ b1 ⊗ (x ⊲ b2) + x1qb ⊗ x2q + x1t ⊗ [b, x2t],
+(F4) δV (x ↼ b) = (x[1] ↼ b) ⊗ x[2] − x1 ⊗ (x2 ⊳ b),
+(F5) ψ(x ↼ b) = (x⟨0⟩ ↼ b) ⊗ x⟨1⟩ + (x ↼ b[1]) ⊗ b[2] − x1 ⊗ (x2 ⊲ b) + x(0) ⊗ [b, x(1)],
+(F6) τψ(x ↼ b) = x⟨1⟩b ⊗ x⟨0⟩ + x(1) ⊗ (x(0) ⊳ b) − (x[1] ⇀ b) ⊗ x[2] − b1 ⊗ (x ⊳ b2),
+36
+
+(F7) γ([x, y]) = [x, y(0)] ⊗ y(1) + y(0) ⊗ (x ⊲ y(1)) − x⟨0⟩ ⊗ (y ⇀ x⟨1⟩) + yx⟨0⟩ ⊗ x⟨1⟩
++ (x ⊳ y1t) ⊗ y2t + y1 ⊗ σ(x, y2) + (y ↼ x1q) ⊗ x2q − x[1] ⊗ ω(y, x[2]),
+(F8) ∆A(x ⊲ b) + t(x ⊳ b) = (x ⊲ b1) ⊗ b2 + b1 ⊗ (x ⊲ b2) + (x⟨0⟩ ⇀ b) ⊗ x⟨1⟩
++ x⟨1⟩ ⊗ (x⟨0⟩ ⇀ b) + bx1q ⊗ x2q − x1q ⊗ bx2q,
+(F9) ∆A(y ⊲ a) + t(y ⊳ a) = −(y ⇀ a[1]) ⊗ a[2] + a[1] ⊗ (y ⇀ a[2]) + (y(0) ⊲ a) ⊗ y(1)
++ y(1) ⊗ (y(0) ⊲ a) − [a, y1t] ⊗ y2t − y1t ⊗ [a, y2t],
+(F10) ∆V (x ⊳ b) = (x[1] ↼ b) ⊗ x[2] − x[1] ⊗ (x[2] ↼ b),
+(F11) ∆V (y ⊳ a) = (y1 ⊳ a) ⊗ y2 + y1 ⊗ (y2 ⊳ a),
+(F12) γ(x ⊳ b) = (x ⊳ b1) ⊗ b2 − x⟨0⟩ ⊗ bx⟨1⟩ + (x⟨0⟩ ↼ b) ⊗ x⟨1⟩ − x[1] ⊗ (x[2] ⇀ b),
+(F13) γ(y ⊳ a) = (y(0) ⊳ a) ⊗ y(1) − y(0) ⊗ [a, y(1)] − (y ↼ a[1]) ⊗ a[2] + y1 ⊗ (y2 ⊲ a),
+(F14) δV (xy) = x[1]y ⊗ x[2] − (y ↼ x⟨1⟩) ⊗ x⟨0⟩ + xy[1] ⊗ y[2] − (x ↼ y⟨1⟩) ⊗ y⟨0⟩
++ y1 ⊗ [x, y2] + y(0) ⊗ (x ⊳ y(1)) + x1 ⊗ [y, x2] + x(0) ⊗ (y ⊳ x(1)),
+(F15) ∆V ([x, y]) = [x, y1] ⊗ y2 + (x ⊳ y(1)) ⊗ y(0) + y1 ⊗ [x, y2] + y(0) ⊗ (x ⊳ y(1))
++ yx[1] ⊗ x[2] − (y ↼ x⟨1⟩) ⊗ x⟨0⟩ − x[1] ⊗ yx[2] − x⟨0⟩ ⊗ (y ↼ x⟨1⟩).
+Conversely, any Poisson bialgebra structure on E with the canonical injection map i : A → E
+both a Poisson algebra homomorphism and a Poisson coalgebra homomorphism is of this form.
+Note that in this case, (A, ·, [, ], ∆A, δA) is a Poisson sub-bialgebra of E = Aσ,ω#q,t V and
+(V, ·, [, ], ∆V , δV ) is a braided Poisson bialgebra.
+Denote the set of all Poisson bialgebraic
+extending datum of type (II) by IB(II)(A, V ).
+In the above two cases, we find that the braided Poisson bialgebra V play a special role
+in the extending problem of Poisson bialgebra A. Note that Ap,s#θ,ν V and Aσ,ω#q,t V are all
+Poisson bialgebra structures on E. Conversely, any Poisson bialgebra extending system E of A
+through V is isomorphic to such two types. Now from Theorem 5.15, Theorem 5.16 we obtain
+the main result of in this section, which solve the extending problem for Poisson bialgebra.
+Theorem 5.17. Let (A, ·, [, ], ∆A, δA) be a Poisson bialgebra, E a vector space containing A
+as a subspace and V be a complement of A in E. Denote by
+HLB(V, A) := IB(I)(A, V ) ⊔ IB(II)(A, V )/ ≡ .
+Then the map
+Υ : HLB(V, A) → BExtd(E, A),
+Ω(I)(A, V ) �→ Ap,s#θ,ν V,
+Ω(II)(A, V ) �→ Aσ,ω#q,t V
+(80)
+is bijective, where Ω(i)(A, V ) is the equivalence class of Ω(i)(A, V ) under ≡.
+37
+
+A very special case is that when ⊲ and ⇀ are trivial in the above Theorem 5.16. We obtain
+the following result.
+Theorem 5.18. Let A be a Poisson bialgebra and V a vector space. An extending datum of
+A by V is Ω(A, V ) = (↼, ⊳, σ, ω, ψ, γ, q, t, ·V , [, ]V , δV , ∆V ) consisting of linear maps
+⊳ : V ⊗ A → V,
+σ : V ⊗ V → A,
+[, ]V : V ⊗ V → V,
+ψ : V → V ⊗ A,
+q : V → A ⊗ A,
+δV : V → V ⊗ V,
+↼: V ⊗ A → V,
+ω : V ⊗ V → A,
+·V : V ⊗ V → V,
+γ : V → V ⊗ A,
+t : V → A ⊗ A,
+∆V : V → V ⊗ V.
+Then the unified product Aσ,ω#q,t V with product
+[(a, x), (b, y)]E =
+�
+[a, b] + σ(x, y), [x, y] + x ⊳ b − y ⊳ a
+�
+,
+(81)
+(a, x) ·E (b, y) =
+�
+ab + ω(x, y), xy + x ↼ b + y ↼ a
+�
+,
+(82)
+and coproduct
+δE(a) = δA(a),
+δE(x) = δV (x) + ψ(x) − τψ(x) + q(x),
+(83)
+∆E(a) = ∆A(a),
+∆E(x) = ∆V (x) + γ(x) + τγ(x) + t(x),
+(84)
+forms a Poisson bialgebra if and only if Aσ,ω#V forms a Poisson algebra, A#q,t V forms a
+Poisson coalgebra and the following conditions are satisfied:
+(G0)
+�
+↼, ω, γ, t) is an algebra extending system of the associative algebra and coassociative
+coalgebra A trough V and
+�
+⊳, σ, ψ, q
+�
+is a Lie extending system of the Lie algebra and
+Lie coalgebra A trough V ,
+(G1) ψ(xy) = x⟨0⟩y ⊗ x⟨1⟩ + xy⟨0⟩ ⊗ y⟨1⟩ + (y ↼ x1q) ⊗ x2q + (x ↼ y1q) ⊗ y2q
++ y1 ⊗ σ(x, y2) + x1 ⊗ σ(y, x2),
+(G2) τψ(xy) = −y(1) ⊗ [x, y(0)] − x(1) ⊗ [y, x(0)] − ω(x[1], y) ⊗ x[2] − ω(x, y[1]) ⊗ y[2]
+− y1t ⊗ (x ⊳ y2t) − x1t ⊗ (y ⊳ x2t),
+(G3) q(x ↼ b) = x1qb ⊗ x2q + x1t ⊗ [b, x2t],
+(F4) δV (x ↼ b) = (x[1] ↼ b) ⊗ x[2] − x1 ⊗ (x2 ⊳ b),
+(G5) ψ(x ↼ b) = (x⟨0⟩ ↼ b) ⊗ x⟨1⟩ + (x ↼ b[1]) ⊗ b[2] + x(0) ⊗ [b, x(1)],
+(G6) τψ(x ↼ b) = x⟨1⟩b ⊗ x⟨0⟩ + x(1) ⊗ (x(0) ⊳ b) − b1 ⊗ (x ⊳ b2),
+(G7) γ([x, y]) = [x, y(0)] ⊗ y(1) + yx⟨0⟩ ⊗ x⟨1⟩ + (x ⊳ y1t) ⊗ y2t + y1 ⊗ σ(x, y2)
++ (y ↼ x1q) ⊗ x2q − x[1] ⊗ ω(y, x[2]),
+(G8) t(x ⊳ b) = bx1q ⊗ x2q − x1q ⊗ bx2q,
+38
+
+(G9) t(y ⊳ a) = −[a, y1t] ⊗ y2t − y1t ⊗ [a, y2t],
+(G10) ∆V (x ⊳ b) = (x[1] ↼ b) ⊗ x[2] − x[1] ⊗ (x[2] ↼ b),
+(G11) ∆V (y ⊳ a) = (y1 ⊳ a) ⊗ y2 + y1 ⊗ (y2 ⊳ a),
+(G12) γ(x ⊳ b) = (x ⊳ b1) ⊗ b2 − x⟨0⟩ ⊗ bx⟨1⟩ + (x⟨0⟩ ↼ b) ⊗ x⟨1⟩,
+(G13) γ(y ⊳ a) = (y(0) ⊳ a) ⊗ y(1) − y(0) ⊗ [a, y(1)] − (y ↼ a[1]) ⊗ a[2],
+(G14) δV (xy) = x[1]y ⊗ x[2] − (y ↼ x⟨1⟩) ⊗ x⟨0⟩ + xy[1] ⊗ y[2] − (x ↼ y⟨1⟩) ⊗ y⟨0⟩
++ y1 ⊗ [x, y2] + y(0) ⊗ (x ⊳ y(1)) + x1 ⊗ [y, x2] + x(0) ⊗ (y ⊳ x(1)),
+(G15) ∆V ([x, y]) = [x, y1] ⊗ y2 + (x ⊳ y(1)) ⊗ y(0) + y1 ⊗ [x, y2] + y(0) ⊗ (x ⊳ y(1))
++ yx[1] ⊗ x[2] − (y ↼ x⟨1⟩) ⊗ x⟨0⟩ − x[1] ⊗ yx[2] − x⟨0⟩ ⊗ (y ↼ x⟨1⟩).
+Acknowledgements
+This is a primary edition, something should be modified in the future.
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+Tao Zhang
+College of Mathematics and Information Science,
+Henan Normal University, Xinxiang 453007, P. R. China;
+E-mail address: [email protected]
+Fang Yang
+College of Mathematics and Information Science,
+Henan Normal University, Xinxiang 453007, P. R. China;
+E-mail address: [email protected]
+40
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+arXiv:2301.11617v1  [math.NT]  27 Jan 2023
+Compairing categories of Lubin-Tate pϕL, ΓLq-modules
+Peter Schneider and Otmar Venjakob
+January 30, 2023
+Abstract
+In the Lubin-Tate setting we compare different categories of pϕL, Γq-modules over
+various perfect or imperfect coefficient rings. Moreover, we study their associated Herr-
+complexes. Finally, we show that a Lubin Tate extension gives rise to a weakly decom-
+pleting, but not decompleting tower in the sense of Kedlaya and Liu.
+Contents
+1
+Introduction
+1
+2
+Notation
+2
+3
+An analogue of Tate’s result
+4
+4
+The functors D, ˜D and ˜D:
+6
+5
+The perfect Robba ring
+19
+6
+The web of eqivalences
+22
+7
+Cohomology: Herr complexes
+24
+8
+Weakly decompleting towers
+27
+References
+30
+1
+Introduction
+Since its invention by Fontaine in [Fo] the concept of pϕ, Γq-modules (for the p-cyclotomic
+extension) has become a powerful tool in the study of p-adic Galois representations of local
+fields. In particular, it could be fruitfully applied in Iwasawa theory [Ben, B, Na14a, Na17a,
+Na17b, V13, LVZ15, LLZ11, BV] and in the p-adic local Langlands programme [Co1]. A
+good introduction to the subject regarding the state of the art around 2010 can be found in
+[BC, FO].
+Afterwards a couple of generalisations have been developed. Firstly, Berger and Colmez
+[BeCo] as well as Kedlaya, Pottharst and Xiao [KPX] extended the theory to (arithmetic)
+1
+
+families of pϕ, Γq-modules, in which representations of the absolute Galois group of a local
+field on modules over affinoid algebras over Qp instead of finite dimensional vector spaces are
+studied. Secondly, parallel to and influenced by Scholze’s point of view of perfectoid spaces
+as well as the upcoming of the Fargues-Fontaine curve [FF] Kedlaya and Liu developed a
+(geometric) relative p-adic Hodge theory [KLI, KLII], in which the Galois group of a local
+field is replaced by the étale fundamental group of affinoid spaces over Qp thereby extending
+an earlier approach by Andreatta and Brinon. In particular, Kedlaya and Liu have introduced
+systematically pϕ, Γq-moduels over perfect coefficient rings, i.e., for which the Frobenius endo-
+morphism is surjective, and they have studied their decent to imperfect coefficient rings, which
+is needed for Iwasawa theoretic applications and which generalized the work of Cherbonnier
+and Colmez [ChCo1].
+Recently there has been a growing interest and activity in introducing and studying
+pϕL, ΓLq-modules for Lubin-Tate extensions of a finite extension L of Qp, motivated again
+by requirements from or potential applications to the p-adic local Langlands programme
+[FX, BSX, Co2] or Iwasawa theory [SV15, BF, SV23, MSVW, Poy]. The textbook [GAL]
+contains a very detailed and thorough approach to the analogue of Fontaine’s original equiv-
+alence of categories between Galois representations and étale pϕ, Γq-modules to the case of
+Lubin-Tate extensions as had been proposed, but only sketched in [KR], see Theorem 4.1.
+In this setting it has been shown in [Ku, KV] that - as in the cyclotomic case due to Herr
+[Her98] - the Galois cohomology of a L-representation V of the absolute Galois group GL of L
+can again be obtained as cohomology of a generalized Herr complex for the pϕL, ΓLq-module
+attached to V , see Theorem 7.1.
+The purpose of this article is to spell out in the Lubin-Tate case concretely the various
+categories of (classical) pϕL, ΓLq-modules over perfect and imperfect coefficient rings (analo-
+gously to those considered in [KLI, KLII] who do not cover the Lubin-Tate situation) such as
+AL, A:
+L, ˜AL, ˜A:
+L, BL, B:
+L, BL, ˜B:
+L, RL, ˜RL to be defined in the course of the main text and to
+compare them among each other. Moreover, we investigate for which versions the generalized
+Herr complex calculates again the Galois cohomology of a given representation. The results
+are summarized in diagrams (6) and (7). Finally, we study in the last section how Lubin-Tate
+extensions fit into Kedlaya’s and Liu’s concept of (weakly) decompleting towers. We show that
+for L ‰ Qp they are weakly decompleting, but not decompleting.
+See [Ste1] for some results regarding arithmetic families of pϕL, ΓLq-modules in the Lubin-
+Tate setting.
+Acknowledgements: Both authors are grateful to UBC and PIMS at Vancouver for
+supporting a fruitful stay. The project was funded by the Deutsche Forschungsgemeinschaft
+(DFG, German Research Foundation) – Project-ID 427320536 – SFB 1442, as well as un-
+der Germany’s Excellence Strategy EXC 2044 390685587, Mathematics Münster: Dynam-
+ics–Geometry–Structure. We also acknowledge funding by the Deutsche Forschungsgemein-
+schaft (DFG, German Research Foundation) under TRR 326 Geometry and Arithmetic of
+Uniformized Structures, project number 444845124, as well as under DFG-Forschergruppe
+award number [1920] Symmetrie, Geometrie und Arithmetik.
+2
+Notation
+Let Qp Ď L Ă Cp be a field of finite degree d over Qp, oL the ring of integers of L, πL P oL a
+fixed prime element, kL “ oL{πLoL the residue field, q :“ |kL| and e the absolute ramification
+2
+
+index of L. We always use the absolute value | | on Cp which is normalized by |πL| “ q´1. We
+warn the reader, though, that we will use the references [FX] and [Laz] in which the absolute
+value is normalized differently from this paper by |p| “ p´1. Our absolute value is the dth
+power of the one in these references. The transcription of certain formulas to our convention
+will usually be done silently.
+We fix a Lubin-Tate formal oL-module LT “ LTπL over oL corresponding to the prime
+element πL. We always identify LT with the open unit disk around zero, which gives us a global
+coordinate Z on LT. The oL-action then is given by formal power series raspZq P oLrrZss. For
+simplicity the formal group law will be denoted by `LT .
+Let Tπ be the Tate module of LT. Then Tπ is a free oL-module of rank one, say with
+generator η, and the action of GL :“ GalpL{Lq on Tπ is given by a continuous character
+χLT : GL ÝÑ oˆ
+L.
+For n ě 0 we let Ln{L denote the extension (in Cp) generated by the πn
+L-torsion points of
+LT, and we put L8 :“ Ť
+n Ln. The extension L8{L is Galois. We let ΓL :“ GalpL8{Lq and
+HL :“ GalpL{L8q. The Lubin-Tate character χLT induces an isomorphism ΓL
+–
+ÝÑ oˆ
+L.
+Henceforth we use the same notation as in [SV15]. In particular, the ring endomorphisms
+induced by sending Z to rπLspZq are called ϕL where applicable; e.g. for the ring AL defined
+to be the πL-adic completion of oLrrZssrZ´1s or BL :“ ALrπ´1
+L s which denotes the field of
+fractions of AL. Recall that we also have introduced the unique additive endomorphism ψL of
+BL (and then AL) which satisfies
+ϕL ˝ ψL “ π´1
+L ¨ traceBL{ϕLpBLq .
+Moreover, projection formula
+ψLpϕLpf1qf2q “ f1ψLpf2q
+for any fi P BL
+as well as the formula
+ψL ˝ ϕL “ q
+πL
+¨ id
+hold. An étale pϕL, ΓLq-module M comes with a Frobenius operator ϕM and an induced
+operator denoted by ψM.
+Let rE` :“ lim
+ÐÝ oCp{poCp with the transition maps being given by the Frobenius ϕpaq “ ap.
+We may also identify rE` with lim
+ÐÝ oCp{πLoCp with the transition maps being given by the
+q-Frobenius ϕqpaq “ aq. Recall that rE` is a complete valuation ring with residue field Fp and
+its field of fractions rE “ lim
+ÐÝ Cp being algebraically closed of characteristic p. Let mrE denote
+the maximal ideal in rE`.
+The q-Frobenius ϕq first extends by functoriality to the rings of the Witt vectors WprEq and
+then oL-linearly to WprEqL :“ WprEqboL0 oL, where L0 is the maximal unramified subextension
+of L. The Galois group GL obviously acts on rE and WprEqL by automorphisms commuting
+with ϕq. This GL-action is continuous for the weak topology on WprEqL (cf. [GAL, Lemma
+1.5.3]).
+By sending the variable Z to ωLT P WprEqL (see directly after [SV15, Lem. 4.1]) we obtain
+an GL-equivariant, Frobenius compatible embedding of rings
+AL ÝÑ WprEqL
+3
+
+the image of which we call AL. The latter ring is a complete discrete valuation ring with prime
+element πL and residue field the image EL of kLppZqq ãÑ rE sending Z to ω :“ ωLT
+mod πL.
+We form the maximal integral unramified extension (“ strict Henselization) Anr
+L of AL inside
+WprEqL. Its p-adic completion A still is contained in WprEqL. Note that A is a complete
+discrete valuation ring with prime element πL and residue field the separable algebraic closure
+Esep
+L
+of EL in rE. By the functoriality properties of strict Henselizations the q-Frobenius ϕq
+preserves A. According to [KR, Lemma 1.4] the GL-action on WprEqL respects A and induces
+an isomorphism HL “ kerpχLT q –
+ÝÑ AutcontpA{ALq.
+Sometimes we omit the index q, L, or M from the Frobenius operator.
+Finally, for a valued field K we denote as usual by ˆK its completion.
+3
+An analogue of Tate’s result
+Let C5
+p together with its absolute value | ¨ |5 be the tilt of Cp. The aim of this section is to
+prove an analogue of Tate’s classical result [Ta, Prop. 10] for C5
+p instead of Cp itself and in
+the Lubin Tate situation instead of the cyclotomic one. In the following we always consider
+continuous group cohomology.
+Proposition 3.1. HnpH, C5
+pq “ 0 for all n ě 1 and H Ď HL any closed subgroup.
+Since the proof is formally very similar to that of loc. cit. or [BC, Prop. 14.3.2.] we only
+sketch the main ingredients. To this aim we fix H and write sometimes W for C5
+p as well as
+Wěm :“ tx P W||x|5 ď
+1
+pm u.
+Lemma 3.2. The Tate-Sen axiom (TS1) is satisfied for C5
+p with regard to H, i.e., there exists
+a real constant c ą 1 such that for all open subgroups H1 Ď H2 in H there exists α P pC5
+pqH1
+with |α|5 ă c and TrH2|H1pαq :“ ř
+τPH2|H1 τpαq “ 1. Moreover, for any sequence pHmqm of
+open subgroups Hm`1 Ď Hm of H there exists a trace compatible system pyHmqm of elements
+yHm P pC5
+pqHm with |yHm|5 ă c and TrH|HmpyHmq “ 1.
+Proof. Note that for a perfect field K (like pC5
+pqH) of characteristic p complete for a multi-
+plicative norm with maximal ideal mK and a finite extension F one has TrF {KpmFq “ mK by
+[Ked15, Thm. 1.6.4]. Fix some x P pC5
+pqH with 0 ă |x|5 ă 1 and set c :“ |x|´1
+5
+ą 1. Then we
+find ˜α in the maximal ideal of pC5
+pqH1 with TrH|H1p˜αq “ x and α :“ pTrH2|H1p˜αqq´1 ˜α satisfies
+the requirement as |TrH2|H1p˜αq|´1
+5
+ď |x|´1
+5
+“ c.
+For the second claim we successively choose elements ˜αm in the maximal ideal of pC5
+pqHm
+such that TrH|H1p˜α1q “ x and TrHm`1|Hmp˜αm`1q “ ˜αm for all m ě 1. Renormalization
+αm :“ x´1˜αm gives the desired system.
+Remark 3.3. Since H is also a closed subgroup of the absolute Galois group GL of L it
+possesses a countable fundamental system pHmqm of open neighbourhoods of the identity, as
+for any n ą 0 the local field L of characteristic 0 has only finitely many extensions of degree
+smaller than n.
+Proof. The latter statement reduces easily to finite Galois extensions L1 of L, which are known
+to be solvable, i.e. L1 has a series of at most n intermediate fields L Ď L1 Ď . . . Ď Ln “ L1
+such that each subextension is abelian. Now its known by class field theory that each local
+field in characteristic 0 only has finitely many abelian extensions of a given degree.
+4
+
+We write CnpG, V q for the abelian group of continuous n-cochains of a profinite group G
+with values in a topological abelian group V carrying a continuous G-action and B for the usual
+differentials. In particular, we endow CnpH, Wq with the maximum norm } ´ } and consider
+the subspace CnpH, Wqδ :“ Ť
+H1⊴H open CnpH{H1, Wq Ď CnpH, Wq of those cochains with
+are even continuous with respect to the discrete topology of W.
+Lemma 3.4.
+(i) The completion of CnpH, Wqδ with respect to the maximum norm equals
+CnpH, Wq.
+(ii) There exist pC5
+pqH-linear continuous maps
+σn : CnpH, Wq Ñ Cn´1pH, Wq
+satisfying }f ´ Bσnf} ď c}Bf}.
+Proof. Since the space CnpH, Wq is already complete we only have to show that an arbitrary
+cochain f in it can be approximated by a Cauchy sequence fm in CnpH, Wqδ. To this end
+we observe that, given any m, the induced cochain Hn
+fÝÑ W
+prm
+ÝÝÑ W{Wěm comes, for some
+open normal subgroup Hm, from a cochain in CnpH{Hm, W{Wěmq, which in turn gives rise
+to fm P CnpH, Wqδ when composing with any set theoretical section W{Wěm
+sm
+ÝÝÑ W of
+the canonical projection W
+prm
+ÝÝÑ W{Wěm. Note that sm is automatically continuous, since
+W{Wěm is discrete. By construction we have }f ´fm} ď
+1
+pm and pfmqm obviously is a Cauchy
+sequence. This shows (i).
+For (ii) recall from Lemma 3.2 together with Remark 3.3 the existence of a trace compatible
+system pyH1qH1 of elements yH1 P pC5
+pqH1 with |yH1|5 ă c and TrH|H1pyH1q “ 1, where H1 runs
+over the open normal subgroups of H. Now we first define pC5
+pqH-linear maps
+σn : CnpH, Wqδ Ñ Cn´1pH, Wq
+satisfying }f ´ Bσnf} ď c}Bf} and }σnf} ď c}f} by setting for f P CnpH{H1, Wq
+σnpfq :“ yH1 Y f
+(by considering yH1 as a ´1-cochain), i.e.,
+σnpfqph1, . . . , hn´1q “ p´1qn
+ÿ
+τPH{H1
+ph1 . . . hn´1τqpyH1qfph1, . . . , hn´1, τq.
+The inequality }yH1 Y f} ď c}f} follows immediately from this description, see the proof
+of [BC, Lem. 14.3.1.]. Upon noting that ByH1 “ TrH|H1pyH1q “ 1, the Leibniz rule for the
+differential B with respect to the cup-product then implies that
+f ´ BpyH1 Y fq “ yH1 Y Bf,
+hence
+}f ´ BpyH1 Y fq} ď c}Bf}
+by the previous inequality, see again loc. cit. In order to check that this map σn is well
+defined we assume that f arises also from a cochain in CnpH{H2, Wq. Since we may make
+5
+
+the comparison within CnpH{pH1 X H2q, Wq we can assume without loss of generality that
+H2 Ď H1. Then
+pyH2 Y fqph1, . . . , hn´1q “ p´1qn
+ÿ
+τPH{H2
+ph1 . . . hn´1τqpyH2qfph1, . . . , hn´1, τq
+“ p´1qn
+ÿ
+τPH{H1
+¨
+˝h1 . . . hn´1
+ÿ
+τ 1PH1{H2
+τ 1
+˛
+‚pyH2qfph1, . . . , hn´1, τq
+“ p´1qn
+ÿ
+τPH{H1
+ph1 . . . hn´1q p
+ÿ
+τ 1PH1{H2
+τ 1pyH2qqfph1, . . . , hn´1, τq
+“ p´1qn
+ÿ
+τPH{H1
+ph1 . . . hn´1q pyH1qfph1, . . . , hn´1, τq
+“ pyH1 Y fqph1, . . . , hn´1q
+using the trace compatibility in the fourth equality. Finally the inequality }σnf} ď c}f} implies
+that σn is continuous on CnpH, Wqδ and therefore extends continuously to its completion
+CnpH, Wq.
+The proof of Prop. 3.1 is now an immediate consequence of Lemma 3.4(ii).
+4
+The functors D, ˜D and ˜D:
+Let RepoLpGLq, RepoL,fpGLq and RepLpGLq denote the category of finitely generated oL-
+modules, finitely generated free oL-modules and finite dimensional L-vector spaces, respec-
+tively, equipped with a continuous linear GL-action. The following result is established in
+[KR, Thm. 1.6] (see also [GAL, Thm. 3.3.10]) and [SV15, Prop. 4.4 (ii)].
+Theorem 4.1. The functors
+T ÞÝÑ DpTq :“ pA boL TqHL
+and
+M ÞÝÑ pA bAL MqϕqbϕM“1
+are exact quasi-inverse equivalences of categories between RepoLpGLq and the category MetpALq
+of finitely generated étale ϕL, ΓLq-modules over AL. Moreover, for any T in RepoLpGLq the
+natural map
+(1)
+A bAL DpTq
+–
+ÝÝÑ A boL T
+is an isomorphism (compatible with the GL-action and the Frobenius on both sides).
+In the following we would like to establish a version of the above for ˜A and prove similar
+properties for it. In the classical situation such versions have been studied by Kedlaya et al
+using the unramified rings of Witt vectors WpRq. In our Lubin-Tate situation we have to work
+with ramified Witt vectors WpRqL. Many results and their proofs transfer almost literally from
+the classical setting. Often we will try to at least sketch the proofs for the convenience of the
+reader, but when we just quote results from the classical situation, e.g. from [KLI], this usually
+means that the transfer is purely formal.
+We start defining ˜A :“ WpC5
+pqL and
+˜A: :“ tx “
+ÿ
+ně0
+πn
+Lrxns P ˜A : |πn
+L}xn|r
+5
+nÑ8
+ÝÝÝÑ 0 for some r ą 0u
+6
+
+as well as ˜DpTq :“ p ˜A boL TqHL and ˜D:pTq :“ p ˜A: boL TqHL.
+More generally, let K be any perfectoid field containing L and let K5 denote its tilt. For
+r ą 0 let W rpK5qL be the set of x “ ř8
+n“0 πn
+Lrxns P WpK5qL such that |πL|n|xn|r
+5 tends to
+zero as n goes to 8. This is a subring by [KLI, Prop. 5.1.2] on which the function
+|x|r :“ sup
+n
+t|πn
+L}xn|r
+5u “ sup
+n
+tq´n|xn|r
+5u
+is a complete multiplicative norm; it extends multiplicatively to W rpK5qLr 1
+πL s. Furthermore,
+W :pK5qL :“ Ť
+rą0 W rpK5qL 1 is a henselian discrete valuation ring by [Ked05, Lem. 2.1.12],
+whose πL-adic completion equals WpK5qL since they coincide modulo πn
+L. Then ˜A: “ W :pC5
+pqL,
+and we write ˜AL and ˜A:
+L for WpˆL5
+8qL and W :pˆL5
+8qL, respectively. We set ˜BL “ ˜ALr 1
+πL s,
+˜B “ ˜Ar 1
+πL s, ˜B:
+L “ ˜A:
+Lr 1
+πL s and ˜B: “ ˜A:r 1
+πL s for the corresponding fields of fractions.
+Remark 4.2. By the Ax-Tate-Sen theorem [Ax] and since C5
+p is the completion of an algebraic
+closure ˆL58 he have that pC5
+pqH “ ppˆL58qHq^ for any closed subgroup H Ď HL, in particular
+pC5
+pqHL “ ˆL5
+8. As completion of an algebraic extension of the perfect field ˆL5
+8 the field pC5
+pqH
+is perfect, too. Moreover, we have ˜AHL “ ˜AL, p ˜A:qHL “ ˜A:
+L and analogously for the rings ˜B
+and ˜B:. It also follows that ˜A is the πL-adic completion of a maximal unramified extension of
+˜AL.
+Lemma 4.3. The rings AL and A embed into ˜AL and ˜A, respectively.
+Proof. The embedding AL ãÑ ˜AL is explained in [GAL, p. 94]. Moreover, A is the πL-
+adic completion of the maximal unramified extension of AL inside ˜A “ WpC5
+pqL (cf. [GAL,
+§3.1]).
+On ˜A “ WpC5
+pqL the weak topology is defined to be the product topology of the valuation
+topologies on the components C5
+p. The induced topology on any subring R of it is also called
+weak topology of R. If M is a finitely generated R-module, then we call the canonical topology
+of M (with respect to the weak topology of R) the quotient topology with respect to any
+surjection Rn ։ M where the free module carries the product topology; this is independent
+of any choices. We recall that a pϕL, ΓLq-module M over R P tAL, ˜AL, ˜A:
+Lu is a finitely
+generated R-module M together with
+– a ΓL-action on M by semilinear automorphisms which is continuous for the weak topol-
+ogy and
+– a ϕL-linear endomorphism ϕM of M which commutes with the ΓL-action.
+We let MpRq denote the category of pϕL, ΓLq-modules M over R. Such a module M is called
+étale if the linearized map
+ϕlin
+M : R bR,ϕL M
+–
+ÝÝÑ M
+f b m ÞÝÑ fϕMpmq
+is bijective. We let M´etpRq denote the full subcategory of étale pϕL, ΓLq-modules over R.
+1In [Ked05] it is denoted by W :pK5qL.
+7
+
+Definition 4.4. For ˚ “ BL, ˜BL, ˜B:
+L we write M´etp˚q :“ M´etp˚1qboLL with ˚1 “ AL, ˜AL, ˜A:
+L,
+respectively, and call the objects étale pϕL, ΓLq-modules over ˚.
+Lemma 4.5. Let G be a profinite group and R Ñ S be a topological monomorphism of
+topological oL-algebras, for which there exists a system of open neighbourhoods of 0 consisting
+of oL-submodules. Consider a finitely generated R-module M, for which the canonical map
+M Ñ S bR M is injective (e.g. if S is faithfully flat over R or M is free, in addition), and
+endow it with the canonical topology with respect to R. Assume that G acts continuously, oL-
+linearly and compatible on R and S as well as continuously and R-semilinearly on M. Then
+the diagonal G-action on S bR M is continuous with regard to the canonical topology with
+respect to S.
+Proof. Imitate the proof of [GAL, Lem. 3.1.11].
+Proposition 4.6. The canonical map
+(2)
+˜AL bAL DpTq –
+ÝÑ ˜DpTq
+is an isomorphism and the functor ˜Dp´q : RepoLpGLq Ñ M´etp ˜ALq is exact. Moreover, we
+have a comparison isomorphism
+(3)
+˜A b ˜AL ˜DpTq –
+ÝÑ ˜A boL T.
+Proof. The isomorphism (2) implies formally the isomorphism (3) after base change of the
+comparison isomorphism (1). Secondly, the isomorphism (2), resp. (3), implies easily that
+˜DpTq is finitely generated, resp. étale. Thirdly, since the ring extension ˜AL{AL is faithfully
+flat as local extension of (discrete) valuation rings, the exactness of ˜D follows from that of D.
+Moreover, the isomorphism (2) implies by Lemma 4.5 that ΓL acts continuously on ˜DpTq, i.e.,
+the functor ˜D is well-defined. Thus we only have to prove that
+˜AL bAL pA boL TqHL
+–
+ÝÑ p ˜A boL TqHL
+s an isomorphism. To this aim let us assume first that T is finite. Then we find an open normal
+subgroup H ⊴HL which acts trivially on T. Application of the subsequent Lemma 4.7 to M “
+pAboL TqH and G “ HL{H interprets the left hand side as
+´
+˜AL bAL pA boL TqH¯HL{H
+while
+the right hand side equals
+´
+p ˜A boL TqH¯HL{H
+. Hence it suffices to establish the isomorphism
+˜AL bAL pA boL TqH
+–
+ÝÑ p ˜A boL TqH.
+By Lemma 4.8 below this is reduced to showing that the canonical map
+˜AL bAL AH boL T
+–
+ÝÑ ˜AH boL T
+is an isomorphism, which follows from Lemma 4.9 below. Finally let T be arbitrary. Then we
+8
+
+have isomorphisms
+˜AL bAL DpTq – ˜AL bAL lim
+ÐÝ
+n
+DpT{πn
+LTq
+– ˜AL bAL lim
+ÐÝ
+n
+DpTq{πn
+LDpTq
+– lim
+ÐÝ
+n
+˜AL bAL DpTq{πn
+LDpTq
+– lim
+ÐÝ
+n
+˜AL bAL DpT{πn
+LTq
+– lim
+ÐÝ
+n
+˜DpT{πn
+LTq
+– ˜DpTq,
+where we use for the second and fourth equation exactness of D, for the second last one the
+case of finite T and for the first, third and last equation the elementary divisor theory for the
+discrete valuation rings oL, AL and ˜AL, respectively.
+Lemma 4.7. Let A Ñ B be a flat extension of rings and M an A-module with an A-linear
+action by a finite group G. Then B bA M carries a B-linear G-action and we have
+pB bA MqG “ B bA MG.
+Proof. Apply the exact functor B bA ´ to the exact sequence
+0
+� MG
+� M
+pg´1qgPG� À
+gPG M,
+which gives the desired description of pB bA MqG .
+Lemma 4.8. Let A be A, Anr
+L , ˜A: or ˜A and T be a finitely generated oL-module with trivial
+action by an open subgroup H Ď HL. Then pA boL TqH “ AH boL T. Moreover, AH and ˜AH
+are free AL- and ˜AL-modules of finite rank, respectively.
+Proof. Since T – Àr
+i“1 oL{πni
+L oL with ni P N Y t8u we may assume that T “ oL{πn
+LoL for
+some n P N Y t8u. We then we have to show that
+pA{πn
+LAqH “AH{πn
+LAH
+(4)
+For n “ 8 there is nothing to prove.
+The case n “ 1: First of all we have A{πLA “ Anr
+L {πLAnr
+L “ Esep
+L . On the other hand,
+by the Galois correspondence between unramified extensions and their residue extensions,
+we have that pEsep
+L qH is the residue field of pAnr
+L qH. Hence the case n “ 1 holds true for
+A “ Anr
+L . After having finished all cases for A “ Anr
+L we will see at the end of the proof that
+pAnr
+L qH “ AH. Therefore the case n “ 1 for A “ A will be settled, too.
+For A “ ˜A we only need to observe that ˜A{πL ˜A “ WpC5
+pqL{πLWpC5
+pqL “ C5
+p and that
+pC5
+pqH is the residue field of pWpC5
+pqLqH “ WppC5
+pqHqL.
+For A “ ˜A: we argue by the following commutative diagram
+pC5
+pqH
+–
+�❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+❙
+–
+� W :ppC5
+pqHqL{πLW :ppC5
+pqHqL
+–
+� p ˜A:qH{πLp ˜A:qH
+�
+˜AH{πL ˜AH
+–
+� p ˜A{πL ˜AqH
+–
+� p ˜A:{πL ˜A:qH.
+9
+
+The case 1 ă n ă 8: This follows by induction using the commutative diagram with exact
+lines
+0
+� AH{πn
+LAH
+–
+�
+πL¨ � AH{πn`1
+L
+AH
+�
+� AH{πLAH
+–
+�
+� 0
+0
+� pA{πn
+LAqH
+πL¨ � pA{πn`1
+L
+AqH
+� pA{πLAqH,
+in which the outer vertical arrows are isomorphism by the case n “ 1 and the induction
+hypothesis.
+Finally we can check, using the above equality (4) for A “ Anr
+L in the third equation:
+AH “
+˜
+lim
+ÐÝ
+n
+Anr
+L {πn
+LAnr
+L
+¸H
+“ lim
+ÐÝ
+n
+pAnr
+L {πn
+LAnr
+L qH
+“ lim
+ÐÝ
+n
+`
+Anr
+L qH{πn
+LpAnr
+L
+˘H
+“ pAnr
+L qH.
+Note that pAnr
+L qH is a finite unramified extension of AL and therefore is πL-adically complete.
+We also see that AH is a free AL-module of finite rank. Similarly, WpC5
+pqH
+L – pWpˆL5
+8qnr
+L qH
+is a free WpˆL5
+8qL-module of finite rank.
+Lemma 4.9. For any open subgroup H of HL the canonical maps
+WpˆL5
+8qL bAL AH
+–
+ÝÑ WppC5
+pqHqL,
+WpˆL5
+8qL b ˜A:
+L p ˜A:qH
+–
+ÝÑ WppC5
+pqHqL
+are isomorphisms.
+Proof. We begin with the first isomorphism. Since AH is finitely generated free over AL by
+Lemma 4.8, we have
+WpˆL5
+8qL bAL AH –
+˜
+lim
+ÐÝ
+n
+WnpˆL5
+8qL
+¸
+bAL AH – lim
+ÐÝ
+n
+´
+WnpˆL5
+8qL bAL AH¯
+.
+It therefore suffices to show the corresponding assertion for Witt vectors of finite length:
+WnpˆL5
+8qL bAL AH{πn
+LAH “ WnpˆL5
+8qL bAL AH
+–
+ÝÑ WnppC5
+pqHqL.
+To this aim we first consider the case n “ 1. From (4) we know that AH{πn
+LAH “ pEsep
+L qH.
+Hence we need to check that
+ˆL5
+8 bEL pEsep
+L qH
+–
+ÝÑ pC5
+pqH
+is an isomorphism. Since the perfect hull Eperf
+L
+of EL (being purely inseparable and normal)
+and pEsep
+L qH (being separable) are linear disjoint extensions of EL their tensor product is equal
+to the composite of fields Eperf
+L
+pEsep
+L qH (cf. [Coh, Thm. 5.5, p. 188]), which moreover has to
+10
+
+have degree rHL : Hs over Eperf
+L
+. Since the completion of the tensor product is ˆL5
+8bELpEsep
+L qH,
+we see that the completion of the field Eperf
+L
+pEsep
+L qH is the composite of fields ˆL5
+8pEsep
+L qH,
+which has degree rHL : Hs over ˆL5
+8. But ˆL5
+8pEsep
+L qH Ď pC5
+pqH. By the Ax-Tate-Sen theorem
+pC5
+pqH has also degree rHL : Hs over ˆL5
+8. Hence the two fields coincide, which establishes the
+case n “ 1.
+The commutative diagram
+ˆL5
+8 bAL AH
+ϕm
+q bid –
+�
+–
+� pC5
+pqH
+ϕm
+q
+–
+�
+ˆL5
+8 bϕm
+q ,AL AH id ϕm
+q � pC5
+pqH
+shows that also the lower map is an isomorphism. Using that Verschiebung V on WnppC5
+pqHqL
+and WnpˆL5
+8qL is additive and satisfies the projection formula V mpxq ¨ y “ V mpx ¨ ϕm
+q pyqq we
+see that we obtain a commutative exact diagram
+0
+� ˆL5
+8 bϕnq ,AL AH
+id ϕn
+q
+�
+V nbid� Wn`1pˆL5
+8qL bAL AH
+can
+�
+� WnpˆL5
+8qL bAL AH
+–
+�
+� 0
+0
+� pC5
+pqH
+V n
+� Wn`1ppC5
+pqHqL
+� WnppC5
+pqHqL,
+from which the claim follows by induction because the outer vertical maps are isomorphisms
+by the above and the induction hypothesis. Here the first non-trivial horizontal morphisms
+map onto the highest Witt vector component.
+The second isomorphism is established as follows: We choose a subgroup N Ď H Ď HL
+which is open normal in HL and obtain the extensions of henselian discrete valuation rings
+˜A:
+L Ď p ˜A:qH “ W :ppC5
+pqHqL Ď p ˜A:qN “ W :ppC5
+pqNqL.
+The corresponding extensions of their field of fractions
+˜B:
+L Ď E :“ p ˜A:qHr 1
+πL s Ď F :“ p ˜A:qNr 1
+πL s
+satisfy F H{N “ E and F HL{N “ ˜B:
+L. Hence F{E and F{ ˜B:
+L are Galois extensions of degree
+rH : Ns and rHL : Ns, respectively. It follows that E{ ˜B:
+L is a finite extension of degree
+rHL : Hs. The henselian condition then implies2 that p ˜A:qH “ W :ppC5
+pqHqL is free of rank
+rHL : Hs over ˜A:
+L “ W :pˆL5
+8qL. The πL-adic completion p´qp of the two rings therefore can be
+obtained by the tensor product with ˜AL “ WpˆL5
+8qL. This gives the wanted
+WpˆL5
+8qL b ˜A:
+L p ˜A:qH “ W :pˆL5
+8qp
+L b ˜A:
+L p ˜A:qH “ W :ppC5
+pqHqp
+L “ WppC5
+pqHqL.
+2See Neukirch, Algebraische Zahlentheorie, proof of Satz II.6.8
+11
+
+Proposition 4.10. The sequences
+0 Ñ oL Ñ A
+ϕq´1
+ÝÝÝÑ A Ñ 0,
+(5)
+0 Ñ oL Ñ ˜A
+ϕq´1
+ÝÝÝÑ ˜A Ñ 0,
+(6)
+0 Ñ oL Ñ ˜A: ϕq´1
+ÝÝÝÑ ˜A: Ñ 0.
+(7)
+are exact.
+Proof. The first sequence is [SV15, (26), Rem. 5.1]. For the second sequence one proves by
+induction the statement for finite length Witt vectors using that the Artin-Schreier equation
+has a solution in C5
+p. Taking projective limits then gives the claim. For the third sequence only
+the surjectivity has to be shown. This can be achieved by the same calculation as in the proof
+of [KLII, Lem. 4.5.3] with R “ C5
+p. 3
+Lemma 4.11. For any finite T in RepoLpGLq the map ˜A boL T
+ϕqbid ´1
+ÝÝÝÝÝÝÑ ˜A boL T has a
+continuous set theoretical section.
+Proof. Since T – Àr
+i“1 oL{πni
+L oL for some natural numbers r, ni we may assume that T “
+oL{πn
+LoL for some n and then we have to show that the surjective map WnpC5
+pqL
+ϕq´id
+ÝÝÝÝÑ
+WnpC5
+pqL has a continuous set theoretical section. Thus me may neglect the additive structure
+and identify source and target with X “ pC5
+pqn. In order to determine the components of the
+map ϕq ´ id “: f “ pf0, . . . , fn´1q : X Ñ X with respect to these coordinates we recall that
+the addition in Witt rings is given by polynomials
+SjpX0, . . . Xj, Y0, . . . , Yjq “ Xj ` Yj ` terms in X0, . . . , Xj´1, Y0, . . . , Yj´1
+while the additive inverse is given by
+IjpX0, . . . Xjq “ ´Xj ` terms in X0, . . . , Xj´1.
+Indeed, the polynomials Ij are defined by the property that ΦjpI0, . . . , Ijq “ ´ΦjpX0, . . . , Xjq
+where the Witt polynomials have the form ΦjpX0, . . . , Xjq “ Xqj
+0 ` πLXqj´1
+1
+` . . . ` πj
+LXj.
+Modulo pX0, . . . , Xj´1q we derive that πj
+LIjpX0, . . . , Xjq ” ´πj
+LXj and the claim follows.
+Since ϕq acts componentwise rising the entries to their qth power, we conclude that
+fj “ SjpXq
+0, . . . Xq
+j , I0pX0q, . . . , IjpX0, . . . Xjqq.
+Hence the Jacobi matrix of f at a point x P X looks like
+Dxpfq “
+¨
+˚
+˝
+´1
+0
+...
+˚
+´1
+˛
+‹‚,
+3For the other see [KLII, Lem. 4.5.3] : There the exactness of corresponding sequences for sheaves on the
+proétale site SpapL, oLqpro´et is shown, which in turn implies exactness for the corresponding sequences of stalks
+at the geometric point SpapCp, oCpq. Note that taking stalks at this point is the same as taking sections over
+it.
+12
+
+i.e., is invertible in every point. As a polynomial map f is locally analytic. It therefore follows
+from the inverse function theorem [pLG, Prop. 6.4] that f restricts to a homeomorphism
+f|U0 : U0
+–
+ÝÑ U1 of open neighbourhoods of x and fpxq, respectively. By the surjectivity of
+f every x P X has an open neighbourhood Ux and a continuous map sx : Ux Ñ X with
+f ˝sx “ id|Ux. But X is strictly paracompact by Remark 8.6 (i) in (loc. cit.), i.e., the covering
+pUxqx has a disjoint refinement. There the restrictions of the sx glue to a continuous section
+of f.
+Corollary 4.12. For T in RepoLpGLq, the nth cohomology groups of the complexes concen-
+trated in degrees 0 and 1
+0
+� ˜DpTq
+ϕ´1
+� ˜DpTq
+� 0 and
+(8)
+0
+� DpTq
+ϕ´1
+� DpTq
+� 0
+(9)
+are isomorphic to HnpHL, Tq for any n ě 0.
+Proof. Assume first that T is finite. For (9) see [SV15, Lemma 5.2]. For (8) we use Lemma
+4.11, which says that the right hand map in the exact sequence
+0
+� T
+� ˜A boL T
+ϕqbid ´1� ˜A boL T
+� 0
+has a continuous set theoretical section and thus gives rise to the long exact sequence of
+continuous cohomology groups
+(10)
+0 Ñ H0pHL, Tq Ñ ˜DpTq
+ϕ´1
+ÝÝÑ ˜DpTq Ñ H1pHL, Tq Ñ H1pHL, ˜A boL Tq Ñ . . .
+Using the comparison isomorphism (3) and the subsequent Prop. 4.13 we see that all terms
+from the fifth on vanish.
+For the general case (for ˜DpTq as well as DpTq) we take inverse limits in the exact sequences
+for the pT{πm
+L Tq and observe that HnpHL, Tq – lim
+ÐÝm HnpHL, T{πm
+L Tq. This follows for n ‰ 2
+from [NSW, Cor. 2.7.6]. For n “ 2 we use [NSW, Thm. 2.7.5] and have to show that the
+projective system pH1pHL, T{πm
+L Tqqm is Mittag-Leffler. Since it is a quotient of the projective
+system pDpT{πm
+L Tqqm, it suffices for this to check that the latter system is Mittag-Leffler. But
+due to the exactness of the functor D this latter system is equal to the projective system of
+artinian AL-modules pDpTq{πm
+L DpTqqm and hence is Mittag-Leffler. We conclude by observing
+that taking inverse limits of the system of sequences (10) remains exact. The reasoning being
+the same for ˜DpTq and DpTq we consider only the former. Indeed, we split the 4-term exact
+sequences into two short exact sequences of projective systems
+0 Ñ H0pHL, V {πm
+L Tq Ñ ˜DpT{πm
+L Tq Ñ pϕ ´ 1q ˜DpT{πm
+L Tq Ñ 0
+and
+0 Ñ pϕ ´ 1q ˜DpT{πm
+L Tq Ñ ˜DpT{πm
+L Tq Ñ H1pHL, T{πm
+L Tq Ñ 0.
+Passing to the projective limits remains exact provided the left most projective systems have
+vanishing lim
+ÐÝ
+1. For the system H0pHL, T{πm
+L Tq this is the case since it is Mittag-Leffler. The
+system pϕ ´ 1q ˜DpT{πm
+L Tq even has surjective transition maps since the system ˜DpT{πm
+L Tq
+has this property by the exactness of the functor ˜D (cf. Prop. 4.6).
+13
+
+Proposition 4.13. HnpH, ˜A{πm
+L ˜Aq “ 0 for all n, m ě 1 and H Ď HL any closed subgroup.
+Proof. For j ă i the canonical projection WipC5
+pq – ˜A{πi
+L ˜A ։ ˜A{πj
+L ˜A – WjpC5
+pq corresponds
+to the projection pC5
+pqi ։ pC5
+pqj and hence have set theoretical continuous sections. Using the
+associated long exact cohomology sequence (after adding the kernel) allows to reduce the
+statement to Prop. 3.1.
+For any commutative ring R with endomorphism ϕ we write ΦpRq for the category of
+ϕ-modules consisting of R-modules equipped with a semi-linear ϕ-action. We write Φ´etpRq
+for the subcategory of étale ϕ-modules, i.e., such that M is finitely generated over R and ϕ
+induces an R-linear isomorphism ϕ˚M
+–
+ÝÑ M. Finally, we denote by Φ´et
+f pRq the subcategory
+consisting of finitely generated free R-modules.
+For M1, M2 P ΦpRq the R-module HomRpM1, M2q has a natural structure as a ϕ-module
+satisfying
+(11)
+ϕHomRpM1,M2qpαqpϕM1pmqq “ ϕM2pαpmqq ,
+hence in particular
+(12)
+HomRpM1, M2qϕ“id “ HomΦpRqpM1, M2q.
+Note that with M1, M2 also HomRpM1, M2q is étale.
+Remark 4.14. We recall from [KLI, §1.5] that the cohomology groups Hi
+ϕpMq of the complex
+M
+ϕ´1
+ÝÝÑ M can be identified with the Yoneda extension groups Exti
+ΦpRqpR, Mq. Indeed, if
+S :“ RrX; ϕs denotes the twisted polynomial ring satisfying Xr “ ϕprqX for all r P R, then
+we can identify ΦpRq with the category S-Mod of (left) S-modules by letting X act via ϕM on
+X. Using the free resolution
+0
+� S
+¨pX´1q � S
+� R
+� 0
+the result follows.
+Remark 4.15. Note that ˜A:
+L Ď ˜AL is a faithfully flat ring extension as both rings are discrete
+valuation rings and the bigger one is the completion of the previous one.
+Proposition 4.16. Base extension induces
+(i) an equivalence of categories
+Φ´et
+f p ˜A:
+Lq Ø Φ´et
+f p ˜ALq
+(ii) and an isomorphism of Yoneda extension groups
+Ext1
+Φp ˜A:
+Lqp ˜A:
+L, Mq – Ext1
+Φp ˜ALqp ˜AL, ˜AL b ˜A:
+L Mq
+for all M P Φ´et
+f p ˜A:
+Lq.
+14
+
+Proof. For the first item we imitate the proof of [KLI, Thm. 8.5.3], see also [Ked15, Lem.
+2.4.2,Thm. 2.4.5]: First we will show that for every M P Φ´et
+f p ˜A:
+Lq it holds that p ˜ALbMqϕ“id Ď
+Mϕ“id and hence equality. Applied to M :“ Hom ˜A:
+LpM1, M2q this implies that the base change
+is fully faithful by the equation (12). We observe that the analogue of [KLI, Lem. 3.2.6] holds
+in our setting and that S in loc. cit. can be chosen to be a finite separable field extension
+of the perfect field R “ ˆL5
+8. Thus we may choose S in the analogue of [KLI, Prop. 7.3.6]
+(with a “ 1, c “ 0 and M0 being our M) as completion of a (possibly infinite) separable field
+extension of R. This means in our situation that there exists a closed subgroup H Ď HL such
+that p ˜A:qH b ˜A:
+L M “ Àp ˜A:qHei for a basis ei invariant under ϕ. Now let v “ ř xiei be an
+arbitrary element in
+˜AL b ˜A:
+L M Ď ˜AH b ˜A:
+L M “ ˜AH bp ˜A:qH p ˜A:qH b ˜A:
+L M “
+à ˜AHei
+with xi P ˜AH and such that ϕpvq “ v. The latter condition implies that xi P ˜AH,ϕq“id “ oL,
+i.e., v belongs to pM b ˜A:
+L p ˜A:qHq X pM b ˜A:
+L
+˜ALq “ M, because M is free and one has
+˜AL X p ˜A:qH “ p ˜A:qHL “ ˜A:
+L. To show essential surjectivity one proceeds literally as in the
+proof of [KLI, Thm. 8.5.3] adapted to ramified Witt vectors.
+For the second statement choose a quasi-inverse functor F : Φ´et
+f p ˜ALq Ñ Φ´et
+f p ˜A:
+Lq with
+Fp ˜ALq “ ˜A:
+L. Given an extension 0
+� M
+� E
+� ˜AL
+� 0 over Φp ˜ALq with M P
+Φ´et
+f p ˜ALq first observe that E P Φ´et
+f p ˜ALq, too. Indeed, ˜AL
+ϕq
+ÝÑ ˜AL is a flat ring extension,
+whence ϕ˚E Ñ E is an isomorphism, if the corresponding outer maps are. The analogous
+statement holds over ˜A:
+L. Therefore the sequence 0
+� FpMq
+� FpEq
+� ˜A:
+L
+� 0
+is exact by Remark 4.15, because its base extension - being isomorphic to the original extension
+- is, by assumption.
+We denote by M´et
+f p ˜A:
+Lq and M´et
+f p ˜ALq the full subcategories of M´etp ˜A:
+Lq and M´etp ˜ALq,
+respectively, consisting of finitely generated free modules over the base ring.
+Remark 4.17. Let M be in M´et
+f p ˜ALq and endow N :“ ˜ALb ˜A:
+L M with the canonical topology
+with respect to the weak topology of ˜AL. Then the induced subspace topology of M Ď N
+coincides with the canonical topology with respect to the weak topology of ˜A:
+L. Indeed for free
+modules this is obvious while for torsion modules this can be reduced by the elementary divisor
+theory to the case M “ ˜A:
+L{πn
+L ˜A:
+L – ˜AL{πn
+L ˜AL. But the latter spaces are direct product factors
+of ˜A:
+L and ˜AL, respectively, as topological spaces, from wich the claim easily follows.
+Proposition 4.18. For T P RepoLpGLq and V P RepLpGLq we have natural isomorphisms
+˜AL b ˜A:
+L
+˜D:pTq – ˜DpTq and
+(13)
+˜BL b ˜B:
+L
+˜D:pV q – ˜DpV q,
+(14)
+as well as
+˜A: b ˜A:
+L
+˜D:pTq – ˜A: boL T and
+(15)
+˜B: b ˜B:
+L
+˜D:pV q – ˜B: bL V,
+(16)
+15
+
+respectively. In particular, the functor ˜D:p´q : RepoLpGLq Ñ M´etp ˜A:
+Lq is exact.
+Moreover, base extension induces equivalences of categories
+M´et
+f p ˜A:
+Lq Ø M´et
+f p ˜ALq,
+and hence also an equivalence of categories
+M´etp ˜B:
+Lq Ø M´etp˜BLq.
+Proof. Note that the base change functor is well-defined - regarding the continuity of the ΓL-
+action - by Lemma 4.5 and Remark 4.15 while ˜D: is well-defined by Remark 4.17, once (13)
+will have been shown. We first show the equivalence of categories for free modules: By Prop.
+4.16 we already have, for M1, M2 P M´et
+f p ˜A:
+Lq, an isomorphism
+HomΦp ˜A:
+LqpM1, M2q – HomΦp ˜ALqp ˜AL b ˜A:
+L M1, ˜AL b ˜A:
+L M2q.
+Taking ΓL-invariants gives that the base change functor in question is fully faithful.
+In order to show that this base change functor is also essentially surjective, consider an
+arbitrary N P M´et
+f p ˜ALq. Again by 4.16 we know that there is a free étale ϕ-module M over
+˜A:
+L whose base change is isomorphic to N. By the fully faithfulness the ΓL-action descends to
+M4. Since the weak topology of M is compatible with that of N by Remark 4.17, this action
+is again continuous.
+To prepare for the proof of the isomorphism (13) we first observe the following fact. The
+isomorphism (3) implies that T and ˜DpTq have the same elementary divisors, i.e.: If T –
+‘r
+i“1oL{πni
+L oL as oL-module (with ni P NYt8u) then ˜DpTq – ‘r
+i“1 ˜AL{πni
+L ˜AL as ˜AL-module.
+We shall prove (13) in several steps: First assume that T is finite. Then T is annihilated
+by some πn
+L. We have ˜D:pTq “ ˜DpTq and ˜A:
+L{πn
+L ˜A:
+L “ ˜AL{πn
+L ˜AL so that there is nothing to
+prove. Secondly we suppose that T is free and that ˜D:pTq is free over ˜A:
+L of the same rank
+r :“ rkoL T. On the other hand, as the functor ˜D: is always left exact, we obtain the injective
+maps
+˜D:pTq{πn
+L ˜D:pTq Ñ ˜D:pT{πn
+LTq “ ˜DpT{πn
+LTq.
+for any n ě 1. We observe that both sides are isomorphic to p ˜A:
+L{πn
+L ˜A:
+Lqr “ p ˜AL{πn
+L ˜ALqr.
+Hence the above injective maps are bijections. We deduce that
+˜AL bA:
+L
+˜D:pTq – lim
+ÐÝ
+n
+˜D:pTq{πn
+L ˜D:pTq
+– lim
+ÐÝ
+n
+˜DpT{πn
+LTq
+– lim
+ÐÝ
+n
+˜DpTq{πn
+L ˜DpTq
+– ˜DpTq
+using that the above tensor product means πL-adic completion for finitely generated ˜A:
+L-
+modules.
+4As γ P ΓL acts semilinearly, one formally has to replace N
+γÝÑ N by the linearized isomorphism ˜AL bγ, ˜
+AL
+N
+γlin
+ÝÝÝÑ N. Upon checking that the source is again a étale ϕ-module with model ˜A:
+L bγ, ˜
+A:
+L M one sees by the
+fully faithfulness on ϕ-modules that the linearized isomorphism descends and induces the desired semi-linear
+action.
+16
+
+Thirdly let T P RepoL,fpGLq be arbitrary and M P M´et
+f p ˜A:
+Lq such that ˜AL b ˜A:
+L M –
+˜DpTq according the equivalence of categories. Without loss of generality we may treat this
+isomorphism as an equality. Similarly as in the proof of Prop. 4.16 and with the same notation
+one shows that p ˜A: b ˜A:
+L Mqϕ“1 “ Àr
+i“1 oLei for some appropriate ϕ-invariant basis e1, . . . , er
+of ˜A: b ˜A:
+L M. Note that r “ rkoL T. Using (3), it follows that
+T “ p ˜A boL Tqϕ“1 – p ˜A b ˜AL ˜DpTqqϕ“1 “ p ˜A b ˜A:
+L Mqϕ“1
+“
+r
+à
+i“1
+˜Aϕq“1ei “
+r
+à
+i“1
+oLei “ p ˜A: b ˜A:
+L Mqϕ“1.
+It shows that the comparison isomorphism (3) restricts to an injective map T ãÑ ˜A: b ˜A:
+L M,
+which extends to a homomorphism ˜A: boL T
+αÝÑ ˜A: b ˜A:
+L M of free ˜A:-modules of the same
+rank r. Further base extension by ˜A gives back the isomorphism (3). Since ˜A is faithfully flat
+over ˜A: the map α was an isomorphism already. By passing to HL-invariants we obtain an
+isomorphism ˜D:pTq – M and see that ˜D:pTq is free of the same rank as T. Hence the second
+case applies and gives (13) for free T and (14). Finally, let T be just finitely generated over oL.
+Write 0 Ñ Tfin Ñ T Ñ Tfree Ñ 0 with finite Tfin and free Tfree. We then have the commutative
+exact diagram
+0
+� ˜AL b ˜A:
+L
+˜D:pTfinq
+–
+�
+� ˜AL b ˜A:
+L
+˜D:pT q
+�
+� ˜AL b ˜
+A:
+L
+˜D:pTfreeq
+–
+�
+� ˜AL b ˜A:
+L H1pHL, ˜A: boL Tfinq
+0
+� ˜DpTfinq
+� ˜DpT q
+� ˜DpTfreeq
+� 0,
+in which we use the first and third step for the vertical isomorphisms. In order to show that the
+middle perpendicular arrow is an isomorphism it suffices to prove that H1pHL, ˜A:boLTfinq “ 0.
+But since Tfin is annihilated by some πn
+L we have
+˜A: boL Tfin – ˜A{πn
+L ˜A boL Tfin – ˜A{πn
+L ˜A b ˜AL ˜DpTfinq,
+the last isomorphism by (3). Thus it suffices to prove the vanishing of H1pHL, ˜A{πn
+L ˜Aq, which
+is established in Prop. 4.13 and finishes the proof of the isomorphism (13).
+Note that this base change isomorphism implies the exactness of ˜D: as ˜D is exact by Prop.
+4.6 and using that the base extension is faithfully flat by Remark 4.15.
+For free T the statement (15) (and hence (16)) is already implicit in the above arguments
+while for finite T the statement coincides with (3). The general case follows from the previous
+ones by exactness of ˜D: and the five lemma as above.
+Corollary 4.19. For a T in RepoL,fpGLq and V in RepLpGLq, the nth cohomology group, for
+any n ě 0, of the complexes concentrated in degrees 0 and 1
+0
+� ˜D:pTq
+ϕ´1
+� ˜D:pTq
+� 0 and
+(17)
+0
+� ˜D:pV q
+ϕ´1
+� ˜D:pV q
+� 0 and
+(18)
+is isomorphic to HnpHL, Tq and HnpHL, V q, respectively.
+17
+
+Proof. The integral result reduces, by (13), Remark 4.14, and Prop. 4.16, to Corollary 4.12.
+Since inverting πL is exact and commutes with taking cohomology [NSW, Prop. 2.7.11], the
+second statement follows.
+Set A: :“ ˜A: XA and B: :“ A:r 1
+πL s as well as A:
+L :“ pA:qHL. Note that B:
+L :“ pB:qHL Ď
+B: Ď ˜B:. For V P RepLpGLq we define D:pV q :“ pB: bL V qHL. The categories M´etpA:
+Lq and
+M´etpB:
+Lq are defined analogously as in Definition 4.4.
+Remark 4.20. There is also the following more concrete description for A:
+L in terms of
+Laurent series in ωLT :
+A:
+L “ tFpωLT q P AL|FpZq converges on ρ ď |Z| ă 1 for some ρ P p0, 1qu Ď AL.
+Indeed this follows from the analogue of [ChCo1, Lem. II.2.2] upon noting that the latter holds
+with and without the integrality condition: ”rvppanq ` n ě 0 for all n P Z” (for r P RzR) in
+the notation of that article.
+In particular we obtain canonical embeddings A:
+L Ď B:
+L ãÑ RL
+of rings.
+Definition 4.21. V in RepLpGLq is called overconvergent, if dimB:
+L D:pV q “ dimL V. We
+denote by Rep:
+LpGLq Ď RepLpGLq the full subcategory of overconvergent representations.
+Remark 4.22. We always have dimB:
+L D:pV q ď dimL V . If V P RepLpGLq is overconvergent
+then we have the natural isomorphism
+(19)
+BL bB:
+L D:pV q –
+ÝÑ DpV q.
+Proof. Since BL and B:
+L are fields this is immediate from [FO, Thm. 2.13].
+Remark 4.23. In [Be16, §10] Berger uses the following condition to define overconvergence
+of V : There exists a BL-basis x1, . . . , xn of DpV q such that M :“ Àn
+i“1 B:
+Lxi is a pϕL, ΓLq-
+module over B:
+L. This then implies a natural isomorphism
+(20)
+BL bB:
+L M – DpV q.
+Lemma 4.24. V in RepLpGLq is overconvergent if and only if V satisfies the above condition
+of Berger. In this case M “ D:pV q.
+Proof. If V is overconvergent, we can take a basis within M :“ D:pV q. Conversely let V
+satisfy Berger’s condition, i.e. we have the isomorphism (20). One easily checks by faithfully
+flat descent that with DpV q also M is étale. By [FX, Prop. 1.5 (a)]5 we obtain the identity
+V “
+´
+B: bB:
+L M
+¯ϕ“1
+induced from the comparison isomorphism
+(21)
+B bL V – B bBL DpV q – B bB:
+L M.
+We shall prove that M Ď D:pV q “ pB: bL V qHL as then M “ D:pV q by dimension reasons.
+To this aim we may write a basis v1, . . . , vn of V over L as vi “ ř cijxj with cij P B:. Then
+(21) implies that the matrix C “ pcijq belongs to MnpB:q X GLnpBq “ GLnpB:q. Thus M is
+contained in B: bL V and - as subspace of DpV q - also HL-invariant, whence the claim.
+5Note that there ¯D actually belongs to the category of pϕ, GF q-modules over ˜BQp b F instead of over ˜BQp
+in their notation.
+18
+
+Remark 4.25. Note that the imperfect version of Prop. 4.18 is not true: the base change
+M´etpB:
+Lq Ñ M´etpBLq is not essentially surjective in general, whence not an equivalence of
+categories, by [FX]. By definition, its essential image consists of overconvergent pϕL, ΓLq-
+modules, i.e., whose corresponding Galois representations are overconvergent.
+Lemma 4.26. Assume that V P RepLpGLq is overconvergent. Then there is natural isomor-
+phism
+˜B:
+L b ˜B:
+L D:pV q – ˜D:pV q.
+Proof. By construction we have a natural map ˜B:
+L b ˜B:
+L D:pV q Ñ ˜D:pV q, whose base change
+to ˜BL
+˜BL b ˜B:
+L D:pV q Ñ ˜BL b ˜B:
+L
+˜D:pV q – ˜DpV q
+arises also as the base change of the isomorphism (19), whence is an isomorphism itself. Here
+we have used the (base change of the) isomorphisms (14), (2). By faithfully flatness the original
+map is an isomorphism, too.
+5
+The perfect Robba ring
+Again let K be any perfectoid field containing L and r ą 0. For 0 ă s ď r, let ˜Rrs,rspKq be
+the completion of W rpK5qLr 1
+πL s with respect to the norm maxt| |s, | |ru, and put
+˜RrpKq “ lim
+ÐÝ
+sPp0,rs
+˜Rrs,rspKq
+equipped with the Fréchet topology. Let ˜RpKq “ lim
+ÝÑrą0 ˜RrpKq, equipped with the locally
+convex direct limit topology (LF topology). We set ˜R “ ˜RpCpq and ˜RL :“ ˜RpˆL8q. For
+geometric interpretation of these definitions, see [Ede]. As in [KLI, Thm. 9.2.15] we have
+˜RHL “ ˜RL.
+Recall from section 2 the embedding oLrrZss Ñ Wp˜EqL. As we will explain in section 8 the
+image ωLT of the variable Z already lies in WpˆL5
+8qL, so that we actually have an embedding
+oLrrZss Ñ WpˆL5
+8qL. Similarly as in [KLI, Def. 4.3.1] for the cyclotomic situation one shows
+that the latter embedding extends to a ΓL- and ϕL-equivariant topological monomorphism
+RL Ñ ˜RL, see also [W, Konstruktion 1.3.27] in the Lubin-Tate setting.
+Let R be either RL or ˜RL. A pϕL, ΓLq-module over R is a finitely generated free R-
+module M equipped with commuting semilinear actions of ϕM and ΓL, such that the action
+is continuous for the LF topology and such that the semi-linear map ϕM : M Ñ M induces
+an isomorphism ϕlin
+M : R bR,ϕR M
+–
+ÝÑ M. Such M is called étale, if there exists an étale
+pϕL, ΓLq-module N over A:
+L and ˜A:
+L (see before Definition 4.4), such that RL bA:
+L N – M
+and ˜RL b ˜A:
+L N – M, respectively.
+By MpRq and M´etpRq we denote the category of pϕL, ΓLq-modules and étale pϕL, ΓLq-
+modules over R, respectively.
+We call the topologies on ˜A:
+L and ˜A:, which make the inclusions ˜A:
+L Ď ˜A: Ď ˜R topological
+embeddings, the LF-topologies.
+19
+
+Lemma 5.1. For M P M´et
+f p ˜A:
+Lq the ΓL-action is also continuous with respect to the canonical
+topology with respect to the LF-topology of ˜A:
+L.
+Proof. The proof in fact works in the following generality: Suppose that ˜A: is equipped with
+an oL-linear ring topology which induces the πL-adic topology on oL. Consider on ˜A:
+L the
+corresponding induced topology. We claim that then the ΓL-action on M is continuous with
+respect to the corresponding canonical topology. By Prop. 6.1 we may choose T P RepoL,fpGLq
+such that M – ˜D:pTq. Then we have a homeomorphism ˜A:boL T – ˜A:b ˜A:
+L M with respect to
+the canonical topology by (15) (as any R-module homomorphism of finitely generated modules
+is continuous with respect to the canonical topology with regard to any topological ring R).
+Since oL Ď ˜A: is a topological embedding with respect to the πL-adic and the given topology,
+respectively, Lemma 4.5 implies that GL is acting continuously on ˜A: b ˜A:
+L M, whence ΓL acts
+continuously on M “
+´
+˜A: b ˜A:
+L M
+¯HL with respect to the induced topology as subspace of the
+previous module. Since all involved modules are free and hence carry the product topologies
+and since ˜A:
+L Ď ˜A: is a topological embedding, it is clear that the latter topology of M
+coincides with its canonical topology.
+We define the functor
+˜D:
+rigp´q : RepLpGLq ÝÑ Mp ˜RLq
+V ÞÝÑ p ˜R bL V qHL,
+where the fact, that ΓL acts continuously on the image with respect to the LF-topology can
+be seen as follows, once we have shown the next lemma. Indeed, (22) implies that for any
+GL-stable oL-lattice T of V we also have an isomorphism ˜RL b ˜A:
+L
+˜D:pTq –
+ÝÑ ˜D:
+rig. Now again
+Lemma 4.5 applies to conclude the claim.
+Lemma 5.2. The canonical map
+(22)
+˜RL b ˜B:
+L
+˜D:pV q –
+ÝÑ ˜D:
+rigpV q
+is an isomorphism and the functor ˜D:
+rigp´q : RepLpGLq Ñ Mp ˜RLq is exact. Moreover, we
+have a comparison isomorphism
+(23)
+˜R b ˜
+RL ˜D:
+rigpV q –
+ÝÑ ˜R boL V.
+Proof. The comparison isomorphism in the proof of (an analogue of) [KP, Thm. 2.13] implies
+the comparison isomorphism
+˜R b ˜
+RL ˜D:
+rigpV q – ˜R boL V
+together with the identity V “ p ˜R b ˜
+RL ˜D:
+rigpV qqϕL“1. On the other hand the comparison
+isomorphism (16) induces by base change an isomorphism
+˜R b ˜B:
+L
+˜D:pV q –
+ÝÑ ˜R boL V.
+Taking HL-invariants gives the first claim. The exactness of the functor ˜D:
+rigp´q follows from
+the exactness of the functor ˜D:p´q by Prop. 4.6.
+20
+
+Let R be BL, B:
+L, RL, ˜BL, ˜B:
+L, ˜RL and let correspondingly Rint be AL, A:
+L, A:
+L, ˜AL,
+˜A:
+L, ˜A:
+L. We denote by ΦpRq´et the essential image of the base change functor R bRint ´ :
+Φ´et,fpRintq Ñ Φ´et,fpRq (sic!).
+Proposition 5.3. Base change induces an equivalence of categories
+Φp ˜B:
+Lq´et Ø Φp ˜RLq´et
+and an isomorphism of Yoneda extension groups
+Ext1
+Φp ˜B:
+Lqp ˜B:
+L, Mq – Ext1
+Φp ˜
+RLqp ˜RL, ˜RL b ˜B:
+L Mq
+for all M P Φp˜B:
+Lq´et.
+Proof. The first claim is an analogue of [KLI, Thm. 8.5.6]. The second claim follows as in the
+proof of Prop. (4.16) using the fact that by Lemma 8.6.3 in loc. cit. any extension of étale
+ϕ-modules over ˜RL is again étale. Note that ˜RL{ ˜B:
+L is a faithfully flat ring extension, ˜B:
+L
+being a field.
+Corollary 5.4. If V belongs to RepLpGLq, the following complex concentrated in degrees 0
+and 1 is acyclic
+0
+� ˜D:
+rigpV q{ ˜D:pV q
+ϕ´1
+� ˜D:
+rigpV q{ ˜D:pV q
+� 0.
+(24)
+In particular, we have that the nth cohomology groups of the complex concentrated in degrees
+0 and 1
+0
+� ˜D:
+rigpV q
+ϕ´1
+� ˜D:
+rigpV q
+� 0
+are isomorphic to HnpHL, V q for n ě 0.
+Proof. Compare with [KLI, Thm. 8.6.4] and its proof (Note that the authors meant to cite
+Thm. 8.5.12 (taking c=0, d=1) instead of Thm. 6.2.9 - a reference which just does not exist
+within that book). Using the interpretation of the Hi
+ϕ as Hom- and Ext1-groups, respectively,
+the assertion is immediate from Prop. 5.3. The last statement now follows from Corollary
+4.19.
+Proposition 5.5. Base extension gives rise to an equivalence of categories
+M´etpB:
+Lq Ø M´etpRLq.
+Proof. [FX, Prop. 1.6].
+Lemma 5.6.
+(i) B:
+L Ď RL are Bézout domains and the strong hypothesis in the sense
+of [Ked08, Hypothesis 1.4.1] holds, i.e., for any n ˆ n matrix A over A:
+L the map
+pRL{B:
+Lqn 1´AϕL
+ÝÝÝÝÑ pRL{B:
+Lqn is bijective.
+Proof. [Ked08, Prop. 1.2.6].
+21
+
+Proposition 5.7. If V belongs to Rep:
+LpGLq, the following complex concentrated in degrees 0
+and 1 is acyclic
+0
+� D:
+rigpV q{D:pV q
+ϕ´1
+� D:
+rigpV q{D:pV q
+� 0,
+(25)
+where D:
+rigpV q :“ RL bB:
+L D:pV q. In particular, the complexes concentrated in degrees 0 and
+1
+0
+� D:
+rigpV q
+ϕ´1
+� D:
+rigpV q
+� 0 and 0
+� D:pV q
+ϕ´1
+� D:pV q
+� 0
+have the same cohomology groups of for n ě 0.
+Proof. This follows from the strong hypothesis in Lemma 5.6 as the Frobenius endomorphism
+on M P M´etpB:
+Lq is of the form AϕL by definition.
+Lemma 5.8. Base change induces fully faithful embeddings ΦpA:
+Lq´et Ď ΦpALq´et and ΦpB:
+Lq´et Ď
+ΦpBLq´et.
+Proof. As in the proof of Prop. 4.16 this reduces to checking that
+´
+AL bA:
+L M
+¯ϕ“id
+Ď M.
+By that proposition we know that
+´
+AL bA:
+L M
+¯ϕ“id
+Ď
+´
+˜AL bA:
+L M
+¯ϕ“id
+Ď ˜A:
+L bA:
+L M.
+Since AL X ˜A:
+L “ A:
+L within ˜AL by definition, the claim follows for the integral version,
+whence also for the other one my tensoring the integral embedding with L over oL.
+Remark 5.9. Note that H0
+: pHL, V q “ H0pHL, V q and H1
+: pHL, V q Ď H1pHL, V q. For the
+latter relation use the previous lemma, which implies that an extension which splits after base
+change already splits itself, together with Corollary 4.12 and Remark 4.14. In general the
+inclusion for H1 is strict as follows indirectly from [FX]. Indeed, otherwise the complex
+0
+� DpV q{D:pV q
+ϕ´1
+� DpV q{D:pV q
+� 0,
+(26)
+would be always acyclic, which would imply by the same observation as in Prop. 7.2 below
+together with [SV23, Thm. 5.2.10(ii)] that H1
+: pGL, V q “ H1pGL, V q in contrast to Remark
+5.2.13 in (loc. cit.).
+6
+The web of eqivalences
+We summarize the various equivalences of categories, for which we only sketch proofs or
+indicate analogue results whose proofs can be transferred to our setting.
+Proposition 6.1. The following categories are equivalent:
+(i) RepoLpGLq,
+(ii) M´etpALq,
+(iii) M´etp ˜ALq and
+22
+
+(iv) M´etp ˜A:
+Lq.
+The equivalences from piiq and pivq to piiiq are induced by base change.
+Proof. This can be proved in the same way as in [Ked15, Thm. 2.3.5], although it seems to be
+only a sketch. Another way is to check that the very detailed proof for the equivalence between
+(i) and (ii) in [GAL] almost literally carries over to a proof for the equivalence between (i)
+and (iii). Alternatively, this is a consequence of Prop. 8.2 by [KLII, Thm. 5.4.6]. See also [Kl].
+For the equivalence between (iii) and (iv) consider the 2-commutative diagram
+M´etp ˜A:
+Lq
+faithfully flat base change � M´etp ˜ALq
+�qqqqqqqqqqq
+RepoLpGLq
+�q
+q
+q
+q
+q
+q
+q
+q
+q
+q
+q
+�▼▼▼▼▼▼▼▼▼▼▼
+,
+which is induced by the isomorphism (13) and immediately implies (essential) surjectivity on
+objects and morphisms while the faithfulness follows from faithfully flat base change.
+Corollary 6.2. The following categories are equivalent:
+(i) RepLpGLq,
+(ii) M´etpBLq,
+(iii) M´etp ˜BLq and
+(iv) M´etp ˜B:
+Lq.
+The equivalences from piiq and pivq to piiiq are induced by base change.
+Proof. This follows from Propositions 4.18 and 6.1 by inverting πL.
+Proposition 6.3. The categories in Corollary 6.2 are - via base change from (iv) - also
+equivalent to
+(v) M´etp ˜RLq.
+Proof. By definition base change is essentially surjective and it is well-defined - regarding
+the continuity of the ΓL-action - by Lemma 5.1 and Lemma 4.5. Since for étale ϕL-modules
+we know fully faithfulness already, taking ΓL-invariants gives fully faithfulness for pϕL, ΓLq-
+modules, too. 6
+6Regarding ϕL-modules cf. [KLI, the equivalence between (e) and (f) of Thm. 8.5.6], see also Thm. 8.5.3 in
+(loc. cit.), the equivalence (d) to (e).
+23
+
+Altogether we may visualize the relations between the various categories by the following
+diagram:
+Rep:
+LpGLq
+RepLpGLq
+Repan
+L pGLq
+M´etpRLq
+M´etpB:
+Lq
+M´etpBLq
+M´etp ˜RLq
+M´etp˜B:
+Lq
+M´etp ˜BLq
+�
+�
+�
+D
+V
+�☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛
+☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛
+�
+˜V
+˜D
+� ✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕
+✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕✕
+�
+D:
+V :
+☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛
+☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛
+�☛☛☛☛
+�
+D:
+rig
+V :
+rig
+�✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸
+✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸
+�
+˜V :
+˜D:
+�✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮
+✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮
+�
+˜V :
+rig
+˜D:
+rig
+�❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+❂
+�
+�
+�
+�
+�
+�
+�
+Here all arrows represent functors which are fully faithful, i.e., embeddings of categories.
+Arrows without label denote base change functors. Under them the functors D, ˜D, D:, ˜D:, D:
+rig,
+and ˜D:
+rig are compatible. The arrows “ą represent equivalences of categories, while the arrows
+´ą represent embeddings which are not essentially surjective in general. We recall that the
+quasi-inverse functors are given as follows
+V pMq “pB bBL Mqϕ“1,
+˜V pMq “ p˜B b ˜BL Mqϕ“1, V :pMq “ pB: bB:
+L Mqϕ“1,
+˜V :pMq “p˜B: b ˜B:
+L Mqϕ“1,
+˜V :
+rigpMq “ p ˜R b ˜
+RL Mqϕ“1 and V :
+rigpMq “ p ˜R bRL Mqϕ“1.
+7 8 9
+7
+Cohomology: Herr complexes
+The aim of this section is to compare the Herr complexes of the various pϕL, ΓLq-modules
+attached to a given Galois representation.
+We fix some open subgroup U Ď ΓL and let L1 “ LU
+8.
+Let M0 be a complete linearly topologised oL-module with continuous U-action and with
+continuous U-equivariant endomorphism f. We define
+T :“ Tf,UpM0q :“ cone
+ˆ
+C‚pU, M0q
+pfq˚´1
+ÝÝÝÝÑ C‚pU, M0q
+˙
+r´1s
+7By [FX, Prop. 1.5 (a)] the third formula holds while by (c) there is an equivalence of categories.
+8For the fourth formula compare with the proof of Propositon 4.16 omitting the index L in ˜A:
+L, etc. to
+conclude that p ˜B bB:
+L Mqϕ“1 “ p ˜B: bB:
+L Mqϕ“1.
+9Since V :pM0q Ď V :
+rigpMq Ď ˜V :
+rigp ˜RLbRL Mq for some model M0 over B:
+L of M we obtain the last formula.
+24
+
+the mapping fibre of C‚pU, f ´ 1q. The importance of this generalized Herr-complex is given
+by the fact that it computes Galois cohomology when applied to M0 “ DpV q and f “ ϕDpV q :
+Theorem 7.1. Let V be in RepLpGLq For DpV q the corresponding pϕL, ΓLq-module over BL
+we have canonical isomorphisms
+(27)
+h˚ “ h˚
+U,V : H˚pL1, V q
+–
+ÝÝÑ h˚pTϕ,UpDpV qqq
+which are functorial in V and compatible with restriction and corestriction.
+Proof. To this aim let T be a GL-stable lattice of V . In [Ku, Thm. 5.1.11.], [KV, Thm. 5.1.11.]
+it is shown that the cohomology groups of Tϕ,UpDpTqq are canonically isomorphic to HipL1, Tq
+for all i ě 0, whence the cohomology groups of Tϕ,UpDpTqqr 1
+πL s are canonically isomorphic to
+HipL1, V q for all i ě 0.
+Note that we obtain a decomposition U – ∆ ˆ U 1 with a subgroup U 1 – Zd
+p of U and
+∆ the torsion subgroup of U. We now fix topological generators γ1, . . . γd of U 1 and we set
+Λ :“ ΛpU 1q. By [Laz, Thm. II.2.2.6] the U 1-actions extends to continuous Λ-action and one has
+HomΛ,ctspΛ, M0q “ HomΛpΛ, M0q. Consider the (homological) complexes K‚pγiq :“ rΛ
+γi´1
+ÝÝÝÑ
+Λs concentrated in degrees 1 and 0 and define the Koszul complexes
+K‚ :“KU1
+‚
+:“ K‚pγq :“
+d
+â
+Λ
+i“1
+K‚pγiq
+and
+K‚pM0q :“K‚
+U1pM0q :“ Hom‚
+ΛpK‚, M0q – Hom‚
+ΛpK‚, Λq bΛ M0 “ K‚pΛq bΛ M0.
+Following [CoNi, §4.2] and [SV23, (169)] we obtain a quasi-isomorphism
+(28)
+K‚
+U1pM0q »
+ÝÑ C‚pU 1, M0q
+inducing the quasi-isomorphism
+(29)
+Kf,U1pM0q »
+ÝÑ Tf,U1pM0q,
+where we denote by Kf,U1pM0q :“ cone
+´
+K‚pM0q
+f´id
+ÝÝÝÑ K‚pM0q
+¯
+r´1s the mapping fibre of
+K‚pfq. More generally, by [SV23, Lem. A.0.1] we obtain a canonical quasi-isomorphism
+(30)
+Kf,U1pM∆q »
+ÝÑ Tf,UpMq,
+i.e., by Theorem 7.1 we also have canonical isomorphisms
+(31)
+h˚ “ h˚
+U,V : H˚pL1, V q
+–
+ÝÝÑ h˚pKf,U1pDpV q∆qq.
+The next proposition extends this result to ˜DpV q, ˜D:pV q and ˜D:
+rigpV q instead of DpV q.
+Proposition 7.2. If V belongs to RepLpGLq, the canonical inclusions of Herr complexes
+K‚
+ϕ,U1p ˜D:pV q∆q Ď K‚
+ϕ,U1p ˜D:
+rigpV q∆q,
+K‚
+ϕ,U1p ˜D:pV q∆q Ď K‚
+ϕ,U1p ˜DpV q∆q and
+K‚
+ϕ,U1pDpV q∆q Ď K‚
+ϕ,U1p ˜DpV q∆q
+are quasi-isomorphisms and their cohomology groups are canonically isomorphic to HipL1, V q
+for all i ě 0.
+25
+
+Proof. Forming Koszul complexes with regard to U 1 we obtain the following diagram of (dou-
+ble) complexes with exact columns
+0
+�
+0
+�
+K‚pDpV q∆q
+�
+ϕ´1
+� K‚pDpV q∆q
+�
+K‚p ˜DpV q∆q
+�
+ϕ´1
+� K‚p ˜DpV q∆q
+�
+K‚pp ˜DpV q{DpV qq∆q
+�
+ϕ´1
+–
+� K‚pp ˜DpV q{DpV qq∆q
+�
+0
+0
+in which the bottom line is an isomorphism of complexes by 4.12, as the action of ∆ commutes
+with ϕ. Hence, going over to total complexes gives an exact sequence
+0 Ñ K‚
+ϕ,UpDpV q∆q Ñ K‚
+ϕ,Up ˜DpV q∆q Ñ K‚
+ϕ,Upp ˜DpV q{DpV qq∆q Ñ 0,
+in which K‚
+ϕ,Upp ˜DpV q{DpV qq∆q is acyclic. Thus we have shown the statement regarding the
+last inclusion. The other two cases are dealt with similarly, now using (24) and 4.19 combined
+with (8). It follows in particular that all six Koszul complexes in the statement are quasi-
+isomorphic. Therefore the second part of the assertion follows from (31).
+In accordance with diagram at the end of subsection 6 we may visualize the relations
+between the various Herr complexes by the following diagram:
+C‚pGL1, V q
+K‚
+ϕ,U1pD:
+rigpV q∆q
+K‚
+ϕ,U1pD:pV q∆q
+K‚
+ϕ,U1pDpV q∆q
+K‚
+ϕ,U1p ˜D:
+rigpV q∆q
+K‚
+ϕ,U1p ˜D:pV q∆q
+K‚
+ϕ,U1p ˜DpV q∆q
+�
+�☛
+☛
+☛
+☛
+☛
+☛
+☛
+☛
+☛
+☛
+☛
+☛
+�
+�
+�
+�
+�
+�
+�
+26
+
+Here all arrows represent injective maps of complexes, among which the arrows “ą repre-
+sent quasi-isomorphisms, while the arrows ´ą need not induce isomorphisms on cohomology,
+in general. The interrupted arrow ´ ´ą means a map in the derived category while ă ´ ´ą
+means a quasi-isomorphism in the derived category. By [SV23, Lem. A.0.1] we have a analogous
+diagram for Tϕ,Up?pV qq with ? P tD, ˜D, D:, ˜D:, D:
+rig, ˜D:
+rigu.
+Remark 7.3. The image of
+hipTϕ,UpD:
+rigpV qqq – hipK‚
+ϕ,U1pD:
+rigpV q∆qq – hipK‚
+ϕ,U1pD:pV q∆qq – hipTϕ,UpD:pV qqq
+in HipL1, V q is independent of the composite (“ path) in above diagram.
+8
+Weakly decompleting towers
+Kedlaya and Liu’s developed in [KLII, §5] the concept of perfectoid towers and studied their
+properties in an axiomatic way. The aim of this section is to show that the Lubin-Tate ex-
+tensions considered in this article form a weakly decompleting, but not a decompleting tower,
+properties which we will recall or refer to in the course of this section. Moreover, we have to
+show that the axiomatic period rings coincide with those introduced earlier.
+In the sense of Def. 5.1.1 in (loc. cit.) the sequence Ψ “ pΨn : pLn, oLnq Ñ pLn`1, oLn`1qq8
+n“0
+forms a finite étale tower over pL, oLq or X :“ SpapL, oLq, which is perfectoid as ˆL8 is by
+[GAL, Prop. 1.4.12].10
+Therefore we can use the perfectoid correspondence [KLII, Thm. 3.3.8] to associate with
+pˆL8, oˆL8q the pair
+p ˜RΨ, ˜R`
+Ψq :“ pˆL5
+8, o5
+ˆL8q.
+Now we recall the variety of period rings, which Kedlaya and Liu attach to the tower, in our
+notation, starting with
+Perfect period rings:
+˜AΨ :“ ˜AL “ WpˆL5
+8qL,
+˜A`
+Ψ :“ Wpo5
+ˆL8qL Ď ˜A:,r
+Ψ :“ ˜A:,r
+L “ tx “
+ÿ
+iě0
+πi
+Lrxis P WpˆL5
+8qL| |πi
+L}xi|r
+5
+iÑ8
+ÝÝÝÑ 0u,
+˜A:
+Ψ :“
+ď
+rą0
+˜A:,r
+Ψ “ ˜A:
+L
+Imperfect period rings:
+To introduce these we first recall the map Θ : Wpo5
+CpqL Ñ oCp, ř
+iě0 πi
+Lrxis ÞÑ ř πi
+Lx7
+i,
+which extends to a map Θ : ˜A:,s
+Ψ Ñ Cp for all s ě 1; for arbitrary r ą 0 and n ě ´ logq r the
+10In the notation of [KLII]: E “ L, ̟ “ πL, h “ r, k :“ oL{pπLq “ Fq, i.e. q “ pr. AΨ,n :“ Ln, A`
+Ψ,n :“ oLn,
+X :“ SpapL, oLq with the obvious transition maps which are finite étale.
+pAΨ, A`
+Ψq :“ lim
+ÝÑnpAΨ,n, A`
+Ψ,nq “ pL8, oL8q
+p ˜AΨ, ˜A`
+Ψq :“ pAΨ, A`
+Ψq^πL´adic “ pˆL8, oˆL8q
+27
+
+composite ˜A:,r
+Ψ
+ϕ´n
+L
+ÝÝÑ ˜A:,1
+Ψ
+Θ
+ÝÑ Cp is well defined and continuous as it is easy to check. It is a
+homomorphism of oL-algebras by [GAL, Lem. 1.4.18].
+Following [KLII, §5] we set A:,r
+Ψ
+:“ tx P ˜A:,r
+Ψ |Θpϕ´n
+q pxqq P Ln for all n ě ´ logq ru,
+A:
+Ψ :“ Ť
+rą0 A:,r
+Ψ , its completion AΨ :“ pA:
+Ψq^πL´adic, and residue field RΨ :“ AΨ{pπLq “
+pA:
+Ψq{pπLq Ď ˜RΨ, R`
+Ψ :“ RΨ X ˜R`
+Ψ.
+Note that ωLT “ trιptqsu P ˜A`
+Ψ :“ Wpo5
+ˆL8qL Ď ˜A:,r
+Ψ
+for all r ą 0 (in the notation of
+[GAL]). [GAL, Lem. 2.1.12] shows
+Θpϕ´n
+q pωLT qq “ Θptrϕ´n
+q pωqsuq “ lim
+iÑ8rπi
+Lsϕpzi`nq “ zn P Ln,
+where t “ pznqně1 is a fixed generator of the Tate module Tπ of the formal Lubin-Tate group
+and ω “ ιptq P Wpo5
+CpqL is the reduction of ωLT modulo πL satisfying with EL “ kppωqq.
+Therefore ωLT belongs to A`
+Ψ :“ AΨ X ˜A`
+Ψ. Then it is clear that first A`
+L :“ oLrrωLT ss Ď ˜A:
+Ψ
+and by the continuity of Θ ˝ ϕ´n
+L
+even A`
+L Ď A:
+Ψ holds. Since ω´1
+LT P ˜A
+:, q´1
+q
+Ψ
+by [Ste, Lem.
+3.10] (in analogy with [ChCo1, Cor. II.1.5]) and Θ ˝ ϕ´n
+L
+is a ring homomorphism, it follows
+that ω´1
+LT P A
+:, q´1
+q
+Ψ
+and oLrrωLT ssr
+1
+ωLT s Ď A:
+Ψ.
+Lemma 8.1. We have R`
+Ψ “ E`
+L and RΨ “ EL.
+Proof. From the above it follows that EL Ď RΨ, whence Eperf
+L
+Ď Rperf
+Ψ
+Ď ˜RΨ “ ˆL5
+8 the latter
+being perfect. Since {
+Eperf
+L
+“ ˆL5
+8 by [GAL, Prop. 1.4.17] we conclude that
+(32)
+Rperf
+Ψ
+is dense in ˜RΨ.
+By [KLII, Lem. 5.2.2] have the inclusion
+R`
+Ψ Ď tx P ˜RΨ|x “ p¯xnq with ¯xn P oLn{pz1q for n ąą 1u
+(*)
+“ E`
+L “ krrωss
+where the equality (*) follows from work of Wintenberger as recalled in [GAL, Prop. 1.4.29].
+Since E`
+L Ď ˜R`
+Ψ by its construction in (loc. cit.), we conclude that R`
+Ψ “ E`
+L.
+Since each element of RΨ is of the form
+a
+ωm with a P R`
+Ψ and m ě 0 by [GAL, Lem.
+1.4.6]11, we conclude that RΨ “ EL.
+Thus for each r ą 0 such that ω´1
+LT P A:,r
+Ψ , reduction modulo πL induces a surjection
+A:,r
+Ψ ։ RΨ. Recall that Ψ is called weakly decompleting, if
+(i) Rperf
+Ψ
+is dense in ˜RΨ.
+(ii) for some r ą 0 we have a strict surjection A:,r
+Ψ ։ RΨ induced by the reduction modulo
+πL for the norms | ´ |r defined by |x|r :“ supit|πi
+L}xi|r
+5u for x “ ř
+iě0 πi
+Lrxis, and | ´ |r
+5.
+We recall from [FF, Prop. 1.4.3.] or [KLI, Prop. 5.1.2 (a)] that | ´ |r is multiplicative.
+Proposition 8.2. The above tower Ψ is weakly decompleting.
+11For α P RΨ there exist m ě 0 such that |ωmα|5 ď 1, i.e., ωmα P R`
+Ψ.
+28
+
+Proof. Since (32) gives (i), only piiq is missing: Since ωLT has rωs in degree zero of its Te-
+ichmüller series, we may and do choose r ą 0 such that |ωLT ´ rωs|r ă |ω|r
+5. Then |ωLT |r “
+maxt|ωLT ´ rωs|r, |ω|r
+5u “ |ω|r
+5. Consider the quotient norm }b}prq “ infaPA:,r
+Ψ ,a”b mod πL |a|r.
+Now let b “ ř
+něn0 anωn P RΨ “ kppωqq with an0 ‰ 0. Lift each an ‰ 0 to ˘an P oˆ
+L and set
+˘an “ 0 otherwise. Then, for the lift x :“ ř
+něn0 ˘anωn
+LT of b we have by the multiplicativity of
+| ´ |r that
+}b}prq ď |x|r “ p|ωLT |rqn0 “ p|ω|r
+5qn0 “ |b|r
+5.
+Since, the other inequality |b|r
+5 ď }b}prq giving by continuity is clear, the claim follows.
+Proposition 8.3. AL “ AΨ.
+Proof. Both rings have the same reduction modulo πL. And using that the latter element is
+not a zero-divisor in any of these rings we conclude inductively, that AL{πn
+LAL “ AΨ{πn
+LAΨ
+for all n. Taking projective limits gives the result.
+Proposition 8.4. A:
+L “ A:
+Ψ.
+Proof. By [KLII, Lem. 5.2.10] we have that A:
+Ψ “ ˜A:
+L X RL. On the other hand A:
+L “
+p ˜A: X AqHL “ ˜A:
+L X A is contained in RL by Remark 4.20, whence A:
+L Ď A:
+Ψ while the
+inclusion A:
+Ψ Ď ˜A: X AL “ A:
+L follows from Prop. 8.3.
+In Definition 5.6.1 in (loc. cit.) they define the property decompleting for a tower Ψ, which
+we are not going to recall here as it is rather technical. The cyclotomic tower over Qp is of this
+kind for instance. If our Ψ would be decompleting, the machinery of (loc. cit.), in particular
+Theorems 5.7.3/4, adapted to the Lubin-Tate setting would imply that all the categories at
+the end of section 6 are equivalent, which contradicts Remark 4.25.
+29
+
+References
+[Ax]
+Ax, J.: Zeros of polynomials over local fields—The Galois action. J. Algebra 15
+(1970), 417–428.
+[BV]
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+34
+

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+Draft version January 16, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+HST Low Resolution Stellar Library
+Tathagata Pal
+,1 Islam Khan
+,1, 2 Guy Worthey
+,1 Michael D. Gregg
+,3 and David R. Silva
+4
+1Washington State University
+1245 Webster Hall
+Pullman, WA 99163, USA
+2Haverford College
+370 Lancaster Ave
+Haverford, PA 19041, USA
+3University of California, Davis
+517 Physics Building
+Davis, CA 95616, USA
+4The University of Texas at San Antonio
+College of Sciences, Dean’s Office, Suite 3.205
+One UTSA Circle San Antonio, TX 78249
+ABSTRACT
+Hubble Space Telescope’s (HST) Space Telescope Imaging Spectrograph (STIS) targeted 556 stars
+in a long-running program called Next Generation Spectral Library (NGSL) via proposals GO9088,
+GO9786, GO10222, and GO13776. Exposures through three low resolution gratings provide wavelength
+coverage from 0.2 < λ < 1 µm at λ/∆λ ∼ 1000, providing unique coverage in the ultraviolet (UV).
+The UV grating (G230LB) scatters red light and this results in unwanted flux that becomes especially
+troubling for cool stars. We applied scattered light corrections based on Worthey et al. (2022a) and
+flux corrections arising from pointing errors relative to the center of the 0.′′2 slit. We present 514
+fully reduced spectra, fluxed, dereddened, and cross-correlated to zero velocity. Because of the broad
+spectral range, we can simultaneously study Hα and Mg II λ2800, indicators of chromospheric activity.
+Their behaviors are decoupled. Besides three cool dwarfs and one giant with mild flares in Hα, only Be
+stars show strong Hα emission. Mg2800 emission, however, strongly anti-correlates with temperature
+such that warm stars show absorption and stars cooler than 5000K universally show chromospheric
+emission regardless of dwarf/giant status or metallicity. Transformed to Mg2800 flux emerging from
+the stellar surface, we find a correlation with temperature with approximately symmetric astrophysical
+scatter, in contrast to other workers who find a basal level with asymmetric scatter to strong values.
+Unsurprisingly, we confirm that Mg2800 activity is variable.
+Keywords: Galaxy: stellar content — stars: abundances — stars: chromospheres — stars: flare —
+stars: fundamental parameters — ultraviolet: stars
+1. INTRODUCTION
+Stellar libraries are important tools used in far-flung
+corners of astronomy and astrophysics.
+They contain
+stellar spectra of a number of pre-selected stars in dif-
+ferent wavelength regimes (UV, visible, NIR), a variety
+of spectral resolutions, and with varied attention to flux
+calibration. Examples include a library of stellar spec-
+tra by Jacoby et al. (1984), XSHOOTER (Verro et al.
+2022a), MILES (S´anchez-Bl´azquez et al. 2006), Indo-
+US (Valdes et al. 2004), IRTF (Cesetti et al. 2013),
+ELODIE (Soubiran et al. 1998; Prugniel et al. 2007),
+Lick (Worthey et al. 1994, 2014), and UVES-POP (Bag-
+nulo et al. 2003) libraries. Such libraries are often incor-
+porated into stellar population synthesis models.
+For
+example, the MILES library (S´anchez-Bl´azquez et al.
+2006) was used to compute simple stellar population
+(SSP) SEDs in the optical wavelength range with com-
+prehensive metallicity coverage (Vazdekis et al. 2010;
+Falc´on-Barroso et al. 2011). There are many other ex-
+amples, such as Bruzual & Charlot (2003); Verro et al.
+(2022b); Le Borgne et al. (2004); Vazdekis et al. (2012);
+Worthey et al. (2022b). On a star by star basis, libraries
+arXiv:2301.05335v1  [astro-ph.SR]  13 Jan 2023
+
+ID2
+Pal et al.
+can be used to infer stellar parameters like Teff, log g,
+and [Fe/H] (e.g., Wu et al. 2011). Stellar libraries also
+find application in study of stellar clusters (Alloin 1996;
+Deng & Xin 2010). One notable example is the BaSeL
+3.1 stellar SED library (Lejeune et al. 1997, 1998; West-
+era & Buser 2003). This library is suitable for study of
+clusters at low metallicities, and has been exploited for
+the study of globular clusters (Bruzual A et al. 1997;
+Weiss & Salaris 1999; Kurth et al. 1999), open clus-
+ters (Pols et al. 1998; Lastennet et al. 1999), and blue
+stragglers (Deng et al. 1999). When well flux-calibrated,
+stellar libraries are also very important for characteriza-
+tion and performance evaluation of observational mis-
+sions like Gaia (Sudzius & Vansevicius 2002; Lastennet
+et al. 2002). Several stellar libraries are built into the ex-
+posure time calculators for HST and JWST. They even
+find use in educational products such as the University of
+Gettysburg’s CLEA and VIREO or New Mexico State
+University’s GEAS laboratory software packages to il-
+lustrate the trends among stellar spectra.
+Spectral resolution and wavelength coverage vary
+among the various existing libraries (c.f. Table 1 of Verro
+et al. 2022a), but none of them extend shortward of 300
+nm into the ultraviolet (UV) regime except those of Wu
+et al. (1983) and Fanelli et al. (1990), who present 172
+and 218 stellar spectra, respectively, observed by the
+International Ultraviolet Explorer (IUE). An important
+motivation for the present HST-based library is to re-
+lieve the relative scarcity of spectral data in the UV.
+Study of integrated spectra in the UV allows us access
+to the hottest stars, which are main sequence turnoff
+stars with some blue straggler (BS) contribution. For
+older stellar populations, UV bright populations include
+blue horizontal branch (BHB) and post-asymptotic gi-
+ant branch (PAGB) stars (Koleva & Vazdekis 2012).
+An important goal is to isolate the various main se-
+quences to chart the star formation history (SFH) of the
+galaxy (Vazdekis et al. 2016). The dlog age/dlog Z =
+−3/2 age/metallicity degeneracy (Worthey 1994) be-
+comes more like ≈ −1/1 in the UV. In UV, we have
+an abundance of strong absorption features that help
+constrain SFH, metallicity, and abundance ratios better
+(Serven et al. 2010; Toloba et al. 2009; Ponder et al.
+1998; Chavez et al. 2007). Needless to say, if we want
+to extend the limit on redshift (z) for stellar population
+studies, the UV regime is of utmost importance (Pettini
+et al. 2000; Daddi et al. 2005; van Dokkum & Brammer
+2010).
+Wu et al. (1983) and Fanelli et al. (1992) gave the
+first large, systematic spectral library in UV using data
+from IUE. The library contained spectra of around 218
+stars with a spectral resolution of 7˚A. Hubble Space
+Telescope’s (HST’s) Space Telescope Imaging Spectro-
+graph (STIS) improves upon IUE in both flux calibra-
+tion and spectral resolution. Forty O, PAGB, and He-
+burning stars were observed with STIS to make a hot
+star spectral library (Khan & Worthey 2018a). Made
+by stitching together spectra from three different grat-
+ings, these spectra have wavelength coverage from ∼
+2000˚A to ∼ 10000˚A with a resolution of R ≈ λ/∆λ ∼
+1000. The hot star library was modeled after an ear-
+lier effort called the Next Generation Spectral Library
+(NGSL, Gregg et al. (2006)) which has not so far been
+completely described in the literature. The NGSL cov-
+ers a wide range of stellar parameters, including metal-
+licity (Heap & Lindler 2010; Koleva & Vazdekis 2012;
+Vazdekis et al. 2016). The original proposal was to ob-
+tain spectra of close to 600 stars via “snapshot” style
+programs (GO9088, GO9786, GO10222, and GO13776)
+in which single orbits left stranded between larger pro-
+grams are exploited for short observations. Spectra of
+more than half of the stars that were observed (around
+374 stars corresponding to proposals GO9088, GO9786,
+and GO10222) were reduced and made publicly avail-
+able by Heap & Lindler (2009). The main intent of this
+paper is to provide a reduction of the full library to the
+community. The spectral quality is improved by apply-
+ing additional corrections such as scattered light, slit
+off-center, and dust corrections.
+We also investigate the Mg II λ2800 feature, which
+is a pair of resonance lines designated by h and k.
+Boehm-Vitense (1981) used high-resolution IUE spec-
+tra on F stars to chart four origins for Mg2800 pro-
+file morphology: the main, broad stellar absorption fea-
+ture, a narrower chromospheric emission core, a rare,
+even narrower self-absorption, and interstellar absorp-
+tion. Fanelli et al. (1990) also noted that, when in emis-
+sion, it probably indicates a chromospheric origin. Lin-
+sky & Ayres (1978) argue that most of the Ca II emission
+(λλ3933, 3968) arises in the lower chromosphere, Mg II
+in the middle chromosphere, and Lyα in the upper chro-
+mosphere, and that, together, these resonance features
+provide the bulk of the radiative cooling that occurs in
+the layers exterior to the photosphere.
+The dynamo action brought about by differential stel-
+lar rotation is one of the most commonly accepted mech-
+anisms for magnetic field generations in main sequence
+stars (Hartmann & Noyes 1987; Fr¨ohlich et al. 2012;
+Quentin & Tout 2018). Chromospheric activity is gen-
+erally associated with strong magnetic fields (Musielak
+& Bielicz 1982; Brown et al. 2022).
+Since stellar ro-
+tation is expected to slow down over the lifetime of a
+star, activity can be presumed to decrease (Barry 1988).
+This leads to the possibility that chromospheric activity
+
+HST Low Resolution Stellar Library
+3
+indicators (CaII, MgII, or Hα) may provide relatively
+precise chronometric information, at least in predefined
+spectral type bands (Barry 1988).
+Although, in gen-
+eral ages would be poorly constrained (Pace 2013). In
+addition, acoustic shocks without a magnetohydrody-
+namic component may also contribute to the chromo-
+spheric activity (Buchholz et al. 1998; Mart´ınez et al.
+2011; P´erez Mart´ınez et al. 2014).
+Connections between Mg2800 strengths and the astro-
+physics of stellar properties are still tenuous, but could
+eventually lead to realistic chromosphere models as a
+function of stellar type and magnetic field strength. In
+the meantime, we have various empirical clues. Houde-
+bine & Stempels (1997) finds that, at spectral type M1,
+metal-deficient stars are also activity-de���cient. Smith
+et al. (1991) compares Mg2800 with the Ca II S index
+(Vaughan & Preston 1980) which measures the width
+of the emission rather than its strength. They also find
+that the available sample of 20 FGK stars can be sep-
+arated into “high activity” and “low activity” groups
+at an approximately 4:16 ratio, but that Mg2800 dis-
+plays a large range of values even amongst the low ac-
+tivity group (Mart´ınez et al. 2011; P´erez Mart´ınez et al.
+2014). Interstellar absorption usually dominates in OB
+stars (Khan & Worthey 2018b). Due to its wide cover-
+age of parameter spaces, the present library can confirm
+or extend these trends.
+This paper is organized as follows. We describe the
+observations and sample in §2. The data reduction pro-
+cess is detailed in §3, and additional corrections that
+affect the continuum shape in §4.
+The format of the
+data catalog is described in §5. In §6, we investigate the
+Mg II 2800 feature and chart the systematics of chromo-
+spheric activity across the H-R diagram. We conclude
+in §7 with a summary of the results and a discussion of
+their implications.
+2. OBSERVATIONS AND SAMPLE
+The stars in the library were selected to cover Teff-
+L-Z space insofar as the Galaxy could provide them.
+For example, given that the metal-poor components of
+the Milky Way are also ancient in age, no luminous,
+low-metallicity stars exist. The parameter coverage is
+shown in Fig. 1 in log g-log Teff space with metallicity in-
+dicated by symbol type. Fig. 2 on the other hand shows
+the distribution of all the stars in different metallicity
+bins. The impact of including stars from GO13776 sig-
+nificantly improves coverage of Teff-L-Z space, as shown
+in Figs. 1 and 2. Several A-type field horizontal branch
+stars were observed to attempt to fill in the warm-and-
+metal-poor gap. Desirable faint stars, such as individual
+Small Magellanic Cloud stars, could not be observed due
+to the one-orbit limit on exposure time. The target list is
+hand-selected, and should not be used for any statistical
+inferences. In addition, HST’s SNAP mode selects from
+a larger input list according to schedulability, leading to
+further randomization.
+Figure 1. NGSL stars are plotted in log Teff, log g space.
+The previously published stars from proposals GO9088,
+GO9786, and GO10222 (Koleva & Vazdekis 2012, pluses)
+and the GO13776 stars (circles) are split by metallicity, metal
+poor (MP): [Fe/H] ≤ −1 (red) or metal rich (MR): [Fe/H]
+> −1 (blue). An approximate Eddington stability line and
+spectral type boundaries are included in the plot as visual
+guides.
+Figure 2. The [Fe/H] distribution of all reduced targets in
+a stacked histogram. Blue corresponds to 345 targets from
+Koleva & Vazdekis (2012) and red corresponds to 169 targets
+from HST proposal GO13776.
+During the orbit in which they were targeted, the
+NGSL stars were observed by cycling through three dif-
+ferent gratings. G230LB sees in UV (central wavelength
+of 2375˚A), G430L sees in blue (central wavelength of
+4300˚A) and G750L sees in red (central wavelength of
+
+0
+B
+A
+G
+K
+M
+2
+Eddington
+3
+6
+4
+5
+十
+Koleva&Vazdekis2012,MP
+十
+Koleva&Vazdekis2012.MR
+6
+GO13776,MP
+GO13776.MR
+4.6
+4.4
+4.2
+4.0
+3.8
+3.6
+3.4
+log Teff120
+Koleva &Vazdekis,2012
+GO 13776
+100
+Frequency
+80
+60
+40
+20
+0
+3
+15
+5
+3
+5
+2
+Metallicity ([Fe/H)4
+Pal et al.
+Figure 3.
+CCD images of HD102212 using (a) G230LB,
+(b) G430L, and (c) G750L. Due to longer exposure time in
+the UV, the topmost panel shows the presence of cosmic ray
+events whereas the bottom two do not have any significant
+ion contamination. It is worth noting for this cool star that
+what appears to be a stellar trace in the UV shortward of
+2500˚A is actually light scattered from the visible portion of
+the spectrum into the UV by grating G230LB.
+7751˚A). The three gratings overlap at 2990˚A-3060˚A and
+5500˚A-5650˚A(Gregg et al. 2006). The CCD detector was
+employed for these observations.
+UV exposure times
+were longer than exposures in the blue or red. A 0.′′2
+slit, equivalent to ±2 pixels (Hernandez & et al. 2012;
+Prichard et al. 2022) was used for all the observations
+and a fringe flat was taken for the G750L grating at the
+end of each sequence of exposures.
+In addition to the usual observational defects (cos-
+mic ray hits, charge transfer efficiency effects, bad pix-
+els, and photon noise) these data suffer from two addi-
+tional sources of error that affect fluxing. Firstly, the
+G230LB grating scatters red light into the UV, creat-
+ing a spurious signal that must be corrected (Lindler &
+Heap 2010; Worthey et al. 2022a). Secondly, scatter in
+telescope pointing plus a narrow slit led to situations in
+which the jaws of the slit sliced off portions of the PSF.
+Because STIS is an off-axis instrument, the PSF is not
+symmetrical, so the resultant attenuation is wavelength-
+dependent.
+Fortunately, both of these effects can be
+modeled, and we give details in §4.
+Of minor note, STIS spectral flux calibrations have
+improved since the previous version of the NGSL library
+was placed at MAST.
+3. REDUCTION AND QUALITY CONTROL
+All 556 targets from proposals GO9088, GO9786,
+GO10222, and GO13776 were reduced from raw obser-
+vation files. Out of these 556 targets, 514 have been re-
+duced completely and additional corrections have been
+applied.
+The remaining 42 targets have not been re-
+duced either because of faulty fringe-flat files or because
+of the absence of one of the observations in UV, blue,
+or red. The raw files for all the observations (which in-
+clude observations in UV, blue, and red as well as CCD
+flats) were downloaded from the Space Telescope Science
+Institute (STScI) archive. The reduction process is car-
+ried on using the stistools Python3 package developed
+by STScI.
+The reduction procedure consisted of several steps
+starting from cosmic rays correction to combining dis-
+parate spectral windows into one continuous spectrum
+for each star.
+3.1. Cosmic Ray Correction
+Cosmic ray corrections are more crucial for observa-
+tions using G230LB grating that was used for longer-
+duration UV observations. This is illustrated in Fig. 3
+where cosmic rays are common in the G230LB expo-
+sure.
+Accordingly, all multiple UV observations were
+run through the ocrreject function of stistools.
+This
+function combined two sets of science observations in
+UV into a single file. In order to run ocrreject, we needed
+to have at least two observations at each pointing. Un-
+fortunately, the UV raw files from proposals GO10222
+and GO13776 did not have multiple UV exposures. For
+these, bad pixels were removed manually from the spec-
+tra.
+3.2. Defringing in the Red
+Fringes are interference patterns caused by photons
+with wavelengths that are integral multiples of the width
+of the CCD layer. In STIS, fringe patterns are promi-
+nent redward of ∼7000˚A and reach peak-to-peak ampli-
+tude of 25% at 9800˚A (Kimble et al. 1998; Malumuth
+et al. 2003).
+The G750L grating produces unwanted
+fringe patterns. Once per orbit, a fringe flat was ob-
+tained using the tungsten lamp on board HST.
+The
+defringing
+process
+was
+carried
+out
+using
+the
+defringe
+tool
+of
+stistools
+(for
+details,
+see
+https://stistools.readthedocs.io/en/latest/).
+The fol-
+lowing three methods were used in sequence for all the
+NGSL observations.
+1. normspflat: this method normalizes the fringe-flat
+that is associated with each observation
+2. mkfringeflat: this method cross correlates the nor-
+malized fringe-flat with that of the observed spec-
+trum to match the fringes between the two.
+It
+minimizes the RMS within a given range of shift
+and scale values to find the best shift and scale
+3. defringe: this method actually defringes the ob-
+served spectrum by removing the fringing pattern
+
+(a)
+0.0005
+-0.0001
+1635
+1842
+2049
+2256
+2463
+2670
+2877
+3084
+(b)
+0.0005
+-0.0001
+2827
+3243
+3659
+4075
+4491
+4907
+5323
+5739
+(c)
+0.0005
+-0.0001
+5122
+5861
+6600
+7339
+8078
+8817
+9556
+10295
+Wavelength (A)HST Low Resolution Stellar Library
+5
+from the observed spectrum using the shifted and
+scaled fringe-flat
+Fig. 4 shows the red spectrum of HD102212 before
+and after defringing.
+Figure 4.
+Extracted, fluxed CCD/G750L spectrum of
+HD102212. The spectrum before defringing (black) is com-
+pared to the same spectrum after (red).
+While defringing the red spectra it was observed that
+no proper fringe-flat is available for 27 targets.
+We
+dropped these stars from further analysis and thus re-
+duced the total number of targets from 556 to 529.
+While trying to defringe red spectra from GO13776. al-
+though some of the targets from GO13776 have 2 or 3 red
+spectra, only one of them defringed properly. Investiga-
+tion yielded an observing irregularity. For run GO13776,
+the G750L (red third of the spectrum) target exposures
+were preceded by a fringe flat through the 0.3 × 0.09
+notch aperture, which is placed near row 512 of the chip
+(the UV and blue spectra were taken at the E1 pseu-
+doaperture around row 900 of the CCD). The telescope
+was slewed to place the target star at row 512 of the
+chip rather than 900, and one exposure taken through
+the nominal 52×0.2 aperture. Due to an oversight, posi-
+tional dithering occurred. The telescope was slewed 0.′′5,
+and an exposure was taken through the 52×0.2 aperture
+followed by an exposure through the 52 × 0.5 aperture.
+This last exposure eliminates edge effects and provides
+the best fluxing, but it cannot be fringe-corrected us-
+ing the data collected on-orbit. Therefore, only one red
+spectrum (for each target) was used for run GO13776.
+For three targets from GO13776 (HD 65589, HD 84035,
+and HD 185264), none of the red observations could be
+satisfactorily defringed. These stars were also dropped
+from further analysis which reduced the total number of
+stars from 529 to 526.
+Also unique to proposal GO13776, the last pair of red
+observations were often a pair obtained through 0.′′2 and
+0.′′5 apertures. Although these could not be defringed
+due to the shift along the aperture center line, they could
+be used to create a relative flux correction, should the
+star have been placed off the central line of the entrance
+aperture. A smoothed division of these two spectra was
+applied to the first, defringed observation in all cases
+where the complete set of observations exists.
+3.3. 1-D Extraction
+The final step in the reduction process was to extract
+the 1-D spectrum for each target and each observation
+in UV, blue, and red using the x1d function of stistools.
+This resulted in a separate file for each UV, blue, or red
+observation for each target.
+Twelve of the remaining
+526 targets did not have one of either UV or blue or red
+observations. These stars were dropped. This reduced
+the total number of available stars to 514. 278 targets
+have 2 observations each of UV, blue, and red.
+189
+targets have 2 observations each of UV and blue, and
+1 of red. The remaining 47 targets have varied numbers
+of observations for UV, blue and red (at least 1 of each).
+3.4. Bad Pixel Handling
+As mentioned in Sec. 3.1, a cosmic ray rejection al-
+gorithm was not applied to blue and red observations.
+Even after applying cosmic ray rejection to the UV ob-
+servations, the UV spectra had leftover wild pixels of
+unusually high and non-astrophysical flux (or counts).
+In order to mitigate this problem, each observation for
+each target was checked manually for bad pixels and
+those pixel locations were flagged. This step generated
+a single text file for each target containing information
+on the number of bad pixels for each observation and
+values for those pixels. Fig. 5 shows an example of pres-
+ence of bad pixels in the spectrum.
+We tried our best to remove as many bad pixels as
+possible from each spectrum, but there are some stars
+for which many bad pixels could not be removed cleanly.
+3.5. Special Flux Scalings
+After fluxing, some spectra appeared to have been
+scaled in comparison with their neighbors. For exam-
+ple, suppose a star has been exposed twice in the UV,
+twice in the blue, and twice in the red. Now and then,
+one of those six exposures appears slightly too strong
+or too weak compared with either the spectral overlap
+region or with its supposedly identical sister spectrum.
+A scaling was applied to these deviant cases, as listed
+in Table 1. The dataset labels relate closely to the ones
+assigned by STScI, but we prepended a short string to
+indicate if the spectrum was UV (uv ), blue (b ), or red
+(r ).
+
+1e-10
+1.6
+1.4
+1.2
+1.0
+0.8
+Flux (
+0.6
+Fringed Spectrum
+0.4
+Defringed Spectrum
+6000
+7000
+8000
+9000
+10000
+Wavelength (A)6
+Pal et al.
+Figure 5. Individual spectra for NGSL star HD190360 in
+the UV (blue and orange) illustrate the presence of bad pix-
+els. After marking, the bad pixels were removed by the algo-
+rithm described in §3.7. The cleaned spectrum is also plotted
+(black). We elevated the errors for the corrected portions of
+the spectrum.
+In addition to sporadic scaling issues, observations for
+HD 1638 may have missed the target altogether, as all
+spectral segments contain mostly noise.
+3.6. Relative Velocities and Template Matching
+The NGSL stars were chosen to encompass a broad
+interval of [Fe/H], log g, and Teff (Gregg et al. 2004).
+Galactic halo stars are mostly metal poor but can pos-
+sess high relative velocity with respect to the local rest
+frame (Du et al. 2018). Thus, some of the stars in NGSL
+have relative velocities > 250 km s−1. This fact called
+for a relative velocity correction before bringing all the
+spectra to rest frame. To be consistent, we applied the
+relative velocity correction to all 514 stars even when the
+effects would be negligible. The nonrelativistic formula
+was used to correct for the relative velocity:
+dλ = v
+c × λ ,
+(1)
+where dλ is correction to the wavelength λ, v is the
+relative velocity of the star in km s−1 and c is the speed
+of light in km s−1. dλ was added or subtracted from
+corresponding λ values depending on the sign of v. The
+values of v were obtained from the SIMBAD astronom-
+ical database (Wenger et al. 2000).
+After correcting for the relative velocities, residual
+shifts to rest frame (vacuum wavelengths) were esti-
+mated by comparing with template spectra. The choice
+of template spectrum was made based on the effective
+temperature of the particular star. The high resolution
+templates were rebinned to match the observed wave-
+Table 1. Special Scalings
+Target
+Deviant
+Clean
+Scale
+Dataset
+Dataset
+Factor
+HD 224801
+b o93a6qk2q flt
+b o93a6qk3q flt
+1.0506
+BD+17 4708
+r o6h03vawq drj
+r o6h03vavq drj
+1.0669
+HD 3712
+r o6h04kf0q drj
+r o6h04kezq drj
+1.2163
+HD 137759
+r o6h04bm3q drj
+r o6h04bm2q drj
+1.1468
+HD 124547
+r o6h038xkq drj
+r o6h038xjq drj
+1.0556
+HD 172506
+r o6h06jp4q drj
+r o6h06jp3q drj
+1.0639
+HD 4128
+r o6h04ynyq drj
+r o6h04ynxq drj
+1.0718
+HD 146233
+r o6h05wb0q drj
+r o6h05wazq drj
+1.0512
+HD 81797
+b o6h03rocq flt
+b o6h03robq flt
+1.1058
+HD 30614
+uv o8ru4c020 crj
+uv o8ru4c010 crj
+0.9720
+HR 753
+b o6h03ntyq flt
+b o6h03ntzq flt
+1.1994
+HD 136442
+b ocr7nwr6q flt
+b ocr7nwrcq flt
+0.9319
+HD 58343
+uv o8ru4s010 crj
+uv o8ru4s020 crj
+0.9668
+HD 217014
+b ocr7pxp7q flt
+b ocr7pxp6q flt
+0.9346
+HD 144608
+r ocr7feacq drj
+b ocr7fea7q flt
+0.9048
+HD 183324
+b o8ruclpqq flt
+b o8ruclprq flt
+1.0501
+BD+37 1458
+b o6h04ti6q flt
+b o6h04ti7q flt
+1.0302
+HD 52089
+uv o8ru46020 crj
+uv o8ru46010 crj
+0.9725
+BD+29 366
+r ocr7aif7q drj
+b ocr7aif6q flt
+0.947
+BD+25 1981
+r ocr7agwlq drj
+b ocr7agwkq flt
+0.9249
+HD 9826
+r ocr7kchgq drj
+b ocr7kcheq flt
+0.9354
+HD 19994
+r ocr7klq6q drj
+b ocr7klq4q flt
+0.852
+HD 21019
+r ocr7koizq drj
+b ocr7koiyq flt
+0.7542
+HD 21770
+r ocr7kpsuq drj
+b ocr7kpssq flt
+0.8409
+HD 25457
+r ocr7ksc9q drj
+b ocr7ksc8q flt
+0.7998
+HD 31128
+r ocr7hxziq drj
+b ocr7hxzgq flt
+0.9685
+HD 34411
+r ocr7kxklq drj
+b ocr7kxkkq flt
+0.9246
+HD 44420
+r ocr7lgwsq drj
+b ocr7lgwrq flt
+0.9174
+HD 48737
+r ocr7liuiq drj
+b ocr7liuhq flt
+0.9549
+HD 52265
+r ocr7lln2q drj
+b ocr7lln1q flt
+0.9594
+HD 57118
+r ocr7cqqaq drj
+b ocr7cqq9q flt
+0.9343
+HD 67523
+r ocr7ien9q drj
+b ocr7ien8q flt
+0.8912
+HD 71369
+r ocr7lrsqq drj
+b ocr7lrspq flt
+0.9432
+HD 82328
+r ocr7lyh7q drj
+b ocr7lyh6q flt
+0.9042
+HD 121370
+r ocr7erjeq drj
+b ocr7erjdq flt
+0.9313
+HD 134169
+r ocr7ezp9q drj
+b ocr7ezp8q flt
+0.9649
+HD 160365
+r ocr7odh7q drj
+b ocr7odh6q flt
+0.9293
+HD 161797
+r ocr7oeobq drj
+b ocr7oeoaq flt
+0.9371
+HD 188510
+r ocr7gff0q drj
+b ocr7gfexq flt
+0.9354
+HD 190390
+r ocr7ghheq drj
+b ocr7ghhdq flt
+0.939
+HD 192718
+r ocr7gkaeq drj
+b ocr7gkadq flt
+0.9066
+HD 217014
+r ocr7pxp8q drj
+b ocr7pxp7q flt
+0.8636
+Note—Additionally, for BD+17 2844 we averaged the red spectra,
+and for HD 183324 we scaled up both the UV spectra by a factor
+of 1.093 to match the blue spectra
+length points, then cross-correlated. The following tem-
+plates were adopted.
+1. Synthetic spectra were used for cool stars (Teff <
+5000 K) and warm stars (5000 K < Teff < 8000 K)
+2. The observed spectrum of α Lyrae was used for
+hot stars (Teff > 8000 K)
+
+1e-12
+1.4
+UV (Obs. 1)
+Bad pixel
+UV (Obs. 2)
+1.2
+Bad Pixel Removed
+Spectrum
+A1.0
+2
+cm
+0.8
+Bad pixel
+Flux
+-Bad pixel
+0.4
+0.2
+0.0
+1700
+1800
+1900
+2000
+2100
+2200
+Wavelength(A)HST Low Resolution Stellar Library
+7
+Figure 6. A part of spectrum for HD115383 (blue) showing
+shift of the spectrum with respect to the template (red)
+The cross correlation function (in ˚A) was fitted with
+a single peak Gaussian function. Fig. 6 shows a part of
+the spectrum for HD 102212 and illustrates the amount
+of shift present in the observed spectrum with respect
+to the template. Correlation value as a function of shift
+is shown in Fig. 7 (for the same star HD 102212). The
+same template was used for all the observations of a par-
+ticular target. To speed convergence, we added initial
+shifts of 3˚A, 9˚A and 14˚A to UV, blue and red obser-
+vations, respectively. This “pre-shift” evidently arises
+because wavelength calibrations were not performed on-
+orbit for NGSL, and so a default wavelength solution
+was assigned.
+Figure 7. Typical cross correlation value as a function of
+pixel shift in ˚A, in this case for the red spectrum of G0 V
+star HD 115383.
+3.7. The Composite Spectrum
+To assemble a single contiguous spectrum, we com-
+bined bad pixel information and shift information from
+template matching to splice all the observations for a
+particular target into one final spectrum. The shift ob-
+tained for each observation was added algebraically to
+the wavelength values.
+While applying the bad pixel
+information, we devised a method for suppressing the
+bad pixels.
+We first divided the range of each obser-
+vation into 50 overlapping boxes of 40 pixels each. For
+each box, we found out the average flux weighted by the
+variance (fbox) using the following formula–
+fbox = f1v1 + f2v2 + ... + f40v40
+v1 + v1 + ... + v40
+,
+(2)
+where fn is the flux at nth wavelength value for a
+particular box and vn is the corresponding variance (de-
+fined by, vn = 1/e2
+n where en is corresponding error in
+flux for that particular wavelength value). These flux
+values were then linearly fitted over the range of obser-
+vation. Now, the flux at the previously identified bad
+pixels was set to a flux value according to this linearly
+extrapolated relation. It is to be noted that the error
+values at the bad pixels were inflated by a factor of 1000
+before calculating fbox.
+This was done to make sure
+that the erroneous pixels do not contribute much to the
+weighted average (as bad pixels generally have very high
+flux values).
+Once the flux values at the bad pixels were set ac-
+cording to the above mentioned algorithm, we then cal-
+culated the weighted average flux value for all the ob-
+servations of a particular type (for eg., UV, blue or red)
+at a particular wavelength value. For eg., if there are 2
+UV observations for a particular target, then the aver-
+age UV flux at nth wavelength value (f UV
+n
+) is given by–
+f UV
+n
+= f 1
+nv1
+n + f 2
+nv2
+n
+v1n + v2n
+,
+(3)
+where f 1
+n and f 2
+n are UV fluxes at nth wavelength value
+for 1st and 2nd observations respectively and v1
+n & v2
+n are
+corresponding variances as defined before. This formula
+can easily be generalized for more than or less than 2 ob-
+servations. Once this operation was performed for all the
+observations of a target, we then combined all the ob-
+servations to make a single spectrum for a target treat-
+ing λ <3057˚A as UV observation, 3057˚A< λ <5679˚A
+as blue observation and λ >5679˚A as red observation.
+This algorithm does not apply without any caveat as
+sometimes the flux values at bad pixels were negative.
+Users are advised to be careful of such artifacts in the
+spectrum by considering the uncertainty we assign.
+4. CONTINUUM CORRECTIONS
+The G230LB grating scatters some red light onto the
+portions of the CCD where UV is expected (Worthey
+et al. 2022a). This is a problem mainly for cool stars
+
+Star
+0.1
+Template
+Y
+0.0
+Normalised Flux
+-0.1
+-0.2
+-0.3
+-0.4.
+6200
+6300
+6400
+6500
+6600
+6700
+6800
+6900
+7000
+Wavelength
+(A)
+(0.30
+0.25
+Correlation Value
+0.20
+0.15
+0.10
+0.05
+0.00
+-0.05
+-2
+0
+2
+4
+6
+8
+Shift
+(A)8
+Pal et al.
+(Teff ≤ 5000 K) where we do not expect significant UV
+flux. This section summarizes the results from Worthey
+et al. (2022a) on scattered light as well as slit off-center
+corrections. We also applied these corrections to the 514
+NGSL stars that we have reduced.
+4.1. Scattered Light Correction
+The scattered light (S(λ)) is approximated by the for-
+mula (Worthey et al. 2022a):
+S(λ) = K0 × (1 + 0.00104 × (λ − 2000)) ,
+(4)
+where K0 is the scattered light count rate at 2000˚A and
+λ is the wavelength. Targets with Teff <5000K, K0 is
+given by the median counts rate around 2000˚A (median
+counts rate for 1950˚A< λ <2050˚A). Two stars in our
+list, HD 124547 and HD 200905, are spectroscopic bi-
+nary stars with Teff <5000K. For these two stars, K0
+calculated using the average counts rate around 2000˚A
+resulted in over correction of the spectra. After visu-
+ally inspecting the spectrum for these two stars, the K0
+values were modified by hand to mitigate the problem
+of over correction. Targets with Teff >5000K and for
+which V magnitudes (mv) are available, K0 is given by–
+K0 = 426 × 10−0.4mv .
+(5)
+But, for some of the targets (with Teff >5000K) mv is
+not available. For such targets, K0 is given by–
+K0 = 1.78 × 10−7 × C ,
+(6)
+where C is the integrated count rate between 2000˚A and
+10000˚A. S(λ) was then subtracted from overall count
+at each λ. Fig. 8 shows an example of scattered light
+correction applied to the spectrum of HD102212.
+After applying the above mentioned formula of S(λ)
+for all the 514 stars, 96 stars (Teff >5000K) were over
+corrected and 8 stars (Teff >5000K) were under cor-
+rected as judged by inspection of the spectra.
+For
+these cases, the coefficient values (426 in Eqn. 5 and
+1.78 × 10−7 in Eqn. 6) was iteratively modified to cal-
+culate K0 until the discrepant star fell among its peers
+in the UV. The updated K0 values were then used to
+calculate S(λ) for those 104 targets.
+4.2. Slit Off-center Correction
+The NGSL targets were observed using the 0.′′2 slit. If
+the target is not placed at the center of the slit, light at
+the edges of the point spread function (PSF) gets atten-
+uated by the slit edges. Because the STIS instrument
+is off-axis, the PSF is asymmetric, and the attenuation
+is wavelength-dependent. To correct for the attenuation
+Figure 8.
+The fluxed spectrum of the star HD102212 in
+the UV region without any scattered light correction (blue)
+and with scattered light correction (red). It is seen that the
+spectrum is a little over corrected in the region around 1800˚A
+effect, we use the attenuation factor (Dλ) which is given
+by (Worthey et al. 2022a):
+Dλ = a + bq + cq2 + dq3 + eq4 + fq5 + gq6 ,
+(7)
+where q =
+�
+λ/4500. The coefficients for the above
+formula at different slit off-center values are given in Ta-
+ble 3 of Worthey et al. (2022a). The slit off-center value
+for each of the 514 NGSL spectra was calculated during
+the defringing process as outlined in §3.2. It is obvious
+that the slit off-center values for our 514 targets were not
+matching the exact values given in Table 3 of Worthey
+et al. (2022a). The Dλ curve (as a function of λ) for
+each of our targets was calculated as linearly interpo-
+lated curve between two nearest Dλ curves (for which
+coefficients are available from Worthey et al. (2022a)).
+Once the Dλ curve was calculated for each target, the
+flux of that target was divided by Dλ at each λ value.
+4.3. Dust
+We compiled interstellar dust extinction data for our
+514-star library sample. Koleva & Vazdekis (2012) gives
+non-negative AV values for around 341 stars. AV for 44
+stars are calculated by us (following Khan & Worthey
+2018b) by matching an observed spectrum with a syn-
+thetic spectrum and then fitting a 1-variable extinction
+law from Fitzpatrick (1999). The rest of the AV values
+are taken from GALExtin website version 1.2 (Amˆores
+et al. 2021) using a three dimensional Galactic extinc-
+tion model by Drimmel et al. (2003). These AV values
+are used to find the E(B-V) values using the following
+equation:
+E(B − V ) = AV
+3.1
+(8)
+
+1e-12
+1.75
+Uncorrected
+Corrected
+1.50
+A
+1.25
+cm
+1.00
+一
+(ergs
+0.75
+Flux
+0.50
+0.25
+0.00
+1800
+2000
+2200
+2400
+2600
+2800
+3000
+Wavelength (A)HST Low Resolution Stellar Library
+9
+Extension
+Description
+Primary
+Contains no data. The header
+contains information about basic
+stellar parameters ([Fe/H], log g,
+etc.) and averaged pointing
+information. Exposure-level
+pointing is available from the
+original MAST archive files.
+Flux Table
+Binary table extension with
+columns for wavelength (in ˚A),
+uncorrected flux, scattered light
+corrected flux, scattered light &
+slit off-center corrected flux, and
+scattered light, slit off-center &
+dust corrected flux (fluxes are in
+erg/s/cm2/˚A). Flux errors are
+also included as separate columns.
+Count Rate Table
+Binary table extension with
+columns for wavelength (in ˚A),
+uncorrected count rate, scattered
+light corrected count rate,
+scattered light & slit off-center
+corrected count rate, and
+scattered light, slit off-center &
+dust corrected count rate.
+Uncertainties are also included as
+separate columns.
+Flux Table (Log Scale)
+This binary table extension
+contains the same information as
+the Flux Table but the
+wavelengths are spaced on log
+scale with log ∆λ = 0.0002
+Count Rate Table
+(Log Scale)
+This binary table extension
+contains the same information as
+the Count Rate Table but the
+wavelengths are spaced on log
+scale with log ∆λ = 0.0002
+Table 2. Brief description of the FITS file structure.
+The extinction law of Fitzpatrick (1999) was used to
+correct the spectra to dust-free versions. Possible self-
+reddening for mass-losing stars was not considered. The
+E(B−V ) values were also used to deredden the observed
+colors that we use for the analysis below.
+5. PRESENTATION OF THE LIBRARY
+5.1. Archived Spectra
+All 514 spectra have been made available at http:
+//astro.wsu.edu/hststarlib/, MAST, and CDS (exact
+phrasing TBD after referee and after the data
+are placed) in 514 separate FITS (Wells et al. 1981)
+files. Each FITS file contains 5 extensions, briefly de-
+scribed in Table 2.
+Table 3 summarizes a mixture of astrophysical and
+reduction-specific metadata for each stellar target.
+5.2. Notable objects
+• Targeted object Gleise 15B, a late M dwarf in a
+visual binary system, was not observed. Due to
+the count rate and spectral shape, it is near certain
+that its primary (Gleise 15A, GJ 15A, HD 1326,
+GX And) was observed instead. Our metadata has
+been updated to reflect this change.
+• Quite a few chemically peculiar stars were in-
+cluded in the library that practitioners wish-
+ing to fit only “normal” stars should exclude.
+HD 319, HD 141851, HD 210111 are λ Bootis
+stars. HD 18769, HD 41357, HD 41770, HD 67230,
+HD 78209, HD 95418, HD 109510, HD 111786,
+HD 140232, HD 141795, and HD 172230 are Am
+stars.
+HD 175640 is a Bp star.
+HD 163641 is
+a Hg-Mn star. HD 103036 has anomalously-low
+Mn.
+CD−62 1346 is a carbon-enhanced metal-
+poor star.
+HD 183915 and HD 101013 are Ba
+stars and spectroscopic binaries. HD 30834 and
+HD 104340 are Ba stars.
+• HD 54361 is a carbon star and it has very little
+Mg2800 emission.
+This might indicate that C-
+stars have abnormal chromospheres. HD 158377
+is also a carbon star and BD+36 3168 is a J-type
+carbon star.
+• HD 37202, HD 58343, HD 109387, HD 138749, and
+HD 142926 are Be stars with strong Balmer emis-
+sion lines, presumably from a disk.
+HD 190073
+is a Herbig Ae star with similar strong emission.
+HD 30614 is a blue supergiant star with strong
+emission for Hα.
+• HD 358,
+HD 15089,
+HD 34797,
+HD 72968,
+HD 78316, HD 108945, HD 112413, HD 137909,
+HD 176232, HD 201601, and HD 224801 are α2
+CVn variable stars, also, broadly, Ap/Bp stars or
+HgMn stars.
+• HD 232078 is a metal-poor long-period variable
+star for which we observe little Mg2800 flux. This
+star appears in most of the large stellar libraries.
+It is a probable Mg2800 variable star, since Dupree
+et al. (2007) give a surface flux of log F = 5.17 erg
+s−1 cm−2. It has also been observed to have Hα
+emission in the wings of the line (Cohen 1976).
+We hypothesize that at some phase range of the
+
+10
+Pal et al.
+Table 3. Stellar Metadata
+Simbad
+Header
+Teff
+log g
+[Fe/H]
+B
+V
+π
+(MV )0
+dSlit
+vr
+K0
+AV
+src
+Name
+Name
+(K)
+(dex)
+(dex)
+(mag)
+mag
+(mas)
+(mag)
+(pixel)
+(km s−1)
+(ADU)
+(mag)
+HD 60319
+HD060319
+5907
+4.03
+-0.82
+9.46
+· · ·
+10.99
+· · ·
+-0.20
+-34.1
+0.2
+0.08
+1
+G 202-65
+G202-65
+6656
+4.25
+-1.37
+· · ·
+· · ·
+3.88
+· · ·
+1.00
+-245.6
+0.0
+0.00
+1
+HD 185351
+HD185351
+4921
+2.95
+0.01
+6.11
+5.17
+24.22
+2.00
+0.80
+-6.6
+5.2
+0.09
+1
+HD 72184
+HD072184
+4643
+2.84
+0.23
+7.01
+· · ·
+14.55
+· · ·
+-0.10
+16.5
+2.4
+0.11
+1
+HD 126614
+HD126614
+5453
+3.87
+0.53
+9.66
+8.79
+13.65
+4.41
+-0.20
+-32.9
+0.2
+0.05
+1
+Note—In this table, B and V are as observed (not dereddened), but (MV )0 is dereddened. The ”src” column is for V -band extinction
+AV : 1 – Koleva & Vazdekis (2012); 2 – Our derivation based on comparison with synthetic templates; or 3 – Drimmel et al. (2003).
+This is a portion of the table, presented to show format and content. The entirety is available online.
+variability cycle, perhaps during heavy mass loss,
+the normal chromosphere structure is disrupted.
+• Variable stars: HD 173819 is a classical Cepheid
+variable star.
+HD 67523 and HD 183324 are δ
+Scuti (dwarf Cepheid) variable stars. HD 344365,
+DH Peg, and SV Hya are RR Lyrae variable stars.
+HD 96446 pulsates and is a Bp star. HD 170756
+is an RV Tauri variable star.
+• Stars with some degree of binary compositeness in-
+clude HD 41357, HD 69083, HD 78362, HD 79469,
+HD 106516, HD 164402, HD 166208, HD 187879,
+HD 193496, HD 210111. Extra UV light from a
+companion can be seen in HD 26630, HD 124547,
+and HD 200905.
+• HD 149382 is a hot subdwarf (sdB) star. The ori-
+gin of these stars is not perfectly clear, but they
+are highly evolved.
+• HD 1638 and LHS 10 have noisy spectra.
+For
+purposes of repeatability, we did not pursue alter-
+native spectral extraction methods, but we note
+that stistools.x1d’s extractions for at least G 63-
+26, G 115-58, G 169-28, G 192-43, G 196-48, and
+BD +66 268 are probably incorrect.
+6. THE MG II 2800 FEATURE AND
+CHROMOSPHERIC ACTIVITY
+In this section, we explore the chromospheric activ-
+ity of the 514 NGSL stars after full reduction, including
+extinction corrections. Wilson & Vainu Bappu (1957)
+showed that the absolute visual magnitudes (MV ) of
+late-type stars correlate linearly with logarithm of H &
+K emission line width of CaII (the Wilson-Bappu effect)
+and Mg2800 h & k share this behavior (Elgarøy et al.
+1999; Cassatella et al. 2001). However, because our spec-
+tra are low resolution we could not reliably compute an
+analogous width for the twin MgII 2800 emission lines.
+We therefore measure overall strength only.
+To summarize the strength of MgII 2800 emission, we
+adopt an equivalent width style index (Mg2800):
+Mg2800 = −2.5 × log10
+�
+F i
+λ dλ
+�
+F c
+λ dλ ,
+(9)
+where F i
+λ is the observed flux within the spectral fea-
+ture band and F c
+λ is the expected flux without the spec-
+tral feature within the same band. We approximate F c
+λ
+by defining a pseudo-continuum from side bands. A line
+is drawn between the central wavelengths and average
+flux values of the two sidebands. The Mg2800 central
+feature band is defined as wavelengths between [2784˚A,
+2814˚A]. The blue side band is [2762˚A, 2782˚A] and the
+red one is [2818˚A, 2838˚A]. These definitions of feature
+and side bands are adopted from Fanelli et al. (1990).
+Figure 9. Mg2800 versus (B-V)0. Dwarfs (red) and giants
+(blue) are given different symbol types to denote metallic-
+ity groups: metal-poor (crosses), intermediate (filled circles),
+and metal-rich (filled triangles). The extremely red point is
+carbon star HD 54361.
+We keep the units (magnitudes) adopted by Fanelli
+et al. (1990). A negative index value signifies net emis-
+sion and a positive value signifies absorption.
+Fig. 9
+
+2
+0
+Dwarfs ([Fe/H]< -1.0)
+Dwarfs (-1.0<[Fe/H]< -0.25)
+Dwarfs ([Fe/H] > -0.25)
+-2 
+Giants ([Fe/H]<-1.0)
+Giants (-1.0<[Fe/H]<-0.25)
+Giants ([Fe/H] > -0.25)
+-0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+(B-V)oHST Low Resolution Stellar Library
+11
+displays Mg2800 as a function of dereddened color for
+the library stars. Hot stars have negligible Mg2800 ab-
+sorption. We also note that, although the sample con-
+tains some strongly-active Be stars, these stars show
+no anomalous Mg2800 absorption or emission. Mg2800
+absorption increases from A0 stars to sunlike stars
+[(B − V )0 = 0.65] and declines thereafter. In cool stars,
+both giants and dwarfs, chromospheric Mg2800 emis-
+sion overtakes photospheric absorption at (B − V )0 ≈ 1
+and dominates for cooler stars. Fig. 9 agrees well with
+Fig. 5c of Fanelli et al. (1990).
+For the plots herein, the distinction between giants
+and dwarfs is approximated via the color-magnitude di-
+agram (CMD) as shown in Fig. 10. Stars warmer than
+(B − V )0 = 0 or fainter than MV = 3.0 were simply
+considered dwarfs regardless of their spectral type. For
+(B−V )0 > 0, any star with MV > 6.25×(B−V )0−2.5 is
+considered a dwarf whereas MV < 6.25×(B −V )0 −2.5
+is considered a giant.
+The Fig. 10 CMD is color-coded by Mg2800 value.
+The verticality of the color bands shows again that both
+cool dwarfs and cool giants have similar Mg2800. Their
+chromospheres are similar by this measure despite vastly
+different size scales (∼ 0.1R⊙ versus ∼ 100R⊙). The
+emission gradually changes to absorption for warm stars
+and declines to near zero for hot stars. Note that some
+distant stars may have extra Mg2800 absorption due to
+warm interstellar material along the line of sight.
+Even given the intentional diversity in sample se-
+lection, outliers are relatively few.
+One is G9 giant
+HD 222093, at (B − V )0 ≈ 1 and MV ≈ 1 in Fig. 10.
+It has a high value for Mg2800 absorption, signified by
+the red color in Fig.
+10.
+The star’s spectrum shows
+emission peaks at the core of a broad absorption feature
+at 2800˚A, normal for a star whose absorption competes
+with emission at (B−V )0 ≈ 1, but this star’s emission is
+weak. HD 222093 also shows up in Fig. 9 as the sole star
+with the highest Mg2800 absorption at (B − V )0 ≈ 1.
+Fig. 11 plots Mg2800 vs. metallicity, color-coded by
+(B − V )0.
+It is clear from this figure that no strong
+correlation exists between these two quantities in any
+color regime, particularly for cools. An anticorrelation
+among cool stars might have been expected from the
+Ca II H & K results of Houdebine & Stempels (1997)
+who found that metal poor stars are activity deficient,
+but we see no such trend. Peterson & Schrijver (1997)
+reports that chromospheric characteristics do not have
+any metallicity dependence.
+A subtle declining trend among medium-temperature
+stars in Fig. 11 deserves a note and an additional figure,
+namely Fig. 12, which restricts the color range to be
+near solar (0.5<(B-V)0 <0.8). Because these are posi-
+Figure 10. CMD for all 514 NGSL stars. The color bar
+shows Mg2800 strength. For dwarfs, Mg II emission fills in
+the absorption redder than B − V = 0.9, whereas emission
+begins to dominate for giants at B − V = 1.2.
+Figure 11. Mg2800 as a function of [Fe/H]. The color bar
+codes (B-V)0 and the symbol type distinguishes dwarfs (tri-
+angles) and giants (circles).
+tive values of Mg2800, indicating absorption, one might
+expect a monotonic increase of Mg2800 with [Fe/H].
+Mg2800 absorption does increase for metal poor stars
+(−2 < [Fe/H] < −1) but then the index value saturates
+and falls for metal rich objects. With the help of syn-
+thetic spectra, two sequences of which are also plotted
+in Fig. 12, the reason appears to be a simple curve of
+growth argument. Mg2800 is a resonance feature that
+scales approximately as the abundance of the Mg II ion.
+It reaches full depth at [Fe/H] ∼ −1, but the flanking
+(in wavelength) absorption features from a plethora of
+atomic species are still weak. From [Fe/H] ∼ −1 and
+higher, these weak features will grow faster than the
+central Mg II absorption pair. As the pseudocontinuum
+drops, the Mg2800 index drops.
+Parenthetically, the
+relatively poor agreement of synthetic spectra and ob-
+
+-7.5
+-5.0
+-2.5
+0
+Mv (mag)
+0.0
+2.5
+-1
+5.0
+7.5
+-2
+10.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+(B-V)o2
+Dwarfs
+Giants
+1.5
+Mg2800 (mag)
+1.0
+0.5
+-2
+0.0
+-2.0
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+[Fe/H]12
+Pal et al.
+Figure 12. Mg2800 as a function of [Fe/H] for a narrowed
+color range of 0.5<(B-V)0 <0.8. Dwarfs (triangles) and gi-
+ants (circles) are color coded by (B-V)0. Black lines indicate
+Mg2800 from synthetic LTE spectra for dwarfs (Teff=5770K,
+log g=4.5, solid) and giants (Teff=5770K, log g=1.5, dashed).
+served spectra in Fig. 12 should be no surprise. The UV
+spectrum is crowded, its lines have not received as much
+attention as optical ones, and for warm and cool stars
+the wavelength regime is on the blue side of the black-
+body curve, exposing defects in the upper layers of the
+model atmosphere due to the absence of backwarming.
+Hα emission is a separate indicator of stellar chromo-
+spheric activity (Montes et al. 1995; Cincunegui et al.
+2007; Gomes da Silva et al. 2014) and also magnetic flare
+activity. An index for the Hα feature is calculated using
+the passband definitions of Cohen et al. (1998) but here
+we convert it to magnitude units (Eqn. 9). The spectral
+feature band is [6548˚A, 6578˚A] and the blue pseudocon-
+tinuum is [6420˚A, 6455˚A] and the red pseudocontinuum
+is [6600˚A, 6640˚A].
+Mg2800 and Hα are plotted against each other in
+Fig. 13. The strongest Hα emitters are Be stars, gen-
+erally assumed to be young stars with disks (Gray &
+Corbally 2009).
+We might also expect to catch some
+flaring M dwarfs, but apparently none of the M dwarfs
+were observed during outbursts, as we see no cool dwarfs
+scattering to negative Hα values. The “triangle” in the
+positive-positive quadrant arises because peak Hα ab-
+sorption occurs among hotter stars than peak Mg2800
+absorption.
+Among cool stars with negative Mg2800,
+the mild correlation is due to expected Hα index ab-
+sorption behavior from species unrelated to Hα itself,
+such as TiO (e.g. Valdes et al. 2004). That is, it is a
+consequence of the strong Mg2800-temperature anticor-
+relation in cool stars, and does not imply Hα emission
+at all.
+Two stars lie at anomalously-negative Hα values.
+They are: HD 126327 (giant) and GL 109 (dwarf). Pre-
+Figure 13. Mg2800 is plotted against Hα for dwarfs (tri-
+angles) and giants (circles). The points are color coded by
+(B-V)0. Three stars to the extreme left of the figure are all
+Be stars: HD 37202, HD 109387, and HD 190073.
+sumably, HST serendipitously observed these objects
+during flare events.
+The correlation between Ca II H & K core emission
+strength (a third stellar activity indicator) and Hα emis-
+sion is also well studied. Some authors report a posi-
+tive correlation between the two (Pasquini & Pallavicini
+1991; Montes et al. 1995), some a lack of correlation,
+and some a negative correlation (Cincunegui et al. 2007;
+Gomes da Silva et al. 2011). Our Mg2800 results shed
+little insight into this uncertain area.
+7. DISCUSSION AND CONCLUSION
+This paper presents a new reduction of the Next
+Generation (HST/STIS low resolution) Spectral Library
+that includes updated flux calibration work, updated
+scattered light corrections, and an increase in sample
+size (345 to 514) due to inclusion of stars from run
+GO13776. This increases the parameter space coverage
+in log g, Teff and [Fe/H] (Figs. 1 and 2).
+After correction for interstellar extinction, the spectra
+were used to explore the chromospheric activity of stars
+using the Mg II 2800 h + k feature and Hα as likely
+indicators.
+Against color, there is a gradual change of sign of
+Mg2800 from positive to negative (signifying absorption
+to emission transition) for both dwarfs and giants within
+0.5<(B-V)0 <1.5.
+From Fig. 9 it is evident that the
+transition happens at (B-V)0=1.0 or spectral class K3
+for dwarfs, and (B-V)0=1.12 or spectral class K4-K5 for
+giants. The color calibration of Worthey & Lee (2011)
+indicates that we expect dwarfs to have B − V bluer
+than giants by about 0.1 mag, so this crossover hap-
+pens at about the same Teff for both dwarfs and giants.
+Largely, this result is consistent with results from Gurza-
+
+1.6
+1.4
+0.75
+1.2
+0.70
+1.0
+0.8
+0.65
+0.6
+0.60
+0.4
+Dwarfs (Synthetic LTE)
+Giants (Synthetic LTE)
+0.55
+0.2
+Dwarfs
+Giants
+0.0
+-2.0
+-1.5
+-1.0
+-0.5
+0.0
+[Fe/H]2
+Dwarfs
+Giants
+1.5
+Mg2800 (mag)
+Be Stars
+1.0
+0
+0.5
+-2 
+0.0
+-3
+-0.4
+-0.2
+0.0
+0.2
+0.4
+Hα (mag)HST Low Resolution Stellar Library
+13
+dian (1975) where it was shown that Mg II 2800 feature
+starts dominating in emission in K2 and later-type stars.
+The photospheric absorption gives way to strong chro-
+mospheric emission as the temperature drops. Temper-
+ature is the emphatic controlling parameter of Mg2800
+emission; the cooler the star, the stronger the emission.
+[Fe/H] and log g have little influence on Mg2800, and
+we see no evidence of flare behavior.
+We chart basic Hα and Hβ behavior in Figs. 14 and
+15. The peaks are the deep absorptions in A stars, and
+strongly negative values indicate that emission has over-
+shadowed absorption. Fig. 14 shows four stars with mild
+flares in progress: GJ 551, GJ 876, and GL 109 are
+dwarfs while HD 126327 is a giant.
+GJ 551 is Prox-
+ima Centauri and it shows up as a flaring dwarf in a
+20 seconds cadence Transiting Exoplanet Survey Satel-
+lite (TESS) monitoring campaign (Howard & MacGre-
+gor 2022). Evidence for flares in GJ 674 is reported in
+Froning et al. (2019). GL 109 is listed as an eruptive
+variable in SIMBAD and categorized as UC Cet-type
+flare star (Gershberg et al. 1999).
+HD 126327 is the
+only cool giant that seems to be flaring. Prominent TiO
+band absorption affects the coolest stars. Cool giants
+saturate at B − V ≈ 1.65 (Worthey & Lee 2011) but es-
+pecially Hβ continues to increase, not because of actual
+Hβ absorption, but because of the increasing influence
+of TiO features. The giant at the extreme right is a car-
+bon star. The hot dwarfs with Hα magnitudes less than
+-0.1 are Be stars.
+The Mg II 2800 line emission in UV is a major probe
+for chromospheric radiative loss(Linsky & Ayres 1978).
+From Fig. 9 it is evident that there is scatter in Mg
+II 2800 line strength for a given temperature, but the
+character of that scatter might be astrophysical. Var-
+ious studies have suggested the existence of a ‘basal’
+flux level for Mg II 2800 that might indicate the level
+of an ongoing, persistent mechanism (acoustic waves are
+often cited) that can be supplemented by a more vari-
+able heating mechanism (such as magnetohydrodynamic
+shocks) that adds Mg emission to some stars but not
+others (Schrijver 1987; Strassmeier et al. 1994; Mart´ınez
+et al. 2011).
+Recast in terms of the Mg II λ2800 flux emerging from
+the star’s surface (Fλ), the above authors find a ‘basal
+level’ that increases with temperature. In order to con-
+firm this, we select NGSL stars with Teff < 5000K and
+recast their emission line strengths as emergent fluxes
+as in Mart´ınez et al. (2011). The scheme follows Oranje
+et al. (1982), but extended to account for interstellar
+extinction. Oranje et al. noted that
+Fλ
+fλ
+= Fbol
+fbol
+,
+(10)
+Figure 14. Hα as a function of (B−V )0 for dwarfs (red) and
+giants (blue) are shown, segregated by metal-poor (crosses),
+intermediate (filled circles), and metal-rich (filled triangles)
+status.
+Be stars scatter to negative values for hot stars
+with (B − V )0 < 0. Any star caught during a flare event
+should also scatter toward negative index values. Four stars
+(3 dwarfs and 1 giant) with Hα < −0.15 and (B − V )0 > 1.5
+are thought to be flaring: GJ 551, GJ 876, and GL 109 are
+dwarfs while HD 126327 is a giant. Noise prevents reliable
+measurement of Mg2800 in GJ 551 (Proxima Centauri) and
+GJ 876. Therefore, these stars do not appear in figures that
+illustrate Mg2800.
+Figure 15. Hβ as a function of (B−V )0 for dwarfs (red) and
+giants (blue), segregated by metal-poor (crosses), intermedi-
+ate (filled circles), and metal-rich (filled triangles) status. Hβ
+is less sensitive to emission than Hα.
+where Fλ is the star’s outbound surface flux (erg cm−2
+s−1) at some wavelength.
+For us, this wavelength is
+2800˚A, and it is chromospheric in origin.
+The lower
+case fλ is then the flux received at earth.
+The right
+hand side are the bolometric versions. This equation is
+only good in the limit of zero extinction. Extinction at
+wavelength λ (Aλ) is defined by:
+Aλ = −2.5log fλ
+f0,λ
+,
+(11)
+
+0.4
+0.2
+(mag)
+0.0
+)H
+-0.2
+Dwarfs ([Fe/H)<-1.0)
+Dwarfs (-1.0<[Fe/H] <-0.25)
+Dwarfs ([Fe/H]> -0.25)
+-0.4
+Giants ([Fe/H] < -1.0)
+Giants (-1.0<[Fe/H] <-0.25)
+Giants ([Fe/H] > -0.25)
+-0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+(B- V)oDwarfs({Fe/Hi<-1.0)
+0.4
+Dwarfs (-1.0<[Fe/Hl< -0.25)
+Dwarfs ([Fe/H] >-0.25)
+Giants ([Fe/H] < -1.0)
+0.3
+Giants (-1.0<[Fe/Hl<-0.25)
+Hβ (mag)
+Giants ([Fe/H] > -0.25)
+0.2
+0.1
+0.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+(B- V)o14
+Pal et al.
+Figure 16. Inferred surface flux from Mg II 2800 (log10,
+cgs units) as a function of Teff for both giants and dwarfs
+with Teff < 5000 K. The green line is the “basal flux” from
+Mart´ınez et al. (2011).
+Three stars with log (Flux) <3.0
+(HD 54361, HD 126327, and HD 232078) are below the
+plot limits.
+The downward black arrows show log10 (Teff)
+for them. For comparison, the blackbody emergent flux in-
+tegrated over the Mg2800 central passband (black line) is
+shown.
+where f0,λ is the extinction corrected version of fλ. This
+equation can be inverted to
+fλ = f0,λ10−0.4Aλ = f0,λ g(Aλ) ,
+(12)
+where the function g(Aλ) is shorthand we introduce. By
+convention, Aλ is positive and thus, fλ is always less
+than f0,λ and 0 < g(Aλ) ≤ 1. Besides g(Aλ), we also
+invent h(Abol) to represent the extinction in bolometric
+quantities, which is more complicated to produce (it re-
+quires the integration of the dust-attenuated flux over
+all wavelengths and thus depends on the spectral type
+of the target star). Putting everything together:
+Fλ = f0,λ
+Fbol
+f0,bol
+g(Aλ)
+h(Abol)
+(13)
+This can be rephrased in terms of Teff by noting that:
+Fbol = σT 4
+eff ,
+(14)
+where σ is the Stephan-Boltzmann constant and
+f0,bol = B10−0.4(V +BCV ) ,
+(15)
+where B is a zeropoint adjustment between physical
+units and the astronomical magnitude scale, V is the ap-
+parent magnitude in V-band, and BCV is the bolomet-
+ric correction for V-band. The B value is obtained by
+noting that, fbol,⊙=1361 Wm−2, V⊙=-26.76 (Willmer
+2018), and BCV,⊙=0.09 (VandenBerg & Clem 2003).
+Known Teff, [Fe/H], and log g values for each star were
+used to interpolate a low resolution synthetic flux from
+Figure 17. Spectra of 5 stars are shown in the λ2800 region.
+TOP: Fluxed spectra are normalized at 2820˚A. BOTTOM:
+Fluxed spectra are normalized such that the continuum-
+subtracted emission scales as the surface-emergent emission
+Fλ derived for Fig. 16. “Normal” HD 136726 and HD 131918
+lie near the green line in Fig. 16 and the remaining three stars
+are low outliers. HD 232078 and carbon star HD 54361 lie
+outside the plot limits in Fig. 16 and HD 126327 was caught
+during a flare event (Fig. 13).
+Worthey (1994). We applied a Fitzpatrick (1999) cubic
+spline extinction curve to this synthetic flux, then in-
+tegrated (with and without extinction) to find h(Abol).
+For the bolometric correction, we used the Worthey &
+Lee (2011) calibration, which also requires T, log g, and
+[Fe/H]. We used these values and our Eqn. 8 to get
+A2800. The quantity f0,bol was calculated by integrat-
+ing the flux over index band for Mg II 2800. A linear
+pseudocontinuum calculated from the Mg II 2800 pass-
+bands was subtracted before the integration.
+Fig. 16 shows the dependence of Fλ as a function of
+Teff in a log-log scale.
+Thus transformed to surface-
+emergent flux, cool dwarfs are seen to emit an order of
+magnitude more Mg2800 flux per unit surface area, with
+two notable low-lying objects.
+As for giants, a num-
+ber of cool giants have lower flux values than the basal
+line given by Mart´ınez et al. (2011) (solid green line in
+Fig. 16). One giant (HD 222093) lies two orders of mag-
+nitudes brighter than typical, and three stars lie offscale
+on the low end. No ready explanation for the difference
+in the morphology of our figure versus Mart´ınez et al.’s
+leaps to mind. Our spectra have lower spectral resolu-
+tion compared to IUE, but continuum subtraction is too
+minor to contribute significant error, our fluxes should
+be reliable, and our treatment of interstellar extinction
+is probably a step better.
+Another giant, HD 126327, lies more than an order
+of magnitude lower than the line but also was caught
+flaring in Hα (Fig. 13). This might indicate that stormy
+
+Giants
+Dwarfs
+Martinez fit
+Blackbody continuum
+6
+log(Flux)
+5
+4
+HD232078
+HD054361&HD126327
+3
+3.70
+3.65
+3.60
+3.55
+3.50
+3.45
+log(Teff)20
+HD232078
+15
+HD054361
+HD126327
+10
+HD136726
+Normalised Flux
+HD131918
+5
+0
+1.5
+1.0
+0.5
+0.0
+2760
+2780
+2800
+2820
+2840
+Wavelength (A)HST Low Resolution Stellar Library
+15
+Figure 18. Mg II 2800 feature in HD 102212 as observed
+by IUE (blue), in the NGSL (red), and by Worthey et al.
+(2022a) (green). The IUE spectrum is at lower resolution
+compared to Worthey et al. (2022a) and NGSL.
+events in the photosphere and lower chromosphere might
+temporarily disrupt the middle chromosphere where the
+Mg2800 arises.
+Fig. 17 elucidates the fact that stars
+lying close to the green line in Fig. 16 in fact have higher
+Mg II λ2800 flux compared to stars lying way below the
+same green line in Fig. 16.
+Fig. 18 shows variation in the MgII 2800 spectral lines
+using observations from International Ultraviolet Ex-
+plorer (IUE), NGSL, and Worthey et al. (2022a) for the
+single star HD 102212. The observations were made in
+1997, 2002, and 2021 for IUE, NGSL, and Worthey et al.
+(2022a) respectively. Mg2800 values for the three cases
+are -1.49±0.05, -1.81±0.003, and -2.26±0.008 for IUE,
+NGSL, and Worthey et al. (2022a) respectively.
+The
+errors in Mg2800 values are calculated by taking into
+consideration the errors in flux at each pixel value and
+then propagating these errors while calculating Mg2800
+values. Even admitting a few percent additional fluxing
+error, it is statistically certain that Mg2800 values show
+a temporal variation in HD 102212.
+Add this to HD 232078, a similar long-period variable
+listed in §5.2 that is probably also variable in Mg2800.
+The sun is known to have a ∼7% Mg2800 variation
+that correlates with the magnetic activity cycle (Deland
+& Cebula 1993). Buccino & Mauas (2008) report cyclic
+chromospheric activity in HD 22049 and HD 128621 us-
+ing IUE spectral data.
+At visible wavelengths, some
+studies show overall variation in chromospheric activity
+from CaII H & K lines. Baliunas et al. (1998) report that
+85% of stars in the 40-year HK Project at Mount Wil-
+son Observatory showed either periodic (60%) or ape-
+riodic (25%) variation in chromospheric activity. Tem-
+poral variation possibly separates magnetically-driven
+chromospheric heating, which can be expected to be
+cyclic, from acoustic wave-driven heating, which might
+be expected to be steadier. In this regard, HD 102212 is
+not an apt test case because it is a long-period variable
+star likely to experience considerable “weather” in its
+gaseous envelope.
+8. ACKNOWLEDGEMENTS
+We acknowledge with thanks the variable star obser-
+vations from the AAVSO International Database con-
+tributed by observers worldwide and used in this re-
+search. This work is based on observations made with
+the NASA/ESA Hubble Space Telescope, program GO
+16188, https://dx.doi.org/10.17909/t9-d42d-z465. Sup-
+port for this work was provided by NASA through grant
+number HST-GO-16188.001-A from the Space Telescope
+Science Institute. STScI is operated by the Association
+of Universities for Research in Astronomy, Inc. under
+NASA contract NAS 5-26555. This research has made
+use of the SIMBAD database, operated at CDS, Stras-
+bourg, France.
+
+1e-12
+2.5
+IUE
+NGSL
+2.0
+Worthey et al. (2022a)
+cm
+1.5
+1.0
+0.5
+0.0
+2760
+2780
+2800
+2820
+2840
+Wavelength (A)16
+Pal et al.
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+Threading light through dynamic complex media
+Chaitanya K. Mididoddi,1, ∗ Christina Sharp,1 Philipp del Hougne,2 Simon A. R. Horsley,1 and David B. Phillips1, †
+1Physics and Astronomy, University of Exeter, Exeter, EX4 4QL. UK.
+2Univ. Rennes, CNRS, IETR – UMR 6164, F-35000 Rennes, France.
+The scattering of light impacts sensing and communication technologies throughout the electromagnetic spec-
+trum. Overcoming the effects of time-varying scattering media is particularly challenging. In this article we
+introduce a new way to control the propagation of light through dynamic complex media. Our strategy is based
+on the observation that many dynamic scattering systems exhibit a range of decorrelation times – meaning that
+over a given timescale, some parts of the medium may essentially remain static. We experimentally demonstrate
+a suite of new techniques to identify and guide light through these networks of static channels – threading op-
+tical fields around multiple dynamic pockets hidden at unknown locations inside opaque media. We first show
+how a single stable light field propagating through a partially dynamic medium can be found by optimising the
+wavefront of the incident field. Next, we demonstrate how this procedure can be accelerated by 2 orders of
+magnitude using a physically realised form of adjoint gradient descent optimisation. Finally, we describe how
+the search for stable light modes can be posed as an eigenvalue problem: we introduce a new matrix operator,
+the time-averaged transmission matrix, and show how it reveals a basis of fluctuation-eigenchannels that can
+be used for stable beam shaping through time-varying media. These methods rely only on external camera
+measurements recording scattered light, require no prior knowledge about the medium, and are independent
+of the rate at which dynamic regions move. Our work has potential future applications to a wide variety of
+technologies reliant on general wave phenomena subject to dynamic conditions, from optics to acoustics.
+Introduction
+Optical scattering randomly redirects the flow of light. It is a
+ubiquitous phenomenon that has wide-ranging effects. Since
+imaging relies on light travelling in straight lines from a scene
+to a camera, scattering prevents image formation through fog,
+and precludes high-resolution microscopy inside biological
+tissue [1, 2]. Scattering also impairs optical communications
+through air and water, and disrupts the transmission of mi-
+crowave and radio signals [3]. Overcoming the adverse effects
+of light scattering is an extremely challenging problem [4].
+Nonetheless, due to its prominence, substantial progress has
+been made over the last decades [5].
+When light propagates through a strongly scattering
+medium (also known as a ‘complex’ medium [1]), the wave-
+front of the incident optical field is distorted, corrupting the
+spatial information it carries. Elastic scattering from a static
+medium is deterministic, meaning that the precise way in
+which light has been perturbed can be characterised and sub-
+sequently corrected. By sending a series of probe measure-
+ments through the medium, a digital model of its effect on
+light can be created: represented by a linear matrix operator
+known as a transmission matrix (TM) [6]. Once measured, the
+linearity of Maxwell’s equations means that the TM describes
+how any linear combination of the probe fields will be trans-
+formed. The TM reveals how to best undo the distortion im-
+parted to a scattered field emerging from a complex medium,
+and the time-reverse: how to pre-distort an input optical field
+so that it evolves into a user-defined state at the output – a
+technique known as wavefront shaping [7].
+Using modern high-resolution spatial light modulators
+(SLMs), it is possible to precisely measure and control the
+∗ [email protected]
+† [email protected]
+relative intensity, phase and polarization of thousands of inde-
+pendent optical spatial modes as they undergo many scattering
+events inside a highly turbid medium [8]. Thus, wavefront
+shaping, and the closely related technique of optical phase
+conjugation [9], have been used to image up to a depth of sev-
+eral hundred microns into fixed biological tissue [10]. TM-
+based approaches have also inspired new forms of ultra-thin
+micro-endoscopy through rigidly-held strands multimode op-
+tical fibre (MMF) [11].
+Despite these successes, control of light through time-
+varying complex media remains a largely open problem [2].
+Evidently, wavefront shaping can only be successfully applied
+if the medium in question remains predominantly stationary
+for the time taken to make probe measurements and apply a
+wavefront correction. Yet many application scenarios feature
+complex media that rapidly fluctuate on a timescale of mil-
+liseconds or faster – rendering wavefront shaping approaches
+exceedingly difficult [12]. Overcoming this challenge offers
+a stepping stone to a potent array of new technologies, in-
+cluding the ability to look directly inside living biological tis-
+sue, to see through fog, and to increase the data-rate of optical
+communications through the turbulent atmosphere and flexi-
+ble fibre-optics.
+So far, the main strategies to control light through mov-
+ing complex media have focused on achieving the task of
+wavefront shaping as quickly as possible [13–17]. In the op-
+tical regime, beam shaping at kiloHertz rates can be imple-
+mented with digital micro-mirror devices (DMDs) [18–20].
+The need for yet higher switching rates has spawned the de-
+velopment of ultra-fast SLMs capable of wavefront shaping
+at hundreds of kiloHertz [21, 22] while megaHertz to giga-
+Hertz modulation-rate SLMs hold future promise [23, 24].
+Spectral multiplexing enables many probe measurements to
+be made simultaneously, speeding up the data gathering part
+of the wavefront shaping process [22, 25]. In addition, the
+arXiv:2301.04461v1  [physics.optics]  11 Jan 2023
+
+2
+number of probe measurements needed to reconstruct a us-
+able TM can be reduced by exploiting prior knowledge about
+the medium itself – such as correlations between elements of
+the TM (known as memory effects), predictions about how
+the power is distributed over the TM elements, or a recent
+but slightly degraded TM measurement [26–33]. Fast optical
+focusing inside biological tissue can be achieved with opti-
+cal phase conjugation guided by ultrasonic guide-stars – re-
+lying on the lower levels of scattering experienced by ultra-
+sound [34–37]. A variety of other methods relying on correla-
+tions between the object of interest and externally measurable
+signals offer alternative routes to image through moving com-
+plex media [38, 39].
+Here we introduce a new way to control the propagation
+of light through dynamic scattering media. Our approach is
+complementary to existing techniques. We begin by classify-
+ing complex media into three categories, based on the level
+and type of motion exhibited over the timescale required for
+wavefront shaping, denoted by τws. Class 1 represents static
+complex media that remain completely fixed over time τws.
+Established TM-based methods can be applied to determin-
+istically control scattered light in this case. Class 2 repre-
+sents moving complex media, which undergo substantial mo-
+tion everywhere over time τws. This class of media eludes
+current wavefront shaping approaches. However, there is an
+opportunity to make progress by considering a third class –
+representing an edge-case between classes 1 and 2.
+Class
+3 comprises partially moving scattering media, which, over
+the timescale τws, exhibit localised pockets with time-varying
+properties embedded within a static medium. Any dynamic
+complex medium possessing a range of decorrelation rates has
+the potential to be classified in this way. For example, this sit-
+uation describes: tissue in which small capillaries conducting
+blood flow represent faster moving regions surrounded by a
+matrix of more slowly changing scattering material; pockets
+of turbulent air above hot chimneys within calmer air over a
+city skyline; and the movement of people modifying the scat-
+tering of microwaves only at floor level throughout a building.
+In this article we focus on how to identify light fields that
+predominantly stay within the static regions of such partially
+moving complex media (i.e. class 3 complex media).
+We
+experimentally demonstrate three new techniques that excite
+largely stable modes within these environments. We show
+how these optimised modes scatter almost entirely around all
+moving pockets. These methods do not rely on prior knowl-
+edge of the location of dynamic regions and only require
+measurements external to the medium. These measurements
+can be made on the same timescale or more slowly than the
+medium is fluctuating – crucial for the practical application
+of these techniques. Our work expands the toolkit of methods
+to overcome dynamic scattering, pointing to a range of future
+applications in the fields of imaging, optical communications,
+and beyond.
+Results
+When a light field u is incident on a time-varying medium,
+the time-dependent transmitted field is given by
+v(t) = T(t)u,
+(1)
+where T(t) is the time-dependent transmission matrix of the
+medium, and here u and v are represented as column vectors.
+Our aim is to find an input u that scatters around dynamic
+regions within the medium, thus minimising the fluctuations
+in the output field v(t).
+To experimentally investigate this new form of light con-
+trol, we emulate a three-dimensional time-varying scattering
+medium using a cascade of three computer controlled diffrac-
+tive optical elements, each separated by a free-space distance
+of δz.
+Cascades of phase planes can emulate atmospheric
+turbulence [40, 41] and have also been shown to mimic the
+complicated optical scrambling effects of a multiple scattering
+sample [42, 43]. In practice this set-up is implemented using
+multiple reflections from a single liquid crystal SLM, allow-
+ing the phase profiles to be arbitrarily digitally reconfigured.
+We choose this test-bed as it is a versatile way to control the
+degree of scattering, and the number and location of dynamic
+regions for proof-of-principle experiments.
+As shown in Fig. 1(e), top row, we display a static random
+phase pattern on each phase screen, spatially distorting
+optical signals flowing through the optical system. On each
+plane we also define an area within which the phase profile
+is programmed to randomly fluctuate in time – these patches
+represent the ‘pockets’ of dynamic material embedded inside
+the scattering sample.
+A second SLM is used to shape
+the light incident onto the dynamic medium, and a camera
+records the level of intensity fluctuations in transmitted light.
+Unguided optimisation: We first explore a straight-forward
+optimisation method: iterative modification of input field u
+to suppress intensity fluctuations at the output. Figure 1(a)
+shows a schematic of this approach. Supplementary Informa-
+tion (SI) §1 shows a full diagram of the optical set-up. The op-
+timisation commences by transmitting an initial trial field u0
+through the sample, and recording the intensity fluctuations
+on the camera. We sample 20 realisations of the fluctuating
+speckle pattern, and the level of fluctuations over these frames
+is quantified by the objective function F = ¯σI/¯I, where ¯σI
+denotes the standard deviation of the fluctuating intensity, av-
+eraged over all illuminated camera pixels, and ¯I is the aver-
+age transmitted intensity. This choice of objective function
+ensures that fluctuations are normalised with respect to trans-
+mitted power.
+The input SLM used to generate the incident fields is sub-
+divided into 1200 super-pixels. The phase delays imparted
+by these super-pixels represents the independent degrees-of-
+freedom we aim to optimise. We begin by setting each super-
+pixel to a random phase value, creating incident field u0, and
+measure the level of output fluctuations. Next, two new test
+fields are sequentially transmitted through the sample. These
+are generated by randomly selecting half of the input SLM
+super-pixels used to create u0, and adding/subtracting a small
+constant phase offset δθ from these pixels, yielding inputs
+
+3
+Figure 1. Unguided optimisation. (a) Schematic of experimental set-up. An input wavefront is iteratively modified to reduce the intensity
+fluctuations in transmitted light. (b) A plot of fluctuation level as a function of iteration number throughout the optimisation procedure.
+Convergence is reached after several thousand iterations: the fluctuation level does not fall to zero, but plateaus when the residual fluctuations
+fall below the experimental noise floor, indicated (approximately) in pink. (c) Fluctuations in the output field for a randomly chosen input field
+used as the starting point of the optimisation. Upper heat maps show the mean intensity of transmitted light at the output plane, and lower
+heat maps show the fluctuation level around the mean, represented as a standard deviation around the mean. The line-plots show line-profiles
+through the output field along the lines marked with white hatched lines, with mean intensity (red line) and fluctuations about the mean (gray
+shading). (d) Equivalent plot to (c) but now showing the optimised transmitted field. We see the fluctuations have been strongly suppressed in
+(d) compared to (c). (e) Measured shape of the optimised field inside the dynamic scattering sample. The top row shows the 3 phase planes
+that form the scattering system, with a fluctuating region on each plane highlighted by a red box. The middle and bottom rows show the optical
+field (middle row) and intensity pattern (absolute square of the field – bottom row) incident on each plane. We see that the optimised field
+arriving at each plane has a low intensity region corresponding to the location of the fluctuating region – highlighted by white arrows – thus
+‘avoids’ these regions.
+u±δθ. We measure the corresponding level of output fluctua-
+tions for these two new trial inputs, and if either exhibit lower
+fluctuations than u0, the optimised input field is updated ac-
+cordingly. This process is repeated until the output fluctuation
+level no longer improves.
+This algorithm relies on accurately capturing the output
+fluctuations on each iteration. However, even in the absence of
+any other sources of noise, there is an uncertainty in the mea-
+surement of ¯σI and ¯I due to the finite number of realisations
+of the dynamic medium sampled. To enhance the algorithm’s
+robustness to this source of noise, on each new iteration we
+re-test the optimum input field from the last iteration and com-
+pare this to the new trial fields – doing so increases the optimi-
+sation time, but crucially prevents a single measurement with
+an erroneously low value of F from blocking the optimiser
+from taking steps in subsequent iterations. Figure 1(b) shows
+a typical optimisation curve throughout our experiment. The
+noise floor is governed by the uncertainty in real fluctuations
+detailed above, along with small variations in the intensity of
+the laser source, camera noise and uncontrolled fluctuations
+in light reflecting from the liquid crystal SLM as it is updated,
+which all add to the apparent level of measured fluctuations.
+Figures 1(c) and 1(d) show examples of the output fluctua-
+tions of an initial trial field (c) and an optimised field (d) using
+this approach. See also Supplementary Movie 1. We see that
+fluctuations of the output field are heavily suppressed after
+optimisation. As we have full control over the test scattering
+medium, we are able to digitally ‘peel back’ the outer scat-
+tering layers to look inside and directly observe the evolution
+of the optimised field as it propagates through the cascade of
+phase planes. Experimentally this is achieved by switching-
+off the aberrating effect of the second and third phase planes,
+and imaging the optimised field that is incident on plane 2.
+We recover the phase of this optical field using digital holog-
+raphy, and reconstruct the fields at planes 1 and 3 by numeri-
+cally propagating the field at plane 2 (see SI §2). We see the
+optimised field scatters through the medium to form a speckle
+pattern that evolves to exhibit near-zero intensity at the loca-
+
+Fluctuating regions
+2元
+(a)
+(b)
+(e)
+Phase masks
+0.25
+Lens
+Optimisation
+2
+3
+ Fluctuation level
+Camera
+ Phase masks
+Phase (rad.)
+Shaped
+input
+wavefront
+Noise floor
+Intensity
+fluctuations
+Dynamic scatterer
+0
+0
+0
+3000
+Iteration number
+Feedback
+2元
+(c)
+Optical field
+Initial transmitted field
+(p)
+Optimised transmitted field
+(pet)
+Phase (
+Intensity
+Intensity
+0.5
+0.5
+Amp.
+0
+0
+Mean
+Intensity (arb.)
+Intensity
+intensity
+Std. intensity
+8z
+fluctuations
+Sz
+0
+0
+0.12
+Speckle evolution
+Std. intensity fluctuations4
+Figure 2. Physical adjoint optimisation. (a) Schematic of experimental set-up. On iteration i an input field u(i) is transmitted through the
+dynamic medium from the left-hand-side (LHS). The output field is time-averaged on the right-hand-side (RHS) – the schematic shows output
+fields recorded at individual times v(t1), v(t2) · · · v(tN) (where N is the total number of recorded output fields). These are averaged to
+yield ⟨v⟩t. Digital optical phase conjugation (DOPC) is carried out to transmit the phase conjugate of ⟨v⟩t back through the medium. The
+resulting field emerging on the LHS is then time-averaged, and used to calculate δu, such that the input of the next iteration (i + 1) is given by
+u(i+1) = u(i) + δu. (b) A plot of fluctuation level as a function of iteration number throughout the optimisation procedure. In this scheme,
+convergence is reached after ∼ 15 iterations. (c) The experimentally recorded intensity of the optimised field arriving at the three phase planes.
+The maximum intensity at each plane is normalised to 1. The white squares indicated the location of the moving region on each plane. We see
+that, once again, the optimised field avoids these moving regions of the sample.
+tions of the fluctuating regions on each plane (Fig. 1(e), bot-
+tom row) – thus avoiding these dynamically changing regions
+and minimising fluctuations in the transmitted field.
+This is an encouraging result, however this form of
+undirected optimisation is a relatively slow process – in this
+case requiring several thousand iterations to converge (see
+Fig. 1(b)). Therefore, we next ask: is there a way to find
+optimised fields more rapidly?
+Physical adjoint optimisation: In our first strategy, on each
+iteration we directly measure how one randomly chosen spa-
+tial component of the input field should be adjusted to re-
+duce the fluctuations in the output field. We now describe
+a more sophisticated technique to simultaneously obtain how
+all spatial components composing the input field should be
+adjusted in parallel. This strategy converges to an optimised
+input beam in far fewer iterations than unguided optimisation.
+Our approach can be understood as gradient descent optimi-
+sation using fast adjoint methods. Adjoint optimisation refers
+to the efficient computation of the gradient of a function for
+use in numerical optimisation. Here, we lack sufficient in-
+formation to numerically perform this adjoint operation, but
+instead we show how it is possible to physically realise it by
+passing light in both directions through the dynamic scattering
+medium.
+SI §3 gives a detailed derivation of this method. In sum-
+mary, to suppress output fluctuations we aim to maximise the
+correlation (i.e. overlap integral) between all output fields over
+time, given by the real positive scalar objective function
+F =
+�����
+T
+�
+t=1
+T
+�
+t′=1
+�
+v†(t) · v(t′)
+�
+�����
+2
+.
+(2)
+To increase F, at each iteration we incrementally adjust the
+complex field of all elements of the input field u, so that the
+input field at iteration i + 1 is given by u(i+1) = u(i) + δu,
+where u(i) is the input field of iteration i, and column vector
+δu = δAeiθ. Here δA is the optimisation step size: a small
+real positive constant, and we find (see SI §3) that column
+vector θ is given by
+θ = − arg
+�
+TT · ⟨v∗⟩t
+�
+,
+(3)
+where ⟨v∗⟩t is the phase conjugate of the time-averaged out-
+put field.
+Our adjoint optimisation scheme is shown schematically in
+Fig. 2(a). Iteration i commences by illuminating the dynamic
+scattering medium from the left-hand-side (LHS) with trial
+field u(i), and time-averaging the transmitted optical field on
+the right-hand-side (RHS), yielding ⟨v⟩t. Equation 3 specifies
+that ⟨v⟩t should be phase conjugated, and transmitted in the
+reverse direction through the dynamic media, from the RHS
+back to the LHS. Measuring the phase of the resulting field on
+
+(a) Physical adjoint optimisiation scheme
+(b)
+0.4
+2元
+Coherent reference
+Fluctuation
+Phase (rad.)
+v(ti)
+level
+u(i+1)
+Camera
+Noise floor
+Shaped
+input
+0
+v(t2)
+Amp.
+0
+field
+u(i)
+0
+30
+Su
+Iteration number
+Time-average
+(c)
+optical field
+LHS
+Dynamic scatterer
+RHS
+Evolution
+Time-average
+optical field
+of optimised field
+.
+v(tn)
+Return
+field
+Sz
+DOPC
+Camera
+Sz
+<v)t
+<v)*
+Intensity (arb.)
+0
+Coherent reference5
+the LHS yields information about how all spatial components
+of the input field should be modified to improve F, enabling
+calculation of the next input u(i+1).
+Experimentally, this adjoint field optimisation strategy re-
+quires a relatively complicated optical setup: two digital opti-
+cal phase conjugation (DOPC) systems – which enable time-
+reversal of optical fields – are arranged back-to-back on ei-
+ther side of the dynamic sample. We use single-shot off-axis
+digital holography to measure the output fields on each side.
+The DOPC systems require very precise alignment, so we im-
+plemented a calibration method that we recently described in
+ref. [44]. Our set-up enables spatial shaping of both the inten-
+sity and phase profile of time-reversed field travelling in both
+directions. We test this approach to guide light through a sim-
+ilar sample dynamic medium to before (see Fig. 1(e), top row)
+and average over N = 5 realisations of the medium in each
+direction. SI §4 shows a schematic of the full optical set-up
+used in this experiment.
+Figure 2(b) shows a typical convergence curve throughout
+the optimisation process.
+After only ∼ 15 iterations, the
+input field converges to a solution with output fluctuations
+suppressed to a similar level than unguided optimisation –
+crucially now achieved in over 2 orders of magnitude fewer
+iterations.
+Supplementary Movie 2 shows the output fluc-
+tuations before and after optimisation. Once again looking
+inside the dynamic sample, we see that we have found a more
+localised optical field that passes almost entirely through
+the static parts of each phase plane and avoids the moving
+regions, as shown in Fig. 2(c).
+The fluctuation-eigenchannels of the time-averaged TM:
+So far we have focused on strategies to find a single optimised
+input field as quickly as possible. We now consider how a set
+of input modes may be determined, that all navigate around
+moving regions of a dynamic medium. Knowledge of such a
+sub-basis would enable a stable shaped output field – such as
+a focussed spot – to be formed from a suitable linear combina-
+tion of these time-independent fields at the output plane. This
+opens up the prospect of imaging through partially dynamic
+scattering media.
+One possibility is to conduct a series of adjoint optimisa-
+tions, each seeded from a different initial field. This would
+lead to a set of stable output fields that can be stored as the col-
+umn vectors of matrix V, and used to generate a target output
+field vtrg by injecting into the medium the field u = V−1vtrg.
+However, this is not an efficient search strategy, since there
+is no way to guarantee the linear independence of the set of
+optimised fields – meaning very similar fields may be inad-
+vertently found.
+To overcome this problem, we now devise a new method ca-
+pable of finding the full set of orthogonal fields that navigate
+around moving regions, for a given input basis. We introduce
+the time-averaged transmission matrix: Tav. To measure Tav,
+a set of M probe fields are sequentially transmitted through
+the dynamic sample, and the time-averaged output field is cal-
+culated in each case, forming the columns of Tav. Figure 3(a)
+shows a schematic of this approach. We illuminate the sample
+with M = 2304 probe fields, and average the output field over
+N = 10 uncorrelated realisations of the scattering medium
+for each input mode. Experimentally this procedure is sim-
+pler than physical adjoint optimisation – although the main
+challenge is that the reference beam required for holographic
+field measurement must be phase-drift-stabilised for the en-
+tire measurement of Tav. In order to achieve this stability, we
+establish a new phase-drift correction protocol. SI §5 gives
+details and the full optical setup for this experiment.
+We aim to discover fields that deliver high levels of time-
+averaged energy to the output plane. Finding these fields can
+be represented as an eigenvalue problem by noting that the to-
+tal intensity P arriving at the output in field v can be expressed
+as
+P = v†v = u†T†
+avTavu.
+(4)
+Therefore, the eigenvectors of matrix T†
+avTav with the largest
+absolute eigenvalues represent input fields that deliver the
+highest time-averaged intensity to the output plane. Assum-
+ing the internal fluctuations of the medium are large enough
+to randomise the phase of scattered light, then the fluctuat-
+ing parts of the output fields will average to near-zero. When
+forward scattering dominates, eigenvectors with high absolute
+eigenvalues also correspond to input fields that interact least
+with the time-varying regions inside the medium. We term
+this basis of eigenvectors the fluctuation-eigenchannels of the
+dynamic medium.
+Figure 3(b) shows the distribution of absolute eigenvalues
+of the matrix T†
+avTav, arranged in ascending order.
+Here
+we compare the eigenvalue distribution resulting from time-
+averaged TMs measured on two independent dynamic sam-
+ples with a different numbers of moving regions: (i) has a
+single dynamic patch on each plane similar to that shown in
+Fig. 1; (ii) has randomly placed fluctuating patches covering
+approximately half of the area of each plane – an example
+is shown in Fig. 3(c). In Fig. 3(b) we see that the magni-
+tude of the eigenvalues decrease more steeply from the maxi-
+mum value in this second case, indicating that the spectrum of
+eigenvectors deliver less time-averaged energy to the output –
+i.e. there are fewer light fields able to circumnavigate a sample
+with more extensive moving regions, as would be expected.
+We
+first
+demonstrate
+excitation
+of
+the
+fluctuation-
+eigenchannels of the more weakly fluctuating sample medium
+(i). Figure 3(d) shows examples of output speckle patterns
+when a selection of fluctuation eigenchannels are excited, with
+some of the highest and lowest absolute eigenvalues. Each
+row shows the output field for a new configuration of the
+dynamic medium (recorded at distinct times t1, t2, t3). The
+transmitted fields corresponding to high index fluctuation-
+eigenchannels remain stable (i.e. largely unchanging), indi-
+cating that the light propagating through the medium in these
+cases is avoiding dynamic regions. Conversely, the transmit-
+ted fields corresponding to low index eigenchannels vary with
+time at the output – as these modes interact strongly with the
+
+6
+Figure 3. Time-averaged transmission matrix. (a) Schematic of experimental set-up. A sequence of orthogonal probe fields are individually
+transmitted through the medium, e.g. u1, u2, u3. For each input, the corresponding time-averaged output field is recorded, e.g. ⟨v1⟩, ⟨v2⟩,
+⟨v3⟩, and arranged column-by-column to build the time-averaged TM Tav. (b) The magnitudes of the eigenvalues of T†
+avTav, for a weakly (i)
+and strongly (ii) fluctuating dynamic medium. Both are arranged in ascending order and normalised to a maximum value of 1. The weakly
+fluctuating medium is the same as used in the earlier experiments. An example of the strongly fluctuating medium is shown in (c), with
+moving regions highlighted in red. (d) Excitation of selected fluctuation-eigenchannels in the weakly fluctuating medium. Each column shows
+the output when the medium is illuminated with different eigenvectors. Each row shows the output at a different time – i.e. for 3 different
+configurations of the dynamic regions of the medium. We see the high index eigenvectors are stable with respect to these movements, while the
+low eigenvectors are not. (e) Eigenvector projection through a strongly fluctuating medium. (f) Enhanced focusing through strongly fluctuating
+scattering media using the time-averaged TM. Left column: an attempt to make a focus using the conventional inverse TM, which is measured
+while the medium fluctuates. We see a poor contrast focus which fluctuates strongly as the medium reconfigures. Right column: An output
+focus created through the same medium, with the input field generated using the top 100 most stable eigenvectors of T†
+avTav. Here we see that
+the contrast and stability of the output focus is significantly improved.
+moving parts of the dynamic sample. Supplementary Movie
+3 shows examples of the stability of output light transmitted
+through a range of different fluctuation-eigenchannels.
+We now investigate light shaping capabilities through
+the more challenging strongly fluctuating medium (ii).
+Figure 3(e) shows the stability of transmitted fields when
+exciting fluctuation-eigenchannels with the highest (left
+column) and lowest (right column) absolute eigenvalues. In
+this case, even light propagating through the most stable
+eigenchannel exhibits non-negligible output fluctuations over
+time, indicating that we have not found any fields that thread
+perfectly around all moving parts of the sample.
+Despite
+this, we find that a significant improvement in focusing
+at the output is possible using the information stored in
+the time-averaged TM. Figure 3(f) shows a focus created
+using a conventional TM approach, where the medium
+freely fluctuates throughout TM measurement (left column)
+compared to using a sub-basis formed from the top 100 most
+stable fluctuation-eigenchannels (right column) – see SI §6
+for details. We see that both the contrast and stability of the
+focus is strongly enhanced using our new approach.
+Discussion and conclusions
+In summary, we have identified a broad new class of partially
+dynamic scattering media that is amenable to deterministic
+light control techniques. We have demonstrated three new
+ways to thread stable light fields through such media, that rely
+on the movement of the medium itself to accomplish:
+
+(b)
+(a) Time-averaged TM measurement
+Eigenvalues of Tt
+(c) Strong fluctuations
+'Iav
+Reference
+Time-averaged
+- Weak (i)
+output fields
+Strong (ii)
+(V1)t
+u1
+u2
+<V2)t
+u3 :
+(V3/t
+Probe fields
+Camera
+0
+Red = fluctuating
+(sequential)
+2302
+Dynamic scatterer
+Tav = [(v1)t, (V2)t,-.. (VM)t]
+Eigenvalue index
+regions
+Strongly fluctuating complex medium
+(d)
+Excitation of fluctuation-eigenchannels
+Fluctuating
+(e) Eigenvector
+(f)Enhanced
+Stable
+focusing
+projection
+output
+output
+ti
+(arb.)
+Time
+Intensity (
+t2
+t2
+t3
+t3
+0
+2302
+2300
+2298
+2296
+20
+10
+2302
+Inverse
+Top 100
+TM
+Eigenvector index
+Eigenvectors
+Eigenvector index7
+The first technique, unguided optimisation, is a straight-
+forward but relatively slow approach, most suited to the case
+where the network of static channels throughout the medium
+remains fixed. Here our brute-force optimisation strategy is
+analogous to the first methods used to shape light through
+static scattering media [7], and as such may be improved us-
+ing more advanced algorithms [45, 46]. This technique is also
+highly flexible: the form of the objective function can be ar-
+bitrarily chosen. For example, intensity shaping terms could
+also be included, to simultaneously reduce fluctuations and
+shape the output.
+The second approach, physical adjoint optimisation, en-
+ables stable light fields to be very rapidly found by passing
+light backwards and forwards through the medium. We phys-
+ically compute the gradient of the objective function with re-
+spect to the optimisation variables (i.e. the field emanating
+from each super-pixel on the SLM). If implemented with fast
+beam shaping, this technique is well-suited to the case where
+a particular configuration of static channels only persist for a
+relatively short time. Our adjoint strategy is reminiscent of it-
+erative time-reversal [47], and recently proposed in-situ meth-
+ods to train photonic neural networks [48]. Indeed our work
+may be considered one of the first real-world implementations
+of a photonic adjoint optimisation routine – a challenging yet
+powerful method to realise experimentally [49]. We note that,
+for our application, the form of objective function is more re-
+stricted than unguided optimisation. For example, we found
+that some choices of objective function require deterministic
+control over the motion of the dynamic parts of the scattering
+medium which is evidently not possible in most cases.
+Our final strategy relies on measurement of the time-
+averaged TM to calculate the fluctuation-eigenchannels of the
+dynamic medium. These channels are excited by an orthog-
+onal set of input eigenfields, ordered in terms of how much
+time-averaged power they deliver to the output plane – thus
+revealing internal fields that minimally interact with the time-
+varying parts of the medium. This new concept is related to
+several previously introduced matrix operators connected to
+physical quantities of interest in scattering media
+[6, 50–
+52].
+As fluctuations in the medium go to zero, the time-
+averaged TM becomes equivalent to the conventional TM,
+and the fluctuation-eigenchannels tend to the transmission-
+eigenchannels of a static scattering medium [53]. The ‘de-
+position matrix’ [52] and the ‘generalised Wigner-Smith op-
+erator’ [50, 54] are both also capable of revealing light fields
+that circumnavigate predetermined regions within a complex
+medium. However, only the time-averaged TM does so with-
+out requiring access to internal fields within the medium [52]
+or the measurement of an entire TM while the medium is held
+static [50]. Here we have demonstrated that the time-averaged
+TM can enhance focusing through partially dynamic scatter-
+ing media. We also expect an improvement in more elaborate
+beam shaping, such as point-spread function engineering [55]
+and arbitrary pattern projection [32].
+We note that previous studies have used localised internal
+motion within scattering media as a guide-star – enabling fo-
+cusing directly onto these moving regions [56, 57] – the op-
+posite of what we have set out to achieve in our study. Recent
+work also investigated the performance of wavefront optimi-
+sation occurring on the same timescale as the medium decor-
+relates – with evidence to suggest that the resulting focus was
+dominated by the most stable modes propagating through the
+medium [58].
+In this study, our experiments have emulated mainly
+anisotropic forward-scattering media, as would be found
+transmitting light through the atmosphere, through multimode
+fibres, or through thin layers of biological tissue. In the fu-
+ture it will be interesting to study how these techniques per-
+form, or indeed need to be adapted, as the level of multi-
+ple scattering increases to the onset of the diffusive [59] or
+chaotic regimes [60]. While we expect the strategies outlined
+here to apply in these domains, strongly scattering environ-
+ments also pose additional challenges, since there are com-
+peting requirements: optical fields must both circumnavigate
+moving regions and also penetrate deeply enough into the
+medium to transmit significant power to the other side. We
+expect a smaller number of internal fields will satisfy both of
+these constraints, since optimised fields will be formed from
+a reduced basis of modes – dominated by high transmission-
+eigenchannels that are weighted to destructively interfere on
+all moving regions [52].
+A further avenue of exploration
+would be to investigate optimal focusing inside partially dy-
+namic media, while simultaneously guiding light around mov-
+ing pockets.
+Finally, it is interesting to note that the problem we have
+addressed in our work, from an optical perspective, is closely
+related to the concept of multi-path fading experienced by ra-
+dio frequency wireless communication channels. In this latter
+case the interaction of transmitted signals with moving me-
+dia in their path is known as mode-stirring, and the Rician
+K-factor quantifies the ratio of ‘unstirred’ to ‘stirred’ paths
+transmitted through a dynamic environment [61]. The tech-
+niques that we have introduced in this article may potentially
+be applied at radio and microwave frequencies, either in the
+spectral domain, or in the spatial domain in conjunction with
+emerging beam-forming systems.
+The concepts that we have introduced here apply generally
+to wave phenomena, and have relevance to a diverse range of
+applications. Possibilities include imaging deep into living
+biological tissue [62], transmission of space-division multi-
+plexed optical communications through turbulent air [63] and
+underwater [64], and propagation noise reduction in acoustic
+beam forming [65] and emerging smart microwave and radio
+environments [66]. Our work adds to the toolbox of methods
+to counteract the adverse effects of dynamic scattering media.
+Acknowledgements
+SARH acknowledges the Royal Society and TATA for fi-
+nancial support through grant URF\R\211033.
+DBP ac-
+knowledges the financial support of the Royal Academy of
+Engineering and the European Research Council (Grant no.
+804626: ‘Rendering the opaque transparent’).
+
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+Draft version January 4, 2023
+Typeset using LATEX twocolumn style in AASTeX63
+Speckle Space-Time Covariance in High-Contrast Imaging
+Briley Lewis
+,1 Michael P. Fitzgerald
+,1 Rupert H. Dodkins
+,2 Kristina K. Davis
+,2 and Jonathan Lin
+1
+1Department of Physics and Astronomy, UCLA, Los Angeles, CA 90024 USA
+2Department of Physics, UCSB, Santa Barbara, CA 93106 USA
+ABSTRACT
+We introduce a new framework for point-spread function (PSF) subtraction based on the spatio-
+temporal variation of speckle noise in high-contrast imaging data where the sampling timescale is
+faster than the speckle evolution timescale. One way that space-time covariance arises in the pupil
+is as atmospheric layers translate across the telescope aperture and create small, time-varying per-
+turbations in the phase of the incoming wavefront. The propagation of this field to the focal plane
+preserves some of that space-time covariance. To utilize this covariance, our new approach uses a
+Karhunen-Lo´eve transform on an image sequence, as opposed to a set of single reference images as in
+previous applications of Karhunen-Lo´eve Image Processing (KLIP) for high-contrast imaging. With
+the recent development of photon-counting detectors, such as microwave kinetic inductance detectors
+(MKIDs), this technique now has the potential to improve contrast when used as a post-processing
+step. Preliminary testing on simulated data shows this technique can improve contrast by at least 10–
+20% from the original image, with significant potential for further improvement. For certain choices
+of parameters, this algorithm may provide larger contrast gains than spatial-only KLIP.
+Keywords: exoplanet detection; high contrast imaging; atmospheric effects; instrumentation: adaptive
+optics; methods: data analysis; methods: statistical; techniques: image processing
+1. INTRODUCTION
+Direct imaging of exoplanets is a challenging endeavor,
+given the extreme contrasts that must be achieved to
+detect faint planets. Although significant starlight sup-
+pression can be achieved through optics and instrumen-
+tation, such as coronagraphs, adaptive optics (AO) sys-
+tems, interferometers, and more, that alone is insuffi-
+cient to detect analogs of planets in our solar system
+(Oppenheimer & Hinkley 2009; Guyon 2005). Improv-
+ing contrast expands the space of the types of planets
+that can be directly detected and characterized.
+Existing instruments, such as the Gemini Planet
+Imager (Macintosh et al. 2008) and VLT’s SPHERE
+(Beuzit et al. 2019) are able to image giant planets and
+brown dwarfs, reaching contrasts (in the astronomical
+sense, meaning the detectable planet-star flux ratio) of
+around 10−6. This is enabled by a combination of wave-
+front sensing, control, and post-processing, which re-
+Corresponding author: Briley Lewis
+[email protected]
+duce the impact of noise by distinguishing between the
+planet signal and residual noise; this noise arises from
+uncorrected wavefront aberrations, resulting in quasi-
+static fluctuations in the focal plane known as “speck-
+les.”
+Generally, these algorithms use the data them-
+selves to create a model of the speckle noise which can
+then be subtracted from the data to recover the tar-
+get planet signal in a process known as point-spread
+function (PSF) subtraction. Previously developed algo-
+rithms include LOCI (Locally Optimized Combination
+of Images; Lafreniere et al. (2007)), KLIP (Karhunen
+Lo´eve Image Processing; Soummer et al. 2012), and
+more (Gebhard et al. 2022).
+Many directly imaged
+planet discoveries to date have relied on such algorithms,
+such as the famous HR 8799 planets (Marois et al. 2008).
+Improvements to data processing pipelines and meth-
+ods are one way in which we can push forward and im-
+prove contrast for future high-contrast imaging instru-
+ments.
+Other approaches to improving high-contrast
+imaging methods focus on wavefront sensing and con-
+trol, such as predictive control techniques, which aim
+to improve adaptive optics corrections (Guyon & Males
+2017; Guyon et al. 2018; Males & Guyon 2018), and sen-
+arXiv:2301.01291v1  [astro-ph.IM]  3 Jan 2023
+
+ID2
+Lewis et. al.
+sor fusion, both currently in development at multiple
+facilities, including Subaru’s SCExAO facility (Guyon
+et al. 2017) and Keck Observatory van Kooten et al.
+(2021); Wizinowich et al. (2020); Jensen-Clem et al.
+(2019); Calvin et al. (2022).
+Other recent work such
+as Guyon & Males (2017) focuses on using on Em-
+pirical Orthogonal Functions (EOFs), a similar math-
+ematical framework, to analyze spatio-temporal correla-
+tions; their work is in the context of predictive control,
+whereas our work applies to image processing. New ad-
+vances in detector technology also affect both wavefront
+sensing and post-processing. High-speed, low-noise de-
+tectors will provide multiple opportunities for improve-
+ments, including focal-plane wavefront sensing, which
+eliminates non-common-path wavefront errors (Vievard
+et al. 2020). Of particular interest are arrayed photon-
+counting devices, such as Microwave Kinetic Inductance
+Detectors (MKIDs) (Schlaerth et al. 2008; Mazin et al.
+2012; Meeker et al. 2018; Walter et al. 2020) and Infrared
+Avalanche Photodiodes (IR APDs) (Goebel 2018; Wu
+et al. 2021). Electron Multiplying CCDs (EMCCDs) are
+a functional equivalent in the optical (Lake et al. 2020).
+Photon arrival times have already been used to distin-
+guish speckles from incoherent signals, such as planets
+(Walter et al. 2019; Steiger et al. 2021), and MKIDs have
+been used for high contrast imaging with the DARK-
+NESS instrument at Palomar (Meeker et al. 2018) and
+with MEC, the MKID Exoplanet Camera for high con-
+trast astronomy at Subaru (Walter et al. 2018).
+This new regime of photon-counting detectors and
+more advanced adaptive optics presents many oppor-
+tunities.
+With the improved temporal resolution of
+next-generation detectors, we will be able to resolve the
+spatial and temporal evolution of atmospheric speckles.
+Some prior work has investigated use of spatio-temporal
+correlations on longer timescales, such as Mullen et al.
+(2019) and Gebhard et al. (2022), but this work focuses
+the shorter timescale changes of atmospheric speckles.
+There is a rich history of theory and measurements
+of space-time atmospheric speckle behavior in the past
+decades, which this work builds off of. Since the 1970s–
+1980s, speckle patterns and intensity distributions have
+been measured (Dainty et al. 1981; Scaddan & Walker
+1978; Goebel 2018; Odonnell et al. 1982), demonstrating
+agreement with models based in Rician statistics (Cagi-
+gal & Canales 2001; Canales & Cagigal 1999) and the
+importance of speckles as the limiting noise source in the
+high-contrast regime (Racine et al. 1999). The space-
+time covariance was even directly measured in Dainty
+et al. (1981), indicating that speckle boiling has a direc-
+tionality related to turbulence. Speckle intensity pat-
+terns have been modeled as a modified Rician distribu-
+tion (Aime & Soummer 2004; Gladysz et al. 2010), and
+speckle lifetimes have been constrained through mod-
+els and direct measurements (Aime et al. 1986; Vernin
+et al. 1991; Glindemann et al. 1993). In fact, models of
+speckle boiling directly relate the lifetime of speckles to
+atmospheric parameters related to wind and turbulence,
+as in Roddier et al. (1982), estimating speckle lifetimes
+on the order of tens of milliseconds.
+This work is a new addition to the variety of time-
+domain algorithms that have been developed in recent
+years. For example, the PACO algorithm uses temporal
+information from the background fluctuations of Angu-
+lar Differential Imaging data (Flasseur et al. 2018), and
+the TRAP algorithm uses temporal information of the
+speckle pattern to improve contrast specifically at close
+separations (Samland et al. 2021). Another algorithm,
+from Gebhard et al. (2022), uses half-sibling regression
+on time-series data. These are all examples of the pos-
+sibilities for temporal information in post-processing, in
+addition to the AO control improvements described ear-
+lier.
+In this work, we aim for a second-order characteriza-
+tion of the statistical behavior of atmospheric speckles
+in the high-contrast regime, described by the space-time
+covariance, which we then leverage for improving con-
+trast in post-processing with the eventual goal of im-
+proving exoplanet detection capabilities. As previously
+mentioned, this goal is not without its challenges — with
+kHz readouts, these detectors can produce large datasets
+and lead to computationally intensive post-processing
+methods. While developing this new technique, we must
+also contend with data storage and computational limi-
+tations.
+In this paper, we first provide an analytical justifica-
+tion for the existence of these covariances in the high-
+contrast regime, observe their occurrence in test simu-
+lations focusing on millisecond time sampling, and then
+present an initial algorithm to exploit these covariances
+for PSF subtraction.
+Specifically, we are testing this
+algorithm in a regime dominated by atmospheric speck-
+les at short exposures (where the timescale of our ex-
+posures is short compared to that of changes in atmo-
+spheric residual wavefront error, so atmosphere is essen-
+tially frozen).
+Here in Section 2, we describe the process of baseline
+Karhunen-Lo´eve Image Processing (KLIP), the origins
+of space-time speckle covariances, and the extension of
+KLIP to space-time covariances. Following, in Section
+3, we describe the models used to create datasets for
+initial testing of this processing framework. Section 4
+presents results of this new algorithm implemented on
+simulated data. Finally, in Sections 5 and 6, we discuss
+
+3
+the promise of this new technique, as well as its current
+challenges/limitations and future work.
+2. SPACE-TIME COVARIANCE THEORY
+Speckles can limit contrast, but can also be subtracted
+to some extent to improve contrast. One of the most
+successful post-processing algorithms has been KLIP,
+described in Section 2.1, which exploits spatial correla-
+tions in long-exposure images. We motivate our exten-
+sion of this technique to include space-time correlations
+on shorter timescales in Section 2.2 by describing how
+these correlations arise in imaging through the atmo-
+sphere. This extension of KLIP, referred to as space-
+time KLIP or stKLIP, is demonstrated in Section 2.3,
+exploiting spatio-temporal correlations between short-
+exposure images.
+2.1. Karhunen-Lo´eve Image Processing
+Karhunen-Lo´eve Image Processing is a data process-
+ing technique that uses principle component analysis
+(PCA), where data are represented by a linear combina-
+tion of orthogonal functions. In high-contrast imaging,
+KLIP is used to build a model, used for PSF subtraction,
+that accounts for spatial correlations between speckles
+and other PSF features, first described in Soummer et al.
+(2012). This technique takes advantage of spatial co-
+variances of the speckles in the image, because strong
+correlations exist in high eigenvalue modes and can be
+suppressed. This is a data-driven approach, which uses
+available information from the data itself to provide an
+approximation of the noise, by using a subset of the data
+as “reference images” from which to build the model of
+the noise while using another subset of the data as the
+“target image” for PSF subtraction.
+To increase readability, all variables for the following
+mathematics are described in Appendix A. As described
+in Soummer et al. (2012), we assume we observe a point
+spread function T(k), where k is the pixel index, that
+contains the stellar point spread function Iψ(k) and may
+also contain some faint astronomical signal of interest
+A(k). Therefore, our target image can be described as
+T(k) = Iψ(k) + ϵA(k),
+(1)
+where ϵ is 0 when there is no astronomical signal of
+interest, or 1 if there is. The goal of PSF subtraction
+is therefore to recreate Iψ(k) in order to isolate A(k).
+Without an infinite number of references, though, we
+cannot exactly infer Iψ(k); instead, we approximate the
+PSF ˆIψ(k). For consistency in our notation, herein we
+represent T(k), A(k), and ˆIψ(k) as vectors t, a and ˆ
+ψ
+respectively.
+In order to approximate
+ˆ
+ψ,
+KLIP computes a
+Karhunen-Lo´eve Transform based on the covariance ma-
+trix of the mean-subtracted reference images.
+A sequence of reference images are first unraveled into
+one-dimensional vectors, each as r. Note: henceforth
+vectors are denoted as bold, matrices with uppercase
+variables and subscript matrix elements. These image
+vectors r are then stacked into an np × ni matrix R,
+where np = nx × ny and ni is the number of images, as
+follows:
+R =
+�
+�����
+R1,1
+R1,2
+. . .
+R1,ni
+R2,1
+R2,2
+. . .
+R2,ni
+...
+...
+...
+...
+Rnp,1 Rnp,2 . . . Rnp,ni
+�
+�����
+(2)
+We then subtract the mean image of the set (summing
+over the matrix columns) from the reference set R, in
+order to produce a set of mean-subtracted images M to
+use throughout the process of KLIP:
+xi = 1
+ni
+ni
+�
+j=1
+Ri,j
+(3)
+Mi,j = Ri,j − xi
+(4)
+The resulting covariance matrix (5) C has size np×np.
+C = MM T =
+�
+�����
+C1,1
+C1,2
+. . .
+C1,np
+C2,1
+C2,2
+. . .
+C2,np
+...
+...
+...
+...
+Cnp,0 Cnp,1 . . . Cnp,np
+�
+�����
+(5)
+Note: in practice, this implementation is computa-
+tionally expensive, so the covariance is instead often
+computed in image space on ni by ni images and then
+re-projected into pixel space, as is done in the Soummer
+et al. (2012) implementation. The ideal implementation
+depends on which dimension is larger / more computa-
+tionally expensive, e.g. Long & Males (2021). In this
+work, the mathematics for KLIP and stKLIP, as written
+here, will be in pixel space.
+An eigendecomposition of the covariance matrix C,
+mathematically described as solutions to the equation
+Cvj = λjvj,
+(6)
+with
+λ1 > λ2 > λ3 > . . . λnp,
+(7)
+produces a length np vector of eigenvalues (λ) and size
+np ×np (or nm ×np if fewer than np eigenvectors/modes
+
+4
+Lewis et. al.
+are used) matrix of eigenvectors/eigenimages (V ) con-
+taining nm rows of individual eigenvectors v each of
+length np, such that Vi,j = (vj)i.
+V =
+�
+�����
+V1,1
+V1,2
+. . .
+V1,np
+V2,1
+V2,2
+. . .
+V2,np
+...
+...
+...
+...
+Vnm,1 Vnm,2 . . . Vnm,np
+�
+�����
+(8)
+The eigenvalues order the eigenimages by their impor-
+tance to rebuilding the original image and are used to
+construct the basis of the new subspace of greatest vari-
+ation onto which we project our images. Assuming the
+vectors are sorted by decreasing eigenvalue, the first co-
+ordinate corresponds to the direction of greatest vari-
+ation. The lowest-order (first coordinate) eigenimages
+are selected to represent ˆ
+ψ, while leaving the high-order
+terms to hopefully contain our astrophysical signal.
+We select a given number nm of the eigenimages as
+our number of modes of variation. The inner product of
+the matrix of eigenvectors V with the one-dimensional
+vector of the target image t (which has length np), is
+described mathematically as
+t =
+�
+�����
+t1
+t2
+...
+tnp
+�
+�����
+(9)
+q = V · t =
+�
+�����
+q1
+q2
+...
+qnm
+�
+�����
+(10)
+and creates a vector of coefficients q of length nm — each
+of these can be thought of as how much of each mode
+(or each eigenvector, vj) is in the image, or equivalently,
+the coordinates in the new rotated principle axis space.
+Lastly, we can project back into our original pixel
+space by taking the product of this vector of coeffi-
+cients with the chosen eigenvectors, recovering a vector
+of length np, the same as our target image:
+ˆ
+ψ = qT · V
+(11)
+The resulting array is our image projected into the sub-
+space of greatest variation, an estimation of the original
+PSF ˆ
+ψ, and what we will subtract from our target image
+for PSF subtraction. Note that the tuneable parameter
+here is the number of eigenvectors used in the basis (the
+number of “modes”).
+The planet signal is also projected onto a distribution
+of these modes, and it is assumed that the planet signal
+is primarily projected onto modes with lower eigenvalue.
+However, as we subtract more modes, the projection of
+the planet onto these modes is also subtracted. There-
+fore, a larger number of modes might lead to oversub-
+traction of a planet signal, but too few may not suffi-
+ciently subtract out the speckle noise. As a result, we
+must correct for this throughput effect and optimize the
+number of modes to attain the largest possible contrast
+gain.
+2.2. Space-Time Covariances
+Whereas KLIP harnesses spatial covariances of speckle
+noise, we propose to expand the scope of such projection
+methods to take advantage of space-time covariances
+in speckle noise.
+For bulk flow in a turbulent atmo-
+sphere, phase errors in the pupil, from atmospheric dis-
+turbances, translate across the telescope with wind mo-
+tion, resulting in changes in phase and amplitude in the
+image plane. Atmospheric perturbations evolve across a
+broad set of spatial frequencies. Since the perturbations
+at these different spatial frequencies are correlated, we
+will illustrate that the speckles at the locations that cor-
+respond to those spatial frequencies in the image plane
+will be correlated as well. Similarly to the above section,
+all variables for the following mathematics are described
+in Appendix B.
+The covariance of intensity in the image plane for
+points separated in space and time is characterized
+through the second moment ⟨I(x1, t)I(x2, t−τ)⟩, where
+I is the intensity in the image. Angle brackets (⟨⟩) de-
+note averaging over a statistical ensemble. Suppose we
+have a perfect coronagraph and only phase aberrations
+are present, ignoring polarization as well as static phase
+errors, and treating electric field as a scalar. Also, we
+presume the phase aberrations are small, a reasonable
+assumption for the high-contrast imaging limit. In this
+case, the pupil amplitude is
+Ψpup(u, t) = P(u)eiφ(u,t),
+(12)
+approximated as
+Ψpup(u, t) ≈ [1 + iφ(u, t)]P(u),
+(13)
+where P(u) is the pupil function, φ is the phase, and u
+is the coordinate in the pupil plane (x is the coordinate
+in the focal plane, related by a Fourier transform). It
+is worth noting that departure from this assumption of
+linearity may affect results. The amplitude in the focal
+plane is
+Ψfoc(x, t) = F {P(u)} + iF {φ(u, t)P(u)} ,
+(14)
+= C(x) + Sφ(x, t).
+(15)
+
+5
+C(x) is the spatially coherent part of the wavefront, and
+Sφ(x, t) comes from phase aberrations – Sφ(x, t) corre-
+sponds to the “speckles” we want to remove (Aime &
+Soummer 2004; Roddier et al. 1982). In the case of a
+perfect coronagraph, C(x) = 0 and the intensity in the
+image is only due to phase aberrations, and can be ex-
+pressed as
+I(x, t) = |Ψfoc(x, t)|2,
+(16)
+= |Sφ(x, t)|2,
+(17)
+= |F {φ(u, t)P(u)} |2.
+(18)
+The covariance of the intensity is
+⟨I(x1, t)I(x2, t−τ)⟩ = ⟨|Sφ(x1, t)Sφ(x2, t−τ)|2⟩. (19)
+If
+we
+assume
+(complex)
+Gaussian
+statistics
+for
+Sφ (Soummer et al. 2007), then by Wick’s theorem (e.g.
+Fassino et al. 2019) we have,
+⟨I(x1, t)I(x2, t − τ)⟩ =
+⟨I(x1, t)⟩⟨I(x2, t)⟩ + |⟨Sφ(x1, t)S∗
+φ(x2, t − τ)⟩|2.
+(20)
+Therefore to compute this covariance, we need the quan-
+tity ⟨Sφ(x1, t)S∗
+φ(x2, t − τ)⟩, which is the covariance of
+the phase-induced aberration in the focal plane.
+Ac-
+counting for the Fourier relationship between the focal
+plane aberration Sφ and the pupil plane phase φ as in
+Equations 14 and 15, we find
+⟨Sφ(x1, t)S∗
+φ(x2, t − τ)⟩ =
+�
+du
+�
+dξ exp[2πiξ · x2 − 2πiu · (x1 − x2)]×
+⟨φ(u, t)φ(u + ξ, t − τ)⟩P(u)P(u + ξ)
+(21)
+where ξ is the coordinate of the displacement in the
+pupil plane. If φ(u, t) is statistically stationary in the
+pupil plane position u (and time), then we can define
+the phase covariance function as
+Bφ(ξ, τ) = ⟨φ(u, t)φ(u + ξ, t − τ)⟩,
+(22)
+independent of u and t.
+Equation 22 for Bφ relates
+space-time covariance in the pupil to space-time covari-
+ance in the image, and can be simplified into the Kol-
+mogorov phase covariance function for turbulence with
+an assumption about time.
+Kolmogorov’s theory of turbulence describes a cas-
+cade of large scale turbulent motions that dissipate en-
+ergy onto smaller scales, following a power spectrum de-
+scribed by Φn(k) ∝ |k|−11/3, where Φn is the variation
+in index of refraction and |k| is the magnitude of the
+turbulence (Kolmogorov 1941; Hickson 2008). Fluctua-
+tions in density correspond to fluctuations in the index
+of refraction. These variations in index of refraction lead
+to differences in path length for the incoming light, cre-
+ating some of the phase and amplitude error that we
+observe. However, we assume the timescale of change
+for this turbulence is generally slow when compared to
+wind speeds, an assumption known as Taylor frozen flow
+(Taylor 1938). This assumption is valid so long as the
+turbulent intensity is low compared to the wind speed,
+generally accepted to be true for astronomical contexts
+with the possible exception of boundary layer turbulence
+(Bharmal 2015). The turbulence can be thought of then
+as a “phase screen” propagating horizontally across the
+telescope with the wind. This phenomenon is described
+mathematically as
+φ(u, t) = φ(u − vwindτ, t − τ)
+(23)
+which states that the phase structure at one time is re-
+lated to the phase structure at a different time, just
+shifted by the wind velocity times the time difference
+(Taylor 1938; Hickson 2008).
+This shows that a single phase screen φ(u, t) (which
+contains Kolmogorov turbulence Φn) under Taylor
+frozen flow is related to a phase screen at a different
+time φ(u, t − τ) via the wind speed vwind. Similarly, we
+can then say
+Bφ(u, t) = Bφ(u − vwindτ, t − τ).
+(24)
+This implies the phase covariance function at one loca-
+tion and time Bφ(ξ, t) in the pupil is related to the phase
+covariance function at that location at a previous time
+Bφ(ξ, 0), where Bφ(ξ, 0) is a covariance related to the
+Kolmogorov phase covariance function. Since we know
+the Kolmogorov phase covariance function is non-zero
+as long as turbulence is present, this demonstrates that
+the phase covariance function at an arbitrary location
+and time Bφ(ξ, τ) is non-zero. Even if frozen flow is vio-
+lated, as long as there is non-zero space-time covariance
+in the pupil, we expect non-zero space-time covariance
+in the image, as shown in Equation 22.
+Rearranging Equation 21,
+⟨Sφ(x1, t)S∗
+φ(x2, t − τ)⟩ =
+�
+dξ exp(2πiξ · x2)Bφ(ξ, τ)
+�
+du exp[−2πiu · (x1 − x2)]P(u)P(u + ξ).
+(25)
+The latter integral is the Fourier transform of the overlap
+of displaced pupils. Defining this function,
+P(r, ξ) =
+�
+du exp(−2πiu · r)P(u)P(u + ξ),
+(26)
+
+6
+Lewis et. al.
+we now have the space-time covariance of speckles as
+the product of the turbulence phase covariance function
+and P, as follows:
+⟨Sφ(x1, t)S∗
+φ(x2, t − τ)⟩ =
+�
+dξ exp(2πiξ · x2)Bφ(ξ, τ)P(x1 − x2, ξ).
+(27)
+This mathematical framework illustrates how the fo-
+cal plane covariance is intimately related to pupil plane
+covariance in the high contrast imaging regime, with
+a perfect coronagraph and small phase errors.
+Look-
+ing at the overlap of displaced pupils, P(x1 − x2, ξ),
+the form of the expression suggests that covariance will
+be strongest at smaller spatial separations. Similarly,
+Equation 24 suggests that covariance will be strongest
+at smaller temporal separations. Overall, if there is non-
+zero space time covariance in the pupil plane, then we
+will have non-zero space time covariance in the focal
+plane. We will test this further with simulations, as de-
+scribed in Section 3.
+2.3. Space-Time KLIP
+Recall that KLIP improves contrast by projecting
+away features that are spatially correlated in image se-
+quences. We can extend the framework of KLIP (Soum-
+mer et al. 2012) to space-time covariances by using an
+image sequence instead of an image.
+Note that for
+the following mathematics we assume discrete time se-
+quences, rather than continuous as in Section 2.2 above.
+Additionally, we assume regular and continuous time
+sampling for this implementation; however, this method
+can be extended easily to block-continuous sampling,
+which may be useful in future work.
+All variables for the following mathematics are also
+described in Appendix A. Baseline KLIP uses an image
+vector of length np (number of pixels in image) as its
+target image and a np × ni matrix as the set of refer-
+ence images to determine covariance between pixels, find
+eigenvectors of covariance, and project out the largest
+eigenvalue modes from the image. Similarly, space-time
+KLIP (referred to as stKLIP) uses an image sequence of
+length ns × np (number of images in the sequence times
+number of pixels per image), as shown in Equation 28,
+to perform those steps.
+Note that this is transposed
+compared to KLIP, which uses np × ns.
+It is then necessary to create a block diagonal covari-
+ance matrix of size ns × np by ns × np, as illustrated
+in Figure 1, from the mean-subtracted image sequence.
+Each block is the covariance at a given time lag, with the
+block diagonal as lag zero (spatial covariance). If only
+lag zero is used, the mathematics here reduces down
+to baseline (spatial) KLIP, as described in Section 2.1.
+Lags should be chosen based on the translation time
+of the smallest relevant feature within the field of view
+at the focal plane up to the full crossing time of the
+wind across the telescope aperture.
+This is an addi-
+tional tuneable parameter to consider when optimizing
+the algorithm, in addition to the number of modes.
+The following computations mirror baseline KLIP,
+but, in practice, are potentially more computationally
+expensive due to the larger size of the covariance matrix
+used in the eigendecomposition. The steps of stKLIP
+are as follows:
+1. Subtract the mean image over the whole refer-
+ence set, then partition the reference set into im-
+age sequences. These image sequences have length
+ns = nl = 2L+1 where L is the largest number of
+timesteps (lags) away from the central image and
+nl is the total number of timesteps (lags) in the se-
+quence. (The following steps will be repeated over
+each image sequence, such that every image, with
+the exception of L images at each end, is at some
+point the central image. Therefore, for ni images,
+there will be ni − 2L image residuals at the end of
+this process.)
+Similarly to KLIP, the reference set/target image
+set S (which in this implementation are the same)
+are unraveled into one-dimensional vectors s of
+length ns × np, as seen below.
+S =
+�
+�����
+S1,1 S1,2 . . . S1,np
+S2,1 S2,2 . . . S2,np
+...
+...
+...
+...
+Sns,1 Sns,2 . . . Sns,np
+�
+�����
+(28)
+s =
+�
+�����������
+S1,1
+S1,2
+...
+S1,np
+...
+Sns,np
+�
+�����������
+(29)
+2. Compute the [nsnp, nsnp] size covariance matrix C
+of the image sequences. In practice, this is more
+straightforward when done by computing the co-
+variance of each image pair (Ci) and then arrang-
+ing them in the block diagonal ordering shown in
+Figure 1.
+3. Perform an eigendecomposition on the covariance
+matrix, obtaining nsnp eigenvalues (λ) and a ma-
+trix eigenvectors (V ) of size [nsnp, nsnp] contain-
+
+7
+Figure 1. Diagram of stKLIP input sequence setup – translating phase screens (top) and resulting image sequence (middle) –
+with the corresponding block diagonal space-time covariance matrix (bottom). Each covariance block Ci is the covariance for a
+single lag, with shape np × np, and together they create a single space-time covariance matrix C with size nsnp × nsnp. The
+covariance matrix takes this form because the 2d images are flattened into 1d vectors, which are then joined to make an np × ns
+1d vector, which is multiplied by its transpose to create this matrix.
+ing individual eigenvectors v.
+Cvj = λjvj
+(30)
+λ1 > λ2 > λ3 > . . . λp
+(31)
+4. Choose a number of modes nm, reducing the vec-
+tor of eigenvalues and matrix of eigenvectors to
+sizes nm and [nm, nsnp] respectively. The matrix
+of eigenvectors contains nm rows of eigenvectors
+each with length nsnp, such that Vi,j = (vj)i.
+V =
+�
+�����
+V1,1
+V1,2
+. . .
+V1,nsnp
+V2,1
+V1,1
+. . .
+V2,nsnp
+...
+...
+...
+...
+Vm,1 Vm,2 . . . Vnm,nsnp
+�
+�����
+(32)
+5. Obtain image coefficients through inner product of
+chosen eigenvectors and image sequence, similar to
+
+Pupil plane view of turbulence, leading to the below image sequence
+Input image sequence with length n,=5, lags=[0,1,2,3,4], niags=5
+Space-time covariance matrix with shape n.
+Xh
+lags
+'pix
+lags'
+'pix
+Each block
+(C,) is the
+covariance for
+that time lag8
+Lewis et. al.
+Equation 10.
+q = V · s =
+�
+�����
+q1
+q2
+...
+qnm
+�
+�����
+(33)
+6. Project the image sequence back into pixel space
+to obtain a reconstructed sequence ˆs with central
+image ˆψk, again mirroring Equation 11. Note: For
+ease of implementation, we have calculated the en-
+tire sequence, but projecting only onto the central
+image may improve efficiency.
+ˆs = ˆqT · V
+(34)
+ˆψk = [ˆsnp((nl+1)/2−1) . . . ˆsnp(nl+1)/2]
+(35)
+7. Perform PSF subtraction using the central image.
+ϵak = sk − ˆψk
+(36)
+8. Iterate through the above steps such that each
+image is the central image of a sequence of
+length ns, resulting in a set of residuals ϵak,j =
+[ϵ0ak,0, ϵ1ak,1, . . . , ϵnsak,ns].
+9. Compute mean of image sequence residuals to out-
+put an averaged residual, rk,avg.
+rk,avg = 1
+ns
+ns
+�
+j=0
+ϵjak,j
+(37)
+Once our image sequence is projected into the new
+subspace in Step 6, we have two options for PSF sub-
+traction: subtract the residuals from the whole sequence
+used, or subtract only from the central “target” im-
+age. We use a central target image to take advantage
+of speckle motions in timesteps both before and after.
+We then iterate through the full data set, as described
+in Step 8, performing stKLIP and PSF subtraction, so
+that each image is the central image of some image se-
+quence with length ns = nl = 2L + 1. This outputs a
+sequence of image residuals that is of length ni − 2L. In
+Step 9, we then average over the number of timesteps to
+output an averaged residual.
+There are possibilities for improving the algorithm,
+such as by exploiting the symmetry in the covariance
+matrix C in order to hasten the process of updating
+the eigenimages; however, we leave this for future work.
+Further improvements are discussed in Section 5.
+3. ALGORITHM DEVELOPMENT
+In Section 2.2, we showed that we expect non-zero
+space-time covariance to exist in speckle noise. In Sec-
+tions 2.1 and 2.3, we showed the mathematical frame-
+work for an algorithm to exploit these statistics for im-
+age processing and PSF subtraction.
+In this section, we illustrate aberrations of increasing
+complexity to examine their covariance structure and
+test the application of stKLIP. These tests and simula-
+tions are described in 3.1, for initial proof of concept.
+Section 3.2 describes the algorithm application to simu-
+lated data and calculations of possible contrast gains in
+the algorithm’s current form; here we also discuss selec-
+tion criteria for the choices of number of modes and lags.
+Analyzing these data sets also requires some computa-
+tional optimization, which is described in 3.3. In the
+following Section 4, we will discuss the results of these
+applications of stKLIP.
+3.1. Foundational Tests
+Our first step was to create and implement simple test
+cases in one and two dimensions to demonstrate that
+our theoretical expectations from Section 2.2 are valid
+and ensure that our algorithm reduced image variance
+as expected.
+A one-dimensional case allows us to di-
+rectly compare a simulated covariance matrix with one
+calculated from the analytic theory in Section 2.2, serv-
+ing as a test of the relationship between pupil plane
+covariance and focal plane covariance.
+Then, a two-
+dimensional case serves as a first in implementing the
+algorithm, ensuring that the algorithm reduces variance
+on a well-understood simple case before moving onto
+more complex atmospheric simulations.
+3.1.1. One-Dimensional Test of Pupil/Focal Covariance
+Relationship
+To begin, we created a simple one-dimensional model
+of two interfering speckle PSFs, which are simply two
+sinusoids with slightly different frequencies in the pupil
+plane. We first use this simple sinusoidal model to com-
+pare the simulated space-time covariance to the pre-
+dicted behavior from theory, to show how a set of input
+aberrations in the pupil plane corresponds with the re-
+sulting focal-plane space-time covariance. Although the
+algorithm does not require pupil plane covariances, this
+test is done to further establish the existence of the focal
+plane covariances that we seek to harness.
+To create the 1-d speckle model, first we must create
+a grid setup for evaluating the wavefront in the pupil
+and focal planes. These are parameterized in units of
+D/λ and λ/D respectively, where λ is our wavelength
+of observation, assuming monochromatic light. Keeping
+
+9
+these units preserves the Fourier duality relationship,
+and they can be converted to more conventional units if
+the focal length is known.
+The next critical piece is to define the entrance aper-
+ture in the pupil plane.
+This pupil function sets the
+amplitude A of the electric field (E = Aeiφ), and is
+simply a top-hat function (Π(u), 1 inside a given region
+and 0 outside). We also apply a translating phase screen
+(shown in the top panel of Figure 2) to the pupil, which
+is where phase aberrations are accounted for. We use a
+simple perturbation of two superimposed sinusoids with
+similar periods/frequencies, so that the wings of their
+PSFs overlap. This set-up is like simulating one layer of
+frozen flow translating across the telescope’s aperture.
+These perturbations are small (≪ 1 radian), consistent
+with the high-contrast regime.
+We then perform the necessary Fourier transform to
+retrieve the focal-plane electric field. By doing this for
+the pupil function with no perturbations, we retrieve
+what we would see in an ideal case for a uniformly illu-
+minated pupil; this is also what would be blocked if we
+had a perfect coronagraph. We subtract this “perfect”
+case from the case with the sinusoidal perturbation, per-
+forming the action of the coronagraph and suppressing
+light from the unaberrated portion of the wavefront.
+A one-dimensional case (Figure 2) illustrates the rela-
+tive evolution of two neighboring speckles created from
+atmospheric perturbations. Atmospheric theory (as in
+Section 2.2), in particular the frozen flow assumption,
+predicts a symmetrical space-time covariance structure,
+which can be computed for a 1-d model with a top-
+hat pupil function (Π(u)), two sinusoidal functions in
+the pupil, and no uniform illumination in the pupil
+(C(x) = 0). We carried out these calculations in two
+ways. First, we solved the integrals in Section 2.2 for the
+simple two sinusoid situation using Fast Fourier Trans-
+forms (FFTs). Second, we began with an array describ-
+ing the sinusoidal “phase screen” and simulated propa-
+gation through an optical system using FFTs.
+The variation in pupil and focal plane covariance over
+various time lags, as shown in Figure 3, can be clearly
+interpreted based on the locations of the two interfer-
+ing speckles. These matrices show a symmetric pattern
+that changes with the number of lags used, due to the
+change in the speckles’ relative locations. At lags 0 and
+100, the peaks are due to the alignment of the speck-
+les’ peaks, as marked in the top panel; lag 25 illustrates
+the lower covariance when the speckles are in slightly
+different places, and lag 50 shows two lower intensity
+peaks when the speckles are separated. Importantly, for
+a given non-zero lag, there are non-zero terms in both
+Figure 2.
+One-dimensional demonstration of speckle in-
+terference. Two sinusoidal perturbations in the pupil plane
+interfere to create moving speckles in the image plane. Top:
+1d phase screen with interfering sinusoids over time. Middle:
+1-d intensity over time without a coronagraph, showing the
+Airy pattern. Bottom: 1-d intensity over time with a coron-
+agraph, with the speckles’ relative evolution appearing more
+clearly due to the lack of coherent light, C(x). This simu-
+lation is used as a test of the space-time speckle covariance
+theory in Section 2.2.
+
+Phase Screens
+100
+0.035
+80
+0.030
+0.025
+60
+Intensity
+Time
+0.020
+40
+0.015
+0.010
+20
+0.005
+0.000
+0
+-1.0
+-0.5
+0.0
+0.5
+1.0
+u (D/入)No coronagraph
+100
+50
+80
+40
+30
+60
+Intensity
+Time
+40 -
+20
+20 -
+10
+-0
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+X (/D)Perfect coronagraph
+100
+10
+80 -
+8
+-09
+6
+Intensity
+Time
+40 -
+4
+20 -
+2
++0
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+x (入/D)10
+Lewis et. al.
+Figure 3. Space-time covariance matrices for pupil plane (middle) and focal plane (bottom) of a 1-d model of two sinusoids
+with different frequencies – as illustrated in the top panel of Figure 2 – with an annotated view of the simulation (top). These
+matrices show a symmetric pattern that changes with the number of lags used, due to the change in the speckles’ relative
+locations. At lags 0 and 100, the peaks are due to the alignment of the speckles’ peaks, as marked in the top panel; lag 25
+illustrates the lower covariance when the speckles are in slightly different places, and lag 50 shows two lower intensity peaks
+when the speckles are separated. Importantly, for a given non-zero lag, there are non-zero terms, indicating that there are
+temporal correlations.
+the pupil and focal plane covariances, indicating that
+there are temporal correlations.
+This simulation further demonstrates the claim that
+a simplified frozen flow scenario in the pupil can create
+calculable space-time covariances in the focal plane, and
+validates our use of this simple test case to test stKLIP.
+3.1.2. Two-Dimensional Test Case for Algorithm
+Development
+In order to ensure that the algorithm is behaving ac-
+cording to our expectations – that it will reduce the
+image variance – we expand this one-dimensional test
+case into two-dimensions to make an image sequence
+of the two time-varying, interfering speckles.
+We use
+
+Perfect coronagraph
+100
+200
+175
+80
+150
+t=l=0
+125
+60 -
+Intensity
+t=l=25
+Time
+t=l=50
+100
+t=l=75
+40 -
+75
+t=l=100
+50
+20
+25
+←0
+¥-2
+-8
+-6
+-4
+0
+2
+4
+6
+8
+X (Λ/D)ld Simulation @
+ ld Simulation @ ld Simulation @ ld Simulation @
+ld Simulation @
+t=0
+t=25
+t=50
+t=75
+t=100
+1d Simulation @ t=0
+Pupil
+[=0
+I=25
+I=50
+[=75
+[=100
+1d Simulation @ t=0
+Focal11
+this idealized test case as a check against our expec-
+tations for our stKLIP implementation, and for a first
+test of efficacy, comparing the reduction in image vari-
+ance between three data processing methods:
+mean-
+subtraction, KLIP, and stKLIP. The setup is the same as
+the above one-dimensional test case, but in two dimen-
+sions, with a circular aperture instead of a top hat as the
+pupil function. We create a series of images at various
+time steps as the input to stKLIP, shown in Figure 4.
+Although there are two tuneable parameters for stK-
+LIP — number of modes (e.g. number of eigenimages
+used in the projection) and number of lags, as described
+in Sections 2.1 and 2.3 — we only test one set of modes
+and lags (10 modes, 2 lags) with this simple test case and
+leave further exploration of these parameters for later
+testing (see Section 3.2). We similarly use 10 modes for
+KLIP to make the comparison fair.
+In this simple test case, KLIP and stKLIP reduce the
+variation in the image by factors of 6.8 and 5.7, re-
+spectively.
+Although stKLIP does not improve upon
+KLIP in this limited test case, it is important to re-
+member that we have not optimized for modes and lags
+in this scenario; determination of performance is left for
+more rigorous and realistic tests in the following section,
+3.2. They both outperform simple interventions, such as
+subtracting the mean of the image, in reducing the to-
+tal variation in the image, as shown in Figure 5.
+To
+summarize, this 2d test was performed to demonstrate
+that the overall image variance decreases after project-
+ing out modes of variation with stKLIP, as qualitatively
+expected, and in that sense the test can be considered
+successful.
+3.2. Simulated AO Residual Tests
+We then wanted to test stKLIP on a more realistic
+atmospheric phase screen and again measure potential
+contrast gains.
+To this end, we created a set of sim-
+ulated observations to represent AO residuals and per-
+formed stKLIP on them for a variety of different modes
+and lags. We measure contrast curves and companion
+SNR for four methods of post-processing in order to un-
+derstand the effectiveness of our new method: stKLIP,
+baseline/spatial KLIP, mean-subtraction, and no post-
+processing.
+Results from these tests are described in
+Section 4 and discussed further in Section 5. In this sec-
+tion, we first detail the methods used to create the simu-
+lated data set, then the methods for computing contrast
+curves and SNR on the processed data.
+To create the simulated data set, we use a simula-
+tor specifically designed for high-contrast imaging with
+next-generation detectors, such as MKIDs, called MEDIS
+(the MKID Exoplanet Direct Imaging Simulator), the
+Figure 4. Two-dimensional test of speckle interference. A
+sinusoidal phase screen (top) produces a speckle pattern im-
+posed on an Airy disk (middle).
+Subtracting the PSF of
+a model without perturbations, we simulate observations of
+this sinusoidal perturbation with a “perfect” coronagraph
+(bottom). All images depict the intensity (I = |E|2). This
+simulation is used as a troubleshooting step for a first imple-
+mentation of the stKLIP algorithm.
+first end-to-end simulator for high contrast imaging
+instruments with photon counting detectors (Dodkins
+2018; Dodkins et al. 2020).
+MEDIS generates atmospheric phase screens with
+HCIPy (Por et al. 2018). These phase screens use mod-
+
+Focal Plane - Sinusoidal Perturbation with Perfect Coronagraph
+15
+1750
+10-
+1500
+5-
+1250
+(Λ/D)
+-0
+1000
+y
+750
+-5-
+500
+-10
+250
+-15
+-15
+-10
+-5
+0
+5
+10
+15
+X (入/D)Sinusoidal Phase Screen
+1.00
+0.75
+0.15
+0.50
+0.10
+0.25
+0.05
+(D/入)
+0.00
+0.00
+y
+-0.25
+-0.05
+-0.50
+-0.10
+-0.75
+-0.15
+-1.00
+-1.0
+-0.5
+0.0
+0.5
+1.0
+X (D/入)Focal Plane - Sinusoidal Perturbation without Coronagraph
+1000
+15
+10
+800
+5.
+600
+(入/D)
+0
+y
+400
+-5 -
+200
+一10
+-15
+0
+-15
+一10
+-5
+0
+5
+10
+15
+X (入/D)12
+Lewis et. al.
+Figure 5. One frame of the input sequence (left) for the simple two-sinusoid test case with a coronagraph, with the residuals
+after PSF subtraction using mean-subtraction, KLIP, and stKLIP, showing a clear reduction in speckle intensity. Both stKLIP
+and baseline KLIP reduce image variance by a factor of at least 5.7 from the original image, an improvement over simple
+interventions like mean-subtraction. Although stKLIP does not improve upon KLIP in this limited test case, it is important
+to remember that we have not optimized for modes and lags in this scenario; this step was intended for troubleshooting, not
+rigorous characterization of the algorithm.
+els of Kolmogorov turbulence, and we use the simplest
+option of a single frozen flow layer. Then, MEDIS uses
+PROPER to propagate the light through the telescope un-
+der Fresnel diffraction, including both near- and far-field
+diffraction effects (Krist 2007). Separate wavefronts are
+propagated for each object in the field — the host star,
+and any companion planets.
+MEDIS also includes op-
+tions to introduce coronagraph optics, aberrations (like
+non-common path errors), and realistic detectors. MEDIS
+outputs the electric field or intensity at specified loca-
+tions in the optical chain, such as the pupil and focal
+planes in our case, as shown in Figure 6.
+Given the wide range of parameters available in MEDIS,
+we had to make decisions on what to use for the MEDIS
+simulations used to test stKLIP. For these simulations,
+we implement a telescope with 10 meter diameter, sim-
+ilar to the Keck Telescopes. We begin with a case with-
+out adaptive optics for simplicity. For this, the sampling
+rate needs to be a few milliseconds, a few times over-
+sampled compared to the smallest temporally resolvable
+features given the field-of-view (FOV) under considera-
+tion. The number of frames is chosen to create a total
+observation time of 30 seconds (6,000 frames at 0.005
+second sampling) to recreate a realistic observation and
+attain a sufficient number of independent samples. The
+grid size is significantly larger than the area of interest
+(256 × 256 pixels) to avoid edge effects. However, we
+choose a region size / FOV that is significantly smaller
+than our whole grid (100 × 100 pixels) to make this
+problem more computationally tractable.
+The simulation includes atmospheric parameters, such
+as the Fried Parameter (r0), a length scale for coherence
+in the atmosphere, and the structure constant (Cn), a
+description of turbulence strength over multiple atmo-
+Figure 6. Examples of MEDIS simulations. (Top) Pupil
+plane, illustrating the phase screen. (Bottom) Focal plane,
+with a clearly bright companion object. These simulations
+are used as a preliminary test of stKLIP’s efficacy and po-
+tential; however, there is a large parameter space to explore
+beyond the scope of this work.
+spheric layers. The atmospheric model we use is a sim-
+ple single layer of extremely mild Kolmogorov turbu-
+lence, with r0 > 10 m, since we want r0 ≫ D to stay in
+the high-contrast regime of small phase errors. Note:
+
+Original
+KLIP
+Mean Subtracted
+stKLIP
+9
+-9
+6 -
+9
+4 
+4 -
+4 -
+4 -
+2 
+2 -
+2 :
+(Λ/D)
+0-
+0-
+-0
+0:
+-2 
+-2 
+-2
+-2 
+-4 -
+-4 -
+-4
+-4 :
+-6 
+-6:
+-6 
+-6 
+-8 
+-8 
+-8 
+-8 
+0
+5
+0
+-5 
+-5
+-5 
+5
+-5
+0
+5
+5
+0
+X (入/D)
+X (Λ/D)
+X (Λ/D)
+X (入/D)Example MEDiS Pupil Plane
+140
+120 
+100
+(pixels)
+80
+60
+40 -
+20 -
+0
+0
+25
+50
+75
+100
+125
+x (pixels)Example MEDiS Focal Plane
+140
+120
+100
+(siaxid)
+80
+y
+60
+40
+20 -
+0
+0
+25
+50
+75
+100
+125
+x (pixels)13
+this simulated atmosphere is not realistic in ground-
+based imaging, but we chose these parameters to ap-
+proximate the high-contrast regime without simulating
+adaptive optics and introducing additional parameters.
+While our numerical experiments will depend on the in-
+put power spectrum, our primary aim was to assess the
+characteristics of a second-order statistical analysis of
+the linearized system (Equation 13), rather than im-
+pacts of the particulars of the wavefront error power
+spectrum.
+It is worth exploring how different atmo-
+spheric conditions (e.g. a smaller r0 value) would change
+the effectiveness of this method, but that is beyond the
+scope of this initial investigation.
+We choose a vortex coronagraph (Mawet et al. 2009),
+since it is the closest to an “ideal” coronagraph of the
+options available in MEDIS (e.g. closest to perfect can-
+cellation of the spatially coherent wave), thanks to its
+small inner working angle (Guyon et al. 2006). We want
+an ideal detector since, for this initial investigation, we
+are not yet interested in how detector noise/error affects
+this method. We also include one companion object that
+would be readily detectable given current capabilities
+(a contrast of 5 × 103), in order to enable SNR mea-
+surements of an injected companion for various post-
+processing methods including stKLIP. As mentioned in
+Section 2.2, lags should be chosen based on crossing
+times and relevant features. In these simulations, this
+ranges from 2 to 10 timesteps (0.01 to 0.05 seconds) for
+a wind speed of 5 m/s and 5 millisecond sampling. Fu-
+ture work should test a further range of lags, up to 400
+timesteps (2 seconds, or one full crossing time), but our
+current method is computationally limited as mentioned
+in Section 3.3. In this investigation, we also test a range
+of modes from 1 to 500.
+Although these simulations are computationally ex-
+pensive, MEDIS is capable of parallel processing, except
+in cases where AO parameters require serialization. We
+take advantage of this capability by using UCLA’s Hoff-
+man2 Cluster. The resultant data sets are quite large,
+and require inventive ways of computing the necessary
+statistics without loading the full array into memory, de-
+scribed further in Section 3.3. These simulations show
+us how realistic space-time covariance differs from the
+idealized case, and allow us to begin to test the effec-
+tiveness of our new method.
+Metrics of efficacy used in this study are measure-
+ments of variance, signal, noise, signal-to-noise ratios
+(SNR), and contrast curves.
+Variance is simply com-
+puted over the whole 100×100 pixel residual image us-
+ing numpy.var. Signal is computed using aperture pho-
+tometry (via photutils), centered on the simulated
+companion.
+Noise is similarly computed using aper-
+ture photometry by taking the standard deviation of
+a series of apertures in an annulus at the same separa-
+tion as the simulated companion. SNR is then the ratio
+of these two measurements. Contrast curves are esti-
+mated using aperture photometry at various distances
+from the image center and dividing by the aperture pho-
+tometry measurement of the unmasked (e.g. no coro-
+nagraph) peak, then adjusting by the signal through-
+put; the throughput here is estimated as the signal after
+processing divided by the signal before data processing.
+These various metrics are computed for the original im-
+ages, as well as different post-processing scenarios, to
+understand the relative efficacy of stKLIP. Results are
+described in Section 4.
+3.3. Iterative Statistics Calculations
+There are two key computational challenges for large
+data sets such as those produced by MEDIS: memory ac-
+cess and computational complexity.
+Simulations with
+MEDIS for a realistic observing sequence based on our
+criteria above can be on the order of 100GB, which
+can pose challenges to RAM-based manipulation for the
+calculation of mean and covariance given our current
+computing resources. To address this problem, we im-
+plemented the framework for iterative statistics calcula-
+tions set forth in Savransky (2015).
+In order to perform a KLIP-style calculation, we first
+need to compute second-order statistical quantities for
+a data set of n samples xi, such as the mean and covari-
+ance. The formula for the calculating mean is:
+µ ≡ 1
+n
+n
+�
+i=1
+xi
+(38)
+When the mean µ is estimated from the data, the sample
+covariance can be calculated as follows:
+C ≡
+1
+n − 1
+n
+�
+i=1
+(xi − µ)(xi − µ)T .
+(39)
+These sums can be broken up into smaller iterative
+steps k, to make the calculation less memory intensive.
+For each step k, the mean can be updated with the for-
+mula
+µk = (k − 1)µk−1 + xk
+k
+(40)
+and the covariance can be updated by
+Sk = k − 2
+k − 1Sk−1 +
+k
+(k − 1)2 (xk − µk)(xk − µk)T . (41)
+
+14
+Lewis et. al.
+However, Equation (41) is only applicable to the spa-
+tial covariance, e.g. a time lag of zero. The space-time
+covariance can be calculated as
+Sl =
+1
+n − l − 1
+n
+�
+i=1
+(xi − µ)(xi−l − µ)T .
+(42)
+Following a similar protocol to Savransky (2015), we
+derived an update formula for the space-time covariance:
+Sl =
+1
+n − l − 1
+�
+n
+�
+i=l
+xixT
+i−l − (n − l)µµT
++ µT
+l−1
+�
+i=1
+xi + µ
+n
+�
+i=n−l−1
+xT
+i − 2lµµT
+�
+(43)
+It is identical to Equation (41), except for the last 3
+additional cross-terms. These cross-terms were directly
+calculated and determined to be negligibly small as the
+sample size becomes large relevant to the maximum lag,
+and thus would only be relevant in edge cases. For 1,000
+samples, the error on the space-time covariance calcula-
+tion is on the order of 10−4% or less. For 10,000 samples,
+the error decreases to 10−6 to 10−7%, indicating a trend
+of decreasing error for an increasing number of samples.
+We do not plan to use fewer than 1,000 samples in a data
+set, so we consider this approximation to the space-time
+covariance acceptable and have implemented it for the
+tests described in Section 3.2.
+Although the mathematics laid out in this section
+make covariance calculations possible, the resulting co-
+variance matrices can be quite large, on the order of
+10GB for even short test cases with small FOVs. Even
+with sufficient RAM for manipulation, these large co-
+variance matrices can lead to long computation times
+for following steps of the algorithm. The image size and
+sequence length of data sets used in our stKLIP method
+is therefore still currently limited by memory require-
+ments and prohibitively long execution times. This is
+mostly due to the eigendecomposition calculations, since
+the full space-time covariance matrix needs to be loaded
+into memory for input into scipy.linalg.eigh. As we
+proceeded with larger data sets, we chose to perform a
+standard eigendecomposition with scipy.linalg.eigh
+using the default backend (C LAPACK evr) but limited
+the maximum number of eigenvalues/eigenvectors com-
+puted, since many of the smaller eigenvalues only cap-
+ture noise and are not necessary for this process. There
+may be more optimal choices for the eigendecomposition
+algorithm, but such optimization is left for future work.
+Another possible solution to mitigate this bottleneck
+would be using an iterative eigendecomposition. This
+could theoretically be done with the NIPALS (Nonlin-
+ear Iterative Partial Least Squares) algorithm (Risvik
+2007). However, applying the NIPALS algorithm is not
+straightforward for this problem; our space-time covari-
+ance matrix is currently assembled from various spa-
+tial covariance matrices, and considerable changes would
+need to be made to NIPALS to accommodate a space-
+time calculation instead of a solely spatial one, since
+the NIPALS algorithm relies on a data matrix as in-
+put instead of a covariance matrix. Future iterations of
+this algorithm could also make use of the dask package
+for parallelization of computations to help speed up run
+time, but as of this writing an eigendecomposition func-
+tion (e.g. dask.linalg.eigh) was not yet implemented,
+although the similar dask.linalg.svd function could
+possibly be used. We leave such improvements in effi-
+ciency for future work.
+4. ALGORITHM PERFORMANCE ON
+SIMULATED AO RESIDUAL DATA
+We have confirmed through theory (§2.2) and simula-
+tion (§3.1) that space-time covariances exist for speckles
+in a simple high-contrast imaging system in the regime
+of small phase errors and short exposures. In Section
+2.3, we defined a new algorithm, similar to Karhunen-
+Loe´ve Image Processing, to take advantage of space-time
+covariances and improve final image contrast, with the
+eventual goal of detecting fainter companion objects. As
+shown in Section 3.2, we have developed an initial imple-
+mentation of this space-time KLIP (stKLIP) algorithm,
+and demonstrated it on simulated data. In this section,
+we present the results of those demonstrations.
+It is
+worth noting that these tests on simulated data only
+explore a small range of parameter space, and are not
+indicative of the absolute potential of using space-time
+covariance in data processing. Instead, we present this
+as a first proof-of-concept for the possibility of this new
+method.
+An example of the images input to and output by
+the stKLIP processing algorithm is shown in Figure 7,
+along with a comparison to two other data processing
+interventions, mean-subtraction (as in Equation 3) and
+KLIP. For this simulated data, mean-subtraction makes
+such a slight improvement that in the following figures
+we omit it from comparison plots, as it would be almost
+precisely coincident with the original image’s metrics.
+To quantitatively measure the efficacy of our stKLIP
+data processing algorithm, we computed total image
+variance, signal-to-noise ratios, and approximate con-
+trast curves, as described in Section 3. To further de-
+termine the utility of this algorithm and characterize
+its dependence on the tuneable parameters, we also in-
+
+15
+Figure 7. One frame of the input sequence (left) from MEDIS, with the residuals after PSF subtraction using mean-subtraction,
+KLIP, and stKLIP. Both stKLIP and baseline KLIP reduce image variance by a factor of ∼1.85 from the original image for the
+listed case of 10 modes and 2 timesteps lag in stKLIP.
+vestigated the relationships between these efficacy met-
+rics, the number of KL modes used, and the number of
+stKLIP lags used. We leave adjustments of the resid-
+ual wavefront error statistics and companion location,
+among other parameters, and their effects on stKLIP’s
+efficacy for future work.
+Image variance is a primary metric for subtraction ef-
+ficiency. Total image variance is reduced by almost half
+for both spatial / baseline KLIP and stKLIP within the
+first 10 modes, and variance approaches 0 around 50
+modes. In this test, spatial and stKLIP are similar in
+their variance reduction abilities, and are both improve-
+ments on mean-subtraction and the original image. Im-
+age variance drops off steeply within the first 20 modes,
+indicating that most of the power is removed with only
+a few eigenimages required in the reconstruction. Given
+that only a small number of modes are required to re-
+move the majority of the variance in the image, future
+applications of this algorithm could exploit this fact to
+reduce the computational burden by only calculating the
+first n eigenvalues/eigenimages.
+For both KLIP and stKLIP on these simulated data,
+signal starts to be lost around 5–10 modes and drops off
+more steeply after ∼30 modes. Space-time KLIP with
+4, 5, 6, or 8 lags in this scenario shows a slight edge over
+baseline KLIP in signal retention, as shown in Figure
+8. It is worth noting that the choice of optimal num-
+ber of lags depends on the wind speed and region in
+the image that we are most interested in. Recall from
+Section 3.2 that this test uses v = 5 m/s, and the com-
+panion location can be seen in Figure 7. Noise reduction
+capabilities appear very similar between KLIP and stK-
+LIP; after about 40–50 modes, so much of the image
+has been removed that noise approaches zero and shows
+small random fluctuations, indicating that these higher
+modes contain less information.
+Figure 8. Companion signal over number of KL modes used
+in the model PSF subtraction; this figure shows that signal
+loss begins around 5 modes, indicating that future iterations
+of this algorithm would benefit heavily from implementing
+measures to prevent self-subtraction. Certain choices of lag
+(4, 5, 6, 8) show a minor improvement in signal retention
+over spatial (lag = 0) KLIP.
+Signal-to-noise ratio (SNR) shows a 10–20% improve-
+ment over the original image within the first 40 modes,
+as shown in Figure 9. The 2nd peak in Figure 9 is pos-
+sibly due to small number statistics (most of the signal
+has been removed by then) and not a real SNR improve-
+ment. It is worth noting that the SNR shown here could
+improve significantly if a method is implemented to re-
+tain signal and improve throughput, which we discuss
+more in the following section. We again see that there is
+a slight advantage for certain lags over spatial (lag=0)
+KLIP on the order of a few percent, indicating that there
+is possibility for properly tuned stKLIP to outperform
+KLIP.
+Contrast curves (as shown in Figure 10) similarly show
+potential for up to 50% improvement depending on the
+number of modes, lags, and region of the image. Within
+20 pixels, we see potential for up to 400% improvement,
+but with the caveat that this close to the coronagraphic
+
+Original
+Mean Subtracted
+KLIP, 10 modes
+stKLIP, 2 lags/10 modes
+0-
+0 -
+0 -
+20 -
+20 -
+20
+20 -
+40 -
+40 
+40
+40-
+Pixels
+0
+60
+60 -
+60
+60.
+80 -
+80
+80
+80
+0
+20
+40 
+20
+0
+80
+20
+60
+80
+40
+60
+80
+20
+40
+60
+0
+40
+60
+80
+0
+Pixels
+Pixels
+Pixels
+Pixels2.550
+KLIP
+2.548
+1lags
+2 lags
+3 lags
+2.546
+4 lags
+5 lags
+2.544
+6 lags
+8 lags
+10 lags
+2.542
+Original
+2.540
+0
+2
+4
+6
+8
+10
+Number of KL Modes16
+Lewis et. al.
+Figure 9. Companion signal-to-noise ratio (SNR) compared
+to the original image SNR over number of KL modes used in
+the model PSF subtraction; this figure shows a 10–20% im-
+provement over the original image using stKLIP and KLIP,
+with stKLIP having a slight edge (on the order of a few per-
+cent) for certain choices of lag.
+mask, measurements of SNR and contrast are less reli-
+able. A slight spread in the contrast curves for various
+lags, such as that seen around 30–40 pixels for 5 modes
+in Figure 10, indicates that it is necessary to strategi-
+cally choose the number of lags used in stKLIP depend-
+ing on the image region in which we want to optimize
+contrast. We will discuss these results and future work
+further in the following section.
+5. DISCUSSION
+Overall, our tests on simulated data (Section 4) show
+that there is a demonstrated contrast gain (or equiva-
+lently, SNR improvement) of at least 10–20% from the
+original image using stKLIP with fewer than 40 modes.
+There is also evidence that stKLIP provides a slight ad-
+vantage over spatial-only KLIP for certain choices of
+number of lags, number of modes, and location in im-
+age. However, the real potential for this method will
+be unlocked when the technique is safeguarded against
+self-subtraction and demonstrated on real data.
+In this section, we first discuss how well the signal is
+retained for this new algorithm, and possibilities for fu-
+ture improvements to better avoid self-subtraction and
+retain signal in Section 5.1. Next, we discuss the rela-
+tionship between the lag parameter and the optimized
+region of the target image in Section 5.2. Then we con-
+sider the addition of quasi-static speckles to our cur-
+rently idealized, only atmospheric speckle regime in Sec-
+tion 5.3.
+Lastly, we propose other considerations for
+future work and implementations of this algorithm in
+Section 5.4.
+5.1. Signal Retention
+The signal clearly decreases beyond ∼5 KL modes as
+shown in Figure 8, indicating that we are not only sub-
+tracting from the noise but also the companion (known
+as self-subtraction). If we can find a way to reduce this
+self-subtraction and retain signal, we could potentially
+further improve the contrast gain. This could possibly
+be accomplished by masking the location of the planet
+or excluding regions with high spatial covariance but low
+temporal covariance, but further development is needed
+to enable this functionality. Depending on the masking
+implementation, this data processing method could be
+used for blind searches or characterization observations.
+In fact, it may be particularly suited to characterization
+observations due to the dependence on a specific image
+region from the nature of atmospheric speckles.
+Based on previous work on LOCI (Locally Optimized
+Combination of Images) (Lafreniere et al. 2007; Marois
+et al. 2014; Thompson & Marois 2021), we can expect
+additional contrast gains once masking is implemented.
+Additionally, there are other techniques used for KLIP
+to differentiate between signal and speckles, such as an-
+gular differential imaging (ADI, Marois et al. (2006a)),
+spectral differential imaging (SDI, Marois et al. (2005)),
+and reference differential imaging (RDI, Marois et al.
+(2003)). Similar efforts to increase the distance between
+the signal and the noise in the eigenimages may be useful
+for stKLIP.
+Additionally, when the number of lags is zero, stK-
+LIP simply reduces to baseline KLIP (Soummer et al.
+2012) as mentioned in Section 2.3, and we have included
+spatial-only/baseline KLIP as a comparison for stKLIP
+in our analyses. It is worth noting, however, that KLIP
+is typically used on long-exposure images, a different
+regime than that for which stKLIP is useful. Addition-
+ally, we are comparing stKLIP to KLIP with no self-
+subtraction mitigation. Most current implementations
+of KLIP, such as pyKLIP (Wang et al. 2015), do have
+some sort of self-subtraction mitigation or method to
+increase spatial diversity implemented, such as forward
+modeling, angular differential imaging, or spectral dif-
+ferential imaging (Pueyo 2016; Marois et al. 2006b; Vi-
+gan et al. 2010).
+Therefore, in practice, KLIP would
+currently have a significant advantage over stKLIP as
+implemented in this work.
+However, future work can
+adapt many of the existing methods and techniques from
+KLIP to improve the implementation of stKLIP and its
+resulting performance.
+5.2. Optimization for Lags and Image Region
+Despite KLIP’s apparent advantages, it appears that,
+depending on the number of lags used and the location
+in the image, stKLIP can outperform KLIP by a few
+percent without self-subtraction implemented for either
+case as is done in our test. This is evident in Figure
+
+1.25
+KLIP
+1.20
+1 lags
+SNR Improvement
+2 lags
+1.15
+3 lags
+4 lags
+5 lags
+1.10
+6 lags
+8 lags
+1.05
+10 lags
+1.00
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Number of KL Modes17
+Figure 10. Contrast curves, as well as contrast improvement (a comparison to the original image’s contrast curve), for three
+cases of KL modes: 5, 7, and 20. Each shows results for the image processed with baseline KLIP (0 lags) as well as stKLIP with
+a variety of lags. stKLIP is consistent with KLIP improvements, and in certain regions may show improvements depending on
+number of lags used.
+10, showing detail of the region with highest contrast
+gain (other than near the central mask). The region of
+highest contrast gain will vary depending on the chosen
+lag as well as the atmospheric conditions creating the
+speckles in question. Optimization of input parameters
+is a notoriously tricky problem for KLIP (Adams et al.
+2021), and it appears stKLIP is subject to the same
+challenges.
+The variation of optimal lag and image region is due
+to the relationship between the wind speed and spatial
+frequency, since wind speed and telescope diameter com-
+bine to determine the crossing time for one cycle of the
+spatial frequency as tcross = dtelescope/vwind. Spatial fre-
+quency in the pupil then corresponds to a location in the
+image plane. The effect of atmospheric parameters on
+speckle properties is further quantified in Guyon (2005)
+and speckle lifetimes are observed on shorter scales in
+Goebel et al. (2018). Empirical investigations of tele-
+scope telemetry and ambient weather conditions are also
+an ongoing area of study, especially with regards to pre-
+dictive control (Guyon et al. 2019; Rudy et al. 2014;
+Hafeez et al. 2021), but that information may addi-
+tionally be useful in determining optimal parameters for
+stKLIP on-sky. Additionally, using this information on
+the temporal/spatial locations of strongest correlations,
+it may be possible to reduce the matrix size or use only
+the most correlated images such as in T-LOCI (Marois
+et al. 2013).
+For this work, we have been operating in the regime
+of milliseconds to track atmospheric speckle motions.
+However, in practice, the full 3D space-time correlation
+matrix will have power on multiple timescales, from that
+of atmospheric speckles to quasi-static speckles.
+It is
+outside the scope of this work to fully explore how space-
+time KLIP could be applied on multiple time domains,
+and there is additionally the caveat that computational
+complexity grows with longer timescales than those we
+have applied here.
+5.3. Including Quasi-Static Speckles
+As mentioned in Section 1, the scenario we have in-
+vestigated is an idealized case — one in which quasi-
+static speckles are absent and our images are dominated
+entirely by atmospheric speckles. We are also working
+on short timescales, where the atmosphere is frozen at
+each time step.
+There is a timescale over which the
+intensity changes, which we are observing in this sce-
+nario, but there is also a timescale for changes in the
+electric field’s phase. These phase changes will only re-
+sult in changes in intensity if superimposed onto a con-
+stant electric field, such as the case of non-coronagraphic
+imaging, or when quasi-static speckles are significant
+(C(x) in Equation 15). This is another regime in which
+to explore algorithm performance, wherein quasi-static
+and atmospheric speckles co-exist and interact, possibly
+even changing the speckle lifetimes (Soummer & Aime
+2004; Fitzgerald & Graham 2006; Bloemhof et al. 2001;
+Soummer & Aime 2004). In this regime, there will likely
+be additional space-time variation as “pinned” speckles
+oscillate. Given that the presence of quasi-static speck-
+les will make visible the additional space-time variations
+in phase, it is possible that stKLIP will operate even
+more effectively with this additional information to ex-
+ploit. However, additional quasi-static speckles will lead
+
+5 KL Modes
+7 KL Modes
+20 KL Modes
+2× 10-4
+2 ×10-
+2 × 10-
+10-4.
+10-4
+rast
+6×10-5.
+6×10-5
+Cor
+4 × 10-5,
+4×10-5.
+4×10-5,
+3 × 10-5
+3 × 10-5
+3×10-5
+2 ×10-
+2×10-
+4.0
+4.0
+3.5
+3.5
+KLIP
+1 lags
+3.0
+3.0
+2 lags
+3 lags
+2.5
+2.5
+2.5
+4 lags
+ 5 lags
+6 lags
+2.0
+2.0
+8 lags
+10 lags
+1.5
+1.5
+1.0
+1.0
+20
+30
+40
+50
+60
+20 
+50
+60 
+20
+30 
+40
+50
+40
+60
+Pixels from Center18
+Lewis et. al.
+to additional photon noise, which may counteract any
+theoretical contrast gains from including phase infor-
+mation. (Note: recent work from Mullen et al. (2019)
+shows that using KLIP on shorter exposures may even
+help remove quasi-static speckles more effectively, fur-
+ther bolstering the case for the stKLIP’s effectiveness
+in this regime.)
+Additionally, the presence of atmo-
+spheric residuals could even provide information about
+the phase of quasi-static speckles, allowing them to be
+effectively nulled with a deformable mirror (Frazin 2014;
+Frazin & Rodack 2021). Future simulations may explore
+this regime and determine if additional contrast gain is
+possible.
+5.4. Considerations for Future Work
+In this idealized test case, we also chose not to simu-
+late adaptive optics corrections, instead leaving an inves-
+tigation of how AO parameters affect space-time corre-
+lations and the resulting stKLIP processing for a future
+investigation. Since AO suppresses low frequencies and
+heaves high frequencies unchanged, although our total
+error is on par with an AO residual scenario, the overall
+shape of the power spectrum would be different. This
+would likely lead to weaker temporal correlations with
+AO. Previous work also shows that AO corrections do
+affect the lifetime of speckles (Males et al. 2021; Males
+& Guyon 2017), so this will be an important factor to
+consider in future work.
+Currently, we have yet to demonstrate the full po-
+tential of this algorithm, in part due to the high com-
+putational costs.
+To run stKLIP on a 100×100 pixel
+window of a simulated 30-second data set (with the pa-
+rameters specified in Section 4) over a range of KLIP
+parameters, we required 128GB of RAM and approxi-
+mately 400 hours of computation time. The high mem-
+ory requirement is due to the eigendecomposition, since
+the space-time covariance matrix can become extremely
+large when including a large number of lags and must
+be loaded in fully to the eigendecomposition. As men-
+tioned in Section 3.3, there are possible solutions to this
+challenge to reduce computational costs in less mem-
+ory intensive implementations, or even analytical gains
+in efficiency that exploit symmetries inherent in the co-
+variance matrix (shown in Figure 1) or focus on only
+the strongest correlations depending on the temporal
+and spatial scales of interest, but those are beyond the
+scope of this paper.
+It may also be possible to reduce the number of eigen-
+values/modes computed, which will reduce computation
+time and possibly memory consumption as well, given
+that we now know that values beyond ∼50 KL modes
+aren’t of much use in our tested scenario, but the ex-
+act threshold will be dependent on the region of interest
+and number of lags used, among other factors. In future
+iterations, this code could also likely be improved by im-
+plementing this algorithm more optimally rather than in
+a high-level language, as the current implementation is
+in Python, and by using parallel processing.
+6. CONCLUSION
+Evolving atmospheric layers lead to time-varying
+speckles in the focal plane of an imaging system; for
+the high-contrast imaging regime, we have shown that
+spatio-temporal covariances in these speckles exist, and
+can be exploited for use in data processing to improve
+contrast.
+Our data processing tool has been imple-
+mented in Python, tested on a simple analytic test case
+to prove viability, and also tested on realistic simula-
+tions to understand the effectiveness of this technique.
+We have shown there is potential for a contrast gain (or
+equivalently, SNR improvement) of at least 10–20% from
+the original image, with significant potential for an even
+larger gain if self-subtraction is adequately addressed.
+Additionally, we have shown evidence that the space-
+time nature of our algorithm, in its current form, may
+provide a slight advantage over spatial-only KLIP in
+certain cases, with significant potential for stronger im-
+provement under different conditions and with improve-
+ments to the algorithm implementation. Although the
+SNR gains for this new method aren’t fully developed,
+this initial work on space-time KLIP opens the door for
+the use of space-time covariances in high-contrast imag-
+ing, especially in the short timescale regime of atmo-
+spheric speckle lifetimes.
+Future work can use our data processing tool to fur-
+ther explore the dependence of the space-time covari-
+ances and the resulting contrast improvements on var-
+ious parameters, such as the type of coronagraph, AO
+performance, strength of quasi-static speckles, and at-
+mospheric conditions.
+It would be particularly inter-
+esting to determine how AO affects these covariances,
+since AO is important in a realistic scenario for exo-
+planet imaging and affects the resulting speckle lifetimes
+and structures.
+Future implementations of this algorithm will also
+need to consider how to minimize self-subtraction of the
+companion object, and overcome the memory and com-
+putational demands in the eigendecomposition. Further
+optimization of the tunable parameters is also necessary
+to optimize algorithm performance and implement this
+as a refined tool for exoplanet imaging. It would also
+be interesting to apply this tool to on-sky data, such as
+that from MEC on SCExAO at Subaru (Walter et al.
+2020, 2018; Jovanovic et al. 2015; Minowa et al. 2010),
+
+19
+to determine potential on-sky contrast gains from this
+technique. Although this current work focuses on the
+use of speckle space-time covariances in post-processing,
+these covariances could even be used in real-time predic-
+tive control (Guyon et al. 2018). Overall, the results in
+this work show that harnessing space-time covariances
+through “space-time KLIP” may be a promising tech-
+nique to add to our toolkit for suppressing speckle noise
+in exoplanet imaging while retaining signal throughput.
+ACKNOWLEDGMENTS
+This work used computational and storage services as-
+sociated with the Hoffman2 Shared Cluster provided by
+UCLA Institute for Digital Research and Education’s
+Research Technology Group. This work was supported
+in part by National Science Foundation award num-
+ber 1710514 and by Heising-Simons Foundation award
+number 2020-1821. This material is based upon work
+supported by the National Science Foundation Grad-
+uate Research Fellowship under Grants No.
+2016-21
+DGE-1650604 and 2021-25 DGE-2034835. Rupert Dod-
+kins is supported by the National Science Foundation
+award number 1710385. Kristina K. Davis is supported
+by an National Science Foundation Astronomy and As-
+trophysics Postdoctoral Fellowship under award AST-
+1801983.
+Any opinions, findings, and conclusions or
+recommendations expressed in this material are those of
+the authors(s) and do not necessarily reflect the views
+of the National Science Foundation. Thanks to Marcos
+M. Flores and Joseph Marcinik for helpful discussions
+on notation and LaTeX.
+Software:
+NumPy (van der Walt et al. 2011),
+IPython (Perez & Granger 2007), Jupyter Notebooks
+(Kluyver et al. 2016), Matplotlib (Hunter 2007), Astropy
+(Astropy Collaboration et al. 2013; Price-Whelan et al.
+2018), SciPy (Jones et al. 2001), h5py (Collette 2013),
+MEDIS (Dodkins 2018; Dodkins et al. 2020), Dask (Dask
+Development Team 2016; Rocklin 2015)
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+
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+Conference Series, Vol. 11763, Society of Photo-Optical
+Instrumentation Engineers (SPIE) Conference Series,
+117631J, doi: 10.1117/12.2586285
+
+23
+APPENDIX
+A. NOTATION GLOSSARY – SECTIONS 2.1 & 2.3
+Symbol
+Definition
+Iψ(k)
+Stellar PSF, as in Soummer et al. (2012)
+k
+Pixel index
+T(k), t
+Target image as in Soummer et al. (2012), and as in
+this work unrolled to 1-d and represented as a vector
+A(k), a
+Faint astronomical signal (as above)
+ϵ
+True/false binary parameter
+ˆIψ(k), ˆ
+ψ
+Approximated PSF as in Soummer et al. (2012) and
+as a vector in this work, respectively
+R
+Matrix of reference images before mean subtraction
+r
+Individual reference image
+X
+Mean image from reference set
+M
+Mean subtracted reference images
+ni
+Number of images in reference set
+np
+Pixel count nx × ny
+nx
+Dimension 1 size
+ny
+Dimension 2 size
+nm
+Number of modes / eigenvectors chosen
+i
+Used as an arbitrary index
+j
+Used as an arbitrary index
+C
+Covariance matrix
+λ, λ
+Vector of eigenvalues, eigenvalue
+V
+Matrix of eigenvectors/eigenimages
+v
+Eigenvector
+q
+Vector of coefficients
+S, s
+Mean subtracted image sequences (in matrix and
+vector form)
+ˆs
+Reconstructed image sequence
+ns
+Number of images in sequence
+L
+Largest number of timesteps/lags in use as measured
+from the central image
+nl
+Total number of timesteps/lags used, equal to ns
+rk,avg
+Averaged residual from stKLIP
+
+24
+Lewis et. al.
+B. NOTATION GLOSSARY – SECTION 2.2
+Symbol
+Definition
+I
+Intensity
+x
+Location in image plane
+u
+Location in pupil plane
+t
+Time
+τ
+Time step
+Ψpup
+Pupil amplitude
+Ψfoc
+Focal amplitude
+ψ(u, t)
+Pupil phase
+P(u)
+Pupil function
+C(x)
+Spatially coherent wavefront
+Sφ(x, t)
+Phase aberrations
+ξ
+Displacement in pupil
+Bφ
+phase covariance function
+vwind
+Wind velocity
+

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@@ -0,0 +1,2210 @@
+Optimal coverage-based placement of static leak
+detection devices for pipeline water supply
+networks
+V´ıctor Blancoa,b and Miguel Mart´ınez-Ant´ona,b
+aInstitute of Mathematics (IMAG), Universidad de Granada
+b Dpt. Quant. Methods for Economics & Business, Universidad de Granada
+Abstract. In this paper we provide a mathematical optimization based
+framework to determine the location of leak detection devices along a
+network. Assuming that the devices are endowed with a given coverage
+area, we analyze two different models. The first model aims to minimize
+the number of devices to be located in order to (fully or partially) cover
+the volume of the network. In the second model, the number of devices
+is given, and the goal is to locate them to provide maximal volume
+coverage. In our models it is not assumed that the devices are located
+in the network (nodes or edges) but in the entire space, which allow to
+more flexible coverage. We report the results of applying our models to
+real-world water supply pipeline urban network, supporting the validity
+of our models.
+1. Introduction
+The design of leak detection systems on water supply networks has at-
+tracted a great interest due to the economic and environmental impact as-
+sociated to the the systematic lost of this resource. Needless to say the im-
+portant role water has in our social and economic system, as in agriculture,
+manufacturing, production of electricity, and to keep humanity healthy. On
+urban networks, were the supply pipelines network is buried, periodically
+lose an average of 20% to 30% of supply water El-Zahab and Zayed [2019].
+This average could rise above 50% in those places less technologically devel-
+oped in which a precarious maintenance makes the system more vulnerable.
+The 70% of the amount of water wasted is due to losses provoked by leaks
+in modern networks El-Zahab and Zayed [2019]. Pipe internal roughness or
+friction factors due to are the main causes of leakage of a water pipeline net-
+work [Walski, 1987, El-Abbasy et al., 2014], and as the pipelines get older,
+they become more susceptible to damage. In developed countries, yearly
+outlays for water leaks in their supply pipelines networks it is expected that
+are close to 10 billion USD of which 2 billion USD would be designated to
+Date: January 13, 2023.
+Key words and phrases. Facility Location, Leak Detection, Coverage Problems, Mixed
+Integer Non Linear Programming, Water Supply Networks.
+1
+arXiv:2301.04707v1  [math.OC]  11 Jan 2023
+
+2
+V. BLANCO and M. MART´INEZ-ANT´ON
+loss water damage cost and 8 billion USD would be devoted to social effect
+cost. Moreover, the International Water Management Institute forecast that
+33% of world population will experience water scarcity by 2025 Seckler et al.
+[1998]. Thus, the efficient management of water supplies should be one of
+the major concerns of water authorities around the world.
+Most efforts concerning the management of water supply networks have
+been focused in the detection of leaks once they occur. The leak location
+is crucial in order to minimize the impact of leaks when occurring. Hamil-
+ton [2009] suggests three different phases in the leak detection problem:
+localization, location and pinpointing. In the localization phase, the goal is
+to detect whether a leak occurred within a given segment of the network
+after the suspicion of a leak. There are several proposed methodologies El-
+Abbasy et al. [2016], Li et al. [2011] where Data Science plays an important
+role, as in the estimation of leak probabilities or supervised classification
+of the event leak/no leak based on historic leakage data. In the location
+phase, the uncertain area where the leak is localized is narrowed to ∼ 30
+cm. Finally, in the pinpoint phase, the exact position of the leak is to be
+determined with a pre-specified accuracy of ∼ 20 cm by using hydrophones
+and/or geophones [Fantozzi et al., 2009, Royal et al., 2011]. Previously to
+the determination of the position of the leak, a vast amount of literature
+have being dedicated to modeling the occurrence of a leak in such a way
+that when a peak in the sound signal alerts about a possible leak, it has
+to be accurately determined if the leak does or does not occur Cody et al.
+[2020a,b].
+Another research line when analyzing leakages in pipeline water networks
+is based on designing control strategies to more accurately and quickly de-
+tect them when they occur. This is the case of the design of devices that
+accurately detect the leak within a restricted area Khulief et al. [2012].
+Nevertheless, these devices are expensive and the placement of the available
+units should be strategically determined. One of the most popular approach
+is by partitioning the network in district metered areas where the flow and
+the pressure are monitorized (leaks can be detected by an increase of flow
+and a decrease the pressure) by means of leak-detection devices at each of
+these areas [see e.g. Puust et al., 2010]. Nevertheless, one still has to decide
+the number of devices and their positions at each of the district metered
+areas.
+There are different types of devices designed to contribute to any of the
+leak detection phases which can be classified into static and dynamic de-
+vices. Static devices, as sensors or data loggers, are usually located over
+the network, at utility holes or directly on-the-ground, they keep a data
+transmission flow with a central server to detect and localize a leak. In con-
+trast, dynamic devices are portable and used in the location and pinpoint-
+ing phases on more specific areas where the leak was suspected to occur.
+Whereas static devices can be automated, dynamic ones must be controlled
+
+Location of Leak Detection Devices
+3
+on-site by humans. Different technologies have been designed for the two
+different types of devices [see e.g. Li et al., 2015, for further details].
+Most of the research on static leak detection systems is focused on the ad-
+equate estimation of the signals transmitted from the devices to the central
+server to detect an actual leak Mohamed et al. [2012], Tijani et al. [2022].
+A few works analyze the optimal placement of a given number of static de-
+vices on a finite number of potential placements based on the capability of
+each of the potential places to detect a leak Venkateswaran et al. [2018],
+or in the use of historic data to place the devices at the more convenient
+places Casillas et al. [2013].
+This paper provides a technological decision support tool to help in the
+design of leak detection systems via the optimal placement of static devices.
+We assume that, instead of assuming that the devices are to be placed in a
+finite set of pre-specified potential places, they can be located in the whole
+space where the network lives, i.e. in the whole town or city. We analyze,
+in this framework, two different strategies to place the devices. On the one
+hand, we derive a method to find the smallest number of devices (and their
+placements) needed to be able to detect any leak in the network.
+Since
+the devices may be costly, and tons of then can be needed to cover the
+whole network, we also derive a method, that fix the number of devices to
+be located based on a budget and find their optimal placements to reach as
+much volume of the network as possible.
+The models that we propose belong to the family of Continuous Covering
+Location Problems. The main characteristic of these problem is that one
+or more services must be located, each of them endowed with a coverage
+area, i.e., a limited region where the service/signal can be provided. Cov-
+ering Location problems are usually classified into (Partial) Set Covering
+Location Problems ((P)SCLP) and Maximal Coverage Location Problems
+(MCLP). The goal of the (P)SCLP is to determine the minimum number of
+services (or equivalently the minimum set-up cost for them) to cover (part
+of) a given demand (usually a finite set if users/demand points), whereas in
+MCLP the number of services is given and the goal is to place them to cover
+as much demand as possible. These problems have been widely studied in
+the literature in case the given demand points to cover are finite and planar
+and the coverage areas are Euclidean disks [see Garc´ıa and Mar´ın, 2015,
+for further information on this problems]. Several extensions of these prob-
+lems have been studied, by imposing connectivity between the services in
+higher dimensional spaces and different coverage areas Blanco and G´azquez
+[2021], multiple types of services Blanco et al. [2022], under uncertainty Hos-
+seininezhad et al. [2013], regional demand Blanquero et al. [2016], or with
+ellipsoidal coverage areas Tedeschi and Andretta [2021].
+We provide versions of the PSCLP and the MCLP, where instead of cov-
+ering single points, the goal is to cover lengths/volumes of a spacial network,
+which may represent the water supply pipeline network whereas the services
+to be located model the devices to detect leaks. The goal is either to find
+
+4
+V. BLANCO and M. MART´INEZ-ANT´ON
+the number of devices and its optimal placement to fully or partially cover
+the whole length of the network (in the case of the PSCLP) or to find the
+placements of a given number of devices to maximize the length of the net-
+work which is covered by the devices. We assume that the coverage areas of
+the devices are ℓτ-norm based balls and that covering a part of the network
+with those shapes implies that the device is able to detect a leak there.
+The rest of the paper is organized as follows. In section 2 we introduce
+the problem under analysis and illustrate some of the solutions that can be
+obtained. Section 3 is devoted to analyze the problem of locating a single
+device, which will be useful for the development of approximation algorithms
+for the multi-device case. In section 4 the general case is analyzed. We
+provide mixed-integer non linear programming formulations for the maximal
+and partial set covering location problems. In Subsection 4.1 two different
+math-heuristic approaches are developed for the problem. The results of
+our computational experiments on real-world urban pipeline networks are
+reported in Section 5. Finally, in Section 6 we draw some conclusions and
+future research lines on the topic.
+2. Length-coverage location of devices
+In this section we detail the problem under study and fix the notation for
+the rest of the sections.
+Let G = (V, E; Ω) be an undirected network with set of nodes V , set
+of edges E and non-negative edge weights Ω. The weights may represent
+the diameter or roughness of a pipeline, that together with its length will
+allows us to compute the covered volume of the network. We assume that
+the graph is embedded in Rd, i.e., V ⊆ Rd and each edge e = {oe, fe} ∈ E
+can be identified with a segment in Rd, with endnodes oe and fe in V .
+Although the edges are undirected, we will call oe as the origin and fe as
+the destination, being its choice arbitrary in our developments. Abusing of
+notation, we will identify the edge e ∈ E with the segment induced by its
+end nodes, i.e., e ≡ [oe, fe].
+A device located at X ∈ Rd is endowed with a ball-shaped coverage area
+in the form:
+BR(X) = {z ∈ Rd : ∥X − z∥ ≤ R}
+where R > 0 is the given coverage radius. We assume that ∥·∥ is an ℓτ-based
+norm with τ ≥ 1 or a polyhedral norm.
+For each edge e ∈ E, and a finite set of positions for the devices X ⊂ Rd,
+we denote by CovWLengthG(e, X) the weighted length of the edge covered
+by the devices. Let us denote by TotWLengthG the total weighted length
+of the network, i.e., TotWLengthG =
+�
+e∈E
+ωe∥oe − fe∥ with ωe ∈ Ω.
+We analyze in this paper two covering location problems for leak de-
+tection devices, the partial set network length covering location problem
+(PSNLCLP) and the maximal network length covering location problem
+
+Location of Leak Detection Devices
+5
+Figure 1. Pipeline urban network of Example 1.
+(MNLCLP). In both cases, the goal is to find the position of different types
+of devices in order to cover all or part of the given network.
+Partial Set Network Length Covering Location Problem (PSNLCLP):
+The goal of this problem is to determine the minimum number of
+devices and their positions in Rd in order to cover at least 100γ% of
+the weighted length of the network, for a given γ ∈ (0, 1].
+The PSNLCLP can be mathematically stated as:
+min
+X⊆Rd:
+�
+e∈E CovWLengthG(e,X)≥γTotWLengthG
+|X|
+Maximal Network Length Covering Location Problem (MNLCLP):
+In this problem the number of devices to locate is given, p ≥ 1, and
+the goal is to find their positions to maximize the weighted covered
+length of the network. While the MNLCLP consists of solving
+max
+X⊆Rd:
+|X|=p
+�
+e∈E
+CovWLengthG(e, X)
+In the following example we illustrate the two problems described above
+analyzed in a real network (see Section 5).
+Example 1. We are given the network drawn in Figure 1. There, each
+edges has a different weight indicating the diameter of the pipeline (as larger
+the weight thicker the line in the picture). A set of five devices with identical
+Euclidean disks coverage areas of radius 0.5 is to be located (the network has
+been adequately scaled to the unit square). In Figure 2 we show the solutions
+of the PSNLCLP for γ = 0.75 (right) and the solution of MNLCLP for p = 5
+(left). There, the centers are higlighted as red stars, the covered segments
+of the network are coloured in blue, and the coverage of the devices are the
+disks.
+Example 2. The models under analysis are defined in a very general frame-
+work in d-dimensional spaces, networks with no further assumptions, and
+
+6
+V. BLANCO and M. MART´INEZ-ANT´ON
+Figure 2. Solutions of MNLCLP (p = 5) and PSNLCLP
+(γ = 0.75) of the network of Example 1.
+general coverage shapes. In Figure 3 we show solutions for the MNLCLP
+with p = 5 obtained in case the coverage areas are induced by ℓ1-norm (left)
+and ℓ∞-norm (right) balls.
+Figure 3. Solutions of MNLCLP with p = 5 for coverage
+areas defined by ℓ1-norm (left) and ℓ∞-norm (right) balls.
+Remark 3. Most covering location problems on networks assume that the
+centers must be located either on the edges or the nodes of the network [Berman
+and Wang, 2011, Berman et al., 2016, see e.g.]. In the problems that we an-
+alyze this condition is no longer assumed, allowing the centers to be located
+at any place in the space where the network lives. This flexibility allows to
+find better positions for the devices implying, in general, a larger coverage of
+the network. In Figure 4 we show the solutions of the edge-restricted (left)
+and node-restricted (right) versions of the MNLCLP, where one can observe
+that the geometrical positions of the devices is different than those obtained
+for our problem.
+Furthermore, we compare the covered lenghts of the three problems (MNL-
+CLP, edge-restricted MNLCLP, and node-restricted MNLCLP) for different
+values of p (2, 5, and 8), and different radii R (0.1, 0.25, and 0.5). In Figure
+5 we show a bars diagram with the average deviations (for each p) of the two
+restricted version with respect to the covered length of the general approach
+that we propose. As can be observed, the solutions of the unrestricted MNL-
+CLP is able to cover more than 6% than the edge-restricted problem and
+
+.Location of Leak Detection Devices
+7
+Figure
+4. Solutions of the edge-restricted and node-
+restricted versions of MNLCLP for p = 5 for the network
+of Example 1.
+Figure 5. Average length coverage deviations between the
+solutions of MNLCLP and the edges/nodes-restricted ver-
+sions of the problem .
+more than 20% than the node-restricted problem. In situations, as the one
+under study, where undetected leaks may produce fatal consequences in an
+urban area, a large coverage, with the available resources, is crucial, being
+then advisable the use of our models.
+3. The single-device Maximal Network Length Covering
+Location Problem
+In this section we provide a mathematical programming model for the
+(MNLCLP) described in the previous section in case p = 1 (a single device
+is located). This model will guide us on the construction of models for the
+general situations, i.e., for the (PSNLCLP) and the (MNLCLP) for p > 1.
+The model is based in the following observation. Let e ∈ E be an edge
+in the network and X ∈ Rd the location of a device. In case the coverage
+area of the device in X, BR(X), does not touch the edge, then the covered
+length is zero. Otherwise, since BR(X) is a compact and convex body in Rd,
+∂BR(X), the border of the ball, will touch the segment in two points (that
+may coincide in case the segment belong to a tangent hyperplane of the ball).
+
+25%
+20%
+15%
+10%
+5%
+0%
+p=2
+p=5
+p=8
+Dev_Edges
+Dev_Nodes8
+V. BLANCO and M. MART´INEZ-ANT´ON
+These points belong to the segment [oe, fe], that can be parameterized as:
+Y 0
+e = λ0
+eoe + (1 − λ0
+e)fe and Y 1
+e = λ1
+eoe + (1 − λ1
+e)fe
+for some λ0
+e, λ1
+e ∈ [0, 1]. We can assume without loss of generality that Y 0
+e is
+closer to oe than Y 1
+e , so we restrict the λ-values to λ0
+e ≤ λ1
+e. With the above
+parameterization, the length of the edge covered by X is (λ1
+e − λ0
+e)Le (here,
+Le denotes the length of the edge e).
+To derive our mathematical programming formulation for the problem,
+we use the following sets of decision variables:
+ze =
+�
+1
+if edge e intersects the device’s coverage area,
+0
+otherwise
+X : Coordinates of the placement of the device.
+Y 0
+e , Y 1
+e : Intersections points of ∂BR(X) with the edge e
+λ0
+e, λ1
+e : Parameterization values in the segment of intersection points Y 0
+e and Y 1
+e , respectively.
+With the above notation, the single-device MNLCLP can be formulated
+as the following Mathematical Programming Model, that we denote as (1-
+MNLCLP):
+max
+�
+e∈E
+ωeLe(λ1
+e − λ0
+e)
+(1)
+s.t. ∥X − Y s
+e ∥ze ≤ R, ∀e ∈ E, s ∈ {0, 1},
+(2)
+Y s
+e = λs
+eoe + (1 − λs
+e)fe, ∀e ∈ E, s ∈ {0, 1},
+(3)
+λ0
+e ≤ λ1
+e, ∀e ∈ E,
+(4)
+λ1
+e ≤ ze, ∀e ∈ E, s ∈ {0, 1},
+(5)
+λ0
+e, λ1
+e ≥ 0, ∀e ∈ E, s ∈ {0, 1},
+(6)
+ze ∈ {0, 1}, ∀e ∈ E,
+(7)
+X ∈ Rd.
+(8)
+Constraints 2 enforce that in case the device intersect the edge, the in-
+tersection points must be in the coverage area of X. This constraint can be
+rewritten as:
+∥X − Y s
+e ∥ ≤ R + ∆(1 − ze), ∀e ∈ E, s ∈ {0, 1}
+where ∆ a big enough constant with ∆ > max
+�
+∥z1 − z2∥ : z1, z2 ∈ {oe, fe :
+e ∈ E}
+�
+. Constraints (3) are the parameterization of the intersection points.
+Constraints (4) force that Y 0
+e is closer to oe than Y 1
+e . In case the device does
+not intersect an edge, we fix to zero the coefficients of the parameterization,
+adding a value of zero to the covered lengths in the objective function. (5)-
+(7) are the domains of the variables.
+
+Location of Leak Detection Devices
+9
+(1-MNLCLP) is a Mixed integer Non Linear Programming problem be-
+cause of the discrete variables z and the nonlinear constraints (2). For ℓτ or
+polyhedral norms, theses constraints are known to be efficiently rewritten as
+a set of second order cone constraints (and in case of polyhedral norms, as
+linear constraints) becoming a Mixed Integer Second Order Cone Optimiza-
+tion (MISOCO) problem that can be solved using the off-the-shelf softwares
+[see Blanco et al., 2014, for further details].
+3.1. Generating feasible solutions of MNLCLP. The single-device ver-
+sion of the MNLCLP is already a challenging problem since it is require to
+obtain a feasible group of edges which is able to be covered by the device. In
+what follows, we derive some geometrical properties and algorithmic strate-
+gies for this problem, that will be useful to derive a integer linear program-
+ming formulation for the problem to generate good quality feasible solutions
+of this problem. The same ideas will be extended to generate feasible solu-
+tions also for the multi-device problem.
+Lemma 4. Let ¯z ∈ {0, 1}|E| be a feasible solution for 1-MNLCLP Denote
+by C = {e ∈ E : ¯ze = 1}, the edges covered by the device. Then, we get that
+(Cov)
+X ∈
+�
+e∈C
+(e ⊕ BR(0)),
+where ⊕ stands for the Minkowski sum in Rd.
+Proof. It follows directly from the verification of constraints (19).
+□
+The above result states that the position of the device, X, must belong
+to the intersection of the extended segments induced by the edges in the
+cluster C . In Figure 6 (left picture) we show an example of the shape of
+e ⊕ BR(0) for a given edge e ∈ E. In Figure 6 (right picture) we show the
+intersection of three of this type of sets, where a device covering the three
+segments should be located.
+One of the main decisions of the models under study, is the determination
+of the edges are touched by the same device, i.e., those for subsets of edges,
+S ⊂ E, such that �
+e∈S(e ⊕ BR(0)). We call the subsets of E verifying this
+condition, compatible subsets, i.e, the set:
+C =
+�
+S ⊂ E :
+�
+e∈S
+(e ⊕ BR(0)) ̸= ∅
+�
+In general, not all the subsets of E belong to C, but only those in C are to
+be constructed in our models.
+In the following result we describe a polynomial set (in |E|) of valid in-
+equalities for our model that filter those non-compatible sets in the solution.
+Lemma 5. The following inequalities are valid for the 1-MNLCLP:
+(9)
+�
+e∈S
+ze ≤ |S| − 1, ∀S ⊂ E with |S| = d + 1 and
+�
+e∈S
+(e ⊕ BR(0)) = ∅
+
+10
+V. BLANCO and M. MART´INEZ-ANT´ON
+Figure 6. Shape of extended edges (left) and intersection
+of three of these compatible shapes (right).
+Proof. It is straightforward to see that non-compatible subsets will not be
+constructed in the models, and then, the following exponential number of
+valid inequalities for the models:
+�
+e∈S
+ze ≤ |S| − 1, ∀j ∈ P, ∀S ⊂ E :
+�
+e∈S
+(e ⊕ BR(0)) = ∅,
+Since the sets in the form (e ⊕ BR(0)) are compact and convex for any e ∈,
+the result follows by applying Helly’s theorem Helly [1923].
+□
+Corollary 6. Let ¯z ∈ {0, 1}|E| be a solution of the system of equations (9).
+Then, ¯z is a feasible solution for the 1-MNLCLP.
+In the classical Maximal Coverage Location Problems, the above observa-
+tion allows one to replace the non-linear covering constraints (in the shape
+of (19)) by inequalities in the shape of (9) and the continuous variables can
+be dropped-out (see [Blanco and G´azquez, 2021, Blanco et al., 2022, e.g.]).
+In our model, it is no longer possible since the λ-variables are also needed
+to compute the covered volume of the network.
+Thus, we propose the following linear integer programming formulation
+to obtain valid compatible subsets for the models.
+max
+�
+e∈E
+ωeLeze
+(10)
+s.t.
+�
+e∈S
+ze ≤ |S| − 1, ∀S ⊂ E(|S| = d + 1) :
+�
+e∈C
+(e ⊕ BR(0)) = ∅,
+(11)
+ze ∈ {0, 1}, ∀e ∈ E.
+(12)
+The above mathematical programming model, are the edge-based versions
+of the classical 1-Maximal Coverage Location Problem, that is known to be
+
+ee
+e
+R
+RLocation of Leak Detection Devices
+11
+NP-hard. Nevertheless, it is a significant simplification of our models which
+is able to be solved for reasonable sizes.
+The main difficulty of this formulation is to determine the intersections
+of d+1 sets in the form e⊕BR(0) is empty, in whose case the corresponding
+inequality is added to the pool of constraints. The general methodology that
+can be applied for any dimension and any ℓτ-based norm, is by applying a
+relax-and-cut approach based on solving the problems above by removing
+constraints (11), separating the violated constraints and incorporate them
+on-the-fly in an embedded branch-and-cut algorithm.
+In what follows we focus on the planar Euclidean case, that is the most
+useful case in practice, and for which the formulations can be further sim-
+plified and strengthened.
+Observe that for d = 2, Constraints (9) are equivalent to:
+ze + ze′ ≤ 1,∀e, e′ ∈ E : (e ⊕ BR(0)) ∩ (e′ ⊕ BR(0)) = ∅,
+ze + ze′ + ze′′ ≤ 2,∀e, e′ ∈ E : (e ⊕ BR(0)) ∩ (e′ ⊕ BR(0)) ∩ (e′′ ⊕ BR(0)) = ∅,
+ze ∈ {0, 1},∀e ∈ E.
+Thus, in order to incorporate these types of constraints one need to check
+two- and three-wise intersections of objects in the form e⊕BR(0). Although
+these shapes can be difficult to handle in general, the planar Euclidean case
+can be efficiently handled by analyzing the geometry of these objects as
+Minkowski sums of segments and disks.
+The following results are instrumental for the development of the Algo-
+rithm that we propose to generate the above sets of constraints. From now
+on, ∥ · ∥ denotes the Euclidean norm in R2.
+Lemma 7. Let e, e′ be two segment in R2, · and δ(e, e′) = min{∥X − X′∥ :
+X ∈ e, X′ ∈ e′}. Then, if δ(e, e′) > 0, there exist X ∈ e and X′ ∈ e′ with
+δ(e, e′) = ∥X − X′∥ such that either X ∈ {oe, fe} or X′ ∈ {oe′, fe′}.
+Proof. The result follows by observing that the minimum distance between
+two segments is always achieved choosing one of the extremes of the seg-
+ments.
+□
+Lemma 8. Let e be a segment in R2, and Q ∈ R2.
+Then, δ(e, Q) :=
+min{∥Q − X∥ : X ∈ e} can be computed as:
+δ(e, Q) = ∥Q − (min{max{0, µ}, 1}(fe − oe) + oe)∥.
+Proof. Let S be the intersection point between the line induced by e, r,
+and its orthogonal line passing through the point Q. We denote by µ the
+parameterization of S in the ray induced by the segment pointed at oe.
+Thus, ∥Q − S∥ = min{∥Q − T∥ : T ∈ r}. Since S ∈ r, one can parameterize
+S as S = (1 − µ)oe + µfe for some µ ∈ R. Let us analyze the different
+possible values for µ:
+
+12
+V. BLANCO and M. MART´INEZ-ANT´ON
+• If µ ∈ [0, 1], one gets that:
+∥Q − (µ(fe − oe) + oe)∥ = ∥Q − S∥ = min{∥Q − T∥ : T ∈ r}
+≤ min{∥Q − T∥ : T ∈ e} = δ(e, Q).
+• If µ < 0, will show that δ(e, Q) = ∥Q − oe∥. Let λ ∈ [0, 1] and
+X = (1 − λ)oe + λfe ∈ e. Then:
+∥Q − oe∥2 = ∥Q − S∥2 + ∥S − oe∥2 = ∥Q − S∥2 + ∥µ(fe − oe) + oe − oe∥2
+= ∥Q − S∥2 + |µ|2∥(fe − oe)∥2 ≤ ∥Q − S∥2 + |(µ − λ)|2∥(fe − oe)∥2
+= ∥Q − S∥2 + ∥µ(fe − oe) + oe − (λ(fe − oe) + oe)∥2 = ∥Q − S∥2 + ∥S − X∥2
+= ∥Q − X∥2.
+• In case µ > 1, let us see that δ(e, Q) = ∥Q − fe∥. Let λ ∈ [0, 1] and
+X = λ(fe − oe) + oe be in e:
+∥Q − fe∥2 = ∥Q − S∥2 + ∥S − fe∥2 = ∥Q − S∥2 + ∥µ(fe − oe) + oe − fe∥2
+= ∥Q − S∥2 + ∥µ(fe − oe) + oe − fe + oe − oe∥2
+= ∥Q − S∥2 + |µ − 1|2∥(fe − oe)∥2
+≤ ∥Q − S∥2 + |(µ − λ)|2∥(fe − oe)∥2
+= ∥Q − S∥2 + ∥µ(fe − oe) + oe − (λ(fe − oe) + oe)∥2
+= ∥Q − S∥2 + ∥S − X∥2
+= ∥Q − X∥2.
+Summarizing, we get that the point in e closest to Q is in the form (1 −
+λ)oe + λfe with
+λ =
+�
+�
+�
+�
+�
+0
+if µ < 0,
+µ
+if 0 ≤ µ ≤ 1,
+1
+if µ > 1.
+that is, λ = min{max{0, µ}, 1}, being then δ(e, Q) = ∥Q−(min{max{0, µ}, 1}(fe−
+oe) + oe)∥.
+□
+The first algorithm (Algorithm 1) starts with a set of edges E and a radius
+R as inputs. We initialize the set M2 = ∅. This set will be sequentially com-
+pleted with the pairs (e, e′) of E ×E verifying (e⊕BR(0))∩(e′ ⊕BR(0)) = ∅
+by checking the distance between the segments, δ(e, e′). In case, δ(e, e′) = 0,
+both segment intersect so also their Minkowski sums by the balls. Other-
+wise, we denote by re and re′ the lines containing the segments e and e′,
+respectively, and by Q0 their intersection point. By Lemma 7 there exists
+a couple X, X′ ∈ R2 with δ(e, e′) being either X or X′ extremes points of
+the segments. Thus, four distances are enough to compute δ(e, e′), namely
+δ1 := δ(oe′, e), δ2 := δ(fe′, e), δ3 := δ(oe, e′) and δ4 := δ(fe, e′), being δ(e, e′)
+the minimum of all of them. In case such a distance exceed the diameter
+
+Location of Leak Detection Devices
+13
+2R, the segment are far enough such that (e ⊕ BR(0)) ∩ (e′ ⊕ BR(0)) = ∅,
+and the tuple (e, e′) is added to M2.
+The second algorithm (Algorithm 2) computes the triplets (e1, e2, e3)
+whose pair-wise intersections are non-empty but their three-wise intersec-
+tion is empty. First, we initialize this set to the empty set, M3 = ∅, and for
+every suitable triplet (e1, e2, e3) whose pairwise intersection is non-empty,
+we solve the following mathematical optimization problem:
+ε∗(e1, e2, e3) := min ε
+(13)
+s.t. Yi = (1 − λi)oei + λifei, i = 1, 2, 3,
+(14)
+∥X − Yi∥ ≤ R + ε, i = 1, 2, 3,
+(15)
+X ∈ R2,
+(16)
+λ1, λ2, λ3 ∈ [0, 1],
+(17)
+ε ∈ R.
+(18)
+In this problem, the goal is to find an intersection point, X, in (e1⊕BR(0))∩
+(e2 ⊕ BR(0)) ∩ (e3 ⊕ BR(0)). If such a point exists, then, is because there
+exists points at each of segments (parameterized by the λ-variables above,
+at distance not exceeding the radius R, being the minimum of the problem
+above ε∗ = 0. Otherwise, ε∗ > 0. The problem above is solvable in poly-
+nomial time since it can rewritten as a Second Order Cone Optimization
+problem.
+4. A general model for (PSNLCLP) and (MNLCLP)
+In this section we provide a general methodology to deal with the optimal
+location of devices in both the PSNLCLP and the MNLCLP. In the two
+models, the covered length of each edge by a set of devices is to be calculated.
+When a single device is located, the coverage of an edge by such a device can
+be computed by parameterizing the intersection of the boundary of the ball
+with the segment, as detailed in the previous section. However, in case more
+than one device touch an edge, then, the covered length does not coincide
+with the sum of the coverages of each single device separately, since a same
+part of the segment may be covered by two or more devices, but the covered
+length must be accounted only once (otherwise the optimal placement for a
+set of devices is the collocation off all of them in the more weighted edge).
+To illustrate the situation, we show in the following example how four
+devices cover a single edge of the network.
+Example 9. Let us consider a single edge e and four planar devices with
+Euclidean ball coverage areas as drawn in Figure 7.
+As can be observed
+the four devices touch the edge. The covered length of the edge is highlighted
+with thicker segments in the picture. Clearly, this length cannot be computed
+by adding up separately, each of the covered lengths of the devices.
+
+14
+V. BLANCO and M. MART´INEZ-ANT´ON
+Algorithm 1: A complete set of 2-wise incompatible edges.
+Data: Set of edges, E, and radius R.
+M = ∅
+for (e, e′) ∈ E × E do
+Set: ¯e = fe − oe.
+Set: ¯e′ = fe′ − oe′.
+Compute the intersection point of the lines oe + ⟨¯e⟩ and oe′ + ⟨¯e′⟩:
+Q0.
+Calculate µ0, µ′
+0 such that Q0 = µ0¯e + oe and Q0 = µ′
+0 ¯e′ + oe′.
+if µ0 or µ′
+0 /∈ [0, 1] then
+(1) Compute the intersection point of the lines oe + ⟨¯e⟩ and oe′ + ⟨¯e⊥⟩:
+Q1.
+Calculate µ1 such that Q1 = µ1¯e + oe.
+Set: δ1 = ∥oe′ − (min{max{0, µ1}, 1}¯e + oe)∥.
+(2) Compute the intersection point of the lines oe + ⟨¯e⟩ and fe′ + ⟨¯e⊥⟩:
+Q2.
+Calculate µ2 such that Q2 = µ2¯e + oe.
+Set: δ2 = ∥fe′ − (min{max{0, µ2}, 1}¯e + oe)∥.
+(3) Compute the intersection point of the lines oe + ⟨¯e′⊥⟩ and oe′ + ⟨¯e′⟩:
+Q3.
+Calculate µ3 such that Q3 = µ3 ¯e′ + oe′.
+Set: δ3 = ∥oe − (min{max{0, µ3}, 1}¯e′ + oe′)∥.
+(4) Compute the intersection point of the lines fe + ⟨¯e′⊥⟩ and oe′ + ⟨¯e′⟩:
+Q4.
+Calculate µ4 such that Q4 = µ4 ¯e′ + oe′.
+Set: δ4 = ∥fe − (min{max{0, µ4}, 1}¯e′ + oe′)∥.
+if min{δ1, δ2, δ3, δ4} > 2R then
+Add (e, e′) to M.
+Result: M = {(e, e′) ∈ E × E : (e ⊕ BR(0)) ∩ (e′ ⊕ BR(0)) = ∅}.
+The positions of the intersection points of the coverage areas of p devices
+with an edge provide a partition of the edge in at most p + 1 subsegments.
+Each of those subsegments is either fully covered or non covered by the
+device.
+Let λ0
+1e, λ1
+1e, . . . , λ0
+pe, λ1
+pe the parameterization of the intersection
+points of the p devices with an edge e (here λ0
+je and λ1
+je stands for the
+parameterizations of the intersection of the coverage area of jth device with
+segment induced by the edge e).
+By convention, we assume that the devices not intersecting the edge will
+have both lambda values equal to zero. Sorting the λ0 and λ1 values one
+get two sorted sequences in the form:
+Λ0
+e := λ0
+(1)e ≤ · · · ≤ λ0
+(p)e
+
+Location of Leak Detection Devices
+15
+Algorithm 2: A complete set of 3-wise incompatible edges (which
+are pair-wise compatible).
+Data: Set of edges, E, and radius R.
+L = {(e1, e2, e3) ∈ E × E × E : (e1 ⊕ BR(0)) ∩ (e2 ⊕ BR(0)) ̸=
+∅, (e ⊕ BR(0)) ∩ (e3 ⊕ BR(0)) ̸= ∅, (e2 ⊕ BR(0)) ∩ (e3 ⊕ BR(0)) ̸= ∅}.
+M3 = ∅.
+for (e1, e2, e3) ∈ L do
+Compute ε∗(e1, e2, e3).
+if ε∗(e1, e2, e3) > 0 then
+Add (e1, e2, e3) to M3.
+Result: M =
+{(e1, e2, e3) ∈ E × E × E : (e1 ⊕ BR(0)) ∩ (e2 ⊕ BR(0)) ∩ (e3 ⊕ BR(0)) = ∅}.
+fe
+oe
+Figure 7. Example of interaction between the coverages of
+different devices.
+Λ1
+e := λ1
+(1)e ≤ · · · ≤ λ1
+(p)e
+Merging both lists one get all the partitions of the segment e by the different
+intersection points:
+Λe := λi1
+(1)e ≤ · · · ≤ λi2p
+(2p)e
+where i1, . . . , i2p ∈ {0, 1} and some of inequalities may be equations, in
+particular for all devices not intersecting e.
+For each l ∈ {1, . . . , 2p}, the intervals [λil
+(l)e, λil+1
+(l+1)e] induce a partition of
+the segment e into 2p + 1 pieces.
+Given the sequence Λe for the p given devices located at X1, . . . , Xp, one
+can easily determine which of the subsegments in the partitions are covered
+by the facilities as stated by the following straightforward observation.
+Lemma 10. A subsegment in the form s = [λil
+(l)e, λil+1
+(l+1)e] is covered by a set
+of devices if and only if s ⊆ [λ0
+je, λ1
+je] for some j = 1, . . . , p with λ0
+je < λ1
+je.
+In the general mathematical programming formulations that we propose,
+we use the following decision variables, where we denote by P = {1, . . . , p}
+
+16
+V. BLANCO and M. MART´INEZ-ANT´ON
+the index set for the devices to locate and by Q = {1, . . . , 2p − 1} the index
+sets for the subsegments in the partition induced by the Λ sequences.
+zje =
+�
+1
+if edge e intersect the jth device’s coverage area,
+0
+otherwise
+∀j ∈ P, e ∈ E.
+Xj1, . . . , Xjd : Coordinates of the placement of the jth device, ∀j ∈ P.
+λ0
+je, λ1
+je : Parameterization in the segment of the two intersection
+points of ∂BR(Xj) with segment e, ∀j ∈ P, e ∈ E.
+weℓ =
+�
+1
+if the ℓ-th subsegment of edge e is covered by some device,
+0
+otherwise
+, ∀ℓ ∈ Q, e ∈ E.
+ξs
+jeℓ =
+�
+1
+if λs
+ej is sorted in ℓth position in the list of Λe,
+0
+otherwise
+∀j ∈ P, ℓ ∈ Q∪{2p}, e ∈ E.
+With the above set of variables, the amount:
+�
+��
+j∈P
+1
+�
+s=0
+λs
+jeξs
+je(ℓ+1) −
+�
+j∈P
+1
+�
+s=0
+λs
+jeξs
+jeℓ
+�
+�
+determines the length of the ℓ-th subsegment in case it is covered by any
+of the devices in P. Note that in case such a subsegment is [λs
+je, λs′
+j′e], the
+above expression becomes Le(λs′
+j′e − λs
+je) which is the desired amount.
+Thus, the overall volume coverage of the network can be computed as:
+�
+e∈E
+�
+ℓ∈Q
+ωeweℓLe
+�
+��
+j∈P
+1
+�
+s=0
+λs
+jeξs
+je(ℓ+1) −
+�
+j∈P
+1
+�
+s=0
+λs
+jeξs
+jeℓ
+�
+�
+In order to adequately represent the decision variables in our model, the
+following constraints are considered:
+(1) Coverage Constraints:
+(19)
+∥(λs
+jee + oe) − Xj∥zje ≤ Rj, ∀j ∈ P, e ∈ E, s = 0, 1
+These constraints enforce that in case a an edge is accounted as
+touched by the jth device (zje = 1), then two intersection points
+(λ0
+jee + oe) and (λs
+jee + oe) must exist in BR(Xj) ∩ e (by the max-
+imization length criterion these intersection points will belong to
+∂BR(Xj)). This constraint can be reformulated as:
+∥(λs
+jee + oe) − Xj∥zje ≤ Rj + ∆(1 − zje), ∀j ∈ P, e ∈ E, s = 0, 1
+where ∆ a big enough constant with ∆ > max
+�
+∥z1 − z2∥ : z1, z2 ∈
+{oe, fe : e ∈ E}
+�
+.
+
+Location of Leak Detection Devices
+17
+(2) Directed Parameterization:
+(20)
+λ0
+je ≤ λ1
+je, ∀j ∈ P e ∈ E.
+In case the coverage area of a device j touches the segment e, the
+segment is oriented in the parameterization.
+(3) Zero parameterizations for untouched edges
+(21)
+λ1
+je ≤ zje, ∀j ∈ P, e ∈ E, s = 0, 1.
+In case the jth device does not touch the segment induced by an edge
+e, the covered length of such an edge by the device will be zero. By
+(19), in that case the device is not restricted to touch the segment,
+but to assure that no length is accounted, we fix both λ-values in
+the fictitious intersection as 0.
+(4) Λ-Sorting Constraints:
+�
+j∈P
+(ξ0
+jeℓ + ξ1
+jeℓ) = 1, ∀e ∈ E, l ∈ Q ∪ {2p},
+(22)
+�
+l∈Q∪{2p}
+ξs
+jeℓ = 1, ∀j ∈ P e ∈ E, s = 0, 1
+(23)
+�
+j∈P
+(λ0
+jeξ0
+jeℓ + λ1
+jeξ1
+jeℓ) ≤
+�
+j∈P
+(λ0
+jeξ0
+je(ℓ+1) + λ1
+jeξ1
+je(ℓ+1)), ∀e ∈ E, ℓ ∈ Q.
+(24)
+These constrains allow to adequately define the variables ξ. Con-
+straints (22) and (23) assure that for each e each λe-value is sorted
+in exactly a single position in Q and that each position is assigned to
+exactly one λe value. Constraint (24) enforces that the ξ-variables
+sort λ-values in non decreasing order.
+(5) Coverage of subsegments:
+weℓ ≤
+�
+j∈P
+�
+��
+i≤ℓ
+ξ0
+jei +
+�
+i>ℓ
+ξ1
+jei − 1
+�
+� , ∀e ∈ E, l ∈ Q,
+(25)
+(26)
+The coverage of a subsegment ℓ ∈ Q is assured by the existence of
+a device j for which its λ0
+ej value is sorted in a previous position to
+ℓ and its λ1
+ej value is sorted in a back position to ℓ. For a given
+device j ∈ Q, �
+i≤ℓ ξ0
+jei = 1 if the λ0 value for j in e is sorted
+in a previous position to ℓ, and zero otherwise. On the other hand,
+�
+i>ℓ ξ1
+jei = 1 indicates if the λ1 value is sorted in a position posterior
+to ℓ, and 0 otherwise. Thus, in case both values are 1, the conditions
+of Lemma 10 are verified, and the subsegment is covered. Otherwise,
+only one of the two expressions can be zero, but not both. Indeed,
+if �
+i≤ℓ ξ0
+jei = 0, then, by (23), �
+i>ℓ ξ0
+jei = 1. Thus, by (20) and
+(24), one has that �
+i>ℓ ξ1
+jei = 1. On the other hand, by a similar
+construction, if �
+i>ℓ ξ1
+jei = 0, one has that �
+i≤ℓ ξ0
+jei = 1. In both
+
+18
+V. BLANCO and M. MART´INEZ-ANT´ON
+cases, �
+i≤ℓ ξ0
+jei + �
+i>ℓ ξ1
+jei − 1 takes value zero, implying that the
+jth device is not covering such a subsegment.
+Apart from the constraints above, we incorporate to our model the fol-
+lowing valid inequalities that allow us to strengthen the model:
+(1) Touched segments and covered subsegments:
+�
+ℓ∈Q
+weℓ ≤ 2
+�
+j∈P
+zje, ∀e ∈ E.
+In case the whole segment is not touched by any device, non of the
+subsegments are covered.
+(2) Symmetry breaking:
+d
+�
+k=1
+Xjk ≤
+d
+�
+k=1
+X(j+1)k, ∀j ∈ P, j < p.
+Since the devices to be located are indistinguishable, any permu-
+tation of the j-index will result in an alternative optimal solution,
+hindering the solution procedure based on a branch-and-bound tree.
+The above inequality prevent for such an amount of alternative op-
+timal.
+(3) Incompatible edges:
+zej + ze′j ≤ 1, ∀j ∈ P, e, e′ ∈ E with min{∥x − x′∥ : x ∈ e, x′ ∈ e′} > 2R.
+Edges that are far enough are not able to be simultaneously touched
+by the same device.
+Mathematical Programming Model for (MNLCLP):
+Using the variables and constraints previously described, the following
+mathematical programming formulation is valid for the MNLCLP:
+max
+�
+e∈E
+�
+ℓ∈Q
+ωeLeweℓ
+�
+��
+j∈P
+1
+�
+s=0
+λs
+jeξs
+je(ℓ+1) −
+�
+j∈P
+1
+�
+s=0
+λs
+jeξs
+jeℓ
+�
+�
+s.t. (19) − (25),
+λs
+je ∈ [0, 1],
+j ∈ P, e ∈ E, s = 0, 1,
+Xj ∈ Rd,
+j ∈ P,
+zje ∈ {0, 1},
+j ∈ P, e ∈ E,
+ξs
+jeℓ ∈ {0, 1},
+j ∈ P, e ∈ E, ℓ ∈ Q, s = 0, 1,
+weℓ ∈ {0, 1},
+e ∈ E, ℓ ∈ Q.
+
+Location of Leak Detection Devices
+19
+Mathematical Programming Model for (PSNLCLP):
+(PSNLCLP) seeks minimizing the number of devices to cover at least a
+portion γ ∈ (0, 1] of the length of the network. Although the above variables
+and constraints can be used to derive similarly a model for this problem,
+the number of devices, p, to locate is unknown in this case. We estimate an
+upper bound for this parameter and consider the following binary variables
+to activate/desactivate them.
+yj =
+�
+1
+if device j is activated,
+0
+otherwise.
+∀j ∈ P.
+Then, the (PSNLCLP) can be formulated as follows:
+min
+�
+j∈P
+yj
+s.t. (19) − (25),
+(27)
+�
+e∈E
+�
+ℓ∈Q
+ωeLeweℓ
+�
+��
+j∈P
+1
+�
+s=0
+λs
+jeξs
+je(ℓ+1) −
+�
+j∈P
+1
+�
+s=0
+λs
+jeξs
+jeℓ
+�
+� ≥ γ
+�
+e∈E
+ωeLe,
+(28)
+�
+e∈E
+zje ≤ yj, ∀j ∈ P,
+(29)
+λs
+je ∈ [0, 1],
+j ∈ P, e ∈ E, s = 0, 1,
+Xj ∈ Rd,
+j ∈ P,
+zje ∈ {0, 1},
+j ∈ P, e ∈ E,
+ξs
+jeℓ ∈ {0, 1},
+j ∈ P, e ∈ E, ℓ ∈ Q, s = 0, 1,
+weℓ ∈ {0, 1},
+e ∈ E, ℓ ∈ Q,
+yj ∈ {0, 1},
+∀j ∈ P.
+Where now, the objective function accounts for the number of activated
+devices, Constraint (28) assure that at least a portion of γ of the coverage
+volume is attained, and (29) prevent covering edges by devices that are not
+activated.
+To avoid multiple optimal solutions due to symmetry, we also incorporate
+to the model the following constraints that avoid activating the j device in
+case the j − 1 device is not activated.
+yj−1 ≥ yj, ∀j ∈ P, j > 1.
+
+20
+V. BLANCO and M. MART´INEZ-ANT´ON
+Remark 11. The upper bound on the number of devices for the (PSNLCLP)
+is calculated as follows. Since we need a generalized method to compute the
+upper bound p must consider that we do not know the network shape so
+we are going to calculate the minimum number of devices necessaries to
+cover each edge of the subset Uγ ⊆ E defined how the minimal set that
+verifies
+�
+e∈U
+ωeLe ≥ γ
+�
+e∈E
+ωeLe where U ⊆ E. This set construction rely on
+initializing Uγ = ∅; sorting the sequence {ωeLe}e∈E in non-increasing way
+(ωe1Le1 ≥ ωe2Le2 ≥ · · · ≥ ωeiLei ≥ ωei+1Lei+1 ≥ · · · ), and appending ei into
+Uγ one-by-one in that order until
+�
+e∈Uγ
+ωeLe ≥ γ
+�
+e∈E
+ωeLe. It is clear the
+minimum number of devices necessaries to cover a single edge e is
+� Le
+2R
+�
+so,
+in sum, the count of p would be p =
+�
+e∈Uγ
+� Le
+2R
+�
+.
+The non linear integer programming models that we develop for (MNL-
+CLP) and (PSNLCLP) have O(p2|E|) variables, O(p|E|) linear contraints
+and O(p|E|f∥·∥) non linear constraints (here, f∥·∥ stand for the number of
+constraints that allow rewriting Constraints 19 as second order cone con-
+straints (see Blanco et al. [2014] for upper bounds on this number for ℓτ-
+norms). Thus, it is advisable in these models to design alternative solution
+strategies for solving them or to provide initial solutions that alleviate the
+search of optimal solutions by providing lower bounds for our problem. In
+the following sections we propose different alternatives taking advantage of
+the geometric properrties of these problems.
+4.1. Constructing initial feasible solutions. The geometric properties
+that we derive in Section 3.1 for the single device problem can be also ex-
+tended to the p-device case.
+Specifically, one can construct solutions of
+MNLCLP by avoiding the computation of covered lengths in the models
+and assuming that once an edge of the network is touched by coverage area
+of a device, the whole is accounted as covered. With these assumptions, we
+construct initial solutions of our problem by solving the following integer
+linear programs:
+max
+�
+e∈E
+�
+j∈P
+ωeLezje
+(30)
+s.t.
+�
+j∈P
+zje ≤ 1, ∀e ∈ E,
+(31)
+�
+e∈S
+zje ≤ |S| − 1, ∀S ⊂ E(|S| = d + 1) :
+�
+e∈C
+(e ⊕ BR(0)) = ∅, j ∈ P,
+(32)
+zje ∈ {0, 1}, ∀e ∈ E, j ∈ P.
+(33)
+
+Location of Leak Detection Devices
+21
+In the problem above, the overall weighted length of the covered edges is
+to be maximized by restricting edges to be covered by the same device to
+those which are feasible for the MNLCLP. The edges are also enforced to be
+accounted at most once in the solution.
+The strategies for generating and separating the constraints of the above
+problem are identical to those detailed in Section 3.1.
+4.2. Math-heuristic approach. This approach that we propose to allevi-
+ate the solution of MNLCLP and PSNLCLP is based on solving the single-
+device location problem (2)-(8) that was described in Section 3.1 in a se-
+quential way. Although this model, in contrast to (30)-(33), is non linear,
+takes into account the covered lengths of the segment, being more accurate
+to approximate our problem.
+Algorithm 3 shows a pseudocode for this math-heuristic approach. As
+already mentioned, the approach is based on solving, sequentially, a single-
+device location device problem until certain termination criterion (which
+depends on the problem to solve, MNLCLP or PSNLCLP) is verified. In
+case the problem is the MNLCLP the algorithm ends when the number of
+devices in the pool reaches the value of p. Otherwise, for the PSNLCLP the
+algorithm ends when the covered length reaches the desired value.
+At each iteration, a device is located, and the network to be covered in
+the next iteration is updated from the previous by removing the segments
+already covered.
+Algorithm 3: Math-heuristic 2.
+Data: Network G = (V, E; Ω), number of devices p and radius R.
+V ′ = V, E′ = E, Ω′ = Ω
+X = ∅
+while Termination Criterion do
+Solve X′, λ0
+e, λ1
+e, ze = arg (1)-(8) for e ∈ E′, ωe ∈ Ω′ and R.
+Update Termination Criterion Add X′ to X.
+for e ∈ E′ do
+if ze = 1 then
+if λ0
+e ∈ (0, 1) then
+Add Y 0
+e to V ′.
+Add {oe, Y 0
+e } to E′.
+Add ωe to Ω′.
+if λ1
+e ∈ (0, 1) then
+Add Y 1
+e to V ′.
+Add {Y 1
+e , fe} to E′.
+Add ωe to Ω′.
+Remove e from E′
+Result: X ∈ R(d×p): Location of the devices.
+
+22
+V. BLANCO and M. MART´INEZ-ANT´ON
+5. Computational Experiments
+In this section we report on the results of a series of computational exper-
+iments performed to empirically assess our methodological contribution for
+the p-MNLCLP and PSNCLP presented in the previous sections. We use
+six real networks obtained from two different sources: one based on the net-
+works developed by the University of Exeter’s (UOE) Centre for Water Sys-
+tems available in https://emps.exeter.ac.uk/engineering/research/
+cws/resources/benchmarks/ and other privately provided by Dr. Ormsbee
+from the University of Kentucky (UKY). These networks, which are called
+gessler, jilin, richmond, foss, rural and zj, have 14, 34, 44, 58, 60 and
+85 edges, respectively. The networks have being scaled to the unit square.
+The networks are drawn in Figure 8.
+(a) gessler
+(b) jilin
+(c) richmond
+(d) foss
+(e) rural
+(f) zj
+Figure 8. Networks used in our computational experi-
+ments.
+We have run the different approaches for the MNCLP and the PSNLCLP
+for disk-shaped coverage areas with radii ranging in {0.1, 0.25, 0.5}. For the
+MNLCLP the number of devices to locate, p, ranges in {2, 5, 8}, whereas for
+the PSNLCLP the values of γ range in {0.5, 0.75, 1}.
+All the experiments have been run on a virtual machine in a physical
+server equipped with 12 threads from a processor AMD EPYC 7402P 24-
+Core Processor, 64 Gb of RAM and running a 64-bit Linux operating system.
+The models were coded in Python 3.7 and we used Gurobi 9.1 as optimiza-
+tion solver. A time limit of 5 hours was set for all the experiments.
+In Tables 1 and 2 we show the average results obtained in our experiments.
+We report average values of the consumed CPU time (in seconds), and per-
+cent of unsolved instances and MIP Gap within the time limit. Both tables
+are similarly organized. In the first block (first three columns), the name
+of the instance together with its number of nodes and edges is provided. In
+
+Location of Leak Detection Devices
+23
+the second block (next two columns) we write the values of p (for the MNL-
+CLP) or γ (for the PSNLCLP) and the radius. The next three blocks are
+the results obtained with each of the approaches. For the MNLCLP we run
+the MISOCO formulation, and also the two solution approaches detailed in
+section 4.1 (MNLCLP 1, for short) and 4.2 (MNLCLP 2). We do not report
+results on the Unsolved instances and MIPGap for the MNLCLP 2 since all
+the instances were solved within the time limit with that approach. In Table
+2 the results are organized similarly for the PSNLCLP, but we do not gen-
+erate initial solutions since that strategy only applies to the MNLCLP, and
+only the strategy PSNLCLP 2. The flag TL indicates that all the instances
+averaged in the row reach the time limit without certifying optimality. The
+flag OoM indicates that the solver outputs Out of Memory at some point
+when solving the instance.
+The first observation from the results that we obtain is that both problems
+are computationally challenging since they require large CPU times to solve
+even the small instances. Actually, the exact MNLCLP was only able to
+solve up to optimality, small instances with small values of p, and the exact
+PSNLCLP only solved a few instances, and in many of them the solver
+outputs Out of Memory when solving them.
+The first strategy, based on constructing initial solutions to the problem,
+had an slightly better performance with respect to those instances that were
+solved with the initial formulation, both in CPU time and MIPGap. Some
+of the instances that were not able to be solved with MNLCLP but were
+able to be solved with the initial solutions that we construct.
+With respect to the heuristic approach, the consumed CPU times are tiny
+compared to the times required by the exact approaches, and was able to
+construct feasible solutions for all the instances, even for those that the ex-
+act approaches flagged Out of Memory. In terms of quality of the obtained
+solutions, in Figure 9 we show the average deviations (for each instance) of
+the alternative approaches with respect to the original one. This measure
+provides the percent improvement of the alternative method with respect
+to the best solution obtained by original formulation of the problem. We
+observed that the solutions that we obtain with the two strategies are sig-
+nificantly better than those obtained with the original formulation for the
+MNLCLP within the time limit. Providing initial solutions to the problem
+allows to obtain solutions with 20% more coverage than the initial formula-
+tion, whereas the heuristic approach get solutions with more than 25% more
+coverage. In case of the PSNLCLP, in most if the instances the solutions
+of the heuristic are better than the ones obtained with the exact approach,
+but in instance jilin, the solutions are 20% worse than the obtained with
+the exact approach.
+
+24
+V. BLANCO and M. MART´INEZ-ANT´ON
+CPU Time (secs)
+Unsolved
+GAP (%)
+instance
+|V | |E| p
+R
+MNLCLP
+MNLCLP 1
+MNLCLP 2
+MNLCLP
+MNCLP 1
+MNLCLP
+MNLCLP 1
+gessler
+12
+14
+2
+0.1
+151.53
+13.69
+0.89
+0%
+0%
+0%
+0%
+0.25
+48.97
+11.87
+1.34
+0%
+0%
+0%
+0%
+0.5
+26.28
+10.59
+0.62
+0%
+0%
+0%
+0%
+5
+0.1
+TL
+TL
+2.26
+100%
+100%
+86%
+84%
+0.25
+TL
+TL
+2.92
+100%
+100%
+69%
+62%
+0.5
+TL
+TL
+1.61
+100%
+100%
+24%
+31%
+8
+0.1
+TL
+TL
+3.54
+100%
+100%
+90%
+87%
+0.25
+TL
+TL
+5.59
+100%
+100%
+74%
+69%
+0.5
+TL
+TL
+2.92
+100%
+100%
+41%
+35%
+jilin
+28
+34
+2
+0.1
+167.25
+39.10
+1.99
+0%
+0%
+0%
+0%
+0.25
+196.56
+144.30
+3.37
+0%
+0%
+0%
+0%
+0.5
+164.83
+152.10
+2.45
+0%
+0%
+0%
+0%
+5
+0.1
+TL
+TL
+2.95
+100%
+100%
+86%
+85%
+0.25
+TL
+TL
+6.64
+100%
+100%
+72%
+64%
+0.5
+TL
+TL
+3.17
+100%
+100%
+40%
+42%
+8
+0.1
+TL
+TL
+6.07
+100%
+100%
+88%
+84%
+0.25
+TL
+TL
+10.34
+100%
+100%
+72%
+73%
+0.5
+TL
+TL
+4.67
+100%
+100%
+70%
+37%
+richmond 48
+44
+2
+0.1
+1180.62
+133.99
+8.75
+0%
+0%
+0%
+0%
+0.25
+717.09
+121.90
+7.47
+0%
+0%
+0%
+0%
+0.5
+184.63
+244.25
+2.32
+0%
+0%
+0%
+0%
+5
+0.1
+TL
+TL
+23.22
+100%
+100%
+78%
+77%
+0.25
+TL
+TL
+13.79
+100%
+100%
+62%
+59%
+0.5
+TL
+TL
+3.70
+100%
+100%
+42%
+41%
+8
+0.1
+TL
+TL
+33.64
+100%
+100%
+88%
+85%
+0.25
+TL
+TL
+23.89
+100%
+100%
+86%
+71%
+0.5
+TL
+TL
+5.82
+100%
+100%
+71%
+56%
+foss
+37
+58
+2
+0.1
+561.98
+39.61
+2.77
+0%
+0%
+0%
+0%
+0.25
+380.54
+38.42
+1.99
+0%
+0%
+0%
+0%
+0.5
+196.92
+86.40
+1.83
+0%
+0%
+0%
+0%
+5
+0.1
+TL
+TL
+6.49
+100%
+100%
+82%
+80%
+0.25
+TL
+TL
+5.46
+100%
+100%
+64%
+62%
+0.5
+TL
+TL
+4.31
+100%
+100%
+61%
+56%
+8
+0.1
+TL
+TL
+9.33
+100%
+100%
+88%
+86%
+0.25
+TL
+TL
+7.99
+100%
+100%
+87%
+71%
+0.5
+TL
+TL
+9.11
+100%
+100%
+78%
+64%
+rural
+48
+60
+2
+0.1
+12263.72
+1169.41
+16.94
+0%
+0%
+0%
+0%
+0.25
+TL
+559.93
+15.69
+100%
+0%
+23%
+0%
+0.5
+5054.64
+1612.73
+13.98
+0%
+0%
+0%
+0%
+5
+0.1
+TL
+TL
+26.46
+100%
+100%
+92%
+91%
+0.25
+TL
+TL
+32.19
+100%
+100%
+83%
+82%
+0.5
+TL
+TL
+21.99
+100%
+100%
+79%
+77%
+8
+0.1
+TL
+TL
+40.89
+100%
+100%
+97%
+94%
+0.25
+TL
+TL
+49.51
+100%
+100%
+91%
+86%
+0.5
+TL
+TL
+40.66
+100%
+100%
+94%
+84%
+zj
+60
+85
+2
+0.1
+TL
+TL
+13.12
+100%
+100%
+49%
+65%
+0.25
+TL
+5235.81
+7.29
+100%
+0%
+51%
+0%
+0.5
+TL
+9603.61
+9.33
+100%
+0%
+5%
+0%
+5
+0.1
+TL
+TL
+25.56
+100%
+100%
+96%
+95%
+0.25
+TL
+TL
+27.48
+100%
+100%
+90%
+89%
+0.5
+TL
+TL
+18.32
+100%
+100%
+87%
+86%
+8
+0.1
+TL
+TL
+37.85
+100%
+100%
+98%
+96%
+0.25
+TL
+TL
+31.05
+100%
+100%
+94%
+90%
+0.5
+TL
+TL
+20.45
+100%
+100%
+91%
+85%
+Table 1. Computational results for the MNLCLP ap-
+proaches.
+
+Location of Leak Detection Devices
+25
+CPU Time (secs)
+Unsolved
+GAP (%)
+instance
+|V |
+|E|
+γ
+R
+PSNLCLP
+PSNLCLP 1
+PSNLCLP
+PSNLCLP
+gessler
+12
+14
+0.5
+0.1
+TL
+19.15
+100%
+96%
+0.25
+TL
+6.28
+100%
+89%
+0.5
+TL
+1.90
+100%
+75%
+0.75
+0.1
+TL
+30.27
+100%
+98%
+0.25
+TL
+10.94
+100%
+93%
+0.5
+TL
+3.39
+100%
+86%
+1
+0.1
+TL
+39.76
+100%
+97%
+0.25
+TL
+13.67
+100%
+93%
+0.5
+TL
+4.74
+100%
+89%
+jilin
+28
+34
+0.5
+0.1
+TL
+26.40
+100%
+96%
+0.25
+TL
+13.29
+100%
+87%
+0.5
+TL
+3.44
+100%
+67%
+0.75
+0.1
+OoM
+54.77
+100%
+-
+0.25
+TL
+20.00
+100%
+95%
+0.5
+TL
+4.60
+100%
+88%
+1
+0.1
+OoM
+78.69
+100%
+-
+0.25
+TL
+24.36
+100%
+97%
+0.5
+TL
+7.05
+100%
+95%
+richmond 48
+44
+0.5
+0.1
+TL
+57.79
+100%
+94%
+0.25
+TL
+14.49
+100%
+92%
+0.5
+TL
+3.90
+100%
+71%
+0.75
+0.1
+OoM
+91.99
+100%
+-
+0.25
+TL
+21.33
+100%
+93%
+0.5
+TL
+5.62
+100%
+91%
+1
+0.1
+OoM
+116.10
+100%
+-
+0.25
+TL
+25.68
+100%
+94%
+0.5
+TL
+7.67
+100%
+96%
+foss
+37
+58
+0.5
+0.1
+TL
+41.95
+100%
+96%
+0.25
+TL
+14.21
+100%
+92%
+0.5
+TL
+6.83
+100%
+75%
+0.75
+0.1
+OoM
+111.96
+100%
+-
+0.25
+TL
+26.74
+100%
+97%
+0.5
+TL
+11.54
+100%
+94%
+1
+0.1
+OoM
+230.93
+100%
+-
+0.25
+OoM
+61.73
+100%
+-
+0.5
+OoM
+19.59
+100%
+-
+Table 2. Computational results for the PSNLCLP ap-
+proaches.
+6. Conclusions and Future Research
+In this paper we study a covering location problem with direct application
+to the determination of optimal positions of leak detection devices in urban
+pipeline networks. We propose a general framework for two different versions
+of the problem. On the one hand, in case the number of devices is known, we
+derive the Maximal Network Length Covering Location problem whose goal
+is to maximize the length of the network for which the device is able to detect
+the leak. On the other hand, in case the number of devices is unknown, the
+Partial Set Network Length Covering Location Problem aims to minimize
+
+26
+V. BLANCO and M. MART´INEZ-ANT´ON
+Figure 9. Average deviations of the cluster and sequential
+approach with respect MNLCLP (left) and sequential ap-
+proach for PSNLCLP.
+the number of devices to locate to be able to detect the leaks in a given
+percent of the length of the network. We derive mathematical optimization
+formulations for the problem and different math-heuristic algorithms. We
+run our models on different real-world urban water supply pipeline networks
+and compare the performance of the different proposals.
+Future research lines in the topic include the incorporation of more so-
+phisticated coverage shapes for the devices, as non-convex shapes obtained
+by the union of different polyhedral and ℓτ-norm balls. It would require
+a further study of τ-order cone constraints, as well as the representation
+of the union by means of disjunctive constraints, being then a challenge to
+provide solutions for real-world networks. In this case, it would be advis-
+able to design efficient heuristic approaches able to adequately scale to large
+networks.
+Acknowledgements
+The authors of this research acknowledge financial support by the Span-
+ish Ministerio de Ciencia y Tecnologia, Agencia Estatal de Investigacion
+and Fondos Europeos de Desarrollo Regional (FEDER) via project PID2020-
+114594GB-C21. The authors also acknowledge partial support from projects
+FEDER-US-1256951, Junta de Andaluc´ıa P18-FR-1422, P18-FR-2369, B-
+FQM-322-UGR20, NetmeetData: Ayudas Fundaci´on BBVA a equipos de in-
+vestigaci´on cient´ıfica 2019, and the IMAG-Maria de Maeztu grant CEX2020-
+001105-M /AEI /10.13039/501100011033. The first author also acknowl-
+edges the financial support of the European Union-Next GenerationEU through
+the program“Ayudas para la Recualificaci´on del Sistema Universitario Espa˜nol
+2021-2023”.
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+Email address: [email protected]
+

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+GeNeDis manuscript No.
+(will be inserted by the editor)
+An Architecture For Cooperative Mobile Health Applications
+Georgios Drakopoulos · Phivos Mylonas ·
+Spyros Sioutas
+Received: date / Accepted: date
+Abstract Mobile health applications are steadily gaining momentum in the modern
+world given the omnipresence of various mobile or WiFi connections. Given that
+the bandwidth of these connections increases over time, especially in conjunction
+with advanced modulation and error-correction codes, whereas the latency drops, the
+cooperation between mobile applications becomes gradually easier. This translates
+to reduced computational burden and heat dissipation for each isolated device at the
+expense of increased privacy risks. This chapter presents a configurable and scalable
+edge computing architecture for cooperative digital health mobile applications.
+Keywords Digital health · Edge computing · Mobile computing · Mobile applica-
+tions · Cooperative applications · Higher order statistics
+Mathematics Subject Classification (2010) 05C12 · 05C20 · 05C80 · 05C85
+1 Introduction
+Mobile smart applications for monitoring human health or processing health-related
+data are increasing lately at an almost geometric rate. This can be attributed to a com-
+bination of social and technolgical factors. The accumulated recent multidisciplinary
+research on biosignals and the quest for improved biomarkers bore fruits in the form
+of advanced bisignal processing algorithms. Smartphone applications are progres-
+sively becoming popular in all age groups, albeit with a different rate for each such
+group and, moreover, mobile subscribers tend to be more willing to provide sensitive
+Georgios Drakopoulos and Phivos Mylonas
+Department of Informatics, Ionian University, Greece
+E-mail: {c16drak, fmylonas}@ionio.gr
+Spyros Sioutas
+Computer Engineering and Informatics Department, University of Patras, Greece
+E-mail: [email protected]
+arXiv:2301.04720v1  [cs.SI]  11 Jan 2023
+
+2
+Drakopoulos et al.
+health data such are heart beat rate, blood pressure, or eye sight status to applica-
+tions for processing. Thus, not only technological but also financial factors favor the
+development of digital health applications.
+The primary contribution of this chapter is a set of guidelines towards a cross-layer
+cooperative architecture for mobile health applications. The principal motivation be-
+hind them are increased parallelism, and consequently lower turnaround or wallclock
+time, additional redundancy, which translates to higher reliability, and lower energy
+consumption. All these factors are critical for mobile health applications.
+The remaining of this chapter is structured as follows. Section 2 briefly summarizes
+the recent scientific literature in the fields of edge computing, mobile applications,
+mobile services, and digital health applications. Section 3 presents the proposed ar-
+chitecture. Finally, section 4 recapitulates the main points of this chapter. The notation
+of this chapter is shown at table 1.
+Table 1 Notation of this chapter.
+Symbol
+Meaning
+△=
+Definition or equality by definition
+{s1,...,sn}
+Set comprising of elements s1,...,sn
+|S| or |{s1,...,sn}|
+Cardinality of set S
+E[X]
+Mean value of random variable X
+Var[X]
+Variance of random variable X
+γ1
+Skewness coefficient
+2 Previous Work
+Mobile health applications cover a broad spectrum of cases as listed for instance
+in Sunyaev et al. (2014) or in Fox and Duggan (2010). These include pregnancy
+as described in Banerjee et al. (2013), heart beat as mentioned in Steinhubl et al.
+(2013), and blood pressure as stated in Logan et al. (2007). Using mobile health
+applictions results from increased awareness of the digital health potential as Rich
+and Miah (2014) claims. A major driver for the latter is the formation of thematically
+related communities in online social media as stated in Ba and Wang (2013). Another
+factor accounting for the popularity as well as for the ease of health applications is
+gamification as found in Lupton (2013) and Pagoto and Bennett (2013), namely the
+business methodologies relying on gaming elements as their names suggest - see for
+instance Deterding et al. (2011a), Deterding et al. (2011b), or Huotari and Hamari
+(2012). Gamification can already be found at the very core of such applications as
+described in Cugelman (2013).
+
+An Architecture For Cooperative Mobile Health Applications
+3
+The processing path of any digital health may take several forms as shown in Ser-
+banati et al. (2011). For an overview of recent security practices for mobile health
+applications see Papageorgiou et al. (2018). Path analysis as in Kanavos et al. (2017)
+play a central role in graph mining in various contexts, for instance in social networks
+as in Drakopoulos et al. (2017). Finally, the advent of advanced GPU technologies
+can lead to more efficient graph algorithms as in Drakopoulos et al. (2018).
+Finally, although it has been only very recently enforced (May 2018), GDPR, the
+EU directive governing the collection, processing, and sharing of sensitive personal
+information, seems to be already shaping more transparent conditions the smartphone
+applications ecosystem is adapting to. In fact, despite the original protests that GDPR
+may be excessively constraining under certain circumstances described in Charitou
+et al. (2018), consumers seem to trust mobile applications which clearly outline their
+intentions concerning any collected piece of personal information as Bachiri et al.
+(2018) found out.
+3 Architecture
+This section presents and analyzes the proposed cooperative architecture for mobile
+digital health applications. Figure 1 visualizes an instance of a mobile health appli-
+cation running on a smartphone and a number of peers which can be reached either
+by WiFi or by regular mobile services.
+client
+peer1
+peer2
+peer3
+peer4
+peer5
+peer6
+peer7
+WiFi
+BS
+analytics
+db
+WiFi connection
+mobile service connection
+local connection
+Fig. 1 Instance of a mobile application surrounded by peers.
+
+4
+Drakopoulos et al.
+As with the majority of mobile architectures, the proposed architecture is concep-
+tually best described with graphs, as concepts such as connectivity and community
+structure can be naturally expressed. To this end, the cell phones, the base stations,
+and the WiFi access points are represented as vertices, each device category being
+represented as a different type. Moreover, connections between these are represented
+as edges, where each edge is also of different type depending on the connection.
+These can be easily programmed in a graph database like Neo4j.
+The general constraints that will be the basis for the subsequent analysis are as
+follows:
+– Assume that a mobile health application monitoring a biomarker or a biosignal
+must deliver results every T0 time units, usually seconds. Additionally assuming
+that the required computation can be split into n+1 parts to be distributed to the
+available n neighbors, then:
+Ta +Tp +Ts +2Tc ≤ T0
+(1)
+Where Ta, Tp, Ts, and Tc denote respectively the time required for analysis, namely
+breaking down the computation and assigning each neighbor a task, processing,
+namely the time of the slowest task, synthesis, namely assemblying the solution
+of each task to create the general solution, and communication. The latter term
+counts twice as the data and the task need to be communicated and then the results
+need to be collected.
+– In mobile communications is of paramount importance the minimization of the
+energy dedicated to a single task. In general the relationship between a given
+task and the energy spent for its accomplishment is unknown. However, given
+that tasks have a short duration, it is fairly reasonable to assume that the same
+function f(·) links the task and the energy at each neighbor. Then the following
+inequality should also be satisfied:
+(n+1) f(Tp)+ f(Ta)+ f(Ts)+2(n+1) f(Tc) ≤ f(T0) ⇔
+f(T0)− f(Ta)− f(Ts)
+f(Tp)+2(Tc)
+−1 ≥ n
+(2)
+Given the fundamental constraints (1) and (2), let us estimate the key parameter Tc,
+since Ta, Ts, and Tp depend on the problem and T0 is a constraint.
+Let ei, j denote the communication link between vertices vi and vj has a given ca-
+pacity Ci, j as well as a propagation delay τi, j. Then, the number of bits bi, j which can
+be transmitted over edge ei, j in a time slot of length τ0 is, assuming the variables are
+expressed in the proper units:
+bi, j = Ci, j(τ0 −τi, j)
+(3)
+If the link delay τi, j is expressed as a percentage 0 < ρτ
+i, j < 1 of the time slot τ0, then:
+bi, j = Ci, jτ0
+�
+1−ρτ
+i, j
+�
+(4)
+Note that the case ρτ
+i, j = 0 represents a near physical impossibility, whereas the case
+ρτ
+i, j = 1 denotes either a useless link or a misconfigured network protocol.
+
+An Architecture For Cooperative Mobile Health Applications
+5
+In a similar way, if C0 is the maximum capacity, then each Ci, j can be expressed as
+a percentage 0 < ρC
+i, j ≤ 1 of the former. Thus:
+bi, j = C0τ0ρC
+i,j
+�
+1−ρτ
+i,j
+�
+= B0ρC
+i, j
+�
+1−ρτ
+i, j
+�
+(5)
+Note that in this case ρC
+i, j can be 1, unless C0 is an asymptotically upper limit. There-
+fore, if for the given task Bi, j bits must be transmitted, then the total number of slots
+for that particular link is:
+Ti,j =
+�Bi,j
+bi, j
+�
+(6)
+At this point, we can estimate Tc as:
+Tc
+△=E[Ti, j]
+(7)
+Furthermore, we can use the distribution of Tc to determine whether a big task
+should be subdivided to smaller tasks. Assuming τ0 is constant, then it can be used as
+a reference point to consider the frequency distribution of Ti, j, which can be treated
+as a probability distribution.
+For any random variable X is possible to define the skewness coefficient γ1 as:
+γ1
+△=E
+�
+X −E[X]
+�
+Var[X]
+�
+= E
+�
+X3�
+−3E[X]Var[X]−E[X]3
+Var[X]
+3
+2
+(8)
+In equation (8) E[X] and Var[X] stand for the stochastic mean and variance of X
+respectively. In actual settings these can be replaced by their sample counterparts and
+they can be updated as new measurements are collected. In the derivation of the right
+hand side of (8) the following properties were used:
+E
+�
+n
+∑
+k=1
+αkXk +α0
+�
+=
+n
+∑
+k=1
+αkE[Xk]+α0
+Var[α1X +α0] = α2
+1Var[X]
+(9)
+The skewness sign indicates the shape of the distribution. When γ1 is negative, then
+X takes larger values with higher probability, whereas when γ1 is positive, then X
+is more likely to take lower values. Finally, in the case where γ1 is zero, then the
+distribution of X is symmetric, as for instance in the case of the binomial distribution.
+Therefore, positive values of the skewness coefficient γ1 for the distribution of Tc of
+the channel delays indicate that it is more likely more time to be available for useful
+information transmission.
+The proposed methodology is summarized in algorithm 1.
+
+6
+Drakopoulos et al.
+Algorithm 1 The proposed scheme.
+Require: Knowledge of T0, Ts, Ta, and Tp.
+Ensure: A cooperative computation takes place.
+1: repeat
+2:
+update estimates for
+�
+Ti, j
+�
+3:
+if equations (2) and (9) are satisfied then
+4:
+break the problem into tasks
+5:
+end if
+6:
+communicate tasks
+7:
+compute tasks
+8:
+collect results
+9:
+compose answer
+10: until true
+4 Conclusions
+This chapter presents a probabilistic architecture for cooperative computation in
+mobile health app settings. It relies on a higher order statistical criterion, namely the
+skewness coefficient of the number of slots which are suitable for communication, in
+order to estimate whether a computation can be broken into smaller tasks and com-
+municated to neighboring smartphones over WiFi or the ordinary cell network. Once
+the tasks are complete, the results are collected back at the controling smartphone
+and an answer is generated using a synthesis of these results.
+In order to find the hard limits of the proposed architecture and to assess its per-
+formance under various operational scenaria, a number of simulations must be run in
+addition to theoretical probabilistic analysis. Additionally, more conditions should be
+added to the architecture, for instance what happens when a neighboring smartphone
+stops working or is moved out of range. Moreover, conditions for duplicating certain
+critical computation must also be created.
+Acknowledgements This chapter is part of Tensor 451, a long term research initiative whose primary
+objective is the development of novel, scalable, numerically stable, and interpretable tensor analytics.
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+

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+page_content='GeNeDis manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  (will be inserted by the editor)  An Architecture For Cooperative Mobile Health Applications  Georgios Drakopoulos · Phivos Mylonas ·  Spyros Sioutas  Received: date / Accepted: date  Abstract Mobile health applications are steadily gaining momentum in the modern  world given the omnipresence of various mobile or WiFi connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Given that  the bandwidth of these connections increases over time, especially in conjunction  with advanced modulation and error-correction codes, whereas the latency drops, the  cooperation between mobile applications becomes gradually easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' This translates  to reduced computational burden and heat dissipation for each isolated device at the  expense of increased privacy risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' This chapter presents a configurable and scalable  edge computing architecture for cooperative digital health mobile applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  Keywords Digital health · Edge computing · Mobile computing · Mobile applica-  tions · Cooperative applications · Higher order statistics  Mathematics Subject Classification (2010) 05C12 · 05C20 · 05C80 · 05C85  1 Introduction  Mobile smart applications for monitoring human health or processing health-related  data are increasing lately at an almost geometric rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' This can be attributed to a com-  bination of social and technolgical factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' The accumulated recent multidisciplinary  research on biosignals and the quest for improved biomarkers bore fruits in the form  of advanced bisignal processing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Smartphone applications are progres-  sively becoming popular in all age groups, albeit with a different rate for each such  group and, moreover, mobile subscribers tend to be more willing to provide sensitive  Georgios Drakopoulos and Phivos Mylonas  Department of Informatics, Ionian University, Greece  E-mail: {c16drak, fmylonas}@ionio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='gr  Spyros Sioutas  Computer Engineering and Informatics Department, University of Patras, Greece  E-mail: sioutas@ceid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='upatras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='gr  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='04720v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='SI]  11 Jan 2023  2  Drakopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  health data such are heart beat rate, blood pressure, or eye sight status to applica-  tions for processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Thus, not only technological but also financial factors favor the  development of digital health applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  The primary contribution of this chapter is a set of guidelines towards a cross-layer  cooperative architecture for mobile health applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' The principal motivation be-  hind them are increased parallelism, and consequently lower turnaround or wallclock  time, additional redundancy, which translates to higher reliability, and lower energy  consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' All these factors are critical for mobile health applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  The remaining of this chapter is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Section 2 briefly summarizes  the recent scientific literature in the fields of edge computing, mobile applications,  mobile services, and digital health applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Section 3 presents the proposed ar-  chitecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Finally, section 4 recapitulates the main points of this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' The notation  of this chapter is shown at table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  Table 1 Notation of this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  Symbol  Meaning  △=  Definition or equality by definition  {s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=',sn}  Set comprising of elements s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=',sn  |S| or |{s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=',sn}|  Cardinality of set S  E[X]  Mean value of random variable X  Var[X]  Variance of random variable X  γ1  Skewness coefficient  2 Previous Work  Mobile health applications cover a broad spectrum of cases as listed for instance  in Sunyaev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' (2014) or in Fox and Duggan (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' These include pregnancy  as described in Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' (2013), heart beat as mentioned in Steinhubl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  (2013), and blood pressure as stated in Logan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Using mobile health  applictions results from increased awareness of the digital health potential as Rich  and Miah (2014) claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' A major driver for the latter is the formation of thematically  related communities in online social media as stated in Ba and Wang (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Another  factor accounting for the popularity as well as for the ease of health applications is  gamification as found in Lupton (2013) and Pagoto and Bennett (2013), namely the  business methodologies relying on gaming elements as their names suggest - see for  instance Deterding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' (2011a), Deterding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' (2011b), or Huotari and Hamari  (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Gamification can already be found at the very core of such applications as  described in Cugelman (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  An Architecture For Cooperative Mobile Health Applications  3  The processing path of any digital health may take several forms as shown in Ser-  banati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' For an overview of recent security practices for mobile health  applications see Papageorgiou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Path analysis as in Kanavos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' (2017)  play a central role in graph mining in various contexts, for instance in social networks  as in Drakopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Finally, the advent of advanced GPU technologies  can lead to more efficient graph algorithms as in Drakopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  Finally, although it has been only very recently enforced (May 2018), GDPR, the  EU directive governing the collection, processing, and sharing of sensitive personal  information, seems to be already shaping more transparent conditions the smartphone  applications ecosystem is adapting to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' In fact, despite the original protests that GDPR  may be excessively constraining under certain circumstances described in Charitou  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' (2018), consumers seem to trust mobile applications which clearly outline their  intentions concerning any collected piece of personal information as Bachiri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  (2018) found out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  3 Architecture  This section presents and analyzes the proposed cooperative architecture for mobile  digital health applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Figure 1 visualizes an instance of a mobile health appli-  cation running on a smartphone and a number of peers which can be reached either  by WiFi or by regular mobile services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  client  peer1  peer2  peer3  peer4  peer5  peer6  peer7  WiFi  BS  analytics  db  WiFi connection  mobile service connection  local connection  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' 1 Instance of a mobile application surrounded by peers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  4  Drakopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  As with the majority of mobile architectures, the proposed architecture is concep-  tually best described with graphs, as concepts such as connectivity and community  structure can be naturally expressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' To this end, the cell phones, the base stations,  and the WiFi access points are represented as vertices, each device category being  represented as a different type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Moreover, connections between these are represented  as edges, where each edge is also of different type depending on the connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  These can be easily programmed in a graph database like Neo4j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  The general constraints that will be the basis for the subsequent analysis are as  follows:  – Assume that a mobile health application monitoring a biomarker or a biosignal  must deliver results every T0 time units, usually seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Additionally assuming  that the required computation can be split into n+1 parts to be distributed to the  available n neighbors, then:  Ta +Tp +Ts +2Tc ≤ T0  (1)  Where Ta, Tp, Ts, and Tc denote respectively the time required for analysis, namely  breaking down the computation and assigning each neighbor a task, processing,  namely the time of the slowest task, synthesis, namely assemblying the solution  of each task to create the general solution, and communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' The latter term  counts twice as the data and the task need to be communicated and then the results  need to be collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  – In mobile communications is of paramount importance the minimization of the  energy dedicated to a single task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' In general the relationship between a given  task and the energy spent for its accomplishment is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' However, given  that tasks have a short duration, it is fairly reasonable to assume that the same  function f(·) links the task and the energy at each neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Then the following  inequality should also be satisfied:  (n+1) f(Tp)+ f(Ta)+ f(Ts)+2(n+1) f(Tc) ≤ f(T0) ⇔  f(T0)− f(Ta)− f(Ts)  f(Tp)+2(Tc)  −1 ≥ n  (2)  Given the fundamental constraints (1) and (2), let us estimate the key parameter Tc,  since Ta, Ts, and Tp depend on the problem and T0 is a constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  Let ei, j denote the communication link between vertices vi and vj has a given ca-  pacity Ci, j as well as a propagation delay τi, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' the number of bits bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' j which can  be transmitted over edge ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' j in a time slot of length τ0 is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' assuming the variables are  expressed in the proper units:  bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' j = Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' j(τ0 −τi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' j)  (3)  If the link delay τi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' j is expressed as a percentage 0 < ρτ  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' j < 1 of the time slot τ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' then:  bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' j = Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' jτ0  �  1−ρτ  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' j  �  (4)  Note that the case ρτ  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' j = 0 represents a near physical impossibility,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' whereas the case  ρτ  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' j = 1 denotes either a useless link or a misconfigured network protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  An Architecture For Cooperative Mobile Health Applications  5  In a similar way, if C0 is the maximum capacity, then each Ci, j can be expressed as  a percentage 0 < ρC  i, j ≤ 1 of the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Thus:  bi, j = C0τ0ρC  i,j  �  1−ρτ  i,j  �  = B0ρC  i, j  �  1−ρτ  i, j  �  (5)  Note that in this case ρC  i, j can be 1, unless C0 is an asymptotically upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' There-  fore, if for the given task Bi, j bits must be transmitted, then the total number of slots  for that particular link is:  Ti,j =  �Bi,j  bi, j  �  (6)  At this point, we can estimate Tc as:  Tc  △=E[Ti, j]  (7)  Furthermore, we can use the distribution of Tc to determine whether a big task  should be subdivided to smaller tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Assuming τ0 is constant, then it can be used as  a reference point to consider the frequency distribution of Ti, j, which can be treated  as a probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  For any random variable X is possible to define the skewness coefficient γ1 as:  γ1  △=E  �  X −E[X]  �  Var[X]  �  = E  �  X3�  −3E[X]Var[X]−E[X]3  Var[X]  3  2  (8)  In equation (8) E[X] and Var[X] stand for the stochastic mean and variance of X  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' In actual settings these can be replaced by their sample counterparts and  they can be updated as new measurements are collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' In the derivation of the right  hand side of (8) the following properties were used:  E  �  n  ∑  k=1  αkXk +α0  �  =  n  ∑  k=1  αkE[Xk]+α0  Var[α1X +α0] = α2  1Var[X]  (9)  The skewness sign indicates the shape of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' When γ1 is negative, then  X takes larger values with higher probability, whereas when γ1 is positive, then X  is more likely to take lower values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Finally, in the case where γ1 is zero, then the  distribution of X is symmetric, as for instance in the case of the binomial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  Therefore, positive values of the skewness coefficient γ1 for the distribution of Tc of  the channel delays indicate that it is more likely more time to be available for useful  information transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  The proposed methodology is summarized in algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  6  Drakopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  Algorithm 1 The proposed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  Require: Knowledge of T0, Ts, Ta, and Tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  Ensure: A cooperative computation takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  1: repeat  2:  update estimates for  �  Ti, j  �  3:  if equations (2) and (9) are satisfied then  4:  break the problem into tasks  5:  end if  6:  communicate tasks  7:  compute tasks  8:  collect results  9:  compose answer  10: until true  4 Conclusions  This chapter presents a probabilistic architecture for cooperative computation in  mobile health app settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' It relies on a higher order statistical criterion, namely the  skewness coefficient of the number of slots which are suitable for communication, in  order to estimate whether a computation can be broken into smaller tasks and com-  municated to neighboring smartphones over WiFi or the ordinary cell network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Once  the tasks are complete, the results are collected back at the controling smartphone  and an answer is generated using a synthesis of these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  In order to find the hard limits of the proposed architecture and to assess its per-  formance under various operational scenaria, a number of simulations must be run in  addition to theoretical probabilistic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Additionally, more conditions should be  added to the architecture, for instance what happens when a neighboring smartphone  stops working or is moved out of range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Moreover, conditions for duplicating certain  critical computation must also be created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  Acknowledgements This chapter is part of Tensor 451, a long term research initiative whose primary  objective is the development of novel, scalable, numerically stable, and interpretable tensor analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  References  Ba S, Wang L (2013) Digital health communities: The effect of their motivation  mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Decision Support Systems 55(4):941–947  Bachiri M, Idri A, Fern´andez-Alem´an JL, Toval A (2018) Evaluating the privacy  policies of mobile personal health records for pregnancy monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Journal of  medical systems 42(8):144  Banerjee A, Chen X, Erman J, Gopalakrishnan V, Lee S, Van Der Merwe J (2013)  MOCA: A lightweight mobile cloud offloading architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' In: Proceedings of the  eighth ACM international workshop on Mobility in the evolving internet architec-  ture, ACM, pp 11–16  An Architecture For Cooperative Mobile Health Applications  7  Charitou C, Kogias DG, Polykalas SE, Patrikakis CZ, Cotoi IC (2018) Use of apps  for crime reporting and the EU General Data Protection Regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Societal Im-  plications of Community-Oriented Policing and Technology pp 55–61  Cugelman B (2013) Gamification: What it is and why it matters to digital health  behavior change developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' JMIR Serious Games 1(1)  Deterding S, Dixon D, Khaled R, Nacke L (2011a) From game design elements to  gamefulness: Defining gamification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' In: Proceedings of the 15th international aca-  demic MindTrek conference: Envisioning future media environments, ACM, pp  9–15  Deterding S, Sicart M, Nacke L, O’Hara K, Dixon D (2011b) Gamification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' using  game-design elements in non-gaming contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' In: CHI’11 extended abstracts on  human factors in computing systems, ACM, pp 2425–2428  Drakopoulos G, Kanavos A, Mylonas P, Sioutas S (2017) Defining and evaluating  Twitter influence metrics: A higher order approach in Neo4j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' SNAM 71(1)  Drakopoulos G, Liapakis X, Tzimas G, Mylonas P (2018) A graph resilience metric  based on paths: Higher order analytics with GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' In: ICTAI, IEEE  Fox S, Duggan M (2010) Mobile health 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Pew Internet and American Life  Project Washington, DC  Huotari K, Hamari J (2012) Defining gamification: a service marketing perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content='  In: Proceedings of the 16th international academic MindTrek conference, ACM,  pp 17–22  Kanavos A, Drakopoulos G, Tsakalidis A (2017) Graph community discovery algo-  rithms in Neo4j with a regularization-based evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' In: WEBIST  Logan AG, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' (2007) Mobile phone–based remote patient monitoring system for  management of hypertension in diabetic patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' American Journal of Hyperten-  sion 20(9):942–948  Lupton D (2013) The digitally engaged patient: Self-monitoring and self-care in the  digital health era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
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+page_content=' IEEE Access 6:9390–9403  Rich E, Miah A (2014) Understanding digital health as public pedagogy: A critical  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Societies 4(2):296–315  Serbanati LD, Ricci FL, Mercurio G, Vasilateanu A (2011) Steps towards a digital  health ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' Journal of Biomedical Informatics 44(4):621–636  Steinhubl SR, Muse ED, Topol EJ (2013) Can mobile health technologies transform  health care?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
+page_content=' JAMA 310(22):2395–2396  Sunyaev A, Dehling T, Taylor PL, Mandl KD (2014) Availability and quality of mo-  bile health app privacy policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNE3T4oBgHgl3EQfygta/content/2301.04720v1.pdf'}
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+Network-theoretic modeling of fluid-structure interactions
+Aditya G. Nair1∗, Samuel B. Douglass1
+1 Department of Mechanical Engineering, University of Nevada, Reno, NV 89557
+Abstract
+The coupling interactions between deformable structures and unsteady fluid flows occur across a wide
+range of spatial and temporal scales in many engineering applications. These fluid-structure interactions
+(FSI) make it challenging to predict flow physics accurately. In the present work, two multi-layer network
+approaches are proposed that characterize the interactions between the fluid and structural layers for an
+incompressible laminar flow over a two-dimensional compliant flat plate at a 35-degrees angle of attack.
+In one approach, the wake vortices and bound vortexlets form the nodes of the network with the edges
+defined by induced velocity. In the other approach, coherent structures (fluid modes) contributing to
+the kinetic energy of the flow and structural modes contributing to the kinetic energy of the compliant
+structure constitute the network nodes. The energy transfers between the modes are extracted using a
+perturbation approach. The network structure of the FSI system is further simplified using the community
+detection algorithm in the vortical approach and by selecting dominant modes in the modal approach.
+Network measures are used to reveal the temporal behavior of the individual nodes within the simplified
+FSI system. Predictive models are then built using both data-driven and physics-based methods. We
+conclude by investigating the controllability of the modal interaction network.
+This work sets the
+foundation for network-theoretic reduced-order modeling of fluid-structure interactions, generalizable to
+other multi-physics systems.
+1
+Introduction
+Fluid-structure interactions (FSI) occur in many engineering applications and over many spatial and temporal
+scales from aircraft and buildings to heart valves and insect wings. In fact, any compliant structure immersed
+in a fluid flow result in fluid-structure interaction. These interactions are often transitory in nature and lead
+to the rich dynamical behavior of the fluid and structural components. For flight systems with compliant
+wings, the structure can extract energy from the air stream leading to an unstable self-excited vibration called
+flutter, which is not only difficult to predict but can have catastrophic effects such as potential structural
+failure [1–3]. In fact, the slender and high aspect ratio wings of High Altitude Long Endurance aircraft are
+highly prone to flutter [4–6]. The situation is similar for wind turbines, where increasing the aspect ratios
+driven by increases in turbine name-plate capacity leads to a higher likelihood of flutter [7]. Furthermore,
+as the utility of a wind turbine is to extract energy from the wind, any energy lost to or because of blade
+distortion is energy that could have been used to turn the generator. Active flutter alleviation systems which
+take advantage of the knowledge of the system interactions are of significant interest as they provide the
+potential for significant weight savings when compared to traditional flutter-resistant structures [8].
+Interest in FSI extends to smaller scales as well. Agile natural flyers such as insects and birds are able to
+maneuver in unsteady aerodynamic environments. Because many insects are unable to fully articulate their
+wings, wing compliance plays a crucial role in the generation of flight forces [9–12]. This provides insights
+into the design and control of autonomous flight vehicles [13, 14], a topic of tremendous engineering interest.
+Because of the prevalence of FSI and the potential for catastrophic phenomena, significant effort has been
+made in modeling and predicting their behavior [15]. Efforts have ranged from simple analytical methods and
+semi-empirical equations of prediction [16] to computationally-intensive high-fidelity numerical simulations
+[17, 18]. Perhaps the most commonly used analytic approach is Theodorsen’s model which was motivated
+∗ Corresponding author ([email protected]).
+1
+arXiv:2301.01314v1  [physics.flu-dyn]  3 Jan 2023
+
+by the importance of understanding wing vibrations and flutter in the early years of flight. Improvements
+have been made to the model in recent years including semi-empirical formulations [19], state-space models
+[20, 21] and insights from careful experiments [22, 23]. Vortex methods can be coupled to low-fidelity
+structural models to build fast solvers, but their speed comes at the expense of ignoring viscous and
+compressible effects in the flow [24–27]. In high-fidelity simulations, it is common to employ partitioned
+solvers for each component physics which are then coupled using implicit or explicit coupling schemes [28].
+This often increases the computational cost and the likelihood of numerical stability issues compared to
+simulating each system separately [29–31]. In the present work, we propose two separate mathematical
+frameworks for modeling coupled fluid-structure systems with a specific focus on capturing the interactions
+between the two systems.
+Network science and graph theory provide a concise and powerful mathematical framework for the
+interactions between actors within a system. In a network representation of a system, actors within the
+system are represented as nodes, and the interaction between the nodes (actors) is represented by edges
+connecting them. Mathematically, a network is represented by a graph G = (V, E, W) where nodes V
+are connected via edges E, each with an associated edge weight W [32]. Despite their widespread use
+in social sciences [33–35], biology [36, 37], computer science [38], network science has not permeated in
+physics and engineering until recently. Aside from promising work in fluid mechanics [39, 40] and the study
+of thermoacoustic combustion instabilities [41], networks have seen little application in systems involving
+multiple physics.
+This work aims to build a scaffolding of network-based approaches for modeling FSI systems. The
+advantage of the network approach is that it naturally allows for the incorporation of physics-based insights
+in data-driven system identification strategies [42] such as those based on proper orthogonal decomposition
+[43], dynamic mode decomposition [44], and eigensystem realization algorithm [45]. The approach also
+naturally lends itself to the systematic reduction of the physical system via community detection [46–48]
+and graph sparsification algorithms [49, 50], along with identifying the key nodes controlling the system
+dynamics [51, 52].
+In this work, we present two approaches for modeling FSI using a network-based framework. The first
+approach characterizes the vortical interactions in FSI with the network nodes in the fluid and structure
+domains defined by discrete point vortices. The edge weights are based on the induced velocity of these
+point vortices [50]. We also introduce a modal network representation of FSI where the network nodes
+are given by coherent spatial modes of the unsteady fluid flow and velocity modes of the structure. Data
+collected from perturbations of the structural modes are used to determine the interaction strengths (edge
+weights) between the nodes [53]. Both approaches not only highlight interactions within each component
+part of FSI but also extracts the cross-coupling interactions in the form of a multilayer network [54], i.e., one
+network layer for the fluid and one for the structure.
+We demonstrate the network modeling approaches for a two-dimensional laminar flow over a compliant
+flat plate at an angle of attack 𝛼 = 35◦. A similar problem was investigated in the work by Hickner et al.[21]
+for developing data-driven system identification models. However, system identification in that work was
+restricted to flows in the steady regime with the angle of attack below the critical angle of attack of 𝛼 = 27◦.
+In this work, we analyze the FSI interactions in the unsteady regime as well as those on the introduction of
+large disturbances to the flow caused by gust encounters. We discuss the numerical setup and methods in
+section 2, results in section 3, and offer concluding remarks in section 4.
+2
+
+2
+Methods
+2.1
+Direct numerical simulation
+We perform direct numerical simulations of two-dimensional incompressible laminar flow over a thin
+deforming flat plate of length 𝑐 at an angle of attack of 𝛼 = 35◦. These simulations are performed using
+the strongly-coupled immersed boundary method [55, 56]. The solver uses a multi-domain technique to
+accelerate the computations [57]. We use five grid levels with the innermost domain fixed at −0.2 ≤ 𝑥/𝑐 ≤ 1.8
+and −1 ≤ 𝑦/𝑐 ≤ 1, with a grid spacing of △𝑥/𝑐 ≈ 0.0077. Grid convergence studies for a similar setup
+were reported in Hickner et al. [21]. Uniform flow with free-stream velocity 𝑈∞ is prescribed at the far-field
+boundaries. An explicit Adam-Bashforth method is used for the discretization of the advection term and
+an implicit Crank-Nicolson scheme is used for the viscous terms of the governing equations. The flat plate
+is evolved using the Euler–Bernoulli equation with a co-rotational finite element discretization. Such a
+co-rotational form allows for large displacements of the structure. The plate is discretized into 65 elements
+(66 points) with the leading edge pinned at (𝑥/𝑐, 𝑦/𝑐) = (0, 0).
+The FSI system is characterized by three non-dimensional parameters: Reynolds number 𝑅𝑒 = 𝑈∞𝑐/𝜈,
+mass ratio 𝑀𝜌 = 𝜌𝑠ℎ
+𝜌 𝑓 𝑐 = 3, and bending stiffness 𝐾𝐵 =
+𝐸𝐼
+𝜌 𝑓 𝑈2∞𝑐3 . Here, 𝜈 is the kinematic viscosity, 𝜌𝑠, and
+𝜌 𝑓 are the densities of the structure and fluid, respectively. Also, ℎ is the thickness, 𝐸 is Young’s modulus,
+and 𝐼 is the second area moment of inertia of the plate. We fix 𝑅𝑒 = 100 and 𝑀𝜌 = 3, unless otherwise
+stated. Data from numerical simulation of three different bending stiffness 𝐾𝐵 = {0.15625, 0.3125, 0.625}
+are collected [21, 58].
+We show a snapshot of vorticity in the top panel of Figure 1(a) and the flow field parameters and domain
+setup of the structure in the bottom panel. The setup also highlights the position of a rigid body at an angle
+of attack of 𝛼 = 35◦ along with the deflected position for a complaint case. The transverse tip displacement
+△𝑦𝑡 is always negative for the cases considered in this work. By convention, we consider positive transverse
+tip displacement of the trailing edge when the plate pitches down compared to the rigid plate position. We
+also show the tip displacements for three different Reynolds numbers and the three bending stiffnesses in
+Figure 1(b). With the choice of the parameters considered, the fluctuation of the tip displacement increases
+with 𝑅𝑒 and 𝐾𝐵 and the mean tip displacement increases with 𝐾𝐵.
+2.2
+Fluid-structure vortical interaction networks
+Due to the different physical nature of the fluid and structural components and their governing dynamics, we
+model each of them into separate vortical network layers and then combine them later to form the multilayer
+network. To construct the network, we collect snapshots of data from direct numerical simulations of the
+FSI problem as described in section 2.1. We then convert this data to a network-based representation as
+illustrated below.
+The fluid network layer is created with the method already described in previous work by Taira et al.
+[51], and Meena et al. [47]. Here, spatial grid points serve as network nodes. The strength of each node is
+determined by the circulation 𝛾 𝑓
+𝑖 corresponding to the grid cell it represents. The superscript 𝑓 indicates
+the nodes in the fluid layer. We only consider nodes in the fluid layer with vorticity values greater than 1%
+of the maximum vorticity of the flow. The influence of these nodes on each other is given by their induced
+velocity. The velocity induced by node 𝑗 on node 𝑖 is given by 𝑢 𝑓
+𝑖← 𝑗 and helps define the fluid layer network
+G 𝑓 . The node 𝑖 does not induce velocity on itself. The network can be neatly summarized with an adjacency
+matrix A 𝑓 as
+𝐴 𝑓
+𝑖 𝑗 =
+�
+𝑢 𝑓
+𝑖← 𝑗
+if 𝑖 ≠ 𝑗 ∈ fluid layer
+0
+otherwise
+where
+𝑢 𝑓
+𝑖← 𝑗 =
+𝛾 𝑓
+𝑗
+2𝜋|r 𝑓
+𝑗 − r 𝑓
+𝑖 |
+.
+(1)
+3
+
+Figure 1: Direct numerical simulation of 2D laminar flow over a compliant flat plate of length 𝑐 at an angle
+of attack 𝛼 = 35◦: (a) Vorticity snapshot (top) and the numerical setup of the flat plate (bottom). (b) The
+transverse tip displacement across different Reynolds number 𝑅𝑒 and bending stiffness 𝐾𝐵.
+where r 𝑓 is the location of the grid cell. The above definition leads to a weighted, directed network. Here,
+we consider 𝑁 fluid nodes after vorticity thresholding to construct the adjacency matrix.
+Vorticity is not a natural quantity to consider when dealing with structural mechanics. However, we
+can represent the structure as a vortex line element formed of bound vortexlets, following the method by
+Mountcastle et al. [12]. In this formulation, for a flat plate, the flow separates tangentially from the trailing
+edge, enforcing the Kutta condition, no-penetration boundary condition, and Kelvin’s circulation theorem.
+We define 𝑛 control points on the structure co-located with every bound vortexlet. To calculate the strength
+of bound vortexlets 𝛾𝑠
+𝑖 (superscript 𝑠 indicates the nodes in the structural layer) corresponding to each point
+on the structure, a linear system of equations is solved using the position and velocity of the structure and
+strength (circulation) of the fluid nodes above obtained from high-fidelity numerical simulations as
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+. . .
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+���������
+�𝑣 𝑝
+1 · ˆ𝑛𝑝
+1
+�𝑣 𝑝
+2 · ˆ𝑛𝑝
+2
+...
+0
+���������
+−
+���������
+𝑀 𝑓 1,𝑝1
+. . .
+𝑀 𝑓 𝑁 ,𝑝1
+𝑀 𝑓 1,𝑝2
+. . .
+𝑀 𝑓 𝑁 ,𝑝2
+...
+...
+...
+1
+. . .
+1
+���������
+����������
+𝛾 𝑓
+1
+𝛾 𝑓
+2...
+𝛾 𝑓
+𝑁
+����������
+������
+�
+(2)
+where the mass matrix is defined as
+𝑀𝑠( 𝑓 )𝑖,𝑝 𝑗 =
+������
+−(𝑦 𝑝
+𝑗 − 𝑦𝑠( 𝑓 )
+𝑖
+)
+2𝜋(𝑟2 + 𝛿2) ,
+(𝑥 𝑝
+𝑗 − 𝑥𝑠( 𝑓 )
+𝑖
+)
+2𝜋(𝑟2 + 𝛿2)
+������
+.
+(3)
+Here, (𝑥 𝑝
+𝑖 , 𝑦 𝑝
+𝑖 ), 𝑣 𝑝
+𝑖 , ˆ𝑛𝑝
+𝑖 , are the position, velocity, and normal vector of each control point along the body,
+respectively. Also, (𝑥𝑠( 𝑓 )
+𝑗
+, 𝑦𝑠( 𝑓 )
+𝑗
+) is the position of bound (fluid) vortexlet 𝑗, 𝑟2 = (𝑥 − 𝑥𝑖)2 + (𝑦 − 𝑦𝑖)2 and
+𝛿 is a smoothing parameter preventing a divide by zero when 𝑟 = 0. We chose 𝛿 = 0.001 as the smoothing
+parameter. Experimentation shows that this value was sufficiently small so as to not significantly impact the
+results and obtain consistent vortical strengths compared to other values. The nodes of the structural layer
+4
+
+(a)
+(b)
+0.5
+0
+-0.5
+-0.3
+200
+>>>>>>
+Re
+Number
+-0.6
+10
+0
+100
+>>>>>
+Deflected plate position
+Q
+Rigid plate position
+Movement bounds
+50
+C
+-0.4
+ Transverse tip displacement
+yt
+0
+0.4
+0.15625
+0.3125
+0.625
+Compliance K Bconsist of bound vortexlets. Once again, we use induced velocity to quantify the interactions between the
+bound vortexlets which leads to the adjacency matrix A𝑠 given by
+𝐴𝑠
+𝑖 𝑗 =
+�
+𝑢𝑠
+𝑖← 𝑗
+if 𝑖 ≠ 𝑗 ∈ structure layer
+0
+otherwise
+where
+𝑢𝑠
+𝑖← 𝑗 =
+𝛾𝑠
+𝑗
+2𝜋|r𝑠
+𝑗 − r𝑠
+𝑖 |,
+(4)
+where r𝑠 are the location of points on the structure.
+An important measure that describes the global influence of the nodes in the network is the node degree
+or strength. The in-degree is defined as 𝑠in
+𝑖 = �𝑁
+𝑗=1 𝐴𝑠( 𝑓 )
+𝑖 𝑗
+, while the out-degree is given by 𝑠out
+𝑖
+= �𝑁
+𝑖=1 𝐴𝑠( 𝑓 )
+𝑖 𝑗
+.
+The nodes with the maximum out-degree influence the network the most, while those with the maximum
+in-degree get influenced the most. With the fluid and structural layers defined, we reduce each network using
+community detection before combining them into a multilayer representation.
+Community detection groups the nodes within a network to form distinct communities. Nodes with a
+community have a higher density of interactions amongst themselves than with nodes in the other commu-
+nities. We utilize the Louvain algorithm [59] to find communities that maximize modularity of the network
+[60] defined as
+𝑄 = 1
+2𝑚
+∑︁
+𝑖 𝑗
+�
+𝐴𝑠( 𝑓 )
+𝑖 𝑗
+−
+𝑠in
+𝑖 𝑠out
+𝑗
+2𝑚
+�
+𝛿(𝐶𝑖, 𝐶𝑗) ∈ [0, 1]
+(5)
+where 𝑚 is the number of nodes and 𝛿 is the Kronecker delta operating on the community labels 𝐶𝑖.
+Modularity provides a measure of the relative connectedness of a group of nodes compared to their expected
+connectedness produced by a null model. As the Louvain algorithm can only be applied to unsigned edge
+weights, we separate the fluid and structural network layers into ones that contain positive or negative edge
+weights and apply community detection.
+The results of community detection applied to one snapshot of the flow field are shown in Figure 2(a).
+The community detection of the structural layer yields 𝑛𝑐 = 3 communities while that of the fluid layer yields
+𝑁𝑐 = 6 communities. For each community, we compute the community centroid shown by the filled black
+circles. The size of the circle indicates the node degree or strength of the community centroid. Through
+community reduction, we achieved a drastic reduction in the dimensionality of the FSI system from 𝑛 = 66
+to 𝑛𝑐 = 3 for the structural layer and from 𝑁 = 67600 to 𝑁𝑐 = 6 for the fluid layer.
+Using the community centroids identified above, we now are ready to define a community-reduced
+adjacency matrix for each layer as well as a combined multilayer adjacency matrix.
+Each community
+centroid 𝑐𝑖 has an associated strength 𝛾𝑠( 𝑓 )
+𝑐𝑖
+and position (𝑥𝑠( 𝑓 )
+𝑐𝑖
+, 𝑦𝑠( 𝑓 )
+𝑐𝑖
+). The community-reduced adjacency
+matrix for the structural layer ˜A𝑠 and the fluid layer ˜A 𝑓 are given by
+˜𝐴𝑠
+𝑐𝑖,𝑐𝑗 =
+�
+𝑢𝑠
+𝑐𝑖←𝑐𝑗
+if 𝑐𝑖 ≠ 𝑐 𝑗 ∈ structure layer
+0
+otherwise
+˜𝐴 𝑓
+𝑐𝑖,𝑐𝑗 =
+�
+𝑢 𝑓
+𝑐𝑖←𝑐𝑗
+if 𝑐𝑖 ≠ 𝑐 𝑗 ∈ fluid layer
+0
+otherwise.
+(6)
+The combined network can be represented with a supra-adjacency matrix, A𝛼 that contains the adjacency
+matrices of both the fluid and structural layers along the block diagonal along with the inter-layer edge
+weight, W𝑖 𝑗 at the off-block diagonal as
+A𝛼 =
+�
+˜A𝑠
+W𝑠← 𝑓
+W𝑓 ←𝑠
+˜A 𝑓
+�
+,
+(7)
+where the inter-layer weights W𝑠← 𝑓 are the velocity induced by the fluid community centroids on the
+structural community centroids and W𝑓 ←𝑠 are the velocity induced by the structural community centroids
+5
+
+on the fluid community centroids. The supra-adjacency matrix is highlighted in Figure 2(b). The edge
+weights are normalized with the maximum edge weight for visualization. We see a lot of interactions among
+the structural nodes and the near wake fluid communities.
+Figure 2: Fluid-structure vortical interaction network for 2D laminar flow over a flat plate (𝑀𝜌 = 3,
+𝐾𝐵 = 0.3125, 𝑅𝑒 = 100): (a) Community reduction of the fluid network layer and the structure network
+layer. (b) Supra-adjacency matrix containing edge weights for the structure layer (top main-diagonal block)
+and fluid (bottom main-diagonal block) and the inter-layer fluid-structure coupling and structure-to-fluid
+coupling on the off-diagonal blocks. The edge weights are normalized with respect to the maximum edge
+weight for visualization.
+2.3
+Fluid-structure modal interaction network
+To construct the modal interaction network, we perform proper orthogonal decomposition (POD) of the flow
+velocity field data obtained from the direct numerical simulations in section 2.1 to extract the most energetic
+coherent structures (modes). In this work, we only extract the modal network for the most compliant case of
+𝐾𝐵 = 0.625. We employ the method of snapshots [61] to decompose the velocity fields 𝒒 𝑓 as
+𝒒 𝑓 (𝑥, 𝑦, 𝑡) = 𝒒 𝑓 (𝑥, 𝑦) +
+𝑁
+∑︁
+𝑗=1
+𝑎 𝑓
+𝑗 (𝑡)𝝓 𝑓
+𝑗 (𝑥, 𝑦).
+(8)
+where 𝑁 is the number of fluid modes, 𝒒 𝑓 (𝑥, 𝑦) is the mean flow, and 𝝓 𝑓
+𝑗 (𝑥, 𝑦) are the fluid modes with
+temporal coefficients given by
+𝑎 𝑓
+𝑗 (𝑡) =
+�
+𝒒 𝑓 (𝑥, 𝑦, 𝑡) − 𝒒 𝑓 (𝑥, 𝑦), 𝝓 𝑗(𝑥, 𝑦)
+�
+.
+(9)
+Here, ⟨·, ·⟩ stands for inner project. We fix 𝑁 = 8 to capture 99.9% of the total energy of the fluid flow given
+by KE = 𝒒 𝑓 · 𝒒 𝑓 ≈ �𝑁
+𝑗=1 𝑎2
+𝑗/2.
+Similarly, principal component analysis (PCA) is performed on the time series of x- and y-velocities,
+𝒒𝑠 = ( �𝒙𝑠, �𝒚𝑠) of each of the structural elements to yield 𝑝 modes 𝝓𝑠 and associated temporal coefficients 𝑎𝑠
+𝑗.
+We fix 𝑝 = 3 to capture 99.9% of the energetics of the structural deformations.
+6
+
+(a)
+Fluid layer
+(b)
+ Multilayer coupling
+Af E RMaM
+Af RNaN
+Ws←f
+As
+Edge
+strength
+Structure
+Structure layer
+0.5
+Community
+Fluid
+O
+1
+O
+2
+Community
+0
+3
+0 4
+Af
+As e Rmam
+5
+13
+O 
+6Fluid flow modes appear in complex conjugate mode pairs. We combine these mode pairs to form an
+oscillator representation of their temporal dynamics as
+𝑧 𝑓
+𝑚(𝑡) = 𝑎 𝑓
+2𝑗−1 + 𝑖𝑎 𝑓
+2 𝑗 = 𝑟 𝑓
+𝑚 exp(𝑖𝜃 𝑓
+𝑚)
+(10)
+where 𝑗 = 1, 2, . . . , 𝑁/2, 𝑟𝑚 = ∥𝑧𝑚∥, and 𝜃𝑚 = ∠𝑧𝑚. The oscillator number 𝑚 is denoted with Roman
+numerals to distinguish them from mode numbering 𝑗 ∈ 1, 2, . . . , 𝑁. We consider 𝑀 = 𝑁/2 fluid oscillators.
+The oscillator representation is akin to the polar decomposition of the temporal coefficients of the mode
+pairs. This helps in building a concise networked oscillator model, similar to the work of Nair et al. [53].
+PCA of the structural velocity data does not yield modes in pairs as in the case of fluid data. To convert
+the temporal coefficients of the structures to oscillator representation, we perform the Hilbert transform
+[62] of the temporal coefficients time-series data. This transformation converts the real data sequence to an
+analytic signal (i.e. complex helical sequence), where the real part is the original data and the imaginary part
+is a version of the real sequence with a 90◦ phase shift. The transformed series, which leads to structural
+oscillator representations, contain the same amplitude, frequency, and instantaneous phase information as
+the original signal. The structural oscillators are given by 𝑧𝑠
+𝑚 = 𝑟𝑠
+𝑚 exp(𝑖𝜃𝑠
+𝑚) corresponding to each temporal
+coefficient with 𝑚 = I, II, . . . , 𝑝.
+Once the fluid and structure oscillator representations are formed, we follow the procedure demonstrated
+in Nair et al. [53] to extract modal interaction networks. In Nair et al. [53], impulse perturbations were
+introduced to the temporal coefficients of the fluid to induce interactions among the modes. However, this
+approach relies on exciting modes of the entire fluid domain, which is infeasible. In this work, impulse
+perturbations are introduced to the structural dynamics, which are both physically meaningful and realistic.
+In particular, we add phase and amplitude impulse perturbations to the structural oscillators The phase
+perturbations in the modes range from −𝜋 to 𝜋 shifts in the phase of the modes relative to the baseline and
+the amplitude perturbation ranges from 0.1 to 100% of total kinetic energy.
+To track the perturbations introduced and the spread among the fluid and structural modes, we normalize
+the oscillator representations for the fluid and structure modes to yield oscillator perturbations as 𝜉 𝑓
+𝑚 =
+𝑧 𝑓
+𝑚/𝑧 𝑓 ,𝑏
+𝑚
+and 𝜉𝑠
+𝑚 = 𝑧𝑠
+𝑚/𝑧𝑠,𝑏
+𝑚 , respectively. Here, 𝑧 𝑓 ,𝑏
+𝑚
+and 𝑧𝑠,𝑏
+𝑚 are the baseline fluid and structure oscillator
+trajectories, respectively. Such a normalization yields zero perturbation amplitude at steady state and a finite
+steady-state phase shift. We collect data corresponding to three periods of baseline oscillation after the
+introduction of impulse perturbation.
+Once the data for the perturbations are tracked and collected, we can form a multilayer network with
+structural and fluid oscillators as nodes. Unlike the vortical network, the modal network lends itself to
+a combined representation automatically. A simple regression is performed on the perturbation datasets
+𝜉𝑚 = {𝜉𝑠
+𝑚; 𝜉 𝑓
+𝑚} with 𝑀 + 𝑝 oscillators. This results in a complex adjacency matrix for both the intra- and
+inter-layer interaction strengths between the structure and fluid oscillator layers as
+𝑑
+𝑑𝑡 𝜉𝑚 =
+𝑀+𝑝
+∑︁
+𝑛=𝐼
+𝐴𝑚𝑛(𝜉𝑛 − 𝜉𝑚) = −
+𝑀+𝑝
+∑︁
+𝑛=𝐼
+𝐿𝑚𝑛𝜉𝑛
+(11)
+where the complex adjacency matrix A and Laplacian matrix L are given by
+𝐴𝑚𝑛 = |𝜔𝑚𝑛| exp(𝑖∠𝜔𝑚𝑛),
+𝐿𝑚𝑛 = 𝑠in
+𝑚 − 𝐴𝑚𝑛
+(12)
+where 𝑠in
+𝑚 is the standard in-degree. As the adjacency matrix is complex-valued, the magnitude of each edge
+|𝜔𝑚𝑛| highlights the overall influence and the ∠𝜔𝑚𝑛 provides the phase relationship between the oscillators.
+To incorporate the insights from different impulse perturbation tests, we separate the data into training and
+test sets and perform model selection on the adjacency matrices obtained.
+7
+
+Figure 3: Fluid-structure modal interaction network for 2D laminar flow over a flat plate (𝑀𝜌 = 3, 𝐾𝐵 = 0.625,
+𝑅𝑒 = 100): (a) Overview of the modal interaction network for fluid and structure oscillators and their inter-
+layer coupling. (b) Magnitude (top) and phase (bottom) of the complex supra-adjacency matrix for modal
+interaction. Note the inter-layer edges between structure nodes I and II and the fluid nodes (corresponding to
+the top of (b)) are omitted for clarity in (a). The magnitude of the edge weights are normalized with respect
+to the maximum edge weight for visualization in (b).
+3
+Results
+3.1
+Vortical interaction network
+For the vortical interaction network described in section 2.2 and illustrated in Figure 2, we elaborate on the
+results in this section. We first look at network metrics that highlight the role of the nodes in the network
+in section 3.1.1. We then develop a data-driven model using nonlinear regression capable of predicting the
+community-reduced FSI vortical network structure over the limit cycle in section 3.1.2. Finally, we present
+results from the physics-based prediction of community centroids in 3.1.3.
+3.1.1
+Network metrics
+To analyze the interactions between the fluid and structural components in the FSI system and how they
+change with time, we analyze the supra-adjacency network structure via network metrics. As we are interested
+in the overall inter-layer influence of the fluid on the structure and vice-versa, we construct the inter-layer
+supra-adjacency Ainter
+𝛼
+as
+Ainter
+𝛼
+=
+�
+0
+W𝑠← 𝑓
+W𝑓 ←𝑠
+0
+�
+,
+(13)
+where the entries on block diagonals corresponding to the structural layer and fluid layer are zero. For the
+structural component, we define total out-degree = �𝑁𝑐
+𝑖
+�𝑛𝑐
+𝑗 W𝑓𝑖←𝑠𝑗 andtotalin-degree = �𝑛𝑐
+𝑖
+�𝑁𝑐
+𝑗
+W𝑠𝑖← 𝑓𝑗.
+This out-degree is the total influence of the structure on the fluid at a particular time and the in-degree is
+the total influence of the fluid on the structure. We also examine Katz centrality of the inter-layer network
+defined as 𝑪 = (𝑰 − 𝛼Ainter
+𝛼
+)−11, where 𝑰 is the identity matrix, 𝛼 is a hyper-parameter to account for nodes
+with zero or low eigenvector centrality, and 1 is a vector of ones. Here, we choose 𝛼 = 0.01. To quantify the
+total strength of the influential community structures, we examine the measure 𝐾 = �
+𝑖 𝐶.
+8
+
+(a)
+Structure layer
+- Multilayer coupling
+(b)
+I[Ag]mn l
+Fluid layer
+Strength
+I
+W
+(VI
+s个
+0.5
+Fluid
+III
+0
+Z[Ag] mn
+VI
+000c
+Phase
+元
+0
+(IV)
+Fluid
+-We show the total in-degree, out-degree, and Katz centrality measure 𝐾 for each snapshot in time for
+the three different bending stiffness in Figure 4(a). On the top, we show the transverse tip displacement. In
+the time between the dashed lines, a “1 − cos" gust encounter is applied with a maximum pitch-down of the
+plate of 5◦. Limit cycle oscillations are observed at other times. We see that there is significant noise in
+degree centrality for the least compliant case of 𝐾𝐵 = 0.15625. As the structure becomes more compliant,
+repeating patterns in the network measures can be seen through the phase progression of the limit cycle. The
+total Katz centrality measure provides the lowest noise response signal during the limit cycle. The square
+wave spike and variations in 𝐾 indicate the detection of new communities caused by vortex shedding.
+For the gust encounter, we see that the total out-degree spikes proportionally increase with compliance.
+For the low and medium compliant cases, a strong in-degree spike is observed just after the gust starts and
+just before it ends and a strong out-degree spike is observed during the middle of the gust encounter. This is
+expected as the structure gets perturbed (influenced) during the gust encounter and as the structure deforms,
+it influences the rest of the flow field. Thus, the in- and out-degree are opposite in phase during the gust
+encounter for the two lesser compliant cases. Strong out-degree spikes are seen just after the gust starts and
+just before it ends for the most compliant case 𝐾𝐵 = 0.625. Also, Katz centrality measure 𝐾 clearly detects
+the gust for the two lesser compliant cases; however, shows only minor changes for the most compliant case.
+This indicates the changes in the vortex shedding events and formation of the new communities for the less
+compliant cases and not many changes in the formation of new communities for the most compliant case.
+The small amplitude of the gust and relatively slow variation gets masked by the oscillation of the structure
+in the most compliant case.
+In addition to the network measures above, we also investigate our system using a participation score vs.
+z-score map (P-Z map) of the community-reduced supra-adjacency A𝛼. This provides a concise and visual
+depiction of the interaction characteristics of nodes within a network. Z-score and participation coefficient
+are defined using the out-degree of the community-reduced supra-adjacency matrix 𝑠𝑖 = 𝑠out
+𝑖
+as
+𝑍𝑖 = 𝑠𝑖 − 𝑠𝑖
+𝜎𝑠𝑖
+,
+𝑃𝑖 = 1 −
+��𝑆𝑠( 𝑓 )
+𝑠𝑖
+�2
++
+∑︁
+𝑘,𝑘≠𝑖
+� 𝑠𝑘
+𝑠𝑖
+�2�
+(14)
+where 𝑆𝑠( 𝑓 ) is the total out-degree strength of the nodes in the structure (fluid) and 𝑠𝑖 is the mean out-degree
+of all centroids and 𝜎𝑠𝑖 is the standard deviation of the out-degree strength.
+The P-Z map provides an intuitive visualization of the role that each community plays in the system as
+seen in Figure 4(b). The corresponding supra-adjacency matrix is shown in Figure 4(c). Nodes with high
+participation scores are called connectors, while those with low-participation scores are called peripherals
+[48]. High z-score indicates hubs that exert maximum influence within their community but have little
+influence over other communities. In fact, both peripherals and hubs do not have much inter-community
+influence. We clearly observe that all of the structure nodes have high participation scores. Also, the
+centroid B which is close to the center of the plate plays the most crucial role in the interaction dynamics.
+This indicates that the structural nodes are the main influencers in the FSI vortical network.
+We also
+see that the first two communities of the fluid have a high z-score and comparatively higher participation
+scores. These near-wake centroids have the most inter and intra-community interactions. As communities
+are advected downstream we see that their influence on the structure and on the fluid diminish in a nearly
+linear fashion with low participation and z-score.
+3.1.2
+Data-based prediction
+In this section, the time-series data of the fluid and structure community centroids 𝑐𝑖 and their associated
+strength 𝛾𝑠( 𝑓 )
+𝑐𝑖
+and position (𝑥𝑠( 𝑓 )
+𝑐𝑖
+, 𝑦𝑠( 𝑓 )
+𝑐𝑖
+) are used to build a predictive dynamical model. We use sparse
+identification of dynamical systems (SINDy) [63] for generating this predictive model as shown in Figure
+9
+
+Figure 4: Network metrics for fluid-structure vortical interaction network: (a) Time evolution of centrality
+measures (in-degree, out-degree, and Katz measure 𝐾 for the structural components) for the inter-layer
+supra-adjacency matrix for three differnt bending stiffnesses during limit cycle and a 5-degree angle of attack
+gust encounter (between dashed lines). (b) P-Z map distribution of supra-adjacency matrix showing the
+structure nodes (□) and fluid (◦) and the (c) corresponding adjacency matrix. The magnitude of the edge
+weights is normalized with respect to the maximum edge weight for visualization in (c).
+5(a). The values predicted by the SINDy model for the circulation of the first structure centroid compared
+to that from direct numerical simulation are presented in Fig 5(b). We see an acceptable agreement between
+the original data and the values predicted by SINDy model. The location and circulation trends for other
+centroids (not shown here) also match reasonably with the DNS data.
+With the SINDy model, we can now predict the evolution of the community-reduced supra-adjacency
+matrix as well. We show the similarity between the predicted network structure of the adjacency matrix
+using the model with that obtained from the direct numerical simulation at three characteristic times in Figure
+5(c). This demonstrates that the relative interaction between the communities is preserved by the predictive
+model. The weights of edge weights are restricted to the same range to show the richness in the interactions
+over the limit cycle. The three structure communities exert maximum influence over the first fluid centroid
+corresponding to the shed positive vortical structure.
+3.1.3
+Physics-based prediction
+In this section, we advect the community centroids from a single flow realization using the potential flow
+code developed by Darakananda et al. [64]. The plate coordinates at the time corresponding to the flow
+realization are provided as input to the solver. The system is then allowed to evolve with the plate coordinates
+being updated at regular intervals.
+The potential flow code is initiated at 𝑡 = 0 with the six vortices at the location of each community
+centroid with strengths corresponding to the total vorticity within each community. The flow was then
+allowed to evolve for one second of simulation time. The starting position of each community centroid
+(vortices) is denoted by an × symbol while the position of each community centroid identified from direct
+numerical simulation is denoted by an empty circle ◦.
+In Figure 6(a), we show the physics-based advection of the community centroids by filled circles and
+compare that with those from direct numerical simulation at characteristic times. We see strong agreement
+between the physics-based advection of the seeded community centroid vortices with that of the community
+10
+
+(a)
+KB = 0.15625
+KB = 0.3125
+KB = 0.625
+(b)
+-0.3
+-
+0.4
+1
+yt
+-0.5
+1
+1
+(1)
+-0.6
+1
+-
+-
+1.0
+· (3)
+(2)
+25
+0.5
+In-degree
+Structure
+(4)
+Out-degree
+ Fluid
+N
+0.0b
+20
+K
+(5)
+/
+-0.5
+(B)
+15
+-1.0
+(A) 
+Strength 
+0.2 0.3 0.4 0.5 0.6
+0.7
+0.8
+P
+10
+(c)
+B
+c
+5
+1
+2
+0.5
+0
+5
+0
+5
+10
+15
+20
+25
+30
+35
+5
+10
+15
+20
+25
+30
+35 5
+10
+15
+20
+25
+30
+35
+Time [s]Figure 5: Data-based prediction of the fluid and structure community centroids of the vortical network: (a)
+Construction of the SINDy model for the evolution of circulation and position of centroids, comparison of the
+predicted (b) trajectories and (c) adjacency matrix of the model with that from direct numerical simulation.
+detection results from DNS data. As seen in the inset of panel (a), the shape of the body is changed at
+regular intervals. Figure 6(b) shows the RMS error associated with the predicted x-position of each of the
+six vortices. We note that the fluid communities that are closest to the structure (1) and (2) have the largest
+error associated with them. This behavior is due to the poor prediction of vortices that have not fully shed.
+The leading and trailing edge suction parameter needs to be tuned when a vortex is shed [65]. The remaining
+four vortices in the far wake show good agreement with the DNS data. The error in the position increases in
+time which can be attributed to the absence of viscosity in the potential flow solution. Both the data-based
+and physics-based strategies are complementary to one another to obtain a fast prediction of FSI interactions
+and the dynamics of centroid communities.
+3.2
+Modal interaction network
+For the modal interaction network described in section 2.3 and illustrated in Figure 8, we elaborate on the
+results in this section. We first discuss the modal decomposition results in section 3.2.1. We then discuss
+the results of predictions from the networked oscillator model of Eq. (11) in section 3.2.2. We conclude by
+looking at the controllability of the modal interaction network in section 3.2.3.
+3.2.1
+Modal decomposition
+The results of the principal component analysis of the time series of velocity of the plate is shown in
+Figure 7(a) and (b). For the structure, the singular values drop off rapidly after the third mode as seen in
+Figure 7(a). This provides a clear threshold for modal truncation. The mode shapes for both the x- and
+y-velocity component are similar to that of the bending modes of a cantilever beam, as seen in the top panel
+11
+
+(a)
+(b)
+X = [Q1 Q2 Q3 ...1 Q = [~s,f,rs,rf
+8 h
+DNS
+Q1 Q2 Q3 .
+[1 Q1 Q2 Q3 Qi Q1Q2 Q1Q3...[51 52 53..
+Model
+0
+8
+t 5
+(c)
+.
+W
+2
+ij
+20
+3 
+3
+3
+Structure
+Fluid-structure 4
+4
+DNS
+Coupling
+5
+5
+15
+6
+6
+6
+Fluid
+7
+Structure-Fluid
+8
+8
+Coupling
+10
+１234
+1_2345678
+123456
+8
+2
+2
+3
+5
+3
+4
+4
+4
+MODEL
+5
+5
+6
+6
+6
+0
+7
+7
+8 
+8
+8
+1234567Figure 6: Physics-based prediction of the fluid community centroids of the vortical network using potential
+flow solver: (a) spatial position of the fluid vortical community centroids at time 𝑡 = 0, 𝑡 = 0.5, and 𝑡 = 1.0
+seconds. Inset shows the starting position, 𝑡 = 0, and current position, 𝑡 = 1.0, of the structure. (b) RMS
+error traces of the x-position for each of the six fluid communities compared to DNS.
+of Figure 7(b). We see typical sinusoidal traces of the temporal coefficients for the first two oscillators in the
+bottom panel of Figure 7(b).
+The singular values from the POD decomposition of the unsteady fluid velocity field snapshots are shown
+in Figure 7(c). We choose eight fluid modes (4 mode pairs) to capture 99.9% of the kinetic energy of the flow.
+Phase portraits of the temporal coefficient of the POD mode pairs along with the spatial modes are shown in
+Figure 7(d). Each of the four mode-pair phase portraits shows a typical circular shape for unsteady laminar
+flows. The modal structures get smaller with increasing mode numbers and the corresponding amplitude of
+the temporal coefficient decreases.
+3.2.2
+Networked-oscillator model
+As discussed in section 2.3, we introduce different ranges of amplitude and phase impulse perturbations
+to the first two structural modes and collect data from direct numerical simulation. We perform simple
+regression on the data to extract the adjacency matrix 𝐴𝑚𝑛 in Eq. (12). We then evolve Eq (11) to predict
+the amplitude and phase perturbation trajectories.
+The predicted amplitude trajectories compared to those extracted from direct numerical simulation for
+the three structural and four fluid oscillators (after the immediate transients in direct numerical simulation
+die out) are shown in Figure 8(a) and (b), respectively. The first two structure oscillators show excellent
+agreement with the simulation data. While the third structure oscillator follows the trace of the true system, it
+has a high-frequency oscillation throughout the 50 seconds of simulation time. This high-frequency vibration
+is possibly due to the low amplitude associated with the third structural oscillator. The fluid oscillators also
+show comparable agreement, however, the results deviate slightly for fluid oscillator IV. Similar agreement
+is observed in the phase of the perturbations (not shown).
+We extract different network models; only considering data from perturbations on structure oscillator I,
+only considering data from perturbations on structure oscillator II, and training from both perturbations. 20%
+12
+
+(a)
+1
+(b)
+0.4
+Centroid label
+0
+0=↑
+-1
+123456
+-2
+0.0
+2.5
+5.0
+7.5
+10.0
+0.3
+1
+X Starting position
+O DNS
+0
+xo
+Ox
+t = 0.5
+OPotentialFlow
+x
+error
+-1
+x
+0.2
+RMS
+0.0
+2.5
+5.0
+7.5
+10.0
+1
+0
+X
+C
+t = 1.0
+xO
+0.1
+-1
+X
+C
+-2
+0.0
+7.5
+10.0
+Starting position
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Current position
+time [s]Figure 7: Modal decomposition of the fluid-structure interaction system (𝑀𝜌 = 3, 𝐾𝐵 = 0.625, 𝑅𝑒 = 100).
+Structure layer: (a) Singular values for the first ten PCA modes of the time-series of the velocity of the
+structure, (b) mode shapes and temporal coefficient traces for the three structural modes selected as nodes
+of the modal interaction network. Fluid layer: (c) Singular values for the first eight POD mode-pairs (16
+modes), (d) vorticity of the spatial modes and temporal coefficient phase portraits for the each mode-pair for
+the four leading mode-pairs selected as nodes of the modal interaction network.
+of the data from all perturbation cases are reserved for testing. For each of the training epochs, perturbations
+on structure oscillator II shows the best agreement with the test data while oscillator one shows only a slight
+increase in error. The aggregate model shows the poorest performance, especially in structure oscillator III.
+All models show similar errors for the two dominant structure modes and the dominant fluid mode pair.
+The amplitude and phase relationship between the modal oscillators of the FSI system is shown in Figure
+8(c). The network structure captures the energy transfers between the modes of the structure and fluid on
+the introduction of impulse perturbations. The associated network centrality measures are shown in Figure
+8(d). The first two structure oscillators have the highest out-degree while the third structure oscillator has
+the highest in-degree. The out-degree for the fluid oscillators decrease with oscillator number while the
+in-degree increases. These results for the fluid oscillators are in agreement with that of Nair et al. [53].
+3.2.3
+Network controllability
+In this section, we perform a controllability analysis of the networked-oscillator model given Eq. (11). Here,
+we intend to control the perturbation dynamics of the model with an addition a control input 𝒗 such that
+�𝝃 = −𝑳𝝃 − 𝑩𝒗
+(15)
+13
+
+(a)
+(b)
+b (1)
+(1)
+(2)
+(3)
+U-velocity
+V-velocity
+0.2
+0.2
+0.2
+100
+Velocity [c/s]
+0.1
+0.1
+0.1
+ (2)
+0.0
+0.0
+0.0
+-0.1
+0.1
+0.1
+· (3)
+10~2
+0.2
+0.2
+0.2
+0.00
+0.25
+0.50
+0.75
+1.00
+0.00
+0.25
+0.50
+0.75
+1.00
+0.00
+0.25
+0.50
+0.75
+1.00
+c
+0.004
+0.0006
+0.002
+ai
+0.0004
+0.000
+0.0002
+0.1
+0.002
+0.0000
+-0.2
+0.004
+0.0002
+0
+2
+4
+6
+0.015
+0
+-0.015
+0
+2
+4
+6
+(c)
+(d)
+Time [s]
+(1, 2)
+Mode 1
+Mode 2
+Mode 5
+Mode 6
+1.0
+0.04
+0.5
+(3, 4)
+0.02
+OOODD
+a2 0.0
+a5 0.00
+10°
+.·(5, 6)
+0.02
+0.5
+-0.04
+-1.0
+(7, 8)
+a1
+a6
+Mode 3
+Mode 4
+Mode 7
+Mode 8
+0.2
+0.010
+0.1
+0.005
+0....
+a3 0.0
+.0000
+0000
+0.000
+0.005
+10~2
+-0.1
+0.010
+-0.2
+a4
+a8Figure 8: Network-oscillator model for fluid-structure interaction system (𝑀𝜌 = 3, 𝐾𝐵 = 0.625, 𝑅𝑒 = 100):
+Trajectories of the three (a) structure and four fluid (b) oscillators for the predictive model (red) and ground
+truth (black) for 50 seconds, (c) performance of the single-oscillator-based models and the aggregate model
+(black). (d) modal interaction model magnitude and phase adjacency matrices after training. (e) In- and
+out-degree for the network nodes.
+where 𝑳 is the Laplacian matrix, 𝒗 ∈ C(𝑀+𝑝)×1 and 𝑩 is the input matrix. Here, 𝝃 = [𝜉𝐼, 𝜉𝐼 𝐼, . . . , 𝜉𝑀+𝑝]𝑇 .
+The optimal full-state feedback controller is obtained with a linear quadratic regulator (LQR) as 𝒗 = −𝑲𝝃 to
+yield
+�𝝃 = (−𝑳 − 𝑩𝑲)𝝃
+(16)
+with the cost function defined as
+𝑱 =
+∫ ∞
+0
+[𝝃(𝑡)𝑇 𝑸𝝃(𝑡) + 𝒗(𝑡)𝑇 𝑺𝒗(𝑡)]𝑑𝑡
+(17)
+where 𝑸 = 𝑰 and 𝑺 = 𝜎𝑰 as the state and input penalty, respectively.
+To assess the controllability of the modal interaction network, we examine the movement of the pole
+of the Laplacian matrix by systematically decreasing the input penalty 𝜎 and changing the input matrix
+𝑩. In the top panel of Figure 9, the input matrix only activates single-structure oscillators. We see that
+the pole trajectories for the first two structure oscillators show similar behavior when control is applied to
+them individually. The third oscillator, however, shows distinct behavior and moves only a single pole when
+control is applied. As seen in the middle panel of Figure 9, applying control simultaneously to structure
+oscillator I and II show the ability to move the poles with the greatest real eigenvalue. We see a similar
+response for all three structural oscillator perturbations, albeit with a higher control input. As seen in the
+bottom panel of Figure 9, the addition of control on the fluid oscillators has little effect on the movement of
+the poles.
+14
+
+(a)
+(b)
+(c)
+4
+1.03
+1.1
+(I)
+(II)
+1.02
+(ΛI)
+(Λ)
+1.1
+1.02
+3.5
+1.05
+1.05
+1.01
+1.01
+3
+1
+1
+0
+t
+50
+0
+t
+50
+0
+t
+50
+0
+t
+50
+△ 2.5
+1.2
+Train:Osc I
+(I)
+(IA)
+1.04
+(IA)
+1.04
+2
+Train:Osc I
+1.1
+DNS
+1.02
++Aggregate
+1.02
+Model
+1
+1.5
+IV
+V
+VIVII
+0
+t
+50
+0
+t
+50
+t
+50
+0
+m
+Z[Ag]mn
+[[Ag] mn]
+(d)
+(e)
+60
+O in-degree
+元
+Strength 
+50
+ out-degree
+ndno
+40上
+■
+ structure nodes
+30
+ fluid nodes
+0.5
+0
+20
+10
+.
+.
+0
+-T
+:
+ob
+.
+.
+1
+IV
+V
+VI
+VII
+Input
+OscillatorFigure 9: Pole trajectories with application of different control inputs to the modal interaction network for a
+range of values of 𝜎.
+4
+Conclusion
+In summary, we develop two reduced-order models of fluid-structure interaction, leveraging a multi-layer
+network framework. The two approaches use distinctive vortical and modal features of the overall FSI system.
+In the vortical approach, grid cells in the Eulerian computational domain with their associated vorticity form
+the nodes of the fluid layer, and bound vortexlets form the nodes of the structural layer. The edge weights
+in this approach are defined using induced velocity. Community detection was used to construct a reduced
+representation of the vortical network. In the second approach, coherent modes from the fluid and structure
+form the nodes of the network. Introducing impulse perturbation to the structural modes and tracking the
+amplitude and phase of the modal perturbations, the modal interaction network model is extracted in a
+data-driven manner.
+Two-dimensional flow over a compliant flat plate at an angle of attack 𝛼 = 35◦ was investigated using the
+network-based approach. Data from direct numerical simulations of three different plate stiffnesses during
+the limit cycle and gust encounters were converted to a community-reduced vortical network. The network
+metrics were able to capture the dynamics of the limit cycle and the influence of gust encounters. A P-Z map
+was constructed to illustrate the unique role of each node of the vortical network in the overall FSI system.
+Prediction of vortex dynamics and the network interactions were performed using two different strategies: a
+pure data-based strategy using SINDy and a physics-based strategy using a potential flow solver which was
+initialized using the data of community centroids. Both methods show acceptable agreement between the
+prediction and ground truth data.
+Then, we demonstrate the extraction of the modal-interaction network for the most compliant structure,
+𝐾𝐵 = 0.625. Using principal component analysis of the velocities of the structure and proper orthogonal
+decomposition of the fluid velocity fields, nodal representations for the network were obtained. Oscillators
+are formed from the fluid conjugate mode-pairs and a Hilbert transform of the structural temporal coefficients.
+15
+
+B= [1000000]T
+B=[0100000T
+B=[0010000T
+10
+10
+10
+5
+5
+5
+(r)s
+6
+0
+0
+0
+10-1
+100
+101102
+103
+-5
+-5
+-5
+10
+-10
+-10
+12-10-8 -6-4-2 0
+12-10-8-6-4-20
+12-10-8-6-4-20
+(入)
+况(入)
+况(入)
+Structure oscillators
+B=[1100000]T
+B=[1010000T
+B=[0110000T
+B=[1110000]T
+10
+10
+10
+10
+5
+5
+5
+5
+(r)S
+(r)s
+(r)S
+0
+L
+0
+米
+0
+0
+Multiple oscillator input
+-5
+-5
+-5
+-5
+10
+12-10-8-6-4-20
+10
+10
+10
+12-10-8 -6-4-2 0
+12-10-8-6-4-20
+12-10-8-6-4-20
+况(入)
+况(入)
+究(入)
+究(入)
+B=[1001000T
+B=[1000100T
+B=[0101000]T
+B=[0100100]T
+Mixed oscillators 
+10
+10
+10
+10
+5
+5
+5
+5
+(r)s
+米
+3
+0
+0
+0
+0
+米
+-5
+-5
+-5
+-5
+-10
+12-10-8-6-4-20
+-10
+-10
+-10
+12-10-8-6-4-20
+12-10-8-6-4-20
+12-10-8-6-4-20
+况(入)
+R(入)
+(入)
+况(入)The dominant two structure modes are perturbed to track the energy transfer in the FSI system. We then
+train our network model with 80% of the perturbation data using simple regression. Oscillator amplitude
+trajectories are predicted from the model and showed close agreement with the retained testing data. A
+controllability assessment of the network indicatesthat applyingcontrolto thetwoleading structureoscillators
+moves the poles with the greatest real eigenvalue.
+We see the possibility for this formulation to be extended into several areas. First, the investigation of
+interactions between multiple bodies in an unsteady fluid flow such as that occurring between the main wing
+and empennage of an airplane. Secondly, the development of a computationally efficient predictive model
+suitable for online control applications is needed for gust alleviation. Lastly, a generalizable approach to the
+characterization and modeling of multiphysics systems.
+Acknowledgements
+AGN acknowledges the support from the Department of Energy Early Career Research Award (Award no:
+DE-SC0022945, PM: Dr. William Spotz) and the National Science Foundation AI Institute in Dynamic
+systems (Award no: 2112085, PM: Dr. Shahab Shojaei-Zadeh). The authors thank Dr. Nitish Arya for his
+insights on the data-driven models.
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